Proceedings of the 4th International Conference on Electrical Engineering and Control Applications: ICEECA 2019, 17–19 December 2019, Constantine, Algeria [1st ed.] 9789811564024, 9789811564031

Table of contents :
Front Matter ....Pages i-xix
Front Matter ....Pages 1-1
Sensorless Speed Control of Induction Motor Used Differential Flatness Theory (Yassine Beddiaf, Fatiha Zidani, Larbi. Chrifi-Alaoui)....Pages 3-21
Multi-UAVs Coverage Path Planning (Abdelwahhab Bouras, Yasser Bouzid, Mohamed Guiatni)....Pages 23-36
Reducing the Collision Checking Time in Cluttered Environment for Sampling-Based Motion Planning (Amine Belaid, Boubekeur Mendil)....Pages 37-46
Random Scaling-Based Bat Algorithm for Greenhouse Thermal Model Identification and Experimental Validation (Mounir Guesbaya, Hassina Megherbi, Ahmed Chaouki Megherbi)....Pages 47-62
Control and Observation of Induction Motor Using First-Order Sliding Mode (Salah Eddine Farhi, Djamel Sakri, Noureddine Goléa)....Pages 63-76
Cuckoo Search Algorithm for Solving the Problem of Unit-Commitment with Vehicle-to-Grid (Amel Terki, Hamid Boubertakh)....Pages 77-92
A Comparative Study of MPPT Techniques for Standalone Hybrid PV-Wind with Power Management (Tarek Boutabba, Hamza Sahraoui, Mouhamed Lamine Bechka, Said Drid, Larbi Chrifi-Alaoui)....Pages 93-108
Adapted Search Equations of Artificial Bee Colony Applied to Feature Selection (Hayet Djellali, Souad Guessoum, Nacira Ghoualmi-Zine)....Pages 109-121
Networked Cooperation-Based Distributed Model Predictive Control Using Laguerre Functions for Large-Scale Systems (Kamel Menighed, Ahmed Chemori, Boumedyen Boussaid, Joseph Julien Yamé)....Pages 123-138
Direct Torque Control Using Fuzzy Second Order Sliding Mode Speed Regulator of Double Star Permanent Magnet Synchronous Machine (Louanasse Laggoun, Brahim Kiyyour, Ghoulemallah Boukhalfa, Sebti Belkacem, Said Benaggoune)....Pages 139-153
Fractional Order Integral Controller Design Based on a Bode’s Ideal Transfer Function: Application to the Control of a Single Tank Process (Chahira Boussalem, Rachid Mansouri, Maamar Bettayeb, Mustapha Hamerlain)....Pages 155-169
Modeling of Torsional Vibrations Dynamic in Drill-String by Using PI-Observer (R. Riane, M. Kidouche, M. Z. Doghmane, R. Illoul)....Pages 171-185
Power System Generator Coherency Identification for Large Disturbances by Koopman Modes Analysis (Zahra Jlassi, Khadija Ben Kilani, Mohamed Elleuch, Lamine Mili)....Pages 187-202
Analysis and Experimental Validation of Single Phase Series Resonance Inverter (Alla Eddine Toubal Maamar, M’hamed Helaimi, Rachid Taleb)....Pages 203-216
Optimal Decentralized State Control of Multi-machine Power System Based on Loop Multi-overlapping Decomposition Strategy (M. Z. Doghmane, M. Kidouche)....Pages 217-231
Large Synchronverter Integration in Power Electrical System: Impacts on SCR and CCT (Raouia Aouini, Khadija Ben Kilani, Mohamed Elleuch, Quoc Tuan TRAN)....Pages 233-246
Immersion and Invariance Based Adaptive Dynamic Surface Control for Parametric Strict-Feedback Nonlinear Systems (Y. Soukkou, S. Labiod, M. Tadjine, Q. M. Zhu, M. Nibouche)....Pages 247-261
Design of an Adaptive Fuzzy Backstepping Synergetic Control Scheme for a Class of Strict-Feedback Nonlinear Systems (Aissa Rebai, Kamel Guesmi, Mohamed Bougrine)....Pages 263-277
Improvement of the Stability Performance of a Quad-Copter Helicopter by a Neuro-Fuzzy Controller (Djalal Baladji, Kheireddine Lamamra, Farida Batat)....Pages 279-291
The Importance of Applying Artificial Intelligence on Unmanned Aerial Vehicle (Amine Mohammed Taberkit, Ahmed Kechida, Abdelmalek Bouguettaya)....Pages 293-304
Fuzzy H∞ Delay-Independent Stabilization of Depth Control for Underwater Vehicle with Input Constraint (Mohamed Nasri, Dounia Saifia, Salim Labiod, Mohammed Chadli)....Pages 305-318
Petri Type 2 Fuzzy Neural Networks (PT2FNN) for Identification and Control of Dynamic Systems—A New Structure and a Comparative Study (Youssouf Bibi, Mohamed Seghiri, Omar Bouhali, Abdelkarim Nemra, Tarek Bouktir)....Pages 319-330
Design Fractional Order PI Controller with Decoupler for MIMO Process Using Diffusive Representation (Sami Laifa, Badreddine Boudjehem, Hamza Gasmi)....Pages 331-346
Direct Sliding Mode Control of Transient Power in Microgrid During Grid Failure ‘‘Unintentional Islanding’’ (I. Ameur, N. Gazzam, A. Benalia)....Pages 347-359
New Virtual Synchronous Generator Control Technique of Distributed Generator Unit to Improve Transient Response of the Microgrid (Yacine Daili, Abdelghani Harrag)....Pages 361-371
Design and Analysis of Robust Nonlinear Synergetic Controller for a PMDC Motor Driven Wire-Feeder System (WFS) (Noureddine Hamouda, Badreddine Babes, Amar Boutaghane)....Pages 373-387
Simulation Study of the Dual Star Permanent Magnet Synchronous Machine Using Different Modeling Approaches (Elyazid Amirouche, Khaldi Lyes, Ghedamsi Kaci, Aouzellag Djamal)....Pages 389-405
Modeling and Experimental Identification of Salient-Pole Synchronous Machine (Khalil-Errahmane Sari, Bilal Sari)....Pages 407-425
Power Quality Improvement of PWM Rectifier-Inverter System Using Model Predictive Control for an AC Electric Drive Application (Abdelkarim Ammar)....Pages 427-439
Novel Smart Air Quality Monitoring System Based on UAV Quadrotor (Mehdi Zareb, Benaoumer Bakhti, Yasser Bouzid, Hamza Kadourbenkada, Kamel Bouzgou, Wahid Nouibat)....Pages 441-454
GA Based Adaptive Fuzzy Sliding Mode for 3 DOF Spatial Arm Manipulator Trajectory Tracking (Rabie Belloumi, Noureddine Slimane)....Pages 455-464
Robust Adaptive Fuzzy Approach with Unknown Control Gain Direction and External Disturbance (Ouassila Bourebia, Nassira Zerari)....Pages 465-476
Adaptive Neural-Network Control Design for Uncertain CSTR System with Unknown Input Dead-Zone and Output Constraint (Zerari Nassira, Chemachema Mohamed)....Pages 477-491
Adaptive Fuzzy Fault-Tolerant Control Using Nussbaum Gain for a Class of SISO Nonlinear Systems with Unknown Directions (Abdelhamid Bounemeur, Mohamed chemachema, Abdelmalek Zahaf, Sofiane Bououden)....Pages 493-510
Front Matter ....Pages 511-511
Optimal Integration of Renewable Distributed Generation Using the Whale Optimization Algorithm for Techno-Economic Analysis (Samir Settoul, Rachid Chenni, Mohamed Zellagui, Hassan Nouri)....Pages 513-532
Enhancement of Power System Transient Stability with a Large Penetration of Solar Photovoltaic Using Facts (Ikram Boucetta, Naimi Djemai, Salhi Ahmed, Zellouma Laid)....Pages 533-547
Minimization of the Energy Consumption of an Aircraft (Abbes Lounis, Kahina Louadj, Mohamed Aidene)....Pages 549-564
Study to Improve the Technical Parameters for the Optimising the Injection of the Photovoltaic Energy into an Electrical Network via a Line 30 KV (Bahriya Badache, Hocine Labar, Mounia Samira Kelaiaia)....Pages 565-578
Second-Order Super-Twisting Control of an Autonomous Wind Energy Conversion System Based on PMSG for Robustness and Chattering Elimination (Abderrahmane Abdellah, Djilali Toumi, M’hamed Larbi)....Pages 579-594
Open-Switch Faults Based on Six/Five-Leg Reconfigurable AC-DC-AC Converter (Sahraoui Khaled, Gaoui Bachir)....Pages 595-613
A Robust Model Predictive Control for a DC-DC Boost Converter Subject to Input Saturation: An LMI Approach (O. Hazil, S. Bououden, M. Chadli)....Pages 615-629
Strategy for Optimization of Energy Management Based on Fuzzy Logic in an FCEV with a Contribution of a Photovoltaic Source (Said Belhadj, Kaci Ghedamsi, Zina Larabi)....Pages 631-644
Modeling and Analysis of an Electromagnetic Vibration Energy Harvester for Automotive Suspension (Mustapha Zaouia, Bachir Ouartal, Nacereddine Benamrouche, Arezki Fekik)....Pages 645-657
Experimental and Numerical Study of Hybrid PV/Thermal Solar Collector Provided with Self Ventilation and Tracking Structure (Mohamed El-Amine Slimani, Rabah Sellami, Achour Mahrane, Madjid Amirat)....Pages 659-670
Maximum Power Point Tracking (MPPT) for a PV System Based on Artificial Neural Network ANN and Comparison with P&O Algorithm (Aicha Djalab, Nassim Sabri, Ali Teta)....Pages 671-682
Development of a Stand-Alone Connected PV System Based on Packe U Cell Inverter (Khaled Rayane, Mohamed Bougrine, Atallah Benalia, Kamel Guesmi)....Pages 683-695
A Novel Hybrid Photovoltaic/Thermal Bi-Fluid (Air/Water) Solar Collector: An Experimental Investigation (Mohamed El-Amine Slimani, Rabah Sellami, Mohammed Said, Amina Bouderbal)....Pages 697-709
A DC/DC Buck Converter Voltage Regulation Using an Adaptive Fuzzy Fast Terminal Synergetic Control (Noureddine Hamouda, Badreddine Babes)....Pages 711-721
Study of Electrical Field Distribution in the Insulation of High-Voltage Cables (Rouini Abdelghani, Kouzou Abdellah, Larbi Messaouda)....Pages 723-734
Self Tuning Fuzzy Maximum Power Tracking Control of PMSG Wind Energy Conversion System (Chaib Housseyn, Mihoub Youcef, Hassaine Said)....Pages 735-749
A Backstepping Controller for Interleaved Boost DC–DC Converter Improving Fuel Cell Voltage Regulation (Ali Dali, Samir Abdelmalek, Maamar Bettayeb)....Pages 751-762
The Impact of the Solar Radiation Profile on Sizing and Performance of Photovoltaic Systems, Case Study Tamanrasset, Algeria (Madjid Chikh, Aicha Degla, Achour Mahrane)....Pages 763-778
Investigation and Prototyping Implementation of a Novel Solar Water Collector Based on Used Engine Oil as HTF (Oussama Touaba, Salah Mohamed AitCheikh, Mohamed El-Amine Slimani, Ahmed Bouraiou, Abderrezzaq Ziane)....Pages 779-789
Piezoelectric Energy Harvesting Based Autonomous Vehicles’ Vibrations (Nadjet Zioui, Sousso Kelouwani, Gilbert Lebrun)....Pages 791-800
Front Matter ....Pages 801-801
Fault Detection of Uncertain Systems Based on Interval Data Driven Approach (Chouaib Chakour)....Pages 803-813
Discrimination of Unbalanced Supply and Stator Interturn Faults in Induction Machines (D. Kouchih, R. Hachelaf, N. Boumalha)....Pages 815-830
Impact of the Stator Winding Topology on the Fault Harmonic Components in Induction Motors (Seddik Tabet, Adel ghoggal, Salah Eddine Zouzou)....Pages 831-842
Sensor Fault Detection for Uncertain T-S DC Model with Descriptor Observer Approach (Moussaoui Lotfi, Aouaouda Sabrina, Righi Ines)....Pages 843-855
Detection of Stator Interturn Fault in Double Star Induction Motors (R. Hachelaf, N. Boumalha, D. Kouchih, M. Tadjine, M. S. Boucherit)....Pages 857-870
Fuzzy-Expert System Fault Tolerant Control of IGBT Open Switch Fault for RSC in Wind Turbine System (Amina Bouzekri, Tayab Allaoui)....Pages 871-883
Multiclass Support Vector Machine Based Bearing Fault Detection Using Vibration Signal Analysis (Issam Attoui, Nadir Fergani, Nadir Boutasseta, Brahim Oudjani, Mohammed Salah Bouakkaz, Ahmed Bouraiou)....Pages 885-895
Processing Signal Parameters Based Fuzzy Inference System Classifier for Analog Circuit Single Parametric Faults (Imad Laidani, Naceredine Bourouba)....Pages 897-913
Line Current Spectrum Analysis as a Technique to Diagnose the Demagnetization Fault in Permanent Magnet Synchronous Motor (Zakaria Gherabi, Djilali Toumi, Noureddine Benouzza, Mostefa Becheikh)....Pages 915-930
Fault-Tolerant Path-Tracking Control with PID Controller for 4ws4wd Electric Vehicles (Youcef Zennir, Sami Allou, J. J. Fernandez-Lozano)....Pages 931-945
Discrete Faults Diagnosis for the Multicellular Converter: Algebraic Approach (K. Haddadi, N. Gazzam, I. Ameur, A. Benalia)....Pages 947-959
Fault Diagnosis of Uncertain Hybrid Actuators Based Model Predictive Control (Zahaf Abdelmalek, Bounemeur Abdelhamid, Bououden Sofiane, Ilyes Boulkaibet)....Pages 961-971
Fault Tolerant Predictive Control for Constrained Hybrid Systems with Sensors Failures (Zahaf Abdelmalek, Mohamed Chemachema, Bououden Sofiane, Ilyes Boulkaibet)....Pages 973-984
Active Adaptive Fuzzy Fault-Tolerant Control for a Class of Nonlinear Systems with Actuator Faults (Abdelhamid Bounemeur, Mohamed Chemachema)....Pages 985-999
Front Matter ....Pages 1001-1001
Simultaneous Localization and Mapping Algorithm based on 3D Laser for Unmanned Aerial Vehicle (Fethi Demim, Abdelkrim Nemra, Oumima Mouali, Mehdia Hedir, Abdenebi Rouigueb, Mustapha Hamerlain et al.)....Pages 1003-1020
Comparison Between Gabor Filters and Wavelets Transform for Classification of Textured Images (Abdelkader Zitouni, Fatiha Benkouider, Fatima Chouireb, Mohammed Belkheiri)....Pages 1021-1031
High Level Modeling and Hardware Implementation of Image Processing Algorithms Using XSG (Kamel Messaoudi, Hakim Doghmane, Hocine Bourouba, E. B. Bourennane, S. Toumi)....Pages 1033-1045
Proposal for a Radar Detection Architecture Based on the Knowledge Based Systems Exploitation (Abdellatif Rouabah, Hamza Zeraoula, M’hamed Hamadouche, Kamal Tourche)....Pages 1047-1060
Comparative Analysis and Simulation of Propulsion and Energy Systems for Satellite Communication (Jalal Eddine Benmansour, Boulanouar Khouane, Tarek Badis)....Pages 1061-1069
Narrowband Sensor-Antenna Based on an Interdigital Capacitor (Rafik Khelladi, Mustapha Djeddou, Farid Ghanem)....Pages 1071-1079
Secure Color Image Transmission Based on the Impulsive Synchronization of Fractional-Order Chaotic Maps Over a Single Channel (Ouerdia Megherbi, Hamid Hamiche, Saïd Djennoune, Maamar Bettayeb)....Pages 1081-1095
Performance of FBMC for Backhauling 5G (Abdelhamid Bounegab, Mohammed Boulesbaa)....Pages 1097-1109
Flexible FPGA-Based RFID Emulation Platform for Experimental Purpose (Ibrahim Mezzah, Omar Kermia, Hamimi Chemali)....Pages 1111-1120
Human Activity Recognition Based on Ensemble Classifier Model (Souhila Kahlouche, Mahmoud Belhocine)....Pages 1121-1132
Security in the Internet of Things: Recent Challenges and Solutions (Hamza Belkhiri, Abderraouf Messai, Mohamed Belaoued, Farhi Haider)....Pages 1133-1145
Characterization of Melanoma Using Convolutional Neural Networks and Dermoscopic Images (Abdelghani Tafsast, Mohamed Laid Hadjili, Ayache Bouakaz, Nabil Benoudjit)....Pages 1147-1155
Advanced Feedforward-and-Feedback Decorrelation Algorithms for Speech Quality Enhancement (Rédha Bendoumia)....Pages 1157-1170
New Log CFAR Radar for Compound Sea Clutter (Nouh Guidoum, Faouzi Soltani, Khadidja Belhi)....Pages 1171-1179
Stochastic Analysis of the Crosstalk Between Transmission Lines (Hemza Gheddar, M. Melit, B. Nekhoul)....Pages 1181-1191
Moving Objects Detection and Tracking with Camera Motion Compensation (Bousta Mohamed Akli, Nemra Abdelkrim, Hamami Fatima, Demim Fethi)....Pages 1193-1210
Potential of Using Adaptive Complex Flower Pollination Algorithm to Interferometric Coherence Optimisation (Sofiane Tahraoui, Mounira Ouarzeddine)....Pages 1211-1220
Intelligent Cooperative Method for Medical Image Segmentation (Karima Benhamza, Hamid Seridi)....Pages 1221-1231
An Image Segmentation Model Using a Level Set Method Based on Improved Signed Pressure Force Function SPF (Larbi Messaouda, Zoubeida Messali, Rouini Abdelghani, Samira LARBI)....Pages 1233-1246
New Convergence Architecture Between 5G Mobile Telecommunication Networks and Satellite Networks to Enhance Their Capacities and Improve Their Performance (Salheddine Sadouni, Malek Benslama, André-luc Beylot, Mohammed Nassim Lacheheub, Ouissal Sadouni)....Pages 1247-1257
Comparative Study of Noise Robustness in Visual Attention Models (Sarra Babahenini, Foudil Cherif, Fella Charif)....Pages 1259-1269
The Fast-NLMS Algorithm for Acoustic Echo Cancellation with Double-Talk Detection (Islam Hassani, Madjid Arezki, Ahmed Benallal)....Pages 1271-1280
Design of Sensor Based on CRLH for Liquid Mixture Application (Sekkache Hocine, Mohamed Lashab, Salim Ouchtati)....Pages 1281-1288

Citation preview

Lecture Notes in Electrical Engineering 682

Sofiane Bououden Mohammed Chadli Salim Ziani Ivan Zelinka Editors

Proceedings of the 4th International Conference on Electrical Engineering and Control Applications ICEECA 2019, 17–19 December 2019, Constantine, Algeria

Lecture Notes in Electrical Engineering Volume 682

Series Editors Leopoldo Angrisani, Department of Electrical and Information Technologies Engineering, University of Napoli Federico II, Naples, Italy Marco Arteaga, Departament de Control y Robótica, Universidad Nacional Autónoma de México, Coyoacán, Mexico Bijaya Ketan Panigrahi, Electrical Engineering, Indian Institute of Technology Delhi, New Delhi, Delhi, India Samarjit Chakraborty, Fakultät für Elektrotechnik und Informationstechnik, TU München, Munich, Germany Jiming Chen, Zhejiang University, Hangzhou, Zhejiang, China Shanben Chen, Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai, China Tan Kay Chen, Department of Electrical and Computer Engineering, National University of Singapore, Singapore, Singapore Rüdiger Dillmann, Humanoids and Intelligent Systems Laboratory, Karlsruhe Institute for Technology, Karlsruhe, Germany Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, China Gianluigi Ferrari, Università di Parma, Parma, Italy Manuel Ferre, Centre for Automation and Robotics CAR (UPM-CSIC), Universidad Politécnica de Madrid, Madrid, Spain Sandra Hirche, Department of Electrical Engineering and Information Science, Technische Universität München, Munich, Germany Faryar Jabbari, Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA, USA Limin Jia, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland Alaa Khamis, German University in Egypt El Tagamoa El Khames, New Cairo City, Egypt Torsten Kroeger, Stanford University, Stanford, CA, USA Qilian Liang, Department of Electrical Engineering, University of Texas at Arlington, Arlington, TX, USA Ferran Martín, Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain Tan Cher Ming, College of Engineering, Nanyang Technological University, Singapore, Singapore Wolfgang Minker, Institute of Information Technology, University of Ulm, Ulm, Germany Pradeep Misra, Department of Electrical Engineering, Wright State University, Dayton, OH, USA Sebastian Möller, Quality and Usability Laboratory, TU Berlin, Berlin, Germany Subhas Mukhopadhyay, School of Engineering & Advanced Technology, Massey University, Palmerston North, Manawatu-Wanganui, New Zealand Cun-Zheng Ning, Electrical Engineering, Arizona State University, Tempe, AZ, USA Toyoaki Nishida, Graduate School of Informatics, Kyoto University, Kyoto, Japan Federica Pascucci, Dipartimento di Ingegneria, Università degli Studi “Roma Tre”, Rome, Italy Yong Qin, State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China Gan Woon Seng, School of Electrical & Electronic Engineering, Nanyang Technological University, Singapore, Singapore Joachim Speidel, Institute of Telecommunications, Universität Stuttgart, Stuttgart, Germany Germano Veiga, Campus da FEUP, INESC Porto, Porto, Portugal Haitao Wu, Academy of Opto-electronics, Chinese Academy of Sciences, Beijing, China Junjie James Zhang, Charlotte, NC, USA

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Sofiane Bououden Mohammed Chadli Salim Ziani Ivan Zelinka •

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Editors

Proceedings of the 4th International Conference on Electrical Engineering and Control Applications ICEECA 2019, 17–19 December 2019, Constantine, Algeria

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Editors Sofiane Bououden Faculty of Sciences and Technology University Abbes Laghrour Khenchela Khenchela, Algeria

Mohammed Chadli University of Evry, IBISC University of Paris-Saclay Evry, France

Salim Ziani Department of Electronics University of Frères Mentouri Constantine 1 Constantine, Algeria

Ivan Zelinka Faculty of Electrical Engineering and Computer Science VŠB TU Ostrava Ostrava, Czech Republic

ISSN 1876-1100 ISSN 1876-1119 (electronic) Lecture Notes in Electrical Engineering ISBN 978-981-15-6402-4 ISBN 978-981-15-6403-1 (eBook) https://doi.org/10.1007/978-981-15-6403-1 © Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Organization

Conference The International Conference on Electrical Engineering and Control Applications (ICEECA) provides a forum for specialists and practitioners to present and discuss their research results in several areas of the conference and also state-of-the-art findings in using the applied electrical engineering and automatic control to solve national problems that face developing countries. The conference ICEECA publishes papers on theoretical analysis, experimental studies, and applications in the domain of automatic control and computer engineering. The objective of the conference is not only the exchange of knowledge and experience, since the conference is an open door to students, but also provides opportunities for researchers to target future collaboration on current issues.

General Chairs Salim Ziani, University of Frères Mentouri Constantine 1, Algeria Mohammed Chadli, University Paris-Saclay, IBISC Laboratory, University of Evry, France

Program Chairs Sofiane Bououden, University of Khenchela, Algeria Hamid Reza Karimi, Polytechnic of Milan, Italy Ivan Zelinka, Technical University of Ostrava, Czech Republic

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Steering Committee Sofiane Bououden, University of Khenchela, Algeria Mohammed Chadli, University Paris-Saclay, UEVE-IBISC Laboratory, France Peng Shi, Victoria University, Australia Salim Ziani, University of Frères Mentouri Constantine 1, Algeria

Local Committee ICEECA established a local committee from the department of electronics as an organizer, faculty of sciences of the technologies, University Frères Mentouri Constantine 1, Algeria. The list below comprises the local committee members. Aris Skander (Chair), University of Frères Mentouri Constantine 1, Algeria Ameur Ikhlef, University of Frères Mentouri Constantine 1, Algeria Kheireddine Cheikh, University of Frères Mentouri Constantine 1, Algeria Kheireddine Lamamra, University of Oum El Bouaghi, Algeria Abdelmalek Zahaf, University of Frères Mentouri Constantine 1, Algeria Merzoug Ammari, University of Frères Mentouri Constantine 1, Algeria Mohamed Chemachema, University of Frères Mentouri Constantine 1, Algeria Mohammed Raslain, University of Frères Mentouri Constantine 1, Algeria Ouassila Bourbia, University of Frères Mentouri Constantine 1, Algeria

International Advisory Committee Abedelfetah Charef, University of Frères Mentouri Constantine 1, Algeria Abedlhamid Tayebi, Lakehead University, Canada Hamid Reza Karimi, Polytechnic of Milan, Italy Ivan Zelinka, Technical University of Ostrava, Czech Republic L. X. Zhang, Harbin Institute of Technology, China N. B. Braiek, Ecole Polytechnique de Tunis, Tunisia Said Mammar, University Paris-Saclay, UEVE-IBISC Laboratory, France

Scientific Program Committee ICEECA established an international committee of selected well-known experts in electrical engineering who are willing to be mentioned in the program and to review a set of papers each year. The list below comprises the scientific program committee members.

Organization

Abdelaziz Hamzaoui, University of Reims, France Abdeldjalil Ouahabi, Polytech Tours, France Abdelhafid Zeroual, University of Skikda, Algeria Abdelhak Bennia, University of frères Mentouri Constantine 1, Algeria Abdellah Kouzou, University of Djelfa, Algeria Abedlhamid Tayebi, Lakehead University, Canada Ahmed Chemori, University Montpellier 2, France Ahmed Hafaifa, University of Djelfa, Algeria Ahmed Rachid, University of Picardie Amiens, France Ali Zemouche, Université de Lorraine, Nancy, France Amar Djouak, Université Catholique de Lille, France Ameur Ikhlef, University of frères Mentouri Constantine 1, Algeria Aouaouda Sabrina, Université Souk Ahras, Algeria Aziz Naamane, Université de Marseille Nord, France Barkat Ouarda, University of frères Mentouri Constantine 1, Algeria Bhekisipho Twala, University of Johannesburg, South Africa Bin Haji Hassan Masjuki, University of Malaya, Malaysia Carlos Astorga Zaragoza, CENIDET, Mexico Chérif Chibane, Massachusetts Institute of Technology, Boston, USA Chériti Ahmed, Université Trois-Rivières, Canada Chokri Ben Salah, Université de Sousse, Tunisie De Lima Neto Fernando, University of Pernambuco, Recife, Brazil Djillali Bouagada, University of Mostaganem, Algeria Faouzi Bouani, Ecole Nationale d’Ingénieurs de Tunis, Tunisia Farid Marir, University of frères Mentouri Constantine 1, Algeria Faycal Megri, Université of Oum El Bouaghi Fella Hachouf, University of frères Mentouri Constantine 1, Algeria Fernando Tadeo, University of Valladolid, Spain Fethi Demim, Ecole Militaire Polytechnique, Algiers, Algeria Fouad Allouani, University of Khenchela, Algeria Fouad Kerrour, University of frères Mentouri Constantine 1, Algeria Fouzi Soltani, University of frères Mentouri Constantine 1, Algeria Gloria Osorio Gordillo, CENIDET, Mexico Hamid Reza Karimi, Ecole Polytechnique de Milan, Italy Hervé Coppier, ESIEE, France Ilyes Boulkaibet, University of Johannesburg, South Africa Ivan Zelinka, Technical University of Ostrava, Czech Republic Jean-Jacques Loiseau, IRCyN, France Kamel Guesmi, University of Djelfa, Algeria Kheireddine Lamamra, University of Oum El Bouaghi, Algeria Lamir Saidi, University of Batna, Algeria Maamer Bettayeb, University of Sharjah, UAE Malika Kandouci, University of Sidi Bel-Abbes, Algeria Manuel Adam Medina, CENIDET, Mexico Mehadji Abri, University of Tlemcen, Algeria

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Organization

Mohamed Boumehraz, University of Biskra, Algeria Mohamed Boutayeb, Centre de Recherche en Automatique de Nancy, France Mohamed Chemachema, University of frères Mentouri Constantine 1, Algeria Mohamed Darouach, University of Lorraine, France Mohamed Elleuch, University of Tunis, Tunisia Mohamed Lahdi Riabi, University of frères Mentouri Constantine 1, Algeria Mohamed Lashab, University of Oum El Bouaghi, Algeria Mohamed Seghir Boucherit, Ecole Nationale Polytechnique, Algeria Mohamed Toufik Benhabiles, University of frères Mentouri Constantine 1, Algeria Mohammed Msaad, University of Caen Normandie, France Mostapha Ziad, University of Suffolk, Boston, USA Nadjet Zioui, Université du Québec à Trois-Rivières, QC, Canada Najib Essounbouli, University of Reims Champagne Ardenne, France Noureddine Slimane, University of Batna 2, Algeria Ouassila Bourbia, University of frères Mentouri Constantine 1, Algeria Peng Shi, Victoria University, Australia Pierre Borne, Ecole Centrale de Lille, France Rachid Illoul, Ecole Nationale Polytechnique de Alger, Algeria Roman Senkerik, Tomas Bata University in Zlin, Czech Republic Salim Ziani, University of frères Mentouri Constantine 1, Algeria Said Benierbeh, University of frères Mentouri Constantine 1, Algeria Said Hassaine, University of Tiaret, Algeria Zidani Mosbah, University of Biskra, Algeria Zoheir Hammoudi, University of Constantine 1, Algeria

Preface

This proceedings book about the advanced control engineering methods in electrical engineering systems conference contains accepted papers presenting the most interesting state of the art on this field of research. Presented topics are focused on classical as well as modern methods for modeling, control, identification, and simulation of complex systems with applications in science and engineering. Topics are (but not limited to) control and systems engineering, renewable energy, faults diagnosis–fault-tolerant control, large-scale systems, fractional-order systems, unconventional algorithms in control engineering, signal and communications, and much more. The control of complex systems dynamics, analysis, and modeling of its behavior and structure is a vitally important problem in engineering, economy, and generally in science today. Examples of such systems can be seen in the world around us and are a part of our everyday life. Application of modern methods for control, electronics, signal processing, and more can be found in our mobile phones, car engines, home devices as, for example, washing machine is as well as in such advanced devices as space probes and communication with them. The main aim of the conference is to create a periodical possibility for students, academics, and researchers to exchange their ideas and novel methods. This conference will establish a forum for the presentation and discussion of recent trends in the area of applications of various modern as well as classical methods for researchers, students, and academics. The accepted selection of papers was extremely rigorously reviewed in order to maintain the high quality of the conference that is supported by organizing universities and related research grants. Regular and student’s papers have been submitted to the conference, and in accordance with review process, have been accepted after a positive review. We would like to thank the members of the program committees and reviewers for their hard work. We believe that this conference represents a high-standard conference in the domain of control, modeling, and analysis of dynamical and electronic systems. We would like to thank all the contributing authors, as well as the members of the program committees and the local organizing committee for their hard and highly valuable work. Their work has definitely contributed to the success of the conference.

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Preface

This event is supported by the University of frères Mentouri Constantine 1, Algeria, and the University Paris-Saclay, IBISC Laboratory, University of Evry, France, and mainly financed by DGRSDT, SONATRACH society, and SONELGAZ society. Khenchela, Algeria Evry, France Constantine, Algeria Ostrava, Czech Republic

Sofiane Bououden Mohammed Chadli Salim Ziani Ivan Zelinka

Contents

Control and Systems Engineering (CSE) Sensorless Speed Control of Induction Motor Used Differential Flatness Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yassine Beddiaf, Fatiha Zidani, and Larbi. Chrifi-Alaoui Multi-UAVs Coverage Path Planning . . . . . . . . . . . . . . . . . . . . . . . . . . Abdelwahhab Bouras, Yasser Bouzid, and Mohamed Guiatni

3 23

Reducing the Collision Checking Time in Cluttered Environment for Sampling-Based Motion Planning . . . . . . . . . . . . . . . . . . . . . . . . . . Amine Belaid and Boubekeur Mendil

37

Random Scaling-Based Bat Algorithm for Greenhouse Thermal Model Identification and Experimental Validation . . . . . . . . . . . . . . . . Mounir Guesbaya, Hassina Megherbi, and Ahmed Chaouki Megherbi

47

Control and Observation of Induction Motor Using First-Order Sliding Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Salah Eddine Farhi, Djamel Sakri, and Noureddine Goléa

63

Cuckoo Search Algorithm for Solving the Problem of Unit-Commitment with Vehicle-to-Grid . . . . . . . . . . . . . . . . . . . . . . Amel Terki and Hamid Boubertakh

77

A Comparative Study of MPPT Techniques for Standalone Hybrid PV-Wind with Power Management . . . . . . . . . . . . . . . . . . . . . . . . . . . Tarek Boutabba, Hamza Sahraoui, Mouhamed Lamine Bechka, Said Drid, and Larbi Chrifi-Alaoui Adapted Search Equations of Artificial Bee Colony Applied to Feature Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hayet Djellali, Souad Guessoum, and Nacira Ghoualmi-Zine

93

109

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Contents

Networked Cooperation-Based Distributed Model Predictive Control Using Laguerre Functions for Large-Scale Systems . . . . . . . . . . . . . . . Kamel Menighed, Ahmed Chemori, Boumedyen Boussaid, and Joseph Julien Yamé

123

Direct Torque Control Using Fuzzy Second Order Sliding Mode Speed Regulator of Double Star Permanent Magnet Synchronous Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Louanasse Laggoun, Brahim Kiyyour, Ghoulemallah Boukhalfa, Sebti Belkacem, and Said Benaggoune

139

Fractional Order Integral Controller Design Based on a Bode’s Ideal Transfer Function: Application to the Control of a Single Tank Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chahira Boussalem, Rachid Mansouri, Maamar Bettayeb, and Mustapha Hamerlain

155

Modeling of Torsional Vibrations Dynamic in Drill-String by Using PI-Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Riane, M. Kidouche, M. Z. Doghmane, and R. Illoul

171

Power System Generator Coherency Identification for Large Disturbances by Koopman Modes Analysis . . . . . . . . . . . . . . . . . . . . . Zahra Jlassi, Khadija Ben Kilani, Mohamed Elleuch, and Lamine Mili

187

Analysis and Experimental Validation of Single Phase Series Resonance Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alla Eddine Toubal Maamar, M’hamed Helaimi, and Rachid Taleb

203

Optimal Decentralized State Control of Multi-machine Power System Based on Loop Multi-overlapping Decomposition Strategy . . . . . . . . . . M. Z. Doghmane and M. Kidouche

217

Large Synchronverter Integration in Power Electrical System: Impacts on SCR and CCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Raouia Aouini, Khadija Ben Kilani, Mohamed Elleuch, and Quoc Tuan TRAN

233

Immersion and Invariance Based Adaptive Dynamic Surface Control for Parametric Strict-Feedback Nonlinear Systems . . . . . . . . . . . . . . . . Y. Soukkou, S. Labiod, M. Tadjine, Q. M. Zhu, and M. Nibouche

247

Design of an Adaptive Fuzzy Backstepping Synergetic Control Scheme for a Class of Strict-Feedback Nonlinear Systems . . . . . . . . . . Aissa Rebai, Kamel Guesmi, and Mohamed Bougrine

263

Improvement of the Stability Performance of a Quad-Copter Helicopter by a Neuro-Fuzzy Controller . . . . . . . . . . . . . . . . . . . . . . . . Djalal Baladji, Kheireddine Lamamra, and Farida Batat

279

Contents

The Importance of Applying Artificial Intelligence on Unmanned Aerial Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Amine Mohammed Taberkit, Ahmed Kechida, and Abdelmalek Bouguettaya

xiii

293

Fuzzy H∞ Delay-Independent Stabilization of Depth Control for Underwater Vehicle with Input Constraint . . . . . . . . . . . . . . . . . . . Mohamed Nasri, Dounia Saifia, Salim Labiod, and Mohammed Chadli

305

Petri Type 2 Fuzzy Neural Networks (PT2FNN) for Identification and Control of Dynamic Systems—A New Structure and a Comparative Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Youssouf Bibi, Mohamed Seghiri, Omar Bouhali, Abdelkarim Nemra, and Tarek Bouktir

319

Design Fractional Order PI Controller with Decoupler for MIMO Process Using Diffusive Representation . . . . . . . . . . . . . . . . Sami Laifa, Badreddine Boudjehem, and Hamza Gasmi

331

Direct Sliding Mode Control of Transient Power in Microgrid During Grid Failure ‘‘Unintentional Islanding’’ . . . . . . . . . . . . . . . . . . . . . . . . I. Ameur, N. Gazzam, and A. Benalia

347

New Virtual Synchronous Generator Control Technique of Distributed Generator Unit to Improve Transient Response of the Microgrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yacine Daili and Abdelghani Harrag Design and Analysis of Robust Nonlinear Synergetic Controller for a PMDC Motor Driven Wire-Feeder System (WFS) . . . . . . . . . . . . Noureddine Hamouda, Badreddine Babes, and Amar Boutaghane Simulation Study of the Dual Star Permanent Magnet Synchronous Machine Using Different Modeling Approaches . . . . . . . . . . . . . . . . . . Elyazid Amirouche, Khaldi Lyes, Ghedamsi Kaci, and Aouzellag Djamal Modeling and Experimental Identification of Salient-Pole Synchronous Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Khalil-Errahmane Sari and Bilal Sari Power Quality Improvement of PWM Rectifier-Inverter System Using Model Predictive Control for an AC Electric Drive Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abdelkarim Ammar Novel Smart Air Quality Monitoring System Based on UAV Quadrotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mehdi Zareb, Benaoumer Bakhti, Yasser Bouzid, Hamza Kadourbenkada, Kamel Bouzgou, and Wahid Nouibat

361

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Contents

GA Based Adaptive Fuzzy Sliding Mode for 3 DOF Spatial Arm Manipulator Trajectory Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rabie Belloumi and Noureddine Slimane

455

Robust Adaptive Fuzzy Approach with Unknown Control Gain Direction and External Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . Ouassila Bourebia and Nassira Zerari

465

Adaptive Neural-Network Control Design for Uncertain CSTR System with Unknown Input Dead-Zone and Output Constraint . . . . . Zerari Nassira and Chemachema Mohamed

477

Adaptive Fuzzy Fault-Tolerant Control Using Nussbaum Gain for a Class of SISO Nonlinear Systems with Unknown Directions . . . . Abdelhamid Bounemeur, Mohamed chemachema, Abdelmalek Zahaf, and Sofiane Bououden

493

Renewable Energy (RE) Optimal Integration of Renewable Distributed Generation Using the Whale Optimization Algorithm for Techno-Economic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Samir Settoul, Rachid Chenni, Mohamed Zellagui, and Hassan Nouri Enhancement of Power System Transient Stability with a Large Penetration of Solar Photovoltaic Using Facts . . . . . . . . . . . . . . . . . . . Ikram Boucetta, Naimi Djemai, Salhi Ahmed, and Zellouma Laid Minimization of the Energy Consumption of an Aircraft . . . . . . . . . . . Abbes Lounis, Kahina Louadj, and Mohamed Aidene Study to Improve the Technical Parameters for the Optimising the Injection of the Photovoltaic Energy into an Electrical Network via a Line 30 KV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bahriya Badache, Hocine Labar, and Mounia Samira Kelaiaia Second-Order Super-Twisting Control of an Autonomous Wind Energy Conversion System Based on PMSG for Robustness and Chattering Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abderrahmane Abdellah, Djilali Toumi, and M’hamed Larbi

513

533 549

565

579

Open-Switch Faults Based on Six/Five-Leg Reconfigurable AC-DC-AC Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sahraoui Khaled and Gaoui Bachir

595

A Robust Model Predictive Control for a DC-DC Boost Converter Subject to Input Saturation: An LMI Approach . . . . . . . . . . . . . . . . . O. Hazil, S. Bououden, and M. Chadli

615

Contents

Strategy for Optimization of Energy Management Based on Fuzzy Logic in an FCEV with a Contribution of a Photovoltaic Source . . . . . Said Belhadj, Kaci Ghedamsi, and Zina Larabi Modeling and Analysis of an Electromagnetic Vibration Energy Harvester for Automotive Suspension . . . . . . . . . . . . . . . . . . . . . . . . . . Mustapha Zaouia, Bachir Ouartal, Nacereddine Benamrouche, and Arezki Fekik Experimental and Numerical Study of Hybrid PV/Thermal Solar Collector Provided with Self Ventilation and Tracking Structure . . . . . Mohamed El-Amine Slimani, Rabah Sellami, Achour Mahrane, and Madjid Amirat Maximum Power Point Tracking (MPPT) for a PV System Based on Artificial Neural Network ANN and Comparison with P&O Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aicha Djalab, Nassim Sabri, and Ali Teta Development of a Stand-Alone Connected PV System Based on Packe U Cell Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Khaled Rayane, Mohamed Bougrine, Atallah Benalia, and Kamel Guesmi A Novel Hybrid Photovoltaic/Thermal Bi-Fluid (Air/Water) Solar Collector: An Experimental Investigation . . . . . . . . . . . . . . . . . . . . . . . Mohamed El-Amine Slimani, Rabah Sellami, Mohammed Said, and Amina Bouderbal

xv

631

645

659

671

683

697

A DC/DC Buck Converter Voltage Regulation Using an Adaptive Fuzzy Fast Terminal Synergetic Control . . . . . . . . . . . . . . . . . . . . . . . Noureddine Hamouda and Badreddine Babes

711

Study of Electrical Field Distribution in the Insulation of High-Voltage Cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rouini Abdelghani, Kouzou Abdellah, and Larbi Messaouda

723

Self Tuning Fuzzy Maximum Power Tracking Control of PMSG Wind Energy Conversion System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chaib Housseyn, Mihoub Youcef, and Hassaine Said

735

A Backstepping Controller for Interleaved Boost DC–DC Converter Improving Fuel Cell Voltage Regulation . . . . . . . . . . . . . . . . . . . . . . . . Ali Dali, Samir Abdelmalek, and Maamar Bettayeb

751

The Impact of the Solar Radiation Profile on Sizing and Performance of Photovoltaic Systems, Case Study Tamanrasset, Algeria . . . . . . . . . Madjid Chikh, Aicha Degla, and Achour Mahrane

763

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Contents

Investigation and Prototyping Implementation of a Novel Solar Water Collector Based on Used Engine Oil as HTF . . . . . . . . . . . . . . . Oussama Touaba, Salah Mohamed AitCheikh, Mohamed El-Amine Slimani, Ahmed Bouraiou, and Abderrezzaq Ziane Piezoelectric Energy Harvesting Based Autonomous Vehicles’ Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nadjet Zioui, Sousso Kelouwani, and Gilbert Lebrun

779

791

Faults Diagnosis-Faults Tolerant Control (FTC) Fault Detection of Uncertain Systems Based on Interval Data Driven Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chouaib Chakour

803

Discrimination of Unbalanced Supply and Stator Interturn Faults in Induction Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Kouchih, R. Hachelaf, and N. Boumalha

815

Impact of the Stator Winding Topology on the Fault Harmonic Components in Induction Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Seddik Tabet, Adel ghoggal, and Salah Eddine Zouzou

831

Sensor Fault Detection for Uncertain T-S DC Model with Descriptor Observer Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moussaoui Lotfi, Aouaouda Sabrina, and Righi Ines

843

Detection of Stator Interturn Fault in Double Star Induction Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Hachelaf, N. Boumalha, D. Kouchih, M. Tadjine, and M. S. Boucherit

857

Fuzzy-Expert System Fault Tolerant Control of IGBT Open Switch Fault for RSC in Wind Turbine System . . . . . . . . . . . . . . . . . . . . . . . . Amina Bouzekri and Tayab Allaoui

871

Multiclass Support Vector Machine Based Bearing Fault Detection Using Vibration Signal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Issam Attoui, Nadir Fergani, Nadir Boutasseta, Brahim Oudjani, Mohammed Salah Bouakkaz, and Ahmed Bouraiou

885

Processing Signal Parameters Based Fuzzy Inference System Classifier for Analog Circuit Single Parametric Faults . . . . . . . . . . . . . Imad Laidani and Naceredine Bourouba

897

Line Current Spectrum Analysis as a Technique to Diagnose the Demagnetization Fault in Permanent Magnet Synchronous Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zakaria Gherabi, Djilali Toumi, Noureddine Benouzza, and Mostefa Becheikh

915

Contents

xvii

Fault-Tolerant Path-Tracking Control with PID Controller for 4ws4wd Electric Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Youcef Zennir, Sami Allou, and J. J. Fernandez-Lozano

931

Discrete Faults Diagnosis for the Multicellular Converter: Algebraic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Haddadi, N. Gazzam, I. Ameur, and A. Benalia

947

Fault Diagnosis of Uncertain Hybrid Actuators Based Model Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zahaf Abdelmalek, Bounemeur Abdelhamid, Bououden Sofiane, and Ilyes Boulkaibet Fault Tolerant Predictive Control for Constrained Hybrid Systems with Sensors Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zahaf Abdelmalek, Mohamed Chemachema, Bououden Sofiane, and Ilyes Boulkaibet Active Adaptive Fuzzy Fault-Tolerant Control for a Class of Nonlinear Systems with Actuator Faults . . . . . . . . . . . . . . . . . . . . . . Abdelhamid Bounemeur and Mohamed Chemachema

961

973

985

Signal and Communications (SC) Simultaneous Localization and Mapping Algorithm based on 3D Laser for Unmanned Aerial Vehicle . . . . . . . . . . . . . . . . . . . . . . 1003 Fethi Demim, Abdelkrim Nemra, Oumima Mouali, Mehdia Hedir, Abdenebi Rouigueb, Mustapha Hamerlain, Mohamed Amine Bendoumi, and Abdelouahab Bazoula Comparison Between Gabor Filters and Wavelets Transform for Classification of Textured Images . . . . . . . . . . . . . . . . . . . . . . . . . . 1021 Abdelkader Zitouni, Fatiha Benkouider, Fatima Chouireb, and Mohammed Belkheiri High Level Modeling and Hardware Implementation of Image Processing Algorithms Using XSG . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033 Kamel Messaoudi, Hakim Doghmane, Hocine Bourouba, E. B. Bourennane, and S. Toumi Proposal for a Radar Detection Architecture Based on the Knowledge Based Systems Exploitation . . . . . . . . . . . . . . . . . . . 1047 Abdellatif Rouabah, Hamza Zeraoula, M’hamed Hamadouche, and Kamal Tourche Comparative Analysis and Simulation of Propulsion and Energy Systems for Satellite Communication . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 Jalal Eddine Benmansour, Boulanouar Khouane, and Tarek Badis

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Narrowband Sensor-Antenna Based on an Interdigital Capacitor . . . . 1071 Rafik Khelladi, Mustapha Djeddou, and Farid Ghanem Secure Color Image Transmission Based on the Impulsive Synchronization of Fractional-Order Chaotic Maps Over a Single Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 Ouerdia Megherbi, Hamid Hamiche, Saïd Djennoune, and Maamar Bettayeb Performance of FBMC for Backhauling 5G . . . . . . . . . . . . . . . . . . . . . 1097 Abdelhamid Bounegab and Mohammed Boulesbaa Flexible FPGA-Based RFID Emulation Platform for Experimental Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111 Ibrahim Mezzah, Omar Kermia, and Hamimi Chemali Human Activity Recognition Based on Ensemble Classifier Model . . . . 1121 Souhila Kahlouche and Mahmoud Belhocine Security in the Internet of Things: Recent Challenges and Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133 Hamza Belkhiri, Abderraouf Messai, Mohamed Belaoued, and Farhi Haider Characterization of Melanoma Using Convolutional Neural Networks and Dermoscopic Images . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147 Abdelghani Tafsast, Mohamed Laid Hadjili, Ayache Bouakaz, and Nabil Benoudjit Advanced Feedforward-and-Feedback Decorrelation Algorithms for Speech Quality Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157 Rédha Bendoumia New Log CFAR Radar for Compound Sea Clutter . . . . . . . . . . . . . . . 1171 Nouh Guidoum, Faouzi Soltani, and Khadidja Belhi Stochastic Analysis of the Crosstalk Between Transmission Lines . . . . 1181 Hemza Gheddar, M. Melit, and B. Nekhoul Moving Objects Detection and Tracking with Camera Motion Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193 Bousta Mohamed Akli, Nemra Abdelkrim, Hamami Fatima, and Demim Fethi Potential of Using Adaptive Complex Flower Pollination Algorithm to Interferometric Coherence Optimisation . . . . . . . . . . . . . . . . . . . . . . 1211 Sofiane Tahraoui and Mounira Ouarzeddine Intelligent Cooperative Method for Medical Image Segmentation . . . . 1221 Karima Benhamza and Hamid Seridi

Contents

xix

An Image Segmentation Model Using a Level Set Method Based on Improved Signed Pressure Force Function SPF . . . . . . . . . . . . . . . . 1233 Larbi Messaouda, Zoubeida Messali, Rouini Abdelghani, and Samira LARBI New Convergence Architecture Between 5G Mobile Telecommunication Networks and Satellite Networks to Enhance Their Capacities and Improve Their Performance . . . . . . . . . . . . . . . . 1247 Salheddine Sadouni, Malek Benslama, André-luc Beylot, Mohammed Nassim Lacheheub, and Ouissal Sadouni Comparative Study of Noise Robustness in Visual Attention Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1259 Sarra Babahenini, Foudil Cherif, and Fella Charif The Fast-NLMS Algorithm for Acoustic Echo Cancellation with Double-Talk Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1271 Islam Hassani, Madjid Arezki, and Ahmed Benallal Design of Sensor Based on CRLH for Liquid Mixture Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1281 Sekkache Hocine, Mohamed Lashab, and Salim Ouchtati

Control and Systems Engineering (CSE)

Sensorless Speed Control of Induction Motor Used Differential Flatness Theory Yassine Beddiaf, Fatiha Zidani, and Larbi. Chrifi-Alaoui

Abstract In this paper, differential flatness theory and reduced order observer are used to develop sensorless control for induction motor. The concept of flatness is based on the description of flat outlets. The first step is showing that the induction motor is a differentially flat system, since all state variables of the circuits describing the motor’s dynamics can be expressed as functions. In the second step, when the chosen flat outputs involve non available state variable measurements the state observer is used to estimate them. Simulation and experimental results are presented to illustrate the effectiveness of the proposed approach for sensorless control of induction motor. Keywords Induction motor · Differential flatness · Sensorless vector control

1 Introduction The induction machine, is the most robust and low cost machine. The progress made in control and the considerable technological advances, both in the field of power electronics and microelectronics, have made possible the implementation of the powerful controls of this machine. However, many problems remain. As the parameters variation, the presence of the mechanical sensor and the degraded functioning are all difficulties that have resulted from the eagerness of the researchers, as evidenced by the ever-increasing number of publications that study the subject. Y. Beddiaf (B) Department of Electrical Engineering, University of Khenchela, Khenchela, Algeria e-mail: [email protected]; [email protected] F. Zidani Laboratory LSPIE, Department of Electrical Engineering, University of Batna-2, Batna, Algeria e-mail: [email protected] Larbi. Chrifi-Alaoui Laboratoire des technologies innovantes (LTI EA 3988), Amiens, France e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_1

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Several strategies have been proposed in the literature to realize the sensorless control of induction motor. A large part of the proposed methods is based on observers depending on the model of the asynchronous machine [1–5, 12]. Other research is on the contribution of artificial intelligence to improve sensorless control of the machine [6–8]. These techniques do not exactly replace the speed sensor, mainly during in the load application. In this context, we propose the sensorless control based on flatness theory. The flatness theory is a more or less recent notion in automatic that was proposed from 1992 by Lévine et al. [9], Dannehl and Fuchs [10], Fliess et al. [11]. This concept makes it possible to control of dynamic system. The concept of flatness theory is based on the highlighting of flat outlets. The first step of control by the flatness method is to generate a desired trajectory that takes into account the model of system. In the second step, this control requires the design of a loopback control allowing the continuation of this trajectory. The validity of the proposed method is verified using simulation and experimentation.

2 Differential Flatness Theory Consider a dynamic system defined by the following state equation: dX = f (X, U ) dt

(1)

with state vector X (t) ∈ Rn input vector U (t) ∈ Rm where f is a smooth vector field, is differentially flat if there exists a vector y(t) ∈ Rm in the form   y(t) = A X (t), U (t), U˙ (t), . . . . . . , U q (t)

(2)

  X (t) = B y(t), y˙ (t), . . . . . . , y r (t)

(3)

  U (t) = C y(t) y˙ (t), . . . . . . , y r +1 (t)

(4)

Such that

where A, B and C are smooth functions. Therefore that the new systems description is given by the m algebraic variables y(t). Since the components of y(t) are differentially independent, the output plate groups all the free variables of the system. But one can also say, through the Eq. (2), which does not depend on the state and the control, what makes an endogenous

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variable system, for example in the state of an observer who is an exogenous variable of the system. Moreover, the notion of differential equivalence in the Lie-Bäcklund [11] sense shows it well, the number of components of y(t) is given by that of the command that is to say dim y(t) = dim U(t). This property allows knowing a priori the number of free variables that must be found on a model to highlight its flatness. To shed more light on the notion of flatness, one considers the following example: 

X˙ 1 (t) = X 2 (t) X˙ 2 (t) = U (t)

(5)

we define the following variables y(t) = X 2 (t) and U (t) = y˙ (t) So, we deduce t X 1 (t) = X 1 (t0 ) +

y(τ )dτ

(6)

t0

So y(t) cannot be considered a flat output because the relation (2) is not verified. If now, we defines, y(t) = X 1 (t) then: X 2 (t) = y˙ (t) and U (t) = y¨ (t) In this case, we can say that X 1 (t) is a flat output, therefore this system is flat and flat output y(t) = X 1 (t).

2.1 Control Law Synthesis Methodology for Flat Systems The synthesis of a control law for a flat system requires a strategy based on the following approaches: • Generation of the trajectories of the reference flat output yr e f . • Generation of the corresponding reference input u r e f trajectory (controls). • Synthesis of a strategy of stabilization of the control around the planned reference trajectories.

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2.2 Trajectory Planning From the relation (4), if one wishes to obtain for the flat system (1), the trajectory: z d (t) for a time t from t0 to t f , it suffices to impose, on the same time segment, the following open-loop control:   (β) Ud (t) = B z d (t), z˙ d (t), . . . . . . . . . . . . z d (t)

(7)

In the hypothesis of a perfect model, we will then have, for t from t0 to t f , y(t) = yd (t), therefore:   X (t) = X d (t) = A z d (t), z˙ d (t), . . . . . . . . . . . . z d(α) (t)

(8)

  (γ ) y(t) = yd (t) = C z d (t), z˙ d (t), . . . . . . . . . . . . z d (t)

(9)

2.3 Stabilization Around Reference Trajectories A control developed from the flatness concept is established and defined for an open-loop trajectory tracking. The perfect chase is ensured when the system is not disturbed. However, non-linear physical systems are subjected on the one hand to disturbances inherent to their working context, and on the other hand to uncertainties about the parameters. It is therefore necessary to provide a solution to stabilize the system around the trajectories if they are disturbed. Figure 1 illustrates the principle of stabilization of the control by insertion of feedback loop.

3 Flatness Control of Im In this section, we present the analyze of the flatness of IM model then the design its control. Fig. 1 Bloc diagram of stabilization

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3.1 Analysis of the Flatness of IM Model The IM modeling stationary reference frame is given by the following equations: d V¯s = Rs i¯s + ϕ¯s dt

(10)

d 0 = Rr i¯r + ϕ¯r dt

(11)

ϕ¯s = L s i¯s + M i¯r e j pθ1

(12)

ϕ¯r = L r i¯r + M i¯s e j pθ1

(13)

 pM ¯ Jm i s · ϕ¯r∗ Lr

(14)

Ce =

where * represents conjugated complex variables. From Eq. (14), one can write: J

 pM ¯ d 2 θ1 = Jm i s · ϕ¯r∗ − Cr − f θ˙ dt 2 Lr

(15)

We set (ρ) the modulus of the rotor flux and (δ) its position in the stationary reference frame (α, β), the rotor flux is defined as: ϕ¯r = ρe jδ

(16)

ϕr = ϕ¯r e j pθ1 = ρe jα1

(17)

From Fig. 2 we can write

Fig. 2 Angle spotting

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We obtain:  1 L r d ϕ¯r ϕ¯r + i¯s = M Rr dt ce =

p 2 ρ α˙ 1 Rr

(18) (19)

From Eq. (15) we define the following equation:

ρ=

  J · R f θ˙1 + θ¨1 + Cr p α˙ 1

(20)

Therefore   ϕ˜r = ρe jα1 = B θ¨1 , θ˙1 , α1, α˙ 1 , Cr

(21)

If we choose y = (α1 , θ1 ) as a flat output, we obtain: ϕ˜r = B(y, y˙ , y¨ )

(22)

1 d ϕ¯r i˜r = − R dt  r = C θ˙1 , θ¨1 , θ1(3) , α˙ 1, α¨ 1 , Cr , C˙ r

(23)

 e j pθ1  i¯s = ϕ¯r − L r i¯r M   = C θ˙1 , θ¨1 , θ1(3) , α˙ 1, α¨ 1 , Cr , C˙ r

(24)

ϕ¯ s = L s i¯s + Me j pθ1 i¯r   = E θ˙1 , θ¨1 , θ1(3) , α˙ 1, α¨ 1 , Cr , C˙ r

(25)

d V¯s = Rs i¯s ϕ¯s dt   = F θ˙1 , θ¨1 , θ1(3) , θ1(4) , α˙ 1, α¨ 1 , α1(3) , Cr , C˙ r , C¨ r

(26)

We note that Eqs. (17)–(26) satisfy the flatness conditions (2)–(4), so we can say that the induction motor is a differentially flat system.

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3.2 Design of Flatness Control Based of IM The mechanical model of the induction motor in the (d–q) reference frame is given by the following equations system: ⎧ dω ⎪ ⎨ dt =

p ρ M I − fJ ω − CJr J Rr Tr sq dρ = −1 ρ + TMr Isd dt Tr dθ1 =ω dt

⎪ ⎩

(27)

where ρ = ϕr = ϕr d . Replace (Isd , Isq , ρ, Cr ) by (Isd−r e f , Isq−r e f , ρr e f , Cr ) We obtain:  1 Cr ρ˙r e f + ρr e f M

(28)

L r J θ¨1−r e f + f θ˙1−r e f + Cˆ r pM ρr e f

(29)

Isd−r e f = Isq−r e f =

where Cˆ r represents the estimated charge torque. To estimate the load torque, we use the following observer [3]:  d ωˆ dt

  = −l1 ωˆ − ω   ˆ = JpM ρ I − Jf ωˆ − CJr + l2 ωˆ − ω L r r e f sq d Cˆ r dt

(30)

where: l1 and l2 are the observer gains. The flux (ρ) and the estimated speed (ω) ˆ are obtained by the reduced order state observer [12]. The dynamic error of the observer (30) is given by: d dt



1 2



 =

0 −l1 − 1J − Jf + l2



1 2

(31)

where: 1 = Cˆ r − Cr and 2 = ωˆ − ω. The observer gain (30) is chosen in such a way that the dynamic error converges to zero. More details about the reduced order state observer applicable to the induction motor are given in [12].

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Finally, the control of stator currents of the reduced model is to ensured by the PI regulators. ∗ Isd

  = k pρ ρˆ − ρr e f + kiρ

t (ρˆ − ρr e f )dτ

(32)

(ωˆ − ωr e f )dτ

(33)

0

∗ Isq

  = k pω ωˆ − ωr e f + kiω

t 0

3.3 Planning Reference Trajectories Applying the flatness approach, reference trajectories are defined by (ρr e f , θ1−r e f ). The objective of this planning is: – Respect the electromechanical constraints of the IM. – Guarantee the existence of bounded derivatives up to order two. Therefore, the methodology chosen is to apply to the speed and flux set points a second-order filter making it possible to obtain the final reference trajectories that are derivable twice.

3.4 Complete Model Control The proposed flatness-based control scheme with the use of reduced observer for estimation of the no measurable parameters of the motor’s state vector is shown in Fig. 3. Equations (10)–(14) can be written as follows: Lsσ

 d I¯s M ˙ e jδ = V¯s − Rs I¯s − ρ˙ + j δρ dt Lr

(34)

Tr (ρ˙ + j α˙ 1 ρ) = −ρ + Me− jδ I¯s

(35)

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Fig. 3 Bloc diagduram of reduced-order state observer

The nominal control planned voltages are defined as follows:  ∗ d I ∗∗ M ∗∗ ∗∗ dδ + sd − − Isq ρr e f Vsd∗ = σ L s γ Isd dt dt σ L s L r Tr  ∗∗ ∗ d Isq M ∗ ∗∗ ∗∗ dδ ∗ Vsq = σ L s γ Isq + Isd + + pω ρr e f dt dt σ Ls Lr where: δr e f = pθ1−r e f + Currents

∗∗ Isd

and

∗∗ Isq

∗∗ Isd

t

(36) (37)

α˙ 1−r e f dτ .

0

are given by:

=

∗ Isd

  + k pρ ρˆ − ρ ∗ + kiρ

t

  ρˆ − ρ ∗ dτ

(38)

0

∗∗ Isq

=

∗ Isq



+ k pω ρˆ − ρ

∗



t + kiω



 ωˆ − ω∗ dτ

(39)

0

From Eqs. (38), (39) and (27), we can write: ∗∗ d Isd d Isd−r e f Tr = + dt dt M



   kiρ ρˆ − ρr e f  +k pρ −Trρˆ + TMr Isd − ρ˙r e f

(40)

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∗ d Isq

Tr = + dt dt M



 k pω

  kiω ωˆ − ω∗ + pM J Lr

∗∗ ρˆ Isq − Jf ω −

Cˆ r J

− ω˙ ∗



 (41)

Finally the control voltages are given by: ⎛



 ∗∗

Vsd = Vsd∗ + σ L s ⎝k p I sd Isd − Isd + ki I sd

t



⎞  ∗∗ Isd − Isd dτ ⎠



⎞  ∗∗ Isq − Isq dτ ⎠

0

⎛



 ∗∗

Vsq = Vsq∗ + σ L s ⎝k p I sq Isq − Isq + ki I sq

t 0

4 Reduced State Observer In this paper, a reduced state observer is used to estimate the rotor speed and rotor flux [12]. Model (42) is used for realize the observer, that’s magnetizing currents i mr α and i mrβ are used as state variables. stator currents i sα and i sβ being the input variables. The correction of the observer is done from the variables Y and Yˆ which depend on the input vectors and its derivative.   ⎧ di R M 1 sα ⎪ + = − i + ϕ + ωϕ sα r α rβ ⎪ σ Ls σ L s L r  Tr ⎪ dt  ⎪ ⎨ disβ = − σ RL s i sβ + σ LMs L r T1r ϕrβ − ωϕr α + dt dϕr α ⎪ ⎪ = TMr i sα − T1r ϕr α − ωϕrβ ⎪ dt ⎪ ⎩ dϕrβ = TMr i sβ − T1r ϕrβ + ωϕr α dt

1 σ Ls 1 σ Ls

Vsα Vsβ

(42)

The observer is synthesized by following equation: 





  X˙ = A Xˆ + BU + G Y − Yˆ Yˆ = DU + Ji U˙ + C Xˆ







(43)

   R 0 k1 k2 ; D = where: A = ; B = ;G = ; Ji = ω −1 0 −1 −k2 k1 0 R Tr Tr           −1 −ω σ Ls 0 i sα Vsα iˆmr α M2 Tr ˆ ; = Lr ;U = ;Y = ; Yˆ = ; X = ˆ ω −1 i sβ Vsβ 0 σ Ls i mrβ Tr   Vˆsα . Vˆsβ −1 Tr

−ω

−1 Tr

0



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In case the speed is not measurable. It is replaced by its estimated value ω, ˆ as in our case, it comes from a speed estimator that will be developed later. The state equation of system (42) is given by: X˙ = AX + BU

(44)

Y = DU + Ji U˙ + C X

(45)

Equation (43) becomes: 



X˙ = (A − GC)X + BU + G Z

(46)

where: Z = C X . Let’s put A − GC = Aobs It comes that: 

X˙ = Aobs Xˆ + BU + G Z

(47)

Aobs Define the observer’s dynamics which is chosen as a function of the convergence speed of the observation error towards the zero value. The gain matrix G is determined using the pole imposition method. the structure of the observer is shown in Fig. 3. For the estimation of speed, several methods have been proposed [12]. We can use the mechanical equation to estimate the rotor speed as shown in Figs. 4 and 5.

Fig. 4 Bloc diagram of speed estimator

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Fig. 5 Schematic diagram the proposed flatness-based control scheme with the use of reduced observer

5 Simulation and Experimental Results The sensorless control used flatness theory is simulated under the environment (Matlab/Simulink).The parameters of the machine are: Rr = 4.2 ; Rs = 5.72 ; Ls = 0.462 H; Lr = 0.462 H; M = 0.44 H; J = 0.0049 kg m2 ; f = 0.003 Nm s/rad; P = 2; Power of asynchronous motor is 1.5 kW. The results of simulation are shown in Fig. 6. These results are obtained by simulation with a load (1, 5 Nm).The speed is shown in Fig. 6a. The estimated speed always followed the reference. The load applied does not affect the performance of the system. Good speed estimation was obtained. We can seen in Fig. 6d that the flux (ϕˆr αβ ) installs correctly and well sinusoidal, Fig. 6c shows that the estimated torque follows the reference value (Figs. 7 and 8).

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Fig. 6 a Simulation result of rotor speed. b Simulation result of rotor speed (zoom). c Simulation result of estimated torque. d Simulation result of estimated rotor flux. e Simulation result of estimated rotor flux (zoom)

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Fig. 6 (continued)

Figure 9 shows the DSP based implementation of the proposed speed sensorless control. The sampling frequency is fixed at 10 kHz and the controller receives the stator currents measurements through two 8-bit A/D converters. Then, using the PWM technique, the reference voltages are sent to the machine via the voltage-source inverter whose switching frequency is fixed at 10 kHz.

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Fig. 7 a Experimental result of rotor speed. b Experimental result of rotor speed (zoom). c Experimental result of estimated torque. d Experimental result of estimated rotor flux. e Experimental result of estimated rotor flux (zoom)

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Fig. 7 (continued)

Figure 7 shows the experimental results obtained for the proposed sensorless control by differential flatness method. A rotor speed reference is imposed with a load torque equal to 1.5 Nm applied at time t = 4.2 s.

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By comparing the results, we can say that both results are very similar. It will be noticed that the estimated and the real rotor speed signals are very close. The results confirm the validity of the proposed method. According to the results we can say that the speed response obtained by the flatness technique is significantly improved compared to that obtained by the classical sensorless vector control, Fig. 8.

Fig. 8 a Experimental result of rotor speed (classical FOC). b Experimental result of rotor speed (classical FOC) (zoom). c Experimental result of Estimated Rotor Flux. d Experimental result of Estimated Rotor Flux (Zoom)

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c

d

Fig. 8 (continued)

Fig. 9 The photograph of the experimental test system

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6 Conclusion The paper has studied sensorless control, for induction motors using differential flatness theory and reduced state observer. The experimental results have shown that an adequate sensorless speed control by differential flatness method of IM drive can be achieved at rated, low, and zero reference speed control. Obtained results show that this control strategy assures a perfect linearization regardless trajectory profiles physically imposed on the induction machine. It can be noticed that the reduced state observer is an efficient approach for the implementation of state estimation-based control of induction motor. Finally the proposed control has given very satisfactory results in terms of load disturbance rejection and tracking rotor speed.

References 1. Morand F (2005) Techniques d’observation sans capteur de vitesse en vue de la Commande des machines asynchrones, Thèse de doctorat, soutenue à l’université de Lyon, France, le 7 janvier 2005 2. Al-Rouh I (2005) Contribution à la commande sans capteur de la machine asynchrone, Thèse de doctorat, Faculté des Sciences & Techniques—54506 Vandœuvre-lès-Nancy, Juillet 2005 3. De Wit PAS, Ortega R, Mareels I (1996) Indirect field-oriented control of induction motors is robustly globally stable. Automatica 32(10):1393–1402 4. Zbede YB, Gadoue SM, Atkinson DJ (2016) Model predictive MRAS estimator for sensorless induction motor drives. IEEE Trans Indus Electron 63(6):3511–3521 5. Comanescu M (2016) Design and implementation of a highly robust sensorless sliding mode observer for the flux magnitude of the induction motor. IEEE Trans Energy Conv 31(2):649–657 6. Chang G-W, Espinosa-Pérez G, Mendes E, Ortega R (2000) Tuning rules for the PI gains of field-oriented controllers of induction motors. IEEE Trans Ind Electr 47(3):592–602 7. Lorenz RD, Lipo TA, Novotny DW (1994) Motion control with induction motors. In: Proceeding of IEEE, power electronics and motion control, vol 82, no 8, pp 1215–1240, August 1994 8. de Doncker RW, Novotny DW (1994) The universal field oriented controller. In: IEEE transactions on industry applications, vol 30, no 1, pp 92–100, January–February 1994 9. Fliess M, Lévine J, Martin P, Rouchon P (2007) Flatness and defect of non-linear systems, introductory theory and examples. Int J Control 61(6):1327–1361 10. Dannehl J, Fuchs FW (2006) Flatness-based control of an induction machine fed via voltage source inverter—concept, control design and performance analysis. In: Proceedings of the 2nd annual. IEEE industrial electronics conference, pp 5125, 5130, 6–10 November 2006 11. Fliess M, Lévine J, Martin P, Rouchon P (1999) A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems. IEEE Trans Autom Control 44:922–937 12. Beddiaf Y, Zidani F, Alaoui LC, Drid S (2016) Modified speed sensorless indirect field-oriented control of induction motor drive. Int J Identif Control 25(4):273– 286

Multi-UAVs Coverage Path Planning Abdelwahhab Bouras, Yasser Bouzid, and Mohamed Guiatni

Abstract In this literature review, we attempt to summarize and analyze some recent works on Coverage Path Planning (CPP) strategies. We are interested by those that consider UAV fleets. Covering a region of interest is like sweeping a space in either 2D or 3D, and visiting specific points just once each, to cover as much space as possible with obstacle prevention. These strategies are classified according to their evolution trends, such as grid-based methods, evolutionary algorithms, sampling based methods, etc. Index Terms Path planning · Multi-robots · Multi-UAVs · Coverage path planning

1 Introduction Autonomous aerial vehicles have become a very attractive field for researchers over the last two decades. Their implication in wide range of applications (e.g. aerial photography, agricultural imaging, search and rescue operations, etc.), brings several benefits. Moreover, the multi-UAVs configurations compensate the limitation of flight autonomy and minimize mission time. This survey deals with the Coverage Path Planning (CPP) strategies using a fleet of UAVs. Usually, each UAV should cover its assigned sub-region by visiting a finite number of points. The locations of these points must be well distributed to ensure the full coverage and satisfy certain constraints such as: flight time, energy consumption, obstacles avoidance, feasibility, etc. Most of published CPP algorithms proceed by four steps that are: A. Bouras (B) · Y. Bouzid · M. Guiatni CSCS Laboratory, Ecole militaire polytechnique, Bordj El Bahri, Algeria e-mail: [email protected] Y. Bouzid e-mail: [email protected] M. Guiatni e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_2

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• (S1) The decomposition of the Region of Interest (RoI) into sub-regions with obstacle isolation using several strategies (e.g. Trapezoidal decomposition, Morse decomposition, Voronoi diagram, etc.). • (S2) Squares, triangles, hexagons, etc. are the usual discretization forms (depends on the footprint of the used sensor) of the sub-regions (obtained in S1) for a good localization and an adaptation to the algorithms of the next step. • (S3) A local path planning algorithm that requires a starting cell and a goal cell is needed to deal with the discrete map provided by S2. Many strategies have been proposed in the literature, which are discussed here after. • (S4) A step of smoothing is used to make the planned trajectory feasible by the drone. Generally, this step does not pose a real problem for UAVs with Vertical Take-Off and Landing (VTOL) ability. The Multi-UAVs CPP can be done in several ways. It depends mainly, on the algorithms and the onboard sensors. A family of these algorithms requires a discretization of the space such as A*, D*, etc. However, this category cannot be apply in the case of multi-UAVs, unless they can be applied for each UAV independently. Active Rapidly-exploring Random Trees (Active RRT), RRT*, Dynamic Domain RRT, Artificial Potential Field (APF), etc. do not require a discretization of the space. A load distribution between UAVs is necessary when an iterative path planning algorithm is adopted. In fact, the updates on the multi case of coverage are rare. Thus, this paper gives a summary of recent works, during the last three years, dealing with the CPP of multiUAVs. It will be a good introduction for the interested researchers on the multi-UAVs CPP field. The remaining of the paper is organized as follows: Sect. 2 provides a review of the literature, presenting the current state of the art on. Sections 3, 4 and 5 describes Grid based methods, Evolutionary Algorithms and Sampling based methods of CPP, whereas Sect. 6 closes the paper with a conclusion and future works.

2 Related Works In the literature, many path planning strategies for one or multi-UAVs have been developed. Herein, we are interested by the second case (multi-UAVs) knowing that the generalization of many algorithms of the first class for a fleet of UAVs is a subject of research, or, simplifying and rendering a problem of multi agent to simple problems already known and optimized for the singular case, for this reason, the works dealing with this multi agent coverage problem must take into account the division of the RoI. Choset [1] has classified the CPP algorithms according to the decomposition class: non-decomposition, exact and approximate decomposition. The author presents for each category the progress of the corresponding methods intended to improve the performance of the coverage. This classification is widely used later in similar works.

Multi-UAVs Coverage Path Planning

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Galceran and Carreras [2] illustrate the most redundant classical offline methods of exact decomposition. They focus on Boustrophdon strategy, which aims to fuse the small adjacent cells resulting from the Trapezoid decomposition method (see Fig. 1a), and reduce the global number of cells (see Fig. 1b). Then, they show some variants of Morse-based cellular decomposition method and other popular approaches of decomposition such as the Grid-based method. After discretization, several algorithms are used for the coverage of the workspace, such as Wave Front algorithm, Spanning Tree Coverage (STC), Backtracking Spiral Algorithm and Neural Networks, etc. For the case of multi-UAVs, the authors discuss the use of Boustrophdon decomposition with a division of roles of UAVs, some of them work as scouts of the borders, while the others make the round trip in the inner space in order to complete an Online coverage mission. The generalization of coverage algorithms from single case to multi such as Spanning Tree Coverage (STC) which has become Multi STC (MSTC), this category fits into the class of approximation decomposition, see Fig. 2. The next point on techniques using the neural networks each robot sees as dynamic

Fig. 1 Variants of Morse decomposition method: a Trapezoid decomposition. b Boustrophdon decomposition

Fig. 2 Approximate cellular decomposition with square cells of different sizes

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obstacles. Then; Graph-based, bioinspired algorithms are limited in practical applications. Finally, multi homogeneous or heterogeneous aerial robots have had a variety of applications and constraints for this task. After three definitions (clear, short and accurate) on the planning and the generation of trajectories, Yang et al. [3] arrange the algorithms of path planning (3D environment) in five categories. These algorithms are well analyzed and discussed. A summary is given in the Table 1. For more details, the reader may refer to [4, 5]. In paper [6], RRT variants have been combined with other strategies in order to improve the computation time, the required memory, etc. Moreover, future challenges have been exposed. In our work, a new criterion for the classification of CPP techniques has been proposed, this choice being in relation to current work trends, such as: grid-based methods, evolutionary algorithms, sampling based methods. These techniques are more adapted to the constraints imposed by this theme. Furthermore, since 2016, this type of update does not exist.

3 Grid Based Methods (Approximate Cellular Decomposition) 3.1 Grid of Elementary Cells of Square Shapes The decomposition of the RoI into elementary cells of the same or different sizes, this is relative to the importance of each sub-region. In general, this is reflected by different flight altitudes for the UAVs, which gives different dimensions of the footprint of the sensors by choosing the right compromise with the coverage resolution. In Kapoutsis et al. [7] propose an offline algorithm named Divide Areas Algorithm for multi-Robot coverage Path planning (DARP), for the balanced sharing of the RoI according to the number of robots. For that, five rules have been established to better fit the multi Coverage Path Planning (mCPP) problem. Indeed, the rules consist of ensuring non-backtracking, full coverage and balancing efforts between the robots, then, the Spanning Tree Coverage (STC) algorithm is used to cover and improve the speed of calculation of robot paths. However, choice of coverage direction for back-andforth (BF), memory capacity and the calculation time are still points to improve. Figure 3 shows a simulation variant of this technique with seven UAVs, and set up several obstacles of our choice. In the same trend [8] have introduced improvements to this algorithm (DARP), with an optimization of Spanning Tree’s generation by Ant Colony Optimization (ACO) algorithm of each sub-region, and the introduction of nodes exchanges between the sub-regions in a second step (ST nodes), which gave a remarkable reduction in the number of turns, and reduce the energy consumed. However, the same problem of computation time persists. In [9], the nodes (vertexes) of STC are distributed to the STCs of the robots by criteria of priorities adopted with the auction (Auction-STC) (auctioneer & bidder), the path traveled by each

Sampling based methods

Dijkstra, A *, Tetha *, Lazy Tetha*, D *, Harmonic Search, etc.

Linear programming and optimal control Linear binary programming Non linear programming Mixed integer linear programming (MILP)

Node based methods

Mathematical methods

Active: RRT, RRT *, DDRRT, Artificial potential field (APF)

Types of algorithms

Passive: PRM, Voronoi, RRG, K-PRM

Category

Table 1 Categories of 3D path planning algorithms Disadvantages

– Do not reproduce the exact form of the environment – Are not suitable for multi agents

– Takes into account all – Calculation time is dynamic and kinematics important – No analytical resolution constraints by methods differential equations – Suitable for working offline

– Easy to be combined with other methods to improve optimization – Suitable to work online

– RRT*, DDRRT, RRG, – RRT does not propose the proposed the issue of issue of re-planning – PRM and Voronoi can not re-planning – Simple and easy structure optimize alone (requires to implement optimization with A* for – Suitable for static and example) dynamic environments, and – Problems of local minima online planning conditions for Artificial potential field (APF)

Advantages

(continued)

• Calculation time: depends on the complexity of the problem • Maybe useful more in advancing in computer technologies

• Requires a pre-knowledge of the environment, then; mapping or rand to find the right planning • Calculation time:   O(n log n) ≤ T ≤ O n2 • Several RRT and PRM series have been developed for the improvement of previous versions • An objective function (cost function) is necessary for the optimization • Calculation time:   O(n log n) ≤ T ≤ O n2

Notes

Multi-UAVs Coverage Path Planning 27

– Are not practical in real situations – Are not suitable for online work

Disadvantages

Road Maps (PRM) optimized, Visibility Graphs optimized, etc.

Multi-fusion-based methods Embedded (mixed): Reactive – Enjoy the advantages of the – Keeps the disadvantages of deformation road maps previous methods the combined algorithms – Possibility to solve a large (RDR), Visibility voronoi part of the problems potential (VVP), etc. – Choice of the combination Ranked (succession depending on the problem algorithms): Probabilistic

NN: Neural networks

Bio-inspired methods

Advantages

Types of algorithms

EA: Evolutionary algorithms – Able to solve NP-hard (ACO), Particle swarm problems, complex optimization (PSO), Shuffled environment – Suitable for work on multi frog leaping algorithm agents (SFLA), Memetic algorithm (MA)

Category

Table 1 (continued)

• Calculation time: O(n log n) ≤ T

• Calculation time:   O n2 ≤ T • Generally simulations work

Notes

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Fig. 3 DARP simulation of seven drones with multiple obstacles

robot is estimated online (in the same time) of this process by quantifying vertexes, including those of big cells, partially filled with obstacles. This approach shows a robust resolution of the problem of mCPP, either in complex environments or with the increase of the number of robots. However, small redundancies of coverage appear, the problem of the initial conditions and the number of turns and/or the power consumption are not discussed.

3.2 Triangles Grid Other work used a considerable tools for a full simulation of the space coverage problem with a fleet of UAVs [10], Balampanis et al. solved the problem in three steps: • The triangulation for the decomposition taking into account the size of the triangles so that it is compatible with the capture (footprint) of the sensor used, and to make the triangles equilateral as possible (Fig. 4); • In a second step, the authors proposed two algorithms called Antagonizing Wave front Propagation and Reverse Watershed Schema (AWP & RWS) for the distribution of sub-regions to be covered by UAVs, taking into account the capabilities of UAVs and getting rid of dead zones by Dead Lock Handling (DLH) algorithm; • Finally; “Waypoint list computation for coverage” is used for coverage.

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Fig. 4 Approximate cellular decomposition of a square area with a circular obstacle using triangular cells of the same size

RWS has shown its advantage in a comparative study. This work was accompanied by an assessment of the complexity of the algorithms, the type of sensor and the flight capabilities of the UAVs. However, the resulting trajectories for the coverage are not discussed in terms of the number of turns and/or energy consumed.

3.3 Hexagons Grid In [11], Azpúrua et al. take advantage of the decomposition of the region of interest in hexagons. Where this type of decomposition offers a minimization of the intercellular distance compared to that of small squares, and also a good orientation of the cells compared to the case of the triangles, as well as this technique takes into consideration the range of communications between the robots. Indeed, the division of the space to cover for each robot is ensured with the K-means algorithm. After that, ensuring coverage using TSP. Finally, a Finite State Machine algorithm (FSM) has been proposed to ensure synchronization between the robots. After the introduction of some dynamic parameters (α, h: Orientation of the hexagons and distance between parallel lines of coverage respectively), and the respect of the feasibility of the mission (size of the hexagons (radius r), autonomy of the robots, obstacles), their simulation work shows that: • The coverage time decreases as the number of robots or the inter-coverage lines (h) increases; • The optimal orientation of the hexagons reduces the intercellular distance. Hence, the reduction of the trajectory length, therefore decreased coverage time.

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However, the comparison of the proposed methodology with cooperative approaches (proposed in [12]) shows that the coverage time in the latter is less than in the proposed technique, either for the case of individual flight or in formation (Multi-robot). This difference is due to the extra-terrain coverage caused by the principle of the space decomposition (exact and non-exact decomposition). On the other hand, this disadvantage becomes an advantage in the case of the evaluation of the quality of detection (magnetic map), this is translated by a collection of more data compared to [12]. An experimental study of the proposed technique was performed.

4 Evolutionary Algorithms (Swarm Intelligence) Among the most used algorithms is ACO, given its efficiency in complex environments, and the possibility of using several ants in parallel for different constraints. Chaotic ACO of Coverage (CACOC) [13], is a version of ACO, the authors use a chaotic process based on Rösler system instead of Random process, this change is introduced when there is an absence of pheromone help with the decision (guidance) of UAVs. The same authors extend this algorithm in [14], by adding a mechanism of Collision Avoidance (CA), which gives CA-CACOC. In this case, except that other drones for each of them that are considered obstacles. This algorithm is simulated in a V-REP (Drone Model Simulation) environment using the Model Predictive Control (MPC), which gives results in terms of coverage-which we are interested in-better than previous versions, but, this type of algorithm is always influenced by the choice of the initial parameters. Other works use a Multi ACO for the CPP problem, in [15], the update of the pheromone of each colony is relative to the fitness of the ten best ants of the previous colony, in another word, each colony benefit from the results of the other colonies, which improves the convergence rate of the algorithm on the one hand, and out of the local minima on the other hand. Nevertheless, computation time and implementation in the real world are the major disadvantage of this algorithm.

5 Sampling Based Methods These methods are based on the use of the algorithms of the first category of Table 1, which need to have a preliminary knowledge on the RoI (in the form of points or cells etc.), both the starting points and the arrival points to conduct an iterative process (sampling) creating a tree of nodes connected to each other, then obtaining the optimal trajectory towards the arrival points. The way in which these nodes are selected or connected at each iteration is the key to the success of the resulting trajectories. Therefore, the calculation time, the narrow passages are the essential points to improve.

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Fig. 5 The expanded trees and a solution path found by the RRT Connect algorithm (a) (9.75 s, 7347 nodes) and LM-RRT approach (b) (1.35 s, 3286 nodes)

Wang et al. [16] propose the Learning-based Multi-RRT algorithm (LM-RRT) as a solution for narrow passages, this approach is structured in four steps, the first one for sampling by the improve Bridge Test algorithm. Then, the hierarchical group obtains the multiple trees in a second step. The third step consists in choosing the tree by an adaptive process. The last step is the extension and the connection of the chosen tree. If the final trajectory is not yet obtained, the algorithm returns to the beginning of the third step. The performances of this algorithm are clear in Fig. 5 in terms of computing time, number of nodes, trajectory quality. In [17], Bouzid et al. use multi-directional algorithm Rapidly-exploring Random Trees Star Fixed Nodes (RRT * FN) to optimize inter-point-of-interest distances, take the optimal path and avoid obstacles. The coverage is solved by the combination of TSP with genetic algorithms for the passage of all selected points. The numerical simulation of a single UAV shows the effectiveness of this solution.

6 Other Techniques The coverage problem is started in [18] in three steps, the approximate reproduction of the terrain (NURBS: Non Uniform Rational B-Spline), this provides the necessary data for the determination of control points, taking into account the coordinates of cell centers and altitude, in order to cover the terrain by Multi-UAVs Vehicle Routing. To minimize the mission time of the UAVs, the authors adopt a criterion of the

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minimization of the distance traveled by each UAV at the same time, instead of minimizing the sum of their traveled distances. This trick makes it possible to reduce the overall time of the mission with similar and/or equivalent traveled distances by UAVs, these distances are adapted to the constraints of the UAVs, that instead of having a minimum sum of the distances traveled with loads unbalanced on the UAVs. Modares et al. [19] introduce a second term of optimization of energy compared to the TSP, in addition to optimizing the energy compared to the distance traveled by the UAV, the authors also take into account the angle described by the path between adjacent cells (0°, 45°, 90°, 135°, and 180°), i.e.; take into account the image of the number of turns (the energy consumed during turns) in the optimization criterion, which gives more meaning to the energy consumed. This term has been introduced in Lin-Kernighan Heuristic (LKH) used in TSP, which has named LKH-D for drones. The comparison with LKH shows that the chosen path is more economical in energy despite small losses in computing time, this time decreases more by increasing the number of agents (UAVs).

7 Conclusion This works presents a recent research dealing the problem of coverage path planning for aerial detection applications using UAV fleets. This works tends towards the combination of several algorithms to benefit from their advantages in order to remedy the constraints posed by the coverage problem. These cell decomposition and connection techniques between cells (Table 2) must take into account the requirements of the application being processed and the available working means such as the flight capabilities of UAVs and the types of on-board sensors. On another side; computation time, required memory space and the diversity of applications are still research topics. Improvements in Sampling based methods have opened up another broad line of research, given the challenges and constraints resulting each time, and the wide scope of their uses. Despite that; the use of these algorithms for coverage is not much used yet. In future works, the research base will be broadened by the introduction of several practical works dealing with this theme, as well as the proposal of the techniques in the same subject by the treatment of the different terrains with several kinds of obstacles, focusing on sampling based methods.

Multi/Single

Multi

Multi

Multi

Multi

Multi

References

Kapoutsis et al. [7]

Balampanis et al. [10]

Gao et al. [8]

Azpúrua et al. [11]

Dentler et al. [14], Cekmez et al. [15]

Approximate cellular decomposition (Squares)

Approximate cellular decomposition (Hexagons)

Approximate cellular decomposition (Squares)

Approximate cellular decomposition (Triangles)

Approximate cellular decomposition (Squares)

Decomposition approaches

2D

2D

2D

2D

2D

2D/3D

Table 2 Categories of path planning approaches with used algorithm

Grid based

K-means

Grid based DARP

Grid based

Grid based DARP

Decomposition algorithm

Offline

CA-CACOC + Multi ACO

Offline

STC + ACO + node exchanges

Offline

Offline

AWP & RWS + Waypoint list computation

TSP

Offline

Offline/Online

STC

Coverage algorithm

• Influenced by the choice of initial parameters • Problems with computing time and implementation in the real world (continued)

• FSM: pour synchronisation • Extra field problem

• Reduction of the number of turns • Requires large memory capacity

• The number of turns and/or energy consumed is not discussed

• Critics of STC persist • Requires large memory capacity

Notes

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Multi/Single

Multi

Multi

Multi

References

Choi et al. [18]

Gao and Xin [9]

Modares et al. [19]

Table 2 (continued) 2D/3D

Approximate cellular decomposition (Squares)

Exact cellular decomposition 2D

2D

NURBS: Non Uniform 3D Rational B-Spline

Decomposition approaches

Grid based

Grid based

NURBS: Non Uniform Rational B-Spline

Decomposition algorithm

STC

Auction-STC

Multi-UAVs Vehicle Routing

Coverage algorithm

Offline

Offline

Offline

Offline/Online

• Take into consideration the number of turns in the optimization of the TSP

• Increase in the number of turns

• Compromise: total distance and time of the mission

Notes

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References 1. Choset H (2001) Coverage for robotics—a survey of recent results. Ann Math Artif Intell 31(1):113–126 2. Galceran E, Carreras M (2013) A survey on coverage path planning for robotics. Robot Auton Syst 61(12):1258–1276 3. Yang L, Qi J, Song D, Xiao J, Han J, Xia Y (2016) Survey of Robot 3D path planning algorithms. J Control Sci Eng 2016:1–22 4. Goerzen C, Kong Z, Mettler B (2010) A survey of motion planning algorithms from the perspective of autonomous UAV guidance. J Intell Robot Syst 57(1–4):65–100 5. LaValle SM (2006) Planning algorithms. Cambridge University Press, Cambridge 6. Noreen I, Khan A, Habib Z (2016) Optimal path planning using RRT* based approaches: a survey and future directions. Int J Adv Comput Sci Appl 7(11) 7. Kapoutsis ACH, Chatzichristofis SA, Kosmatopoulos EB (2017) DARP: divide areas algorithm for optimal multi-robot coverage path planning. J Intell Robot Syst 86(3–4):663–680 8. Gao C, Kou Y, Li Z, Xu A, Li Y, Chang Y (2018) Optimal multirobot coverage path planning: ideal-shaped spanning tree. Math Prob. Eng 2018:1–10 9. Gao G-Q, Xin B (2019) A-STC: auction-based spanning tree coverage algorithm formotion planning of cooperative robots. Front Inf Technol Electron Eng 20(1):18–31 10. Balampanis F, Maza I, Ollero A (2017) Coastal areas division and coverage with multiple UAVs for remote sensing. Sensors 17(4):808 11. Azpúrua H, Freitas GM, Macharet DG, Campos MF (2018) Multi-robot coverage path planning using hexagonal segmentation for geophysical surveys. Robotica 36(8):1144–1166 12. Maza I, Ollero A (2007) Multiple UAV cooperative searching operation using polygon area decomposition and efficient coverage algorithms. In: Alami R, Chatila R, Asama H (eds) Distributed autonomous robotic systems 6. Springer, Tokyo, Japan, pp 221–230 13. Rosalie M, Danoy G, Chaumette S, Bouvry P (2016) From random process to chaotic behavior in swarms of UAVs. In: 6th ACM symposium on development and analysis of intelligent vehicular networks and applications, Malta, pp 9–15 14. Dentler J et al (2019) Collision avoidance effects on the mobility of a UAV swarm using chaotic ant colony with model predictive control. J Intell Robot Syst 93(1–2):227–243 15. Cekmez U, Ozsiginan M, Sahingoz OK (2018) Multi-UAV path planning with multi colony ant optimization. In: Abraham A, Muhuri PK, Muda AK, Gandhi N (eds) Intelligent systems design and applications, vol 736. Cham: Springer International Publishing, pp 407–417 16. Wang W, Zuo L, Xu X (2018) A learning-based multi-RRT approach for robot path planning in narrow passages. J Intell Robot Syst 90(1–2):81–100 17. Bouzid Y, Bestaoui Y, Siguerdidjane H (2017) Two-scale geometric path planning of quadrotor with obstacle avoidance: First step toward coverage algorithm. In: 2017 11th international workshop on robot motion and control (RoMoCo), Wasowo Palace, Poland, pp 166–171 18. Choi Y, Chen M, Choi Y, Briceno S, Mavris D (2019) Multi-UAV trajectory optimization utilizing a NURBS-based terrain model for an aerial imaging mission. J Intell Robot Syst. https://doi.org/10.1007/s10846-019-01027-9 19. Modares J, Ghanei F, Mastronarde N, Dantu K (2017) UB-ANC planner: energy efficient coverage path planning with multiple drones. In: 2017 IEEE international conference on robotics and automation (ICRA), Singapore, pp 6182–6189

Reducing the Collision Checking Time in Cluttered Environment for Sampling-Based Motion Planning Amine Belaid and Boubekeur Mendil

Abstract In the Sampling-based motion planning, the collision checking procedure called many times and this operation takes the most planning time. The time spend in the collision checking increases with the number of obstacles. This paper aims to reduce the number of calls of the collision checking procedure in environment containing high number of obstacles. Our approach shows that the complexity of collision checking does not depend on the number of obstacles. The idea is to perform a pre-procedure that localizes the obstacles by segmenting the workspace into a grid and save it in a matrix form. This information is used in the collision checking procedure. We have used the Probabilistic Roadmaps (PRM) motion planner with a punctual robot and polygonal obstacles in 2D environment. Simulation results show the efficiency of our algorithm in reducing the complexity of collision checking and minimizing the planning time. Keywords Path planning · Probabilistic roadmap · Collision checking · Motion planner

1 Introduction Motion planning is widely used in robotics, games, manufacturing and many related areas, to find free collision path of objects, characters and robots. Samplingbased motion planning methods, such as Probabilistic Roadmaps [1], and Rapidly Exploring Random Tree (RRT) [2], are developed especially for problems with high degree of freedom. These algorithms avoid explicit computation of obstacle boundaries in the configuration space, C-space. Instead it uses sampling techniques to compute paths in the free space Cfree . A. Belaid (B) · B. Mendil Industrial Technologies and Information Laboratory, Faculty of Technology, Bejaia University, 06000 Bejaia, Algeria e-mail: [email protected] B. Mendil e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_3

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Collision checking procedure in sampling-based path planners is considered like a black box called each time needed [3]. The first presented sampling motion planning method uses the uniform distribution to generate samples in the configuration space. However, this strategy has bad performance in complex spaces, like the presence of narrow passages. To deal with this problem, many sampling strategies have been proposed like Gaussian sampling [4], Bridge sampling, etc. These strategies use an intensive number of calls to collision checking procedure, and as a result, the sampling phase takes a long time. Many algorithms have been proposed to reduce the number of calls to collision checking procedure and, thus, to minimize the planning time in workspaces with high number of obstacles. The most of them use prior information about the sampled configurations [6, 7] or collision queries [5, 8, 9]. In other words, they modify the sampling strategy to reduce the complexity of the approach. A wide variety of approaches and techniques for geometric modeling exist. The choice usually depends on the application and the difficulty of the problem. Two models, with different collision checking procedures are often used in path planning: Bitmap and Polygonal or Polyhedral representation for 2D or 3D environments [3]. For the polygonal representation, the obstacles and the robot are represented by polygons, each polygon is described by its vertices. in the other hand, for the Bitmap representation, the environment is discretized into rectangular cells that may or may not be occupied. This information is saved in two or three-dimensional matrix. Each geometric modeling technique presents advantages and drawbacks. But when the workspace is large the bitmap needs a huge storage space [3]. In this paper we present a new technique to address the problem of collision checking in 2D cluttered environment for sampling-based methods. Unlike the previous presented approaches, we don’t use prior information about the sampled configurations, and we don’t modify the sampling strategy. The main contribution of our method is to disassociate the dependence between the number of obstacles and the complexity of the algorithm. The idea is to localize the obstacles by discretizing the environment into cells in the beginning of the motion-planning algorithm and to save them as a matrix, which is used in the collision checking procedure. This preprocedure allows the avoidance of the collision checking with all obstacles existing in the workspace. In order to test the performance of our method, we integrated it with the PRM path planner. The 2D environment is considered. The obstacles are modeled by polygons and the robot is punctual. The results show the efficiency of our approach in environment with high number of obstacles, comparing with the traditional techniques. The rest of this paper is organized as follows. The Sect. 2 presents the Probabilistic Roadmap path planner. Section 3 gives the details of our approach. The simulation results and discussion are presented in Sect. 4. Section 5 concludes this paper.

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2 Probabilistic Roadmap The planners based on an explicit representation of the configuration space cannot solve problems with high degree of freedom. Because, as the dimension of the configuration space grows, these planners become impractical, due to the required time. The sampling-based planners are developed to solve this kind of problems [10]. The probabilistic roadmap [1] was the first proposed sampling based-algorithm. It proceeds in two phases: learning phase and query phase. In the learning phase, the sampled configurations are stored in a graph data structure called a roadmap. The latter is constructed in a probabilistic way. The roadmap is an unidirectional graph R = (V, E), where V is a set of free configurations sampled randomly and E is a set of edges corresponding to local paths. For every configuration q ∈ V, a set N q of k closest neighbors to the configuration q according to some metric distance, dist, is chosen from V. The local planner is called to connect q to each configuration q0 ∈ N q . If the local planner succeeds in computing a feasible path between q and q0 , the edge (q, q0 ) is added to the roadmap. Algorithm 1 Learning phase of the Probabilistic Roadmap Input: n : number of nodes to put in the roadmap k : number of closest neighbors to examine for each configuration Output: A roadmap G(V,E) E ←{} V ← set of free configurations generated by a sampling method for all q V do Nq ← the k closest neighbors of q chosen from V according to dist for all q0 Nq do if (q,q0) E and ∆(q,q0) ≠ NIL then E ← E {(q,q0)} end if end for end for

In the query phase, the initial and the end configurations qinit , qgoal are connected to the roadmap using the local planner. Then, an algorithm of finding the shortest path in the graph, like A∗ or Dijkstra, is used to find the path between the start and the end configuration. A post-processing step can be performed to improve the quality of resulting path by checking whether nonadjacent configurations q and q0 along the path can be connected with the local planner.

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3 Collision Checking in Cluttered Environment Collision testing takes the most of time in path planning, especially with high number of obstacles, since the temporal complexity increases with the number of obstacles. Our main contribution in this paper is to disassociate the dependence of collision testing complexity with the number of obstacles in the workspace. To avoid the collision test with all the obstacles, we add a pre-processing to localize the obstacles in the workspace by decomposing the workspace into rectangular cells. We give for each obstacle a rectangular region by taking the maximum and the minimum of X and Y abscise for the obstacle vertices (Fig. 1). Then, we construct the matrix M representing the workspace. Each matrix element contains the number of obstacle regions overlapping the corresponding cell. Once the construction of the matrix M is done, it can be used now in the collision checking procedure. After localizing the sampled configuration in the workspace, we do the collision checking with just the obstacles found in the cell of the matrix M that corresponding to the location of the sampled configuration.

Fig. 1 A non-convex obstacle A with his rectangular region found by taking the maximum abscises according to X and Y

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Algorithm 2 Query phase of the Probabilistic Roadmap Input: qinit : the initial configuration qgoal : the goal configuration k : number of closest neighbors to examine for each configuration G(V,E) : the roadmap constructed by the Algorithm 1 Output: A path from qinit to qgoal or failure V ← V {qinit} {qgoal} for q ←{qinit,qgoal} do Nq ← the k closest neighbors of q chosen from V according to dist q0 ← the closest neighbor of q from Nq while Nq is not empty do if ∆(q,q0) ≠ NIL then E ← E (q,q0) N ← N −{q0} else N ← N −{q0} end if q0 ← the closest neighbor of q from Nq end while end for

3.1 Pre-processing (Obstacles Localization) The idea of a pre-processing algorithm is to perform the most of collision checking tests in the beginning. In this paper, we consider the environment is in 2D, and the obstacles are non-convex polygons, so the matrix M that represent the environment is of dimension (n × m). First of all, the algorithm divides the environment into a grid as is shown in Fig. 2. The number of the grid cells along the X axis is n and the number of the grid cells along the Y axis is m. Then, the algorithm creates the (n × m) empty matrix M which have the same dimension of the grid. The obstacles are

Fig. 2 2D environment divided into grid with its corresponding matrix M. the bold rectangles are the obstacles regions. One cell can be covered by more than one obstacle region. NIL means the cell is not cover by any obstacle’s region

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numbered from 1 to k. The algorithm gives for each obstacle Oi a rectangular region (Fig. 1), where the vertices coordinates of this region in clockwise direction are X Oir egion = [min(X Oi) min(X Oi) max(X Oi) max(X Oi)], Y Oir egion = [min(Y Oi) max(Y Oi) max(Y Oi) min(Y Oi)] In the workspace, each obstacle region covers some cells of the grid, so for each obstacle region, the algorithm will write its number in all the elements of the matrix M that correspond to the cells of the grid covered by this obstacle region. Note that a cell of the grid can be covered by more than one obstacle region. An example of creating the matrix M for a 2-dimension environment is shown in Fig. 2.

3.2 Collision Checking Procedure By considering the robot is punctual, in this procedure, a configuration is sampled, then, the algorithm finds the cell of the grid that contain this configuration. Next, the collision checking will perform with just the obstacles whose number are found in the element of the matrix M that correspond to this cell. By applying this strategy, we avoid the collision checking with all obstacles in the workspace. This strategy can be used with polygonal robot, where we find all the cells that contain the robot then we perform the collision checking with the union of obstacles numbers found in the elements of the matrix M corresponding to those cells. And this algorithm can be generalized to three dimensions environment, which is our perspective. Algorithm 3 pre-processing (obstacles localization) Input: Obs : obstacles vertices coordinate Output: The matrix M for A in Obs do Imin,Imax ← indices of cells that contain min(XA), max(XA) respectively according to X axis Jmin,Jmax ← indices of cells that contain min(YA), max(YA) respectively according to Y axis for i ← Imin to Imax do for j ← Jmin to Jmax do if N(A) in M(i,j) then Continue else M(i,j) ← M(i,j)&N(A) end if end for end for end for

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Algorithm 4 Collision checking procedure Input: Obs : obstacles vertices coordinate conf : the configuration to test for collision M : the matrix that produced by algorithm 3 Output: true: if the configuration is in collision ; false: otherwise i,j ← indices of cells that contain conf according to X axis and Y axis Obst ← M(i,j) for A in Obst do if conf in collision with A then return true end if end for return false

4 Simulation and Results To state the performance of our approach, we integrated it with the probabilistic roadmap planner. Then, we compared its performance with PRM using a traditional collision checking. A 2D environment (Figs. 3, 4, and 5) and a punctual robot model are considered where the scene is taken to be [−10, 10] along both, X axis and Y axis. The obstacles are convex or non-convex polygons generated randomly. We compared the simulation results of just the learning phase of the PRM. The Gaussian sampling

Fig. 3 Execution time in function of number of obstacles for learning phase of PRM path planning with traditional collision checking, using the uniform sampling and the Gaussian sampling strategies

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Fig. 4 Execution time in function of number of obstacles for learning phase of PRM path planning with our collision checking approach, using the uniform sampling and the Gaussian sampling strategies

Fig. 5 The 2D environment considered with 300 obstacles generated randomly

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strategy with σ = 0.2, and the uniform sampling strategy are chosen. The number of configurations to generate in each simulation is set to n = 500. In each simulation, we take the average time of 3 attempts. All the simulations have done in MATLAB 2014a with 2.3 GHz DualCore Intel PC. Figure 3 shows the variation of execution time of learning phase in function of number of obstacles for PRM with a traditional collision checking procedure. For the Gaussian sampling the time execution increase according to the number of obstacles until t = 112 s with around 150 obstacles, then it decreases to stay roughly constant t = 96 s, this decreasing is because the scene is cluttered of obstacles, so the free space will narrow, and this gives high chances in finding the second configuration for the Gaussian sampling. On the other hand, in Fig. 4 the execution time of the learning phase with Gaussian sampling and using our algorithm decrease according to the number of obstacles from t = 6.6 to t = 2.3 s with 300 obstacles, and here is the power of our algorithm. In other comparison with uniform sampling, the execution time with the traditional collision checking increase linearly according to the number of obstacles until t = 7.96 s with 300 obstacles. In other hand, when we used our algorithm with the uniform sampling, the execution time stay roughly constant, around t = 0.28 s.

5 Conclusion This paper presents a new collision-checking algorithm for sampling-based motion planning in environment with high number of obstacles. We have integrated our algorithm with the PRM path planner and compared its execution time of the learning phase with that of PRM using a traditional collision checking. Gaussian and uniform sampling strategies are used to test the performance of our algorithm. Simulation results show the efficiency of our approach in collision checking especially with high number of obstacles. In future works, we will investigate our approach in the case where the robot is a polygon in 2D environment then in 3D where the robot and the obstacles are modeled by polyhedrons.

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5. Kim D, Kwon Y, Yoon S (2018) Adaptive lazy collision checking for optimal sampling-based motion planning. In: 2018 15th international conference on ubiquitous robots (UR). IEEE 6. Hauser K (2015) Lazy collision checking in asymptotically-optimal motion planning. In: 2015 IEEE international conference on robotics and automation (ICRA). IEEE 7. Bialkowski J et al (2013) Efficient collision checking in sampling-based motion planning. In: Algorithmic foundations of robotics X. Springer, Berlin, Heidelberg, pp 365–380 8. Pan J, Manocha D (2016) Fast probabilistic collision checking for sampling-based motion planning using locality-sensitive hashing. Int J Robot Res 35(12):1477–1496 9. Daee P, Taheri K, Moradi H (2014) A sampling algorithm for reducing the number of collision checking in probabilistic roadmaps. In: 2014 22nd Iranian conference on electrical engineering (ICEE). IEEE 10. Choset HM (2005) Principles of robot motion: theory, algorithms, and implementation. MIT Press, Cambridge

Random Scaling-Based Bat Algorithm for Greenhouse Thermal Model Identification and Experimental Validation Mounir Guesbaya, Hassina Megherbi, and Ahmed Chaouki Megherbi

Abstract In this paper, we propose a Random Scaling-based Bat Algorithm (RSBA) for parametric identification of a greenhouse thermal model. The proposition includes modifying the exploitation of the standard BA by making the scaling parameter changes randomly over the iterations. The proposed thermal model identification method has been assessed first on a simulated greenhouse thermal model with known parameters. The simulation results have shown the superiority of the proposed RSBA compared to the standard BA in term of convergence and performance accuracy. To experimentally investigate the proposed identification method, we used a greenhouse prototype under arid climate conditions located in M’ziraa, Biskra, Algeria. The obtained prediction results are found to be in a good agreement with the measured ones which show the effectiveness of the proposed RSBA in identifying the real greenhouse thermal model. Keywords Bat Algorithm · Exploitation and exploration · Metaheuristic optimization · Greenhouse thermal modelling · Parameter identification

1 Introduction Greenhouse systems have become a prominent mean in the agricultural field. It is constantly developed in order to reach the quantity and quality of crops that meet the world growing population and the strict standards of the competitive markets today. The inside air temperature of greenhouse plays an important role in affecting M. Guesbaya · A. C. Megherbi Laboratory of Identification, Command, Control and Communication, University of Mohamed Kheider, Biskra, Algeria e-mail: [email protected] A. C. Megherbi e-mail: [email protected] H. Megherbi (B) Department of Electrical Engineering, University of Mohamed Kheider, Biskra, Algeria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_4

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the yield qualitatively and quantitatively. Analysing and controlling this physical process needs an optimized model that describes it with sufficient accuracy and low complexities and costs [1]. Provision of such a model will increase the implementation of the new intelligent control approaches to improve agriculture production in greenhouses. The Bat Algorithm (BA) is a popular population-based and stochastic algorithm developed by Yang [2]. It shares the feature of swarm intelligence with the most popular nature-inspired algorithms. BA is characterized by quick convergence at an earlier stage of the search process, frequency-tuning which increases the diversity of solutions during the search and the parameter control feature of BA that leads to achieve smart moves from explorative to local intensive exploitative search steps. However, the standard BA includes problems with local and global dimensional search space. Among these problems; the low accuracy and the slow convergence rate when the search is near to optimal solution, which are indicated in many research studies [3–5]. In this paper, we propose a random scaling-based bat algorithm (RSBA) to identify the static parameters of a greenhouse thermal model under arid desert climate conditions. In the RSBA, the scaling parameter that determines the step size changes randomly over iterations. The idea behind this proposition is to update the scaling parameter in a way that makes the step size hit relatively large and small values to maintain an effective step size control in relative to the closeness of the optimal solution. This mechanism aims to affect and enhance the exploitation mechanism especially in the final stage of the search. The greenhouse thermal model used in this work is proposed in [6]. This model is devoted to predict the inside air temperature. It describes this phenomenon as a set of heat exchanges generated by the differences in energy content between the inside and outside air. The rest of this paper is organized as follows. In Sect. 2, the used grey-box thermal model is described. Section 3 includes the details of the RSBA-based identification method of the static parameters of the used thermal model. In Sect. 4, simulation and experimental results of the thermal model identification process are discussed. Section 5 concludes this paper.

2 Greenhouse Thermal Model The grey-box thermal model used in this work is presented in [6]. Greenhouse models represent the heat exchange processes between all the different subsystems of the greenhouse. However, in the proposed model only two components are studied; the cover and the internal air. It is considered as a mean that quantitatively describes the energy exchanges. The temperature dynamic behaviour is a combination of physical interactions including the conduction, the convection, the solar and thermal radiation and the infiltration as depicted in Fig. 1. These processes are basically affected by the external environmental conditions and the structure of the greenhouse [7]. The constructed greenhouse is used as a nursery. Hence, the canopy which is constituted

Random Scaling-Based Bat Algorithm for Greenhouse Thermal …

49

Fig. 1 Heat exchanges affecting the inside air temperature

of seedlings has a small effect on inside air temperature due to its small reserved area. The nursery includes trays full of treated soil without planted seedlings. The following assumptions have been taken into account: • The temperature of the canopy is neglected. • The temperature of the treated soil in seedling trays is neglected because it has a small effect compared with the other elements due to the reserved small surface. • The heat exchange between external soil and the inside air is neglected, due to the role of the wooden floor as a separator. • Transfer coefficient of the convective heat and the absorbed solar radiation are uniform throughout the cover. • There is no stratification in greenhouse air temperature. • The heat storage of greenhouse elements can be neglected. • The greenhouse is East-West oriented. The inside heat balance equation [6] is given by: ρa

V dTin = Q solar − Q cnv,cnd − Q loss − Q ther mal S f dt

(1)

where t is the time in seconds, Tin is the inside air temperature (K), V is the volume of the greenhouse (m3 ), S f is the floor surface (m2 ) and ρa is the inside air density (kg m−3 ) which is given by: ρa = γ0 + Ha

(2)

where, γ0 is the inside dry air density (kg m−3 ) and Ha is the absolute humidity (kg m−3 ), which has been obtained by converting the relative humidity as follows:

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Ha =

(0.62198 Pw ) Patm − Pw

(3)

where, Patm (kPa) is the atmospheric pressure, Pw is the instant vapour pressure, calculated as follows: Pw = Pws ·

Hr 100

(4)

where, Hr is the relative humidity of greenhouse (%) and Pws is the saturation vapour pressure (kPa °C−1 ); it is defined by: Pws = 0.61078 e

(17.2694(Tin −273.15)) Tin

(5)

Q solar is the shortwave radiation absorbed by the greenhouse given by: Q solar = Csolar I

(6)

where, Csolar is the solar energy coefficient and I is the external global solar radiation (W m−2 ). Q cnv,cnd , is the convection and conduction heat exchange rate given by: Q cnv,cnd = Ccnd,cnv U (Tin − Ta )

(7)

where, Ccnd,cnv is the convection and conduction energy coefficient, Ta is the ambient temperature (K) and U is the overall heat transfer coefficient through the greenhouse cover calculated as follows: 

1 1 + U= ho hi

−1 (8)

where, h o and h i are respectively, the convective heat transfer coefficients of the outside and inside greenhouse cover (W m−2 K−1 ). They are calculated by: h o = 2.8 + 1.2 Wv

(9)

hi = 5.2 · |Tin − Ta |0.33

(10)

where, Wv presents external wind velocity (m s−1 ). Q loss is the leakage rate of air through the greenhouse given by: Q loss = Closs ρa (Tin − Ta )

(11)

Random Scaling-Based Bat Algorithm for Greenhouse Thermal … Table 1 Input parameters used for computation

Symbol

51

Value

Unit

Description

Sf

0.68

m2

Greenhouse floor surface

V

0.76

m3

Greenhouse volume m−3

γ0

1.205

Kg

Patm

101

kPa

The inside dry air density The atmospheric pressure

where, Closs is the heat loss coefficient. Q ther mal is the longwave radiation absorbed by the greenhouse given by:   Q ther mal = Cther mal h o Tin − Tsky

(12)

where, Cther mal is the thermal energy coefficient and Tsky is the sky temperature (K). The linearly dependent static parameters: Csolar , Ccnv,cnd , Closs and Cther mal have to be accurately known, which highlights the need for their identification. The other static parameters are fixed and given in Table 1.

3 Thermal Model Identification Based on Bat Algorithms This work concerns the application of the proposed RSBA in an off-line parametric identification process. The problem of identification in this paper consists of finding the optimal static parameters (Csolar , Ccnv,cnd , Closs and Cther mal ) of the thermal model described in Eqs. (1)–(12), that fit the data samples of the inside air temperature of the greenhouse. BA is a bio-inspired population-based metaheuristic algorithm. It has been developed based on bats behaviour of how they search on their targets using the echolocation capability [2]. Virtual bats are used in simulation and a set of bases are defined for clarification as follows: • All bats sense distance and differentiate between prey and background barriers by using echolocation. • Bats fly randomly (Random walks technique) with velocity vi at position xi . The frequency (or wavelength) of their emitted pulses is automatically adjusted. • The rate of pulse emission r can also be tuned automatically according to the closeness of target, or considered as a constant. • The loudness can be considered variant, starting from a large positive value A0 to a minimum value Amin or it can also be a constant. Bats position (solution) xit and velocity vit in a predefined d-dimensional search space are updated according to the following equations: f i = f min + ( f max − f min )β

(13)

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  vit+1 = vit + xit − x∗ f i

(14)

xit+1 = xit + vit+1

(15)

where, f i ∈ [ f min , f max ] is the randomly assigned frequency to each bat. β ∈ [0, 1] is a random variable drawn from a uniform distribution, x∗ is the current global best solution which is selected after comparing all the solutions of all the n bats. When a solution is chosen among the current best solutions, a new solution for some bats (according to pulse emission rate) is generated locally during the exploitation stage using random walk based on the following equation: xnew = xold + σ  t At

(16)

where, xnew and xold are respectively the new and the old best local solutions,  t ∈ [−1, 1] is a random number and σ ψ is a scaling parameter to control the step size. The scaling parameter is declared constant for the standard BA. It should be linked to the scalings of the design variables of the problem under consideration [2]. However, this is not enough to reach the optimality, because when the optimal solution is near to be reached, the constant step size of search (local random walk) remains relatively large even with the effect of loudness on local search. It results in a reduction in convergence speed and low-accurate solution as a sub-stagnation state at the end. Based on this fact, we proposed to make the scaling parameter σ: 1. Fully responsible about step size control by eliminating the effect of loudness on local search equation given by:

xnew = xold + σ t  t

(17)

2. Dynamically updated over iterations based on random scaling parameter mechanism, given by:

σ t+1 = σmin + (σmax − σmin )β

(18)

where the scaling parameter σ ∈ [σmin , σmax ]. As in standard BA, the loudness Ai and the pulse emission rate ri have to be updated as the iterations proceed. The loudness generally decreases once a bat finds the target, whereas the rate of pulse emission increases. They are updated based on the following equations: Ait+1 = α Ait

(19)

Random Scaling-Based Bat Algorithm for Greenhouse Thermal …

  rit+1 = ri0 1 − ex p(−γ t)

53

(20)

where α and γ are constants. For any 0 < α < 1 and γ > 0, the change in loudness and pulse rate is directed as follows: Ait → 0, rit → ri0 , as t → ∞ In order to replace the parameter values xit j out of the search range   max x j , min x j with random ones, an algorithm has been implemented, it is given as follows:     I f xit j > max x j or xit j < min x j then   xit j = min x j + max x j − min x j · rand(0, 1) The objective function to be minimized by BAs is the commonly used least squares criterion, defined as: J=

N  

yri eal − y ipr e

2

(21)

i=1

where, yri eal and y ipr e are respectively the real and the predicted data samples and N is the number of data samples. Based on the aforementioned assumptions and rules the essential stages of the proposed RSBA are summarized as the schematic pseudo code presented in Fig. 2.

4 Simulation and Experimental Validation The aim of this section is two folds. Firstly, it is to investigate the efficiency of the RSBA against the standard BA in identifying simulated greenhouse thermal model with known parameters. Secondly, it is to use the RSBA to identify the thermal model of an experimental greenhouse prototype. The obtained model will be then validated experimentally for predicting the inside air temperature of this prototype. A. Identification setup and specifications The searching ranges of the model parameters are listed in Table 2. The common control parameters between standard BA and RSBA are: the number of population is n = 100 bats, the minimum and maximum frequency respectively are f min = 0 and f max = 1.5, the loudness of the initial bats is A0i = 1, the rate of pulse emission

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Random Scaling-based Bat Algorithm Initialize the bat population and Initialize frequencies , pulse rates and the loudness while (Current iteration < Maximum number of iterations) Generate new solutions (positions) by frequency tuning Update velocities and solutions Eqs. (14-15) if (rand > ) Select a solution among the best solutions Update scaling parameter value Eq. (18) Generate a local solution around the selected best solution Eq. (17) end if Generate a new solution by flying randomly < ) if (rand < and Accept the new solutions Increase and reduce Eq. (19-20) end if Find the current best end while Fig. 2 Pseudo code of RSBA

Table 2 The search range of the parameters to be identified Parameter Range

Csolar   0, 10−4

Ccnv,cnd   0, 10−3

Closs   0, 10−3

Cther mal   0, 10−4

of the initial bats is generated randomly r i ∈ [0, 1] and ri0 = 0.1, and the constants ψ ← ψ ← 0.9. The scaling parameter of the standard is constant σ = 10−2 .   BA −7 −2 Whereas the range of its variations regarding RSBA is 10 , 10 . B. Analysis of search dynamics The analysis is done based on ten runs for each BA. The results of identification are analysed according to a limited number of iterations (500 iterations) and achieved using different random initial populations. The evolution of the best fitness function values for all the runs of BAs along the iterations are illustrated in Fig. 3. It can be noticed that standard BA seems to have a weak performance for all the runs in term of solutions precision. In contrast, RSBA runs have achieved better performance in term of convergence speed at late stage and solution accuracy. Table 3 includes all the fitness function values and illustrates that RSBA scored the best fitness value, as a result of the influence of random scaling parameter on the local search step size. Moreover, it shows a good balance between convergence speed and result quality. As a result, RSBA proves to be more efficient than the standard BA in finding the parameters of the thermal model.

Random Scaling-Based Bat Algorithm for Greenhouse Thermal …

55

Fig. 3 Evolution of the best fitness function values

Table 3 Best fitness function values of identification processes

Identification process

Bat Algorithms Standard BA

RSBA

1st run

3.4003e−04

9.7949e−08

2nd run

1.4187e−03

2.6286e−05

3rd run

3.6587e−04

1.0008e−06

4th run

2.1736e−04

1.4771e−04

5th run

2.0633e−04

1.3245e−05

6th run

2.2142e−04

2.9300e−05

7th run

2.8384e−05

2.2254e−04

8th run

3.7457e−05

7.1692e−07

9th run

1.7585e−04

3.5900e−04

10th run

1.0056e−04

1.5055e−06

C. The evaluation of parameter values found by RSBA The relative error criterion is used to evaluate the best identified parameter values by RSBA against the optimal ones. It is given by: RE =

| p R S B A − pr eal | · 100 pr eal

(22)

where, pr eal is the assumed parameter value and p R S B A is the parameter value identified by the proposed RSBA

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The evolution of the relative error of every identified parameter value from the best RSBA execution is shown in Fig. 4. It is obvious that the RSBA succeeded at the end to find the parameter values. The best parameter values are listed in Table 4, next to their corresponding relative errors. It shows that the value of Closs is the less accurate identified parameter due to its small effect on the output, in contrast to Cther mal and Csolar which are the most accurate identified parameters because of their strong effect on the inside air temperature. It is recognized that the acceptable identification result corresponds to the relative errors of less than 0.5%. Hence, the results show that the parameters have been identified with satisfactory accuracy. D. Experimental setup (a) Description of the greenhouse prototype A small-size single span wooden-structured greenhouse has been constructed as a nursery prototype covered by polyethene with 0.2 mm of thickness as it appears in Fig. 5. It is located in the municipality of M’ziraa, affiliated to Biskra province in Algeria (34°43 19.7 N 6°17 39.2

Fig. 4 Evolution of the relative error of every parameter from the best identification run of the RSBA

Table 4 Comparison between the assumed and the identified model parameter values using rsba

Parameter Assumed value Identified value Relative error (%) Csolar

5.8e−05

5.800004e−05

6.007544e−05

Ccnv,cnd

5.3e−04

5.300057e−04

1.067788e−03

Closs

1.7e−04

1.699894e−04

6.251072e−03

Cther mal

2.2e−05

2.199999e−05

5.369802e−05

Random Scaling-Based Bat Algorithm for Greenhouse Thermal …

57

Fig. 5 External and internal view of greenhouse prototype

E); it is an arid region with a moderate desert climate. The experimental nursery has a wooden soilless floor that embraces three seedling bunches full of treated soil. (b) Data acquisition system A low-cost data acquisition system has been designed based on two programmable boards; Arduino Mega 2560 and NodeMCU v0.1. It gathers measurements from all the sensors then send them to the PC via Wi-Fi. The data is treated by a MATLAB code to create the database used in this study. The internal and external climate variables are measured using the following sensors: – Two DHT22 equipped with a low-cost design of a plate-shaped radiation shield [8]. They are installed indoor the greenhouse at high of 0.3 m and outdoor the greenhouse at high of 1.25 m to measure air temperature and relative humidity. The accuracy of humidity and temperature respectively are ± 5%RH and < ±0.5 °C. – A low-cost pyranometer based on BPW34 silicon photodiode with an accuracy of 2.3% error [9]. It is installed outdoor the greenhouse at a height of 1.4 m, to measure the global solar irradiation. – A low-cost anemometer based on DC motor is installed outside at a height of 1.55 m [10]. The sensitiveness of each of the sensors is listed in Table 5. (c) Greenhouse database The database consists of measurements of the inside and the outside air temperature, the inside relative humidity, the outside global solar radiation and the wind velocity. These measurements are recorded in five consecutive days in the winter season (26th to 30th of January 2019) as it appears in Table 5 Sensors sensitiveness Sensor Sensitiveness

Temperature 0.1 °C

Humidity 0.1%

Pyranometer 9.6

w/m2

Anemometer 0.47 m/s

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Figs. 6 and 7. The sampling time was 1 min. The data has been divided into two parts; the data of 2nd and 3rd days are devoted to the identification process. The diversification of their climate (calm and turbulent) will ensure

Fig. 6 Inside and outside air temperature and inside relative humidity

Fig. 7 External global solar radiation and wind velocity

Random Scaling-Based Bat Algorithm for Greenhouse Thermal … Table 6 The identified parameters of the real greenhouse model

Parameter

Experimentally identified value

Csolar

6,83455e−05

Ccnv,cnd

4,14191e−04

Closs

2,55726e−04

Cther mal

2,55627e−05

59

an effective selection of model parameter values and prediction flexibility for various climate states. whereas, the data of the other three days are kept for the experimental validation. E. Experimental results We aim in this section to validate the simulation outcomes of the proposed thermal model identification strategy, using an experimental database obtained from the greenhouse prototype. The RSBA-based identification of the experimental thermal model has been achieved using the same control parameters as for the identification of the simulated thermal model. Whereas, in this section, the target is the real measured inside air temperature. The identified parameter values using the data of the 2nd and 3rd days of the experimental database are listed in Table 6. The results of thermal prediction using these identified parameter values with the measured inside temperature of the 2nd and 3rd days are shown in Fig. 8. The validation of the identified model is achieved by predicting the measured inside air temperature of the 1st, 4th and 5th days. The variation of predicted and measured inside air temperature are illustrated in Fig. 9. The performance of the identified model is evaluated based on five criteria: Mean Absolute Error (MAE), Max

Fig. 8 Measured and predicted inside temperature of the 2nd and 3rd days

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Fig. 9 Measured and predicted inside temperature of the 1st, 4th and 5th days

Table 7 Statistical evaluation of predicted inside air temperature Criterion

MAE

R2

EF

MaxAE

Value

0.793692

0.995624

0.995127

3.391057

Absolute Error (MaxAE), Coefficient of determination (R2 ) and Model Efficiency (EF). They evaluate the difference between predicted and measured air temperature in different techniques based on Eqs. (23)–(26). M AE =

n 1  Tm (i) − T p (i) n i=1

MaxAE = max Tm (i) − T p (i)

R = 2

2   Tm (i) − Tm T p (i) − T p  n   n  i=1 Tm (i) − Tm · i=1 T p (i) − T p

(24)

n  i=1

n  EF =

(23)

i=1

2 n  2 Tm (i) − Tm − i=1 T p (i) − Tm (i) 2 n  i=1 Tm (i) − Tm

(25)

(26)

where Tm is the measured temperature, Tm is the mean value of the measured temperature, T p is the predicted temperature, T p is the mean value of the predicted temperature and n is the number of data samples. Table 7 gives the results of this statistical analysis as an evaluation of the identified model. The MAE shows a satisfactory evaluation represented by a very small

Random Scaling-Based Bat Algorithm for Greenhouse Thermal …

61

difference between the measured and predicted inside air temperature. We can also see that R 2 gives a very high value for regression analyses meaning that 99.5% of the variance in the measured inside air temperature of the greenhouse prototype has been predicted by the real thermal model. Moreover, EF shows a very good evaluation where the closer the value is to “1” the more accurate the model is. The MaxAE shows that the biggest error value among all the data samples equals to 3.39 °C which is logically small. The aforementioned evaluation criteria illustrate that the model is a promising tool to study this phenomenon in term of usefulness, efficiency and applicability.

5 Conclusion In this paper, we proposed a variant of BA called RSBA to identify the thermal model of a greenhouse. The proposed RSBA differ from the standard BA in using a random scaling parameter mechanism. A comparative study has been conducted between the proposed RSBA and the standard BA on a simulated thermal model with known parameters. By examining the evolution of the search and the results of the optimization, we found that the RSBA outperforms the standard BA and its capability of the exploitation has been enhanced. Specifically, the accuracy of the solution and the speed of convergence when the optimal solution is near to be found have been effectively increased. The RSBA was then applied to identify a real greenhouse thermal model. The results of the identified model validation using the experimental database exhibited a very good fit between the measured and predicted inside air temperature. This study paves the way for future investigation on developing a novel mechanism that will make the scaling parameter changes adaptatively with the closeness of the optimal solution.

References 1. Ma D, Carpenter N, Maki H, Rehman TU, Tuinstra MR, Jin J (2019) Greenhouse environment modeling and simulation for microclimate control. Comput Electron Agric 162:134–142 2. Yang XS (2014) Nature-inspired optimization algorithms, 1st edn. Elsevier 3. Lyu S, Li Z, Huang Y, Wang J, Hu J (2019) Improved self-adaptive Bat Algorithm with stepcontrol and mutation mechanisms. J Comput Sci 30:65–78 4. Ramli MR, Abas ZA, Desa MI, Abidin ZZ, Alazzam MB (2018) Enhanced convergence of Bat Algorithm based on dimensional and inertia weight factor. J King Saud Univ Comput Inform Sci (In press) 5. Yilmaz S, Kucuksille EU (2013) Improved Bat Algorithm (IBA) on continuous optimization problems. Lect Notes Softw Eng 1(3):279–283 6. Guesbaya M, Megherbi H (2019) Thermal modeling and prediction of soilless greenhouse in arid region based on particle swarm optimization: experimentally validated. In: International conference on advanced electrical engineering, Algiers, Algeria

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7. Rodríguez F, Berenguel M, Luis Guzmán J, Ramírez-Arias A (2015) Modeling and control of greenhouse crop growth. Springer International Publishing, Switzerland 8. Holden ZA, Klene AE, Keefe RF, Moisen GG (2013) Design and evaluation of an inexpensive radiation shield for monitoring surface air temperatures. Agric Meteorol 180:281–286 ˇ 9. Cekon M, Slávik R, Juráš P (2016) Obtainable method of measuring the solar radiant flux based on silicone photodiode element. Appl Mech Mater 824:477–484 10. Horsey J (2016) DIY Arduino wind speed meter anemometer project. Geeky Gadgets. https:// www.geeky-gadgets.com/arduino-wind-speed-meter-anemometer-project-30032016/

Control and Observation of Induction Motor Using First-Order Sliding Mode Salah Eddine Farhi, Djamel Sakri, and Noureddine Goléa

Abstract A sliding mode controller is one of the non-linear techniques that have advantages of being robust and simple to implement. In this paper, a First Order Sliding Mode (FOSM) is chosen to provide decoupled control of the induction machine. In addition and in order to eliminate the use of sensors, flux and speed observers based on First Order Sliding Mode are introduced. The proof of stability of the algorithm is also presented using Lyapunov theory. The obtained results through simulations verifies that the designed control scheme have strong robustness toward the variation of the speed, load torque and parameter uncertainty. Keywords Induction Motor (IM) · Lyapunov · Observer · Sensors · Sliding mode controller · First order sliding mode (FOSM)

1 Introduction The asynchronous or induction motor is the most widely used electrical propulsion motor, his invention at the end of the last century gave a strong impetus to the transition from DC to AC motor [1]. It is the most popular machine in many fields such as chemistry, medicine, aerospace and various industries, the induction motor has many advantages more than the other types of electrical machines, among which: robustness, low cost, power maximum rating, maximum top speed, ease of maintenance. Another aspect is more important in the realization of electric motors is the notion of robustness. The models used are approximate and the variation of electrical and S. E. Farhi (B) · D. Sakri · N. Goléa Laboratory of Electrical Engineering and Automatic , Department of Electrical Engineering, LGEA, Université Larbi Ben M’hidi, Oum El Bouaghi, Algeria e-mail: [email protected] D. Sakri e-mail: [email protected] N. Goléa e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_5

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mechanical parameters degrades the performance of the controls and can lead, in some cases, to unstable operations. Powerful nonlinear control techniques have been presented widely in electric drives. Among them, we mention sliding mode control [2]. Sliding mode control theory, given its ability to circumventing the effects of parameter variation along with low implementation complexity [3] and disturbance rejection, strong robustness and simplicity of execution, is one of the future control methodologies for electrical machines [4, 5]. Sliding mode setting which by its nature is a nonlinear control, is a powerful tuning technique is basically a method that forces the response to slide along a predefined path. For getting high performance with an asynchronous machine, requiring reliable information from process control [6], Most of the research in induction motors control during the last years has been on sensorless solutions [7]. Observer design is an important topic in control system [8]. Sensorless control means that an encoder is replaced by estimated speed and flux from an observer or an estimator in order to reduce cost and increases the reliability of the system [9]. Sliding mode observers have unique characteristics, that a produces a set of state estimates that are strictly proportional with the actual output of the plant [10] and makes the flux and speed estimation robust to all variations, such as sensors and measurement noises, off set in the measurement system, phase shift in the measured value [11]. High-performance controllers are achieved when speed and flux are readily available, it seems better to estimate the speed and flux [12]. To replace the sensor, the information is extracted on the rotor speed and flux from measured stator voltages and currents at the ends of the motor [13]. In this paper, the sliding mode approach is introduced both for control and for observers design basing only on the measurements of stator voltages and currents.

2 Model of Induction Motor The mathematical model of IM in the stationary reference (α, β) can be written as follows [14]: 1 M ϕr α − ωr ϕrβ + i sα Tr Tr 1 M = ωr ϕr α − ϕrβ + i sβ Tr Tr

ϕ˙r α = − ϕ˙rβ

1 Vsα i˙sα = a1 ϕr α + a1 Tr ωr ϕrβ − a2 i sα + σ Ls 1 Vsβ i˙sβ = −a1 Tr ωr ϕr α + a1 ϕrβ − a2 i sβ + σ Ls

(1)

Control and Observation of Induction Motor …

65

The torque equation that describes the relationship between the motor torque and the rotor speed is depicted as: ˙ = Tm − TL − F J

(2)

where the electromagnetic torque of motor is given as follows: Tm =

 3 M p i sβ ϕr α − i sα ϕrβ 2 Lr

(3)

where Lr Ls M2 M 1 , Ts = ,σ = 1 − , a1 = , a2 = Tr = Rr Rs Ls Lr σ Tr L s L r σ



1 1−σ + Ts Tr



3 Speed and Flux Control The method of speed and flux control uses the sliding mode technique to design discontinuous controls without explicitly involving current control. The advantage of this approach is its robustness with respect to the mechanical parameters [15]. The method of selecting the sliding surfaces in the majority of controls is the errors of controlled states and according to the relative degree required. The expression of the speed and flux control surface is defined by:   s1 = c1 ϕr∗ − ϕr + (ϕ˙r∗ − ϕ˙r )   s2 = c2 ωr∗ − ωr + (ω˙ r∗ − ω˙ r )

(4)

where ϕr =



2 ϕr2α + ϕrβ

(5)

Equation (5) is the magnitude of the rotor flux and the sign (*) in (4) represents the references inputs. c1 , c2 are positive constants determining the motion performance in sliding mode [15]. The temporary derivative of (4) is:     s˙1 = c1 ϕ˙r∗ − ϕ˙r + ϕ¨r∗ − ϕ¨r     s˙2 = c2 ω˙ r∗ − ω˙ r + ω¨ r∗ − ω¨ r After calculates and reductions, s˙1,2 written on the matrix form simplified as:

(6)

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s˙1 s˙2



 =

   F1 Vsα − A1,2 F2 Vsβ

(7)

where 

s1,2 = [s1 s2 ] , F1,2 = [F1 F2 ] , T

T

A1,2

a cos θ a sin θ = −b sin θ b cos θ



where F1,2 are function of the state variables, the reference inputs and their time ϕ derivatives. But they do not depend on the controls Vsα , Vsβ ; θ = tan−1 ϕrβra and a, b are defined by: a = Rr

M Ls Lr − M 2

b=

M 3 ϕr P 2 J Ls Lr − M 2

With L s L r − M 2 > 0 We extract relation between s˙1,2 and the switching controls Vabc : s˙1,2 = F1,2 − D1,2 Vabc

(8)

abc With D1,2 = A1,2 Aabc αβ , matrix Aαβ is defined as the Clarke transformation. Equation (8) is the starting point of the sliding mode control design. Since matrix D1,2 is time-varying. The controls defined in Vabc cannot perform their switching depending on s1,2 . In this case a transformed vector of new switching function is required to achieve a time-invariant control matrix. Design the transformation as [15]: + s1,2 s∗ = D1,2

(9)

T  where s∗ = s∗1 s∗2 s∗3 is the vector representing the transformed switching + function, D1,2 is the pseudo inverse of matrix D1,2 given by:   3 + T T −1 D1,2 D1,2 = D1,2 = (Aabc )T (A1,2 )T Q D1,2 2 αβ

(10)

+ With Q is a positive definite time-varying matrix, Q and D1,2 being defined by

 Q=

1 a2

0

0 1 b2



⎡1

+ D1,2

⎤ cos θa − b1 sin θa = ⎣ cos θb − b1 sin θb ⎦ cos θc − b1 sin θc a 1 a 1 a

(11)

where θa = θ , θb = θ − 23 π , θc = θ + 23 π , parameter b is proportional to ϕr . The control Vabc law is given by:

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  Vabc = V0 sign s∗

(12)

  T    With sign(s∗ ) = sign(s∗1 ) sign s∗2 sign s∗3 and V0 is the DC link voltage. To ensure the operation of control well, we need to prove the stability, by the Lyapunov function we provide the format: V =

1  T s1,2 Qs1,2 2

(13)

 T 1  T ˙ V˙ = s1,2 Q˙s1,2 + s1,2 Qs 1,2 2 = (s∗ )T (D1,2 )T Q F1,2 − (s∗ )T (D1,2 )T Q D1,2 Va,b,c +

1  T ˙ s1,2 Qs1,2 2

(14)

Can be simplified:  T abc   Aαβ V0 sign s∗ V˙ = (s∗ )T F ∗ − (s∗ )T Aabc αβ   − f ϕra , ϕrβ , ϕ˙ra , ϕ˙rβ s22

(15)

where  T F ∗ = F1∗ F2∗ F3∗ = (D1,2 )T Q F1,2   1  T ˙ s1,2 Qs1,2 = − f ϕra , ϕrβ , ϕ˙ra , ϕ˙rβ s22 2 T abc  (D1,2 )T Q D1,2 = Aabc Aαβ αβ   f ϕra , ϕrβ , ϕ˙ra , ϕ˙rβ =



3 J Ls Lr − M 2 2p M

2

(16)

ϕra ϕ˙ra + ϕrβ ϕ˙rβ

2 2 ϕr2α + ϕrβ

According to (15) there exists a DC link voltage V0 high enough such that V˙ < 0. This means that system (6) converges to the origin s1,2 = 0 in finite time under control (12) and transformation (9) [16]. This means is system asymptotic stable.

4 Sliding Mode Observer At present, we will proceed to design an observer to facilitate the maximum control and neglected the sensors and to establish a good compromise between the stability and the simplicity of control, it is necessary to take a reference axis linked to the stator (α, β), we suggest observer submitted in [15]. The observer model is described

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by the following state equation: 1 M ϕ˙ˆr α = − ϕˆra −ωˆ r ϕˆrβ + i sα Tr Tr 1 M ϕ˙ˆrβ = ωˆ r ϕˆr α − ϕˆrβ + i sβ Tr Tr ˙iˆ = a ϕˆ + a T ωˆ ϕˆ − a iˆ + 1 V sα 1 ra 1 r r rβ 2 sα sα σ Ls 1 Vsβ i˙ˆsβ = −a1 Tr ωˆ r ϕˆra + a1 ϕˆrβ − a2 iˆsβ + σ Ls

(17)

ωˆ r is discontinuous and given by: ωˆ r = ω0 sign(s3 )

(18)

where ω0 is a positive control gain, the sliding surface s3 is defined as: s3 = i sβ ϕˆra − i sα ϕˆrβ

(19)

where i sα = iˆsα − i sα ; i sβ = iˆsβ − i sβ Defines the errors: ϕ¯r α = ϕˆr α − ϕr α ; ϕ¯rβ = ϕˆrβ − ϕrβ ; ω¯ r = ωˆ r − ωr The temporary derivative of s3 is:     2 s˙3 = a1 Tr ωr ϕˆr α ϕr α + ϕˆrβ ϕrβ − a1 Tr ωˆ r ϕˆr2α + ϕˆrβ     − ωˆ r ϕˆrβ i¯sβ + ϕˆr α i¯sα + a1 ϕˆr α ϕ¯rβ − ϕˆrβ ϕ¯r α   1 − s3 + M i sβ i¯sα − i sα i¯sβ − a2 s3 Tr

(20)

So s˙3 can be simplified: s˙3 = −a1 Tr ωr e1 − a1 Tr ω¯ r ϕr2 − ωˆ r e3 + a1 e2   1 s3 + M i sβ i¯sα − i sα i¯sβ − a2 s3 − Tr where e1 , e2 and e3 is defined as:   2 − ϕˆr α ϕr α + ϕˆrβ ϕrβ e1 = ϕ¯r α ϕˆr α + ϕ¯rβ ϕˆrβ = ϕˆr2α + ϕˆrβ e2 = ϕ¯rβ ϕˆr α − ϕ¯r α ϕˆrβ = −(ϕˆr α ϕrβ − ϕˆrβ ϕr α )

(21)

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  e3 = ϕˆrβ i¯sβ + ϕˆr α i¯sα

(22)

Replaces ωˆ r in (21) with (18) where the condition of ω0 is    −a T ω e + a T ω ϕ 2 + a e − a s     1 r r 1 1 r r r 1 2  2 3  e3 + a1 Tr ϕr2 ω0 >   1 ¯ ¯ − Tr s3 + M i sβ i sα − i sα i sβ  

(23)

where   e3 + a1 Tr ϕr2 > 0

(24)

If condition (23) check such that s3 s˙3 < 0, this means that s3 enforced equal a zero after a finite time interval and this means the estimated current converges to reference current. The condition (24) for sliding mode to occur on the surface is s3 = 0 not very restrictive because the stator currents i sα and i sβ are measurable. We can always choose the initial conditions iˆsα (0) and iˆsβ (0) close enough to the true stator currents i sα (0) and i sβ (0) such that the initial errors i¯sα = 0 and i¯sβ (0) and hence e3 are small enough to guarantee that this condition holds [16]. Sliding mode trajectories lie in the manifold s3 = 0 and the equivalent control ωˆ r _eq is a solution to the equation s˙3 = 0 it is necessary to replace discontinuous control with such continuous control that the state velocity vector lies in the tangential manifold [17]. From (19) see that if i¯sα = 0 and i¯sβ = 0 is a solution of s3 = 0 replaces in (21), the equivalent control ωˆ r _eq is given by:   ωˆ r _eq = ωϕ 2r ϕˆr α ϕr α + ϕˆrβ ϕrβ + r 2 ϕr2 = ϕˆr2α + ϕˆrβ = 0∀t

1 e Tr ϕr2 2

(25)

The Eq. (25) implies that if the estimated rotor flux converges to the true flux then the equivalent value of ωˆ r _eq is equal to the true rotor speed, lim ωˆ r _eq = ωr . ϕ→ϕ ˆ

ωˆ r _eq can be obtained through a low pass filter with the discontinuous value [15]: ωˆ r _eq =

1 ωˆ r 1 + tc P

(26)

where tc = f1r is the time constant of the filter, and P is the Laplace variable. Where the time constant of the low-pass filter tc should be small compared with the slow component of ω0 sign(s3 ) but large enough to filter out the high rate component [16]. The control structure of in IM is (Fig. 1):

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Fig. 1 Induction motor controls with sliding mode

5 Simulation Results To assess the performance of control studied, the simulation is applied for the IM drives. The software used is Simulink Matlab 2016b. The following figures shows the behavior of the induction motor at different values of speeds and load torque variation. Figure 2 show the speed response, at startup the speed is changed from 0 rad/s to their reference value 148.7 rad/s at t = 0.3 s change to −148.7 rad/s at second t = 0.6 Back to their reference value. Note that the real speed and estimated have the same behavior, the tracking performance is good that means the observer can follow the speed command strictly. When the load is suddenly increased at t = 0.8 s valued at Cr = 10 N m there is no change on the speed and the estimated value and which is illustrated by speed zoom.

Fig. 2 Speed response

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Figure 3 show the time response of the actual stator current components, we note the classic call of the current when powering up the induction motor. The current reaches a maximum value of about 8–10 times the nominal current of the machine as well as when the speed change then stabilizes at the value of off-load current about 4 A once the load is applied an increase is noted the current reaches about 5.35 A and which is illustrated by current zoom, due to the increase of the motor torque. Figure 4 is the results of the rotor flux, we note the components are perfectly in quadrature and their amplitudes are identical. At seconds to change values of speeds

Fig. 3 Stator current response

Fig. 4 Rotor flux response

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there is only a slight change on the flux and it goes back to steady state quickly. See that the convergence between the rotor flux and the estimated of this value it occurs at the startup and good tracking performance the motor torque with reference load torque and good robustness to external load torque variation, which is illustrated by Fig. 5. Which is proves ability and strength to this control and observer. Figures 6, 7, and 8 give the result when stator resistance is increase 50% in motor model. The sliding mode control and observer was tested with difference speed and the load torque:

Fig. 5 Zoom flux and torque response

Fig. 6 Speed response in case of increased 50% stator resistances

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Fig. 7 Rotor flux response in case of increased 50% stator resistances

Fig. 8 Flux zoom and torque response in case of increased 50% stator resistances

In this test speed response for sliding control and observer there is a slight difference through a slow response during speed changes and then a good tracking followup of its reference we note their robustness with respect to the variation of the load torque this is what appears in speed zoom. The estimate flux is bigger than the actual value at seconds to change values of speeds and the real value are lowers than their actuals values and their return to steady state, note that the load torque does not affect them this is what appears in flux zoom. As a result, the control and observer

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small error in transient speeds only when stator resistance change, and good tracking performance the motor torque with reference load torque.

6 Nomenclature Vsα , Vsβ V0 i sα , i sβ ϕra , ϕrβ ωr Ls , Lr , M Rs , Rr Tr p, F J, σ Tm , TL ϕˆra , ϕˆrβ iˆsα , iˆsβ r e f , mes , est

Stator voltages DC link voltage Stator currents Rotor fluxes Electrical rotor speed Stator, rotor and mutual inductance Stator, rotor resistance Rotor time constant Number of pole pairs, coefficient of friction Moment of inertia, coefficient of dispersion Motor torque and load torque. Estimated for rotor flux Estimated for stator currents Reference, measured and estimate speed.

7 Conclusion In this paper, First order sliding mode strategy was applied to ensure a decoupled control for induction motor. To avoid the need of sensors, the speed and the rotor flux were estimated using non-linear observers based on sliding mode. Based on the Lyapunov theory, the convergence of control and observer was proved. The simulation results show the robustness of the proposed control scheme and confirm that the FOSM is more efficient and present a low sensitivity to disturbances and motor parameters variation.

Appendix Table 1presents the parameters utilized in the control:

Control and Observation of Induction Motor … Table 1 Parameters of induction motor and control

Parameters of induction motor [18]

75 Parameters of sliding mode control

Pn = 1500 W n = 1420 rpm c1 = 200 c2 = 100 ω0 = 1200 fr = 45 Hz Rs = 4.85  Rr = 3.805  L s = 0.274 H L r = 0.274 H M = 0.258 H p = 2 V0 = 537 V F = 0.001136 N ms

References 1. Leonhard W (2001) Control of electrical drives, 3rd edn. Springer, Berli 2. Ammar A, Bourek A, Benakcha A (2017) Nonlinear SVM-DTC for induction motor drive using input-output feedback linearization and high order sliding mode control. ISA Trans 67:428–442 3. Alsmadi YM, Utkin V, Haj-Ahmed M, Xu L, Abdelaziz AY (2018) Sliding-mode control of power converters: AC/DC converters & DC/AC inverters. Int J Control 91(11):2573–2587 4. Derdiyok A, Guven M, Utkin V (2001) A sliding mode speed and rotor time constant observer for induction machines. In: IECON’01, 27th annual conference of the IEEE industrial electronics society (Cat. No. 37243), vol 2, pp 1400–1405 5. Oliveira CMR, Aguiar ML, Monteiro JBA, Pereira WC, Paula GT, Santana MP (2015) Vector control of induction motor using a sliding mode controller with chattering reduction. In: 2015 IEEE 13th Brazilian power electronics conference and 1st southern power electronics conference (COBEP/SPEC), pp 1–6 6. Dris A, Bendjebbar M (2016) Different technics of observation of the flux and the speed of an induction motor (KUBOTA and MRAS observers). In: 2016 8th international conference on modelling, identification and control (ICMIC), pp 61–66 7. Bullo D, Ferrara A, Rubagotti M (2011) Sliding mode observers for sensorless control of current-fed induction motors. In: Proceedings of the 2011 American control conference, pp 763–768 8. Bernat J (2018) Multi observer structure for rapid state estimation in linear time varying systems. Int J Control Autom Syst 16(4):1746–1755 9. Dong CST, Tran CD, Ho SD, Brandstetter P, Kuchar M (2008) Robust sliding mode observer application in vector control of induction motor. In: 2018 ELEKTRO, pp 1–5 10. Fridman L, Li S, Man Z, Wang X, Yu XH (2017) Advances in variable structure systems and sliding mode control—theory and applications. Springer 11. Nadh G, Syamkumar U, Jayanand B (2016) Sliding mode observer for vector control of induction motor. In: 2016 international conference on next generation intelligent systems (ICNGIS), pp 1–6 12. Ghanes M, Zheng G (2009) On sensorless induction motor drives: sliding-mode observer and output feedback controller. IEEE Trans Indus Electron 56(9):3404–3413 13. Holtz J (2002) Sensorless control of induction motor drives. Proc IEEE 90(8):1359–1394 14. Rao S, Buss M, Utkin V (2009) Simultaneous state and parameter estimation in induction motors using first- and second-order sliding modes. IEEE Trans Indus Electron 56(9):3369– 3376 15. Utkin V, Guldner J, Shijun M (1999) Sliding mode control in electro-mechanical systems. CRC Press, Philadelphia 16. Utkin V, Guldner J, Shi J (2009) Sliding mode control in electro-mechanical systems. CRC Press, Boca Raton 17. Utkin VI (1993) Sliding mode control design principles and applications to electric drives. IEEE Trans Indus Electron 40(1):23–36

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18. El hani S, Essaaidi E (2019) Recent advances in electrical and information technologies for sustainable development. In: Proceedings of the 3rd international conference on electrical and information technologies — ICEIT 2017 Morocco. Springer International Publishing

Cuckoo Search Algorithm for Solving the Problem of Unit-Commitment with Vehicle-to-Grid Amel Terki and Hamid Boubertakh

Abstract This paper addresses the optimization of unit commitment (UC) with Vehicle to Grid (V2G) problem using a Binary-real coded Cuckoo Search (BCS) algorithm. The problem lies in the efficient planning of gridable vehicles in restricted constrained parking lots. Cuckoo Search (CS) algorithm is an intelligent metaheuristic algorithm that implement local and global search mechanisms via Lévy flights approach. It has been effectively used to solve several optimization problems. In order to verify the efficiency of the described algorithms; numerical simulation are implemented for 10-unit system with V2G in constraints parking lots to minimize the operating costs and has compared to the other existing algorithms. Keywords Unit commitment problem · Power system · Renewable energy · Vehicle to grid (V2G) · Cuckoo Search algorithm

1 Introduction The Unit Commitment (UC) problem is one of difficult and complex optimization problem in power system. The main role of the UC is to determine when to start-up or shut-down of each generating unit for a scheduling horizon in order to minimize the operating costs and satisfy the prevailing constraints, including minimum up and down time limits, ramp rate limits, generator power output limits and proper spinning reserves [1, 2]. Vehicle to grid (V2G) technology is collecting a large research attention in the generation of energy storage units; it’s one of the many energy storage technologies

A. Terki (B) · H. Boubertakh Laboratoire d’Automatique de Jijel-LAJ, University MSB of Jijel, BP 98 Ouled Aissa 1800, Jijel, Algeria e-mail: [email protected] H. Boubertakh e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_6

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[3]. The V2G have the ability of minimizing the cost of the power generation in small expensive units, and they could provide a distributed storage to power [4]. The Unit Commitment with Vehicle to Grid (UC-V2G) is the perspectives that involve the efficient planning of gridable vehicles in restricted constrained parking lots to meet out the demand. In this context, the UC-V2G problem is among of the most complex optimization problem in power systems than classic UC problem [5], because they should meet the forecasted load demand and spinning reserve requirements at each time interval such that the total cost is minimum [6]. Various optimization methods have been employed to solve the UC problem, such as Priority List (PL) [7] Dynamic Programming (DP) [8], and Lagrangian Relaxation (LR) [9]. These methods have a simple representation and fast convergence but the solution quality is not guaranteed. Recently, intelligent meta-heuristic optimization approaches attract much attention to solve the UC problem. This attraction is due to the parallel search mechanism of such approaches which offers a best way to search the global optimum. Among them, one find, Genetic Algorithm (GA) [10], Differential Evolution (DE) [11], Evolutionary Programming (EP) [12], Gravitational Search Algorithm (GSA) [13], Ant Colony Optimization (ACO) [14], Particle Swarm Optimization (PSO) [15, 16]. Moreover, since each method has its own advantages over the others, so, in order to combine the advantages of these approaches, some hybridized approaches have been proposed; Hybrid GA–PSO [17], Genetic Algorithm Based on Priority List (GABPL) [18] and Hybrid LRGA [19]. The Cuckoo Search (CS) approach, has been inspired by the natural behaviour of cuckoos, is one of the most powerful emerging approaches, initially proposed by Yang and Deb [20]. The CS has been applied successfully to different optimization problems [21–28], particularly to the UC problem [29–32]. Binary Cuckoo Search (BCS) algorithm it’s a binary version of the CS algorithm. BCS has never been used to solve the UC-V2G problem until now. In [29, 31] the On/Off status of a unit is based on a signed integer representation. Chandrasekaran and Sishaj [30] developed a fuzzy integrated binary real coded cuckoo search Lagrangian (BCSL) algorithm, which is used for multi-objective UC problem. In [32] weight pattern based cuckoo search algorithm is used for 3 and 6 units system. In this paper, we propose a new CS algorithm for solving the UC-V2G problem by using a binary code to deal with the units scheduling and a real code to deal with the produced power levels of the committed units. The CS algorithm is the only approach which implements separately a local and global search mechanism. The rest of paper is organized as follows. Section 2 provides the description of the UC-V2G problem, the CS and the BCS algorithm will be successively detailed in Sect. 3, the technique of handling UC-V2G constraints is described in Sect. 4. Results are presented in Sect. 5 and the conclusions in Sect. 6.

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2 Unit Commitment with V2G Problem To improve the quality level of power system by integrating new generation and storage opportunities, the electric vehicles will be seen as a charge in the unit commitment. The objective in the UC problem with the V2G is to find the optimal schedule of committed/uncommitted units and generated power per each unit over a study period of time while satisfying a variety of constraints. The objective function F to minimize comprises the fuel costs Cit ( pit ) of the generating units, the start-up costs STit of the committed units and the V2G cost during the scheduling hours. F=

T  T N        Cit ( pi ) + STit 1 − u i,t−1 u it + Pv N V 2G,t t=1 i=1

(1)

t=1

where u it is the On/off status of unit i at hour t, pit is the power output of unit i at hour t in (MW), Pv is the Capacity of each vehicle in (MW), N V 2G,t is the number of vehicles connected to the grid at time t, N is the number of units and T the time horizon. Cit ( pit ) = ai + bi pit + ci pit2

(2)

where ai , bi and ci the fuel cost coefficients of unit.  STit =

of f

di , i f idown ≤ τit ≤ idown + f i of f ei , i f τit > idown + f

(3)

where di and ei are the hot and cold startup cost of unit i, in ($), f i is the cold start up hours of unit i, in h (hour), idown and i are the Minimum down and up time of of f unit i, in h and τit is the continuously off time of unit i up to time t. A. Constraints The system and unit constraints plus the V2G constraints must be satisfied during the optimization process are as follows. (1) Power balance constraint: The total generated power from the gridable vehicles and all committed units at each hour must be equal to the load demand plus the system losses at hour t.

N  i=1

pit u it + Pv N V 2G,t = Dt + Losses

(4)

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where Dt is the load demand at hour t, in (MW) (2) Spinning reserve constraint: To assure reliable operation, the total generating unit and the gridable vehicles capacity must be greater than or equal to the load demand and the specified spinning reserve.

N 

pimax u it + Pvmax N V 2G,t ≥ Dt + Rt

(5)

i=1

where Rt is the spinning reserve at hour t, in (MW) (3) Minimum up/down time: A unit must remain on or off for a certain number of hours before it can be, respectively, shut off or put on. The constraints are given by  on  up  τi,t−1 − i u i,t−1 − u it ≥ 0    of f τi,t−1 − idown u it − u i,t−1 ≥ 0

(6)

where τiton is the continuously on time of unit i up to time t (4) Power generating limit: The active powers that can be generated by the generating units have minimum and maximum values.

pimin ≤ pit ≤ pimax

(7)

where pimin and pimax Minimum and Maximum capacity of unit i, in (MW). (5) Gridable vehicle balance in V2G: The total numbers of vehicles are registered to charge from renewable sources and discharge to the grid during 24 h is predefined for V2G technology.

T  t=1

N V 2G,t = N Vmax 2G

(8)

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where N Vmax 2G is the maximum number of vehicle in the system. (6) Vehicle parking limits: The number of vehicles will discharge at a given time is limited. The constraints are given by

N V 2G,t ≤ N Vmax 2G,t

(9)

(7) Charging-discharging frequency: The daily charging/discharging frequency is assumed as 1 per day. Each vehicle should have a suitable departure State of charge Soc level. Charging and inverter Efficiencies Effi should be considered.

3 Overview of the Cuckoo Search Metha-Heuristic A. Standard Cuckoo Search The CS is a swarm-intelligence optimisation method which has been initially developed by Yang and Deb [20]. It is inspired by the unavoidable brood parasite of cuckoo species. As it is well-known, the cuckoo bird never builds its personal nest and lays their eggs in the nest of another host’s. The implementation of CS algorithm follows the rules given bellow [21]. – Each cuckoo lays one egg at a time, and puts it in a random nest; – The best nests with high quality of eggs are carried to the next generations; – A host bird can discover a foreign egg with probability pa ∈ [0, 1]. In this case, they discard it away or abandon the nest to build a new nest. The main steps of the CS algorithm are recapitulated in the Algorithm 1. The first nest population is placed randomly and the generation of the new solution for the ith cuckoo is updated using random walk via Lévy flights [28] such as ´ xi(t+1) = xi(t) + α ⊕ L evy(λ)

(10)

where, xi for i = 1, 2, . . . , N is the position of a nest i, N the population size, α > 0 the scaling factor of the step size t and ⊕ the entry-wise product. The Lévy flights provide a random walk which is drawn from a Lévy distribution given by L evy ´ ∼ u = t−λ , (1 < λ ≤ 3)

(11)

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The new solution xi(t+1) can be generated by local and global search. The CS is used to represent N × T real variables of the pit quantity of power to be generated by  committed units, where pit assigned a random real value in the range pimin , pimax . B. Binary Cuckoo Search Binary Cuckoo Search BCS algorithm is developed by [33]. It is a binary version of the CS algorithm, in which the search space is defined as a binary n-bit string. The BCS algorithm represents each nest as a binary vector. Equations (12) and (13) are used in order to build binary vectors. In this study, we adopt the BCS algorithm to determine the On/Off status of the generating units u it over each interval of the time horizon with 0/1 decision variables with equal probability [33].  1 S xi(t) =  (t) 1 + e−xi   1 i f s xi(t) xi(t+1) = 0 other wise

(12)

(13)

where, σ ~ U (0, 1) and xi(t+1) stands for the new eggs.

4 Procedure for Handling the UC-V2G Problem Constraints In this paper, a new binary- real coded CS algorithm is proposed to solve the UCV2G problem. In fact, the problem is divided in two sub-problems. A procedure to handle constraints and an optimization method to solve the UC-V2G problem. A. Handling Constraints Several methods have been proposed in the literature for handling infeasible solutions; rejecting methods, repairing methods and penalty methods [10]. Unfortunately these algorithms cannot satisfy all the problem constraints. In this paper, the repairing and penalty methods are applied to solve UC-V2G constraint problems. In fact, when the constraints are not satisfied after the repairing methods, the penalty method is added to discourage the invalid solutions.

Cuckoo Search Algorithm for Solving the Problem …

(10)

83

(11)

(1) Gritable vehicle balance in V2G and Vehicle parking limits: A randomly distribution is applied to satisfy (10) and (11) during 24 h. (2) Spinning reserve constraint repairing: To satisfy spinning reserve constraint, a heuristic-based repair method is applied, through which de-committed units are committed in an ascending order of their average full load costs αi , i = 1, 2, …, N; until the spinning reserve constraint is satisfied [34]. αi =

Cit ( pit ) ai = max + bi + ci pimax max pi pi

(14)

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(3) Minimum up/down time constraints repairing: The minimum uptime/downtime constraints impose a minimum number of time periods that must elapse before the unit can change its status. The continuous On/Off times of units at hour t can be decided as follows. on + 1 i f u i,t = 1 τ on = i,t−1 τi,t 0 other wise of f + 1 i f u i,t = 0 τ of f (15) τi,t = i,t−1 0 other wise To satisfy these time periods, a heuristic technique procedure is summarized in [35]. Its purpose is to handle the minimum up-time and down-time constraints. (4) De-commitment of excess units: The high operation cost leads to undesirable excessive spinning reserve. In this case, a heuristic algorithm based on priority list is applied to de-commit some units one by one, in a descending order of their average full load costs αi , without violating the minimum uptime/downtime and spinning reserve constraints, which were checked after de-committing a unit [35, 36]. (5) Power balance constraint repairing: Each unit must generate power within the interval given by Eq. (7), but should lead to a higher production cost and satisfy the load demand constraint of Eq. (4). The quantity of divergence in the generated power with respect to the power demand at time t is given by et =

N 

u it pit − Dt

(16)

i=1

To satisfy the power balance constraint the algorithm is demonstrated in [34]. B. The Proposed Approach for Solving UC-V2G Problem In the proposed method the decision variables include binary variables for generating unit, real variables for power output and integer variables for gridable vehicle. The proposed algorithm is shown in Algorithm 2.

Cuckoo Search Algorithm for Solving the Problem …

85

5 Simulation Results To demonstrate the efficiency and feasibility of the proposed algorithm, we consider a basic system of 10 units with 50,000 gritable vehicles which are charged from renewable source for 24 h scheduling horizon. The basic unit data and the load

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demand D t of system are taken from [15]. In all cases, the spinning reserve requirement is set to 10% of the total load demand. The parameters’ settings of proposed algorithm are a population size of 100, the Capacity of each vehicle Pv is 15 kWh, the maximum and the minimum battery capacity are 25 kWh and 10 kWh respectively, the maximum parking lot capacity at hour t N Vmax 2G is 10% of total vehicles, Effi is 85% and SOC is 50%. Load demand is dispersed among generating units and gridable vehicles. At each hour t, the power from vehicles is E f f i × Soc × N V 2G,t × Pv and capacity of each vehicle Pv is constant. All simulations have been run on Matlab2016a environment. Tables 1 and 2, show the obtained results for all the test system without and with gritable vehicles respectively. The total cost (fuel cost plus start-up cost) is $554,01660 when gritable vehicles are considered in the UC system and is $561,28456 when gritable vehicles are not considered in the UC system. Therefore, it is observed that the total cost is reduced by $7.2680 for only gritable vehicles, this reduction due to, addition of vehicle power to the grid On the other hand the reserve of unit is increased when using V2G technology. It is obvious from Tables 2 and 3 that there is a saving of $7,2680 of the total cost comparison of the generating units with and without the V2G parking lot during 24, which shows the effectiveness of the proposed method for UC-V2G problem. It can be concluded that theV2G technology maintaining reliability of the power system. The best results obtained with other methods are compared to confirm the performance of the proposed algorithm. Table 3 shows a numerical comparison of proposed algorithm and other existing methods for the same number of unit system. It is clear that the total cost obtained by the proposed algorithm is less than the other approaches.

6 Conclusion In this paper, a new binary-real coded Cuckoo Search BCS algorithm is proposed for solving UC–V2G problem. The CS algorithm has been proposed to determine the real variables representing the power generated by committed units and a binary version of the CS algorithm is used to determine the binary variables representing the status of the generating units during each interval of the scheduling period. Results of the proposed algorithm have been compared with other methods reported in the literature. The production costs of the proposed algorithm are less than the existing algorithms. In conclusion, we have shown through simulations that the proposed algorithm outperforms other methods in the minimization of the total generation cost of the UC and the effective V2G scheduling can reduces this generation cost, increase reserve and improve the power system reliability.

455

455

455

455

455

455

455

455

455

13

14

15

16

17

18

19

20

455

8

12

455

7

455

455

6

11

455

5

455

455

4

455

455

3

10

455

9

455

2

1

455

455

455

404.37

356.16

455

455

455

455

455

455

455

455

455

360.57

448.16

455

370

295

245

2

0

0

0

0

0

130

130

130

130

130

130

130

130

85

129.42

0

0

0

0

0

3

130

130

130

115.62

115

130

130

130

130

130

130

130

130

130

130

71.83

0

0

0

0

4

Power generated by units (MW)

1

Time (h)

162

160

60

25

25

30

85

162

162

162

162

85

30

25

25

25

40

25

0

0

5

80

0

0

0

0

0

20

43

80

73

33

20

0

0

0

0

0

0

0

0

6

85

0

0

0

0

0

25

25

58

25

25

25

0

0

0

0

0

0

0

0

7

33

0

0

0

0

0

0

0

10

10

10

0

0

0

0

0

0

0

0

0

8

Table 1 Dispatch schedule and reserve power of generating units without V2G

0

0

0

0

0

0

0

0

10

10

0

0

0

0

0

0

0

0

0

0

9

0

0

0

0

0

0

0

0

10

0

0

0

0

0

0

0

0

0

0

0

10

1422

1202

1202

1202

1202

1332

1497

1497

1662

1607

1552

1497

1332

1332

1332

1202

1072

1072

910

910

Max. capacity (MW)

1400.0

1200.0

1100.0

1000.0

1050.0

1200.0

1300.0

1400.0

1500.0

1450.0

1400.0

1300.0

1200.0

1150.0

1100.0

1000.0

950.0

850.0

750.0

700.0

Demand (MW)

22

2

102

202

152

132

197

97

162

157

152

197

132

182

232

202

122

222

160

210

(continued)

Reserve (MW)

Cuckoo Search Algorithm for Solving the Problem … 87

455

455

455

23

24

345

455

455

455

2

Total Cost ($) = 5.6128e + 05

455

22

1

0

0

0

0

3

0

0

0

130

4

Power generated by units (MW)

21

Time (h)

Table 1 (continued)

0

0

145

162

5

0

0

20

73

6

0

0

25

25

7

0

0

0

0

8

0

0

0

0

9

0

0

0

0

10

910

910

1237

1367

Max. capacity (MW)

800.0

900.0

1100.0

1300.0

Demand (MW)

110

10

137

67

Reserve (MW)

88 A. Terki and H. Boubertakh

455

455

455

455

455

455

455

455

455

13

14

15

16

17

18

19

20

455

8

12

455

7

455

455

6

11

455

5

455

455

4

455

454

3

10

455

9

455

2

1

455

455

455

455

455

455

455

455

455

455

455

455

455

455

455

454.98

454.70

374.70

282.60

237.74

2

130

129.90

51.72

20

20

124.29

130

130

130

130

130

130

114.77

77.35

24.27

0

0

0

0

0

3

130

130

130

58.18

113.91

130

130

130

130

130

130

130

130

130

130

58.81

32.26

0

0

0

4

Power generated by units (MW)

1

Time (h)

0

0

0

0

0

25

97.49

157.26

162

162

162

97.49

25

25

25

25

0

0

0

0

5

80

20.06

0

0

0

0

20

20

80

53.49

25.70

20

0

0

0

0

0

0

0

0

6

85

0

0

0

0

0

0

25

33.08

25

25

0

0

0

0

0

0

0

0

0

7

Table 2 Dispatch schedule and reserve power of generating units with V2G

43.12

0

0

0

0

0

0

0

10

10

10

0

0

0

0

0

0

0

0

0

8

0

0

0

0

0

0

0

0

10

10

0

0

0

0

0

0

0

0

0

0

9

0

0

0

0

0

0

0

0

10

0

0

0

0

0

0

0

0

0

0

0

10

3431

1574

1298

1853

955

1680

1967

4351

3908

3059

1145

1961

3173

1200

1683

973

1259

3183

1945

1138

No.of vehicle

21.87

10.03

8.27

11.81

6.08

10.71

12.53

27.73

24.91

19.50

7.29

12.50

20.22

7.65

10.72

6.20

8.02

20.29

12.39

7.25

Capacity of vehicles (MW)

1411.9

1260

1178.3

1181.8

1176.1

1342.7

1424.5

1524.7

1686.9

1626.5

1559.3

1424.5

1352.2

1339.7

1.42.7

1208.2

1048

930.3

922.4

917.3

Max. capacity (MW)

1400.0

1200.0

1100.0

1000.0

1050.0

1200.0

1300.0

1400.0

1500.0

1450.0

1400.0

1300.0

1200.0

1150.0

1100.0

1000.0

950.0

850.0

750.0

700.0

Demand (MW)

(continued)

11.87

60.03

78.27

181.81

126.08

142.71

124.53

124.73

186.91

176.50

159.29

124.50

152.22

189.65

242.72

208.20

98.02

80.29

172.39

217.25

Reserve (MW)

Cuckoo Search Algorithm for Solving the Problem … 89

455

455

455

23

24

332.74

435.51

455

455

2

0

0

129.85

130

3

Total Cost ($) = 5.5402e + 05

455

22

1

0

0

0

130

4

Power generated by units (MW)

21

Time (h)

Table 2 (continued)

0

0

0

0

5

0

0

26.03

80

6

0

0

25

28.17

7

0

0

0

0

8

0

0

0

0

9

0

0

0

0

10

1922

1488

1430

3424

No.of vehicle

12.25

9.48

9.11

21.82

Capacity of vehicles (MW)

922.3

919.5

1214.1

1356.8

Max. capacity (MW)

800.0

900.0

1100.0

1300.0

Demand (MW)

122.25

19.48

114.11

56.82

Reserve (MW)

90 A. Terki and H. Boubertakh

Cuckoo Search Algorithm for Solving the Problem … Table 3 Comparison between proposed algorithm and other algorithm for 10 unit systems

Method

91 Total cost ($) Without V2G

With V2G

HSA, [37]

573,87934

554,13459

CRO, [38]

–

564,72787

PM, [39]

564,714

557,5549

PSO, [6]

563,74183

559,08136

Proposed

561,28456

554,01660

References 1. Abookazemi K, Mustaf MW, Ahmad H (2009) Structured genetic algorithm technique for unit commitment problem. Int J Recent Trends Eng Technol 1(3), November 2009 2. Park HG, Lyu JK, Kang YC et al (2014) Unit commitment considering interruptible load for power system operation with wind power. Energies 7:4281–4299 3. Zhile Y, Kang L, Xiandong X (2016) A hybrid meta-heuristic method for unit commitment considering flexible charging and discharging of plug-in electric vehicles. In: 2016 IEEE Congress on Evolutionary Computation (CEC), Vancouver, BC, 2016, pp 2014–2020. https:// doi.org/10.1109/CEC.2016.7744035 4. Mohammad EK, Lei W, Shahidehpour M (2012) Hourly coordination of electric vehicle operation and volatile wind power generation in SCUC. IEEE Trans Smart Grid 3(3):1271–1279 5. Ghanbarzadeh T, Goleijani S, Parsa M (2011) Reliability constrained unit commitment with electric vehicle to grid using hybrid particle swarm optimization and ant colony optimization. In: 2011 IEEE Power and Energy Society General Meeting, Detroit, MI, USA, 2011, pp 1–7. https://doi.org/10.1109/PES.2011.6039696 6. Saber AY, Venayagamoorthy GK (2009) Unit commitment with vehicle-to-grid using particle swarm optimization. In: IEEE Bucharest power tech conference, Romania 7. Manisha G (2016) Solution to unit commitment with priority list approach. Int J Adv Res Electr Electron Instr Eng (IJAREEIE) 3(6):5114–5121 8. Snyder WL, Powell HD, Rayburn JC (1987) Dynamic programming approach to unit commitment. IEEE Trans. Power Syst. 2(2):339–347 9. Chuang CS, Chang GW (2013) Lagrangian relaxation-based unit commitment considering fast response reserve constraints. Energy Power Eng 5:970–974 10. Farag MA, El-Shorbagy MA, El-Desoky IM et al (2015) Binary-real coded genetic algorithm based k-means clustering for unit commitment problem. Appl Math 6:1873–1890 11. Jeong YW, Lee WN, Kim HH et al (2009) Thermal unit commitment using binary differential evolution. J Electr Eng Technol 4(3):323–329 12. Juste KA, Kita H, Tanaka E et al (1999) An evolutionary programming solution to the unit commitment problem. IEEE Trans Power Syst 14:1452–1459 13. Yuan X, Ji B, Zhang S et al (2014) Anew approach for unit commitment problem via binary gravitational search algorithm. Appl Soft Comput 22:249–260 14. Zand A, Bigdeli M, Azizian D (2016) A modified ant colony algorithm for solving the unit commitment problem. Adv Energy: Int J (AEIJ) 3, July 2016 15. Guo YR, Wu YH, Zhang JR et al (2015) Improved PSO approach for the solution of unit commitment problem. In: International conference on artificial intelligence and industrial engineering (AIIE), pp 510–513 16. Kumar S, Singh HD (2017) Optitimization of unit commitment problem using classical soft computing technique (PSO). Int Res J Eng Technol (IRJET) 4(11), November 2017 17. Archana N (2015) Unit commitment and economic load dispatch using hybrid genetic—particle swarm optimization algorithm. Int J Appl Res 1(6):109–113

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18. Sarjiya B, Arief M, Andi S (2015) Unit commitment solution using genetic algorithm based on priority list approach. J Theor App Inform Technol (JATIT) 72(3), February 2015 19. Chuan-Ping C, Chih-Wen L, Chun-Chan L (2000) Unit commitment by lagrangian relaxation and genetic algorithms. IEEE Trans Power Syst 15(2):107–114 20. Yang XS, Deb S (2009) Cuckoo search via levy flights. In: World congress on nature and biologically inspired computer (NaBIC), pp 210–214 21. Jiang M, Song H et al (2016) A cuckoo search-support vector machine model for predicting dynamic measurement errors of sensors. IEEE Access 4:5030–5037 22. Svetlana A, Jussi S, Olli P et al (2015) Cuckoo search for wind farm optimization with auxiliary infrastructure. Wiley, pp 1–21 23. Feng Y, Zhou J, Mo L et al (2018) A gradient-based cuckoo search algorithm for a reservoirgeneration scheduling problem. MDPI 11(36):1–21 24. Normansyah A, Erick R et al (2017) Cuckoo search algorithm for environmental/economic dispatch problem. IOSR-JEEE 12:59– 63 25. Intissar K, Taoufik L, Faouzi M et al (2018) Cuckoo search approach for parameter identification of an activated sludge process. Comput Intell Neurosci 1–9, January 2018 26. Andrés I, Akemi G, Patricia S (2018) Cuckoo search algorithm with Lévy flights for globalsupport parametric surface approximation in reverse engineering. MDPI 10(8):1–25 27. Liu L, Liu X, Wang N et al (2018) Cuckoo search algorithm with variational parameters and logistic map. MDPI 11(30):1–11 28. Zhao J, Liu S, Wang Y (2017) Cuckoo search algorithm with interactive learning for economic dispatch. In: 2017 36th Chinese control conference (CCC), pp 2904–2909, July 2017 29. Chitra J, Ravichandran CS (2014) Cuckoo and Levy flights algorithm applied to unitcommitment problem. Int J Adv Res Electr Electron Instr Eng (IJAREEIE) 3(12):13670–13677 30. Chandrasekaran K, Sishaj PS (2012) Multi-objective unit commitment problem using Cuckoo search Lagrangian method. Int J Eng Sci Technol (IJEST) 4(2):89–105 31. Chitra J, Ravichandran CS (2014) Performance comparison of integer coded cuckoo and Levy flights algorithm applied to unit-commitment problem. J Theor Appl Inform Technol(JATIT) 66(3):876–883 32. Sharma S, Mehta S, Verma T (2015) Weight pattern based cuckoo search for unit commitment problem. Int J Res Adv Technol 3(5):102–110 33. Rodrigues D, Pereira LA, Almeida T et al (2013) BCS: a binary cuckoo search algorithm for feature selection. In: 2nd International Conference on Knowledge Engineering and Applications (ICKEA), London, pp 6–12. https://doi.org/10.1109/ICKEA.2017.8169893 34. Datta D (2013) Unit commitment problem with ramp rate constraint using a binary-real-coded genetic algorithm. Appl Soft Comput, Elsevier 13:3873–3883 35. Yuan X, Su A, Nie H (2009) Application of enhanced discrete differentia evolution approach to unit commitment problem. Energy Convers Manag 50:2449–2456 36. Yuan X, Su A, Nie H (2011) Unit commitment problem using enhanced particle swarm optimization algorithm. Soft Comput 15:139–148 37. Pavithra Priya R, Sivaraj N, Muruganandam M (2015) A solution to unit commitment problem with V2G using Harmony Search algorithm. Int J Adv Res Electr Electron Instr Eng 4(3):1208– 1214 38. Yu JJQ, Li VOK, Lam AYS (2013) Optimal V2G scheduling of electric vehicles and unit commitment using chemical reaction optimization. In: IEEE congress on evolutionary computation (CEC), Cancun, Mexico, pp 392–399, June 2013 39. Hossein B, Zareian M, Jahromi M, Rashidinejad M (2012) A combinatorial artificial intelligence real-time solution to the unit commitment problem incorporating V2G. Electr Eng, Springer, 341–355, December 2012

A Comparative Study of MPPT Techniques for Standalone Hybrid PV-Wind with Power Management Tarek Boutabba, Hamza Sahraoui, Mouhamed Lamine Bechka, Said Drid, and Larbi Chrifi-Alaoui

Abstract In this work a comparative study of maximum power point tracking techniques for a standalone hybrid generation power system with power management is presented, this system comprising a double fed induction generator branch (DFIG) based wind turbine and photovoltaic generator branch compared to the conventional perturbation and observation (P&O) and the incremental conductance algorithms respectively. The fuzzy logic controller is designed to vary the duty-cycle of the DC– DC converter automatically such that to maintain the load voltage constant. MPPT provides a high precision in current transition and keeps the voltage without any changes. In addition, to improve the performance of the hybrid system, a strategy of energy management has been used, as well as the control of the charging and discharging of the battery and the setting of the priority of the load and the control of the secondary charge. The proposed scheme ensures optimal use of the photovoltaic (PV) array and DFIG wind proves its efficacy in variable load conditions, unity and lagging power factor at the inverter output (load) side. A dynamic and steady-state mathematical model and simulations for the entire scheme is presented. The model is implemented in the MATLAB/Simulink platform.

T. Boutabba (B) · H. Sahraoui LSPIE Batna Laboratory, University of Abbès Laghrour Khenchela, Khenchela, Algeria e-mail: [email protected] H. Sahraoui e-mail: [email protected] Faculty of Technology, University of Hassiba Benbouali University of Chlef, UHBC, Ouled Fares, Algeria M. L. Bechka · S. Drid LSPIE Laboratory, University of Batna 2 (Mostefa Ben Boulaïd), Fesdis, Algeria e-mail: [email protected] L. Chrifi-Alaoui University of Picardie Jules Verne, LTI (EA 3899), 13 av F. Mitterrand, 02880 Cuffies, France e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_7

93

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Keywords MPPT · Hybrid energy · Stand alone system · Photovoltaic · Wind energy · Power management · Fuzzy logic control

1 Introduction The constantly increasing consumption of energy, the exhaustible nature of fossil fuel and its ever increasing cost and global environment worsening have created interest in green power generation systems. However, due to the random nature of the wind speed and solar radiation, the reliable power supply cannot be provided with only the use either a wind turbine generator (WTG) or a photovoltaic generator (PV) [2]. However, it can be improved by using a hybrid association of photovoltaic and wind systems, where the storage is the solution for standalone applications [3–10]. Moreover, In addition, numerous studies and economic analyzes have indicated that hybrid PV-wind and PV-wind-diesel systems with or without battery backup for power generation are now considered cost-effective technologies [7, 9, 11. The major limitation for these hybrid systems is the control requirement for optimal efficiency. For that, the control systems used a maximum power point tracking (MPPT) control algorithm to track the maximum available power from the high penetrating renewable sources (PV-wind). In Solar conversion system, MPPT is achieved with the DC to DC converter which operates PV module at its maximum power point by using the conventional algorithms such us P&O (Perturb and Observe) and incremental conductance algorithm. This algorithm may fail to track the point of maximum power since performance depends largely on the choice of the perturbation size value used. The value of this disturbance is usually calculated by testing and/or simulation tests [5]. In recent years, some control techniques not conventional and artificial such as the fuzzy logic algorithm and the neural networks (ANN) have been developed [5, 6]. The application of these algorithms varies with their complexity and convergence speed of the maximum power point. In order to improve energy conversion efficiency of the photovoltaic generator, this paper presents a comparative study between an classical method of maximum power point tracking (MPPT) and intelligent controller as fuzzy logic controller (FLC), under various climatic conditions. Also, the power management strategy that enhances the hybrid system performance by introducing battery charging and discharging limit control, load priority setting and secondary load control is discussed. This proposed strategy has been implemented on a PV-wind hybrid energy system model with battery backup.

A Comparative Study of MPPT Techniques for Standalone Hybrid …

95

2 Modeling of Hybrid System The hybrid scheme with wind turbine and photovoltaic system, doubly fed induction generator models proposed is presented in this section.

2.1 Wind Turbine Modeling Wind turbines convert the kinetic energy present in the wind into mechanical energy by means of producing torque. Since the energy contained by the wind is in the form of kinetic energy, its magnitude depends on the air density and the wind velocity. The wind power developed by the turbine is given by the Eq. (1) Pw =

1 C P Av3 2

(1)

where C P is the Power Co-efficient, ρ is the air density in kg/m3 , A is the area of the turbine blades in m2 and v is the wind velocity in m/s. The power coefficient C P gives the fraction of the kinetic energy that is converted into mechanical energy by the wind turbine. It is a function of the tip speed ratio λ and depends on the blade pitch angle for pitch-controlled turbines. The tip speed ratio may be defined as the ratio of turbine blade linear speed and the wind speed λ=

Rω V

(2)

Substituting (2) in (1), we have: Pw =

 3 1 R C P (λ)ρ A ω3 2 λ

(3)

The output torque of the wind turbine is calculated by the following Eq. (4). Ttur bine =

1 C P ρ AV /λ 2

(4)

where R is the radius of the wind turbine rotor (m) There is a value of the tip speed ratio at which the power coefficient is maximum [1]. Variable speed turbines can be made to capture this maximum energy in the wind by operating them at a blade speed that gives the optimum tip speed ratio. This may be done by changing the speed of the turbine in proportion to the change in wind speed.

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2.2 Photovoltaic Array Model Figure 1 shows the equivalent circuit of photovoltaic (PV) cell. A PV cell can be represented by an equivalent circuit [9] as shown in Fig. 1. The characteristics of this PV cell can be obtained using standard Eq. (5).     V +Rs I V + Rs I Vt a −1 − I = I pv − I0 e Rp

(5)

I PV = photovoltaic current I O = saturation current V t = N S k T/q, thermal voltage of array Ns = cell connected in series T = is the temperature of the p–n junction k = Boltzmann constant q = electron charge RS , RP = equivalent series/parallel resistance of the array. The general equation of a PV cell describes the relationship between current and voltage of the cell. Since the value of shunt resistance RP is high compared to value of series resistance RS the current through the parallel resistance can be neglected. The light generated current of the photovoltaic cell depends linearly on the solar irradiation and is also influenced by the temperature [10] given by the Eq. (6).  G I pv = I pv,n + K I T Gn

(6)

I PV = is the light generated current at nominal condition (25 °C and 1000 W/m2 ) Fig. 1 Equivalent circuit of PV cell

A Comparative Study of MPPT Techniques for Standalone Hybrid … Table 1 Parameter of KPC-12075 solar array at 25 °C,1000 W/m2

97

I mp

4.40 A

V oc

21.20

V mp

17.00 V

a

1.3

Pmax

74.8 W

Rsc

0.511 

I sc

5.02 A

Rsh

44.25 

Ns

36

Kv

−74.7 mv/°C

I 0, n

9.83 × 10−8

KI

2.80 mA/°C

ΔT = T −Tn

(7)

T, T n = actual and nominal temperature [K] K I = current coefficients G = irradiation on the device surface [W/m2 ] Gn = nominal irradiation. The current and voltage coefficients KV and KI are included as shown in Eq. (8) in order to take the saturation current IO which is strongly dependent on the temperature (Table 1). Isc,n + K I T

Io =



e

Voc,n +K V T aVt



(8)

−1

K V = voltage coefficients K I = current coefficients.

2.3 Doubly-Fed Induction Generator Dynamic Model The application of Concordia and Park’s transformation to the three-phase model of the DFIG permits to write the dynamic voltages and fluxes equations in an arbitrary d–q reference frame: ⎧ ⎪ Vds ⎪ ⎪ ⎨ Vqs ⎪ V dr ⎪ ⎪ ⎩V qr

= = = =

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+ + + +

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− ωψqs + ωψds − ωψqr + ωψdr

(9)

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⎧ ψds ⎪ ⎪ ⎨ ψqs ⎪ ψ ⎪ ⎩ dr ψqr

= = = =

L s Ids L s Iqs L r Idr L s Iqr

+ M Idr + M Iqr + M Ids + M Iqs

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The stator and rotor angular velocities are linked by the following relation: ωs = ω + ωr

(11)

where the electromagnetic torque c e can be written as a function of stator fluxes and rotor currents: Γe = p

 M ψqs Idr + ψds Iqr Ls

(12)

3 DC–DC Boost Converter A dual stage power electronic system comprising a boost type DC–DC converter and an inverter is used to feed the power generated by the PV array to the load. To maintain the load voltage constant a DC–DC step up converter is introduced between the PV array and the inverter. The block schematic of the proposed scheme is shown in Fig. 2. In this scheme a PV array feeds DC–DC converter used in step-up configuration. For a DC–DC boost converter, by using the averaging concept, the input–output voltage relationship for continuous conduction mode is given by 1 Vo = Vin (1 − D)

Fig. 2 A MPPT controller in a PV system

(13)

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where, D = duty cycle.

4 The Control of Hybrid System In this section the strategy of controls of the photovoltaic system and the DFIG are discussed.

4.1 Control Strategy of DFIG By choosing a reference frame linked to the stator flux, rotor currents will be related directly to the stator active and reactive power. An adapted control of these currents will thus permit to control the power exchanged between the stator and the grid. If the stator flux is linked to the d-axis of the frame we have ψds = ψs and ψqs = 0

(14)

and the electromagnetic torque can then be expressed as follows. The active power and consequently the torque depend on the rotor current component of the q-axis. If the statorique resistance per phase is neglected, which is a realistic approximation used in wind energy conversion, the statorique vector voltage is in quadrature. The stator voltages are Vds = 0 and Vqs = Vs = ωs ψs

(15)

Using Eqs. (10), (14) and (15) the stator active and reactive power can then be expressed only versus these rotor currents as 

P = Vs Iqs = −Vs LMs Iqr Q = Vs Ids = VLs ψs s − VLs M Iqr s

(16)

In steady state, the second derivative terms of the two equations in (16) are nil. The third term, which constitutes cross-coupling terms, can be neglected because of their small influence. Knowing relations (9) and (16), it is possible to synthesize the regulators and establish the global block-diagram of the controlled system (Fig. 3).

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Fig. 3 Block-diagram of the controlled system

4.2 Control Strategy of Photovoltaic System Several MPPT control algorithms in which the classical algorithms are the technique of disturbance and observation (P & O) and the technique of incremental conductance where they have been developed in many research works. This work involves applying the Fuzzy Logic Controller (FLC) as an intelligent MPPT method for PV system control.

4.2.1

MPPT Using Fuzzy Logic Controller

Fuzzy logic has been used for tracking the MPP of PV systems [7] because it has the advantages of being robust, relatively simple to design and do not require the knowledge of an exact model. The proposed FLC for MPPT in PV systems is based on the model developed by [11] but with a few modifications made in the fuzzification and rule base. The main components in a FLC are fuzzification, rule-base, and inference and defuzzification as shown in Fig. 4. The input variables to the FLC are the change in PV array voltage (ΔV pv ) and change in current (ΔI pv ) whereas the output of FLC is the duty cycle (ΔD). Using seven fuzzy subsets in terms of their linguistics variables for the inputs (ΔV pv ) and (ΔI pv ) to define Their universe of discourse. Figures 5, 6, and 7 shown inputs and output variable membership functions. The fuzzy system rule base is created as shown in Table 2 with (ΔV pv ) and (ΔI pv ) as inputs while D is the output. The fuzzy inference of the FLC is based

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Fig. 4 Components of a fuzzy logic controller

VSS VS S M B VB VVB

Fig. 5 Membership functions of the 1st input variable 1st (V pv )

VSS

VS

S

M

B

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Fig. 6 Membership functions of the 2nd input variable 2nd (I pv )

on the Mamdani’s method which is associated with the max–min composition. The defuzzification technique is based on the centroid method which is used to compute the crisp output, D.

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VS S M B VB

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Fig. 7 Membership functions of the output variable (D)

Table 2 Fuzzy rule V P V

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5 Hybrid Power Management Strategy To operate the stand alone PV-wind hybrid energy system a customized control strategy is implemented ensuring higher reliability of this system under different atmospheric conditions. The control strategy proposed is developed based on the state of charge (SOC) of the batteries which includes AC and DC load control, secondary AC and DC load control during higher generation and most importantly the battery charge and discharge limit control. The proposed control, subdivided into five different stages, considers all possible operating conditions to run a hybrid system efficiently and reliably under different weather conditions. The block diagram representation of the suggested control strategy is shown in Fig. 8. The major five control stages are: 1. Initialization 2. Normal operating condition 3. Lower limit of battery SOC control (High load or low generation)

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Fig. 8 Block diagram representation of the diagram of the management strategy

4. Upper limit of battery SOC control (Low load or high generation) 5. Wind energy extreme case control.

6 Simulation Results The dynamic simulation model for the system is described. The system consists of various units, PV power and wind power units as primary sources of energy, battery bank unit as auxiliary source of energy, DC–DC and DC–AC converters, load unit and control unit. The function of controller unit is to ensure the management of the power, which is delivered by the hybrid system to satisfy the load and to charge the battery. The inverter unit is used to convert the DC generated power from renewable energy sources to feed the load with the required AC power. The excessive charge from the battery will be dumped to the dump load unit. The dump load in this case is the battery storage Figs. 8 and 9. In Fig. 10 we can see the two characteristics (I-V) and (P-V) with the application of three different techniques to maximize the power of panels through a boost converter. Figures 11 and 12 shows respectively that the output voltage of the photovoltaic panel is well controlled to follow the optimal value according to the variation of the sunlight and the load, and while the panel always produce a maximum power under all conditions. Our controller has optimized the production of the photovoltaic system. The report named “Gate” is controlled to ensure proper operation of the converter “bidirectional chopper” voltage applied to the battery (Fig. 13). Until the moment of 1.5 s, the climatic conditions deteriorate “a kiss of the sunshine with a weak wind”, but the requested power remains in surplus the battery

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ONDULEUR

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ΔP Convertisseur Buck/BOOST bidirectionnel

Fig. 9 Block diagram representation of PV-wind hybrid power system with supervisor

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Fig. 10 (I-V) and (P-V) MPPT characteristics

is discharged (S1 = 1) because the load request exceeds the renewable power, and when this power is sufficient, the battery will be deactivated (S2 = 1) in order to use all the renewable power produced (Fig. 14). In the case where the load power exceeds the power supplied, the battery must compensate for this difference. In the 1 s to the 1.5 s, the wind speed increases, at the same time the load demand decreases, the surplus becomes too big (Ppv + Peel-Pdem = 3.2 kW), then to avoid the gaseous release of the battery, the load shed resistance is switched on (Gate = 1) to dissipate the excess energy Fig. 12.

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DC bus Voltage Vdc(V)

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Fig. 12 Power of the PV panel

The voltage quality is always respected (f = 50 Hz, V = 380 V) in all climatic and operating conditions. These simulations show that our controller has good results. It has the load demand, despite the variations in weather conditions, with good power while respecting the charging process of the battery. Other simulations with other more or less important variations made it possible to verify these good results (Fig. 13).

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Fig. 14 Renewable power, battery power and load power

7 Conclusion This method is capable of extracting maximum power from each of the independent PV arrays connected to DC link voltage The hybrid power management strategy is developed to control the power flow of the system and maintain battery charging and discharging limits under any operating conditions. Fuzzy Logic Maximum Power Point Tracking (MPPT) algorithms, that provide maximal voltage extracted from the PV system and fed the inverter, have been included.

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We have introduced a power generation system for sites that cannot be connected to the grid. Indeed, we used a continuous bus which receives the energy produced by the photovoltaic and wind power sources and then delivered to the consumer using an inverter. One of the advantages of this structure is the use of the necessary batteries in case of need to compensate for a possible lack of power. We have also used a load shedding resistor which allows to dissipate the surplus energy in the event of a drop in demand and full batteries. To have optimal behavior of the installation from a power flow point of view, we have developed a supervisor or. It enables efficient and rational energy management to meet the energy needs of the consumer. Several simulation results have been presented to illustrate the performance of our facility in the presence of climate change and changes in energy consumption. Based on the above analysis, it can be concluded that the proposed hybrid scheme offers a reliable and alternative solution of renewable energy to meet the increasing power demand, overcoming any intermitting challenges and allowing widespread development of such systems especially in locations where no grid is available.

For Reviewer • Lack of explanation on the control of the DFIG and the rectifier, and Why the boost chopper is used to control the power produced by the DFIG. Static converters are indispensable in the structure of such a system. They make it possible not only to operate at variable speed but also to extract the maximum of the power produced. A structure often used is that which uses a diode rectifier bridge associated with a controllable converter (of the Boost type, buck or buck_boost). • Lack of explanation on the operation and control of the DC/AC converter supplying the load. For simplification reason, the operation and control of the DC–AC converter feeding the three-phase load will be applied to prove that our controller is working well.

References 1. Lei Y, Mullane A, Lightbody G, Yacamini R (2006) Modeling of the wind turbine with a doubly fed induction generator for grid integration studies. IEEE Energy Conv 21:257–264 2. Messaïf I, Berkouk E-M, Saadia N (2010) A study of DTC-power electronic cascade fed by photovoltaic cell-three-level NPC inverter. Smart Grid Renew Energy 1:109–118 3. Vijayalakshmi R, Nazar Ali A (2012) Hybrid power generations (wind/solar by PV)—an efficient output with reduced total harmonics distortions using multi level inverter. Int J Commun Eng 4(4), 4 March 2012

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4. Balasubramanian G, Singaravelu S (2012) Fuzzy logic based controller for a stand-alone hybrid generation system using wind and photovoltaic energy. Int J Adv Eng Technol, May 2012. ISSN 2231-1963 5. Jain SK, Agrawal P, Gupta HO (2002) Fuzzy logic controlled shunt active power filter for power quality improvement. IEE Proc Electr Power Appl 149(5):317–328 6. Esram T, Chapman PL. Comparison of photovoltaic array maximum power point tracking techniques. IEEE Trans Energy Conver 22(2):439–449 July 2007 7. Boutabba T, Drid S, Chrifi-Alaoui L, Mehdi D, Benbouzid MH, Alibi A, Ouriagli M (2018) dSPACE—Real-time implementation of sliding mode maximum power point tracker for photovoltaic system. In: 7th international conference on systems and control, ICSC 2018, València, Spain, 24–26 October 2018 8. Tarek B, Drid S, Benbouzid MEH (2013) Maximum power point tracking control for photovoltaic system using adaptive neuro-fuzzy “ANFIS”. In: 2013 ecological vehicles and renewable energies (EVER), 8th international conference and exhibition on ecological vehicles and renewable energies (EVER). IEEE, Monaco 9. Tarek B, Drid S, Benbouzid MEH (2014) A multi-output boost converter (MOB) controlled by fuzzy logic technique supplied by a photovoltaic system with grid-connected and fed by three level inverter. In: International conference on electro-energy, ICEE’14, Skikda, Algeria, 10–11 November 2014 10. Boutabba T, Drid S, Benbouzid MH (2014) A hybrid power generations system (wind turbine/photovoltaic) to driving a DFIG fed by a three inverter. In: STA’2014, the 15th international conference on sciences and techniques of automatic control & computer engineering, Hammamat, Tunisia 11. Boutabba T, Drid S, Chrifi-Alaoui L, Ouriagli M, Benbouzid MEH (2016) dSPACEs—Realtime implementation of maximum power point tracking based on ripple correlation control (RCC) structure for photovoltaic system. In: 2016 IEEE ICSC, Marrakesh, Morocco, pp 371– 376

Adapted Search Equations of Artificial Bee Colony Applied to Feature Selection Hayet Djellali, Souad Guessoum, and Nacira Ghoualmi-Zine

Abstract This article proposes a Novel Search Equation of Artificial Bee Colony called NSABC. This study introduces new search equations for employed and onlooker bees involving more parameters for each food source (more features) and avoid local optima. NSABC outperforms various ABC variants and particle swarm optimization PSO in term of reduced size of feature and accuracy. Experimental results validate the efficiency of NSABC method on machine learning UCI data. Keywords ABC · Feature selection · Search equation · Exploitation · Exploration

1 Introduction Feature selection (FS) aim to select the best collection of features that ensures the highest accuracy and minimal number of features. Four categories of FS approaches are described in literature: filters, wrappers, embedded and hybrid approaches [1–3]. To solve feature selection task, optimization algorithms such as Artificial Bee Colony [4] (ABC), Genetic algorithm [5] (GA); Ant Colony Optimization (ACO) [6]; Particle swarm optimization [7] (PSO), and differential evolution [8] (DE) are efficient methods. However, they often stuck in local optima. ABC is considered as wrapper method because of using machine learning algorithms (classifier) to assess the proposed solutions quality defined as fitness. Employed, Onlooker and Scoot bees are different types of Artificial Bee colony ABC ensuring different functions. Employed bees convey via waggle dance the position and quality of food sources to onlooker. Onlooker bees select food source to be exploited using received information. If the source is exhausted, the employed become scout and start to explore a new food source. ABC is simple to implement with few parameters, ABC benefits from high exploration compared to GA, and ACO [6]; PSO [7] However, it converges slowly because of the poor local exploitation. H. Djellali (B) · S. Guessoum · N. Ghoualmi-Zine Department of Computer Science, LRS Laboratory, Badji Mokhtar University, Annaba, Algeria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_8

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The search equation of ABC focuses on exploration, but it neglects exploitation. ABC does not profit from the prior information of the best solution and the promising evolution direction. To enhance ABC, new solutions search equations were proposed in ABC variants to realize a tradeoff between exploration and exploitation [9, 10]. Various previous search equations exploited the information of the Current best solution. For instance, Kiran and Findik [11] design five search strategies and different solutions update rules in ABC. Influenced by PSO and searching to improve the exploitation ability of ABC, Zhu and Kwong [10] developed a new search equation called Gbest-guided ABC (GABC) which benefit from prior information of the actual good solution. The comparative study proved that GABC is better than standard ABC. The best so far ABC [12] algorithm conveyed the best solution to the whole colony and improved the solutions of onlooker bees. Evolutionary based similarity search process is introduced in binary ABC for feature selection investigated by Hancer et al. [1]. It is compared to PSO, GA, discrete ABC and the experiments carried out concluded that the accuracy is better on 10 datasets. A modified onlooker bee is described by karaboga [13] where a new search equation is developed and benefit from the information of the best solution. The algorithm is called quick ABC (qABC). Gao and Liu [14] developed a novel search equation with crossover operation of GA (called CABC) improving the local search of ABC. The experiments validated that CABC is effective. ABC algorithm generates a new food source affecting only one parameter (feature) of the corresponding previous solution. This modification makes the new solution disturbed weakly and founded in the neighborhood of the previous solution. We propose a novel ABC method called Novel search Algorithm ABC (NSABC) to enhance the performance of ABC and realize a tradeoff between the exploration and the exploitation. Designed search equations create better candidate solutions involving more features for each food source of employed and onlooker bees. Our contributions focus on: – Reducing the features space to the efficient features realizing high accuracy; – Adapting the search equations to ameliorate the exploitation process; – Enhancing the ABC convergence and comparing the proposed NSABC with PSO and different variants of ABC. The manuscript is organized as follows. Section 2 describes original ABC and related work. Section 3 explains the proposed FS methods: Novel search ABC. Experimental results and discussions are carried out in Sects. 4 and 5. finally conclusion in Sect. 6.

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2 Methods and Related Works 2.1 Artificial Bee Colony In Artificial bee colony (ABC) [4], foraging behaviors of honey bee colony is simulated. The candidate solutions represent food sources and their fitness is the nectar amounts.

Algorithm 1. ABC algorithm[4]. 1: Initialize the candidate solutions of the population Xi, i =1, ..., NBsource. 2: Evaluate the population Xi using fitness function, i, i = 1, .., NBsource. For it = 1 to Max iterations do Employed Bees(); Onlooker_bees(); Scout _Bees(): Memorize the best solution realized. iter =iter + 1 End for.

Employed Bees() { For each Employed Bees i Compute and evaluate new solutions Vi using Eq. (1). V(i,j) = X(i,j) + phi(i,j) * (X(i,j) – X( p, j))

(1)

Where p=1…NBsource and p≠i; parameter j ϵ [1..NBfeatures]; 5: Apply the greedy selection. Update trial counter. Endfor }

Onlooker Bees(){ For each Onlooker Bee i Compute the probability Pi for the solution Xi by Eq. (2). P(i) = fitness (i) / ∑ fitness(t) , t=1..N.

(2)

Calculate and evaluate new solutions Vi for the solution Xi depending on Pi. Apply the greedy selection. Update trial counter}

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Scout Bees () { The abandoned solutions are replaced with a new Xi described by Eq. (3). X(I,j) =X(min, j) + rand(0,1) * ( X(max,j) – X(min, j)) Reset trial to zero. }

(3)

2.2 Related Work Gao et al. [15] developed a new algorithm combining DE with Gbest guided ABC (GABC) proposing a new strategy. The information of the previous search exploration is applied to enhance the exploitation. The experimental results validated that DGABC outperforms the other algorithms. Authors [16] developed a hybrid artificial bee colony algorithm. During the search process, each bee can reproduce or die leading to dynamic size of bee population. The search equation is redesigned with insertion of global best. Experiments consolidate the effectiveness of this algorithm tested on the CEC 2014 benchmarks compared to other mimetic algorithms. This study [17] introduces swap and crossover operators in a new binary version of ABC algorithm. The improvement of local and global search of ABC is realized. The obtained results confirm that GBABC is suitable algorithm in binary optimization. Gao et al. [18] developed a new ABC algorithm named Enhanced EABC. The search equations of employed and onlooker bees are modified. Experiments validate that EABC significantly produces faster global convergence and better performance compared to traditional FS methods. Mao et al. [19] implemented a modified artificial bee colony algorithm (ABCEM) with extended memory in employed and onlooker bees. From Experiments, ABCEM algorithm converges faster and realizes a tradeoff between exploration and exploitation compared with other ABC versions. Authors [20] developed an approach named depth first search (DFS). The elite information is added to two novel search equations in employed bee phase. The experiments conducted consolidate that DFS enhance both the convergence and accuracy of ABC. Authors [21] propose two hybrid FS approaches based on particles swarm and artificial bee colony ABC-PSO and ABC with Genetic algorithm ABC-GA. Mutation operators and PSO are implemented in ABC phases. This architecture attempts to improve the search ability of Bees. To increase exploitation and found better solutions, mutation is introduced in both Onlookers and Scout Bees and PSO particles are inserted in Employed Bees. It has been found that the proposed method ABCGA outperforms the proposed ABC-PSO method and other approaches on WDBC, hepatitis, and DLBCL. Akay and Karaboga [22] enhanced the search ability of onlookers by incorporating the parameters: modification rate (MR) and scaling factor (SF) to adjust the frequency and the perturbation.

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3 Proposed Novel Search Equations of ABC Applied to Feature Selection To overcome the insufficiency of poor solution search equation at exploitation; we propose two new search equations. The solution search equation of employed and onlooker bees of (ABC) algorithm are adapted to change more features. In this paper, we propose a novel ABC method called Novel search ABC (NSABC) to improve the performance of ABC in Fig. 1. New search equations are designed for the employed and onlooker bees involving more features for each source. In fact, ABC generates the new food source modifying one parameter of the current solution. This change conducts to disturb weakly the new solution and it will be located in the neighborhood of the associated solution. We increase the number of parameters chosen randomly set to parameter called parlimit to achieve better exploitation. Firstly, the solution search equation generated concerns more than one parameter j  (1…,NBfeatures) to increase the search space in the neighborhood for the employed bees.

- Pseudo code A For c=1: parlimit Generate j randomly between [1… NBFeatures]; V(i,j) = X(I, j) + φ(i,j) (X(r1,j) – X(r2,j) )

(4)

End Fig. 1 Novel search equations NSABC algorithm

Begin

Initialize_ABC() ; cycle=1 ; Computeparlimit() ;

Cycle =Max Iter

Enhanced Employed Bee Enhanced Onlooker Bee Scout Bees: Save best solution ; cycle=cycle+1;

Best More feature Best solution

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Where parlimit is the maximum number of features allowed to change. r1 and r2 are two different neighbors for the same parameters j. Example: Source food vector V(3,:) is modified on 3 values of j. index 1; 3 and 6 after Pseudocode A executed. J = 1..NBfeatures. where NBfeatures = 7; V(3,j); j = 1..7 before Pseudocode A 1

1

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Parlimit is a maximum number of parameters (features) to change depending on the feature size of dataset. We choose various dataset with different size of features which means that the value of parlimit must defined according to the size of feature datasets. Parlimit parameter must be subject to deep investigation according to the size of features in each data. We adopt the strategy to vary the index c of parlimit (pseudocode A) between 1 to parlimit randomly. High value of parlimit means that exploration is prioritized than exploitation. Nbfeatures is the size of feature

Pseudo code Computeparlimit(Nbfeatures){ If (Nbfeatures 0). It should be noted that u j (k + l − 1|k) is excluded in the performance index, since it is independent of the future control sequence of S j .

4 Cooperation Based Distributed MPC Problem In order to find an explicit solution to the first version of DMPC problem, each subcontroller Ci is decomposed into three connected function blocks: an optimizer, a state predictor and an interaction predictor. Assumptions 1 • the prediction and control horizons are the same for all sub-controllers, i.e., m i = m j =, pi = p j = p, ∀i, j = 1, 2, . . . , N , j = i; • all the sub-controllers are synchronous; • all the sub-controllers communicate only once within a sampling interval; • all the communication channel introduces a delay of one sampling period. To simplify the mathematical expressions, the following notations are adopted • 0a×b is the a × b null matrix; • Ia (0a ) is the a × a identity (null) matrix;

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• diaga {A} is a diagonal block matrix made by a blocks equal to A. A. Interaction Prediction Under Assumptions 1, at step k, the predictions of the interaction vectors are given by Wˆ i (k, p|k − 1) = A˜ i Vˆ i (k, p|k − 1) = C˜ i

Xˆ (k, p|k − 1) + B˜ i Γ˜ i U (k − 1, m|k − 1) Xˆ (k, p|k − 1)

(7)

H˜ i  [diag p {Hi,1 } . . . diag p {Hi,i−1 } 0 diag p {Hi,i+1 } . . . diag p {Hi,N }] where H˜ i ∈ { A˜ i , B˜ i , C˜ i }. B. State Predictor Under Assumptions 1 at step k, the local state prediction for the sub-controller Ci is expressed by xˆ i (k + l|k) = Alii xˆ i (k|k) +

l 

s−1 Aii Bii u i (k + l − s|k) +

s=1

l 

s−1 Aii wˆ i (k + l − s|k − 1)

s=1

(8) C. Optimal Control Sequence Under Assumptions 1, at step k, based on the exchanged information, the interaction prediction together with the local measurement is used by the optimizer to solve the MPC optimization problem (without constraints). Once computed the optimal control sequence [u i (k|k), . . . , u i (k + m − 1|k)], which minimize the local cost function (6), only the first element of the optimal sequence u i (k|k) is selected and the control action u i (k) = u i (k − 1) + u i (k|k) is computed and applied as control action to the subsystem Si , following the receding horizon strategy. Generally, the parameters of the above optimization problem are p and m; these parameters are directly affecting computational load in MPC. One of the MPC formulation is the classic approach presented in [5]. In this approach, for the case of rapid sampling, complicated process dynamics and/or high demands on closed-loop performance, satisfactory approximation of the control signal u(k) may require a very large number of parameters (large m), leading to poorly numerically conditioned solutions and heavy computational load when implemented on-line. Instead, a more appropriate technique would be to use Laguerre network in the design of MPC presented in [9, 16].

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5 Description of the Control Signal Trajectory The Z-transforms of the discrete-time Laguerre networks are written as follows n (z) = n−1 (z) with 1 (z) =

z −1 − a n = 2, 3, . . . , M 1 − az −1

(9)

√ 1−a 2 . 1−az −1

where M is the number of Laguerre functions in the network and a is the pole of the Laguerre network. The scaling factor a is required to be selected by the user, where 0 ≤ a < 1 for the stability of the network. Note that the Laguerre networks are well known for their orthonormality. With the relation (9), the Laguerre network is illustrated in Fig. 1. The discrete-time Laguerre functions are obtained through the inverse Z-transform of the Laguerre network. The set of Laguerre functions can be expressed as L(k) = [l1 (k)l2 (k) . . . l M (k)]T , where li (k) denotes the inverse Z-transform of i (z, a). Taking advantage of the network realization (1), the set of discrete-time Laguerre functions satisfies the following difference equation L(k + 1) = Al L(k)

(10)

where Al ∈  M×M and is a function of the parameters a and β = (1 − a 2 ), and the initial condition is given by L(0)T =



β[1 − a a 2 − a 3 . . . (−1) M−1 a M−1 ]

The orthonormality can be expressed by ∞ 

li (k)l j (k) = 0 for i = j

(11a)

li (k)l j (k) = 1 for i = j

(11b)

k=0 ∞  k=0

The orthonormal property of the Laguerre functions will be used in the design of MPC. The key idea in MPC based on Laguerre functions lies in the approximating member of control sequence by a set of Laguerre functions as

Fig. 1 Illustration of a discrete Laguerre network

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⎧ M  ⎪ ⎪ ln (l)cn (k) = L(l)T η(k) ⎨ u(k + l|k) = n=1

⎪ for l ⎪ ⎩ where

= 0, 1, . . . , m − 1 u(k|k) = u(k|k) − u(k − 1|k)

(12)

with k being the initial time of the moving horizon window and l being the future sampling. The parameter vector η comprises M Laguerre coefficients: η = [c1 c2 . . . c M ]T and L(l)T is the transposed Laguerre function vector as defined in the difference Eq. (10). By using this approximation, the optimization problem (6) can be expressed in terms of coefficient vector η, instead of u(k) as in the classic approach. Thus, the coefficient vector η will be optimized and computed in the design. With this design framework, the control horizon m from the classical MPC approach has vanished. Instead, the number of terms M(M < m) is used to describe the complexity of the trajectory in conjunction with the free parameter a. Furthermore, a long control horizon m can be achieved without using a large number of parameters, leading to low computational burden and memory storage. In this paper, the MPC based on Laguerre functions is used in the proposed DMPC scheme.

6 Use of Laguerre Functions in DMPC Design 6.1 Modified-State Predictor (1) For SISO subsystem Si , i = 1, 2, . . . , N Under Assumptions 1, at step k, the future state prediction over the horizon p are given by xˆ i (k + l|k) =

Alii

xˆ i (k|k) +

l 

Aiis−1 Bii u i (k + l − s|k)

s=1

+

l 

Aiis−1 wˆ i (k + l − s|k − 1)

(13)

s=1

 let us denote u i (k) = u i (k−1)+u i (k) and u i (k+l−s) = u i (k−1) + l−s r =0 u i (k+ r ) for l = 1, 2, . . . , p. Then, the prediction of the future state variables at time l becomes xˆ i (k + l|k) = Alii xˆ i (k|k) +

l  s=1

Aiis−1 Bii u i (k − 1)

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+

 l l   s=1

131

L i (s − 1)T ηi +

Al−h ii Bii

l 

Aiis−1 wˆ i (k + l − s|k − 1)

s=1

h=s

f or l = 1, 2, . . . , p

(14)

where the function u i (k + s − 1) is replaced by L i (s − 1)T ηi for s = 1, 2, . . . , l. The prediction of the plant output will be yˆ i (k + l|k) =

Cii Alii

xˆ i (k|k) + Cii

l 

Aiis−1 Bii u i (k

− 1) + Cii

s=1

L i (s − 1)T ηi + Cii

l 

 l l   s=1

Al−h ii Bii

h=s

Aiis−1 wˆ i (k + l − s|k − 1) + vˆ i (k + l|k − 1)

s=1

(15) With this formulation, both predictions of state and output variables are expressed in terms of the coefficient vector ηi of the Laguerre network, instead of u i as in the classical approach. Thus, the coefficient vector ηi will be optimized and computed in the control design. To compute the prediction, the convolution sum Sci (l) =

 l l   s=1

T Al−h ii Bii L i (s − 1)

(16)

h=s

needs to be computed. To this end, note that Sci (1) Sci (2)

= Bii L i (0)T = [Aii Bii + Bii ]L i (0)T + Bii L i (1)T = [Aii + I ]Bii L i (0)T + Bii L i (0)T AliT = Sci (1) + Aii Sci (1) + Sci (1)AliT Sci (3) = Sci (2) + Aii2 Sci (1) + Aii Sci (1)AliT + Sci (1)(Ali2 )T Sci (4) = Sci (3) + Aii3 Sci (1) + Aii2 Sci (1)AliT + Aii Sci (1)(Ali2 )T +Sci (1)(Ali3 )T

(17)

Continuing the recursion in (10) reveals that for l = 2, 3, . . . , p ⎧ ⎨ ⎩

Sci (l) = Sci (l − 1) +

l  h=1

with

Aiih−1 Sci (1)(Ali(l−h) )T

Sci (1) = Bii L i (0)

(18)

T

The difference equation L i (k + 1) = Ali L i (k) is used for generating the set of Laguerre functions, where Ali ∈  Mi ×Mi and is a function of the parameters ai and βi = (1 − ai2 ).

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(2) For MIMO subsystem Si , i = 1, 2, . . . , N To extend the description to MIMO subsystem, with full flexibility in the choice of ai and Mi parameters, let u i (k) be T

n u i (k) = u i1 (k) u i2 (k) · · · u i ui (k) and the input matrix be partitioned to

n  Bii = Bii1 Bii2 · · · Biiui g

where n u i is the number of inputs and Bii for g = 1, 2, . . . , n u i is the g th column g of the Bii matrix. We express the g th control signal u i (k) by choosing a scaling g g g g factor ai and order Mi , where ai and Mi are selected for this particular input, such that g

g

g

u i (k) = L i (k)T ηi g

for g = 1, 2, . . . , n u i

g

where ηi and L i (k) are the Laguerre network description of the g th control, namely g

g

g

g

L i (k)T = [li(1) (k) li(2) (k) . . . li(M g ) (k)] i

Based on the partition of the input matrix and given the state variable information at xˆ i (k), the prediction of the future state at time l can be written as xˆ i (k + l|k) = Alii xˆ i (k|k) + i (l)u i (k − 1) + i (l)T ηi + i (l)

(19)

with

i (l) = i (l) =

l  s=1 l  s=1

Aiis−1 Bii , i (l)T = Aiis−1

 l l   s=1 h=s

 T Al−h B ii L i (s − 1) ii

wˆ i (k + l − s|k − 1)

where the parameter vector ηi and the data matrix i (l)T consist of the following individual coefficient vectors nu

ηiT = [ηi1 ηi2 . . . ηi i ]T i (l) = T

 l l   s=1

1 1 T Al−h ii [Bii L i (s)

Bii2 L i2 (s)T

...

nu Bii i

h=s

for i = 1, 2, . . . , N and g = 1, 2, . . . , n u i .

nu L i i (s)T ]

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Note that the kth block matrix ik (l)T =

l  l 

k k T Al−h ii Bii L i (s − 1)

s=1 h=s

has an identical structure as the single-input case defined by Sci (l), thus it can be computed recursively using (18). From here on, the convolution sum in each multi-input subsystem Si is decomposed into computing the subsubsystems, and the computed results are put together block by block to form the multi-input structure. It is worth stressing that in the formulation of the multivariable problem, the scaling g g factors ai and the number of terms Mi can be chosen independently for each input signal of subsystem Si .

6.2 Modified-Optimizer Under Assumptions 1, since the discrete Laguerre functions are orthonormal for a sufficiently large prediction horizon p, the cost function (6) is equivalent to the modified local cost function defined as p    d  y (k + l|k) − yˆ i (k + l|k)2 + η T R L ηi Ji = i i i Qi

(20)

l=1

where R L i ∈  Mi ×Mi is a diagonal matrix with the weighting matrix Ri on its diagonal. Therefore, the objective is to find the coefficient vector ηi that minimizes Ji . Without constraints, by substituting (15) into (20), the optimal solution of the parameter vector ηi is ηˆ i =

opt ηi

=

 p  l=1

−1 i (l)Q i i (l) + R L i T

×(

p 

i (l)Q i (yid (k + l|k)

l=1

− i (l) xˆ i (k|k) − i (l)u i (k − 1) − i (l) − vˆ i (k + l|k − 1)))

(21)

with

i (l) = Cii Alii , i (l) = Cii i (l) i (l)T = Cii i (l)T , i (l) = Cii i (l) Upon obtaining the optimal parameter vector ηi , the control increment at time k is as follows

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⎡

L i1 (0)T 02T ⎢ ⎢ 01T L i2 (0)T u i (k) = ⎢ .. ⎢ .. ⎣ . . 02T 01T

. . . 0nTu i . . . 0nTu i .. .. . . nu . . . L i i (0)T

⎤ ⎥ ⎥ ⎥ηi ⎥ ⎦

(22)

where 0rT , for r = 1, 2, . . . , n u i , represents a zero block row vector with identical dimension of L ri (0)T . Consequently, the control signal u i (k) can be calculated as u i (k) = u i (k − 1) + u i (k)

(23)

6.3 Modified Interaction Prediction Under Assumptions 1, at step k, the predictions of the new interaction vectors are given by Wˆ i (k, |k − 1) = A˜ i Xˆ (k|k − 1) + B˜ i U˜ (k − 1|k − 1) Vˆ i (k, |k − 1) = C˜ i Xˆ (k|k − 1)

(24)

F˜ i  [Fi,1 . . . Fi,i−1 0 Fi,i+1 . . . Fi,N ] where F˜ i ∈ { A˜ i , B˜ i , C˜ i }.

7 NC-MPC Algorithm For the ith sub-controller Ci , where the desired output yid (k + l|k) is provided by a proper reference generator, the algorithm for the novel distributed MPC is outlined in detail: 1. Set k = 1. 2. Acquire by network the predicted future state trajectories Xˆ j (k|k −1) and control inputs U j (k − 1|k − 1) from sub-controllers C j . 3. Build Xˆ (k|k − 1) and U (k − 1|k − 1) by combining the local state trajectory xˆ i (k|k − 1) and control input u i (k − 1|k − 1) with the acquired information, and compute the corresponding predictions of the interactions according to (24). 4. Acquire the measures xˆ i (k) and the desired trajectory yid (k +l|k) over the horizon p. 5. Compute the optimal control u i (k) and broadcast it by network to sub-controllers C j , cf. (21), (22) and (23).

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6. Apply the control input u i (k) to subsystem Si . 7. Compute the future state trajectory of subsystem Si over the horizon p and broadcast it by network to sub-controllers C j , cf. (19). Increment the sample time index k ← k + 1 and iterates by going to step 2.

8 Simulation and Results In this section, the performance of the proposed NC-DMPC is investigated and compared to the centralized MPC. Consider the following unstable-non minimum phase plant S introduced in [12] discretized with a sampling time Ts = 0.2 s. A state-space realization of S has the form (1), with the following matrices  A=

     A11 0 B11 0 C11 C12 ,B = ,C = C21 C22 0 A22 0 B22

The examination of the process model S leads to decomposition into two interconnected subsystems S1 and S2 . The corresponding state-space realizations of S1 and S2 have the form (2), with matrices {A11 , B11 , C11 } and {A22 , B22 , C22 }, respectively. The constant parameter α is used to study the effect of the interactions between S1 and S2 . ⎡

⎡ ⎤ ⎤ 2.859 −1.335 0.409 0 0.063 ⎢ 2 ⎢ ⎥ 0 0 0 ⎥ ⎥, B11 = ⎢ 0 ⎥ A11 = ⎢ ⎣ 0 ⎣ ⎦ 1 0 0 0 ⎦ 0 0 0 0.819 0.125 ⎡ ⎡ ⎤ ⎤ 0.819 0 0 0.125 A22 = ⎣ 0 1.637 −0.67 ⎦, B22 = ⎣ 0.250 ⎦ 0 1 0 0     C11 = −0.0153 −0.029 −0.007 0 , C12 = α 0.145 0 0     C22 = 0 −0.193 0.292 , C21 = α 0 0 0 0.145 In the proposed test, the stability performances depend on the choice of the tuning parameters α and p. The control performance of the resulting closed-loop system is plotted in Figs. 2, and 3, where the black lines correspond to the desired outputs, the red solid lines correspond to the system outputs using Centralized MPC (CMPC) based on Laguerre functions, and the blue solid lines represent the system outputs using the proposed NC-DMPC. In Fig. 2, the performance of ND-DMPC is comparable to the one of C-MPC. However, for strong interactions with α = 3.5,

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Fig. 2 Control performance with α = 1, p= 10

Fig. 3 Control performance with α = 3.5, p = 10

Fig. 3 shows that the ND-DMPC strategy can achieve a better global performance of the closed-loop system than C-MPC. To sum up, for the given example, the NC-DMPC can achieve a satisfactory global performance even whether the interactions among subsystems are strong or not. Furthermore, the cost of computation is very small as compared with the classical centralized controller.

9 Conclusion In the present study, a formulation of a novel distributed model predictive control for a class of large-scale systems is proposed, in which the whole system is divided into many small scale subsystems interacting with each other by both their states

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and inputs. The NC-DMPC solution, was proposed for improving the global performance of closed-loop system; it is based on Laguerre functions used in MPC formulation. The proposed methodology is demonstrated on an example for a set-point tracking. Compared to the C-MPC scheme, the proposed NC-DMPC allowed to achieve an improved performance of the whole system through using a local index in optimization.

References 1. Christofides PD, Scattolini R, Muñoz de la Peña D, Liu J (2013) Distributed Model Predictive control: A Tutorial Review and Future Research Directions. Comput Chem Eng 51:21–41 2. Ikeda M, Šiljak DD (1986) Overlapping decentralized control with input state and output inclusion. Control Theory Adv Technol 2:155–172 3. Jia D, Krogh BH (2001) Distributed model predictive control. In: Proceeding of the American control conference, arlington, pp 2767–2771 4. Lunze J (1992) Feedback control of large-scale systems. Prentice-Hall, London, UK 5. Maciejowski JM (2002) Predictive control with constraints. Prentice-Hall, London, U.K 6. Menighed K, Aubrun C, Yamé JJ (2009) Distributed state estimation and model predictive control: application to fault tolerant control.In: th IEEE international conference on control and automation, pp 936–94 7. Negenborn RR, Maestre JM (2014) Distributed model predictive control: an overview and roadmap of future research opportunities. IEEE Control Syst 34(4):87–97 8. Patton RJ, Kambhampati C, Uppal FJ (2005) Challenges of networked control systems: autonomy, reconfiguration and plug and play. In: Proceeding of 1st workshop on network control systems and fault-tolerant control. Ajaccio, Corsica 9. Qian X, Yin Y, Zhang X, Sun X, Shen H (2016) Model predictive controller using laguerre functions for dynamic positioning system. In: th Chinese control conference, pp 4436–4441 10. Razavinasab Z, Farsangi MM, Barkhordari M (2017) State estimation-based distributed model predictive control of large-scale networked systems with communication delays. IET Control Theory Appl 11(15):2497–2505 11. Vaccarini M, Longhi S, Katebi MR (2009) Ünconstrained networked decentralized model predictive control. J Process Control 19(2):328–339 12. Vaccarini M, Longhi S, Katebi MR (2006) Stability analysis tool for tuning unconstrained decentralized model predictive controllers. In: Proceedings of the international conference control. Glasgow, Scotland, UK. August 2006 13. Vaccarini M, Longhi S, Katebi MR (2006) State space analysis of unconstrained decentralized model predictive control systems. In: Proceeding of the American control conference. Minneapolis, Minnesota, pp 159–164 14. Ventak AN, Hiskens IA, Rawlings JB, Wright SJ (2006) Distributed output feedback MPC for power system control. In: Proceedings of the 45th IEEE conference on decision and control, pp 4038–4045. San Diego, California 15. Ventak AN, Rawlings JB, Wright SJ (2005) Stability and optimality of distributed model predictive control. In: Proceedings of the 44th IEEE conference on decision and control, and the European control conference, pp 6680–6685 16. Wang L (2009) Model predictive control system design and implementation using MATLAB. Advances in industrial control. Springer

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17. Zeˇcevi´c AI, Šiljak DD (2005) A new approach to control design with overlapping information structure constraints. Automatica 41:265–272 18. Zheng Y, Li S, Li N (2011) Distributed model predictive control over network information exchange for large-scale systems. Control Eng Pract 19(7):757–769 19. Zheng Y, Li S, Qiu H (2013) Networked coordination-based distributed model predictive control for large-scale system. IEEE Trans Control Syst Technol 21(3):991–998

Direct Torque Control Using Fuzzy Second Order Sliding Mode Speed Regulator of Double Star Permanent Magnet Synchronous Machine Louanasse Laggoun, Brahim Kiyyour, Ghoulemallah Boukhalfa, Sebti Belkacem, and Said Benaggoune Abstract The conventional Direct Torque Control (DTC) strategy using PI regulators has certain disadvantages such as significant flux, torque ripples and sensitivity to parametric variations. To overcome these drawbacks, we apply a new type with more robust regulators such as the Second Order Sliding Mode Control (SOSMC) based on Super Twisting algorithm and fuzzy second order sliding mode control. This work deals with the modeling and performance improvement study of the DTC of a Double Star Permanent Magnet Synchronous Machine (DSPMSM) using a hybrid FUZZY-SOSMC speed regulator. The torque ripple, speed and currents will be evaluated and compared by the classical PI-DTC and SOSMC-DTC. Simulation results demonstrate the feasibility and validity of the proposed FUZZY-SOSMCDTC system by effectively accelerating system response, reducing torque and a very satisfactory performance has been achieved. Keywords Direct torque control (DTC) · Double star permanent magnet synchronous machine (DSPMSM) · Second order sliding mode control (SOSMC) · Fuzzy logic control

1 Introduction In recent years, variable speed drives consisting of an alternative machine associated with a static converter, have attracted much attention from research groups and industry. They are more and more present in the fields of high-power industrial L. Laggoun Department of Industrial Engineering, University Abbas Laghrour, LSTEB Laboratory, Khenchela, Batna, Algeria e-mail: [email protected] B. Kiyyour (B) Department of Electrical Engineering, University of Biskra, Biskra, Algeria e-mail: [email protected] G. Boukhalfa · S. Belkacem · S. Benaggoune Department of Electrical Engineering, University of Batna2, LEB Laboratory, Batna, Algeria © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_10

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applications. However, the constraints on the power components limit the switching frequency, and consequently reduce its performances. To enable the use of higher switching frequency components, the power should be divided to realize this, one of the solutions is to use machines with a large number of phases (multi-phase (n > 3) or (multi-star) [1–3]. One of the most common examples is the double-star permanent magnetic synchronous machine (DS-PMSM), which is the most widely used in industry [3–5]. In recent years, several techniques have been developed to improve the performance of these electrical machines. The Direct Torque Control (DTC) proposed by Depenbrock and Takahashi in the middle of the eighties is a vector control solution [6]. It was introduced especially for three-phase machines and several studies allowed to apply this control technique on multi-phase machines. Like every command, the DTC has advantages and disadvantages, it has to be less dependent on the parameters of the machine (the stator resistance is theoretically the only parameter of the machine, which intervenes in the command) and to provide a faster response of the torque. Despite these advantages, this control scheme also has significant disadvantages, the instability problem such as the lack of control of the switching frequency of the inverter and the use of hysteresis bands generating electromagnetic torque ripples and noise in the machine [1, 3, 7]. Several research works has discussed this subject, initially based on the control principles set out in AL. Takahashi and giving rise to various developments of DTC type strategies. This structure of control is based on PI type controllers, these regulators suffer from sensitivity to variations in the motor parameters. This requires a good identification of the parameters. To ensure the robustness and good performance of the direct torque control using a PI controller, several approaches have been recently proposed were adopted to change the PI controllers by other controllers [1, 8]. The sliding mode control (SMC) is a modern control strategy, with higher robustness against load and parameter variations, faster response and higher level of energy efficiency. It relies on fast switching, which made it difficult to implement it that time [1, 8, 9]. Nevertheless, the classic sliding mode (standard sliding modes) has a significant disadvantage, like the phenomenon of chattering. The main cause of chattering phenomenon has been identified as the presence of unmodelled parasites in the switching device [3]. These disadvantages have been minimized by the introduction of the higher order sliding mode control in the 80 s. The second order sliding mode control (SOSMC) keeps the main advantages of standard sliding modes and has the additional advantage that it can be used to remove chattering effect. In fact, during the last decade, the fuzzy logic control (FLC) has been selected as suitable control solution in the field of power electronics and drives. Among the advantages provided by this control approach over the conventional controllers in other hand it does not require accurate mathematical model. It can thus work with inaccurate inputs, handle nonlinear model systems and easily reach performances of PI controllers. Accordingly, this paper aims at combining the

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advantages of Fuzzy and SOSMC. This approach has been successful applied in wide area [10]. In order to develop a robust DTC, current researchers have proposed the use of a DTC with a speed controller based on hybrid fuzzy-Second order sliding mode control to obtain a more flexible control. This last solution allowed the reduction or even the attenuation of the chattering phenomenon while keeping the properties of robustness [3, 11]. The strategy proposed in this work is the study of the dynamic behavior of doublestar permanent magnets synchronous machine (DSPMSM) controlled by a DTC during a speed adjustment by conventional regulators (PI), SOSMC and by regulators based on fuzzy-second order sliding mode. Simulation results reveal that the FUZZYSOSMC-DTC has a very robust behavior against the SOSMC-DTC and PI-DTC.

2 Modeling of the Double Star Permanent Magnet Synchronous Machine By applying the Park transformation to the model of the DSPMSM, the equations are expressed in a reference frame linked to the rotating field as follows [12]: d Ψ ds1 − w Ψ qs1 dt d = R s1 i qs1 + Ψ qs1 − w Ψ ds1 dt

Vds1 = R s1 i ds1 + Vqs1

d Ψ ds2 − w Ψ qs2 dt d = R s2 i qs2 + Ψ qs2 − w Ψ ds2 dt

(1)

Vds2 = R s2 i ds2 + Vqs2

(2)

And the expressions for stator flux are: Ψds1 = L ds1 i ds1 + M ds2 i ds2 + Ψ P M Ψqs1 = L qs1 i qs1 + M qs2 i qs2 Ψds2 = L ds2 i ds2 + M ds1 i ds1 + Ψ P M Ψds2 = L ds2 i ds2 + M qs1 i qs1

(3)

For studying the dynamic behavior, the following equation of motion was added: J

dΩ = Te − Tr − f r Ω dt

(4)

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Fig. 1 Schematic representation of DSPMSM windings

Sb1 isb2 Sb2

isb1 Vsb1

Vsb2

Vsa2 isa2

ψ PM

θ

Sa2 Vsa1

γ=30°

isa1 Sa1

Vsc1 isc1 S c1

Vsc2 isc2

where: J is the moment of inertia, fr is the friction coefficient, Te is the electromagnetic torque, Tr is the load torque and  is the mechanical rotation speed of the rotor [3]. The model of the DSPMSM has been completed by the expression of the electromagnetic torque Te given below [3]: T e = p(Ψ ds1 i qs1 − Ψ qs1 i ds1 + Ψ ds2 i qs2 − Ψ qs2 i ds2 ) = p(Ψ P M ( i qs1 + i qs2 ) + (L d − L q ) − ( i ds1 i qs1 + i ds2 i qs2 ) + (M d + M q )( i ds1 i qs2 + i ds2 i qs1 ))

(5)

where: p is the number of pole pairs, ψ P M permanent magnet. The structure of the DSPMSM is represented in the electrical space by Fig. 1.

3 Direct Torque Control (DTC) of the DSPMSM The principle of the DTC is based on the direct determination of the command sequences applied to the switches of a voltage inverter to deliver the stator voltage vectors. These vectors are chosen from a selection table as a function of the flow and torque errors as well as the position of the stator flow vector. Two comparators control the status of system control variables, stator flux and electromagnetic torque. The hysteresis corrector is the simpliest and best suited to DTC. Its role is to maintain the error between the value to be set and its reference in a hysteresis band. For a two-state controller, the choice of the voltage vector depends only on the sign of the error and does not depend on its amplitude. However, a hysteresis band is added around zero to avoid unnecessary switching when the flow error is very small [13]. A two-level voltage inverter allows to have 7 distinct positions in the phase plane, corresponding to the 8 voltage vectors of the inverter. These positions are illustrated in Fig. 2.

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Fig. 2 Voltage vector

Table 1 Vectors voltage localization Cflx

CTe

1

2

3

4

5

6

Comparator

1

1

V2

V3

V4

V5

V6

V1

Two levels

1

0

V7

V0

V7

V0

V7

V0

Three levels

Two levels

Three levels

1

−1

V6

V1

V2

V3

V4

V5

0

1

V3

V4

V5

V6

V1

V2

0

0

V0

V7

V0

V7

V0

V7

0

−1

V5

V6

V1

V2

V3

V4

Moreover, Table 1 presents the sequences corresponding to the position of the stator flux vector to the different sectors (see Fig. 2). The flux and the torque are controlled by two hysteresis comparators at 2 and 3 levels, respectively, in the case of a two-level voltage inverter. Direct torque control is based on the orientation of the stator flux. The expression of the stator flux in the park frame of reference is described by [2, 3, 13]: t Ψ ds1.2 =

(V ds1.2 − R s i ds1.2 )dt 0

t Ψ qs1.2 =

(V qs1.2 − R s i qs1.2 )dt

(6)

0

where Vsd1.2 and Vsq1.2 are the estimated components of the vector voltage. They are expressed from the model of the inverter. Thus, the stator flux module becomes:

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Fig. 3 Block diagram of the proposed PI-DTC of DSPMSM

Ψ s1 = Ψ s2 =

 

2 2 Ψ ds1 + Ψ qs1 2 2 Ψ ds2 + Ψ qs2

(7)

The electromagnetic torque can be estimated from the estimated magnitudes of the flux, and the measured magnitudes of the line currents, by the equation below: Tˆ e = p(Ψ ds1 i qs1 − Ψ qs1 i ds1 + Ψ ds2 i qs2 − Ψ qs2 i ds2 )

(8)

he block scheme of the investigated DTC for a voltage source inverter fed DSPMSM is presented on Fig. 3.

4 Second Order Sliding Mode Control of DSPMSM The proposed control strategy is based on the Super Twisting Algorithm. This algorithm is an exception that only requires information about the sliding surface [14–18]. The application of this control strategy begins with the determination of the relative degree of the variable to be regulated. This variable is the speed, so we choose a surface that is sufficient to make the command appear. We define the following sliding surface: S  =  ref − 

(9)

Direct Torque Control Using Fuzzy Second Order Sliding Mode …

Te − Tr − f r  d = dt J S∗ =  r∗e f − ∗ =  r∗e f −

Te − Tr − f r  J

145

(10) (11)

If we define the functions AΩ as follows: A =  r∗e f −

1 ( f r − Tr ) J

(12)

∗∗ Thus: S t = A∗ − 1J T e∗ The second-order sliding mode controllers contain two parts:

Ter e f = T eeq + T esut

(13)

where: T esut = I 1 + I 2 I 1∗ = −α sin g (S Ω ) 1

I 2 = −λ|S Ω | 2 sin g (S Ω )

(14)

The corresponding sufficient conditions for finite time convergence to the sliding manifold are:  α > KCm0 (15) M (α+C 0 ) λ2 ≥ 4CK0 3K(α−C 0) m

λ, α, K m , K M and C 0 are positive gains used to adjust the Super Twisting controller. [3, 13]

5 Fuzzy Second Order Sliding Mode Direct Torque Control (FUZZY-SOSMC-DTC) SOSMC has proven itself in many studies and research applications of its effectiveness in minimizing chattering effect is mainly caused by the presence of a discontinuous control term containing the sign function [19–21]. To improve the SOSMC-DTC of the DSPMSM and more and more decrease the negative effect caused by the sign function, Recently, fuzzy logic controllers (FL) have generated a lot of interest in some applications. In this work we propose to employ a fuzzy-second order sliding mode. FUZZY-SOSMC is a hybrid development of second order sliding mode control and fuzzy logic control, where the switching controller term sign, has been replaced by an inference fuzzy system [3, 22], membership functions are chosen to represent

146 Fig. 4 Membership functions for e and ec

L. Laggoun et al. NB

1

NM

NS

ZE

PS

PM

PB

0.8 0.6 0.4 0.2 0 -1.5

Fig. 5 Membership functions for Tem

-1

-0.5

0

0.5

1

NB

NM

NS

ZE

PS

PM

0 -15

- 10

-5

0

5

10

1

1.5

PB

0.8 0.6 0.4 0.2 15

the linguistic variables for the inputs and outputs of the controllers. in the FUZZYSOSMC the Eq. (14) becomes: I 1∗ = −α f uzzy(S Ω ) 1

I 2 = −λ |S Ω | 2 f uzzy(S Ω )

(16)

5.1 Fuzzification The inputs to the Fuzzy-GA have to be fuzzified before being fed into the control rule and gain rule determinations. The triangular membership functions (MFs) used for the input (e, ec and, ΔT em ) are shown in Figs. 4 and 5.

5.2 Inference and Defuzzification The present paper uses MIN operation for the calculation of the degree μ(ΔT em ) associated with every rule, for example, μ(ΔT em ) = Min[μ(e),μ(ec )]. In the defuzzification stage, a crisp value of the electromagnetic torque is obtained by the normalized output function as:

Direct Torque Control Using Fuzzy Second Order Sliding Mode …

147

m    μ  T em j T em j

du =

j=1

(17)

m    μ  T em j j=1

where: m is the total number of rules (7*7), μ(ΔT em ) is the membership grade for the n rule, ΔT em is the position of membership functions in rule n in U(−15,−10,−5,0,10,15).Fig. 4.

5.3 Control Rule Determination The logic of determining this rule matrix is based on a global knowledge of the system operation. As an example, we consider the following two rules: if e is PB and ec is PB then Tem is PB if e is ZE and ec is ZE then Tem is ZE They indicate that if the speed is too small compared to its reference (e is PB), so a big gain (Tem is PB) is required to bring the speed to its reference and if the speed reaches its reference and is established (e is ZE and ec is ZE) so impose a small gain Tem is ZE. (Table 2) Figure 6 illustrates the general structure of the FUZZY-SOSMC-DTC of DSPMSM.

6 Simulation Results and Discussion In order to have a better appreciation of the results obtained through the direct torque control DTC based on the three types of speed regulators PI and SOSMC Table 2 Inference rules of fuzzy logic controller Tem e

ec NB

NM

NS

ZE

PS

PM

PB

NB

NB

NB

NB

NB

NM

NS

ZE

NM

NB

NB

NB

NM

NS

ZE

PS

NS

NB

NB

NM

NS

ZE

PS

PM

ZE

NB

NM

NS

ZE

PS

PM

PB

PS

NM

NS

ZE

PS

PM

PB

PB

PM

NS

ZE

PS

PM

PB

PB

PB

PB

ZE

PS

PM

PB

PB

PB

PB

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Fig. 6 Block diagram of the proposed FUZZY-SOSMC-DTC of DSPMSM

and FUZZY-SOSMC applied to the DSPMSM, it is necessary to compare the static and dynamic characteristics of the three speed controllers under the same operating conditions (reference, disturbance loads) and in the same simulation configuration. A series of numerical simulations under the MATLAB/Simulink environment have been undertaken for the following two modes of operation: – Operation with speed reversal. – Operation with variation of the load torque.

6.1 Robustness Test for Speed Reversal The purpose of this test is to validate the robustness of the controllers PI-DTC and SOSMC-DTC and FUZZY-SOSMC for speed reversal. Figure 7 depicts the waveforms of the improved performances of speed control in the case of a no-load. It can be noticed that the use of the FUZZY-SOSMC controller allows the speed to judiciously follow its reference value of +100 to −100 rad/s at time t = 0.15 s and at 50 rad/s at t = 0.3 s. In fact, this behavior represents a clear improvement in dynamic response with a FUZZY-SOSMC-DTC, contrary to a drive with a standard PI-DTC and SOSMC-DTC. Performance with each controller is also analyzed through these of Integral Squared Error (ISE), Integral Absolute Error (IAE) and Integral Time Squared Error (ITSE), and the results described in Appendix Table 3 confirm the improved performance with the FUZZY-SOSMC-DTC.

Direct Torque Control Using Fuzzy Second Order Sliding Mode …

149

120

PI

90

SOSMC FUZZY-SOSMC

Speed (red/s)

60 30 0 -30 -60 -90 -120

0

0.05

0.1

0.15

150

0.25

0.2 t(s)

0.3

0.35

0.4

100

Torque(N.m)

100

90

20

80

10

50

0,299

0

0

0,314

-20 -40 -60

-50

PI

-80

-100

0.15

0

0.05

0.1

0.15

0.2 t(s)

SOSMC FUZZY-SOSMC

0.175

0.25

0.35

0.3

0.4

30

Current iq1(A)

20 10 0 -10 -20

PI SOSMC

-30

FUZZY-SOSMC

0

0.05

0.1

0.15

0.2 t(s)

0.25

0.3

0.35

0.4

Fig. 7 Comparison of the speed regulation of the PI- DTC, SOSMC-DTC and FUZZY-SOSMCDTC with speed reversal Table 3 Comparison of Performance Index

Controllers

IAE

ISE

ITSE

PI-DTC

0.056

0.1331

3.33 × 10−3

SOSMC-DTC

0.0093

0.0125

5.375 × 10−5

FUZZY-SOSMC-DTC

0.0076

0.001503

4.856 × 10−5

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6.2 Robustness Test for a Variation of the Load Torque Figure 8 shows the simulation result (speed, the torque and the current isq1 ) during a load torque setpoint variation of 10 N.m at time t = 0.12 s with 20 Nm and at time t = 0.2 then 10 N.m at time t = 0.28 s and 0 N.m at t = 0.35 s. In this case of simulation, we realize that the torque is perfectly following the setpoint value, the speed reaches its reference after a small deformation for the case of PI-DTC 120

Speed (red/s)

100

102.5 98

80

0

0.05

0.1

0.2

0.15

0.25

0.3

40

0.2 PI SOSMC FUZZY-SOSMC

0

0

60

0.05

0.1

0.15

0.25

0.2 t(s)

0.3

55

0.4

FUZZY-SOSMC

45

40

0.35 PI SOSMC

50

50

Torque(N.m)

0.4

60

20

40 35

30

30

20

0

0.01 24 22 20 18 16

10 0

0.12

0

0.05

0.1

0.15

25

0.2 t(s)

0.14

0.16

0.25

0.3

0.35

0.4

PI SOSMC

20 Current iq1(A)

0.35

FUZZY-SOSMC

15 10 5 0 -5

0

0.05

0.1

0.15

0.2 t(s)

0.25

0.3

0.35

0.4

Fig. 8 Comparison of the speed regulation with variation of the load torque, PI-DTC, SOSMC-DTC and FUZZY-SOSMC-DTC

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and with a negligible influence which is recovered quickly with its reference for the SOSMC-DTC, the use of the FUZZY-SOSMC-DTC control presents a high dynamic performance. Moreover, the tracking performances are improved by the use of the FUZZY-SOSMC-DTC control, in comparison with others controllers. The electromagnetic torque produced by the DSPMSM controlled by PI-DTC, SOSMCDTC and FUZZY-SOSMC-DTC is presented in Fig. 8, it can be noticed that the ripple is not the same for the three techniques. A zoom of the electromagnetic torque for the three strategies is shown. It’s clear that the classical PI-DTC suffer from two problems: steady state error and high torque ripple. Howere, the SOSMC-DTC and FUZZY-SOSMC-DTC corrects the steady state error and reduces the torque ripple.

7 Conclusion In this paper, a comparative study between the conventional PI-DTC, SOSMC-DTC and FUZZY-SOSMC-DTC has been carried out for speed controller of DSPSM. Simulation results reveal an improvement in the control performance of the torque including ripple, steady state error reduction and a satisfactory performance with the use of the FUZZY-SOSMC-DTC controller. Furthermore, the effectiveness of the proposed algorithms is evaluated and justified from performance indices IAE, ISE and ITSE. So, this algorithm is suitable for applications requiring a high tracking accuracy when external disturbances occur.

Appendix (Table 4) Table 4 DSPMSM parameters

Parameter

Symbol

Nominal voltage

250 V

Stator resistance Rs1 = Rs2

0.12 

Stator inductance Ls

0.8 mH

Mutual inductance Lm

0.3 mH

Flux linkage ψPM

0.394 Wb

Pole pairs P

4

Machine inertia J

5 10−5 kg . m2

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References 1. Naas B, Nezli L, Naas B, Mahmoudi MO, Elbara M (2012) Direct torque control based three level inverter-fed double star permanent magnet synchronous machine. Sciverse Sci direct Energy Procedia 18(1):521–530 2. Moubayed N (2009) Speed control of double stator synchronous machine supplied by two independent voltage source inverters. Wseas Trans Syst Control 4(6): 253–258 3. Laggoun L, Youb L, Belkacem2 S, Benaggoune S, Craciunescu A (2018) Direct torque control using second order sliding mode of a double star permanent magnet synchronous machine. U.P.B Sci Bull Ser C 80(4): 93–106 4. Moubayed N, Bernard B (2009) Comparison between two double stator synchronous machine supplying strategies. 7th Int Conf Electromech Power Syst, pp 143–147. Iasi, Romania. 8–9 October 2009 5. Dieng A, Benkhoris MF, Mboup AB, Aït-Ahmed M, Le Clairel JC (2016) Analysis of fivephase permanent magnet synchronous motor. Revue Roumaine des Sciences Techniques - Serie Électrotechnique et Énergétique 61(2):116–120 6. Boudjema Z, Taleb R, Djeriri Y, Yahdou A (2017) Novel direct torque control using second order continuous sliding mode of a doubly fed induction generator for a wind energy conversion system. Turk J Electr Eng Comput Sci 25(2): 965–975 7. Vivek D, Rohtash D (2011) Comparative study of direct torque control of induction motor using intelligent techniques. Can J Electr Electro Eng 2(11): 550–556 8. Humod AT, Abdullah MN, Faris FH (2016) A comparative study between vector control and direct torque control of induction motor using optimal controller. Int J Sci Eng Res 7(4):1362– 1371 9. Tria FZ, Srairi K, Benchouia KT, Mahdad B, Benbouzid M (2016) An hybrid control based on fuzzy logic and a second order sliding mode for MPPT in wind energy conversion systems. Int J Electri Eng Inf 8(4):711–726 10. Kairous D, Belmadani B (2015) Robust fuzzy-second order sliding mode based direct power control for voltage source converter. (IJACSA) Int J Adv Comput Sci Appl 8(8): 167–175 11. Elawady WM, Lebda SA, Sarhan AM (2015) Continuous second order sliding mode control with on line tuned pid. In: Proceedings of the world congress on engineering and computer science, pp 864–869. San Francisco, USA 12. Benyoussef E, Meroufel A, Barkat S (2015) Three-level DTC based on fuzzy logic and neural network of sensorless DSSM using extende kalman filte. Int J Power Electron Drive Syst (IJPEDS) 5(4):453–463 13. Kiyyour B, Naimi D, Salhi A, Laggoune L (2019) Hybrid fuzzy second-order sliding mode control speed for direct torque control of dual star induction motor. J Fundam Appl Sci 11(3):1440–1454 14. Bounasla N, Hemsas KE (2013) Second order sliding mode control of a permanent magnet synchronous motor. In: The international conference on sciences and techniques of automatic control & computer engineering, STA’2013, pp 535–539. Sousse. Tunisia, 20–22, Dec. 2013 15. Meghni B, Dib D, Azar AT (2017) A second-order sliding mode and fuzzy logic control to optimal energy management in wind turbine with battery storage. Nat Comput Appl 28(6): 1417–1434 16. Guangping Zh, Hostettler JD, Patrick G, Wang X (2016) Robust sliding mode control of permanent magnet synchronous generator-based wind energy conversion systems. Sustainability 8(12):1–20 17. Hostettler J, Wang X (2015) Sliding mode control of a permanent magnet synchronous generator for variable speed wind energy conversion systems. Syst Sci Control Eng: Open Access J 3(1–3): 453–459 18. Hafiane M, Sabor J, Taleb M (2017) Optimal speed control based on adaptive second order sliding mode and modified Hsc Mppt algorithm for wind turbine. ARPN J Eng Appl Sci 12(21):5891–5902

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19. Boudjema Z, Taleb R, Yahdou A (2016) A new DTC scheme using second order sliding mode and fuzzy logic of a DFIG for wind turbine system. (IJACSA) Int J Adv Comput Sci Appl 7(8): 49–56 20. Yuan X, Chen Z, Yuan Y, Huang Y (2015) Design of fuzzy sliding mode controller for hydraulic turbine regulating system via input state feedback linearization method. Energy Elsevier. 93(1): 173–187 21. Ullah N, Shaoping W, Khattak MI, Shafi M (2015) Fractional order adaptive fuzzy sliding mode controller for a position servo system subjected to aerodynamic loading and nonlinearities. Aerosp Sci Technol 43(6): 381–387 22. Ardjoun SAEM, Abid M (1686) Fuzzy sliding mode control applied to a doubly fed induction generator for wind turbine Turk J Electr Eng Comput Sci 23(6): 1673–1686

Fractional Order Integral Controller Design Based on a Bode’s Ideal Transfer Function: Application to the Control of a Single Tank Process Chahira Boussalem, Rachid Mansouri, Maamar Bettayeb, and Mustapha Hamerlain Abstract The paper deals with the water level control of a single tank process associated feedback linearization (FL) with a novel fractional order integral controller (FOIC) which is based on the Bode’s ideal transfer function. The first controller is used to cancel the nonlinearities of the single tank process and the latter used to solve the tracking problem. The new controller is implemented on a single tank process and the results are compared with a Linear Quadratic Regulator (LQR). Keywords Fractional calculus · Nonlinear systems · Single tank · Feedback linearization · Fractional integral controller · Bode’s ideal transfer function

1 Introduction The first references to the Fractional calculus refer to Leibniz and L’hopital in 1695, where a differentiation of order 0.5 was discussed. This theory attracted increasing interests in the world and research is still underway in all fields, especially in automatic control for modeling of physical systems, stability questions and control. C. Boussalem (B) · R. Mansouri Automatic Department, University of Tizi-Ouzou, Tizi-Ouzou, Algeria e-mail: [email protected] R. Mansouri e-mail: [email protected] C. Boussalem · M. Hamerlain Productive and Robotics Division, Center for Development of Advanced Technologies, Algiers, Algeria e-mail: [email protected] M. Bettayeb Center of Excellence in Intelligent Engineering Systems, King Abdulaziz University, Jeddah, Saudi Arabia e-mail: [email protected] Electrical Engineering Department, University of Sharjah, Sharjah, United Arab Emirates © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_11

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Designing and tuning fractional controllers appear to be very important for their superior results in control. In fact, many methods have been developed. In [1, 2], the authors developed the CRONE (Commande Robuste d’Ordre Non Entier) and the fractional order PID controllers and demonstrated the superior performance and advantage of both controllers over the classical PID controller. An optimization method to tune the fractional PI controller based on a nonlinear function minimization was used in [3]. The authors also show better performance of this controller in comparison with a conventional PI controller. In [4], A fractional order fuzzy control is developed and its parameters are tuned with a particle swarm optimization algorithm. This controller shows better performance over the classical PID, and the integer order fuzzy PID controller. Most physical systems are nonlinear therefore their control attracted attention of many researchers around the world. Analysis and control of nonlinear systems are in general extremely difficult problems in both integer and non integer control, where we find several control methods based on the linearized models: some methods used the usual Jacobian linearization [5, 6], and some other used feedback linearization [7–9]. In the fractional control, several works have been addressed for validating the developed control strategies through the experimental setup. Recentlly, in [10, 11], the authors proposed a fractional PI-state feedback controller for a linearized system by using the pole placement technique and it is implemented on an inverted pendulumcart system to stabilize the pendulum angle and the cart position to the setpoint zero. A novel control structure based on the state feedback with fractional integral control is proposed in [12] if the setpoit is not zero. The controller is also implemented on an inverted pendulum-cart system. The aim of this paper is to develop a method of determining the integral gain of the fractional controller in SISO linearized system (transfer function of a simple integrator) to solve the output tracking problem and to validate the method by an experimental setup of a single tank system. We use the feedback linearization (FL) technique to cancel the nonlinearities of the system and a fractional order integral controller (FOIC) to the tracking. The proposed method is based on the Bode’s ideal transfer function. The latter has very interesting properties. The first property refers to the location of the poles for values of γ in the range [1, 2]. In fact, the poles are complex and the step response have the behavior of an underdamped second order system with the two important parameters such as overshoot and time constant which depend directly on the values of the fractional order γ and the gain crossover frequency ωc , respectively. The second property can be exploited in the design of the integral gain kI of the fractional integral controller. Another interesting property is the iso-damping robustness property [13]. The closed-loop systems is represented as an ideal closed-loop system whose open-loop is given by Bode’s ideal transfer function. Several control techniques use this type of transfer function [13–15]. Many works have shown that fractional systems presents best qualities [13] and very interesting structure for its implementation [12, 15].

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Single or multiple tanks systems are featured as a bench-mark problem in the category of nonlinear control systems. Control of liquid level in tanks and fluid flow between tanks is a fundamental requirement in almost all process industries such as waste water treatment, chemical, petrochemical, pharmaceutical and beverages. Single Tank System is used for the experimental verification of designed controllers. The system is found to be very simple to operate and it is a digital control system which works on MATLAB/Simulink platform. A simulation study of fractional order model reference adaptive control applied to level control in a conic tank was presented in [16]. In [17], modelling and designing of controllers on real time single tank system and simulation were presented. Three controller tuning methods such as Relay Auto-Tunning, Ziegler-Nichols, and Tyreus-Luyben are used. A simulation and experimental results of PI/PID controllers of integer order as well as of fractional order for level control on a conical tank have been presented in [18]. The contribution of this paper is twofold. We first present a new fractional integral control strategy based on the Bode’s ideal transfer function for a single tank system to solve the problem of tracking. Second, we show the effectiveness of the proposed controller by simulation, experimental results and comparison with an optimal Linear Quadratic Regulator (LQR). The rest of the paper is organized as follows: Sect. 2 describes the single tank system and its model. Section 3 describes the input-output feedback linearization theory and a fractional integral controller design. Simulation and experimental results using the proposed controller, and the experimental results using the LQR controller applied to single tank system are carried out in Sect. 4. We conclude with Sect. 5.

2 Single Tank Modeling A. Description of Experimental Setup The experimental setup consists of four cylindrical tanks of uniform cross sections for water circulation, and a rectangular tank for water storage. It can be configured into several types of experiments (SISO, MIMO, SIMO, MISO). In this paper only one tank is considered as shown in Fig. 1. The latter consists of a tank of cross Section A for water circulation, and a rectangular tank for water storage. The water level in tank is measured by a pressure sensor. Water is pumped from the storage tank to the tank by submersed pump. The water flows freely to the storage tank through the orifices. The system is controlled via the computer and Matlab has been used to simulate the model. An advantech peripheral communication interface (PCI1711) card is used for interfacing between the plant and the controller. The water level information is transferred to the PC via power supply unit and power amplifier (PSUPA) [17, 19].

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Fig. 1 Schematic diagram of a single tank experimental setup

B. Single Tank Modeling As shown in Fig. 2, the water is pumped from the storage tank to tank by pump via valve MVB providing a flow Q0 . The water is flowing out of the tank through an orifice of area a with a flow Q1 . The input is the voltage u to the pump, and the output is the water level H. Let V be the volume of liquid in the tank, the mass balance is given by: d H (t) d V (t) =A = Q0 (t) − Q1 (t) dt dt

(1)

Consider the Bernoulli’s law which defines the outflow rate as the product between the outlet cross sectional area a and the outflow velocity v. The outflow rate from tank is given by:  Q1 (t) = av(t) = a 2g H (t)

Fig. 2 Schematic diagram of a single tank system

(2)

Fractional Order Integral Controller Design Based …

159

The flow rate generated by pump is given by: Q 0 (t) = ηu(t)

(3)

where, η is a constant relating the control voltage with the water flow from pump. Substituting (2) and (3) into (1), we obtain a nonlinear state equation which describe the system dynamics of the single tank process: a d H (t) = ηu(t) − 2g H (t) dt A

(4)

Defining water level x = H(t) of tank as state variable, the voltage u = u(t) to the pump as the input to the process and y = H(t) as the output, a nonlinear state space model of the single tank system can be obtained as follows:  d x(t) dt

√ = − aA 2gx(t) + ηu(t) y(t) = x(t)

(5)

The single tank parameters and variables are: A = 0.01389 is the cross-sectional area of the tank (m2 ), a = 50.265.10−6 is the outlet area at tank (m2 ), η = 2.4.10−3 is the constant relating the control voltage with the water flow from the pump, Q0 is the input flow rate of the tank (m3 /s), Q1 is the output flow rate of tank (m3 /s), u is the voltage applied to Pump (V), H is the water level of tank (m), g = 9.81 is the gravitational acceleration (ms−2 ) [17].

3 Input-Output Feedback Linearization and Fractional Integral Controller Design A. Input-Output Feedback Linearization The theory of the input-output feedback linearization (IOFL) has been presented in [20, 21]. In this subsection, our goal is the application of the IOFL to the single tank process (5). For this, the following preliminaries are given. Consider a SISO nonlinear system given by: 

x˙ = f (x) + g(x)u y = h(x)

(6)

where x ∈ Rn is the state vector, u ∈ R is the control input, y ∈ R is the output, f and g are smooth vector fields, and h is the smooth scalar function. The basic approach of input-output Feedback linearization is simply to differentiate the output function

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y repeatedly until the input u appears, and then design u to cancel the nonlinearity [9]. The r time derivatives of y can be expressed as ⎡

⎤

⎡

⎤ L f h(x) ⎢ ⎥ ⎢ ⎥ L 2f h(x) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ .. ⎣ ⎦ ⎣ ⎦ . r −1 r r L f h(x) + L g L f h(x)u y y˙ y¨ .. .

(7)

where Lf h is the Lie derivative of h with respect to f. The number r of differentiations required for the input u to appear is called the relative degree of the system [21]. The control law: u = α(x) + β(x)υ = −

L rf h(x) L g L rf−1 h(x)

+

1 L g L rf−1 h(x)

υ

(8)

applied to y r = L rf h(x) + L g L rf−1 h(x)u yields the simple linear relation: yr = υ

(9)

In this form, the system is readily seen to be completely controllable and thus can be stabilized using a linear feedback. In the following, the input-output linearization √ concept is applied to the single tank nonlinear system (5), where, f (x) = − aA 2gx(t), g(x) = η and h(x) = x. We consider the voltage u to the pump as the control input and we want the tracking of the desired output Href . The relative degree of the single tank system is r = 1. The first time derivatives of y is: y˙ = L f h(x) + L g h(x)u = −

a 2gx(t) + ηu(t) A

(10)

By using (8) the control law is given by u(x) =

a√ 2gx(t) A

η

+

1 υ η

(11)

By replacing u in (10), the system can be transformed into a simple linear system: y˙ = υ

(12)

B. Fractional Integral Controller Design and Stabilization of the Single Tank Process Design of fractional-order controllers has been the subject of many researchers for their superior results in control [1–3]. In this paper, we use the fractional integral

Fractional Order Integral Controller Design Based …

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controller to control a system with a pure integrator. The fractional aspect of the control law give an ideal closed-loop system whose open-loop is given by the Bode’s ideal transfer function. • Bode’s ideal transfer function: Bode [22] has suggested an ideal shape of the open-loop transfer function of the form:

ω γ c L(s) = , γ ∈ R+ (13) s where ωc is the gain crossover frequency which satisfies |L(jωc )| = 1. The parameter γ is the slope of the magnitude curve, on a log-log scale, and may assume integer and non-integer values. In fact, the transfer function L(s) is a fractional order differentiator for γ < 0, and a fractional order integrator for γ > 0. In the Bode diagrams of L(s) (1 < γ < 2), the amplitude curve is a straight line of constant slope −20γ db/dec, and the phase margin of L(s) is π − γπ/2 rad. That means the phase margin will not change along with the change of ωc . The closed-loop system whose open-loop transfer function is L(s) is robust to gain variations and step responses exhibiting an iso-damping property [13, 23]. The closed-loop transfer function of the Bode’s ideal transfer function is ωc γ 1 L(s) (ωc )γ = ωc sγ = γ = γ F(s) = s 1 + L(s) s + (ωc )γ +1 +1 s ωc

(14)

where γ ∈ R + , and 1/ωc is the time constant of the system. For 1 < γ < 2, the poles of F(s) in the s-plane are given by s1,2 = ωc e± jπ/γ

(15)

A detailed study of this function is given in [13, 23]. • Fractional Order Controller Design Consider the linearized model of the single tank system described by (12) and the fractional integral controller of the form

υ(t) = k I Iα Hr e f (t) − y(t)

(16)

where kI ∈ R1×1 , is the designed gain to achieve a desired closed-loop characteristic polynomial. Iα is the non-integer integration operator of order α. Href is the desired setpoint.

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Fig. 3 Block diagram of input-output feedback linearization with fractional order integral controller

Let e(t) = Href − y(t) be the error between the desired setpoint and the output. Figure 3 shows the block diagram of the input-output feedback linearization with the fractional integral controller applied to the single tank process. The result of the fractional order integral controller design is given by the following proposition. Proposition Let’s consider an integrator pur system described by (12) and the control law given by (16). The gain kI that can achieve a desired closed-loop characteristic polynomial as that of the Bode’s ideal transfer function written in the form d (s) = sγ + ωc γ is given by kI = ωc γ. Proof Using the control law (16), the resulting closed loop transfer function is obtained from (12) and (16) by: kI y(s) = α+1 Hr e f (s) s + kI

(17)

This closed loop transfer function is similar to the closed loop transfer function of the Bode’s ideal transfer function (14) (with γ = α + 1), witch is involves two tuning parameters. The first one is the fractional order γ and the second is the integral gain kI . The tuning of both parameters consists in the choice of the closed loop reference specifications such as the gain crossover frequency and the phase margin. The fractional order γ = α + 1 is deduced by the formula phase margin = π − γπ/2 rad. Let ωc and γ be the desired gain crossover frequency and the fractional order of the closed loop reference respectively. The desired characteristic polynomial of the closed-loop system is then d (s) = s γ + ωcγ

(18)

The identification term by term between the characteristic polynomial of the closed loop transfer function (17) and the desired characteristic polynomial d (s) (18), give: k I = ωcγ

(19)

Water level (m)

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0.14

H ref

0.12

H:

H: H: H:

0.1

γ γ γ γ

= 1.1 = 1.33 = 1.5 = 1.66

0.08 0.06 0.04 0.02 0

0

2

4

6

8

10

14

12

Time (s) 60

γ γ γ γ

Control (V)

50

= 1.1 = 1.33 = 1.5 = 1.66

40 30 20 10 0 -10 0

4

2

6

8

10

14

12

Time (s)

Fig. 4 Evolution of the output H and control u for a step reference and for several values of order γ = 1.1, 1.33, 1.5, 1.66, using FL combined with FOIC

Table 1 Fractional order integral controller parameters

γ

ωc = 1.75 rad/s kI

α

1.1

1.8507

0.1

1.33

2.1049

0.33

1.5

2.315

0.5

1.66

2.5319

0.66

ωc (rad/s)

M = 60◦ (γ = 1.33) kI

α

0.25

0.1582

0.33

0.75

0.6821

0.33

1.25

1.3455

0.33

1.75

2.1049

0.33

The following algorithm presents the steps for the design of the fractional order integral controller: • Apply the Input-Output Feedback Linearization method presented in Sect. 3-A to the SISO nonlinear system to get a simple linear model with a pure integrator.

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• Choose the desired phase margin and the gain crossover frequency of the desired closed-loop transfer function. Deduce the fractional order γ, where γ = α + 1. We only take (1 < γ < 2) into consideration which realizes the closed loop transfer function of the Bode’s ideal transfer function. • Deduce the fractional order α, (0 < α < 1) of the controller (16), and then the feedback gain kI by using (19).

4 Simulation and Experimental Results To evaluate the performance of the proposed control experimentally, we first provid a series of simulations in the case of step and square references to single tank system and then an implementation experimentally.

0.12

Water level (m)

0.1 H ref H: ω c = 0.25

0.08

H: ω c = 0.75 H: ω c = 1.25

0.06

H: ω c = 0.75

0.04 0.02 0

0

5

10

15

20

25

30

35

Time (s) 50

ω c = 0.25

Control (V)

45

ω c = 0.75

40

ω c = 1.25

35

ω c = 1.75

30 25 20 15 10 5 0 0

5

10

15

20

25

30

35

Time (s)

Fig. 5 Evolution of the output H and control u for a step reference and for several values of the gain crossover frequency ωc = 0.25, 0.75, 1.25, 1.75, using FL combined with FOIC

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A. Simulation Results Consider the linear model of the single tank system given by (5). We use steps 2 and 3 of the algorithm summarized in Sect. 3-A. We use one test for square reference and two different tests for the step reference. For the latter, in the first test, we choose a fixed value of the gain crossover frequency ωc = 1.75 and we vary the phase margin as M = 30°, 45°, 60°, 81° or the fractional order γ as γ = 1.66, 1.5, 1.33, 1.1. In the Second test, we use a fixed value of the fractional order γ = 1.33 (M = 60°) and we vary the gain crossover frequency ωc as ωc = 0.25, 0.75, 1.25, 1.75. The controller parameters are listed in Table 1. Simulation results of those tests are shown in Figs. 4 and 5 respectively. The system responses and control for a square reference signal for the proposed controller and by using a fixed value of the fractional order γ = 1.1 (M = 81°) and a fixed value of the gain crossover frequency ωc = 1.75 are shown in Fig. 6. From Figs. 4, 5 and 6, the output H tracks the setpoint Href with different overshoots and almost constant rise time by varying the fractional order γ and for a fixed coefficient ωc . In the other hand, by varying the coefficient ωc and maintaining 0.14 H ref

Water level (m)

0.12

H:

γ = 1.1

0.1 0.08 0.06 0.04 0.02 0 0

100

200

300

400

500

600

Time (s) 45 40

Control (V)

35 30 25 20 15 10 5 0

0

100

200

300

400

500

600

Time (s)

Fig. 6 Evolution of the output H and control u for a square reference signal using FL combined with FOIC

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a fixed order γ, the output H tracks the setpoint Href with different rise times and a constant overshoot. The dynamic of the system changes according to the choice of the gain crossover frequency ωc and the fractional order γ. This shows that the output tracking is achieved and the stability and convergence of the error to zero is guaranteed. A. Experimental Results To show the pratical effectiveness of the proposed controller, performances and robustness are tested experimentally to single tank process and results are compared with a robust and optimal Linear Quadratic Regulator (LQR). For the fractional controller, the fractional order γ or phase margin are chosen as: γ = 1.1 (M = 81°) and a coefficient ωc is chosen as ωc = 1.75. For the LQR controller, the weighting matrix Q and the weighting factor R are chosen as 10 and 0.001 respectively. Figure 7 shown the experimental results of the system responses and control for a square reference for both controllers. From the experimental results shown in Fig. 7, it can be observed that both types of controllers can guarantee the closed-loop water level stability and realize well setpoint tracking.

Water level (cm)

15

H ref H: FL & FOIC: H: FL & LQR

γ = 1.1

10

5

0

0

50

100

150

200

250

300

350

400

450

Time(s) 6

FL & FOIC: γ = 1.1 FL & LQR

5

Control (V)

4 3 2 1 0

0

50

100

150

200

250

300

350

400

450

Time (s)

Fig. 7 Experimental results of the water level H and control u in single tank system for a square reference signal using FL combined with FOIC and LQR controller

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Water level (cm)

15

10 H ref H: FL & FOIC H: FL & LQR

5

0

0

50

100

150

200

250

300

350

400

450

Time (s) 6

FL & F0IC: γ = 1.1 FL & LQR

Contrl (V)

5 4 3 2 1 0

0

50

100

150

200

250

300

350

400

450

Time (s)

Fig. 8 Experimental results of the water level H and control u in single tank system for a square reference signal with disturbance at time t = 150 s and t = 250 using FL combined with FIC and LQR controller

For the proposed control, the dynamics of the system changes according to the choice of the gain crossover frequency ωc and the fractional order γ and changes according to the choice of the weighting matrix Q and the weighting factor R for LQR. In order to test the robustness of both controllers, we first added one liter of water in tank 1 at t = 150 s, and then valve MV1 opened at t = 250 s during 20 s. The experimental results for square reference are shown in Fig. 8. This figure shows the robustness of both controllers. Indeed, after adding one liter of water in tank 1 at t = 150 s, the water level in FOIC and LQR increase instantly to 12.16 cm, and then it is rejected quickly and reaches the reference value. We also see that after opening valve MV1 at t = 250 s, the water level in FOIC and LQR decrease to almost 9.7 cm and 9.2 cm, respectively and then it is rejected and reaches the reference value.

5 Conclusion In this paper, FL with a novel FOIC for setpoint tracking is presented and successfully implemented to a single tank process. From the simulation and experimental results,

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it is shown that the proposed controller is very effective. This shows the feasibility of the designed controller in practice. From the comparison of the experimental results, both controllers have a good performance regarding robustness and tracking effect.

References 1. Oustaloup A (1991) La commande CRONE. Herme‘s, Paris 2. Podlubny I (1999) Fractional-order systems and P I α Dα controllers. IEEE Trans Aut Control 44:208–214 3. Monje CA, Caldero´n J, Vinagre BM, Chen YQ (2004) On fractional P I λ controllers: Some tuning rules for robustness to plant uncertainties. Nonlinear Dyn 38: 369–381 4. Pan I, Das S (2016) Fractional order fuzzy control of hybrid power system with renewable generation using chaotic PSO. ISA Trans 62:19–29 5. Owczarkowski A, Horla D, Zietkiewicz J (2018) Introduction of feedback linearization to robust lqr and lqi control analysis of results from an unmanned bicycle robot with reaction wheel. Asian J Control 21:1–13 6. Grognard F, Sepulchre R, Bastin G (1999) Global stabilization of feedforward systems with exponentially unstable Jacobian linearization. Syst Control Lett 37:107–115 7. Noah DM, Laheeb M, Roger CF (2017) Using feedback linearization to improve the tracking performance of a linear hydraulic-actuator. J Dyn Syst Meas Control 8. Moradi H, Hajikolaei KH, Motamedi M, Vossoughi GR (2009) Motion control of a nonlinear single-rod hydraulic actuator via feedback linearization In: ASME 2009 international mechanical engineering congress and exposition, American society of mechanical engineers 9. Cesáreo R, Alejandro FV, Antonio B (2014) Adaptive tracking in mobile robots with inputoutput linearization. J Dyn Syst Meas Control 136 10. Bettayeb M, Boussalem C, Mansouri R Al Saggaf UM (2014) Stabilization of an inverted pendulum cart system by fractional PI State feedback. ISA Trans 53: 508–516 11. Mansouri R, Bettayeb M, Boussalem C, Al Saggaf UM (2015) Linear integer order system control by fractional PI-state feedback. Chapter In: RAZ Daou, X Moreau (eds) Fractional calculus applications 12. Al-Saggaf UM, Mehedi IM, Mansouri R, Bettayeb M (2015) State feedback with fractional integral control design based on the Bode’s ideal transfer function. Int J Syst Sci 1–13 13. Barbosa R, Machado JAT, Sneddon IN (2004) Tuning of PID controllers based on Bode’s ideal transfer function. Nonlinear Dyn 305–321 14. Amoura K, Mansouri R, Bettayeb M, Al-Saggaf UM (2016) Closedloop step response for tuning PID-fractional-order-filter controllers. ISA Trans 64: 247–257 15. Bettayeb M, Mansouri R (2014) IMC-PID-Fractional order filter controller design for integer order systems. ISA Trans 53:1620–1628 16. Balaska H, Ladaci S, Zennir Y (2018) Conical tank level supervision using a fractional order model reference adaptive control strategy. In: 15th International conference on informatics in control, automation and robotics (ICINCO). Portugal 17. Jagnade SA, Pandit RA, Bagde AR (2015) Modeling, simulation and control of flow tank system. Int J Sci Res 657–669 18. Jáuregui C, Duarte-Mermoud MA, Oróstica RA, Travieso-Torres JC, Beytía O (2016) Conical tank level control using fractional order PID controllers: a simulated and experimental study. J Control Theory Technol 14: 369–384 19. Mahapatro SR (2012) Control algorithms for a two tank liquid level system: an experimental study. Thesis for the award of master of technology by research in electrical engineering. India. (2012–2014)

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Isidori A (1995) Nonlinear control system, 3rd edn. Springer, Berlin ESlotine JJ, Li. Weiping (1991) Applied nonlinear control. Prentice Hall Bode HW (1945) Network analysis and feedback amplifier design. Van Nostrand, New York Liu L, Zhang S (2018) Robust fractional-order PID Controller tuning based on Bode’s optimal loop shaping. Hindawi 2018:1–14

Modeling of Torsional Vibrations Dynamic in Drill-String by Using PI-Observer R. Riane, M. Kidouche, M. Z. Doghmane, and R. Illoul

Abstract Torsional vibrations, appeared in drill string of oil well, are one of the major cause of drilling failures, in most of cases the penetration rate reduction and drilling costs expansion. The driller’s intervention is basically based on parameters manipulation to mitigate torsional vibrations, precisely variation of the angular velocity of the Top drive. The weight on bit variation is also proven to be practically effective in the reduction of such vibration but with a considerable time delay. In this paper, we modeled the nonlinear dynamic of the down hole using a PI Observer, thus, we estimate the downhole angular speed and perceive the torsional vibrations as soon as they appear so that the time delay is minimized. The obtained model is useful to design robust controller to mitigate the torsional vibrations in real-time. Keywords Torsional vibration · Drill string · Top drive · PI observer · Real-time

1 Introduction Rotary drilling systems are designed to drill a hole on different layers of the earth to reach its target called reservoir. They are generally composed of two main important parts: mechanical part that ensures the propagation of energy delivered by the top drive to the bit, and hydraulic part to take material from the bottom hole to the surface and cool the bit. The mechanical system contains a Top drive to rotate the drillstring, the latter is composed of many pipes and it transfers the energy to the bit which cuts the rocks and creates the borehole. The hydraulic system ensures the circulation of drilling fluid, by using pumps and transport channels. During drilling process, the system can face many types of vibrations, they can be categorized according the direction of their propagation, we can find: torsional, axial, and lateral [1, 2], they R. Riane (B) · M. Kidouche · M. Z. Doghmane Laboratory of Applied Automatic, University M’hamed Bougara, Boumerdes, Algeria e-mail: [email protected] R. Illoul Department of Automation, National Polytechnic School, Algiers, Algeria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_12

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can appear at the same time or separately. Drilling systems, considered in this study, contain drag bits; it contains a fixed blades and cutter at the biggest surface of bit body. This type of bit configuration in most of cases leads to torsional vibrations for which the angular velocity of the bit starts decreasing till it completely stop (stick phase) the cumulating energy will, after some threshold, free the tool that slips with angular velocity much higher than the angular velocity of the top drive (slip phase). This stick-slip phenomenon affects the drilling efficiency, the early damage of drag bits, and may cause system failure. In this study, the main objective is to model the nonlinear dynamic of the drillstring under torsional vibration in order to detect as quickly as possible these vibrations and mitigate them in real-time. The torsional pendulum equivalent system with different degree of freedom has been considered to describe mathematically the drilling system [3, 4]. Like [3, 5, 7, 8], a two degree of freedom mathematical model is considered. Besides the reliability of the system, a PI observer is proposed to estimate the nonlinearity with the unmeasured states and inputs.

2 Drilling System Description Figure 1 shows the main parts of drilling system. One of the most important components is the drillstring, it is of drill pipes that are used to guarantee the circulation of the mud using pumps, and transmission of torque generated by the top drive to the drill bit. In addition to drill pipes, the drillstring also contains 9 m slender tubes jointed by threaded connections. The drillstring bottom end is called the borehole assembly (BHA), it involves a thick walled tubes called drill collars used to avoid buckling of the drillstring [1]. At the BHA bottom end, there is the drilling tool, named the bit. Typical bit design is a steel body Traditional bits consist of a steel body set with three rotating conical cylinder, it contains teeth made of tungsten that cuts the rocks. The cutters of the bit contained the fixed steel body and enclosed by diamond tooth that can cut the rock, they have been used in rotary drilling since early the 19th. The bit diameter can vary from 0.1 to 0.9 m, it depends on the diameter of the section, and the deeper sections are drilled with smaller bit diameters [1]. The top drive is generally composed of an AC/DC electric motor, a rotary table and a gearbox. The role of the Kelly is to transmit the torque from the electric motor to the drillstring through the rotary table. The newest rigs are supported by high technology top drive that contains multi-functional unit providing the rotary movement of the drillstring. As a result of the importance of the TD in oil drilling industry, many oil equipment producers have launched their project for developing new generation of the automated TDs. Such projects may lead to new top drive motors that can generate high torques at low speed automated based on the torque on bit real-time variation.

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Fig. 1 Descriptive schema of rotary drilling system

3 Mathematical Model The mathematical model of the drilling system can be derived by representing the behavior of the system as the behavior of torsional pendulum, for which the pipes behaves as torsional springs and the collars as a rigid body, we suppose that the top drives rotates at constant speed (Fig. 2).

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Fig. 2 Representation of the mechanical part under torsional behavior of the drill-string

For simplification of this study, we suppose that the interaction between the drillstring and the formation is in the bit, thus, no lateral of axial vibrations exist. that no lateral or The dry frictional model is considered [1, 8, 9], the Interaction between the bit and the formation is a mixture of cutting process and frictional forces [10–12]. The equation of motion can be given as follows.

3.1 The Dynamic of BHA The BHA dynamic is given by the following equation [13]. Jb ϕ¨b = k(ϕt − ϕb ) − Cb ϕ˙b − T ob(ϕ˙b )

(1)

Let φ = ϕt − ϕb and, ϕ˙b = b , then Jb Ω˙ b = kφ − Cb b − T ob(b )

(2)

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where: (ϕb ) (b ) (Jb ) (Cb ) are, the angular displacement, angular velocity, equivalent of mass moment of inertia, equivalent viscous damping coefficient at the bottom of the drill-string respectively, (ϕt ) is the angular displacement at the top of the drillstring, (k) is the torsional stiffness coefficient, and Tob is a nonlinear function which will be referred to the torque on bit.

3.2 The Dynamic of Drillstring The components that defines the mechanical dynamic of the drillstring are: the gearbox characterized by an (n:1) ratio, and the DC excited electric motor. The equations that describe this behavior are given as follows.

3.2.1

Mechanical Behavior Jt t = T − Ct t − kφ

(3)

where (Ω t ) (J t ) (C t ) are the angular velocity, equivalent of mass moment of inertia, equivalent viscous damping coefficient at the top of the drill-string respectively, T is the torque delivered by the motor to the system multiplied by the gearbox ratio (n):T = nTm .

3.2.2

Electrical Equation v=l

di + ri + vcem dt

(4)

where (l), (r), (i) and (v) are respectively defined as motor current, motor resistance, motor inductance and motor input voltage, (vcem ) is the counter-electromotive force (back-emf), (K) is the motor constant multiplied by the gearbox ratio such as K = n Km . The motor speed and current are connected to counter-electromotive force, and the torque, by the coefficient K respectively. vcem = K t , And, T = K i Finally, we can find Jt Ω˙ t = K i − Ct t − kφ

(5)

di = v − ri − K t dt

(6)

l

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As explained earlier, the Tob is of frictional contact between the bit and the rock as well as the cutting operation [12]. T ob = Tc + T f

(7)

By exploring conclusion in [14], we can say that the cutting torque can be written as Tc =

1 2 R εd 2 bit

(8)

ε is the amount of energy required to cut a unit volume of rock, and d the cut depth, and (Rbit ) is the radius of the bit. In [6, 10, 12] the frictional torque is considered continuous static function, however, in this study it has been taken as a dynamic discontinuous function for dry friction. The discontinuity between stick and slip phase caused the model to be more complex especially for low velocity values of the slip phase [15]. During the slip phase, the relative movement is almost zero, thus, the friction can be seen as constraints that preserve the initial condition of zero velocity between rubbing surfaces and the rock. Coulomb model, shown in Fig. 3, has been lengthily used even it reveals discontinuity problem, consequently KARNOOP [16] proposed a zero velocity constraints in his model in which where a transition interval from stick to the slip phases is set up, the Coulomb model is still valide outside that interval [17]. The Karnopp model can be written as (9). ⎧ i f b ≺ Dv and Te ≺ Ts ⎨ Te T f = Ts sign(Te ) i f b ≺ Dv and Te  Ts ⎩ Td sign(ϕb ) i f b  Dv Fig. 3 PI Observer diagram for unknown input torque

(9)

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T e: is the external torque that must be applied to conquer the static friction torques T s . T d is the dynamic friction torque, Dv is the zero-velocity band, where: Ts(d) =

1 W ob · g · μs(d) · Rbit 2

(10)

μs(d): is the static (or dynamic) dry frictional coefficient, Wob is the Weight on Bit. In conclusion, the mathematical model of the entire system can be written as ⎧ ⎪ φ = t − b ⎪ ⎪ ⎨ Ω˙ = − k φ − Ct  + n K i t t Jt Jt Jt Cb k 1 ˙ ⎪ Ω = φ −  − T ob(b , W ob) b b ⎪ J J J b b b ⎪ ⎩ di nK r 1 = − l t − l i + l v dt

(11)

4 PI Observer Design The PI observer is a system in which the added terms are proportional and integral of unknown variable to be estimated so that the robustness and performance is improved. In this study, the unknown variable is the input torque which represents the state and the unknown nonlinear torque on bit as shown in Fig. 5, [5, 18]. The mathematical model given in (11) can be rewritten the state space model in (12), it represents the torsional behavior of the drillstring. 

X˙ = AX + Bu + Ed Y = CX

(12)

X = ( Ωt Ωb i)t is the state vector, Y = (Ωt i)t is the output vector, u = v is the known input (voltage), d = Tob is an unknown input torque on bit). A, B, C, and E are known matrices with appropriate dimensions. ⎛

⎞

⎛ ⎛ ⎞ 0 0 ⎟ ⎜ 0 ⎜0⎟ ⎟ ⎜ ⎟ ⎟, B = ⎜ ⎝ 0 ⎠, E = ⎜ ⎠ ⎝ − J1b 1 0 l

 0100 C= 0001

0 1 −1 0 ⎜− k −C 0 k ⎜ Jt Jt Jt A=⎜ k ⎝ Jb 0 −Cb 0 0 − Kl 0 − rl

⎞ ⎟ ⎟ ⎟ ⎠

The PI observer can designed and analyzed based on the new augmented system given by (13).

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Z˙ = A z Z + Bz u Y = Cz Z

(13)

 AE 0 0 By checking the observability (Az, C z) , the observer is proposed for the system (12) as shown in (14). ⎧   ⎨ X˙ = A X˜ + Bu + E d˜ + K p Y − C X˜   (14) ⎩ d˜ = K I Y − C X˜

 T  T   where Z = X d , Bz = B 0 , C z = C 0 , and A z =

K P and K I are respectively proportional and integral gains, the observer in (14) can now be rewritten in the following form [17]. 

  Z˙ = A z Z˜ + Bz u + K z Y − C z Z˜ Y˜ = C Z Z˜

(15)

  T  where Z˜ = X˜ d˜ , and K z = K p K I The error of estimation of the unknown input torque is given as: e = Z − Z˜ Thus e˙ = Z˙ − Z˙˜ = (A z − K z C z )e

(16)

It is noticeable that the error in (16) converges asymptotically to zero if we prove that the AO = Az – Kz Cz is Hurwitz. Because the observability (Az, Cz) is proven, we can use many techniques to find the gain Kz that forces the observer error toward zero, in this study, pole placement technique has been used.

5 Results and Discussion The effectiveness and robustness of the proposed observer have examined by simulation of the model with different scenarios, they includes random variation of Wob and voltage of the motor, then estimate the torque on bit for these each variation. The simulations have been carried out in LabView environment (S/N: M76X07294), all model parameters are given in Tables 1 and 2. Practically, to mitigate torsional vibrations the drillers manipulate the drilling parameters such as the Wob, the top drive torque, and the mud viscosity. It is based

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Table 1 Numerical values of the drilling system Parameter

Description

Value

Unit

Jt

Mass moment of inertia at the top of the drill-string

1000

kgm2

Jb

Mass moment of inertia at the bottom of the drill-string

127

kgm2

Ct

Viscous damping coeffi-cientat the top of the drill-string

51

Nms/rad

Cb

Viscous damping coeffi-cientat the bottom of the drill-string

40

Nms/rad

k

Drill-string stiffness

481

Nm/rad

r

Motor resistance

10

m

l

Motor inductance

5

mH

Km

Motor constant

6

Nm/A

n

Gearbox ratio

7.20

–

Table 2 Numerical values of the Bit-Rock interaction Parameter

Description

Value

Unit

e

Intrinsic specific energy of the rock

130

MJ/m3

d

Depth of cut

4

mm/rev

μs

Static dry frictional coefficient

0,6

–

μd

Dynamic dry frictional coefficient

0,4

–

Rbit

Bit radius

0.10

m

on their recommendations; the main drawback of such practice is that the time delay between the beginning of the oscillation phenomenon and the time of observing it and manipulating parameters and its effect [19, 20]. In order to show the effectiveness of the proposed observer for minimizing this time delay, the same methodology used in the field is considered in this simulation, the only difference is that we replace the human observation by PI observer.

5.1 Weight on Bit and Top Drive Speed Variations It has been recommended by drillers practice that severity of torsional vibrations can be minimized by reducing the Weight on bit (WOB) and increasing the Top drive angular velocity (by increasing the voltage of DC motor), this recommendation has been simulated as given by Figs. 4 and 5.

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Fig. 4 First scenario: increase of DC motor voltage with Wob = 15T

Fig. 5 Second scenario: Wob decreasing with constant top drive angular velocity (v = 125 V)

5.2 Drilling Fluid Viscosity Variation The third scenario for torsional vibrations mitigation is by manipulating the drilling fluid viscosity; the increase of this characteristic leads to amplification of the damping coefficient in the bottom hole at the bit. Figure 6 demonstrates the effect of increasing the equivalent damping coefficients C b and C t.

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Fig. 6 Third scenario: Variation of drilling fluid viscosity with Wob = 15T and (v = 125 V)

As the results of simulation that indicate the same dynamic behavior of field practical scenario, we can say that the model is representative and valid (Fig. 7). For the second part of the simulation, we design the PI observer to estimate the unknown Tob as well as the angular velocity of the BHA. Pole placement technique has been considered for gain calculation for which the optimal Eigen values are found at (-10-50-70-100-130). Figure 8 demonstrate the effectiveness of the PI observer estimation for BHA angular velocity, however, Fig. 9 is for Tob estimation. The Wob and Motor voltage (proportional to the Top drive speed) used for this scenario are shown in Fig. 7. Figure 10 represents the Wob under perturbations; the latter can be caused by many factors. One of them is the presence of axial vibration stimulated by the slip Fig. 7 weight-on-bit and motor voltage used with PI observer model

182 Fig. 8 The BHA angular velocity and its estimation, and estimation error (on bottom)

Fig. 9 Torque-on-bit and its estimation, estimation error (on bottom)

Fig. 10 The weight-on-bit variation under perturbation

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phase of torsional vibrations. The designed PI observer has tested under such types of disturbances. The Eigen values has been recalculated to improve the estimation performance of our observer under structured and unstructured perturbations, the optimal values are (−10 −100 −110 −120 −130) (Figs. 11 and 12). Fig. 11 BHA angular velocity and its estimation under perturbations, and estimation error (bottom)

Fig. 12 Torque-on-bit and its estimation under perturbation, and estimation error (on bottom)

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6 Conclusion This study proposed the use of PI observer to estimate the nonlinear torsional dynamic of drillstring in petroleum wells. The mode that describes the behavior of the system under these vibrations has been detailed. The bit-rock interaction term is also used to characterize the frictional forces during drilling process, even though it may vary instantly due the change of rock mechanical characteristics. The proposed PI observer is designed to estimate the instantaneous angular velocity variation of the BHA, as well as the variation of unknown disturbances in the Torque on bit, the good estimation is used to define the compensation value to be applied in real-time to suppress (or mitigate) the torsional vibrations and protect drilling equipment.

References 1. Riane R (2015) Contribution to the adaptive observer synthesis with unknown input of rotary drilling system: simulation under LabView Environment. Magister thesis, University M’hamed Bougara of Boumerdes 2. Mihajlovi´c N (2005) Torsional and lateral vibrations in flexible rotor systems with friction. Doctoral thesis, Technical University of Eindhoven 3. Abdulgalil F, Siguerdidjane H (2004) Nonlinear friction compensation design for suppressing stick-slip oscillations in oil-well drill-strings. In: 7th IFAC DYCOPS, Massachusets, USA. https://doi.org/10.1016/S1474-6670(17)31903-1 4. Doghmane MZ, Kidouche M (2018) Decentralized controller Robustness improvement using longitudinal overlapping decomposition-application to web winding system. Elektronika ir Elektronika 24(5):10–18. https://doi.org/10.5755/j01.eie.24.5.21837 5. Koenig D, Mammar S (2002) Design of Proportional-Integral Observer for Unknown Input Descriptor Systems. IEEE Trans Autom Control 47(12):2057–2062. https://doi.org/10.1109/ TAC.2002.805675 6. Detournaya E, Richard T, Shepherd M (2008) Drilling response of drag bits: theory and experiment. Int J Rock Mech Min Sci 45:1347–1360. https://doi.org/10.1016/j.ijrmms.2008. 01.010 7. Detournay E, Defourny P (1992) A phenomenological model for the drilling action of drag bits. Int J Rock Mech Min Sci Geomech Abstr 29(1):13–23. https://doi.org/10.1016/0148-906 2(92)91041-3 8. Navarro-López EM, Suárez R (2004) Modelling and analysis of stick-slip behaviour in a drill-string under dry friction. In: The congress of the Mexican Association of Automatic Control 9. Li L, Zhang Q-Z, Rasol N (2011) Time-varying sliding mode adaptive control for rotary drilling system. J Comput 6(3):564–570. https://doi.org/10.4304/jcp.6.3 10. Richard T, Germay C, Detournay E (2000) Self-excited stick–slip oscillations of drill bits. C R Mecanique 332:619–626. https://doi.org/10.1016/j.crme.2004.01.016 11. Besselink B, van de Wouw N, Nijmeijer H (2011) Model-based analysis and control of axial and torsional stick-sliposcillations in drilling systems. In: IEEE international conference on control applications (CCA). https://doi.org/10.1109/CCA.2011.6044505 12. Bayliss MT, Panchal N, Whidborne JF (2012) Rotary steerable directional drilling stick-slip mitigation control. In: The proceedings of the 2012 IFAC workshop on automatic control in offshore oil and gas production, Norwegian University of Science and Technology, Trondheim, Norway, May 31–June 1, 2012. https://doi.org/10.3182/20120531-2-no-4020.00001

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13. Riane R, Kidouche M, Doghmane MZ (2018) Kalman filter for unknown input estimation application to a rotary drilling system modeled with simscape simulink environment. In: International conference on technological advances in electrical engineering, 10–12 December, 2018, Skikda, Algeria 14. Detournay E, Defourny P ()1992 A phenomenological model for the drilling action of drag bits. Int J Rock Mech Min Sci Geomech Abstr 29(1):13–23. https://doi.org/10.1016/0148-906 2(92)91041-3 15. Bousaada I, Saldivar B, Mounier H, Mondié S, Cela A, Niculescu S-L (2016) Delay system modeling of rotary drilling vibrations. In: E Witrant et al (eds) Recent results on timedelay systems, advances in delays and dynamics, 5th edn. Springer International Publishing Switzerland, pp 23–43. https://doi.org/10.1007/978-3-319-26369-4_2 16. Karnopp D (1985) Computer simulation of stick-slip friction in mechanical dynamic systems, asme journal of dynamics systems. ASME J Dyn Syst Meas Control 107(1):100–103. https:// doi.org/10.1115/1.3140698 17. Iamandei A, Miloiu G (2013) Motor drives of modern drillingand servicing rigs for oil and gas wells. Springer Science+Business Media Dordrecht. https://doi.org/10.1007/978-94-0076558-0_50 18. Doghmane MZ (2019) Conception de commande décentralisée des systèmes complexes en utilisant les stratégies de décomposition et optimisation par BMI. Doctoral thesis, University M’hamed Bougara of Boumerdes, Algeria 19. Vromen TGM (2015) Control of stick-slip vibrations in drilling systems. Technische Universiteit Eindhoven, Eindhoven 20. Dan S, Roar N, Vahid A (2013) Real-time optimization of rate of penetration during drilling operation. In: 10th IEEE international conference on control and automation (ICCA) Hangzhou, China, June, pp 12–14. https://doi.org/10.1109/icca.2013.6564893

Power System Generator Coherency Identification for Large Disturbances by Koopman Modes Analysis Zahra Jlassi, Khadija Ben Kilani, Mohamed Elleuch, and Lamine Mili

Abstract This paper presents an effects analysis of large disturbances on power system generator coherency. The analysis is based on a comprehensive parametric study combining fault parameters with system operating conditions. Generator coherency is identified via nonlinear Koopman modal analysis technique applied on generators rotor speed dynamics following large disturbances. The Koopman operator captures the highly nonlinear spatiotemporal dynamics that cannot be assessed with standard linear coherency identification techniques. Faults are short-circuits of variable durations and locations. System operating parameters include loading levels and on line generators. The study methodology is demonstrated on the Tunisian electric network comprising 153-buses 26-machines with a comparison to the conventional slow coherency method. The results show that large disturbance location and system loading levels, alter the coherent grouping determined by weak connections topology. Slow coherent areas may degenerate into smaller coherent groups under large disturbances. Such information is particularly useful for emergency control where intentional islanding actions ought to be taken to mitigate large blackouts. Keywords Electric power system · Coherency · Koopman modes analysis · Large disturbances

Z. Jlassi · K. Ben Kilani · M. Elleuch (B) University of Tunis El Manar, ENIT-LSE. B.P 37- le Belvédère 1002, Tunis, Tunisia e-mail: [email protected] Z. Jlassi e-mail: [email protected] K. Ben Kilani e-mail: [email protected] L. Mili Department of Electrical and Computer Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, USA e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_13

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1 Introduction Unit coherency of power system generators is a property describing similarity in their dynamic responses following disturbances. Synchronous generators are coherent if they swing together in phase and frequency with constant angular difference over a given time interval, and within a specific tolerance. Coherency importance emanates from its dynamic equivalency implications. Its applications range from model aggregation needed in off line system security assessment and control design, to on line security assessment, and emergency control. In recent works, the coherency concept has been utilized to develop suitable corrective actions for intentional power grid islanding [1–4]. One important aspect in coherency identification is the set of disturbances considered and the identification technique. In the literature, many generator coherency determination techniques have been proposed and advanced techniques are still being developed. In [2, 5–8], the authors used data of generators swing curves or rotor angle evolution with time following the disturbance. From these time responses, coherent generators can be identified using a clustering algorithm with specific coherency tolerance value. This technique is able to capture large signal dynamics, but is computationally burdensome and its performance become poor as the power system size increases. The large number handled of generators makes the correspondence between modes and machines invisible and hard to study. In [9], linear time simulation has been used to identify the coherent machines groups. Linear time simulations are advantageous over nonlinear in their time frames. In [10, 11] the authors used the slow coherency method to determine coherent generators groups of a linearized model by selecting the slowest modes. The disadvantage of this method is the linear approximation of modes and the need of power system modeling. Large disturbance excite various high nonlinear low frequency oscillations and would pull the operating point of the power system far away from its stable equilibrium [12]. Post disturbance oscillations are a complex phenomenon involving all power system generators which may swing in phase or oscillate against each other. Loss of synchronism may result among machines in very few seconds following large disturbances. Power system stability is affected, power transfers may be limited and potentially widespread blackouts can be developed. Hence, the linear approximation may be invalid for large disturbances and not effective for emergency control and system partitioning under severe disturbances. Recently, a mathematical method of decomposing nonlinear dynamics into modes called Koopman Modes Analysis (KMA) has gained an increased attention as recent technique applied to electric power systems [4, 13, 14]. This method keeps all system nonlinear dynamics and doesn’t employ explicit system analytical models. The Koopman operator captures the highly nonlinear spatiotemporal dynamics that cannot be assessed with standard linear coherency identification techniques. The presented work aims at conducting an effects analysis of large disturbances on generator coherency. Generator coherency is identified via nonlinear Koopman modal

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analysis technique applied on generators rotor speed dynamics. Faults are shortcircuits of variable durations and locations. System operating parameters include loading levels and on line generators. The study is demonstrated on the Tunisian electric network with a comparison to the conventional slow coherency method.

2 Koopman Modes Analysis 2.1 Koopman Modes Definition For a given certain measurement, a Koopman modes set is defined. Consider the dynamical discrete-time system evolving on a given finite-dimensional manifold M: xk+1 = f (xk )

(1)

where k is an integer index and f is a map from the manifold M to itself. The linear Koopman operator U acts on scalar-valued functions evolving on M as follow: for a scalar-valued function g : M → R, the operator U maps g into a novel function Ug given by: U g(x) = g( f (x))

(2)

The idea is the analysis of nonlinear dynamics governed by (1) via the eigenfunctions and the eigenvalues of the operator U. from available data collected either experimentally or numerically. To this end, let λ j ∈ C denote the Koopman operator eigenvalues and ϕ j : M → R denote its eigenfunctions, U ϕ j (x) = λ j ϕ j (x) j = 1, 2, . . .

(3)

Consider x ∈ M and a vector-valued observable g : M → R p . For instance, if x contains the full information at a particular time about a flow field, g(x) denote a vector of any interest quantities, such as generator frequency measurements. If all the p components of g lie within the Koopman eigenfunctions span, then the vector-valued g is expanded in terms of these eigenfunctions as: g(x) =

∞ 

ϕ j (x)v j

(4)

j=1

We typically think of (4) as expanding g(x) as a linear combination of Koopman vectors v j . But, alternatively we may think of (4) as expanding g(x) as a linear combination of Koopman operator eigenfunctions ϕ j , where v j are the vector coefficients in this expansion. We will refer in this paper to the eigenfunctions ϕ j as the

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Koopman eigenfunctions and the corresponding v j vectors in (4) as Koopman modes of f , corresponding to the study observable g. Each Koopman eigenvalue λ j characterizes the temporal behavior of each corresponding Koopman mode v j : the magnitude of the eigenvalue determines its growth rate and the phase determines its frequency.

2.2 Koopman Modes Computation from Snapshots The commonly used Arnoldi algorithm actually produces approximations to the Koopman operator eigenvalues and their corresponding modes [4, 13, 14]. The Ritz values λ˜ j and Ritz vectors v˜ j behave in exactly the same manner as the Koopman operator eigenvalues λ j and modes v j . Consider the sequence {g(x0 ), . . . , g(xm−1 )} where g(x j ) is a vector measuring the observables at a certain time t j . The empirical Ritz vectors v˜ j and values λ˜ j of this sequence are defined by this following algorithm: (i)

Define K by:   K = g(x0 )g(x1 )g(x2 ) . . . g(xm−1 )

(5)

(ii) Find the constants c j where a residual r is defined as: r = g(xm ) −

m−1 

  c j g x j = g(xm ) − K C,

j=0

r ⊥span{g(x0 ), . . . , g(xm−1 )}

(6)

(iii) Define the companion matrix by: ⎡

0 0 ... ⎢1 0 ... ⎢ ⎢ C = ⎢0 1 ⎢. ⎣ ..

0 0 0 .. .

c0 c1 c2 .. .

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(7)

0 0 . . . 1 cm−1 (iv) Find the companion matrix eigenvalues: C = T −1 T,  = diag(λ˜ 1 , . . . , λ˜ m ) where T is the Vandermonde matrix defined as:

(8)

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⎡

λ˜ 1 λ˜ 2 λ˜ 3 .. . 1 λ˜ m

1 ⎢1 ⎢ ⎢ T = ⎢1 ⎢. ⎣ ..

λ˜ 21 λ˜ 22 λ˜ 23 .. . λ˜ 2m

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⎤ λ˜ m−1 1 ⎥ λ˜ m−1 2 ⎥ ⎥ ˜λm−1 3 ⎥ ⎥ .. ⎦ . m−1 . . . λ˜ m ... ... ... .. .

(9)

(v) Define the Ritz vectors v˜ j to be the columns of: V = K T −1

(10)

The eigenvalues of the defined companion matrix C are approximations to Koopman eigenvalues, called Ritz values. The corresponding eigenvectors given by (10), called Ritz vectors, are approximations to Koopman modes. The Koopman mode growth rate «GR» assesses the sampled dynamics damping. It is defined for a given mode j as the modulus of the Koopman eigenvalue λ˜ j given by: GR = λ˜ j

(11)

Largest growth rates of Koopman modes denote its smallest damping ratios [4, 13, 14]. The frequency of each Koopman mode is computed by (12), where f s is the chosen sampling frequency of sampled observables data: 

 f K M = I m ln λ˜ j ∗ f s /2π

(12)

The norm of a Koopman Mode j defined by (13) quantifies its contribution magnitude in the dynamics: v˜ j =



v˜ Tj v˜ j

(13)

3 Coherency in Power Systems Power system generators coherency is a property describing similarity in their dynamic responses following disturbances. Two synchronous machines are considered to be coherent, if they swing together in phase and frequency with constant angular difference over a given time interval, and within a specified tolerance [4]. The coherency condition in general simplifies to: δi (t) − δj (t) ≤ constant

(14)

where δi (t) is the i th generator rotor angle and δj (t) is the j th generator rotor angle.

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3.1 Conventional Slow Coherency Method Slow coherency is coherency arising from the slower inter-area modes, which are oscillatory modes due to groups of machines oscillating against each other across power transfer interfaces. From the slow coherency theory, the eigen-subspace of the inter-area modes can be used to identify the slow coherent machines [10]. The weak coupling and the singular perturbation theories may be used to separate variables into slow and fast ones [11]. The slow variables are the interarea oscillatory modes variables. The local modes of oscillations within the areas described the fast variables. Generators coherency identification based on this linear technique is independent of applied disturbance.

3.2 Coherency in the Koopman Modes The concept of coherency in the context of Koopman modes has been defined in various works [4, 13, 14]. This concept is applied on the generators dynamic responses following arbitrary disturbances. For a given oscillatory KM v j , or Mode j, to identify coherent generators swing dynamics in frequency and phase, it is quite sufficient to check the initial phases α ji of mode j defined by (15) [4, 14]. A set of oscillatory observables I ⊆ [1 . . . p] is a coherent group with respect to Mode j, if all i ∈ I have similar initial phases behaviors [4].     tan α ji = I m ϕ j (x0 )v j i /Re ϕ j (x0 )v j i

(15)

  The notation Re ϕ j (x0 )v j i denotes the i th component of the vector real part of ϕ j (x0 )v j . This vector stands for initial amplitudes of the modal dynamics. The imaginary part vector I m[ϕ j (x0 )v j ] affects the initial phases of the modal dynamics.

4 Study Case 4.1 System Description The proposed technique is conducted on the Tunisian power system comprising 153-bus 26-machines for load peaks and average load cases and comprising 137-bus 10-machines for load dips case. Synchronous generators are equipped with automatic voltage and speed regulators. They are geographically grouped into three main groups: North, Center and South. The North contains 12 generators for load peaks and average load cases and 7 generators for load dips case.

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Table 1 Study cases data Loading level

Disturbance Location/Duration (ms) North

Center

South

50

150

250

50

150

250

50

250

250

Load peaks

x

x

x

x

x

x

x

x

x

Average load

x

x

x

x

x

x

x

x

x

Load dips

x

x

x

x

x

x

x

x

x

The Center contains 7 machines for load peaks and average load cases and only two machines for load dips case. The South contains 7 generators for load peaks and average load cases and only one generator for load dips case.

4.2 Study Cases The KM analysis is performed on sampled generators angular speeds dynamics following a short-circuit fault. Generators coherency identification based on Koopman modes is performed and compared to the conventional slow coherency method. Three loading levels are studied: an average load (3000 MW), a load peaks (4200 MW) and a load dips (1600 MW). For each loading level, three fault durations (50 ms, 150 ms and 250 ms) and three locations of applied fault are tested: in the North (in RadesII bus), in the Center (in SousseC bus) and in the South (in Gannouch bus). The different study cases are defined in Table 1. The simulations are obtained using the software PSS/E: Power System Simulator for Engineering.

4.3 Simulation Results Table 2 summarizes the slow modes and the KM analysis results for three dominant Koopman modes. For all fault parameters and system loading levels, the modes are oscillatory with low frequencies, corresponding to inter-are modes. For the standard linear modal analysis of the system, one inter-area oscillatory mode is indicated, independently of the fault. Intrinsic linear modes are captured by the KMA method and nonlinear intrinsic modes of the power system are identified as shown in Table 2.

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Table 2 Frequencies results of Koopman and linear modes for the different study cases Fault parameter

Interarea mode frequency (Hz)

Loading level

Location

Average

North

Center

South

Load peaks

North

Center

South

Load dips

North

Center

South

Duration (ms)

Koopman mode

Linear mode 0.706

50

0.703-0.755-0.649

150

0.698-0.749-0.647

250

0.695-0.745-0.640

50

0.640-0.727-0.671

150

0.767-0.728-0.668

250

0.772-0.721-0.658

50

0.686-0.694-0.766

150

0.709-0.655-0.767

250

0.710-0.656-0.767

50

0.726-0.747-0.652

150

0.728-0.760-0.659

250

0.723-0.765-0.663

50

0.776-0.730-0.660

150

0.773-0.733-0.661

250

0.764-0.737-0.664

50

0.728-0.763-0.659

150

0.718-0.751-0.651

250

0.718-0.758-0.590

50

0.827-0.771-0.721

150

0.830-0.774-0.721

250

0.831-0.777-0.723

50

0.830-0.776-0.725

150

0.835-0.782-0.728

250

0.838-0.785-0.732

50

0.828-0.776-0.725

150

0.829-0.776-0.724

250

0.833-0.779-0.726

0.706

0.706

0.726

0.726

0.726

0.780

0.780

0.780

5 Faults Effects Analysis Based on phase-coherency results in nonlinear Koopman modes, coherent generators groups are defined by applying K-means clustering algorithm. Combining the grouping results of dominant Koopman interarea modes, a fine partition into coherent machines of a power system can be determined. Coherency is identified for all study cases. The effect of each case parameter is studied.

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5.1 Fault Duration For the average load case, the simulation results of generators angular speeds following northern short-circuit with various durations (50, 150 and 250 ms) is depicted in Fig. 1. For each fault duration case, northern generators have coherent dynamics, central generators have other coherent dynamics and southern generators have another coherent dynamics as shown in Fig. 1. The three coherent motions are depicted by combining the coherent generators partitions for dominant Koopman modes given in Fig. 2.

Fig. 1 Generators responses for average load case following northern fault with various durations. a Northern generators. b Central generators. c Southern generators

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(a) Mode of frequency 0.755 Hz

Mode of frequency 0.703 Hz

Mode of frequency 0.649 Hz

4

4

4 North

South

-4

0

0.2

0.4

0.6

0.8

Center

0

-4 0.2

1

South

0.4

0.6

0.8

Phase (rad)

0

North

Phase (rad)

Phase (rad)

North Center

Center

0 South

-4

1

0

0.2

0.4

0.6

0.8

1

Amplitude

Amplitude

Amplitude

(b) Mode of frequency 0.647 Hz

Mode of frequency 0.749 Hz

Mode of frequency 0.698 Hz

4

4

4

North

North

0 South

-4

0

0.2

0.4

0.6

0.8

Center

0

South

-4 0.2

1

0.4

0.6

0.8

Phase (rad)

Center

Phase (rad)

Phase (rad)

North

Center

0 South

-4

1

0

0.2

Amplitude

Amplitude

0.4

0.6

0.8

1

Amplitude

(c) Mode of frequency 0.640 Hz

Mode of frequency 0.745 Hz

Mode of frequency 0.695 Hz

4

4

4

North North

South

-4

0

0.2

0.4

0.6

0.8

1

Amplitude

Center

0

-4

South

0

0.2

0.4

0.6

Amplitude

0.8

1

Phase (rad)

Center

0

Phase (rad)

Phase (rad)

North

Center

0

-4

South

0

0.2

0.4

0.6

0.8

1

Amplitude

Fig. 2 Coherency in Koopman modes for average load case following northern fault with various durations. a Duration of 50 ms. b Duration of 150 ms. c Duration of 250 ms

5.2 Fault Location For each loading level, the effects of short-circuit location on generators coherency is analysed. Results led to the same conclusion that depending on fault location, the power system may exhibit distinct generator coherent groups. For example, for load peaks study case, a fault of duration 150 ms is studied for three cases of locations: in the north, in the center and in the south. The coherency results in dominant Koopman modes for this study case are given in Fig. 3. For northern fault location, the dominant Koopman mode of frequency 0.830 Hz captures three coherent groups: northern generators group, central generators group and southern generators group.

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(a) Mode of frequency 0.830 Hz

Mode of frequency 0.721 Hz

Mode of frequency 0.774 Hz

4

4

4

0 South

Center

0 South

North

Phase (rad)

Center

North

Phase (rad)

Phase (rad)

North

Center

0

South

-4

-4

-4 0

0.2

0.4

0.6

1

0.8

0

0.2

Amplitude

0.4

0.6

1

0.8

0

0.2

Amplitude

0.4

0.6

0.8

1

Amplitude

(b) Mode of frequency 0.835 Hz

Mode of frequency 0.782 Hz

Mode of frequency 0.728 Hz

4

4

4

0 South

Center

0 South

Phase (rad)

Center

North

Phase (rad)

Phase (rad)

North

North

Center

0

South

-4

0

0.2

0.4

0.6

0.8

-4

1

0

0.2

Amplitude

0.4

0.6

0.8

-4

1

0

0.2

Amplitude

0.4

0.6

0.8

1

Amplitude

(c) Mode of frequency 0.776 Hz

Mode of frequency 0.829 Hz

Mode of frequency 0.724 Hz

4

4

4

0 South

Phase (rad)

Phase (rad)

North

Center

Center

0 South

Phase (rad)

North

North

Center

0

South

-4

0

0.2

0.4

0.6

Amplitude

0.8

1

-4

-4 0

0.2

0.4

0.6

Amplitude

0.8

1

0

0.2

0.4

0.6

0.8

1

Amplitude

Fig. 3 Coherency in Koopman modes for load dips case following a fault of duration 150 ms with various locations. a In the North. b In the Center. c In the South

For the Koopman modes of frequencies 0.774 Hz and 0.721 Hz, two coherent groups are captured as depicted in Fig. 3. The northern generators form a first group, the central and southern generators form a second group. By combining the coherent generators partitions for all dominant Koopman modes, a partition of the Tunisian test system with three coherent groups is obtained: North, Center and South. The same partition is captured by dominant Koopman modes for central fault location as given in Fig. 3. For southern fault location, the three dominant Koopman modes capture two coherent motions as shown in Fig. 3. A first coherent motion for southern generators is captured, a second coherent motion for northern and central generators

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is captured. Thus, for this test fault location, the power system is decomposed into two coherent zones instead of three zones.

5.3 System Loading For the average load case, following a central fault of duration 250 ms, the coherency in identified dominant Koopman modes are given in Fig. 4. For mode of frequency

(a) Mode of frequency 0.658 Hz

Mode of frequency 0.721 Hz

North

4

4 North

Phase (rad)

Phase (rad)

Center

Southern-

0

Eastern

-4

Southern-

0

0.5

Center

0 South

North

Phase (rad)

Mode of frequency 0.772 Hz

4

-4

1

0.2

0

0.4

0.6

0.8

South

-4

1

Center

0

0

0.2

0.4

0.6

0.8

1

Western

Amplitude

Amplitude

Amplitude

(b) Mode of frequency 0.737 Hz

Mode of frequency 0.764 Hz

Mode of frequency 0.664 Hz

4

4

4 North

North

South

Center

0 South

Phase (rad)

0

Phase (rad)

Phase (rad)

North

Center

-4

-4 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

South

-4

1

Center

0

0

0.2

0.4

0.6

0.8

1

Amplitude

Amplitude

Amplitude

(c) Mode of frequency 0.838 Hz

Mode of frequency 0.732 Hz

Mode of frequency 0.785 Hz

4

4

4

0 South

-4

0

0.2

0.4

0.6

Amplitude

0.8

1

North

Center

0 South

-4 0

0.2

0.4

0.6

Amplitude

0.8

1

North

Phase (rad)

Center

Phase (rad)

Phase (rad)

North

Center

0

South

-4

0

0.2

0.4

0.6

0.8

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Fig. 4 Coherency in Koopman modes following a central fault with duration 250 ms for various system loading levels. a Average load. b Load peaks. c Load dips

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0.772 Hz, two coherent motions are captured. Northern and southern-western generators show a first coherent motion. Central and southern-eastern generators show a second coherent motion. The mode of frequency 0.721 Hz captures other two coherent groups: southern and central generators group and northern generators groups. The mode of frequency 0.658 Hz captures three coherent groups: north, center and south. Combining the grouping results of these three Koopman modes, a partition into four coherent zones is determined: northern generators group, central generators group, southern-eastern generators group and southern-western generators group. Nonetheless, following the same applied central fault of duration 250 ms, for load dips and load peaks cases, a partition into three coherent zones is captured by combining the grouping results of dominant Koopman modes given in Fig. 4. The two Koopman modes of frequencies (0.737 Hz and 0.664 Hz), (0.785 Hz and 0.732 Hz) for load peaks case and load dips case respectively, capture two coherent groups: northern and central generators group and southern generators group. The Koopman mode of frequency 0.838 Hz for load dips case captures three coherent groups: north, center and south. Southern and central generators group and northern generators group are captured by mode of frequency 0.764 Hz for load peaks case. Thus depending on loading level, different coherent dynamics may be captured.

6 Coherency Identification Comparison Based on linear modal analysis of the study Tunisian power system, generators slow coherency for each loading level case is identified. Figure 5 defines the mode shapes for each case. The results show that for any system operating condition, the northern and central generators show coherent dynamics and the southern generators have other coherent dynamics. Therefore, a partition with two disjoint parts (north and south) is defined based on slow coherency method as depicted in Fig. 6. But based on Koopman modes analysis, depending on fault location and power system loading level, the Tunisian power grid may be decomposed into three or four coherent machines zones instead of two topological zones as shown in Fig. 6. The northern slow coherent generators zone is decomposed into two coherent groups for load peaks case and average load case under any applied fault in the north, in the center or in the south. The southern slow coherent generators zone is decomposed into two coherent groups (southern-eastern and southern-western) for average load case under central fault. Only, for load dips case following a southern short-circuit fault, the two topological slow coherent zones (northern and southern) are not decomposed into other coherent groups. Thus, coherency property based on linear and topological approximations may be invalid under large disturbances. Several generators may switch from a group to another. A slow coherent area may degenerate into smaller coherent zones.

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Average load (0.706 Hz) 120

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0.4 0.2

0

180

North Center South

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210 240 270

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Load dips (0.780 Hz) Fig. 5 Coherency in the Tunisian slow mode for various loading levels

7 Conclusion This paper proposes a comprehensive parametric study combining fault parameters with power system loading levels to analyze their effects on generators coherency. Faults are short-circuits of variable durations and locations. Coherency is identified via the nonlinear Koopman modes analysis method to capture the highly nonlinear spatiotemporal dynamics that cannot be assessed with standard linear coherency identification techniques. The nonlinear modal analysis is applied on generators rotor speed dynamics following large disturbances. The study is demonstrated on a large scale industrial electrical network: the Tunisian power system. A comparison of the nonlinear KMA based coherency identification method is carried out with the conventional slow coherency method. The results have shown that the tested power system has exhibited distinct generator coherent groups, emerging under different severe system faults depending on their location and the power system loading level. Predetermined Tunisian slow coherent areas based on weak connections topology and linear approximations may be decomposed into two coherent groups under large

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North

Center

SouthernWestern

SouthernEastern

South Legend Gas Turbine generator Thermal Turbine generator Combined Cycle Turbine generator

Groups based on slow coherency & KMA for load dips case following southern fault. Groups based on KMA for load peaks case following northern, central and southern faults & KMA for load dips case following northern and central faults & KMA for average load case following northern and southern faults. Groups based on KMA for average load case following central fault. Fig. 6 Coherent generators groups of the Tunisian power system under study cases

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disturbances. Such information is particularly useful for emergency control where intentional islanding actions ought to be taken to mitigate large blackouts.

References 1. Lin ZZ, Wen FS, Zhao JH, Xue Y (2016) Controlled islanding schemes for interconnected power systems based on coherent generator group identification and wide-area measurements. J. Mod. Power Syst Clean Energy 4(3):440–453 2. Koochi MHR, Esmaeili S, Dehghanian P (2018) Coherency detection and network partitioning supported by wide area measurement system. In: 2018 IEEE Texas power and energy conference (TPEC), College Station, TX, USA, IEEE, Mar 2018 3. Ab Salam AN, Hasmaini M, Dahlan NY, Raza S (2017) Performance of multiple passive islanding detection technique for synchronous type of DG. J Electric Syst 13(3):568–578 4. Raak F, Susuki Y, Hikihara T (2015) Data-driven partitioning of power networks via nonlinear Koopman mode analysis. IEEE Trans Power Syst 31(4):2799–2808 5. Khalil AM, Iravani R (2016) A dynamic coherency identification method based on frequency deviation signals. IEEE Trans Power Syst 31(3):1779–1787 6. Yang S, Zhang B, Hojo M (2018) A dynamic generator coherency identification method based on phase trajectory vector. In: 2018 IEEE innovative smart grid technologies-Asia (ISGT Asia), Singapore, IEEE, Sept 2018 7. Lin Z, Wen F, Ding Y, Xue Y (2018) Data-driven coherency identification for generators based on spectral clustering. IEEE Trans Ind Inf 14(3):1275–1285 8. Ul Banna H, Iqbal T, Khan A, Zahra Z (2018) Generators coherency identification using relative correlation based clustering. In: 2018 International conference on engineering and emerging technologies (ICEET), Lahore, Pakistan, IEEE, Apr 2018 9. Kyriakidis T, Cherkaoui R, Kayal M (2013) Generator coherency identification algorithm using modal and time-domain information. In: EUROCON, Zagreb, Croatia, IEEE 10. Verdejo H, Montes G, Olgui X (2014) Identification of coherent machines using modal analysis for the reduction of multimachine systems. Latin Am Trans IEEE 12(3):416–422 11. Stadler J, Renner H, Köck K (2015) An inter-area oscillation based approach for coherency identification in power systems. In: 2014 power systems computation conference, Wroclaw, Poland, IEEE, Feb 2015 12. Zaid MM, Malik MU, Bhatti MS, Razzaq H, Aslam MU (2017) Detection and classification of short and long duration disturbances in power system. J Electric Syst 13(4):779–789 13. Susuki Y, Mezic I, Raak F, Hikihara T (2016) Applied Koopman operator theory for power systems technology. Nonlinear Theory Appl IEICE 7(4):430–459 14. Susuki Y, Mezic I (2011) Nonlinear Koopman modes and coherency identification of coupled swing dynamics. IEEE Trans Power Syst 26(4):1894–1904

Analysis and Experimental Validation of Single Phase Series Resonance Inverter A 3–7 KHz Full-Bridge Inverter Alla Eddine Toubal Maamar, M’hamed Helaimi, and Rachid Taleb

Abstract In this paper, analysis and modelling of a single-phase cascaded fullbridge resonance inverter are considered. The control of proposed inverter by full-wave phase method at high frequency is implemented. In the first step, MATLAB/Simulink environments are used to simulate the model and show obtained results of waveforms. The second step is the experimental validation of simulation results by a realization of the inverter and generated signals. The Root-Mean-Squared voltage of a capacitor waveform between 3 and 7 kHz frequency with simulation and realization are presented, comparatively, for a comparison. The results obtained are satisfactory and show the efficiencies of analysis and the proposed model of the full-bridge resonance inverter. Keywords Power converters · Cascaded full bridge · High-frequency control · Resonant inverters · MATLAB/Simulink

1 Introduction In recent years, cascaded full-bridge inverter have received more attention and have drawn great research interest for several applications, some examples include high power induction motor, resonant DC-AC converters, microgrid with solar panels and storage batteries [1–3]. Resonance DC-AC inverter is a type of electric power converter, which contains a resistor (R), inductance (L) and capacitor (C), there has been a great increase in the use of this converter in a number of applications, some examples include: induction heating systems, radio transmitter, heat treatment of metals in industrial processes by melting and welding [3, 4].

A. E. Toubal Maamar (B) · M. Helaimi · R. Taleb Laboratoire Génie Electrique et Energies Renouvelables (LGEER), Electrical Engineering Department, Hassiba Benbouali University of Chlef, BP. 78C, 02180 Ouled Fares, Chlef, Algeria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_14

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A lot of resonant converters have been developed, as (SRI) Series Resonant Inverter, (PRI) Parallel Resonant Inverter, (ZVS) Zero Voltage Switching Resonant Converter, (ZCS) Zero Current Switching Resonant Converter, Two Quadrant ZVS Resonant Converter, (CERC) Class E Resonant Converter, Resonant DC-Link Inverter. The series resonant inverter is being more popular because it simplified topology, operation and control [5]. Many projects used the Arduino technologies for their advantages, as the design of digital filter, power measurement, control of motor for robotic systems [6–8]. In this work, the control of electronic switches is ensured by Arduino Mega 2560 Microcontroller, and we interest to control the output power by varying the frequency with a fixed duty cycle. The paper is organized as follows. In Sect. 2, Analysis of the inverter operation and simulation of a series resonant inverter are discussed. Some experimental results using a prototype are given in Sect. 3. Results comparison, simulation and experimental, are presented in Sect. 4. Finally, conclusions and future trends of work are presented in Sect. 5.

2 Analysis and Simulation of a Series Resonant Inverter A. Analysis of the Inverter Operation A full-bridge inverter has two arms, and each arm with two electronic switches, with series RLC Load, Resistor, Inductor, and Capacitor, the inverter powered by a DC power supply, the topology shown in Fig. 1. The operation of the switches is periodic with adjustable period T, and the switching devices must be controlled in a complementary mode to avoid the shortcircuiting of the DC power supply, a time delay is inserted between the signals to avoid short-circuit, so depending on the states of the switches S1, S2, S3, S4, four operating sequences of the full-bridge inverter can be distinguished during a switching period, and we work with two sequences, positive and negative. Figures 2, 3, 4, and 5, shows the operation sequences of the inverter. Fig. 1 The structure of a full-bridge series resonance inverter

UE

+ -

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CONTROL

R

L

C UC

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(a) Sequence 1: The switch S1, S3 was closed and switch S2, S4 was opened, the output voltage URLC = 0. (b) Sequence 2: The switch S1, S4 was closed and switch S2, S3 was opened, the output voltage URLC = +UE . (c) Sequence 3: The switch S1, S3 was opened and switch S2, S4 was closed, the output voltage URLC = 0. (d) Sequence 4: The switch S1, S4 was opened and switch S2, S3 was closed, the output voltage URLC = −UE B. Simulation of a Series Resonant Inverter Simulation of the electrical circuit of the system is done in MATLAB environment (SIM/POWER/SYSTEMS). The simulated circuit is a MOSFET based RLC series Fig. 2 Sequence 1 of the inverter operation

UE

+ -

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Fig. 3 Sequence 2 of the inverter operation

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Fig. 4 Sequence 3 of the inverter operation

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resonance inverter with four switches. Figures 6 and 7, shows the model of inverter in MATLAB environement and the control block. To have a similar model to a real system, and identification, we have measured the electronic components values, resistor, inductor, and capacitor. RRESISTOR = 1004 [Ohm]; RINDUCTOR = 200 [Ohm]; RTOTAL = RRESISTOR + RINDUCTOR = 1204 [Ohm]; L = 310e−3 [H]; C = 3.13e−9 [F], and use these values in simulation. Figures 8, 9, 10, 11, and 12, shows the simulation results, a permanent and temporary regime is obviously observed, a temporary regime caused by the non-linearity of electronic components, saturation and hysteresis. From the figures, we can see that Fig. 10 is a very special one, the wavelength is increasing and regular because the control frequency is near to the circuit resonance frequency. The resonance frequency of a circuit is determined by the following relationship:

Fig. 6 The Resonance inverter using MATLAB

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Fig. 7 The control block of resonance inverter Fig. 8 The output waveforms, MATLAB, with f = 3 kHz, D = 50%

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Fig. 9 The output waveforms, MATLAB, with f = 4 kHz, D = 50%

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Fig. 10 The output waveforms, MATLAB, with f = 5 kHz, D = 50%

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f0 =

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x 10

-3

1 √ 2π LC

by numerical application with the values previously declared, RTOTAL = 1204 []; L = 310e−3 [H]; C = 3.13e−9 [F], the resonance frequency f0 = 5.1 kHz. f0 =

1 = 5.1120e+03 √ 2π 310e−3 × 3.13e−9

Analysis and Experimental Validation … Fig. 11 The output waveforms, MATLAB, with f = 6 kHz, D = 50%

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Fig. 12 The output waveforms, MATLAB, with f = 7 kHz, D = 50%

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3 Experimental Results The experimental prototype of the system is shown in Fig. 13. The full-bridge inverter consists of four MOSFET switches IRFZ44N. The MOSFET drivers use TLP 250 photocouplers. The load consists of a resistor, inductor and capacitor in series. The generalized control signals have been implemented using the programmable board (ARDUINO Mega 2560 Microcontroller) and PC with open source software (Arduino IDE). Figures 14, 15, 16, 17 and 18, shows the output voltages of RLC Load and the capacitor voltage, with different frequency from 3 to 7 [KHz] and fixed duty cycle D = 50%.

210 Fig. 13 The Experimental prototype of a series resonant inverter. A: Pc + ARDUINO IDE software. B: Inverter (driver circuits + MOSFET switches) + ARDUINO Mega. C: Series RLC (Resistor, Inductor, Capacitor) circuit. D: Digital oscilloscope

Fig. 14 The output waveforms, with f = 3 kHz, D = 50%

Fig. 15 The output waveforms, with f = 4 kHz, D = 50%

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Analysis and Experimental Validation … Fig. 16 The output waveforms, with f = 5 kHz, D = 50%

Fig. 17 The output waveforms, with f = 6 kHz, D = 50%

Fig. 18 The output waveforms, with f = 7 kHz, D = 50%

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The first step is the realization of the Experimental prototype, and next step is the implementation of the digital control through ARDUINO Mega 2560 and calculates the RMS values of capacitor voltage with the variation of frequency. To evaluate the obtained results, the comparison is necessary, and this is what we will present in the following section.

4 Comparison and Discussions After the implementation of the resonance inverter and calculates the RMS value with the variation of frequency, the collected data is shown in the following Table 1. From presented data, two curves (simulation with MATLAB and experimental) are generated, shown in Fig. 19. From these two comparison curves, it is observed that the peak value of the curves indicates the resonance frequency, also there is a shift between the two curves (simulation and experimental) because of the perturbations of electrical equipment and electronic components, but generally the obtained results show the good concordance existing between the simulation model and the real system, and it is important to note that the output power of the resonant inverter can be controlled by adjusting the frequency of the switches or adjusting the duty cycle (D), some testing results with variable duty cycle are presented in Table 2. Figures 20, 21 and 22, shows the obtained results with variable frequency and duty cycle.

5 Conclusion The simulation and realization of a single-phase cascaded Full-bridge resonance inverter are discussed in this paper. The effectiveness of the system is verified by obtained results, simulation and experimental. The resonance is a very attractive physical phenomenon used in many industrial applications, with the periodic control there is energy exchange between the inductor and capacitor, if the inductive reactance equals that of the capacitive reactance (XL = XC), then a physical phenomenon obtained, with a generation of higher output voltages. This work opens new ways for future research with other topologies and other control cards. For an effective control system for the resonant inverter, the uses of closed-loop circuit to adjusting the duty cycle and frequency is very important.

Analysis and Experimental Validation … Table 1 RMS voltage in capacitor, simulation and reaisation

213

Frequency [KHz]

UCRMS [V] experimental

UCRMS [V] MATLAB

3

13.563

13.74

3.1

14.033

14.22

3.2

14.535

14.76

3.3

15.072

15.36

3.4

15.712

16.04

3.5

16.425

16.81

3.6

17.242

17.67

3.7

18.153

18.66

3.8

19.195

19.8

3.9

20.42

21.11

4

21.810

22.64

4.1

23.56

24.45

4.2

25.56

26.6

4.3

28.27

29.19

4.4

31.17

32.35

4.5

34.92

36.26

4.6

39.84

40.13

4.7

45.29

47.21

4.8

52.24

53.59

4.9

58.92

62.8

5

64.01

69.89

5.1

64.87

72.41

5.2

61.27

68.42

5.3

54.88

60.25

5.4

48.09

51.41

5.5

41.48

43.66

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36.26

37.37

5.7

31.75

33.38

5.8

27.92

28.36

5.9

24.96

25.14

6

22.440

22.5

6.1

20.06

20.31

6.2

18.554

18.47

6.3

16.887

16.91

6.4

15.630

15.57

6.5

14.496

14.41 (continued)

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UCRMS [V] experimental

UCRMS [V] MATLAB

6.6

13.453

13.39

6.7

12.522

12.5

6.8

11.740

11.71

6.9

10.977

11

7

10.347

10.37

Fig. 19 The comparison of resonance frequency results, MATLAB and experimental

80 Experimental MATLAB

70

RMS Voltage [v]

60 50 40 30 20 10 0

2

3

4

5

6

7

Frequency [KHz]

Table 2 RMS voltage with variable duty cycle

Frequency [KHz]

D = 25%

D = 50%

D = 75%

5.0

UCRMS = 45.37 [v]

UCRMS = 64.01 [v]

UCRMS = 45.22 [v]

5.1

UCRMS = 46.27 [v]

UCRMS = 64.87 [v]

UCRMS = 46.13 [v]

5.2

UCRMS = 43.56 [v]

UCRMS = 61.27 [v]

UCRMS = 43.42 [v]

8

Analysis and Experimental Validation … Fig. 20 The output waveforms, with f = 5.1 kHz, D = 25%

Fig. 21 The output waveforms, with f = 5.1 kHz, D = 50%

Fig. 22 The output waveforms, with f = 5.1 kHz, D = 75%

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References 1. Taleb R, Benyoucef D, Helaimi M, Boudjemaa Z, Saidi H (2015) Cascaded H-bridge asymmetrical seven-level inverter using THIPWM for high power induction motor. Energy Procedia 74:844–853 2. Helaimi M, Benghanem M, Beldamani B (2012) An improved PIλ controller for resonant inverter induction heating systems under load and line variations. Stud Inform Control 21(4):423–430 3. Abdar HM, Chakraverty A, Moore DH, Murray JM, Loparo KA (2012) Design and implementation a specific grid-tie inverter for an agent-based microgrid. In: 2012 IEEE energytech conference 4. Feng W, Mattavelli P, Lee FC (2013) Pulsewidth locked loop (PWLL) for automatic resonant frequency tracking in LLC DC–DC transformer (LLC-DCX). IEEE Trans Power Electron 28(4):1862–1869 5. Kazimierczuk MK, Czarkowski D (2011) Resonant power converters. Wiley, New York 6. Toubal Maamar AE, Helaimi M, Taleb R, Chabni F (2018) Analysis and implementation of half-bridge series resonant inverter using Arduino. In: ICCEE’18 international conference on communications and electrical engineering, El-Oued, Algeria 7. Adel Z, Hamou AA, Abdellatif S (2018) Design of real-time PID tracking controller using Arduino Mega 2560 for a permanent magnet DC motor under real disturbances. In: 2018 international conference on electrical sciences and technologies in Maghreb (CISTEM), Algiers, pp 1–5 8. Fannakh M, Elhafyani ML, Zouggar S (2019) Hardware implementation of the fuzzy logic MPPT in an Arduino card using a Simulink support package for PV application. IET Renew Power Gener 13(3):510–518

Optimal Decentralized State Control of Multi-machine Power System Based on Loop Multi-overlapping Decomposition Strategy M. Z. Doghmane and M. Kidouche

Abstract Most of large scale systems are described by their complex mathematical model’s structures which caused by the overlapping between its inputs and outputs variables. In many cases, the complexity of the model may lead to lose some part of the input-output signals. Besides, it obscures study and analysis of such type of systems for researchers. Many studies have been discussed the design of simpler controller based on the mathematical structure of the systems’ model, wherein the aim was to decrease control complexity. The main objective of this manuscript is to benefit from the multi-overlapping structure of complex systems in order to design a simpler controller using loop approach; the latter allows highlighting the advantages of applying the overlapping decomposition strategy in order to improve the robustness of the designed controller. Therefore, it can be considered as an efficient strategy suitable for such type of large scale systems. A multi-machine power system with loop structure is considered in order to demonstrate the usefulness of the proposed design. Keywords Large scale systems · Multi-overlapping decompositin · Loop approach · Multi-machine power system

1 Introduction The study of large scale systems is one of the biggest challenges facing researchers in the last few years, the mains reason for that is the complexity that exist in the mathematical model and also in its analysis. Many studies have been conducted in order to overcome such kind of difficulties [1–5] (Bochen and Stankovic [6]; Zecvic and Siljack [7]); these researchers have developed decomposition technique named “Overlapping” in order to decouple the original system into subsystems with no M. Z. Doghmane (B) · M. Kidouche Laboratory of Applied Automatic (LAA), Department of Automation, University M’hamed Bougara of Boumerdes, Boumerdes, Algeria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_15

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overlapped variables between them. Despite its usefulness, there are many necessary and sufficient mathematical conditions that should be verified in order to apply such technique. Lately, Francisco et al. [8] has extended the application of this technique for systems composed of more than two subsystems namely multi-overlapping decomposition, it has been used to overcome the difficulties of controlling large scale systems globally with convex optimization constraints. Multi-machine power system is very known industrial system for which many control strategies have been applied; the structure of the nonlinearity of its model and the interactions between its variables present a challenge for researchers. Many papers have treated the described problematic [9]. One of the main reasons of studying this system is that its mathematical model verifies the necessary and sufficient conditions, thus it can be used to illustrate the construction of the decomposed subsystems by using an overlapped original system. Our objective in this paper is to design an optimal decentralized state feedback controller based on multi-overlapping decomposition strategy in order to reduce the complexity of the design with preserving the global stability of the original system.

2 Control Structure and Complex Systems A. Centralized Controller In this structure, the control system uses all input variables to generate one control law for all the outputs, this means that all the states and their interconnections are considered as one single dynamic system. In the case of large-scale systems, the order of the multivariable controller becomes high, which complicated its real time implementation. The feedback control law has been used to improve performance as well as reject external disturbances. Control designs, such as PID, loop shaping, gain-scheduling [10], multivariable control [11] are industrially used with centralized structure. Unlike the centralized configuration, in the decentralized configuration scheme, each manipulation variable is connected to one controlled variable. B. Decentralized Controller In decentralized control of large scale systems, the interactions terms between states and control inputs are the main obstacle, these terms are either neglected or analyzed for design simplification reasons [6]. The main contribution in such design is how to find a controller that reduces the influence of the interactions terms without losing the dynamic of the original system from one side, and ensures the stability of the total original system from the other side.

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C. Advantages of Decentralized Controller The advantage of the decentralized structure is that there are no any measurements of the coupling quantities even if it is required to identify the active quantities of coupling for each subsystem. Even though, the obtained controller is robust against external parametric variation in a limited interval [6]. Thus, we can set the advantages of decentralized controller, in comparison to centralized controller, for large scale system as follows • More robust control system with high degree of stability for the decomposed system. • Less complication for control implementation, the frequency of regulation interval is wider.

3 Multi-overlapping Decomposition Strategy In literature, there are many definition of Large-scale dynamic systems, the most used one is that any system that is represented by large number of variables that are strongly interconnected can be considered as complex systems [3]. Stability analysis of the complex systems has also been very interesting research topic in the last decades (Karacanias and Wilson [12]; Wu [13]; Doghmane and Kidouche [14, 15]). The analysis and control of such systems can be achieved by either studying the whole mathematical model of the system which is very difficult because of the interaction terms, or by decomposing the model into less complicated subsystems. A. Decomposition principle Complex systems’ dynamic can be composed of overlapped subsystems that share common variables [6]. This structure can be expanded into larger state space representation in which the subsystems are separated from each other; consequently, higher order system is obtained. The latter includes the necessary information of the original system; the expanded system is then contracted by applying standard disjoint decomposition strategy [16]. Consider the large scale system given by (1) 

x(t) ˙ = A(t)x(t) + B(t)u(t) , y(t) = C(t)x(t)

(1)

where ⎧  T T T T m ⎪ ⎨ u = u 1 , u 2 , u 3  , u i ∈  i , i = 1, 2, 3 T T T T x = x1 , x2 , x3 , xi ∈ ni , i = 1, 2, 3 . ⎪ T  ⎩ y = y1T , y2T , y3T , yi ∈ li , i = 1, 2, 3 and consider the system given by (2)

(2)

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x˙e = Ae xe + Be u e , ye = Ce xe

(3)

xe ∈ n e , u e ∈ m e and ye ∈ le , are, state, input and output vectors of system (3) respectively. The decomposition principle is based on the idea of finding a full-rank transformations matrices: T : n → n e , N : m → m e , S : l → le , for any initial state xe ∈ n e , of the system (1) and any input u e (t) ∈ m e , so that the system in (3) can be expansion of the system (1) (or system (1) is contraction of system (3)), which means   xe (t; xe0 ; u e ) = T x(t; x0 ; u) xe0 = T x0 ⇒ , ∀t ≥ 0 (4) ye (t, xe ) = Sy(t; x) u(t) = N u e (t) If condition in (4) is satisfied, then we can say that system (3) is an expansion of system (1) or system (1) is contraction of (3) [8]. B. Restrictions on Decomposition Principle Restriction on applying decomposition principle can be summarized by the necessary and sufficient conditions in Theorem 1. Theorem 1 The system in (3) is an expansion of the system in (1) if and only if there exist full-rank matrices T ∈ n e × n , N ∈ m × m e , S ∈ le × l such that ⎧ ⎨ T A = Ae T T B N = Be , ⎩ SC = Ce T

(5)

The two systems are related by (6) ⎧ ⎨ Ae = T AT I + E A B = T B N + EB ⎩ e Ce = SC T I + E C

(6)

T I is the pseudo-inverse of T, satisfying (7) T I T = In

−1 I TI = TIT T .

(7)

E A ∈ n e ×n e , E B ∈ n e ×m e , and E C ∈ le ×n e , are the complementary matrices [17]. They must satisfy conditions given by (8) so that Theorem 1 is verified [16]. Consider that the matrices in (1) and (3) satisfy (6); then theorem is verified if and only if (8) is verified [18].

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E A T = 0, E B = 0, E C T = 0.

(8)

C. Contraction Contraction is an equivalent principle of expansion, reversibly, the system (1) is contraction of system (3) if, for any initial state x0 of (4), there is a contraction matrix T with full row rank such that xe0 = T I x0 ⇒ x(t, x0 ) = T xe (t, xe0 ), ∀t ≥ 0.

(9)

A complex system has an overlapping structure if at least two of its subsystems share some common states, inputs, and outputs. Consider the system (1) where A, B has the form given in (10). (10)

The system (10) can be now expanded into a larger system that preserves all necessary dynamic of the original system; the system can be then decomposed into subsystems controlled separately, the obtained controller is contracted for implementation in the original system [1, 2]. The state vector x in (1) is composed of three

T vectors x1 , x2 and x3 with dimensions n 1 , n 2 and n 3 , so that x = x1T , x2T , x3T and n = n 1 + n 2 + n 3 , the dimensions of sub-matrices Ai j are n i × n j , i, j = 1, 2, 3 [2]. The structure of the gain matrix of the controller always pursues the structure of the mathematical model of the original system. Consider the matrices T : n → n e , N : m → m e , such that {T A = Ae T, T B N = Be ,

(11)

Our goal is derive the overlapping dynamic matrices (10) into matrices structure given by (12). ⎡

⎤ A13 A23 ⎥ ⎥, A23 ⎦ A33 0 A12 ⎢ 0 A22 E A = 21 ⎢ ⎣ 0 −A22 0 −A32

A11 ⎢ A21 Ae = ⎢ ⎣ A21 A31

where (12) can be written as

A12 A22 0 0

0 0 A22 A32 ⎡

⎡

B11 ⎢ B21 Be = ⎢ ⎣ B21 B ⎤ 31 −A12 0 −A22 0 ⎥ ⎥ A22 0 ⎦ A32 0

⎤ B12 B22 ⎥ ⎥, B22 ⎦ B32

(12)

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Ae = T AT I + E A . Be = T B N + E B

(13)

By rearranging the components of the state x into two overlapping subsys T

T tems xe1 = x1T , x2T , and xe2 = x2T , x3T , which compose a new vector T T T xe = xe1 , xe2 , and persuade the decomposition of A as shown by dashed lines in (12). T is an n e × n matrix defined by (14).

(14)

The expanded system Se can now be conducted into two interconnected subsystems Se1 and Se2 defined by (15). 

Se1 : x˙e1 = Ae11 xe1 + Ae12 xe2 + Be11 u e1 + Be12 u e2 , Se2 : x˙e2 = Ae21 xe1 + Ae22 xe2 + Be21 u e1 + Be22 u e2

(15)

Ae12 , Ae21 : are the interconnection matrices. System in (15) indicates that the decoupled matrices of the expanded matrix Ae correspond to the decomposed part of the original matrix A [19, 3]. The class of expanded systems Se which corresponds to the system S can be found by selecting the appropriate transformation matrix T and its complementary matrices. E A , E B must satisfy the condition {E A T = 0, E B = 0. The matrix in (12) has identity diagonal blocks, thus, the resultant expanded system contains two identical subsystems [17]. System in (15) can now be stabilized by the control law given in (16) [4]. ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ e1 ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ e2 ⎪ ⎪ ⎩

  z˙ e1 = Fe1 z e1 + G 1e1 G 1e2 ye1  1  1 1  : He1 K 11 K 12 ⎩ u e1 = z ye1 + e1 1 1 1 He2   2K 21 2K 22 ⎧ , Fe2 z e2 + G e1 G e2 ye2 ⎨ z˙ e2 =  2  2 2  : He1 K 11 K 12 ⎩ u e2 = z ye2 + e1 2 2 2 He2 K 21 K 22 ⎧ ⎨

(16)

(17)

The controller in (16) is contracted to decentralized controller in (17) using Theorem 1 for implementation on the system (1). The necessary and sufficient conditions for this transformation are all verified and respected.

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4 Multi-overlapping Decomposition Multi-overlapping decomposition strategy has been widely used to solve many problems in different domains such as: applied mathematics, structures’ construction, robotics, and electric power systems. The mathematical framework of decomposition is essentially based on expansion-contraction principle detailed before. The principle is to expand the state space model in away the common variables appear as disjoint, then design control law for each subsystem alone and contract the obtained controllers to the original system [8]. Satisfaction of the inclusion conditions is essential for transferring properties of the expanded system to the original one. In loop topology, the common variable are shared by all the subsystems, in the case of complex system composed of three subsystems, each two subsystems share a common overlapped variables different from the other pairs [6]. An application of designing a loop structure controller of system based on a multi-machine power system is given in the next subsection. A. Loop structure of Multi-overlapping system The system in (1) can be rewritten in the form of (18)  Si :

x˙i = Aii xi + Bii u i i = 1, 2, . . . , N . yi = Cii xi

(18)

The structure of matrix A can be longitudinal, radial or loop. In this study, the loop structure is considered in order to apply the proposed decomposition strategy to multi-machine power system [12]. The following steps are followed to achieve the objective of this paper First, the transformation matrices are given by (19) 

T = blockdiag [I11 , I11 ]T , . . . , [I N N , I N N ]T , N = 0.5 blockdiag ([I11 , I11 ], . . . , [I N N , I N N ])

(19)

where E A is proposed as in (20). ⎡

A11 ⎢ −A 11 ⎢ ⎢ −A ⎢ 21 ⎢ −A21 E A = 0.5⎢ ⎢ . ⎢ . ⎢ . ⎢ ⎣ AN1 AN1

−A11 A11 A21 A21 .. .

−A N 1 −A N 1

A12 A12 A22 −A22 .. .

−A12 −A12 −A22 A22 .. .

0 0

0 0

Thus, the expanded matrix is found as in (21).

. . . −A1N . . . −A1N ... 0 ... 0 .. ··· .

· · · AN N · · · −A N N

A1N A1N 0 0 .. .

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ −A N N ⎦ AN N

(20)

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(21)

Second, the permutation matrices (22) are applied to (21). ⎡ T A = T A12 T A23 · · · T A(2N −1)(2N ) =

2N −1 k=1

⎢ ⎢ T Ak(k+1) = ⎢ ⎣

⎤ 0 · · · 0 I11 I11 · · · 0 0 ⎥ ⎥ .. . . .. .. ⎥ . . . . ⎦ 0 . . . IN N 0

(22)

So that, the expanded system becomes:

(23)

Theorem 1 discussed early in this paper, has been used to decompose the matrices of the expanded system (23), they have multi-overlapping structure with loop topology. For each subsystem Sei , a local control law given by (24) is proposed [8]. 

ui uj



   K ii K i j xi =− , K ji K j j xj

(24)

The gain matrix for this control law is written as   K ii K i j ,... , K e = blockdiag . . . , K ji K j j 



(25)

The conditions of contraction are verified in [10, 11, 15], and its contracted controller is found as: p V p . K = QpK B. Loop Structure of Multi-machine power system

(26)

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A power system, composed of N machines, is seen as complex system with N subsystems with nonlinear interconnection terms. Let xi = [δi (t) ωi (t) δmi (t) δei (t)]T be the state vector of the ith machine with i = 1, . . . , N . Its dynamic is described by (27) [9]. 

x˙i = Ai xi (t) + Bi u i (t) + h i (x, t) , yi = Ci xi (t)

(27)



N h i (x, t) = j=1, j=i pi j G i j gi j x i , x j is the nonlinear interaction terms between machine i and all other machines. The dynamic matrices of the ith subsystems are given by (28). ⎡

0 1 0 0 ω0 ⎢ 0 − Di 0 2Hi 2Hi ⎢ Ai = ⎢ K mi 1 − − 0 ⎣0 Tm i Tm i 1 0 − Te KReii ω0 0 − Te i i

⎤

⎡

0

⎢ ω0 Eqi Eq j Bi j ⎥ ⎢ − 2Hi ⎥ ⎥, G i j = ⎢ ⎣ ⎦ 0 0

⎤ ⎥ ⎥ ⎥ ⎦



The interactions between



the subsystems are defined as gi j xi , x j sin δi (t) − δ jt − sin δi0 − δ j0 . Parameters of system (27) are described by (29) ⎧ ⎨ δi (t) = δi (t) − δi0 Pmi (t) = Pmi (t) − Pmi0 , ⎩ X ei (t) = X ei (t) − X ei0

(28)

=

(29)

u i (t) is the control vector of the ith subsystem, yi (t) is the output vector of the ith subsystem; δi (t) is the rotor angle for the ith machine (rad); ωi (t) is the relative speed for the ith machine (rad); Pmi (t) is the per unit mechanical power for ith machine [9] (Fig. 1). Fig. 1 Three-machine power system diagram

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5 State Feedback Control Using the developed algorithm, solve for K i matrix for each subsystem; by applying the expansion–contraction theorem, the control design can be expressed by finding matrices Fe , G e , He and K e of the controller e that minimizes the cost function Je such that the closed loop system in (30) 

x˙ Sec : e z˙ e



 =

Ae + Be K e Ce Be He G e Ce Fe



 xe , ze

(30)

is asymptotically stable [5, 13, 14], with



Je (Fe , G e , He , K e , xe0 , z e0 ) ≤ Je F e , G e , H e , K e , xe0 , z e0 , ∀ F e , G e , H e , K e ∈ e

(31)

Je F e , G e , H e , K e , xe0 , z e0 represents the cost function value for the controller  T T T (26) in the closed-loop system (30) with initial state xe0 z e0 . The conditions for controller design and system stability are achieved by transforming (30) with its performance function into LQR standard form [7, 20]. Consider the system described by (32).  

Se :

υ x˙ˆe = Ae xˆe + i=1 B ei uˆ ei , i = 1, 2, . . . , υ yˆei = C ei xˆe 





(32)





K ei (i = 1, 2, . . . υ) are gain matrices for the system S e that minimize the cost function   ∞ υ  uˆ eiT R ei uˆ ei dt xˆeT Q e xˆe + J e = Je = 



0



i=1

The semi-decentralized controllers can be now found by solving recursively the four parts of (33) [21]. ⎧ ⎪ T P + P + Q + C T K T R K C = 0 ⎪ ⎪ ⎨ ∇ J = B T P LC T + R K C LC T = 0 Ki i i i i i

i −1 . ⎪ K i = −RiT BiT P LCiT Ci LCiT ⎪ ⎪ ⎩ L + LT + X 0 = 0

(33)

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6 Simulation Results As shown in Figs. 2, 3, 4, 5 and 6, the input reference for multi-machine power system are tracked, • For state responses, the controller tracks the input for machine 1 (Fig. 2), machine 2 (Fig. 3), and machine 3 (Fig. 4). The cost function values are found from solving second equation in (33) J01 = 1.54 × 103 , J02 = 1.5 × 102 , and J03 = 0.76 × 103 for machine 1, 2, and 3 respectively. • For the errors of overlapped zone, the quadratic error for machine 1 is e1 = 2.7%, and for machine 2 is e2 = 1.68%, and for machine 3 is e3 = 4.05%, the overlapped response error of state 3 in very high (Fig. 6). Using another input reference, the desired output feedback control signals for multi-machine power system are reached as it is shown in Fig. 7, where

Fig. 2 State response for multi-power machine system after decomposition—Machine 1

Fig. 3 State response for multi-power machine system after decomposition—Machine 2

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Fig. 4 State response for multi-power machine system after decomposition—Machine 3

Fig. 5 State response of overlapped system

Fig. 6 Error of responses of overlapped system

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Fig. 7 State response for multi-power machine system after decomposition—Machine 3

Fig. 8 Error of responses for multi-power machine system after decomposition—Machine 3

• For state responses, the controller tracks the input for machine 1, machine 2 (Fig. 7), and machine 3. The cost function values J01 = 2.35 × 102 , J02 = 1.25 × 103 , and J03 = 1.73 × 103 for machine 1, 2, and 3 respectively. • For the errors, the quadratic error for machine 1 e1 = 2.64%, and for machine 2 is e2 = 3.04% (Fig. 8), and for machine 3 is e3 = 5.75%.

7 Conclusion In this paper, a decomposition strategy based on multi-overlapping structure with loop topology has been studied for complex multi-machine power system. The mathematical background of the proposed strategy has been detailed, theorem of inclusioncontraction principle has been considered for designing decentralized controller based on the decomposition, and for global stability analysis of the contracted system. Necessary and sufficient conditions for application of such techniques for complex systems with loop topology have been checked. Furthermore, power system with

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multi-overlapping loop topology is considered to demonstrate the effectiveness of the algorithm; optimal decentralized state feedback controller has been designed for the studied system. Based on the obtained results, it has been established that multi-overlapping decentralized controller could be advantageous in comparison to other classical techniques, because it reduces the complexity of interaction terms and controller implementation simplification. Moreover, this method has been improved by introducing an optimization technique. The advantages of the proposed approach can be showed by comparing obtained results with recent controller based on new approaches such as sliding mode. Several other types of topologies can be discussed for future scientific works.

References 1. Iftar A (1991) Decentralized optimal control with overlapping decompositions. In: IEEE international conference on system engineering, Dayton, OH, USA, pp 299–302. https://doi.org/ 10.1109/icsyse.1991.161138 2. Iftar A (1993) Decentralized estimation and control with overlapping input-state and output decomposition. IFAC Proc 23(8):99–104. https://doi.org/10.1016/S1474-6670(17)52078-9 3. Ikeda M, Siljack DD, White DE (1981) Decentralized control with overlapping information sets. J Optim Theory Appl 34:279–310. https://doi.org/10.1007/bf00935477 4. Ikeda M, Siljack DD (1986) Overlapping decentralized control with input, state, and output inclusion. Control Theory Adv Technol 2(2):155–172 5. Stankovic S, Stanojevic M, Siljak DD (2000) Decentralized overlapping control of a platoon of vehicles. IEEE Trans Control Sys Technol 8:816–832. https://doi.org/10.1109/87.865854 6. Bochen X, Stankoviç S (2005) Decomposition and decentralized control of systems with multi overlapping structure. Automatica 41:1765–1772. https://doi.org/10.1016/j.automatica.2005. 01.020 7. Zecevic AI, Šiljak DD (2005) A new approach to control design with overlapping information structure constraints. Automatica 41:265–272. https://doi.org/10.1016/j.automatica.2004. 09.011 8. Francisco PQ, Rodellar J, Rosell JM (2010) Sequential design of multi-overlapping controllers for longitudinal multi-overlapping systems. Appl Math Comput 207(3):1170–1183. https://doi. org/10.1016/j.amc.2010.01.130 9. Tlili AS, Baiek NB (2009) Decentralized observer based guaranteed cost control for nonlinear interconnected systems. Int J Control Autom 2(2):29–45 10. Koç H, Knittel D, Mathelin MD, Abba G (2000) Robust gain-scheduled control of winding systems. In: IEEE conference decision and control, Sidney, Australia. https://doi.org/10.1109/ cdc.2000.912360 11. Koç H, Knittel D, Mathelin MD, Abba G (2002) Modeling and robust control of winding systems for elastic webs. IEEE Trans Control Syst Technol 10(2). https://doi.org/10.1109/87. 987065 12. Karcanias N, Wilson DR (1989) Decentralized diagonal dynamic stabilization of linear multivariable systems. Linear Algebra Appl 121:455–474. https://doi.org/10.1016/0024-379 5(89)90716-7 13. Wu HS (1999) Decentralized stabilizing state feedback controllers for a class of large-scale systems including state delays in the interconnections. J Optim Theory Appl 100:59–87. https:// doi.org/10.1023/A:1021764814535 14. Doghmane MZ (2011) Optimal decentralized control design with overlapping structure. Magister Thesis, University M’hamed Bougara of Boumerdes, Algeria

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15. Doghmane MZ (2019) Conception de commande décentralisée des systems complexes en utilisant les stratégies de décomposition et optimisation par BMI. PhD Thesis, University M’hamed Bougara of Boumerdes, Algeria. http://dlibrary.univ-boumerdes.dz:8080/handle/123 456789/5432 16. Doghmane MZ, Kidouche M (2018) Decentralized controller robustness improvement using longitudinal overlapping decomposition-application to web winding system. Elektronika ir Elektrotechnika 24(5):10–19. https://doi.org/10.5755/j01.eie.24.5.21837 17. Doghmane MZ, Kidouche M, Habbi H et al (2015) A new decomposition strategy approach applied for a multi-stage printing system control optimization. In: 4th international conference on electrical engineering (ICEE), Boumerdes Algeria, May 2015. https://doi.org/10.1109/intee. 2015.7416751 18. Garcia CE, Prett DM, Morari M (1989) Model predictive control: theory and practice, a survey. Automatica 25:335–348. https://doi.org/10.1016/0005-1098(89)90002-2 19. Iftar A, Ozguner V (1999) Contractible controller design and optimal control with state and input inclusion. Automatica 26(03):593–597. https://doi.org/10.1016/0005-1098(90)90031-C 20. Chen J, Hailing H, Tongguang Y (2019) Robust decentralized H∞ control for a multi-motor web-winding system. IEEE Access 7:41241–41249. https://doi.org/10.1109/ACCESS.2019. 2906223 21. Hailiang H, Xiaohong N, Chen J, Dengfeng X (2018) Decentralized coordinated control of elastic web winding systems without tension sensor. ISA Trans 80:350–359. https://doi.org/ 10.1016/j.isatra.2018.06.006

Large Synchronverter Integration in Power Electrical System: Impacts on SCR and CCT Raouia Aouini, Khadija Ben Kilani, Mohamed Elleuch, and Quoc Tuan TRAN

Abstract This paper investigates the impacts of large synchronverters (SV) integration on transient stability and strength of electric power systems. Large Synchronverters account for large amounts of electrical energy injected into the power grid through power converters, as in PV and off shore wind power sources. Potential stability problems may occur when large power converters are connected to weak power systems (PES). An SV is an inverter that mimics a synchronous generator (SG). Methodologically, the SV performances are contrasted to the standard SG of like parameters in terms of system strength and transient stability. These criteria are quantified respectively by the Short-circuit Ratio (SCR) and the critical clearing time (CCT). The penetration ratio of SV generation is increased to the detriment of SG power. Due to their current controllers’ limitations, the SV participation in short-circuit currents is reduced to nominal values, diminishing the network strength. Although SVs have similar properties as SGs and have variable virtual inertia, this study reveals many limitations that must be taken into account when planning their integration into the network. In opposition, SGs enhance the transient stability margin even in weaker systems. Hence, strong buses in a PES indicated by large SCR and CCT feature larger amounts of SG power compared to the power generated by SVs. Keywords Synchronverter · Network strength · Inertia · SCR · CCT

R. Aouini · K. Ben Kilani · M. Elleuch (B) Université de Tunis El Manar, ENIT, L.S.E, LR11ES15, BP 37-1002 Tunis le Belvédère, Tunisie e-mail: [email protected] R. Aouini e-mail: [email protected] K. Ben Kilani e-mail: [email protected] Q. T. TRAN INSTN, CEA–INES, Saclay, France e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_16

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1 Introduction Electric power systems are undergoing significant changes in the generation mix and source types. Power electronic converters have evolved in number and capacity, being an essential equipment in renewable energy sources (RES), such as wind farms, solar PV plants, storage batteries and HVDC interconnection. Although, mostly are equipped with fast-acting controls, many stability issues may arise when large power converters are connected to weak power grids [1]. System strength is one of the important concerns in the integration of RES. Technically, a weak AC system may be evaluated using several measures: low ratio of inductance over resistance, high impedance, and low inertia [2]. The shortcircuit ratio (SCR) at the point of interconnection (POI) has evolved as an indicator of system strength. It is defined as the ratio of system MVA short-circuit capacity (Scc ), to the MW rated power of the interconnected device [3], which could absorb or inject power at the POI. The POI bus is considered strong if its SCR is above three. As in indicator of system strength, the SCR has originally been used for HVDC systems [3], then extended to RES featuring similar architecture and control as HVDC devices [4, 5]. As a stability indicator, the strength of a power system at the POI affects its ability to maintain its voltage and power quality under variable generation levels of RES. Connecting RES to weaker portions of the grid may tremendously harm voltage stability and quality at the POI [6], in particular transient stability. For instance, a major fault occurring in an interconnected PES through weak transmission lines may lead to transient instability. The criteria used for transient stability evaluation is the Critical Clearing Time. This is because, in AC power grids integrating RES, large disturbances are mainly compensated by large conventional synchronous generators. However, as electronic power converters lack inertia, their increasing number diminishes the overall system inertia, which affects the dynamic and transient stability of the power grid [7]. Consequently, power system operators have set rules requiring that RES actively take part in power regulation of the grid, just like conventional SGs. Based on this idea, the concept of Synchronverters was proposed. The aim is to render grid-connected inverters emulate the essential behavior of SGs, including the droop mechanism and inertial properties [8]. As a result, the SV should have the ability of providing grid support by automatically adjusting its active and reactive powers to comply with frequency and voltage grid requirements [11]. Meanwhile, the inertial characteristics emulated by the SV contribute to the total inertia of the grid, enhancing thereafter the system strength. Hence, the SV provides a promising technology for various applications, such as HVDC transmission [9], STATCOM [10], and wind power systems. One important advantage of the SV is that some system parameters, such as inertia, can be suitably chosen to improve the system dynamic performances. In comparison, SGs inertia are constant, and prime movers may have large time constants in droop control loops.

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In this paper, we investigate the transient stability of a four-bus power system, considering five configurations of SV integrations. The SV parameters are adapted to the conventional SG parameters. Dynamic responses of the SV and the SG are compared under different operating conditions. The SV benefits its operation frequency and voltage drooping mechanisms for load sharing. For each configuration, strength at POI is evaluated by its SCR. Also, transient stability of the systems is assessed in terms of the CCT, for different levels of the SCR at the load bus.

2 Overview of the Synchronveretr Technology In this Section, the synchronverter concept is briefly reviewed. The synchronous generator (SG) is the dominant type of generator in conventional electric power systems [12]. SGs offer many advantages such as (i) supplying inertial response out of the kinetic energy stored in the rotating masses; (ii) they can supply and absorb reactive power; (iii) they have simple control structures with guaranteed grid stabilizing responses. Based on these proprieties, an inverter based on the synchronverter controls is developed to mimic a SG [8]. A synchronverter consists of a power part, which is the same as a conventional power electronic converter depicted in Fig. 1a, and an electronic part, which consists of the measurement and the control circuits. The power part of the SV is the standard hardware of a three-phase pulse width modulation (PWM) inverter, with an LCL filter at the output of each phase. Following the terminology defined in [8], the three inductors with parameters Ls and Rs play the role of stator coils in a SG. The input of the power part is the value of the desired synchronous internal voltage e from the electronic part. The three phase inverter then generates a high-frequency switching voltage signal eabc = [ea eb ec ]T which average value over a switching period is e. The basic mathematical model of an SV is given in Eqs. (1–6). 2H

dω = (Pm − Pe )/ω dt

(1)

dω dt

(2)

θ=

Pm = Pr e f + D p (ωn − ω)

(3)

Pe = ωE m (i a sin θ + i b sin(θ − 2π/3) + i c sin(θ + 2π/3))

(4)

Em =

1 (Dq (Ur e f − Um ) + Q r e f − Q e ) Ks

Q e = −ωE m (i a cos θ + i b cos(θ − 2π/3) + i c cos(θ + 2π/3))

(5) (6)

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a)

b)

Fig. 1 Organization of the synchronverter. a the power part of SV, b the electronic part of SV

where H is the inertia constant, Pref and Qref are the given values of active and reactive powers, Pe and Qe are the actual output values of active power and reactive power, ωn and ω are the rated and actual values of electric angular velocity, θ is the rotor angle, Dp and Dq are the droop coefficients of active and reactive loops, Um and Uref are the actual and given values of grid voltage amplitude, Em is the internal potential amplitude of the SV, K is the integral coefficient. The electronic part of a three-phase SV, as shown in Fig. 1b, includes the mathematical model (1–6) of a three-phase round-rotor synchronous machine as the core. The SV controller strategy processes the electric measurements (vabc and iabc ) as well as the Pref and the Qref which purpose is to generate e, using the following expression, eabc = ωE m [sin θ sin(θ − 2π/3) sin(θ + 2π/3)]T

(7)

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The latter expression is passed through a PWM generation block to generate six pulses to drive the power semiconductors in Fig. 1a. The DC-DC converter and the PV panel or any other renewable source, presented in Fig. 1a, can be represented by one equivalent constant voltage source noted VDC . It is noted that there is a difference in hardware between a SV and SG: some modest energy storage is required on the DC bus of the inverter to mimic the effect of rotor inertia. The mechanical equation of the SV rotor is expressed by (1). The SV control can be divided into active loop control and reactive loop control as described by (3) and (5) respectively. Its active loop includes active frequency droop control and inertial response control, which mainly realizes the function of independent frequency response. The reactive loop consists of reactive power-voltage droop control and endvoltage closed-loop control, which achieve automatic voltage regulation and voltage amplitude control of the SV [8].

3 Short-Circuit Ratio and System Strength The Short-circuit Ratio is an indicator of the system strength, which is defined as the ratio of system MVA short-circuit power to the MW rating power of the interconnected device [3]. The strength of a power system at the POI of the device is viewed as the ability of the system at the point to maintain its voltage stability and quality. In this section, the relationship between the SCR and the system strength at the POI of RES is explained. For an HVDC interconnection, the SCR is used to quantify the strength at the POI and its value determines how strong the AC system to support the stable operation of HVDC converter. Moreover, the RES can either be on grid or off grid. When connected to the grid, RES are typically interfaced to the distribution, far from the main grid. The points of interconnection of these RES are generally weak, and voltage stability problems are likely to occur [5, 6]. Therefore, it is necessary to measure the system strength at all possible interconnection points. Like HVDC devices, the commonly used SCR has been extended to quantify the strength at the POI of RES. The SCR at bus i can be expressed as: SC Ri =

|Vi |2 1 Sac,i = . Pd,i Pd,i |Z i |

(8)

where Sac,i is the short-circuit capacity of the system at bus i; Pd,i is the MW rated power of the RES or the load connected to bus i; |Vi | and |Z i | are respectively the Thevenin voltage and impedance viewed at bus i. From the expression of the SCR, it is clear the system strength is highly dependent on the ratio of short-circuit capacity to the power of the interconnected device at a specific point. The short-circuit capacity may also be expressed as:

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Sac = |V0 |.|Icc |

(9)

where |V0 | and |Icc | are the pre-fault bus voltage magnitude and fault current, respectively.

4 The Testedfour Area Power System The synchronverter control strategy is applied to the system described in Fig. 2. The system is built based on MATLAB/Simulink toolbox. A series of time domain simulations are performed to investigate the transient stability of the tested system, considering five configurations. The impact of the SV integration on load point strength and on its transient stability is assessed in terms of the CCT and SCR. The proposed tests consist of four control areas designated as Area 1, Area 2, Area 3 and Area 4, feeding a common load area at POI, via AC lines as in Fig. 2. Two different generation sources are studied: synchronverters and synchronous generators. SVs are modeled by (1–7), SGs of the same power rating are equipped with power frequency and voltage reactive power controls described in (3) and (5), respectively. The study cases are defined as follows:

Fig. 2 Four areas and four SV/SG interconnected generation

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– Case 1: Areas 1, 2, 3 and 4 are SVs designated as SV1, SV2, SV3 and SV4. This case is named 4SV with SV integration ratio equals to 100%. – Case 2: SV in Area 1 has been replaced byan equivalent SG (marked as SG1 in Fig. 2). Areas 2, 3 and 4 arekept as SVs. – Case 3: Areas 1 and 2 are SGs named SG1 and SG2, respectively. This case is called 2SG-2SV with SV power integration ratio equal to 50%. – Case 4: Areas 1, 2 and 3 are SGs. Area 4 is an SV. This case is designed as 3SG-1SVwithSV power integration ratio equals 25%. – Case 5 named 4SG: Area 1, 2, 3 and 4 are SGs designated SG1, SG2, SG3 and SG4with no SV power integration. The generation sources are chosen to have identical generator, with voltage, power and frequency ratings given respectively by V = 100 kV, Sn = 200 MVA, f = 50 Hz. The nominal current of the SV and the SG is expressed by: Sn 200 =√ In = √ = 1.15 kA 3Vn 3.100

(10)

For the SV parameters, they are firstly adapted to match standard SG parameters [9, 12]: a 5% governor droop (1/Dp-SV and 1/Dp-SG ), inertia constant H = 4 s, and3% voltage regulation gain (1/DQ-SV and 1/DQ-SG ).

5 Simulation Results The main advantage of the SVs is to feature the structure of classic SGs, which controls are well known. The SV operation is verified on the test power system shown in Fig. 2. Dynamic and transient performances of SV are compared with SG of same capacity under different faults. A. Load share of the SV versus SG An important mechanism for SVs and SGs for even load sharing is to adjust the real power delivered to the grid according to the grid frequency. In this Section, the dynamic performances of the cases 4SV and 3SV-1SG are compared. The responses of these cases to a load change at t = 10 s are shown in Figs. 3 and 4. In Fig. 3, SV1, SV2, SV3 and SV1 responses of the 4SV system are presented. Figure 4 depicts the 3SV-1SG responses. The active powers, the speeds, the voltages and the reactive powers dynamics are similar. These results confirm that the SV performances are nearly identical to SG ones, for steady state and small signal stability. The SV and the SG coupled via an AC lines as shown in Fig. 2 are set in a parallel arrangement and are initially at nominal frequency f0 = 1 p.u with power outputs, respectively, PSV1 = PSG1 = 0.8 p.u (Figs. 3a, b and 4a, b show, a slow down of the speed response to an active load increase ΔPL = +0.8 p.u.

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Fig. 3 Responses of the 4SV system toa +0.8 p.u step in the load. a Responses of the SVs active powers in (p.u), b Responses of the SV speeds in (p.u), c Responses of the SVs reactive powers in (p.u), d Responses of the SV voltages in (p.u)

The governors increase their power output until they reach a new common operating frequency f . The frequency of the SG followed the SV frequency, and the real and reactive powers tracked their set points. For example, for the 3SV-1SG system, the amount of load picked up by each unit depends on the droop characteristics, such that:  f = f  − f0 =

PL 0.8 = 0.01 p.u. = D SG1 + D SV 2 + D SV 3 + D SV 4 4 × 20

(11)

The SG1, SV2, SV3 and SV4 increased their real powers output by 0.2 p.u PSG1 =  f × D P-SG1 = 0.01 × 20 = 0.2 p.u PSV 2 = PSV 3 = PSV 4 =  f × D P-SV = 0.01 × 20 = 0.2 p.u

(12) (13)

The amount of load increase corresponds to a 0.1% drop of the frequency. Hence, P-SV = DDP-SG1 , where DP-SV and DP-SG1 are respectively, the static droop of SV and SG, which are equal to 1/5%. PSV PSG1

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Fig. 4 Responses of the SV1 and the SG1 to +0.8 p.u step in the load. a Responses of the SVs active powers in (p.u), b Responses of the SV speeds in (p.u), c Responses of the SVs reactive powers in (p.u), d Responses of the SV voltages in (p.u)

Due to the load increase, the local terminal voltage Vabc decreased by about 0.6% from nominal (as in Figs. 3d and 4d). In response, the reactive power output of SV1 and SG1increased by about 0.195 p.u (see Figs. 3c and 4c), depending on the droop control value Q SG1 = −

VSG1 0.992 − 0.998 = 0.195 p.u. =− Dq-SG1 0.03

(14)

B. Evaluation of the SCR and the CCT for different ratio of SV power penetration In this Section, the impact of large SVs integration is evaluated in terms transient stability, and system strength at the POI, using Fig. 2. Transient stability is quantified by the Critical Clearing Time (CCT), which is the maximum time duration that a short- circuit may be applied without losing the system capacity to recover to a steadystate stable operation. To evaluate the CCT, a three phase short-circuit was simulated near the load. The power at the POI may be considered as a load, or RES power injection. Different levels of SCRs are used to investigate the impact of the system strength on transient stability. To do so, the transmission lines lengths (L) of the test

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system in Fig. 2 are varied: (400, 300, 200 and 50 km) to change the SCR values at POI. Tables 1, 2, 3 and 4 and Fig. 6 depict the CCT and the SCR for SV integration cases and for different transmission lines lengths. The following methodology is adopted. First, the test power system in Fig. 2 is replaced by an equivalent Thevenin generator Eth in series with its impedance Zth (Fig. 5), as viewed from the terminals a and b (at the POI) with all the generation sources (SV and SG). The SCR at POI is evaluated as follows: • Base Case: the system in Fig. 5 is in normal operating condition, and governed by: Table 1 CCTs, Current default Icc , Eth , Zth and the SCRs at POI for different ratios of SV integration with a line length L = 400 km Length

400 km

Cases

CCT (s)

   I cc  (p.u)

   E th  (p.u)

   Z th  (p.u)

SCR

4SG

2.1

2.9

0.95

0.32

1.09

3SG-1SV

1.75 s

2.12

0.95

0.44

0.80

2SV-2SG

0.9 s

1.65

0.949

0.57

0.64

3SV-1SG

0.75 s

1.3

0.948

0.72

0.49

4SV

0.6 s

1.27

0.948

0.74

0.48

Table 2 CCTs, Current default Icc , Eth , Zth and the SCRs at POI for different ratio of SV integration with a line length L = 300 km Length

300 km

Cases

CCT (s)

   I cc  (p.u)

   E th  (p.u)

   Z th  (p.u)

SCR

4SG

2.5

3.53

0.965

0.27

1.52

3SG-1SV

1.85

2.82

0.965

0.34

1.08

2SV-2SG

1

1.72

0.964

0.56

0.84

3SV-1SG

0.8

1.55

0.964

0.62

0.59

4SV

0.75

1.52

0.964

0.63

0.58

Table 3 CCTs, Current default Icc , Eth , Zth and the SCRs at POI for different ratio of SV integration with a line length L = 200 km Length

200 km

Cases

CCT (s)

   I cc  (p.u)

   E th  (p.u)

   Z th  (p.u)

SCR

4SG

2.8

4.6

0.97

0.21

1.77

3SG-1SV

2.3

3.7

0.97

0.26

1.42

2SV-2SG

1.1

2.4

0.969

0.4

0.92

3SV-1SG

0.9

2

0.969

0.48

0.77

4SV

0.8

1.90

0.969

0.51

0.73

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Table 4 CCTs, Current default Icc , Eth , Zth and the SCRs at POI for different ratio of SV integration with a line length L = 50 km Length

50 km

Cases

CCT (s)

   I cc  (p.u)

   E th  (p.u)

   Z th  (p.u)

SCR

4SG

3.4

10.6

0.99

0.09

4.18

3SG-1SV

2.8

7.76

0.99

0.12

3.06

2SV-2SG

2.2

4.6

0.989

0.21

1.81

3SV-1SG

1.75

3.5

0.989

0.28

1.38

4SV

1.45

3.4

0.989

0.29

1.34

Fig. 5 Thevenin equivalent circuit as seen from POI

V L = E th − Z th .I L

(15)

    from which  I L  and V L  have been measured. • Contingency Case:  athree phase short-circuit fault is applied at the POI bus, and the fault current  I cc  is measured for different cases as summarized in Tables 1, 2, 3 and 4. The fault current is given by:

I cc =

E th Z th

(16)

Then, the Thevenin voltage is computed as: E th =

V L .I cc I cc − I L

(17)

Finally, the SCRs in Tables 1, 2, 3 and 4, are given by: scr poi =

sac, poi pd, poi

(18)

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Fig. 6 The CCTs and the SCRs of different configurations and line length. a CCTs, b SCRs

where Sac,POI is the short-circuit capacity and Pd,POI is the load power at POI. As indicated in Eq. (9), Sac,POI is calculated by:    Sac,P O I =  E th . I cc 

(19)

The load power Pd, POI is equal to 500 MW which may also be considered as the RES power to be injected at the POI. C. The impact of the SVs integrations on the CCT and the SCR First, it can be observed from Tables 1, 2, 3 and 4 (column 2) that the best CCT is for case 4SG. The CCT decreased with increasing ratios of SV integration, and with the decoupling between generators by increasing their tie lines. Figure 6a illustrates the worst CCTs for the case with full SV integration and no SG. Second, it is confirmed that the strength of the grid at POI is well reflected by its SCR and its fault current Icc . In fact, the  I cc  which is almost proportional to the SCR, has higher values when the more power is generated by conventional SG with less SV integration. It is obvious that SGs supply fault currents up to several times their rated current (Table 4), whereas converters have to reduce their fault current up to almost their nominal values [13], which results in low participation in the fault current Icc . This leads to the decrease of SCR, and network strength reduction. Tables 1, 2, 3 and 4 and Fig. 6b confirm the increase of Icc , SCR, and the CCT with the generation sources proximity to the POI. The best CCT and SCR values are obtained for the shortest tie-line (50  As shown in column 3 of Tables 1, 2, 3 and 4, the  km). fault current magnitudes  I cc  are reduced as the line length increased, specifically, as the Thevenin impedance magnitude Z th increased (as in column 5). In addition, as it can be seen from Fig. 6b and Table 4 (column 3 and 5), with a line length L = 50 km, higher fault current values are depicted, because of the reduction of the Thevenin equivalent impedance Z th seen from the POI bus which is considered strong. A higher fault current value reflects a strong response of the generation in a PES, which produces, at the fault location, a higher current and a voltage drop. Nonetheless, in Table 1 with a

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Table 5 The CCTs for different inertia constant Constant inertia H (s)

CCT(s) 0.5

2

4

10

15

20

3SG-1SV

1.5

2.1

2.4

2.5

2.8

Unstable

2SV-2SG

0.95

1

1.2

1.8

2

Unstable

3SV-1SG

0.75

0.8

1

1.15

1.5

Unstable

4SV

0.22

0.45

0.98

1.1

1.4

Unstable

  line length L = 400 km, the fault current levels  I cc  decreased when the POI bus is considered weak, since the SCR is smaller than 2. Similar observations for the CCT can be made based on data in Tables 1, 2, 3 and 4 and Fig. 6a, where the CCT tends to overestimate the system strength in terms of transient stability. The lower CCT is observed with weaker POI bus; the higher CCT indicates strong POI bus. Thus, strong buses in a PES indicated by large SCR and CCT values feature higher power quantity from SGs and less from SVs. The higher values of fault current at theses buses tend to maintain stable operation during disturbances. D. The effect of the SV inertia constant on the CCT One important advantage of SVs over SGs is the possibility to vary their parameters, like the inertia constant, to improve power system stability. In this section, the inertia constant H of cases with SV integration is varied to enhance the transient stability in term of CCT. The SG inertia constant H is fixed to 4 s, while the SV inertia constant is increased up to 20 s. The results summarized in Table 5, confirm the improvement of transient stability with increased inertia, limited by a threshold value, beyond which the system becomes instable. For the system strength, the inertia constant has no significant effect on the SCR, since it doesn’t affect the fault current Icc . This is another limitation of SV performance.

6 Conclusion This paper dealt with a comparative analysis of a SV/SG in four-areas network. The performance criteria used are CCT and SCR. With appropriate controls, it is shown that SVs are able to recover many performances of SGs in terms of frequency and voltage droop control. In addition, thanks to its variable virtual inertia, SV has improved the transient stability of the tested network. However, the SV has some limitations highlighted along this work which consisted of: – Low participation in the fault current leading to low SCR and consequently a reduction of the network strength;

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– Maximum value of the inertia constant beyond which the system loses its stability; – The increase of the integration ratio of SVs power has a negative impact on network performances (by reducing the SCR and CCT) as opposed to SGs.

References 1. Urdal H, Ierna R, Zhu J, Ivanov C, Dahresobh A, Rostom D (2015) System strength considerations in a converter dominated power system. IET, Renew. Power Gen 9(1):10–17 2. Gavrilovic A (1991) AC/DC system strength as indicated by short-circuit ratios. In: International conference on AC and DC power transmission, September 17–20, 1991, pp 27–32 3. Jia C, Wen J, Bie X (2012) Study of the effect of AC system strength on the HVDC startup characteristics. In: International conference on sustainable power generation and supply, September 8–9, 2012 4. D. Wu, G. Li, M. Javadi (2018) Assessing impact of renewable energy integration on system strength using site-dependent short-circuit ratio. IEEE Trans Sustain Energy 9:1072–1080 5. Zhang Y, Huang SH, Schmall J, Conto J, Billo J, Rehman E (2014) Evaluating system strength for large-scale wind plant integration. In: IEEE power and energy society general meeting, July 2014, pp 1–5 6. Huang SH, Schmall J, Conto J, Zhang Y, Li Y, Billo J (2015) Voltage stability of large-scale wind plants integrated in weak networks: an ERCOT case study. IEEE power and energy society general meeting, July 2015, pp 1–5 7. Nkechi AI, Howlader AM, Yona A (2018) Integration of photovoltaic energy to the grid, using the virtual synchronous generator control technique. J Energy Power Eng 12:329–339 8. Zhong Q-C, Weiss G (2011) Synchronverters: inverters that mimic synchronous generators. IEEE Trans Ind Electron 58(4):1259–1267 9. Aouini R, Marinescu B, Ben Kilani K, Elleuch M (2016) Synchronverter-based emulation and control of HVDC transmission. IEEE Trans Power Syst (1):278–286 10. Aouini R, Nevzi I, Ben-Kilani K, Elleuch M (2016) Exploitation of synchronverter control to improve the integration of renewable sources to the grid control. J Electr Syst 12–3:515–528 11. Nguyen P-L, Zhong Q-C, Blaabjerg F, Guerrero J-M (2012) Synchronverter-based OPERATION OF STATCOM to mimic synchronous condensers. In: 2012 7th IEEE conference on industrial electronics and applications (ICIEA) 12. Kundur P (1994) Power system stability and control. Mc Graw-HillInc 13. Callavik M, Blomberg A, Häfner J, Jacobson B (2012) The hybrid HVDC breaker—aninnovation breakthrough enabling reliable HVDC grids, Technical Paper, ABBGrid Systems

Immersion and Invariance Based Adaptive Dynamic Surface Control for Parametric Strict-Feedback Nonlinear Systems Y. Soukkou, S. Labiod, M. Tadjine, Q. M. Zhu, and M. Nibouche

Abstract This paper presents an immersion and invariance based adaptive dynamic control scheme for a class of parametric strict-feedback nonlinear systems. By using dynamic surface control, the problem of explosion of complexity inherent in the conventional backstepping control method is avoided. The uniformly ultimately boundedness (UUB) of all signals in the closed-loop system is performed based on the Lyapunov stability analysis theory. Simulation results for the control of a one-link manipulator actuated by a brush DC motor are provided to demonstrate the effectiveness of the proposed adaptive control method. Keywords Adaptive control · Immersion and invariance · Dynamic surface control · Lyapunov stability theory · One-link manipulator actuated by a brush DC motor

1 Introduction Over the past few decades, adaptive backstepping control method is one of the most popular and effective control approaches for a class of parametric strict-feedback nonlinear systems, which has received much attention [1–5]. Nevertheless, the main drawback in this approach is the overparametrization problem. Tuning functions Y. Soukkou · M. Tadjine LCP, Department of Automatic Control, National Polytechnic School (ENP), 10, Av. Hassen Badi, BP. 182, Algiers, Algeria Y. Soukkou (B) Research Center in Industrial Technologies CRTI, P. O. Box. 64, 16014 Cheraga, Algiers, Algeria e-mail: [email protected] S. Labiod LAJ, Faculty of Science and Technology, University of Jijel, BP. 98, Ouled Aissa, Jijel, Algeria Q. M. Zhu · M. Nibouche Department of Engineering Design and Mathematics, University of the West of England, Frenchay Campus, Coldharbour Lane, Bristol BS16 1QY, UK © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_17

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based adaptive backstepping control design is proposed to avoid the overparametrization problem [2–5]. However, adaptive backstepping control and tuning functions based adaptive backstepping control designs suffer from the problem of explosion of complexity, which is caused by the repeated derivations of virtual control inputs [6–8]. Recently, a dynamic surface control has been proposed to eliminate the problem of explosion of complexity by introducing a first-order low-pass filter at each step of the conventional backstepping control method [6–8]. In [6, 9], the dynamic surface control method has extended to adaptive control. For parametric strict-feedback nonlinear systems, adaptive dynamic surface control has been widely applied [10–13]. In recent years, immersion and invariance based adaptive control of nonlinear systems was proposed by Astolfi and Ortega [14]. It has been established for controlling of class of parametric strict-feedback nonlinear systems [15, 16]. It has been also a great attention in the few recent years, which has been widely used to the control of a class of parametric strict-feedback nonlinear systems [17, 18]. During the last few years, the control methodologies of the one-link manipulator actuated by a brush DC motor have drawn considerable interest [11, 19–22]. Several nonlinear control methodologies and adaptive approaches have been also studied and applied to electromechanical systems. This paper focuses on an immersion and invariance based adaptive dynamic surface control for a class of parametric strict-feedback nonlinear systems. Immersion and invariance based adaptive dynamic surface control is introduced to avoid the problem of explosion of complexity. The uniformly ultimately boundedness (UUB) of all signals in the closed-loop system is proven based on the Lyapunov stability theory. Simulation results for a one-link manipulator actuated by a brush DC motor are provided to demonstrate the effectiveness of the proposed adaptive control scheme. The remaining of the paper is organized as follows. The problem formulation and preliminaries are presented in Sect. 2. Section 3 is devoted to the design of immersion and invariance based adaptive dynamic surface control. Stability analysis is discussed in Sect. 4. Simulation results are included in Sect. 5. Conclusions are given in Sect. 6.

2 Problem Formulation and Preliminaries We will consider the following nonlinear system in parametric strict-feedback form: x˙1 = g1 (x1 )x2 + ϕ1T (x1 )θ + ψ1 (x1 ) x˙i = gi (x¯i )xi+1 + ϕiT (x¯i )θ + ψi (x¯i ), i = 2, · · · , n − 1 x˙n = gn (x)u + ϕnT (x)θ + ψn (x)

(1)

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T ∈ Rn and u ∈ R are system states and the control x1 x2 · · · xn T  and θ ∈ R p are unknown constant parameter vector. input. x¯i = x1 x2 · · · xi T The nonlinear functions ϕi , ψi and gi = 0 are known and continuous. The control objective is to design the controller u such that the output x1 tracks the desired trajectory x1d and to ensure the uniformly ultimately boundedness (UUB) of all signals in the closed-loop system.

where, x =



Assumption 1 There exists a positive constant g0 such that |gi (x¯i )| ≥ g0 , i = 1, · · · , n. Assumption 2 The desired trajectory x1d and its derivatives x˙1d and x¨1d are known, continuous and bounded.

3 Immersion and Invariance Based Adaptive Dynamic Surface Control The design procedures of the proposed adaptive control consisted of a general two steps, while the first step is to design an estimator and the second step is to design a control law. A. Estimator Design We define the estimation errors as [16]: 

 θi = θ i − θ + βi (x¯i )

(2)



where, θ i are the estimator states and βi : Ri → R p are functions yet to be specified. Then, the estimation dynamics are given by [16]:     ∂βi  ∂βi  gk xk+1 + ϕkT θˆi + βi −  x˙k = θ˙ˆi + θi + ψk θ˙˜i = θ˙ˆi + ∂ xk ∂ xk k=1 k=1 i

i

(3)

where, xn+1 = u. Selecting the adaptive laws θˆi as: θ˙ˆi = −

i     ∂βi  gk xk+1 + ϕkT θ i + βi + ψk ∂ xk k=1 

(4)

yields the error dynamics: i

˙θ˜ = −  ∂βi ϕ T  θi i ∂ xk k k=1

(5)

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where, the functions βi are selected as [16]: xi βi = i

ϕi (χ )dχ + εi

(6)

0

where, i > 0 are constants and εi are functions with ε1 = 0. Assumption 3 There exist functions εi satisfying the partial differential (matrix) inequality [16]: Fi + FiT ≥ 0

(7)

for, i = 2, · · · , n − 1, where, ⎛ x ⎞ i i−1  ∂ ⎝ ∂εi T Fi = i ϕi (χ )dχ ⎠ϕkT + ϕ ∂ xk ∂ xi i k=1

(8)

0

Remark 1 In the special case when ϕi (·) is a function of xi only, the partial differential (matrix) inequality (7) admits the trivial solution εi = 0 for, i = 2, · · · , n [16]. Lemma 1 Consider the system (1), where the functions βi are given by (6) and functions εi exist which satisfy (7). Then, the system (1) has a uniformly globally stable θi ∈ L 2 (square integrable). In equilibrium at the origin,  θi ∈ L ∞ (bounded) and ϕiT  θi converges to zero. addition, ϕi and its time derivative ϕ˙i are bounded, then ϕiT  Proof We consider the Lyapunov function W as: 1  T −1  θ   θi 2 i=1 i i n

W =

(9)

The derivative of the Lyapunov function W˙ becomes: n n     1   ˙ T −1  θi +  θiT i−1 i ϕi ϕiT + Fi + FiT  θ˜i i  θi θiT i−1 θ˙˜i = − W˙ = 2 i=1 i=1

≤−

n 

 θiT ϕi ϕiT  θi ≤ −

i=1

n   T 2 θi ϕi 

(10)

i=1

By integrating of inequality (10) over [0, ∞], we obtain: ∞  n 0

i=1

 T 2 θi dτ ≤ W (0) − W (∞) < ∞ ϕi 

(11)

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θi ∈ L 2 . Furthermore, ϕi ∈ L ∞ and As a result, W ∈ L ∞ ,  θi ∈ L ∞ and ϕiT  θi = 0. ϕ˙i ∈ L ∞ . By using Barbala’s lemma [23], we get limt→∞ ϕiT  B. Controller Design The controller design presented here includes adaptive dynamic surface control. Step 1(i = 1): Define the first surface error as S1 = x1 − x1d , and the time derivative of S1 is defined as: S˙1 = g1 x2 + ϕ1T θ + ψ1 − x˙1d

(12)

The virtual control x¯2 is chosen as: x¯2 =

  1  T −ϕ1 θ 1 + β1 − ψ1 + x˙1d − K 1 S1 − g1 S1 g1 

(13)

where, K 1 > 0. We introduce a new variable x2d and let x¯2 pass through a first order filter with time constant τ2 to obtain x2d as:

x˙2d

τ2 x˙2d + x2d = x¯2 , x2d (0) = x¯2 (0)

(14)

    1 1  T −x2d + −ϕ1 θ 1 + β1 − ψ1 + x˙1d − K 1 S1 − g1 S1 = τ2 g1

(15)



Step i(i = 2, · · · , n − 1): Define the ith surface error Si = xi − xid , and it’s time derivative is defined as: S˙i = gi xi+1 + ϕiT θ + ψi − x˙id

(16)

The virtual controllers x¯i+1 are chosen as: x¯i+1 =

  1  T −ϕi θ i + βi − ψi + x˙id − K i Si − gi Si gi 

(17)

where, K i > 0. We introduce a new variable x(i+1)d and let x¯i+1 pass through a first order filter with time constant τi+1 to obtain x(i+1)d as: τi+1 x˙(i+1)d + x(i+1)d = x¯i+1 , x(i+1)d (0) = x¯i+1 (0) x˙(i+1)d =

1 τi+1

 −x(i+1)d +

  1  T −ϕi θ i + βi − ψi + x˙id − K i Si − gi Si gi 

(18)  (19)

Step n: Define the nth surface error Sn = xn − xnd , and its time derivative is defined as:

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S˙n = gn u + ϕnT θ + ψn − x˙nd

(20)

The actual control input u is given as follow: u=

  1  T −ϕn θ n + βn − ψn + x˙nd − K n Sn gn 

(21)

where, K n > 0.

4 Stability Analysis Define the boundary layer error as [9, 10, 12, 13]: yi = xid − x¯i , i = 2, · · · , n

(22)

and the parameter estimation errors as [16]: 

 θi = θ i − θ + βi , i = 1, 2, · · · , n

(23)

Then, the closed-loop dynamics can be expressed in terms of the surface errors θi . Si , the boundary layer errors yi , and the parameter estimation errors  The surface errors dynamics are expressed, for i = 1, as: S˙1 = x˙1 − x˙1d = g1 x2 + ϕ1T θ + ψ1 − x˙1d = g1 S2 + g1 x2d + ϕ1T θ + ψ1 − x˙1d = g1 S2 + g1 y2 + g1 x¯2 + ϕ1T θ + ψ1 − x˙1d θ1 − g12 S1 = g1 S2 + g1 y2 − K 1 S1 − ϕ1T 

(24)

For i = 2, · · · , n − 1: S˙i = x˙i − x˙id = gi xi+1 + ϕiT θ + ψi − x˙id = gi Si+1 + gi x(i+1)d + ϕiT θ + ψi − x˙id = gi Si+1 + gi yi+1 + gi x¯i+1 + ϕiT θ + ψi − x˙id θi − gi2 Si = gi Si+1 + gi yi+1 − K i Si − ϕiT 

(25)

For i = n: θn S˙n = x˙n − x˙nd = gn u + ϕnT θ + ψn − x˙nd = −K n Sn − ϕnT  The boundary layer errors dynamics are expressed, for i = 2, as:

(26)

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y˙2 = x˙2d − x˙¯2 =

253

1 1 (−x2d + x¯2 ) − x˙¯2 = − y2 − x˙¯2 τ2 τ2

(27)

1 1 (−xid + x¯i ) − x˙¯i = − yi − x˙¯i τi τi

(28)

For i = 3, · · · , n: y˙i = x˙id − x˙¯i =

We consider the following Lyapunov function as: Vi S =

1 2 1 1 T −1 θ   θi S , Vi y = yi2 , Viθ =  2 i 2 2 i i

(29)

where, V =

n 

Vi S +

i=1

n 

Vi y +

i=2

n 

Viθ

(30)

i=1

Then, for i = 1, · · · , n − 1, by some simple computations, one has, θi − gi2 Si2 V˙i S = Si S˙i = gi Si Si+1 + gi Si yi+1 − K i Si2 − Si ϕiT 

(31)

Applying of the following Young’s inequalities: gi Si Si+1 ≤

1 2 2 1 2 g S + Si+1 2 i i 2

(32)

gi Si yi+1 ≤

1 2 2 1 2 g S + yi+1 2 i i 2

(33)

The following Young’s inequalities hold: 1 2 1 2 θi + yi+1 − Si ϕiT  V˙i S ≤ −K i Si2 + Si+1 2 2

(34)

θn V˙nS = Sn S˙n = −K n Sn2 − Sn ϕnT 

(35)

and, for i = n:

one has, x˙1 = S˙1 + x˙1d and x˙i = S˙i + y˙i + x˙¯i , then from:   T     ˙x¯2 = 1 − ∂ϕ1 x˙1 θ 1 + β1 − ϕ1T θ˙ˆ1 + β˙1 g1 ∂ x1    ∂g1 ∂ψ1 ˙ ˙ x˙1 + x¨1d − K 1 S1 − g1 S1 − x˙1 S1 − ∂ x1 ∂ x1 

(36)

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and by induction, it is easy from: ⎛ ⎛ ⎞ i−1 T      ∂ϕi−1 1 T ⎝−⎝ x˙ j ⎠ θ i−1 + βi−1 − ϕi−1 x˙¯i = θ˙ˆi−1 + β˙i−1 gi−1 ∂x j j=1 ⎛ ⎞ ⎞ i−1 i−1   ∂ψi−1 ∂g i−1 x˙ j + x¨(i−1)d − K i−1 S˙i−1 ⎠ − gi−1 S˙i−1 − ⎝ x˙ j ⎠ Si−1 − ∂ x ∂ x j j j=1 j=1 

(37) one has,   T     ∂ϕ1 1 1 − y2 − x˙1 θ 1 + β1 − ϕ1T θ˙ˆ1 + β˙1 τ2 g1 ∂ x1    ∂g1 ∂ψ1 x˙1 + x¨1d − K 1 S˙1 + g1 S˙1 + x˙1 S1 − ∂ x1 ∂ x1 

y˙2 = −

(38)

Since all terms in (38) can be dominated by some continuous functions, it follows that:      1 1   (39) y˙2 + y2 ≤  y˙2 + y2  ≤ B2 S1 , S2 , y2 , θ 1 , β1 , x1d , x˙1d , x¨1d τ2 τ2 

where, B2 (.) is a continuous function. Thus: y2 y˙2 +

1 2 y ≤ B2 (.)|y2 | τ2 2

(40)

By using Young’s inequality, it gives: y2 y˙2 ≤ −

B 2 (.) 1 2 y22 + 2 y2 + τ2 2 2

(41)

as similarly for i = 3, · · · , n, we obtain: yn y˙n ≤ −

B 2 (.) 1 2 yn2 + n yn + τn 2 2

(42)

  where, Bn (.) = Bn S1 · · · Sn , y2 · · · yn , θ 1 · · · θ n , β1 · · · βn , x1d , x˙1d , x¨1d is a continuous function. From (41) and (42), we can write: 



B 2 (.) y2 1 V˙i y ≤ − yi2 + i + i , i = 2, · · · , n τi 2 2

(43)

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From (10), one has,  2 θi V˙iθ ≤ − ϕiT 

(44)

Therefore, the derivative of the Lyapunov function V˙ becomes: V˙ =

n 

V˙i S +

i=1

n  i=2

V˙i y +

n 

V˙iθ

i=1

 n−1   1 2 1 2 2 T −K i Si + Si+1 + yi+1 − Si ϕi θi − K n Sn2 − Sn ϕnT  θn ≤ 2 2 i=1   n  n   T 2 y2 B 2 (.) 1 − yi2 + i + i − θi ϕi  + τi 2 2 i=2 i=1

(45)

Applying of the following inequality: θi ≤ −Si ϕiT 

1 2 1  T 2 S + ϕi  θi 2 i 2

(46)

We obtain:   n−1 1 2  V˙ ≤ − K 1 − S1 − (K i − 1)Si2 − (K n − 1)Sn2 2 i=2  n  n  1 1   T 2 θi ϕ  − − 1 yi2 + ϕ − τi 2 i=1 i i=2 where, ϕ =

n  i=2

Mi 2

(47)

and Mi are the maximums of Bi (.). By choosing the design

parameters such that, K 1 > 21 , K i > 1, K n > 1 and τi < 1, it can be concluded that, θi are bounded. Furthermore, all signals in the closed-loop system, V , Si , yi and ϕiT  i.e., xi , xid , x˙id , x¯2 , · · · , x¯i+1 , u and θ i + βi are also bounded. 

Remark 2 To robustify the adaptive laws, we introduce a σ-modification term into the adaptive laws (4). Then, we choose the adaptive laws for θ i as: 

θ˙ˆi = −

i       ∂βi  gk xk+1 + ϕkT θ i + βi + ψk − i σθi θ i + βi ∂ xk k=1 



(48)

where, σθi are small design positive constants. The estimation dynamics are given by:

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i

  ˙θ˜ = −  ∂βi ϕ T  i k θi − i σθi θ i + βi ∂ xk k=1 

(49)

one has, V˙i S , V˙nS and V˙i y are expressed in (34), (35), (43), and,   2  θi − σθi  θiT θ i + βi V˙iθ ≤ − ϕiT  

(50)

By using of inequality (46), the derivative of the Lyapunov function V˙ becomes:    n−1 n   1 1 2  S1 − V˙ ≤ − K 1 − − 1 yi2 (K i − 1)Si2 − (K n − 1)Sn2 − 2 τi i=2 i=2   1   T 2  θi − θiT θ i + βi ϕi  σθi  2 i=1 i=1 n

+ϕ−

n



(51)

From Young’s inequality, it is easy to show that:   1 T 1 θi  θi + θ 2 − θiT θ i + βi ≤ −  2 2 

(52)

Substituting (52) into (51) yields:    n−1 n   1 1 2  S1 − V˙ ≤ − K 1 − − 1 yi2 (K i − 1)Si2 − (K n − 1)Sn2 − 2 τi i=2 i=2  σθ 1   T 2  σθi T i  θ 2 θi  θi − θi + ϕi  +ϕ− 2 i=1 2 2 i=1 i=1 n

n

n

≤ −π V + μ

(53)

      where, π = min 2 K 1 − 21 , 2(K i − 1), 2(K n − 1), 2 τ1i − 1 , i σθi and μ = n  σθi θ 2 . ϕ+ 2 i=1

Theorem 1 Consider the closed-loop system composed of plant described by (1). Suppose that Assumptions 1 and 2 are satisfied. Then, the virtual controllers (13) and (17), the actual control input (21) and the parameter adaptive laws (48) guarantee that all signals in the closed-loop system are uniformly ultimately bounded (UUB) and the surface errors converge to a sufficiently small neighborhood of the origin by appropriately adjusting the design parameters. Proof Multiplying both sides in (53) by eπt yields:

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 d V (t)eπt ≤ μeπt dt

(54)

Moreover, by integrating (54) over [0, t], we obtain: 0 ≤ V (t) ≤ Since

μ π

μ  μ  −πt e + V (0) − π π

(55)

> 0, it can be obtained that: 0 ≤ V (t) ≤ V (0)e−πt +

μ π

(56)

θi and  θi are UUB. Furthermore, all signals in Therefore, we know that, Si , yi , ϕiT  the closed-loop system, i.e., ϕi , ϕ˙i , xi , xid , x˙id , x¯2 , · · · , x¯i+1 , u and θ i + βi are n  also UUB. In addition, from (30) and (56), it follows that: S = Si2 ≤ i=1   √ −0.5πt 2V (0)e + 2μ π . Accordingly, when t → ∞, it is easy to show that:   S ≤ 2μ π . This completes the proof. 

5 Simulation Results The dynamics of a one-link manipulator actuated by a brush DC motor are described as follows [20–22]: D q¨ + B q˙ + N sin(q) = I M I˙ + H I + K m q˙ = V

(57)

Denote, x1 = q, x2 = q˙ and x3 = I , then, (57) can be rewritten as: x˙1 = x2 x˙2 = f 2 (x¯2 ) + b2 x3 x˙3 = f 3 (x) + b3 u

(58)

H x3 , b3 = M1 where, f 2 (x¯2 ) = − ND sin(x1 ) − DB x2 , b2 = D1 , f 3 (x) = − KMm x2 − M and u = V . The parameters of the one-link manipulator actuated by a brush DC motor are given by [11, 19, 22]: D = 1, B = 1, M = 0.05, H = 0.5, N = 10 and K m = 10. The values of unknown constant parameters are assumed that: θ2 = ND and θ3 = KMm . The objective of this simulation is to design the proposed adaptive controller

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u in such a way that the link  angular position q tracks the reference trajectory [20]: π −0.1t 2 . x1d = 2 sin(t) 1 − e We define the signals x2d and x3d by: x˙2d = x˙3d

1 (−x2d + x˙1d − K 1 S1 − S1 ) τ2

(59)

      B 1 1 −x3d + sin(x1 ) θ 2 + β2 + x2 + x˙2d − K 2 S2 − b2 S2 = (60) τ3 b2 D 

The actual control input u is given by:     H 1 u= x2 θ 3 + β3 + x3 + x˙3d − K 3 S3 b3 M 

(61)

The adaptive laws are defined as:   B   ∂β2 ∂β2 b2 x3 − sin(x1 ) θ 2 + β2 − x2 θ˙ˆ2 = − x2 − ∂ x1 ∂ x2 D   − 2 σθ2 θ 2 + β2 



    ˙θˆ = − ∂β3 x − ∂β3 b x − sin(x ) θ + β − B x 3 2 2 3 1 3 3 2 ∂ x1 ∂ x2 D     H   ∂β3 b3 u − x2 θ 3 + β3 − x3 − 3 σθ3 θ 3 + β3 − ∂ x3 M

(62)







(63)

where, β2 = −2 x2 sin(x1 ) and β3 = −3 x2 x3 . The initial conditions are chosen  T as: x(0) = 0 0 0 , θ 2 (0) = θ 3 (0) = 0 and x2d (0) = x3d (0) = 0. The control parameters are chosen as: K 1 = K 2 = 50, K 3 = 500, 2 = 3 = 0.5, σθ2 = σθ3 = 0.01 and τ2 = τ3 = 10−3 . The simulation results are shown in Figs. 1, 2, 3 and 4. From these Figures, we can obviously see that all states asymptotically converge to their desired values, the ultimate surface errors converge to zero, and the convergence of the parameter estimates to their true values is guaranteed. We can conclude then that the proposed adaptive control method provides better trajectory tracking performances. 



6 Conclusion In this paper, a new design of immersion and invariance based adaptive dynamic surface control for parametric strict-feedback nonlinear systems is proposed. The proposed adaptive control method is designed to avoid the explosion of complexity

Immersion and Invariance Based Adaptive Dynamic Surface Control …

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Angular position [rad]

2

x1d

1.5

x1

1 0.5 0 -0.5 -1 -1.5 -2

0

5

10

15

20

25

Angular velocity [rad/sec]

Time [sec] 2

x2d

1.5

x2

1 0.5 0 -0.5 -1 -1.5 -2

0

5

10

15

20

25

Motor armature current [A]

Time [sec] 10 8 6 4 2 0 -2 -4 -6 -8 -10 0

x3d x3

5

15

10

20

25

Time [sec]

Fig. 1 Trajectories of the output variables 0.3

S1

Surface errors

0.25 0.2

S2

0.15

S3

0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 0

5

10

15

Time [sec]

Fig. 2 Trajectories of the surface errors

20

25

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Parameter estimate

12 10 8 6

θ1 θˆ1

4 2 0

0

5

15

10

25

20

Time [sec] Parameter estimate

250 200 150 100

θ2 θˆ

50 0

2

0

5

10

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Time [sec]

Fig. 3 Trajectories of the parameter estimates

Input control voltage [V]

25

u

20 15 10 5 0 -5 -10 -15 -20 -25

0

5

15

10

20

25

Time [sec]

Fig. 4 Trajectory of the control input

problem. Based on the Lyapunov stability theory, it has been proven that the proposed adaptive control method guarantee the uniformly ultimately boundedness (UUB) of all signals in the closed-loop system and the surface errors converge to a small neighborhood of the origin. The simulation results clearly show the effectiveness of the proposed adaptive control scheme.

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References 1. Kanellakopoulos I, Kokotovi´c PV, Morse AS (1991) Systematic design of adaptive controllers for feedback linearizable systems. IEEE Trans Autom Control 36(11):1241–1253 2. Krsti´c M, Kanellakopoulos I, Kokotovi´c PV (1995) Nonlinear and adaptive control design. John Wiley & Sons, New York 3. Krsti´c M, Kanellakopoulos I, Kokotovi´c PV (1992) Adaptive nonlinear control without overparameterization. Syst Control Lett 19(3):177–185 4. Soukkou Y, Labiod S (2015) Adaptive backstepping control using combined direct and indirect adaptation for a single-link flexible-joint robot. Int J Ind Electron Driv 2(1):11–19 5. Zhou J, Wen C (2008) Adaptive backstepping control of uncertain systems: nonsmooth nonlinearities, interactions or time-variations. Springer-Verlag, Berlin/Heidelberg 6. Hedrick JK, Yip PP (2000) Multiple sliding surface control: theory and application. J Dyn Syst Meas Contr 122(4):586–593 7. Swaroop D, Hedrick JK, Yip PP, Gerdes JC (2000) Dynamic surface control for a class of nonlinear systems. IEEE Trans Autom Control 45(10):1893–1899 8. Swaroop D, Gerdes JC, Yip PP, Hedrick JK (1997) Dynamic surface control of nonlinear systems. In: Proceedings of the American control conference, Albuquerque, New Mexico, pp 3028–3034 9. Yip PP, Hedrick JK (1998) Adaptive dynamic surface control: a simplified algorithm for adaptive backstepping control of nonlinear systems. Int J Control 71(5):959–979 10. Hou MZ, Duan GR (2011) Robust adaptive dynamic surface control of uncertain nonlinear systems. Int J Control Autom Syst 9(1):161–168 11. Li TS, Wang D, Feng G, Tong SC (2010) A DSC approach to robust adaptive NN tracking control for strict-feedback nonlinear systems. IEEE Trans Syst Man Cybern Part B Cybern 40(3):915–927 12. Soukkou Y, Labiod S, Tadjine M (2018) Composite adaptive dynamic surface control of nonlinear systems in parametric strict-feedback form. Trans Inst Meas Control 40(4):1127– 1135 13. Soukkou Y, Labiod S (2017) Adaptive backstepping control using combined direct and indirect σ-modification adaptation. Lecture Notes in Electrical Engineering, vol 411. Springer, Cham, pp 17–30 14. Astolfi A, Ortega R (2003) Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems. IEEE Trans Autom Control 48(4):590–606 15. Astolfi A, Karagiannis D, Ortega R (2008) Nonlinear and adaptive control with applications. Springer-Verlag, London 16. Karagiannis D, Astolfi A (2008) Nonlinear adaptive control of systems in feedback form: an alternative to adaptive backstepping. Syst Control Lett 57(9):733–739 17. Han C, Liu Z, Yi J (2019) Immersion and invariance adaptive finite-time control of air-breathing hypersonic vehicles. Proc Inst Mech Eng Part G J Aerosp Eng 233(7):2626–2641 18. Han C, Liu Z, Yi J (2018) Immersion and invariance adaptive control with σ-modification for uncertain nonlinear systems. J Franklin Inst 355(5):2091–2111 19. Bechlioulis CP, Rovithakis GA (2013) Reinforcing robustness of adaptive dynamic surface control. Int J Adapt Control Signal Process 27(4):323–339 20. Carroll JJ, Dawson DM (1995) Integrator backstepping techniques for the tracking control of permanent magnet brush DC motors. IEEE Trans Ind Appl 31(2):248–255 21. Dawson DM, Carroll JJ, Schneider M (1994) Integrator backstepping control of a brush DC motor turning a robotic load. IEEE Trans Control Syst Technol 2(3):233–244 22. Sun G, Wang D, Li X, Peng Z (2013) A DSC approach to adaptive neural network tracking control for pure-feedback nonlinear systems. Appl Math Comput 219(11):6224–6235 23. Slotine JJE, Li W (1991) Applied nonlinear control. Prentice Hall, Englewood Cliffs, New Jersey

Design of an Adaptive Fuzzy Backstepping Synergetic Control Scheme for a Class of Strict-Feedback Nonlinear Systems Aissa Rebai, Kamel Guesmi, and Mohamed Bougrine

Abstract The aim of this paper is to construct an adaptive fuzzy controller for strict feedback nonlinear uncertain disturbed systems. The backstepping and synergetic control techniques are combined to develop a robust controller for the considered class of systems. In this paper, fuzzy systems are utilized to estimate the unknown behaviors of the system. The proposed approach guarantees that all signals in the closed-loop adaptive control system are semi-globally uniformly ultimately bounded, and the tracking error eventually converges to a small neighbourhood of the origin. Numerical simulations are provided to show the effectiveness of the presented control technique. Keywords Nonlinear systems · Backstepping control · Synergetic control · Fuzzy systems · Adaptive control

1 Introduction During the last decades, the control of nonlinear systems has been widely studied to deal with nonlinear characteristics existing in practical systems. In recent years, various advanced control approaches have been proposed for realistic applications in the literature such as sliding mode control [1–5], backstepping control [6–8], fuzzy logic and neural network control [9–12].

A. Rebai Département Maintenance et Sécurité Industrielle, Institut Algérien de Pétrole, Hassi Messaoud, Algérie e-mail: [email protected] K. Guesmi (B) CReSTIC, IUT de Troyes, 09 rue de Quebec, 10026 Troyes, France e-mail: [email protected] M. Bougrine LACoSERE Lab, Amar Telidji, University of Laghouat, Laghouat, Algeria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_18

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During the last years, an increased attention is paid to backstepping-based control of nonlinear systems due to its efficiency [13, 14]. To cope with the highly uncertain and complex nonlinear systems, neural networks and fuzzy logic theories are used to approximate the nonlinear behaviors and to deal with the tracking and regulation control problems. Up to now, various interesting results have been presented in the literature [15–20]. In [15] a fuzzy adaptive backstepping controller is developed for uncertain nonlinear systems with triangular form using the dynamic surface control technique. This technique is used to construct an adaptive fuzzy backstepping controller for input-delay nonlinear systems [16]. In [17], the authors proposed a fuzzy adaptive output feedback control scheme for switched nonlinear systems with unknown nonlinearities and immeasurable states. The authors in [18] addressed the problem of robust adaptive fuzzy funnel control for nonlinear strict-feedback systems with dynamic uncertainties. Furthermore, in [19, 20], the backstepping and small gain approaches are combined to design adaptive fuzzy controller for uncertain nonlinear systems with unmodeled dynamics. In spite of the good tracking control performances, the controller robustness remains questionable. Since its introduction by the Russian researcher Kolesnikov [21], synergetic control theory has been used in several fields such as power systems [22], electrical systems [23], robotics [24], and other nonlinear systems [25–27]. Compared to sliding mode control approach, the synergetic control theory presents better control performance outside the manifold, and the chattering phenomenon is avoided. Despite the application of synergistic control theory in several research axes, the use of this theory for the development of robust control approaches has always been a challenging yet a rewarding task, which motivate this study. In this work, the backstepping and synergetic control theory are combined to design a new robust adaptive fuzzy backstepping synergetic controller for a class of strict-feedback nonlinear systems. The key contributions of this work lie in the following: (i) a systematical approach to control a class of non-linear strictfeedback systems with external disturbances and parameters uncertainties; (ii) the synergetic control theory to improve the tracking control performances; (iii) fuzzy systems to approximate the nonlinear unknown dynamics and to overcome the design complexity; (iv) a fuzzy system to estimate the external disturbance and to ensure the controller robustness. The outline of this paper is summarized as follows: the problem formulation and some assumptions are given in Sect. 2. The proposed adaptive fuzzy backstepping synergetic control scheme is presented in Sects. 3 and 4. Finally, some simulation results are given in Sect. 5.

2 Problem Formulation and Preliminaries We consider the class of uncertain strict-feedback nonlinear SISO systems represented by:

Design of an Adaptive Fuzzy Backstepping Synergetic Control …

⎧ ⎨ x˙i = xi+1 , i = 1, . . . , n − 1 x˙ = [ f (x) +  f (x)] + [g(x) + g(x)]u + d(t) ⎩ n y = x1

265

(1)

where x = [x1 , x2 , . . . , xn ]T ∈ R n , u ∈ R and y ∈ R are the state vector, the control input, and the system output, respectively. f (x) and g(x) are two unknown non-linear continuous functions to be estimated using fuzzy systems. d(t),  f (x) and g(x) represent the unknown external disturbance and the unknown bounded uncertainties, respectively. Assumption 1 For controllability requirement, the function g(x) must verify gx / = 0 and must be strictly positive or strictly negative. Without loss of generality, we assume that g(x) > 0. For all x ∈ R n , there exists some unknown constants g1 and g2 such that: 0 < g1 < g(x) < g2

(2)

Assumption 2 The desired reference yd and its n first derivatives exist, are continuous and bounded. i.e.    (i)  (3)  yd  ≤ ym i , i = 1, . . . , n Assumption 3 The unknown quantities  f (x), g(x), and d(t) are bounded : ⎧ ⎨ ||d(t)|| ≤ dm || f (x)|| ≤  f m ⎩ ||g(x)|| ≤ gm

(4)

Assumption 4 There exists a large positive constant u m such that ||u(t)|| ≤ u m

(5)

The system (1) can be transformed to the following form: ⎧ ⎨ x˙i = xi+1 , i = 1, . . . , n − 1 ¯ x˙ = f (x) + g(x)u + d(t) ⎩ n y = x1

(6)

¯ is considered as a global disturbance and is given by where d(t) ¯ =  f (x) + g(x)u(t) + d(t) d(t) From Assumptions 3 and 4, we have

(7)

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  d(t) ¯  ≤  f m + gm u m + dm

(8)

The control objective of this work is to design an adaptive fuzzy sliding mode control law to ensure: • all signals in the closed-loop system are semi-globally uniformly ultimately bounded. • the desired trajectory yd (t) can be tracked by the system output y(t) with a neglected tracking error. In this paper, fuzzy logic systems (FLS) are used to identify the unknown nonlinear continuous functions f, g and d¯ defined on some compacts. Consider the following form of fuzzy rules: j

j

j

Ri I f x1 is F1 and x2 is F2 . . . and xn is Fn , then ω is B j ( j = 1, . . . , N )

(9)

where x = [x1 , x2 , . . . , xn ] ∈ R n and ω ∈ R are the FLS input and output, respectively. F ji and B j are the fuzzy sets associated with the membership functions μ F jl (xi ) and μ B j (y), respectively. Since the strategy of singleton output fuzzification, center average defuzzifier and product inference is used, the fuzzy system output can be expressed as:  n y¯ l i=1 μ Fil (x i ) ω(x) =   N n l=1 i=1 μ Fil (x i ) N

l=1

(10)

where ω¯ l = maxω∈R μ B j (ω), μ Fil (xi ) is the membership function value at xi for the fuzzy rule l. The FLS (10) can be reformulated as follows: ω(x) = θ T φ(x)

(11)

T where θ = ω¯ 1 , ω¯ 2 , . . . , ω¯ N is ideal constant weight vector, and φ(x) =

1 T 2 N φ (x), φ (x), . . . , φ (x) is the basis function vector defined by: n

i=1 μ F l (x i ) φ(x) =   i N n l=1 i=1 μ Fil (x i )

(12)

If all membership functions are chosen Gaussian form, the following lemma points out that the FLS (12) can approximate any nonlinear continuous function over a compact set  in any accuracy. Lemma 1 Wang [28]. Let f (x) be a continuous nonlinear function defined on a compact set  ∈ R n . Then, there exists an FLS of form (12) such that

Design of an Adaptive Fuzzy Backstepping Synergetic Control …

  sup  f (x) − θ T φ(x) ≤ ζ

267

(13)

x∈R n

where ζ is a given positive constant.

3 Adaptive Fuzzy Backstepping Synergetic Controller Design The backstepping design procedure consists of n steps [29]. For this procedure, the system (6) is regarded as a series of subsystems. In each design step, we use a virtual control variable αi to stabilize a subsystem using an appropriate Lyapunov function. Then, we calculate the derivative of the tracking error e˙i . The detailed design is described step-by-step as follows. Step 01: The tracking errors e1 and e2 are defined as e1 = x1 − yd e2 = x2 − α1

(14)

The Lyapunov function is chosen as V1 =

1 2 e 2 1

(15)

The derivative of V1 is V˙1 = e1 e˙1 = (x1 − yd )(α1 − y˙d )

(16)

Then, the virtual control variable α1 and the derivative of e1 are given by α1 = −k1 e1 + y˙d e˙1 = e2 − k1 e1

(17)

where k1 is a positive constant. Step 02: The tracking errors e2 and e3 are e2 = x2 − α1 e3 = x3 − α2 The Lyapunov function is chosen as

(18)

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V2 =

1 2 e1 + e22 2

(19)

The derivative of V2 is V˙2 = e1 e˙1 + e2 e˙2 = e1 (e2 − k1 e1 ) + e2 (α2 − α˙ 1 )

(20)

Then, the virtual control variable α2 and the derivative of e2 are obtained as α2 = −e1 − k2 e2 + α˙ 1 e˙2 = e3 − e1 − k2 e2

(21)

where k2 is a positive constant. Step i, i ≤ n − 1: The tracking errors ei and ei+1 are ei = xi − αi−1 ei+1 = xi+1 − αi

(22)

The Lyapunov function is selected as 1 2 e 2 k=1 k i

Vi =

(23)

Then, the virtual control variable αi and the derivative of ei are obtained as αi = −ei−1 − ki ei + α˙ i−1 e˙i = ei+1 − ei−1 − ki ei

(24)

where ki is a positive constant. Step n: The design procedure of a synergetic controller begins by selecting a macrovariable σ which is a function of the system state variables [30]. The control law forces the trajectories to operate on the attractor σ (x) = 0. The desired dynamic evolution of σ is generally chosen such that τ σ˙ + σ = 0, τ > 0

(25)

For our controller design, the tracking error at step n and the macro-variable σ are chosen as en = xn − αn−1 σ = cen−1 + en

(26)

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where c is a positive constant. Then, the control input u is obtained as 1 −c(en − en−1 − kn−1 en−1 ) − f (x) − d¯ + α˙ n−1 g(x)  1 −en−1 − σ τ

u=

(27)

To obtain the adaptive fuzzy backstepping synergetic controller (AFBSC), we ¯ by their fuzzy logic estimations. replace the unknown functions f (x), g(x) and d(t) Then, the proposed control law is expressed as 1  −c(en − en−2 − kn−1 en−1 ) − fˆ(x) − dˆ¯ + α˙ n−1 g(x) ˆ  1 −en−1 − σ τ

u=

(28)

ˆ¯ are given by where fˆ(x), g(x) ˆ and d(t) 

fˆ x, θ f = θ Tf φ f (x)

(29)



gˆ x, θg = θgT φg (x)

(30)



dˆ¯ x, θd¯ = θdT¯ φd¯ (x)

(31)

Finally, the new AFBSC is summarized through Theorem 1. Theorem 1 Consider the nonlinear strict-feedback system (1) with unknown bounded uncertainties and external disturbances. The AFBS controller is designed as described in (28). If the adapting laws are chosen as follows: θ˙ f = γ1 σ φ f (x)

(32)

θ˙g = γ2 σ φg (x)u

(33)

θ˙d¯ = γ3 σ φd¯ (x)

(34)

where γ1 , γ2 and γ3 are the positive learning rates, then the tracking error converges asymptotically to a small neighbourhood of the origin and all signals in the closed loop system are bounded.

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4 Stability Analysis For the stability analysis of the closed system, we consider the Lyapunov functional candidate given by:   n−1 1  2 1 T 1 T 1 T 2 V (t) = e + σ + θ˜ f θ˜ f + θ˜g θ˜g + θ˜d¯ θ˜d¯ 2 l=1 l γ1 γ2 γ3

(35)

where θ˜ f = θ ∗f − θ f θ˜g = θg∗ − θg θ˜d¯ = θd∗¯ − θd¯ where θ ∗f , θg∗ and θd∗¯ are the optimal parameter vectors expressed, respectively, as θ ∗f = arg min



θ f ∈ f

θg∗

 = arg min

θd∗¯

θg ∈g

 

   sup  fˆ x|θ f − f (x, t)

(36)

 sup |g(x|θ ˆ g ) − g(x, t)|

(37)

x∈R n

x∈R n

 = arg min

θd¯ ∈d¯

ˆ¯ ¯ sup |d(x|θ d¯ ) − d(t)|

 (38)

x∈R n

where  f , g and d¯ are constraint sets for θ f , θg and θd¯ , respectively. The time derivative of V (t) is obtained as V˙ =

n−1  l=1

el e˙l + σ σ˙ −

1 T 1 1 θ˜ θ˙ f − θ˜gT θ˙g − θ˜dT¯ θ˙d¯ γ1 f γ2 γ3

(39)

From (26) we have σ˙ = ce˙n−1 + e˙n Using (22), (24) and (26) in (40) we obtain ¯ − α˙ n−1 σ˙ = ce˙n−1 + f (x) + g(x)u + d(t)  ¯ − g(x)u = ce˙n−1 + f (x) + g(x)u + d(t) ˆ + ce˙n−1 + fˆ(x)  1 +en−1 + σ + dˆ¯ τ

(40)

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= −



271



  f (x) − fˆ(x) + g(x) − g(x) ˆ u + d¯ − dˆ¯ − en−1

1 σ τ

(41)

Substituting (41) in (39), it results in V˙ = −

n−1 

kl el2 + en−1 en + σ



f (x) − fˆ(x) + (g(x)

l=1

   1 1 1 ˆ ¯ ¯ −g(x) ˆ u + d − d − en−1 − σ − θ˜ Tf θ˙ f − θ˜gT θ˙g τ γ1 γ2 1 − θ˜dT¯ θ˙d¯ γ3

(42)

The minimum approximation error is described as ε = ε1 + ε2 + ε3 with 

ε1 = f (x) − fˆ x, θ f

(43)





ε2 = g(x) − gˆ x, θg u

(44)

 ε3 = d¯ − dˆ¯ θd ¯

(45)

From (43)–(45), it can be written





f (x) − fˆ x, θ f = θ˜ Tf φ f (x) + ε1

(46)

 g(x) − gˆ x, θg u = θ˜gT φg (x)u + ε2

(47)

 d¯ − dˆ¯ θd¯ = θ˜dT¯ φd¯ + ε3

(48)

Using (46)–(48) in (42) gives that V˙ = −

n−1 

kl el2 + en−1 en − σ en−1 −

l=1

+

θ˜gT φg (x)u

 1 2 σ + σ θ˜ Tf φ f (x) τ

1 1 1 + θ˜dT¯ φd¯ + ε − θ˜ Tf θ˙ f − θ˜gT θ˙g − θ˜dT¯ θ˙d¯ γ1 γ2 γ3

(49)

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Substituting the adapting laws of (32)–(34) in (49) yields that V˙ = −

n−1 

kl el2 + en−1 en − σ en−1 −

l=1

1 2 σ + σε τ

(50)

Replacing (26) in (50) results that V˙ = −

n−1 

2 kl el2 − cen−1 −

l=1

1 2 σ + σε < 0 τ

(51)

The inequality (51) is the optimum result that we can obtain because ε is the minimum approximation error. As a result, we can conclude that all the signals of the closed-loop adaptive control system are semi-globally uniformly ultimately bounded.

5 Numerical Simulations Consider the following inverted pendulum system ⎧ x˙1 =  x2 ⎪ ⎪    ⎪ ⎨ mlx 2 cosx1 sinx1 cosx1 gsinx1 − 2m c +m c +m u + d(t)  x˙2 = +  4 mm(cosx m(cosx )2 )2 ⎪ l 43 − m c +m1 l 3 − m c +m1 ⎪ ⎪ ⎩ y = x1

(52)

where x1 and x2 are the swing angle and swing rate respectively. g = 9.8 m/s2 , mc = 1 kg, m = 0.1 kg, l = 0.5 m are the gravity, mass of the cart, mass of the pendulum and length of pendulum, respectively. u(t) represents the control input. The disturbance in this example is chosen as dt = 0.4 ∗ sin(π/100t). The membership functions of the FLS are given as follows:   (xi + π/6 − ( j − 1)π/12) 2 , μ F j (xi ) = exp − i π/24 i = 1, 2; j = 1, . . . , 5 The parameters of design are chosen as: c = 100, k1 = 10 and τ = 0.01. The initial conditions are defined as [x1 (0), x2 (0)]T = [π/60, 0]T . The desired signal is defined as: yd = 0.1 sin(t). The simulation results are shown on Figs. 1, 2, 3, 4 and 5. Figure 1 plots the responses of output y(t) and the desired reference yd (t). Figure 2 gives the trajectories of the states x1 (t) and x2 (t). It can be seen that the closed loop system is stable.

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0.15 Desired Approach in [31] Proposed approach

0.1

System output y(t)

0.08

0.06

0.05

0.04

0.02

0 0 0

0.2

0.4

0.6

0.8

-0.05

-0.1

-0.15

1

0

2

3

4

5

6

7

8

9

10

Time (sec)

Fig. 1 Trajectories of output y(t) and the desired reference yd (t) 0.5 x (t) 1

x (t) 2

0.4 0.3

System state x(t)

0.2 0.1 0 -0.1 -0.2 -0.3 -0.4

-0.5

0

1

2

3

4

5

6

Time (sec)

Fig. 2 Trajectories of states x1 (t) and x2 (t)

7

8

9

10

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Fig. 3 Trajectory of control input u(t) 0.06 Approch in [31] Proposed approach

Tracking error e(t)

0.05

0.04

0.03

0.02

0.01

0

-0.01

0

1

2

3

4

5

Time (sec)

Fig. 4 Trajectory of tracking error e(t)

6

7

8

9

10

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Fig. 5 Trajectory of macro-variable σ (t)

Figures 3, 4 and 5 show the control signal u(t), the response error e(t) and the macrovariable σ (t), respectively. From these figures, it can be seen that all variables of the considered system are bounded. It is clear from the obtained results that the proposed approach presents good tracking performance with a setting time of 0.033 s compared to 0.467 s for the approach in [31]. The performance index is I T AE = 0.2543 for the proposed approach and I T AE = 14.7487 for the approach in [31]. According to the obtained simulation results, the effectiveness of the proposed control approach is proven.

6 Conclusion In this paper, an adaptive fuzzy control approach has been presented for a class for strict-feedback nonlinear systems. Fuzzy logic systems are used to describe the unknown nonlinear functions existing in the system. Furthermore, based on backstepping and synergetic control strategies, the proposed adaptive fuzzy control scheme guarantees that all signals of the closed-loop system remain bounded and the tracking error converges asymptotically to the origin. Finally, simulation results have provided the effectiveness the proposed control approach.

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References 1. Ray PK, Paital SR, Mohanty A, Eddy FY, Gooi HB (2018) A robust power system stabilizer for enhancement of stability in power system using adaptive fuzzy sliding mode control. Appl Soft Comput 73:471–481 2. Xiuxiang C, Ting W, Yongkun Z, Wen Q, Xinghua Z (2018) An adaptive fuzzy sliding mode control for angle tracking of human musculoskeletal arm model. Comput Electr Eng 72:214– 223 3. Zhao Y, Wang J, Yan F, Shen Y (2019) Adaptive sliding mode fault-tolerant control for type-2 fuzzy systems with distributed delays. Inf Sci 473:227–238 4. Nguyen SD, Vo HD, Seo T-I (2017) Nonlinear adaptive control based on fuzzy sliding mode technique and fuzzy-based compensator. ISA Trans 70:309–321 5. Farahani M, Ganjefar S (2017) Intelligent power system stabilizer design using adaptive fuzzy sliding mode controller. Neurocomputing 226:135–144 6. Baigzadehnoe B, Rahmani Z, Khosravi A, Rezaie B (2017) On position/force tracking control problem of cooperative robot manipulators using adaptive fuzzy backstepping approach. ISA Trans 70:432–446 7. Sadek U, Sarjas A, Chowdhury A, Svecko R (2017) Improved adaptive fuzzy backstepping control of a magnetic levitation system based on Symbiotic Organism Search. Appl Soft Comput 56:19–33 8. Qiao J-F, Hou Y, Zhang L, Han H-G (2018) Adaptive fuzzy neural network control of wastewater treatment process with multiobjective operation. Neurocomputing 275:383–393 9. Wu Y, Huang R, Li X, Liu S (2019) Adaptive neural network control of uncertain robotic manipulators with external disturbance and timevarying output constraints. Neurocomputing 323:108–116 10. Jin L, Li S, Yu J, He J (2018) Robot manipulator control using neural networks: A survey. Neurocomputing 285:23–34 11. de Jesús Rubio J (2018) Discrete time control based in neural networks for pendulums. Appl Soft Comput 68:821–832 12. Sui S, Chen CLP, Tong S (2019) Fuzzy adaptive finite-time control design for non-triangular stochastic nonlinear systems. IEEE Trans Fuzzy Syst 27(1):172–184 13. Wang W, Xie B, Zuo Z, Fan H (2018) Adaptive backstepping control of uncertain gear transmission servosystems with asymmetric dead-zone nonlinearity. IEEE Trans Ind Electron 66(5):3752–3762 14. Roy TK, Mahmud MA, Oo AMT (2019) Robust adaptive backstepping excitation controller design for higher-order models of synchronous generators in multimachine power systems. IEEE Trans Power Syst 34(1):40–51 15. Peng J, Dubay R (2019) Adaptive fuzzy backstepping control for a class of uncertain nonlinear strict-feedback systems based on dynamic surface control approach. Expert Syst Appl 120:239– 252 16. Zhou Q, Wu C, Jing X, Wang L (2016) Adaptive fuzzy backstepping dynamic surface control for nonlinear input-delay systems. Neurocomputing 199:58–65 17. Hou Y, Tong S (2017) Command filter-based adaptive fuzzy backstepping control for a class of switched nonlinear systems. Fuzzy Sets Syst 314:46–60 18. Wang H, Zou Y, Liu PX, Liu X (2018) Robust fuzzy adaptive funnel control of nonlinear systems with dynamic uncertainties. Neurocomputing 314:299–309 19. Su H, Zhang T, Zhang W (2017) Fuzzy adaptive control for SISO nonlinear uncertain systems based on backstepping and small-gain approach. Neurocomputing 238:212–226 20. Zhang X, Liu X, Li Y (2017) Adaptive fuzzy tracking control for nonlinear strict-feedback systems with unmodeled dynamics via backstepping technique. Neurocomputing 235:182–191 21. Kolesnikov AA (2014) Introduction of synergetic control. In: American control conference, Portland, OR, USA, June 4–6 2014, pp 3013–3016 22. Nechadi E, Harmas MN, Essounbouli N, Hamzaoui A (2016) Optimal synergetic control based Bat Algorithm for DC-DC boost converter. IFAC-PapersOnLine 49(12):698–703

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23. Wang C, Zhang D, Zhuang H, Lu B (2018) Coordinated synchronization control of multimotor system based on synergetic control theory. In: Chinese control and decision conference, Shenyang, China, June 9–11 2018, pp 160–164 24. Sklyarov AA, Veselov GE, Sklyarov SA, Pohilina EE (2017) Synthesis of the synergetic control law of the transport robotic platform. In: IEEE II international conference on control in technical systems, St. Petersburg, Russia, October 25–27 2017, pp 285–288 25. Rastegar S, Araújo R, Sadati J, Mendes J (2017) A novel robust control scheme for LTV systems using output integral discrete-time synergetic control theory. Eur J Control 34:39–48 26. Djennoune S, Bettayeb M (2013) Optimal synergetic control for fractional-order systems. Automatica 49:2243–2249 27. Rebai A, Guesmi K, Hemici B (2016) Adaptive fuzzy synergetic control for nonlinear hysteretic systems. Nonlinear Dyn 86:1445–1454 28. Wang LX (1994) Adaptive fuzzy systems and control: design and stability analysis. PrenticeHall, Englewood Cliffs, NJ 29. Zhou J, Wen C (2008) Adaptive backstepping control of uncertain systems: nonsmooth nonlinearities, interactions or time-variations. Springer, Berlin, Heidelberg 30. Rastegar S, Araujo R, Sadati J (2018) Robust synergetic control design under inputs and states constraints. Int J Control 91(3):639–657 31. Wang J, Rad AB, Chan PT (2001) Indirect adaptive fuzzy sliding mode control: part I: fuzzy switching. Fuzzy Sets Syst 122(1):21–30

Improvement of the Stability Performance of a Quad-Copter Helicopter by a Neuro-Fuzzy Controller Djalal Baladji, Kheireddine Lamamra, and Farida Batat

Abstract UAVs are quad-copter unmanned helicopters, They been the subject of growing importance and are widely used in several fields; for this reason and to improve their performance, in this article we have proposed an algorithm based on the ANFIS. The genetic algorithms is used in order to optimize the ANFIS parameters and thereby ensure its learning, which maintains the ability for self-organization and self-learning. The proposed control scheme aims to implement good capabilities such as the description of qualitative knowledge, a learning mechanism and direct processing of quadcopter helicopter quantitative data. This control is adopted the precision and time of adjustment. On the other hand, If the position and attitude deviation becomes relatively smaller, The PID command will be used to limit this error. Experimental results indicate that the proposed ANFIS control algorithm has good performance in the flight process. Keywords Quad-rotor helicopter · Stability performance · Neural network control (ANFIS) · Fuzzy control · Flight angle deviation

1 Introduction Recently, The quad-copter helicopter has attracted attention because of its very good flexibility, its ability to perform various air missions even in difficult conditions. The quad-copter helicopter can adopt a variety of flight attitudes, which is attributed to

D. Baladji (B) · K. Lamamra · F. Batat Department of Electrical, Engineering, University of Oum El Bouaghi, Oum El Bouaghi, Algeria e-mail: [email protected]; [email protected] K. Lamamra e-mail: [email protected]; [email protected] F. Batat e-mail: [email protected]; [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_19

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the efficient engine speed controller with about four propellers. In order to enable the quad-copter helicopter to better complete its flight mission, flight stability performance becomes particularly important. A neuron-fuzzy control algorithm (ANFIS) is proposed in this article to ensure the stability performance of the quadcopter helicopter. In recent years, some current research has focused on the fuzzy and neural network; for example, Qu et al. [8] has addressed the design of an autopilot for an autonomous UAV using a draft genetic algorithm for an evolution on the rules of fuzzy limb functions. Roopashree et al. [11] explained the design of a compact, accurate and economical system. Fuzzy logic controllers (FLC) and fuzzy inference systems (FIS), which estimated the attitude of drones. Sabo and Cohen [13] proposed a methodology for two-dimensional data management [1–3]. In order to improve the system’s performance characterization, a recurrent online neural network modeling for the dynamic uncertainty of the four-rotor unmanned aerial (QUAV) was used to develop an under-operated sliding mode control based on a recurrent neural network (Hwang 2012) [5, 21]. A new adaptive neural control scheme for stabilizing quadcopter helicopters in the presence of sinusoidal turbulence has been proposed (Boudjedir et al. 2012) [6, 8]. Due to the excellent performance of the fuzzy system and the neural network, fuzzy control of the neural network has been used in several areas; for example, He and Dong [2] conducted research for adaptive control of the neural fuzzy network (NN) by learning [12, 17, 18]. The above control strategies, with the aim of suggesting a flexible control scheme based on piloting errors, which uses the advantages of the ANFIS system, so that regulation is effective on the four rotors of the four helicopter quad-copters.

2 Quad-Copter Four-screw helicopters are four-engine quad-copters mounted on a cross, usually made of carbon fibre, hence their name quad-copters. Examples are shown in Fig. 1. In a quad-copter, the front and rear engines rotate clockwise while the right and left engines rotate counter-clockwise. The propellers used are fixed-pitch. The pitch is obtained by a difference in the rotational speed of the front and rear rotors. The roll is obtained in a similar way with the speed difference of the lateral engines. Yaw is achieved by increasing the speed of the front and rear engines while reducing the speed of the side engines. There is also a version of X4 in which all rotors rotate in the same direction. In the latter, two rotors are tilted horizontally to create a yaw torque. The inclination can be fixed or variable and allows to control the movement of the yaw. Since quadri-rotors are controlled by the difference in rotor speed, it is important that the motor speed can be varied quickly. For this purpose, very light blades and ratios should be used. • The structure of the quad-copter is assumed to be rigid and symmetrical, which means that the inertia matrix will be assumed to be diagonal.

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Fig. 1 Four-propeller helicopters

• Les hélices sont supposées rigides pour pouvoir négliger l’effet de leur déformation lors de la rotation. • Le centre de masse et l’origine du repère lié à la structure coïncident. • The lift and drag forces are proportional to the squares of the rotors’ rotational speed, which is a very close approximation of aerodynamic behaviour. To evaluate the mathematical model of the quad-copter, two benchmarks are used, a fixed benchmark linked to the ground Rb and another mobile Rm. The passage between the moving reference frame and the fixed reference frame is given by a matrix called transformation matrix T which contains the orientation and position of the moving reference frame with respect to the fixed reference frame. The following axis convention is chosen: (Fig. 2)  T =

Rξ 0 1



with R the rotation matrix (describes the orientation of the moving object), ζ = [x y z]T is the position vector. To determine the elements of the rotation matrix R, Euler angles are used. A. Euler’s Angles: At the beginning the moving reference mark is coincident with the fixed reference mark, after the moving reference  mark makes a rotational movement about the x-axis by a roll angle − π2 < φ < π2 , followed by rotation about the y-axis at a pitch angle   π − 2 < φ < π2 , followed by a rotation about the z axis of angle of (−π < φ < π ). so we have the formula of the rotation matrix R : R = Rotz (ψ) + Roty (θ ) + Rotx (φ)

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Fig. 2 Géométrie du quadcopter

⎡

⎤ ⎡ ⎤ ⎡ ⎤ cψ −sψ 0 cθ 0 sθ 1 0 0 = ⎣ sψ cψ 0 ⎦ × ⎣ 0 1 0 ⎦ × ⎣ 0 cφ −sφ ⎦ 0 0 1 −sθ 0 cθ 0 sφ cφ ⎤ ⎡ cψcθ sφsθ cψ − sψcφ cφsθ cψ + sψsφ R = ⎣ sψcφ sφsθ sψ + cψcφ cφsθ sψ − cψsφ ⎦ −sθ sφcθ cφcθ with : c = cos and s = sin B. Angular speeds: Rotation speeds 1, 2, 3 in the fixed reference frame are expressed as a function of the rotation speeds ψ, θ, φ in the moving reference frame, we have: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 0 0 φ˙ −1 ⎣ ˙ ⎦ −1 ⎣ ⎣ ⎦ ⎣ ⎦  = 2 = 0 + Rotx (φ) θ + (Rot y (θ )Rotx (φ)) 0⎦ 0 ψ˙ 3 0 Indeed, the rotation in roll takes place when the marks are still confused. Then, with regard to pitch, the vector representing rotation must be expressed in the fixed reference frame: it is therefore multiplied by Rot x (φ)−1 . Similarly, the vector representing the yaw rotation must be expressed in the fixed reference frame which has already undergone two rotations. This brings us to:

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283

⎤ . ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ . ⎤ ⎡ 1 φ˙ 0 −ψsθ φ˙ − ψsθ ⎥ ⎢ ⎥ ⎢ ˙ ˙ − ψsφcθ  = ⎣ 2 ⎦ = ⎣ 0 ⎦ + ⎣ θ˙ cφ ⎦ + ⎣ ψsφcθ ˙ ⎦ = ⎣ φcφ ⎦ ˙ ˙ 3 0 −θ˙ sφ ψcφcθ ψcφcθ − θ˙ sφ ⎡ ⎤ ⎡ ⎤ 1 0 −sθ φ˙ ⎣ ⎦ ⎣  = 0 cφ sφcθ × θ˙ ⎦ 0 −sφ cφcθ ψ˙ When the quadcopter makes small rotations, the following approximations can be made: cθ + cφ + cψ = 1 and sθ + sφ + sψ = 0 so the angular velocity will be:

 = φ˙ θ˙ ψ˙ C. Linear speeds: Linear speeds Vxb , Vyb , Vzb in the fixed reference mark as a function of linear speeds ⎡ b⎤ ⎡ m⎤ Vx Vx Vxm , Vym , Vzm in the moving reference frame are given by: V = ⎣ Vyb ⎦ = R× ⎣ Vym ⎦ Vzb

Vzm

3 ANFIS Commande (Adaptive Network-Based Fuzzy Inference System) ANFIS (Adaptive Network Based Fuzzy Inference System) it is a neuro-vague adaptive system of inference that consists in using a network neuron of type MLP in 5 layers for who every layer corresponds to the realisation of a stage of a system of vague inference of type Takagi Sugeno. For simplicity, we assume that the vague system of inference at two entries x and there, and in as an exit fr. Assume that the usual base contains two vague rules of type Takagi-Sugeno. Rule 1: SI x is A1 and y is B1 THEN f1 = p1 x + q1 y + r1 Rule 2: SI x is A2 and y is B2 THEN f2 = p2 x + q2 y + r2 L’ANFIS à une architecture posée par cinq couches comme représenté sur la Fig. 3.

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Fig. 3 Example ANFIS with 2 entries with 9 rules

The ANFIS system applies the learning mechanism of neural networks to fuzzy inference techniques. Another term ANFIS is a fuzzy inference system (SIF) whose parameters of the membership functions are adjusted using the backpropagation learning algorithm, or in combination with another type of algorithms such as the least square. In the ANFIS architecture proposed in Fig. 3, the overall output can be expressed as linear combinations of the consequent parameters. More precisely, the conclusion (the output) in Fig. 3 can be rewritten as: W2 W1 f1+ f2 w1 + w2 w1 + w2             = W 1 x p1 + W 1 y q1 + W 1 r 1 + W 2 x p2 + W 2 y q2 + W 2 r 2

f=

The output is a linear function of the consequence parameters (p, q, r). ANFIS is parametric representation two sets of parameters: S1 and S2 ANFIS uses a learning cycle of two passages: • Passage forward: S1 is fixed and S2 is calculated using the least square error algorithm (LSE). (The LSE is applied only once when starting to get the initial values of the consequent parameters). • Passback: S2 is fixed and S1 is calculated using the Retro-Propagation algorithm Ek error measurement for the given Kiemme input learning: Ek =

n(L)   2 di − xL,i i=1

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4 Application of ANFIS on Quadcopter The ANFIS controllers are taught along the three axes by the least squares method to estimate the consequent parameters and the gradient descent algorithm to determine the parameters of the premises (adjustment of the parameters related to the belonging functions). The genetic algorithms is used here to choose the learning parameters of the ANFIS. This is called “blended learning”. The ANFIS network used in this work was programmed by the logic toolbox fuzzy Toolbox with which we built the fuzzy rules (Si-Then) with their appropriate membership functions to four together, while respecting the following steps: 1. Using the genetic algorithms to choose the ANFIS learning parameters • The number of iterations = 20. • The number of membership functions = 5. • Error tolerance = 10−4 . 2. Start learning using the ANFIS parameters. 3. Stop when the tolerance is satisfactory. 4. Test the results with another data set. The equivalent neural structure proposed to generate the vertical flight control is presented in Fig. 4 we have the same structure for the other two axes: x and y to generate the commands u2x and u2y respectively. The characteristics of the ANFIS model used to control the XSF UAV in all three axes are as follows:

Fig. 4 Neural structure of the proposed ANFIS controller to generate u3

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Ɵme(s) diplacement follow (z)

Ɵme(s)

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diplacement follow (y)

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Fig. 5 Reference and actual path of the straight connections (ANFIS command)

• • • • • •

Number of nodes: 75. Number of parameters of the premises: 30. Total number of parameters: 105. Number of learning data pairs: 25,001. Number of test data pairs: 25,001. Number of fuzzy rules: 25.

We can see from Fig. 5 that the trajectory of the XSF UAV follows the desired trajectory (the reference) and this is shown in Fig. 6 whose errors tend towards zero. From Fig. 7, we can see that the ANFIS method ensures the flight of our UAV according to the desired trajectory (at equilibrium u 3 = mg and the commands u 2x = u 2y = 0) but nevertheless it does not keep the readability of the rules.

4.1 Robustness Study Figures 8 and 9 illustrate the simulation results for an opposite drag force of 2 N and 4 N respectively, applied after take-off at time t = 8 s in the direction of vertical flight. The zoom of the vertical flight shows very well the influence of the disturbance at t = 8 s. From Figs. 8 and 9, we can see that the variation of the error is proportional to the variation of the disturbance: the more the tained force increases, the more the static error increased with ocsillations. From Figs. 10 and 11, we notice that the error of the ANFIS controller is much smaller than that of the fuzzy PD controller but nevertheless the control by the ANFIS controller presents ocsillations which increases with the increase in drag force which can disturb the drone.

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Ɵme(s)

verƟcal flight (m)

Fig. 6 The commands u 3 , u 2x and u 2y for the next movement x, y and z

Ɵme(s)

Ɵme(s)

Ɵme(s)

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Fig. 7 The influence of wind with a drag force, F_(trn) = 2 N (ANFIS command)

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Ɵme(s)

Ɵme(s)

Ɵme(s)

Ɵme(s)

Fig. 8 The influence of wind with a drag force, F_(trn) = 2 N (ANFIS command)

Ɵme(s) Fig. 9 The influence of wind with a drag force of 2 N on the error of the fuzzy PD and ANFIS controller

Ɵme(s) Fig. 10 The influence of wind with a drag force of 4 N on the error of the fuzzy PD and ANFIS controller

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diplacement follow (y)

diplacement follow (x)

Fig. 11 Cone ascent

4.2 Generation of a Complex Trajectory (a Cone) In this part, we generate a trajectory based on rational fractions in order to have trajectories without decoupling from an initial point, with a zero speed, and reach a well determined final point with the same zero speed. Next, we explicitly present the expressions of the position variables that generate a cone, as well as a figure illustrating its shape. x r (t) = ρ(t)sin(α(t)) y r (t) = −ρ(t)sin(α(t)) t5 z r (t) = h dz  5 t5 t f − t with: t f = 28 s and hdz = 5 m. α=

αr e f t 5  5 t5 t f − t

with: αr e f = 4 ρ = hr

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Fig. 12 Realization of a cone by the ANFIS controller

where hr = 2 m. Figure 12 shows that our control system ensures the continuation of the complex trajectory. We notice that the errors tend towards zero so the desired position is reached.

5 Conclusion It appears from this work that the use of an architecture called “ANFIS” is a simple and effective way to obtain by learning a powerful controller whose behaviour can be interpreted in the form of decision rules. We have found that the use of a simplified structure allows us to obtain an efficient command law by online learning. The main role of the genetic algorithm, is to choose the learning parameters of the ANFIS. The controller results used (ANFIS) shows the validity of our approach for ordering our system. So these techniques have been successfully applied to design control algorithms that move the drone from an initial position to a desired equilibrium position. The simulation results confirmed the controller performance used.

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References 1. Ali MS, Gunasekaran N, Zhu Q (2017) State estimation of T-S fuzzy delayed neural networks with Markovian jumping parameters using sampled-data control. Fuzzy Sets Syst 306:87–104 2. He W, Dong Y (2017) Adaptive fuzzy neural network control for a constrained robot using impedance learning. IEEE Trans Neural Netw Learn Syst 29(4):1174–1186 3. Jin Z, Chen J, Sheng Y, Liu X (2017) Neural network based adaptive fuzzy PID-type sliding mode attitude control for a reentry vehicle. Int J Control Autom Syst 15(1):404–415 4. Lin M-Y, Zheng C-D (2017) Novel stability conditions of fuzzy neural networks with mixed delays under impulsive perturbations. Optik 131:869–884 5. Mi B, Liu D (2017) A fuzzy neural approach for vehicle guidance in real–time. Intell Autom Soft Comput 23(1):13–19 6. Mushage BO, Chedjou JC, Kyamakya K (2017) Fuzzy neural network and observer-based fault-tolerant adaptive nonlinear control of uncertain 5-DOF upper-limb exoskeleton robot for passive rehabilitation. Nonlinear Dyn 87(3):2021–2037 7. Prasad M, Lin C-T, Li D-L, Hong C-T, Ding W-P, Chang J-Y (2015) Soft-boosted selfconstructing neural fuzzy inference network. IEEE Trans Syst Man Cybern Syst 47(3):584–588 8. Qu Y, Pandhiti S, Bullard KS, Potter WD, Fezer KF (2011) Development of a genetic fuzzy controller for an unmanned aerial vehicle. In: International conference on industrial, engineering and other applications of applied intelligent systems, pp 328–335 9. Raffo GV, Ortega MG, Rubio FR (2015) Robust nonlinear control for path tracking of a quad-rotor helicopter. Asian J Control 17(1):142–156 10. Rodríguez-Canosa GR, Thomas S, Del Cerro J, Barrientos A, MacDonald B (2012) A real-time method to detect and track moving objects (DATMO) from unmanned aerial vehicles (UAVs) using a single camera. Remote Sens 4(4):1090–1111 11. Roopashree S, Deepika KM, Bhat S (2011) Design and implementation of fuzzy controller for estimating the attitude of unmanned aerial vehicles. In: International joint conference on advances in signal processing and information technology, pp 255–261 12. Ruan J, Chen X, Huang M, Zhang T (2017) Application of fuzzy neural networks for modeling of biodegradation and biogas production in a full-scale internal circulation anaerobic reactor. J Environ Sci Health Part A 52(1):7–14 13. Sabo C, Cohen K (2012) Fuzzy logic unmanned air vehicle motion planning. Adv Fuzzy Syst 2012:13 14. Sun L, Zuo Z (2015) Nonlinear adaptive trajectory tracking control for a quad-rotor with parametric uncertainty. Proc Inst Mech Eng Part G J Aerosp Eng 229(9), 1709–1721 15. Sun W, Gao H, Kaynak O (2010) Finite frequency H_∞ control for vehicle active suspension systems. IEEE Trans Control Syst Technol 19(2):416–422 16. Sun W, Gao H, Kaynak O (2014) Vibration isolation for active suspensions with performance constraints and actuator saturation. IEEE ASME Trans Mechatron 20(2):675–683 17. Yao J, Deng W, Jiao Z (2015) Adaptive control of hydraulic actuators with LuGre model-based friction compensation. IEEE Trans Ind Electron 62(10):6469–6477 18. Yao J, Jiao Z, Ma D, Yan L (2013) High-accuracy tracking control of hydraulic rotary actuators with modeling uncertainties. IEEE ASME Trans Mechatron 19(2):633–641 19. Yao J, Jiao Z, Ma D (2014) Extended-state-observer-based output feedback nonlinear robust control of hydraulic systems with backstepping. IEEE Trans Ind Electron 61(11):6285–6293 20. Yao J, Jiao Z, Ma D (2015) A practical nonlinear adaptive control of hydraulic servomechanisms with periodic-like disturbances. IEEE ASME Trans Mechatron 20(6):2752–2760 21. Zhao D-J, Yang D-G (2016) Model-free control of quad-rotor vehicle via finite-time convergent extended state observer. Int J Control Autom Syst 14(1):242–254

The Importance of Applying Artificial Intelligence on Unmanned Aerial Vehicle Amine Mohammed Taberkit, Ahmed Kechida, and Abdelmalek Bouguettaya

Abstract Unmanned Aerial Vehicles (UAVs) are used in several applications and they are growing in popularity. Recent progress in unmanned aerial vehicles and artificial intelligence constitutes a new chance for an autonomous operation and flight. Nowadays, artificial intelligence and deep learning are driving the evolution of UAVs and fueling their autonomous future. Computer vision achieved very important progress in image classification and segmentation, and object detection, which make it very attractive research field when it is applied on unmanned aerial vehicle. Artificial intelligence is not only important and benefic, but can be rather, dangerous and serious matter since the UAVs learns through algorithms, and use that for future decision making. This work is a survey, where we present works, challenges and dangerous part of using artificial intelligence on UAVs. Keywords UAV · Machine learning · Artificial intelligence · System · Drone

1 Introduction An unmanned aerial vehicle (UAV) is defined as an aircraft, without any presence of a pilot on board. UAVs can be used to execute observation or detection (objects and/or persons) missions [1] through automatic or remote control. Taking aerial records have two main advantages, low cost and high movability. The reference standards developed by the UAV community are based on several parameters such as: flight altitude, endurance, speed, maximum payload and others [2]. The three main components of an UAV system are: the aircraft with common or other sensor features, Ground control station, and operator. There are a wide collection of UAV depending on their shapes, functionalities, configurations, and characteristics. Their hardware A. M. Taberkit (B) · A. Kechida · A. Bouguettaya Research Center in Industrial Technologies CRTI, 16014 Cheraga, Algiers, Algeria e-mail: [email protected] A. Bouguettaya e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_20

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Fig. 1 Some quadcopter UAVs developed in CRTI research center

and software design can be modified with the task requirements. Quadcopter UAVs are one of the most favored kinds of small unmanned aerial vehicles because of their very simple mechanical construction and propulsion principle [3]. The civilian and military applications in several fields have got a drastic increase during the last years [4]. Some examples encompass the inspection of power line [5], wildlife conservation [6], building inspection [7], precision agriculture [8] and military surveillance [9]. In our research center we developed many Quadcopter UAVs, two among them were presented in the university Salah BOUBNIDER, Constantine 3 (Fig. 1).

2 Machine Learning Machine Learning is defined as the capability allowing Artificial Intelligence (AI) systems to learn through data. A suitable definition for what learning covers is as follow: “A computer program is said to learn from experience E with respect to some of class of tasks T and performance measure P if its performance at tasks in T, as measured by P, improves with experience E” [4–10]. It can be defined as an evolving field of computational algorithms that are designed to mimic the human intelligence by learning through the surrounding environment [11]. This ability is the main key to develop successful machine learning. Historically, the inception of machine learning can be traced to the seventeenth century. Pascal and Leibniz [12] developed machines that can emulate ability of add and subtract. Arthur Samuel, from IBM, invented the term “Machine Learning” and demonstrated the computer ability of playing checkers [13]. In 1958, Rosenblatt

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developed the perceptron and it was one of the early neural networks [14]. A great development was made in 1975 with the development of Werbos multilayer perceptron (MLP) [15], and development of decision trees by Quinlan in 1986 [16]. Several works based on Machine learning algorithms have been proposed, including Adaboost [17] and random forests [18]. Deep learning (DL) in artificial intelligence (AI) has recently won a significant interest. Deep learning is applied in many fields such as autonomous systems, facial recognition, classification, and object detection. Among the most promising systems that can exploit deep learning are unmanned aerial vehicles (UAVs). Convolutional Neural Networks (CNNs) is class of deep neural networks, and mostly applied to analyze visual imagery. CNN is considered as very good and powerful tool in the classification and object detection [19]. The structure of a CNN typically contains a feature extractor stage followed by a classifier [20]. Many object detectors have been proposed by the deep learning researchers, including R-CNN [21], R-FCN [22], YOLO [23] and SSD [24]. Many auspicious CNN architectures were recently proposed, such as CaffeNet [25] and GoogLeNet [26]. In Fig. 2, we represent the evolution of some object detectors over the years. CNN architecture includes many layers of different types [27]: • Convolutional layers, as their name indicates, they compute the convolution of the input image with the weights of the network [27]. • Pooling layers, the role of these layers is to diminish the size of the input layer using some local non-linear operations. • Normalization layers, the aim of using those layers is to improve generalization of the CNN. Neurons used in these layers are sigmoid [28]. • Fully-connected layers, these layers are used in the last levels of the network [28]. We distinguish three different categories: supervised, unsupervised, and reinforcement learning (Fig. 3). The reinforcement learning (RL) methods allow an agent to learn suitable actions with little or without knowledge about its environment, and it can be used to adapt to randomly changing environmental conditions [29, 30]. Reinforcement learning is actually very used in Robotics [31]. Despite the fact that deep learning has been extremely successful in many applications; it has remained limited to applications in which useful features can be created manually [32] or applications with fully observed low dimensional state space. Volodymyr et al. [32] used deep neural network to develop a new artificial agent, termed a deep Q-network, this agent can learn successful policies from high-dimensional input. In this work they demonstrated that a single architecture with very essential prior knowledge such as pixels and the game score as inputs can successfully learn control policies at human-level control. Alejandro et al., developed a deep reinforcement learning strategy for UAV autonomous landing on a moving platform [33]. Compared to supervised learning, the amount of feedback the learning system obtains in reinforcement learning is much less [34]. We can measure the learning

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Fig. 2 Development of some objects detectors. Main developments in chronological order are: R-CNN, SPPNet, Fast-RCNN, Faster RCNN, RFCN, FPN, Mask RCNN [66]

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Fig. 3 Different fields and sub-fields of machine learning

performance by the number of correct answers, resulting in predictive accuracy. The difficulty comes from the possibility that learning can generalize new unclassified examples. In supervised and unsupervised learning, the data is usually considered static, and it is not the case in RL, where we notice a stochastic nature, this issue can be solved thought a constant interaction between evaluation and improvement of policies, and the use of learning rate adaption schemes [34]. Additionally, in front of some problems, it can be useful to provide agent with rewards for reaching intermediate sub-goals [34]. Christiano and et al., reported that we need to communicate complex goals to get sophisticated reinforcement learning systems, in their study they show that’s possible to solve complex RL tasks without access to the reward function [35]. The most universally used machine learning methods are supervised learning methods, and we found them in several applications such as spam e-mail filtering, face recognition and identification, and medical helping diagnosis [36]. In supervised learning, the agent observes some input-output, and learns a map. The agent thought knowledge of the input; achieve the output labeling [37]. If the output is a discrete number of possible “classes”, this is called a classification problems, if the output is continue it is called a regression [38]. In unsupervised learning, we haven’t labeled output the agent focuses on observing patterns without labeled data. The aim is to search pattern structure and features embedded with data [39]. Clustering is an example of unsupervised learning. By eyeballing data, the agent will be able to

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discover the existence of many apparent clusters, then classify each data into the corresponding one [38].

3 Machine Learning Applied to UAVs for Autonomous Flight UAV can be used in many different services, in the military context, the research are focused on enhancing the autonomy, with different kinds of missions [40]: • • • •

Suppression of enemy air defense. Air-to-ground targeting scenario. Surveillance and reconnaissance. Avoidance of danger zones. In the civilian context, UAVs could be used for purposes such as:

• • • • • •

Weather forecast/Disaster management [41, 42, 67]. Urbane police surveillance. Agriculture production management [43]. Border surveillance and inspection of infrastructures [44]. Road Traffic Monitoring (RTM) [20]. Emergency Response (ER) system [20].

Let us first cite what John Wyndham said: “The man and machine are naturally complementary” [45]. One of the current issues in AI is to make the link between heuristics used by human and programming. Another question is to know how to use the large data collected from various drone sensors. Many questions are pushing UAV research: how does a machine fly an aircraft like a human can, can a machine really “think” enough to fly autonomously? [46]. Beside those questions, UAVs have some challenges for control, real-time path planning and object recognition beneath uncertain environments [37]. To solve these problems, many approaches have been proposed such as negotiation approach, a heuristic approach, and graph theory [47]. Machine learning is an attractive approach to overcome these challenges for autonomous flight. It allows recognizing patterns or predicting from data [37]. Machine learning has contributed several fields of UAVs applications. Figure 4 shows the time-line of previous studies covered below [37]. Among the control strategies applied to an autonomous flight, we recognize the parameter tuning and real time path planning and navigation. To compensate the limitations of the Proportional, Integral Derivative (PID) control systems technique, and operate in unpredictable and harsh environments, reinforcement learning (RL) is an active and successful area of research [48]. In reinforcements learning (RL) an agent is given a reward for every action it makes in

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Fig. 4 Timeline of previous studies on machine learning applied on UAVs [37]

an environment with the objective to maximize the rewards. When we use reinforcement learning, it is possible to develop some optimal control policies for a UAV, and avoid making any assumptions about the aircraft dynamics [48]. Real-time path planning and navigation have been considered as elementary and required in UAVs for autonomous flight [37]. Many researchers have exploited and work on this area of research [49, 50, 68]. Loquerico et al. focused on civilian drones, because controlling the UAVs can become difficult when we work in urban areas, in those cases, the autonomous agent is not only expected to navigate while avoiding collisions, but to interact safely with other agents such as pedestrians or cars [51]. In this research they proposed a model that learns to navigate by initiating cars and bicycles and respecting the traffic rules. Collision avoidance is also a very important research track, which can lead to new and very interesting technologies. Lei et al. [52] proposed a restructured Q-value learning algorithm based on reinforcement learning. The autonomous navigation for large unmanned aerial vehicles (UAVs) is not problem; it is the micro and small UAVs, which fly at low altitude in crowded environments that meet many challenges in this area [53]. Ross et al., presented a small quadrirotor that navigates autonomously at low altitude through natural forest environments, They presented a MAV that autonomously fly at speeds of up to 1.5 m/s and altitudes of up to 4 m above the ground. They developed model based on the knowledge of a human pilot avowing collision with trees and they succeeded in more than 680 trees [53]. However, this method is limited when we need to perform longer flights and denser forests or any cluttered environments, The reactive method is limited in case of narrow field view, and causes most of time failures. Junell et al., proposed a high level reinforcement learning algorithm for UAVs to avoid collisions [54], they studied autonomous flight of UAV in unknown or uncertain environments, by taking pictures from disaster site. This method is efficient to solve the conflict between limited battery life and the big number of required iterations. However, it is

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limited in finding an efficient rout to the destination, and learning the best recharging point, in the aim of having the highest level of battery when it reaches the destination. Nusrsultan.I and al, achieved experimental study of autonomous navigation by using real-time model based reinforcement learning [29], which solved some limitations of the previous method [54]. Moreover, deep learning contributed in many applications, such as construction of database for aerial image classification in emergency response, and development of suitable CNN training strategy with low-computational and low-cost low-power [55]. Dmitriy et al., worked also on emergency response, but this time was about wildfires detection using unmanned aerial vehicles and they compared several methods in term of speed and accuracy [56]. Cherian et al. proposed a semi-supervised machine learning (SSL) algorithm to estimate the altitude of UAVs using top down aerial images [57]. The basic idea of this study is to learn mapping between the texture information contained in an image with a possible altitude. However this approach is only suited for low altitudes and low speeds. Manjia Wu et al., proposed an approach to detect intruding drones in sensitive areas, in real time using deep learning [58]. Most of cited studies are based on supervised learning, which control known problems; it is suggested to aim researches on unsupervised learning methods [37]. Reducing time spent on classification is another issue, Gonzalo-Martin et al., proposed a strategy to reduce the time spent on the classification trough another method called “superpixel” segmentation. The results evince that while the classification accuracy was identical to the results generated by pixel-based approach, the time spent is dramatically decreased [28]. In the work of Gonzalo et al. [28], we didn’t notice the tests of this method on unmanned aerial vehicle unlike the work of Taro Suzuki et al. [59] who used this method in vegetation classification. In classification approach based on CNN, training is achieved using big dataset [29], which require high performance equipment for processing. None of actual commercial drones, own sufficient control autonomy to achieve missions without human skills, which makes the missions slow, dangerous and not scalable [60]. In an interview with BBC, Bill Gates discussed AI and he told: “I am in the camp that is concerned about super intelligence … That should be positive if we manage it well. A few decades that thought the intelligence is strong enough to be a concern.” Tesla’s Elon Musk believes that we should be very concerned when it comes to AI. According to Stephen Hawking AI could be harmful to humans: “I think the development of full artificial intelligence could spell the end of the human race”. Legal regulation should give to autonomous flight, great attention to eliminate the danger for people and things [61]. The existence of moving obstacles push the researchers to equip the drone with additional sensors such as: “Sonars” and “Passive infrared sensors” [61]. At the same level of concern, drones damages during flight couldn’t be acceptable, and a check of components state and weather conditions should be performed. A mistake and danger come from flights under bridges, inside tunnels, or near high-voltage power lines, which could lead to GPS data errors, and by result drone flight errors [62].

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In the other hand, the scope of permissible self-help in defending the privacy should be broad-ranging [63]. The worst scenario is when a drone escapes with intrusive recordings that can be a major harm. Another issue concerns the images of detectable individuals captured by aerial surveillance which should not be retained or shared except if it contains criminal suspicion activity [2]. Froomkin and Colangelo, suggested measures including forbidding weaponized robots, and mandating RFID chips and serial numbers, in the aim of identifying the robot’s owner [63]. The United States has had a monopoly over the use of drones, but cannot maintain that much more [64]. These new weapons will not transform the international system, as did the fast increase of nuclear weapons and ballistic missiles, they still be highly dangerous and deadly [64]. In USA, the Federal Aviation Administration (FAA) predicted that 30,000 drones could be flying in sky in the next 20 years [65]. When the numbers of UAVs increase, many accidents can happen in the sky, that is why several models was proposed to make the process easy and safe [65].

4 Conclusion UAVs are expected to be more used, as there is considerable demand in all sectors (private and public). UAVs have the potential to be used in huge number of applications. Many solutions are suggested such as: adding sensors, enhancements of data processing and others can expand their use. Artificial intelligence is the most important solution, to boost their performances and allowing them to be autonomous. However, the privacy, security and cognitive aspect should not be ignored. In this work we presented the UAVs, their application, than we presented machine learning and its application on UAVs, finally we spoke about the aspects of security and privacy which are solvable.

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Fuzzy H∞ Delay-Independent Stabilization of Depth Control for Underwater Vehicle with Input Constraint Mohamed Nasri, Dounia Saifia, Salim Labiod, and Mohammed Chadli

Abstract This paper investigates the depth control of unmanned underwater vehicle (UUV) in the presence of external disturbances, time delay, and actuator saturation. Firstly, a Takagi–Sugeno model is elaborated based on the speed variation. Then, using the polyquadratic representation with Lyapunov-Krasovskii function, an H∞ criteria is applied to design a new LMI based stabilization to calculate PDC controller gains taking into consideration the input limitation. Finally, simulation results demonstrate the superiority of the proposed approach. Keywords Takagi-Sugeno model · Delay-independent stability · H∞ performance · Actuator saturation · Lyapunov Krasovskii function

1 Introduction Over last decade, unmanned underwater vehicle have take a growing attention. Indeed, the high ability of UUV devices to replace humans tasks, allowing to various sectors such as: military, scientific research …etc. [1, 2] to use it. Particularly in missions where human life is in danger. However, these works cannot be achieved without implementation of precise depth controller for UUV. For this raison, many studies have developed various depth control strategies such as: sliding mode controller [3], fuzzy controller [4], adaptive controller [2]. M. Nasri (B) · D. Saifia · S. Labiod Faculty of Science and Technology, LAJ, University of Jijel, Ouled Aissa, 18000 Jijel, Algeria e-mail: [email protected] D. Saifia e-mail: [email protected] S. Labiod e-mail: [email protected] M. Chadli IBISC Lab, University of Paris-Saclay, 40 rue Pelvoux, Evry 91020, France e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_21

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Among depth control strategies, the approaches based on Takagi-Sugeno models [5], have been largely utilized [6, 7], either parallel distributed compensation (PDC) or static output feedback (SOF) controllers. In any case, T-S-based approaches have many advantages; especially have facilitated maneuvering in control by the implementation of dedicated techniques for linear systems. The depth control challenge consists at stabilizing the UUV under deferent obstacle like: sea waves, water current, time delay…etc. for this raison, the previous studies, have been focused on solving these problems (for example, see [4, 6]. However, the input limitation, which is the hydroplane angle, has never been taken into account. Indeed, actuator saturation is an important factor to the performances degradation of closed-loop systems; in some cases, it can drive the system to instability [8]. This influence motivate the scientists to research a control approach that can guarantee the closed-loop performances in the presence of input constraints [9, 10]. In this context, the saturation nonlinearity can be modeled by sector nonlinearity, by dead zone representation, or by polytopic representation [8]. Among control solutions, the nicknamed saturated controller [11], gives better stabilization performances, even though, it can cause chattering. Besides, the stabilization problem of T-S systems with time delay is very challenging. Indeed, time delay is often encountered in many practical systems [6, 12]. Generally, the studies for time delay systems can be classified into two categories, the first one uses delay-independent criteria, and the second one uses delay-dependent criteria [13]. For our case, we rely on the first criterion to deduce the stabilization conditions of the closed-loop system. Motivated by the above discussion, this paper investigates the depth control for UUV system in the presence of external disturbances, time delay, and actuator saturation. The nonlinear model of the UUV is first expressed as a Takagi-Sugeno model. Using polyquadratic form with Lyapunov Krasovskii function and H∞ criterion, a new LMI stabilization conditions are derived. Finally, simulation results are given to demonstrate the effectiveness of the proposed approach. The rest of this paper is organized as follows. Section 2 gives the system modeling. Section 3 gives the T-S representation of the UUV system. Saturated controller for delayed UUV system is developed in Sect. 4. In Sect. 5, simulation results are given. A conclusion in Sect. 6 finishes the paper.

2 Mathematical Modeling of Under Actuated UUV System The body fixed coordinate system and the earth coordinate system are depicted in Fig. 1, where x, y, z are the Cartesian position of UUV, and φ, θ , ψ are the vehicle altitude with respect to the earth fixed coordinate system. u, v, w are surge, sway, heave velocities. p, q, r are the angular velocities (roll, pitch, yaw).

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Fig. 1 The reference of the UUV

Based on the works in [6], the mathematical model of the depth control system can be simplified as follows Heave equation of motion:   m w˙ − uq − x G q˙ − z G q 2 = Z q˙ q˙ + Z w˙ w˙ + Z uq uq+ Z uw uw + (W − B0 ) cos(θ ) + Z uu u 2 δs

(1)

Pitch equation: I yy + m[x G (uq − w) ˙ + z G wq] = Mq˙ q˙ + Mw˙ w˙ + Muq uq +Muw uw − (x G W − x B B0 ) cos(θ )− (z G W − z B B0 ) sin(θ ) + Muu u 2 δs

(2)

Pitch and depth: z˙ = w cos(θ ) − u sin(θ ) θ˙ = q

(3)

where (x G , yG , z G ) and (x B , y B , z B ) are the centre of gravity and buoyancy. I yy , Z {.} , M{.} are the constant parameters of UUV, B0 is the buoyancy of the UUV; W is the weight of the UUV; u is the speed of UUV; δs is the hydroplane angle. Now, let us define: ze = zd − z

(4)

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with z d is the desired depth value. By assuming that: • (x G , yG , z G ) = (x B , y B , z B ) = 0 • The heave velocity w has little effect on the depth control sin(θ ) = θ, cos(θ ) = 1 From the above equations and assumptions, we can give the mathematical model of depth control system as follow: ⎧ ⎨ z˙ e (t) = u(t)θ (t) θ˙ (t) = θ (t) ⎩ q(t) ˙ = a(t)q(t) + b(t)δs (t) M u(t)

with: a(t) = I yyuq−Mq˙ , b(t) = T  Let x = z e θ q Therefore,

(5)

Muu u 2 (t) I yy −Mq˙

x(t) ˙ = A(t)x(t) + B(t)δs (t)

(6)

where ⎡

⎤ 0 u(t) 0 A(t) = ⎣ 0 0 1 ⎦, 0 0 a(t)

⎡

⎤ 0 B(t) = ⎣ 0 ⎦ b(t)

A. Dynamic model with time delay Based on the time-delay model of Network Control Systems NCS in [14], the mathematical model of depth control can be expressed as follows [6]: ⎧ ˜ ˜ ˙ = Ax(t) + A˜ d x(t − τ ) ˙ = Ax(t) + A˜ d x(t − τ ) + Bδs (t) + Bϕ ϕ(t)x(t) ⎨ x(t) + Bδs (t) + Bϕ ϕ(t) ⎩ x(t) = η(t), t ∈ [−τ, 0] (7) where A˜ = β A(t) and A˜ d = (1 − β)A(t), β is the steady coefficient of the depth control system, ϕ(t) is the external perturbations, and η(t) is the initial state of the system.

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3 Takagi-Sugeno Model for the Depth Control System In this section, a T-S model is developed based on the sector nonlinearity approach [15]. In this context, unlike previous works [6], the UUV speed u is supposed time varying. The T-S fuzzy representation of a delayed continuous system is complies with the following IF-THEN form: Rulei : if ς1 (t) is about M1i and … and ςq (t) is about M1q 

x(t) ˙ = A˜ i x(t) + A˜ di x(t − τ ) + Bδs (t) + Bϕ ϕ(t) y(t) = C x(t)y(t) = C x(t)

(8)

In our case the nonlinearity in the depth control system is the UUV speed which is time varying and bounded (u min ≤ u(t) ≤ u max ). Consequently, by applying the sector nonlinearity method, the T-S fuzzy model of the depth control system is expressed as follows: ⎧  8

⎪ ⎪ x(t) ˙ = μ A˜ i x(t) + A˜ di x(t − τ )+ (ς (t)) i ⎪ ⎪ ⎨ i=1  Bδs (t) + Bϕ ϕ(t) ⎪ ⎪ ⎪ y(t) = C x(t) ⎪ ⎩ ψ(t) = C1 x(t)

(9)

where y(t) is the output and ψ(t) is the controlled output. μi (ς (t)) are the membership functions, which are calculated as follows: Let us write, V1min ≤ u(t) = V1 ≤ V1max V2min ≤ u 2 (t) = V2 ≤ V2max

(10)

Using sector nonlinearity method, we get F11 = F21 =

V1 −V1min , V1max −V1min V2 −V2min , V2max −V2min

F12 = 1 − F11 F22 = 1 − F21

(11)

Consequently, the membership functions can be written as: μ1 = F11 F21 , μ2 = F11 F22 μ3 = F12 F21 , μ4 = F12 F22 with

(12)

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⎤ 0 0 βV1max ⎥ ⎢ 1 A˜ 1 = ⎣ 0 0 ⎦; Muq 0 0 β I yy −Mq˙ V1max ⎤ ⎡ 0 0 βV1max ⎥ ⎢ 1 A˜ 2 = ⎣ 0 0 ⎦; Muq 0 0 β I yy −Mq˙ V1max ⎤ ⎡ 0 0 βV1min ⎥ ⎢ 1 A˜ 3 = ⎣ 0 0 ⎦; Muq 0 0 β I yy −Mq˙ V1min ⎤ ⎡ 0 0 βV1min ⎥ ⎢ 1 A˜ 4 = ⎣ 0 0 ⎦; Muq 0 0 β I yy −Mq˙ V1min ⎤ ⎤ ⎡ ⎡ 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 B1 = ⎣ ⎦; B2 = ⎣ ⎦; Muu Muu V V I yy −Mq˙ 2max I yy −Mq˙ 2min ⎤ ⎤ ⎡ ⎡ 0 0 ⎥ ⎥ ⎢ ⎢ 0 0 B3 = ⎣ ⎦; B4 = ⎣ ⎦; Muu Muu V V 2max 2min I yy −Mq˙ I yy −Mq˙ ⎤ ⎡ 0 0 (1 − β)V1max ⎥ ⎢ 0 1 A˜ d1 = ⎣ 0 ⎦; Muq 0 0 (1 − β) I yy −Mq˙ V1max ⎤ ⎡ 0 0 (1 − β)V1max ⎥ ⎢ 0 1 A˜ d2 = ⎣ 0 ⎦; Muq V 0 0 (1 − β) I yy −M 1max q˙ ⎤ ⎡ 0 0 (1 − β)V1min ⎥ ⎢ 0 1 A˜ d3 = ⎣ 0 ⎦; Muq 0 0 (1 − β) I yy −Mq˙ V1min ⎤ ⎡ 0 0 (1 − β)V1min ⎥ ⎢ 0 1 A˜ d4 = ⎣ 0 ⎦. Muq V 0 0 (1 − β) I yy −M 1min q˙ ⎡

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4 Fuzzy H∞ Delay-Independent Stabilization of UUV System Under Input Constraint In this section, the goal is to design a PDC controller to guarantee the closed-loop stability of the depth control system in the presence of time delay, actuator saturation, and external disturbances. A. Saturation modeling In this work, the polytopic representation of saturation will be used. Let us consider that the input saturation is σ (U ), where is a nonlinear function of its input defined as σ (U ) = sat(U ) = sat(δs )

(13)

  Now, let us define the set ℵ H j as follows        j  ℵ H j = x(t) ∈ R n / h i x  ≤ U¯ i

(14)

j

with, H j is a matrix m × n, h i is the ith row of the H j , n is the number of the state variables, m is the number of the control inputs and U¯ i is the saturation level of the control input signal. The polytopic representation of the saturation is described as follows: ⎧ 2m  

⎪ ⎪ σ = ηs E s U + E¯ s ν (u) ⎪ ⎪ ⎪ s=1 ⎪ ⎨ r

μ j (ς (t))H j x(t) ν(t) = ⎪ j=1 ⎪ ⎪ ⎪ 2m

⎪ ⎪ ⎩ ηs = 1, 0 ≤ ηs ≤ 1

(15)

i=1

with, E s represents all components of E, E¯ s = I − E s and E s ∈ {0, 1} In this case, the PDC control law is given by U (t) =

r 

μi (ς (t))K i x(t)

(16)

i=1

Therefore, the closed-loop depth control system is given by ⎧ ⎨

   x(t) ˙ = A T + B2T E T K T + E¯ T HT x(t) + AdT x(t − τ ) + BϕT ϕ(t) ⎩ ψ(t) = z(t) = C1T x(t)ψ(t) = z(t) = C1T x(t) with

(17)

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AT = BT =

r

i=1 r

μi (ς (t))Ai , BϕT = μi (ς (t))Bi , C1T =

i=1

r

μi (ς (t))Bϕi ,

i=1 r

μi (ς (t))C1i

i=1

B, H∞ performance The H∞ criterion allows us to give the results attained. It is defined by T

T ψ (τ )ψ(τ )dτ < γ T

ϕ T (τ )ϕ(τ )dτ

2

0

(18)

0

It is equivalent to the following optimization problem: 

minδ V˙ (t) + ψ T (t)ψ(t) − γ 2 ϕ T (t)ϕ(t) < 0

(19)

With V˙ (t) is the derivative of the Lyapunov-Krasovskii function (LKF), which is chosen based on the delay-independent stability technique. The contribution here is that the LKF is depending on the membership function, which gives less conservatism stability conditions and more relaxation. In this context, the new LKF is described as  r −1 t  μi Pi x(t) + x T (s)x(s)ds V (t) = x (t) T

i=1

(20)

t−τ

with Pi = PiT > 0,  = T > 0. For a constant ρ > 0 and a symmetric positive definite matrix P, we define an ellipsoid as   ε(P, ρ) = x ∈ n / x T P x ≤ ρ

(21)

Lemma 1 Henrion [16] The ellipsoid ε(P, ρ) is included in the   and Tarbouriech polyhedron  = x/ m iT x ≤ λi , i ∈ Im if and only if  (m i )

T

P ρ

−1

m i ≤ λi2

(22)

Theorem 1 For all k = 1, . . . , r − 1; μ˙ k (t) ≥ φk , the closed-loop system (17), is asymptotically stable with an attenuation level of disturbances γ . if there exist matrices Q i = Q iT > 0, S = S T > 0, matrices F j ∈ R m×n , Z j ∈ R m×n , satisfying the following LMIs:

Fuzzy H∞ Delay-Independent Stabilization of Depth Control …

min ⎡ γ r −1   φk (Q k − Q r ) ⎢ Ai Q i + Bi E s F j + Bi E¯ s Z j + (∗) + S − ⎢ k=1 ⎢ Q i AdTi ⎢ ⎢ ⎣ BϕT C1 Q i ⎤ ∗ ∗ ∗ −(1 − α)S ∗ ∗ ⎥ ⎥ 0

(12)

The system (12) is the diffusive canonic realization of operator H. The impulse response h(t) is expressed from h(t) by: +∞ h(t) = e−ξ t μ(ξ )dξ

(13)

0

The transfer function of the operator H is given by: +∞ H ( p) = 0

μ(ξ ) dξ p+ξ

(14)

The transfer function associated with the diffusive representation. In other words, μ(ξ) the representation is obtained directly by the inverse Laplace transformation of the impulse response h(t) La The diffusive representation is therefore a “countersense” use of the Laplace transform: Lξ

Lt

μ(ξ ) → h(t) → H ( p), μ(ξ ) = L−1 (h(t))

(15)

For example, the diffusive symbol of the fractional integrator s of order α is H

1 ∂ ou´ H ( p) = a for 0 < a < 1 ∂t p

The diffusive symbol is expressed as (see [31]): μ(ξ ) = ξ −α

sin(απ) ξ >0 π

(16)

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where α is the order of integration. And the diffusive realization of the controller P I a with input u and output y may expressed by [21]: ⎧ ⎨ ∂t ψ(ξ, t) = −ξ ψ(ξ, t) + u(t) ⎩

+∞

y(t) = ∫ v(ξ )ψ(ξ, t)dξ

(17)

0

the transfer function of the fractional controller P I a by means of diffusive representation are: +∞

C(s) = ∫

−∞

v(ξ ) dξ p+ξ

(18)

where v the diffusive representation know as follows:

1 sin(απ) −α ξ v(ξ ) = K P δ(ξ ) + Ti π

(19)

C. Fractional Order P I a Controller The differential equation of the fractional order P I a (FO-PI or P I a ) is given by:

1 u(t) = K p e(t) + D −a e(t) Ti

(20)

where: D → is the derivative operation, K P → the proportional gain TI → is the integral time constant, α → is the integration order and the transfer function of the fractional order PI (FO-PI) controller is given by: C(s)frational = K P 1 +

1 TI s a

= KP +

KI = K P + K I s −a sa

as: K I = K P /TI

(21)

If a = 1. we get classical PI controller

1 KI = KP + C(s)classical = K P 1 + = K P + K I s −1 TI s s

(22)

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4 Roposed Design Method Figure 1 shows the diagram of the decentralized fractional P I a controller implemented on a multivariable system (a Pilot Plant Binary Distillation Column) in the presence of a decoupler. The mathematical model is identified by Vinaya and Arasu for a Pilot Plant Binary Distillation Column [27] is expressed by the following transfer matrix:  G( p) =

−0.13e−0.03s 0.18e−0.03s 1.14s+1 0.64s+1 −0.34e−1.22s 0.18e−0.03s 1.23s+1 0.32s+1

 (23)

The transfer function matrix of decoupler is:  D(s) =

1 1.888(0.32s+1)e−1.19s 1.23s+1

1.3846(1.14s+1) 0.64s+1



1

(24)

The resulting diagonal elements of H (s) = G(s)D(s) = diag(h 11 (s), h 22 (s)) are: h 11 (s) =

0.3398(0.32s + 1)e−1.22s −0.13e−0.03s + 1.14s + 1 (0.64s + 1)(1.23s + 1)

(25)

h 22 (s) =

0.47076(1.14s + 1)e−1.22s 0.18e−0.03s + 0.32s + 1 (1.23s + 1)(0.64s + 1)

(26)

The decoupled systems (h 11 (s) and h 22 (s)) is controlled by two fractional PI controllers F O − P I1 and F O − P I2 (see in Fig. 2). The transfer functions of the two fractional controllers are given by (21).

Fig. 1 Schematic diagram of the decentralized fractional P I a controller designed with decoupler for a pilot plant batch distillation column

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Fig. 2 Diagram illustrating the proposed design method

Thus the transfer function of the two fractional P I a1 and P I a2 controllers by means of diffusive Approach are: +∞

F O − P I1 = ∫

−∞ +∞

F O − P I2 = ∫

−∞

v1 (ξ ) dξ p+ξ

(27)

v2 (ξ ) dξ p+ξ

(28)

Our objectives are to setting the six optimal parameters (K p1 , K i1 , K p2 , K i2 α1 , α2 ) of the decentralized fractional PI controllers that minimizing the performance index error defined by objective function that may be given as follows

 ∞ JIAE, ISE and ITAE K p1 , K i1 , K p2 , K i2 α1 , α2 = ∫ f (e(t))dt

(29)

0

With f is a function of the error e(t) that based to the criteria (ISE, IAE and ITAE). These three commonly used measures: Integral Squared Error (ISE), Integral Absolute Error (IAE) and Integral Time weighted Absolute Error (ITAE), and are defined as: ∞

∞

∞

0

0

0

I AE = ∫ |e(t)|, I T AE = ∫ t|e(t)|dt, I S E = ∫ e(t)2 dt

(30)

The steps to determine the fractional PI controllers parameters can be summarized as follows:

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• Implement the feedback control system of controlled system with fractional PI controller in Matlab/Simulink including diffusive realization of the fractional operator through Simulink model. • Select the appropriate function of Matlab optimization toolbox to minimize the objective function J. • The stability margin based Non-Dimensional Tuning method (NDT) [10] is used for determine the initial controller parameters. • Calculate the error (IAE, ISE and ITAE). For more details of proposed design method (see in Fig. 2). The clasical PI controller based on the NDT-PI and SIMC-PI methods is designed using Non-Dimensional Tuning method and Simplified Internal Model Control technique respectively [10–32], the methods can be summarized in the following steps: • decoupled subsystems (h 11 (s) and h 22 (s)) are reduced to first order plus dead time (FOPDT) model by fitting of frequency response For each FOPDT reduced subsystem.

Q iiFOPDT (s) =

K ii e−L ii s Tii s + 1

(31)

where : i = 1, 2; K ii = process gain; L ii = delay time; Tii = time constant • The Classical PI controllers (C − P I1 and C − P I2 ) are given as follows

C − P I1 = K p1 + K i1

1 1 , C − P I2 = K p2 + K i2 s s

The Parameters of the PI controllers are tuned by NDT method as follows. K pii K ii =

Tii 1 + 2L ii 14

(32)

TI ii 1 L ii L ii = min 1 + ,9 Tii 7 Tii 14

(33)

• also for determining PI parameters by SIMC technique as follows  K pii =

Tii 1 K ii Tcii + L ii

 (34)

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TI ii = min(Tii , 4(Tcii + L ii ))

(35)

For more details of NDT and SIMC design methods (see in [10, 11]).

5 Simulation and Results As an simulation studie, we consider the mathematical model identified by Vinaya and Arasu for a Pilot Plant Binary Distillation Column [11, 26, 27] given by matrix transfer function, see in Eq. (23). The results and performance of proposed fractional PI controller compared with Conventional PI controller such NDT-PI [10] and SIMCPI [32], is discussed using MATLAB/Simulink. The optimized parameters of the fractional PI controller (FO-PI) based on ISE criterion were organized in Tables 1 and 2. Attached with the classical controllers (NDT-PI and SIMC-PI). The results of the performance (IAE, ISE, ITAE, Overshoot (%) and Settling time (sec)) were organized in Table 3. The two Figs. 3 and 4, shows the Step response of the first and second outputs (Y1 and Y2) of the controlled system based on design method of decentralized fractional and classical controllers. Figures 5 and 6 shows the result for output response (first and second outputs (Y1 and Y2)) of closed loop with different setpoint for a Distillation Column. With fractional PI (FO-PI) and classical PI (NDT and SIMC) controllers. The photographic view of the experimental setup [11] of a pilot plant binary distillation column (MIMO process) is as shown in Fig. 7. To demonstrate the efficiency of the proposed design method, we compare our results with those obtained using deentralized classical PI controller NDT-PI and Table 1 Tuning parameter for first output (Y1) P I1 controller

Parameter K P1

KI1

α1

FO-PI1

1.9278

0.010479

0.13567

NDT-PI1

1.9473

1.19

–

SIMC-PI1

1.6071

1.1904

–

Table 2 Tuning parameter for second output (Y2) P I2 controller

Parameter K P2

KI2

α2

FO-PI2

−0.9786

−0.01038

1.1451

NDT-PI2

−1.0665

−1.116

–

SIMC-PI2

−0.8202

−1.116

–

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Table 3 Performance analysis results for FO-PI, NDT-PI, and SIMC-PI controllers Performance

Input(u)–Output(y)

Method

IAE

u 1− y1 u 2− y2

3.55

3.951

4.431

ISE

u 1− y1

4.561

4.797

4.945

u 2− y2

3.725

3.948

4.063

FO-PI

ITAE

NDT-PI

SIMC-PI

5.946

6.375

5.123

u 1− y1

15.06

u 2− y2

6.14

23.22

Overshoot %

u 1− y1

6.2

u 2− y2

5.01

23.2

23.95

Settling time (s)

u 1− y1

8.02

15.12

17.15

u 2− y2

7.45

8.11

8.54

8.134 26

28.55 11.44 27.42

1.5

Y1

1

0.5

FO-PI Controller NDT-PI Controller

0

-0.5

SIMC-PI Controller

0

20

40

60

80

100

Time (s)

Fig. 3 Step responses for first output (Y1) of the controlled system with fractional PI (FO-PI) and classical PI (NDT and SIMC) controllers 1.5

Y2

1

0.5

FO-PI Controller NDT-PI Controller SIMC-PI Controller

0

-0.5

0

20

60

40

80

100

Time (s)

Fig. 4 Step responses for second output (Y2) of the controlled system with fractional PI (FO-PI) and classical PI (NDT and SIMC) controllers

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100 80

Y1 and Y2

60 40 FO-PI Controller method of Output Y1

20

FO-PI Controller method of Output Y2 NDT-PI Controller of Output Y1

0

NDT-PI Controller of Output Y2 SIMC-PI Controller of Output Y1

-20 -40

SIMC-PI Controller of Output Y2

0

20

60

40

80

100

Time (s) Fig. 5 Step responses for the first and second outputs (Y1 and Y2) of controlled system with Setpoint-reference1 = 50 and Setpoint-reference2 = 70 100

Y1 and Y2

80 60 40 FO-PI Controller method of Output Y1 FO-PI Controller method of Output Y2 NDT-PI Controller of Output Y1 NDT-PI Controller of Output Y2 SIMC-PI Controller of Output Y1 SIMC-PI Controller of Output Y2

20 0 -20 -40

0

20

60

40

80

100

Time (s) Fig. 6 Step responses for the first and second outputs (Y1 and Y2) of controlled system with Setpoint reference1 = 64 and Setpoint-reference2 = 76

SIMC-PI) are given as follows: Figs. 3, 4 and Table 1 show clearly that the decentralized fractional controllers give a good performance (settling time and Overshoot) in comparison with classical one. It is observed also that different performance indices (IAE, ISE and ITAE) as objective functions, with fractional PI controller gives very good and superior results When compared to the classical controller (C-PI), see Table 3. It is clear that error indices and peak overshoot are less in proposed method. Figures 5 and 6 shows the result for output response of closed loop with different setpoint for controlled system (a Distillation Column). We can say that the use of the decentralized fractional PI controller with decoupler provides good performance and robustness.

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Fig. 7 Laboratory setup of binary distillation column

6 Conclusion In this paper, decentralized fractional PI controller with decoupler is designed and implemented for a Pilot Plant Binary Distillation Column (multivariable process), The problem of assigning the optimum possible fractional controllers parameters for multivariable system using Matlab optimization toolbox is addressed based on minimizing the performance indices criterion (IAE, ISE and ITAE). simulation results show clearly that the time response performances (settling time and maximum overshoot) given by of the controlled system with fractional-order controller are give a good performance and acceptable in comparison with the classical PI controller such (Non-Dimensional Tuning (NDT) method and Simplified Internal Model Control (SIMC) technique).We can say that the performance of the fractional PI controller based on the diffusive Approach is clearly superior to the performance of classical PI controllers.

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References 1. Lee J, Kim D, Edgar T (2005) Static decouplers for control of multivariable processes. Am Inst Chem Eng 51(10):2712–2720 2. Wang QG, Huang B, Guo X (2000) Auto-tuning of TITO decoupling controllers from step tests. ISA Trans 39:407–418 3. Nordfeldt P, Hagglund T (2006) Decoupler and PID controller design of TITO systems. J Process Control 16(9):923–936 4. Ram DV, Chidambaram M (2015) Simple method of designing centralized PI controllers for multivariable 155 systems based on SSGM. ISA Trans 56:252–260 5. Majaaz V, Bhat S, Thirunavukkarasu I, Shanmuga Priya S (2015) Centralized controller tuning for MIMO process with time delay. In: IEEE Xplorer, 4th international conference on renewable energy research and applications, Palermo, Italy, pp 660–664 6. Hajare VD, Patre BM (2015) Decentralized PID controller for TITO systems using characteristic ratio assignment with an experimental application. ISA Trans 59:385–397 7. Hu W, Cai WJ, Xiao G (2010) Decentralized control system design for MIMO processes with integrators/differentiators. Ind Eng Chem Res 49(24):12521–12528 8. Labibi B, Marquez HJ, Chen T (2009) Decentralized robust PI controller design for an industrial boiler. J Process Control 19(2):216–230 9. Maghade DK, Patre BM (2012) Decentralized PI/PID controllers based on gain and phase margin specifications for TITO processes. ISA Trans 51(4):550–558. https://doi.org/10.1016/ j.conengprac.2005.06.006 10. Tavakoli S, Griffin I, Fleming PJ (2006) Tuning of decentralised PI (PID) controllers for TITO processes. Control Eng Pract 14(9):1069–1080. https://doi.org/10.1016/j.conengprac. 2005.06.006 11. Santhosh Kumar PL, Selva Kumar S, Thirunavukkarasu I, Bhat VS (2018) Decentralized PI controllers with decoupler for the distillation column. Int J Pure Appl Math 118(20):9–14 12. Vinagre BM, Feliú V (2002) In: Proceedings: 41st IEEE conference on decision and control, Las Vegas, 9 December 2002 13. Podlubny I, Dorcak L, Kostial I (1997) On fractional derivatives, fractional-order dynamic system and PID-controllers. In: Proceedings of the 36th conference on decision & control, vol 5, pp 4985–4990 14. Çelik V, Demir Y (2010) Effects on the chaotic system of fractional order PI α controller. Nonlinear Dyn 59(1–2):143–159 15. Sabatier J, Agrawal OP, Tenreiro Machado JA (2007) Advances in fractional calculus: theoretical developments and applications in physics and engineering. Springer, the Netherlands 16. Monje CA, Chen YQ, Vinagre BM, Xue DY, Feliu V (2010) Fractional-order systems and controls: fundamentals and Applica-tions. Springer, London 17. Podlubny I (1999) Fractional order systems and PI λ Dμ controllers. IEEE Trans Autom Control 44:208–214 18. Boudjehem D, Sedraoui M, Boudjehem B (2013) A fractional model for robust fractional order smith predictor. Nonlinear Dyn 73:1557–1563 19. Boudjehem B, Boudjehem D (2013) Fractional order controller design for desired response. J Syst Control Eng 227:243–251 20. Aydogdu O, Korkmaz M (2019) Optimal design of a variable coefficient fractional order PID controller by using heuristic optimization algorithms. Int J Adv Comput Sci Appl (IJACSA) 10(3) 21. Boudjehem B, Boudjehem D (2016) Fractional PID controller design based on minimizing performance indices. IFAC-PapersOnLine 49(9):164–168 22. Cajo R, Muresan CI, Ionescu R, Keyser D (2018) Plaza-multivariable fractional order pi autotuning method for heterogeneous dynamic systems. IFAC-PapersOnLine 51-4:865–870 23. Edet E, Katebi R (2018) On fractional-order PID controllers. In: 3rd IFAC conference on advances in proportional integral-derivative control Ghent, Belgium, 9–11 May, IFACpapersOnline 51-4:739–744

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24. Lakshmanaprabu SK, Sabura Banu U, Hemavathy PR (2017) Fractional order IMC based PID controller design using Novel Bat optimization algorithm for TITO process. Energy Proc 117:1125–1133 25. Gargi B, Somanath M, Chitralekha M (2018) Auto-tuning of FOPI controllers for TITO processes with experimental validation. Int J Autom Comput. https://doi.org/10.1007/s11633018-1140-0 26. Bhat VS, Thirunavukkarasu I, Janani R (2017) Design and implementation of MSC based multi-loop pid controller for pilot plant binary distillation column. In: International conference on circuits power and computing technologies [ICCPCT]. IEEE 27. Bhat VS, Thirunavukkarasu I, Priya SS (2016) Design and implementation of decentralized pi controller for pilot plant binary distillation column. Int J ChemTech Res 10(2):284 28. Podlubny I (1999) Fractional differential equations. Academic Press, California 29. Oldham KB, Spanier J (1974) The fractional calculus, theory and applications differentiation and integration to arbitrary order. Elsevier. ISBN 0486450015 30. Montseny G (2004) Simple approach to approximation and dynamical realization of pseudo differential time operators such as fractional ones. Int EEE Trans Circuits Syst II 51:613–618. https://doi.org/10.1109/TCSII.2004.834544 31. Laudebat L, Bidan P, Montseny G (2004) Modeling and optimal identification of pseudo deferential dynamics by means of diffusive representation part i: modeling. Int EEE Trans Circuits Syst 51:1801–1813. https://doi.org/10.1109/TCSI.2004.834501 32. Skogestad S (2003) Simple analytic rules for model reduction and PID controller tuning. J Process Control 13:291–309

Direct Sliding Mode Control of Transient Power in Microgrid During Grid Failure “Unintentional Islanding” I. Ameur, N. Gazzam, and A. Benalia

Abstract An important feature of a Microgrid is the ability to disconnect or connect to the main grid with a transparent transfer of power between the two modes. During a sudden grid loss (due to defects on the main grids), the Microgrid passes to the Unintentional Islanding Mode (UIM). In this case, a transient power in the AC bus increase the voltage at the DC link and it can turn off the inverter thus reduces the reliability of the Microgrid. This article proposes a new control strategy based on the Sliding Mode Control (SMC) in order to regulate the DC voltage and improve the performance of the Microgrid when the UIM occurs. Finally, the simulation results in the MATLAB/ Simulink environment made it possible to check the robustness of the proposed controllers. Keywords Microgrid · Unintentional islanding · DC link voltage · Sliding mode controller

1 Introduction Today, the rising fuel prices and the environmental constraints (the global warming, CO2 emissions) encourage various governments around the world to integrate Renewable Energy Sources (RES) in the electrical networks. From this, the concept of a Microgrid has emerged and started to develop especially in order to reduce the cost and the losses of energy in the transport. A Microgrid comprises a set of renewable energy systems (solar energy, wind energy …) or non-renewable energy (diesel …), in the form of a distributed generation (DG), and a storage means (battery, I. Ameur (B) · N. Gazzam · A. Benalia Faculty of Technology, LACoSERE Lab, University of Laghouat, Laghouat, Algeria e-mail: [email protected] N. Gazzam e-mail: [email protected] A. Benalia e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_24

347

348 PV array

I. Ameur et al. DC/DC converter Ipv_dc DC- Link

Ipv

DC/AC inverter

+ Vdc -

+ Vpv -

U_inv1

photovoltaic Controller

Voltage Controller Vac_ref

Secondary Control

photovoltaic Controller U_PV2

DC/AC inverter

DC- Link

Ipv_dc

+ Vdc -

U_bat1

P2

Load

Vac

Voltage Controller Vac_ref

U_bat2

Battery Controller

LCL Filter

U_inv2

DC/DC bid.converter

+ Vbat Battery

Utility Grid

Primary Control

+ Vpv -

Ibat

S.T.S

Droop Controller

P 1*

Secondary Control Ipv

P1 Vac

U_PV1

PV array

LCL Filter

P 2*

Droop Controller Primary Control

AC-Link

Secondary Control

Fig. 1 Typical structure of microgrid adopted

storage tank of H2…) associated with charges (small of consumers) [1]. Our Microgrid is composed of three distributed generation units, two photovoltaic systems and a battery bank as an Energy Storage System (ESS). These three elements are connected to the grid via a Static Transfer Switch (STS), as illustrated in Fig. 1. The reliability of Microgrid is afected when one or several renewable energy sources are used, due to their low and intermittent efficiency, since they are associated with climate change (temperature, sunlight …), and therefore storage (ESS) is necessary to balance any mismatch between load demand and the available power of DG sources. In the Microgrid, it is important to share the power to satisfy the load demand. The droop control is used to improve the power sharing between the inverters of the distributed generators that connect in parallel in a Microgrid. This sharing is made it without any physical communication between DG units [2, 3]. There are many strategies in the control of droop, according to the type of grid and line impedance. In a medium/high voltage grid, the Q/V strategy is used, but in the case of low voltage grid, which is the case here, the active power/voltage (P/V) droop strategy is used. On the other hand, if the dominant line impedance is indicative, the active power/frequency (P/f) is used, and

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respectively, if the dominant line impedance is resistive, the active power/voltage (P/V) droop strategy is used [3–5]. In normal operation, the Microgrid can operate in two modes via a Static Transfer Switch (STS). The grid-connected mode: where the Microgrid operates in conjunction with the main grid in order to feed the load or inject the excess energy produced in the main grid. The stand-alone mode: the Microgrid operates without the main grid, the droop control works to correct the deviation of the voltage and frequency of the AC-link [2]. In the abnormal (unintentional) case, the Microgrid operates in unintentional islanding mode (UIM), this mode occurs during a sudden grid loss and the Supervisory Controller (SC) is unable to select quickly the power setpoints, due to the lack of knowledge about the state of the grid because of the inherent delay of the communication channels [6, 7]. According to IEEE 1547 standards, the anti-islanding controller that detects the failure of the grid must not exceed 2s to inform the supervisory controller for opening the STS, and set the appropriate power setpoints for all inverters. The Major problem occurs before the activation of the anti-islanding controller when the transient power may flow to the inverters that received a low power setpoint, which lead to increase the DC bus voltage (instability), and the inverter may close to prevent damage [8, 9]. There are a number of publications discussed the seamless continuous transfer of power in the Microgrid, some have addressed the problem of stability of the DC link voltage in the case of unintentional islanding like Issa et al. [10] who proposed a controller that prevent the shutdown of the inverter. However, this control strategy has certain limitations: it is used only for two inverters and the results are validated without load. The authors in [8] studied the current flow effect between several inverters in the Microgrid, they proposed a regulator to control the discharge circuit switch (resistor), and the limitation in this regulator is the dissipation of power and the disturbance of the DC source. This article proposes a controller based on the sliding mode technique for each DC source incorporated with droop control for regular DC bus especially in the unintentional islanding mode, thus, we improve the stability and the reliability of the Microgrid. The rest of the article embodies five sections. The next section discusses the operation of droop control. Analyzing the phenomenon of power import into the inverters is presented in Sect. 2. Accordingly, Sect. 3 explores the sources of energy used in the microarray. Furthermore, the proposed controllers are disclosed in Sect. 4. Hence, the simulation results that prove the performance of the proposed control strategy are presented in Sect. 5.

2 Operation Modes of a Microgrid and Problem Statement The energy management in the supervisory controller is charged to generate the setpoint of each inverter in the Microgrid. During grid connected mode the supervisory delivered different setpoint power to the inverters, in order to provide different

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amount of energy to the Microgrid according to the load requirement, the voltage and frequency of the AC bus are fixed by the utility grid. When the Microgrid is operating in islanded mode, all inverters have a constant setpoint power. Before the transition toward islanded mode, the supervisory controller operates all the inverter in their power rating. The unintentional islanding mode occurs when the utility grid is in failure situation, while the STS remains in grid-connected mode. The different value of the set-point power operations the inverters may cause the importation of power in the inverter. For further clarification, we use the droop control method to explain the issue of the unintentional islanding where each inverter has a droop control. The droop control equations can be expressed as: [3].   ωr e f = ωn − K ω Pavg Pi − Pi∗

(1)

  Vr e f _i = Vn − K V Q avg Q i − Q i∗

(2)

where ωn and Vn are the nominal frequency and nominal voltage respectively, where K ω and K V is the frequency and voltage proportional drooping coefficient, respectively.Pavg and Q avg are the average measured active and reactive powers, Pi∗ and Q i∗ are the active and reactive set points power. The K ω and K V droop slopes are determined according to the power rating of the inverter and according to the maximum allowable variations in output frequency and voltage [11]. Figure 2 show the droop control characteristics, we assumed that the two inverters have the same power ratings (Pn1 = Pn2 ) which means they have the same drooping gains (K ω1 = K ω2 = K ω ). During grid-connected mode, the frequency of the Microgrid   is fixed and maintained by the utility grid and equals to the nominal frequency ω M.g = ωgrid = ωn .

Fig. 2 Structure (P - ω) characteristics of two inverters in droop control

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(P1+P2)>PL

Mode 1

Mode 2

Gridon

Gridoff

=

=

(P1+P2) 0). The conditions leading to the switching between the different operating modes of the Microgrid can summarized in Fig. 3 [10]. The importing power occurs in the inverter that receives lower setpoint power resulting high DC voltage to last, if this value exceeds the maximum voltage limit allowable for the DC bus, the inverter will shut down by protection circuit element. This process can affect the reliability of this Microgrid.

3 Dynamic Controllers Design To overcome the problem of reliability of the Microgrid, the system needs the regulation of the DC bus voltage. Two controllers can be used in this study. The Primary Control designed to control the inverter and the Secondary Control designed to regulate the DC link voltage by controlling the DC-DC converter. The design and the synthesis of the two controllers are given in the following subsections. A. The Primary Control The primary controller consists of two control loops: the inner loop represented by the voltage control and the outer loop represented by droop controller.

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The outer control loop The inverter controllers received the signal from the supervisory controller (SC) about the state of STS (gridon or gridoff). In case of grid-connected mode, the active and reactive power of inverters (Pi , Q i ) with i = 1, 2 are regulated at their reference values (Pi∗ , Q i∗ ) that are delivered by the SC. In case of standalone mode, the nominal power values (Pn , Q n ) become the reference values, which ensure the regulation of the frequency. When the unintentional islanding mode happens, the reference values become the reference DC bus voltage in order to detect and prevent the circulating of the transit power among the inverters as illustrated in Fig. 4. The power error is the input of the droop controller, this controller consists of the Eqs. (1, 2), the output of this controller is the reference voltage signal of AC link Vac_r e f . The inner control loop In order to ensure the stabilization of the output voltage of the inverter (Vac ) around its desired value (Vac_r e f ) that is obtained by the outer loop, a Proportional-Integral (PI) controller with PWM techniques is used, as well which is given, by 



Iac_r e f = K P Vac_r e f − Vac + K I





 Vac_r e f − Vac dt

Fig. 4 The unintentional islanding controller in the outer control loop

(3)

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Fig. 5 The inner control loop

where K P and K I are the parameters of PI controller, obtained by using the PSO algorithm. It should be noted that we can use the Feedforward control instead of the integral action in the PI controller (Fig. 5). B. The Secondary Control (Proposed Controller) This section discusses the proposed Dc-bus voltage controller. In this paper, the regulation of the DC-bus voltage is accomplished by the photovoltaic system (structure1 in Fig. 1) and the storage system—battery (structure2 in Fig. 1). The control of photovoltaic system The PV boost controller has two functions: extract the maximum power available from the PV array through MPPT algorithm, and regulate the DC link voltage by reducing the injection of the current from the DC source (PV) when the Vdc increased especially if the inverter imports the power (unintentional islanding case) and cannot handle with the Vdc . The Second Order Sliding Mode (SOSM) approach can used in order to track the reference of DC bus voltage (Vdc_r e f ) and improve the transient response with a fast stabilization of the system against parametric variation and large range of load variation. In order to determine the control law, let us first recall the dynamic model of the system PV-boost converter which is highly nonlinear [12]: ⎧ x˙1 = λ1 x3 − λ1 x2 (1 − u) ⎪ ⎪ ⎨ x˙2 = λ2 x1 (1 − u) − λ3 x2 ⎪ x ˙ = λ4 I pv − λ4 x1 ⎪ ⎩ 3 y = s(x, t)

(4)

where λ1 = L1 , λ2 = C1dc , λ3 = RC1 dc , λ4 = C1pv , x1 = i L , x2 = Vdc , x3 = V pv

t and x = x1 x2 x3 R3 is the state vector and, uR is the bounded input control, s(x, t) is the smooth output- feedback function . The control objective is making output function s = 0.

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Supposing that the entire variable can be measured, the sliding mode surface is combined with the integration of the error is chosen as: 

e = x2 − x2−r e f s(x, t) = e + k e dt

(5)

where k is a constant. we choose to apply the second order sliding mode control. The control law depends on the switching function as follows: u = u1 + u2

(6)

u˙ 1 = −λ1 sign(s) u2 = −λ2 sγ

(7)

With 

where λ1 > 0, λ2 > 0 and 0 < γ ≤ 1/2. Where λ1 and λ2 are positive real constant parameters that can be selected properly and aθ = |a|θ sign(a), ∀a  R, θ > 0 [Lagh15]. In order to ensure that the real sliding order is a maximum, we choose γ = 1/2 and can reach the second order sliding [79-thes sahraoui]. The reader may refer to [13–18] for a detailed proof (Fig. 6).

Fig. 6 The proposed control structure of the PV-boost converter

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The control of battery system The bidirectional DC-DC converter can work in the two direction as a boost in one direction (discharging mode operation), or as buck in the other direction (charging mode operation), depending on the battery current orientation. The main objective to control the battery is to regulate the DC-bus voltage, dualloop feedback control strategy can be used (outer and inner loops). The first can realized by the PI control to generate the reference of the battery current which is given by Eqs. (8, 9) and illustrate in Fig. 7. maintain the stability of the DC bus voltage which can be expressed as:  Idc−r e f = K p .(Vdc−r e f − Vdc ) + ki .



 Vdc−r e f − Vdc dt

I Bat−r e f = 1/Vbat (Idc−r e f .Vdc + Iinv .Vdc − I pv .V pv )

(8) (9)

and the latter loop can be achieved through a sliding mode controller (SMC) to control the charging/discharging current value. In order to design the Sliding Mode controller, let us first recall the dynamic model of the battery-bidirectional converter that can be defined as [20]: For buck mode: ⎧ ⎨ x˙1 = λ1 Vbat − λ1 x2 u bat1 (10) x˙ = −λ2 x1 u bat1 + λ3 x2 ⎩ 2 y = s(x, t) For boost mode: ⎧ ⎨ x˙1 = λ1 Vbat − λ1 x2 (1 − u bat2 ) x˙ = λ2 x1 (1 − u bat2 ) − λ3 x2 ⎩ 2 y = s(x, t)

(11)

where λ1 = L1 , λ2 = C1dc , λ3 = RC1 dc , λ4 = C1bat , x1 = Ibat , x2 = Vdc , and

t x = x1 x2 R2 is the state vector and, (u bat1 , u bat2 )R is the bounded input control, s(x, t) is the smooth output- feedback function. The control objective is making output function s = 0. Supposing that the entire variable can be measured, the sliding mode surface is chosen as: s(x, t) = x1 − x1−r e f where x1−r e f is the reference of input current. The control law depends on the switching function as follows:

(12)

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Fig. 7 The proposed control structure of the battery converter

• For the charging battery process “buck mode”: the control law corresponding this mode is: u bat1 =

1 (1 − sign(s)) 2

(13)

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357

• For the discharging battery process “boost mode”: the control law corresponding this mode is: u bat2 =

1 (1 + sign(s)) 2

(14)

For a more detailed, the reader may refer to [19, 20].

4 Simulation Results The proposed control of the Microgrid, presented in Fig. 2, is validated through simulations for a typical converter. We care about the steady state of the modes. The performance and robustness of the sliding mode controller will be checked through unintentional islanding scenario. The Fig. 8 illustrates the continuous state evolution of the DC Microgrid, when the power of the inverter 300 W to the grid, while the inverter 2 is exporting 600 W. At t = 3 s, the grid was lost and the transient power phenomena is occurred, which is corresponding to the circulating of the power from inverter 2 to the inverter1, and this results the increase of the DC bus voltage. The primary controller of the inverter 1 and 2 responses to prevent the circulating of the power. While the secondary controller is interested to the DC voltage regulation, the PV-1 controller involve by reducing the PV generated power. On the other hand, the DC voltage of the inverter 2 increase also and need to regulate. Since the SOC of the battery is low, the battery controller intervenes to regulate the DC voltage by adjusting the output power of the inverter because the islanding mode has not been detected yet. At t = 3.25 s, the DC voltage was regulated and the disturbance is eliminated.

5 Conclusion In this paper, a new control scheme is developed for controlling a Microgrid in case of transient power between paralleled inverters during unintentional islanding mode. Different order of Sliding Mode Control strategies are used in the system, which is advantageous to regulate the DC link with excellent dynamic responses compared to the classical control method. The simulation results confirm the effectiveness and the robustness of the suggested controllers under different modes operation of the Microgrid with the seamless transfer of power between them. As a future work, a complete study of the voltage and frequency restoration in Islanded Microgrid will be investigated, to ensures the power sharing and the perfect control of frequency restoration.

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a.

Power inverters

b.

DC bus voltage

c.

d.

e.

PV 1 power

f.

g.

PV2 power

Battery power

PCC voltage

Grid current

Fig. 8 State evolution of the DC Microgrid when Pinv2 > Pinv1 under different mode operation

References 1. Gao M, Chen M, Jin C, Guerrero J, Qian Z (2012) Analysis, design, and experimental evaluation of power calculation in digital droop-controlled parallel microgrid inverters. J Zhejiang Univ Sci C (Computers & Electronics). https://doi.org/10.1631/jzus.c1200236 2. Issa W, Abusara M, Sharkh S (2014) Control of transient power during unintentional islanding of microgrids. IEEE Trans Power Electron 30(8):4573–4584 3. Abusara MA, Guerrero JM, Sharkh SM (2014) Line-interactive UPS for microgrids. IEEE Trans Ind Electron 61(3):1292–1300

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4. Laaksonen H, Saari P, Komulainen R (2005) Voltage and frequency control of inverter based weak LV network microgrid. In: Int Conf Future Power Syst. IEEE, p 6 5. Hossain MA, Azim MI, Mahmud MA, Pota HR (2015) Active power control in an islanded microgrid using dc link voltage status. In: IEEE Innovation Smart Grid Technology-Asia (ISGT ASIA), pp 1–6 6. He J, Lu C, Jin X, Li P (2008) Analysis of time delay effects on wide area damping control. In: IEEE Asia Pacific conference on circuits system, APCCAS 2008. IEEE, pp 758–761 7. Liu F, Gao H, Qiu J, Yin S, Fan J, Chai T (2014) Networked multirate output feedback control for setpoints compensation and its application to rougher flotation process. IEEE Trans Ind Electron 61(1):460–468 8. Hossain M, Pota H, Haruni A, Hossain MJ (2014) DC-link voltage regulation of inverters to enhance microgrid stability during network contingencies. IEEE Trans Ind Electron 61(1):460– 468 9. IEEE Standard for interconnecting distributed resources with electric power systems, IEEE Standard 1547–2003, New York, Jun 2003 10. Issa W, Abusara M, Sharkh S (2015) Control of transient power during unintentional islanding of microgrids. IEEE Trans Power Electron. https://doi.org/10.1109/tpel.2014.2359792 11. Mohamed YR, Zeineldin HH, Salama M, Seethapathy R (2012) Seamless formation and robust control of distributed generation microgrids via direct voltage control and optimized dynamic power sharing. IEEE Trans Power Electron 27:1283–1294 12. Mahmood H, Michaelson D, Jin J (2012) Control strategy for a standalone PV/battery hybrid system. In: 38th annual conference on IEEE industrial electronics society—IECON 2012, pp 3412–3418 13. Mu C, Sun C, Qian C, Zhang R (2013) Super-twisting sliding mode control based on Lyapunov analysis for the cursing flight of hypersonic vehicles. In: 10th IEEE international conference on control and automation (ICCA), Hangzhou, China, 12–14 June 2013 14. Koo B, Yoo Y, Won S (2012) Super-twisting algorithm-based sliding mode controller for a refrigeration system. In: 12th international conference on control, automation and systems, In ICC, 17–21 October 2012, Jeju Island, Korea 15. Utkin V (2014) Mechanical energy-based Lyapunov function design for twisting and supertwisting sliding mode control. IMA J Math Control Inf, 1 of 13 https://doi.org/10.1093/ima mci/dnu010 16. Chalanga A, Kamal S, Fridman M, Bandyopadhyay AM (2016) Implementation of supertwisting control: super-twisting and higher order sliding mode observer based approaches. IEEE Trans Ind Electron. https://doi.org/10.1109/tie.2016.2523913 17. Derafa L, Benallegue A, Fridman L (2011) Super twisting control algorithm for the attitude tracking of a four rotors UAV. J Franklin Inst. https://doi.org/10.1016/j.jfranklin.2011.10.011 18. Sahraoui H, Drid S, Alaoui L, Ouriagli M, Bussy P (2014) Robust control of the boost converter applied in photovoltaic systems using second order sliding mode. In: 15th international conference on sciences and techniques of automatic control & computer engineering—STA’2014, Hammamet, Tunisia, 21–23 December 2014. IEEE 978-1-4799-5907-5/14/$31.00 19. Ameur K, Hadjaissa H, Cheknane A, Essounbouli N (2017) DC-bus voltage control based on power flow management using direct sliding mode control for standalone photovoltaic system. Electric Power Comp Syst. https://doi.org/10.1080/15325008 20. Tsang KM, Chan WL (2013) Model based rapid maximum power point tracking for photovoltaic. Energy Convers Manage 70:83–89

New Virtual Synchronous Generator Control Technique of Distributed Generator Unit to Improve Transient Response of the Microgrid Yacine Daili and Abdelghani Harrag

Abstract To allow a high penetration of the renewable energy into the grid, Distributed Generator (DG) based Virtual Synchronous Generator (VSG) control technique has been proposed. DG units are often connected to low and medium line voltage, where the impedance line is not purely inductive. The active and reactive powers are nonlinear functions, which makes the control of active and reactive power more complicated. Thus, dramatic oscillations of the delivered power and frequency of the DG based VSG may occurred in dynamic steady state. The oscillation of frequency and power could be suppressed by increasing the VSG damping coefficient. However, the response time of the system is increased. The main contribution of this work is to propose a new VSG control technique to improve the dynamic performances of the conventional one. With the proposed VSG technique the damping coefficient is auto-adjustaed to eliminate the power and frequency oscillation without scarifying the response time of the system. The effectiveness of the proposed VSG control strategy is verified and compared with basic VSG control technique by simulation, the results demonstrate the superior performances of the proposed VSG control strategy. Keywords Virtual synchronous generator (VSG) · Micro-grid · Power electronics converter · Renewable energy sources · Distributed generator · Grid stability

Y. Daili (B) · A. Harrag Mechatronics Laboratory (LMETR) - E1764200, Optics and Precision Mechanics Institute, University of Ferhat Abbas – Sétif 1, 19000 Sétif, Algeria e-mail: [email protected] A. Harrag e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_25

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1 Introduction With rapid growth penetration of renewable energy sources, such as wind turbines, fuel cell and photovoltaic into the grid, the conventional Synchronous Generators (SGs) are more and more substituted by renewable energy sources usually interfaced to the main grid through power electronic inverters. The absence of inertia with inverters leads to decrease the total inertia of the power system. Therefore, high penetration of the renewable energy into the grid can cause a stability problem of the grid [1–5]. The droop-based control method and Virtual Synchronous Generator (VSG) control method are widely employed in Distributed Generator (DG) units [6–8]. These control techniques can ensure both grid connected and island modes operating of the system. With a droop-based control method, the injected active and reactive powers into the grid are performed by imitating static operation characteristics of SGs. The problem associated with DG based this controller is the absence of the inertia [9]. To solve this issue, virtual synchronous generator known also as synchronverters was introduced by [8, 10]. The idea of the VSG is to control the DG based inverter to emulate the essential dynamic behavior of synchronous generators, the inertia characteristic emulated by the VSG controller participates in providing the necessary frequency support for the grid. The authors of [3] prove that the VSG control technique and droop control strategy have the same dynamic if an appropriate low-pass filter is added to the droop controller. DG units are often coupled to the grid via a low and medium voltage line, where the line impedance is not dominantly inductive and the power angle is not enough small. The active and reactive powers are nonlinear functions and strongly coupled [11]. As results, using the conventional VSG control technique could introduce an oscillatory mode of the system, which causes low frequency oscillations of the inverter output power and frequency. These oscillations deteriorate the dynamic performance of VSG, and it may even result in unnecessary trips of DG units, because inverters have very limited over-current capability compared to real synchronous generator [1]. Consequently, the parameters design the VSG controller to fulfil the requirements of stability, oscillation damping and dynamic performances of the system is challenging task [11–14]. The main contribution of this paper is to propose a new VSG control method to improve the transient response of the DG system. With the proposed technique, the damping coefficient of VSG controller is auto-adjustated to eliminate the power and frequency oscillations without scarifying the response time of the system. The effectiveness of the proposed VSG control strategy is tested and compared with basic VSG controller by simulation. The paper is organised as follows: basic operation principal of VSG controller is discussed in Sect. 2. The principal of the proposed VSG controller is presented in Sect. 3. The simulation results with proposed and conventional VSG controller are discussed in Sect. 3 to verify the theoretical analysis. Finally, this paper is summarized in Sect. 4.

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2 Principle and Modelling of a VSG Control Strategy A. Principal of a VSG control technique The control circuit diagram of the VSG is illustrated in Fig. 1. To simplify studies, the DC source is used in place of distributed generation (DG). The three phases voltage source inverter (VSI) is connected to the grid at the point of common coupling (PCC) through an LC filter, where L F and C F are its elements and line impedance Z L , which is included the resistance and inductance line respectively RL and L L . Where vm and vg represents the terminal voltage of a VSG and the grid voltage respectively. In this schema, the delivered active power Pout and reactive power Qout to the grid are computed from vg and ig in αβ reference frame as follows: 

Pout = Vgd Igd + Vgq Igq Q out = Vgβ Igα − Vgα Igβ

(1)

PCC LF

RL

VSI

LL

CF

abc Active power controller

Voltage reference synthese (Eq 5)

Vg

Reactive power controller

Fig. 1 Control block diagram of VSG

Pout

PLL

Vg

Active & Reactive power calculation (Eq 1)

Capacitor voltage controller

PWM

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The VSG mimics the swing equation and damping characteristic of a synchronous generator. Thus, the power equation can be expressed as [2, 15]: 

Pin −Pout = ωg dθm = ωm dt

  m D ωg − ωm + J dω dt and

dθg dt

= ωg

(2)

where θ g and θ m represent respectively the grid voltage phase and VSG virtual phase, J is the rotational inertia, D is the damping coefficient, Pin is the input power (as same the prime mover power in a conventional SG), Pout is the electrical output power, ωg is the grid angular velocity and ωm is the virtual rotor angular velocity. The grid angular velocity ωg is detected at the PCC by the phased looked loop PLL system, and then the virtual rotor speed ωm is computed by resolving the Eq. (2). By passing ωm through an integrator, the virtual mechanical phase angle θ m is produced. The input power Pin is generated by frequency droop regulator according to Eq. (3), which simulates the governor of the conventional SG [16]:   Pin = Pr e f + K p ω0 − ωg

(3)

Pref is the reference value of active power, ω0 is the rated angular velocity, K p is the active droop coefficient. The reactive power controller emulates the conventional excitation circuit of a real SG. The control loop includes a reactive power droop mechanism Eq. (4) and a voltage regulator (5) [17]: 

  Q in = Q r e f + K q V0 − Vg Vm = KSi (Q in − Q out ) + V0

(4)

where V g is the amplitude value of the grid voltages, V 0 is the no-load voltage, K q is the reactive droop coefficient, K i is the integral coefficient of the voltage regulator and Qref is the reference value of reactive power. According to (3) and (4) the active and reactive power delivered by the DG unit change with respect to the PCC frequency and the amplitude of its voltage The amplitude of electromotive force V m and the virtual mechanical phase angle θ m information are used for synthesizing the references voltages (vma , vmb ,vmc ), which are the inputs of voltages controllers. The voltage references can be written as: √ ⎧ ⎨ vma = √2Vm sinθ m v = 2V sin θ − ⎩ mb √ m  m vmc = 2Vm sin θm +



2π 3  2π 3

(5)

Finally, the voltages controller is introduced to track the voltage references. The outputs of the voltage controller are fed into the PWM modulator to the drive signals of the inverter.

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It worth noting that, the coefficient K p and K q in (3) and (4) are selected based on the local grid standard [11, 18]. B. Modelling of a VSG control strategy Considering the equivalent model of the system illustrated in Fig. 1 where tow nodes V m and V g separated by a line impedance Z L . The delivered active and reactive powers to the grid are expressed as [2]: ⎧ ⎨ Pout = ⎩ Q out =

R L Vm2 −R L Vm Vg cos δ+X L Vm Vg sin δ R 2L +X 2L X L Vm2 −X L Vm Vg cos δ−R L Vm Vg sin δ R 2L +X 2L

(6)

where δ is the load angle has the following expression:   δ = θg − θm = ∫ ωg − ωm dt

(7)

For a large synchronous generator coupled to the grid through a high voltage line, the line impedance is purely inductive. Thus, the active power is controlled by the angle and the reactive power by amplitude output voltage V m . However, for a small and medium DG systems coupled to a low or medium voltage line, the impedance line is not dominantly inductive, and as a consequence, the active and reactive powers are nonlinear functions. Additionally, they are not independent, changes of V m and affect both Pout and Qout variations. This behaviour makes the control of active and reactive more complicated. Considering small perturbations of the active and reactive powers denoted by ΔPout , ΔQout resulting from small variations of the input variables δ and ΔV m . By linearizing (6) around an equilibrium point, the following functions are obtained: ⎧



2 ⎨ Pout = P0 + R2L Vm02 Vm + −Q0 + X2L Vm02 δ = K P−V Vm + K P−δ δ m RL +XL

RL +X L Vt0 2 R V Q X V L m0 L m0 0 ⎩ Q out = + R2 +X2 Vm + P0 − R2 +X2 δ = K P−Vm Vm + K P−δ δ Vt0 L

L

L

L

(8) P0 and Q0 are the active and reactive powers equilibrium point respectively. The small signal of the active and reactive controllers (2)–(4) could be obtained as:   Pin − Pout = D ωg − ωm + J s ωm ωg0

(9)

Pin = Pr e f − K p ωg

(10)

Q in = Q r e f − K q Vg

(11)

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Vm =

Ki ( Q in − Q out ) s

(12)

From (8) to (10), the transfer functions of active and reactive powers have the following expressions: Pout (s) =

K P−δ J ωg0 s 2 K P−Vm V Pr e f (s) + (s) m K P−δ J ωg0 s 2 + 1 K P−δ J ωg0 s 2 + 1

+

K P−δ J ωg0 s 2 K P−δ J D K p s 2 ω δ(s) − (s) g K P−δ J ωg0 s 2 + 1 K P−δ J ωg0 s 2 + 1

(13)

Ki Kq Ki K Q−δ s Q r e f − Vg (s) + δ(s) s + Ki s + Ki s + Ki

(14)

Q out (s) =

The frequency and voltage amplitude deviations of the micro-grid could be a result of connection/disconnection of other DG systems or a transit of the micro-grid load. The parameters determining the dynamic behavior of the active and reactive powers loop are the rotational inertia J, the damping coefficient D and the integral coefficient of the voltage regulator K i respectively.

3 Proposed VSG Control Technique As demonstrated in the previous section, both active and reactive powers loops are strongly coupled, which makes the the parameters design of active and reactive loops to fulfil the requirements of stability, oscillation damping and dynamic performances of the system is not straightforward task. Assuming that, the coupling effect on the active power control loop is neglected, one can observe that the transfer function from reference power to the output power (Eq. 13) that the overshoot and the response time for a fixed inertia J depend only on the damping coefficient D. So that, a small damping coefficient leads to large oscillations of the DG output power. The output power and frequency oscillations could be considerably reduced by choosing a large value of damping coefficient. However, the system takes a long time to reach the set point. In order to improve the response time without causing large oscillations of the output power, a new technique is proposed in this paper. It consists in adjusting automatically the VSG damping coefficient. Figure 2 summarizes how this coefficient is adjusted. This figure shows the reference power in pointed line with power measured in solid line. From this figure, we can distinguish two phases, a first one is when the system is divergent from the set point and the second phase is the divergent of the system from the set point. To improve a system response time and weaken the overshoot and oscillations of output power and frequency, this coefficient is selected

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Measured power Reference power

Small D

Big D

Small D

Big D

Small D

Fig. 2 Rules of damping coefficient selection

small when the system is convergent to the set point, while in the divergence phase the coefficient is chosen big to damp rapidly the kinetic energy stored in the virtual energy and reduce oscillations of the output power. In order to detect the divergence and the convergence of the system from the set point, both power error and derivative of the measured power are used. So that, the system is convergent in two cases. The first one is when the active power derivative is positive and the error is negative. Secondly, when the derivative of the measured active power is negative and the error is positive. While, it is divergent in case of positive error and positive derivative or when the error is negative and also the derivative is negative. The synoptic schema of the proposed VSG control technique is illustrated in Fig. 3, the additional damping coefficient D´ is added or removed by switching S

Fig. 3 Control diagram of the proposed VSG technique

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according to either the system is divergent or convergent from the set point. The switch S is controlled by two signals, the derivative of the measured power and the power error. The hysteresis comparator are used instead of a simple comparator to make the system insensitive to the noise of measurement signals.

4 Simulation Results To verify the improvement of the proposed VSG compared with a conventional one, the distributed generation system shown in Fig. 4 with a proposed and conventional VSG are simulated in Matlab/Simulink environment under several power scenario are tested. The system under study is presented in Fig. 4. The connected load is powered by the DG system and the grid, the DG is coupled to the PCC trough resistive-inductive line impedance to get more realistic case. The system is running in grid connected mode. The essential parameters of the system are listed in Table 1. Throughout the simulation validation, the comparative study between the conventional and proposed VSG techniques is performed. Figure 5 shows the dynamic response of the system illustrated in Fig. 4 controlled with proposed and conventional VSG techniques. This figure presents the behavior of the DG system with both proposed and conventional VSG techniques under step up and steep down of the reference active power. The active power increases from 2000 to 7000 W in the right column and decreases from 7000 to 2000 W in the lift column, in both cases the reactive power reference is set to 0 VAR, and it is remained unchanged. As can be observed from this figure by applying the conventional algorithm, the injected active power at the PCC presents large oscillations around its reference with an overshoot about 33.7% and the setting time is equal to 0.347 s. By using the proposed algorithm, the active power reacts with excellent dynamic. Comparing with the conventional technique, the overshoot and setting time are reduced by almost half. So that, the proposed algorithm detects successfully the

+ -

VSI

RL

LL

Vg

PCC

Fig. 4 The simulated micro-grid platform

Grid Load : P=8kW, Q=4 kvar

New Virtual Synchronous Generator Control Technique … Table 1 System parameters

369

Parameters

Values

Nominal utility line-line voltage (rms)

220 V

Nominal utility frequency

50 Hz

DC link voltage

450 V

Inverter switching frequency

4 kHz

Inverter output filter inductance

5 mH

Inverter output filter capacitor

80 μF

Inverter rated power

10 kW

Line resistance

0.641

Line inductance

0.32 mH

Virtual inertia

0.5 kg m2

Damping coefficient

10

Active droop coefficient

7957

Reactive droop coefficient

1250

divergence/convergence of the active power to set point in all cases (see Fig. 5b), and it is switch on/off the additional coefficient D’ to improve a system response time and weaken the overshoot. By observing waveforms of the inverter output frequency by applying both VSG control approaches (Fig. 5c), one can note that the two techniques present the same overshoot in dynamic steady state. However, the system with the proposed controller is less oscillatory, the system reaches the steady state rapidly with lower setting time. Consequently of the oscillations of the output power with the conventional VSG technique, the inverter output current is oscillatory and presents a considerable overshoot (see Fig. 5d). It worth to note that currents oscillations may result in unnecessary trips of the DG unite, because the inverter have very limited over-current capability compared to the conventional synchronous generator. Thanks to the proposed technique, the overshoot and low frequency oscillations of the output current are reduced and system reaches the steady state with lower setting time (Fig. 5e).

5 Conclusion This work interests on the development of a new VSG control strategy applied to DG unit to improve transient response of the conventional VSG, the proposed control schema consists in auto-adjusting the damping coefficient of VSG controller to eliminate the power and frequency oscillation without scarifying the response time of the system, So that, the dumping coefficient is selected small when the system is convergent to the active power set point, while in the divergence phase the coefficient is chosen big to reduce oscillations of the output power.

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8000

Active Power [W]

Tr-Pop=0.184 S

6000

9000

Reference power Conventional VSG Proposed VSG

(a)

7000

6000

4000

1680 W

6

2000 6.05

6.1

6.15

6.25

6.2 Time [Sec]

6.3

6.35

6.4

6.4

1000

(b)

1

8.5

Control Signal

0.6

6.05

6

6.1

6.15

6.2 Time [Sec]

6.25

6.3

6.35

8.75 Time [Sec]

8.8

8.85

8.9

8.95

9

(b)

0.6 0.4

6.4

8.5

6.4

8.55

8.6

8.65

8.7

50.06

50.02

Conventional VSG Proposed VSG

(c)

50 49.99 49.98 49.97

6.2 Time [Sec]

6.25

6.3

6.35

6.4

50

8.55

8.6

8.65

8.7

6.4

20

20 (d)

15

9

Conventional VSG Proposed VSG

(c)

8.5 6.15

8.95

50.01

49.98 6.1

8.9

50.02

49.99

6.05

8.85

50.03

49.95 6

8.8

50.05

49.96

49.94 5.95

8.75 Time [Sec]

50.04

Frequency [Hz]

50.01

8.75 Time [Sec]

8.8

8.85

8.9

8.95

9

8.8

8.85

8.9

8.95

9

8.8

8.85

8.9

8.95

9

(d)

15 10 Current [A]

10 Current [A]

8.7

0

0

5 0 -5

5 0 -5

-10

-10

-15

-15 6

6.05

6.1

6.15

6.2 Time [Sec]

6.25

6.3

6.35

6.4

6.45

-20

8.5

8.55

8.6

8.65

8.7

8.75 Time [Sec]

20

20 (e)

15

(e)

15 10 Current [A]

10 Current [A]

8.65

0.2

0.2

5 0 -5

5 0 -5 -10

-10

-15

-15 -20 5.95

8.6

1

0.4

-20 5.95

8.55

0.8

0.8 Control Signal

Tr-Conv=0.349 S

3000

866 W

0 5.95

Frequency [Hz]

Tr-Pop=0.192 S

4000

1000

5.95

Reference power Conventional VSG Proposed VSG

5000

3000 2000

870 W

7000

Tr-Conv=0.347 S

5000

(a)

8000 1685 W

6

6.05

6.1

6.15

6.2 Time [Sec]

6.25

6.3

6.35

6.4

6.45

-20

8.5

8.55

8.6

8.65

8.7

8.75 Time [Sec]

Fig. 5 Dynamic performance comparison between proposed and conventional VSG under step up (right column) and step down (lift column) of the reference active power. a DG output active power with both techniques, b switching signal of the proposed algorithm, c DG output frequency with both techniques, d DG output current with conventional technique, e DG output currents with proposed technique

The proposed VSG has been implemented and compared to the basic VSG scheme using Matlab/Simulink environment. Simulation results show the superior performances of the proposed VSG control strategy, Thus, based on the proposed new VSG controller, the overshoot and setting time are reduced by almost half for tested reference active power steps. All these results demonstrate the effectiveness of the proposed VSG control technique.

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Acknowledgment The Algerian Ministry of Higher Education and Scientific Research via the DGRSDT supported this research (PRFU Project code: A01L07UN190120180005).

References 1. Liu J, Miura Y, Ise T (2019) Fixed-parameter damping methods of virtual synchronous generator control using state feedback. IEEE Access 7:99177–99190 2. Daili Y, Harrag A (2019) New model of multi-parallel distributed generator units based on virtual synchronous generator control strategy. Energy Ecol Environ, 1–11 3. Liu J, Miura Y, Ise T (2016) Comparison of dynamic characteristics between virtual synchronous generator and droop control in inverter-based distributed generators. IEEE Trans Power Electr 31:3600–3611 4. Zhong Q-C (2016) Virtual synchronous machines: a unified interface for grid integration. IEEE Power Electron Mag 3:18–27 5. Huang L, Xin H, Wang Z (2019) Damping low-frequency oscillations through VSC-HVDC stations operated as virtual synchronous machines. IEEE Trans Power Electron 34(6):5803– 5818 6. Monica P, Kowsalya M (2016) Control strategies of parallel operated inverters in renewable energy application: a review. Renew Sustain Energy Rev 65:885–901 7. Shuai Z, Hu Y, Peng Y, Tu C, Shen ZJ (2017) Dynamic stability analysis of synchronverterdominated microgrid based on bifurcation theory. IEEE Trans Ind Electron 64:7467–7477 8. Zhong Q-C, Weiss G (2011) Synchronverters: inverters that mimic synchronous generators. IEEE Trans Ind Electron 58:1259–1267 9. Zhong Q-C, Konstantopoulos GC, Ren B, Krstic M (2018) Improved synchronverters with bounded frequency and voltage for smart grid integration. IEEE Trans Smart Grid 9:786–796 10. Beck H-P, Hesse R (2007) Virtual synchronous machine. In: Proceedings of 9th international conference on electrical power quality and utilization, pp 1–6 11. Li B, Zhou L, Yu X, Zheng C, Liu J (2017) Improved power decoupling control strategy based on virtual synchronous generator. IET Power Electron 10:462–470 12. Wu T, Liu Z, Liu JJ, Wang S, You ZY (2016) A unified virtual power decoupling method for droop-controlled parallel inverters in microgrids. IEEE Trans Power Electron 31(8):5587–5603 13. Bin L, Lin Z, Xirui Y, Chen Z, Jinhong L, Bao X (2016) New control scheme of power decoupling based on virtual synchronous generator. In: Proceedings of IEEE power and energy conference at Illinois (PECI), pp 1–8 14. Li M, Wang Y, Xu N et al (2016) A power decoupling control strategy for droop controlled inverters and virtual synchronous generators. In: Power electronics motion control conference. IEEE, pp 1713–1719 15. Yibin H, Xiangwu Y, Dongxue L, Xinxin L, Bo Z (2017) Experimental study of micro sources inverter based on virtual synchronous generator. In:Procedings of IEEE conference on electrical and electronic engineering (ICEEE), pp 2433–2438 16. Meng J, Shi X, Wang Y, Fu C (2014) A virtual synchronous generator control strategy for distributed generation. In: Proceedings of IEEE China international conference on electricity distribution, pp 495–498 17. D’Arco S, Suul JA, Fosso OB (2015) Small-signal modeling and parametric sensitivity of a virtual synchronous machine in islanded operation. Int J Electr Power Energy Syst 72:3–15 18. Dong S, Chen YC (2017) Adjusting synchronverter dynamic response speed via damping correction loop. IEEE T Energy Conver 32:608–619

Design and Analysis of Robust Nonlinear Synergetic Controller for a PMDC Motor Driven Wire-Feeder System (WFS) Noureddine Hamouda, Badreddine Babes, and Amar Boutaghane

Abstract In this paper a robust non-linear synergetic control (NSC) algorithm was developed for the speed control of a permanent magnet DC motor (PMDC) driven wire-feeder systems (WFSs) of gas metal arc welding (GMAW) process. The proposed control approach allows the PMDC motor to track desired trajectories by the output voltage of a full bridge converter with respect to the large torque disturbances, unexpected changes in rotor inertia and makes the wire-feeder unit (WFU) has fast and stable starting process as well as excellent dynamic characteristics. The effectiveness of the proposed control scheme is illustrated by numerical simulations within the MATLAB/SIMULINK environment software. The simulation results clearly demonstrate the significant improvement rendered by the proposed controller in the wire-feeder system’s reference tracking performance, torque disturbance rejection capability, and transient recovery time. Keywords Component · GMAW process · Wire-feeder system (WFS) · Non-linear synergetic control (NSC) algorithm · Simulation results

1 Introduction Nowadays, joining metals is a fundamental aspect of modern industrialized operations such as the ship building, automotive, and construction industries. It can be accomplished by different arc welding techniques. Among the various types of welding the typical GMAW is the most frequently employed and economically important welding process for joining metals. It is preferred for its flexibility, rapidity and can be utilized for both manual and automatic modes of welding for wide range of ferrous and non-ferrous metal pieces [1]. N. Hamouda · B. Babes (B) · A. Boutaghane Research Center in Industrial Technologies (CRTI), Algiers, Algeria e-mail: [email protected] N. Hamouda e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_26

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Consistency and high quality welding procedures are the key issues to maintain and increase the overall product quality. During GMAW, the electrode wire is melted and liquid droplets are formed at the tip of the electrode. When detaching from the electrode, the droplets transfer both mass and heat into the weld pool. In order to achieve quality welds, the transfer process must be controlled. One of the strategies to control the quality of the weld is to maintain the set values of welding current and arc length to achieve the preferred values of heat and mass transfer to the work-piece. The control variables selected are, the wire feed rate and the open circuit voltage, which are utilized to control the current and arc length of the GMAW process [2]. Large swing in wire feed rate results in large increased stress in PMDC motor and in welding current during welding process. This changes in wire feed rate causes the arc breaking, affecting the arc stability which ceases the welding operation. Therefore, wire-feeder system (WFS) is an important subsystem of typical automatic GMAW process. It should not only prevent the fracture and vibration of wire from occurring, but also guarantee the high speed of wire feed to meet the need of high rate of automatic welding production. Therefore, the control of wire feed rate for the WFS is the key technologies of the wire-feeder units (WFUs), which has a strong impact on the welding quality. The available WFUs are designed for constant wire feed rate and feature a large inertia and static friction due to the reduction gearbox and the eccentricities in the wire roller mechanism, and also due to the wire spool and the important frictions of the wire feed path [3]. Thus, this mechanical dynamic is very slow as compared to the arc welding melting process [4]. Many solutions have been introduced in the literature to improve the wire feed speed responses of the WFUs either by developing new mechanisms with various types of permanent magnet DC (PMDC) motors or by designing robust wire feeder controllers [4, 1]. The accurate design of wire feeder controller is essential to provide productivity, wide range welding capability, and comfort level to the welder user. However, the design of the accurate wire feeder controller for PMDC motor of the WFUs using the traditional PID method require accurate modeling of PMDC motor that considers the non-linear dynamics. It is possible to use a traditional PID regulator to control the wire feed speed of the WFUs [5]. However, traditional PID controllers do not yield reasonable performance over a wide range of working conditions because of the fixed gains utilized. Thus, the PID parameters need to be automatically adjusted by a fuzzy set [6]. Accordingly, many non-linear control techniques have been introduced to solve the wire feeder control problem of WFUs, and hopefully realize certain performance objectives such as variable structure sliding mode control. Paul [7] used a sliding mode to control the wire feeding rate to achieve a desired constant welding current easily and quickly. A similar method with variable structure sliding mode control has been applied by Ngo et al. [8] to ensure robustness in the GMAW process. Another application of sliding mode control function was also elaborated by Paul [9], who developed an accurate speed control of wire feeding using back-EMF as feedback signal. However, their studies typically needed large time-consuming computations.

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Although the non-linear synergetic control (NSC) is recently proposed, numerous applications of these techniques are successfully reported in various domains of engineering. These controllers are successfully applied to a number of domestic and industrial manufacturing. The nonlinear power system stabilizers [10], power converters for pulse current charging [11], DC-DC boost converters [12] and control of chaotic oscillation in power systems [13] are such of these applications. In this paper, the new way to controlling the wire feed speed of WFUs with nonlinear synergetic control (NSC) strategy has been investigated. More specifically, the model of PMDC motor is simplified which makes the NSC algorithm simple to design. Moreover, the NSC controller performance in presence of parametric uncertainties and disturbance is discussed.

2 Dynamics of GMAW Process In the GMAW process, the welding inverter controls the open circuit voltage V oc (V ) between the contact tip tube and the work-piece. In addition, the wire feed servo motor rotates a set of pinch rollers, which force the wire into the torch head and through the contact tube whereupon the wire is consumed by the GMAW process as illustrated in Fig. 1. The wire feed servo motor is in itself a feedback controlled system which is capable of delivering wire to the weld process at a controlled wire feed rate; increasing or decreasing the wire feed speed V f (m/s) on the wire feeder servo motor increases or decreases the welding current I w (A) as well as the metal transfer mode [1]. In majority of cases, the value of V f is reserved constant at desired value [14].

Pinch roller

Fig. 1 Physical representation of WFS for GMAW process

Guides Rs

Vf Liner

L1 I W Contact tube ls

Voc

CT larc

Varc

Workpiece

Wire spool

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For the purposes of this derivation, the wire feed rate is considered to be the input. The arc dynamics form the plant and the arc current I arc (A) is taken to be the output. CT = larc + ls = constant

(1)

where CT (m) is the contact tube height, larc (m) is the arc length, and ls (m) is the stick-out length. The dynamic equation for the electrical circuit of the GMAW process, is Voc = L1

dIW + Rs IW + Varc dt

(2)

where V oc (V ) represents the open circuit voltage of arc welder power supply, I w (A) is the instantaneous welding current, Rs (Ω) is the Thevenin resistance of arc welder power supply plus cabling resistance and L 1 (mH) is the inductance of arc welder power supply. The dynamic equation of arc voltage V arc (V ) is expressed as Varc = Earc larc + Rarc IW + Vp

(3)

where E arc (V /m) is the electric field intensity of the arc plasma, V p (V ) is the output voltage of power source, and Rarc (Ω) is the arc resistance. The arc resistance can be written as a function of the arc length Rarc =

ρ ρ ls = [CT − larc ] A A

(4)

where ρ(Ω.m) is the resistivity of the weld wire and A(m2 ) is the cross-sectional area of the weld wire. The dynamic equation of arc length larc , is dlarc = Vm − Vf dt

(5)

where V m (m/s) represents the wire melting rate may be expressed as Vm = km IW

(6)

where k m indicates the coefficient of wire melting rate. The dynamic equation of the power source V p (V ), is Vp = (Ru + k1 IW )k0

(7)

where Ru (V ) is the control input of the power source, k 0 is gain of power source and k 1 is feedback gain. Taking the Laplace transform of above equations and connecting

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k1 Ru(s)

IW (s)

Voc(s) _ Vp(s) + 1 + k0 + + L1s+Rs

377

Vf (s) Km

_ +

1 s

larc(s)

Rarc Earc Fig. 2 Block diagram of a GMAW for a constant wire feed-rate system

them according to their physical relationship, one can obtain the configuration of the control unit as shown in the block diagram of Fig. 2.

3 Dynamics of PMDC Motor Driven WFSs Advances in permanent magnet equipments and attractive characteristics such as light weight, low cost, low speed, fast response etc., enlarged the exploitation of permanent magnet DC (PMDC) motor particularly for wire-feeding units (WFUs) in GMAW process [4]. The graphic illustration of power circuit of the wire-feed servo motor control is depicted in Fig. 3. The PMDC motor control variables are DC input voltage V a (V ) and load torque T L (N.m). In WFUs, T L depends on diameter d(cm) of electrode wire and its material composition. For a particular application it could be considered as constant. The PMDC motor output variables are the wire feed speed V f (m/s), the angular displacement of the motor shaft θ (rad) and the armature current ia (A).

D1 Vac

D3 C

D2

D4

Va A

S1 PMDC motor La Ra

S2

Fig. 3 Power circuit for the wire-feed servo motor control

Ea

S3 B S4

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The PMDC motor’s dynamic equations can be developed based on the Kirchhoff’s voltage law around the armature circuit and the Newton’s moment law using Eqs. (8)– (10) Va = La J

dia dθ + Ra ia + k2 dt dt

d 2θ dθ +f = k3 ia dt dt Vf = k4

dθ dt

(8)

(9) (10)

where, V a (V ) is the armature voltage, ia (A) is the armature current, Ra (Ω) is the armature resistance, L a (mH) is the armature inductance, θ (rad) is the angular displacement of the motor shaft, V f (m/s) is the wire feed rate, J(kg·m2 ) is the moment of inertia of the motor and mechanical load converted to the motor shaft, f (Nm·s) is the coefficient of viscosity of the motor and mechanical load converted to the motor shaft, k 2 , k 3 and k 4 are constants. Substitution of (8) into (9) and (10) yields a state space model of the PMDC motor dynamics x˙ = Ax + Bu

(11)

where ⎡

⎤ ⎡ ⎤ 1 − RLaa − k4kL2 a 0 La   ⎢ k4 k3 ⎥ ⎥ ⎢ x = ia Vf θ , A = ⎣ J − kJ4 f 0 ⎦, B = ⎣ 0 ⎦ 1 0 0 0 k4

(12)

Based on the state space model of PMDC motor of formula (12), manifold and NSC algorithm can be designed for wire feed speed controller [15].

3.1 Torque Disturbance Model The wire feed servo motor turns a set of spring-loaded pinch rollers which pull wire from a large spool, through a series of guides, and down to the torch head (See Fig. 1). The resulting torque load on the PMDC motor shaft has two principal components. There is a large constant load due to the stiffness of the wire and the friction of the guides, rollers, and bearings. In addition, mechanical imperfections in the pinch roller assembly cause an eccentricity that gives rise to an approximately sinusoidal term [3]. The total torque load on the PMDC motor shaft is modeled as [3]

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TG (t) = A + B sin(2π fD t)

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(13)

This torque at the gearbox is a product of the torque load on the PMDC motor, the torque efficiency of the gearbox, and the gear reduction ratio. Reflect the gearbox torque back through the reduction gears to obtain the torque load on the PMDC motor shaft [3] TL (t) = (0.66)−1 (56.5)−1 · TG (t) = 0.0231TG (t)

(14)

The frequency of the sinusoidal component is proportional to the wire feed rate fD =

1 1revolution 1min Vf = 3.26 × 10−3 Vf π 1.626 in 60 s

(15)

where 1.626 inches (4.13 cm) is the diameter of each pinch roller and the wire feed rate V f is given in inches per minute (IPM). Extensive studies [3] demonstrate that the typical values for constants A and B in Eq. (13) are 141 and 15.0 (oz.in) respectively. Reasonable wire feed rates lie in the 200–400 (IPM) range, resulting in disturbance frequencies of 0.65–1.3 (Hz). These facts yield a typical torque load at the PMDC motor shaft of TL (t) ≈ 3.27 + 0.35 sin(2π fD t) (oz.in)

(16)

where 0.65 Hz ≤ fD ≤ 1.3 Hz The scaling factor k 4 converts motor angular velocity ω(rad/s) into wire feed rate V f (IPM) k4 =

IPM.s 1 π 1.626 in 1 revolution 60 s = 0.7447 65.5 1revolution 2π rad 1 min rad

(17)

4 Wire Feed Servo Controller Design Generally, the wire feed servo controller is designed to realize accurate and robust tracking of the preferred speed from no load to full load conditions. The level of uncertainty on the PMDC motor shaft could be huge. There is a perception that NSC algorithm concept could be capable to achieve accurate and stable tracking in finite time under such conditions. Hence, a simple NSC control law is employed in

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this paper to achieve the design objectives as closely as possible. The design procedure based on NSC algorithm follows the procedure design of aggregated controller methods [15]. Given that Eq. (12) is in the structure described by (18) x˙ = f (x, u, t)

(18)

where x is the state vector, u is the control input vector, and t is time. Our objective now is to calculate the control signal u such that the closed-loop system is exponentially stable at the operating point in finite time. To this end, we define the macro function σ = σ (x, t). The macro function σ (x, t) contains all the information of PMDC motor control index. Since the control parameters of the PMDC motor consist of wire feed rate, angular displacement and, armature current, consider the macro function of the NSC algorithm as





σ (x, t) = λ1 Vf − Vf∗ + λ2 θ − θ ∗ + λ3 ia − ia∗

(19)

where, λ1 , λ2 and λ3 for weight coefficient, θ * reference value of angular displacement, Vf∗ is the reference value of wire feed speed, ia∗ is the reference value of armature current. The control signal u will force the PMDC motor to operate on the manifold σ (x, t) = 0. When the PMDC motor is in the manifold σ (x, t) = 0, the stability after the introducing of control parameter can be ensured at the same time, and it will converge to the manifold balance. The preferred dynamic evolution of the manifold is defined by τ σ˙ (x, t) + σ (x, t) = 0

(20)

where τ is a positive constant which imposes the speed of convergence of macro function to the manifold σ (x, t) = 0. The chain rule of differentiation yields σ˙ (x, t) =

d σ (x, t) · x˙ dx

(21)

Substitution of (18) and (20) into (21) gives τ

dσ f (x, u, t) + σ (x, t) = 0 dx

(22)

Resolving (22) for u provides the NSC control law as u = g(x, σ (x, t), τ, t)

(23)

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Combined system state space model with synergetic control theory, it can obtain the NSC control algorithm in Eq. (24).    λ2 La λ1 fLa λ1 k3 La k2 Vf + Ra − ia . . . − + u= k4 λ3 k4 λ3 k4 J λ3 J 



 La 

− λ1 θf − θf∗ + λ2 Vf − Vf∗ + λ3 ia − ia∗ τ λ3 

(24)

To prove stability, we chose the following Lyapunov function candidate V (t) =

1 σ (x, t)2 ≥ 0 2

(25)

After differentiation, this leads to 1 V˙ (t) = σ (x, t) σ˙ (x, t) = − σ (x, t)2 τ

(26)

Because (25) is satisfied and τ > 0, stability is therefore guaranteed for we now have V˙ (t) ≤ 0

(27)

By substituting the selected control parameter λ1 , λ2 , λ3 and τ into (24), with the Lyapunov stability theory, one can obtain that speed of controlled PMDC wire feeding system is asymptotic stability in the equilibrium state. Figure 4 shows block diagram of PMDC motor control WFS with NSC algorithm.

TL PMDC Motor

Converter

u Vf*

NSC Algorithm

Vf θf - θf* ia - ia*

Fig. 4 Block diagram of PMDC motor control WFS with NSC algorithm

Vf* θf* ia*

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5 Simulation Results In this section, a PMDC motor simulation model is carried out by using MATLAB/SIMULINK software in order to verify the possibility and capability of the controlled PMDC wire feeding system using the proposed NSC control law. The system parameters are set as shown in Table 1 when process simulating. The tuning parameters required for the proposed controller are λ1 = 0.01, λ2 = 0. 1, λ3 = 100 and τ = 0.000001. A set of simulation tests is considered, which are detailed in Table 2.

5.1 Tracking Performance Without Load Torque Disturbance Figure 5 shows the simulation responses for the step change of wire feed speed (0– 280 IPM) without load torque disturbance. It is observed that excellent trajectory tracking responses are obtained by using the proposed NSC control law due to the proper selection of the control parameters. Table 1 Specifications of the PMDC motor used in simulation work Specifications of the PMDC motor Major parameters

Parameter value

Armature resistance Ra

0.25

Armature inductance L a

0.5 mH

Rotary inertia J

1 × 10−4 kg m2

Viscous damping coefficient f

1.29 × 10−3

Back EMF constant k 2

N.m.s/rad

Electromagnetic torque constant k 3

0.0573 V.s/rad 0.0573 N.m/A

DC input voltage V a

24 V

Table 2 Simulation tests Test

Details

Test 1

Phase of start-up: a step change in V f (0 − 280 IPM) Load torque disturbance: none

Test 2

Phase of start-up: a step change in V f (0 − 280 IPM) Load torque disturbance: TL = 3.27 + 0.35 sin(2π fD t)

Test 3

Phase of start-up: a step change in V f (0 − 280 IPM)  J 0 ≤ t ≤ 0.05 s Rotor inertia variations: J = 5 ∗ J t > 0.05 s

Test 4

Steady-state response of the proposed NSC, SMC and PI controllers

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250

282

200

()

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300

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150 278

100 0.066 0.068 0.07 0.072 0.074 0.076

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Time (s) 8 6 4 2 0

0

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0.1

Time (s) Fig. 5 Wire feed speed, armature current and angular displacement tracking responses of PMDC motor in test 1

5.2 Tracking Performance Under Load Torque Disturbance In order to demonstrate the effectiveness of the proposed NSC controller in the presence of external disturbances, a torque disturbance is included in this test. The torque disturbance applied to the PMDC motor is chosen as in Eq. (16). The wire feed speed, armature current and angular displacement of PMDC motor in association

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with the desired trajectories under load torque disturbance are shown in Fig. 6, respectively. It is obvious that the ripples of the wire feed speed are small under the action of the proposed NSC controller, this validating the better disturbance rejection ability of the proposed control method.

Wire feed rate (IPM)

300 290

200 280 270

100

260 0.1

0.12

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0.16

0.18

0 0

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80 60 40 20 0

0

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0.04

0.06

Angular displacement (rad)

Time (s) 8 6 4 2 0 0

0.02

0.04

0.06

Time (s) Fig. 6 Wire feed speed, armature current and angular displacement tracking responses of PMDC motor in test 2

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5.3 Recovery Performance Under Rotor Inertia Variation Considering that the sudden rotor inertia variation may be involved in the running automatic GMAW process, we can assume that the rotor inertia of PMDC motor increases to 5*J after t ≥ 0.05 s. The responses obtained are shown in Fig. 7, which

Wire feed rate (IPM)

300

281

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280 279

100

278 277 0.04

0

0

0.02

0.04

0.045

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Time (s) 80 60 40 20 0

0

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Angular displacement (rad)

Time (s) 8 6 4 2 0

0

0.02

0.04

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Time (s) Fig. 7 Wire feed speed, armature current and angular displacement tracking responses of PMDC motor in test 3

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Wire feed rate (IPM)

350 300 250 200

PI SMC NSC

150 100 50 0 0

0.05

0.1

0.15

0.2

Time (s) Fig. 8 Wire feed speed tracking responses of PMDC motor in test 4

demonstrates the high tracking performance and the strong robustness of the proposed NSC control strategy against the model uncertainties.

5.4 Steady-State Response of the Proposed NSC, SMC and PI Controllers To better show the advantage of the considered technique, two other types of wire feed speed controllers are also considered in the simulations for the purpose of comparison, which are sliding mode control (SMC) of Paul [9] and conventional PI control of Chaouch et al. [5]. The comparative simulations are performed with the same conditions given above without torque disturbance. It can be noticed from the obtained results that the control performance of the system using the proposed NSC algorithm was better than that using the SMC and conventional PI controllers with regard to, not only the settling time, but also the overshoot, rising time, and steady error (Fig. 8).

6 Conclusion In this paper, a key solution to the problem of wire feed rate regulation in GMAW process has been introduced using a robust non-linear synergetic control theory. Based on the desired manifold and the Lyapunov function method, a digital NSC control law has been developed to ensure the rapidity and accuracy of tracking control of WFUs. The proposed NSC algorithm assumes no knowledge of the inertia of the PMDC motor and is thus unconditionally robust with respect to this mechanical parametric uncertainty. The major advantage of the proposed NSC control law is its robustness with respect to torque disturbances, inertia uncertainty and change of the

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operating point. Numerical simulations show that global asymptotic tracking can be obtained by the proposed control scheme. In the future work, the authors intend to carry out the experimental part to complete and finalize the project.

References 1. Naidu DS, Ozcelik S, Moore KL (2003) Modeling sensing and control of gas metal arc welding. Elsevier 2. Kahla S, Boutaghane A, Abdallah L, Dehimi S, Hamouda N, Babes B, Amraoui R (2017) Grey wolf optimization of fractional PID controller in gas metal arc welding process. In: Proceedings of 5th international conference on control engineering and information technology CEIT, Sousse, Tunisia 3. Greene BW (1988) Arc current control of a robotic welding system: modeling and control system design. University of Illinois, Champaign 4. Jiluan P (2003) Arc welding control. Woodhead Publishing, CRC Press, Cambridge 5. Chaouch S, Hasni M, Boutaghane A, Babes B, Mezaache M, Slimane S, Djenaihi M (2018) DC-motor control using arduino-uno board for wire-feed system. In: Proceedings of the IEEE, 3rd CISTEM’18—Algiers, Algeria, 29–31 October 2018 6. Truong DQ, Ahn KK (2009) Force control for hydraulic load simulator using self-tuning grey predictor–Fuzzy PID. Mechatronics 19:233–246 7. Paul AK (2014) Experimental design approach to explore suitability of PI and SMC concepts for power electronic product development. Int J Power Electron 6(1):42–65 8. Ngo MD, Duy VH, Phuong NT, Kim HK, Kim SB (2007) Development of digital gas metal arc welding system. J Mater Process Technol 189:384–391 9. Paul AK (2019) Robust PMDC motor control for accurate wire feeding in GMAW using back EMF. IEEE Trans Industr Electron. https://doi.org/10.1109/TIE.2019.2896131 10. Jiang Z (2009) Design of a nonlinear power system stabilizer using synergetic control theory. Electric Power Syst Res 79:855–862 11. Jiang Z, Dougal RA (2004) Synergetic control of power converters for pulse current charging of advanced batteries from a fuel cell power source. IEEE Trans Power Electron 19(4):1140–1150 12. Santi E et al (2003) Synergetic control for DC-DC boost converter: implementation options. IEEE Trans Ind Appl 39(6):1803–1813. https://doi.org/10.1109/TIA.2003.818967 13. Ni J et al (2014) Variable speed synergetic control for chaotic oscillation in power system. Nonlinear Dyn 78(1):681–690 14. Wu GD, Richardson RW (1989) The dynamic response of self-regulation of the welding arc. In: Proceedings of recent trends in welding science & technology, Tennessee, USA, pp 929–933 15. Kolesnikov A, Veselov G (2000) Modern applied control theory: synergetic approach in control theory. TSURE Press, Moscow-Taganrog

Simulation Study of the Dual Star Permanent Magnet Synchronous Machine Using Different Modeling Approaches Elyazid Amirouche, Khaldi Lyes, Ghedamsi Kaci, and Aouzellag Djamal

Abstract Four mathematical models of a dual star permanent magnet synchronous machine (DSPMSM) are presented, one in the natural reference frame, and three others in a synchronously rotating d-q reference frame. In order to simplify the machine analysis, a proper transformation is needed to transform the time varying quantities of the machine into constant variables, many transformations are reported in the literature. In this paper, three transformations are selected and applied to a DSPMSM. The models are implemented and simulated using Matlab/Simulink environment, the differences between the models are discussed. Finally, the limits of these transformed models are highlighted. Keywords Dual star permanent magnet synchronous machine · Machine modeling · Different modeling methods · Limits of simplified d-q models

1 Introduction Nowadays, power electronic converters are used to supply alternative current machines, consequently they are decoupled from the grid, so the number of the machine’s phases is not limited to three anymore, making possible the exploitation of multiphase machines in domains requiring some specific advantages, such as lower

E. Amirouche (B) · K. Lyes · G. Kaci · A. Djamal Laboratoire de Maitrise des Energies Renouvelables, Faculté de Technologie, Université de Bejaia, 06000 Béjaïa, Algerie e-mail: [email protected] K. Lyes e-mail: [email protected] G. Kaci e-mail: [email protected] A. Djamal e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_27

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torque repulsion [1] and fault tolerance [2]. The voltage source inverter fed double star machine is one of the widely discussed topics in the literature [3, 4]. Because the machine’s variables are not constant in time in steady state, the use of an adequate transformation in order to obtain constant variables presents many advantages, constant variables are easier to analyze, and the rotor position dependency of inductances that characterizes salient pole machines is eliminated, simplifying the model as well as the control of the machine. Many modeling technics are proposed in the literature, the double winding approach represents the machine with two pairs of d-q windings with a mutual coupling between each other [5], so this model is an extension of the conventional three phase machine model. In order to eliminate this coupling, a second consecutive transformation is proposed by [6], called decoupled d-q model, which simplifies the model analysis and implementation. An extended park transformation is reported by [7], it proposes a model where the stator circuits are completely decoupled by applying only one transformation. This paper focuses on the double star permanent magnet synchronous machine modeling, many modeling approaches will be presented, simulated and compared, and the limitation of these models will be presented at the end of this article.

2 Double Star Electrical Machines Multiple star machines are a particular case of multiphase machines, where the phases are divided into multiple three phase sets, with isolated neutral points. In a double star machine, the angle between the sets can have any value from 0° to 60°, the study of [1] shows that the torque characteristics of a double star with a displacement angle of 30° between the sets is better than any other value, the torque ripple is greatly reduced, also the predominant frequency has been shifted 12 times the supply frequency [1]. The use of multiphase machines instead of conventional three phase machines presents multiple advantages, in addition to the torque ripple reduction, as stated above, the machine’s power is divided among more inverter legs, so the power electronics rating is reduced, also, the reliability of the machine is increased in case of loss of one or more machine phases [8]. Because of the fault tolerant behaviour of multiphase machines, they are intensively used in applications requiring continuous service even with partial power, for example, in ship propulsion systems, aerospace applications, electrical vehicles, and in renewable energy conversion systems [8].

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3 Natural Model (ABC) of DSPMSM Before doing any transformation, the abc model of the machine must be developed, Fig. 1 shows the coils configuration of a dual star synchronous motor. In order to simplify the modeling of the machine, the following assumptions are made: all the sets are identical, and the windings are sinusoidally distributed around the air gap, mutual leakage inductances, saturation and eddy current are not considered.

3.1 Electric Model Starting from the natural reference frame, the electric equation of the system can be written: vabc = r i abc +

dϕabc dt

(1)

where  t vabc = va1 vb1 vc1 va2 vb2 vc2

(2)

t  i abc = i a1 i b1 i c1 i a2 i b2 i c2

(3)

t  ϕabc = ϕa1 ϕb1 ϕc1 ϕa2 ϕb2 ϕc2

(4)

⎤ rs · · · 0 ⎥ ⎢ r = ⎣ ... . . . ... ⎦ 0 · · · rs

(5)

⎡

Fig. 1 Dual star synchronous machine

C1

C2

d

A1 A2 q

B2 B1

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ϕabc is the resultant air gap flux, r a 6 × 6 matrix with rs the armature resistance. The resultant air gap flux can be written according to the inductance matrix and permanent magnet flux vector as follow:  [L 1 ] [M12 ] i abc + ϕ P Mabc = [M12 ]t [L 2 ]

ϕabc

(6)

where [L i ] the inductance matrix of the same three phase set windings, [M12 ] the mutual inductance matrix between the different sets, and ϕ P Mabc the flux produced by the permanent magnets. ⎡

ϕ P Mabc

⎤ cos(θ ) ⎢ cos θ − 2π ⎥ ⎢ ⎥ 3 ⎢ ⎥ ⎢ cos θ + 2π ⎥ 3 = ψP M ⎢ ⎥ ⎢ cos(θ − α) ⎥

⎥ ⎢ 2π ⎣ cos θ − 3 − α ⎦

−α cos θ + 2π 3

(7)

With α the electric displacement angle between the sets, and Ψ pm the magnitude of the permanent magnet flux.

3.2 Inductance Matrix Considering the rotor position dependency of the inductances, the self and mutual inductances are generally expressed by Fourier expansions [8]. Taking into account higher order harmonics, the self inductance of winding i is written as follow: L i (θ ) = L s0 +

∞ 

L S2n cos 2n(θ + θi ) + ϕin

(8)

n=1

where L s0 is a constant average value, L s2n is the coefficient of inductance harmonics (2nd, 4th,…), θi is the electric displacement angle of the ith winding magnetic axis from the reference axis (chosen to be the magnetic axis of coil A1 as shown in Fig. 1), while ϕin is the displacement angle of the corresponding harmonic component. The constant term L s0 can be expressed as the sum of the stator leakage inductance L Sl and the stator magnetizing inductance m [7]. L s0 = L sl + m Similarly, the mutual inductance between the phases can be expressed as:

(9)

Simulation Study of the Dual Star Permanent Magnet Synchronous … ∞



Mi j (θ ) = mcos θi − θ j + Ms2n cos n(2θ + θi + θ j + ϕin )

393

(10)

n=1

Considering only the first inductance harmonic (n = 1), the self and mutual inductance equations become: L i (θ ) = L s0 + L s2 cos(2θi )

(11)



Mi j (θ ) = mcos θi − θ j + Ms2 cos 2θ + θi + θ j

(12)

Comparing to the general equation given by [7], the coefficients L s2 and Ms2 are equal. The full inductance matrix of the windings is given by (13). ⎡

L a1 ⎢M ⎢ a1 b1 ⎢ ⎢M L s (θ ) = ⎢ a1 c1 ⎢ M a1 a2 ⎢ ⎣ Ma1 b2 Ma1 c2

Ma1 b1 L b1 Mb1 c1 Ma2 b1 Mb1 b2 Mb1 c2

Ma1 c1 Mb1 c1 L c1 Ma2 c1 Mc1 b2 Mc1 c2

M a1 a2 Ma2 b1 Ma2 c1 L a2 Ma2 b2 Ma2 c2

Ma1 b2 Mb1 b2 Mc1 b2 Ma2 b2 L b2 Mb2 c2

⎤ Ma1 c2 Mb1 c2 ⎥ ⎥ ⎥ Mc1 c2 ⎥ ⎥ Ma2 c2 ⎥ ⎥ Mb2 c2 ⎦ L c2

(13)

The four matrices presented earlier in (6) can be recognised in this matrix. Note that every single term of this matrix is θ dependent, as a consequence, the inverse of this matrix must be calculated at every time step, making this model very complex, time consuming, and requires a high processing power.

3.3 Electromagnetic Torque The electromagnetic torque is derived starting from the electromagnetic power divided by the angular mechanical speed. Pe = ea1 i a1 + eb1 i b1 + ec1 i c1 + ea2 i a2 + eb2 i b2 + ec2 i c2

(14)

With ei the induced electromotive force of winding i, expressed by: dϕabc dt

(15)

ei = vi − ri i

(16)

[e] = − Or

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Then the electromagnetic torque is given by (17): te =

Pe ω

(17)

4 Double d-q Winding Model In order to simplify the analysis of the dual star machine behaviour, the park transformation is applied to the abc system, leading to a couple of d-q winding with a mutual coupling between theme [9]. The standard park transformation matrix for a three phase system is defined as:  T (θ ) =

⎡



⎤ cos θ + 2π cos(θ ) cos θ − 2π 3 3

2⎢ 2π 2π ⎥ ⎣ −sin(θ ) −sin θ − 3 −sin θ + 3 ⎦ 3 √1 √1 √1 2

2

(18)

2

Starting from this, the transformation matrix T1 for a dual star system can be deduced [9].

T1 =

T (θ ) 03,3 03,3 T (θ − α)

 (19)

With 03,3 a 3 × 3 null matrix. The transformation is given by (20). X dq = T1 X abc

(20)

where X can be any phase variable vector (current, voltage or flux). Applying (20) to the stator flux (21) will result in (22) ϕabc = L s i abc + ϕ P Mabc

(21)

ϕdq = T1 L s T1t i dq + ϕ P Mdq

(22)

With ϕ P Mdq =

√

6 ψ pm 2

t

√

00

6 ψ pm 2

00

(23)

After several lines of trigonometric simplifications, the transformed inductance matrix can be obtained.

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⎡

L T1

L d1 ⎢ 0 ⎢ ⎢ ⎢ 0 t = T1 L s T1 = ⎢ ⎢ Md2 ⎢ ⎣ 0 0

0 L q1 0 0 Mq2 0

0 0 L 01 0 0 0

Md1 0 0 L d2 0 0

0 Mq1 0 0 L q2 0

395

⎤ 0 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ L 02

(24)

In this matrix, we can see four square 3 × 3 sub-matrices, related to two couples of orthogonal windings dq1 and dq2 , with a mutual coupling between theme expressed by Md and Mq . L d1 = L d2 = L s0 +

3 m + L s2 2 2

(25)

L q1 = L q2 = L s0 +

3 m − L s2 2 2

(26)

L 01 = L 02 = L s0 − m

(27)

Md1 = Md2 =

3 3 m + L s2 2 2

(28)

Mq1 = Mq2 =

3 3 m − L s2 2 2

(29)

As can be seen, the resultant matrix inductance is not diagonal, adding some complexity to the model simulation and analysis. Also, Eq. 23 shows that the magnet flux is present in both sub-machines, so the total developed torque is the sum of the torques developed by each sub-machine. Despite this complexity, it can be used in a current control scheme such proposed by [10] in which a decoupled voltage equations are introduced to improve the PI controllers’ performance, but these equations induce more errors since the current derivatives are ignored.

4.1 The Current Model in the Coupled d-q Reference Frame The machine electric model can now be obtained by applying the transformation matrix to the electric Eq. (1) and combining the equations aforementioned, leading to the electric equation in the dq reference frame (30). vdq = r i dq +

dϕdq + ωjϕdq dt

The current model of the machine (31) is obtained by developing (30).

(30)

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dϕ P Mdq di dq v = L −1 − r + ωj L − ωjϕ − i dq P Mdq T1 dq T1 dt dt

(31)

where ⎡

0 ⎢1   ⎢ ⎢ 1 dT1−1 ⎢0 j= T1 =⎢ ⎢0 ω dt ⎢ ⎣0 0

−1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0

0 0 0 −1 0 0

⎤ 0 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎦ 0

ϕdq = L T1 i dq + ϕ P Mdq

(32)

(33)

4.2 Electromagnetic Torque Starting from the instantaneous output power, the electromagnetic power as well as torque can be derived. Pout = vd1 i d1 + vq1 i q1 + vd2 i d2 + vq2 i q2

(34)

By replacing the voltage by it’s equation given by (30), ignoring the resistive voltage drop, and dividing by the angular speed, the electromagnetic torque is obtained:

te = P ϕd1 i q1 − ϕq1 i d1 + ϕd2 i q2 − ϕq2 i d2

(35)

5 Decoupled D-Q Model The matrix of the further transformation required to eliminate the coupling between the two d-q windings, is given by T2 , leading to a final decoupled D1 -Q1 and D2 Q2 windings, note the introduction of a coefficient √12 to get an invariant power [6].

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⎡

1 ⎢ 0 ⎢ 1 ⎢ ⎢ 0 T2 = √ ⎢ 0 2⎢ ⎢ ⎣ −1 0

0 1 0 1 0 0

0 0 1 0 0 0

1 0 0 0 1 0

0 1 0 −1 0 0

397

⎤ 0 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎦ 1

(36)

The application of T2 to (24) yields ⎡

L T2

L D1 ⎢ 0 ⎢ ⎢ ⎢ 0 t = T2 L T1 T2 = ⎢ ⎢ 0 ⎢ ⎣ 0 0

0 L Q1 0 0 0 0

0 0 l0∗ 0 0 0

0 0 0 L D2 0 0

0 0 0 0 L Q2 0

0 0 0 0 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(37)

l0∗∗

where L D1 = L d1 + Md1

(38)

L Q 1 = L q1 + Mq1

(39)

L D2 = L q 1 − M q 1

(40)

L Q 2 = L d1 − Md1

(41)

L 01 2

(42)

L ∗0 = L ∗∗ 0 =

The final transformation matrix can be found by combining the two transformation matrices.

 1 T (θ ) π T (θ − α)π T3 = T2 T1 = √ (43) 2 T θ −α+ 2 T θ −α− 2 In this system, the fundamental wave and (12 k ± 1)th harmonics (k = 1, 2, 3,…) in the original quantities are mapped into D1 -Q1 pair quantities, resulting as constant terms, and 12 kth harmonics, while the (6(2 k + 1) ± 1)th harmonics (k = 0, 1, 2,…) are mapped into D2 -Q2 pair quantities, resulting as (6(2k + 1))th harmonics [6].

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5.1 The Current Model in the Decoupled D-Q Reference Frame Same as in Sect. 4.1, applying the transformation matrix T3 to (1), yields the current model of the machine in the decoupled d-q reference frame (44).  

dϕ P M D Q di D Q v = L −1 − r + ωj L − ωjϕ − i D Q P M T D Q DQ 2 T2 dt dt

(44)

where ⎡

0 ⎢1 ⎢   ⎢ 1 dT3−1 ⎢0 J= T3 =⎢ ⎢0 ω dt ⎢ ⎣0 0

−1 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 1 0

0 0 0 −1 0 0

⎤ 0 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎦ 0

(45)

ϕ P M D Q can be easily deduced by applying the transformation matrix T3 to ϕ P Mabc ϕ P M D Q = T3 ϕ P Mabc =

√

3ψ pm 0 0 0 0 0

t

(46)

5.2 Electromagnetic Torque The electromagnetic torque is derived starting from the instantaneous power divided by the mechanical angular speed , yielding an equation similar to the one derived for the coupled d-q model.

te = P ϕ D 1 i Q 1 − ϕ Q 1 i D 1 + ϕ D 2 i Q 2 − ϕ Q 2 i D 2

(47)

6 Extended 0dq Model Another transformation is proposed by [7] where a unique transformation matrix is used to fully decouple the inductance matrix. Starting from the following three phase park transformation matrix T (θ ), the full transformation matrix T p (θ ) for the dual star system can be constructed.

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⎡

⎤ cos(θ ) −sin(θ )



⎥ 2⎢ −sin θ − 2π cos θ − 2π T (θ ) = ⎣ 3 3 ⎦ 3 −sin θ + 2π cos θ + 2π 3 3

 1 T (θ ) T (θ ) T p (θ ) = √ T − α) −T (θ (θ − α) 2 

√1 2 √1 2 √1 2

399

(48)

(49)

Note the use of the same coefficient as in T2 to get an invariant power. The application of this transformation to the inductance matrix L s (θ ) will result in the following matrix: ⎡

L n0 ⎢ 0 ⎢ ⎢ ⎢ 0 t L T = Tp L s Tp = ⎢ ⎢ 0 ⎢ ⎣ 0 0

0 L nd 0 0 0 0

0 0 L nq 0 0 0

0 0 0 L a0 0 0

0 0 0 0 L ad 0

⎤ 0 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ L aq

(50)

where L n 0 = L a0 = L ad = L aq = L s0 − m

(51)

L n d = L s0 + 3L s2 + 2m

(52)

L nq = L s0 − 3L s2 + 2m

(53)

This transformation splits the machine into two decoupled systems, one called normal system, and the other is called anti system, in the normal system, the current support each other, creating a normal rotational field in the air gap, and in the anti system, the currents oppose each other leaving only leakage fluxes in the stator [7].

6.1 The Current Model in the Extended 0dq Reference Frame As done in Sect. 4.1, the current model of the machine in the extended 0dq reference frame is given by:   dϕ P M0dq di 0dq v = L −1 − + ωj L − ωjϕ − (r )i 0dq P M0dq T 0dq T dt dt With

(54)

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⎡

⎤ 00 0 00 0 ⎢ ⎥   ⎢ 0 0 −1 0 0 0 ⎥ t−1 ⎢ ⎥ dT p 1 ⎢0 1 0 0 0 0 ⎥ j= T pt =⎢ ⎥ ⎢0 0 0 0 0 0 ⎥ ω dt ⎢ ⎥ ⎣ 0 0 0 0 0 −1 ⎦ 00 0 01 0 t  √ ϕ P M0dq = T p ϕ P Mabc = 0 3Ψ pm 0 0 0 0

(55)

(56)

6.2 Electromagnetic torque As done in Sect. 4.2, the electromagnetic torque is given by (57).

te = P ϕnd i nq − ϕnq i nd + ϕad i aq − ϕaq i ad

(57)

7 Simulation Results All the three models have been simulated using Matlab/Simulink environment, the machine is simulated as a generator driven with constant speed, and feeding a resistive load. The simulation results are presented and discussed below. The parameters of the considered machine are shown in Table 1. The Figs. 2, 3 and 4 present the currents in the transformed reference frames, the decoupled and extended models are similar, they describe the machine in the same way, but they are different from the double winding model, which describes the machine differently. In the coupled reference frame, the d components (i d1 and i d2 ) are collapsed together, and have a non null value, also the q (i q1 and i q2 ) and homopolar components, Table 1 Machine parameters [11]

Description

Value

Stator resistance

0.53

Average self inductance L s0

18.93 mH

Second harmonic coefficient L s2

−3.03 mH

Mutual inductance m

10.53 mH

Permanent magnet flux

1.8328 Wb

Number of pole pairs

8

Rotational speed

350 rpm

Simulation Study of the Dual Star Permanent Magnet Synchronous … Fig. 2 Currents in the coupled reference frame

Fig. 3 Currents in the decoupled reference frame

Fig. 4 Currents in the extended reference frame

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but in the decoupled and extended models, only two components are active, i D1 and i Q 1 for the decoupled model, and i n d and i nq for the extended model, all the other component are null, meaning that they do not actively affect the machine comportment. After applying the invers transformation, the currents appear identical and match the current of the natural frame model, confirming the validity of the simulated models. Because the currents are similar, the Fig. 5 is used to show the currents of the transformed models, and the Fig. 6 represents the current of the natural frame model (Fig. 7). The torque is shown in Fig. 8, a zoom into the plot shows an error of about 0.0063% of the transformed models compared to the natural frame model in the steady state, this error in negligible and do not influence the comportment of the models. Fig. 5 The currents of the transformed models

Fig. 6 The currents of the natural frame model

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Fig. 7 Electromagnetic torque of the four models

Fig. 8 Electromagnetic torque of the double winding model

In the double winding model, each sub machine generates half of the total machine torque, like shown in Fig. 8, and the total developed torque is the same as for the other models.

8 Electrical Machine d-q Model Limitations The transformed model of an electrical machine simplifies not only the analysis of the machine behaviour, but also the control, for example, in a predictive current control strategy [12], since it deals with constant parameters, but it presents some limits and inaccuracies.

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Such a model ignores the higher order harmonics in voltage and flux, assuming pure sinusoidal waveforms, but in faulty conditions, harmonics are present, and the pure sinusoidal variation is removed, making the model mathematically incorrect [13]. Also, the simplified model ignores the effect of saturation, magnetic circuit geometry, and assume that all the coils are identical, also, mutual inductances are supposed to be equal [14]. All these assumptions give inaccuracies to the model.

9 Conclusion To simplify the analysis and control of double star machine, an adequate mathematical model must be constructed, for this purpose, many models are reported in the literature. The natural frame model gives more details about the machine since it includes all the voltage harmonics, and can easily take into account saturation effect and other phenomena, but it’s very complex to solve and requires a high computational power. Simplified models have been proposed, the double winding model is very simple to compute, and allows a separate control of the two windings, but the mutual coupling between the two winding couples decrease the dynamic performance of the control scheme. The use of decoupled equations in order to minimize the effect of such coupling would induce more errors because the derivatives of the currents can not be considered. Also, in this model, all the current components are active, so the analysis of the machine behaviour becomes more complex comparing to other simplified models. To solve the problems encountered with the model aforementioned, a second transformation is proposed to fully decouple the windings, this new approach is called decoupled d-q approach, it’s comparable to another model called extended d-q model, these two models require less processing power. Used in a PI controller based current control strategy give better results, since the performance of the PI controllers is higher when the controlled variable is decoupled from other variables in the system. However, these models don’t permit a separate control of the two winding sets. The major drawback of the decoupled d-q model is the use of two consecutive transformations. Despite of their differences, they describe the machine behaviour in a rotating reference frame, eliminating the dependency of machine inductances on the rotor position, the main difference between all these models is the complexity and the difficulty of implementation.

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References 1. Nelson RH, Krause PC (1974) Induction machine analysis for arbitrary displacement between multiple winding sets. IEEE Trans Power Appar Syst PAS-93(3):841–848 2. Levi E, Bojoi R, Profumo F, Toliyat HA, Williamson S (2007) Multiphase induction motor drives—a technology status review. IET Electr Power Appl 1(4):489–516 3. Star D, Magnet P, Machine S, Order S, Mode S (2018) Direct torque control using second order sliding mode of a double star permanent, vol. 80 4. Zaimeddine R, Berkouk EM (2012) A novel DTC scheme of double-star induction motors using three-level voltage source inverter 5. Barrero F, Arahal MR, Gregor R, Toral S, Durán MJ (2009) A proof of concept study of predictive current control for VSI-driven asymmetrical dual three-phase AC machines. IEEE Trans Ind Electron 56(6):1937–1954 6. Kallio S, Andriollo M, Tortella A, Karttunen J (2013) Decoupled d-q model of double-star interior-permanent-magnet synchronous machines. IEEE Trans Ind Electron 60(6):2486–2494 7. Knudsen H (1995) Extended Park’s transformation for 2 × 3-phase synchronous machine and converter phasor model with representation of AC harmonics. IEEE Trans Energy Convers 10(1):126–132 8. Cardinale D (2016) Double star Pm machine : analysis and simulations. Politecnico di Milano 9. Fuchs EF, Rosenberg LT (1974) Analysis of an alternator with two displaced stator windings. IEEE Trans Power Appar Syst PAS-93(6), 1776–1786 10. Karttunen S, Kallio S, Peltoniemi P, Silventoinen P, Pyrhonen O (2012) Dual three-phase permanent magnet synchronous machine supplied by two independent voltage source inverters. In: International symposium on power electronics power electronics, electrical drives, automation and motion, pp 741–747 11. Karttunen J, Peltoniemi P, Kallio S, Silventoinen P, Pyrhönen O (2014) Determination of the inductance parameters for the decoupled d-q model of double-star permanent-magnet synchronous machines. IET Electr Power Appl 8(2):39–49 12. Zhang Y, Gao S, Xu W (2016) An improved model predictive current control of permanent magnet synchronous motor drives. In: Conference proceedings—IEEE applied power electronics conference and exposition - APEC, 2016, vol. 2016, pp 2868–2874 13. Mohammed OA, Liu S, Liu Z (2004) Phase-variable model of PM synchronous machines for integrated motor drives. IEE Proc Sci Meas Technol 151(6):423–429 14. Mohammed OA, Liu S, Liu Z (2005) Physical modeling of PM synchronous motors for integrated coupling with machine drives. IEEE Trans Magn 41(5):1628–1631

Modeling and Experimental Identification of Salient-Pole Synchronous Machine Khalil-Errahmane Sari and Bilal Sari

Abstract This paper is a complete procedure to determine a valid mathematical model for the salient-pole synchronous machine. It represents: (i.) a development of a suitable model for the synchronous machine formed as a lumped-parameter equivalent circuit, and (ii.) an experimental identification of the different machine’s parameters. Taking into account the presence of damper windings, an electrical equivalent circuit of the machine is found in the dq-stator reference frame. In order to accurately measure the machine’s parameters: X d , X d , X d , X q , Td , and Td , etc., experimental identification tests including: the open-circuit test, the sustained short-circuit test, the slip test, and the sudden three-phase short-circuit test were performed. The reduction factor of the field winding is obtained directly from the short-circuit test and without any knowledge of design information of the synchronous machine. The accuracy of the developed model is validated by comparing model simulation results to practical measurements. Keywords Salient-pole synchronous machine · Lumped-parameter equivalent circuit · Experimental identification tests · Sudden three-phase short-circuit test

1 Introduction Synchronous machines are very important electro-mechanical energy-conversion devices. They are widely used in both: electricity production as generator units, and in motor applications. In fact, most power plants rely on synchronous machines to provide power to the grid. For that, modeling of the machine is considered essential for power systems simulation and analysis, and control purposes [1, 2]. The appropriate representation of a synchronous machine is subjected to the physical construction of its rotor, the availability of data, and computational considerations [3]. The author in [3] also stated that: “A complete synchronous machine model K.-E. Sari (B) · B. Sari DAC HR Laboratory, Department of Electrical Engineering, Faculty of Technology, University Ferhat Abbas Sétif 1, Sétif 19000, Algeria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_28

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consists of the combination of a model structure and a set of parameter values.” This definition can be broken into two interconnected distinct parts: (i) the construction of the model structure, which is the basic form for machine representation, and (ii) the evaluation of the model parameters. The model structure can be formed as a lumped-parameter equivalent circuit, differential equation representation, etc., whereas, the model parameters can be obtained from manufacturers’ data, experimental test results, or based on other analytical techniques [3]. Synchronous machines are usually acknowledged to be accurately modeled by a lumped-parameter equivalent circuit representing the d-axis and the q-axis [1], where the stator and the rotor windings are included without using the same magnetic circuit. In this work, we consider a more general method in which the same magnetic circuit is used. This is done by referring rotor quantities to the stator reference frame. An extra work, however, is in the determination of the reduction factors [2]. To study the machine’s behavior during transient and steady-state conditions, accurate knowledge of the machine’s equivalent circuit parameters is required [4]. IEC standard 34-4-1 [5] and IEEE standard 115 [6] include the laboratory tests procedures for the practical determination of these parameters. The present work is carried out in five main sections. After the introduction given in Sect. 1, the rest is as follows: Sect. 2 covers the modeling of the synchronous machine. Starting from the classical machine equations, a mathematical model is developed in the stator reference frame and an electrical equivalent circuit of the machine is found. Three reduction factors: K f , K D and K Q are used in reducing rotor quantities to the stator. Section 3 describes the experimental determination of the machine’s parameters. The laboratory procedures of the open-circuit test, the sustained shortcircuit test, the slip test, and the sudden three-phase short-circuit test are explained and analyzed, whereby, the values of the required parameters are extracted. Section 4 sums the relationships between the developed model parameters and the parameters extracted experimentally. Based on these relationships, it is found that K D and K Q do not affect the terminal behavior of the machine, and therefore, finding their exact values is outside the scope of this paper. The value of K f , on the other hand, is of great importance and is found directly from the short-circuit test. Section 5 represents the validation of the deduced model under MATLAB and Simulink. A comparison between practical measurements and the corresponding simulation results is done. This work finally ends with some concluding remarks.

2 Synchronous Machine Modeling The schematic of a three-phase synchronous machine conventional model structure is illustrated in Fig. 1. The machine consists of two essential elements: armature windings and excitation, or field winding. The magnetic axis of the machine is defined as the direct- or d-axis, and an orthogonal quadratic- or q-axis is located 90 electrical degrees ahead of the d-axis [7]. The machine is modeled with one damper winding for the d-axis and one damper winding for the q-axis [2, 7].

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Fig. 1 Synchronous machine conventional structure, with one damper winding for the d- axis and one damper winding for the q-axis

2.1 Mathematical Formulation and dq-Model The model is to be formed as a set of equations that completely describes the electrical behavior of the synchronous machine. Before proceeding to develop a suitable model for the machine, some assumptions are made [7, 8]: • Magnetic saturation and hysteresis are neglected. • Stator currents are assumed to set up magneto-motive force sinusoidally distributed in space around the air-gap. Therefore, the effect of space harmonics is neglected. Applying Maxwell’s equation to the configuration shown in Fig. 1, the phasevoltage equations in the natural-reference frame are simply [7, 8]: dψa dt dψb dt dψc dt dψ f − dt dψ D − dt dψ Q − dt

= i a Rs + va = i b Rs + vb = i c Rs + vc = if Rf − vf = i D RD = i Q RQ ,

where: • vabc , iabc : The stator voltages and the stator currents.

(1)

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• • • • •

K.-E. Sari and B. Sari

vf , if : The field voltage and the field current. iD , iQ : The dampers currents. ψ abc : The stator total fluxes. ψ f : The field total flux. ψ D , ψ Q : The dampers total fluxes.

The above equations completely describe the electrical behavior of the synchronous machine. These equations, however, contain parameters which vary with θ, the rotor position, which varies with time [8]. Further development of the machine’s model can be done using Park’s transformation to mathematically transform the three-phase time-varying stator quantities into time-invariant d- and q-axes quantities under steady-state conditions [1]. The rotor windings: f , D, and Q are already aligned along with the d- and q-axes. That is, only stator equations that need to be transformed. The transformation from abc- to dq-variables can be done by applying the transformation matrix given by (2) [7, 8]: P(θ ) =

  2 cos(−θ ) cos(−θ + 23 π ) cos(−θ − 23 π ) . 3 sin(−θ ) sin(−θ + 23 π ) sin(−θ − 23 π )

(2)

Therefore: vdq = P(θ )vabc , i dq = P(θ )i abc and, ψdq = P(θ )ψabc . The analysis of the machine’s model in terms of dq-variables is considerably simpler for the following reasons [8]: • For balanced steady-state operations, the stator quantities have constant values. • For balanced conditions, the zero sequence quantities disappear. • The parameters associated with d- and q-axes may be directly measured from terminal tests. The inverse transformation is [7, 8]: [P(θ )]−1 =

3 [P(θ )]T . 2

(3)

Applying the dq-transformation, the following equations in terms of transformed components of voltages, currents, and flux linkages result [7, 8]: dψd = i d Rs + vd + ωr ψq dt dψq = i q Rs + vq − ωr ψd dt r dψ f = i rf R rf − vrf − dt

Modeling and Experimental Identification of Salient-Pole …

dψ Dr = i rD R rD dt dψ Qr − = i rQ R rQ , dt

411

−

(4)

with: ψd = −L d i d + M f i rf + M D i rD ψq = −L q i q + M Q i rQ 3 3 M f i d + M f D i rD 2 2 3 3 r + L Dm )i D − M D i d + M f D i rf 2 2 3 r + L Qm )i Q − M Q i q , 2

ψ rf = (L rf l + L f m )i rf − ψ Dr = (L rDl ψ Qr = (L rQl

(5)

and: dθ = ωr . dt

(6)

2.2 dq-Model in the Stator Reference Frame To simulate the machine’s model presented by (4) (5), the different mutual inductances: M f , M D , M Q , M fD , etc., need to be evaluated. By reducing the rotor variables to the stator, the number of inductances reduces, and the electrical equivalent circuit of the machine is found [2, 7]. Rewriting the machine’s model equations referring all variables to the stator, the following result [7]: dψd dt dψq dt dψ f − dt dψ D − dt dψ Q − dt

= i d Rs + vd + ωr ψq = i q Rs + vq − ωr ψd = if Rf − vf = i D RD = i Q RQ .

The flux-current relations are now [7]:

(7)

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ψd = −L sl i d + L dm (−i d + i D + i f ) ψq = −L sl i q + L qm (−i q + i Q ) ψ f = L f l i f + L dm (−i d + i D + i f ) ψ D = L Dl i D + L dm (−i d + i D + i f ) ψ Q = L Ql i Q + L qm (−i q + i Q ),

(8)

where: • • • •

L dm , L qm : The direct and the transverse stator main leakage inductances. L sl : The stator leakage inductance. L fl : The field leakage inductance, referred to the stator. L Dl , L Ql : The dampers leakage inductances, referred to the stator.

Comparing (5) to (8), the following definitions of current reduction factors are considered valid [7]: i f = i rf K f , i D = i rD K D and, i Q = i rQ K Q ,

(9)

with: Mf : The reduction factor of the field winding L dm MD : The reduction factor of the d - axis damper winding KD = L dm MQ : The reduction factor of the q - axis damper winding. KQ = L qm Kf =

Using the reduction factors, and from (5), the following is obtained [7]: 2 L dm = ψ f = L f l i f + L dm (−i d + i D + i f ) 3 Mf 2 L dm = ψ D = L Dl i D + L dm (−i d + i D + i f ) ψ Dr 3 MD 2 L qm ψ Qr = ψ Q = L Ql i Q + L qm (−i q + i Q ). 3 MQ ψ rf

From (10), it follows that: 2 3K 2f 2 = L rDl 3K D2

L f l = L rf l L Dl

(10)

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413

Fig. 2 Electrical equivalent circuit, with rotor quantities referred to the stator

L Ql = L rQl

2 . 3K Q2

(11)

Reducing the rotor circuit resistances and the field voltage to the stator is done by power equivalence [7], this yields: 2 3K 2f 2 R D = R rD 3K D2 2 R Q = R rQ , 3K Q2 R f = R rf

(12)

and: v f = vrf

2 . 3K f

(13)

Using the above equations, the electrical equivalent circuit of the machine with rotor quantities referred to the stator is deduced [7, 2] (See Fig. 2).

3 Experimental Determination of the Machine’s Equivalent Circuit Parameters In order to obtain the synchronous machine’s electrical equivalent circuit parameters, different experimental identification tests need to be performed [4]. A LabVolt salientpole synchronous machine of 1.5 KVA with damper windings is used in this work. Figure 3 shows the experimental test bench.

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Fig. 3 Experimental test bench

3.1 Open-Circuit Test The open-circuit test is performed on the synchronous machine at no-load to obtain the no-load saturation curve, or the open-circuit characteristic [4]. The synchronous machine being tested is firstly driven, using a prime mover, at its rated speed with no excitation being applied. The field winding current is then supplied and gradually increased while measuring the terminal voltage until reaching the saturation condition [4]. Figure 4 shows the obtained I-V curve. Fig. 4 Open-circuit characteristic

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415

3.2 Sustained Short-Circuit Test The objective of sustained short-circuit test is to obtain the short-circuit characteristic of the machine. This last is used with the previously obtained open-circuit characteristic curve to determine the synchronous reactance X d [4]. During the test, the machine is rotating at its rated speed with its stator terminals permanently short-circuited. The excitation current is supplied and gradually increased while measuring the short-circuit current flowing in armature windings. The short-circuit current is plotted versus the field winding current to obtain the shortcircuit characteristic curve (See Fig. 5). While performing this test, it is important to keep track of the stator current, it might exceed the rated value by only a small tolerance to prevent stator windings damage. The d-axis unsaturated impedance Z d(unsat) can be obtained as a quotient of the voltage on the open-circuit characteristic curve at a field current point, and the short-circuit armature current on the short-circuit characteristic curve corresponding to the same field current point. That is [4]: VL−L . Z d(unsat) = √ 3Isc

(14)

The d-axis unsaturated synchronous reactance is therefore: X d(unsat) =



2 Z d(unsat) − Rs2 .

where: • Rs : The stator resistance, and it is measured directly. Fig. 5 Short-circuit characteristic

(15)

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3.3 Slip Test The aim of the slip test is to obtain the saliency ratio. Using this ratio and the value of X d obtained previously, X q can be determined [4]. During the test, the synchronous machine is driven at a speed slightly different from the rated speed of about 1% to achieve a very small slip. With no excitation, a balanced three-phase voltage of about 25% of the rated value is applied across the machine’s terminals. Figure 6 shows a recording of the armature current and voltage. Clearly, both quantities oscillate, having maximum and minimum values. It is remarkable also that when the armature voltage is of maximum amplitude, the armature current is minimum value. This is, in fact, the time when the d-axis is in line with the armature mmf, and thereby the offered air-gap reactance represents the d-axis synchronous reactance. Similarly, when the voltage is of minimum amplitude and the current is maximum value, the q-axis is aligned with the armature mmf, and the presented air-gap reactance is the q-axis synchronous reactance [4]. Using the obtained current and voltage recordings, the following calculations can be made: Vt (L−L)max √ 3Imin Vt (L−L)min = √ . 3Imax

X d(sli p) = X q(sli p)

(16)

The saliency ration is: 

Finally, X q is obtained: Fig. 6 Real current and voltage waveforms from slip test

Xq Xd

 = sli p

X q(si p) . X d(sli p)

(17)

Modeling and Experimental Identification of Salient-Pole …

 Xq = Xd

Xq Xd

417

 .

(18)

sli p

3.4 Sudden Three-Phase Short-Circuit Test Many transient and subtransient parameters can be extracted from a suitable oscillogram of a short-circuit current recorded during the sudden three-phase short-circuit test [5, 6]. The interest here is to calculate the d-axis transient and subtransient synchronous reactances, X d and X d respectively. The values of the q-axis transient and subtransient synchronous reactances are beyond the scope of this paper. The synchronous machine is firstly driven at its rated speed, and is excited so that its terminal voltage is equal to the rated value at no-load condition. A sudden threephase short-circuit is then applied to the machine’s terminals. The short-circuit armature currents are recorded [5, 6]. Figure 7 shows the short-circuit current waveform of phase a. The current waveform consists of: an ac component and a dc component. The ac component can be resolved into three different components: subtransient I  , transient I  and steady-state I s components [4]. Also, it is clear that the current envelope obeys an exponential decay. The time constant controlling the rapid decay during the subtransient period is called the d-axis subtransient time constant Td , whereas, the time constant controlling the slower decay during the transient period is called the d-axis transient time constant Td [5, 6]. The rms amplitude of the total ac component of the short-circuit current in one phase can be expressed as [4]: Fig. 7 Phase-a real short-circuit current

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Fig. 8 Typical polynomial curve fitting of a three-phase short-circuit current envelope

 −t   −t  Isc (t) = I  − I  e Td + I  − Is e Td + Is

(19)

or,  Isc (t) = E 0

 −t   −t  1 1 1 1 1 Td Td e e . − + − + X d X d X d Xd Xd

(20)

By extrapolating the current envelope back to the zero time, or the time of the short-circuit occurrence (See Fig. 8), the transient and subtransient time constants and currents, Td , Td , I  and I  respectively, can be evaluated [4, 5]. Finally, the values of X d and X d result in [4]: E0 I E0 X d =  . I X d =

(21)

where: • E 0 : The rms line-to-neutral open-circuit prefault terminal voltage.

3.5 Results from Experimental Identification Tests The previously discussed experimental identification tests were realized for several tries. For each try, data were recorded, and the mean value was taken. This is done in order to accurately identify the different machine’s parameters. Table 1 summarizes the results from experimental identification tests, as well as, the nominal values of the synchronous machine used in this work.

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419

Table 1 Parmeters of the Labvolt Ems 8557-1 Salient-Pole Synchronous Machine Measured parameters X d () 75.443 Rrf () 127



X q () 46.556 T ’d (s) 0.0776

X ’d () 10.309  T d (s) 0.0147

X d () 8.5298 T ’d0 (s)a 0.235

Rs () 2.2

Voltage V n 220 V

Current I n 4A

fn 50 Hz

tr/min 1500

Nominal values Power S n 1.5KVA a Measured

having the synchronous machine under a short-circuit then a sudden open-circuit is

made [2]

4 Mathematical Relationships Between Model Parameters and Measured Parameters The machine’s model presented by (7) (8) requires the determination of leakage inductances. The parameters of Table 1, however, are in terms of reactances and time constants. For that, mathematical relationships between these parameters and leakage inductances need to be found. According to IEC standard [5], these relationships can be written as follows: ⎧ X = Lωr ⎪ ⎪ ⎪ ⎪ ⎪ X d = X sl ⎪ ⎪ ⎪ ⎪ X q = X sl ⎪ ⎪ ⎪ ⎪ ⎪ X  = X sl ⎪ ⎪ d ⎪ ⎪  ⎪ ⎪ X d = X sl ⎪ ⎪ ⎪  ⎪ ⎪ ⎨ X q = X sl

+ X dm + X qm X X + X dmdm+Xf lf l

X Dl X f l X Dl +X f l X X + X qmqm+XQlQl

 X sl Td = ωr1R f X f l + XXdmdm+X sl ⎪   ⎪ ⎪  ⎪ = ωr1R f X dm + X f l Td0 ⎪ ⎪

 ⎪ ⎪ X l X sl ⎪ ⎪ Td = ωr 1R D X Dl + X f lf +X ⎪ sl  ⎪

⎪ ⎪ X f l X dm 1  ⎪ ⎪ X = + T Dl ⎪ d0 ωr R D X f l +X dm ⎪ ⎪ ⎪ ⎪ T  = 1 X Ql + X qm X sl ⎪ ⎪ q ωr R Q X qm +X sl ⎪ ⎪ ⎩ T  = 1  X + X  qm Ql q0 ωr R Q

+

(22)

Based on the above expressions, and after some mathematical manipulation, relationships between model parameters and measured parameters are deduced [2, 9]:

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 ⎧    ⎪ X = Td0 ωr R f X d − X d dm ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ X sl = X d − X dm ⎪ ⎪ X qm = X q − X sl ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ X f l = Td0 ωrR − X dm ⎪ ⎪ X X d −X sl ) ⎪ ⎪ X Dl = X f l (+X  ⎨ fl sl −X d X qm ( X q −X sl ) X Ql = X qm +X sl −X  ⎪ ⎪ q

 ⎪ ⎪ X l X dm ⎪ ⎪ R D = ωr 1T  X Dl + X f lf +X ⎪ ⎪ d0   dm ⎪ ⎪ 1 ⎪ X = + X R  ⎪ Q qm Ql ωr Tq0 ⎪ ⎪  ⎪ ⎪   X d ⎪ T = T ⎪ d0 d X d ⎪ ⎪ ⎩ T  = T  X q q X  q0

(23)

q

Using the relationships of (23), the model parameters can now be computed. Notice that the value of X dm requires the determination of Rf , the rotor resistance referred to the stator. This implies the determination of the field winding reduction factor K f .

4.1 Determination of the Reduction Factors The reduction factors used to reduce rotor quantities to the stator may be calculated through analytical or numerical methods, and may also be measured [2, 7]. (1) The dampers reduction factors: The dampers’ parameters of (23) are computed without using the dampers reduction factors K D and K Q . In fact, K D and K Q are essentially needed to find the dampers currents I D and I Q [2]. Practically, these currents are not measured, and since it is possible to simulate the machine’s electrical behavior without using these factors, finding their exact values is outside the scope of this work. (2) The field winding reduction factor: The value of the field winding reduction factor K f can be extracted directly from the short-circuit test [2]. This is done by simply applying:

If Kf = √ . 3Isc

(24)

The obtained values of K f at different values of field winding currents are illustrated in Fig. 9. By taking the mean value, K f is finally found [2].

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Fig. 9 Reduction factor of the field winding, K f

5 Validation of the Developed Model Having the synchronous machine’s parameters of Table 1, the mathematical relationships of (23), and the value of the field winding reduction factor K f , the simulation of the machine’s electrical behavior is now possible. The system of equations given by (7) (8) can be written as:   di f vd = −Rs i d + L sl + L qm ωr i q − L qm ωr i Q + L dm dt di D di d − (L sl + L dm ) + L dm dt dt vq = −Rs i q − (L sl + L dm )ωr i d − L dm ωr i f + L dm ωr i D  di q  di Q + L dm − L sl + L qm dt dt   di d di D + L dm v f = −R f i f + L f l + L dm ωr i d − L dm dt dt di f di d di D + L dm − L dm 0 = −R D i D + (L Dl + L dm ) dt dt dt  di Q  di q − L qm . 0 = −R Q i Q + L Ql + L qm dt dt

(25)

In matrix form: ⎡

vd ⎢v ⎢ q ⎢ ⎢vf ⎢ ⎣0 0 with:

⎡

⎤ ⎡ ⎤ id id ⎢i ⎥ ⎥ ⎢i ⎥ ⎢ q ⎥ ⎥ q ⎥ d⎢ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥ = Z ⎢ i f ⎥ + L ⎢ i f ⎥, ⎢ ⎥ ⎥ dt ⎢ ⎥ ⎣ iD ⎦ ⎦ ⎣ iD ⎦ iQ iQ ⎤

(26)

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⎤   −Rs L sl + L qm ωr 0 0 −L qm ωr ⎥ ⎢ −(L + L )ω −Rs L dm ωr L dm ωr 0 ⎥ ⎢ sl dm r ⎥ ⎢ Z =⎢ 0 0 0 0 Rf ⎥ ⎥ ⎢ ⎦ ⎣ 0 0 0 0 RD 0 0 0 0 RQ ⎡ ⎤ −(L sl + L dm )  0 L 0 L dm dm  ⎢ ⎥ 0 0 L qm 0 − L sl + L qm ⎢ ⎥ ⎢ ⎥ L=⎢ −L dm 0 L f l + L dm L dm 0 ⎥, (27) ⎢ ⎥ ⎣ ⎦ 0 L dm L Dl + L dm 0 −L dm 0 0 L Ql + L dm 0 −L qm ⎡

in state-space representation: X˙ = AX + BU Y = C X,

(28)

with: A = −L −1 Z , B = L −1 , T  U = vd vq v f 0 0 .

T  X = id iq i f i D i Q and,

Validation of the synchronous machine’s model given by (25) is done using MATLAB and Simulink software. To check the fitness of the mathematical model developed and the accuracy of the identified parameters, a sudden three-phase shortcircuit test is performed under Simulink. With the help of the Simscape library, a three-phase short-circuit is applied at 55% rated terminal voltage. The armature currents, as well as, the field winding current obtained by experimental tests and those obtained by simulation are plotted on the same graphs (please refer to Figs. 10 and 11). Clearly, the resulting currents have approximately the same waveforms. At the moment of the sudden three-phase short-circuit, and in the subtransient-state, it is noticeable that the real currents decrease more rapidly than the model currents. This is because the measured subtransient time constant Td is defined long. In the transient-state, the real and model currents tend to converge. However, in the steadystate, a very good agreement between real and model currents is achieved. This is also the case during the prefault period. The synchronous machine’s model developed closely reflects the actual machine’s electrical behavior. Hence, successful validation of the model is achieved. The presence of some discrepancies is mainly due to the neglected effects of saturation, hysteresis, and harmonics of the main field mmf, as well as measurements and rounding errors.

Modeling and Experimental Identification of Salient-Pole …

Fig. 10 Real and model armature short-circuit currents

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Fig. 11 Real and model field winding short-circuit current

6 Conclusion A complete procedure to determine a valid model for the salient-pole synchronous machine is presented in this work. Taking into account the presence of damper windings, an electrical equivalent circuit of the machine is developed in the dq-stator reference frame. Experimental identification tests are performed to measure the exact constants of the machine’s electrical equivalent circuit, as well as, the field winding reduction factor. Validation is done using MATLAB and Simulink software where a sudden three-phase short-circuit test is performed on the simulation model developed. The machine states before and after the sudden short-circuit are presented. The simulation current graphs show a very good agreement to those measured practically. The small difference is essentially due to modeling assumptions and measurement errors.

References 1. Zhang Y (2018) Advanced synchronous machine modeling. Diss, University of Kentucky 2. Barakat A, Tnani S, Champenois G, Mouni E (2010) Analysis of synchronous machine modeling for simulation and industrial applications. Simul Model Pract Theory 18(9):1382–1396 3. IEEE Guide (2003) Synchronous generator modeling practices and applications in power system stability analyses. In: IEEE Std 1110–2002 (Revision of IEEE Std 1110-1991), pp 1–80, 12 Nov 2003. https://doi.org/10.1109/IEEESTD.2003.94408 4. Ghanim DFD (2012) Experimental determination of equivalent circuit parameters for a laborator salient-pole synchronous generator. Diss. Memorial University of Newfoundland 5. IEC 60034-4-1 (2018) Rotating electrical machines - Part 4-1: methods for determining electrically excited synchronous machine quantities from tests 6. IEEE Guide (2010) Test Procedures for Synchronous Machines Part I—Acceptance and Performance Testing Part II—Test Procedures and Parameter Determination for Dynamic Analysis. In: IEEE Std 115-2009 (Revision of IEEE Std 115-1995), pp 1–219, 7 May 2010. https://doi. org/10.1109/IEEESTD.2010.5464495 7. Boldea I (2018) Synchronous generators. CRC Press

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8. Kundur P (1994) Power system stability and control. McGraw-hill, New York 9. Kirtley Jr, J (2013) 6.685 Electric Machines. Fall 2013. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA

Power Quality Improvement of PWM Rectifier-Inverter System Using Model Predictive Control for an AC Electric Drive Application Abdelkarim Ammar

Abstract The AC–DC–AC conversion system is extensively used in a variety of applications, such as power system, industry applications, and transportation systems. The back-to-back converter with adjustable DC-link provides a bi-directional power flow between the load and the grid. The grid-side-converter (GSC) can control independently active and reactive power. Moreover, it functions as an active power filter to compensate harmonics and provide a sinusoidal grid current. The Finite state model predictive control (FS-MPC) has been widely used in power electronics and electrical drives field due to its promising results. This technique incorporates the power converter model in the control design and does not need any modulation unit. In this work, the predictive model control will be applied to both back-to-back converter stages. Predictive power control and predictive torque control are applied respectively on the rectifier and the inverter which fed the motor. MPC evaluates the control variables in a cost-function to generate optimal switching states in a sampling period. This optimization reflects on power quality and harmonics elimination. The effectiveness of the presented algorithms is investigated by simulation using MATLAB/Simulink software. Keywords Back-to-back converter · PWM-rectifier · Induction motor · Direct power control · Model predictive control

1 Introduction The AC-DC-AC conversion system in electrical drives application uses a full-diode bridge for the grid side rectifying where the power transfer cannot be reversible. In addition, the diode-bridge causes high grid current distortion and thereby it reduces A. Ammar (B) Signals and Systems Laboratory (LSS), Institute of Electrical and Electronic Engineering, University of M’hamed BOUGARA of Boumerdes, Boumerdes, Algeria e-mail: [email protected] LGEB Laboratory, Electrical Engineering Department, Biskra University, Biskra, Algeria © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_29

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the power factor and the quality of the energy [1]. Recently, and due to breakthrough advancement in the power devices developments, the application of three-phase pulse width modulation (PWM) rectifier has been risen quickly [2]. The back-to-back converter is consisting of a PWM-rectifier (AC-DC) and an inverter a (DC-AC) with an intermediate DC-link. This converter is widely used in many applications such as electrical drives and renewable energy conversion systems. It offers plenty of advantages, such as: a bi-directional energy flow ability, a controllable DC-link voltage and reactive power and four-quadrant operation [3]. Several strategies have been proposed either for the rectifier power control or for the inverter. They can be classified into two groups, indirect methods with cascade current control like Field/Voltage Oriented Control (i.e. FOC and VOC) [4, 5] or direct methods like the Direct Power/ Torque Control (i.e. DTC and DPC) [6, 7], which directly controls the input active and reactive power. Despite that the Direct power/torque control methods show high performance over the indirect methods such as the fast dynamic and the eliminating of the current loops, they present some drawbacks. The high power/torque ripples and variable switching frequency were the two most notable disadvantages [8]. Therefore, another method has been appeared and attracted wide attention in the area of power electronics which is known by the Model Predictive Control (MPC) [9]. Nowadays, MPC has become very popular research topic. It can be classified into two main categories, named by the continuous model predictive control and finitestate model predictive control (FS-MPC) [10]. The continuous MPC like the classical generalized predictive control (GPC) can demonstrate good performance. However, it has high order of complexity and needs a modulator in control design [11]. Contrariwise, the finite-state MPC, known as finite-control-set in other reference, (FCSMPC), eliminates the use of modulation block, it incorporates the converter model in the control design and respects its discrete nature [9]. The converter switching states are considered in order to minimize a predefined cost function which consists of the errors between references and predicted measured control variables. Due to its high flexibility, MPC has been well applied in power electronics and drives field where MPC can suit the different power converters structures. Among them, there are predictive control strategies for active front-end-rectifier like predictive power control (PPC) [12] and predictive current control (PCC) [9]. Others for inverters that feed motors like the predictive torque control (PTC) [13]. In particular, PPC and PTC are featured by stationary reference frame unlike PCC. Moreover, they keep a similar structure of traditional DPC or DTC, where the switching table is replaced by the online optimization procedure [14]. In finite-state MPC, the control variables such as active and reactive power in case of PPC or torque and flux in case of PTC are predicted for the finite number of possible switching states of a power converter. The optimal selection of switching states which is obtained by actuating the cost function can reduce power and torque ripples. Then, the selected state is directly applied to the converter in the next sampling time [15]. This paper presents a model predictive control strategy applied to the back-toback fed induction motor. This topology has been proposed for the sake of power quality improvement and performance enhancement for the induction motor and the

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grid-tied converter. A comparative study has been done between the MPC controlled rectifier-inverter system and other traditional structures that don’t employ grid-side control or employ a traditional DPC strategy. The simulation results have obtained using been MATLAB/Simulink in order to examine the effectiveness of the presented algorithm.

2 System Modeling and Configuration The main circuit’s topology of two-level back-to-back PWM converter is shown in Fig. 1. it can by divided to a rectifier with RL-filter in the grid side, DC-link and an inverter which drive the induction motor in the motor side.

2.1 Active-Front-End Rectifier Model The three PWM rectifier is a full controlled bridge consisting of six power transistors (IGBTs). The grid delivers three-phase voltages vg using the L-filter inductances and resistances. Transforming the three-phase mathematical model into two-phase stationary αβ frame, the model of PWM rectifier can be expressed as: eg = Ri g + L g

di g + vr ec dt

(1)

where eg , ig and vrec , present, grid voltage and current vectors and rectifier voltage vector. with eg =

 2 ega + egb e j(2π/3) + egc e j(4π/3) 3

(2)

ig =

 2 ega + egb e j(2π/3) + egc e j(4π/3) 3

(3)

Fig. 1 Circuit’s topology of the PWM Rectifier-Inverter system driving and induction motor

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vr ec =

  2 Vdc Sa + Sb e j(2π/3) + Sc e j(4π/3) 3

(4)

V dc : is the DC-link voltage. The rectifier control bases on the logic values S i , where: S i = 1, Ti is ON and Ti is OFF. S i = 0, Ti is OFF and Ti is ON. with: i = a, b, c. The DC-link dynamics of the back-to-back converter can be modeled as follows Switching states:  1 d Vdc (t) 1 = Idc (t) = Ig (t) − Im (t) dt C C

(5)

where I g and I m represent the DC-link components of the grid and load side (machine) currents. There are eight possible positions from the combinations of switching states. Six are active vectors (V1, V2 … V6) and two are zero vectors (V0, V7).

2.2 Voltage Source Inverter Model In this work, a two-level voltage source inverter (VSI) fed the controlled IM. The voltage vector is generated by the following equation: vinv =

  2 Vdc Sa + Sb e j(2π/3) + Sc e j(4π/3) 3

(6)

2.3 Induction Machine Model The dynamic model of the induction machine can be employed in the stationary frame (i.e. stator-fixed reference frame) (α, β) in complex matrix from by:    1    d i¯s i¯ = A ¯s + σ L s v¯ s ψs dt ψ¯ s 1

(7)

where: A=

  − σ1τs +

1 σ τr

−Rs



+ jω

1 σ Ls



1 τr

− jω 0



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M2

τs = LRss ; τr = LRrr ; σ = 1 − L s Lsrr v¯ s is the stator voltage vector; ψ¯ s is the stator flux vector. i¯s is the current. Rs and Rr are the stator and rotor resistances. L s , L r and M sr are stator, rotor and mutual stator-rotor inductance respectively. ω is the electrical velocity The electromagnetic torque expressed in terms of stator current and flux is given by:

 Te = p · Im ψ¯ s · i¯s

(8)

p is the number of pole pairs.

3 Conventional Direct Power and Torque Contral The direct torque control uses a switching table to select an appropriate voltage vector. This selection is related directly to the variation of the stator flux and the torque and the position of the flux vector. The relationship between the stator voltage and the stator flux change can be established as: ψs = Vs Tz

(9)

From (9), it can be deduced that the flux can be varied by the application of stator voltage during a sampling period T z . Then, the torque of the induction motor can be expressed in terms of stator and rotor flux vectors, the rotor flux vector is supposed invariant. |Te | = p

Msr |ψs ||ψr | sin(δ) σ Ls Lr

(10)

where: δ load angle between the stator and rotor flux vectors. According to (10), the torque can be controlled by adjusting the load angle δ if the stator and rotor flux amplitudes are maintained constant. The hysteresis comparators are used for flux and torque control. The twelve sector switching divides the circular flux locus into 12 regions instead of 6. This table makes all the six states used per sector to solves the ambiguity in torque control. Therefore, it is necessary to define small and large torque variations which require to divide the hysteresis band of the torque into four parts instead of three. Then, all six voltage vectors can be used.

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Table 1 12-sectors switching table Flux Torque

S1

S12

Increase

V1, V2 and V6

V1, V2 and V6

Decrease

V3, V4 and V5

V3, V4 and V5

Increase

V2, V3 and V4

V1, V2 and V3

Decrease

V5, V6 and V1

V4, V5 and V6

Table 1 shows the voltage vectors of the first and the 12th sectors, the rest sectors can be deduced. The control scheme of basic direct torque control strategy is shown in Fig. 1. In similar way, the basic principle of direct power control for PWM rectifiers is similar to direct torque control (DTC) in motor drives. It selects the voltage vector using a switching table, according to the grid voltage position [8]. The used regulators are hysteresis comparators for instantaneous errors of active and reactive power [16] (Fig. 2). In DPC control design, six sectors switching table has been chosen (Table 2). k indicates to the sector number.

Fig. 2 Block diagram of basic direct torque control algorithm

Table 2 6-sectors switching table for DPC strategy

Sector Active power Reactive power

Increase

V k+3 , V k-1

Decrease

V k+1 , V k

Increase

V k+3 , V k+1

Decrease

V k-1 , V k

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4 Model Predictive Control Strategies 4.1 Grid Side Converter (GSC) Control To predict currents into the next step the Euler forward discretization equation is used: x(k + 1) − x(k) dx ≈ dt Tz

(11)

The predicted input current is given by:

¯i g (k + 1) = 1 − Rg Tz · i¯s (k) + Tz + i¯g (k) Lg Ls   + vg (k) − vr ec (k)

(12)

The predicted instantaneous active and reactive power can be expressed by equations in term of the input voltage and current vectors as: 

P(k + 1) = Re eg (k + 1).i g (k + 1)

(13)

 Q(k + 1) = Im eg (k + 1).i g (k + 1)

(14)

For a high sampling frequency with respect to the grid fundamental frequency, it can be assumed that eg (k + 1) ≈ eg (k). The control variable for the rectifier are the input active and reactive powers. In order to select the optimal voltage vector, the cost function that summarizes the desired behavior of the rectifier is given as follows:     g1 =  P ∗ − P(k + 1) + λ1  Q ∗ − Q(k + 1)

(15)

λ1 is the weighting factor between active power and reactive power, P* and Q* are the active and reactive power references. The reference value of the reactive power Q* is usually set to zero to obtaining unity power factor operation.

4.2 Machine Side Converter (MSC) Control FS-PTC algorithm includes three main steps, the estimation of the stator flux and the torque is the first step. The second step is the prediction of the next-instant of

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current i¯s (k + 1), flux ψ¯ s (k + 1) and torque Te (k + 1). The cost function design is done finally [17]. From the IM model, the stator current and the stator flux can be described as follows: 

d ψ¯ s = v¯ s − Rs i¯s dt

¯ ¯i s = − 1 L σ · d i s − kr · 1 − jω ψ¯ r − v¯ s Rσ dt Tr

(16)

(17)

where: k r =M sr /L r, Rσ = Rs + kr2 · Rr , L σ = σ.L s . After discretization with the sampling time T z , the stator flux and current prediction can be obtained as: 



ψ¯ s (k + 1) = ψ¯ s (k) + Tz v¯ s (k) − Rs · Tz i¯s (k)

¯i s (k + 1) = 1 − Tz · i¯s (k) + Tz · Tσ Tσ

1 1 ¯ kr · − jω(k) ψr (k) + v¯ s (k) Rσ Tr

(18)

(19)

with Tσ = σ L s /Rσ By the predictions of the stator flux and current, the electromagnetic torque can be predicted as following:

 Te (k + 1) = p · Im ψ¯ s (k + 1) · i¯s (k + 1)

(20)

The cost function in the MPC strategy compares the predicted and reference values [10]. The classical cost function for the PTC method is:     g2 = Te∗ − Te (k + 1) + λψ¯ s∗ − ψ¯ s (k + 1)

(21)

Te∗ is the reference torque and Tˆe (k+1) is the predicted torque for a given switching state, ψ¯ s∗ is the reference stator flux and ψ¯ s (k + 1) is the predicted stator flux. λ is the weighting factor which defines a trade-off between the torque and flux tracking [18]. Figure 3 illustarate the global control scheme of the model predictive power/torque control for the back-to-back converter that feed in induction motor.

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Fig. 3 Diagram of predictive power/torque control for an induction motor drive using back to back converter

5 Simulation Results This section presents the simulation results of the proposed model predictive control for a back-to-back converter that feed an induction motor. The following figures displayed the performance analysis of the grid-side-converter and the motor-sideconverter as a comparative study. The predictive power/torque control (PPC-PTC) is compared to another two topologies. A conventional topology (DTC) that consists of an uncontrolled rectifier (full diode bridge) in the grid-side and a traditional DTCcontrolled inverter in the motor-side. The second topology (DPC-DTC) consisted of DPC-controlled rectifier in the grid side and DTC controlled inverter in the motorside. The figures are specified by ‘a’ for DTC, ‘b’ for DPC-DTC and ‘c’ for PPC-PTC strategies. The grid-side-converter (GSC) control perfromance analysis is shwon in Fig. 4. The figure presents from top to buttom, the active power, reactive power, grid currents and the DC-link voltage. Generally, the active power depends to the load demand. In

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0 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.2

0.4

0.6

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1

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2

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0.6

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1

1.2

1.4

1.6

1.8

2

Q[VAR]

4000

Q[VAR]

P[W]

2000

Q[VAR]

P[W]

P[W]

4000

2000 0

ig[A]

ig[A]

ig[A]

5

0

Vdc [V]

Vdc [V]

Vdc [V]

-5 540

520

500

Times [s]

Times [s]

Times [s]

Fig. 4 Grid-side-converter performance: Active power “P”, Reactive power “Q”, Grid current “ig ”, DC-link Voltage “V dc ”

4

4

2

2

2

0

iga[A]

4

iga[A]

iga[A]

Fig. 4a, the case of the uncontrolled rectifier, it can be seen that active power shows high ripples, while the reactive power and DC-link voltage cannot be controlled. In Fig. 4b, the direct power control has been applied in GSC, in this case, the active power provides reduced ripples. In addition, the control of reactive power and DCvoltage has been achieved. the reactive power has been set to zero in order to have a unity power factor. The control of DC-link rises the system stability during the speed variation, moreover, this topology is featured by a bi-directionality which gives the ability to control the motor in four quadrants. The predictive power control (PPC) in Fig. 4c, provide a higher performance compared to the two previous topologies. The smooth powers waveform and the minimized voltage overshoot/undershoot is clearly observed. Next, Figs. 5 and 6 present the stator current and its FFT analysis, it can be seen that the traditional topology with diode rectifier in Figs. 5a and 6a shows highly distorted input current, where the total harmonics distortion (THD) is >31% and this can be

0

-2

-2

-4

-4 1.6

1.61

1.62

1.63

1.64

1.65

-4

1.6

1.66

0

-2

1.61

1.62

1.63

1.64

1.65

1.6

1.66

1.61

1.62

Times [s]

Times [s]

1.63

1.64

1.65

1.66

Times [s]

Fig. 5 ZOOM of grid current waveform iga

100

100

90

90

Mag (% of Fundamental)

Mag (% of Fundamental)

90 80 70 60 50 40 30 20 10

Mag (% of Fundamental)

100

80 70 60 50 40 30 20

0

50

100

150

200

250

300

Frequency (Hz)

350

400

450

500

70 60 50 40 30 20 10

10

0

80

0

0 0

50

100

150

200

250

300

350

400

450

500

0

50

Frequency (Hz)

Fig. 6 Spectrum of total harmonics distortion (THD) for the grid current

100

150

200

250

300

Frequency (Hz)

350

400

450

500

0 0

0.6

0.8

1

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5 0 0.2

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0 0

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10 5 0 0

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-10 0.2

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1 0.5

isa[A]

-10 0.2

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1

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0

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2

10 5 0

10 0

2

1

0.5

0

0

100

-10 0

2

ψs [Wb]

0

0

ψs [Wb]

0

437

2

10

10

isa[A]

isa[A]

0.4

10

0

ψs [Wb]

0.2

Te[rN.m] W[rpm]

100

Te[rN.m] W[rpm]

Te[rN.m] W[rpm]

Power Quality Improvement of PWM Rectifier-Inverter System …

0

0.2

0.4

0.6

0.8

1

Times [s]

1.2

1.4

1.6

1.8

2

1

0.5

0

Times [s]

Fig. 7 Motor-side-converter performance: Rotor speed “W ”, Torque “T e ”, stator current “isa ”, Flux Linkage “ψ s ”

easily observed for, current waveform. Furthermore, the THD in cases of DPC-DTC and PPC-PTC strategies is 6.28% and 0.82% respectively. These results show the importance of the PWM-rectifier over an uncontrolled rectifier in one hand and the merit of the MPC application in the other hand. the optimum voltage selection of PPC strategy reflect on the quality of signals specially the current. Then, Fig. 7 displayed the motor side converter control performance. The figure shows from top to bottom, rotor speed, torque, stator phase current and stator flux magnitude. The machine behavior of DTC and DPC-DTC in Fig. 7a, b is the same since both topologies use the traditional direct torque control in the motor side. The high toque and flux ripples can be clearly observed However, in Fig. 7c, PPC-PTC has a reduced torque and flux and smoother current waveform due to the optimized MPC.

6 Conclusion This paper presents a model predictive strategy for PWM-rectifier-inverter that feed induction motor drive. MPC has been employed for grid and motor-sides. The proposed algorithms have been compared to conventional techniques that use a lookup switching table. The simulation results have been obtained under different conditions in order to investigate the performances such as steady state, speed and load variation. The use of PWM-rectifier offers different advantages, in particular, current quality improvement and unity power factors. Moreover, the application of MPC can enhance the control performance and reduce power/torque ripples and current distortion. Generally, the finite-state model predictive control techniques are presented as an alternative solution in electrical drives and power electronics field. They preserve numerous good proprieties of conventional drives and offer more advantages. Nevertheless, they still a challenging subject for researches concerning performance enhancement.

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Appendix The different parameters for simulation in SI units are: Grid-side-converter: Line voltage

vg = 220 V

Line resistance

R = 0.4

Line inductance

Lg = 0.015 H

Frequency

f = 50 Hz

DC-link voltage

Vdc = 500 V

DC-link capacitor

C = 2040 μF

Motor-side-converter: Motor’s power

P = 1.1 kW

Motor’s rated current

I = 2.5 A

Rated stator flux

ψs = 0.95 Wb

Rated Torque

Te = 6 N·m

Rated speed

ωr = 1450 rpm

Stator resistance

Rs = 6.75

Rotor resistance

Rr = 6.21

Stator inductance

Ls = 0.5192 H

Rotor inductance

Lr = 0.5192 H

Mutual inductance

Msr = 0.4957 H

Number of pole pairs

p=2

Friction coefficient

fr = 0.002 N·m·s

The moment of inertia

J = 0.01240 kg·m2

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Novel Smart Air Quality Monitoring System Based on UAV Quadrotor Mehdi Zareb, Benaoumer Bakhti, Yasser Bouzid, Hamza Kadourbenkada, Kamel Bouzgou, and Wahid Nouibat

Abstract In tomorrow’s society called Smart community, the air quality and pollutants will take the attention of researchers and leaders to develop intelligent systems to better protect tomorrow’s citizens from the risks of Smart Risk that can take place in Smart City. Also, Unmanned Aerial Vehicles (UAVs) due to their flexibility, mobility in 3d and ability to carry air low-cost sensing units have become a cheap alternative to monitor pollution values in a given area. However, enough still needs to be learned as these types of studies are expected to increase in the future years. In this paper, we propose an electronic system based on low-cost sensors designed for air quality monitoring. This system is embedded in a quadrotor type UAV in which, the mobility of the quadrotor is controlled with an autopilot. Experiment tests are made on the campus of Mascara University, where the obtained results have proved the effectiveness of the proposed UAV-based air quality monitoring system. Keywords UAV · Quadrotor · Air quality monitoring · AQM · AQI · Path planning · Smart risk · Smart city · IoT M. Zareb (B) Electronic Department, USTO-MB, University of Mascara, Mascara, Algeria e-mail: [email protected] B. Bakhti Physics Department, University of Osnabrueck, University of Mascara, Osnabrueck, Germany e-mail: [email protected] Y. Bouzid CSCS Laboratory, Ecole Militaire Polytechnique, Bordi El Bahri, Algeria e-mail: [email protected] H. Kadourbenkada Electronic Department, University of Mascara, Mascara, Algeria e-mail: [email protected] K. Bouzgou · W. Nouibat Electronic Department, LEPESA Laboratory, USTO-MB University, Oran, Algeria e-mail: [email protected] W. Nouibat e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_30

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1 Introduction For many families in the word, bad air quality is causing more and more health issues and dangerous conditions for all kinds of communities. This is due to the pollutants in the air. The pollutant arising from artificial sources contaminating are the most relevant with regard to human/animal health. It can be grouped into first, stationary sources including as example power stations, where coal, natural gas, or petroleum are burned. Second, mobile sources including gasoline and diesel-fueled vehicles. However, air pollutants bear similarities based on which are categorized into gaseous pollutants cased directly by fossil fuel combustion as Carbon Monoxide CO, Nitrogen Oxide NO, ozone O3 , nitrogen dioxide NO2 , Volatile Organic compounds VOCs, Sulfur Dioxide SO2 , etc. Heavy metals include of mercury Hg, lead Pb, cadmium Cd and silver Ag, etc. Can be transported by air and their most common sources are an industrial and mining waste, vehicle emissions, fertilizers, and acid batteries, etc. Other important categories of pollutants are the Particulate matter PM. Waste incineration facilities, dust, fires, vehicles are a major source of PM. The particle size ranges between 2.5 μm (PM2.5) and 10 μm (PM10). The literature has well-established the link between long-term exposure to air pollution and rate of morbidity and mortality in different country as mentioned in [1–3, 4]. As an example, in [5] they estimate 1.1 million annual premature deaths from PM2.5 in India between 2012 and 2018. The relationship between the pollution levels around the days and death for 22 million adults aged 65 and older based on death records from 2000 to 2012 in the USA is investigated in [6, 7]. In fact, short-term exposure to Air Pollutants can cause pulmonary and cardiovascular diseases neurodegeneration, neurological disorders, etc. as studied in [8–10]. Also, the different parts of the respiratory system are affected by PM depends upon the size of the particle as presented in [11]. Also, it may affect to people with allergic diseases [12]. The first idea in this topic was to use vision UAV base guide solution by analyzing imagery interpretation in the investigation of environment pollution by Zang et al. [13] and Yang et al. [14]. Where, they demonstrate the effectiveness of applying UAVs than the use of satellite or manned aerial remote sensing, which is too expensive and take time. Then, Gas Sensing System (GSS) based on the low-cost sensor is developed, as an example, in [15] a GSS system was developed to investigate the Gas leakage. Another example of ecological monitoring with the prime price is proposed by Danilov et al. [16]. Where a GSS was developed and mounted on the Small DJI hexacopter UAV to build 3D models of the pollution of the atmosphere. This approach is carried out in two steps, first, one the trajectory of flight is presented on the spiral of Archimedes, then the trajectory of the flight on the second stage is proposed on a helicoidal trajectory. The application of the UAV-based GSS system for Smart Cities was investigated: first for multiple air pollutants by Gu and Jia [17] and by Alvear et al. [18, 19]. Where the experiments tests have verified the feasibility and precision of the system. Where they found that the on-board sensor devices did not affect the UAV’s power consumption and flight time. However, the UAV during flight operations produced an important

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noise into the sensors and the data acquisition modules, which resulted in the variation of sensor readings as the case of NO2 sensor for example. To overcome this problem, they propose as a solution, the use of a separate power source for the UAV and the onboard devices. Secondly, UAV-based GSS system for a specific pollutant as in [20] where they trend relating PM concentration with elevation and traffic conditions near the specific urban area. Also, in where Volatile organic compounds VOCs in the air was analyzed. The design methodology problems for the UAV-based AQM system can be summarized: First, the location where the sensors should be mounted. Second, the influence of sensor orientation on the captured values. Third, the influences of incomplete measurement. Also, noise caused by the UAVs propellers during the flight. In this paper, we propose a UAV-based air quality monitoring system AQM. Where design methodology that we follow consider the AQM problems already cited. First, based on results of the work [21], for example, the AQM system is positioned between the central tile positions of quadrotor UAV (we make the choice on this type of UAV for this study). A data acquisition module based on SD card is added to AQM electronic system to overcome the influences of incomplete measurement. Also, a separate power source for the UAV and the onboard AQM devices are used to reduce the noise caused by the UAVs propellers during the flight. Finally, the trajectories of flight are presented on the Boustrophedon, and helicoidal. The rest of the paper is organized as follow: Section two presentation of the developed AQM UAV-based system, section three results and discussion. Finally, conclusion.

2 Presentation of the Developed AQM UAV-based System In general, drones are equipped with low-cost micro sensors that have the advantage of capturing the spatial and temporal variability of air pollutants. In addition, drones provide spatial resolution and vertical profiling near the surface of air pollution due to their mobility in both horizontal and vertical dimensions. In this report, we design an AQM electronic system assembled on a quadrotor UAV rotor as shown in Fig. 1. The objective of the system is to inspect and monitor air quality in 3D. This system has been tested in the University of Mascara, whose results have proven the effectiveness of using the drone as an AQM system. The proposed system is a two-tier system: • The electronic air quality measurement system responsible for data acquisition and analysis • Unmanned aerial vehicles (UAVs) responsible for flying the system in a threedimensional space

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Fig. 1 Photo of the realized UAV-based AQM system taken during experiments in Mascara university

2.1 The Electronic Air Quality Measurement System The AQM System is composed of different elements ranging from the microcontroller to a variety of sensors and an external power supply, etc. Where the sensor elements are tested to test their reliability. The Arduino Mega 2560, a microcontroller board based on Atmega2560, is chosen for program processing because it contains enough digital and analog I/O pins. The MQ-7 sensor is chosen to detect the concentration of carbon monoxide CO. Just as the MQ-7 MQ131 is a semi-conductor gas sensor, the sensitive material of this gas sensor is O3 . To detect temperature and humidity the DHT22 is used as a low-cost basic digital temperature and humidity sensor. It uses a capacitive humidity sensor and a thermistor to measure ambient air. For the particle mater, Shenyei PPD42NS is chosen, it’s a low-cost PM sensor that operates according to the light scattering method, it continuously detects airborne particles. It is possible to obtain a pulse output corresponding to the concentration per unit volume of particles, using an original detection method based on a light scattering principle similar to that of the particle counter, this sensor is very sensitive to very fine particles. The rest of the hardware used includes Arduino SD card shield and a real-time clock for data

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storage. An LCD module for data display. An external power supply. By also adding other elements such as assembly box, wires, connection card, etc. Figure 2 presents the electrical schematic of the AQM measurement system that shows the connection of each sensor to the Arduino Mega. Thus, it represents the photo of the AQM electronic system produced. In Fig. 3 the flowchart of the program that manages the operation of the AQM system is shown, where the sampling time chosen to take the concentration of O3, CO and PM equal to 60 s, whose measured data are displayed and recorded at the same time in an SD card. For the software part of the project, we have programmed a Final Sketch (in C Arduino program) that groups all the other programs of each sensor with modification and alteration taking into account the different properties of each element (response time, calibration, etc.). However, for implementation, each sensor is tested and verified on its own. Starting with the MQ-7 sensor where the sensor’s test process was performed by lighting a burning paper and exposing the sensor to smoke, then MQ-131 and Shenyei PPD42NS sensors. Figure 4a shows the test result obtained from the MQ-7 sensor for the concentration of CO. The concentration of O3 obtained from the MQ-131 sensor is presented in Fig. 4b. In addition, the PM2.5 and PM10 concentrations are presented in Fig. 4c, d. The calibration of the gas sensors, MQ-7, MQ-131 is an important procedure, where we have followed the method represented by Kularatna et al. [22].

Fig. 2 Electrical schematic of the AQM measurement system

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Fig. 3 The flowchart of the AQM program

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2.2 UAV System An overview of the UAV system is presented. The Quadrotor or Quadcopter which is defined as a UAV with four rotors [23] (Fig. 1). Where, according to [24] this type of UAV has the largest use, for this reason, we have made the choice to use it. The UAV is equipped with an autopilot that contains the planned trajectory (as proposed

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by [25]). Figure 9 presents the schematic block of the Quadrotor UAV autopilot. Where the first bock contains the autonomous trajectory generation (path planer) for UAV-AQM. We will focus on the Bous-trophedon trajectory (as Figs. 5 and 6 present) that follows the Lawnmower search and the Archimedes Spiral trajectory that follows the Spiral Search as Fig. 7. Second, in the schematic block of the UAV (Fig. 9) a PID controllers are used to guide and maintain the UAV Quadrotor to a specific location (x, y and z) with a three desired orientation φ, θ and ψ that represent respectively the roll, pitch and yaw angles of the UAV. The desired states (xd yd z d and ψ) are calculated by the path planer block to follow the proposed trajectory. The Fig. 5 Boustrophedon trajectory with a circular foot print

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3 Results and Discussion According to their official website, the University of Mas-cara has 16875 undergraduate and graduate students, 293 Ph.D. students and about 1746 staff members (professors and agents). It is a large number of people who enter and exit through the main door daily, we saw this as an opportunity to test our system to determine the air quality to which all these people are exposed daily. That is why we have organized a series of tests at different locations on campus to measure pollutant concentration levels, which helps us determine the overall air quality in the region and we have conducted these tests at times when we suspected the presence of air pollutants. All these tests will be used to validate the system we propose.

3.1 First Test For the first test, we chose a trajectory that most students follow when they enter university, as shown in Fig. 10, point A is the starting point and point B is the ending point, where the UAV flights at an altitude equal to 15 meters. The measurements obtained from the first test are presented from Fig. 8a, d. First, the AQM electronic system has successfully detected all required pollutant levels. We notice the presence of a peak of ozone (a secondary pollutant), high temperature and humidity contribute to its formation, carbon monoxide CO was low because this pollutant intervenes in the event of a gas leak or a form of combustion now. Second, the AQI (Air Quality Index) of the area inspected for the first test can be calculated by using pollutant concentration data with the Eq. 3: Ip =

(P Mobs − P Mmin ) ∗ (Imax − Imin ) + Imin (P Mmax − P Mmin )

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where, PM obs is the observed average concentration μg/m3 , PM min is the maximum concentration of AQI color category that contains PM obs , PM min is the minimum concentration of AQI color category that contains PM obs , I max is the maximum AQI value for color category that corresponds to PM obs and Imin is the minimum AQI value for color category that corresponds to PM obs (These values can be taken from the Table 1). The AQI value tells us how clean or unhealthy your air is. Table 1 presents the AQI six levels of health concern. Let’s calculate the AQI around the

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Fig. 10 Mapping of UAV air quality monitoring at the main entrance of the University of Mascara

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This AQI value of the fine particles (PM) means that the air quality conditions are unhealthy of sensitive groups around the main entrance of the campus we thought it

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was the high presence of motor vehicles and their emission rate that caused these bad levels of p.m. At the end of the trajectory, most levels have decreased, which means that both indoor places and classes have better air quality, so we advise students not to stay too long at the main entrance. Figure 10 presents the mapping of the air quality index for the first test, where the orange circle means that the air quality is unhealthy for sensitive groups, close to bad (the main door of the University of Mascara). The green circle means that the air quality is good (zone B) in Fig. 10.

3.2 Second Test For the second test, we followed a similar trajectory, but this one is horizontal to the other. As shown in Fig. 11, point A is the starting point and point B is the endpoint. Figure 12 presents the PM concentration during the second test UAV trajectory, where it reaches 65 μg/m3 with an AQI value equals to 85 at the starting point (A) (Fig. 11) which corresponds to moderate AQI category. At the end of the trajectory (point b), the PM concentration reaches 42 g = m3 with an AQI value equals to 39 which corresponds to good air quality. Also, in Fig. 11 yellow and green circles present respectively the moderate and good air quality zone. Finally, Fig. 13 presents the trajectory made by the UAV quadrotor during the AQM tests at the points A and

Fig. 11 Mapping of UAV air quality monitoring of the second test

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4 Conclusion The UAV equipped with low-cost micro-sensors that have the advantage of capturing the spatial and temporal variability of air pollutants has enabled spatial resolution and vertical profiling near the surface of air pollution due to their mobility in horizontal and vertical dimensions. In this paper, the pro-posed AQM electronic system assembled on a quadrotor UAV have properly inspected and monitored the 3D air quality at Mascara University. Unlike the AQM systems already existing in the literature, it measures several pollutants such as PM 10 and 2.5, bad ozone O3 , carbon monoxide at the same time. According to the tests made, the air quality is moderate, close to bad at the main entrance to the campus of Mascara University indicated by

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an orange zone. While the green zone (good air quality) is located in the center of the university. So, the results obtained proved the effectiveness of using the UAV as an AQM system. In future work, we propose to carry out wireless communication with a base station to do air monitoring online. Also, Adaptive and automatic monitoring techniques that ensure better battery consumption of the UAV, with sufficient accuracy of air quality index will be investigated.

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GA Based Adaptive Fuzzy Sliding Mode for 3 DOF Spatial Arm Manipulator Trajectory Tracking Rabie Belloumi and Noureddine Slimane

Abstract In this paper a trajectory tracking control law for an arm manipulator of three degrees of freedom is developed using an adaptive fuzzy sliding mode control (AFSMC) with a proportional integral derivative (PID) sliding surface. The introduction of this controller in the control loop permits that the joints track the desired trajectory despite external disturbances. The developed control laws enable us to note that the trajectories tracking performances of the manipulator are directly related to the choice of the used adjustable parameter values. For that a genetic algorithm optimization is used for the determination of the PID surface parameters and the normalization gains of the fuzzy controllers. This approach allows us not only to force the behavior system to a desired state very fast, but makes also the torques functions smooth without chattering. The obtained results show an improvement of the trajectory tracking with this approach and implied that this strategy is feasible, effective and robust with regard to disturbances. Keywords Genetic algorithm optimization · PID sliding surface · Fuzzy sliding mode controller · 3 DOF manipulator · Adaptive fuzzy control

1 Introduction The sliding mode control has shown its application simplicity and efficiency through the reported theoretical studies; the advantage that gets such a command and which makes it so important, is the robustness towards the model uncertainties and disturbances.

R. Belloumi Faculty of Technology, Department of Electronics, Batna 2 University, Batna, Algeria e-mail: [email protected] N. Slimane (B) Advanced Electronics Laboratory, Faculty of Technology, Electronics Department, Batna 2 University, Batna, Algeria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_31

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Several control approaches are proposed in the literature concerning the manipulators with two degrees of freedom but limited work is carried out in the case of manipulators to three degrees of freedom. In [1] a fuzzy adaptive control is applied for a 2 DOF manipulator. Ngo et al. [2] propose a robust adaptive neural-fuzzy network tracking control for robot manipulator. In [3] a PID fuzzy controller, considering uncertainties of the model parameters, is proposed for the trajectory tracking. In [4] a control by fuzzy sliding mode is proposed for MIMO systems by taking account of the disturbances. Choi and Kim [5] use a fuzzy sliding mode controller for robust tracking of robotic manipulators. Chen et al. [6] propose a fuzzy sliding mode control of robot manipulators using the reaching law. In [7] fuzzy sliding mode controller and particle swarm optimization are used for tracking the position of 2 DOF arm manipulator, Shao and Ma [8] introduce a new method of backstepping sliding mode control for tracking control of multi-joint robot manipulators. In the present paper, genetic algorithm is proposed to optimize the gain parameters of sliding surface and fuzzy controllers. Because of the need for treating several problems varieties, the optimization algorithms received an increased attention of the engineers treating these types of problems, which could not be solved with conventional techniques. The latter is inspired by the natural evolution of the human beings and the mechanisms of evolution: crossing, mutation, selections ….etc. The genetic algorithms try to optimize an objective function (fitness) from the evaluation of a population. The best individuals are chosen to constitute the appropriate parents giving birth to better descents children, among who will be fired the acceptable solutions of the handled problem [9–11]. In this work a genetic algorithm (GA) is proposed to optimize off line the parameters of the proposed PID sliding surface and the parameters of the fuzzy controllers introduced in the control scheme. Automatically obtaining of these various parameters enables us to apply effectively on line this obtained adaptive fuzzy sliding mode control. Simulation works are undertaken for trajectory tracking of the articular variables of an arm manipulator with three degrees of freedom, under these two optimization algorithms by taking account of the external disturbances. The obtained results are used to check the tracking performances in both cases and a comparison is generated. This paper is structured as follows. In Sect. 2, the dynamic equation of the manipulator is presented. In Sect. 3, an adaptive fuzzy sliding mode controller with a PID surface and a supervisory fuzzy control is proposed using the Lyapunov approach to guarantee the system stability. To illustrate the performance of the proposed approach, simulation works are undertaken in Sect. 4. The conclusions are given in Sect. 5.

2 Design of Manipulator Model To define manipulator kinematics and dynamics and for the development of control laws to be able to control it suitably, it is necessary to describe the various mathematical relations and to give a precise mathematical model, which makes it possible to

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define the movements of this last in space. In general the manipulator model depends on the considered application: geometrical, kinematic or dynamic. The manipulator dynamic is defined by (1). M(θ )θ¨ + C(θ, θ˙ )θ˙ + G(θ ) = U + P

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Posing x 1 = θ , x2 = x˙1 et y = x 1 

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  where: f a = −M(θ )−1 C(θ, θ˙ )θ˙ + G(θ ) , ga = M(θ )−1 et b = M(θ )−1 P With |b| < α, α represents the maximum value of the disturbance (Fig. 1). Fig. 1 Manipulator structure

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3 Adaptive Sliding Mode Control The control by sliding mode rests on the principle of bringing in the first time the dynamics of the system to reach a preset surface, called sliding surface, once arrived at this surface it slips towards the equilibrium point. The first phase is called convergence mode and the second it is the slip mode. The equation of surface is of the form:  s(t) = k p e(t) + ki

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s˙ (t) = k p e˙ + ki e + kd (θ¨d − θ¨ )

(7)

s˙ (t) = k p e˙ + ki e + kd (−ga U − f a − b + θ¨d )

(8)

The resolution of the equation s˙ (t) = 0 without disturbance enables us to have the following equivalent control:   u eq = (kd ga )−1 kd (θ¨d − f a ) + (k p e˙ + ki e)

(9)

This equivalent control cannot ensure the performances wanted in the presence of disturbances and/or of parametric uncertainties, then it is necessary to add a control signal, which will eliminate the disturbances effect, by ensuring the stability and the robustness of the system. The choice is: u = u eq + u c = u eq + k f u f

(10)

where k f is the standardization factor of the output variable of dimensions 3 × 3 and uf is the output vector of the FSMC of dimensions 3 × 1 ˙ k f = f (e, e)

(11)

u f = [F S MC(s1 , s˙1 ), F S MC(s2 , s˙2 ), F S MC(s3 , s˙3 )]

(12)

The design of the proposed controller requires two inputs and one output, which are respectively s, s˙ and the control signal uf , with the choice to use the same linguistic variables for the three fuzzy units. [(s, μ(s)), (˙s, μ(˙s)), (uf , μ(uf ))] (Fig. 2). The used linguistic variables are: NB: negative big, NM: negative middle, NS: negative small, Z: zero, PS: positive small, PM: positive middle, PB: positive big.

GA Based Adaptive Fuzzy Sliding Mode for 3 DOF Spatial Arm …

459

Fig. 2 Diagram of adaptive fuzzy sliding mode (AFSMC) with surface PID

Fig. 3 Linguistic variables of s and uf

The range of input s and output uf of the controller is standardized and respectively presented on Fig. 3. The rules are arranged in the following table, for each entry of the controller there are seven output fuzzy wholes, which gives 49 fuzzy rules (Table 1). The used inference mechanism is of Mamdani type and the minimum function of intersection is employed for the fuzzy implication. For the defuzzification, the gravity center method is taken to calculate the values of the output vector uf . The following control surface is obtained (Fig. 4). The design of the supervisory controller requires two inputs and one output, which are e, de/dt and the gain signal k f . The used linguistic variables for inputs are: NB: negative big, NM: negative middle, NS: negative small, Z: zero, PS: positive small, PM: positive middle, PB: positive big. For the output: VVS: very very small, VS: very small, S: small, M: medium, B:big, VB: very big, VVB: very very big (Fig. 5). The rules are arranged in the following table (Table 2). The obtained surface is given below (Fig. 6).

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Table 1 Fuzzy rules uf ds/dt

S PB

PM

PS

Z

NS

NM

NB

PB

NB

NB

NB

Z

Z

Z

Z

PM

NB

NB

NB

Z

Z

Z

PS

PS

NB

NB

NM

Z

Z

PS

PM

Z

NB

NM

NS

Z

PS

PM

PB

NS

NM

NS

Z

Z

PM

PB

PB

NM

NS

Z

Z

Z

PB

PB

PB

NB

Z

Z

Z

Z

PB

PB

PB

Fig. 4 Control surface of uf

Fig. 5 Linguistic variables of e and k f

GA Based Adaptive Fuzzy Sliding Mode for 3 DOF Spatial Arm …

461

Table 2 Fuzzy rules supervisory kf

S

ds/dt

NB

NM

NS

Z

PS

PM

PB

NB

M

S

VS

VVS

VS

S

M

NM

B

M

S

VS

S

M

B

NS

VB

B

M

S

M

B

VB

Z

VVB

VB

B

M

B

VB

VVB

PS

VB

B

M

S

M

B

VB

PM

V

M

S

VS

S

M

B

PB

M

S

VS

VVS

VS

S

M

Fig. 6 Fuzzy gain surface

The sliding surface written in Eq. (5) is selected with a Lyapunov function V = ½ s2 . with the derivative:   V˙ = s s˙ = s k p e˙ + ki e + kd e¨

(13)

  V˙ = s k p e˙ + ki e + kd (θ¨d − ga u − f a − b)

(14)

Substituent u and ueq by their values in (14) we obtain:   V˙ = s −kd ga k f F S MC(s, s˙ ) − kd b(t)

(15)

With |b| ≤ α, α represents the terminals ends of uncertainties and disturbances.   V˙ ≤ −skd ga k f + α To ensure the convergence of the state towards the sliding surface: V˙ ≤ 0

(16)

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  −skd ga k f + α ≤ 0

(17)

1 It is necessary that k f ≥ -g− α α, thus the closed system is asymptotically stable, and the error will converge towards zero.

4 Simulation Results Throughout this work one will simulate the worked out algorithms and the proposed controllers on a model of arm manipulator to 3 DOF by using the matlab software. The parameters of the used manipulator are in the Table 3 with the following considerations: the distance between the gravity center of the second and third link is r = 0.25; and stands for viscous friction of joints is c = 0.15. The wished trajectory for the trajectory tracking of the three articulations is:  θd (t) =

1 + sin(π(t + 3/4)) i f t ≥ 0 0 if t < 0

The disturbance is taken as follows: P(t) = 5 + 5sin(20(t − 1)) + 6sin(10(t − 0.5)) + 5u(t − 1) + 5u(t − 0.5) The simulation results obtained by the use of these two optimization algorithms, concerning the trajectories tracking of the three articulations, are illustrated in Fig. 7. The trajectories tracking errors for the three joints are represented in Fig. 8. Figure 9 represents the various torques delivered by the actuators of the three articulations. The simulation results are depicted in Figs. 7, 8, 9. Figure 7 shows the robot manipulator angular positions Θ 1 , Θ 2 , Θ 3 and their desired references Θ 1d , Θ 2d , Θ 3d , respectively. Figure 8 represents the evolution of the tracking errors. Figure 9 depicts the generated torques of the three joints. The proposed control law shows its improvement in tracking accuracy with generated minimal torque, without chattering. Table 3 Manipulator Characteristics

Joint 1

Joint2

Joint 3

Mass

3 kg

2 kg

1 kg

length

0.5 m

0.5 m

0.3 m

inertia

0.052 kg.m2

0.052 kg.m2

0.052 kg.m2

GA Based Adaptive Fuzzy Sliding Mode for 3 DOF Spatial Arm … Fig. 7 Trajectory tracking for the three joints

Fig. 8 Trajectory tracking error for the three joints

Fig. 9 The generated torques of the three joints

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5 Conclusion The genetic algorithm is used to obtain the parameters of the sliding surface and the normalization gains of the fuzzy controllers, which are injected in real-time into the fuzzy sliding mode controller suggested for the manipulator. This control enabled us to reduce the chattering phenomenon existing in the traditional sliding mode control and to minimize the torque values generated by the actuators. The presence of strong disturbance forces the system to react negatively, but the suggested control with the fuzzy supervisory makes it possible to preserve the trajectory tracking of the various articulations with no increase in the torques generated by the actuators. The normalization gains are not fixed any more, they vary and adapt automatically to the fluctuations of the disturbances.

References 1. Guo Y, Woo PY (2003) Adaptive fuzzy sliding mode control for robotic manipulators. In: Proceedings of the 42nd IEEE conference on decision and control. Hawaii 2. Ngo T, Wang Y, Mai TL, Nguyen MH, Robust JC (2012) Adaptive neural-fuzzy network tracking control for robot manipulator. Int J Comput Commun Control 7: 341–352 3. Li W, Chang XG, Wahl FM, Farrell J (2001) Tracking control of a manipulator under uncertainty by fuzzy PID controller. Fuzzy Sets Syst 122: 125–137 4. Aloui S, El Hajjaji A, Chaari A, Koubaa Y (2011) Improved fuzzy sliding mode control for a class of MIMO nonlinear uncertain and perturbed systems. Appl Soft Comput 11:820–826 5. Choi S-B, Kim J-S (1997) A fuzzy sliding mode controller for robust tracking of robotic manipulator. Mechatronics 7:199–216 6. Chen Z, Meng W, Wang H (2008) Fuzzy reaching law sliding mode control of robot manipulators. In: Proceedings of 2008 Pacific-Asia workshop on computational intelligence and industrial application 7. Soltanpour MR, Khooban MH (2013) A particle swarm optimization approach for fuzzy sliding mode control for tracking the robot manipulator. Nonlinear Dyn 74(1–2): 467–478 8. Shao K, Ma Q (2014) Global fuzzy sliding mode controlfor multi-joint robot manipulators based on backstepping. Found Intell Syst 277: 995–1004 9. Schmitt LM (2004) Theory of genetic algorithms II: models for genetic operators over the string-tensor representation of populations and convergence to global optima for arbitrary fitness function under scaling. Theoretical Comput Sci 310(1–3):181–231 10. McCall J (2005) Genetic algorithms for modelling and optimisation. J Comput Appl Math 184(1):205–222 11. Konak A, Coit DW, Smith AE (2006) Multi-objective optimization using genetic algorithms. Reliab Eng Syst Saf 91(9):992–1007

Robust Adaptive Fuzzy Approach with Unknown Control Gain Direction and External Disturbance Ouassila Bourebia and Nassira Zerari

Abstract This paper studies a fuzzy indirect adaptive approach to control uncertain nonlinear system with unknown control gain sign (direction), in presence of external disturbance. In the design procedure the unknown nonlinear functions are approximated by fuzzy logic system in view of their universal approximation properties. We proposed using projection algorithm to avoid singularity problem and introducing the Nussbaum Function to solve control direction problem. Then a robust adaptive control is employed to build the overall controller architecture and deal with approximation errors and disturbance. Furthermore, stability of the closed-loop system is guaranteed using Lyapunov theory. Finally, simulated studies have demonstrated the effectiveness of the proposed approach. Keywords Fuzzy function · Adaptive control · NL system · Nussbaum function

1 Introduction Fuzzy control has been successfully applied to many commercial products and industrial systems, where controllers using fuzzy logic are generally considered applicable to models that are mathematically complex, non-linear, where human experiences are available to provide qualitative knowledge [1]. Therefore, fuzzy controllers are supposed to work in situations where there is a great deal of uncertainty or unknown variation in the parameters and structures of the system. Generally, the basic purpose of adaptive control is to maintain the logical performance of a system in the presence of these uncertainties [2, 3]. Fuzzy adaptive control provides a tool for using fuzzy O. Bourebia (B) Automatic and Robotics Laboratory, Faculty of Sciences and Technology, Department of Electronics, Constantine1 University, Constantine, Algeria e-mail: [email protected] N. Zerari Signal Processing Laboratory SP-Lab, Faculty of Sciences and Technology, Department of Electronics, Constantine1 University, Constantine, Algeria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_32

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information in a systematic and effective way. Several adaptive fuzzy control schemes have been developed for uncertain nonlinear systems in [4]. In [5] the stability of the underlying control systems has been investigated using a Lyapunov approach. A common assumption in the schemes proposed in [4, 5] is that the sign of the control gain is known apriori. However, in the general case, this assumption is by no means realistic as pointed out in [6]. When there is no a priori knowledge about the control gain sign, this problem has been solved by incorporating the Nussbaum-type function [7, 8]. Inspired by [8, 9], we propose in this paper a robust adaptive fuzzy controller for uncertain nonlinear systems to achieve the control objective. In the proposed controller, a robustifying control term is added to the basic fuzzy controller to deal with approximation errors. The Nussbaum gain function is introduced to solve the problem of unknown control gain sign and the projection algorithm is applied to cope with the problem of singularity. The stability of the closed-loop system is studied using Lyapunov method. In Sect. 2, we present basic ideas, in a constructive manner, of how to construct indirect adaptive fuzzy controllers based on the fuzzy systems, and how to use the projection algorithm to meet the control objectives. In Sects. 3 adaptive fuzzy controllers are used to control the inverted pendulum to track a trajectory. Section 4 concludes the paper.

2 Construction of Fuzzy Adaptive Approach (A) Problem formulation: Consider a class of nonlinear systems modeled by the following differential equation: x (n) = f (x) + g(x)u + d(t) y = x1

(1)

T  where x n = [x1 , x2 , . . . , xn ]T = x x˙ · · · x (n−1) ∈ Rn is the state vector of the system which is assumed to be available for measurement, uR, and yR are the input and output of the system, respectively. d(t) is the external disturbance which is assumed bounded. f (x) and g(x) are uncertain continuous functions. Assumption 1 ([10]) The sign of the control gain g(x) is also unknown. And there ¯ exist two unknown positive constants g¯ and g such us 0 < g ≤ |g(x)| ≤ g. We can rewrite (1) in the state space as follows: x˙ = Ax + b( f (x) + g(x)u) + d

(2)

Robust Adaptive Fuzzy Approach with Unknown Control …

467

⎡ ⎤ ⎤ 0 0 ⎢0⎥ 0⎥ ⎢ ⎥ ⎥ .. ⎥ b = ⎢ .. ⎥ ⎢.⎥ ⎥ .⎥ ⎢ ⎥ ⎣0⎦ ⎦ 0 ··· 0 1 1 0 0 ··· 0 0 The control objective is to design an adaptive fuzzy controller u(t) for system (1) such that the system output y(t) follows a desired trajectory xd (t), while all the signals of the closed-loop system are bounded. ⎡

0 ⎢0 ⎢ ⎢ With A = ⎢ ... ⎢ ⎣0

1 0 .. .

0 1 .. .

··· 0 .. .

Throughout this paper we make the following assumption: = x d (t) Assumption 2 The desired trajectory xd  T (n−1) xd (t), x˙d (t), · · · , xd (t) and its derivative are known and bounded. Let define the tracking errors as  T e = xd − x, x˙ d − x˙ · · · x d(n−1) − x (n−1) T  = e e˙ · · · e(n−1)

=

(3)

Then, from (2) we get 

e˙ = Ae + b xd(n) − x (n)

 = Ae + b − f (x) − g(x)u + xd(n) − d

(4)

While the feedback control is given by: u∗ =

 1

− f (x) + xd(n) (t) − k T e g(x)

(5)

  Where k T = k0 , k1 , . . . , k(n−1) ∈ R n is the real number vector. The coefficients k j , j = 0, . . . , n are selected so that the polynomial: S (n) +k(n−1) S (n−1) +· · ·+k0 = 0 are in the open left half plane (Hurwitz), [5] which implies that the tracking error will converge to zero, i.e.limt→∞ |e(t)| = 0. The control law (5) of system (1) can be easily implemented if f (x) and g(x) are well known, the functions are generally uncertain, the goal is then to approach them by fuzzy logic system. (B) Development of the control law: We propose to use fuzzy systems in this approach 1. Fuzzy logic system [6] Lets a fuzzy system of q inputs v1……vq. The defuzzification by center of gravity gives the output:

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m q y=

l=1 m l=1

 

j=1 μ Alj v j wl  

q j=1 μ Alj v j

(6)

where: m is the number of fuzzy ruler, Alj is the jth fuzzy set that corresponds to the fuzzy ruler, μ Alj is the membership function of the fuzzy set Alj , and wl are the centers of the lth fuzzy set of the fuzzy system. The output is rewritten by: y=

m  l=1

   μ Alj v j wl   wl m q l=1 j=1 μ Alj v j

 q

j=1

(7)

T  Such that: w = w1 w2 w3 · · · wm and T  ξ = ξ1 ξ2 ξ3 · · · ξm with  x=

q

 

j=1 μ A1j v j   m q l=1 j=1 μ Alj v j

q

···

 

m j=1 μ A j v j   m q l=1 j=1 μ Alj v j

T (8)

Exponent (7) by: y = w T ξ C. Modeling unknowns: f (x) and g(x) are unknown nonlinear functions. We can approximate using fuzzy systems by w Tf ξ f (x) and wgT ξg (x) repectively, given w∗f and wg∗ constant and optimal vectors such as:   w∗f = arg minwf {supx  f (x) − wTf x f (x)}

(9)

  w∗g = arg minwg {supx g(x) − wgT xg (x)}

(10)

Parameters w∗f and w∗f are unknown constants. From the above analysis, we can write:   f (x) = fˆ x, w∗f + Φ f

(11)

  g(x) = gˆ x, w∗g + Φg

(12)

  f (x) − fˆ x, w∗f = w˜ Tf ξ f (x) + Φ f

(13)

Robust Adaptive Fuzzy Approach with Unknown Control …

469

  g(x) − gˆ x, w∗g = w˜ gT ξg (x) + Φg

(14)

w˜ f = w f − w∗f

(15)

w˜ g = wg − wg∗

(16)

With

Assumption 3 The errors Φ f and Φg are bounded for all    fuzzy approximation x ∈ Ωx and Φ f  ≤ Φ¯ f and Φg  ≤ Φ¯ g , where Φ¯ f and Φ¯ g are unknown positive constants. This assumption is reasonable, since we assume that fuzzy systems used for approximating unknown functions have the universal approximation property. Remark 1 Development that follows does not require knowledge of the sign of the control gain, because the Nussbaum function technique will be used to tackle this problem. The Nussbaum function technique was originally proposed in [7]. A function is called a Nussbaum-type function if it has the following properties [7, 11] 

s

(1) limt→∞ sup(1/s)

N (ζ)dζ = ∞

(17)

0



s

(2) limt→−∞ inf(1/s)

N (ζ)dζ = −∞

(18)

0

Here, an even function ζ2 cos(ζ) satisfying the above equalities will be employed to handle unknown signs of control gain functions in this paper.  Lemma 1 ([12]) Let V (t) and ζ(t)are smooth functions on the interval [0, t f . Suppose that V (t) > 0, ∀t ∈ [0, t f and N (ζ) is an even smooth Nussbaum-type function. If the following inequality holds: 

t

V (t) ≤ c0 +

(g(τ )N (ζ(τ )) + 1)ζ˙ (τ )dτ

(19)

0

where c0 and c1 are positive constants, and g(.)Is atime-varying parameter that / I , then V (t), ζ(t), takes values in the unknown closed intervals I = l, l¯ with 0 ∈ t and g(τ )N (ζ)ζ˙ dτ are bounded on [0, t f ) with t f = ∞. 0

Proof of Lemma 1 To proof this lemma, see [12] Controller design and stability analysis based on the above approximations and the control law given by (5), we propose the following adaptive control law: u(t) = u a (t) + u s (t)

(20)

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The control law (20) is a summation of an adaptive control term, u a (t), which attempts to approximate the control law (5) by using the estimates of the system’s nonlinearities, and additional control term u s (t), which is introduced to compensate for reconstruction errors. The adaptive control term: ua =



wgT ξg wgT ξg + 

−wTf ξ f + xd(n) − ke − αwgT ξg N (ζ )e T Pb

 (21)

Where  is a small positive constant, α positive constant. This additional control term u s is called supervisory control, is given by: u s = u ns N (ζ)

(22)

Where u ns =

  T e P B  

   Φˆ f¯ + Φˆ g u a − α N (ζ)e T Pb + u¯

eT P B    u¯ = T −wTf x f + xd(n) − ke − αwgT xg N (ζ)e T Pb wg xg + 

(23) (24)

N (ζ) = ζ2 cos(ζ)

(25)

  ζ˙ = e T Pbu ns + α e T Pb2 

(26)

With the following parameter adaption laws with   w˙ f (t) = Pr ojw f Γ f e T Pbξ f

(27)

   w˙ g (t) = Pr ojwg Γ g e T Pbξ g u a − α N (ζ )e T Pb

(28)

Where Γ f Rr f ×r f and Γ g Rrg ×rg are diagonal matrices positives, the matrix P the solution of Lyapunov equation AmT P + P Am = − Q for a given arbitrary n×n matrix Q = Q T > 0. We can check, using the definition of Proj for each I.w fi ≤ w∗fi ≤ w¯ fi And w gi ≤ wg∗i ≤ w¯ gi , it means that:





    T T ≤0 Γ − e Pr oj e Pbξ Pbξ wTf Γ −1 w f f f f f 

T T wTg Γ −1 g Pr ojwg Γ g e Pbξ g u a − α N (ζ )e Pb





− e T Pbξg u a − α N (ζ )e T Pb

(29) 

≤0

(30)   Φ˙ˆ f¯ = γ f e T Pb

(31)

Robust Adaptive Fuzzy Approach with Unknown Control …

471

   Φ˙ˆ g = γg e T Pbu a − α N (ζ )e T Pb

(32)

Where γ f > 0, γg > 0. D. Adaptive law derivation: From (1) and (13)–(14) and (20), we get   e˙ = Ak e + b[−w˜ Tf ξ f (x) − Φ f − w˜ gT ξg (x) u a − α N (ζ)e T Pb   − Φg u a − α N (ζ)e T Pb − g(x)u s + u¯ − αwgT xg N (ζ)e T Pb − d]

(33)

Where Φ f¯ = Φ f + d ⎡

0 0 .. .

1 0 .. .

0 1 .. .

··· 0 .. .

0 0 .. .

⎤

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Ak = ⎢ ⎥ ⎢ ⎥ ⎣ 0 0 ··· 0 1 ⎦ −kn −kn−1 · · · · · · −k1 Choose the Lyapunov function candidate as:   1 T 1 2 1 2 T −1 T −1 ˜ ˜ V = e Pe + w˜ f Γ f w˜ f + w˜ g g w˜ g + Φ + Φg 2 γ f¯ γg

(34)

where Φ˜ f¯ = Φ f¯ − Φˆ f¯ , Φ˜ g = Φg − Φˆ g , and Φ˙˜ f¯ = −Φ˙ˆ f¯ , Φ˙˜ g = −Φ˙ˆ g , w˜ i = wi − wˆ i , w˙˜ i = −w˙ i , i = f, g. The time derivative of V is given by  1 V˙ = e˙ T Pe + e T P e˙ − w˜ Tf Γ f−1 w˙ˆ f − w˜ gT Γg−1 w˙ˆ g 2 1 1 − Φ˜ f¯ Φ˙ˆ f¯ − Φ˜ g Φ˙ˆ g γ f¯ γg  T   1 T ≤ − e Qe + −w˜ f ξ f (x) − Φ f¯ − w˜ gT ξg (x) u a − α N (ζ )e T Pb 2  − Φg u a − α N (ζ )e T Pb − g(x)u s + u¯ − αwgT xg N (ζ )e T Pb] 1 1 Φ˜ ¯ Φ˙ˆ ¯ − Φ˜ g Φ˙ˆ g − w˜ Tf Γ f−1 w˙ˆ f − w˜ gT Γg−1 w˙ˆ g − (35) γ f¯ f f γg By substituting (23)–(26), and with the law adaptive (29)–(30), V˙ can be written as follows: 1 V˙ ≤ − e T Qe + (1 + g(x)N (ζ ))ζ˙ 2

(36)

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We can obtain the following inequality 

t

V (t) ≤ V (0) +

(g N (ζ(τ )) + 1)ζ˙ (τ )dτ

(37)

0

 t Using Lemma 1, we can conclude  from (36) the bounded of V (t), ζ(t) et ˙ (τ )dτ with [0, t f . ζ N + 1) (g (ζ(τ )) 0 According to [13], since no finite-time escape phenomenon may happen, then t f = ∞. Therefore, e, w˜ i i = f, g, Φ˜ f¯ , Φ˜ g , x(t) and u(t) are bounded. Moreover, by invoking Barbalat’s lemma [14], we can conclude the asymptotic convergence of e(t).

3 Application to Inverted Pendulum Tracking Control In this section, we use our adaptive fuzzy controller to control the inverted pendulum system to tack a sin wave trajectory. The inverted pendulum system defined here is shown in Fig. 1, which is formed from a cart, a pendulum and a rail for defining the position of the cart. The pendulum is hinged in the center of the top surface of the cart and can rotate around the pivot in the same vertical plane with the rail. The cart can move right or left on the rail freely. It is given that no friction exists in the system between the cart and the rail or between the cart and the pendulum [7, 8]. Let x1 = θ and x2 = θ˙ , the dynamic equation of the inverted pendulum system can be expressed as [8], Fig. 1 The inverted pendulum system

Robust Adaptive Fuzzy Approach with Unknown Control …

⎧   θ¨ = f θ, θ˙ + g(θ )u + d(t) ⎪ ⎪ ⎪ ⎨ y=θ   gsin(θ)−mlaθ θ˙2 −sin(2θ )/2 ˙ f θ, θ = ⎪ ⎪ 4l/3−mlacos 2 (θ) ⎪ ⎩ a cos(θ) g(θ ) = − 4l/3−mlacos 2 (θ)

473

(38)

Where θ is the angular position of the pendulum, g = 9.8m/s 2 is the acceleration due to gravity, M, is the mass of cart, m is the mass of pole, l is the half length of pole, and u isthe applied force (control). We chose M = 1kg, m = 0.1 kg, and l = 0.5m, = 1/m + M, d(t) = 0.1 sin(2 ∗ t). It is easy to verify that g(θ) takes positive values and far from zero for all −

π π ≤θ ≤ 2 2

  As the denominator of f θ, θ˙ is positive for any − π2 ≤ θ ≤ π2 and assume that   θ˙ is limited, and we have limited f θ, θ˙ . We chose the reference signal θd (t) to be a sinusoid with amplitude of π/12 and a frequency of 1 Hz. In our simulation we assume that the function f and g are unknown, we used two fuzzy systems to approximate f and g with the same fuzzy sets, shown in Fig. 2. We choose 4 fuzzy sets for θ, θ˙ respectively which gives us 16 fuzzy rules possible: Rule1 : if θ is NB and θ˙ is NB, Then y = w1 Rule2 : if θ is NB and θ˙ is NS, Then y = w2

Fig. 2 Fuzzy sets for θ and θ˙

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.. . Rule16 : if θ is PB and θ˙ is PB, Then y = w16 With wi , i = 1, . . . , 16 are adaptation parameters, the limits w f and wg respectively w¯ fi = 100, w fi = −100, w¯ gi = 1 and w gi = 0.1. all the initial values of the components of w f are put to 0, and wg from to 0.9. We choose d f = dg = 5, and all the elements of Γ f and Γg respectively 1000, 20, N (ζ ) = ζ 2 cos(ζ) 1 = 0.75, 2 = 159  and  3 = 0.25 We simply choose the vector k = 1 2 , to place the poles of Am at −1, and  1.5 0.5 Q = 0.25I 2 we solve Lyapunov equation to obtain P = with Eigen 0.5 0.5 √ values P = 1 ∓ 0.5 for the different initial values used θ0 = 0.5, θ0 = −0.5 . Figure 3 shows the simulation results of the first adaptive fuzzy controller for initial conditions θ0 = 0.5, we see that the initial parts of control are apparently improved after incorporating these fuzzy rules. The result shows that the angle θ (t) followed the desired trajectory after 3 s, and we notice that the system is stabilized. The same remark for Fig. 4, where starting with, θ 0 = − 0.5 rad, which shows the effectiveness of this approach. Figure 5 depicts the control input signal, the error tends to 0, shown in Fig. 6. In order to show the effectiveness of the proposed method a comparison with [15] is considered. It is clear that the results obtained by the proposed approach are better to that in the [15] in terms of performance since the cost of command signal in [15] is more energetic [50–250](N). variation de l'angle x1 pour x0=0.5

0.5

xd x1

0.4 0.3

rad

0.2 0.1 0 -0.1 -0.2 -0.3 0

5

10

15

Fig. 3 Response of the variable x1: θ 0 = 0.5

20 Temps (s)

25

30

35

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15 10 5 0 -5 -10 -15 -20 0

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Fig. 5 Control input signal

4 Conclusion In this work we have proposed an approach to solve control problem of nonlinear system with unknown control direction, to track a sin wave trajectory. We used fuzzy indirect adaptive control to build a controller that guarantees the stability and robustness of the system, whose approach based on the Lyapunov stability to build the adaptation law, and fuzzy logic to approximate unknown functions to a robust and stable controller. Compared with previous approaches, the proposed approach obtains more compact control structure. The results are satisfactory.

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References 1. Zadeh LA (1967) Fuzzy algorithms. Inf Control 12(2): 94–102 2. Becerikli Y, Chik BK (2006) Fuzzy control of inverted pendulum and concept of stability using Java application. Mathematical and computer Modeling 3. Wang HQ, Chen B, Liu XP, Liu KF, Lin C (2013) Robust adaptive fuzzy tracking control for pure-feedback stochastic nonlinear systems with input constraints. IEEE Trans Cybern 43(6):2093–2104 4. Chang YC (2001) Adaptive fuzzy-based tracking control for nonlinear SISO systems via VSS and H∞ approaches. IEEE Trans Fuzzy Syst 9:278–292 5. Boulkroune A, Tadjine M, M’saad M, Farza M (2009) Adaptive fuzzy controller for non-affine systems with zero dynamics. Int J Syst Sci 40(4): 367–382 6. Nussbaum RD (1983) Some remarks on the conjecture in parameter adaptive control. Syst Control Lett 3:243–246 7. Liu Y-J, Li Y-X (2010) Adaptive fuzzy output-feedback control of uncertain SISO nonlinear systems. Nonlinear Dyn 61:749–761 8. Boulkroune A, M’saad M (2010) On the design of observer-based fuzzy adaptive controller for nonlinear systems with unknown control gain sign. Fuzzy Sets Syst 201 (2012): 71–85 9. Nassira Z, Mohamed C, Najib E (2015) Stable indirect adaptive HONN control for a class of non affine SISO nonlinear systems. In: 3rd Inernational conference on control, engineering & information technology (CEIT), pp 1–6. Tlemcen, Algeria 10. Liu L, Huang J (2008) Global robust output regulation of lower triangular systems with unknown control direction. Automatica 44:1278–1284 11. Ye X, Jiang J (1998) Adaptive nonlinear design without a priori knowledge of control direction. IEEE Trans Aut Control 43:1617–1621 12. Wang LX (1996) Stable adaptive fuzzy controllers with application to inverted pendulum tracking. IEEE Trans Syst Man Cybern part b Cyber 26(5) 13. Boulkroune A (2016) A fuzzy adaptive control approach fornonlinear systems with unknown controlgain sign. Neurocomputing 14. Slotine JJE, Li W (1991) Applied nonlinear control. Prentice-Hall, USA 15. luo L, luo F, Xu Y (2014) Direct adaptive fuzzy control of nonlinear systems with unknown control direction. Inf Technol J 13(4): 697–702

Adaptive Neural-Network Control Design for Uncertain CSTR System with Unknown Input Dead-Zone and Output Constraint Zerari Nassira and Chemachema Mohamed

Abstract In this paper, the control problem for Continuously Stirred Tank Reactor (CSTR) systems with input nonlinearity and unknown disturbances is addressed. To ensure constraints satisfaction on the input and the output of CSTR we employ a system transformation technique to transform the original constrained model of CSTR into an equivalent unconstrained model, whose stability is sufficient to achieve the tracking control of the original CSTR system with a priori prescribed performance. In the proposed control methods, NNs are employed to approximate the unknown dynamics of CSTR and additional adaptive compensators are introduced to cope with NNs approximation errors and external disturbance; a robust control term is introduced to overcome the effects of the unknown input dead-zone. The proposed adaptive neural tracking controllers are designed with only one adaptive parameter by using the second Lyapunov stability method. In comparison with the traditional back-stepping based techniques, usually used in CSTR control, the structures of the proposed controllers are much simpler with few design parameters since the causes for the problem of complexity growing are completely eliminated. Simulation results are presented to show the effectiveness of the proposed controllers. Keywords Neural networks (NNs) · Adaptive control · Continuously stirred tank reactor (CSTR) · Input nonlinearity · Prescribed performance

1 Introduction The continuously stirred tank reactor (CSTR) is a typical nonlinear, and unstable dynamic system that is applied in chemical industries. Control of the CSTR is used to verify that a new control method has a strong ability to address nonlinear and Z. Nassira · C. Mohamed (B) Signal Processing Laboratory, Department Electronics, Faculty of Sciences of Technology, Constantine 1 University, Constantine 2500, Algeria e-mail: [email protected] Z. Nassira e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_33

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instability problems. The control design of CSTR systems have been widely studied in the literature, by using control techniques ranging from classical approaches to advanced ones. In [1, 2] robust PI stabilizing controllers are designed. State feedback controllers with state estimator and dynamic output feedback controllers are designed in [3, 4] respectively. Standard nonlinear approaches are applied on CSTR using feedback liearization and based on high gain and variable structure observers [5–7]. More elaborated controllers are developed to deal with uncertainties in CSTR systems such as fuzzy PID controller in [8] and fault tolerant control approach in [9] to deal with CSTR uncertainties and faults occurrence, respectively. In [10] fractional order PID based on firefly algorithm for optimal parameters settings is used in the control of CSTR systems. The merits of the above mentioned approximation-based control are that the adaptive laws of adjustable weights of neural networks or fuzzy logic systems can be obtained on the basic of Lyapunov theorem, which guarantees the stability of closed-loop system. Despite these efforts, the existing results [5–10] were focused on uncertain CSTR systems without considering the input nonlinearity. Practical engineering often requires the proposed control schemes to satisfy certain quality of the performance indices such as overshoot, convergence rate, and steady-state error. To address this problem, numerous existing works, including the Barrier Lyapunov Function (BLF), demonstrate that it can guarantee the constraints satisfaction. In [11, 12], a robust adaptive controller is designed for CSTR where the barrier Lyapunov function is employed in the synthesis to prevent the velocity constraint violation. On the other hand, an error transformation method, which was originally presented in the prescribed performance control (PPC), has been used to deal with constraints [13, 14]. Despite these efforts, the existing results [12–14] are focused on uncertain CSTR systems without considering any input nonlinearities. It is well known that input nonlinearities exist in real world. CSTR systems are not an exception for which, in this paper, dead-zone nonlinearity is considered. The existence of input dead-zone may lead to the instability of the closed-loop system and hence makes the stabilization problem more difficult. Valuable works on adaptive fuzzy/neural based controller design for CSTR dynamics preceded by unknown dead-zone are proposed [15–17]. In [15] an adaptive NNs controller is designed based on “n” NNs for the online learning of the CSTR dynamics. In [16, 17], the non-affine pure-feedback discrete time nonlinear systems with non-symmetric dead-zone inputs are transformed into a n-step ahead predictors where adaptive NNs compensative terms are used to compensate for the dead-zone parameters. However, from a point of view of practical applications, the methods developed in [15–17] are computationally expensive because of the NNs and fuzzy approximators are used at every step to online approximate the unknown dynamics, and hence render the control laws computationally heavy and the stability analysis more complicated. Motivated by the preceding remarks, in this paper, we are inspired to propose a robust adaptive control approach for CSTR systems with input nonlinearity, output constraints and unknown disturbance. To ensure constraints satisfaction with a priori prescribed performance, we employ a system transformation technique to transform the original constrained system into an equivalent unconstrained model whose stability is sufficient to achieve the tracking control of the original system, and guaranteeing the prescribed performance. Input dead-zone nonlinearity along with

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unknown external disturbance are considered in this tracking control design. In order to compensate for the effects of dead-zone input and external disturbance, we introduced an adaptive robust control structure. In the proposed control designs, NNs are used to approximate the unknown nonlinearity based on their universal approximation properties. The adaptive NNs tracking controllers are designed with only one adaptive parameter by using the common Lyapunov stability theorem. It is shown that the proposed adaptive control methods can guarantee that all the signals in the closedloop system are bounded, and the tracking errors converge to a small neighborhood of the origin. The main contributions of this paper are summarized as follows. (a) The proposed approach is much simple since it avoids the well known explosion of complexity problem of the back-stepping based methods caused by the repeated differentiations of virtual controllers. This simplicity is explained by simpler derivation of the adaptive laws without virtual controllers and hence simpler stability analysis. (b) The number of parameters of the proposed adaptive laws is highly reduced due to the use of only one NNs, however, the traditional back-stepping based methods, as in [11, 15–17], require tow NNs units with tow virtual controllers and their derivatives leading to more complicated control laws with high computation burden. (c) The stability analysis of the closed loop and the derivation of the adaptive learning laws of the proposed controllers are highly simplified. (d) The simplified proposed work permits to cope with both of input nonlinearities and output constraints along with unknown disturbances. In the contrary, the back-propagation based approaches with recursive design procedures deal only with input nonlinearities of CSTR in [15–17] and only output constraints of CSTR in [11, 12]. (e) Furthermore, the proposed controllers guarantee the stability of the closed loop of CSTR systems with a priori prescribed performance imposed by the designer as shown theoretically and in a simulation example which is not the case for most of the approaches in the literature. The rest of the paper is organized as follows. The dynamics of the CSTR is described in Sect. 2. In Sect. 3, we design the controls for the unknown input deadzone, while in Sect. 4, simulation results are presented to show the effectiveness of the approach. Finally, conclusions are drawn in Sect. 5.

2 Problem Statement 2.1 Model Description The dynamic of the process of CSTR considered in this paper can be written in the following form [18, 19]:

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⎧ x˙1 = −2x1 + Da (1 − x1 )ex p[x2 /(1 + (x2 /ϕ))] + x1 ⎪ ⎪ ⎨ x˙n = −x2 (1 − δ) + B Da (1 − x1 )ex p[x2 /(1 + (x2 /ϕ))] ⎪ δu(v) + d(t) ⎪ ⎩ y = x1

(1)

where x1 and x2 denote the dimensionless reactant concentration and mixture temperature, respectively; u ψ is the dimensionless coolant flow rate; Da , ϕ, B ψ and δ ψ denote the Damokhler number, the activated energy, heat of reaction, and heat transfer coefficient, respectively. Then, we can rewrite (1) as ⎧ ⎨ x˙1 = f 1 (x1 , x2 ) + x1 x˙ = f 2 (x1 , x2 ) + δu(v) + d(t) ⎩ 2 y = x1

(2)

with f 1 (x¯2 ) = −x1 + Da (1 − x1 )ex p[x2 /(1 + (x2 /ϕ))] − x2 f 2 (x¯2 ) = −x2 (1 − δ) + B Da (1 − x1 )ex p[x2 /(1 + (x2 /ϕ))]

(3)

where d(t) is the unknown external disturbance term, which is bounded; f i (., .) is the unknown smooth function. The input nonlinearity u(v) which includes dead-zone. To facilitate the control design, the following standard Assumptions and Lemmas are introduced for system (2). Assumption 1 For the system (2), there exists an unknown positive constant d¯ such ¯ that |d(t)| < d. Lemma 1 [7]: For ∀ς > 0 and ∀q ∈ R, the following inequality holds 0 ≤ |q| − q tanh

  ς ≤ κς q

(3)

where κ = 0.2785 Lemma 2 (Young’s inequality [20]) For ∀(x, y) ∈ R 2 , the following inequality holds: xy ≤

εp p 1 |x| + q |y|q , p pε

where ε > 0, p > 1, q > 1, and ( p − 1)(q − 1) = 1.

(4)

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In this paper, the control objective is twofold. Design a control input u(t) for the CSTR system in (1) such that: P1. The system output y(t) follows the desired output signal yd (t), while all the signals of the closed-loop system are bounded. P2. Both prescribed transient and steady state performance of the tracking error e(t) = x1 (t) − yd (t) are achieved.

2.2 Dead-Zone Model and Its Properties The control input with a dead-zone, is described by [15] as follows: ⎧ ⎨ m r (v(t) − br ) v(t) ≥ br u(v) = 0 br ≤ v(t) ≤ bl ⎩ m l (v(t) − bl ) v(t) ≤ bl

(5)

where m r and m l are the general slope of the dead zone characteristic. It is assumed that m = m r = m l ,br and bl denote the right and left dead zone breakpoints, respectively. Assumption 2 ([15]) The strictly positive constants m, dr and dl are unknown, but their signs are known (m > 0, dr ≥ 0 and dl ≥ 0 respectively). Assumption 3 ([15]) The dead zone parameters are bounded  by known constants m min ,m max , drmin , drmax , dlmin , dlmax such that m ∈ m min , m max , dr ∈ [drmin , drmax ] and dl ∈ [dlmin , dlmax ]. As a result, the output of the dead zone can be reformulated as u(v) = mv(t) + h(v(t))

(6)

where h(v(t)) can be calculated from (5) and (6) as follows ⎧ v(t) ≥ br ⎨ −m r br h(v(t)) = −m(t)v(t) br ≤ v(t) ≤ bl ⎩ −m l bl v(t) ≤ bl

(7)

From Assumption 2 and 3 we can conclude that h(v(t)) is bounded and satisfies |h(v)| ≤ ρ, where ρ isthe upper-bound, which can be chosen as ρ = max m max drmax , −m min dlmin .

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2.3 Neural Networks Neural networks (NNs) has been frequently used as function approximators, in this study, a continuous function f (Z ) : R + → R will be approximated by neural networks (NN) with the output given by:

fˆ Z /Wˆ = Wˆ T ∅(Z )

(8)

ˆ ∈ RN is called the weight vector, and where Z ∈ Z ∈ R m is the input to NN, W N is the number of network nodes, ∅(Z ) = [∅1 (Z ), . . . , ∅ N (Z )]T is the so-called vector of activation function chosen as the hyperbolic tangent function in our study     ∅i (Z ) = 1 − e−Z / 1 + e−Z , i = 1, . . . , N

(9)

According to the approximation property of the NNs in [21], any continuous function over a compact set Ωz , there exists an integer N and optimal synaptic weight vector W ∈ R N such that:        (10) W ∗ = argmin W supz∈Ωz  fˆ Z /W ∗ − f (Z ) In other terms, if the nodes number N is sufficiently large and the nodes number terms are appropriately selected, then there exists a weight vector W such that W T ∅(Z ) can approximate f (Z ) to any  degree of accuracy, in a given compact set. In   ˆ general, supz∈Ωz  f (Z /W ) − f (Z ) becomes smaller as N increases. More precisely, we may substitute, with no loss of generality, the unknown function by a linear in the weights neural network structure, plus a modeling error term ∀Z ∈ z , obtaining: f (Z ) = W ∗T ∅(Z ) + ε(Z )

(11)

where W ∗ is the optimal weights vector which minimizes the approximation error ε(Z ). Owing to the approximation capabilities of the linear in the weights neural network, there exists an unknown constant ε¯ > 0 such that: |ε(Z)| ≤ ε¯ , ∀Z ∈ z

(12)

From the universal approximation results for neural networks stated in [22], it is known that the constant ε¯ can be made arbitrarily small by increasing the NN nodes number N.

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3 Design of Adaptive Neural Prescribed Performance Control with Input Nonlinearities 3.1 Prescribed Performance Function To achieve the second objective P2, as defined in [13], a positive decreasing smooth function μ(t) : R + → R + with lim μ(t) = μ∞ > 0 will be used as the prescribed t→∞

performance function (PPF). As stated in [13, 14], the tracking error e(t) = x1 (t) − yd (t) is satisfied using the following predefined inequality: ¯ ∀t > 0 −δμ(t) < e(t) < δμ(t),

(13)

where δ and δ¯ > 0 are constants selected by the designer. In this study, the performance function μ(t) is set as [13]: μ(t) = (μ0 − μ∞ )e−κt + μ∞

(14)

where μ0 > μ∞ and κ > 0 are design parameters. ¯ 0 defines the upper bound of the maximum In (13) and (14), we know that δμ overshoot and −δμ0 defines the lower bound of the undershoot, the decreasing rate κ introduces a lower bound on the convergence speed, and μ∞ denotes the allowable steady-state tracking error [13]. Hence, the transient and steady state performances ¯ δ, κ, μ0 and μ∞ . can be designed a priori by tuning the parameters δ, To solve the control problem with the prescribed performance (13), and according to [13], an output error transform will be introduced by transforming condition (13) into an equivalent unconstrained one. In this study, we define a smooth, strictly increasing function T (ξ1 ) of the transformed error ξ1 ∈ R, such that: ¯ ∀ξ1 ∈ L∞ (1) − δ < T(ξ1 ) < δ, (2)

lim T (ξ1 ) = δ¯

ξ1 →+∞

lim T (ξ1 ) = δ

ξ1 →−∞

From the properties of T(ξ1 ), condition (13) leads to: e(t) = μ(t)T (ξ1 )

(15)

Since T (ξ1 ) is strictly monotonic increasing and the fact that μ(t) ≥ μ∞ > 0 holds according to (14), the inverse function of T (ξ1 ) exists and can be deduced as ξ1 = T

−1



e(t) μ(t)

 (16)

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¯ δ, κ, μ0 and μ∞ Note that PPF (14), T (ξ1 ) and the associated parameters δ, ¯ μ0 and are all a priori designed. For any initial condition e(0), if parameters δ, ¯ and ξ1 can be controlled to be δ.are selected such that −δμ(0) < e(0) < δμ(0) bounded (i.e., ξ1 ∈ L ∞ , ∀t > 0), then −δ < T (ξ1 ) < δ¯ holds; thus, the condition ¯ is guaranteed. Consequently, the tracking control problem −δμ(t) < e(t) < δμ(t) of system (1) is now transformed to stabilize the transformed system (16). T −1 (.) denotes the inverse function of T (.). A typical choice of T (ξ1 ) can be given by T (ξ1 (t)) =

¯ ξ1 − δe−ξ1 δe eξ1 + e−ξ1

(17)

Then, from (16), the transformed error ξ1 is derived as ξ1 = T

−1



 e(t) 1 ρ(t) + δ = ln μ(t) 2 δ¯ + ρ(t)

(18)

where ρ(t) = e(t)/μ(t) To stabilize the error system ξ1 and thus to achieve the guaranteed performance of error e, we further deduce    1 1 eμ˙ ∂ T −1 (.) 1 e˙ ρ(t) ˙ = − − ∂ρ(t) 2 ρ(t) + δ ρ(t) − δ¯ μ μ2   eμ˙ = ϕ x2 + f 1 (x1 , x2 ) − y˙d − μ

ξ˙1 =

 1 can be calculated based on e(t) and μ(t) and fulfills − ρ(t)− δ¯    ¯ . 0 < ϕ < ϕ M = δ + δ¯ /μ∞ δδ Moreover, one may obtain that where ϕ =

1 2μ



(19)

1 ρ(t)+δ

  eμ˙ ¨ξ1 = ϕ˙ x2 + f 1 (x1 , x2 ) − y˙d − μ   eμμ ¨ ∂ f 1 (x¯2 ) ∂ f 1 (x¯2 ) e˙μ˙ eμ˙ 2 − 2 + 2 x˙1 + x˙2 − y¨d − + ϕ x˙2 + ∂ x1 ∂ x2 μ μ μ   eμ˙ = ϕ˙ x2 + f 1 (x1 , x2 ) − y˙d − μ   eμμ ¨ ∂ f 1 (x¯2 ) ∂ f 1 (x¯2 ) e˙μ˙ eμ˙ 2 − 2 + 2 −ϕ x˙1 + x˙2 − ∂ x1 ∂ x2 μ μ μ + ϕ[ f 2 (x1 , x2 ) + δu(v) + d(t) − y¨d ]

(20)

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Define the filtered error as T  es = [ 1] ξ1 , ξ˙1

(21)

where > 0 is a positive constant such that the tracking error ξ1 is bounded as long as es is bounded. Consequently, we have 

 eμ˙ ˙ x2 + f 1 (x1 , x2 ) − y˙d − e˙s = ( ϕ + ϕ) μ + ϕ[ f 2 (x1 , x2 ) + δu(v) + d(t) − y¨d ]   eμμ ¨ ∂ f 1 (x¯2 ) ∂ f 1 (x¯2 ) e˙μ˙ eμ˙ 2 − 2 + 2 −ϕ x˙1 + x˙2 − ∂ x1 ∂ x2 μ μ μ = ϕ[F(x1 , x2 , y˙d , y¨d , e, ϕ) + δu(v) + d(t) − y¨d ]

(22)

  − ˙ x2 + f 1 (x1 , x2 ) − y˙d − eμμ˙ ( + ϕ/ϕ)  2 ∂ f 1 (x¯2 ) ¨ eμ˙ + f 2 (x1 , x2 ) denotes the unknown x˙1 + ∂ f∂1 x(x2¯2 ) x˙2 − e˙μμ˙ − eμμμ 2 + μ2 ∂ x1 nonlinear function, which are approximated by the NNs (11). where 

F(x1 , x2 , y˙d , e, ϕ)

=

F(x1 , x2 , y˙d , e, ϕ) = W ∗T ∅(Z ) + ε(Z ),

(23)

where Z = [x1 , x2 , y˙d , e, ϕ]T

3.2 Stability Analysis Consequently, taking (23) into (22) yields  e˙s = ϕ W1∗T ∅(Z ) + ε1 (Z ) + δu(v) + d(t) − y¨d

(24)

Substituting (25) into (27) yields  e˙s = ϕ W1∗T ∅(Z ) + δmv(t) − y¨d + δh(v) + D1 (t)

(25)

where D1 (t) = ε1 (Z ) + d(t). It follows from Assumptions 1 that there exists an unknown upper bound τ ∗ such that: |D1 | ≤ τ1∗ . Furthermore, the adaptive NNs controller is chosen to be:   α1 es2 τˆ12 1 T ˆ − es + y¨d − W1 ∅(z) − v(t) = δm ϕ es τˆ1 tanh(es /) +  + vc (t)

(26)

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where vc (t) is a compensative control term for h(v) and it is chosen as vc (t) = −κ ∗ sign(es )

(27)

where κ ∗ is a control gain satisfying κ ∗ ≥ ρ/m min . with the corresponding adaptive control laws .

Wˆ = ϕ1 es ∅(Z ) + σ1 Wˆ 1

(28)

.  τˆ = ϕγ1 −|es | + σ2 τˆ1

(29)

1

1

where Wˆ 1 and τˆ1 are the estimate values of the unknown parameters W1∗ and τ1∗ , respectively, and Γ1 ,γ1 , σ1 , σ2 , α1 are positives constants. The stability of the closed-loop system will be proved via Lyapunov analysis. Theorem 1 Consider the CSTR system (1) subject to the input dead-zone (6). Based on Assumptions 1–4, consisting the controller (26), the compensative control term (27), the adaptive laws (28)–(29), while the design parameters are chosen appropriately, we can conclude that all the signals in closed loop of the CSTR system are bounded, and the tracking errors will converge to a small compact. Proof Define the following Lyapunov function candidate: V1 =

1 2 1 ˜ T −1 ˜ 1 2 es + W1 Γ1 W1 + τ˜ 2 2 2γ1 1

(30)

where τ˜1 = τ1∗ − τˆ1 and W˜ 1 = W1∗ − Wˆ 1 are the parameter estimation errors. The derivative of (30) is given by: . . 1 V˙1 = es e˙s − W˜ 1T Γ1−1 Wˆ − τ˜1 τˆ 1 γ1 1  ∗T = es ϕ W1 ∅(Z ) + δmv(t) − y¨d + δh(v) + D1 (t) . . 1 − W˜ 1T Γ1−1 Wˆ − τ˜1 τˆ 1 γ1 1

(31)

According to the control law (26) and parameter adaptation laws (28)–(29), one has V˙1 = −α1 es2 + ϕδmes {vc + h(v)/m} + ϕes D1 (t)  .  ϕes2 τˆ12 1 . −1 ˆ T ˜ − + W1 ϕes ∅(Z ) − Γ1 W − τ˜1 τˆ 1 es τˆ1 tanh(es /) +  γ 1 2 2 ϕes τˆ1 + ϕ|es |τ1∗ + ϕ|es |τ˜1 ≤ −α1 es2 − es τˆ1 tanh(es /) +  − σ2 ϕ τ˜1 τˆ1 − σ1 ϕ W˜ 1T Wˆ 1

(32)

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Using Young’s inequality and invoking Lemma 2, we have   2   ϕ M σ1    T ˆ ∗ 2  ˜ W1 − W˜ 1  −σ1 ϕ W1 W1 ≤ 2  ϕ M σ2  ∗2 −ϕσ2 τ˜1 τˆ1 ≤ τ1 − τ˜12 2

(33) (34)

In addition, using the following inequality, 0 ≤ xtanh(x/a) ≤ |x|, ∀x ∈ R, a > 0, and by substituting (33) and (34) into (32), we obtain:  ϕ M σ1  ϕ M ε|es |τˆ1  ˜ 2 ϕ M σ2 2 − τ˜ V˙1 ≤ −α1 es2 +  W1  − |es |τˆ1 +  2 2 1  ϕ M σ1  W ∗ 2 + ϕ M σ2 τ ∗2 + 1 2 2 1 Then, using the inequality 0
0, it yields

 ϕ M σ1   ˜ 2 ϕ M σ2 2 V˙1 ≤ −α1 es2 − τ˜  W1  − 2 2 1   ϕ M σ1  ∗ 2 ϕ M σ2 ∗2 W1 + τ + ϕM  + 2 2 1 ≤ −θ1 V1 + β1  where θ1 = min 2α1 , λ

ϕ M σ1 ,ϕ σ γ max (Γ1−1 ) M 2 1

β1 =

(35)

(36)



 ϕ M σ1  W ∗ 2 + ϕ M σ2 τ ∗2 + ϕ M  1 2 2 1

Moreover, by multiplying eθ1 t and integrating on both sides of (36) over [0, t], it follows that:   β1 −θ1 t β1 β1 e + V1 |t=0 − ≤ V1 |t=0 + (38) V1 (t) ≤ θ1 θ1 θ1 Combining (30) with (38), we can conclude that: |es | ≤



2( V1 |t=0 + θ1 /β1 )

     ˜  W1  ≤ 2( V1 |t=0 + θ1 /β1 )/λmin Γ1−1 |τ˜1 | ≤



γ1 2( V1 |t=0 + θ1 /β1 )

(39)

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Consequently, the parameter estimate errors W˜ 1 , τ˜1 and tracking error es are bounded from (39), and remain in the compact sets Ωτ˜1 , ΩW˜ 1 and Ωes defined by:  √  Ωes = es ∈ R||es | ≤ Ω      −1   ˜  ˜ = W1 ∈ R W1  ≤ Ω/λmin Γ1 

ΩW˜ 1

   Ωτ˜1 = τ˜1 ∈ R||τ˜1 | ≤ γ1 Ω where Ω = 2( V1 |t=0 + θ1 /β1 ) with θ1 and β1 defined as (33). The proof is completed.

4 Simulations In order to show the effectiveness of the proposed method, simulation study is described in this section. Consider the CSTR system excited by boundary disturbance d(t, x) = cos(t)cos(x2 ). The parameters of CSTR model are chosen as Da = 0.072, δ = 0.3, ϕ = 20 and B = 8.The objective is to apply the proposed adaptive NN controller such that the system output y tracks the desired trajectory yd = 0.5 + 0.1sin(t). In the simulation study, one NN system in the form of (8) is used for each case to generate the unknown nonlinearities of the nonlinear CSTR system. The NNs input vector is Z = [x1 , x2 , y˙d , e, ϕ]T , with N = 2 network nodes, and the activation function ∅(.) is chosen as the hyperbolic tangent function in our study. The design parameters used in this simulation are chosen by trial and error as follows:Γ1 = 20, Γ2 = 35, σ1 = σ3 = 0.1, σ2 = σ4 = 20, γ1 = γ2 = 10,  = 0.1, α1 = 1.2,α2 = 2.8, κ ∗ = 4.5, λ = 10. The initial conditions are chosen as x1 (0) = 0.6, x2 (0) = 0.5, τˆ1 (0) = τˆ2 (0) = 0, Wˆ 1 (0) = Wˆ 2 (0) = 0. The dynamic responses of the CSTR system are simulated for two cases. In the first case, the CSTR dynamics are simulated with the following input dead zone parameters: mmin = 0.6, mmax = 0.7, br = 2.09, bl = −1.5. The transient and steady state tracking error bounds are prescribed through the performance function μ(t) = (μ0 − μ∞ )e−κt + μ∞ with the parameters μ0 = 1.5, μ∞ = 0.05, κ = 0.09, δ = δ¯ = 1. The trajectories of CSTR output along with the reference signals when applying the proposed controllers are shown in Fig. 1. The boundedness of the control signals v(t) and N (v(t)) is illustrated in Fig. 2. From the simulation results, we can observe that the proposed methods are able to control CSTR system with effective performance in the presence of dead-zone input nonlinearities and preserving smooth trajectories for all the states and the control input signals. Indeed, the tracking errors are guaranteed to be inside the imposed bounds (14) induced by the prescribed performance function (13) as shown in Figs. 2.

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0.7

y

yd

0.6 0.5 0.4 0.3 0.2 0.1 0

0

5

10

15 Time (sec)

20

25

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Fig. 1 Tracking performance: desired trajectory yd , output y 20

v(t)

u(v(t))

10 0 -10 -20 -30 -40

0

5

10

15 Time (sec)

20

25

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Fig. 2 Trajectory of v(t) and N (v(t))

To further show the effectiveness of the proposed controller, a comparison is conducted with traditional back-stepping CSTR control with input nonlinearities and without considering the output constraints [15]. The obtained results when applying the controller defined in [15] with input dead zone and without output constraints are shown in Fig. 3. It is very clear that our approach outperform the approach proposed in [15] since the obtained response in Fig. 3 shows that the prescribed performance (13) is guaranteed.

5 Conclusions In this paper, adaptive NN controller with dynamic robust control terms was proposed for CSTR systems subjected to input dead-zone nonlinearity, output constraint and external disturbance. Compared to the traditional back-stepping control, it was proved that the developed controller exhibited more simplified control algorithms

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0.1 Proposed method

0.08

Back-stepping method

PPF bound

0.06 0.04 0.02 0 -0.02 -0.04 -0.06

0

5

10

15 Time (sec)

20

25

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Fig. 3 Tracking error of different controllers

with reduced design parameters thanks to the elimination of the causes of explosion dimensionality problem. Then, the proposed adaptive neural tracking controller was designed with only one adaptive parameter by using the second Lyapunov stability method. Adaptive robust control term was used to deal with neural networks approximation errors and external disturbance term. It was proved that the proposed method was able to effectively control CSTR systems with guaranteed, a priori, prescribed performances. Simulation results showed the effectiveness of the proposed method.

References 1. Alvarez-Ramirez J, Femat R (1999) Robust PI stabilization of a class of chemical reactors. Syst Control Lett 38(4–5):219–225 2. Pérez M, Albertos P (2004) Self-oscillating and chaotic behaviour of a PI-controlled CSTR with control valve saturation. J Process Control 14(1):51–59 3. Viel F, Jadot F, Bastin G (1997) Robust feedback stabilization of chemical reactors. IEEE Trans Automatic Control 42(4):473–481 4. Antonelli R, Astolfi A (2003) Continuous stirred tank reactors: easy to stabilise? Automatica 39(10):1817–1827 5. Biagiola SI, Figueroa JL (2004) A high gain nonlinear observer: application to the control of an unstable nonlinear process. Comput Chem Eng 28(9):1881–1898 6. Jana AK, Samanta AN, Ganguly S (2005) Globally linearized control on diabatic continuous stirred tank reactor: a case study. ISA Trans 44(3):423–444 7. Daaou B, Mansouri A, Bouhamida M, Chenafa M (2012) Development of linearizing feedback control with a variable structure observer for continuous stirred tank reactors. Chin J Chem Eng 20(3):567–571 8. So GB, Jin GG (2018) Fuzzy-based nonlinear PID controller and its application to CSTR. Korean J Chem Eng 35(4):819–825 9. Wang ZY, Wang GX (2017) Temperature fault-tolerant control system of CSTR with coil and jacket heat exchanger based on dual control and fault diagnosis. J Central South Univ 24(3):655–664 10. Alamdar Ravari M, Yaghoobi M (2019) Optimum design of fractional order pid controller using chaotic firefly algorithms for a control CSTR system. Asian J Control

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11. Li DJ, Li DP (2015) Adaptive controller design-based neural networks for output constraint continuous stirred tank reactor. Neurocomputing 153:159–163 12. Li DJ (2015) Adaptive neural network control for a two continuously stirred tank reactor with output constraints. Neurocomputing 167:451–458 13. Bechlioulis CP, Rovithakis GA (2008) Robust adaptive control of feedback linearizable MIMO nonlinear systems with prescribed performance. IEEE Trans Autom Control 53(9):2090–2099 14. Bechlioulis CP, Rovithakis GA(2009) Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems. Automatica, 45(2): 532–538 15. Li D (2014) Adaptive neural network control for a class of continuous stirred tank reactor systems. Sci Chin Inf Sci 57(10):1–8 16. Li S, Gong M, Liu Y (2016) Neural network-based adaptive control for a class of chemical reactor systems with non-symmetric dead-zone. Neurocomputing 174:597–604 17. Li DJ, Tang L (2014) Adaptive control for a class of chemical reactor systems in discrete-time form. Neural Comput Appl 24(7–8):1807–1814 18. Zerari N, Chemachema, M, Essounbouli N (2018) adaptive neural-network output feedback control design for uncertain cstr system with input saturation. In 2018 (CISTEM), pp 1–6. IEEE 19. Henson MA, Seborg DE (1990) Input-output linearization of general nonlinear processes. AIChE J 36(11):1753–1757 20. Polycarpou MM, Ioannou PA (1993) A robust adaptive nonlinear control design. In: 1993 American control conference, pp 1365–1369. IEEE 21. Park J, Sandberg IW (1991) Universal approximation using radial-basis-function networks. Neural Comput 3(2):246–257 22. White DA, Sofge DA (eds) (1992) Handbook of intelligent control: neural, fuzzy, and adaptative approaches. Van Nostrand Reinhold Company

Adaptive Fuzzy Fault-Tolerant Control Using Nussbaum Gain for a Class of SISO Nonlinear Systems with Unknown Directions Abdelhamid Bounemeur, Mohamed chemachema, Abdelmalek Zahaf, and Sofiane Bououden

Abstract This paper treatsan indirect adaptive fuzzy fault-tolerant control using fuzzy systems for a class of uncertain SISO systems with unknown control gain sign and actuator faults. The uncertain nonlinearities of the systems and the actuator faults are approximated by fuzzy systems that have been proven to be universal approximators and the Nussbaum-type function is used to deal with the unknown control gain sign. The proposed control scheme completely overcomes the singularity problem that occurs in the indirect adaptive feedback linearizing control. Projection in the estimate parameters is not required and the stability analysis of the closed-loop system is performed using Lyapunov approach. Simulation results are provided to verify the effectiveness of the proposed design. Keywords Adaptive fuzzy control · Nonlinear systems · Lyapunov stability

1 Introduction Universal approximators have been successfully applied to adaptive control problems such as fuzzy systems [1, 2] and neural networks [3–5] because they need no accurate mathematical models of the system under control. Fuzzy systems have been proven to be able to approximate continuous nonlinear functions [6]. Based on this property A. Bounemeur (B) · M. chemachema · A. Zahaf Laboratory of Automatic, Robotic and Control Systems, Faculty of Engineering, Department of Electronics, University of Constantine 1, Constantine, Algeria e-mail: [email protected] M. chemachema e-mail: [email protected] A. Zahaf e-mail: [email protected] S. Bououden Faculty of Sciences and Technology, University Abbes Laghrour, Khenchela, Algeria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_34

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many researchers have presented adaptive control architecture for uncertain nonlinear systems [2, 7–9]. Conceptually, there are two distinct approaches that have been formulated in the design of a fuzzy adaptive control system: direct and indirect schemes. In the direct approach which consists to approximate the ideal control law by a fuzzy system [7, 8]. However, in the indirect approach the nonlinear dynamics of the system are approximated by fuzzy systems to develop a control law based on these systems [7, 9, 10]. However, all the previous results need the assumption that the input gain, which is the function of the system states in general, is away from zero and its sign is known a priori. The sign, called control direction, represents motion direction of the system under any control, and knowledge of this sign makes adaptive control design much easier. In the adaptive control literature, the unknown control direction is mainly solved by using the Nussbaum-type function [11–14], and it has been successfully applied for robust control design [12, 13], and adaptive control design for nonlinear systems [15–18]. An alternative method, the so-called correction vector approach [19], has been applied to first-order nonlinear systems [20]. Authors in [21], have investigated an indirect adaptive fuzzy fault-tolerant scheme for a class of nonlinear systems with both actuator and sensor faults. A combination of fuzzy systems and backstepping technique allowed the online estimation of all adaptive parameters and ensured the boundedness of all signals involved in the closed-loop system. Inspired by [20, 21], we propose in this paper an indirect adaptive fuzzy controller for a class of SISO nonlinear systems to achieve tracking of a desired output. New learning algorithms are proposed in the presented controller which permits superior control performance compared to the same class of controllers. In the proposed controller, a robustifying control term is added to the basic fuzzy controller to deal with approximation errors. The Nussbaum gain function is introduced to solve the problem of unknown control gain sign and the regularized inverse is employed to solve problem of singularity. The stability of the closed-loop system is studied using Lyapunov method. The outline of the paper is as follows. Section 2 presents a brief description of the fuzzy system. In Sect. 3, a new control law and adaptive algorithms are proposed and stability analysis is given. Simulation examples are illustrated in Sect. 4. The conclusion is finally given in Sect. 5.

2 Description of Fuzzy Systems It is shown that fuzzy systems are capable of approximating any real continuous function on a compact set with arbitrary precision given by [6]. Sugeno and the employee have proposed a class of fuzzy system that allows to represent knowledge that is expressed in analytical form, describing the internal structure of system. This class of fuzzy system is called Fuzzy systems Takagi-Sugeno (TS).

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Note by x = [x1 , . . . , xn ]T is the input of fuzzy system and its output. For each j xi is associated m i fuzzy sets Fi in X i , as for xi ∈ X i there is at least one degree of membership μ F j (xi ) = 0 where i = 1, 2, . . . , n et j = 1, 2, . . . , m i . The basic i n m i fuzzy rules of the form: rules of the fuzzy system has N = i=1 k

k

Rk : i f x1 is F 1 and . . . and xn is F n T hen y = f k (x), k = 1, . . . , N   k where F i ∈ Fi1 , . . . , Fim i et f k (x) is a numerical function on the output space in general, f k (x) is a polynomial function depending on variable inputs, but it can also be an arbitrary function so that it can properly describe the behavior of the studied system if f k (x) is a function: f k (x) = a0k +

n 

aik xi

(1)

i=1

Then it’s the first order Tackagi-Sugeno (TS1). If against. f k (x) is a polynomial of zero order. f k (x) = a k

(2)

We have the Tackagi-Sugeno zero order (TS0). In this work we will consider a fuzzy zero order (TS0). Each rule has a numerical conclusion, the total output of the fuzzy system is obtained by calculating a weighted average, and in this manner the time consumed by the procedure of défuzzification is avoided. Then the output of fuzzy system is given by following relationship: N y(x) =

with μk (x) =

k=1 μk (x) f k (x) N k=1 μk (x)

(3)

 1 k mi  k, F . μ i ∈ Fi , . . . , Fi i=1 

n

Fi

Which represents the degree of confidence or activation rule Rk . We can simplify the output of fuzzy system as follows: N y(x) = k=1 N

μk (x)a k

k=1

μk (x)

(4)

By introducing the concept of fuzzy basis functions [1], the output of fuzzy system TS0 can be written as:

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y(x) = w T (x)θ

(5)

with

  • θ = a 1 . . . a N : Vector of parameters of the conclusion of rules fuzzy part. • w(x) = [w1 (x) . . . w N (x)]T , where μk (x) , k = 1, . . . , N w N (x) =  N j=1 μ j (x)

(6)

3 Controller Design and Stability Analysis 1. Problem Formulation Consider the class of nonlinear systems that can be described by the following deferential equations ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

x˙1 = x2 x˙2 = x3 .. .

⎪ ⎪ ⎪ ⎪ x˙ = f (x1 , x2 , . . . xn ) + g(x1 , x2 , . . . xn )u ⎪ ⎩ n y = x1

(7)

Or equivalently

y (n) = f (x) + g(x)u y = x1

(8)

where x = [x1 , . . . , xn ]T ∈ Rn , is the vector of the system; u ∈ R is the scalar control input; y ∈ R is the scalar system output; f (x) and g(x) are unknown smooth nonlinear functions. 2. Actuator Faults Model The following Table 1 describe the studied faults. where t f i denotes the time instant of failure of the  ith sensor/actuator and  bidenotes  its accuracy coefficient such that b −b¯0i , b¯0i , where b¯0i > 0. k k¯i , 1 , where in which bi and k are k¯i > 0 denotes the minimum sensorand actuator  effectiveness,   slowly varying respectively within −b¯0i , b¯0i and k¯i , 1 .

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Table 1 Actuator faults Actuators

Faults kinds

Conditions

Faults names

u(t)

u(t) + b

b(t) = 0,   b t f i = 0

bias (Lock in place)

u(t) + b(t)

dri f t

u(t) + b(t)

|b(t)| = λt, 0 0, γg > 0, η f > 0, ηg > 0, σ0 > 0 and δ(0) > 0. Theorem 1 Consider the system (9) with the sign of the control gain assumed unknown, we assume that the Assumptions 1–5 Are satisfied. The control law defined by (23) and (24) with adaptions law (30–37) ensures the following properties: • The tracking error and its derivatives converge to zero, e(i) (t) → 0 when for i = 0, 1, . . . n − 1. • The output of the system, and its derivatives up to the order (n −1) and the control signal are bounded: . Proof

e = yd − y

(38)

e(n) = yd(n) − y (n) = yd(n) − f (x) − f a (x) − g(x)u

(39)

e(n) = yd(n) − f (x) − f a (x) − g(x)u c − g(x)u r

(40)

using (21)

by adding and subtracting fˆ(x, θ ), fˆa (x, θ ) and g(x, ˆ θ )u c ,

(41)

becomes:   e(n) = yd(n) − f (x) − fˆ(x, θ )     − f a (x) − fˆa (x, θ ) − g(x) − g(x, ˆ θ ) uc − g(x, ˆ θ )u c − fˆ(x, θ ) − fˆa (x, θ ) − g(x)u r

(42)

using (17) and (18), (38) can be simplified to: θ f − ε f (x) − w Tf a (x) θ f a − ε f a (x) e(n) = yd(n) − w Tf (x) T  − wg (x)θg u c − εg (x)u c − g(x, ˆ θ )u c − fˆ(x, θ ) − fˆa (x, θ ) − g(x)u r

(43)

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using the adaptive control law (22), (39) can be simplified to: (n)

e(n) = yd

− w Tf (x) θ f − ε f (x) − w Tf a (x) θ f a − ε f a (x) − wgT (x) θg u c − εg (x)u c    g(x, ˆ θ) ˆ(x, θ ) − fˆa (x, θ ) + y (n) + k T e + α g(x, − f − g(x, ˆ θ) ˆ θ)N (τ )G d ε0 + gˆ 2 (x, θ ) ˆ (44) − f (x, θ) − fˆa (x, θ ) − g(x)u r

by adding and subtracting k T e and α g(x, ˆ θ )N (τ )G then θ f − ε f (x) − w Tf a (x) θ f a − ε f a (x) e(n) = yd(n) − w Tf (x) − wgT (x) θg u c − εg (x)u c +

gˆ 2 (x, θ ) fˆ(x, θ ) gˆ 2 (x, θ ) fˆa (x, θ ) + ε0 + gˆ 2 (x, θ ) ε0 + gˆ 2 (x, θ )

gˆ 2 (x, θ )yd(n) gˆ 2 (x, θ )k T e − − fˆ(x, θ ) − fˆa (x, θ ) − fˆa (x, θ ) ε0 + gˆ 2 (x, θ ) ε0 + gˆ 2 (x, θ ) ˆ θ )N (τ )G − α g(x, ˆ θ )N (τ )G (45) − g(x)u r + k T e − k T e + α g(x, −

using (27), one can rewrite (42) as θ f − ε f (x) − w Tf a (x) θ f a − ε f a (x) − wgT (x) θg u c e(n) = −w Tf (x) − εg (x)u c − g(x)u r − k T e + u¯ − α g(x, ˆ θ )N (τ )G

(46)

by adding and subtracting αg(x)N (τ )G θ f − ε f (x) − w Tf a (x) θ f a − ε f a (x) − wgT (x) θg (u c − α N (τ )G) e(n) = −w Tf (x) − εg (x)(u c − α N (τ )G) − g(x)u r − k T e + u¯ − αg(x)N (τ )G

(47)

Then the dynamics of the error can be written as follows:  e(t) ˙ = Ae(t) + B −w Tf (x) θ f − ε f (x) − w Tf a (x) θ f a − ε f a (x) − wgT (x) θg (u c − α N (τ )G)  − εg (x)(u c − α N (τ )G) − g(x)u r + u¯ − αg(x)N (τ )G (48)

⎡

0 0 .. .

1 0 .. .

⎢ ⎢ ⎢ A=⎢ ⎢ ⎣ 0 0 −kn −kn−1

0 0 ... 1 0 ... .. .. . . . . . 0 0 ... ··· ··· ...

⎤

⎡ ⎤ 0 ⎥ ⎢ .. ⎥ ⎥ ⎢ ⎥ ⎥ ⎥B = ⎢ . ⎥ ⎥ ⎣0⎦ 0 1 ⎦ 1 · · · −k1 0 0 .. .

0 0 .. .

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Until (|s I − A|) = s (n) + k1 s (n−1) + · · · + kn is stable (A stable), we know that there exists a symmetric positive definite matrix P(n, n) that satisfies the Lyapunov equation: A T P + P A = −Q

(49)

where Q is a symmetric positive definite matrix of arbitrary dimensions (n × n). Whether V , is the Lyapunov function, then 1 T 1 T 1 T 1 2 1 2 1 T    e Pe + θf  θf + θ f a θfa + θg  θg +  εf +  ε 2 2γ f 2γ f 2γg 2η f 2η f f a 1 2 1 2 +  ε + δ (50) 2ηg g 2σ0

V =

where ⎧ εf = εf − εf ⎨   εfa = εfa − εfa ⎩  εg = ε g − ε f 



(51)



The time derivative of V is given by 1 V˙ = e˙ T Pe + 2 1 ˙ −  ε f εˆ f ηf

1 T 1 T˙ 1 T ˙ 1 T˙  θ θ˜ f + θ θ˜ f a +  θ θ˜g e P e˙ +  2 γf f 2γ f f a γg g 1 1 ˙ 1 ˙ −  ε f a εˆ˙ f a −  εg εˆ g + δδ ηf ηg σ0

(52)

Substituting (41), and using the equalities θ˙˜ f = −θ˙ f , θ˙˜ f a = −θ˙ f a and θ˙˜g = −θ˙g it follows that   1  θ f − ε f (x) − w Tf a (x) θfa V˙ = e T A T P + P A e + e T P B −w Tf (x) 2 − ε f a (x) − wgT (x) θg (u c − α N (τ )G) − εg (x)(u c − α N (τ )G) − g(x)u r + u¯ 1 T 1 T 1 T 1 ˙ θ f θ˙ f −  θ f a θ˙ f a −  θg θ˙g −  − αg(x)N (τ )G] −  ε f εˆ f γf γf γg ηf 1 1 ˙ 1 ˙ (53) −  ε f a ε˙ˆ f a −  εg εˆ g + δδ ηf ηg σ0 From (42) one can obtain

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 1 V˙ = − e T Qe + e T P B −w Tf (x) θ f − ε f (x) − w Tf a (x) θ f a − ε f a (x) − wgT (x) θg (u c − α N (τ )G) 2  1 T 1 T   θ θ˙ f − θ θ˙ f a − εg (x)(u c − α N (τ )G) − g(x)u r + u¯ − αg(x)N (τ )G − γf f γf fa −

1 T 1 1 1 ˙ 1 ˙  θ θ˙g −  ε f ε˙ˆ f −  ε f a ε˙ˆ f a −  εg εˆ g + δδ γg g ηf ηf ηg σ0

(54)

1 V˙ = − e T Qe − αe T P Bg(x)N (τ )G + V˙1 + V˙2 2

(55)

where 

1 · θ f +w f (x)e T P B γf  1 · T  θ f a +w f a (x)e T P B − θfa γf  1 ·  θg +e T P Bwg (x)(u c − α N (τ )G) θgT γg

V˙1 = − θ Tf

(56)

Using (26) and (27), (49) can be simplified to V˙1 = 0

(57)

V˙2 = −e T P Bg(x)u r − e T P Bε f (x) − e T P Bε f a (x) − e T P Bεg (x)(u c − α N (τ )G) + e T P B u¯ 1 ˙ 1 1 ˙ 1 ˙ −  ε f εˆ f −  ε f a εˆ˙ f a −  εg εˆ g + δδ ηf ηf ηg σ0

(58)

using Assumption 5, V˙2 can be upper bounded by:   V˙2 ≤ −e T P Bg(x)u r + e T P B u¯ + e T P B ε f      1  + e T P B ε f a + e T P Bu c (u c − α N (τ )G)ε g − ε f − ε f ε˙ˆ f ηf    1  1 1 ˙ − ε f a − ε f a ε˙ˆ f a − ε g − ε g ε˙ˆ g + δδ ηf ηg σ0 





(59)

with (28), (29) and (31), one can rewrite V˙2     V˙2 ≤ −e T P Bg(x)u r + e T P B u¯ + ε f e T P B  + ε f a e T P B    + ε g e T P B(u c − α N (τ )G) − δ 2 





(60)

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using (24), (53) can be simplified to   V˙2 ≤ −e T P Bg(x)u r + e T P B ϕ − δ 2

(61)

by adding and subtracting e T P Bu r b   V˙2 ≤ −e T P Bg(x)u r + e T P B ϕ − δ 2 + e T P Bu r b − e T P Bu r b

(62)

using (22) one obtains V˙2 ≤ −e T P Bg(x)u r + e T P Bu r b

(63)

Substituting the results given by (50) and (56), V˙ can be bounded by 1 V˙ ≤ − e T Qe − αe T P Bg(x)N (τ )G − e T P Bg(x)u r + e T P Bu r b 2

(64)

from (21), one can obtain 1 V˙ ≤ − e T Qe − αe T P Bg(x)N (τ )G − e T P Bg(x)u r b N (τ ) + e T P Bu r b 2

(65)

Thus, V˙ can be rewritten as    2 1 V˙ ≤ − e T Qe − g(x)N (τ ) α e T P B  + e T P Bu r b + e T P Bu r b 2

(66)

 2 by adding and subtracting αe T P B     2 1 V˙ ≤ − e T Qe − g(x)N (τ ) α e T P B  + e T P Bu r b 2  2  2 + e T P Bu r b + α e T P B  − α e T P B 

(67)

Substituting (30)  2 1 V˙ ≤ − e T Qe − α e T P B  − g(x)N (τ )(τ˙ ) + τ˙ 2

(68)

where α > 0. We can obtain the following inequality  V (t) − V (0) ≤ o

t

−(g(v)N (v) + 1)τ˙ (v)dv

(69)

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Equation (63) can be simplified to  V (t) ≤ V (0) +

t

−(g(v)N (v) + 1)τ˙ (v)dv

(70)

o

 t Using Lemma 1, we can conclude  (64) the boundedness of V (t), N (t), and  from . −(g(v)N (v) + 1) τ ˙ (v)dv for t 0, t f o According to (Liu06 and Liu08), since no finite-time escape phenomenon may happen, then t f → ∞. θg (t), ε f (t), ε g (t), δ(t), x(t) and u(t) are bounded. As Therefore,  θ f (t), e(t),  an intermediate result e(t) is square integrable and e(t) ˙ is bounded. Moreover, by invoking Barbalat’s lemma, we can conclude the asymptotic convergence of e(t). 



4 Simulation Results In this section, to demonstrate the effectiveness of the indirect adaptive fuzzy control algorithms, we consider the tracking control of the second-order system. The dynamic equations of such system are given by [22–24]. ⎧ ⎨

x˙1 = x2 x˙2 = f (x) + f a (x, u) + g(x)u ⎩ y = x1

(71)

  f (x) = 1.5 1 − x12 x2 − x1

(72)

g(x) = 1 + x12 + x22 + x1 x2

(73)

f a (x) = 1.5 ∗ r ect(1 − u ∗ t) − t + u

(74)

where

The control objective is to force the system output y(t) = x1 (t) to track the desired trajectory yd (t) = sin(t). The actuator faults are applied from the t ≥ 0 until the end of the simulation. Within this simulation, two fuzzy systems in the form of (5) are used to approximate the unknown smooth functions f (x) and g(x). The input variables of the used fuzzy systems are x1 and x1 , and we added the control input u as an input variable of the used fuzzy systems for approximating f a (x). For each input variable we have define five Gaussian membership functions with center −1, −0.5, 0, 0.5 and 1, and a

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variance equal to 4. The initial conditions are x(0) = [0.5, 0]T , and the initial values of the adjustable parameters θ f (0) and θg (0) are set equal to zero. p = [13 0.25; 0.25 0.52], Q = diag(10, 10), k = [20; 10], ε f (0) = 0, εg (0) = 0, τ (0) = 0, δ(0) = 1, σ0 = 5, ε0 = 0.1, γ f = γg = 20, η f = ηg = 0.001. Simulation results for the case of an unknown positive control direction (sgn(g(x)) = 1) are shown in Figs. 1 and 2. To show the controller behavior in the case of a negative control direction, we consider the case  g(x) = − 1 + x12 + x22 + x1 x2 . Using the same simulation design parameters, simulation results are shown in Figs. 3 and 4. We can see that actual trajectories converge to the desired ones and that the control input signal is bounded. 1.5 X1

X2

1 0.5 0 -0.5 -1 -1.5

0

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4

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8

10 12 Time (s)

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Fig. 1 System’s response (g(x) > 0): actual (blue lines); desired (red lines) 9 8 7 6 5 4 3 2 1 0 -1

0

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6

Fig. 2 Control input signal (g(x) > 0)

8

12 10 Time (s)

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X1

1 0.5 0 -0.5 -1 -1.5

0

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10 12 Time (s)

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Fig. 3 System’s response (g(x) < 0): actual (blue lines); desired (red lines) 2 0 -2 -4 -6 -8 -10

0

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Fig. 4 Control input signal (g(x) < 0)

5 Conclusion In this paper, an indirect adaptive fuzzy control for class of unknown (SISO) nonlinear systems with unknown control sign is developed. The scheme consists of an adaptive fuzzy controller with a robuste control term used to compensate for approximation errors. New adaptive parameters update law are used, besides the Nussbaumtype function, is used to overcome the problem of unknown control gain sign. The proposed adaptive schemes allow initialization to zero of all adjustable parameters of the fuzzy systems, guarantee the boudedness of all signals in the closed-loop, and the tracking error has been shown to be asymptotically stable. Simulation results have verified the effectiveness of the proposed adaptive control approach.

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References 1. WangL X (1994) Adaptive fuzzy systems and control. Prentice-Hall, EnglewoodCliffs, NewJersey 2. Passino KM, Yurkovich S (1998) Fuzzy control. Addison-Wesley Longman Inc. 3. Chemachema M, Belarbi K (2011) Direct adaptive neural network controller for a class of nonlinear systems based on fuzzy estimator of the control error. Int J Syst Sci 42(7):1165–1173 4. Chemachema M (2012) Output feedback direct adaptive neural network control for uncertain SISO nonlinear systems using a fuzzy estimator of the control error. Neural Netw 36:25–34 5. Ge SS, Hong F, Lee TS (2004) Adaptive neural control of nonlinear time-delay systems with unknown virtual control coefficients. IEEE Trans Syst Man Cybern Part B 34(1):499–516 6. Wang LX, Mendel JM (1992) Fuzzy basis functions, universal approximation and orthogonal least-squares learning. IEEE Trans Neural Netw 3(5):807–814 7. Essounbouli N, Hamzaoui A (2006) Direct and indirect robust adaptive fuzzy controller for a class of nonlinear systems. Int J Control Autom Syst 4(2):146–154 8. Labiod S, Guerra TM (2007) Directe adaptive fuzzy control for a class of MIMO nonlinear systems. Int J Syst Sci 38(8):665–675 9. Boukezoula R, Galichet S, Foulloy L (1998) Apprentissage de lois de commande floues pour des systèmes non linéaires (synthèse directe et indirecte). Actesdes rencontresfrancophones sur lalogiquefloueetsesapplications, Rennes, France, pp 19–27 10. Labiod S, Boucherit MS (2006) Indirect fuzzy adaptive control of a class of SISO nonlinear systems. Arab J Sci Eng 31(1B):61–74 11. Nussbaum RD (1983) Some remarks on the conjecture in parameter adaptive control. Syst Control Lett 3:243–246 12. Liu L, Huang J (2006) Global robust stabilization of cascade-connected systems with dynamic uncertainties without knowing the control direction. IEEE Trans Autom Control 51:1693–1699 13. Liu L, Huang J (2008) Global robust output regulation of lower triangular systems with unknown control direction. Automatica 44:1278–1284 14. Ye X, Jiang J (1998) Adaptive nonlinear design without a priori knowledge of control direction. IEEE Trans Autom Control 43:1617–1621 15. Zhang TP, Ge SS (2007) Adaptive neural control of MIMO nonlinear state time-varying delay systems with unknown dead-zones and gain signs. Automatica 43:1021–1033 16. Liu YJ, Wang ZF (2009) Adaptive fuzzy controller design of nonlinear systems with unknown gain sign. Nonlinear Dyn 58:678–695 17. Boulkroune A, Tadjine M, M’Saad M, Farza M (2010) Fuzzy adaptive controller for MIMO nonlinear systems with known and unknown control direction. Fuzzy Sets Syst 161:797–820 18. Li X, Wang D, Yao Y, Lan W, Peng Z, Sun G (2010) Adaptive fuzzy control for a class of nonlinear time-delay system with unknown control direction. In: Proceedings of the 29th Chinese control conference, pp 2593–2597 19. Lozano R, Collado J, Mondie S (1990) Model reference adaptive control without a priori knowledge of the high frequency gain. IEEE Trans Autom Control 35:71–78 20. Labiod S, Bouberyakh H, Guerra TM (2011) Indirect adaptive fuzzy control for a class of uncertain nonlinear systems with unknown control direction. Int J Fuzzy Syst Appl 1(4):1–17 21. Bounemeur H, Chemachema M, Essounbouli N (2018) Indirect adaptive fuzzy fault-tolerant tracking control for MIMO nonlinear systems with actuator and sensor failures. ISA Trans. https://doi.org/10.1016/j.isatra.2018.04.014 22. Abdelhamid B, Mouhamed C, Najib E (2017) Optimal indirect robust adaptive fuzzy control using PSO for MIMO nonlinear systems. In: International conference on electrical engineering and control applications. Springer, Cham, pp 208–224 23. Bounemeur A, Chemachema M, Essounbouli N (2014) Robust indirect adaptive fuzzy control using Nussbaum gain for a class of SISO nonlinear systems with unknown directions. In: 2014 15th international conference on sciences and techniques of automatic control and computer engineering (STA). IEEE, pp. 748–754

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Renewable Energy (RE)

Optimal Integration of Renewable Distributed Generation Using the Whale Optimization Algorithm for Techno-Economic Analysis Samir Settoul, Rachid Chenni, Mohamed Zellagui, and Hassan Nouri

Abstract A new method for optimal integration of renewable energy sources based on photovoltaic solar panels and wind turbines in the distribution network is presented with the objective of reducing the Active Power Loss (APL) index, to improve the Total Voltage Variation (TVV) index and the Total Operating Cost (TOC) index. The objectives are achieved by optimal integration of Distributed Generation (DG) based solar photovoltaic (PV) and wind turbine (WT) renewable sources using a novel optimization algorithm, namely the Whale Optimization Algorithm (WOA) for determining the optimal DG location and sizing subject to the constraints such as power conservation, distribution line constraints, DG capacity limits, and DG penetration limit. The proposed algorithm is evaluated on standard IEEE 12, 33, and 69 bus distribution networks. A numerical simulation including comparative studies is presented to demonstrate the performance and applicability of the Whale Optimization Algorithm (WOA). The validity of the proposed WOA technique is demonstrated by comparing the obtained results with those reported in literature using other optimization algorithms. Keywords Renewable energy · Optimal integration · Power distribution systems · Active power loss index · Voltage variation index · Total operating cost index · Whale optimization algorithm

S. Settoul (B) · R. Chenni Department of Electrotechnic, Mentouri University of Constantine 1, Constantine, Algeria e-mail: [email protected] M. Zellagui Department of Electrical Engineering, École de Technologie Supérieure, Montréal, Canada e-mail: [email protected] H. Nouri Department of Engineering Design and Mathematics, University of West of England, Bristol, UK e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_35

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1 Introduction The rapid development of Distributed Generation (DG) in different forms and capacities is transforming the conventional planning of power distribution networks. Despite the benefits offered by renewable DG technologies, several economic and technical challenges can result from the inappropriate integration of DG in existing power distribution networks. Therefore, the optimal planning of DG is of paramount importance to ensure that the performance of the distribution network can meet the expected power quality, voltage stability, power loss reduction, reliability and profitability [1]. Decentralized power generation from renewable energy sources (RES) is a long-term solution that addresses the present environmental threats because of its widespread availability, sustainability, nonpolluting generation and eco-friendliness. The most widely used renewable distributed generation types are WT and PV systems, of which their generation is intermittent. The rapid development of DG technologies in various forms and capacities is significantly reshaping the conventional planning of power distribution networks [2]. DGs are a reliable and economic solution for supplying electricity to customers and are normally directly connected to the distribution network or on the customer site. It is therefore necessary to allocate DGs optimally (size, placement and type) to obtain commercial, technical, environmental and regulatory advantages of power systems [3]. There are numerous algorithms to address the optimal DG planning problem. Metaheuristic algorithms are often used as they offer more flexibility, particularly for objective function planning problems without the pursuit of a globally optimized solution. For example, the Particle Swarm Optimization (PSO) algorithm for optimal placement of WT and PV sources based DGs for power loss minimization and voltage stability improvement [4], using PSO and nonlinear programming for the optimal multiples DG planning on the power loss sensitivity [5]. Applied Bat Algorithm (BA) for optimal DG based planning to maximize the Voltage Stability Index (VSI) and also to minimize the total active power losses [6]. Analytical approach based Sensitivity Analysis Technique (SAT) for optimal allocation of DG and shunt capacitor [7]. Applied Backtracking Search Optimization Algorithm (BSOA) for Optimal WT and PV sources based DGs for power loss minimization [8]. Flower Pollination Algorithm (FPA) to determine the optimal DG-unit’s size and location in order to minimize the total system real power loss and improve the bus voltage [9]. Krill Herd Algorithm (KHA) for the solution of optimal placement and sizing of DG in order to minimize the line losses considering various technical constraints [10]. Applied Ant Lion Optimization Algorithm (ALOA) for optimal integration of PV and WT sources and applied in various systems [11]. Optimal multiples DGs placement by multi-objective opposition based Chaotic Differential Evolution (CDE) algorithm for techno-economic analysis [12]. Improved Chaos Stochastic Fractal Search (CSFS) algorithm for optimal determination of location, size, and quantity of DGs in complex distribution systems [13]. A newly effective Stochastic Fractal Search Algorithm (SFSA) for optimal DGs integration into the distribution systems

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based on three objectives involving active power loss, voltage deviation, and VSI [14]. Simulated Annealing (SA) algorithm for a novel approach to identify optimal access point and sizing of multiple DGs in a small, medium and large scale radial distribution systems [15]. Loss Sensitivity Factor (LSF) that determines the most sensitive bus for DG installation and Bacterial Foraging Optimization Algorithm (BFOA) for minimization of power loss, voltage deviation and TOC of the distribution system [16]. Teaching-Learning Based Optimization (TLBO) and Grey Wolf Optimizer (GWO) algorithm for multi-objective placement of renewable DGs with objective to reduce losses and improve reliability [17]. Symbiotic Organism Search (SOS) algorithm for Optimal number, location, and size determination of DGs in order to minimize the power loss [18] and finally the Spider Monkey Optimization (SMO) algorithm for optimal allocation of renewable DG to the demand side management, using distribution system stability indicator [19]. This paper presents a new approach to optimal integration of renewable energy source based DG in electrical power systems using the WOA algorithm for technoeconomic analysis. In this study, optimal integration of renewable DG based solar photovoltaic and wind turbine source was installed in three different standard distribution systems using the Whale Optimization Algorithm (WOA). The optimal integration has been selected to minimize the total active power losses, total voltage variation and annual losses cost value using the proposed Whale Optimization Algorithm. The validity of the proposed WOA algorithm is demonstrated by comparing the obtained results with those reported in literature, which used other optimization techniques. Analysis of the obtained results shows that the proposed algorithm is able to achieve better minimization of the active power loss index and the total operating cost index when compared with existing optimization techniques.

2 Problem Formulation A. Objective function The proposed objective function (OF) is used to reduce the Active Power Loss (APL) index, to improve the Total Voltage Variation (TVV ) index and the Total Operating Cost (TOC) index. It presents a multi-objective function for the distribution system. The DG locations and their sizing are optimally obtained by solving the following objective function: O F = min

bus N bus N   i=1

α1 .A P L i j + α2 .T V Vi + α3 .AT OCi j

(1)

j=2

where, α 1 , α 2 and α 3 are weighting factors. The sum of the absolute values of the weights assigned to all impacts should add up to one as shown in the following

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equation: 3 

αm = |α1 | + |α2 | + |α3 | = 1

(2)

m=1

In this paper, α 1 is taken as 0.5 while α 2 and α 3 are taken as 0.25. The value of these factors is based on the practical, technical and economic indicators. The three indices are presented in Eqs. (4), (6) and (8) as described in the following: A f ter DG

PLoss

A P L =

(3)

Be f or e DG

PLoss

and, PLoss =

N bus N bus   i=1

 Ri j

j=2

Pi2j + Q i2j Vi2

 (4)

where, PLoss is the total active loss on the distribution system. Pij and Qij are active and reactive powers that flow through branches from bus i to bus j. Nbus is the number of buses, Rij is the resistance of the distribution line and V i is the voltage at bus i. T V V =

T V V A f ter DG T V VBe f or e DG

(5)

and, TVV =

N bus 

|1 − Vi |

(6)

i=2

The Total Operating Cost (TOC) is another DGs installation merits in the distribution networks [20]. The operational cost is composed of two components. The first component is related to the active power supplied from the substation, which can be reduced by minimizing the total real power loss of the system. The second is the cost of active power supplied by those DGs units that are integrated into the network. This cost could be reduced by minimizing the amount of real power drawn from the DGs. The Total Operating Cost (TOC) index is calculated by: T OC =

T OC A f ter DG T OC Be f or e DG

(7)

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The TOC before and after the integration of the DG is calculated by the following expressions: Be f or e DG

T OC Be f or e DG = K 2 × PLoss T OC A f ter

DG

 A f ter = K 1 × PLoss

DG



  Be f or e DG + K 2 × PLoss

(8) (9)

where, K 1 is the cost coefficient of real power supplied by the substation and DG and K 2 represents the maintenance and installation costs of DGs units. K 1 and K 2 are taken to be 4 and 5 $/kW [20]. B. Power conservation constraint Active and reactive power injected into each bus should equate the total load bus power. The power flow equations are defined as equality constraints in the optimal DG problem. The mathematical representation is given by [12–18]: PG + PDG = PD + PLoss

(10)

Q G + Q DG = Q D + Q Loss

(11)

where PG , and PDG , PD and PLoss are active power of generator, DG, load and total active losses respectively. Similarly, QG , QDG , QD and QLoss are respectively the reactive power of the generator, DG, load and total losses respectively. Bus voltage limits: The magnitude of voltage at each bus must be limited by the following equation: Vmin ≤ |Vi | ≤ Vmax

(12)

where, V min and V max are the minimum and maximum specified voltage at each bus, respectively. Voltage drop limit: The limits of the voltage drop (V ) can be considered as the flowing equation: |V1 − Vi | ≤ Vmax

(13)

where, V 1 is the voltage at the generating station is equal to 1.0 p.u. and V max is the maximum voltage drop at each branch. Line capacity constraint: Power flow through any distribution feeder must comply with the thermal capacity of the line as given by the following equation.    Si j  ≤ |Smax |

(14)

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where, S ij represents the apparent power in the distribution line between bus i and S max denotes its corresponding maximum power. C. DG constraints DG capacity limits: The used DG units must have the allowable size in their following range: min ≤ P ≤ P max PDG DG DG

(15)

max Q min DG ≤ Q DG ≤ Q DG

(16)

where, Pmin and Pmax are the minimum and maximum active power output limits of the DG respectively; Qmin and Qmax are the minimum and maximum reactive power output limits of the DG respectively. In the case of PV, the DG source only delivers active power. While, in the case of WT, the DG source delivers active and reactive power. Power Factor of DG: The PF DG injected by the DG unit at each bus must be limited between the minimum and maximum value as represented by the following equation: min ≤ P F ≤ P F max P FDG DG DG

(17)

where, P FDG = 

PDG 2 PDG + Q 2DG

(18)

Position of DG: The DG should not be positioned at bus 1 as it represents the substation or slack bus. The position of DG is varied between the second bus and all the buses in the distribution power system: 2 ≤ DG Position ≤ Nbus

(19)

Number of DG: The number of DGs installed in the distribution power system is limited by the maximum DGs (N DG.max ) as governed by the following equation: N DG ≤ N DG.max

(20)

Location of DG: This constraint presented one DG installed in the distribution power system at one location or bus: n DG,i /Location ≤ 1

(21)

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Table 1 Limits values of constraints Parameters

Cases studies

Bus number PDG (kW) QDG (kVar) V i (p.u.)

69-bus

12

33

69

10

10

10

Max

3000

3000

3000

Min

10

10

10

Max

2000

2000

2000

Min

0.95

0.95

0.95

Max

1.05

1.05

1.05

5

5

5

0.80

0.80

0.80

Min Max

Number of DGs

33-bus

Min

V max (%) PF DG

12-bus

1.00

1.00

1.00

1

2

3

The limits of constraint values after installation of DG based photovoltaic source (PV-DG) and DG based wind turbine source (WT-DG) is shown in Table 1.

3 Whale Optimization Algorithm Mirjalili developed the WOA approach in 2016 [21] as a novel nature-inspired heuristic technique to solve problems related to engineering and different mathematical optimization issues. The common behaviors of humpback whales are the basis of the WOA. This optimization technique is inspired by the bubble net hunting approach of humpback whales as they follow a circular shaped route for hunting small fish near the surface. This feeding process is a distinctive behavior of humpback whales, making this optimization unique among other nature-inspired optimization methods. To design the mathematical model of the WOA, three steps are involved in the bubble-net hunting process. All steps are described in detail as follows [21, 22]. A. Encircling Prey Humpback whales can recognize the location of prey and en-circle them. Since the position of the optimal design in the search agent is not known in advance, hence, the WOA algorithm initially assumes that the best position of the target prey is a place close to the position of the optimal design. Thereafter, the other search agents will try to update the position towards the best location. This procedure mathematically is represented by the following equations: − →  →− → − → − D =  C . X ∗ (t) − X (t)

(22)

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− → − → − →− → X (t + 1) = X ∗ (t) − A . D

(23)

where, D is the distance of ith whale to the prey, t indicates the current iteration, A and C are the coefficient vector. X * is the position vector of the best solution obtained so far, X is the position vector. It is worth mentioning here that X * should be updated in each iteration if there is a better solution. In addition the vectors A and C are calculated from equations: − → A = 2. a . r − a

(24)

− → C = 2. r

(25)

where, a is linearly decreased from 2 to 0 over the course of iterations (in both exploration and exploitation phases) and r is a random vector generated with uniform distribution in the interval of 0 and 1. According to Eq. (23) the search agents (whales) update their position according to the position of the best known solution (prey). B. Bubble-Net Attacking Method To model this strategy, the two techniques employed by humpback whales are presented as follows: (1) Shrinking Encircling Mechanism This behavior is achieved by decreasing the value of vector a in Eq. (24). Note that the fluctuation range of A is also decreased by a. In other words A is a random value in the interval [−a, a] where a is decreased from 2 to 0 over the course of iterations. Setting random values for A in [−1, 1], the new position of a search agent can be defined anywhere in between the original position of the agent and the position of the current best agent. Figure 1 shows the possible positions from (X, Y ) towards (X * , Y * ) that can be achieved by 0 ≤ A ≤ 1 in a 2D space. (2) Spiral Updating of Position In this process, the separation distance between the whale positioned at (X, Y ) and prey positioned at (X * , Y * ) is determined. Then, for mimicking the spiral-shaped movement of humpback whales, a helix condition is composed amongst the whale location and prey location as: − → − → − → X (t + 1) = D  .ebt . cos(2πl) + X ∗ (t)

(26)

where, D determines the location or distance of whales i to the prey, b is a constant to represent the state of the logarithmic helix, and l is an arbitrary number in the range −1, and 1. In view of the concurrent swimming of humpback whales around

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the prey in a contracting loop and following a helix formed path, the equal likelihood of choosing either the shrinking surrounding technique or the helix strategy can be summarized as: − →∗ − →− → − → X (t) − A . D i f p 1. The mathematical model can be expressed as: →− → − → − − → D =  C . X rand − X 

(28)

− → − → − →− → X (t + 1) = X rand − A . D

(29)

Fig. 1 Exploration mechanism implemented in WOA [21]

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where, X rand is an arbitrary position vector (a random whale) which is chosen from the existing population. D. The Steps of the WOA The principal procedures of the WOA approach are as follows. The WOA starts with an established set of randomly created results. The locations of pursuit agents, considering an arbitrary determination of the hunt agent or the given best result, are then updated. For exploration and exploitation, parameter a is reduced from 2 to 0. When |A| > 1, an arbitrary hunt agent is chosen, and when |A| < 1, the best result is chosen for updating the positions of the pursuit agents as depicted in Fig. 1. This enhancement strategy can modify the movement among spiral and circular movements depending on the measure of p. Finally, the WOA approach is terminated once the termination criterion is satisfied [22]. The pseudo code of the WOA technique is presented in Algorithm 1 shown below. Algorithm 1 Pseudo-code of whale optimization algorithm 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

Generate the initial population X i (i = 1, 2, …, np ) Evaluate the fitness for each candidate solutions in Xi X * is the best candidate solutions while (t < maximum number of iterations) for i = 1 to np (for each search agent) Update a, A, C, l and p if (p < 0.5) if (|A| < 1) Update the position of the search agent by Eq. (23) else if (|A| ≥ 1) Select a random search agent (X rand ) Update the position of the search agent by Eq. (29) end if else if (p ≥ 0.5) Update the position of the current agent by the Eq. (26) end if end for Check if any search agent goes beyond the search space Calculate the fitness of each search agent Update X * if there is a better solution t=t+1 end while Return X *

4 Results, Discussion and Comparison The proposed WOA algorithm is developed using MATLAB 2017.b and simulations are carried out on a PC that poses Intel i5, 2.7 GHz, and 8 GB RAM.

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The first test system used in this paper is a standard IEEE 12-bus radial distribution system as shown in Fig. 2a which is composed of 12 busses, 11 distribution lines and branches with an active power load of 435 kW and reactive power load of 405 kVar with nominal voltage of 11 kV.

Fig. 2 Single line diagram of distribution systems. a. IEEE 12-bus, b IEEE 33-bus, c IEEE 69-bus

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The second test system is a standard IEEE 33-bus radial distribution system and composed of 33 busses, 32 distribution lines branches with an active power load of 3715 kW and reactive power load of 2300 kVar with nominal voltage of 12.66 kV as represented in Fig. 2b. To demonstrate the applicability of the proposed WOA approach in large-scale distribution systems, a standard IEEE 69-bus radial distribution system, which is composed of 69 busses, 68 distribution lines and branches with active power load of 3791.89 kW and reactive power load of 2694.10 kVar with nominal voltage of 12.66 kV is considered as shown Fig. 2c. Fig. 3 Convergence curve of WOA for first case study. a PV-DG, b WT-DG

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The convergence optimization characteristics of the proposed WOA algorithms for the three considered IEEE distributions systems in the presence of PV-DG and WT-DG are presented in Fig. 3. The parameters of all distribution systems before DG integration is represented in Table 2, and optimization results after integration of PV-DG and WT-DG is represented in Table 3. From Table 3, for the first test system (IEEE 12-bus), the best location for installation of the PV-DG with size 236.60 kW and WT-DG with size 245.10 kW is bus number 9. It can be observed that the PV-DG is capable of minimizing the APL by 51.9987% whereas the WT-DG by 15.9708%. For the second test system (IEEE 33-bus), the identified sizes for PV-DGs are 850.3 and 1158.6 kW and for the WT-DG are 828.0 kW with power factor of 0.8944 (lagging) and 1243.3 kW with power factor of 0.8 (lagging). The DGs are connected at bus number 13 and 30. The results show that APL is minimized by 41.3132% for PV-DG and 13.9037% for WT-DG. Table 2 Parameters of distribution systems before DG   PLoss (kW) QLoss (kVar) V min (p.u.) Case study



TVV (p.u)

TOC (k$)

IEEE 12-bus

20.8311

9.9489

0.9405

0.4466

104.1555

IEEE 33-bus

210.9875

143.1284

0.9038

1.8047

1054.9375

IEEE 69-bus

224.9480

102.1406

0.9092

1.8704

1124.74

Table 3 Optimization results after integration of DG Parameters 

PV-DG

WT-DG

IEEE 12-bus PLoss (kW)

PV-DG

WT-DG

IEEE 33-bus

PV-DG

WT-DG

IEEE 69-bus

10.8319

3.3269

87.1657

29.3351

69.4215

4.5367

APL (%)

51.9987

15.9708

41.3132

13.9037

30.8611

2.0168

V min (p.u.)  TVV (p.u)

0.9809

0.9897

0.9685

0.9804

0.9790

0.9943

0.2945

0.0849

0.6775

0.1862

1.0277

0.1087

TVV (%)

65.943

19.010

37.541

10.318

54.946

5.812

TOC (k$)

147.48

117.46

1403.60

1172.28

1402.43

1142.89

TOC (%)

141.599

112.777

133.051

111.123

124.689

101.613

DG

Bus

9

9

13–30

13–30

11–21–61

12–21–61

PDG (kW)

236.60

245.10

850.3 1158.6

828.0 1243.3

576.1 339.3 1716.8

432.7 315.8 1662.2

PF DG

1.0000

0.8000

1.0000 1.0000

0.8944 0.8000

1.0000 1.0000 1.0000

0.8000 0.8000 0.8000

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For the third test system, the IEEE 69-bus is considered. The three PV-DGs with the sizes of 576.1, 339.3, and 1716.8 kW are placed at buses 11, 21, and 61 respectively, which minimizes the APL to 30.8611%. Similarly, the WT-DG with sizes of 432.7, 315.8, and 1662.2 kW are installed at buses 11, 21 and 61 respectively and lead to minimization of APL to 2.0168%. Figure 4 indicates the bus voltage profiles for different case studies performed for all distribution systems. Voltages of all the buses are within allowable limits. The best profile is justifiably obtained after PV-DG and WT-DG installation. This is because active power is higher than the reactive power and hence compensation of real power has greater influence in controlling losses and voltage in the distribution system. Figure 5 illustrates the impact of DG installation on active power losses for the three test systems. Analysis of Fig. 5 reveals that after installation of the DG, there is significant minimization on active power losses. For example, the WT-DG installation contributes to more minimization of the power losses than the installation of PV-DG. This is related to the contribution of the reactive power injected from WT-DG. Figure 6 shows the influence of DGs’ installation on voltage variation on all three-test systems. In general, the installation of the two types of DGs i.e. PV-DG and WT-DG will reduce the voltage variation in all the buses. However, from the

(a). IEEE 12-bus.

(b). IEEE 33-bus.

(c). IEEE 69-bus.

Fig. 4 Bus voltages profile of standard distribution system

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(a). IEEE 12-bus.

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(b). IEEE 33-bus.

(c). IEEE 69-bus.

Fig. 5 Active power loss of standard distribution system

voltage variation curve, one can suggest that the WT-DG provides more improvement than PV-DG in terms of system voltage. Table 4 presents the comparison results of various optimization algorithms against the proposed WOA algorithm in presence of DG based photovoltaic and wind turbine sources. Figures 7 and 8 show the bar chart form representation of TOC and APL for the three IEEE test scenarios in presence of PV-DG and WT-DG respectively. The results shown in Figs. 7 and 8 suggest that in all the three IEEE test scenarios, the proposed WOA technique performed well in maximizing the active power loss index and total operation cost index up to an optimal value. Furthermore, the overall objective function gain accomplished by the suggested the WOA exhibits the predominance and favorable circumstances of WOA over the algorithms specified in the references.

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(a). IEEE 12-bus.

(b). IEEE 33-bus.

(c). IEEE 69-bus.

Fig. 6 Voltage variation value of standard distribution system

5 Conclusions This paper has presented optimal integration of renewable energy source based distributed generations in electrical distribution networks using WOA approach for techno-economic analysis. In this paper, a multi-objective function has been introduced for optimal integration of solar photovoltaic source and wind turbine based DG units and tested for three different electrical power distribution networks to demonstrate the effectiveness of the proposed optimization algorithm. The optimal integration of PV-DG and WT-DG units has been selected to the active power loss index, the total voltage variation index and total operation cost index value while satisfying system operational constraints within a predefined tolerance to

Optimal Integration of Renewable Distributed Generation … Table 4 Results comparison of various algorithms  PLoss (kW) Test system Algorithms IEEE 12-bus

PV-DG

WT-DG

IEEE 33-bus

PV-DG

WT-DG

IEEE 69-bus

PV-DG

WT-DG

529

APL (%)

TVV (%)

TOC (%)

70.5676

65.7859

156.454

10.9200

65.5750

66.9727

152.460

13.6600

52.4216

68.8983

141.937

WOA

10.8319

51.9987

65.9427

141.599

PSO [4]

7.9000

37.9241

43.1930

130.339

CS

3.5120

16.8594

16.9727

113.488

SCA

3.3764

16.2085

14.6216

112.967

PSO [4]

14.7000

BA [6] SAT [7]

WOA

3.3269

15.9708

19.0103

112.777

BSOA [8]

89.3400

42.3437

41.0040

133.875

FPA [9]

89.2000

42.2774

35.5738

133.822

KHA [10]

87.4260

41.4366

37.4577

133.149

WOA

87.1657

41.3132

37.5409

133.051

PSO [5]

39.1000

18.5319

10.6721

114.826

BSOA [8]

31.9800

15.1573

32.6148

112.126

ALOA [11]

30.9251

14.6573

9.2037

111.726

WOA

29.3351

13.9037

10.3175

111.123

CDE [12]

69.4360

30.8676

54.9401

124.694

CSFS [13]

69.4300

30.8649

54.9401

124.692

SFSA [14]

69.4280

30.8640

54.9348

124.691

WOA

69.4215

30.8611

54.9455

124.689

SA [15]

16.2600

7.2283

15.0075

105.783

BFOA [16]

12.9000

5.7347

25.9356

104.588

GWO [17]

7.3100

3.2496

7.4476

102.600

WOA

4.5367

2.0168

5.8116

101.613

avoid constraints violation. Using the proposed WOA algorithm, the optimal size and location of renewable energy sources are identified which results in total power loss reduction and satisfaction of the permissible voltage limits. The simulation results show that the proposed WOA algorithm provides better results when compared with other existing optimization algorithms.

530 Fig. 7 Comparison of WOA with other algorithms in the presence PV-DG. a IEEE 12-bus, b IEEE 33-bus, c IEEE 69-bus

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Fig. 8 Comparison of WOA with other algorithms in the presence WT-DG. a IEEE 12-bus, b IEEE 33-bus, c IEEE 69-bus

References 1. Ehsan A, Yang Q (2018) Optimal integration and planning of renewable distributed generation in the power distribution networks: a review of analytical techniques. Appl Energy 210:44–59 2. Abdmouleh Z, Gastli A, Ben-Brahim L, Haouari M, Al-Emadi NA (2017) Review of optimization techniques applied for the integration of distributed generation from renewable energy sources. Renew Energy 113:266–280 3. Zubo RHA, Mokryani G, Rajamani HH, Aghaei J, Niknam T, Pillai P (2017) Operation and planning of distribution networks with integration of renewable distributed generators considering uncertainties: a review. Renew Sustain Energy Rev 72:1177–1198

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4. Kayal P, Chanda CK (2013) Placement of wind and solar based DGs in distribution system for power loss minimization and voltage stability improvement. Int J Electr Power Energy Syst 53:795–809 5. Kaur S, Kumbhar G, Sharma J (2014) A MINLP technique for optimal placement of multiple DG units in distribution systems. Int J Electr Power Energy Syst 63:609–617 6. Remha S, Chettih S, Arif S (2017) A novel multi-objective bat algorithm for optimal placement and sizing of distributed generation in radial distributed systems. Adv Electr Electron Eng 15(5):736–746 7. Naik SG, Khatod DK, Sharma MP (2013) Optimal allocation of combined DG and capacitor for real power loss minimization in distribution networks. Int J Electr Power Energy Syst 53:967–973 8. El-Fergany A (2015) Optimal allocation of multi-type distributed generators using Backtracking search optimization algorithm. Int J Electr Power Energy Syst 64:1197–1205 9. Oda ES, Abdelsalam AA, Abdel-Wahab MN, El-Saadawi MM (2017) Distributed generations planning using flower pollination algorithm for enhancing distribution system voltage stability. Ain Shams Eng J 8(4):593–603 10. Chithra Devi SA, Lakshminarasimman L, Balamurugan R (2017) Study krill herd algorithm for multiple DG placement and sizing in a radial distribution system. Eng Sci Technol Int J 20:748–759 11. Ali ES, Elazim SA, Abdelaziz AY (2017) Ant lion optimization algorithm for optimal location and sizing of renewable distributed generations. Renew Energy 101:1311–1324 12. Kumar S, Mandal KK, Chakraborty N (2019) Optimal DG placement by multi-objective opposition based chaotic differential evolution for techno-economic analysis. Appl Soft Comput J 78:70–83 13. Nguyen TP, Tran TT, Vo DN (2018) Improved stochastic fractal search algorithm with chaos for optimal determination of location, size, and quantity of distributed generators in distribution systems. Neural Comput Appl 30(11):3545–3564 14. Nguyen TP, Ngoc Vo D (2018) A novel stochastic fractal search algorithm for optimal allocation of distributed generators in radial distribution systems. Appl Soft Comput J 70:773–796 15. Injeti SK, Kumar NP (2013) A novel approach to identify optimal access point and capacity of multiple DGs in a small, medium and large scale radial distribution systems. Int J Electr Power Energy Syst 45(1):142–151 16. Mohamed Imran A, Kowsalya M (2014) Optimal size and siting of multiple distributed generators in distribution system using bacterial foraging optimization. Swarm Evolut Comput 15:58–65 17. Nowdeh SA, Davoodkhani IF, Moghaddam MJH, Seifi Najmi E, Abdelaziz AY, Ahmadi A, Razavi SE, Gandoman FH (2019) Fuzzy multi-objective placement of renewable energy sources in distribution system with objective of loss reduction and reliability improvement using a novel hybrid method. Appl Soft Comput J 77:761–779 18. Nguyen-Phuoc T, Vo-Ngoc D, Tran-The T (2017) Optimal number, location, and size of distributed generators in distribution systems by symbiotic organism search based method. Adv Electr Electron Eng 15(5):724–735 19. Deb G, Chakraborty K, Deb S (2019) Spider monkey optimization technique based allocation of distributed generation for demand side management. Int Trans Electr Energy Syst 29:1–17 20. Imran AM, Kowsalya M (2014) Optimal size and siting of multiple distribution generators in distribution system using bacterial foraging optimization. Swarm Evolut Comput 15:58–65 21. Mirjalili S, Andrew L (2016) The whale optimization algorithm. Adv Eng Softw 95:51–67 22. Elaziz MA, Mirjalili S (2019) A hyper-heuristic for improving the initial population of whale optimization algorithm. Knowl Based Syst 172:42–63

Enhancement of Power System Transient Stability with a Large Penetration of Solar Photovoltaic Using Facts Ikram Boucetta, Naimi Djemai, Salhi Ahmed, and Zellouma Laid

Abstract In this paper, a study on the transient stability of power system with a large amount of solar photovoltaic (PV) penetration was performed on the famous 30 bus IEEE test system, under the Power System Analysis Toolbox (PSAT) simulation software. Based on the critical clearing time (CCT) as a stability index. The simulation results showed a significant decrease in network stability performance after solar PV penetration, this degradation of the power system stability is due to the non-participation of photovoltaic sources in the regulation of the voltage plan as a consequence of the absence of reactive power, which leads us to have recourse to the FACTS systems. the simulation results after integration of the two famous types of FACTS (SVC, STATCOM) clearly showed the improvement of network stability with a slight superiority in favor of STATCOM. Keywords Renewable energy · Power system · Photovoltaic · PSAT · CCT · FACTS

1 Introduction The main objective of integrating renewable sources into power system is to contribute to the reduction of toxic gases emitted by fossil fuel power plants, the maintenance of fossil fuel reserves and the reduction of dependence on fossil resources I. Boucetta (B) · Z. Laid Echahid Hamma Lakhdar University, LEVRS, Eloued, Algeria e-mail: [email protected] Z. Laid e-mail: [email protected] N. Djemai · S. Ahmed Mohamed Khider University, LGEB, Biskra, Algeria e-mail: [email protected] S. Ahmed e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_36

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threatened with exhaustion in the short term. The influence of the integration of large quantities of these renewable sources on the state of the transmission grid is currently the subject of several scientific publications [1]. In this paper, a study is presented concerning the improvement of the integration of a renewable source widely known by the scientific community is photovoltaic solar energy. This research focuses on the impact of this integration on the transient stability of the power system [2] where the index of this stability is represented by the maximum time during which the grid can withstand the fault without losing its stability, this time is known as the critical clearing time (CCT) [3]. The use of the FACT system to improve network performance in the presence of this type of renewable source represented by PV source is justified by the opportunity offered by power electronics [3]. A three-phase short circuit is applied to a 30-bus IEEE test system; all simulations are performed under the Power System Analysis Toolbox software (PSAT) [4]. This paper is organized as follows. After a shirt introduction, transient stability is presented in second section, the facts devices are described briefly in Sect. 3; also, the aim of Sect. 4 is the description of solar PV penetration. Then, the test system, results and discussion are presented respectively in Sects. 5 and 6. Finally the paper concludes with a summary in Sect. 7.

2 Transient Stability A. Definition Power system stability has been a major concern in the last few decades. Transient stability of an electrical network is defined as the ability to ensure synchronous operation of its generators when it is subjected to severe transient disturbances such as a short circuit affecting a network element or a loss of a significant part of the load or generation. Thus, practically the entire system remains intact [3, 5]. Consequences of these defects can be very serious and can lead to the total collapse of the network (Black-out) (Fig. 1). B. Swing equation For a system to be transiently stable during a disturbance, it is necessary for the rotor angle, to oscillate around an equilibrium point. If the rotor angle increases indefinitely, the machine is said to be transiently unstable as the machine continues to accelerate and does not reach a new state of equilibrium. So, for maintain this equilibrium, the power system must at any time keep one’s balance between all torques applied on rotor of synchronous generators, as its behavior described by the swing equation [5].

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Fig. 1 Classification of power system stability [12]

    2 2H d δ(t) D dδ(t) = P w pu (t) (t) − P (t) − mpu epu wsyn d 2 (t) wsyn dt

(1)

where t is time in seconds, P mpu is mechanical power supplied by prime mover minus mechanical losses, per unit and P epu is electrical power output of the generator plus electrical losses, per unit. w pu and ws yn represent the per unit frequency and the synchronous radian frequency respectively, as to δ is rotor position with respect to synchronously rotating reference in rad. D represents a damping torque anytime the generator deviates from its synchronous speed. Finally, it’s convenient to work with a normalized inertia constant, called the H constant [5]. The swing equation is a second-order differential equation and it is also nonlinear (Fig. 2). In steady-state operation, all synchronous machines in the system rotate at the same electrical angular speed. The mechanical torque Tm has the same direction as the rotation of the generator axis. The electrical torque Te is in the opposite direction to that of rotation [6]. Pe Electromagnetic power Pm Mechanical power W Angular speed. C. Critical Clearing Time (CCT)

Fig. 2 Illustration of the couples adjusting on the rotor

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This is the most decisive parameter in the analysis of the transient stability of an electrical grid, mathematically it is the solution of the second order non-linear differential equation known as the Swing equation, which physically represents the maximum time during which our network can withstand a fault (short-circuit, over load, over current…) without losing its stability [6]. This is the stability assessment criterion adopted in this article.

3 Flexible AC Transmission System (Facts) Transient stability is one of the most important key factors during power transfer at high levels. One of the powerful methods for enhancing the transient stability is to use flexible AC transmission system (FACTS) devices [7]. The FACTS controllers offer a great opportunity to regulate the transmission of alternating current (AC), increasing or diminishing the power flow in specific lines and responding almost instantaneously to the stability problems. The potential of this technology is based on the possibility of controlling the route of the power flow and the ability of connecting networks that are not adequately interconnected, giving the possibility of trading energy between distant agents [8]. Flexible Alternating Current Transmission System (FACTS) is static equipment used for the AC transmission of electrical energy. It is meant to enhance controllability and increase power transfer capability. It is generally power electronics-based device [8]. A. Classification of Facts Devices There are different classifications for the FACTS devices can be divided in three groups, depending on the type of connection to the network: In this paper, we focus on shunt compensators (SVC, STATCOM) [9] (Fig. 3). • SSSC: Static Synchronous Series Compensator • TCSC: Thyristor Controlled Series Capacitor • SVC: Static Var Compensator Fig. 3 Block diagram of FACTS controllers [10]

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Fig. 4 SVC connected to a transmission line [10]

Fig. 5 STATCOM connected to a transmission line [10]

• STATCOM: Static Compensator • UPFC: Unified Power Flow Controller • IPFC: Inter line Power Flow Compensator (Figs. 4 and 5).

4 Solar PV Penetration Renewable energy sources, such as photovoltaic (PV) systems, wind turbines, and fuel cells, are integrated into conventional power systems to address fossil fuel deficiency, intensifying energy demand, and environmental pollution. Among all types of renewable energy resources, solar PV receives major attention for its promising energy resources and low-cost installation. The fundamental operation system of solar PV differs from other generating systems. Solar PV converts sunlight into DC power using semiconductor solar cells. The DC power is then converted into AC power through a DC-to-AC converter [11]. PV penetration level is defined as the ratio of total PV generation to total system generation, as expressed in the following equation:

538

P V penetration (%) f or generation based =

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T otal P V generation (M W ) T otal generation (M W ) (2)

5 Test System The simulation analysis was established on the famous 30 bus IEEE test system by PSAT/MATLAB, which gives access to an extensive library of grid components. A. Simulation Environment Several simulation tools were used to analyze the stability of electrical networks, such as: (MATLAB, Power World, PSAT…, Etc.). In the study presented, PSAT was chosen as a simulation tool: • PAST combines traditional statistical testing and data visualization functions that are generally only found in specialized software. • PSAT is compatible with the databases of most electrical network analysis software. • PSAT is an “Open Source” software that allows for modification as required by the study [4]. Figure 6 shows the famous 30 bus IEEE test system simulated by PSAT under MATLAB.

Fig. 6 IEEE 30 bus model with PSAT

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Table 1 Technical data of the study model Number of bus

Number of charges

Number of generators

Number of transformers

Number of line of transmissions

30

21

6

7

33

B. Characteristics of the Staudy Model. Table 1 represents the characteristics of the Staudy Model (Technical data of the study model).

6 Results and Discussion A. Choice of fault Type Three-phase short-circuit is chosen as a result of its most dangerous consequences against power system stability. B. Analysis Strategy A comparative study has been carried out with respect to the values of the critical clearing time corresponding to each simulation. This study is divided into three parts: Part 1: initial state • Creating a fault at bus “i”, except the reference bus. • Calculation of critical clearing time with minimum singular value method, in order to identify the most sensitive bus in the presence of the fault. The same study will be performed for all selected buses. Part 2: penetration of the solar photovoltaic (PV) Part 3: improvement of transient stability using an SVC, STATCOM. C. Simulation Results The performance of the proposed method was tested in the MATLAB/Simulation environment. a. Steady State Analysis (Before creating the default) Steady state analysis of the power grid represents no anomalies were essentially the voltage profile presented in Fig. 7 shows that the voltage values at all buses are within their acceptable limits (0.90–1.1 p.u).

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Fig. 7 Voltage profile

b. Initial State. We took bus 8 as excemple to show the location of the fault, this one is represented in Fig. 8. T Critical clearing time (CCT) (Fig. 9) According to Fig. 10, we can see that the challenge of just 1 (ms) further of the critical clearing time that can cause the instability of the system. The study performed at bus 8 will be done in the same way for all selected buses. The results are shown in Table 2. According to Table 2, we find that the critical time varies from one bus to another; this is due to the ability of each bus to support the fault.

Fig. 8 Fault location at bus 8

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Fig. 9 Voltage profile at the presence of a fault at bus 8

Fig. 10 Generator rotor speeds

Table 2 Critical clearing time for different fault location Bus

Bus 2

Bus 4

Bus 5

Bus 8

Bus 12

Bus 17

Bus 18

Bus 27

CCT (ms)

15

19

20

40

20

48

30

20

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c. Penetration of the Solar Photovoltaic (PV) A fault has been considered at bus 18 with integration a solar PV plant. The power injected by solar PV plant into the power grid around to 50 MW. Figure 11 shows the Penetration of a solar PV with presence of the fault at bus 18. Decreased is observed in Fig. 12 when the solar PV is connected to power system grid. It can be seen that the change of just 1 (ms) can cause the instability of the power grid. Figure 13 shows the simulation results obtained in MATLAB. According to Fig. 14, it is clear that the penetration of solar PV has enormously reduced the CCT and consequently the ability of the electrical network to maintain its stability during the fault in all the cases studied. This is due to the inability of the

Fig. 11 Penetration of a solar PV with presence of the fault at bus 18

Fig. 12 Rotor speeds corresponding

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Fig. 13 Voltage profile with penetration of solar PV at bus 18

Fig. 14 CCT comparison histogram with and without PV integration

PV to participate in the setting of the voltage plan in the absence of reactive energy in such sources. Additionally, PV penetration rate is inversely proportional to the corresponding critical time values. This is shown by the results of the Table 3. d. Improvement of transient stability using an SVC, STATCOM The use of the FACTS system is justified to improve the network’s ability to withstand faults for longer periods of time.

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Table 3 Variation in critical clearing time with percentage penetration of solar PV Bus

Bus 2

Bus 4

Bus 5

Bus 8

Bus 12

Bus 18

Active power (MW)

0.2

0.2

0.05

0.05

0.06

0.06

CCT (ms)

3

4

8

9

7

3

PV penetration %

23

23

6

6

7

7

In this section, an SVC and STATCOM respectively has been introduced in order to checking their influence on solar power penetration in the same localization. After several simulations important results are achieved: • SVC According to Table 4, the integration of the SVC has improved the critical clearing time of the network. We take bus 4 as an example: from 4 to 90 (ms) (Fig. 15). Instead of SVC, another FACTS device (STATCOM) of the same type has been introduced in same bus.

Table 4 CCT for different rates of solar power penetration with and without SVC Bus

Bus 4

Bus 5

Bus 8

Bus 12

CCT (ms)without SVS

19

20

40

20

CCT(ms) with PV without SVC

4

8

9

7

CCT(ms) with PV and SVC

90

60

130

419

Fig. 15 Localization of SVC

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• STATCOM After the integration of FACTS (SVC, STATCOM) it can be seen that a considerable improvement in the CCT has been noticed, which will provide an opportunity to integrate more energy from PV sources. Figure 16 shows the location of the SVC in the test system. Table 5 represents the critical clearing time for different rates of solar power penetration with and without STATCOM. STATCOM as shown in the histogram in Fig. 17 shows a modest superiority over SVC.

Fig. 16 Localization of STATCOM

Fig. 17 CCT comparison histogram

Table 5 CCT for different rates of solar power penetration with and without STATCOM Bus

Bus 4

Bus 5

Bus 8

Bus 12

CCT (ms) without PV and STATCOM

15

20

40

20

CCT (ms) with PV penetration

4

8

9

7

CCT (ms) with PV and STATCOM

100

123

140

421

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7 Conclusion This paper has mainly focused on the assessment of power system transient stability by determinate a critical clearing time (CCT) for the several cases by observing the behavior simulation of a test system during grid faults using a Power System Analysis Toolbox (PSAT). According to previously simulations, we find that: • The penetration of solar PV has reduced the CCT and thus the ability of the electrical network to maintain its stability during the fault in all the cases studied. • A considerable improvement in the CCT has been noticed after the integration of FACTS (SVC, STATCOM). STATCOM shows a modest superiority over SVC. • The simulation results clearly showed that there is an opportunity to integrate more energy from PV sources into the power grid through FACTS.

References 1. Yagami M, Ishikawa S, Ichinohe Y, Misawa K, Tamura J (2015) Transient stability analysis of power system with photovoltaic systems installed. J Energy and Power Eng 9 2. Remon D, Cantarellas AM, Mauricio JM, Rodriguez P (2017) Power system stability analysis under increasing penetration of photovoltaic power plants with synchronous power controllers. IET Renew Power Gener 11:733–741 3. Damor KG, Patel DM (2014) Improving power system transient stability by using facts devices. Int J Eng Res Technol (IJERT) 3(7). ISSN: 2278-0181 4. Milano F (2006) Power system analysis toolbox documentation for PSAT version 2.0.0 β1, July 9, 2006 5. Naimi D, Bouktir T, Ahmed S (2013) Improvement of transient stability of algerian power system network with wind farm. IEEE Conference: irsec’13, At Ouarzazet morroco, March 2013 6. Ayasun S, Liang Y, Nwankpa CO (2006) A sensitivity approach for computation of the probability density function of critical clearing time and probability of stability in power system transient stability analysis. Appl Math Comput, 563 7. Damor KG, Patel DM, Agrawal V, Patel HG (2014) Improving power system transient stability by using facts devices. Int J Eng Res Technol (IJERT) 3(7). ISSN: 2278-0181 8. Abido MA (2009) Power system stability enhancement using facts controllers: a review. Arab J Sci Eng 34(2):153–172 9. Gupta S, Tripathi RK (2014) Transient stability enhancement of multimachine power system using robust and novel controller based CSC-STATCOM. Adv Power Electron, 12. Article ID 626731 10. Ramakrishna Rao BT, Chanti P, Lavanya N, Chandra Sekhar S, Mohan Kumar Y (2014) Power system stability enhancement using fact devices. J Eng Res Appl 4(4)(Version 1):339–344. ISSN: 2248-9622

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11. Kouro S, Leon JI,Vinnikov D, Franquelo LG (2015) Grid-connected photovoltaic systems: an overview of recent research and emerging PV converter technology. IEEE Ind Electron Mag 9:47–61 12. Kamaruzzaman ZA, A. Mohamed A, Shareef H (2015) Effect of grid- connected photovoltaic systems on static and dynamic voltage stability with analysis techniques – a review. Przegl˛ad Elektrotechniczny, ISSN0033-2097, R. 91 NR 6/2015

Minimization of the Energy Consumption of an Aircraft Abbes Lounis, Kahina Louadj, and Mohamed Aidene

Abstract In this article, we studied a problem of minimizing an aircraft energy flow, the model used is obtained from the fundamental principle of dynamics. We solved this problem with the method of relaxation coupled with the method of shooting. Keywords Free final time · Shooting method · Pontryaguin principle · Relaxation method

1 Introduction Today, one of the major problems of aeronautics is the excessive consumption of fuel by aircraft. As we all know, excessive fuel consumption has a real impact on the environment because these planes pollute our atmosphere, increases the costs of travel and maintenance, so the minimization of Kerosene is a major challenge that has great interest in our daily life. Fuel consumption depends on altitude and speed. When climbing, the device consumes more obviously, downhill, it consumes less. It is therefore realistic to consume on average compared to our cruise consumption. When climbing, a fraction of the weight is added to the drag. The resultant is no longer balanced by the thrust. We must therefore increase the thrust or risk seeing the speed decrease. The thrust cannot be increase indefinitely, so if we continue to increase the incline, the speed will decrease. To maintain lift equal to the apparent weight, it will increase the impact to offset the decrease in speed. If the rate continues A. Lounis (B) · K. Louadj · M. Aidene Laboratoire de Conception et Conduite des systèmes de Production, University of Tizi Ouzou ( UMMTO), Tizi-Ouzou, Algeria e-mail: [email protected] K. Louadj e-mail: [email protected] M. Aidene e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_37

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to fall, it will reach the incidence of dropping out, and the plane will fall. In descent, a fraction of the weight is added to the thrust. Maintain a constant speed therefore requires reducing power. If too accentuates the slope, the trail will not be enough to balance the resulting «push more weight.” The aircraft will continue to accelerate and eventually end up in over speed. In this work, we studied a problem of minimizing the energy of an airplane; the model used is obtained from the fundamental principle of dynamics. We solved this problem with the method relaxation coupled with the method of shooting. We applied these results to a Boeing 747 [1–8].

2 Statement of the Problem • • • • • • • • •

V: flight speed. θ: angle between horizontal and flight path. α:Angle of attack (angle between flight path and chord line). w: aircraft weight. L: Lift, force normal to flight path generated by air acting on aircraft. D: Drag, force along flight path generated by air acting on aircraft. M: Pitching moment. T: Propulsive force supplied by aircraft engine/propeller. α T : Angle between thrust and flight path. To derive the equations of motion, we apply   

F = m a dV dt = T cosαT − D − wsinθ

FI I = ma I I = m FI T

T cosαT − D − wsinθ = m

(1)

dV dt

(2)

Applying (1) in ⊥ direction to flight path 

F⊥ = ma⊥ = m

V2 rc

rc : radius of curature of flight path: 

F⊥ = L + T sinαT − wcosθ = m

V2 rc

Minimization of the Energy Consumption of an Aircraft

L + T sinαT − wcosθ = m 

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V2 rc

(3)

dθ dt  F⊥ = L + T sinαT − wcosθ F⊥ = ma⊥ = mV

L + T sinαT − wcosθ = mV

(4)

dθ dt

We have: dh dx = V cosθ = V sinθ dt dt

(5)

= V cosθ = V sinθ dV 1 = cosα (T T − D − wsinθ ) dt m ⎪ dθ 1 ⎪ ⎪ = + T sinαT − wcosθ ) (L ⎪ Vm ⎪ ⎩ dt dm = −Q dt

(6)

The equations are: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

dx dt dh dt

The optimal control problem considered is to find the control Q(t) that minimizes the performance index: t f Q(t)dt → Min Q

(7)

0

where Q is mass flow the prescribed initial condition at the initial time t 0 : x(0) = x 0 ; h(0) = h0 ; V(0) = V 0 ; θ (0) = θ 0 ;m(0) = m0 ; and prescribed final condition at the final time t f : x(t f ) = x tf ; h(t f ) = htf ; V(t f ) = V 0 ; θ (t f ) = θ tf ;m(t f ) = mtf . The aim of this paper is to determine a trajectory (x(t);Q(t)) leading an initial state x(0) = x 0 ; h(0) = h0 ; V(0) = V 0 ;θ (0) = θ 0 ;m(0) = m0 ; to the final state: x(t f ) = x tf ; h(t f ) = htf ; V(t f ) = V 0 ; θ (t f ) = θ tf ;m(t f ) = mtf which minimize a criteria (7) tf is not fixed (tf is free libre) [4, 2, 3].

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We will solve the problem as follows: t f Q(t)dt → Min Q ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

0

= V cosθ = V sinθ dV 1 = cosα (T T − D − wsinθ ) dt m ⎪ dθ 1 ⎪ ⎪ = V m (L + T sinαT − wcosθ ) ⎪ ⎪ ⎩ dt dm = −Q dt dx dt dh dt

With a boundary conditions: x(0) = x 0 ; h(0) = h0 ; V(0) = V 0 ; θ (0) = θ 0 ;m(0) = m0 . x(t f ) = x tf ; h(t f ) = htf ; V(t f ) = V 0 ; θ (t f ) = θ tf ;m(t f ) = mtf . With constraint of control: −1 ≤ Q(t) ≤ +1 The Hamiltonian H is defined by: H = p1 (t)V cosθ + p2 (t)V sinθ + p3 (t) m1 (T cosαT − D − mgsinθ) + p4 (t) V1m (L + T sinαT − mgcosθ) − p5 (t)Q(t) − Q(t) The Euler-Lagrange equations are given by: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

= ∂∂ pH1 = ∂∂ pH2 = ∂∂ pH3 = ∂∂ pH4 m˙ = ∂∂ pH5 x˙ h˙ V˙ θ˙

·

p1 · p2 · p3 · p4 · p5 −1 ≤

= − ∂∂Hx = − ∂∂hH = − ∂∂ HV = − ∂∂θH = − ∂∂mH Q(t) ≤ +1

We have the system of the equations follows:

(8)

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⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

553

x˙ = V cos θ h˙ = V sin θ

V˙ = m1 (T cos αT − D − mg sin θ ) θ˙ = V1m (L + T sin αT − mg cos θ ) m˙ = −Q · p1 = 0 · p2 = 0

⎪ ⎪ ⎪  ⎪ · ⎪ p4 (t) ⎪ ⎪ p = −[ p (t) cos θ + p (t) sin θ − − mg cos θ + T sin α ) (L 3 1 2 T 2 ⎪ mV ⎪ ⎪ mg ⎪ ⎪ = −[− p (t)V sin θ + p (t)V cos θ − p3 (t) mg cos θ + p4 (t) mV sin θ p ˙ 4 1 2 ⎪ m

⎪ ⎪ · p4 (t) ⎪ ⎪ p5 = − − pm3 (t) ⎪ 2 (T cos α T − D) − m 2 V (L + T sin α T ) ⎪ ⎪ ⎩ −1 ≤ Q(t) ≤ +1 From the Hamiltonien H, the control Q is given by: Q(t) = max H Q(t)∈[−1.1] = si gn(− p5 (t) − 1)

3 Shooting Indirect Method The shooting indirect method is used to obtain the value of p(0) necessary to the solution of the problem characterized by the Pontryaguin principle. If it is possible, from the condition of minimization of the Hamiltonian to express the control extremal function of (x(t); p(t)) then the extremal system is a differential system of the form z˙ (t) = G(t, z(t)) where z(t) = (x(t), p(t)). With a numerical integrator from z0 we obtain:z i∼z0 ∼z(ti ) where the t i , i = 1, 2… are the time moments discretized by the integrator. But in z0 = (x 0 ; p0 ); the value x 0 is given by the initial condition. Then, by doing some obtain the different z i∼z0 . Which interests us are the  variations on p0 , we∼z0 ∼z0 ∼z0 z N ∼z t f (at final time); else z N = (x N∼z0 , p ∼z0 N ) and only the x N are significant. ∼P0 Since they depend only on p0 , note that x N . Let G be the implicit function giving p0 by numerical calculation using an integrator returns x N∼P0 − x f : G: Rn → Rn and G( p0 ) = x N∼P0 − x f Here G is an implicit nonlinear system of n equations and n unknowns: G(p0) = 0 [9–12].

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For the solution, we used the Newton’s method. The principle of the Newton’s method is described as follows: in the k–th step, let p0k be an approximation of the zero p0 of G; therefore p0 can be written p0 = p0k + p0k and then:  ∂G  k    0 = G( p0 ) = G p0k + p0k = G p0k + p p0 − p0k + o p0 − p0k ∂ p0 0 This leads to the solution of:  ∂G  k  −G p0k = p0 p0 − p0k ∂ p0  k where ∂∂G p0 is the Jacobian matrix of the application p0 → G( p0 ) computed when p0 k p0 = p0 ; note that the mapping p0 → G( p0 ) is not explicitly known but is known numerically p0 → G( p0 ) is not explicitly known but is known numerically So we will use a method ofnumerical derivation based on the finite difference.  k To avoid p0k it suffices to find an approximation of ∂∂G p0 , according the calculation of ∂∂G p0 p0 to [13], we will use one of the following finite difference approximations. 

j j−1   ∂G i  k 1 k k [G i ( p0 + h ik e ) − G i p0 + h ik e ) p ≈ ∂ p0 j 0 hi j k=1 k=1 Or else  ∂G i  k 1 [G i p0 + h i j e j − G i ( p0 )] p0 ≈ ∂ p0 j hi j where the hij are the given discretization step of the i-th equation with respect to the j-th variable, and ek are the k-th vector of the canonical basis; note that, classically, we can always choose the values of hij equal each other at a common value h. Let ij (p0 ; h) be a finite difference approximation, we have lim i j ( p0 , h) =

h→0

∂G i ( p0 ), i, j = 1, 2 . . . ∂ p0 j

Let, ( p0 , h) = i j ( p0 , h) which is an approximation of the Jacobian matrix; then the approximate Newton’s method can be written as follows:

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−1   p0k+1 = p0k − J p0k , h k .G p0k The problem of convergence of this iterative process is ensured by using a result of the book of Ortega and Rheinboldt [14]; indeed if the discretization step hij are small and tend to zero, the convergence is ensured. A. Transversality condition on p Generally, when the terminal cost is considered in the cost functional, the functional to be minimized can be written as follows:  J = φ t0 , t f +

t f L(t, x, u)dt

(9)

t0

Let M 0 and M 1 be two subsets of Rn ; then to minimize the cost functional one should find a trajectory between M 0 and M 1 . Moreover if M 0 and M 1 are two varieties of Rn having the tangent spaces T x0 M 0 and T x(tf) M 1 respectively x 0 ∈ M 0 and x tf ∈ M 1 ; then the vector p(t) must verify the transversality conditions: p(0) ⊥ Tx0 M0

(10)

  p t f − p 0 ∇x φ t f , x f ⊥Tx f M1

(11)

where p0 is a real such that p0 < 0 leads to the Pontryaguin’s maximum principle and p0 > 0 leads to the Pontryaguin’s minimum principle. If M 0 = x 0 ; the condition (10) becomes empty and the variety M 1 can be written as follows:   M1 = x ∈ Rn /F1 (x) = F2 (x) = . . . . = Fq (x) = 0 where F i are functions of class C 1 on Rn ; then the tangent space to M1 at a point x∈ M1 is defined by:   Tx M1 = ν ∈ Rn /∇ Fi (x(t f ))ν = 0i = 1, 2, . . . q and the condition (11) is written as follows    νi ∇x Fi (x(t f )) + + p 0 ∇x φ t f , x f ∃ν1 , ν2 , . . . . . . . . . ..νq ∈ R/ p t f = q

i=1

where ν i are the Lagrange multipliers.

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The transversality condition of Hamiltonian is defined by:    H t f , x f , p t f , p0 , u t f , = 0 corresponding to the fact that the Hamiltonian vanishes at final time.

4 Numerical Solution For the numerical solution, we used the shooting indirect method to determine a final time. Then we have to solve the following system: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

v˙ 1 = v3 cosv4 v˙ 1 = v3 sinv4 1 v˙ 3 = v5 (T cosαT − D − v5 gsinv4 ) ·

v4 =

1 v3 v5 (L

+ T sinαT − v5 gcosv4 ) ·

v5 = −Q · v6 = 0 · v7 = 0

· ⎪ ⎪ v8 = −[v6 cosv4 + v7 sinv4 − vvv9 2 (L + T sinαT − v5 gcosv4 ) ⎪ ⎪ 5 3 ⎪  ⎪ ⎪ v5 .g v5 .g ⎪ = v sinv − v v cosv +v cosv − v sinv v [v ⎪ 9 6 3 4 7 3 4 8 4 9 4 v5 v5 v3  ⎪ ⎪

⎪ ⎪ v8 v9 ⎪ − D)+ cosα + T sinα (T (L = v 2 T 2 T) ⎪ 10 v5 v5 v3 ⎪ ⎪ ⎪ ⎪ ⎪ μ = sign(−v10 − 1) ⎪ ⎪ ⎪ ⎪ −1 ≤ Q(t) ≤ +1 ⎪ ⎪ ⎩ vi (0) ∈ R, i = 1, . . . .10



Let υ(t, x 0 , h0 ,v0 , θ 0 ,m0 , p1 , p2 , p3 , p4 , p5 ) be the solution of the previous system at time t with the initial condition υ 1 (0), υ 2 (0), υ 3 (0), υ 4 (0), υ 5 (0), υ 6 (0), υ 7 (0), υ 8 (0), υ 9 (0), υ 10 (0). We construct a shooting function which is a nonlinear algebraic equation of the variable ψ at time t = 0; This shooting function is computed by a numerical procedure of integration of ordinary differential equation (using Euler method, Runge-Kutta method,…); the shooting function is defined by: ⎛

 v1t f , x0 , h 0 , v0 , θ0 , m 0 , p1 , p2 , p3 , p4 , p5 − xt f ⎜v t ,x ,h ,v ,θ ,m , p , p , p , p , p − h ⎜ 2 f 0 0 0 0 0 1 2 3 4 5 tf ⎜ ϕ(p) = ⎜ v3 t f , x0 , h 0 , v0 , θ0 , m 0 , p1 , p2 , p3 , p4 , p5 − Vt f ⎜  ⎝ v4 t f , x0 , h 0 , v0 , θ0 , m 0 , p1 , p2 , p3 , p4 , p5 − θt f  v5 t f , x0 , h 0 , v0 , θ0 , m 0 , p1 , p2 , p3 , p4 , p5 − m t

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

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The problem to solve is then written: Find p(0) such that ‘(p(0)) gives the desired value of x(T). The algorithm for numerical solution of this problem will then be completely defined if one gives oneself: 1. The integration algorithm of a differential system with initial condition (e.g., Euler or Runge-Kutta procedure) to compute the shooting function G (implemented in ‘ode45’ of Matlab which is a method of Runge-Kutta 4/5 with variable pitch). 2. The solution algorithm G(z) = 0 which in our case uses the method quasi- newton (implemented in ‘fsolve’ of Matlab).

5 The Steps of the Algorithm (Relaxation Methodcoupled with Shooting Method) 1. Approximate an initial control u0 (t); t ∈ [t 0 ; t f ], and adjoint state p0 (0). 2. r ← 0 3. while convergence > ε do • Determine a state x r (t) and adjoint state pr (t) component by component sequentially to t∈ [t 0 ; t f ] by numerical integration, the increasing time ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

d xr dt dh r dt

= V r cosθ = V r sinθ dV r 1 = m r (T cosαT − D − wsinθ ) dt ⎪ dθ r 1 ⎪ ⎪ = + T sinαT − wcosθ ) ⎪ V r m r (L r ⎪ ⎩ dt dm = −Q dt x(0) = x 0 ; h(0) = h0 ; V(0) = V 0 ; θ (0) = θ 0 ;m(0) = m0 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

dp1r dt dp2r dt r

pr = − p1r (t) cos θ r + p2r sin θ − m r V4 r 2 (L + T sin αT − m r g cos θ r )]   r r = − − p1r (t)V r sin θ r + p2r (t)V r cos θ r − p3r (t) mm rg cos θ r + p4r (t) mmr Vgr sin θ r dp3r dt

dp4 ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ r

=0 =0

dp5r dt

=

p3r (T cosαT mr 2

pr

− D)+ m r 24V r (L + T sinαT ) pr (0)

where pr (0) is found by shooting method [15].

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• Determine a control ur+1 (t)  u r+1 (t) ← sign − p5r (t) − 1   • Convergence: u r+1 (t) − u r (t) • Determine a shooting function:  ϕ( p) = x r t ∗ − x ∗ • Solution of the shooting equation is determine by Newton method and determine a new values of p(0) p r+1 (0) ← p r (0) + cor r ect i on • r ← r + 1. while end

6 Numerical Application Aircraft Boeing 747, Cruise Stage. The boundary conditions are: x 0 = 12 km; h0 = 6.092 km; V 0 = 920 km/h; θ 0 = 0.0349066 rad, m0 = 90,000 kg. x tf = 775 km, htf = 12.192 km, V tf = 955 km/h, θ tf = 0.0523599 rad, mtf = 4000 kg. Shooting Method (Determine final time t f ) Final Time: t f = 0:95 h, iterations Number: 25 iteration, execution time: 2.74 s. Shooting Method (Determine p(0)) execution time: 1.32 s, Iterations Number: 03. Relaxation method execution time: 5.23 s, Iteration Number: 03 (Figs. 1, 2, 3, 4, 5 and 6).

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Fig. 1 a The position x(t). b Altitude h(t)

7 Conclusion In this paper, we have solved a problem of minimization problem of an energy flow of an aircraft with free final time using relaxation method’s coupled with shooting method’s based Pontryaguin’s maximum principle. We used the shooting indirect method to find the transversality conditions in both cases where the state is submitted. We concluded that, in the numerical procedure, the convergence is fast and the computational time are small.

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Fig. 3 a The adjoint state p1 (t). b The adjoint state p2 (t)

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Fig. 4 a The adjoint state p3 (t). b The adjoint state p4 (t)

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Fig. 5 a The adjoint state p5 (t). b The mass m(t)

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Fig. 6 The angular acceleration θ(t)

References 1. Efstathios Bakolas-Panagiotis Siotras (2013) Optimal synthesis of the Zermelo-MarkovDubins, problem in a constant drift field, J Opt Theory Appl 156:469–492 (Springer) 2. Pontryagin et al L (1974) Mathematical theory of optimal processes. Mir, Moscou 3. Lee EB, Markus L (1967) Foundations of optimal control theory. The SIAM series in applied mathematics. Wiley, New-York, London-Sydney 4. Hagelauer P, Mora Camino F (2014) A soft dynamic programming approach for on-line aircraft 4D-trajectory optimization. LAAS, Toulouse 5. Titouche S, Spiteri P, Messine F, Aidene M (2015) Optimal control of a large thermic. J Control Syst 25:50–58 6. Sethi SP, Thompson GL (2000) Optimal control theory: applications to management science and economics, 2nd edn. Springer, Berlin, pp 504 7. Demim F, Nemra A, Louadj K (2016) Solution of an optimal control problem of the robot with Bellman Principle’s. In: Conférence Internationale en Sciences et Technologies Electriques, October 26–28, CISTEM’16, Morocco, Marakech 8. Demim F, Nemra A, Louadj K, Hamelain M (2018) Simultaneous localization, mapping, and path planning for unmanned vehicle using optimal control. Adv Mech Eng 10(1):1–25 9. Hull DG (2003) Optimal control theory for applications. Springer, New York 10. Sethi SP, Thompsonn GL (2000) Optimal control theory, applications to management science and economics, 2nd edn 11. Sager S, Bock HG, Reinelt G (2009) Direct methods with maximal lower bound for mixedinteger optimal control problems. Math Program Ser A 118:109–149 12. Ternovskii VV, Khapaev MM (2008) Direct numerical method for solving optimal control problems. Doklady Math 77(3):428431 13. Trelat E (2005) Contrôle optimal: théorie et applications. Vuibert, collection Mathématiques Concrètes, France 14. Ortega JM, Rheinboldt WC (1970) Iterative solution of nonlinear equations in several variables. Academic Press, New York 15. Demim F, Louadj K, Aidene M, Nemra A (2016) Solution of an optimal control problem with vector control using relaxation method. Autom Control Syst Eng J 16(2) (ISSN 1687-4811, ICGST LLC, Delaware, USA)

Study to Improve the Technical Parameters for the Optimising the Injection of the Photovoltaic Energy into an Electrical Network via a Line 30 KV Bahriya Badache, Hocine Labar, and Mounia Samira Kelaiaia Abstract The aim of this paper is study to improve the technical parameters for the optimizing the injection of the photovoltaic energy into an electrical network via a line 30 kV, the study of three variants to improve the integration of the solar energy of the OUED KEBRITE located in the EST of Algeria power station to the electrical network. This investigation is based on geological topology of the region essential to the grid design and the economical payback of the whole project. Concerning the investment of the construction of the photovoltaic power plant, which will enable it to produce 15 MWp to be fed into the electricity grid, the 30 kV line used to evacuate its production has a thermal limit that does not allow all the energy produced to pass through. There for we have carried out a study to find a solution in order to overcome the constraint of limiting the energy transit of the 30 kV line. Keywords Photovoltaic systems · Grid connected · Simulation · Improvement · Technical parameters

1 Introduction In this research, we approached three variants: the rehabilitation of the existing 30 kV line, the construction of a new line and the construction of an underground cable line in order to improve the injection of photovoltaic energy into the electricity grid. Renewable energy is a key challenging problem increasingly gaining attention in worldwide [1]. Solar energy is the most abundant, inexhaustible and clean of all renewable energy resources [2]. In the twenty first century the energy demands are B. Badache (B) · H. Labar · M. S. Kelaiaia Department of Electrical Engineering, University Badji Mokhtar Annaba, Annaba, Algeria e-mail: [email protected] H. Labar e-mail: [email protected] M. S. Kelaiaia e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_38

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rapidly growing-up [3]. Renewable energy can reduce greenhouse gas emissions by providing renewable electricity [4]. The renewable energy sources (RES) are an important part of the energy system, with the features of wide distribution capability and great development potential. RES are environmentally friendly and sustainable compared with conventional energy sources. Therefore, the exploitation and utilization of RES has been an important measure in the world to ensure energy security and to tackle climate change [5]. Renewable energy can supply two-thirds of the total global energy demand, and contribute to the bulk of the greenhouse gas emissions reduction that is needed between now and 2050 for limiting average global [6]. Furthermore, the PV panels produce sometimes an excess of power and in other times a lack of power [3]. The photovoltaic generator is the only direct converter to transform light into electrical energy [7], a photovoltaic generator is a generator whose characteristic I = f(U) is highly non-linear. Consequently, for the same amount of sunlight, the power delivered will be different depending on the load. A maximum power point tracking (MPPT) controller controls the static converter, connecting the load (a battery for example) and the photovoltaic panel, in order to permanently provide the maximum power to the load. The Perturbation and Observation (P&O) method is the most widely used maximum power search method; it is an iterative method to obtain the maximum power point (MPP) [8]. The PV system works around its MPP according the preset voltage step [9]. Photovolatic systems normally use a MPPT technique to continuously deliver the highest possible power to the load when variations in the insolation and temperature occur [10]. The principle of the Perturbation and Observation method consists in disturbing the voltage (VPV) of the generator with a small amplitude around the initial values and analysing the behaviour of the resulting power variation (PPV). If a positive voltage increment (VPV) results in an increase in power (PPV), this means that the operating point is to the left of the PPM Fig. 1, if, on the contrary, the power decreases, this means that the system has exceeded the PPM. A similar reasoning can be applied when the voltage decreases. To ensure the photovoltaic generator operates at optimal efficiency, requires the insertion of static converters (chopper) between the generator and receiver (load) at a stable (constant) voltage can be optimized by adjusting the duty cycle α so that the generator can operate at optimal speed. However, for connection to the grid it requires an inverter converter [8] (Fig. 2). Fig. 1 Continuation of the maximum power point

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Fig. 2 The real photovoltaic power station production according to the radiation day of 24-10-2018

The photovoltaic power plant is located in the eastern region of Algeria Fig. 3, was commissioned in 2014. The plant has a capacity of 15 “Megawatt (MW)” and is built on an area estimated at 30 “hectares (ha)”, this first station of its kind in eastern Algeria, will have photovoltaic solar cells. The Photovoltaic plant is composed of approximately 48.000 photovoltaic modules with a power of 250 “Watt peak (Wp)” in Polycrystalline silicon for an actual installation power of 15 “Mega Watt peak (MWp)”. Each subfield consists of approximately 4.000 photovoltaic modules with an installed capacity of approximately 01MWp. Each subfield is equipped with two centralized inverters and a step-up transformer. These inverters, with an output voltage of 315 “Volt (V)”, are connected on the low voltage side via alternating current cables to the 1.250 “kilo volt ampere (kVA)” step-up transformer, which raises the voltage to 31.5 “kilo volt (kV)”. Fig. 3 Renewable energies in Algeria

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This power plant will support the electricity grid and fight against power outages, often recorded during the summer period. The operation will also help to generate new jobs, particularly in the technical specialities related to this field, such as renewable energy sectors. The purpose of this work is to examine the impact of the integration of the plant Photovoltaic of 15 MWp in Eastern Algeria from Shariket Kahraba wa Taket Moutadjadid (SKTM) on the distribution network. The evacuation of production to the 30 kV floor of the 220/90/30 kV substation in EL AOUINET. The connection mode proposed by the Electricity Distribution Company (SDE), the plant will flow all its production on the 30 kV busbar on the 30 kV floor of EL AOUINET substation. The study will determine the optimal injection point on EL AOUINET transformers [8].

2 Economical Impact of Renewable Energy The electric power generated by a solar array fluctuates depending on the solar radiation value and temperature [10]. Therefore, a change in the spatial arrangement of energy generation systems is taking place. Traditional locations for energy generation (e.g. lignite or hard coal mining areas) may lose their substantial significance in favor of renewable energy generation locations, if the former do not possess suitable local conditions for the generation of renewable energy (RE), which leads to economic losses [11]. Cross-regional connection lines are cost-effective methods of increasing renewable energy consumption [12]. The incorporation of renewable energy in a power system is determined by certain factors, including maximum system load, daily load curve and peak-valley difference power source mix, capacity of crossregional lines, as well as power transmission curve and demand side response of the system Fig. 2, [12].

3 Case Study Modeling and Features In recent years, the production of photovoltaic (PV) systems and integration into the electricity grid has become the most important [13]. In many countries, photovoltaic (PV) solar systems are considered one of the best renewable energy (RE) sources in terms of installation costs [14]. There has been an increased attention to the photovoltaic (PV) energy systems during the last decade owing to the many advantages that these systems have such as: it is a worldwide available energy source, it is pollution free, it has noiseless operation [15]. The revision of the national program focuses mainly on the development of photovoltaic and wind power on a large scale, the introduction of biomass (waste recovery),

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cogeneration and geothermal energy and also on the postponement of the development of Concentrated Solar Power (CSP) to 2021 [8]. This program underwent a first phase devoted to carrying out pilot projects and tests of the various available technologies Fig. 3, during which relevant elements concerning the technological developments of the sectors under consideration appeared on the energy scene and led to the revision of this program. In accordance with current regulations, the implementation of this national program is open to domestic and foreign public and private sector investors [8]. The investment of the photovoltaic power plant Fig. 3, is planned to produce a capacity of about 15 MWhp, nevertheless the production of the power plant is limited to 12 MWp because it transits energy on a 30 kV line whose thermal limit is 120 A/HT, see the characteristics at level Table 1, because of the dilapidated state of the 30 kV OUED KEBRIT/EL AOUINET line Fig. 4, an available energy of 03 “Mega Watt peak (MWhp)”not produced and not consumed, it is for this reason that a reliability study of the line is necessary. This line crosses the two lines 220 kV Table 1 Characteristics of the 30 kV Kebrit/El Aouinet line Tower type

Tower height

Conductor cable section

Type of insulators

Transit limit

Length

Poles 95BS66 and concrete Poles

Between 10 m et 12 m

98.30 mm2

CTV 175

120 A

6 km

Fig. 4 Representing the different types of supports of the 30 kV line

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KHROUB/EL AOUINET and EL AOUINET/AIN BEIDA, the line is crossed on the 30 kV span ex M’DAOUROUCHE. A. Characteristic of line 30 kV Oued Kebrit/El Aouinet: B. Simulation and results: (1) VARIANT N°1: Study for the renovation of the existing line to ensure a higher thermal limit (Fig. 6).

Fig. 5 The 30 kV line from the photovoltaic power plant to El Aouinet substation

Fig. 6 The line layout on GOOGLE EARTH, option 1

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Replacement of the 98.3 mm2 conductive cable by 116 mm2 cable of section type ALMELEC see length profile shown on the AutoCAD software Fig. 7. Using the same existing pylons see the layout located on a uneven ground Fig. 6. This study requires checking the crossing distances or to cross (regulatory vertical distances) Fig. 8 and calculating the electrical parameters according to the formulae (1–7).  X 0 = 0, 144 log

Dmoy rcond

 + 0, 016

Fig. 7 The profile along the 30 kV line on AUTOCAD, option 1

Fig. 8 The result of the crossing on NC/EtudLig software, option 1

(1)

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X L = X0 × L

(2)

R = R0 × L

(3)

b0 = c0 × ω =

A =1+

7, 58 × 10−6   Dmoy log rcond

Y×Z =D 2

(4)

(5)

B=Z

(6)

  Y×Z C = 1+ ×Y 4

(7)

C. Simulation and results: This variant consists in studying the construction of a new 30 kV line, choosing a new layout shorter than that of the existing line and a level-free ground Fig. 9. Using larger and more robust supports to ensure proper adjustment of the conductor cables see the long profile Fig. 10, in order to improve the crossing distance or to cross (regulatory vertical distance) Fig. 11, which reduces the frequency of aggression of the latter and reduces the number of its activations and improves its quality of service parameters by using anti-pollution insulators.

Fig. 9 The line layout on GOOGLE EARTH, option 2

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Fig. 10 The profile along the 30 kV line on AUTOCAD, option 2

Fig. 11 The result of the crossing on NC/EtudLig software, option 2

4 Analyze and Discussion In order to ensure the transit of the available power of the OUED KEBRIT 15 MWp power plant instead of 12 MWp. The construction of a new line (variant n°2) is the most economical solution. Variant n°3 consists of the construction of an underground cable line (Fig. 12) this solution presents a considerable improvement in electrical parameters and the transit limit is the highest. For the last variant the loss of revenue will be saved by the cost of maintenance and operation.

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Fig. 12 The line layout on GOOGLE EARTH, option 3

From the above, choosing the third variant is necessary Table 2 for the following reasons: • Improvement of Service quality standards following the commission of a new underground line since it is not affected by climatic conditions or external aggressions. • Incident recovery: current time, quick access to fault point. • Aesthetically, the underground line does not hinder aesthetically. • The underground line also provides the optical fibre link to connect the OUED KEBRIT power plant to SONELGAZ’s optical fibre network. • No impact on the environment and health • Cost is the only disadvantage of this variety, nevertheless the latter will be saved since the underground line requires no maintenance, even the number of interventions is very reduced comparing with variants N°1 and 2.

5 Conclusion The objective of this work is to study the integration of renewable energies into the networks because the improvement of the voltage of the electricity grid is a concern of the electricity distributor. In the Medium Voltage (MV) network, the quality of the voltage will be unacceptable when the voltage drop exceeds 10%. In addition to the study to improve the integration of solar energy injected by the photovoltaic power plant into the 30 kV grid. It was necessary to find an answer to the question: what is the revenue loss of the National Electricity and Gas Company (Sonelgaz) and the losses suffered caused by the thermal limit of the 30 kV Oued Kebrit/ El Aouinet line. In this work three (03) cases are considered and studied, using four (4) software:

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Table 2 Comparison of the three (03) variants Variants

Advantages

Variant n°1 Section of conductor: 116 mm2 ALMELEC Works duration: Short (about 12 days without refund)

Disadvantages Drawing of the line: Reuse the existing line layout Length: Number of tower: 120 Longest: 06 km Ground: Uneven level, Rough Undistributed Energy (MWh): Extended High Tower: 10 m to 12 m, type 95BS66 Poles and concrete poles Restoring situation in the event of an incident: Important time, Long line Type of Insulator: CTV 175 normal Line and plant logging: (about 12 days), Throughout the period of the works Presentation of the most dangerous risk to the third part Showing the effect of electromagnetic fields from overhead lines on human health, animals and the environment The mechanical strength of the existing towers does not allow the installation of the guard cable to protect the line against lightning strikes, which causes permanent disruption of the network The line is exposed to adverse weather conditions, strong winds, lightning, pollution, etc. The line does not have an optical fibre link with the SONELGAZ network Cost: In order of 25,000,000.00 DA

Variant n°2 Undistributed energy (MWh): low

Length: Number of Tower: 84 Shorter: 05 km Drawing of the line: New layout, Realization of a new line

High tower: 14 m, type Metallic tower 126BS77

Works duration: Long (more than 03 months)

Section of conductor: 116 mm2 ALMELEC

Important deadlines and times for the regularization of landowners’ objections

Type of insulator: U80AS Anti-pollution

The cost of compensating landowners for the installation of new towers

Line and plant logging: One day only for connecting conductive cables

Restoring situation in the event of an incident: Less time, shorter line

Cost: in order of 44,000,000.00 DA

Presentation of the most dangerous risk to the third part (continued)

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Table 2 (continued) Variants

Advantages

Disadvantages Showing the effect of electromagnetic fields from overhead lines on human health, animals and the environment The line not protected from adverse weather conditions The line does not have an optical fibre link with the SONELGAZ network

Variant n°3 Construction of an underground cable line, Shorter: 3.50 km

Cost: In order of 34,000,000.00 DA

No implementation constraints such as oppositions and no line maintenance Undistributed Energy (MWh): low Section of conductor: 120 mm2 Armored type Recovery in the event of an incident: current time, quick access to the fault point Lack of danger to third parties No impact on the environment and health Line and plant logging: One day only for connecting conductive cables Capacitive current is not very important for the 30 kV voltage level The underground line also provides the optical fibre link to connect the OUED KEBRIT power plant to SONELGAZ’s optical fibre network The underground line is protected against adverse weather conditions Works duration: Long (1.5 months)

MATLAB SIMULINK, AutoCAD, Google Earth, NC/EtudLig, the advantages and disadvantages of each variant are well addressed. The state of the art of renewable energies (photovoltaic) plays a major role in such a study. This work is initiated by a theoretical study of the characteristics of a primary source (sunshine), followed by an examination of how a photovoltaic system is integrated into the electricity grid. The aim is therefore to optimize the overall treatment of electrical energy within the system and in particular, at the generator level, by placing it at all times at its optimal operating point thanks to a MPPT-type tracking system. In addition, due to their switching operations, these static converters generate disturbances that can affect the proper functioning of the generator both electrically and energetically.

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On the energy level, the ripples of the voltage or current at the input of some converters result in an oscillation of the operating point on the characteristic around the optimal power point, which implies a degradation of the photovoltaic conversion. Variant n°03 seems to be the best solution to be adopted by Sonelgaz company so that it can allows a recovery of about 100% from its investment [8]. Annexes: Data Option N°1: • • • • • • •

Length: 06 km X0 = 0.326 “ohrm ()” XL = 1.956 “ohm.kilometre (.Km)” B0 = 3.51710−6 “siemens (S)” A = D = 0.99 + j 1.79 × 10−5 B = 1.69 + j 1.95 C = 1.88 × 10−10 + j 2.11 × 10−5

Data Option N°2: • • • • • • • • •

Length: 05 km X0 = 0.326  XL = 1.63 .Km R = 1.41 .Km Z = 1.41 + j 1.63 B0 = 3.51710−6 S A = D = 0.99 + j 1.24 × 10−5 B = 1.41 + j 1.63 C = −1.09 × 10−10 + j 1.75 × 10−5

Data Option N°3: • • • •

Length: 3.5 km R = 0.325 .Km C = 0.182 μF.Km L = 0.415 mH.Km

References 1. Harrou F, Sun Y, Taghezouit B, Saidi A, Hamlati M-E (2017) Reliable fault detection and diagnosis of photovoltaic systems based on statistical monitoring approaches. Renew Energy (Accepted manuscript) 2. Ali MH, Rabhi A, El hajjaji A, Tina GM (2016) Real time fault detection in photovoltaic systems. Energy Proced 111:914–923 (Elsevier, Sept 2016, Digests 8th International Conference Sustainability in Energy and Buildings, SEB-16, Italy, 2016, p 914)

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3. Djelailia O, Kelaiaia MS, Labar H, Necaibia S, Merad F (2018) Energy hybridization photovoltaic/diesel generator/pump storage hydroelectric management based on online optimal fuel consumption per kWh. Sustain Cities Soc (Accepted manuscript) 4. Azhgaliyeva D (2019) Energy storage and renewable energy deployement: empirical evidence from OECD countries. Energy Proced 158:3647–3651(Digests 10th International Conference Applied Energy “ICAE2018” Hong Kong China, p 1, 2019) 5. Zeng M, Duan J, Wang L, Zhang Y, Xue S (2015) Orderly grid connection of renewable energy generation in China: management mode, existing problems and solutions. Renew Sustain Energy Rev 41:14–28 6. Gielen D, Boshell F, Saygin D, Bazilian MD, Wagner N, Gorini R (2019) The role of renewable energy in the global energy transformation. Energy Strategy Rev 24:38–50 7. Khemliche M, Djeriou S, Latreche S (2012) Diagnostic de Défauts dans le Système Photovoltaïque par les Réseaux de Neurones, Diagnostic fault in photovoltaic system for Neurones networks, SIENR, pp 1–9, Oct 2012 (Digests The 2 nd International Seminar on New and Renewable Energies Ghardaïa Algérie, p 1, 2012) 8. http://biblio.univ-annaba.dz/ingeniorat/wp-content/uploads/2018/10/Merzougui-Nour-ElHouda-Badache-Bahriya.pdf, 2018 9. Hocine L, Samira KM (2018) Real time partial shading detection and global maximum power point tracking applied to outdoor PV panel boost converter. Energy Convers Manag 171:1246– 1254 10. Hua C, Lin J, Shen C (1998) Implementation of a DSP-controlled photovoltaic system with peak power tracking. IEEE Trans Ind Electron 45(1):1–9 11. Jenniches S (2018) Assessing the regional economic impacts of renewable energy sources. Renew Sustain Energy Rev 93:35–51 12. Zhang D, You P, Liu F, Zhang Yu, Zhang Ya, Feng C (2018) Regulating cost for renewable energy integration in power grids. Global Energy Interconnect 1(5):544–551 13. Al-Shetwi AQ, Sujod MZ, Blaabjerg F, Yang Y (2019) Fault ride-through control of gridconnected photovoltaic power plants. Sol Energy 180:340–350 14. Ahmad NI, Ab-Kadir MZA, Izadi M, Azisa N, Radzi MAM, Zainia NH, Nasir MSM (2017) Lightning protection on photovoltaic systems (in press) 15. Mellit A, Tina GM, Kalogirou SA (2018) Fault detection and diagnosis methods for photovoltaic systems. Renew Sustain Energy Rev 91:1–17

Second-Order Super-Twisting Control of an Autonomous Wind Energy Conversion System Based on PMSG for Robustness and Chattering Elimination Abderrahmane Abdellah, Djilali Toumi, and M’hamed Larbi Abstract This article focuses on improving the performance of the autonomous wind energy conversion system based on a Permanent Magnet Synchronous Generator (PMSG). There are several problems related to the control of this system, such as chattering phenomenon, static error and coupling between d-q currents, associated with the use of conventional regulators like the PI controller. To remedy these problems, a comparative study is performed between the Super Twisting Sliding Mode Controller (STSMC) and Adaptive Fuzzy Sliding Mode Controller (AFSMC) that are used to drive the rotor speed to the optimal speed to get the maximum power. The simulation study shows that the proposed STSMC is more effective at different operating conditions. Keywords Wind turbine · PMSG · Adaptive fuzzy controller · Sliding mode controller · Super twisting algorithm

1 Introduction Fossil fuels do much more damage than renewable energy, including air and water pollution, public health, loss of wildlife and habitat, water use, global warming emissions. Moreover, the demand for this source of energy continues to increase as industrial and economic development depends on these energy needs. Faced with this challenge, we opted for alternative solutions, including the use of renewable energies. A. Abdellah (B) · D. Toumi · M. Larbi Laboratory of Energy Engineering and Computer Engineering (L2GEGI), Ibn Khaldoun University, Tiaret, Algeria e-mail: [email protected] D. Toumi e-mail: [email protected] M. Larbi e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_39

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However, Wind energy is one of the cleanest and most sustainable ways to produce energy because it produces no toxic pollution or global warming emissions. The wind is also abundant and inexhaustible, and affordable, making it a viable and large-scale alternative to fossil fuels [1]. Currently, the generators most used in the production of wind energy are based on the Doubly-Fed Induction Generator (DFIG) and PMSG. Faced with the problems of producing wind energy, the synchronous generator with permanent magnet has advantages that are among others: elimination of DC excitation system, low maintenance, …etc. There are several methods of having been proposed for controlling PMSGs, including the conventional PI controller may not give a satisfactory performance under challenging operating conditions like a parameter variation. To remedy this problem, research has turned to control techniques which ensure the robustness and improve the operation performance of the PMSG. In [2–4], the author worked on an induction machine using the PI controller, Sliding Mode Controller (SMC) and Adaptive Fuzzy Sliding Mode Controller (AFSMC) in order to compare between them and found that the AFSMC is better than the others in time response, performances, robust to parameter variations and the steady-state chattering elimination. However, there is an interesting robust method that has better performances, it called Super-Twisting Sliding Mode control (STSMC) that is studied in [5–8], their results show that it gets better performance in the test of robustness and chattering elimination. The idea of this article is concerning the comparison between three propositions of control as follow (PI–PI), (AFSMC–SMC) and (STSMC–SMC) the first controller corresponded to the speed control and the second to the d-q currents control. The purpose of this comparison is to see the control performances and chattering elimination at each proposition. The paper is planned as follows: in Sect. 2 the studied system modeling including PMSG and wind turbine is presented. The control of speed and currents of the PMSG using the three propositions of control is synthesized in Sect. 3. Section 4 presents the results to compare the three propositions of control. Finally, the main conclusion is presented in Sect. 5.

2 System Description The proposed structure of an autonomous wind energy conversion system is composed of a wind turbine, a PMSG, a PWM (Pulse Width Modulation) converter, and a battery, as it is shown in Fig. 1.

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Fig. 1 The wind energy conversion system

A. Wind Turbine Model The power produced by the wind is given by (1): Pw =

1 ρ π R3 V 3 2

(1)

The aerodynamic power captured by the wind turbine is dependent on the produced power, and it’s given by (2): Paer =

1 ρ π R 3 V 3 C p (λ, β) 2

(2)

where, ρ: the air density, R: the turbine radius, V: the wind speed, λ: the tip speed ratio, β: the pitch angle. Thus, the general form of the power coefficient C p is [1]:  C p = C1

 −C5 C2 − C3 β − C4 e λi + C6 λi λi

(3)

where, C1 = 0.5176, C2 = 116, C3 = 0.4, C5 = 21C6 = 0.0068 and

1 λi

=

1 λ + 0.08β

−

0.035 β3 + 1

The tip speed ratio is given by (4): λ=

Rt V

(4)

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where t denotes the turbine speed. The Maximum Power Point Tracking (MPPT) algorithm is used to maximize the wind power extraction. For that, we set the tip speed ratio and the blade pitch angle to their optimal values λopt = 8.1 and βopt = 0, respectively. Thus, the optimum speed is given by: opt =

λopt V R

(5)

The aerodynamic torque expression is given by (6): Caer =

Paer t

(6)

In order to adapt the speed of the turbine with the generator one, a gearbox is installed. Therefore, the turbine and the generator speeds are linked by a coefficient noted G and called the gear ratio, as it is given in (7).  = G · t

(7)

Thus, the torque of the generator is presented by (8): Cg =

Caer G

(8)

B. Model of PMSG The mathematical model of the PSMG is established under the d and q axis synchronization reference frame [9]. It is simplified as: Lq Rs did 1 = − i d + ωe i q + Vd dt Ld Ld Ld   diq Ld 1 Rs 1 = − i q − ωe id + ϕf + Vq dt Lq Lq Lq Lq

(9) (10)

where subscripts d and q refer to the physical quantities that have been transformed into the d − q synchronous rotating reference frame, Rs is the stator resistance [], L d and L q are the inductances [H] of the generator on the d and q axis, ϕ f is the permanent magnetic flux [Wb] and ωe is the electrical rotating speed [rad/s] of the generator, defined as ωe = p where p is the number of pole pairs and  is the mechanical rotating speed of the generator. The electromagnetic torque equation of PMSG is described by this equation:

Second-Order Super-Twisting Control of an Autonomous …

Te =

583

   3 iq p L d − L q id + ϕ f = K t iq 2

(11)

  3  p L d − L q id + ϕ f 2

(12)

Kt =

The mechanical equation which connects the generator with the wind turbine is described by: Jeq

d = Te + Tl − Bm  dt

(13)

where Tl is the torque applied by the turbine to the generator. Jeq is the equivalent moment of inertia; Bm and is the viscous turn coefficient. Substituting (11) and (12) in (13) yields Kt Tl Bm d = iq + −  = Ai q + DTl + B dt Jeq Jeq Jeq where A =

Kt , Jeq

B = − BJeqm , D =

(14)

1 . Jeq

3 Controller Synthesis In this section, different types of controllers are examined and presented to regulate the current and speed. A. Classical PI Controller Figure 1 shows the scheme of the generator side control [9]. It consists of two loops, the first is used to control the d-q axis components of the stator currents of the PMSG. The outer-loop is the speed control that tracks the optimal speed reference given in (5) to generate the maximum power from the wind generator (Fig. 2). B. Adaptive Fuzzy Sliding Mode Controller design The structure comprises a speed regulation loop by AFSMC which generates the ∗ ∗ current reference Iq∗ . While regulating the currents Id−q imposes the Vd−q commands. In this work, the regulating of currents Id−q are done by first order SMC. The choice of surfaces for each loop is described as follows: (1) The Speed Control of PMSG by AFSMC The expression of speed error [2, 3]: e (t) = ∗ (t) −  (t)

(15)

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Fig. 2 Generator side control scheme of the direct drive wind turbine

Considering the following expression of switching surface: S (t) = h (C e (t) + e˙ (t))

(16)

where C and h are strictly positive constant. Substituting (14) in (16), the first derivative of S(t) is obtained as     ∗ ¨ ∗ − A Iq − D Tl − B  ˙ ˙ − A i q − D Tl − B  +  S˙ (t) = h C 

(17)

During the sliding mode and the steady-state, S(t) = 0 and S˙ (t) = 0. From (17) we get the equivalent control law as: Iqeq =

  1 ˙ ∗ − D Tl − B ) +  ¨ ∗ − D Tl − B  ˙ C ( (1 + C) A

(18)

We must add a discontinuous term to the equivalent control part across the sliding surface S(t). The discontinuous term is called reaching control part of control effort and it is given as [3, 4]. Iqr = −(Ah)−1 k (t) sgn (S (t))

(19)

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where k(t) > 0 Which give us the Iq∗ output control of the speed regulator. Iq∗ = Iqeq + Iqr

(20)

Stability condition can be obtained from Lyapunov stability theorem as [4], V˙ (t) = S (t) . S˙ (t) ≤ −η |S(t)|

(21)

where, η is a strictly positive constant.   Substituting (16) and (17) in (21) and considering T˙l  < m, the stability of the system is guaranteed by the following equation. K (t) ≥ h m + η

(22)

We know that sliding mode techniques can generate undesirable chattering. To eliminate the chattering phenomenon, we replace a discontinuous “sgn” function by a fuzzy system. Then, the reaching control can be defined as Iqr = −(A h)−1 k (t) I f smc

(23)

where I f smc is the output fuzzy system: I f smc = F S MC (S (t), S (t))

(24)

Iqr e f = Iqeq − (A h)−1 k (t) I f smc

(25)

The input membership functions for the switching variable S(t) and the alteration of the switching variable S(t) are fuzzy sets negative (N), zero (Z), and positive (P). The output membership functions for the control variable I f smc are divided into five fuzzy segments namely negative big (NB), negative medium (NM), negative small (NS), zero (ZE), positive small (PS), positive medium (PM), positive big (PB) on the common universe of discourse is taken as [−1 1]. The selected membership function for the input and output variables have triangular shapes as shown in Figs. 3 and 4. Corresponding to these Figs, the thickness of the boundary layer can be changed by modifying the universe of discourse of the fuzzy sets “Z” and “ZE” respectively in the input and output. The rule base expressed by linguistic terms is shown in Table 1. The switching gain is adapted using the following algorithm according to S(t) k˙ (t) = λk |S (t)|

(26)

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Degree of mumbership function

Fig. 3 Membership functions of the inputs (S(t),

S(t))

1 N 0.8 0.6 0.4 0.2 0 -1

Degree of membership function

Fig. 4 Membership functions of the output (I f smc )

1 NB

-0.5

0

0.5

NM

NS ZE PS

PM

-0.5

0

0.5

1

PB

0.8 0.6 0.4 0.2 0 -1

Table 1 Fuzzy rules used

P

Z

dS

S N

Z

P

N

NB

NS

PM

Z

NB

ZE

PB

P

NM

PS

PB

1

where, λk is a strictly positive constant. In fact, k(t) acts like an adaptive filter to minimize the control effort. (2) The Current Control of PMSG by SMC The surfaces of the inner loops responsible for controlling the d-q axis stator currents are described by [10]. 

S (t) = Iq∗ − Iq S (t) = Id∗ − Id

(27)

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The derivative of the surfaces is given by. 

S˙ I q (t) = I˙q∗ + S˙ I d (t) = I˙d∗ +

Rs Lq Rs Ld

Iq + Id −

Ld Lq Lq Ld

Id ωe + Id ωe −

ϕf Lq 1 Ld

ωe −

1 Lq

Vq

Vd

During the sliding mode and the steady-state, we have. S I dq (t) = 0 and S˙ I dq (t) = 0 Hence we deduce. ⎧ ⎨ V qeq = L q I˙∗ + Rs Iq + L d Id ωe + Lq Lq q

⎩ V deq = L d I˙∗ + Rs Id − L q Id ωe d Ld Ld

ϕf Lq

ωe

(28)

(29)

In this work, the reaching control, it is given as. 

  Vqr = kiq tanh S I q (t) Vdr = kid tanh (S I d (t))

(30)

Since the hyperbolic tangent function is continuous and smooth function, the hyperbolic tangent function tanh(x) can be used to reduce the chattering in the sliding mode control by using the hyperbolic tangent function tanh(x) instead of the sgn function sgn(x) [5], where kiq , kid is the positive gain to be set. The outputs control of the regulators Vq∗ , Vd∗ for the d-q axis stator currents are given by 

Vq∗ = Vqeq + Vqr Vd∗ = Vdeq + Vdr

(31)

The block diagram of the speed regulation by AFSMC and the d-q axis stator currents regulation by SMC based on PMSG is shown in Fig. 5. C. Super-Twisting Sliding Mode Controller design In this section, a second-order SMC controller based on the Super-twisting algorithm, and first-order SMC d-q axis stator currents controller are designed. The STSMC consists of two terms: the equivalent control Iqeq and super twisting control Iqr ST . We use the speed error and the surface that is shown in (15), (16), then during the sliding mode and the steady-state the equivalent control law is the same as like as (18). Therefore, the STSMC speed controller can be designed as [6, 8]: Iq∗ = Iqeq + Iqr ST

(32)

Iqr ST = I1 + I2

(33)

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Fig. 5 Control system structure



1

I1 = −λ |S (t)| 2 sgn (S (t)) I˙2 = −γ sgn (S (t))

(34)

where λ and γ are the positive constants, in order to ensure the convergence in finite time, the constants λ and γ can be chosen as follows [7]. ⎧ ⎨λ >



⎩γ2 ≥

ϕ

min 4ϕ 2

max (λ−ϕ) (λ−ϕ)

(35)

min min

The block diagram of the Super-twisting sliding mode controller is illustrated in Fig. 6.

Fig. 6 The block diagram of the STSMC

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4 Simulation Results and Discussion The system studied was simulated in Matlab/Simulink environment with the parameters of the PMSG and the turbine used in the simulation are shown respectively in Tables 2 and 3. The system is initially simulated with the wind speed profile shown in Fig. 7. The Fig. 8 shows that the overshoot of the AFSMC and STSMC controllers are better than that of the PI controller and this last shows that there is a static error in the steady-state, whereas the AFSMC and STSMC follow their reference speed but the STSMC shows that less chattering than AFSMC and PI controller. Figures 9 and 10 show respectively the response of the d-q axis stator currents of the PMSG, which are controlled by the three propositions of control that we studied in the previous section, are named by (PI–PI), (AFSMC–SMC), (STSMC–SMC). From these Figs, it can be noted that there are large peaks in the q axis stator current based on the PI controller and AFSMC at the transient regime and speed variation (at 1 s). Also, the PI controller shows in d axis stator current a peak at the moment of speed variation. On the other, hand the STSMC show that it has small peaks in the q axis and no peaks in the d axis compared to the conventional PI controller and AFSMC. It means that the AFSMC and STSMC ensure a perfect decoupling between the two axes. We notice that there is less chattering in the SMC which has the speed STSMC compared to PI controller and AFSMC. Also using the harmonic spectrum of one Table 2 PMSG Parameters

Table 3 Turbine parameters

Parameter

Value (units)

Rated power

2500 (W)

Rated line current

11 (A)

Rated speed

3000 (rpm)

Mutual inductance in q-axis

7.5 (mH)

Mutual inductance in d-axis

7.5 (mH)

Stator resistance

0.45 ()

Number of pole pairs

3

Permanent magnet flux

0.52 (Wb)

Friction coefficient

0.017 (Nm/rad/s)

Inertia moment

0.00208 (Kg.m2 )

Parameter

Value (units)

Rotor radius

3 (M)

Gear box ratio

5.4

Inertia moment

0.042 (Kg.m2 )

Friction coefficient

0.017 (Nm/rad/s)

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Fig. 7 Wind profile

7

wind (m/s)

6.5 6 5.5 5 4.5 4 0

Mechanical speed (rad/s)

100

0.5

1 time (s)

1.5

2

W* W AFSMC

80

W PI W STSMC

60

94.85

70 94.8

40

65

20

94.75 60

0

0.005 0.01 0.015 0.02

0

0.5

1.5

1.505

1

1.51

1.5

2

time (s)

Fig. 8 Mechanical speed 30

-6

Iq AFSMC Iq PI Iq STSMC Iq*

current Iq (A)

20 -6.5 10

-7

0

-7.5 0.5

20 10 0.55

0 -10

-10 -20

1

0

0.5

1

Time (s)

Fig. 9 PMSG stator current in the q-axis

1.005

1.5

2

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4

Current Id (A)

Id AFSMC Id PI Id STSMC Id*

2

3

0

2

-2 0.99

1

1

1.01

0 -1 -2 -3

0

0.5

1

1.5

2

Time (s)

Fig. 10 PMSG stator current in the d-axis

Fundamental (45.2493Hz) = 11.76 , THD= 2.78% 12 10

Mag

8 6 4 2 0

0

100

200

300

400

500

Frequency (Hz)

Fig. 11 Spectrum harmonic of one phase stator current for PI

phase stator current of the PMSG obtained using the FFT technique for the three controllers which are presented in Figs. 11, 12 and 13. It can be seen that the THD is reduced for STSMC (THD = 1.95%) when compared to AFSMC (THD = 2.29%) and PI controller (THD = 2.78%). Therefore, the STSMC presents a great advantage to ensure the lifetime of PMSG, it means that it protects the machine against the over-current and the vibration which are generated by the chattering phenomenon. In order to test the robustness of the used regulators in speed and currents, the parameters of the PMSG are modified by doubling the values of Inertia moment and friction coefficient. The result presented in Fig. 14 shows that STSMC and PI are robust to the variation of the moment of inertia and viscous friction but the response time of the STSMC is faster than the PI regulator, and for the AFSMC we notice that there are a static error and a large uncertainty to external disturbances in the system. It also found in Figs. 15 and 16 when we varied the parameters of the PMSG that the

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chattering phenomenon of the generator with the proposed (STSMC–SMC) is also less as compared to the other proposed controllers.

5 Conclusion In this paper, the control of autonomous wind energy conversion system based on a PMSG has been studied. However, three types of regulators (PI–PI), (AFSMC– SMC), (STSMC–SMC) to control PMSG has been proposed. Our objective is to make a comparative study between them concerning the control performance and chattering elimination. The simulations show that the STSMC has a better performance in the transient and steady-state. Moreover, it is robust against parameter variations compared with the two other regulators. Besides, the STSMC eliminates the chattering phenomenon in speed and currents and ensures a perfect decoupling

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between the two axes that leads to a great advantage by ensuring the lifetime for the PMSG against vibrations and over-currents. On the other hand, The AFSMC shows that there is a large uncertainty in the system appears in the form of an undesirable error-static when we varied the parameters of the PMSG.

References 1. Barkia A et al (2016) A comparative study of PI and Sliding mode controllers for autonomous wind energy conversion system based on DFIG. In: 2016 17th international conference on sciences and techniques of automatic control and computer engineering (STA), Sousse, Tunisia, pp 612–617 2. Boubzizi S, Abid H, Kharrat M et al (2015) Comparison of adaptive input-output linearization & fuzzy sliding-mode control for induction motor. In: 2015 16th international conference on sciences and techniques of automatic control and computer engineering (STA), Monastir, Tunisia, pp 746–751 3. Saghafinia A et al (2015) Adaptive fuzzy sliding-mode control into chattering-free IM drive. IEEE Trans Ind Appl 51(1):692–701 4. Saghafina A, Ping HW, Uddin MN, Gaied KS (2012) Adaptive fuzzy sliding-mode control into chattering-free induction motor drive. In: 2012 IEEE industry applications society annual meeting, Las Vegas, NV, USA, pp 1–8 5. Li Z, Zhou S, Xiao Y, Wang L (2019) Sensorless vector control of permanent magnet synchronous linear motor based on self-adaptive super-twisting sliding mode controller. IEEE Access 7:44998–45011 6. Meghni B, Dib D, Azar AT (2017) A second-order sliding mode and fuzzy logic control to optimal energy management in wind turbine with battery storage. Neural Comput Appl 28(6):1417–1434 7. Azzaoui ME et al (2017) FPGA implementation of super twisting sliding mode control of the doubly fed induction generator. In: 2017 14th international multi-conference on systems, signals & devices (SSD), Marrakech, pp 649–654 8. Belabbas B, Allaoui T, Tadjine M, Denai M (2017) high order sliding mode controller simulation by a wind turbine for DFIG protection against overcurrent. Electrotehnica, Electronica, Automatica 65:142–147 9. Ayadi M, Naifar O, Derbel N (2017) Sensorless control with an adaptive sliding mode observer for wind PMSG systems. In: 2017 14th international multi-conference on systems, signals & devices (SSD), Marrakech, pp 33–37 10. Larbi M, Gherabi Z, Doudar K (2017) A robust sensorless control of PMSM based on sliding mode observer and model reference adaptive system. IJPEDS 8(3):1016

Open-Switch Faults Based on Six/Five-Leg Reconfigurable AC-DC-AC Converter Sahraoui Khaled and Gaoui Bachir

Abstract The purpose of this paper is the study by simulation of a fault-tolerant control with Pulse Width Modulation (PWM) ac-dc-ac converter supplying a threephase rotor field-oriented induction motor. Before fault occurrence, the fault-tolerant converter operates like a conventional back-to-back six-leg converter and after the fault; it becomes a five-leg converter. Fault-tolerant topology of ac-dc-ac converter that without redundancy have been studied and associated with affective and fast method of fault detection and compensation to guarantee the continuity of service, in the presence of a possible open circuit failure on the level of one of their legs. Although of the presence of open circuit on the level of one of the converter legs the control based on the Zero Sequence Signal (ZSS) assure the service continuity, the simulation results obtained prove that it is possible to maintain the good performance of the drive without redundancy leg (6/5 topology). Keywords Asynchrouns machine · Fault diagnosis · PWM power convertors · Service continuity · Zero sequence signal

1 Introduction Back-to-Back converters are widely used in various industrial areas such as electric machine drives, uninterruptible power supply, unified power quality conditioner and grid connected renewable energy systems [1, 2]. A research survey has reported that the failure rates of these converter are 30%, 26%, 21% and 13% due to their capacitors, printed circuit boards, semiconductors, and soldering, respectively [2, 3]. Therefore, the reliability and performance of PWM converters have been paid a great deal of attention. Recently, fault detection and tolerant control techniques for power S. Khaled (B) · G. Bachir LACoSERE Laboratory, University of Amar Telidj, 03000, BP 37G Laghouat, Algeria e-mail: [email protected] G. Bachir e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_40

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converters have attracted a lot of interest due to their higher reliability and lower maintenance. Recently, the using of power electronic increase dramatically in industrial applications, in the most industrial applications, induction motors are predominantly fed from pulse width modulation voltage source inverter (PWM) for variables speed operation [3]. Voltage three-phase structure static converters are essential elements of many power electronic systems such as inverters machines alternatives, uninterruptible power supplies, and active filters [4, 5]. The security of these systems, their reliability, performance, power quality and continuity of service are today the major concerns in the field of energy [6]. PWMinduction motors are usually more reliable than those supplied directly online. For instance, the problem of faults can occur in the machine or in the converter [7–10]. The use of converters has some drawbacks because the power electronic converters came with an increased possibility of component failures [6]. The failures of a power converter, whether from drivers, its close control, a controllable power components, lead to loss total or partial control of currents in phases. These failures can cause serious system malfunctions. Indeed, it would endanger the system in some cases if the defect is not detected quickly and compensated [2, 10]. Several researchers in the fault-tolerant converter applications have obtained promising results by using an additional leg to guarantee the continuity of service, in [2], the authors studied the fault-tolerant six-leg converter for a wind energy conversion system (WECS) with a doubly-fed induction generator (DFIG). In [11–14], the authors treated the fault detection, isolation, and compensation as an open-circuit and short-circuit tolerant motor drive system and discusses fault-tolerant ac-dc-ac single-phase to three-phase converter. The approach introduced here minimizes the time interval between the fault occurrence and its diagnosis. This paper demonstrates the possibility to detect a faulty switch in less than 40 µs by using a time criterion instead of voltage criterion. In this paper, a more general fault-tolerant converter without redundancy is proposed for the back-to-back three-phase two-level converters based on a 6/5 leg topology. More general, fault-tolerant converter without redundancy leg is proposed for the induction motor drives. The studied system is described briefly in Sect. 2. In Sect. 3, a modeling and control of the two sides of our converter and the induction machine are devoted. The fault detection and control schemes method are used as in [6, 15, 16] discussed in Sects. 4 and 5. Finally, fully simulation results are provided in Sect. 6. The obtained simulation results here prove that the continuity of service is assured although the presence of faults in the switches.

2 System Description A technical advantage of the voltage converter PWM from the decoupling capacitor placed between the network-side converter and the machine side of the converter. The network side converter provides sinusoidal currents and voltage regulation of the

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DC bus and the machine side of the converter which assure the control thereof. This will provide some protection and separate controls for the two sides. By against the important inconvenience of this voltage converter PWM lies in the switching losses and reduce the overall life of the system [1, 14] (Figs. 1 and 2). converter which assure the control thereof. This will provide some protection and separate controls for the two sides. By against the important inconvenience of this voltage converter PWM lies in the switching losses and reduce the overall life of the system [1, 14].

Electrical grid

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3 Modeling and Control of Studied System A. Modeling of converter side grid A PWM rectifier is simply a voltage inverter used in reverse. It produces a DC voltage from an AC supply, It is built based on bidirectional semiconductor components. Also it preferred for frequently applications operating in regenerative mode such as electric drives especially where the AC rectifier is part of the controller. According to the closing or opening of the switches, kij branch voltages can be equal to Uc or 0. Introduces another variable S11, S21 and S31 which taking 1 if the switch ki1 is closed or −1 if the switch opened [17, 18]. ⎤ ⎤ ⎡ ⎤⎡ V1N 2 −1 −1 S11 U c⎣ ⎥ ⎢ −1 2 −1 ⎦⎣ S21 ⎦ ⎣ V2N ⎦ = 6 S31 −1 −1 2 V3N ⎡

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(2)

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B. Modeling of converter side machine The inverter is a power electronics converter that allows conversion from continuous to alternative. It works by forced commutation and designed generally based transistors ⎡ ⎤ ⎤ ⎡ ⎤⎡ V1N 2 −1 −1 f 11 ⎢ ⎥ Uc ⎣ (3) −1 2 −1 ⎦⎣ f 21 ⎦ ⎣ V2N ⎦ = 6 f 31 −1 −1 2 V3N The input current of inverter can be expressed by if =

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C. Modeling of the machine The equations which describe the dynamic behavior of asynchronous machine [6]. Stator voltage equations 

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4 Fault Tolerant System In case of fault occurrence in sensors, actuators or other parts of a complex system, conventional systems may produce undesirable results. In critical applications where the need for safety, reliability and fault tolerance is high, it is necessary to anticipate the fault tolerance ability against unpredicted faults to increase the reliability and availability of the system while providing acceptable performances [10, 13]. These types of systems are known as Fault-Tolerant Systems (FTS). In other words, an FTS can maintain stability and acceptable performance after a fault occurrence. The performance of the system in normal and post-fault conditions may be different. An FTS may be based on redundancy or not. In an FTS with redundancy, a redundant part will replace the faulty part, after fault detection and upon reconfiguration. In this case, by correctly designing the FTS, the same operation capability can be obtained. However, in some cases, some degree of performance degradation is accepted. In these cases, a suitable reconfiguration might be enough to assure the minimum required performance [10, 18, 19]. After fault detection and isolation, a reconfiguration is necessary. Reconfiguration is required in both hardware and software parts of the system. Reconfiguration in the software part consists of modifying the references and the controller characteristics. The system is composed of three-phase electrical sources and an induction machine, with an ac-dc-ac fault tolerant converter between them. All different parts are explained further in detail. D. 6/5 leg converter topology The studied converter is shown in Fig. 4 [14]. In normal operation (before the fault occurrence), the converter is a conventional back-to-back converter with three additional bidirectional switches (Fig. 4a). These bidirectional switches are used for the converter reconfiguration after the fault isolation, i.e. before the fault occurrence, these switches are all off. The converter configuration after a fault in one of the inverter legs is shown in Fig. 4b. In this case, the fault has been occurred at either B31 or B32. Depending on the application, each side of the converter can be connected to a source, load or machine. For example, in drive application, one side might be connected to a three-phase source and the other will supply a three-phase achine. For a WECS with DFIG, side 2 will supply the rotor of the machine. In this paper, it is assumed that side 1 of the converter is connected to a three-phase balanced sinusoidal source and the second side of the converter is connected to a balanced three-phase load. It should be noted that the voltage producing capability in the five-leg converter is lower than the six-leg converter, meaning that with the same dc-link voltage, it can produce smaller three-phase voltages at its AC terminals, compared to the six-leg

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Fig. 4 6/5 leg fault tolerant converter a pre-fault b post-fault (after reconfiguration)

converter. Also, the current rating in the shared leg of the converter is larger than in the other legs. E. Reconfigurable Control of the converter Before the fault occurrence, the fault-tolerant converter is performing like a conventional six-leg converter and any PWM method can be used for its control. After the fault detection and reconfiguration, one leg will be shared between the two sides of the converter and the faulty leg will be disconnected. So, the converter will become a five-leg structure. Among the several PWM approaches studied in the literature for this 5- leg post-fault topology, it seems that the suggested method in [15, 16]. Therefore this approach is used in this paper.

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In this method, a so-called double zero-sequence injection method is performed. ∗ (x ∈ {a, b, c}, i ∈ {1, 2}) for both sides of the converter Voltage reference signals vxi are calculated using the appropriate methods depending on the targeted applications. Then, a Zero Sequence Signal (ZSS) is added to these values to form the modulation signals (10). ZSS does not change the output line-to-line and phase voltages, therefore it is used as a degree of freedom to reduce the current harmonics and improve the dc-bus utilization. ∗ (t) + vzssi (t) vxi (t) = vxi

(10)

where vzssi (i ∈ {1, 2}) is the ZSS for side i. ZSS signal for a three-phase system is computed as follows:

∗ ∗ 1 max(vai (t) + vbi (t) + vci∗ (t))+ vZSSi (t) = − ∗ ∗ (t) + vbi (t) + vci∗ (t)) 2 min(vai

(11)

Since there are 6 v references and only 5 legs, a reduction in the number of voltage references is required. Reduction of the number of voltage references can be made by using an inverse lookup table [2, 11], this is realized by adding another ZSS in accordance to the converter configuration in five-leg mode. The new set of voltage references assuming that two c legs of the two sides are connected in the five-leg mode are calculated as: v A1 = va1 + vc2 ; v B1 = vb1 + vc2 v A2 = va2 + vc1 ; v B2 = vb2 + vc1 vC = vc1 + vc2

(12)

Since the same signal is added to all three reference values of the converter, the fundamental output voltage will not be affected. Figure 5 shows the principle of this method. In Fig. 5a the ZSS for a conventional converter is shown. This ZSS injection will be repeated for both sets of three-phase voltage references at the two sides of the converter, then by using (11) a new set of five voltage references are produced and sent to the PWM unit, as shown in Fig. 5b. F. Voltage reference generation In fact, in healthy or faulty cases, the controls of both converters are not the same and must be changed as quickly as possible to avoid any discontinuity or transient in the currents The control unit of the 6/5 leg converter produces the reference voltages and sends them to the PWM blocks [21, 22]. For each application case, depending on the source/load connected at two sides of the converter, there are different methods for producing the reference voltages.

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Fig. 5 a Principle of PWM module for one converter; b for 5-leg converter when B3 legs are shared

5 Fault Detection and Reconfiguration Fast fault detection is essential for fault-tolerant systems. Here the method detailed in our earlier contributions is used for detection of fault and its location [20]. Using this method, it is possible to detect the fault quickly and efficiently. In [20] it is shown that the fault occurrence in each leg can be diagnosed by comparing the measured and estimated pole voltages. However, in reality, due to measurement and discretizing errors, and mainly because of non-ideal behaviors of switches and drivers (turn-off and turn-on propagation time and dead time generated by the drivers), the voltage error is not zero during normal operation [23, 24].

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To avoid false fault detections, two adjustments are employed to compensate for the effect of the measurement errors and delays. For this purpose, first, the absolute value of the error between measured and estimated pole voltage is calculated. Then, this value applied to a comparator with a threshold value h, to determine if the difference between the measured and estimated voltages is large enough to be considered as an error. Then, this signal is applied to an upcounter that computes the number of pulses while the output of the first comparator is high. In the other words, the output of the up-counter corresponds to the time during which vknm (measured voltage) and vknes (estimated voltage) are different [7, 13]. Consequently, the fault occurrence is detected using simultaneously a time criterion and a ‘voltage criterion’. To do this, the up-counter output is applied to a second comparator with a threshold value of ‘N’. In this way, false fault detection due to semiconductor switching is avoided and fault can be detected very fast. Figure 6 shows one leg of the converter, while there is an open-circuit fault in the upper switch. Note that in some cases, based on the direction of the current ik, there might be a condition that Dk conducts instead of Sk; therefore, in this case, the converter operates normally and the fault cannot be detected. For example, while Tk = 1 and ik < 0, Dk is on. Here Tk is the command signal for the upper switch of the leg k. Tk = 0 indicates that the switch is commanded to be open, whilst Tk = 1 means that the switch is commanded to be closed. The switch commands in each leg are complementary. Figure 7 shows the mentioned fault detection principle. The stateflow diagram of the fault detection scheme is presented in Fig. 8. For a short-circuit fault of a switch, the very fast acting fuses isolate the faulty leg. Lets now consider two converter legs as shown in Fig. 9 and suppose that there is a short circuit fault in Sk. When Tk = 1 the leg k operates normally. However when the Tk = 0, the dc-link is short-circuited. Figure 9b shows the short circuit path. The short circuit current is limited by only the short circuit resistance, which is a small value. That is why suitable protection Fig. 6 One leg of converter during an open-circuit fault in the upper switch

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Fig. 7 Fault detection scheme

Fig. 8 State-flow diagram of the fault detection scheme

is needed in this case. In this case, if the fuse clearing time is smaller than the fault detection time, the faulty leg will be isolated by the fuses. If the fault is detected before the fuses act, the command of the faulty leg switches is set at zero and the corresponding bidirectional switch will be turned on. In this case, there is still at least one fuse in the short circuit path; therefore the short circuit will be cleared after fuse operation (Fig. 9c).

6 Simulation Results We simulated the system studied by the following test: At first, we start our machine unloading ref = 100 rad = s then we inserting and removing a torque load 12 N·m in instants t = 0.3 s and t = 0.7 s respectively, after we change direction of rotation ref = −100 rad = s at t = 1 s, after that we inserting and removing a torque load −12 N·m in instants t = 1.4 s and t = 1.7 s respectively. We considered that the fault has occurred in the converter side machine in the leg B3 at t = 0.5 s. The fault-tolerant control is activated after 40 µs, by the homopolar injection method (ZSS) which ensures the continuity of service of our system.

606 Fig. 9 Equivalent circuit for the leg ‘k’. a A short circuit fault in Sk. b Short circuit path after the fault. c Short circuit path when the fuse clearing time is larger than the fault detection time

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The obtained results are presented in Fig. 10. We observed that the converter operates normally, as a healthy system. DC bus voltage control stays stable (Fig. 11c) 40

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(we constrained to increase the DC bus voltage to ensure the necessary capacity in the production of the three alternating voltages). Figure 11a presents the line current in the input system, we can observe converter side grid before and after the fault detection and compensation, it is in phase with the voltage quasi sinusoidal and does not mark a significant peak. The other controlled quantities suit regularly their references (rotor speed, torque, and flux). Also, we can note that no noticeable deformation appears on the waveforms, because the time of fault detection and composition is shorter (40 µs) than the commutation period of the switches. The ac drive system configuration employs a five-leg converter and can handle the failure of all the power switches connected to the positive or negative rail of the dc-link. The operation of drive in the post-fault is intended for short time periods to avoid problems related to losses and magnetization. However, the magnetization of the machine as resulted from its operation in the post-fault condition can be removed once the drive is repaired by designing a suitable pulse width modulation pattern. The simulation results have demonstrated the feasibility of the proposed strategy for improving the reliability of a three-phase ac motor drive system.

7 Conclusion In this paper, we devoted our study of the topology without redundancy leg converter “back-to-back”, named in this paper “6/5 leg converter”, with a very fast fault detection scheme and suited reconfigurable control is studied. Moreover, a specific and appropriate “fault-tolerant” controller was necessary to guarantee the fast and efficient reconfiguration of the system, not only at the level of the converter topology but also at the level of its control, which must be modified during the commutation a 6-leg topology with a 5-leg topology. However, we do not know at which leg possible defect would occur and which leg would be shared. Similarly, an increase in the DC bus voltage reference may be required after reconfiguration to maintain the converter’s rated capacity if system operation requires it. It is, therefore necessary to dimension accordingly the entire 6-leg “fault-tolerant” converter if it is desired to guarantee operation in normal mode after defect dimensioning. Finally, we have proposed methods to reduce the number of voltage sensors required for fault detection. This ability to reduce the number of voltage sensors reduces the additional cost of fault tolerance.

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References 1. Extremiana G, Abad G, Arza J, Torre I (2013) Rotor flux oriented control of induction machine based drives with compensation for the variation of all machine parameters. Pol Acad Sci Tech Sci 61(2):309324 2. Shahbazi JM, Zolghadri MR, Poure P, Saadate S (2013) Wind energy conversion system based on DFIG with open switch fault tolerant six-legs AC-DC-AC converter. In: 2013 IEEE international conference on industrial technology (ICIT). IEEE, pp 1656–1661 3. Dehghan SM, Amiri A, Mohamadian M, Andersen MAE (2013) Modular space-vector pulsewidth modulation for nine-switch converters. IET Power Electron 6(3):457467 4. Jones M, Dujic D, Levi E (2008) A performance comparison of PWM techniques for five-leg VSIs supplying two-motor drives. Proc 5. Frapp E, De Bernardinis A, Bethoux O, Marchand C, Coquery G (2010) Action palliative par le convertisseur statique en cas de dfaillance dun gnrateur PAC modulaire de puissance, Electron, puissance du Futur 6. Sahraoui K, Gaoui B, Mokrani L, Belarbi K (2016) Fault tolerant control of redundant topology of pwm ac-dc-ac converter supplying an electromechanical drive. In: 2016 8th international conference on modelling, identification and control (ICMIC). IEEE, pp 374–380 7. Jacobina CB, De Freitas IS, Da Silva ERC, Lima AMN, De Araujo Ribeiro RL (2006) Reduced switch count DC-link ACAC five-leg converter. IEEE Trans Power Electron 21(5):13011309 8. Sae-Kok W, Grant DM, Williams BW (2010) System reconfiguration under open-switch faults in a doubly fed induction machine. IET Renew Power Gener 4(5):458–470 9. Peng T, Dan H, Yang J, Deng H, Zhu Q (2016) Openswitch fault diagnosis and fault tolerant for matrix converter with finite control set-model predictive control. IEEE Trans Ind Electron 63(9):59535963 10. Sahraoui K, Kouzi K, Ameur A (2017) A robust sensorless iterated extended Kalman filter for electromechanical drive state estimation. Electrotehnica, Electronica, Automatica 65(2) 11. Daigavane MB, Vaishnav SR, Shriwastava RG (2015) Sensorless field oriented control of PMSM drive system for automotive application. In: 2015 7th international conference on emerging trends in engineering & technology, no 2, pp 106–112 12. Karimi S, Gaillard A, Poure P, Saadate P (2008) FPGA based real-time power converter failure diagnosis for wind energy conversion systems. In: IEEE transactions on industrial electronics, vol 55, No 12, pp 4299–4308 13. Shahbazi M, Poure P, Saadate S, Zolghadri MR (2011) Five-leg converter topology for wind energy conversion system with doubly fed induction generator. Renew Energy 36(11):3187– 3194 14. Jacobina CB, Araujo Ribeiro RL, Lima AMN, da Silva ERC (2003) Fault-tolerant reversible AC motor drive system. In: IEEE transactions on industry applications vol 39, pp 1077–1084 15. Jones M, Vukosavic SN, Dujic D, Levi E, Wright P (2008) Five-leg inverter PWM technique for reduced switch count two-motor constant power applications. IET Electr Power Appl 2(5):275– 287 16. Jones M, Dujic D, Levi E (2008) A performance comparison of PWM techniques for five-leg VSIs supplying two-motor drives. IECON, pp 508–513 17. Rodrigues PLS, Jacobina CB, de Freitas NB (2019) Singlephase universal active power filter based on acdcac converter with eight controlled switches. IET Power Electron 18. Malinowski A, Yu H (2011) Comparison of embedded system design for industrial applications. IEEE Trans Industr Inf 7(2):244–254 19. Shahbazi M, Poure P, Saadate S, Zolghadri MR (2013) FPGAbased reconfigurable control for fault-tolerant back-to-back converter without redundancy. IEEE Trans Ind Electron 60(8):3360–3371 20. Shahbazi M, Saadate S, Poure P, Zolghadri M (2016) Open-circuit switch fault tolerant wind energy conversion system based on six/five-leg reconfigurable converter. Electr Power Syst Res 137:104–112. End the bibliography

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A Robust Model Predictive Control for a DC-DC Boost Converter Subject to Input Saturation: An LMI Approach O. Hazil, S. Bououden, and M. Chadli

Abstract The design of robust model predictive controller (RMPC) for nonlinear system under actuator saturation is a challenging topic in the area of systems control. Yet. A new RMPC algorithm is developed here for control of a DC-DC boost converter subject to actuator saturation. Based on the saturating actuator property, an appropriate linear feedback control law is designed that minimises an infinite horizon cost function within the framework of linear matrix inequalities. In particular, it is shown that the solution of the optimisation problem can stabilise the system under actuator nonlinearity. The extensive simulation results are presented to illustrate the efficiency of the proposed control scheme. Keywords Rrobust model predictive controller · DC-DC boost converter · Actuator saturation · Linear matrix inequalities

1 Introduction Power converters are used extensively in most of the power supply systems such as personal computers, laptops, aircrafts and electronic equipment. A DC-DC converter is a switching circuit, which transform a certain electrical voltage to another level

O. Hazil Centre de Développement des Energies Renouvelables CDER, B.P. 62, Route de l’observatoire, Bouzaréah, 16340 Alger, Algérie e-mail: [email protected] Faculty of Technology, Department of Electronics, University of Constantine 1, Campus A. Hamani, Route Ain El Bey, 2500 Constantine, Algeria S. Bououden (B) Department of Industrial Engineering, University Abbes Laghrour, Khenchela, Algeria e-mail: [email protected] M. Chadli IBISC Lab.-UEVE, University of Paris-Saclay, 40 rue Pelvoux, Evry, France e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_41

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of voltage, this is obtained by switches operating (open or closed) at high frequencies; the control objective of such devices is to maintain regulation of the output voltage at the desired value [1, 2]. Generally, the dynamic behavior of DC-DC converters is usually described by bilinear models with non-minimum phase, under saturating control signal. In spite of nonlinearities, the linear feedback control of power converters based on the small signal control theory proposed by Middlebrook and Cuk [3] is widely applied at present. For instance, the proportional–integral (PI) controllers [4–6] and PID controllers [7, 8]. The models used to derive the control strategy usually neglect the high-frequency ripple considering that the switching time is small enough [9]. In DC-DC converters, the duty ratio is bounded in the interval [0, 1]. The classical control techniques applied to power converters do not take into account input saturation constraints, which can lead to degradation of performance and even to destabilization of the closed-loop system. In this work, we interest to exploit the ability of the linear matrix inequality (LMI) based MPC algorithm for the control of systems with actuator non-linearity, in which a new approach consists of a saturation model based on a deadzone nonlinearity is used to handle the saturation non-linearity. At each time step, a minimization problem of an infinite horizon performance index is solved to obtain a state feedback control law. The stability of the system is guaranteed in the sense of Lyapunov, and the adequate state feedback synthesis condition is obtained by solving an LMI optimisation problem, which directly incorporates input saturation. By this means a computationally cost effective algorithm is proposed, which is implementable for a wide range of systems with actuator non-linearity, This paper is organized as follows. Section 2 gives the averaged model of DCDC boost converter and the mathematical formulation of the problems. Section 3 describes the mathematical formulation of the proposed RMPC problem with guaranteed stability as an LMI-based optimization problem. Finally we give some simulation results to test the robustness of the proposed control law in Sect. 4.

2 Averaged Model of Basic PWM Boost Converter A. State-space of the boost converter The schematic circuit diagram of boost DC-DC converter under consideration is shown in Fig. 1, and the relevant control signal. In Fig. 1, vs is the output voltage and vin the line voltage. The output voltage must be controlled using the duty ratio d in order to keep the given constant vref value, the diode D is on inverse polarization, R models the converter load, while C and L represent, respectively, capacitor and inductor values, the switch sw was a power transistor controlled by a binary signal ub . The binary signal that triggers on and off the switches is controlled by a fixed-frequency pulse width modulation (PWM) circuit (Fig. 2). The constant switching frequency is 1/T s , with T s the switching period is given by the sum of T on (when ub = 1) and T off (when ub = 0) and the ratio

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iL L

vin

D

sw

C

vC

R

vs

d (t )

PMW Fig. 1 Schematic of the boost converter

v (t )

u b (t )

d (t )

T on

T off

T Fig. 2 Waveforms of the PWM process

of T on /(T on + T off ) is the duty cycled d(t). Duty cycle is compared with a sawtooth signal v(t) of amplitude equals to 1. Consequently 0 ≤ d ≤ 1. We assume that the converter operates in continuous conduction mode (CCM) and that the inductor current is always larger than zero. So, the studied converter has two working topologies corresponding to its switch states (Fig. 3). Choosing the state vector as x = [i L , vC ]T where vc is the capacitor voltage; iL is the inductor current, the general equation that governs the operation of the boost converter is: 

x˙ = Ai x + Bi y = Ci x

(1)

where: i = 1 to the first configuration described in Fig. 2a and: i = 2 the second configuration shown in Fig. 2b. Where: iL

iL

vin

L vC

(a)

C

R

vs

vin

L

vC C

(b)

Fig. 3 Equivalent circuit of the boost converter. a Switch on. b Switch off

R

vs

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     0 0 1/L A1 = , B1 = , C1 = 0 1 0 −1/RC 0       0 −1/L 1/L A2 = , B2 = , C2 = 0 1 1/C −1/RC 0 So the combination of state-space representation of mode 1 (on mode) and mode 2 (off mode) induces the following linear state space representation: 

x˙ = Ad x + Bd u y = Cx

(2)

With Ad = d A1 +d  A2 ,Bd = d B1 +d  B2 ,Cd = dC1 +d  C2 and d  = 1−d, u = d. This gives:  Ad =

   0 −(1 − d)/L 1/L , Bd = , Cd = [ 0 1 ] (1 − d)/C 1 − /RC 0

B. Discrete-time state-space model The MPC is to be fed with a discrete time model which is easily obtained from the continuous model assuming that the switching period T s is much smaller than the time constants associated with the circuit, the following discrete-time model is obtained using the forward Euler approximation, from the continuous time version (2): x(k + 1) = (I + Ts Ad )x(k) + Ts Bd d(k)

(3)

Using this assumption, the discrete-time state-space model of the buck-boost converter can be written as:        x1 (k + 1) 1 −(1 − d(k))Ts /L x1 (k) Ts /L = + d(k) x2 (k + 1) (1 − d(k))Ts /C 1 − (Ts /RC) x2 (k) 0 (4) The system in (4) is a nonlinear system since there are products of two input signals. It is therefore mandatory to make some sort of linearization at some point. We can rewrite the nonlinear model of boost converter as follows: x(k) = f (x(k), sat(u(k)))

(5)

where sat(u(k)) is the saturated control input u(k) of the boost converter, constrained between 0 and 1 and is defined as:

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⎧ if u(k) < 0 ⎨0 sat(u(k)) = u(k) if 0 ≤ u(k) ≤ 1 ⎩ (0,1) 1 if u(k) > 1

(6)

The total instantaneous quantities can be presented as the sum of the DC and AC components, x = x˜ + X

(7)

d = d˜ + D

(8)

where X and D represent the equilibrium values and x˜ and d˜ are the perturbed values of state and duty cycle, The linearized averaged model of the DC-DC converter can be written as ˜ + K X) x(k + 1) = A x(k) ˜ + AX + B sat (K x(k) (0,1)

(9)

where the matrices A and B are given by:



    δ f

δ f

1 −(1 − D)Ts /L Vc Ts /L ,B = = = A= (1 − D)Ts /C 1 − Ts /RC −I L Ts /C δx x=X δd x=X d=D

d=D

The control input is subject to a no symmetric actuator saturation, we can use the method proposed in [9] to transform it to a symmetric saturation. The function sat(u(k)) can be can be written as follows: ⎧ if K x(k) ˜ + KX 1

(10)

Which can be rearranged in the following form: ⎧ if K x(k) ˜ < −K X ⎨0 ˜ = u(k) if K X ≤ K x(k) sat (K x(k)) ˜ ≤1− KX ⎩ (K X,1−K X ) 1 if K x(k) ˜ + KX >1− KX

(11)

Both saturation functions are equivalent if we add KX to (11), that is: ˜ + K X) = sat (K x(k)

(0,1)

sat

(K X,1−K X )

(K x(k)) ˜

(12)

Assume that the operating point is an equilibrium point which the steady state part of the model is equal to zero

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f (X, U ) = AX + K X = 0

(13)

The steady state control signal is KX = 0.5. The system described in (9) can be written as follows: ˜ x(k ˜ + 1) = A x(k) ˜ + Bsat(δ(k)) ˜ where sat(δ(k)) =

sat

(K X,1−K X )

(14)

(K x(k)) ˜ the new control input and the new model can

take into account the symmetric saturation model as in [10]: −u 0 < u(k) < u 0 ↔ −K X < δ(k) < K X

(15)

We defined a new decentralized deadzone nonlinearity ˜ ˜ ˜ φ(k) = sat(δ(k)) − δ(k)

(16)

The closed-loop system is rewritten as: ˜ x(k ˜ + 1) = (A + B K )x(k) ˜ + B φ(k)

(17)

The saturation obtained in (18) is a symmetric saturation and the following lemma can be used to handle the saturation nonlinearity Lemma 1 ([11]) Assume that the saturation constraint is defined as (18), let: φ = sat(δ) − δ

(18)

Then, there exists a real number ε ∈ (0, 1) such that φ T φ ≤ εδ T δ

(19)

where,ϕ = [ϕ1 , ϕ2 , . . . , ϕm ]T ∈ R m and ϕi = (i = 1, 2, . . . , m) is the dead-zone nonlinearity function. Proof See [12]. Moreover, we have the following technical lemmas for later use. Lemma 2 (Schur complements) The LMI 

 Q(x) S(x) >0 S(x)T R(x)

(20)

In which, Q(x) = Q(x)T , R(x) = R(x)T and S(x) are affine functions of x, and is equivalent to

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R(x) > 0, Q(x) − S(x)R(x)−1 S(x)T > 0

(21)

Q(x) > 0, R(x) − S(x)Q(x)−1 S(x)T > 0

(22)

or, equivalently

Proof See [13]. Lemma 3 For any matrices (or vectors) X and Y with appropriate dimensions, we have: X T Y + Y T X ≤ α X T Y + α −1 Y T Y

(23)

where α > 0 C. Uncertainty Model We consider that the load R at the operating point is uncertain or time-varying parameter. Then, matrice A depend on such uncertain which have been grouped in a vector p, and we can express (17) as a function of these parameter ˜ x(k ˜ + 1) = (A( p) + B K )x(k) ˜ + B φ(k)

(24)

Generally, the vector p consists of N uncertain parameters p =(p1 , …, pN ), where each uncertain parameter pi is bounded between a minimum and a maximum value p¯ i and P i  pi ∈ pi , pi

(25)

The admissible values of vector p are constrained in an hyperrectangle in the parameter space ɌN with L = 2 N Vertices {ʋ1 , …, ʋL }. The images of the matrix A(p) for each vertex ʋi corresponds to a set {j 1 , …, j L }. The components of the set {j 1 , …, j L }are the extrema of a convex polytope, noted Co{j 1 , …, j L }, which contains the images for all admissible values of p if the matrix A(p) depends linearly on p, that is A( p) ∈ Co{ς1 , . . . , ς L } =

L 

λi ς1 , λi ≥ 0,

i=1

L 

 λi

(26)

i=1

In this context, we consider that N = 2 and the parameter vector p∈[1/R] where: 1/R ∈ [1/Rmax , 1/Rmin ]

(27)

Since the matrice A depend linearly on the uncertain parameter 1/R, we can define a polytope of L = 2 Vertices that contains all the possible values of the uncertain

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matrices. The vertices of the polytopic model are:    1 −(1 − D)Ts /L 1 −(1 − D)Ts /L , A2 = A1 = (1 − D)Ts /C 1 − Ts /Rmin C (1 − D)Ts /C 1 − Ts /Rmax C 

3 Proposed Robust Model-Based Predictive Control Using LMIs In this section, a synthesis approach to robust MPC using LMIS is presented for a class of systems under saturating control signal in the presence of uncertainties and then, we incorporate an input constraint. A. The proposed RMPC ˜ + i|k ) = K x(k In order to design a state-feedback control law δ(k ˜ + i|k ), i ≥ 0 for (24), one may consider the minimisation problem of the infinite horizon quadratic performance index as follows: J (k) =

∞ 

˜ + i|k ))T Rsat(δ(k ˜ + i|k )) (28) x(k ˜ + i|k )T Q x(k ˜ + i|k ) + sat(δ(k

i=0

where R(i), Q(i) are two positive definite states and control weights respectively. Moreover, the input constraints are considered as: 2   ˜ 2 ,i ≥ 0 δ(k + i|k ) ≤ δmax 2

(29)

where δmax is the maximum control input. Let us introduce a quadratic function V (x) = x T Px, P > 0 of the state x(k|k) of the system (24), with V (0) = 0. At sampling time k, suppose the following inequality is satisfied ˜ + i|k ) V (k + i + 1|k ) − V (k + i|k ) ≥ −(x(k ˜ + i|k )T Q x(k T ˜ + i|k )) ˜ + i|k )) Rsat(δ(k + sat(δ(k

(30)

Summing (30) from i = 0 to i = ∞, we have ˜ ) − x(k|k ˜ )T P x(k|k ˜ ) ≥ −J x(∞|k ˜ )T P x(∞|k

(31)

If the resulting closed-loop system for (24) is stable, x(∞|k) must be zero and result in

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J ≤ x(k|k ˜ )T P x(k|k ˜ ) ≤ −γ

(32)

where γ is a positive scalar and is regarded as an upper bound of the objective in (28) ∞ 

˜ + i|k ))T Rsat(δ(k ˜ + i|k )) ≤ γ x(k ˜ + i|k )T Q x(k ˜ + i|k ) + sat(δ(k

(33)

i=0

Theorem 1 At each sample time k, consider the discrete-time system (3) and assume x(k|k ˜ ) is the measured state of x(k). ˜ The state feedback control law that minimizes the upper bound γ on the infinite horizon quadratic performance index J (k)and the closed-loop system within an invariant ellipsoid

robustly stabilizes  ˜ + i|k ) = K x(k ˜ + i|k ), i ≥ 0, G = γ P−1 ε = x˜ x˜ T Q −1 x˜ ≤ 1) , is given by δ(k −1 and K = Y G Where G,G > 0 and Y are the solutions to the following LMIs: γ subjecte to   I x(k|k ˜ )T ≥0 x(k|k ˜ ) G ⎡ G √ ⎢ (1 + α1 )(Ai + B K ) ⎢ √ ⎢ QG ⎢ √ ⎢ + α2 )RY (1 ⎢  ⎢   ⎢ ε 1 + α1−1 BY ⎣    ε 1 + α2−1 Y  2  δmax I Y ≥0 YT G min

G,Y,γ ,ξ1 ,ξ2

∗ G 0 0

∗ ∗ γI 0

∗ ∗ ∗ γI

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

⎤

⎥ ⎥ ⎥ ⎥ ⎥≥0 ⎥ ⎥ 0 0 0 ζ1 I ∗ ⎥ ⎦ 0 0 0 0 ζ2 I

G − ξ1 I > 0 G − ξ2 I > 0

(34)

The symbol ‘*’ denotes symmetric elements in the matrix Proof To obtain (34), the modified quadratic function V (x) ˜ is required to satisfy: ˜ + i|k ) V (k + i + 1|k ) − V (k + i|k ) ≥ −(x(k ˜ + i|k )T Q x(k T ˜ + i|k )) ˜ + i|k )) Rsat(δ(k + sat(δ(k

(35)

Now, by substituting the state space (24), in inequality (35) results in ˜ + i|k)] ˜ + i|k)]T P[(A + B K )x(k ˜ + i|k) + B φ(k [(A + B K )x(k ˜ + i|k) + B φ(k ˜ + i|k ))T Rsat(δ(k ˜ + i|k ) + x(k ˜ + i|k)T Q x(k ˜ + i|k) + sat(δ(k − x(k ˜ + i|k)T P x(k ˜ + i|k) ≤ 0

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˜ as Defining the function g(x, ˜ δ) ˜ = [(A + B K )x(k ˜ + i|k)]T g(x, ˜ δ) ˜ + i|k) + B φ(k ˜ + i|k)] P[(A + B K )x(k ˜ + i|k) + B φ(k = [(A + B K )x(k ˜ + i|k)]T P[(A + B K )x(k ˜ + i|k)] ˜ + i|k)] + [(A + B K )x(k ˜ + i|k)]T P[B φ(k ˜ + i|k)]T + [(A + B K )x(k ˜ + i|k)]P[B φ(k ˜ + i|k)]T P[B φ(k ˜ + i|k)] + [B φ(k ˜ becomes Applying Lemma 3, the upper bound of g(x, ˜ δ) ˜ ≤ (1 + α1 )[(A + B K )x(k g(x, ˜ δ) ˜ + i|k)]T P[(A + B K )x(k ˜ + i|k)] ˜ + i|k)]T P[B φ(k ˜ + i|k)] + (1 + α1−1 )[B φ(k Consider P ≤ λ1,max I ≤ μ1 I , then ˜ ≤ (1 + α1 )[(A + B K )x(k ˜ + i|k)]T P[(A + B K )x(k ˜ + i|k)] g(x, ˜ δ) −1 T T ˜ ˜ + (1 + α1 )μ1 B B φ(k + i|k) φ(k + i|k) ˜ + i|k) in the above equation is bounded as φ(k ˜ + The term involving φ(k ˜ + i|k) ≤ εδ(k ˜ + i|k)T δ(k ˜ + i|k), then i|k)T φ(k ˜ ≤ (1 + α1 )[(A + B K )x(k g(x, ˜ δ) ˜ + i|k)]T P[(A + B K )x(k ˜ + i|k)] −1 T ˜ T˜ + ε(1 + α1 )μ1 B B δ(k + i|k) δ(k + i|k) ˜ as Moreover, we defined the function h(δ) ˜ + i|k ) ˜ = sat(δ(k ˜ + i|k ))T Rsat(δ(k h(δ) ˜ + i|k )) − δ(k ˜ + i|k ) + δ(k ˜ + i|k )]T = [sat(δ(k ˜ + i|k )) − δ(k ˜ + i|k ) + δ(k ˜ + i|k )] × R[sat(δ(k T ˜ + i|k ) + δ(k ˜ + i|k )] ˜ + i|k ) + δ(k ˜ + i|k )] R[φ(k = [φ(k ˜ + i|k ) + φ(k ˜ + i|k )T R δ(k ˜ + i|k ) ˜ + i|k )T R φ(k = φ(k ˜ + i|k )T R δ(k ˜ + i|k ) ˜ + i|k )R δ(k ˜ + i|k )T + δ(k + φ(k ˜ becomes Applying Lemma 3, the upper bound of h(δ) ˜ + i|k )T R δ(k ˜ + i|k ) + (1 + α2−1 )φ(k ˜ + i|k )T R φ(k ˜ + i|k ) ˜ ≤ (1 + α2 )δ(k h(δ) Consider R ≤ λ2,max I ≤ μ2 I ,then

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˜ + i|k )T φ(k ˜ + i|k ) ˜ ≤ (1 + α2 )δ(k ˜ + i|k )T R δ(k ˜ + i|k ) + (1 + α2−1 )μ2 φ(k h(δ) ˜ + i|k) in the above equation is bounded as φ(k ˜ + The term involving φ(k T ˜ T˜ ˜ i|k) φ(k + i|k) ≤ εδ(k + i|k) δ(k + i|k), then ˜ + i|k )T δ(k ˜ + i|k ) ˜ + i|k )T R δ(k ˜ + i|k ) + ε(1 + α2−1 )μ2 δ(k ˜ ≤ (1 + α2 )δ(k h(δ) ˜ and h(δ) ˜ in the inequality (54), the following condition holds By replacing g(x, ˜ δ) for all i ≥ 0 ˜ + i|k)]T P[(A + B K )x(k ˜ + i|k)] (1 + α1 )[(A + B K )x(k −1 T ˜ T˜ ˜ + i|k) + ε(1 + α1 )μ1 B B δ(k + i|k) δ(k + i|k) + x(k ˜ + i|k)T Q x(k −1 T ˜ + i|k )T δ(k ˜ + i|k ) ˜ + i|k ) R δ(k ˜ + i|k ) + ε(1 + α2 )μ2 δ(k + (1 + α2 )δ(k − x(k ˜ + i|k)T P x(k ˜ + i|k) ≤ 0 ˜ + i|k ) = K x(k Consequently, by replacing δ(k ˜ + i|k ), the inequality can be expressed as: (1 + α1 )[(A + B K )T P[(A + B K )] + ε(1 + α1−1 )μ1 B T B K T K + Q + (1 + α2 )K T R K + ε(1 + α2−1 )μ2 K T K − P ≤ 0 Pre- and post-multiplying by G > 0, and substituting G = γ P−1 , Y = KG, 1 = γ μ1 and 2 = γ μ2 , and then applying Schur complements, (19) becomes ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

√

G (1 + α1 )(Ai + B K ) √ QG √ + α2 )RY (1    ε 1 + α1−1 BY    ε 1 + α2−1 Y

∗ G 0 0

∗ ∗ γI 0

∗ ∗ ∗ γI

∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

⎤

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Setting K = Y G−1 then δ˜ = K x˜ = Y G −1 and (38) can be written as: 2   ˜ ˜ + i|k ) 22 δ(k + i|k ) = K x(k 2  2 = Y G −1/2 (G 1/2 x(k ˜ + i|k ))2  2 ≤ Y G −1/2  2

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4 Simulations and Results The performances of the proposed robust MPC control design applied to the boost converter are illustrated through simulations. The parameters nominal values of the buck-boost converter are given by R = 20 , C = 20 μF, L = 3.5 mH, vs = 24 V, vin = 12 V, f s = 20 kHz, D = 0.5. We perform two simulation tests to verify the performance of the proposed control law and its robustness according to the presence of the input constraint and the parametric uncertainty. Figure 4 illustrates the transient simulation of the boost converter without any change in the converter parameters, we assume that the saturation limit is δ max = 0.5. The waveforms depicted in the Fig. 4 are the duty-cycle d capacitor voltage vc and inductor current iL . One can conclude that the output voltage settle to their desired value without any overshoot and the settling time of the circuit as clearly seen in the graph is t = 80 s. These results demonstrate that our RMPC controller guarantees better stabilization performance under harder actuator saturation constraint. The best performances are obtained with controllers parameters α 1 = 0.01, α 2 = 0.211, Q = 0.1, R = 0.001. In Fig. 5, we take into account not only the actuator saturation but also the load uncertainty in order to show the robustness of the proposed controller. Figure 5 show that our controller is able to stabilize the system on the desired output voltage in presence of load change. These results demonstrate the effectiveness of proposed

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method and its robustness, in particular its aptitude to quickly compensate the system dynamic variations even for significant parameters change.

5 Conclusions In this paper, a RMPC approach based on LMI method has been proposed for the nonlinear systems subject actuator nonlinearity. The LMI-based RMPC controller is applied to regulate the output voltage of a boost DC-DC converter; we can extend this work to other converters (buck converter, buck-boost converter, Cuk converter, etc.). The saturation effect is represented by a model based on a deadzone nonlinearity. The state feedback control law is obtained by minimizing the upper bound of the infinite

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horizon cost function at each time instant. The internal stability of the closed-loop system is guaranteed in the sense of Lyapunov. Finally, the simulation results on a DC-DC converter have demonstrated the effeteness of the proposed control.

References 1. Wu Y et al (2019) A strong robust DC-DC converter of all-digital high-order sliding mode control for fuel cell power applications. J Power Sources 413:222–232 2. Tian Z, Lyu Z, Yuan J, Wang C (2019) UDE-based sliding mode control of DC–DC power converters with uncertainties. Control Eng Pract 83:116–128 3. Middlebrook RD, Cuk S (1977) A general unified approach to modeling switching-converters. Int J Eletron 42:521–550 4. Fakham H et al (2011) Power control design of a battery charger in a hybrid active PV generator for load-following applications. IEEE Trans Ind Electron 58(1):85–94 5. Eghtedarpour N, Farjah E (2012) Control strategy for distributed integration of photovoltaic and energy storage systems in dc micro-grids. Renew Energy 45:96–110 6. Elgendy Mohammed A, Zahawi Bashar, Atkinson David J (2012) Assessment of perturb and observe MPPT algorithm implementation techniques for PV pumping applications. IEEE Trans Sustain Energy 3(1):21–33 7. Montagner VF, Oliveira RCLF, Leite VJS, Peres PLD (2005) LMI approach for H1 linear parameter-varying state feedback control. In: IEE Proceedings control theory and applications, vol 152(2), pp 195–201 8. Shirazi Mariko, Zane Regan, Maksimovic Dragan (2009) An autotuning digital controller for DC–DC power converters based on online frequency-response measurement. IEEE Trans Power Electron 24(11):2578–2588 9. Olalla C, Queinnec I, Leyva R, El Aroudi A (2011) Robust optimal control of bilinear DC-DC converters. Control Eng Pract 19(7):688–699 10. Gomes da Silva JJM, Tarbouriech S (2005) Antiwindup design with guaranteed regions of stability: an lmi-based approach. In: IEEE transactions on automatic control, vol 50, no 1, pp 106–111 11. Sun L, Wang Y, Feng G (2015) Control design for a class of affine nonlinear descriptor systems with actuator saturation. IEEE Trans Autom Control 60(8):2195–2200 12. Boyd S, ElGhaoui L, Feron E, Balakrishnan V (1994) Linear matrix inequalities in systems and control theory. Studies in applied and numerical mathematics, vol 15. SIAM, Philadelphia 13. Sitbon M et al (2015) Disturbance observer based voltage regulation of current-mode-boostconverter-interfaced photovoltaic generator. In: IEEE transactions on industrial electronics pp 0278–0046 (c)

Strategy for Optimization of Energy Management Based on Fuzzy Logic in an FCEV with a Contribution of a Photovoltaic Source Said Belhadj, Kaci Ghedamsi, and Zina Larabi

Abstract In this article, we will study how to increase fuel economy, in a fuel cell/battery (FC + B) and fuel cell/battery/photovoltaic source (FC + B + PV) configuration in a hybrid vehicle with fuel cell. Hybrid vehicle models (FC + B) and (FC + B + PV) will be designed under Simulink/Matlab use a new hybrid approach based on the battery’s soc based on the availability of sunshine so the availability of the PV source. The results show that the proposed control of this strategy can meet the energy needs for good optimization in energy management. Comprehensive comparisons of this energy management strategy based on the control and monitoring of the fuel consumption of the fuel cell according to the chosen driving cycles. Therefore, the proposed strategy will provide a novel approach for advanced of energy management optimization system. Keywords Hybrid vehicle · Fuel cell · PV source · Battery · Fuel economy · Energy management · Fuzzy logic

1 Introduction The performance of a vehicle and the fuel consumption of hybrid vehicles are significantly influenced by the strategy of power management and the dimensioning of components due to the multiplicity of power sources and the difference in their S. Belhadj (B) · K. Ghedamsi Laboratoire de Maitrise des Énergies Renouvelables, University of Bejaia, Bejaia, Algeria e-mail: [email protected] K. Ghedamsi e-mail: [email protected] Z. Larabi Departement d’Electrotechnique, University M. Mammeri of Tizi-Ouzou, Tizi-Ouzou, Algeria e-mail: [email protected] Departement d’Ingenierie et Systemes Electriques, University M. Bougara of Boumerdes, Boumerdes, Algeria © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_42

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characteristics [1]. ICE technology can be considered quite mature as ICE-equipped vehicles have been around for more than 100 years. This type of vehicle generates during their use, two major drawbacks that are air pollution and the greenhouse effect. Thus, HEVs were developed to overcome the disadvantages of ICE vehicles and pure BEVs [2, 3]. According to [4], Globally, the transportation sector is the second largest energy consuming sector after the industrial sector and accounts for 30% of the world’s total delivered energy. Fuel cells convert chemical energy directly into electrical energy, usually characterized by the electrolyte material used. Currently, five types of fuel cells are on the market [5]. The benefits of this technology are high efficiency, almost at partial load, low emissions, no noise and free adjustable ratio (50 kW–3 MW) of electric and heat generation. A high conversion efficiency, between 40 and 85% and even more, depending on the type of fuel cell and the application of use [6]. In this article, in addition to these two sources which are the fuel cell and the battery, we have added a PV source. The power assignment between the fuel cell system, the PV source and the auxiliary energy storage device, which is the battery in our case, is the energy management in the hybrid vehicle, representing an important technique. Various energy management control strategies have been used for hybrid vehicles in recent years. Chen et al. [7] proposed an adaptive control approach with fuzzy logic parameter tuning (AFLPT) for the energy management of electric vehicles that are using fuel cell battery hybrid systems. Yu et al. [8] had used an innovative strategy for optimal power allocation for fuel cell, battery and supercapacitor hybrid electric vehicle. Zheng et al. [9] had used the Minimum Principle to reduce fuel consumption by designing an optimal controller in the power management of fuel cell hybrid vehicles. Kamal et al. [10] have proposed an intelligent energy management strategy for a hybrid hydraulic-electric vehicle in order to minimize its total energy consumption. In Refs. [7–12] the proposed control strategies are not sufficiently detailed and the influence of fuel economy and hybrid vehicle dynamics did not take into account for a balancing purpose. Regarding the limitations of previous work in the field of energy management using fuzzy logic controllers to optimize energy in FCEV; we can mention the decrease in the life of the battery, because of its frequent solicitations, and the risk of exceeding or not to reach the reference values due to the use of limiters. In this paper, a hybrid vehicle model is designed and implemented based on subsystems with FC + B configuration in Simulink/Matlab for simulation validation. After we has developed a second model by adding a subsystem with a PV source with its command. In order to improve the fuel economy of hybrid vehicles, increase the mileage of the continuation of the journey, and increase the lifespan of the FC, a fuzzy logic control is used to design a relevant strategy for the management of energy for the FC + B Hybrid Vehicle and FC + B + PV Hybrid Vehicle. According to the NEDC driving cycle conditions, the proposed control strategy is contrasted with the power tracking control strategy, which is wide adopted in Simulink in terms of the indexes of fuel economy and dynamic property, as well as the duration of life of the FC. This paper is decomposed as follows. First, the Fuel cell/Battery (FC + B) and Fuel cell/battery/ultra-capacitor (FC + B + PV) structures, dynamic modeling of the FC, PV system modeling, and the modeling of battery pack were presented. After,

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Fig. 1 Global scheme of the FC + B + PV Hybrid Electric Vehicle

Energy Management Strategy for Hybrid Vehicle was explained. Before finishing, the simulation results were showed. Finally, discussions and conclusions were given (Fig. 1).

2 Drive Structure of Hybrid Vehicle A. Fuel cell/battery (FC + B) structure There is a growing interest in the use of fuel cells for hybrid vehicles, given the great progress of fuel cell technology. In Fig. 2 is presented an FC + B structure with its command, it should be noted the wide use of this hybrid structure [13, 14].

Fig. 2 FC + B drive structure of fuel cell electric vehicle

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Fig. 3 FC + B + PV drive structure of fuel cell electric vehicle

B. Fuel cell/battery/ultra-capacitor (FC + B + PV) structure In order to improve the performance of the hybrid vehicle, according to the structure FC + B, a new structure FC + B + PV is proposed as shown in Fig. 3. In this structure, the PV source supplies the battery without interruption (except at night) constituting a significant additional supply, necessary for the phases of starting, acceleration and rise of the vehicle.

3 Modeling of Hybrid Vehicle The overall system constituting the vehicle consists of two (FC + B) or three sources (FC + B + PV) depending on the configuration concerned. The three energy sources (FC, B, and PV) exchange their power with a DC bus. For the FC and the PV are connected to the DC bus, each via a unidirectional up converter in the current. The two inductances added as filters and to respect the alternations of sources. The DC bus powers the traction motor via a three-phase inverter to convert DC power into an alternative. This system is controlled by the torque of the electric motor acting on both rear wheels with high precision and high stability of the HEV, reducing the clutter caused by mechanical parts such as the mechanical differential and the drive shafts, releasing the space in the vehicle for the PV, the battery and the hydrogen tank. The traction machine used is of the three-phase pmsm type, widely used in transport, which is, apparently, the researchers’ favorite topology, since it presents several advantages: a good power/weight ratio, high efficiency level, continuous decrease of PMs price [15].

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C. Dynamic modeling of the FC The fuel cell represents the main source of energy of the vehicle, it is of the PEMFC type. The voltage of its cell and its total power are given by the equations according to the references [16, 17]. According to JC Amphlett, the expression of Nernst’s equation is as follows [18]:     E Nernst = 1.229 + 0.8510−3 (TFC − 298.15)4.308510−5 TFC 0.5ln PH2 ln PO2 (1) The power demand of the vehicle PV will be provided by the FC taking into account the efficiency of ηFC [19]. There is a relationship of proportionality between the partial pressure and the molar flow of a gas (hydrogen or oxygen) through a certain valve, the expression of this relation is given in reference [20]. D. PV system modeling The equivalent circuit of the PV module is modeled under Matlab/Simulink while representing the dynamic interactions between the parameters. The PV module is described using various mathematical equations. The configuration of a PV cell is an ideal equivalent circuit having a current source parallel to a diode [21–24]. The ideal solar cell can be designed as a single diode connected in parallel with a light generated Iph current source. With regard to the electrical characteristics of a solar cell, it can be described by a four-component electrical circuit only. The resistances of the circuit are respectively Rs and Rsh which are the series resistance and shunt of the solar cell. In order to estimate the parameters of the equivalent circuit of a PV module, a method is developed by Batzelis et al. [25]. E. Modeling of battery pack In this study, a lithium-ion battery is used to power the vehicle and also to recover energy storage, and especially to be powered by the PV source especially in the shutdown phase. In this section, the dynamic model of the lithium-ion battery is presented [26].

4 Energy Management Strategy for Hybrid Vehicle Since the hybrid vehicle is non-linear and multivariable, the fuzzy model Logic Controller (FLC) is more suitable for energy management. A list of IF-THEN rules links the output to the controller inputs. A degree of adhesion is assigned to the variables according to the definition of the membership functions [7]. Several studies have developed controllers based on fuzzy logic to determine power sharing between different sources, and achieve the maximum in fuel economy and system efficiency.

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In this article, a fuzzy logic controller according to the power demanded by the vehicle and the SOC of the battery is presented (Figs. 4 and 5). F. Fuzzy logic control for FC+ B To restart the power of the fuel cell and the battery to meet the demand of the electric motor, a fuzzy treatment between two inputs that are the requested power and the state of charge of the battery. The output variable is the reference current of the fuel cell. When the vehicle is started, the battery drives the vehicle alone. The control provides for the fuel cell intervention in the particular case of a start where the power demand is high and the SOC of the battery is at a minimum. The relation (2) defines the powers concerned. Fig. 4 Fuzzy logic controller for power split

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G. Fuzzy logic control for FC+ B+ PV To manage the power supply between the three sources fuel cell, battery and PV source (FC + B + PV), according to the power demand of the electric motor two sub-control systems are designed in the fuzzy controller according to the phases of driving the vehicle. When starting the vehicle, the battery is solicited alone but with a contribution of the PV source. Depending on the power demand and SOC of the battery and after preheating, the fuel cell system shall be activated in accordance with the relationship expressed as follows: 

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where PPV is the photovoltaic power, and η2 is the efficiency of the control of the loop generating the reference current of the fuel cell. In fuel cell-only fuel-based driving mode, the battery is discharged and the PV source is disabled because the SOC value of the battery is lower than the battery reference SOC value and the command of the PV source is disabled and the power required at operating conditions is within the maximum and minimum power range of the fuel cell system, so the output power of the fuel cell system must satisfy the requested power, and charge the battery. In the case of delivering the peak load power to the electric motor quickly, the fuel cell will provide much of the charge until the battery that will assist the FC and afterward will take over by activating the PV source. The relation below defines the balance of the different powers. PFC η1 = Pm + PB + PPV η2

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In the case where the requested power is greater than the nominal power of the fuel cell, the latter provides its maximum power and the rest will be provided by the PV source. If the requested power is still not satisfied, the battery should discharge even though the SOC is not higher than the SOC. The equilibrium relation is indicated as follows:  PFC = PFC,max (9) PFC η1 + PB + PPV η2 = Pm

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5 Results and Discussion In this paper, the development of a SIMULINK model is implemented based on the FC + B hybrid vehicle system architecture. Sub-models reconfigured to load a model hybrid vehicle FC + B + PV. In this article, the NEDC cycle is selected as the standard cycle conditions for testing vehicle performance. The choice of this cycle is chosen with respect to the parameters of the vehicle, namely the degree of hybridization, the initial SOC values and the reference SOC, SOC_max and SOC_min. To highlight the relevance and effectiveness of the FLC intelligent controller placed in the FC + B + PV configuration system compared to FC + B, we compared the fuel economy indices according to the dynamic property of the hybrid vehicle. In addition, according to the NEDC standard cycle conditions chosen, the proposed control strategies contrast with the power tracking control strategy that is widely adopted in SIMULINK for FC + B. The FLC speed curves for FC + B, FLC for FC + B + PV can correspond to the speed curves required in the NEDC cycle conditions. Therefore, the FLC controller designed for the FC + B + PV configuration can meet the speed requirements for all driving phases with standard NEDC cycle conditions. The velocity curves are illustrated in Fig. 7 (Fig. 6). For optimum fuel economy values, under different cycle conditions, the FLC for FC + B has a higher fuel consumption than FLC for FC + B + PV. Even the latter controller has a better fuel economy compared to other methods already designed. Therefore, the conclusion can be drawn that the FLC for FC + B + PV has better performance in terms of fuel economy, and even in terms of fuel cell lifetime, due to the demolition of fuel cell interventions. The FC. The optimal values of the dynamic property of the FLC for FC + B + PV are the most efficient than the other methods, namely acceleration time, distance traveled in 5 s and acceleration time for a distance of 400 m. However, the FLC for FC + B performs better than the other methods

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compared to acceleration time of 100 km/h, an acceleration time of 60–100 km/h, and maximum speed (Fig. 8). Figures 9, 11 and 12 show the power curves of the three sources supplying the hybrid vehicle FC + B + PV. The PV source can provide the remaining power for the acceleration conditions to lighten the fuel cell system and assist the battery. In regenerative braking condition, the PV source will either deactivate or charge the battery according to the real SOC and reference of the battery. Therefore, the PV source in the FC + B + PV structure can assist in powering and decreases the charges and fast discharges of the battery especially in phases of the urban journey at reduced speed. In Fig. 10, the comparison of hydrogen consumption in the NEDC cycle is shown. The FLC for FC + B + PV has the lowest hydrogen consumption. Therefore, this method can produce a better fuel economy. Referring to Fig. 11, it represent the 4

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power of the PV source according to the command used, the PV source directly feeds the DC-DC bus as the battery and the FC, and it is a continuous supply of the PV source, which charges the battery in phases of stops of the vehicle.

6 Conclusions In this paper, a fuzzy logic control method is implemented under Simulink Matlab environment, used as an energy management optimization strategy of hybrid vehicles FC + B and hybrid vehicle FC + B + PV for the economy of consumption. Hydrogen and increase the vehicle’s autonomy and the life of the fuel cell. Using the algorithm,

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the parameters’ values of the fuzzy controller and the motor sizing have been inserted in order to optimize energy management. A particular development on Simulink as being the second model is implemented based on the architecture of the hybrid vehicle system FC + B. Then, sub-models are loaded in order to reconfigure, a model of hybrid vehicle FC + B + PV is developed. The results obtained indicate that the proposed control strategy at the required power demand for the NEDC driving cycle. In the different phases of the NEDC cycle, the FLC for FC + B has a lower consumption (L/100 km). The FLC for FC + B + PV has the lowest consumption (L/100 km) than other methods. Therefore, the conclusion can be deduced that FLC for FC + B + PV has better performance in terms of fuel economy. In addition, FLC for FC + B + PV has better performance in terms of dynamic property under driving conditions than NEDC cycle standards. Therefore, the proposed strategy gives a new approach to the advanced hybrid vehicle energy management system.

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24. Hayrettin C (2013) Model of a photovoltaic panel emulator in MATLAB–Simulink. Turkish J Electr Eng Comput Sci 21(2):301–308 25. Batzelis EI, Papathanassiou SA (2015) A method for the analytical extraction of the singlediode PV model parameters. IEEE Trans Sustain Energy 7(2):504–512. https://doi.org/10.1109/ tste.2015.2503435 26. AlSharif A, Das M (2013) A time-varying transfer function model for modeling the charging process of a Lithium-ion battery. In: 2013 IEEE energytech. https://doi.org/10.1109/energy tech.2013.6645288

Modeling and Analysis of an Electromagnetic Vibration Energy Harvester for Automotive Suspension Mustapha Zaouia, Bachir Ouartal, Nacereddine Benamrouche, and Arezki Fekik

Abstract The aim of this work is to present a model for the study and analysis of an Electromagnetic vibration energy harvester for automotive suspension. The Electromagnetic energy harvester (EEH) transforms the kinetic energy due to vibrations created by roads irregularities to electrical energy. The model is based on the electromagnetic equation solved using finite element method (FEM) coupled to the mechanical one obtained through the modeling a quarter of the automotive system. The coupling with mechanical phenomena is carried out principally by magnetic force. The displacement of EEH moving part is simulated using the Macro-Element (ME) technique. The cycle of vibration is transmitted to electromagnetic energy harvester device as vertical displacement. The obtained results are mainly the output electrical power, the suspension deflection and the induced electromotive force, for an imposed cycle of vibration. The effect of the weight of the sprung mass on the induced electromotive force and the electrical power will be presented. Keywords Electromagnetic energy harvester · Automotive · Energy conversion · Finite elements · Macro-Element · Electrical power

1 Introduction Research in alternative energies field is constantly evolving particularly in automotive applications. The kinetic energies due to the vibrations produced by roads irregularities can be recuperated and transformed into the electrical energy in order to improve the efficiency of the automotives particularly hybrids and electric one. To recover these kinetic energies and transform them in electrical energy, types of generators are used and integrated in the suspension of automotive. In literature, M. Zaouia · B. Ouartal · N. Benamrouche LATAGE Laboratory, Mouloud Mammeri University, B.P17 RP, 15000 Tizi-Ouzou, Algeria e-mail: [email protected] A. Fekik (B) Faculty of Science and Applied Science, Akli Mohand Oulhadj University, Bouira, Algeria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_43

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several authors have proposed different studies. The systems based on rotary actuators energy harvesting are used in several domains as civil structures [1, 2] and automotives [3, 4]. The authors have presented in [4], systems constituted of rotary and ball-screw mechanism for harvesting the energy of vehicle suspensions. However, using these actuators requires motion converters which have many disadvantages. Nowadays, research has focused on the use of electromagnetic actuators based on linear tubular topologies because they do not require a motion converter [5–8]. The objective of our work is the modelling and analysis of the electromagnetic energy harvester which operates as a generator. The EEH transforms the kinetic energy of vibrations caused by several irregular forms of roads into electrical energy to improve the efficiency of vehicles. This electrical energy recovery will be used to supply different electrical components constituting the automotives and also to restore the energy to the batteries. The EEH is a permanent magnets (PM) tubular linear generator. The permanent magnets are necessary to product a magnetic flux. The magnets are placed in way to alternate the magnetic polarities on the surface of the moving part in the axial direction. The analysis of the EEH is carried out by the application of an electromagnetic – mechanical coupled model. The developed model is based on electromagnetic equation expressed in terms of Magnetic Vector Potential (MVP) solved using the FEM. The mechanical equation is obtained after modelling a quarter of the automotive [8, 9] and solved to calculated the displacement, the suspension deflection and the velocity. The mechanical system representing the quarter of the automotive is constituted principally by a sprung mass that represents the mass of chassis, and the unsprung mass including the mass of the wheel or tyre, the suspension and the brake system. The electromagnetic and the mechanical equations are coupled through the magnetic force and the modified magnetic flux distribution due to the displacement of the moving part. To take into account the moving parts the Macro-Element (ME) technique is used [10, 11]. The ME technique is based on the resolution of Laplace equation analyticallay in the unmeshed region which separates the meshed fixed and moving regions. The ME technique presents advantage which is to manage variable displacement steps obtained through the mechanical equation and allows us to compute accurately the quantities such as velocities, displacements of the moving part without any constraints on the meshed topology. The geometrical characteristics of the electromagnetic energy harvester are given by [12] and [13]. The developed electromagnetic—mechanical model is implemented under Matlab environment and applied to study and analyse the global system of the quarter automotive including the EEH. The important obtained results are principally, the suspension deflection, electromotive force and the output electrical power, for an imposed cycle of vibration introduced as vertical displacement. On the other hand the influence of the sprung mass that presents the chassis of the automotive on the electromotive force and the principally output electrical power is studied when varying the values of the sprung mass.

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Fig. 1 Model of the quarter automotive including the EEH

2 Description of the EEH-Quarter Automotive Model Figure 1. shows the mechanical system which represents a quarter automotive model [8, 9] including the EEH. The system is composed by two distinct masses. The sprung mass represents the mass of chassis, and the unsprung mass represents the mass of the wheel with the suspension and brake equipment. The road position reference represents a cycle of vibration.

3 Electromagnetic-Mechanical Modeling A. Electromagnetic modeling Using Maxwell equations, governing electromagnetic equation in terms of MVP is given as follow:  → σ  ∧ υ∇  ∧− ∇ A + r

 − →   DA − →  ∧ bm = J s + υpm ∇ Dt

(1)

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− → Js=



 N i · n Sc

− → − →  DA ∂A − → = − v ∧ B Dt ∂t

(2)

(3)

− → where υ is the magnetic reluctivity of the material, υ0 the vacuum reluctivity, A the − → MVP, σ the electrical conductivity. B the flux density, bm remanent induction of − → the permanent magnet, J s is the current density. Sc and N are the cross-sectional area of the coil, and the number of turns respectively. n is the unit vector along the direction of the exciting current i and v is velocity of the moving part. The Eq. (1) is discredised in space using the Galerkin finite element method. When the displacement is implicitly taken into account using ME technique, the Eq. (1) is written in space in the 2D cylindrical coordinates (r, ϕ, z) and in time as follows:  ME    ¨

αi ∂A σ ∂A ∂αi ∂ A ∂αi ∂ A dr dz + + dr dz dΓME υ αi υo ∂r ∂r ∂z ∂z r r ∂t r ∂n Ω Ω ΓME      ¨

¨ ∂bm z αi ∂ A ∂bm r dΓ = dr dz − υ υpm · αi − (αi Js )dr dz + r ∂n ∂r ∂z ¨



Γ ∩ΓME

Ωs

ΩPM

(4) where, A = r Aϕ is the modified magnetic vector potential, bm z and bm r are respectively the axial and radial components of the remanent induction, ΓME the MacroElement boundary and Γ the boundary surrounding the meshed regions area Ω. The conductors and permanent magnet regions are respectively Ωs and ΩPM . The weighted function is αi . The MVP is given by means of the approximation functions α j (r, z) and α ME j (r, z) for each finite element j respectively of the meshed node regions and the unmeshed Macro-Element boundary nodes:

A(r, z) =

⎧ nn  ⎪ ⎪ α j (r, z)A j ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

j=1 ME n t j=1

in the meshed regions (5)

α ME j (r, z)A j in the Macro-Element

are the numbers of the meshed region nodes and Macro-Elements With nn and n ME t boundary nodes respectively. The contribution of the Macro-Element to the global stiffness matrix is considered by the term of the matrix SiME j and is given by [10] and [11] as:

SiME j = ΓME

αiME ∂α ME j ΓME r ∂n

(6)

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z

Stainless Steel shaft

r

Steel spacer Magnet

Coil

wc

Nylon Spacer Tube support

ri

re Symmetry axis

(a)

(b)

Fig. 2 Geometry of the Electromagnetic energy harvester (a) and study domain (b)



SiME j

αiME (r2 , z) α ME j (r 2 , z) dz r ∂r 2 0

b ME αi (r1 , z) α ME j (r 1 , z) dz − r1 ∂r 0

=

b

(7)

r1 and r2 are the boundaries of the Macro-Element (Fig. 4). The introduction of the approximation functions given by Eq. (5) in the MVP for each node, leads to the following algebraic equation system:       Δt [M] + [M ME ] At+Δt = Δt.({F} + {G}) − [K ] At+Δt − At

(8)

where: Where Δ t is the step time, [M ME ] = υ0 SiME j is the global ME Matrix. ¨ Mi j = Ω

   j dr dz  i ∇.α υ ∇.α r

(9)

¨

Fi =

(αi Js ) dr dz Ω

(10)

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Fig. 3 Electrical circuit model of the EEH

Rc

i

E

VL

Energy regeneration

RL

Lc

Ki j = Ωs



¨ Gi =

υpm ΩPM

  dr dz σ αi α j r

 ∂αi ∂αi bmz − bmr dr dz ∂r ∂z

(11)

(12)

B. Electrical Circuit Model of the Electromagnetic Energy Harvester The equivalent electrical circuit of electromagnetic energy harvester is given by Fig. 3, where Rc and L c are the resistance and inductance of the coil respectively, and R L is the external resistance characterising the energy regeneration. The electrical load connected to EEH determines the relationship between the circuit current and induced electromotive force E when the load is R L . The relation is given by: E = (Rc + R L )i + L c

di dt

(13)

The maximum output electrical power occurs as the external resistance is set equal to the internal resistance. The output electrical power is given by the following equation: pe =

E2 Rc + R L

(14)

C. Mechanical Equation The mechanical system representing the quarter of automobile, given by Fig. 1, is governed by the following equation system:

Modeling and Analysis of an Electromagnetic Vibration … Table 1 Mechanical parameters of quarter model automotive



651

Quantity/ Symbol

Value

Sprung mass (m s )

290 kg

Unsprung mass (m u )

59 kg

Spring stiffness (ks )

16,000 N/m

Tyre stiffness (ku )

190,000 N/m

Damping coefficient (bs )

1000 N/m/s

m s z¨ s = −ks (z s − z u ) − bs (˙z s − z˙ u ) + Fa m u z¨ u = ks (z s − z u ) + bs (˙z s − z˙ u ) − ku (z u − zr ) − Fa

(15)

where, z˙ s , z˙ u and z¨ s , z¨ u are the velocities and accelerations of sprung mass and unsprung mass respectively. (z s − z u ) and (z u − zr ) are the deflection of suspension and tyre or wheel respectively. ks is spring stiffness, ku Tyre stiffness and bs is the damping coefficient. The mechanical parameters are presented in Table 1. The Lorentz low is used to calculate the force exerted on the mover due to current flowing through the coils. The magnetic force is given by:

Fa = N

− → i d l ∧ B r

(16)

− → where, d l is the tangential direction to the coil turns, and B r the magnetic flux density through the coil in the radial direction.

4 Results and Discussion Due to the axial symmetry of the EEH, the device can be simplified to a 2D axial symmetry model in cylindrical coordinates. Only one fourth of the structure is studied. The geometrical and the study domain of the electromagnetic energy harvester are presented in Fig. 2 [12, 13]. The permanent magnets used are of NdFeB type with axially magnetized. The dimensions and characteristics are given by Table 2. Figure 4. shows the finite element mesh including the Macro-Element of the studied domain. The homogeneous Dirichlet condition of the MVP is imposed at the boundary (Γ − ΓME ) and at the symmetry axis. The Figs. 5 and 6 show respectively the calculated and given measured values of radial and axial flux density at 3.8 mm above moving part. The model is applied to analyse the harvester electromagnetic energy. The remanent flux density of the permanent magnet of NdFeB is 1.2 T and the relative permeability is unity. The steel magnetic permeability is considered as infinite [12]. Figure 7

652 Table 2 Dimensions and characteristics of EEH

Fig. 4 Finite element mesh including the Macro-Element

M. Zaouia et al. Quantity/ Symbol

Value (m)

Diameter of stainless steel shaft

0.02

Diameter of steel spacer

0.1

Diameter of permanent magnet

0.1

Permanent magnet width

0.025

Steel width

0.025

Inter. Diameter of coil

0.1

Outer diameter of coil

0.155

Inter. diameter of tube support

0.155

Inter. diameter of tube support

0.180

Air gap

0.005

Coil width (wc )

0.05

Number of turns of each coil (N)

690

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Fig. 5 Radial flux density at 3.8 mm above moving part: Calculated and given measured values

Fig. 6 Axial flux density at 3.8 mm above moving part: Calculated and given measured values

Fig. 7 Road position reference versus time

presents wave form of the imposed reference cycle of vibration Z r . This vibration describes the imposed road irregularities reference position versus time. The maximum amplitude is of 0.05 m. Figure 8 presents the suspension deflection from equilibrium, which is the difference between the position of the sprung and the unsprung mass for the considered

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Fig. 8 Suspension deflection versus time

cycle of vibration. Figure 9 depicts the magnetic force versus time corresponding to the imposed cycle of vibration.

Fig. 9 Magnetic force versus time

Fig. 10 Electromotive force versus time

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Figure 10 gives the induced electromotive force versus time corresponding to the cycle of vibration. The electromotive force depends of the relative velocity between the moving part and the coils of the harvester electromagnetic energy. The relative velocity corresponds also to the difference between the sprung and the unsprung mass velocity. The obtained output electrical power versus time is shown in Fig. 11. It is computed using the Eq. (14) governing the equivalent electrical circuit given in Fig. 3. The peak value of the electrical power corresponding to the imposed cycle of vibrations is of 68.73 W. In Fig. 12 and Fig. 13, one shows respectively the electromotive force versus time and the output electrical power versus time for several values of sprung mass of the automotive. By visualizing the curves of Figs. 12 and 13, one notices that the weight of the sprung mass has a great influence on the electromotive force and output electrical power.

Fig. 11 Output electrical power versus time

Fig. 12 Electromotive force versus time for different values of sprung mass

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Fig. 13 Electrical power versus time for different values of sprung mass

5 Conclusion In this work we have presented a model to analyse and study a harvester electromagnetic energy for automotive suspensions. This device is used to recover and transform the kinetic energy, due to the vibration of road irregularities, to the electrical one. The model is based on the coupling of the electromagnetic equation and mechanical one obtained by considering the model of quarter of the automotive. The developed model is implemented under Matlab. The principal results as the suspension deflection, electromotive force and the output electrical power are obtained after application of the model for an imposed cycle of vibration. We notice that the effect of the variation of the sprung mass has a great influence on the electromotive force and electrical power. This type of device is very interesting to recover the kinetic energy due to the vibrations created by the irregularities of the roads, and transform it into electrical energy in order to improve the efficiency of automobiles, particularly electric and hybrid ones.

References 1. Cassidy IL, Scruggs JT, Behrens S (2011) Design of electromagnetic energy harvesters for large-scale structural vibration applications. In: Proceedings of. SPIE, vol 7977, pp 79770P 2. Tang X, Zuo L (2011) Simulation and experiment validation of simultaneous vibration control and energy harvesting from buildings via tuned mass dampers,” presented at the American Control Conference, San Francisco, CA, USA, Jun. 29–Jul. 1 3. Gupta A, Jendrzejczyk JA, Mulcahy TM, Hull JR (2006) Design of electromagnetic shock absorbers. Int J Mech Mater Des 3:285–291 4. Kawamoto Y, Suda Y, Inoue H, Kondo T (2008) Electro-mechanical suspension system considering energy consumption and vehicle maneuver. Vehicle Syst Dyn 46:1053–1063 5. Weeks DA, Bresie DA, Beno JH, Guenin AM (1999) The design of an electromagnetic linear actuator for an active suspension. In: Proceedings of the SAE steering and suspension technology Symposium, Detroit, MI, pp 1–11

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6. Lequesne B (1996) Permanent magnet linear motors for short strokes. IEEE Trans Ind Applicat 32(1):161–168 7. Murty BV, Henry RR (1992) Active vehicle suspension with brushless dynamoelectric Actuator. U.S. Patent 5 091 679, 25 Feb 1992 8. Martins I, Esteves J, Marques GD, Pinada Silva F (2006) Permanent-Magnets linear actuators applicability in automobile active suspensions. IEEE Trans Vehicular Technol 55(1):86–94 9. Chaves M, Maia J, Esteves J (2008) Analysis of an electromagnetic automobile suspension system. In: Proc. ICEM08, Portugal, Paper ID 1285 10. Mohellebi H, Latrèche ME, Féliachi M (1998) Coupled axisymmetric analytical and finite element analysis of induction devices having moving parts’. IEEE Tran Magn 34(5):3308–3310 11. Azzouz F, Bendjima B, Féliachi M (1999) Application of macro-element and finite element coupling for the behaviors analysis of magnetoforming systems. IEEE Tran Magn 35(3):1845– 1848 12. Baker NJ (2003) Linear generators for direct drive marine renewable energy converters. PHD. Thesis, University of Durham, U.K, July 2003 13. Baker NJ, Mueller MA, Spooner E (2004) Permanent magnet air-cored tubular linear generator for marine energy converters. In: Second international conference on power electronics, machines and drives, PEMD (Conf. Publ. No. 498) vol 2, pp 862–867

Experimental and Numerical Study of Hybrid PV/Thermal Solar Collector Provided with Self Ventilation and Tracking Structure Mohamed El-Amine Slimani, Rabah Sellami, Achour Mahrane, and Madjid Amirat Abstract This paper presents the results of an experimental and numerical study of a hybrid photovoltaic/thermal (PV/T) solar air collector provided with fins, assisted PV ventilation, and tracking support. This work aims to study the effective influence of the various key elements (fins, assisted PV ventilation, and tracking support) on the thermal and electrical behaviors of a hybrid PV/T air collector. The addition of fins placed in the dynamic air vein aims to improve the evacuated heat from the PV panel. The purpose of the assisted PV ventilation system is to control the flow of air according to the solar radiation incident on the PV panel. A comparison between simulation and experimental was done in order to validate the numerical model developed for this study. The results show that the three elements provided to PV/T collector give interesting results. The outlet air temperature from the solar device (23 °C) is adequate for residential applications (building air conditioning). Electrical, thermal, and overall average efficiencies have reached significant values of around 13.5, 30, and 70%, respectively. Keywords Fins · PV/T hybrid collector · Modeling · Energy efficiency · Solar thermal conversion · Solar photovoltaic conversion

1 Introduction The conventional hybrid photovoltaic/thermal air solar collectors have low thermal and electrical performance. To increase the performance of these hybrid collectors, various methods have been studied by several researchers [1–7]. M. E.-A. Slimani (B) · M. Amirat Theoretical and Applied Fluid Mechanics Laboratory, Department of Energetic and Fluid Mechanics, University of Science and Technology Houari Boumediene (USTHB), 16111 Bab Ezzouar, Algiers, Algeria e-mail: [email protected]; [email protected] R. Sellami · A. Mahrane Unité de Développement des Equipements Solaires (UDES), EPST Centre de Développement des Energies Renouvelables (CDER), 42415 Tipaza, Algeria © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_44

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In the study of Mortezapour et al. [8] a comparison between two hybrid collectors, one of which the PV module cells are covered with a layer of glass and for the other; one PV module face is covered with a glass layer, and the other side is covered with a layer of Tedlar. They noticed that the thermal efficiency of the second hybrid collector is the highest. Srinivas and Jayaraj [2] designed and studied a hybrid solar collector with double air passage with fins in order to evaluate its thermal and electrical performance. This study has shown that the loss of electrical energy production due to the glazing is compensated by the thermal gain of the collector; which makes this hybrid collector very useful for energy conversion. Mojumder et al. [9] studied two types of hybrid solar collectors to see the effects of solar collector fins on system performance. They used three algorithms SVM, FFA, and WT. Thin flat metal fins were introduced as a heat sink in the collector. Othman et al. [4] have developed several prototypes of hybrid PV/T collectors; a double-pass PV/T collector with fins (in two configurations with and without concentration), and a PV/T collector with a grooved absorber, and a PV/T collector with rectangular tunnel collector absorber. Each design has its specific way to cool the solar cells to increase their yield. Shan et al. [10] did a study on a PV/T solar collector equipped with baffles using several working heat transfer fluids such as air, water, and refrigerant R410a. The results of the simulation show the influence of the meteorological parameters and the evaporation temperature of the coolant on the performance of the PV/T solar collector. Hybrid PV/T collectors can be used in several application fields; in residential and agricultural activities [11–18]. From the previous literature, several investigations have been carried out on the PV/T collectors design and structure for a single or dual heat transfer fluid but without combined fins and tracking structure. This work aims to improve the thermal and electrical performance of a hybrid PV/T collector provided with fins, assisted PV ventilation, and tracking support. This system is to be integrated into building heat and air conditioning in winter season. A numerical model was developed and experimentally validated for this purpose.

2 System Description The hybrid PV/T collector presented for this study based on a monocrystalline photovoltaic module (JT-185 M). The PV/T solar collector composed mainly of the following components; a cover glass, PV cells immersed in a layer of EVA polymer, a protective layer of Tedlar below the PV cells, a metal plate below the Tedlar sheet, another metal plate placed on the thermal insulation of the hybrid collector, and Baffles that are stuck between the two metal plates. A Thermal insulation, to minimize thermal losses on the side and rear sides of the hybrid solar collector. The air circulates in the sub-channels formed by the baffles to allow the heat transfer fluid to convey the thermal energy produced. The experimental prototype is provided with assisted PV ventilation system for PV cooling and tracking support to track the

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Outlet Air

Glass

PV cells PV panel

Tedlar Metal plates

Insolation

Cover

Length, L Air inlet

Fig. 1 Schematic representation of hybrid PV/T collector with fins

Table 1 The main electrical characteristics of the used PV panel at standard test conditions (STC) [19]

Parameter PV Module type

Value

Monocrystalline silicon (JT-185 M)

Cells number, N

12 × 6

Short-circuit current, Isc,ref

5.76 A

Open-circuit voltage, Voc,ref

43.2 V

Current at MPP, Imp,ref

5.14 A

Voltage at MPP, Vmp,ref

36 V

Maximum power, Pm,ref

185 W

Electrical conversion efficiency, ηref

14.4%

movement of the sun in order to receive maximum solar radiation throughout the day (Fig. 1). During the operation of the hybrid collector, the solar energy reaches the outer surface of the collector in the form of solar radiation. A portion of this radiation is converted into electrical energy and another part into thermal energy so that the heat produced by the cells is evacuated through the coolant flowing between the fins and the two metal plates (upper and lower). The hybrid air collector electrical and design parameters used in this document are given in Table 1 and Table 2, respectively.

3 Numerical Modeling During the modeling and for the energy balance equations of the system through each constitutive layer of the hybrid collector, a few simplifying hypotheses are taken into account:

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Table 2 The main geometrical characteristics of the PV/T collector

PV/T geometry Parameter

Characters

Values

Length

L

1.30 [m]

Width

l

0.65 [m]

Thickness of fluid vein

e

0.02 [m]

Number of fins

N

12

Length

L fin

1.30 [m]

Width

h

0.02 [m]

Fins geometry

Height

efin

0.002 [m]

Space between two fins

p

0.05 [m]

• The thermo-physical properties of the system components are assumed to be constant. • Each layer of the hybrid collector is considered a temperature node (average of the input to the output). • Heat losses on the lateral sides of the hybrid collector are negligible compared by the losses of the front and rear faces of the collector. The energy conservation principle is used for each element (i) of the PV/T collector in order to predict the temperature in each layer of the collector. Mi Ci

  dTi Qi − Qi = dt e s

(1)

The thermal balance takes into account the phenomena of heat transfer by conduction, convection, and radiation; interacting at the system level and the thermo-physical proprieties of each component. Energy balance equations for glass layers, PV cells, Tedlar, and thermal insulation are given and detailed by Ref. [20]. The heat transfer balance for the top metal plate is given by the following equation. M p1 .C p1 .

    dT p1 = h c, p−t .S. Tt − T p1 + h v, f − p1 .S  . T f − T p1 dt     + h r, p2− p1 .S  . T p2 − T p1 + h c,ch− p1 .S  . Tch − T p1

(2)

where S = L.l the collector area, S = N .e.L and S = (N + 1).p.L; e is the thickness of a baffle, and p is the pitch (distance) between two baffles. The heat transfer balance for the fins is given by the following equation. Mch .Cch .

    dTch = h c,ch− p1 .S  . T p1 − Tch + h v, f −ch .Sch . T f − Tch dt   + h c,ch− p2. S  . T p2 − Tch

(3)

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where Sch = 2.N.L.h; N is the fins number, and h is fin height. The heat transfer balance for the coolant fluid (air) is given by the following equation. M f .C f .

  dT f = h v, f − p1 .S  . T p1 − T f + h v, p2− f .S  .(T p2 − T f ) dt   + h v, f −ch .Sch . Tch − T f − Q u,th

(4)

The heat transfer balance for the lower metal plate is given by the following equation. M p2 .C p2 .

dT p2 dt

    = h r, p1− p2 .S  . T p2 − T p1 + h v,c− p2 .S  . T f − T p2     + h c,c− p2 .S  . Tch − T p2 + h c, p2−i .S. Ti − T p2

(5)

The conduction, convection, and radiation heat transfer coefficients at the hybrid collector layers are given in detail in our previous works [20, 21]. Several correlations have been proposed by the researchers in this field to predict the variation of the electrical efficiency of the PV module. The correlation proposed by EVANS [19] given by: ηele

     G = ηref 1 − βT . TC − TC,ref + γ .ln G ref

(6)

G and G ref which are the illumination and the reference illumination (G ref = 1000 W/m2 ); Tc and Tc,ref which are the temperature and the reference temperature of the cells (Tc,ref = 25 o C); where β T is the temperature coefficient, and γ is the solar radiation coefficient. Another rather important parameter that shows the overall energy efficiency of the PV/T collector is given as the sum of the thermal yield (ηth ) and the equivalent of the electrical yield in thermal (ηele,th ) using the thermal conversion factor (Cf ) [22, 23]. ηT = ηth + ηele,th = ηth +

ηele Cf

(7)

where:   ˙ f . T f,O − T f,i ηth = mC

(8)

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4 Experimental Tests As presented in Fig. 2, the experimental designed system is mainly composed of a single pass unglazed PV/T collector provided with fins linked to a PV-assisted forced air circulation device. in the first step, and for numerical validation, the PV/T collector is ventilated by a constant airflow assured by a DC electric alimentation instrument and the PV/T collector has been fixed (without tracking device) on a properly insulated aluminum housing using polyurethane foam. The air ventilation system consists of a brushless type DC fan that can be directly coupled to small PV modules (N*4Wp) for different arrangements. In order to allow us to evaluate the thermal performance of the designed collector, a bench of measurements has been set up for the simultaneous measurement of the different parameters such as the temperature of the fluids (air and water), the temperature of the layers and components of the collector, solar illumination and air velocity. For this purpose, many measuring instruments have Fig. 2 Photograph of the designed hybrid PV/T system

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been used (type K thermocouples, hot wire anemometer, pyranometer, and data acquisition) to conduct simultaneous measurements of all necessary parameters with a time step of five minutes. It is important to note that meteorological parameters were collected from the meteorological station installed at the UDES site. The mathematical and numerical model developed for this study has been tested and validated with the experimental data accorded for a typical day (of Oct. 04, 2017). The comparison between experimental and numerical results is given by Fig. 3. The mean absolute percent error (MAE) given by Eq. 9 [24] calculates the difference between the simulated results found by the model used in this study and the experimental ones in the same operating conditions. The values of the mean absolute percent error are 3.10% for Outlet air temperature, 6.31% for Tedlar temperature, 4.1% for Fin temperature, and 3.81% for Electrical Power. The calculated MAE shows that there is a good agreement between the results of the simulation and the experimental data with a maximum value of 6.31%. n 100  X sim,i − X exp,i M AE(%) = n 1 X exp,i

(9)

T t (°C)

80 60 40

Theo Exp

20 8

9

10

11

12

13

14

15

16

17

13

14

15

16

17

13

14

15

16

17

50

Tfo (°C)

40 30 Theo

20 10

Exp

8

9

10

11

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60 50 40

Theo

30

Exp

20 10

8

9

10

11

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Time (hr)

Fig. 3 The experimental and theoretical hourly evolutions of the Tedlar (Tt ), the outlet air (Tfo ), and the fins (Tfin ) temperatures for the day of 04/10/2017

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5 Results and Discussions

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1000

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G (W/m²) Tamb (°C)

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Solar radiation (W/m²)

The simulations were made under the weather conditions of UDES (Unité de Développement des Equipements Solaires) located at Bousmail, Tipaza, Algeria for a cold day of the winter season (Feb. 24, 2018). The Meteorological data for the test day of Feb. 24, 2018, are given by Fig. 4. The weather data include solar radiation, ambient temperature, and wind speed. The evolution compared between the different temperatures of PV/T layers shows compliance and logic in the temperature distribution of the hybrid collector elements. Figure 5 shows the temperature evolution for PV cells and outlet air during the test day. The PV cells’ temperature is relatively stable between 35–45 °C during the test day. The outlet air temperature stabilizes between 20 and 23 °C. This range of air temperature is suitable and comfortable for building heating. Stabilization in temperature is caused by two key factors; PV-assisted ventilation system and solar tracking system. The temperature of the solar cells represents the highest temperature in the hybrid collector; it reaches a value of about 46 °C. This is explained by the fact that the photovoltaic cell is a heat generator converting the received solar radiation into electricity and heat. Figure 6 and shows the evolution of the temperature of the coolant and the cells, respectively.

Wind Speed (m/s)

6

4

2

0 08:00

09:00

10:00

11:00

12:00

13:00

14:00

15:00

16:00

17:00

Time (hr) (Feb.24,2018)

Fig. 4 Evolution of solar irradiance, ambient air temperature, and wind speed during the test day (Feb. 24, 2018)

Experimental and Numerical Study of Hybrid PV/Thermal Solar …

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Temperatue (°C)

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0 08:00

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Fig. 5 Temperature evolution for PV cells and outlet air during the test day

25 23

Air temperature (°C)

21

Heating Range

19 T

17

fo

-T

T

amb

T

fo amb

15 13 11 9 7 5 08:00

09:00

10:00

11:00

12:00

13:00

14:00

15:00

16:00

17:00

18:00

Time (h) (Feb. 24, 2018)

Fig. 6 Temperature evolution of inlet (Tamb ) and outlet (Tfo ) air during the test day

From Fig. 6 it can be noted that for this configuration, the temperature difference between PV/T collector inlet and outlet is important and appreciable. The temperature difference range would be as high as 10 °C in a relatively clear sky and cold conditions. This difference is favorable for comfort improvement inside a building space when exterior temperature (PV/T collector inlet) is not very low. This indicates that in such cloudy conditions, the operating flow rate is sufficiently low to allow noticeable heat gain by the air circulating through the collector duct. Figure 7 illustrates the evolution of electrical and thermal powers produced from the hybrid PV/T collector.

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The

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350 300 250 200 150 100 50 0 08:00

09:00

10:00

11:00

12:00

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Time (h)

Fig. 7 Evolution of electrical and thermal powers during the test day

15

40

14

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13

20

12

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11 08:00

09:00

10:00

11:00

12:00

13:00

14:00

Fig. 8 Evolution over time of thermal and electrical efficiencies

15:00

16:00

0 17:00

Thermal efficiency (%)

Electrical efficiency (%)

From Fig. 7, the electrical and thermal powers reached important and stabilized values about 170 W for the electrical power and between 350 and 450 W for the thermal power. The fins have a significant effect on the energy performance of a hybrid collector, resulting in large heat dissipation of the hybrid collector. Figure 8 presents the evolution of electrical and thermal efficiencies from the hybrid PV/T collector. It is noted that under the studied operating conditions the thermal efficiency of the hybrid collector quickly increases between 08h00 and 09h00 then stabilizes from 09h00 to 17h00 around a value of 32%.

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The variation of electrical efficiency is presented in the same figure. It is clearly perceived from this figure that interesting values of electrical efficiency are obtained although conditions are stable and very favorable. This fact is explained by the fact that under these stable conditions (solar radiation and airflow), the PV cell temperature is also stable (not very high) and still within the acceptable range. These results can be confirmed by returning in Fig. 5, the PV cell temperature did not exceed 45 °C and still stable between 35 and 45 °C throughout the whole day. The simultaneous improvement of thermal and electrical efficiencies leads to an increase in the overall efficiency of the hybrid collector.

6 Conclusion In this study, the energy performance of a hybrid PV/T collector provided with fins and equipped with a direct PV air ventilation device and tracking support was investigated and analyzed, this configuration suitable for winter cold periods to air conditioning in a building. A mathematical model was established based on electrical and thermal parameters and characteristics of the solar device system with the use of an empirical correlation for airflow prediction as a function of solar irradiance. The numerical model was successfully validated and compared with experimental data. The evaluation of the energy performance of the PV/T collector has been carried out using the numerical model of this study. Interesting results are obtained from this work. • The validation results given by the proposed model coincides with the experimental ones. • The solar system provides a suitable and comfortable air temperature (20–23 °C) for spaces building heating. • The developed configuration of PV/T collector makes the electrical and thermal energy efficiencies appreciable and stable.

References 1. Slimani MEA, Amirat M, Bahria S (2015) Study and modeling of heat transfer and energy performance in a hybrid pv/t collector with double passage of air. Int J Energy Clean Environ 16(1–4):235–245 2. Srinivas M, Jayaraj S (2013) Performance study of a double pass, hybrid-type solar air heater with slats. Int J Energy Eng 3(4):112 3. Slimani MEA, Amirat M, Bahria S (2015) Analysis of thermal and electrical performance of a solar PV/T air collector: Energetic study for two configurations. In: 3rd International conference on control, engineering and information technology, CEIT 2015, pp 1–6 4. Othman MY, Ibrahim A, Jin GL, Ruslan MH, Sopian K (2013) Photovoltaic-thermal (PV/T) technology–the future energy technology. Renew Energy 49:171–174

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5. Boumaaraf B, Touafek K, Ait-cheikh MS, Slimani MEA (2020) Comparison of electrical and thermal performance evaluation of a classical PV generator and a water glazed hybrid photovoltaic–thermal collector. Math Comput Simul 167:176–193 6. Sarhaddi F, Farahat S, Ajam H, Behzadmehr A (2010) Exergetic performance assessment of a solar photovoltaic thermal (PV/T) air collector. Energy Build 42(11):2184–2199 7. Hegazy AA (2000) Comparative study of the performances of four photovoltaic/thermal solar air collectors. Energy Convers Manag 41(8):861–881 8. Mortezapour H, Ghobadian B, Khoshtaghaza MH, Minaee S (2012) Performance analysis of a two-way hybrid photovoltaic/thermal solar collector. J Agric Sci Technol 14(4):767–780 9. Mojumder MSS, Uddin MM, Alam I, Enam HK (2011) Study of hybrid photovoltaic thermal (PV/T) solar system with modification of thin metallic sheet in the air channel. J Energy Technol Policy 3(5):47–55 10. Shan F, Tang F, Cao L, Fang G (2014) Dynamic characteristics modeling of a hybrid photovoltaic–thermal solar collector with active cooling in buildings. Energy Build 78:215–221 11. Chaouch WB, Khellaf A, Mediani A, Slimani MEA, Loumani A, Hamid A (2018) Experimental investigation of an active direct and indirect solar dryer with sensible heat storage for camel meat drying in Saharan environment. Sol Energy 174:328–341 12. Slimani ME-A (2017) Etude d’un séchoir solaire agricole muni d’un capteur solaire de type” PV-THERM”-réalisation d’un prototype et caractérisation,” UNIVERSITE DES SCIENCES ET DE LA TECHNOLOGIE HOUARI BOUMEDIENE 13. Parker GJ (1976) A forced circulation system for solar water heating. Sol Energy 18(5):475–479 14. Gholampour M, Ameri M (2016) Energy and exergy analyses of photovoltaic/thermal flat transpired collectors: experimental and theoretical study. Appl Energy 164:837–856 15. Boumaaraf B, Boumaaraf H, Slimani ME-A, Tchoketch_Kebir S, Ait-cheikh MS, Touafek K (2020) Performance evaluation of a locally modified PV module to a PV/T solar collector under climatic conditions of semi-arid region. Math Comput Simul 167:135–154 16. Slimani ME-A, Sellami R, Mahrane A, Amirat M (2019) Study of hybrid photovoltaic/thermal collector provided with finned metal plates: a numerical investigation under real operating conditions. In: 2019 International conference on advanced electrical engineering (ICAEE), pp 1–6 17. Touaba O et al (2020) Experimental investigation of solar water heater equipped with a solar collector using waste oil as absorber and working fluid. Sol Energy 199:630–644 18. Slimani ME-A, Sellami R, Mahrane A, Amirat M (2019) Experimental study of a Glazed Bi-Fluid (water/air) solar thermal collector for building integration. In: 2019 International conference on advanced electrical engineering (ICAEE), pp 1–6 19. Sellami R, Amirat M, Mahrane A, Slimani ME-A, Arbane A, Chekrouni R (2019) Experimental and numerical study of a PV/Thermal collector equipped with a PV-assisted air circulation system: configuration suitable for building integration. Energy Build 190:216–234 20. Slimani MEA, Amirat M, Bahria S, Kurucz I, Aouli M, Sellami R (2016) Study and modeling of energy performance of a hybrid photovoltaic/thermal solar collector: configuration suitable for an indirect solar dryer. Energy Convers Manag 125:209–221 21. Slimani MEA, Amirat M, Kurucz I, Bahria S, Hamidat A, Chaouch WB (2017) A detailed thermal-electrical model of three photovoltaic/thermal (PV/T) hybrid air collectors and photovoltaic (PV) module: comparative study under Algiers climatic conditions. Energy Convers Manag 133:458–476 22. Luque A, Hegedus S (2011) Handbook of photovoltaic science and engineering. Wiley 23. Joshi AS, Tiwari A, Tiwari GN, Dincer I, Reddy BV (2009) Performance evaluation of a hybrid photovoltaic thermal (PV/T) (glass-to-glass) system. Int J Therm Sci 48(1):154–164 24. Nadji Maachi I, Mokhtari A, Slimani ME-A (2019) The natural lighting for energy saving and visual comfort in collective housing: a case study in the Algerian building context. J Build Eng 24:100760

Maximum Power Point Tracking (MPPT) for a PV System Based on Artificial Neural Network ANN and Comparison with P&O Algorithm Aicha Djalab, Nassim Sabri, and Ali Teta

Abstract For the optimal operation of photovoltaic system, The MPPT (Maximum Power Point Tracking) control unit is an essential part for the photovoltaic system. In addition to the protection function, this command ensures the continuation of the maximum power point (MPPT) and allows the PV generator to deliver its maximum power regardless of the variation in climatic conditions (sunshine and) temperature). This work intends to provide an artificial neural network (ANN) maximum power point tracking (MPPT) method which is fast and precise in finding and tracking the maximum power point (MPP) in photovoltaic (PV) applications, under rapidly changing of solar irradiation, and the P&O algorithm. ANN and P&O MPPT algorithms, and other components of the MPPT control system which are PV module and DC-DC boost converter, are simulated in MATLAB/ Simulink, we used in The proposed ANN two inputs which are irradiation and ambient temperature, and one output is the optimum voltage of the PV system. The proposed ANN was analyzed under different irradiation conditions. The response of the proposed ANN for MPPT controllers found to be lesser oscillation at MPP and faster tracking response compared with the P&O algorithm. Comparisons of MPPT with P&O algorithm and without MPPT tracker are also shown in this paper. It is demonstrated that the neural network based MPPT tracking require less time and provide more accurate results than the P&O algorithm based MPPT. Keywords Photovoltaic system · Tracking the maximum power point MPPT · Artificial neural network (ANN) · The P&O algorithm

A. Djalab (B) · A. Teta Applied Automation and Industrial Diagnostics Laboratory, Djelfa University, Djelfa, Algeria e-mail: [email protected] A. Teta e-mail: [email protected] N. Sabri Department of Electrical Engineering, Medea University, Medea, Algeria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_45

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1 Introduction Currently, looking for alternative energy resources has become essential and a significant growth worldwide., due to increasing demand for energy and depletion of fossil fuel reserves, in order to find sustainable energy resources, various researches have been carried out, mainly on renewable energy sources [1]. There are many sources to produce, but there are also constraints related to its production, such as the effect of pollution and global warming… etc. These constraints lead research towards the development of renewable and non-polluting energy sources; the use of renewable energy such as solar energy has shown that they could contribute on a large scale to find a solution to the above-mentioned problems [2]. Renewable energy means that the energy of the Sun, wind, Earth, water or biomass heat is regarded as a clean alternative energy source. Among the different types of renewable energy, solar energy has been exploited to produce calories, thermal energy and lastly the electricity which is the main subject of this article. However, the characteristics of the outputs of PV systems are not linear and change according to the temperature and irradiation, so a MPPT controller is needed to extract the maximum power to the terminals of PVG. Therefore, the MPPT techniques are used to maintain the PV works at its MPPT [3]. Several MPPT techniques have been proposed in the literature; For example, Perturb and Observe (P&O) method [4, 5] incremental conductance (INC) method and fuzzy logic based methods [6], etc. Many MPPT algorithms have been developed so far, and these MPPT algorithms vary in application, complexity, precision, sensors, cost, popularity, etc. Among them, P&O method has drawn significant attention of the researchers due to its simplicity. However, its deviation from MPP under rapidly changing of solar irradiation, and oscillation problem in steady states is unavoidable [7]. New approach in MPPT systems is to apply artificial intelligent systems, such as fuzzy logic and artificial neural network (ANN) as MPPT controller or as a part of integrated MPPT system [7]. These methods are fast in tracking the MPP, and give more stable response in compare with conventional algorithms. This paper presents a PV model with MATLAB/SIMULINK, and tracks the MPP using an Artificial Neural Network (ANN) and the P&O technique., And the comparison between them. The ANN uses the irradiance and temperature as inputs. The PV model was simulated to generate training and testing data for PV module at various conditions to train and test the proposed neural net-work. We started this work by PV module modeling is described in Sect. 2, followed by DC-DC converter in Sect. 3 and maximum power point tracking (MPPT) in Sect. 3. Section 4 includes simulation results, and finally a conclusion is given in Sect. 5.

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2 Photovoltaic System Modeling and Simulation Photovoltaic system consists of four blocks as shown in Fig. 1. The first block is the source of energy (solar panel), the second block is a static converter DC-DC, the third block represents the load and the fourth block represents the control system. The main role of the static converter is to an impedance matching so that the Panel delivers maximum energy. To obtain a desired output such as the power, the output current and output voltage [3, 8], the photovoltaic system which consists of a set of basic photovoltaic cells can be connected in series and/or parallel, where the Photovoltaic cells are the main components of the module. A. Model of a Photovoltaic Cell The simplest circuit of the photovoltaic cell model is represented by a current source in parallel with an ideal diode, which will be adopted in this study [9]. The equivalent circuit of a solar cell is given in Fig. 2. This equivalent circuit is composed of a current source controlled models which the photovoltaic effect (the generated current is controlled by the Sun’s rays). The diode represents the effect of the junction semiconductor of the cell. Two resistors (Rs, Rsh) represent the series resistor Rs and a shunt resistor Rsh represent the power losses by connections in series and in parallel [8]. The basic equations that express the I-V characteristic of the photovoltaic model are given in the following equations [9, 10]: Iph = Id + Ish + I

Fig. 1 Photovoltaic system

(1)

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Fig. 2 Equivalent circuit of a solar cell

The current in the diode Id is given by:     V + Rs I −1 Id = I0 exp Vt a

(2)

The current in the RP resistance is given by:  Ish =

V + Rs I Rsh

 (3)

From Eq. (1), we obtain the expression of current I: I = Iph − Id − Ish

(4)

Replacing (4) in the Eqs. (2) and (3), the characteristic equation becomes:       V + Rs I V + Rs I −1 − I = Iph − I0 exp Vt a Rsh

(5)

where the junction thermal voltage Vt is defined by Vt =

Ns K T q

where: I Id Iph Imp Vmp Ish

Output current [A], Current of parallel diode [A], Photo current; [A], Current corresponds to maximum power point, Voltage corresponds to maximum power point, Shunt current; [A],

(6)

Maximum Power Point Tracking (MPPT) for a PV System … Table 1 Ideality factor (A)

Isc Io Voc V Rs Rsh Vt Ns T q K a

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Ideality factor

Si-mono

1.2

Si-poly

1.3

a-Si-H

1.8

a-Si-H tandem

3.3

a-Si-H triple

5

cdTe

1.5

CTs

1.5

AsGa

1.3

Short-circuit current [A], Saturation current of the diode [A], PV module open-circuit voltage [V], The cell voltage [V], The resistance series cell [], The shunt resistance cell [], The thermal voltage of the module [V], The number of cells connected in series, The temperature of the cell [°K], Electron’s charge e = 1.6 *10−19 C, The Boltzmann constant (1.3854*10−23 J°K−1 ), Diode quality (or ideality) factor, it can be classified by the PV technology according to Table 1 [11].

Figure 3 shows a nonlinear characteristic of the photovoltaic cell. This characteristic varies with the change in metrological terms. As the optimal power point varies broadly according to weather conditions, a power converter switch should be controlled by a specific algorithm to track the maximum power point [12]. B. DC-DC Boost Converter DC-DC converters which convert a DC voltage from one level in input to another DC level at its output are composed of switching devices, inductors and capacitors. The switch is usually an IGBT or MOSFET. A PWM signal driven at the gate of either of them, switches it on and off. In MPPT systems, this signal is controlled by MPPT controller. DC-DC converter used in this work is a boost converter which steps up the input voltage to a desired voltage at its output. Input voltage of the boost is output voltage of the PV module. Figure 4 shows the electrical circuit of the DC-DC boost converter [1]. The duty cycle, D = ton /T where T is the period equal to ton + toff . The relation between the input and output voltage is given from. vo =

1 vi 1− D

(7)

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Fig. 3 I-V and P-V characteristics of a photovoltaic cell

Fig. 4 DC-DC boost converter

The input voltage of the converter is the solar module output voltage that is changes all the time. However, its output voltage must be kept at a desired value. This is done by controlling the duty ratio, so that the operating point of the PV system can be adjusted to realize MPPT algorithm [13]. C. PV Array Characteristics The characteristics of the PV array used in this paper are presented in Table 2.

Maximum Power Point Tracking (MPPT) for a PV System … Table 2 Electrical characteristics of the kc 130 ght PV module

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Electrical Characteristics Power at maximum power point (Pmpp)

130 W

Optimal operation voltage (Vmpp)

17.6 V

Current at maximum power point (Impp)

7.39 A

Open circuit voltage (Voc)

21.9 V

Short circuit current (Isc)

8.02 A

Number of cells connected in series

36

Number of cells connected in parallel

1

3 Maximum Power Point Traking Contrellers The photovoltaic system to work at maximum power points of their characteristics, there are specific laws that meet this need. This command is named in the “Maximum Power Point Tracking” (MPPT) literature. The principle of these commands is to seek the maximum power point (MPP) by keeping a good adaptation between the generator and the load to ensure the transfer of maximum power. The technique of control so to act on the duty cycle in an automatic way to bring the point of operation of the generator at its optimum value whatever weather instabilities or brutal changes in load [4]. For such reason, three MPPT control techniques will be discussed. A. P&O Controller The perturbation and observation (P&O) algorithm is probably the most frequently used in practice, mainly due to its easy implementation [4]. As the name suggests it is based on the perturbation of system by the increase or decrease in Vref where acting directly on the duty cycle of the converter DC-DC, then observation of the effect on the output power of the panel. If the current value of the power P(k) panel is greater than the previous value P(k-1) is then retains the same direction of previous disturbance or we reverse disruption of the previous cycle. The Fig. 5 shows the flowchart of this algorithm. B. MPPT Method Based on Artificial Neural Network—ANN Artificial neural networks (ANNs) are a family of statistical learning models inspired by biological neural networks (the central nervous systems of animals, in particular the brain) and are used to estimate or approximate functions that can depend on a large number of inputs and are generally unknown. Artificial neural networks are generally presented as systems of interconnected “neurons” which send messages to each other. The connections have numeric weights that can be tuned based on experience, making neural nets adaptive to inputs and capable of learning [14]. The ANN model is modeled in MATLAB/Simulink environment. In our work, Neural network used is multilayer perceptron (MLP) type of architecture. with two neurons in input layer, five neurons in output layer and one in output layer is constructed. The model of network is as below (Fig. 6).

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Yes

P(k)-P(k-1)=0 No

No

Yes

V(k)-V(k-1)>0

Decrease Vref

P(k)-P(k-1)>0

No

No

Increase Vref

V(k)-V(k-1)>0

Decrease Vref

Return

Fig. 5 Flowchart perturbation and observation algorithm

Fig. 6 Architecture of a typical ANN

Yes

Yes

Increase Vref

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Best Training Performance is 2.3274e-05 at epoch 119

10 2

Mean Squared Error (mse)

Train Best

10 0

10 -2

10 -4

0

20

40

60

80

100

119 Epochs

Fig. 7 Training performance curve

The inputs to neural network are temperature and irradiance and the output is the voltage at the maximum power point. In hidden layer the basis function is weighted linear sum and activation function tansig. For output layer basis function is linear weighted sum whereas activation function is Linear. The algorithm used is Levenberg-Marquardt, and performance function of the network is mean square error (MSE) which is given by Eq. (8) as below: E=

N 

(t (k) − O(k))2

(8)

k=1

The network which was fully trained with the lowest error is capable to be used in the testing process. Performance of ANN to minimize the MSE error is shown in the training performance curve of Fig. 7. Network was trained until it achieved a very small MSE typically 2.3274e-05 which reached after 119 epochs. which implies that the network parameters (weight and bias) are optimized (Fig. 7).

4 Simulation and Results In this section, we begin by assessing the photovoltaic system by simulation with MATLAB/Simulink simulation tool. Then, two further MPPT methods are studied:

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Power (w)

ANN / G=1000w/m2, T=25°C

100

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130 120 110

50

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Fig. 8 PV power by P&O, and ANN method in standard conditions

the method (P&O), and the ANN method. The two systems are simulated under standard environmental conditions and many changes of irradiance conditions. A. Operation Under Constant and Uniform Irradiance In this test PV system was simulated with uniform irradiation on all the solar cells. The irradiance and temperature are held constant. It takes the values of the standard conditions: the temperature T = 25 °C and irradiance G = 1000 W/m2 on the PV module. Only one MPP occurs in this case (130 W). Figure 8 shows that P&O take more time to converge to MPP estimated at 6.5 ms. However, the ANN method track the MPP with fast time response estimated at 2.5 ms. Moreover, sustained oscillations are present in PV output power of P&O around MPP. Figures 8 show the results of the output power under standard conditions. B. Operation Under Partial Shading Conditions In order to evaluate the performance of the P&O and ANN MPPT under changing of irradiation, they will be tested under a constant temperature of 25 °C and changing irradiance. the signal presented in Fig. 9 is applied as irradiation to the PV module in SIMULINK. Figure 10 shows the simulation results for P&O and ANN methods. As Fig. 10 depicts, ANN method can rapidly track the MPP under rapidly changing irradiation, while although the advantage of the P&O simple but it causes an oscillation around the MPP and has a quite low response under sudden change in irradiance leading to the power loss in steady state. Finally, it is obvious through the simulation that the ANN gives better results compared to the P&O under changing irradiance.

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Irradince (W/m2)

1000 800 600 400 200 0 0

0.2

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1

1.2

1.4

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Time (s)

Fig. 9 partial shading conditions

150 G = 1000 W/m2

ANN / T=25°C

G = 850 W/m2

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Power (w)

100 100

80

G = 650 W/m2

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G = 420 W/m2

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20 0.18

0

0

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0.6

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0.24

0.8

0.26

0.28

0.3

1

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1.4

1.6

1.8

2

Time (s)

Fig. 10 PV power by P&O, and ANN method under partial shading conditions

5 Conclusion The optimal operation of a photovoltaic system is depending on the MPPT controllers. These controllers are designed to pursue the MPP and minimize the error between the operation power and the maximum power. In this article, we have described the main elements of the PV system. Then, we illustrated the principle of a two MPPT techniques namely P&O, and ANN. Finally, we finished by a simulation of the different techniques. Simulation results for ANN and P&O methods presented in this paper, show that ANN method is very fast and precise in finding and tracking the MPP in case

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of changing of solar irradiation. Furthermore, this method can stably extract the maximum power point under changing of solar irradiation. On the contrary, P&O method has high oscillation around MPP under slowly changing of solar irradiation, which leads to high power loss in long term. From this results, we can to say that the ANN method is the better method compared to the P&O.

References 1. Yongchang Y, Chuanan Y (2012) Implementation of a MPPT controller based on AVR Mega16 for photovoltaic systems. Int Conf Futur Electr Power Energy Syst 17:241–248 2. Safari A, Mekhilef S (2010) Simulation and hardware implementation of incremental conductance MPPT with direct control method using cuk converter. IEEE Tran Ind Electro. 58(4), 1154–1161 3. Bensaci W (2012) Modélisation et simulation d’un système Photovoltaïque adapté par une commande MPPT, Thèse de doctorat“, Université Kasdi Merbah–Ouargla 4. Abbes H, Abid H, Loukil K, Toumi A, Abid M (2013) Comparative study of five algorithms of MPPT control for a photovoltaic system (CIER’13) Sousse, Tunisie. ISSN 2356-5608 –2013 5. Bun L, (2011) Détection et Localisation de Défauts pour un Système PV. Université de Grenoble Thèse de Doctorat en Génie Electrique 6. Rezgui W, Mouss LH, Mouss MD (2013) Modeling of a photovoltaic field in malfunctioning. In: The Proceedings of the 2013 IEEE CoDIT (international conference on control, decision and information technologies), pp 788-793. Hammamet Tunisia. (06–08 May 2013) 7. Khanaki R, Radzi M, Hamiruce M (2013) Marhaban comparison of ANN and P&O MPPT methods for PV applications under changing solar irradiation. In: IEEE conference on clean energy and technology (CEAT) 8. Drir N, Barazane L, Loudini M (2013) Fuzzy logic for tracking maximum power point of photovoltaic generator. Revue des Energies Renouvelables 16(1): 1–9 9. Shareef H, Mohamed A, Mutlag AM (2014) A current control strategy for a grid connected PV system using fuzzy logic controller. In: IEEE International Conference on Industrial Technology (ICIT), pp. 890–894. Busan, Korea (February 2014) 10. Abbes H, Abid H, Loukil K (2015) An improved MPPT incremental conductance algorithm using T-S Fuzzy system for photovoltaic panel. IJRER 5(1) 11. Bellia H, Youcef R, Fatima M (2014) A detailed modeling of photovoltaic module using MATLAB. NRIAG J Astron Geophys 12. Harsha P, Dhanya PM, Karthika K (2013) Simulation and proposed hardware implementation of MP controller for a solar PV system. Int J Adv Electr Electron Eng (IJAEEE) 2(3) 13. Abdelkader HI, Hatata AY, Hasan MS (2015) Developing intelligent MPPT for PV systems based on ANN and P&O algorithms. Int J Sci Eng Res 6(2). ISSN 2229-5518 14. Mathur D (2014) Maximum power point tracking with artificial neural network. Int J Emerg Sci Eng (IJESE) 2(3). ISSN: 2319–6378

Development of a Stand-Alone Connected PV System Based on Packe U Cell Inverter Khaled Rayane, Mohamed Bougrine, Atallah Benalia, and Kamel Guesmi

Abstract In this paper, we investigate a competitive Multilevel Inverter (MLI) named by Packed U Cell (PUC). This converter is capable to generate more levels with fewer components compared with existing MLI topologies. The system consists of a PV arrays supply connected to a single-phase PUC5 inverter. A boost converter is employed to extract the Maximum Power (MPP) controlled by an MPP tracking (MPPT) algorithm. Then, A nonlinear control strategy with Pulse Width Modulation (PWM) is used to control the 5-level PUC (PUC5) to ensure good tracking and highquality convention. Some simulations were performed in MATLAB/SIMULINK to demonstrate the effectiveness of this topology with the proposed control scheme. Keywords Photovoltaic · MPPT · Packed U cell (PUC) · PUC5 · Multilevel converters

1 Introduction The use of renewable energy has increased significantly in the last decade in order to reduce environmental pollution and satisfy the dramatically increased energy demand. Among others, photovoltaic energy became more favorable due to its simplicity of allocation, high dependability and lower cost. However, due to their DC nature, photovoltaic generators require the use of an inverter to convert DC to K. Rayane · M. Bougrine · A. Benalia LACoSERE Lab, University of Laghouat Laghouat, Laghouat, Algeria e-mail: [email protected] M. Bougrine e-mail: [email protected] A. Benalia e-mail: [email protected] K. Guesmi (B) Department of Electrical Engineering, University of Djalfa, Djelfa, Algeria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_46

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Table 1 Components counts of different multilevel topologies Inverter

DC source

Clamped diode

Flying capacitor

Active switch

Total

NPC

1

10

0

12

23

FC

1

0

6

12

19

CHB

2

0

0

8

10

PUC7

1

0

1

6

8

AC. Conventional inverters have many drawbacks, such the high power losses and harmonics. In this context, multilevel topologies are becoming preferable for their high-quality conversion and low harmonics content that can provide [1–3]. Drawbacks of multilevel inverters are not trifling, the most salient one is their topological complexity. The most common structures developed during the last decades are the Neutral point clamped (NPC), Cascade H Bridge (CHB), Flying capacitors (FCI) [4–9]. Recently a new topology has been introduced by [10] called Packed U Cells (PUC) that can generate more levels with lower components requirement. PUC can generate up to 7-level voltage with 6 active switches, one capacitor and one DC source [11, 12]. Compared to other topologies, the PUC7 decreases significantly the required components for the same number of levels (see Table 1). In spite of these, the PUC7 has a big flaw, it requires a complicated control, and more than that, the obtained wave lacks symmetry, which is caused by the uncontrollability of the capacitor voltage. To solve this problem a 5-level PUC inverter has been proposed [13–15] were in this case, the capacitor voltage should be stabilized at E/2 instead of E/3, where E is the DC source voltage. This will give some redundant states that make the capacitor voltage perfectly controllable. In this work, we aim to study the PUC5 converter when supplied by a PV system. First. The studied system composed of PV arrays supplies an AC load through PUC5 inverter. A DC-DC boost converter is employed to extract the maximum power from the solar panels. Perturb and Observe (P&O) is used to control the boost converter to track the maximum power point [16, 17]. Then, a nonlinear feedback control based on a linearizing PI regulator is designed to control the PUC5 inverter to convert DC to AC with an attempt to provide high-quality conversion with low harmonics. The rest of the paper will be organized as follows. Section 2. Will briefly describe the PV system used for this work. Section 3 presents the PUC5 with a description of its state space averaged model. Section 4 discusses the nonlinear control schemes applied to the PUC5. Section 5 provides some simulation results. Finally, we achieve the paper with a conclusion.

2 The Proposed Power Converter System The PV system proposed in this work is constructed of three main parts, the first part is the PV array composed of three Sun Power SPR-305-WHT PV modules in series and two parallel strings. The second part is the DC/DC boost converter used

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Fig. 1 Equivalent Circuit of PV cell

to extract the maximum power from the PV array. The third part is a PUC5 inverter to convert DC power to AC in order to supply an AC load. A. PV Array Modeling A photovoltaic cell converts solar light into electrical energy. When photons hit the surface of the photovoltaic cell an electron is released from the atom, which results in generating a flow of electricity. The equivalent circuit of the PV cell is shown in Fig. 1. The output current of photovoltaic generator (PVG) is given by  (V +I Rs )  V + IR s I = I ph − Io e ηK T /q − 1 − Rsh

(1)

where I ph is the photo-current, I 0 the saturated current voltage across the diode, q the electron charge, T the junction temperature, K is the Boltzman constant, ï the ideality factor, and Rs and Rsh are respectively the series and shunt cell resistors. B. Perturb and Observe (P&O) Algorithm This part explains the proposed algorithm that controls the boost converter to extract the maximum power from the PV array, P&O is the most common technique for achieving the Max Power Point Tracking (MPPT) due to its simplicity and efficiency. Using the measurement of the PV voltage/current and comparing it with the previous one. If the power increase the perturbation direction take the same direction. Otherwise, the perturbation takes the reverse direction. The flowchart of the P&O algorithm is shown in Fig. 2.

3 Five-Level Packed U Cell Converter Single-phase PUC inverter is shown in Fig. 3 is constructed of six switches, one DC source cell (E, S1 , S 1 ), one capacitor cell (C, S2 , S 2 ) and two additional switches (S p , S p ) to generate the negative output voltage. There are eight switching possibilities where the corresponding output voltages are listed in Table 2. In fact, the PUC

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Fig. 2 The flow chart of P&O control

is capable of generating seven levels in output voltage, this can be achieved by regulating the voltage across the capacitor to E/3. However, this configuration will cause the loss of capacitor voltage contractility. To overcome this problem its preferable to control the capacitor voltage at E/2, this configuration will generate only five levels, but gives us a redundancies that give different dynamics for the capacitor voltage (see Table 2). C. Dynamical Model By using Kirchhoff law the average mathematical model of PUC5 is expressed as follow 1 d V2 = (u 2 − u 1 ) dt C  vo = V1 u p − u 1 + V2 (u 1 − u 2 )

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Fig. 3 The PUC inverter

Table 2 Output voltage of all switching states State

Sp

S1

S2

vo

V˙2

1

0

0

0

0

0

2

0

0

1

−V2

–

3

0

1

0

V2 − V1

+

4

0

1

1

−V1

0

5

1

0

0

V1

0

6

1

0

1

V1 − V2

+

7

1

1

0

V2

–

8

1

1

1

0

0

1 di o = (vo − Ri o ) dt L

(2)

where V1 , V2 and vo are the source, capacitor and output T-average voltages respectively. i o is the load current. C, L and R are the cell capacitor, load inductor and load

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resistance respectively. u 1 , u 2 and u p are the T-average inputs, which stand for the duty cycles of the switches S1 , S2 and S p . It can be seen from Table 2 S p and S p are responsible of the positive and negative path of the current and for that they are considered known variables that depend on the positive and negative cycle of the voltage reference.S p is expressed according to vor e f as follows  Sp =

1 vor e f ≥ 0 0 vor e f ≤ 0

(3)

4 Control Design of PUC5 Figure 4 illustrate the proposed closed-loop control of the PUC5 inverter. The control strategy consists of two sensors and two linearizing PI regulator for the capacitor voltage and load current. The two controllers are designed to achieve the desired current and stabilize the capacitor voltage at half the DC Source. The average model is given as: x2 (u 2 − u 1 ) = w1 C  vo = E u p − u 1 + x1 (u 2 − u 1 ) = w2

x˙1 =

(4)

where x1 = V2 , x2 = i o and 



w1 = K p x1r e f + x1 + K i





w2 = vor e f

x1r e f + x1

 (5)

We recall that our objective is to track a desired reference of current, using Eq. (2) we define vor e f as: vor e f = Ri o + Lw3

Fig. 4 Control strategy of PUC5

(6)

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689

where w3 is a PI action of the form w3 =

  x2r e f + K p x2r e f + x2 + K i dt





x2r e f + x2



(7)

From Eq. (4) we define the input vector as follow: 

u1 u2

=

 x2

− xC2 x1 V1 − x1

−1 

C

w1 w2 + V1 u p

(8)

5 Simulation Results In order to examine our proposed scheme, simulation has been performed in MATLAB/SIMULINK environment. The parameters used in the simulation are listed on the Table 3. The PV system application for PUC5 converter in standalone operation is shown in Fig. 5. The PV system consists of three series and three parallel PV modules of type Sun Power SPR-305-WHT. The characteristics of the used PV array are shown in Fig. 6. In order to evaluate the MPPT and the proposed control for the PUC5, three irradiation steps change was made, begins with 1KW/m2 until t = 2 s were than it to become equal to 300 W/m2 , another change occur at t = 4 s were the irradiation becomes equal to 700 W/m2 as is shown in Fig. 7. The PV voltage and current behavior are depicted in Fig. 8 and Fig. 9, respectively. were the PV power is shown in Fig. 10, from those results we could say that the P&O algorithm to obtain MPP has achieved its goal to extract the maximum power from the PV panel for all the three irradiation scenarios. Now in order to evaluate the performance of PUC5 a simulation was performed for the proposed control that has been discussed in Sect. 4. Figure 11 shows the voltages of the capacitors, it is clear that the capacitor voltage V2 maintains tracking its reference value (V1/2) with fast response. The load voltage and current wave are depicted in Figs. 12 and 13. Using the zoom effect it can be clearly seen that the output voltage is constituted of five levels for every different DC bus voltage case. Table 3 Parameters used in the simulation

Parameters

Values

Fundamental frequency

50 Hz

Load Inductance

1 mH

Load resistor

15 

DC capacitance

100 µF

PWM switching frequency

10 kHz

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Fig. 5 PUC5 Supplied by MPPT PV System for stand-alone application

The load current in Fig. 13 takes a sinusoidal waveform following perfectly its reference i o = i dc sin(wt) with every change condition. Such results show highquality conversion with low harmonic contents, which proves the PUC5 can be a challenging topology with high potentials in photovoltaic application uses.

6 Conclusion In this work, we discuss a recently developed topology known as Packed U Cell (PUC) converter. This has a high potential of use on renewable energy conversion such as photovoltaic applications. In order to examine this topology a photovoltaic system based on PUC5 has been designed. The system consists of a boost converter connecting the PV arrays to a single-phase 5-level PUC (PUC5) inverter to supply an AC load. We proposed a control strategy for the PUC5 to provide highquality convergence with low harmonic contents harnessing the five levels voltage it provides. Therefore, there is no need to use bulky output filters. Simulation results have proved the effectiveness of this topology with the employed control strategy making it suitable for photovoltaic applications.

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Fig. 6 PV array characteristics of sun power SPR-305-WHT

Fig. 7 Irradiation levels

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Fig. 8 PV voltage

Fig. 9 PV current

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Fig. 10 PV power

Fig. 11 DC source and capacitor

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Fig. 12 Evolution of the output voltage

Fig. 13 Evolution of the output current

References 1. Lai J-S, Peng FZ (1995) Multilevel converters-a new breed of power converters. In: IAS’95 conference record of the 1995 IEEE industry applications conference thirtieth IAS annual meeting, vol 3, pp 2348–2356. IEEE 2. Al-Haddad K, Ounejjar Y, Gregoire L-A (2016) Multilevel electric power converter. May 3 2016, US Patent 9,331,599 3. Abu-Rub H, Malinowski M, Al-Haddad K (2014) Power electronics for renewable energy systems, transportation and industrial applications. Wiley

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4. Kouro S, Malinowski M, Gopakumar K, Pou J, Franquelo LG, Wu, Rodriguez J, Pe´rez MA, Leon JI (2010) Recent advances and industrial applications of multilevel converters. IEEE Tran Ind Electron 57(8): 2553–2580 5. Rodr´ıguez J, Bernet S, Wu B, Pontt JO, Kouro S (2007) Multi-level voltage-source-converter topologies for industrial medium-voltage drives. IEEE Tran Ind Electron 54(6): 2930–2945 6. Benmiloud M, Benalia A, Defoort M, Djemai M (2016) On the limit cycle stabilization of a Dc/Dc three-cell converter. Control Eng Pract 49:29–41 7. Benmansour K, Benalia A, Djemai M, de Leon J (2007) Hybrid control of a multicellular converter. Nonlinear Anal Hybrid Syst 1(1):16–29 8. Rodriguez J, Lai J-S, Peng FZ (2002) Multilevel inverters: a survey of topologies, controls, and applications. IEEE Trans Indus Electro 49(4):724–738 9. Rodriguez J, Franquelo LG, Kouro S, Leon JI, Portillo RC, Prats MAM, Perez MA (2009) Multilevel converters: an enabling technology for high-power applications. Proc IEEE 97(11):1786–1817 10. Ounejjar Y, Al-Haddad K, Gregoire L-A (2011) Packed u cells multi- level converter topology: theoretical study and experimental validation. IEEE Trans Ind Electron 58(4):1294–1306 11. Trabelsi M, Bayhan S, Ghazi KA, Abu-Rub H, Ben-Brahim L (2016) Finite-control-set model predictive control for grid-connected packed- u-cells multilevel inverter. IEEE Trans Ind Electron 63(11):7286–7295 12. Sheir A, Orabi M, Ahmed ME, Iqbal A, Youssef M (2014) A high efficiency single-phase multilevel packed u cell inverter for photovoltaic applications. In: 2014 IEEE 36th International Telecommunications Energy Conference (INTELEC), pp 1–6. IEEE 13. Vahedi H, Labbe´ P-A, Al-Haddad K (2016) Sensor-less five-level packed u-cell (puc5) inverter operating in stand-alone and grid- connected modes. IEEE Trans Ind Inf 12(1): 361–370 14. Vahedi H, Al-Haddad K (2016) Puc5 inverter-a promising topology for single-phase and three-phase applications. In: IECON 2016-42nd Annual Conference of the IEEE Industrial Electronics Society, pp 6522–6527. IEEE 15. Abarzadeh M, Vahedi H, Al-Haddad K (2019) Fast sensor-less voltage balancing and capacitor size reduction in puc5 converter using novel modulation method. IEEE Tran Ind Inf 16. Ahmed J, Salam Z (2015) An improved perturb and observe (p&o) maximum power point tracking (mppt) algorithm for higher efficiency. Appl Energy 150:97–108 17. Oubbati B, Boutoubat M, Belkheiri M, Rabhi A (2018) Extremum seeking and p&o control strategies for achieving the maximum power for a pv array. In: International Conference in Artificial Intelligence in Renewable Energetic Systems, pp 233–241. Springer

A Novel Hybrid Photovoltaic/Thermal Bi-Fluid (Air/Water) Solar Collector: An Experimental Investigation Mohamed El-Amine Slimani, Rabah Sellami, Mohammed Said, and Amina Bouderbal

Abstract This work deals with the construction, and the experimentation of a Bifluid (air/water) hybrid photovoltaic/thermal solar collector intended for the residential sector. The hybrid PV/T collector consists mainly of à 185Wp mono-crystalline photovoltaic module, a copper radiator for the circulation of water, and a ribbed metal absorber with fins. The copper radiator and the absorber were glued to the backside of the PV module. This structure was mounted in an insulated box with reservation of a rectangular duct for the air circulation between the absorber and the insulation panel of the box. An experimental bench was set up to evaluate the energy performance of the designed collector under real operating conditions. The obtained results showed that the proposed configuration of the bi-fluid PV/T collector both allows the air to be heated to temperatures appropriate for the intended application (space heating) and to recover additional calories by a second fluid (water) at appropriate temperatures (40–45 °C). Hot water can be used for a double purpose; a direct consumption in a building, and to prolong the production of hot air by a simple recirculation of the stored hot water in the radiator. The average of thermal, electrical, and overall energy efficiencies evaluated for this configuration are very important with values of 56.96%, 13.6%, and 94.53%, respectively. Keywords Photovoltaic cell · Hybrid solar collector · Bi-fluid · Energy performance · Photovoltaic and thermal conversion M. E.-A. Slimani (B) Theoretical and Applied Fluid Mechanics Laboratory, Department of Energetic and Fluid Mechanics, University of Science and Technology Houari Boumediene (USTHB), 16111 Bab Ezzouar, Algiers, Algeria e-mail: [email protected] R. Sellami Unité de Développement des Equipements Solaires (UDES), EPST Centre de Développement des Energies Renouvelables (CDER), 42415 Tipaza, Algeria M. Said · A. Bouderbal Department of Energetic and Fluid Mechanics, Faculty of Physics, University of Science and Technology Houari Boumediene (USTHB), 16111 Algiers, Algeria © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_47

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1 Introduction The overall performance of a photovoltaic module is strongly influenced by the solar radiation received on the plane of the latter, the ambient temperature, and the wind speed. In fact, the operating temperature of the module is the most important parameter which plays a key role in the photovoltaic conversion process and which thus influences its electrical efficiency. The rise in the operating temperature of the photovoltaic module is due to the part of the solar radiation absorbed by the cells and converted into heat. This undesirable effect can be reduced or eliminated by the introduction of a suitable heat extraction device using one or two heat transfer fluids to maintain, at the same time, the electrical efficiency of the PV module at a minimum reasonable level and recover from low-temperature heat. This combination of thermal and photovoltaic effects in the same device was introduced for the first time by Wolf, Kern and Russell in the 70s [1] of the last century giving rise to hybrid collects or PV/T type collectors. A hybrid photovoltaic/thermal (PV/T) collector is the combination of photovoltaic module and solar thermal collector to produce both electricity and heat at the same time. About 15–20% of the solar radiation received by the photovoltaic panels is converted into electrical output, and the rest of the solar energy ends up heating the PV cells [2]. Several kinds of research have been carried out in this field in order to study and improve the performance of hybrid PV/T system [3–8]. Browne et al. [9] compared the performance of a hybrid air collector to a thermal collector. They found that the PV/T air collector can produce a significant amount of energy on an equivalent surface. Geographical location and catchment area have a strong influence on the productivity of the hybrid component. The study of a PV/T water-based collector with glazed and unglazed to evaluate its performance has been the subject of several studies [10–13]. The results showed that glazed collectors show an increase in thermal efficiency and a reduction in electrical performance. And amorphous cells are the most suitable for thermal applications because they are less sensitive to temperature variations. Sarhaddi et al. [14] Contributed to the development of a simulation program for the parametric study of a PV/T air system in order to improve its thermal and electrical efficiency. Elsafi and Gandhidasan [15] conducted a comparative study of a dual-pass PV/T air collector, with and without concentrator and fins. They found the following points; performance has improved by using fins, aluminum fins are more efficient than copper or nickel, and rectangular fins are more efficient. Su et al. [16] developed a dynamic model for a PV/T water system using the thermosiphon effect. The results showed that the connection mode of the PV/T system has a significant effect on the energy outputs. Hybrid PV/T collectors can be used in several sectors; especially in building and agricultural activities [8, 17–21]. In the previous literature, several investigations have been carried out on the PV/T sensors for a single heat transfer fluid (either water or air).

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The aims of this work are the realization and the study of a hybrid photovoltaic/thermal bi-fluid solar collector intended for the residential sector. The main objective of this work is to offer users a hybrid collector configuration that will achieve high overall efficiency (Bi-fluid) and extend its usage time by integrating a heat storage device. For this purpose, this paper is comprised of several sections: system description; construction and experimental work; energy performance equations; experimental results and discussions; and finally a conclusion as a summary part of the work.

2 Structure and Description The hybrid PV/T collector presented for this study based on a monocrystalline photovoltaic module (JT-185 M). The PV/T solar collector composed mainly of the following components; a cover glass, PV cells immersed in a layer of EVA polymer, a protective layer of Tedlar below the PV cells, a metal plate below the Tedlar sheet, copper tubes bonded to the back of the module (on the top metal plate), another metal plate placed on the thermal insulation of the hybrid collector, and Baffles that are stuck between the two metal plates. A thermal insulator to minimize heat losses on the side and rear sides of the hybrid solar collector. The air circulates in the subchannels formed by the baffles to allow the heat transfer fluid to convey the thermal energy produced. The experimental prototype can be provided with an assisted PV ventilation system. Figure 1 represents a prototype of the studied hybrid collector and schematic design of its component. During the operation of the hybrid collector, the solar energy reaches the outer surface of the collector in the form of solar radiation. A portion of this radiation is converted into electrical energy and another part into thermal energy so that the heat produced by the cells is evacuated through the coolant flowing between the fins and the two metal plates (upper and lower). The hybrid air collector electrical and design parameters used in this document are given in Table 1.

3 Construction and Experimentation A. Experimental design As presented in Fig. 1, the experimental designed system is mainly composed of an unglazed PV/T collector provided with fins and tube radiator. The tow fluids follow with forced circulation. The air in PV/T collector is ventilated by a constant airflow assured by a DC electric alimentation instrument and the water circulate through the radiator by using AC pump PV/T collector has been fixed on a properly insulated aluminum housing using polyurethane foam. The air ventilation system consists of a brushless type DC fan that can be directly coupled to small PV modules.

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(a)

(b)

PV Cells Copper tube

Fin

(c) Fig. 1 Designed hybrid PV/T collector: a front view; b back view; c schematic section with fins and copper tubes

The realization of the PV/T solar collector took place in several stages taking into account the dimensions and the arrangement of the PV cells in the used photovoltaic module. It is, in fact, a module of 72 PV cells arranged in six parallel rows of twelve cells in series. For this purpose, it was decided to make a radiator consisting of 12 tubes in parallel so that each row of cells is cooled by two tubes as shown in Fig. 2. Once the radiator was completed, further work was started as follows:

A Novel Hybrid Photovoltaic/Thermal Bi-Fluid … Table 1 The main design parameter and characteristics of the hybrid PV/T collector

701 Parameter

PV module type

Value

Monocrystalline silicon

Cells number, N

12 × 6

Short-circuit current, Isc,r e f

5.76 A

Open-circuit voltage, Voc,r e f

43.2 V

Current at MPP, Imp,r e f

5.14 A

Voltage at MPP, Vmp,r e f

36 V

Maximum power, Pm,r e f

185 W

Electrical conversion efficiency, ηr e f

14.4%

PV/T geometry Parameter

Characters

Values

Length

L

1.60 (m)

Width

l

0.75 (m)

Thickness of fluid vein

e

0.04 (m)

Fins geometry Number of fins

N

Length

L f in

1.60 (m)

6

Width

h

0.04 (m)

Thickness

e f in

0.002 (m)

Number of tubes

N

12

Tubes length

L tube

1.62 (m)

Outside diameter

Do

0.012 (m)

Tubes length

Di

0.01 (m)

Radiator geometry

• Fixing the copper radiator on the back of the PV module by using a polyester resin reinforced with aluminum filings. The filings are used for a dual purpose which is to ensure both the reinforcement of the resin and to ensure that good thermal contact is done between the tubes and the surface of the PV module. • Making and bonding the fins on the radiator tubes and on the back of the PV module using an epoxy type resin. It is important to note that these fins have a dual role to play. They make it possible to extract both the heat transmitted by the copper tubes and the rear surface of the PV cells. This heat is then transferred to the air flowing through the PV/T collector. • Characterization by IR thermography of the reliability of the method of fixing the radiator and the fins on the rear part of the PV module. For this purpose, IR images have been taken before and after the passage of cold water inside the radiator tubes as shown in Fig. 3. • Installation of the fan at the end of the collector isolation box to allow control of the airflow circulating in forced convection. In order to guarantee a homogeneous

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Fig. 2 The different stages of construction and assembly of the PV/T collector: a radiator design; b fixation of the radiator on the back of the PV module; c bonding the fins on the radiator tubes and on the back of the PV module

distribution of air in the air circulation channel, a flow control grille (perforated grid with different opening diameters) is installed at the outlet of the channel. • Manufacture of a rectangular shaped tank for water storage. This rectangular shape of the tank was chosen for purely aesthetic reasons adapted to the realized system. • Installation of the water circulation pump, the flow control device, and the thermal insulation of the piping. As well as the installation of the various measuring instruments. B. Instruments of measure In order to allow us to evaluate the thermal performance of the collector realized under real operating conditions, a bench of measurements has been set up for the simultaneous measurement of the different parameters such as the temperature of the

A Novel Hybrid Photovoltaic/Thermal Bi-Fluid …

703 T ambiante (0C) solar Radiation (W/m 2 ) Wind Speed (m/s)

20

1000

9 8

19

800

18

600

7 6

17

400

5 4 3

16

200

2 1

15 0

:0

0

0

20

18

:0

0

0 :0 16

0 :0 14

0 :0 12

10

:0

0

0 :0 08

06

:0

0

14

Time (h)

Fig. 3 Evolution of solar irradiance, ambient air temperature, and wind speed during the test day (May. 14, 2019)

fluids (air and water), the temperature of the layers and components of the collector, solar illumination and air velocity. For this purpose, many measuring instruments have been used (type K thermocouples, hot wire anemometer, pyranometer, data acquisition) to conduct simultaneous measurements of all necessary parameters with a time step of five minutes. It is important to note that meteorological parameters were collected from the meteorological station installed at the UDES site.

4 Energy Performance The thermal power produced from the hybrid PV/T bi-fluid collector by its tow fluids (air and water) can be calculated using the following equation.     Q th = m˙ a × C p,a Ta,o − Ta,i + m˙ w × C p,w Tw,o − Tw,i

(1)

The electrical power generated by a PV or PV/T collector is calculated using the following correlation.      G Q ele = G · A · ηref 1 − βT · TC − TC,ref + γ · ln G ref

(2)

where G and G ref 2 which are the illumination and the reference illumination G ref = 1000 W /m ; which are the temperature and the reference temperature of the cells  Tc and Tc,ref Tc,ref = 25 ◦ C ; β T is the temperature coefficient, and γ is the solar radiation coefficient.

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The following relations give the thermal and the electrical efficiencies, respectively, this later is calculated from the correlation proposed by Evans [22]. ηth = ηele

Q th G·S

(3)

     G Q ele = ηref 1 − βT · TC − TC,ref + γ · ln = G·S G ref

(4)

Another rather important parameter that shows the overall energy efficiency of the PV/T collector is given as the sum  thermal yield (ηth ) and the equivalent of  of the the electrical efficiency in thermal ηele,th using the thermal conversion factor (C f ). η O = ηth +

ηele Cf

(5)

5 Results and Discussions The experimentation was made under the weather conditions of UDES (Unité de Développement des Equipements Solaires) located at Bousmail, Tipaza, for the test day of 14 May 2019. Table 2 gives the imposed operating conditions used for the experimentation step, including water and air flows and the volume of the storage tank. The Meteorological data for the test day of 14 May 2019 are given by Fig. 3. The weather data include solar radiation, ambient temperature, and wind speed. The evolution between the different key temperatures of PV/T collector that are; the PV cells temperature, Inlet and outlet water temperatures, inlet and outlet air temperatures. The evolution shows compliance and logic in the temperature distribution in different PV/T collector elements (Fig. 4). All temperatures of the collector being higher than the ambient temperature. The temperature of the solar cells shows the highest temperature in the hybrid collector; it reaches a value of about 47 °C at 13h30. This is explained by the fact that the photovoltaic cell is a generator converting the received solar radiation into electricity and heat at the same time. The temperatures of the collector elements increase as a function of time until solar noon, then decrease. The temperature of the fluid at the output of the collector passes through a maximum of 42.5 °C at 14h00 for water, and 32.5 °C for air. This range of temperature is very acceptable and comfortable for residential and building applications. Table 2 Gives the imposed operating conditions

Water flow (l/min)

Airflow (m3 /s)

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1

0.003535

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T O-w T Cell T i-w T amb T O-a

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:3 6 21

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:4 8 16

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04

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Fig. 4 Temperature evolution during the test day

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From Fig. 4 it can be noted that for this configuration, the temperature difference between the air inlet and air outlet is important and appreciable. The temperature difference range would be as high as 12 °C in relatively clear sky. Figure 5 shows the evolutions of the electrical and thermal powers for the considered day test. The graphs in Fig. 5 show, respectively, the evolution of the overall thermal and electric powers for the test day of 14/05/2019 as a function of time. We observe that these energies grow until reaching maximum values at solar noon. Note that the maximum value of the thermal power reaches 736.91 W and that of the electric power 800 600 400 200 0 -200 06:00

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Fig. 5 Evolution of electrical and thermal powers during the test day

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133.09 W. It is noted that the evolution of the electric power is substantially related to the meteorological conditions. The thermal power as for it; is naturally influenced by the weather conditions but also parametric. The evolution of the thermal power is related and influenced by the storage volume because by increasing it, the temperature difference of the coolant (water) increases and consequently the thermal power. Figures 6, 7, and 8 illustrate the evolution of electrical, thermal, and overall energy efficiencies, respectively, during the test day. At the beginning of the day, the thermal efficiency is zero because of the absence of solar radiation. Then, it increases to relatively stable values between 60 and 70% and then decreases rapidly towards the end of the day until it is canceled. Peaks can be observed due to disturbances of solar radiation. An average value of 56.96% was noted. The electrical efficiency at the beginning of the day recorded its maximum with a value of 15.03%. This efficiency decreases when the solar illumination increases until solar noon with a minimum value of 13%, and this is explained by the increase in the temperature of the module. 100

Thermal Effeciency, η th (%)

Fig. 6 Evolution of the thermal efficiency during the test day

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It is observed that the overall energy efficiency increases to its maximum value (about 110%). The yield remains almost stable, then they start to decrease rapidly towards the end of the day. The peaks are due to weather disturbances (cloudy passage). The average value of the overall efficiency calculated is 94.53%.

6 Conclusion The temperatures and the energy performance of an unglazed hybrid PV/T bi-fluid (air/water) solar collector provided with fins, radiators, and storage tank were investigated and analyzed. The developed configuration of the hybrid collector is designed to be suitable for winter cold periods to providing both hot water and air conditioning in a building. An experimental bench was set up and the energy performance of the solar device was evaluated for typical test day under a mild Mediterranean climate. The results obtained showed that the proposed configuration of the bi-fluid PV/T collector both allows the air to be heated to temperatures appropriate for the intended application (space heating or drying) and to recover additional calories by a second fluid (water) at appropriate temperatures (40–45 °C). Hot water can be used for a double purpose; a direct consumption in a building (domestic hot water), and to prolong the production of hot air by a simple recirculation of the stored hot water in the radiator and thanks to the fins which promote the thermal exchange between water and the air. The average of thermal, electrical, and overall energy efficiencies evaluated for this configuration are very appreciable with values of 56.96%, 13.6%, and 94.53%, respectively.

References 1. Kern MC Jr, Russell EC (1978) Combined photovoltaic and thermal hybrid collector systems. In: The 13th IEEE photovoltaic specialists, 5–8 June 1978, vol. 79, pp COO-4577-3; CONF780619-24

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2. Slimani MEA, Amirat M, Kurucz I, Bahria S, Hamidat A, Chaouch WB (2017) A detailed thermal-electrical model of three photovoltaic/thermal (PV/T) hybrid air collectors and photovoltaic (PV) module: Comparative study under Algiers climatic conditions. Energy Convers Manag 133:458–476 3. Slimani MEA, Amirat M, Bahria S (2015) Study and modeling of heat transfer and energy performance in a hybrid Pv/T collector with double passage of air. Int J Energy Clean Environ 16(1–4):235–245 4. Srinivas M, Jayaraj S (2013) Performance study of a double pass, hybrid-type solar air heater with slats. Int J Energy Eng 3(4):112 5. Slimani MEA, Amirat M, Bahria S (2015) Analysis of thermal and electrical performance of a solar PV/T air collector: Energetic study for two configurations. In: 3rd international conference on control, engineering and information technology, CEIT 2015, pp 1–6 6. Othman MY, Ibrahim A, Jin GL, Ruslan MH, Sopian K (2013) Photovoltaic-thermal (PV/T) technology – the future energy technology. Renew. Energy 49:171–174 7. Boumaaraf B, Touafek K, Ait-cheikh MS, Slimani MEA (2020) Comparison of electrical and thermal performance evaluation of a classical PV generator and a water glazed hybrid photovoltaic–thermal collector. Math Comput Simul 167:176–193 8. Gholampour M, Ameri M (2016) Energy and exergy analyses of Photovoltaic/Thermal flat transpired collectors: experimental and theoretical study. Appl Energy 164:837–856 9. Browne MC, Norton B, McCormack SJ (2016) Heat retention of a photovoltaic/thermal collector with PCM. Sol Energy 133:533–548 10. Chow TT, Pei G, Fong KF, Lin Z, Chan ALS, Ji J (2009) Energy and exergy analysis of photovoltaic–thermal collector with and without glass cover. Appl Energy 86(3):310–316 11. Fujisawa T, Tani T (1997) Annual exergy evaluation on photovoltaic-thermal hybrid collector. Sol Energy Mater Sol Cells 47(1–4):135–148 12. Fraisse G, Ménézo C, Johannes K (2007) Energy performance of water hybrid PV/T collectors applied to combisystems of Direct Solar Floor type. Sol Energy 81(11):1426–1438 13. Boumaaraf B, Boumaaraf H, Slimani ME-A, Tchoketch_Kebir S, Ait-cheikh MS, Touafek K (2020) Performance evaluation of a locally modified PV module to a PV/T solar collector under climatic conditions of semi-arid region. Math Comput Simul 167:135–154 14. Sarhaddi F, Farahat S, Ajam H, Behzadmehr A, Mahdavi Adeli M (2010) An improved thermal and electrical model for a solar photovoltaic thermal (PV/T) air collector. Appl Energy 87(7):2328–2339 15. Elsafi AM, Gandhidasan P (2015) Comparative study of double-pass flat and compound parabolic concentrated photovoltaic–thermal systems with and without fins. Energy Convers Manag 98:59–68 16. Su D, Jia Y, Huang X, Alva G, Tang Y, Fang G (2016) Dynamic performance analysis of photovoltaic–thermal solar collector with dual channels for different fluids. Energy Convers Manag 120:13–24 17. Slimani ME-A (2017) Etude d’un séchoir solaire agricole muni d’un capteur solaire de type” PV-THERM”-réalisation d’un prototype et caractérisation. Universite Des Sciences Et De La Technologie Houari Boumediene 18. Othman MY, Hamid SA, Tabook MAS, Sopian K, Roslan MH, Ibarahim Z (2016) Performance analysis of PV/T Combi with water and air heating system: an experimental study. Renew Energy 86:716–722 19. Xiang B et al (2018) A novel hybrid energy system combined with solar-road and soilregenerator: Sensitivity analysis and optimization. Renew Energy 129:419–430 20. Sathe TM, Dhoble AS (2017) A review on recent advancements in photovoltaic thermal techniques. Renew Sustain Energy Rev 76:645–672 21. Slimani ME-A, Sellami R, Mahrane A, Amirat M (2019) Study of hybrid photovoltaic/thermal collector provided with finned metal plates: a numerical investigation under real operating conditions. In: 2019 international conference on advanced electrical engineering, ICAEE’19, pp 1–6

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22. Sellami R, Amirat M, Mahrane A, Slimani ME-A, Arbane A, Chekrouni R (2019) Experimental and numerical study of a PV/Thermal collector equipped with a PV-assisted air circulation system: Configuration suitable for building integration. Energy Build 190:216–234

A DC/DC Buck Converter Voltage Regulation Using an Adaptive Fuzzy Fast Terminal Synergetic Control Noureddine Hamouda and Badreddine Babes

Abstract In this paper, an adaptive fuzzy fast terminal synergetic voltage regulation for DC/DC buck converter is designed based on recently developed synergetic theory and a terminal attractor method. The advantages of presented synergetic control include the characteristics of finite time convergence, insensitive to parameters variation and chattering free phenomena. Rendering the design more robust, fuzzy logic systems are used to approximate the unknown parameters in the proposed controller without calling upon usual model linearization and simplifications. Taking the DC/DC buck converter in continuous conduction mode as an example, the algorithm of proposed synergetic control is analyzed in detail. All the simulation results demonstrate the effectiveness and the high dynamic capability of the proposed AF-FTSC control technique over the FTSC strategy. Keywords Component · Synergetic control · Fuzzy logic system · Terminal technique · Finite time convergence · DC/DC buck converter

1 Introduction Many nonlinear control approaches such as variable structure technique has attracted community attention in the last decades. Due to its inherent robust features, this control technique has been extensively used in many nonlinear control systems such as voltage regulation of DC/DC buck converter. Problems of variable structure control are the chattering phenomenon, steady state errors caused by high frequency switching actions, and the fact that it needs fairly high bandwidths for the controller, which makes digital control realization unfeasible [1]. On the others hand, this article introduces the fast terminal synergetic control (FTSC) method which is utilized on N. Hamouda · B. Babes (B) Research Center in Industrial Technologies (CRTI), Algiers, Algeria e-mail: [email protected] N. Hamouda e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_48

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the DC/DC buck converter voltage control scheme. This approach has numerous advantages such as: it’s excellent suitable to digital control, it functions at fixed switching frequency which reduces the burden of filtering design, and it shows low chattering as well as the inherent robustness features of sliding mode control [2]. However, the robustness of synergetic control structure is not assured pending transient action when system states trajectories have not achieved the attractor. Therefore, a new technique to avoid this difficulty intrinsic to fast terminal synergetic control, an adaptive fuzzy fast terminal synergetic control (AF-FTSC) structure is suggested thus reducing the reaching phase. The rest of the manuscript is structured as follows. In Sect. 2, the fundamental concepts related to synergetic control theory and fuzzy logic systems are reviewed briefly. In Sect. 3, the basic model of the DC/DC buck converter in continuous conduction mode (CCM) is introduced. In Sect. 4, the adaptive fuzzy fast terminal synergetic control (AF-FTSC) is developed for the DC/DC buck converter system. In Sect. 5, simulation is performed on a DC/DC buck converter to verify the effectiveness and the applicability of the proposed AF-FTSC control technique over the FTSC strategy. In Sect. 6, the conclusions are given.

2 Preliminaries 2.1 The Basis of Synergetic Control (SC) Strategy In this section, we briefly introduce the fundamentals of synergetic control synthesis for a nonlinear dynamic system expressed as x˙ = f (x, u, t)

(1)

Basically, the synergetic control design structure begins with the definition of a macro-variable σ as a function of the state variables [3] σ = σ (x, t)

(2)

The control purpose is to oblige the system to work on a pre-chosen manifold = 0. Design a synergistic controller that would force the system states to exponentially approach the preferred manifold with an evolution constraint which can be declared as τ σ˙ + σ = 0

(3)

where τ is a positive constant which imposes the designer select speed convergence to the specified attractor.

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Taking into account the chain of differentiation which is given by σ˙ =

dσ · x˙ dx

(4)

Substituting (1) and (3) into (4) gives (5) τ

dσ f (x, u, t) + σ = 0 dx

(5)

Resolving (5) for u provides the synergetic control law as u = g(x, σ (x, t), τ, t)

(6)

The obtained control law (6) forces state trajectories to satisfy (3). Appropriate choice of the macro-variable (2) and judicious manifolds guarantees the performance and the pursued stability [4].

2.2 The Basis of Fuzzy Logic Systems It is known that fuzzy logic systems (FLSs) are universal estimators and have exceptional functions in controller design and identification. A type-1 FLS involves four main parts: fuzzifier, rule base, inference engine and defuzzifier. The main feature of the FLS is the aptitude to express the human-like thinking and deal with the experience effectively through the conditional fuzzy If-Then rules, which can be formulated as Rl : If x1 is F1l , x2 is F2l , . . . , xn is Fnl Then y1 is G l1 , y2 is G l2 , . . . , ym is G lm

(7)

where l = 1, 2,…, L, with L indicating the total number of rules, x = [x1 , x2 , . . . , xn ]T and y = [yl , y2 , . . . , ym ]T represent the input and output parameter vectors of fuzzy logic system, respectively. F il and Gjl are the linguistic variables of the fuzzy sets, described by their membership function vectors μ Fil (xi ) and μG lj (y j ). Using the singleton fuzzifier, product inference engine, and centre-average defuzzifier, the inferred output of the FLS can be expressed as [5]  L n

 l l (x i ) y μ j l=1 i=1 Fi  = θ jT ξ(x), j = 1, 2, . . . , m. y j (x) =   L n l=1 i=1 μ Fil (x i ) and:

(8)

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n ξi (x) =

i=1

μ Fil (xi )

L   n i=1 l=1

μ Fil (xi )

(9)



where y lj is the point in V j at which μG lj (y j ) achieves its maximum value (assuming μG lj (y j ) = 1).  T θ j = y 1j , y 2j , . . . , y Lj ∈ R L is called the adjustable parameter vector and  1 T ξ(x) = ξ (x), ξ 2 (x), . . . , ξ L (x) ∈ R L is the vector of fuzzy primary function or the antecedent function vector. T  ξ(x) = ξ 1 (x), ξ 2 (x), . . . , ξ L (x) ∈ R L

(10)

If y lj are preferred as the free parameter, the above fuzzy system (7) becomes an adaptive fuzzy system. Then, it can be rewritten as y(x) = θ T ξ(x)

(11)

where θ is (L × m) matrix, θ j and indicates the (L × 1) j-th column of the matrix θ .

3 Modeling of the DC/DC Buck Converter The corresponding circuit of the DC/DC buck converter is displayed in Fig. 1. The positive constants R, L, C, V in and V o are respectively load resistor, self-inductor, capacitor, input voltage and output voltage. The dynamic equations of a DC/DC buck converter in the continuous conduction mode (CCM) are expressed as in [6] ⎧ di 1 Vin L ⎪ = − Vo + u ⎨ dt L L ⎪ ⎩ d Vo = 1 i − 1 V L o dt C RC

(12)

iL

Fig. 1 The DC/DC buck converter

L

Q

Vin

+ _

D u

C

R

Vo

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where iL denotes the inductor current and u is the duty ratio of the DC/DC buck converter. Choosing the output voltage and its derivative as system state variables, we get

x1 = Vo x2 = x1 =

d Vo dt

(13)

The state space average model of the DC/DC buck converter can be rewritten as fellow x˙1 = x2 (14) Vin x1 x2 x˙2 = − LC − RC + LC u

4 Adaptive Fuzzy Fast Terminal Synergetic Control (AF-FTSC) Design If we suppose the reference tracking voltage to be V ref , then the tracking error and its derivative are as follows e = x1 − Vr e f

(15)

e˙ = x2 − V˙r e f

(16)

To allow the application of AF-FTSC algorithm to a DC/DC buck converter, a special variable called the nonlinear macro-variable function needs to be introduced as [7] σ = e˙ + αe + βe p/q

(17)

where α and β are the macro-variable constants, p and q are positive odd numbers, that assure the following condition 1 < p/q < 2. A constraint imposing desired dynamics to the macro-variable is selected as τ σ˙ + σ = 0

(18)

where τ is a positive constant which imposes the designer decide speed convergence to the preferred attractor. When the macro-variable reaches σ = 0, the system dynamics (17) can be expressed as follows e˙ + αe + βe p/q = 0

(19)

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On integrating (19), and through setting suitable parameters α, β, p and q, the convergence time of the system from any given initial condition e(0) = 0 to the attractor σ = 0 is finite-time t r ≥ 0 and is given by

 α|e(0)|1−( p/q) + β p ln tr = α( p − q) β

(20)

The time derivative of σ can be rewritten as σ˙ = e¨ + α e˙ + β

d p/q e dt

(21)

Combining (18) and (21), leads to (22) e¨ + α e˙ + β

p ( p/q)−1 1 e e˙ = − σ q τ

(22)

For the controller design we set the following simplification g(x) =

Vin x1 x2 , and f (x) = − − LC LC RC

Solving for the terminal synergetic control algorithm uFTSC , guides to (23) u F T SC =

  p 1 1 V¨r e f − f (x) + σ − α e˙ − β e( p/q)−1 e˙ g(x) τ q

(23)

Generally, the nonlinear system functions ƒ(x) and g(x) are difficult to determine precisely. So, a fuzzy logic system using the universal approximation theorem is used to estimate the nonlinear and unknown functions f (x) and g(x) in (23). Let ˆ x|θg ) be the fuzzy approximates of the vector functions ƒ(x) and fˆ( x|θ f ) and g( g(x), respectively. Then, the controller (23) is improved as follows u F T SC

  p ( p/q)−1 1 1 ˆ ¨ = e˙ Vr e f − f ( x|θ f ) − σ − α e˙ − β e g( ˆ x|θg ) τ q

(24)

where fˆ( x|θ f ) = θ Tf ξ f (x)

(25)

g( ˆ x|θg ) = θgT ξg (x)

(26)

where ξ f (x) and ξg (x) are the fuzzy basis functions vector, θ f and θg are the parameters vector of fuzzy system.

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Define the optimal estimation parameters of fuzzy system as θ ∗f

     ˆ  = arg min sup  f x|θ f − f (x)

(27)

θg∗

        = arg min sup gˆ x|θg − g(x)

(28)

θf ∈ f

x∈R n

θg ∈ g

x∈R n

where Ω f and Ω g represent constraint sets for θ f and θ g , respectively. Defining the minimum estimation error as         ε = fˆ x|θ ∗f − f (x) + gˆ x|θg∗ − g(x) u

(29)

Substituting (29) into (18), the dynamic of macro-variable is obtained as follows           1 σ˙ = fˆ x|θ ∗f − fˆ x|θ f + gˆ x|θg∗ − gˆ x|θg u − σ + ε τ

(30)

Leading after some simple manipulations to    ∗T  1 T T σ˙ = θ ∗T f − θ f ξ(x) + θg − θg ξ(x)u − σ + ε τ

(31)

Or in a more compact form 1 σ +ε τ

(32)

  1 1 1 2 σ + θ˜ Tf θ˜ f + θ˜gT θ˜g 2 r1 r2

(33)

σ˙ = θ˜ Tf ξ(x) + θ˜gT ξ(x)u − where θ˜ f = θ ∗f − θ f and θ˜g = θg∗ − θg Define the Lyapunov function candidate V as V =

where r 1 and r 2 are positive constants. The time derivative of V can be written as 1 1 V˙ = σ σ˙ + θ˜ Tf θ˙˜ f + θ˜gT θ˙˜g r1 r2

(34)

Substituting (32) into (34), we can obtain (35)   1 1 1 T T ˙ ˜ ˜ V = σ θ f ξ(x) + θg ξ(x)u − σ + ε + θ˜ Tf θ˙˜ f + θ˜gT θ˙˜g τ r1 r2

(35)

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Fig. 2 Block diagram for proposed AF-FTSC controller

In which we make use of θ˙˜ f = θ˙ f and θ˙˜g = θ˙g to obtain (36)  1   1  1 V˙ = − |σ | + σ ε + θ˜ Tf r1 σ ξ(x) + θ˙ f + θ˜gT r2 σ ξ(x)u + θ˙g τs r1 r2

(36)

To make V˙ ≤ 0, we choose the adaptive laws as θ˙ f = −r1 σ ξ(x)

(37)

θ˙g = −r2 σ ξ(x)u

(38)

1 1 V˙ ≤ − |σ | + σ ε ≤ − |σ | ≤ 0 τ τ

(39)

Then (36) becomes

The block diagram of the designed AF-FTSC controller is presented in Fig. 2.

5 Simulation Results In this section, some computer simulations are conducted using MATLAB/Simulink software to show the control performance of the designed AF-FTSC control technique over the FTSC strategy [7]. The DC-DC buck converter utilized in simulation study has the following component values: L = 5 mH, C = 1100 µF, R = 20–5 , V in = 50 V. The main parameters used in designing controllers are: α = 100, β = 150, p/q = 7/9, τ = 0.005, r 1 = 2.106 and r 2 = 3.5.105 .

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The start-up transient response of V o and iL during the changes in the desired output voltage from 0 to 30 V is shown in Fig. 3a, b, respectively. It is observed that the proposed AF-FTSC algorithm has better performance in terms of the settling time relatively to the FTSC controller.

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Fig. 5 Simulation results of output voltage for a reference trajectory tracking, b sinusoidal-wave tracking, c triangular-wave tracking

Figure 4 shows the simulation responses of V o when the load resistance steps from nominal value of 20 to 5  and vice versa. It is evident from Fig. 4 that under load resistance changes the AF-FTSC algorithm can respond faster than the FTSC algorithm. Also, due to the independence of AF-FTSC control law to load resistance

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R, the proposed controller shows a robust behavior against drastic changes of the load resistance. Figures 5a–c depicts the output voltages tracking performance of the proposed AF-FTSC and FTSC controllers. It is intuitive to see that under reference changes the proposed AF-FTSC controller can respond faster than the FTSC controller. Also, the above simulation comparisons effectively confirm the advantage of the proposed AF-FTSC algorithm over FTSC algorithm.

6 Conclusions In this manuscript, an AF-FTSC control law has been developed and implemented to deal with the voltage tracking problems of the DC/DC buck converter. The proposed control algorithm has been designed by using techniques of the synergetic theory of control and the terminal attractor technique, which assures finite time convergence of the output voltage to the desired voltage. The unknown parameters in the proposed controller can be approximated precisely by using the fuzzy system with suitable adaptive laws. The stability of the system is also proved using direct Lyapunov theorem. Simulation results demonstrate the robustness and effectiveness of the proposed AF-FTSC control technique over the FTSC strategy.

References 1. Son YD, Heo TW, Santi E, Monti A (2004) Synergetic control approach for induction motor speed control. In: The 30th annual conference of the IEEE industrial electronics society, 2–6 November 2004, Busan, Korea 2. KolesnikovA, Veselov G, Kolesnikov A, Monti A, Ponci F, Santi E, Dougal R (2002) Synergetic synthesis of DC-DC boost converter controller: theory and experimental analysis. In: IEEE APEC 2002, 2 DilZ, pp 409–415 3. Santi E, Monti A, Li D, Proddutur K, Dougal RA (2003) Synergetic control for DC-DC boost converter: implementation options. IEEE Trans Ind Appl 39(6):1803–1813 4. Bouchama Z, Essounbouli N et al (2016) Reaching phase free adaptive fuzzy synergetic power system stabilizer. Electr Power Energy Syst 77:43–49 5. Yoo BK, Ham WC (2000) Adaptive control of robot manipulator using fuzzy compensator. IEEE Trans Fuzzy Syst 8:186–199 6. Nettari Y, Kurt S (2018) Design of a new non-singular robust control using synergetic theory for DC-DC buck converter. Electrica 18(2):292–299 7. Zerroug N, Harmas MN, Benaggoune S, Bouchama Z, Zehar K (2018) DSP-based implementation of fast terminal synergetic control for a DC–DC buck converter. J Frankl Inst 355:2329–2343

Study of Electrical Field Distribution in the Insulation of High-Voltage Cables Rouini Abdelghani, Kouzou Abdellah, and Larbi Messaouda

Abstract Insulations are the most important part of high voltage equipment such as cables and machines. Therefore the study of the condition and mechanism of failure of high voltage insulations is important. This paper studies the electrical constraints within an XLPE insulated cable containing micro-cavities. The purpose of this work is to determine by simulation the distortions caused by micro-cavities on the electric field distribution in the insulating layer of cable. The calculation of these constraints is performed by the resolution of LAPLACE equation using Finite Difference Method (FDM) because of its simplicity in the symmetrical and twodimensional systems, which corresponds perfectly in the case of cables which can be considered as cylindrical systems with axial symmetry. Keywords High-voltage cables · LAPLACE equation · Finite difference method (FDM) · Air microcavities

1 Introduction The chemically cross linked polyethylene PRC is the most used material in the industry of electric cables (high voltage). However, the crosslinking processes applied to polyethylene to obtain the PRC all lead to the formation of microcavities within the material [1, 2]. The sizes of these defects vary from 1 to 20 μm and they are concentrated in the part located at three quarter of the thickness of the cable insulation starting from the core [3]. The heterogeneity of the dielectric medium that presents R. Abdelghani (B) · K. Abdellah Electrical Engineering Department, Applied Automation and Industrial Diagnostic Laboratory, Djelfa, Algeria e-mail: [email protected] K. Abdellah e-mail: [email protected] L. Messaouda Department of Science and Technology, University of Batna, Batna, Algeria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_49

723

724 Table 1 Characteristic of the cable

R. Abdelghani et al. Cable features

Value for PRC

Type

185 mm2 , Al, 30/50 kV

Ray of the core

7.65 mm

Thickness of the semi-layer conductor on the core

1.0 mm

Thickness of the insulation

11.3 mm, PRC

Thickness of the semi-layer conductive on the insulation

1.2 mm

Metal screen made of copper wire

0.8 mm

PVC protective sheath

5.3 mm

Dielectric loss factor

4.10-3

Electrical conductivity

10–12 ( cm) -1

Thermal conductivity

0.286 W/m °C

Volume heat

2.08 J/cm3 °C

the portion containing the microcavities leads to the distortion in the lines of electric field in the vicinity of these defects. When the intensity of the electric field inside the microcavities reaches the dielectric strength of the gas, a partial electric discharge arises. The magnitude of these distortions depends on the permittivities of the two environments, the shape of the microcavity and its position relative to the cable core [4, 5]. In the industry, cable breakdown tests are necessary, however during their manufacturing, it is difficult to avoid the penetration of foreign particles within the insulation (defects), and we can hardly realize such case to study the influence of these particles. The size of the fault, its nature (permittivity) and its position in the cable insulation as well as the effects of the electromechanical pressure and the temperature, considerably influence the behavior of the insulation of the cable and thus limiting its lifetime. The purpose of this article is based on the study of the effect of the presence of an air cavity, its position and size on the distribution of the electric field, the electromechanical pressure and the dielectric losses in an insulator. PRC type used in the insulation of medium voltage cables (18/30 kV) manufactured by ENICAB-Biskra. This cable has the characteristics listed in Table 1.

2 Influence of Microcavities on the Distribution of the Electric Field The effects of a permittivity cavity ε2 on a host dielectric of ε1 permittivity are first evaluated in terms of potential perturbation [5–10]. We will determine for the case of a spherical cavity the expressions of the potential and the electric field in the insulating medium and in the cavity. Assuming that the density of free charges ρ in

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725

the host dielectric is small enough to be neglected, the Poisson equation describing the potential in both media is reduced to the Laplace equation such that: ∇2 V = 0

(1)

Given the cylindrical shape of the cable, the lines of the electric field are radial and divergent, on the other hand the dimensions of the cavity are quite small in front of the thickness of the insulation [11–18]. Given the symmetry of the system at turn of the cable axis (Z), the Eq. (1) can be expressed by the following form:     1 ∂ ∂V ∂ r2 ∂V + sin θ =0 ∂r ∂r sin θ ∂θ ∂θ The general solution of (2) is of the form V (r, θ ) = f (r )g(θ ) With f (r )

(2) 

An r n

n

and g(θ ) = Pn (cos θ ) or Pn is the Legendre polynomial. This solution is valid in both environments. The expression of the potential for the case of a spherical cavity is of the form:   Bi (3) Vi = Ai + 2 cos θ r The index i takes the values 1 or 2 respectively corresponding to the insulator and the cavity. The integration constants Ai , Bi are determined by the boundary conditions. The expressions of the potential in the insulator and in the spherical cavity are obtained as follows:   ε1 − ε2 r23 · 2 cos θ V1 = E 0 r + 2 ε1 − ε2 r 3 ε1 r cos θ 2 ε1 + ε2

(5)

3 ε1 ∂ V2 = E0 ∂x 2ε1 + ε2

(6)

V2 = E 0 E2 =

(4)

or: r 2 : Is the radius of the cavity. r: Is the position of a given point in the insulation relative to the center of the cavity. It is noted that the potential V 2 inside the cavity depends on E 0 (rcosθ), Knowing that:   Rt ln ri V1 E0 = =   (7) Rt ri ln rc

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where V 1 is the conductor voltage, Rt is the outer radius of the cable, r c is the radius of the conductive core and r i is a given radial position. It follows that the intensity of the proportional E 2 field at E 0 depends on the values of the permittivities ε1 and ε2 of the two environments [2].

2.1 Approximation of the Electrical Potential at Each Node Generally, when dealing with a problem with two environments, it is better to use the finite difference method, especially if the two environments have a symmetry and a fairly simple geometry. This method consists in replacing the continuous medium, in which the equation to be studied is applicable at all points by a set of points to which the discretized equation applies (Fig. 1) [5, 7–10]. For such reason this method has been chosen to be used in this article. The equation of the potential obtained by the series development of TAYLOR: 1 2 2 V (i, j + 1) + ri2 · h 2t · h r2 V (i, j) =  2 2 2  . ri 1·h t V (i, j − 1) + 2. ri · h t · h r r 2 ·h 2t i

(2·ri +h r ) V (i 2·ri ·h r2 2·(ri +h r ) V (i 2·ri ·h r2

+ 1, j)+ − 1, j)

This formula is applicable for each node in both environments [7, 8]. Fig. 1 Discritization of the cavity

(8)

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2.2 Approximation of the Electric Field in Each Node Since the electric field is deduced by calculation of the potential gradient, it is given by: E =



grad V = − ∂∂rV · er − r1 · Er · er + E θ · eθ

∂V ∂θ

· eθ

(9)

Thus, electric field module will be:

E=

Er2 + E θ2

(10)

However, the radial E r and tangential E θ components of the electric field will be written as follows:

Er (i, j) = 2h1 r [V (i + 1, j) − V (i − 1, j)] (11) E= E θ (i, j) = 2h1t ri [V (i, j + 1) − V (i, j − 1)] The module in each point will be given by: E(i, j) =



Er2 (i, j) + E θ2 (i, j)

(12)

2.3 Approximation of Electrostatic Pressure and Dielectric Losses The electromechanical pressure and the dielectric losses are connected directly to the electric field by the following expressions [19–21] P(i, j) =

1 ε0 εr E 2 (i, j) 2

Pdiel (i, j) = ωε0 εr tgδ E 2 (i, j) + σ E 2 (i, j) P(i, j): is the electromechanical pressure (N/m2 ) in the point (i, j). Pdiel (i, j): is the dielectric loss (W/m3 ) in the point (i, j).

(13) (14)

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3 Results and Interpretations 3.1 Influence of Air Cavity on Electric Field Distribution In Fig. 2 we have presented the radial distribution of the electric field according to the radius of the cable. This distribution is marked by increased distortion at the cavity-insulator interface, which can be explained by the accumulation of polarization charges that strengthens the field inside the cavity and limits it in the insulator. This distortion quickly recovers over a distance of about three times the diameter of the cavity. It is found that the distribution of the electric field which passes through the center of the cavity and on the walls follows the same pace with values much higher than the center of the cavity. Figure 3 show the values of the electric field according to the position of a1μm radius of cavity. The field in the cavity center has a value of 8.5 kV/mm, it decreases depending the position of the cavity and reaches a value of 3.5 kV/mm in a cavity near the sheath. In Fig. 4 we presented the values of the electric field in terms of the radius of a cavity located at a distance of 8.35 mm from the conductive core. Note that the electric field in the center of the cavity is proportional to its radius.

On the line passing through the walls On the line passing through the center of the cavity Case of a homogeneous insulation

4.5

Electric field E (KV/mm)

4

3.5

3

2.5

2

1.5 17.305

17.31

17.315

17.32

17.325

17.33

17.335

17.34

radial position ri (mm)

Fig. 2 Distribution of electric field under the influence of a cavity located 17.325 mm from the center of the conductive core

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729

9 On the line passing through the walls On the line passing through the center of the cavity Case of a homogeneous insulation

Electric field E (KV/mm)

8 7 6 5 4 3 2 1

8

10

12

14

16

radial position ri (mm)

Fig. 3 Electric field inside the cavity according to its position

Fig. 4 Electric field inside the cavity according to its radius

18

20

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3.2 Influence of the Air Cavity on the Electromechanical Pressure Distribution We elaborate in Fig. 5 the electromechanical pressure in accordance with the radius of the cable. The electromechanical pressure in the center of the cavity is more intense than in the homogeneous case because of the permittivity changing, and on the other hand the electromechanical pressure in the walls of the cavity is greater than in the cavity center because of the accumulation of the charges on the walls of the cavity, which exerts an additional pressure on its walls. It is noted in Fig. 6 that the electromechanical pressure is greater in the case of a cavity close to the conducting core, and each time the distance between the core and the cavity is enlarged the electromechanical pressure decreases, since the Electromechanical pressure is proportional to the electric field. It can be seen in Fig. 7 that the electromechanical pressure increases as the radius of the cavity increases because of the charges in the walls which are in a very large quantity as the size of the cavity increases.

On the line passing through the walls On the line passing through the center of the cavity Case of a homogeneous insulation

180

electrostatic pressure P (N/m2)

160 140 120 100 80 60 40 20 17.305

17.31

17.315

17.32

17.325

17.33

17.335

17.34

radial position ri (mm)

Fig. 5 Distribution of the electromechanical pressure under the influence of a cavity located 17.325 mm from the center of the conductive core

Study of Electrical Field Distribution …

731

700 On the line passing through the walls On the line passing through the center of the cavity Case of a homogeneous insulation

electrostatic pressure P (N/m 2)

600

500

400

300

200

100

0

8

10

12

14

16

18

radial position ri (mm)

Fig. 6 Influence the position of the cavity on the electromechanical pressure

Fig. 7 Influence of cavity radius on electromechanical pressure

20

732

R. Abdelghani et al. 450 On the line passing through the walls On the line passing through the center of the cavity Case of a homogeneous insulation

dielectric losses Pdié l (W/m3)

400 350 300 250 200 150 100 50 17.305

17.31

17.315

17.32

17.325

17.33

17.335

17.34

radial position ri (mm)

Fig. 8 Distribution of dielectric losses under the influence of a cavity located at 17.325 mm center of the conductive core

3.3 Influence of the Air Cavity on the Dielectric Loss Distribution We have shown in Fig. 8 the distribution of dielectric losses according to the radial position. It is noted that the dielectric losses on the line passing through the walls of the cavity are very important as the line passing through the center of the cavity. In Fig. 9 we present the distribution of the dielectric losses within the position of the cavity, it is found that the dielectric losses rises to an important stage in the case of a cavity (1 μm) close to the conductive core, and drops each time the cavity approaches the outer sheath. Figure 10 shows the dielectric losses depending the radius of the cavity, a linear increase is observed because of the charges in the walls, which are in a very large quantity as the size of the cavity increases.

4 Conclusion The main goal of this work is the study of the behavior of insulators used in unipolar high voltage (HV) cables in the presence of cavities. We found that defects disturb the electric field within the insulation and strongly depend on their positions and sizes. This field disturbance causes a dielectric losses increasing within the insulator and leads to the acceleration of the degradation thereof. This disturbance does not

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1600 On the line passing through the walls On the line passing through the center of the cavity Case of homogeneous insulation

dielectric losses Pdiél (W/m 3)

1400 1200 1000 800 600 400 200 0 8

10

12

14

16

18

20

radial position ri (mm)

Fig. 9 Influence of the position of the cavity on the distribution of dielectric losses

Fig. 10 Influence of the radius of the cavity on the distribution of dielectric losses

exceed a certain zone called zone of influence, this allowed us to reduce the size of the calculation procedure, and reduce the execution time.

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References 1. Harisha KS, Gouthami N, Harshitha V, Madhu C (2016) Partial Discharge Analysis of a Solid Dielectric Using MATLAB Simulink. Int J Innov Res Electr Electron Instrum Control Eng 4(6) 2. Gupta K, Yadav NK, Rattewal PK (2015) Partial discharge within a spherical cavity in solid dielectric material. Int Res J Eng Technol (IRJET) 02(08) 3. Ramachandra B, Manohara HC (2016) PD analysis in cylindrical void with respect to geometry of the void. Int J Innov Res Electr Electron Instrum Control Eng 4(10) 4. Srinivasa DM, Harish BN, Harisha KS (2017) Analysis study on partial discharge magnitudes to the parallel and perpendicular axis of a cylindrical cavity. Int J Eng Trends Technol (IJETT) 45(7) 5. Hadjadj M, Mokhtari B, Mahi D (2014) Prediction of the physical parameters change inside aspherical cavity located in a material XLPE of a medium voltage cable by non-stationary modelling. J Electr Eng 14(1):184–190 6. Karmakar S, Sabat A (2011) Simulation of partial discharge in high voltage power equipment. Int J Electr Eng Inform 3 7. Arief YZ, Izzati WA, Adzis Z (2012) Modeling of partial discharge mechanisms in solid dielectric material. Int J Eng Innov Technol (IJEIT) 1(4) 8. Sharma P, Bhanddakkar A (2015) Simulation model of partial discharge in power equipment international. J Electr Electron Res 3(1):149–155 9. Manani HT, Dudani KK (2014) Analysis of partial discharge signal by FDTD technique. Int J Adv Eng Res Dev (IJAERD) 1(3) 10. Khan Taimur, Haleem Adnan (2018) A novel method for insulation testing of high voltage electrical equipment. Int J Eng Works 5(2):32–36 11. Kumar CS, Ramachandra B (2016) Comparison of PD activity in cylindrical and cubical void using MATLAB simulink. Int J Innov Res Electr Electron Instrum Control Eng 4(10) 12. Illias HA, Lee ZH, Bakar AHA, Mokhlis H (2012) Distribution of electric field in medium voltage cable joint geometry. In: 2012 IEEE international conference on condition monitoring and diagnosis, 23–27 September 2012, Bali, Indonesia 13. Illias H, Chen G, Lewin PL (2011) Modeling of partial discharge activity in spherical cavities within a dielectric material. In: IEEE electrical insulation magazine, vol 27, pp 38–45 14. Illias HA, Chen G, Lewin PL (2012) Partial discharge within a spherical cavity in a dielectric material as a function of cavity size and material temperature. In: IET science, measurement and technology, vol 6, pp 52–62 15. Agarawal SK, Mittal LK, Jafri H (2017) Simulation of partial discharge in high voltage power equipment. Int J Electron Electr Comput Syst IJEECS 6(8) 16. Callender G, Rapisarda P (2016) Investigating the dependence of partial discharge activity on applied field structure. In: IEEE electrical insulation 17. Dabbak SZ, Illias HA, Ang BC (2017) Effect of surface discharges on different polymer dielectric materials under high field stress. In: IEEE transactions on dielectrics and electrical insulation 18. Altay Ö, Kalenderli Ö, Merev A (2009) Preliminary partial discharge measurements with a computer aided partial discharge detection system. In: ELECO 19. Bhandakkar J (2017) Analysis and simulation of partial discharge for different insulation material. Int J Innov Eng 20. Illias H, Chen G, Lewin PL (2011) Modeling of partial discharge activity in spherical cavities within a dielectric material. In: IEEE electrical insulation 21. Tanmaneeprasert T, Lewin PL (2017) Analysis of degradation mechanisms of silicone insulation containing a spherical cavity using partial discharge detection. In: Conference (EIC)

Self Tuning Fuzzy Maximum Power Tracking Control of PMSG Wind Energy Conversion System Chaib Housseyn, Mihoub Youcef, and Hassaine Said

Abstract This paper proposes a chart to compare several ways to control the Wind Energy Conversion System (WECS) connected to the grid, using permanent magnet synchronous generator (PMSG). To overcome the limitation of classical fixed gain based proportional-integral (PI) witch requires knowledge of system parameters, a Fuzzy Logic Controller (FLC) is witch doesn’t rely on mathematical model and integrates human experience. In this study, we used several methods to control the Maximum Power Point Tracking algorithm (MPPT): PI controller, FLC and Self Tuning Fuzzy Logic Control (ST-FLC). The system contains two controllers for the machine side (MSC) and grid side (GSC) converters. On the machine side, the rotation speed control is obtained by assuring the algorithm MPPT to follow the maximum power of the wind. In the grid side, the DC voltage and (d q) currents control is used to ensure active and reactive power. The system analysis is realized in volatile wind conditions using the Matlab/simulink® with SimPowerSystems block. Obtained results show that fuzzy self-tuning technique appears superior to the others, as it guarantees in general very good performances in the set-point and load disturbance step responses. It requires a modest implementation effort; therefore its practical implementation in industrial environments appears to be very promising. Keywords Wind energy conversion system (WECS) · Maximum power point tracking (MPPT) · Permanent magnet synchronous generator (PMSG) · Proportional integral (PI) controller · Fuzzy logic controller (FLC) · Self-tuning fuzzy logic controller (ST-FLC)

C. Housseyn (B) · M. Youcef · H. Said Laboratory of Energy Engineering and Computer Engineering (L2GEGI), University of Tiaret, Tiaret, Algeria e-mail: [email protected] M. Youcef e-mail: [email protected] H. Said e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_50

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1 Introduction Winds are a renewable energy resource for electricity generation because of their productive capacity compared to other types of renewable energy resources. It is harmless to the environment and is abundant in nature. Thus, wind energy can be used by mechanically converting it into electrical energy using wind turbines (WTs) [1]. The development of wind turbine concepts and the creation of several different types of wind generators is a consequence of the fast-paced evolution of wind-transfer technology. There is two types of wind turbines, fixed and variable speed. Fixed-speed wind turbine is based on a standard squirrel-cage induction generator (SCIG) that uses a multi-stage gearbox. Variable-speed wind turbines rely on two different types of generators. The first type is a doubly-fed induction generator (DFIG) that uses a multi-stage gearbox. The second type is a direct-drive generator or a low-speed high-torque synchronous generator without gear [2]. In recent years, wind turbines have developed to use a mixed type with gearbox and PMSG giving a solution for modern wind systems due to the different advantages to other generators [3, 4]. Control of wind turbines must be permanent and continuous so as to maintain the output power when the wind speed is varying. PI classical control requires knowledge of system parameters [5–7], and presents several problems when switching to low frequency, as well as a slow response. To overcome this limitation; several advanced control methods based on artificial intelligence control have been proposed such as fuzzy logic [8], genetic algorithms [9, 10] and neural networks [10, 11]. FLC based control systems offer much-improved performance to run the wind system at maximum power under changing environmental conditions and uncertainties [12]. The main interest of Fuzzy Logic Controllers is their ability to integrate experience, intuition, and heuristics into the system instead of relying on mathematical models, making them more efficient in system applications especially where existing models are poorly defined and insufficiently reliable [13]. Fuzzy logic controllers are suitable for wind systems characterized by high non-linearity [14]. A new technique ST-FLC, in which the controller gain is continuously, adjusted using the gain update factor is proposed in this study. We focused only on setting the output scale factor gain (Gu), considering it as equivalent to the gain of the controller. The highest priority is given to setting the Gu because of its considerable influence on the performance and stability of the system self tuned by a gain update factor β determined from a mathematical algorithm [15]. Updating instantaneously the output scale is the major contribution to improve the FLC in witch this gains is fixed. The purpose of this article is to present a comparative study of control for a maximum power tracking strategy for variable speed wind turbines between a classical PI, FLC and ST-FLC controllers. The proposed algorithms are applied to control the PMSG rotation speed which regulates the output voltage to the predetermined value to extract the maximum possible wind power. The paper is classified into five sections: Sect. 2 details the dynamic modelling of each element of WECS, Sect. 3 describes the MSC design, Sect. 4 describes the

Self Tuning Fuzzy Maximum Power Tracking Control …

737

GSC design, Sect. 5 presents the simulation results, and final section dedicates to the conclusion and perspectives.

2 Wind Energy Conversion System Description Figure 1 shows the system structure of a variable speed wind turbine based on a PMSG, two full-size back-to-back PWM (pulse width modulation) converters and a power grid. The modeling of each system component is presented as following.

2.1 Wind Turbine The mechanical power extracted from the wind can be expressed as follows pm = 1/2ρπ R 2 VW3 C P (λ, B)

(1)

where Pm is the extracted power from the wind, ρ is the air density (Kg/m3 ), R is the blade radius (m), V w is the wind speed (m/s) and C p (λ, B) is the power coefficient and λ is the tip speed ration. The term λ is defined as λ=

ωm · R VW

(2)

The blade pitch angle B is always constant during MPPT control (B = 0), ωm is the mechanical speed of the turbine. The input torque for the generator is obtained from: Tm =

Fig. 1 Wind energy conversion system

Pm ωm

(3)

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2.2 PMSG Model The generator is modeled by the following voltage equations in the rotor reference frame (dq-axes): Vsd = Rs · Isd +

dφsd + ωe · φsq dt

(4)

Vsq = Rs · Isq +

dφsq + ωe · φsd dt

(5)

where isd and isq are the currents in the (dq) reference frame, Rs is the stator resistance, ωe is the electrical rotating speed, ϕ sd and ϕ sq are the d and q-axes stator flux linkages given by: ϕsd = −L d · i sd + φm

(6)

ϕsq = −L q · i sq

(7)

where L d and L q are the inductances in the (dq)-axis and φ m is the rotor magnetic flux produced by the generator. The expression of the electromagnetic torque T e is provided by: Te =

 3  P L d − L q i sd i sq + φm i sq 2

(8)

where, P is the number oh the pole pairs. The mechanical equation which connects the generator with the wind turbine is described by: jeq

dwg = T e + Tl − Bm wg dt

(9)

where T l is the torque applied by the turbine to the generator. jeq is the equivalent moment of inertia; and Bm is the viscous turn coefficient.

2.3 DC Line Modeling The dc line is the capacitor located between the convertors; its function is to keep the DC voltage stable, the model of the dc line is expressed as Cd Vdc = is − i g dt

(10)

Self Tuning Fuzzy Maximum Power Tracking Control …

739

where is the DC line voltage, is the current of DC line at the generator side and ig the courant of DC line at the grid side, C is the capacity of DC-line capacitor. Let us suppose that the convertor is ideal does not consume power; the voltage of the capacitor is written as C

d Vdc Ps − ig = dt Vdc

(11)

where, Ps is the output power of generator. We can write it as: Ps = 3Vs i s = Vdc i dc

(12)

We rephrase it as: Ps =

3 E max Imax = Vdc i dc 2

(13)

where V s is the voltage of generator side and E max is its amplitude, I max is the amplitude of the generator current, and idc is the current through the capacitor.

2.4 Grid Model The dynamic model of the grid connection when selecting a reference frame rotating synchronously with the grid voltage space vector is u gd = −R g i gd −

L g di gd + wgr L g i gq + egd dt

(14)

L g di gq + wgr L g i gd dt

(15)

u gq = −R g i gq −

where Rg and L g are the resistance and inductance respectively of the filtre, this latter is located between the converter and the grid. ugd and ugq are the inverter voltage components, wgr is the electrical angular velocity of the grid.

3 Control of the Machine Side Converter To control the PMSG while the wind speed is variable, we use the MPPT algorithm. It consists in operating the PMSG at the speed corresponding to the point of maximum power when the wind speed is lower than the nominal wind speed as shown in Fig. 2.

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Fig. 2 Diagram of proposed control system of the machine side converter

3.1 Proportional-Integral Control (PI) The mathematical model of the control of the mechanical speed of PMSG performed by the PI controller is given by the following relationship: Iq∗ (s)

 =

Ki Kp + S



  × ωm∗ (s) − ωm (s)

(16)

where ω*m is the reference speed given by MPPT and ωm is the speed of measured PMSG, K p is the proportional gain and K i is the integral gain.

3.2 Fuzzy Logic Control Design The FLC code consists of three parts: preprocessing, fuzzy rules and interface engine, and post-processing. In the preprocessing part, the input variables of the fuzzy controller are speed error (e) and change of error (e). The input scale and input error scale gain factors are Ge and Ge respectively. The final step of the FLC system is post-processing. The output signal I q is multiplied by the scaling factors du(t) as shown in Fig. 3.

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Fig. 3 FLC control structure

e(t) = ωm∗ (t) − ωm (t)

(17)

e(t) − e(t − 1) tsamp

(18)

Iq∗ = Iq∗ (t − 1) + du(t)

(19)

e(t) =

Using Mamdani’s method, the fuzzy input variables are fuzzified into suitable linguistic values, then processed in the fuzzy set region which includes membership function where an appropriate fuzzy output is acquired using fuzzy rules, and thereafter the fuzzy output is converted into a crisp value in the defuzzification using the centroid method, also known as the center of gravity method specified by: n 

U =

Ci .μi

i=1 n 

(20) μi

i=1

where C i is discrete element of an output fuzzy set, μi is its membership function and n is the number of fuzzy rules. Once the inputs are converted into linguistic variables the FLC gives a manageable change in the reference voltage to reach the maximum power, based on the rules displayed in Table 1. Table 1 Set of fuzzy rules E

dU E

NB

NS

Z

PS

PB

NB

NB

NB

NS

NS

Z

NS

NB

NS

NS

Z

PS

Z

NS

NS

Z

PS

PS

PS

NS

Z

PS

PS

PB

PB

Z

PS

PS

PB

PB

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Fig. 4 Inputs/Output membership functions of the F-PI controller

The membership functions for the inputs and the output variables are triangular and trapezoidal membership functions named: Negative Big (NB), Negative Small (NS), Zero (Z), Positive Small (PS), and Positive Big (PB) (Fig. 4).

3.3 Self-tuning Fuzzy Logic Controller ST-FLC is used to adjust instantaneously the output scale factor (Gu ), according to the speed input change error (Ge ). The proposed automatic tuning mechanism focused on the on-line adjustment of the fuzzy system output as a function of the shift error. The block diagram of the proposed ST-FLC is shown in Fig. 5. The proposed self tuning bloc calculation used a simple algorithm reducing the system complexity and the computational burden. The ST-FLC scaling factors (Ge , Ge and Gu ) are related to the inputs and outputs (e, e, u) in the following equations:

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Fig. 5 Self-tuning fuzzy logic controller structure

e = G e × e.

(21)

e = Ge × e

(22)

u = (β ∗ G e ) × u

(23)

where:  β = K1

 1 + |e| m

(24)

The variable β formulated based on expert Knowledge of the system according to this concept: ((If the system is moving faster towards its desired operating-point (small Δe), then output action (Δu) needs to be reduced (reduce Gu ) to prevent big overshoot and/or undershoot. In contrast, if the system is rapidly moving away from the desired operating-point (big e), then output action (u) needs be increased (increase Gu ) for limiting these deviations for a faster recovery of the system to its desired operating point)). In other words, if the value of e is small, then Gu need to be reduced and if e is big, then Gu need to be increased. Hence, the β value is formulated based on this concept by adding the e to the fraction (1/m) to avoid lower gain multiplication (Gu ) when the e is very small. Lower multiplication of gain may result in oscillation and not stable condition during steady state operation. The value of m is chosen to be equal to the number of uniform input (e and e) fuzzy partition (number of MFs). The value of K1 is chosen to make the possible variation in β which set to 4 based on tuning process [15]. The ther fuzzy parameters were kept unchanged, similar to the standard fixed parameters FLC.

4 Control of the Grid Side Converter The grid side converter control has to adjust the DC (Direct Current) line voltage to the desired value, and ensure that the output voltage converter has a similar amplitude

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Fig. 6 Diagram of the loop control of DC line

and frequency with the grid, but the main objective is to control the active and reactive power.

4.1 DC Line Control The control block diagram of the DC-line is presented in Fig. 6. The current loop is considered to be fast than the voltage loop and supposed equal to 1. The PI controller is designed with the pole placement method.

4.2 Control of Active and Reactive Power Then active and reactive power are transmitted through the grid side converter, they are expressed as follow: P=

3 u gd i gd 2

(25)

Q=

3 u gd i gq 2

(26)

To control active and reactive power is achieved by acting on the direct and quadrature current as shown in Fig. 7. The first channel controls the DC voltage which is used to provide the d-axis current reference for the active power control. This assures that all the power coming from the PMSG converter is instantly transferred to the grid through the converter. The second channel controls the reactive power by setting the q-axis current reference to the current control loop similar to the previous one. The current controllers will provide a voltage reference for the converter that is compensated by adding rotational EMF (Electromotive Force) compensation terms. All controllers are PI and are tuned using the pole placement method.

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Fig. 7 Bloc diagram of the proposed control system of the grid side converter

5 Results and Discussion To test the performances of the proposed control system, several simulation test with a variable wind speed profile indicated in Fig. 9 were developed under Matlab/simulink® with SimPowerSystems block. The used wind turbine and the PMSG parameters are given respectively in Tables 2 and 3. The main measurements include speed, torque, three-phase voltage, and stator current. The speed response of the ST-FLC controller has better tracking characteristics than other controllers, as shown in Fig. 8, in which the speed monitoring controller operates in a critical situation, the response time of the PMSG speed is very short, there is no overshoot and no steady-state error as shown in Fig. 9. There is a slight difference in response time to reach the reference DC voltage between the ST-FLC and FLC, But there is a lag of about a classical PI control near to 20 ms without no steady-state error as shown in Fig. 10. Table 2 wind turbine parameters

Parameter

Value

Units

Air density ρ

1.22

Kg/m3

Gear ration Gr

5.4

Rotor radius R

3

Maximum power coefficien C p

0.48

Equivalent inertia (turbine + generator)

0.042

M Kg m2

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C. Housseyn et al. Parameter Rated power Rated line courant Rated speed

Value 2.5 11 3000

Units kW A r/m

Mutual inductance in d-axis L d

7.5

mH

Mutual inductance in q-axis L q

7.5

mH

Stator resistance Rs

0.45



Number of pole pairs P

3

Permanent magnet flux ψf

0.52

Wb

viscous friction Bm

0.017

Nm/rad/s

Fig. 8 Wind speed profile

Figure 11 shows the active power of the grid and the active power of PMSG. Note that the response time for the ST-FLC and FLC controllers is small compared to the Classical PI. The difference between the active power of the grid and the active power of the PMSG is due to the resistance of the filter which causes the voltage to decrease in the steady-state. The comparison between the three controllers at the estimated wind speed of 6 m/s is shown in Table 4. We notice that the ST-FLC controller has a short response time compared to the other controllers. The ST-FLC and FLC controllers have no overshoot compared to the PI. Finally, we can say that the ST-FLC controller has a smaller rise time and a settling time than the last two controllers.

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Fig. 9 PMSG speed controlled by different methods

Fig. 10 DC voltage controlled by different methods

6 Conclusion In this paper, a comparison between classical and fuzzy based strategies to control WECS using PMSG and integrated into the grid. FLC offer much-improved performance. It is efficient where the model is poorly defined and can use human experience. The purpose of this study is to test a two advanced control of the MPPT of variable speed wind turbines: FLC using the error and the change of error to increase or decrease the control using fixed gain scale factors and ST-FLC where the output gain is instantaneously calculated according to the system conditions.

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Fig. 11 Power of grid and power of PMSG controlled by different methods

Table 4 Comparison performances at rated wind speed for the different controllers

PI

FLC

ST-FLC

Rising time (s)

0.017

0.017

0.016

Settling time (s)

0.020

0.019

0.017

Overshoot (%)

2.03

0

0

Steady state error (%)

0.57

0

0

Simulation results confirm that FLC performance is better compared with a classic PI controller for response speed and ability to track maximum wind power. The STFLC strategy, using a simple bloc calculation, guarantees robust performances and gives satisfactory results compared to other strategies used. Further studies include the testing of ST-FLC control algorithms on the full structure of WECS to control DC voltage. Finally, the developed simulation algorithms can be validated through an experimental implementation.

References 1. Polinder H, Ferreira JA, Jensen BB, Abrahamsen AB, Atallah K, McMahon RA (2013) Trends in wind turbine generator systems. IEEE J Emerg Sel Top Power Electron 1(3):174–185 2. Chen Z, Li H (2008) Overview of different wind generator systems and their comparisons. IET Renew Power Gener 2(2):123–138 3. Slah H, Mehdi D, Lassaad S (2016) Advanced control of a PMSG wind turbine. IJMNTA 05(01):1–10 4. Durán MJ, Kouro S, Wu B, Levi E, Barrero F, Alepuz S, Six-phase PMSG wind energy conversion system based on medium-voltage multilevel converter, p 10

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5. Wei C, Zhang Z, Qiao W, Qu L (2016) An adaptive network-based reinforcement learning method for MPPT control of PMSG wind energy conversion systems. IEEE Trans Power Electron 31(11):7837–7848 6. Liu J, Zhou F, Zhao C, Wang Z (2019) A PI-type sliding mode controller design for PMSG-based wind turbine. Complexity 2019:1–12 7. Tripathi SM, Tiwari AN, Singh D (2016) Optimum design of proportional-integral controllers in grid-integrated PMSG-based wind energy conversion system: optimum design of PI controllers. Int Trans Electr Energy Syst 26(5):1006–1031 8. Suganthi L, Iniyan S, Samuel AA (2015) Applications of fuzzy logic in renewable energy systems – a review. Renew Sustain Energy Rev 48:585–607 9. Daraban S, Petreus D, Morel C (2014) A novel MPPT (maximum power point tracking) algorithm based on a modified genetic algorithm specialized on tracking the global maximum power point in photovoltaic systems affected by partial shading. Energy 74:374–388 10. Chavero-Navarrete E, Trejo-Perea M, Jáuregui-Correa JC, Carrillo-Serrano RV, Ríos-Moreno JG (2019) Expert control systems for maximum power point tracking in a wind turbine with PMSG: state of the art. Appl Sci 9(12):2469 11. Karrari M, Rosehart W, Malik OP (2008) Hierarchal control system for a variable speed cage machine wind generation unit using neural networks. Asian J Control 7(3):286–295 12. Asri A, Mihoub Y, Hassaine S, Logerais P, Allaoui T (2019) Intelligent maximum power tracking control of a PMSG wind energy conversion system. Asian J Control asjc.2090 13. Harrabi N, Kharrat M, Aitouche A, Souissi M (2018) Control strategies for the grid side converter in a wind generation system based on a fuzzy approach. Int J Appl Math Comput Sci 28(2):323–333 14. Farh HM, Eltamaly AM (2013) Fuzzy logic control of wind energy conversion system. J Renew Sustain Energy 5(2):023125 15. Farah N et al (2019) A novel self-tuning fuzzy logic controller based induction motor drive system: an experimental approach. IEEE Access 7:68172–68184

A Backstepping Controller for Interleaved Boost DC–DC Converter Improving Fuel Cell Voltage Regulation Ali Dali, Samir Abdelmalek, and Maamar Bettayeb

Abstract The aim of the present work is to design a robust nonlinear controller which ensures a regulated output voltage for a Proton-Exchange Membrane Fuel Cell (PEMFC) fed interleaved boost DC–DC converter. In this regard, a particle swarm-based optimization algorithm is used for tuning parameters of the nonlinear controller allowing greater flexibility in guaranteeing system specifications compared to the classical Proportional Integral (PI) controller. The effectiveness of the proposed controller for the studied system was validated under various operating conditions of the PEMFC and load perturbations using Sim Power System (SPS) toolbox and MATLAB/Simulink environment. Simulations and comparison of results show that the proposed robust nonlinear controller offers several advantages for smoother tracking, smaller overshoot, faster response and better stability. Keywords Interleaved boost converter · Robustness · Conventional PI · Fuel cell · Nonlinear controller

A. Dali (B) Centre de Développement des Energies Renouvelables, CDER, BP 62 Route de l’Observatoire, 16340 Bouzaréah, Algiers, Algeria e-mail: [email protected] S. Abdelmalek Department of Electrical Engineering, Faculty of Technology, University of Medea, Medea, Algeria e-mail: [email protected] M. Bettayeb Department of Electrical and Computer Engineering, University of Sharjah, United Arab Emirates, and Center of Excellence in Intelligent Engineering Systems (CEIES), King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_51

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1 Introduction In the past few decades, Fuel Cells (FCs) have emerged as a potential alternative solution to fossil fuel based energy sources [1]. In addition, they are considered as one of the most promising technology for electrical power generation [2]. The Fuel cell is capable of converting chemical energy directly into electrical energy, without any intermediate step. Therefore, the electrical efficiency of FCs is considerably higher than the other traditional sources for generating electrical power. Fuel cells are widely used in several applications for military, automotive and low power generation as they offer many advantages such as low operating temperature, fast response to load changes, high power density, low corrosion, very low noise and environmental friendliness [1–3]. In general, DC–DC converters are employed to adapt the voltage and current levels between sources and loads while guarantee a low power loss in the power generation systems [4, 5]. The DC–DC converter is an essential part of the fuel cell power generation to control the delivered power/voltage. Incorporating PEMFC dynamics with the classical DC/DC converter makes the voltage regulation task quite challenging [6]. In this context, in order to deal with this drawback, the interleaved boost DC–DC converter can be used [7]. This kind of converters has the advantages of reducing the inductors size, increasing the power level and minimizing the current stress on the power switches [8]. With regard to the literature, different kinds of control strategies are widely employed for interleaved boost DC–DC converters, such as adaptive controller [9], robust control [10], nonlinear model predictive control (MPC) [11], state feedback control [12], sliding mode fuzzy PID controller [12], robust adaptive neural network control [13], adaptive sliding mode controller [14] and intelligent controllers [15]. However, among the aforementioned control techniques, the problem of robust nonlinear controller design based on optimization techniques was not studied up to now. Therefore, this work aims to propose a robust nonlinear control law based on the Particle Swarm Optimization (PSO) algorithm. The novelty and highlights of the paper can be stated as follows: (i) Using a new architecture which is based on an interleaved boost DC–DC converter, (ii) A Backstepping controller design for output voltage regulation under varying loading conditions, (iii) Inclusion of PSO algorithm for tuning of the proposed controller parameters and iv) Validation and comparison of the proposed controller design with those of the conventional PI controller in Sim Power System (SPS) toolbox and MATLAB/Simulink environment. The remaining sections are organized as follows. First, Sect. 2 introduces the model of the interleaved boost DC–DC converter. Section 3 then presents the fuel cell mathematical model, followed by the design of the nonlinear controller in Sect. 4. The parameters optimization using the PSO algorithm for the robust nonlinear and traditional PI controllers is deployed in Sect. 5. The simulation results with robustness tests are presented in Sect. 6. The last section summarizes the key aspects of this paper and presents some conclusions.

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2 Interleaved Boost DC–DC Converter Figure 1 shows the schematic of an interleaved boost DC–DC converter (IBC), the main advantage of the IBC is the reduction of the current ripples compared to the classical boost converter. where V c is the output voltage and V fc is the Fuel Cell (FC) source voltage. The average model of the interleaved boost DC–DC converter is the same as the traditional boost DC–DC converter with except that it contains two current inductors [16]. The dynamics of the converter are represented by: ⎧ d IL ⎪ ⎨ L 1 dt 1 = V f c − Vc (1 − D1 ) dI L 2 L 2 = V f c − Vc (1 − D2 ) ⎪ ⎩ C d Vdtc = I (1 − D ) + I (1 − D ) − I L1 1 L2 2 R dt

(1)

⎧ ⎨ I f c = IL1 + IL2 D = D2 = 1 − u ⎩ 1 L1 = L2 = L

(2)

such that

where L and C are respectively the inductance and the capacitor, D1 and D2 are the duty cycles of each phase (which one is shifted by half of period from the other). I L 1 and I L 2 are the inductances currents, I f c is the fuel cell current, I R = V c /R is the resistor (R) load current and Ic is the capacitor current. Then, the average model of the system (3) may be expressed as follows 

dIfc dt dVf c dt

= L2 V f c − L2 Vc u 1 = C2 I f c u − RC Vc

Fig. 1 Voltage regulation of the fuel cell interleaved boost DC converter

(3)

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where u is the average duty cycle.  The system state space vector χ = [Vc , I f c , − Vc ]T includes the measurable output voltage (Vc ) and the fuel cell current (I f c ), the third state is introduced to improve the robustness of the controller design, the state vector of the augmented model is then described as: χ(t) ˙ = Aχ(t) + Bu(t) + 

(4)

where the associated matrices are given by: 2

T ⎤ ⎧ 1 0 0 0 ⎪ ⎨ B = − L x2 C x1 0 1 A = ⎣ 0 − RC 0 ⎦, χ = [x1 x2 x3 ]T ⎪

0 1 0 ⎩  = L2 V f c 0 0 T ⎡

3 Fuel Cell Mathematical Model This section develops a mathematical model that represents a fuel cell stack by a controlled voltage source in series with a constant resistance as illustrated in Fig. 2. The controlled voltage source (E) is expressed as 

E = E oc − N A0 ln



V f c = E − Rohm I f c where,

Fig. 2 Simplified fuel cell stack model

Ifc Io

 s

1 +1

Td 3

(5)

A Backstepping Controller for Interleaved …

E oc N Ao Io Td Rohm I fc Vfc

755

open circuit voltage (V); number of cells; Tafel slope (V); exchange current (A); the response time (at 95% of the final value); internal resistance (); fuel cell current (A); fuel cell voltage (V).

For more details, authors can read Ref. [17].

4 Backstepping Controller Design A Backstepping controller is designed for satisfying a good trajectory tracking of the desired voltage (V cref ) by acting on the duty cycle (u) of the interleaved boost DC–DC converter switch. The important steps to design the Backstepping controller are described in this section. Let us define the following tracking errors:  ⎧ ⎨ e1 = e2 dt e = Vcr e f − Vc ⎩ 2 e3 = I f cd − I f c

(6)

with V˙cr e f = 0. The control law is designed in three steps. A first Lyapunov function (V 1 ) and its derivative with respect to time is defined respectively by: 

V1 = 21 e12 V˙1 = e1 e2

(7)

Then, a new variable e2d is introduced for stabilizing e1 : e2d = −k1 e1

(8)

where, k 1 is a positive constant. Then, V 1 is a negative definite function of e1 : V˙1 = −k1 e12 < 0, k1 ∈ ∗+

(9)

In the following step, a second augmented Lyapunov function is selected as follows: 1 V2 = V1 + (e2 − e2d )2 2

(10)

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Differentiating V 2 with some mathematical manipulations yield to,   2u 1 2 ˙ Ifc + Vc V2 = −k1 e1 + (e2 − e2d ) e1 − e˙2d − C RC

(11)

such that: I f cd

  C 1 e1 − e˙2d + Vc + k2 (e2 − e2d ) = 2u RC

(12)

Based on the Eqs. (11) and (12) and with a mathematical development, one can get: V˙2 = −k1 e12 − k2 (e2 − e2d )2 < 0, (k1 , k2 ) ∈ ∗+

(13)

Finally, with the same procedure we choose the third augmented Lyapunov positive function described as follows: V3 = V2 +

1L 2 e 2C 3

(14)

Using Eqs. (9) and (13), with some developments V 3 can be derived as follows: L V˙3 = −k1 e12 − k2 (e2 − e2d )2 − k3 e32 < 0, k3 ∈ ∗+ C

(15)

Then, the control input (u) is given by: u=

  L 2 − I˙f cd + V f c − k3 e3 2(Vcd − e2d ) L

(16)

The PSO is an optimization technique inspired from the social behavior of biological organisms, such as birds flock to a food source with the help of a combination expressing the influence on the position of the body through the historical positions of itself and its neighbors. This algorithm has been developed to solve nonlinear equations. The basic idea is to build a population of particles and to move them according to specific criteria to achieve the final goal. The movement of a particle can be represented as follows [18–21]: X j (t + 1) = X j (t) + V j (t + 1)

(17)

V j (t + 1) = ω ∗ V j (t) + C1r1 (ρi − X j (t)) + C2 r2 (ρg − X j (t))

(18)

A Backstepping Controller for Interleaved … Table 1 PSO parameters

PSO parameters

O-PI

O-NL

C 1, C 2

3.0, 1.0

3.0, 1.0

ω

Table 2 Search range of the parameters

Table 3 Controllers parameters

757

0.8

0.8

Population size

10

20

Maximum iteration

20

100

Controller parameters

Min

K = [k 1 k 2 k 3 ]

[10−1 ,

Max

[K p K i ]

[0, 0]

Controller

Parameters

1, 1]

[200 100 1000] [10 10]

O-PI

K p = 0.077, K i = 5.65

O-LQRO-LQR

K = [137.16 1.57 728.57]

For each particle X j , the speed V j is calculated as a linear combination of three elements: its own speed, its deviation from its ρi neighborhood, and compared to the best ρg particle. Then, the current position of the particle is updated through the coefficients ω, C 1 and C 2 to get the new position. The effectiveness of the method focuses in particular on the randomness of the last two coefficients (r 1 and r 2 ). The PSO technique is used to get the optimal controller parameters by minimizing the following objective function 

tf

e=

  Vcr e f (t) − Vc (t)dt 

(19)

0

Equation (19) represents the performance index used for each controller technique (the Backstepping and the conventional PI). The PSO parameters for each controller and the search range are listed in Tables 1 and 2, respectively. The optimization results for each controller are listed in Table 3.

5 Simulation Results In order to verify the advantages of the designed nonlinear controller that adopts the improved tracking performance in the PEMFC control system, the comparison between the optimized nonlinear (O-NL) controller and the optimized conventional PI (O-PI) controller on the dynamic output characteristic of PEMFC will be discussed. The interleaved DC–DC boost converter parameters used in the simulation are listed in Table 4.

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Table 4 DC–DC interleaved boost converter

Parameters L

500 μH

C

1000 μF

R

50 

F

100 Khz

Simulations are performed for two different cases for different situations including normal operating conditions and a load variation. Test 1: The reference voltage (V cref ) changes as follows:

Vcr e f

⎧ ⎨ Vcr e f0 − 10%Vcr e f0 0 ≤ t ≤ 0.1 s = Vcr e f0 0.1 s ≤ t ≤ 0.2 s ⎩ Vcr e f0 + 10%Vcr e f0 0.2 s ≤ t ≤ 0.3 s

with Vcr e f 0 = 75. Figures 4 and 5 present a comparative study between the proposed controller and the optimized PI controller. It is clear that V c (O-NL) reaches the desired set point (V cref ) in the settling time of 0.05 s. In addition, the O-NL controller presents a minimal tracking error, a good transition response and a very fast system reaction against set point change. Therefore, one can deduce that the proposed approach is better than the O-PI control approach. The reference tracking capability of the proposed COSF controller is investigated. The obtained results are compared with those O-PI and O-LQR controllers with the same conditions. The responses of the three controllers are given in Figs. 3 and 4. The results show that the proposed COSF controller has better performance in terms of the settling time relative to the O-PI and the O-LQR controllers. It can be concluded Fig. 3 Fuel cell and load current

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Fig. 4 Voltage tracking

that the proposed COSF controller ensures better robustness and has good satisfactory response during disturbances and abrupt changes [22–24] (Fig. 5). Test 2: in this test, a load variation of −50% −10% is considered at t = 0.05 s, then the system response is given in Figs. 5 and 6. The results confirm that the output response of the O-PI controller is affected seriously by the load resistance variations. It can be concluded that the proposed controller ensures better robustness and has satisfactory response in the presence of load variation. The proposed O-NL controller has higher accuracy and reliability in tracking reference also in the presence of load variation. Figure 7 shows the values of performance indexes of each controller. This figure confirm the simulation results carried out for the Test 1. It is evident that performance Fig. 5 Fuel cell and load currents response to load variation

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Fig. 6 Voltage tracking response to load current variation

Fig. 7 Performance index

index

  tf   e = ∫Vcr e f (t) − Vc (t)dt with the proposed controller is smaller than 0

O-PI controller.

6 Conclusion In this work, a robust nonlinear control design based on the PSO algorithm is proposed. The main goal of the control was to ensure a well-regulated output voltage for a Proton-exchange membrane fuel cell fed interleaved boost DC–DC converter. The simulations and comparison of results showed the superior control performance and the advantage of the proposed nonlinear compared to the optimized PI controller

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in terms of tracking, smaller overshoot, faster response and better stability. In addition, the designed nonlinear controller is simple to implement in real time. As a part of this works, it will be extended with T-S fuzzy technique to treat the problem of fault diagnosis and fault tolerant control for various class of faults (actuator, sensor, uncertainties, system, …) based on references [25–30].

References 1. Bankupalli P, Ghosh S, Kumar L (2018) Fractional order modeling and two loop control of PEM fuel cell for voltage regulation considering both source and load perturbations. Int J Hydrog Energy 43(12):6294–6309 2. Grötsch M, Mangold M, Kienle A (2009) Analysis of the coupling behavior of pem fuel cells and dc- - dc converters. Energies 2(1):71–96 3. Li Q, Chen W, Wang Y (2011) Parameter identification for pem fuel-cell mechanism model based on effective informed adaptive particle swarm optimization. IEEE Trans Ind Electron 58(6):2410–2419 4. Dali A, Abdelmalek S, Bettayeb M (2018) A new combined observer-state feedback (cosf) controller of PWM buck converter. In: CISTEM 5. Abdelmalek S, Dali A, Bettayeb M (2018) An improved observer-based integral state feedback (OISF) control strategy of flyback converter for photovoltaic systems. In: CISTEM 6. Choe S-Y, Lee J-G, Ahn J-W (2007) Integrated modeling and control of a pem fuel cell power system with a pwm dc/dc converter. J Power Sources 164(2):614–623 7. Choe S-Y, Lee J-G, Ahn J-W (2014) Analysis, design and performance of a zero-currentswitching pulse-width-modulation interleaved boost dc/dc converter. IET Power Electron 7(9):2437–2445 8. Somkun S, Sirisamphanwong C, Sukchai S (2015) A dsp-based interleaved boost dc–dc converter for fuel cell applications. Int J Hydrog Energy 40(19):6391–6404 9. Su J-T, Liu D-M, Liu C-W (2009) An adaptive control method for two-phase dc/dc converter. In: International conference on power electronics and drive systems, pp 288–293 10. Ramakumar R, Chiradeja P (2004) Distributed generation and renewable energy systems. In: 2002 37th intersociety energy conversion engineering conference, pp 716–724 11. Kirubakaran A, Jain S, Nema RK (2009) The pem fuel cell system with dc/dc boost converter: design, modeling and simulation. Int J Recent Trends Eng 1(3):157 12. Guo L, Hung JY, Nelms RM (2011) A dsp-based interleaved boost dc–dc converter for fuel cell applications. Int J Hydrog Energy; Comparative evaluation of sliding mode fuzzy controller and PID controller for a boost converter. Electr Power Syst Res 81(1):99–106 13. Abbaspour A, Khalilnejad A, Chen Z (2016) Robust adaptive neural network control for PEM fuel cell. J Hydrog Energy 41(44):20 385–20 395 14. Romero A, Martínez-Salamero L, Valderrama H, Pallás O, Alarcón E (1998) General purpose sliding-mode controller for bidirectional switching converters. In: Proceedings of the 1998 IEEE international symposium on circuits and systems. ISCAS’98, vol 6. IEEE, pp 466–469 15. Navarro-López EM, Cortés D, Castro C (2009) Design of practical sliding-mode controllers with constant switching frequency for power converters. Electr Power Syst Res 79(5):796–802 16. Habiba M, Khoucha F, Abdelghani H (2017) Ga-based robust LQR controller for interleaved boost dc–dc converter improving fuel cell voltage regulation. Electr Power Syst Res 152:438– 456 17. Souleman NM, Olivier T, Dessaint L-A (2009) A generic fuel cell model for the simulation of fuel cell vehicles. Electr Power Syst Res 1722–1729 18. Dali A, Bouharchouche A, Diaf S (2015) Parameter identification of photovoltaic cell/module using genetic algorithm (ga) and particle swarm optimization (PSO). In: 2015 3rd international conference on control, engineering & information technology (CEIT). IEEE, pp 1–6

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19. Rezazi S, Hanini S, Si-Moussa C, Abdelmalek S (2016) Modeling and optimization of the operating conditions of marrubium vulgare l. essential oil extraction process: kinetic parameters estimation through genetic algorithms. J Essent Oil Bear Plants 19(4):843–853 20. Abdelmalek S, Belmili H (2015) A new robust h control power. In: Handbook of research on advanced intelligent control engineering and automation, pp 601–623 21. Baadji B, Bentarzi H, Bakdi A (2019) Comprehensive learning bat algorithm for optimal coordinated tuning of power system stabilizers and static VAR compensator in power systems. Eng Optim. https://doi.org/10.1080/0305215X.2019.1677635 22. Bakdi A, Bounoua W, Mekhilef S, Halabi LM (2019) Nonparametric Kullback-divergencePCA for intelligent mismatch detection and power quality monitoring in grid-connected rooftop PV. Energy. https://doi.org/10.1016/j.energy.2019.116366 23. Abdelmalek S, Rezazi S, Bakdi A, Bettayeb M (2019) Voltage dips effects detection and compensation for doubly-fed induction generator based wind energy conversion system. Revue Roumaine Des Sciences Techniques-Serie Electrotechnique Et Energetique 64(3):199–204 24. Bakdi A, Kouadri A, Mekhilef S (2019) A data-driven algorithm for online detection of component and system faults in modern wind turbines at different operating zones. Renew Sustain Energy Rev 103:546–555 25. Abdelmalek S, Barazane L, Larabi A (2016) Fault diagnosis for a doubly fed induction generator. Revue Roumaine Des Sciences Techniques-Serie Electrotechnique Et Energetique 61(2):159–163 26. Bendoukha S, Abdelmalek S, Abdelmalek S (2018) A new combined actuator fault estimation and accommodation for linear parameter varying system subject to simultaneous and multiple faults: an LMIs approach. Soft Comput 1–14 27. Abdelmalek S, Barazane L, Larabi A (2017) An advanced robust fault-tolerant tracking control for a doubly fed induction generator with actuator faults. Turk J Electr Eng Comput Sci 25(2):1346–1357 28. Abdelmalek S, Barazane L, Larabi A, Belmili H (2015) Contributions to diagnosis and fault tolerant control based on proportional integral observer: application to a doubly-fed induction generator. In: 2015 4th international conference on electrical engineering (ICEE). IEEE, pp 1–5 29. Abdelmalek S, Barazane L, Larabi A, Bettayeb M (2016) A novel scheme for current sensor faults diagnosis in the stator of a dfig described by a TS fuzzy model. Measurement 91:680–691 30. Abdelmalek S, Azar AT (2018) A novel actuator fault-tolerant control strategy of dfig-based wind turbines using takagi-sugeno multiple models. Int J Control Autom Syst 16(3):1415–1424

The Impact of the Solar Radiation Profile on Sizing and Performance of Photovoltaic Systems, Case Study Tamanrasset, Algeria Madjid Chikh, Aicha Degla, and Achour Mahrane

Abstract The sizing of photovoltaic installations requires a correct control of the solar radiation profile; incorrect sizing leads to significant additional costs. To overcome this problem, a comparative conceptual analysis of two design techniques based on different solar radiation profiles was required. In this paper, we have defined two different sizing techniques based on two solar radiation profiles; the first one is based on hourly step and the second with daily step model. The sizing results were established and analyzed in detail, and followed by a comparative analysis of the performance of the photovoltaic systems under study. This analysis procedure is applied to Tamanrasset region located in the south of Algeria. Keywords Solar · Radiation · Sizing of photovoltaic system · Hourly profile · Daily profile

1 Introduction Sizing and simulation of stand-alone photovoltaic (PV) systems requires a good knowledge of the solar system variability at a given site, including the global solar radiation models received on the PV module level. In order to design stand-alone PV systems, several techniques have been developed and described in the scientific literature. Among these sizing techniques, there M. Chikh (B) · A. Mahrane UDES, Solar Equipment Development Unit, Route Nationale n°11, BP386, 42400 Bousmail, Tipaza, Algeria e-mail: [email protected] A. Mahrane e-mail: [email protected] A. Degla Renewable Energy Development Center, CDER, BP 62 Route de l’observatoire Bouzareah, 16000 Algiers, Algeria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_52

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are two main techniques of major importance widely used by designers and users of PV installations. The first method requires a solar profile radiation with hourly time step variation; the second one considers the amount of solar energy accumulated over a day. However, these two techniques, which certainly lead to different sizes and dimensions of the PV system, have very different torque values (Ps , C s ), [Ps : Peak power of the PV system and C s : nominal battery capacity]. The objective of this work is to highlight the impact of these solar radiation profiles on the design of PV systems and to see how the variability of this solar energy will affect the performance and reliability of these PV systems. This analysis procedure is applied to Tamanrasset region located in the south of Algeria. Tamanrasset is an oasis city located in the south of Algeria, in the Ahaggar Mountains. It has an altitude of 1320 m. Tamanrasset has a warm desert climate, with very hot summers and mild winters. There is very little rain throughout the year, although occasional rain falls at the end of summer because it is located in the northern extension of the inter-tropical convergence zone.

2 Global Solar Radiation Estimation for Photovoltaic Modules 2.1 Global Solar Radiation with Hourly Steps The general problem when calculating hourly radiation on sloping surfaces when the global radiation on a horizontal surface is the only known. For that purpose, we require directions from which direct and diffuse components reach the surface in question. Incident solar radiation is the sum of a set of radiation fluxes including direct radiation, the three components of diffuse sky radiation and radiation reflected from various surfaces as viewed from the sloped surface. The total incident radiation on this area can be expressed as follows: IT = IT,b + IT,d,iso + IT,d,cs + IT,d,hz + IT,r e f l

(1)

where: iso, cs, hz and refl refer to isotropic, circumsolar, horizon and reflected radiation respectively. The global radiation on sloping surface is composed of three elements: Direct, diffuse and reflected from the ground. The geometric conversion factor Rb is the ratio of direct hourly (or instantaneous) radiation on an inclined surface to radiation on a horizontal surface: Rb = Ib,T /Ib = cosθ/cosθ Z

(2)

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Fig. 1 Direct, diffuse and reflected radiation of the ground on a sloped surface

Radiation reflected from the ground is assumed diffuse and obtained by [1].  Ig,T = Iρg (1 − cos β) 2

(3)

where ρg is the ground albedo, and β is the slope angle. Diffuse radiation is difficult for modelling, as its spatial distribution is usually unknowns and time-dependent. Adding direct radiation and reflection terms from the ground to the anisotropic model modified by Reindl, see Fig. 1, results in an HDKR model (Hay, Davies, Klucher, and Reindl model) which can compute the overall hourly solar radiation on a sloped surface Eq. (4).      (1 + cos β)  3 × 1 + f sin β 2 IT = (Ib + Id Ai )Rb + Id (1 − A I ) 2   1 − cos β + Iρg 2

(4)

where Ai is the Anisotropy Index.

2.2 Global Solar Radiation with Daily Steps The global solar radiation on a sloped surface is evaluated by using corresponding components calculated for a horizontal surface. The monthly daily average of the diffuse radiation, the monthly daily average of direct radiation and the reflected component from the ground are used to calculate the daily average global solar radiation on a sloped surface is given by [2];

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Hg = Hb Rb + Hd Rd + Hρ ((1 + cos β)/2)

(5)

    Hd H = 0.741 − 0.552 n N   2   3 − 0.208 n N + 0.099 n N

(6)

where:

The average monthly direct solar radiation may be calculated using this equation: Hb = H − Hd

(7)

Rb is the ratio of direct radiation on sloped surface to direct radiation on a horizontal surface, which can be calculated from [3]: Rb =

cos(ϕ − β)∂ sin ωs + (π/180)ωs sin(ϕ − β) sin ∂ cos ϕ cos ∂ sin ωs + (π/180)ωs sin ϕ sin ∂

(8)

ωs is the hourly angle of sunrise for a sloped surface and can be determined as follows:

−1 cos (− tan ϕ tan ∂ (9) ωs = min cos−1 (− tan(ϕ − β) tan ∂ The conversion factor of the diffuse radiation Rd can be determined using the following equations [4]. Rd =

1 (1 + cos β) 2

(10)

The total solar radiation on a sloped surface was calculated using the angle of inclination β from 0° to 90° with an interval of 1 to determine the optimal angle of inclination. The angle at which H g produced the maximum value was considered the optimal angle of inclination. This procedure was followed to obtain an optimal angle of inclination for each month of the year, for the four seasons of the year, every six months and for a full year.

3 Sizing Study of a Stand-Alone PV System The main parts of the PV system are shown in Fig. 2. The system is controlled as follows [5]: the battery imposes its voltage on the system, storage in batteries has priority, and the battery is controlled between two state of charge limits: maximum SOCmax and minimum SOCmin, corresponding respectively to the maximum Vbh and min Vbl voltages of the battery.

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Fig. 2 Main parts of the photovoltaic system

3.1 Description of the Sizing Method Based on the Hourly Model of Solar Radiation The electrical energy consumption profile used is as shown in Fig. 3. It is almost the same for every day in the year, and is consistent with the profile usually found in isolated sites. In other words, there is a shortage when the power demand is higher than the power supplied by the PV module and the battery is at the lowest level, so the management forbids any energy supply to users. The design criteria selected are the shortage and the cost of installation. At each moment when this occurs, the simulation program counts it as a shortage time. The problem consists in determining the peak power Pc of the PV field and the nominal capacity C s of the electrochemical storage system, corresponding to the minimum total cost of the installation. The method adopted to Fig. 3 Consomption profile of a group of houses located in Tamanrasset

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perform the sizing of the PV system components considers that the peak power Pc and the nominal capacity C s are variable, the other subsystems remaining at constant cost. For this purpose, the overall cost of the installation of a PV system can be expressed as follows: C T = A Pc + B Q s + C T0

(11)

A: cost per PV peak watt (e/Wp), B: cost per kilowatt-hour of the battery (e/kWh), Q s : the energy amount in the battery (kWh), C T0 is the sum of the costs of all other subsystems, design studies, on-site system installation (e). The simulation of the PV system operation allowed us to determine a series of (Pc , C s ) corresponding to a certain number of shortages αe fixed in advance. Figure 4 shows the overall cost of the PV installation, the optimal point of (Pc , C s ), giving the optimal size of the PV field and the nominal capacity of the electrochemical storage battery, corresponds to the estimated minimum cost for the entire PV system. This PV sizing procedure gives the optimal size of the PV field and battery capacity and is repeated for several values of the number of shortages, namely 0, 100, 500 and 1000 h of shortages. Table 1 shows the optimized values of the PV system and the corresponding minimum cost.

Fig. 4 The analytical method for determining the minimum cost

Table 1 Optimal values of the PV system sizing for different number of shortage hours Nbre of shortage hours αe

Peak power (kWp)

Battery capacity (Ah)

Minimal cost of PV system (ke)

0

15.4

2300

72.6

100

10.3

2000

56.3

500

9.9

1200

42.0

1000

8.3

1000

35.0

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3.2 Description of the Sizing Method Based on the Daily Model of Solar Radiation As measured data on diffuse solar radiation are not readily available for the study site, well-known and generally adapted correlation methods were used to estimate the components of daily diffuse solar radiation on a horizontal surface. Then, different components of solar radiation on a sloped surface are determined in order to estimate the overall solar radiation on that surface. Finally, the total radiation on different sloped surfaces were obtained to estimate the optimal angle of inclination. The PV field size is characterized by its peak power Pc (W c ) calculated on the basis of a consumption profile (see Fig. 2) and by the global average daily radiation taking into account a K r ratio (quality factor). This factor depends on multitude of elements: sensor distribution, orientation and inclination of PV panels, quality of diodes, cables and inverter… It generally ranges between 0.55 and 0.65 for welldesigned conventional installations, according to the International Energy Agency (IEA). The peak power of the PV system is given by the following expression: Pc =

D E × HST C Hg × K r

(12)

DE is the daily energy demand (wh); HSTC: solar radiation at STC conditions (1000 w/m2 ); Hg: The average daily radiation, from the PV module of the worst month (wh/m2 ) and estimated by the Eq. (5). The nominal storage capacity of the battery is calculated from the number of days of system autonomy with a preset depth of discharge (DOD = 80%). This capacity is given by the following expression: Cb =

As × D E D O D × Vb

(13)

As : Autonomy of the storage system (days) and V b is the battery voltage (V), Table 2 shows the sizes of the PV system and the costs of the PV installation for 03 values of the quality factor K r . Table 2 Sizing of the PV system with a daily solar radiation profile

K r factor

Peak power (kWp)

Battery capacity (Ah)

Cost of PV system (ke)

0.55

23.3

4000

118.4

0.60

21.3

4000

114.0

0.65

19.7

4000

110.4

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4 Sensitivity Analysis 4.1 Hourly Time Step Model Figure 5 shows the variation of the installation cost as a function of the allowable shortage duration. Table 3 presents the optimal sizing and costs corresponding to different levels of shortages.

C T is the absolute savings achieved if a certain shortage; C T /C T represents relative value of the economy. We can see the slope of the cost increasing as well as the number of shortage hours decreases; the increase is sudden between 0 and 100 h of shortage. This is perfectly illustrated in Table 4; we can observe that while the cost of erasing shortage hour increases with decreasing shortage duration, it suddenly increases when we move from 100 to 0 h of shortage. Table 5 shows the relative and absolute savings corresponding to some levels of shortage, taking as a reference the level α e = 100 h. The ratio α e /(α e , 100) corresponds to the factor of increase (or decrease) of the shortage compared to the reference level 100 h shortage. It can be seen that around this reference level, an increase in investment of 29% decreases the shortage by a factor of 100 and a decrease of 25% leads to an increase in the shortage by a factor of 5. The dimensioning corresponding to this shortage level could serve as a realistic reference case for choosing the definitive size of the installation. Fig. 5 Installation cost versus the number of shortage hours

Table 3 Investment costs and savings related to some levels of shortages PC (kWp)

1

15.4

2300

72.6

0.00

0.00

100

10.3

2000

56.3

16.30

22.45

500

9.9

1200

42.0

30.60

42.15

1000

8.3

1000

35.0

37.60

51.79

Cb (Ah)

CT (ke)

CT

C T CT %

Nbre of shortage hours αe

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Table 4 Erasing cost of the shortage time Shortage interval

Cost of erasing shortage hours (ke/h)

0–100

0.16

100–500

0.04

500–1000

0.02

Table 5 Relative and absolute economies related to some levels of shortage Shortage hours αe (h)

CT (ke)

CT

C T CT %

αe αe,100

1

72.6

16.30

28.95

100

56.3

0.00

0.00

1/100

500

42.0

−14.30

−25.40

5

1000

35.0

−21.30

−37.83

10

1

Indeed, some users could admit 100 h of shortage; they correspond to 16 min of daily shortage.

4.2 Daily Time Step Model Figure 6 shows the variation of the installation cost as a function of the quality factor K r . It can be seen that the slope of the cost curve remains almost constant regardless of the K r value; reflecting a linear variation in the system’s investment cost. Table 6 shows the dimensions, costs and savings corresponding to different Kr values. We notice that by increasing the value of K r , the savings become more important; passing from K r = 0.6–0.65, the savings double in size. Fig. 6 Installation cost versus the quality factor K r

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Table 6 Investment costs and savings related to some Kr values K r factor

PC (kWp)

C b (Ah)

C T (ke)

CT

C T CT %

0.55

23.3

4000

118.4

0.0

0.00

0.60

21.3

4000

114.0

4.4

3.86

0.65

19.7

4000

110.4

8.0

7.25

Table 7 Main reference configurations selected

Configuration

PC (kWp)

C b (Ah)

C T (ke)

Hourly profile of solar radiation

10.3

2000

56.3

Daily profile of solar radiation

19.7

4000

110.4

This means that more K r increases, savings become more significant; however, according to the study done by [6], if K r is greater than 0.65 it does not mean that the system is efficient. There is risk of under-designing i.e. energy demand may not be met at every moment. However, it appears that a K r value closer to 0.65 characterizes an optimized system. Finally, the PV system performance analysis will be performed for the two structures based on two profiles (hourly and daily) of solar radiation profiles. The two selected configurations are presented in Table 7.

5 Performance Analysis In this section, we will see how the solar radiation profile can influence the performance of PV systems. By using performance indices and selected indicator coefficients published in IEA PVPQS Task2 [7, 8], we will proceed to quantify the performance gaps between the two design techniques already described above. The performance indicators generally used in the evaluation of a stand-alone PV are: • The efficiency of the PV field related to the energy produced:

Ya = (E P V p,d )/P0 (h/KWp,d )

(14)

• The reference efficiency related to received energy of PV modules:

Yr =

24 1

IT /IT,ST C (KWh/KW)

(15)

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• The final efficiency of the energy used by the consumer:

Y f = E P Vu,d /P0

  KWh/KWp,d

(16)

• The solar radiation pick-up losses by the PV array

L c = Yr − Ya

  KWh/KWp,d

(17)

• System losses caused by the different equipment composing the PV system.

L S = Ya − Y f

  KWh/KWp,d

(18)

• The Performance ratio

P R = Y f /Yr

(19)

A stand-alone PV system, which is not working properly, will present a lower PR; however, a higher PR value does not mean the system is working correctly. To have an overview of the system’s use of the potential of solar energy, a new factor has been implemented. It has been defined as a ratio of the PV power used by the load to the PV power produced by the PV field. This factor is called UF or the use factor. It also reflects the notion of self-consumption. It is calculated by the following expression: UF =

E P Vu,d E P V p,d

(20)

More this factor is high, by reaching the unit; more the system presents very low energy losses, therefore better performances. If the UF factor can provide an idea of the overall system, it seems necessary to define another factor to evaluate the performance of the PV field. It is the factor of production that is the ratio between two performance indices Y a and Y r .

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5.1 Variation of the PR Performance Ratio as a Function of the UF Use Factor Figure 7 shows the correlation of PR and UF for the two reference sizing cases (100 h shortage and K r = 0.65). PR is more or less a linear function of UF. As the UF is higher, the system uses the maximum of its solar potential. It is clear that the tendency curve for the PR variation for the 100 h shortage case (hourly solar radiation profile) has a higher directional coefficient than the K r case = 0.65 (daily solar radiation profile); this shows that the sizing based on an hourly solar profile is more or less consistent.

Fig. 7 Variation of PR as a function of UF

Fig. 8 Number of occurrences and cumulative frequencies of the PR performance ratio

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Fig. 9 Variation of the PR performance ratio as a function of solar radiation

Figure 8 illustrates histograms variation of occurrence in defined PR ranges and corresponding cumulative frequencies. We observe that more than 68% of the occurrence number registered for 100-hour of shortage is in a PR range between 0.6 and 0.8. These values provide information on the correct performance of the PV system. On the other hand, for the reference system (K r = 0.65), 45% of recorded values are in the PR range (0.4–0.5). This indicates that the PV system is oversized, causing additional costs in the initial investment capital.

5.2 Variation of the PR Performance Ratio as a Function of the Global Solar Radiation Figure 9 shows that the PR is clearly dependent on the amount of energy received in terms of PV modules. For the case of 100 h shortage, more solar radiation increases, more the PR follows an increasing curve, whereas for the case of K r = 0.65, we can see the opposite, which means that the system is oversized. In other words, not all PV production was consumed; a significant part remained unusable. It also shows how the PV system for the 100-hour shortage reference case efficiently uses its potential for solar energy.

5.3 Calculation of Performance Gaps To clearly demonstrate the effect of both solar radiation profiles on the performance of PV systems, we have chosen to quantify this influence by identifying the gaps between the PV system performances of the two sizing techniques.

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Fig. 10 Variation in PR/PF system yield deviations

5.4 Gaps in PR/PF System Performance Figure 10 shows the system’s performance gaps for each month of the year. There are positive and significant differences over all months, exceptionally the one recorded in February, which remains relatively low. It should also be noted that a peak difference is reached in August, which is estimated at almost 35%, a significant value. For the other months, the differences range from about 12 to 22%.

5.5 Differences in System Losses Ls A similar observation is seen in Fig. 11, where the gaps between the losses in the system caused by both reference systems are displayed (Case of 100 h shortage and case of K r = 0.65). The negative peak is reached in August and it is estimated at almost 02 h equivalent to the maximum energy produced by the system. It is clear that these differences result in a gain, including energy produced by the PV system and subsequently used by the consumer.

6 Conclusion This paper highlights the effect of the solar radiation profile on the design and even the simulation of energy systems, particularly PV systems. The most commonly used profile in the design of PV systems is the daily solar radiation profile, it is found that installations made are frequently oversized, generating significant cost.

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Fig. 11 Differences in losses of the system Ls

To overcome this problem, we have established a comparative conceptual analysis of two design techniques based on different solar radiation profiles. In our case, we have established two sizing techniques based on two profiles of solar radiation, one with hourly steps and the other based on a daily model. The sizing results were established and interpreted in a detailed manner, followed by a comparative analysis of the performance of the PV systems studied. The first sizing technique using the hourly solar radiation model is based on accounting for the number of shortage hours determined over a one-year period. The design results showed that the sensitivity of the system for 100 h of shortage is the most important with the lowest initial investment cost. Similarly, the sizing technique using the daily profile of solar radiation is based on the establishment of a quality factor noted K r . For this purpose, we have taken three values of this factor, namely K r = 0.55, 0.60 and 0.65. The sizing results for this technique revealed that for K r = 0.65, the sensitivity of the system is better. The simulation of the PV system established for both sizing has shown that the sizing based on the hourly model of solar radiation has better performance. The PR performance ratio per 100 h of shortage (hourly solar radiation profile) changes in a way that is consistent with the use of solar energy potential. The system’s efficiency for 100 h of shortage is significantly higher compared to that of the daily profile (K r = 0.65). In general, the sizing technique based on the time profile of solar radiation is the most consistent, with a low initial investment cost and a minimum of energy produced by the PV field that is not used by the consumer. Considering all this, modelling solar radiation at the hourly scale and even at the minute scale is essential for the evaluation of solar energy systems.

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Acknowledgments The National Office of Meteorology located in Tamanrasset, south of Algeria, sponsored this study.

References 1. Aras H, Balli O, Hepbasli A (2006) Estimating the horizontal diffuse solar radiation over the Central Anatolia Region of Turkey. Energy Convers Manag 47:2240–2249 2. El-Sebaii A, Al-Hazmi F, Al-Ghamdi A, Yaghmour SJ (2010) Global, direct and diffuse solar radiation on horizontal and tilted surfaces in Jeddah, Saudi Arabia. Appl Energy 87:568–76 3. Duffie JA, Beckman WA (2013) Solar engineering of thermal processes. John Wiley & Sons Inc, Hoboken 4. Jiang Y (2009) Estimation of monthly mean daily diffuse radiation in China. Appl Energy 86:1458–1464 5. Chikh M (1994) Analytical and conceptual study of power PV systems in Algeria, magister thesis, CDER 1994 6. Chikh M, Mahrane A, Chikouche A (2008) A proposal for simulation and performance evaluation of standalone PV systems. In: 23rd European photovoltaic solar energy conference and exhibition, 1–5 September 2008, Valencia, Spain, pp 3579–3584 7. Analysis of photovoltaic systems. Report IEA- PVPS T2-01 (2000) 8. Operational performance, reliability and promotion of photovoltaic systems. Report IEA-PVPS T2-03 (2002)

Investigation and Prototyping Implementation of a Novel Solar Water Collector Based on Used Engine Oil as HTF Oussama Touaba, Salah Mohamed AitCheikh, Mohamed El-Amine Slimani, Ahmed Bouraiou, and Abderrezzaq Ziane Abstract This work is based purely on experimental results, where a Water Heater Collector with Storage Tank (WHCST) was realized using used engine oil as a basic material in the principle operating. After applying several experiments on the type of used oil type S15 W40t was confirmed that it can be a good absorber of sunlight and heat transfer simultaneously and it can be used in heating purposes. Through experimental thermal properties of the proposed thermal collector. this water heat system provides a significant efficiency since the temperature reaches values greater than 40 °C for a 50 L water tank in less than four operating hours under a moderate sunlight (less than 700 W/m2 ) and external temperatures less than 18 °C (In February). Keywords Solar water heater · Solar energy · Waste engine oil · Solar thermal collector O. Touaba (B) · A. Bouraiou · A. Ziane Unité de Recherche en Energies Renouvelables en Milieu Saharien (URERMS), 01000 Adrar, Algeria e-mail: [email protected] A. Bouraiou e-mail: [email protected] A. Ziane e-mail: [email protected] O. Touaba Ecole Nationale Polytechnique d’Alger, 16000 Alger, Algeria S. M. AitCheikh Laboratoire des Dispositif de Communication et de Conversion Photovoltaïque, Ecole Nationale Polytechnique d’Alger, 16000 Algiers, Algeria e-mail: [email protected] M. E.-A. Slimani Departement of Energetic and Fluid Mechanics, Faculty of Physics, University of Science and Technology Houari Boumediene (USTHB), 16111 Algiers, Algeria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_53

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1 Introduction The renewable energy sector became an important sector due to GHG emission reduction and decreasing fossil sources dependence [1]. Besides, imported fossil fuels account for more than 95% of their primary energy sources in the world [2], resulting in high electricity bills, energy system foreclosures, economies of scale, low economies and low energy security. In the interest of sustainable development to minimize the impact of pollution on the environment, we proposed using used motor oils in heating system. The RE application based on solar energy like photovoltaic and thermal is widely used in the world in recent years [3, 4]. Solar heating is one of this energy application that uses thermal energy from sunlight to heat the water or other fluids via a collector [5–7]. Several studies concerning various kind of solar heater system were performed in different locations [6–13]. The improvement and development of this solar converter is to be exploited in several application and fields; in housing and agricultural activities [3, 4, 14–18]. The solar water heater is generally composed of one or more solar collectors, a storage tank and a heat transfer fluid circulation system which transfers the heat generated by the solar collectors to the storage tank as presented in Fig. 1. In this paper, novel solar water heater employing waste engine oil as absorber and heat transfer fluid for water heating, is described. This device is based on two systems which are thermal and photovoltaic, we will focus on the characteristics of the thermal panel part. The work is organization as follow: – Materials and methods: proposed prototype and operation Mode and mathematical formulation of energy performance. – Experimental measures and results: Contain the Platform measurement description and heat thermal panel characteristic, discussion results of power and thermal efficiency. Fig. 1 Individual domestic solar water heater

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Fig. 2 Diagram illustrating of all system components

2 Materials and Methods 2.1 Prototype and Operation Mode In this work, it is a matter of combining photovoltaic system and a thermodynamic system to release a heating system which is mounted on a structure of aluminum. The basic components of the system components presented in this study are illustrated in Fig. 2. The prototype thermodynamic collector proposed in this study consists of two plates of tempered glass in which circulates the liquid (used oil) between them and the distance between the two plates is 5 mm, an electric pump transfers the hot oil to the water tank, and this system behaves like a greenhouse that absorbs and increase the heat that converted in high thermal energy. The heat obtained is transmitted through pipeline containing a hot fluid which is, in our case, the used oil engine to other liquid (water). The heat exchanger made from copper, and it is responsible of heat exchange inside the tank storage. For this, the completion of a prototype has yielded considerable results; so many experiments have been done on the prototype as shown in Fig. 3.in order to evaluate this idea and the possibility of its future use in the industrial field. The design parameters of the prototype are summarized in Table 1.

2.2 Energy Performance The performance indices like thermal gain and thermal yield are expressed below. The useful thermal output of the system is given by following relationship:

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Fig. 3 Photograph of the proposed prototype

Table 1 Design parameter of the system Symbol

Characteristic

Value

Sc

Thermal surface exposed to light

1 m2

Ip

Pipes length

400 cm

Se

Heat exchanger surface

0.17 m2

Ve

Volume of oil in the exchanger

0.3 L

Vt

Volume of oil in pipes with thermal insulation

0.4 L

Vc

Volume of oil in heat collector exposed to light

3.4 L

Vo

Total oil volume

4.1 L

Qth = mwt Cpw

 mwt Cpw  Twt = Tw,j − Tw,i t tj − ti

(1)

The instantaneous thermal efficiency is calculated using the following relationship:   Tw,j − Tw,i dqth   = mwt Cpw ηth = Gi .A Gi .A tj − ti

(2)

While the overall thermal efficiency is calculated using the following relationship:  tf

ηth =

dqth dt  tf A ti Gi dt ti

(3)

Investigation and Prototyping Implementation … Table 2 Monthly Average day weather parameters (2016) of Tipaza city

783 G (KWh/m2 )

TAverage (°C)

January

118.55

15.66

February

119.01

15.1

March

157.36

16.12

April

146.55

19.2

May

144.04

20.75

Jun

161.28

26.95

July

166.84

22.5

August

176.29

21.36

September

166.14

26.27

October

153.48

20.5

November

106.85

18.32

December

119.30

16.55

3 Experimental Results 3.1 Experimentation Sites The experiments tests were held at the Solar Equipment Development Unit UDES in the northern Algeria in Bouismail city, Table 2 provides monthly Average day of weather parameters (Solar Irradiation G, Tmin and maximum Tmax temperatures, relative humidity RH). The values of this table were calculated from the daily measurements of the meteorological station along all the days of the year.

3.2 Used Measurement Materials The data acquisition equipment used in the experimental prototype for several consecutive days to perform the necessary parameters (temperature, irradiation, level heating water), and all the measurement which is exploited in this study, is shown in Fig. 4 and presented as follow: 1. 2. 3. 4.

Data acquisition equipment Weather station model meteorological station (NEAL) installed in UDES Thermocouples type k Computer.

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Fig. 4 Data acquisition equipment used in the experimental prototype

4 Results and Discussion In the beginning, we made an experimental test on the prototype at September with a quarterly heat rate about 23 °C. In order to assess the performance and effectiveness of the proposed solar thermal collector, the following scenarios and steps have been developed. • The first step is to test the prototype in typical sun radiation for a full day for heating the water in order to know the level of storage water temperature and the variation in the temperature of the used oil (heat transfer fluid) inside in the thermal collector. • The second step consists to measure the oil temperature in the inlet and outlet of the thermal collector and comparing the temperature levels reached by the two fluids (Storage water and spent oil), under a typical day and then calculate the thermal efficiency of the prototype thermal collector • We compare the effectiveness of the system by testing the prototype in two different days with different levels of solar radiation by monitoring the level of effectiveness for each day of measurement

4.1 Evolution of Water and Oil Temperatures In this step, we carried out the tests on the prototype. We tested the prototype by experimenting with heating water twice in a day, one in the morning (see Fig. 5) and the afternoon (Fig. 6). This test for order to observe the variation of the temperature of the water compared to oil temperature and this happen by the exchange of heat between them.

Investigation and Prototyping Implementation …

785 Input Oil temperature Ambient temperature 1000

Temperatue (°C)

100 80

800

60

600

40

400

20

200

0

Power (W )

Output oil temperature Water temperatue

09:54 10:06 10:18 10:30 10:42 10:54 11:06 11:18 11:30 11:42 11:54 12:06 12:18 12:30 12:42 12:51 13:01

0

Time of day Fig. 5 Temperature compared to the temperature of heat transfer oil in the morning

Output oil temperature Ambient temperature

70

700

60

600

50

500

40

400

30

300

20

200

10

100

Power(W)

Temperatue(°C)

Input Oil temperature Water temperatue Thermal power

0 15:31

15:43

15:19

15:07

14:43

14:55

14:31

14:19

14:07

13:43

13:55

13:31

13:19

13:07

0

Time of day Fig. 6 Water temperature compared to the temperature of heat transfer oil in the afternoon

The observation of the difference between the temperature of the hot oil coming from the thermodynamic sensor to the water (blue Curve) and the temperature of the oil coming out of the water after the exchange heat with water (red curve), see Fig. 5. As shown in Fig. 5 at 9:54, the oil pump is triggered and the prototype begins to operate. We note that the temperature of the water in the tank (Curve in green) increases in parallel with that of the oil and that the oil (Curve in red) is heated because of the heat absorbed by the sun. When the oil comes out of the water tank (in blue), its temperature is close to the temperature of the water. We conclude that the oil has to exchange most of its

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90 80 70 60 50 40 30 20 10 0

Thermal efficiency

13:07 13:16 13:25 13:34 13:43 13:52 14:01 14:10 14:19 14:28 14:37 14:46 14:55 15:04 15:13 15:22 15:31 15:40

Efficiency(%)

temperature to water. We also note that the temperature difference between the input oil and the output oil is 14 °C. Where we observe that the thermal power (Curve in black) that the oil gives is increases and reaches 700w maximum. This is due to the large difference between the heat of the water and then the thermal power decreases to 100 W, due to saturation of the system despite the presence of solar radiation. As shown in Fig. 6 we observe almost the same results obtained in the morning experiment of Fig. 5 with a difference in the temperature of water and oil (the temperature of oil and water in the first experiment is greater than the second) the water temperature has reached to 58 °C and the temperature of oil in the thermal collector output to 80 °C, but regarding in the experiment of Fig. 6, we note that the water temperature does not exceed 46 °C and the temperature of oil 65 °C. This is caused by the power of solar radiation in the period between 10: 00 and 13: 00; of course, the thermal energy is less in the second experiment than the first experiment as observed in (Curve in black) so that it is initially at 650 watts and then decreased rapidly to reach 350 watts within one hour. From this experience we conclude that the proposed system can be reheat the water for domestic use twice a day and also it is concluded that the used oil, which is used in this system, is the good absorbent solar thermal and good heat transfer fluid. The average efficiency η (%) of the system of the experiment which detailed in the previous figure (Fig. 7) reach an important value of 65%. To obtain a high temperature of water heating we increase the space of the thermodynamic capture exposed to the sun.

Time of day Fig. 7 The evolution of thermal efficiency during the test day

Radia on(W/m²)

Radiation A Water temperature A

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Radiation B Water temperature B 70

1000

60

800

50 40

600 30 400 200 0

Temperatue(°C)

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20 10 0

Fig. 8 The variation of water heating temperature for two days: (B) a day with low solar irradiation; (A) a day with high solar irradiation

4.2 Effect of Solar Irradiation on Water Temperature These measurements allow observing the efficiency of the system in the heating of water through climate change. Figure 8. shows that the water temperature (red curve) inside the tank in the prototype increases when the solar radiation is large (black curve) and we note that also the value of radiation in the curve exceeds 1000 w/m2 and the water temperature reached 60 °C. On the other hand, we took the same heating time for another day, when the solar radiation is weak as shown in (blue curve) and its value in the rate of 600 w/m2 , in addition, note that the temperature of water (green curve) inside the tank does not exceed 43 °C.

5 Conclusion In this paper, based on the experimental superficial study reached in this work. A new water heater prototype based on used oil was presented. Several experimental tests were conducted in the city of Tipasa to show the feasibility of this device. Focusing on the thermal collector section of the prototype. The results obtained were so satisfactory that we reached an acceptable heating temperature of 60 °C and 65% heating efficiency rate, which is a very acceptable result so this innovative system

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can be helpful. In the development of the field of solar heaters and conservation of the environment successfully.

References 1. Sahouane N, Dabou R, Ziane A, Neçaibia A, Bouraiou A, Rouabhia A, Mohammed B (2019) Energy and economic efficiency performance assessment of a 28 kWp photovoltaic grid-connected system under desertic weather conditions in Algerian Sahara. Renew Energy 143:1318–1330 2. Bouraiou A, Necaibia A, Boutasseta N, Mekhilef S, Dabou R, Ziane A, Sahouane N, Attoui I, Mostefaoui M, Touaba O (2020) Status of renewable energy potential and utilization in Algeria. J Clean Prod 246:119011 3. Boumaaraf B, Boumaaraf H, Slimani ME-A, Tchoketch_Kebir S, Ait-cheikh MS, Touafek K (2020) Performance evaluation of a locally modified PV module to a PV/T solar collector under climatic conditions of semi-arid region. Math Comput Simul 167:135–154 4. Sellami R, Amirat M, Mahrane A, Slimani ME-A, Arbane A, Chekrouni R (2019) Experimental and numerical study of a PV/Thermal collector equipped with a PV-assisted air circulation system: configuration suitable for building integration. Energy Build 190:216–234 5. Harmim A, Boukar M, Amar M, Haida A (2019) Simulation and experimentation of an integrated collector storage solar water heater designed for integration into building facade. Energy 166:59–71 6. Smyth M et al (2019) Experimental performance characterisation of a hybrid photovoltaic/solar thermal façade module compared to a flat integrated collector storage solar water heater module. Renew Energy 137:137–143 7. Li B, Zhai X, Cheng X (2018) Experimental and numerical investigation of a solar collector/storage system with composite phase change materials. Sol Energy 164:65–76 8. Al-Kayiem HH, Yassen TA (2015) On the natural convection heat transfer in a rectangular passage solar air heater. Sol Energy 112:310–318 9. Tewari K, Dev R (2019) Exergy, environmental and economic analysis of modified domestic solar water heater with glass-to-glass PV module. Energy 170:1130–1150 10. Garnier C, Muneer T, Currie J (2018) Numerical and empirical evaluation of a novel building integrated collector storage solar water heater. Renew Energy 126:281–295 11. Slimani ME-A, Sellami R, Mahrane A, Amirat M (2019) Study of Hybrid Photovoltaic/Thermal Collector Provided With Finned Metal Plates: A Numerical Investigation under Real Operating Conditions. In: 2019 International Conference on Advanced Electrical Engineering, ICAEE 2019, pp 1–6 12. Seddegh S, Wang X, Henderson AD, Xing Z (2015) Solar domestic hot water systems using latent heat energy storage medium: a review. Renew Sustain Energy Rev 49:517–533 13. Touaba O, Cheikh MSA, Slimani MEA, Bouraiou A, Ziane A, Necaibia A, Harmim A (2020) Experimental investigation of solar water heater equipped with a solar collector using waste oil as absorber and working fluid. Solar Energy 199:630–644 14. Chaouch WB, Khellaf A, Mediani A, Slimani MEA, Loumani A, Hamid A (2018) Experimental investigation of an active direct and indirect solar dryer with sensible heat storage for camel meat drying in Saharan environment. Sol Energy 174:328–341 15. Slimani MEA, Amirat M, Bahria S (2015) Study and modeling of heat transfer and energy performance in a hybrid PV/T collector with double passage of Air. Int. J. Energy a Clean Environ. 16(1–4):235–245 16. Bahria S, Amirat M, Hamidat A, El Ganaoui M, Slimani M (2016) Parametric study of solar heating and cooling systems in different climates of Algeria – A comparison between conventional and high-energy-performance buildings. Energy 113:521–535

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17. Slimani ME-A, Sellami R, Mahrane A, Amirat M (2019) Experimental study of a glazed bi-fluid (water/air) solar thermal collector for building integration. In: 2019 international conference on advanced electrical engineering, ICAEE 2019, pp 1–6 18. Hmida A, Chekir N, Laafer A, Slimani MEA, Brahim A B, (2019) Modeling of cold room driven by an absorption refrigerator in the south of Tunisia: A detailed energy and thermodynamic analysis. J Clean Prod 211:1239–1249

Piezoelectric Energy Harvesting Based Autonomous Vehicles’ Vibrations Nadjet Zioui, Sousso Kelouwani, and Gilbert Lebrun

Abstract This work aims to analyze the impact of using carbon fibers based materials in the process of generating electrical power based energy harvesting. The harvester is a piezoelectric element exposed to vibrations of a vehicle in motion. The combination of fiber carbon material associated to the piezoelectric transducer are used as a source to generate electrical energy to power cars onboard wireless sensors. Many works treated the possibility to implement the vibrations energy harvesting using piezoelectric components. The approaches are oriented towards flexible piezoelectric materials subjected to flexion efforts for instance. In this work, we choose to use a carbon based material to which we fix a mass placed on a piezoelectric material. This combination is equivalent to a spring mass system. This paper highlights the advantages of such a solution compared to a classic spring mass system. Keywords Energy harvesting · Piezoelectric transducer · Carbone fiber material · Smart wireless sensors · Autonomous vehicles

1 Introduction and Recent Trends Many sources of energy are available in nature. Some of them are unused and most of them wasted every day. For many years, human being has been thinking about ways to harvest energies. Sun panels are used to generate electrical energy from solar source. Sun ovens are widely used in some isolated areas. Wind turbines are used to extract the wind power in order to provide electricity or to pump water. Water flow is used to drive turbines and generate electrical power. Sun, water, wind, hydrogen, Bio sources … are all examples of successful attempts to harvest energy instead of wasting it. Vibrations are practically present in every mechanical system in motion. Application to onboard sensors energy generation. N. Zioui (B) · S. Kelouwani · G. Lebrun Department Mechanical Engineering, Université du Québec à Trois-Rivières, Trois-Rivières, QC, Canada e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_54

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They are a huge potential source of energy. A vehicle in motion vibrates due to several reasons, but mainly due to the engine and the mechanical structure of the vehicle itself, as well as the contact with the road. This work addresses the harvesting of this source of energy and its use to generate electrical power for vehicle’s onboard wireless sensors. The paper explores more specifically, the use of carbon fiber in the physical implementation for energy extraction. Energy harvesting EH is a current interest for lots of industries. With the emerging wireless technologies, phones, wearable devices, wireless sensors …etc., the need of reliable and autonomous ways to provide energy, ideally wirelessly, is a fact. Most of wireless and embedded devices rely on batteries to operate. However, the intelligence growing and the sophisticated processing and applications that are involved come out of a cost: energy. Batteries are starting to show limitations in terms of fulfilling smart devices needs for processing and communicating, and people are starting to consider alternative energy resources to compensate these limitations: solar chargers, solar cellphones [1–4] … etc. Even some new industries are emerging using EH. Many works on the literature highlighted the need of harvesting energy, especially for wireless sensors applications. Song and Tan [5] treats the estimation of the energy needed from wireless sensors in several situations, in order to evaluate the need of energy for EH applications. Ruan et al. [6] focuses on an algorithm that deals with the mismatch between the sensors demand and the available harvested energy from plane aisle vibrations, using a piezoelectric device. Jushi et al. [7] presents the wind prediction for the nearest future in order to harvest wind energy for wireless sensors. Obaid and Fernando [8] highlights the importance of EH for rural communities. Piezoelectric materials are very promising as potential source of energy [9]. In fact, many works based on piezoelectric materials have been successfully conducted and showed very promising results. Sharma and Baredar [3], Yesner et al. [10] introduced piezoelectric material in shoes to generate power depending on the applied mass. A very interesting work in [4] treats the energy harvesting based piezoelectric panels do generate energy from the floor under the effect of people and cars motion. In this paper, we briefly present the principle of EH based piezoelectric materials to put into perspective the implementation solution that we analyze. In the second section, we introduce a brief abstract of the analysis of energy consumption for one type of wireless sensors based ZigBee network for design and implementation purpose. The analysis concerns two parts of the device: the sensing part and the wireless network part. In the third section, we present the modal vibration overview available in a car in motion. This leads to the presentation and the design of the carbon fiber material for the physical implementation. In section four, we recall the piezoelectric model and highlight the electrical-mechanical equivalence model method. The fifth section presents the results using the fiber carbon implementation. We end this paper with a conclusion as a sixth and last section.

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2 Energy Consumption of a Wireless Sensor One of the most important steps in designing harvesting systems for smart wireless sensors is the knowledge of the amount of energy needed by the sensor. In fact, there are several cases to consider while estimating the energy consumption: the sensor behaving as an end device or as a router. For these categories, many works consider the situations of the device powering up, transmitting data, receiving data, with or without acknowledgement requirements. This last configuration is very important. In fact, one of the important reasons of consuming all the battery energy or other energy source is the phase where the device is looking for a network to join by iterating a process of sending data and waiting for a response. Many works investigated the power consumption for wireless networks. Song and Tan [5] analyzed the energy consumption of ZigBee based wireless sensors. The paper also presented several methodologies to simulate and estimate energy consumption for a wireless sensor. Most of the studies rely mainly on the wireless device needs rather than the wireless sensors’ needs in terms of energy. In fact, every sensor has its own operating principle, its own conditioning circuitry and therefore its own needs of energy. This makes it very challenging or even impossible to estimate with precision the amount of needed energy for all kind of wireless sensors. In this work, we determine an estimate of the energy needed for a particular wireless part of the sensor which is ZigBee based network, according to the analysis of technical documentation and experimental data for some of them from several manufacturers.

2.1 Consumption Analysis for the Wireless Layer In order to estimate the energy consumption of the wireless devices, we are interested in looking into the technical data of a few wireless devices which some have been experimentally validated. The selected devices are 2.4 GHz ZigBee protocol devices. We motivate the ZigBee Radiofrequency technology choice by the low energy consumption and a large range compared to other technologies for the same amount of exchanged data. The data that we seek concern the energy consumption of the wireless devices during their different operating modes. Table 1 summarizes the technical data collected from different ZigBee wireless based Radiofrequency devices, from different manufacturers [11–16]. Apart from some odd values, for a typical operating voltage of 3.3 V, the sleeping mode implies approximately a 1.2 µA current. The Reception mode needs around 26 mA and the Transmission mode uses around 33 mA. We base our design of the harvester on these collected values. However, the current might reach much higher values during the particular phase of powering up. Therefore, we need to consider this important parameter during the components’ choices and the design step as well. We can accomplish this part by adding a properly sized capacitor.

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Table 1 Energy consumption data for different RF ZigBee modules, for the sleeping, transmission and reception modes Devices (manufacturers)

Typical consumption data Operating voltage (V)

Typical Rx current (mA)

Typical Tx current (mA)

Typical sleep current (µA)

Device 1 (M1)

3.3

19

23

2

Device 2 (M2)

2.1–3.6

17

40

2

Device 3 (M3)

2.1–3.6

26

31

1

Device 4 (M4)

2.0–3.6

15.3

17

0.7

Device 5 (M4)

2.0–3.6

35

22

0.7

Device 6 (M4)

2.0–3.6

175

22

0.7

Device 7 (M5)

2.0–3.6

24

34

0.4

Device 8 (M5)

2.0–3.6

27

175

1.3

Device 9 (M6)

3.3

32

175

0.65

Device 10 (M1)

1.87–3.8

9.8

8.2

2.5

Device 11(M7)

2.1–3.6

30

44

1

Device 12 (M7)

2.1–3.6

34

100

2.4

Device 13 (M8)

2.1–3.6

28

67

0.4–225

2.2 The Sensing Layer The sensing layer is composed of a transducer that generates electrical energy from variations of force, or vibrations and a conditioner circuitry that is in charge of the signals conditioning and adaptation. We assume the transducer as an ideal generator and neglect the dissipations if any. The only part that is supposed to dissipate any energy is the conditioner. With a good components sizing and design, the conditioning circuit for the harvester can reach around 90% efficiency with a 100 mA load [17]. We assume for this study that the conditioner board will draw approximately 10% of the available energy. Finally, the data and analysis show that a typical wireless sensor needs in terms of energy are about 100 mW for the wireless module and about 20 mW for the harvester device. Which leads to total energy needs of about 120 mW for the wireless sensor.

3 Vibrations Analysis for Vehicles There are many works in the literature about the analysis, the modeling and the simulation of the vibrations of vehicles. Most of the papers if not all of them concentrate the study on the vibrations transmitted to the body based people comfort or for the improvement of the vehicle’s structure [18]. The works focus mainly on the suspension response as a main source of vibration, and highlight the first mode of vibration

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Fig. 1 Quarter, half and full car model for vehicles

as a main source. Some papers illustrate the analysis of the vibrations based amplitude and frequency, but still consider the second mode and above as noise. In this work, we make a synthesis of the common results of the modeling and simulation works, regarding the available vibrations according to the road speed, the frequency and the constraints directions. The idea from this is to bring out some insights on the vibrations available as a source for energy harvesting in one hand. On the other hand, this gives a tool to support the simulations thus the harvester design. The main models that the researchers use are quarter, half or full models [19] as illustrated in Fig. 1. Most of the models consider masses, springs and dampers models, with more or less complex equations. The models are usually based on Euler and Lagrange formulations, but there are some other available techniques like pseudo excitation, decoupling and bound graph methods [20–22]. Barbosa [23] uses a half car model and considers four degrees of freedom. The paper analyzes the vertical and angular movement of the body. The paper points out some important frequency analysis facts like the body’s frequencies that are around 1 to 2 Hz while the suspension’s modes are around 12 Hz, which represents a difference of about a decade. Another important result is that the frequency response closely depends on the speed of the vehicle. In fact, the higher the vehicle speed is, the higher the frequencies are where the peaks occur. [20] uses a decoupling method to highlight the several modes of a car in terms of vibrations. The paper adds an additional degree of freedom considering a man’s seat, thus focuses on the comfort in the development work. The paper uses computation tools in a comparative way and points out five modes. Three of the modes have values around 1 to 1.5 Hz while two of them around 11 and 12 Hz, which joins the analysis results of [23]. Nahvi et al. [24] evaluates the vibrations according to several parameters and road types. The paper presents results for several speeds. The author underlines the random character of the vibrations and points out the Gaussian aspect under some considerations such as low IRI (International Roughness Index). Unlike the other works found in the literature, [21] highlights an interesting point of the speed variations during the tests instead of multiple constant speeds tests. Senthil Kumar et al. [18] presents the modal analysis of the vibrations of a car using numerical methods. On the light of the several works,

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we will consider two main frequencies for the car vibrations to be 1 and 12 Hz. We base our design on these values.

4 Piezoelectric Element Model There are several modeling methods for the piezoelectric element. In fact, this element can be studied from the mechanical side approach or from the electrical behavior side. The piezoelectric element produces mechanical force if an electrical voltage is applied to it, and inversely, it can produce electrical energy if subjected to a variable force. The mechanical model can be established using several methods. We will consider the electrical-mechanical equivalence in this section. The electrical model states that the voltage produced by the piezoelectric element is proportional to the strain applied to it [25]. U = −kx

(1)

where: U the voltage (V) k the piezoelectric constant (V/m) x the displacement (m) The energy conservation states that: U q = −F x

(2)

where: F the force (N) q the electric charge (C) From (1) and (2), it appears that the electrical charge is proportional to the applied force for thickness mode vibration [25]. Moreover, if the harmonic excitations are considered, then the voltage and the force will be expressed as (3) and (4), respectively. Equation (5) is then obtained.

v the velocity (m/s) I the electrical current (A)

U =−

k v jω

(3)

F =−

k I jω

(4)

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Fig. 2 Piezoelectric model



U F



 =

Z e −Z em Z em −Z m

  I v

(5)

Z e Electrical impedance (V/A) Z m Mechanical impedance (kg/s) Z em Electromechanical coupling impedance (Vs/m) Figure 2 presents the combination of the electrical and electrical-mechanical equivalent model for the piezoelectric element.

5 Experimentation and Initial Results One of the advantages in using carbon fiber material is the fact that the vibrations last longer in the time, which is a huge advantage since the piezoelectric needs varying constraints in the time. Moreover, the physical implementation is more optimal in terms of effort direction. In fact, using a classical spring mass system makes it hard to guarantee the effort’s direction, thus losses of energy is implied. Another advantage of using carbon material instead of metal spring is the resistance of the material against fatigue. Our experimental platform is composed of a carbon fiber slat under a variable charge due to the car vibrations. The experimental setup is illustrated in Fig. 3. It has been implemented on a GMC Terrain 2011 car. Figure 4 illustrates the harvester design that is composed of a ceramic piezoelectric plate associated carbon fiber slat. The electrical power that is generated from the piezoelectric plate is limited using the conditioner to a maximum of 5 VDC voltage. The experiences were run on a flat asphalt road. The harvester is fixed on the front of the vehicle and wired up to an acquisition card that is connected to a portable computer. We lead the experience with several car speeds and observed the generated energy for each one of them.

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Fig. 3 The experimental setup

Fig. 4 The carbone fiber slat

Figure 5 summarizes the experiment results of the generated voltage wave form with regard to the several speed values. The results highlight a potential of energy harvesting using a car in motion vibrations. The generated power depends on the piezoelectric plates that are used and the characteristics of the material and its flexibility. We intend on conducting comparative analysis of several types of piezoelectric materials in future works.

6 Conclusion and Future Work This work presented a design and experimental setup of a harvester that aims to provide energy to wireless sensors for autonomous vehicles. The work considers a particular application of a vehicle on an asphalt road. The design of the harvester is presented as a combination of two components that are the transducer and the

Fig. 5 Output observed with the experimental protocol

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conditioner. Experimental setup is used to analyze results depending on the car speed. The setup considered for the implementation uses a carbon fiber slat based design in order to improve the results and benefit from the several advantages of these materials such as the material resistance and flexibility. In future work, we intend to implement additive and multi modal configurations in order to include all potential frequencies signals in order to improve the amount of the harvested energy. We also intend to study the distribution of the results according to several types of vehicles and roads. Also, and more precisely, we will analyze the impact of each communication mode of the sensor on the energy consumption, as well as several types of piezoelectric materials.

References 1. http://downloadcenter.samsung.com/content/UM/200912/20091210082503578/GT-S7550_ UM_France_Fre_Rev.1.1_091105.pdf. Accessed 30 Oct 2019 2. https://www.amazon.ca/s?k=solar+cellphone&__mk_fr_CA=%C3%85M%C3%85%C5% BD%C3%95%C3%91&ref=nb_sb_noss_2. Accessed 08 July 2019 3. Turkmen AC, Celik C (2018) Energy harvesting with the piezoelectric material integrated shoe. Energy 150:556_564. Elsevier 4. Elhalwagy AM, Ghoneem MY, Elhadidi M (2017) Feasibility study for using piezoelectric energy harvesting floor in buildings’ interior spaces. Energy Proc 115:114–126. Elsevier 5. Song J, Tan YK (2012) Energy consumption analysis of zigbee based energy harvesting wireless sensor networks. In: 2012 IEEE International conference on communication systems (ICCS). IEEE, pp 468–472 6. Ruan Z, Chew J, Zhu M (2017) Energy-aware approaches for energy harvesting powered wireless sensor nodes. IEEE Sensors J 17(7):2165–2173 7. Jushi A, Pegatoquet A, Le TN (2016) Wind energy harvesting for autonomous wireless sensor networks. In: 2016 Euromicro conference on digital system design (DSD). IEEE, pp 301–308 8. Obaid A, Fernando X (2017) Wireless energy harvesting from ambient sources for cognitive networks in rural communities. In: 2017 IEEE Canada international humanitarian technology conference (IHTC). IEEE, pp 139–143 9. Sharma PK, Baredar PV (2017) Analysis on piezoelectric energy harvesting small scale device—a review. J King Saud Univ Sci 10. Yesner G, Jasim A, Wang H, Basily B, Maher A, Safari A (2019) Energy harvesting and evaluation of a novel piezoelectric bridgetransducer. Sensors Actuators A Phys 285:348_354, Elsevier 11. https://www.digi.com/pdf/dsxbee_3_zigbee_3:pdf. Accessed 08 July 2019 12. https://www.silabs.com/documents/public/data-sheets/TG-PM-0516-ETRX35x.pdf2f. Accessed 08 July 2019 13. https://www.silabs.com/documents/public/datasheets/MGM111DataSheet:pdf. Accessed 08 July 2019 14. http://ww1.microchip.com/downloads/en/DeviceDoc/70329b.pdf. Accessed 08 July 2019 15. https://www.nxp.com/docs/en/data-sheet/JN5168-001-MXX.pdf, accessed: 2019-07-08 16. https://radiocrafts.com/uploads/RC2400RC2400HP_ZNMDataSheet:pdf. Accessed 08 July 2019 17. https://www.analog.com/media/en/technical-documentation/datasheets/35881fc.pdf. Accessed 08 July 2019 18. Senthil Kumar M, Naiju C, Chethan Kumar S, Kurian J (2013) Vibration analysis and improvement of a vehicle chassis structure. In: Applied mechanics and materials, vol 372. Trans Tech Publ, pp 528–532

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19. Raju B, Venkatachalam R (2013) Analysis of vibrations of automobile suspension system using full-car model. International Journal of Scientific & Engineering Research 4(9):2105–2111 20. Xu B, Bao J, Wu J, Wang H (2015) Model analysis of car five degree of freedom vibration system based on energy decoupling method. In: 2015 International conference on electromechanical control technology and transportation. Atlantis Press 21. Guo L-X, Zhang L-P (2010) Vehicle vibration analysis in changeable speeds solved by pseudoexcitation method. Mathematical problems in engineering 22. Tan D, Wang Q (2016) Modeling and simulation of the vibration characteristics of the in-wheel motor driving vehicle based on bond graph. Shock Vib 23. Barbosa RS (2012) Vehicle vibration response subjected to longwave measured pavement irregularity. J Mech Eng Autom 2(2):17–24 24. Nahvi H, Fouladi MH, Nor MM (2009) Evaluation of whole-body vibration and ride comfort in a passenger car. Int J Acoust Vib 14(3):143–149 25. Staworko M, Uhl T (2008) Modeling and simulation of piezoelectric elements-comparison of available methods and tools. Mech AGH Univ Sci Technol 27(4)

Faults Diagnosis-Faults Tolerant Control (FTC)

Fault Detection of Uncertain Systems Based on Interval Data Driven Approach Chouaib Chakour

Abstract Kernel principal component analysis (KPCA) is an effective and efficient data-driven approach for nonlinear processes monitoring. However, the KPCA algorithm cannot be applied for monitoring uncertain processes data that is bounded by intervals. The key idea of the present work is to propose an extended methodology of conventional kernel PCA to handle nonlinear interval-valued data. This improves the FDI decisions by considering the uncertainties in the analysis. The results for applying this algorithm on the nonlinear processes of the Tennessee Eastman benchmark show its feasibility and advantageous performances. Keywords KPCA · Symbolic PCA · Interval data · Fault detection

1 Introduction Modeling and identification of an uncertain system are mainly concerned with characterizing the unknown system on the basis of measured input–output data in an uncertain environment. It is of fundamental importance in predictive control, fault diagnosis, and decision analysis since most real-life systems are associated with uncertainties due to noises, measurement errors, uncertain physical parameters, incomplete knowledge, etc. [1]. This gives an uncertain form of input-output data. The PCA is one of the most known and used linear data driven approach for process monitoring. Thus a great difficulty in applying, the traditional PCA identification technique is dealing with those uncertainties. Each imprecise data can be identified by a couple of values. However, the values are intervals, delimited by ordered couples of bounds referred to as minimum and maximum [2]. The measurement data represented by mono-valued data is one of the admissible measures that belong to a possible range of data. In fact, it is more C. Chakour (B) Department of Electronics and Communications, Kasdi Merbah University, Route de Ghardaia, Ouargla, BP.511 30 000, Algeria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_55

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appropriate to study variation and variability of phenomena that are better described by interval-valued data. In its standard form, PCA method is not designed to handle interval type data. Including uncertainty of the sensors measurements in the analysis requires extending the PCA methodology to the Symbolic Data Analysis (SDA) [8]. The Midpoints-Radii PCA method, a pioneer in symbolic PCA, is one of the best known examples of this family of methods [5]. For modeling and fault diagnosis of some complicated cases in industrial processes with nonlinear characteristics, PCA performs poorly due to its assumption that the processes are linear. In fact, one of the most used nonlinear generalization of the linear PCA is the kernel PCA method (KPCA) [3, 4]. Actually, extensive studies have been conducted for effectively identifying linear uncertain systems, especially with the symbolic PCA techniques [7], but are even non-existent for the nonlinear case. The motivation of this paper is to construct reliable symbolic kernel PCA model for the identification of uncertain nonlinear systems. The symbolic kernel PCA methodology is developed on the basis of midpoints and radii of the uncertain and/or interval nonlinear process data. The conventional kernel PCA is extended to handle nonlinear interval-valued data. In other words, the linear symbolic PCA methodology is developed and extended to the nonlinear case. This paper is organized as following. In Sect. 2 the modeling of nonlinear interval data with kernel PCA method is presented. Section 3 gives the application of the proposed method for fault detection. Experimental results of the proposed KPCA for fault detection of Tennessee Eastman process are given in Sect. 4. Finally, conclusions are given Sect. 5.

2 Modeling Uncertainties as Intervals 2.1 Interval Data (Radii and Midpoints) There are several measurement scales (or types of data), which are simply ways to categorize different types of variables. The epistemic uncertainty, which is naturally modeled by intervals, is called incertitude [6]. Let the matrix X ∈ Rn×m be the conventional data matrix of measurements, where n is the number of samples and m is the number of process variables collected under normal operating condition. Consider the vector of measurements at instant k: x(k) = {x1 (k), x2 (k), . . . , xm (k)} where, x j (k) is the k − th observation of the j − th variable.

(1)

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805

  Let us note that an j − th interval-valued variable x j (k) is just a standard variable but its values are intervals, delimited by ordered couples of bounds referred to as minimum and maximum:     (2) x j (k) ≡ x j (k), x j (k) , where, x j (k) < x j (k). Indeed, the interval data matrix [X], having 2m columns and n rows, is formalized as a set of interval-value variables for different instants of observation.   ⎞ ⎛  x 1 (1), x 1 (1) · · · · · ·  x m (1), x m (1) ⎜ x 1 (2), x 1 (2) · · · · · · x m (2), x m (2) ⎟ ⎟ ⎜ ⎟ .. .. .. (3) [X] = ⎜ ⎟ ⎜ . . . ⎠ ⎝    .  x (n), x (n) · · · . . x (n), x (n) 1

1

m

m

  The generic interval-valued variable x j can also be expressed by the couple x cj , x rj , where x cj represents the center of the interval and x rj its radius, x cj (k) = and, x rj (k) =

x j (k) + x j (k) 2 x j (k) − x j (k) 2

(4)

(5)

  The interval x j is given, from its center and radius, as follows: 

   x j (k) ≡ x cj (k) − x rj (k), x cj (k) + x rj (k)

(6)

In fact, the interval data matrix [X] can be written in terms of midpoints and radii:   [X] ≡ Xc , Xr .

(7)

2.2 Kernel PCA to Modelling Uncertain Nonlinear Systems Consider a set of measurements on a system, in good working order, of n observations {x1 , x2 , . . . , xn } in a given space R of dimension m. The KPCA algorithm allows the mapping of the data from the input space to a feature space F and the application of

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PCA in F. However, the training samples {x1 , …, xn } ∈ Rm mapped to the feature space F are represented by its images {φ(x1 ), …, φ(xn )}. xi → φ (xi )

(8)

An uncertain nonlinear system is defined by intervals measurement data {[x1 ], [x2 ], . . . , [xn ]} ∈ [X]. The interval-valued data [X] is implicitly transformed into a functional space called feature space F, where the classical PCA is applied. Let φ be the non-linear transformation of the interval observations [X] to the functional space F, where for each interval [xi ] an interval image [φ (xi )] is associated. [xi ] → [φ (xi )]

(9)

As indicated before, the interval nonlinear data [xi ] can be written in terms of midpoints and radii:   (10) [xi ] ≡ xic , xri .  c r xi , xi into the feaThen the transformation of each interval observation  r    c [xi ] ≡ ture space F is discribed by its image [φ (xi )] ≡ φ xi , φ xi .   xic → φ xic   xri → φ xri

(11) (12)

The PCA is then performed to find the principal components of mapped samples φ(xi ). Then, the sample covariance matrix in the feature space is given by: Cf =

n 1 φ(xi )φ(xi ) n i=1

(13)

Assuming {φ(x1 ), …, φ(xn )} have been mean-centered, the PCA is performed by finding the eigenvalues  > 0, and eigenvectors Pφ ∈ F\{0} satisfying P = C f P. An important property of the feature space F is that it allows us to compute the value of the inner product without having to carry out the nonlinear map φ explicitly. However, the computational difficulties in the higher dimensional feature space can be simplified into finding the dot products of vectors in the feature space by using a kernel function K [10]. K(xi , x j ) =< φ(xi ), φ(x j ) >   where K is the matrix of size n × n and of general term Ki j = k xi , x j .

(14)

Fault Detection of Uncertain Systems Based on Interval Data Driven Approach

⎛

k (x1 , x1 ) k (x1 , x2 ) ⎜ k (x2 , x1 ) k (x2 , x2 ) ⎜ K=⎜ .. .. ⎝ . . k (xn , x1 ) k (xn , x2 )

⎞ · · · k (x1 , xn ) · · · k (x2 , xn ) ⎟ ⎟ ⎟ .. .. ⎠ . . · · · k (xn , xn )

807

(15)

For interval observations, the centers and the midranges kernel matrices of [X] in the feature space are defined separately as:

and,

⎛  c c  c c ⎞ k x1 , x1  k x1 , x2  · · · k (x1 , xn ) ⎜ k xc2 , xc1 k xc2 , xc2 · · · k xc2 , xcn ⎟ ⎜ ⎟ Kc = ⎜ ⎟ .. .. .. .. ⎝ ⎠ .  c. c   c. c   c. c  k xn , x1 k xn , x2 · · · k xn , xn

(16)

 ⎞ ⎛  r r  r r k x1 , x1  k x1 , x2  · · · k xr1 , xrn  ⎜ k xr2 , xr1 k xr2 , xr2 · · · k xr2 , xrn ⎟ ⎟ ⎜ r K =⎜ ⎟ .. .. .. .. ⎠ ⎝ .  r. r   r. r   r. r  k xn , x1 k xn , x2 · · · k xn , xn

(17)

The interval covariance matrix, or the intervel kernel matrix, can be expressed in terms of kernel matrices of midpoints and radii:   [K] ≡ Kc , Kr .

(18)

The global uncertain kernel matrix, from which an uncertain kernel PCA model will be constructed, is given by the hybridization between the midpoints kernel matrix and the midranges kernel matrix. Kg = Kc + Kr − 2Kcr Aφ

(19)

  where Kicrj = k xic , xrj , and Aφ = QPT . So that, Kcr = Pcr QT

(20)

The eigenvalues and eigenvectors decomposition of the matrix Kg will be calculated to obtain a KPCA representation of the nonlinear uncertain data [X]. The solution is given by the following eigensystem: Kg Pφg = nφg Pφg φ

(21) φ

φ

where g is the eigenvalues matrix of Kg and Pφg = [ p1 , . . . , pn ]T is the corresponding eigenvectors.

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Kernel functions that have been used successfully in the literature is the Radial Basis Functions (RBF):  2 K(xi , x j ) = exp(− xi − x j  /2δ 2 )

(22)

where 2δ 2 = c is the width of the gaussian kernel.

3 Fault Detection The PCA method models the correlations between the historical data when the process is in normal operation. A violation of the correlation indicates an unusual situation because the data do not maintain their normal relations. The quadratic prediction error is defined as the norm squared of the residual vector [10, 14], SPE(k) = x(k) − xˆ (k)2 =

m 

(ei (k))2

(23)

i=1

In fact, the interval [SPE] is defined by:   [SPE] ≡ SPE, SPE

(24)

SPE(k) = x(k) − xˆ (k)2

(25)

2 ˆ SPE(k) = x(k) − x(k)

(26)

where,

and,

The process is considered in normal operation if, SPE ≤ δ 2

(27)

where δ 2 is the detection threshold of the SPE.   δ 2 = (θ2 /θ1 ) χα2 θ12 /θ2 .

(28)

m m λi , θ2 = i=l+1 λi2 and λi is the eigenvalue of the covariance where θ1 = i=l+1 matrix. The KPCA based monitoring method is similar to that using the PCA, the square of prediction error (SPE) in the feature space F. The SPE indicates the extent to which each sample conforms to the KPCA model. In the feature space, the SPE is defined as [15]:

Fault Detection of Uncertain Systems Based on Interval Data Driven Approach n l     (x)2 = SPEφ = φ(x) − φ t2j − t2j j=1

809

(29)

j=1

where, n is the number of nonzero eigenvalues and: φT

t = Pl φ(x)

(30)

  The interval SPEφ , in the feature space, is defined as:

where,

and,

  SPEφ ≡ SPEφ , SPEφ

(31)

  (x)2 SPEφ (k) = φ(x) − φ

(32)

  (x)2 SPEφ (k) = φ(x) − φ

(33)

If it is considered that φ(x) is zero mean and normally distributed, the confidence limit for the SPE can be computed from its approximate distribution: S P E lim ≈ γ χh2

(34)

where γ = b/2a and h=2a 2/b, a and b are the estimated mean and variance of the SPE [11, 12].

4 Application to Tennessee Eastman Process The TEP was developed by Downs and Vogel of the Eastman Company to provide a realistic simulation for evaluating process control and monitoring methods. There are five major units in TEP simulation (Fig. 1) a reactor, separator, stripper, condenser, and a compressor. The process has 12 manipulated variables, 22 continuous process measurements, and 19 composition measurements sampled less frequently [10]. The collection of uncertain measurements of the different variables from the system to be monitored, in normal operation, makes it possible to obtain the interval data matrix. The TEP was run for one hour, and we collected 1000 samples from 22 measurements. Considering an uncertainty δxi of the order of 15% of the measurement variation range taken for each variable xi . Then we construct a new interval data matrix of the process, with δxi being the radius of the data intervals (shown in Fig. 2). In the modeling steps using MRKPCA and KPCA methods, the first 500 samples of X were utilized to build the KPCA and MRKPCA models, thus, the rest 500 samples are used for the test. In fact, the matrix of interval observations [X] can,

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Fig. 1 Tennessee eastman process Measurement lower bound

Measurement upper bound 70

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Reactor pressure

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20 Upper SPE Lower SPE Control limit 95%

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Fig. 4 Evolution of the MRKPCA SPE for interval data under normal operation

therefore, be estimated from the l selected principal components (l = 6) corresponding to the l largest eigenvalues of the covariance matrix. For the MRKPCA, the covariance matrix is constructed from the Eq. (19). In order to show that classical KPCA model is not appropriate to monitor uncertain processes, Fig. 3 shows the monitoring performances of the PCA method when the process is operating under normal condition. However, the false alarm rate provided is undesirable. In this case, the detection index SPE based KPCA shows that the system is faulty, knowing that the system works properly in this simulation part.

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SPE

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SPE

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Fig. 5 Evolution of the MRKPCA SPE for inaccurate acquired measurements Table 1 False Alarms Rate (FAR) of KPCA and MRKPCA methods Fault detection metric FAR(%) 95% FAR(%) 99% PCA SPE lower PCA SPE upper MRPCA SPE lower MRPCA SPE upper

37 34 09 11

20 22 5 6

Figure 4 shows that the SPE based MRKPCA method is more robust to false alarms due to uncertainties. Figure 5 shows the SPE index based MRPCA of inaccurate acquired measurements, varying within the initial data intervals, in fault-free case and faulty case. It is well illustrated that the robust MRPCA monitoring model is more robust to uncertainties and sensitive to faulty measurement. The false alarms rate of both MRPCA and PCA methods is shown in Table 1. A fault affecting the variable x2 is simulated between the sample 700 and the sample 1000 with the magnitude of 40% of the variation range of the D feed sensor. The control limits are calculated at the confidence level of 95%. After time point k = 700, it was found that the monitoring index, MRPCA-SPE, continuously exceeds its threshold, which indicates a fault has been successfully detected.

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5 Conclusion In this paper, a new algorithm of kernel PCA (MRKPCA), based on midpoints and radii, is proposed. Our study shows the advantages of the proposed method, in comparison with the classical one, to model interval data and their impact on the fault detection and isolation (FDI) steps. The Simulation results show that the proposed method is more robust to uncertainty compared to the classical KPCA method.

References 1. Chena Yu-Wang, Yang Jian-Bo, Pan Chang-Chun, Dong-Ling Xu, Zhou Zhi-Jie (2015) Identification of uncertain nonlinear systems: constructing belief rule-based models. Knowl-Based Syst 73:124–133 2. Lauro NC, Verde R, Irpino A (2008) Principal component analysis of symbolic data described by intervals. In book: Symbolic data analysis and the SODAS software, pp 279–311 3. Kramer MA (1991) Nonlinear principal component analysis using autoassociative neural networks. AICHE J 37(2):233–243 4. Scholkopf B, Smola A, Muller K-R (1998) Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput 10:1299–1319 5. Palumbo F, Lauro NC (2003) A PCA for interval-valued data based on midpoints and radii. In: Yanai H, Okada A, Shigemasu K, Kano Y, Meulman J (eds) New developments in psychometrics, Tokyo 6. Le-Rademacher J, Billard L (2012) Symbolic covariance principal component analysis and visualization for interval-valued data. J Comput Graph Stat 21(2):413–432 7. Chakour C, Benyounes A, Boudiaf M (2018) Diagnosis of uncertain nonlinear systems using interval kernel principal components analysis: application to a weather station. ISA Trans 83:126–141 8. Lauro NC, Verde R, Irpino A (2008) Principal component analysis of symbolic data described by intervals. In: Symbolic data analysis and the SODAS software, pp 279–311 9. Chakour C, Harkat MF, Djeghaba M (2015) New adaptive kernel principal component analysis for nonlinear dynamic process monitoring. Appl Math 9(4):1833–1845 10. Chakour C, Harkat M-F, Djeghaba M (2015) Neuronal principal component analysis for nonlinear time-varying processes monitoring. In: Safe process 9th IFAC symposium on fault detection, supervision and safety of technical processes. September 2–4, Paris 11. Box G (1954) Some theorems on quadratic forms applied in the study of analysis of variance problems, I. Effect of inequality of variance in the one-way classification. Ann Math Stat 25:290–302 12. Joe Qin S (2003) Statistical process monitoring: basics and beyond. J Chemom 17:480–502 13. Dunia R, Qin SJ, Edgar TF, McAvoy TJ (1996) Identification of faulty sensors using principal component analysis. AICHE J 42(10):2797–2812 14. Dunia R, Qin SJ (1998) Joint diagnosis of process and sensor faults using principal component analysis. Control Eng Pract 6(4):457–469 15. Alcala Perez CF (2011) Fault diagnosis with reconstruction-based contribution for statistical process monitoring. Thesis of the University of Southern California

Discrimination of Unbalanced Supply and Stator Interturn Faults in Induction Machines D. Kouchih, R. Hachelaf, and N. Boumalha

Abstract This paper describes an efficient approach which is used for the discrimination of unbalanced power supply and stator interturn fault in the star connected induction motors (IM) using space harmonic components. For this purpose, an analytical method for the modeling of asymmetrical star connected IM is presented using coupled circuits theory and taking into account the increase of the common point voltage for unbalanced conditions. The equations which describe the state model as well as the calculation of machine inductances are presented. The calculation of inductances is based on the magnetomotive force (MMF) distribution through the machine air-gap. It will be deduced that the sideband components around the space harmonics contain useful information and they can be considered as a good interesting tool for the discrimination of the stator asymmetries. Simulation results show the consistency and the applicability of the proposed approach. Keywords Component · Discrimination of faults · Unbalanced conditions · Interturn faults · Induction machines

1 Introduction From a number of surveys, the stator faults of IM drives account approximately 40% of all failures and induce severe impacts on performances of induction motors [1–4]. Faults detection is really important for improved system design, protection, and conception of fault tolerant control. For this purpose, many techniques have D. Kouchih (B) · R. Hachelaf Department of Electronics, University Saad Dahlab, Blida, Algeria e-mail: [email protected] R. Hachelaf e-mail: [email protected] N. Boumalha Department of Automation, National Polytechnic School, Algiers, Algeria e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_56

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been proposed in the literature [5, 6]. The well-known method is based on the Motor Current Signature Analysis (MCSA) [7, 8]. The classical MCSA has been widely used due to its inherent advantages. In the literature, two kinds of harmonic components in the stator current spectrum are proposed as faults indicators. The most references focus on the sideband components around the fundamental of the stator current. Others references use the sideband components around the space harmonics [9–11]. The principal disadvantage of the MCSA technique is to clearly discriminate the unbalanced power supply from stator faults and the saturation of magnetic material where all cases exhibit similar kind of current signatures [12, 13]. In this study, we are interested to the analysis of the common point voltage of the star connected induction machines. For the discrimination of stator asymmetries, it will be concluded that the sideband components around space harmonics are considered as an efficient tool. For the modeling of asymmetrical IM and the calculation of the common point voltage, a new approach is adopted using coupled circuits theory. This approach extracts unbalance signatures on electromagnetic torque, stator current, rotor current and common point voltage. Simulation results show the consistency and the applicability of the proposed approach.

2 Calculation of Machine Inductances 2.1 Stator Inductances The spatial harmonics are caused according to: (i) The windings distribution in a finite number of slots (distribution harmonics). (ii) The slots opening (slotting effect) that alter the effective length of the air gap (permeance harmonics) [14, 15]. The effect of permeance harmonics is neglected in comparison with distribution harmonics. To illustrate the calculation of the machine inductances, it is appropriate to consider the elementary 2-poles, star connected three phase IM. The stator windings are concentric with consequent poles. Considering initially only one coil with Nj turns of a generic phase « j » and followed by current ij . The MMF distribution, through the machine air-gap, is illustrated by Fig. 1. Fig. 1 MMF distribution of a generic coil

Discrimination of Unbalanced Supply and Stator Interturn …

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The harmonic mutual inductance between the two phases « i » and « j » is expressed L ji h

qj qi  π  π 4 μ0 r L 1 = cosh(α j − αi ) N j sin(hβ j ) Ni sin(hβi ). π g h2 2 2 j=1 i=1

(1)

As a result, the total mutual inductance will be given by qj qi ∞  π  π 4 μ0 r L  1 Li j = cosh(α j − αi ) N j sin(hβ j ) Ni sin(hβi ). π g h=1 h 2 2 2 j=1 i=1

(2)

For i = j, we obtain the magnetizing inductance. It’s expressed by L ms h αs αj μ0 Ni L r

qi qi ∞ π  π 4 μ0 r L  1  = Ni sin(hβ j ) Ni sin(hβi ). π g h=1 h 2 i=1 2 i=1 2

(3)

Harmonic order Angular position which locates any point along the circumference of the air-gap from a fixed reference. Value of αs in through the center of coil. is the air magnetic permeability and g is the air-gap length. Number of turns for coil b; Magnetic length of the rotor; Average radius of the air-gap.

The harmonic mutual inductance between phase « j » and the rotor loop of order « k » is L jr

qj ∞  π 4 μ0 r L  1 = cosh(θk − α j − δ) N j sin(hβ j ). π g h=1 h 2 2 j=1

(4)

θk Angular position of the rotor loop of order « k»; αr Rotor loop pitch

δ=

αr . 2

θk = θr + (k − 1)αr . θr Angular position of the first rotor loop.

(5) (6)

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3 Machine Equations 3.1 Stator Equations For unbalanced conditions, we employ the known line to line voltages as inputs in the state model of the machine. The stator voltage equations can be expressed by: [u s ] = [Rs ][i s ] +

d[Φs ] . dt

(7)

Using phase voltages, the line to line voltages are expressed: [u s ] = [T ][vn ].

(8)

[u s ] = [u ab u bc u ca ]T [vn ] = [van vbn vcn ]T [i s ] = [i as i bs i cs ]T uab , ubc and uca are the line to line voltages; van , vbn and vcn are the phase voltages of the network; ias , ibs and ics are the line currents. ⎡

⎤ +1 −1 0 [T ] = ⎣ 0 1 −1 ⎦. −1 0 +1

(9)

The matrix of resistances is expressed ⎡

⎤ ras −r bs 0 [Rs] = ⎣ 0 r bs −r cs ⎦. −ras 0 r cs

(10)

ras , rbs and rcs are the resistances of stator windings. The stator flux is defined by [Φs ] = [T ][φs ].

(11)

[φs ] = [φas φbs φcs ]T [φs ] = [L sr ][ir ] + [L ss ][i s ]. [L sr ] is the matrix of stator-rotor mutual inductances.

(12)

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3.2 Rotor Equations The rotor is modeled by an equivalent circuit containing q magnetically coupled circuits. Each rotor loop consists of two adjacent bars and the two portions of the end ring connect them. The rotor currents comprise of the q rotor loop currents plus a circulating current in one of the end rings, ie . In a motor with a complete end rings, ie would be equal zero. The rotor voltage equation is expressed: [0] = [Rr ][ir ] +

d[φr ] . dt

(13)

With [φr ] = [L r s ][i s ] + [L rr ][ir ].

(14)

[L r s ] = [L sr ]T

(15)

[ir ] is the rotor vector current; [L r s ] is the matrix of rotor-stator mutual inductances; [Rr ] is the n by n symmetric matrix of the rotor resistances. In the case of healthy rotor, it can be demonstrated that [16, 17]: ⎡

R0 ⎢ −r ⎢ b ⎢ . ⎢ ⎢ [Rr ] = ⎢ . ⎢ ⎢ . ⎢ ⎣ 0 −rb

−rb 0 . . 0 R0 −rb . . 0 . . . . . . . . . . . . . . . 0 . . −rb R0 0 . . . −rb

⎤ −rb 0 ⎥ ⎥ . ⎥ ⎥ ⎥ . ⎥. ⎥ . ⎥ ⎥ −rb ⎦ R0

(16)

With R0 = 2(rb + re ).

(17)

re is the end ring segment resistance and rb is the total bar resistance. [L rr ] is the qxq symmetric matrix of the rotor inductances. In the case of healthy rotor, it can be verified that [16, 17]:

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⎡

L kk + L 0 L km − lb L km ⎢ L −l L + L L −l b kk 0 km b ⎢ km ⎢ . . . ⎢ ⎢ [L rr ] = ⎢ . . . ⎢ ⎢ . ⎢ ⎣ L km L km . L km − lb L km .

. . . .

L km L km . .

L km L km . .

L km − lb L km . .

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥. (18) ⎥ ⎥ ⎥ . L km − lb L kk + L 0 L km − lb ⎦ . . L km − lb L kk + L 0

With L 0 = 2(lb + le ).

(19)

Lkm : Mutual inductance between two healthy loops of orders « k » and « m » . Lkk is the magnetizing inductance of each healthy loop, lb is the rotor bar leakage inductance and le is the rotor end ring leakage inductance. Applying the same method used for the calculation of rotor inductances, we obtain: L kk =

μ0 rl αr (1 − αr /2π ). g

(20)

μ0 rl −(αr )2 /2π . g

(21)

L km =

3.3 Electromagnetic Torque The mechanical equation is J

dΩ = Te − Tl − f v Ω. dt

(22)

J is the inertia of the rotor and the connected load, Te the electromagnetic torque, Tl the load torque, Ω the mechanical angular speed and f v is the viscose friction coefficient. The electromagnetic torque is expressed [18–20] Te =

∂[L sr ] P [i s ]t [ir ]. 2 ∂θ

(23)

P is the number of poles pairs and θ is the electrical angular displacement of the rotor.

Discrimination of Unbalanced Supply and Stator Interturn …

821

3.4 Calculation of the Common Point Voltage The machine common point voltage can be calculated using the stator voltages and phase voltages of the network (Fig. 2). Ts is the sampling time. The same voltages are expressed in terms of phase voltages of the network and the common point voltage by ⎧ ⎪ ⎨ vao = van − von vbo = vbn − von . ⎪ ⎩ vco = vcn − von

(24)

The common point voltage is expressed:    v jn − v jo von =

j=a,b,c

3

.

(25)

« n » is the reference neutral point of the network. The stator voltages are calculated using the numerical derivation of the stator flux: ⎧ Δϕas ⎪ ⎪ ⎪ vao = ras i as + T ⎪ s ⎪ ⎪ ⎨ Δϕbs vbo = rbs i bs + . ⎪ Ts ⎪ ⎪ ⎪ ⎪ Δϕcs ⎪ ⎩ vco = rcs i cs + Ts

(26)

Ts is the sampling time. Fig. 2 Star connected IM

(a) Common point (o)

(b)

von

(c) Network neutral point (n)

822

D. Kouchih et al.

3.5 Modeling of Interturn Fault To illustrate this interturn fault, we consider the following example, where the coil U-V has four turns and occupied two slots. When, a short circuit occurred between the contact points c1 and c2 , three turns in series are obtained. In addition, a new shortcircuited turn which we call the short circuited phase D is created and magnetically coupled with all the other circuits (Fig. 3) Generally, when N j f short-circuited turns are created, the fault current reaches high values causing the destruction of the short-circuited turns and thus their setting off during a short transitional period. Consequently, the interturn fault causes the reduction of the number of turns in faulty phase and the alteration of the magnetizing and mutual inductances. To characterize the Interturn fault, we define the fault factor: k=

Njf . Ns

(27)

In interturn fault condition, the effective turns by coil can be expressed in terms of k. Consequently, all the machine inductances will be updated using the effective turns.

4 Simulation Results The proposed state model has been implemented in the MATLAB environment. The induction machine used in this study is three phases 4 kW, 230/400 V, 50 Hz, 14.2/8.2 A, 2840 rpm, 2 poles, 30 rotor bars. Fig. 3 Short-circuited coil

C1 C2

U

V

Discrimination of Unbalanced Supply and Stator Interturn … 10

823

2

Current ias (A)

fs

10

10

10

0

fsh1

fsh2

-2

-4

0

500

1000

1500

2000

Frequency (Hz)

Fig. 4 Spectrum analysis of the stator current ias in healthy condition

4.1 Healthy Condition The induction machine is supplied by balanced three phase power supply 230/400 V, 50 Hz. To show the impact of space harmonics effect, we take h = 55. The following figure shows the spectrum analysis of the stator current ias (Fig. 4). In healthy conditions, we note the presence of specific harmonics through the spectrum analysis of the stator current. These harmonics vary particularly with the slip of the machine. For the stator current, the space harmonics are characterized by the frequency [15]  f sh =

 λq (1 − s) ± 1 f s . p

(28)

λ = 1, 2, 3, . . . ., q is the number of rotor bars. For λ = 1 and s = 5.33%, the first space harmonics are: fsh1 = 1370 Hz and fsh2 = 1470 Hz.

4.2 Unbalanced Supply Using the National Electrical Manufacturer Association (NEMA) definition of voltage unbalance [19, 20], we consider a starting up with rated load under balanced power supply of the studied IM. At time t = 1 s, an unbalance of +15% is occurred on phases « b » and « c » . Te machine characteristics are shown as follows (Figs. 5, 6, 7, 8 and 9). When IM operates under unbalanced power supply, a backward rotating field will be produced. The interaction of the backward rotating field with the forward one produces pulsating electromagnetic torque, decrease and ripple in speed. The

824

D. Kouchih et al. 100

Torque (N.m)

80 60 40 20 0 -20 -40

0

0.5

1

1.5

1

1.5

Time (s)

Fig. 5 Electromagnetic torque for unbalanced supply

Current ias (A)

100

50

0

-50

-100 0

0.5

Time(s)

100

100

50

50

Current ics (A)

Current ibs (A)

Fig. 6 Stator current ias for unbalanced supply

0

-50

-100

0

-50

-100 0

0.5

1

1.5

Time (s)

Fig. 7 Stator current ias and ibs for unbalanced supply

0

0.5 Time (s)

1

1.5

Discrimination of Unbalanced Supply and Stator Interturn … x 10

Rotor loop current (A)

1

825

4

0.5 0 -0.5 -1

0

0.5

1

1.5

Time (s)

Common Point Voltage (V)

Fig. 8 Rotor loop current for unbalanced supply 30 20 10 0 -10 -20 -30

0

0.5

1

1.5

Time (s)

Common Point Voltage (V)

(a) 10

10

10

10

2

0

-2

-4

0

500

1000

1500

2000

Frequency (Hz)

(b) Fig. 9 Common point voltage for unbalanced supply: a common point voltage and b spectrum analysis of common point voltage

826

D. Kouchih et al.

common point voltage increases significantly with only the fundamental frequency. All these characteristics are traduced by severe impacts such as vibrations, noise and overheating of the machine, so a progressive destruction of this last.

4.3 Interturn Fault Condition A short circuit of 20% is occurred on phase a. The induction machine is supplied by balanced three phase power supply 230/400 V, 50 Hz. We simulated a starting up mode with rated load. The machine characteristics are represented as follows (Figs. 10, 11, 12, 13 and 14). For interturn fault, the IM is subject to oscillating torque, increase of the common point voltage, decrease and ripple speed. Therefore, the interturn fault causes severe impacts such as vibrations, noise and overheating of the machine. The common point voltage increases significantly with the fundamental frequency and sideband components at ±2 f s around the space harmonics as indicated. 100

Torque (N.m)

80 60 40 20 0 -20 -40

0

0.5

1

1.5

1

1.5

Time (s)

Fig. 10 Electromagnetic torque for interturn fault

Current ias (A)

100

50

0

-50

-100

0

0.5

Time (s)

Fig. 11 Stator current ias for interturn fault

Discrimination of Unbalanced Supply and Stator Interturn … 100

Current ics (A)

100

Current ibs (A)

827

50

0

-50

50

0

-50

-100

-100 0

0.5

1

0

1.5

0.5

1

1.5

Time (s)

Time (s)

Fig. 12 Stator current ias and ibs for interturn fault

Rotor loop current (A)

1

x 10

0.5

0

-0.5

-1

0

0.5

1

1.5

Time (s)

Fig. 13 Rotor loop current for interturn fault

5 Conclusion In this paper, a new approach for the discrimination of the stator interturn fault and unbalanced power supply in star connected induction motors has been developed using space harmonic components and the common point voltage. An analytical method for the stator asymmetry is presented using coupled circuits theory. With this method, different characteristics of unbalanced IM can be highlighted: pulsating torque; stator and rotor currents, common point voltage and etc. For unbalanced supply, the common point voltage is characterized only by its fundamental frequency. The sideband components around the space harmonics at high order are considered as good indicators of the interturn fault. Simulation results show the consistency and the applicability of the proposed approach for the discrimination of stator asymmetries. Extensive experimental studies are necessary to full assess usefulness of the proposed approach. This work must be performed to take into account the saturation

D. Kouchih et al. Common Point Voltage (V)

828 50

0

-50 0

0.5

1

1.5

Time (s)

Common Point Voltage (V)

(a) 10

2

fs fsh2

fsh1 10

10

10

0

fsh1-2fs

fsh2+2fs

-2

-4

0

500

1000

1500

2000

Frequency (Hz)

(b) Fig. 14 Common point voltage for interturn fault: a common point voltage and b spectrum analysis of common point voltage

of magnetic material, skin effect, skewed rotor, and slotting effects in the calculation of the machine inductances. Simulated Machine Parameters [16] Stator phase resistance Rotor phase resistance Effective air-gap Stack length Rotor radius Stator leakage inductance Rotor leakage inductance Drive inertia Friction coefficient Stator phase turns Rotor bar resistance Rotor end ring segment resistance Rotor bar leakage inductance End ring leakage inductance

rs = 1.5950 rr = 1.3053 g = 0.35 mm L = 125 mm r = 37.35 mm Lls = 0.0040 H Llr = 0.0033 H J = 0.045 kg.m2 fv = 0.0038 kg.m2 .s−1 Ns = 124 rb = 3.04E − 4 re = 8.75E − 7 lb = 5.16E − 7 H le = 1.59E − 9 H

Discrimination of Unbalanced Supply and Stator Interturn …

829

References 1. Bonnett AH, Soukup GC (1988) Analysis of rotor failures in squirrel-cage induction motors. IEEE Trans Indus Appl 24(6):1124–1130. https://doi.org/10.1109/28.17488 2. Tolyat HA, Lipo TA (1995) Transient analysis of cage induction machines under stator, rotor bar and end ring faults. IEEE Trans Energy Conv 10(2):241–247. https://doi.org/10.1109/60. 391888 3. Stefan G, Jose MA, Bin L, Thomas GH (2008) A survey on testing and monitoring methods for stator insulation systems of low-voltage induction machines focusing on turns insulation problems. IEEE Trans Indus Electron 55(12):4127–4136 https://doi.org/10.1109/tie.2008.200 4665 4. Pinjia Z, Yi D, Thomas GH, Bin L (2011) A Survey of condition monitoring and protection methods for medium-voltage induction motors. IEEE Trans Ind Appl 47(1):34–46. https://doi. org/10.1109/tia.2010.2090839 5. Rangarajan MT, Sang BL, Greg CS, Gerald BK, Jiyoon YA, Thomas GH (2007) Survey of methods for detection of stator-related faults in induction machines. IEEE Trans Industry Appl 43(4):920–933. https://doi.org/10.1109/tia.2007.900448 6. Arfat S, Yadava GS, Bhim S (2005) A review of stator fault monitoring techniques of induction motors. IEEE Trans Energy Conv 20(1):106–114. https://doi.org/10.1109/tec.2004.837304 7. Benbouzid MH, Vieira M, Theys C (1999) Induction motors faults detection and localization using stator current advanced signal processing techniques. IEEE Trans Power Elec 14(1):14– 22. https://doi.org/10.1109/63.737588 8. Henao H, Capolino GA, Razik H (2003) Analytical approach of the stator current frequency harmonics computation for detection of induction machine rotor faults. In: 4th IEEE International symposium on diagnosis for electric machines, power electronics and drives, Atlanta. https://doi.org/10.1109/DEMPED.2003.1234583, pp. 259–264 9. Fudeh HR, Ong CM (1983) Modeling and analysis of induction machines containing space harmonics, part III: three-phase cage rotor induction machines. IEEE Trans Apparatus Syst PAS-102(8):2621–2628. https://doi.org/10.1109/tpas.1983.317783 10. Neto LM, Camacho JR, Salerno CH, Alvarenga BP (1999) Analysis of a three phase induction machine including space and time harmonic effects: the A, B, C reference frame. IEEE Trans Energy Conv 14(1):80–85. https://doi.org/10.1109/60.749151 11. Khezzar A, Kaikaa MY, Oumaamar MEK, Boucherma M, Razik H (2009) On the use of slot harmonics as a potential indicator of rotor bar breakage in the induction machine. IEEE Trans Ind Electron 56(11):4592–4605. https://doi.org/10.1109/tie.2009.2030819 12. Lashkari N, Poshtan J (2015) Detection and discrimination of stator intertum fault and unbalanced supply voltage fault in induction motor using neural network. In: The 6th international power electronics drive systems and technologies conference, Shahid Beheshti University, Tehran, Iran. https://doi.org/10.1109/PEDSTC.2015.7093287, pp. 275–280 13. Refaat SS, Abu-Rub H, Saad MS, Aboul-Zahab EM, Iqbal A (2012) Detection, Diagnoses and Discrimination of Stator Turn to Turn Fault and Unbalanced Supply Voltage Fault for Three Phase Induction Motors. In: 2012 International conference on power and energy, Kota Kinabalu Sabah, Malaysia. https://doi.org/10.1109/pecon.2012.6450347, pp. 910–915 14. Nandi S (2004) Modeling of induction machines including stator and rotor slot effects. IEEE Trans Industry Appl 40(4):1058–1065. https://doi.org/10.1109/tia.2004.830764 15. Kouchih D, Boumalha N, Tadjine M, Boucherit MS (2016) New approach for the modeling of induction machines operating under unbalanced power system. Int Trans Electric Energy Syst 26(9):1832–1846. https://doi.org/10.1002/etep.2171 16. Houdouin G, Barakat G, Dakyo B, Destobbeleer, E (2003) A winding function theory based global method for the simulation of faulty induction machines. In: IEEE electric machines and drives conference, vol. 1, Madison, Wisconsin, USA. https://doi.org/10.1109/iemdc.2003.121 1279, pp 297–303

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17. Xiaogang L, Yuefeng L, Toliyat HA, El-Antably A, Lipo TA (1995) Multiple coupled circuit modeling of induction machines. IEEE Trans Industry Appl 31(2):311–318. https://doi.org/10. 1109/28.370279 18. Krause PC (1987) Analysis of electric machinery. McGraw-Hill Book Company 19. Kandli S, Amol B (2010) Performance analysis of 3-phase asynchronous motor under various voltage conditions. Archiv Appl Sci Res 2(2):380–387 20. Sobhan J (2013) Performance Evaluation of Three-Phase Induction Motor Fed by unbalanced voltage using complex voltage unbalance factor. J Eng Interdiscip Res 64(1):31–37

Impact of the Stator Winding Topology on the Fault Harmonic Components in Induction Motors Seddik Tabet, Adel ghoggal, and Salah Eddine Zouzou

Abstract This paper deals with analysis of the impact of the type of the induction motor’s (IM) stator windings on the fault harmonic components (FHC). The model is based on the know modified winding function approach (MWFA) to evaluate the machine inductances. Both broken rotor bars and air-gap eccentricity faults are considered. The numerical investigations and simulations set conclusions as far as the winding topology impact is concerned. Keywords SCIMs · MWFA · PSH · Slot skewing · FFT · Distribution and winding function · BRB · Eccentricity · Concentric winding at a full step layer · Total-step two-ply nested winding · Concentric winding with two short-pitch layers · Chain winding · Computer simulation

1 Introduction Three-Phase IMs are widely used for the many industry applications such as mills, electric heaters, conveyor belts, air conditioners, pumps, etc. This is because of its robust construction and easy maintenance. For this reason, condition monitoring techniques to detect incipient faults and prevent IMs from serious damages are particular concern. The published researches on the fault diagnosis show that the majority of these studies have been focused on detection of the electrical faults, including the stator windings and the rotor bars faults [1]. As classified, broken rotor bars (BRB) have various impacts on the motor performance. This paper provides an analytical model of IM able to take into account various severity degrees of BRB fault. To achieve such a task, the multiple coupled-circuit model (MCCM) is applied, the rotor bars skew effect is taken into account by using 2D winding function theory. S. Tabet (B) · A. ghoggal · S. E. Zouzou Electrical Engineering Laboratory of Biskra LGEB, University of Biskra, Biskra, Algeria e-mail: [email protected] A. ghoggal e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_57

831

832 Table 1 The meaning of letters and symbols

S. Tabet et al. FFT

Fast Fourier transformer

SCIMs

Squirrel cage induction motors

MWFA

Modified winding function approach

BRB

Broken rotor bars

SE

Static Eccentricity

DE

Dynamic Eccentricity

ME

Mixed eccentricity

PSH

Principal slot harmonic

S

Single layer winding (concentric winding)

T

Double-layer lap winding (full pitch)

R

Double-layer concentric winding

C

Chain winding

E

Stator slot

FHC

Fault harmonic components

It is known that the rotor slot number and the magneto-motive force (MMF) have a great impact on the inductances values and shapes and it on the inductances [2]. In this work, a 2-D model analytical of IM is called up the comparison between various winding types and their effect on the FHC, simulation results confirm the paper out comes (Table 1).

2 Analytical Model for MWFA We extend the approaches in [3] by using the axial dimension. Therefore, the Gauss’s law, the integral of the magnetic flux density on closed surface S of a cylindrical volume defined in comparison to the average radius of the air-gap r is null.  Bds = 0

(1)

s

By defining, at any of coordinates (ϕ, z), the effective air-gap function g the magneto motive force F, the magnetic field intensity H, such as B = μ0 H and H = Fg , the Eq. (1) can be written as μ0 r

2π  l 0

0

F(ϕ, z, θr ) dz dϕ = 0 g(ϕ, z, θr )

Of the Amper’s law, where l is the length of the air-gap.

(2)

Impact of the Stator Winding Topology on the Fault Harmonic …

833



 H (ϕ, z, θr )dl =

J ds

(3)

Ω

abcda

J the current density and  is a surface enclosed by the closed path abcda. According to the MMF and the number of turns enclosed by the closed path abcda and traversed by the same current i, (3) becomes Fab (0, 0, θr ) + Fbc + Fcd (ϕ, z, θr ) + Fda = n(ϕ, z, θr )i

(4)

By taking into account the permeability of the iron is infinity, Fbc and Fda are null. Where n(ϕ, z, θr ) is called the 2-D spatial winding distribution [4]. The Eq. (4) gives Fcd (ϕ, z, θr ) = n(ϕ, z, θr )i − Fab (0, 0, θr )

(5)

We enter the average value of the reverse air gap function  g −1 (ϕ, z, θr )  with  g −1 (ϕ, z, θr ) =

1 2π

2π 0

1 { l

l

g −1 (ϕ, z, θr )dz}dϕ

(6)

1

And using (2) and (5), the expressions give Fcd (ϕ, z, θr ) Fcd (ϕ, z, θr ) = n(ϕ, z, θr ) −

1 n(ϕ, z, θr ) g −1 (ϕ, z, θr )idzdϕ 2πlg −1 (ϕ, z, θr ) (7)

By dividing the members of (7) by the current i, we can be obtained the 2-D winding function 1 N (ϕ, z, θr ) = n(ϕ, z, θr ) − 1 2πlg (ϕ, z, θr )

2π  l n(ϕ, z, θr ) g 1 (ϕ, z, θr )dzdϕ 0

0

(8) It is to be noticed that this new expression does not hold any restriction as for the axial uniformity, in particular in term of skewed slots and axial air-gap non uniformity [5].

834

S. Tabet et al.

3 The Type of Winding The SCIMs studied in this paper is a three-phase, 4-pole motor. The different structure of the stator winding is presented in the Figs. 1, 2, 3 and 4. Where only the phase A is considered.

Fig. 1 Single layer winding (concentric winding)of the stator phase A

Fig. 2 Double-layer lap winding (full pitch) of the stator phase A

Fig. 3 Double-layer concentric winding of the stator phase A

Fig. 4 Chain winding of the stator phase A

Impact of the Stator Winding Topology on the Fault Harmonic …

835

A. Single layer winding (concentric winding). The three planes winding generally used for a q pair of notches per pole and per phase, which allows to divide a group of reels, belonging to a phase zone. This winding was used in some engines [6]. B. double-layer lap winding (full pitch). On the winding diagram in Fig. 2, we can clearly see the phase zones that make up the half-phases of the winding. all six groups of connected reels inscribed in the same line and forming the half-phase of the winding have resulting FEM equal in value and are in phase; for this reason the given winding allows to connect the half-phases not only in series as shown in Fig. 2, but also in two and in four parallel groups [6]. C. double-layer concentric winding. The following stator winding arrangement. D. chain winding. The pattern of such windings shown in Fig. 4. Winding by the shape of its frontal parts is called chain winding [6].

4 Simulation Results A. The distribution and the winding function The figures below represent the winding and distribution function of the first stator phase A. The machine is supposed to be symmetrical. The winding consists of four coils per pole and per phase and a beam of w = 25 turns per slot. The calculation gives us an average value of the distribution function equal to 1.5w in (a), (b), (d), and equal to 3w in (c) (Fig. 5). The inductances are the key points in a successful simulation of induction machines. These inductances are under the influences of different factors, such as winding distribution, stator and rotor slots, skewed slots of rotor, and asymmetries caused by eccentricities faults. The Figs. 6 and 8 illustrate the impact of different winding upon the inductances of the SCIMs. There are three types of eccentricity fault: SE, DE and ME faults. Eccentricity fault varies the air gap length and therefore changes the inductances of the motor. In the SE fault, the air gap length is non-uniform but the minimum angular position is constant. This fault happens because of the careless in fixing the rotor. Oval rotor also leads to the SE fault. In the DE fault, the minimum air gap is a function of the rotor position and rotates around the rotor. This may be arisen from misalignment, shaft bending, bearing erosion or mechanical resonance at critical speed. This type of eccentricity fault causes UMP in one direction and this result in shaft bending and bearing tear. Presence of both SE and DE fault leads to the ME [7]. The Figs. 7 and 9 illustrate the impact of different winding upon the inductances of the SCIMs. The four cases are considered with taking into account the slot opening and the rotor bars

836

S. Tabet et al. (b)

(a)

2π

2π

φ

(c)

φ

(d)

2π

2π

φ

φ (a)

1.5W

(b)

1.5W 2π

2π

φ

φ

-1.5W

-1.5W (d)

(c)

3W

1.5W 2π

2π

φ

φ -1.5W

-3W

Fig. 5 Turns functions n(φ) and winding functions N(φ) of stator phase a and rotor loop 1 x 10

-4

4

(a)

2

L (mH)

L (mH)

4

0

-4

(b)

2 0 -2

-2 -4

x 10

4

2

0

-4

6

0

2

x 10

-4

4

(c)

2

L (mH)

L (mH)

4

0

6

x 10

-4

(d)

2 0 -2

-2 -4

4

teta (rad)

teta (rad)

0

4

2

teta (rad)

6

-4

0

2

4

6

teta (rad)

Fig. 6 a The inductance of Single layer winding (concentric winding), b double-layer lap winding (full pitch), c the inductance of double-layer concentric winding, d the inductance of chain winding

Impact of the Stator Winding Topology on the Fault Harmonic … x 10

-4

4

(a)

2

L (mH)

L (mH)

4

0 -2 -4

x 10

837

-4

(b)

2 0 -2

0

2

4

-4

6

0

2

teta (rad) x 10

-4

4

(c) L (mH)

L (mH)

10 5 0 -5

4

6

teta (rad) x 10

-4

(d)

2 0 -2

0

4

2

6

teta (rad)

-4

0

2

4

6

teta (rad)

Fig. 7 Mutual inductance between rotor loop 1 and stator phases in different winding with 20% SE and 20% DE a Single layer winding (concentric winding), b double-layer lap winding (full pitch), c double-layer concentric winding, d the chain winding

skewing. We noticed that inductance (c) is the closest to the sinusoidal (Figs. 8 and 9). It is now clear that the presence of PSH in line current of three-phase IM is primarily dependent on the number of rotor slots and number of fundamental pole pairs of the machine. It was also shown in previous works that only PSH associated to a nontriplen pole pair can be seen ideally with a balanced power supply and in symmetrical conditions. For more learning about this problem, the effect of pole pair and rotor slot numbers on the presence of these harmonics under healthy and eccentric conditions was mainly studied in [8] (Fig. 10). B. Bar faults in different types of winding In this context, the analysis of the simulation tests consists in particular of inspecting the presence of lines characterizing default in the stator current spectrum. Since the stator coupling adopted is stationary with free neutral. The Figs. 11, 12, 13 and 14 represent the broken bar default. C. Evolution of default to different winding By comparing the types of winding we conclude in broken rotor bars that there is a small difference in the amplitude, the tendency and harmonic of default. And in mixed eccentricity we notice that there is a difference in amplitude that we find the concentric two-ply shortned winding has a weak harmonic compared to the other

S. Tabet et al. 1

x 10

-3

(a)

0.5 0 -0.5 -1

0

2

4

dL/dteta (mH/rad)

dL/dteta (mH/rad)

838

6

x 10

1

-3

(b)

0.5 0 -0.5 -1

x 10

-4

(c)

2 0 -2 -4

0

2

6

teta (rad)

4

6

dL/dteta (mH/rad)

dL/dteta(mH/rad)

4

4

2

0

teta(rad) 1

x 10

-3

(d)

0.5 0 -0.5 -1

0

2

4

6

teta (rad)

teta (rad)

0.05

(a)

0 -0.05

0

2

4

6

dL/dteta (mH/rad)

dL/dteta (mH/rad)

Fig. 8 a The inductance derivative of Single layer winding (concentric winding), b the inductance derivative of double-layer lap winding (full pitch), c the inductance derivative of double-layer concentric winding, d the inductance derivative of chain winding

0.05

(b)

0 -0.05

0

(c)

0 -0.05

0

2

4

teta (rad)

6

dL/dteta (mH/rad)

dL/dteta (mH/rad)

0.05

2

4

6

teta (rad)

teta (rad) 0.05

(d)

0 -0.05

0

2

4

6

teta (rad)

Fig. 9 Mutual inductance derivative between rotor loop 1 and stator phases in different winding a Single layer winding (concentric winding), b double-layer lap winding (full pitch), c double-layer concentric winding, (d) the chain winding with 20% SE and 20% DE

windings and we also note that the amplitude of the harmonic at 20% SE and 30% DE is large compared to 40% SE and 10% DE (Figs. 15, 16).

0

(a)

-100 -200

1000

0

2000

3000

4000

Amplitude (dB)

Amplitude (dB)

Impact of the Stator Winding Topology on the Fault Harmonic … 0

(b)

-100 -200

1000

0

0

1000

2000

3000

4000

Amplitude (dB)

Amplitude (dB)

(c)

-100 -200

2000

3000

4000

Fréquence (Hz)

Fréquence (Hz) 0

839

0

(d)

-100 -200

0

1000

2000

3000

4000

Fréquence (Hz)

Fréquence(Hz)

0

(a)

-44.7382

-45.0724

-50 -100

0

50

100

Amplitude (dB)

Amplitude (dB)

Fig. 10 Simulated normalized spectra of the stator current at steady state with different winding a Single layer winding (concentric winding), b double-layer lap winding (full pitch), c double-layer concentric winding, d chain winding

0

(b)

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0

-45.9572

-44.8983

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0

50

Fréquency (Hz)

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100

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Amplitude (dB)

(c)

-45.7759

50

Fréquency (Hz) 0

-45.6603

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0

(d)

-43.1445

-44.0548

-50 -100

0

50

100

Fréquency (Hz)

Fig. 11 Partial break of a bar default

5 Conclusions In this study, different winding topologys are considered to check the harmonic content of the stator current of IM suffering from BRB and eccentricity fault. To somme up no obvious difference can be classified here and the winding types can not be considered as an influential parameter.

0

(a)

-44.7062

-44.9191

-50 -100

50

0

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S. Tabet et al. Amplitude (dB)

840 0

(b)

-50 -100

Fréquency (Hz)

-45.9399

-44.815

-50 -100

0

50

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Amplitude (dB)

(c)

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Fréquency (Hz) 0

-45.6578

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0

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-43.1282

-44.2214

-50 -100

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Fréquency (Hz)

Fréquency (Hz)

0

(a)

-37.2794

-40.4136

-50 -100

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Amplitude (dB)

Fig. 12 A Broken bar default 0

(b)

-100

-40.99

-38.3918

-50 -100

0

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Fréquncy (Hz)

Fig. 13 Default of two bars broken

100

50

0

Fréquency (Hz)

100

Amplitude (dB)

Amplitude (dB)

(c)

-39.5883

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Fréquency (Hz) 0

-41.614

0

(d)

-20

-38.38

-38.37

-40 -60 -80

0

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Fréquency (Hz)

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0

(a)

-32.9158

-33.6953

-50 -100

0

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Amplitude (dB)

Amplitude (dB)

Impact of the Stator Winding Topology on the Fault Harmonic … 0

(b)

-35.2542

-100

-34.0212

-37.3822

-50 -100

0

100

50

0

Fréquency (Hz)

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100

Amplitude (dB)

Amplitude (dB)

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-32.4428

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Fréquency (Hz) 0

841

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-34.4673

-36.0319

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Fréquecy (Hz)

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-30

S

T

R

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Amplitude (dB)

Amplitude (dB)

Fig. 14 Default of three bars broken

-35 -40 -45 -50

80

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T

R

C

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S

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Fig. 15 Evolution of the default to the left (1 − 2 g) and right (1 + 2 g). For different levels of broken rotor bars

Amplitude (dB)

-30 -40

R

S

20%SE 10%DE

-50 30

20%SE 30%DE

C

T 20%SE 30%DE

40%SE 20%DE 40%SE 30%DE

40%SE 10%DE

40

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70

Amplitude (dB)

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R

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20%SE 20%SE 30%DE 10%DE

-50 -60 30

C

T 40%SE 20%DE

20%SE 30%DE

40%SE 30%DE

40%SE 10%DE

40

50

60

70

default level (%)

Fig. 16 Evolution of the default to the left (1 − 2 g) and right (1 + 2 g) for different levels of mixed eccentricity

Appendix Machines Parameters 3 phase; p = 3 KW; E = 36; Nb = 28; rs = 1.8; Ws = 25; p = 2; f = 50; J = 0.023976; fv = 0.00014439.

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References 1. Ojaghi M, Yazdandoost N (2015) Oil-Whirl fault modeling, simulation and detection in sleeve bearings of squirrel cage motors. IEEE Trans Energy Conv 30(4):1537–1545 2. Gyftakis KN, Kappatou J (2013) The impact of the rotor slot number on the behavior of the induction motor. Adv Power Electron 3. Al-Nuaim NA, Toliyat H (1998) A novel method for modeling dynamic air-gap eccentricity in synchronous machines based on modified winding function theory. IEEE Trans Energy Conv 13(2):156–162 4. Bossio G et al (2004) A 2-D Model of the induction machine. Extension of the modified winding function Approach. IEEE Trans Energy Conv 19(1):144–150 5. Ghoggal A, Sahraoui M, Aboubou A, Zouzou SE, Razik H (2005) An improved model of the induction machine dedicated to faults-detection—extension of the modified winding function. In: Proceeding of IEEE ICIT 2005, Hong-Kong, China, pp 14–17 6. Kotenko G, Piotrrovski M (1979) Machines électrique. Edition MIR; Moscou 7. Faiz J, Ghasemi-Bijan M (2015) Estimation of induction machine inductances using threedimensional magnetic equivalent circuit. IET Electric Power Appl 8. Nandi S, Ahmed S, Toliyat H (2001) Detection of rotor slot and other eccentricity related harmonics in a three-phase induction motor with different rotor cages. IEEE Trans Energy Conv 16(3):253–260

Sensor Fault Detection for Uncertain T-S DC Model with Descriptor Observer Approach Moussaoui Lotfi, Aouaouda Sabrina, and Righi Ines

Abstract In this paper we present a sensor fault detection approach of a DC motor described by a Takagi-Sugeno (T-S) multimodel with parameter uncertainties based on a descriptor observer. Indeed the observer is synthesized with a guaranteed H∞ performance. A descriptor design approach is used by considering the sensor fault as an auxiliary state variable. Then, the Lyapunov stabilization conditions are expressed as a linear matrix inequality (LMI) formulation. Keywords Fault detection · Takagi-Sugeno (T-S) model · Sensor faults · Descriptor observer

1 Introduction Diagnostic, monitoring and fault detection methods play an attractive role in industrial processes [1, 2]. Recently, several fault diagnosis methods have been developed for linear models [3]. However, linear models do not give an accurate representation of real systems and almost physical dynamic systems cannot be represented by linear differential equations. Moreover, the increasing demand for consistency, availability and high system performance has led to the use of nonlinear models to represent the systems. Hence obtained models are very complex and task of modelbased fault diagnosis becomes more difficult to achieve. However, the complexity of the nonlinear models obtained present difficulties for the diagnostic and control based on nonlinear models. Recently, nonlinear systems described by T-S models have been considered actively and specially in the fields of control, state estimation and diagnosis of nonlinear systems. The popularity of T-S modeling framework is due, on the one M. Lotfi (B) · A. Sabrina · R. Ines Faculty of Science and Technology, University of Souk Ahras LEER, BP.1553, Souk Ahras 41000, Algeria e-mail: [email protected] A. Sabrina e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2021 S. Bououden et al. (eds.), Proceedings of the 4th International Conference on Electrical Engineering and Control Applications, Lecture Notes in Electrical Engineering 682, https://doi.org/10.1007/978-981-15-6403-1_58

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hand, to the property of universal approximation [4] and, on the other hand, to its ease of manipulation from the mathematical point of view compared to the original nonlinear models. In the literature, various diagnostic approaches have been proposed for systems represented by T-S models [5, 6]. Unfortunately, there is little literature dealing with the problem of diagnosing sensor or actuator faults when the system model is subject to uncertainties or time-varying parameters. In general and in most works, the uncertainties are defined as parameters of the model [7, 8], or as inputs of the model [9]. Recent work has focused on estimating uncertain T-S system parameters based on fault diagnosis [10]. In this paper we propose a descriptor observer to estimate the sensors faults of uncertain T-S model with unmeasurable premise variables. The observer is realized with a multi-objective requirement for a T-S model subjected to sensor faults, external disturbances and uncertainties. For the estimation procedure, the idea is to add the sensor faults to the state vector which will be augmented and this system is called descriptor system [11]. The observer gains are obtained by minimizing a norm of H∞, and the stability conditions has been developed by minimizing the Lyapunov quadratic error in the form of a linear Matrix Inequality (LMI). The outline of this paper is organized as follows: in Sect. 2, a description of uncertain T-S system is given. In Sect. 3, a robust descriptor observer for fault detection is given with unmeasurable premise variables. Section 4 presents an example to illustrate the effectiveness of the proposed methods. Concluding remarks are given in Sect. 5.

2 System Description The change of the system environment or the change of the system itself can cause errors in the parameters of the model describing this system, this error is called the uncertainties of the model [12]. The modeling of this uncertainty can be represented as follows: x(t) ˙ = (A + A)x(t) + (B + B)u(t)

(1)

where A, B represents the matrices of the nominal model and A, B the uncertainties of the system. The global uncertain continuous T-S model affected by sensor faults f s (t) and external disturbances d(t) is represented as follows: x(t) ˙ =

r  i=1

μi (ξ (t))((Ai + Ai )x(t) + (Bi + Bi )u(t) + Bd d(t))

y(t) = C x(t) + Ds f s (t)

(2)

Sensor Fault Detection for Uncertain T-S DC Model …

845

where x(t) ∈ R n is the state vector, u(t) ∈ R m is the input vector and y(t) ∈ R p is the output vector. Ai , Bi , C, Bd and Ds are constant real matrices with appropriate dimensions. ζ j ( j = 1, . . . , g) are the premise variables. Mi j (i = 1, . . . , r ; j = 1, . . . , s) are the fuzzy sets and r is the number of rules. Where: wi (ξ (t)) μi (ξ (t)) = r i=1 wi (ξ (t)) wi (ξ (t)) =

r 

  Mi j ξ j (t)

j=1

r

i=1 μi (ξ (t)) = 1, and μi (ξ (t)) > 0 for i = 1, . . . , r The uncertainties Ai ∈ Rn×n , Bi ∈ Rn×m are defined by:

Ai = AF A (t)E A Bi = BF B (t)E B where A, E A , B, E B : are known constant matrices. And F A (t), F B (t): are the matrices uncertainties supposed unknown and bounded verifying the following condition: (F A (t))T F A (t) ≤ I, (F B (t))T F B (t) ≤ I In order to estimate state vector x and sensor fault f s , an augmented system is constructed using the descriptor technique. Then the T-S fuzzy system (2) can be written as follows: ˙¯ = E¯ x(t)

r  i=1

  ¯ i x¯ (t) + B¯ i u(t) + Nxs + B¯ d d(t) μi (ξ(t)) A y(t) = C¯x(t) ¯ = C0 x¯ (t) + xs

where  x xs = Ds f s (t), x¯ = , xs         Ai 0 Bi ¯ Bd I 0 ¯ , B¯ i = , Bd = , Ai = E¯ = n 0 0 0 0 0 −Ip     0 , C0 = C 0 , C¯ = C Ip N= Ip 

(3)

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Vector xs is considered as an auxiliary state of the augmented system (3). So if the state estimation of the augmented system (3) exists, then the state estimation of the original system (3) and the fault estimation exist too. Based on this model, the following fuzzy observer structure is adopted E z˙ (t) =

r 

  μi (ξ(t)) Fi z(t) + Bi u(t)

i=1

ˆ¯ x(t) = z + Ly

(4)

ˆ¯ where z(t) ∈ R n+ p is an auxiliary state vector of observer and x(t) ∈ R n+ p is the state estimation of (3). Lemma 1 ([13]). If there exist positive definite symmetric matrices P1 and P2 , Matrix R, non singular matrix M and scalar γ > 0 such that the following LMI is satisfied for i = 1, . . . , r ⎡

⎤ ∗ ∗ P1 A1 + (P1 A1 )T + Z 1 C + (Z 1 C)T + In ⎣ −Z 2 − Z 2T ∗ ⎦ < 0 Z 1T − P2 C Ai − Z 2 C −C T P2 −γ 2 I P1

(5)

Then there exist a fuzzy observer in the form (4) to asymptotically estimate the state and fault sensor for fuzzy system (2). The observer parameters are given by:  Ai 0 Fi = −C −I p   0 L= IP   In + RC R E= MC M 

(6)

(7)

(8)

where −1  M = P2−1 Z 2 − C P1−1 Z 1

(9)

R = P1−1 Z 1 M

(10)

Sensor Fault Detection for Uncertain T-S DC Model …

847

3 Design of Robust Descriptor Observer with Un-Measurable Variables for Uncertain T-S System In this section, we assume that the weighting functions that depend on the system state, then the T-S uncertain system (3) become: x(t) ˙ =

r 

μi (x(t))((Ai + Ai )x(t) + (Bi + Bi )u(t)) + Bd d(t)

i=1

(11)

y(t) = C x(t) + Ds f s (t)

We obtain the following equivalent system with the weighting functions depending on the estimated state. x(t) ˙ =

  ˆ μi x(t) ((Ai + Ai )x(t) + (Bi + Bi )u(t)) + Bd d(t) + w(t)

r  i=1

(12)

y(t) = C x(t) + Ds f s (t)

Then global system can be rewritten as follows: x(t) ˙ =

r 

  ˆ μi x(t) ((Ai + Ai )x(t) + (Bi + Bi )u(t)) + B¯ d v(t)

i=1

(13)

y(t) = C x(t) + Ds f s (t)

where:   T B¯ d = Bd I and v = d T w T And the augmented system becomes: r    ¯ i x¯ (t) + B¯ i u(t) + B¯¯ d v(t) + Dx ¯ s (t) ˙¯ = μi (ξ(t)) A E¯x(t) i=1

y(t) = C¯x(t) ¯ = C0 x¯ (t) + xs (t)  ¯ T = x(t)T xs (t)T ∈ R n+ p xs (t) = Ds f s (t) ∈ R p , x(t)       Ai 0 Bi I 0 ¯ , B¯ i = , , Ai = E¯ = n 0 0 0 0 −Ip   ¯d  B ¯ Bd = , B¯ d = Bd I 0     0 D¯ = , C0 = C 0 , C¯ = C Ip Ip Ai = Ai + Ai , Bi = Bi + Bi ,

(14)

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Let us consider the following fuzzy E z˙ (t) =

r 

  μi (ξ(t)) Fi z(t) + Bi u(t)

i=1

ˆ¯ x(t) = z + Ly

(15)

ˆ¯ ∈ R n+ p is the where z(t) ∈ R n+ p is an auxiliary state vector of observer and x(t) state estimation of (14). By substituting (14) into differential algebraic equation of the observer (4), we obtain: r 

E x˙ˆ¯ − E L C¯ x˙¯ =

    μi (ξ (t)) Fi xˆ¯ − LC0 x¯ − L xs + B¯ i u

(16)

i=1

Subtracting (16) from (14):  

r   μi (ξ (t)) E¯ + E L C¯ x˙¯ − E x˙ˆ¯ = B¯¯ d v(t) + i=1



A¯ i + Fi LC0 x¯ − Fi xˆ¯ + (N + Fi L)xs + Ai x¯ + Bi u

 (17)

ˆ¯ and suppose that: Let the error estimation be defined by e¯ = x¯ − x, E = E¯ + E L C¯

(18)

Fi = A¯ i + Fi LC0

(19)

N = −Fi L

(20)

The error dynamic can then be written as follows: E e˙¯ =

r 

  μi (ξ (t)) Fi e¯ + B¯¯ d v +  A¯ i x¯ +  B¯ i u

(21)

i=1

E e˙¯ =

r 

¯ i μi (ξ (t))Fi e¯ + M ¯

(22)

i=1

  ¯ i = B¯¯ d  A¯ i  B¯ i M

(23)

Sensor Fault Detection for Uncertain T-S DC Model …

849

⎤ v(t)  ¯ = ⎣ u(t) ⎦ x(t) ¯ ⎡

By simplifying e˙¯ =

r 

μi (ξ (t))Si e¯ + G¯¯  ¯

(24)

Si = E −1 Fi

(25)

i=1

where:

¯i G¯¯ = E −1 M V = e¯ T P e¯ > 0

(26)

With P > 0 and a positive scalar γ such the following condition: ¯ T ¯ ≤0 V˙ + e¯ T P e¯ − γ 2 

(27)

With: e˙¯ =

r 

μi (ξ (t))Si e¯ + G¯  ¯

(28)

i=1

And e¯˙ T =

r 

T μi (ξ (t))e¯ T SiT +  ¯ T G¯¯

(29)

i=1 T  ¯ T G¯¯ P e¯ + e¯ T P G¯¯  ¯ V˙ = e¯ T SiT P + P Si e¯ + 

(30)

And from (27), and (30) we have: T  ¯ T G¯¯ P e¯ + e¯ T P G¯¯  ¯ + e¯ T e¯ − γ 2  ¯ T ¯ ≤0 V˙ = e¯ T SiT P + P Si e¯ + 

(31)

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⎡

From which we deduce that   SiT P + P Si + I P G¯¯ ≤0 T −γ 2 I G¯¯ P

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(32)

⎤ sym(Ai P1 ) + sym(Z 1 C) + I ∗ ∗ ∗ ∗ ∗ ⎢ Z T − P C A − Z C + I −Z − Z T + I ∗ ∗ ∗ ∗ ⎥ 2 i 2 2 ⎢ ⎥ 1 2 ⎢ ⎥ 2 −C Bd P2 −γ I ∗ ∗ ∗ ⎥ Bd P1 ⎢ ⎢ ⎥≤0 2 ⎢ −C P2 0 −γ I ∗ ∗ ⎥ P1 ⎢ ⎥ ⎣ −CAi P2 0 0 −γ 2 I ∗ ⎦ Ai P1 −CBi P2 0 0 0 −γ 2 I Bi P1 (33) ⎡ ⎤ 0 0 0000 ⎢ 0 0 0 0 0 0⎥ ⎢ ⎥ ⎢ ⎥ 0 0 0 0 0⎥ ⎢ 0 Q(t) = ⎢ ⎥ ⎢ 0 0 0 0 0 0⎥ ⎢ ⎥ ⎣ Ai P1 −CAi P2 0 0 0 0 ⎦ Bi P1 −CBi P2 0 0 0 0 ⎤T ⎤T ⎡ ⎡ ⎤⎡ ⎤⎡ 0 0 0 0 0 0 0 0 ⎢ 0 0 ⎥⎢ 0 0 ⎥ ⎢ 0 0 ⎥⎢ 0 0 ⎥ ⎥ ⎥ ⎢ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎥ ⎢ ⎢ ⎥⎢ ⎥⎢ 0 ⎥⎢ 0 0 ⎥ 0 ⎥⎢ 0 0 ⎥ −1 ⎢ 0 −1 ⎢ 0 T Q (t) + Q(t) < ε1i ⎢ ⎥ + ε2i ⎢ ⎥ ⎥⎢ ⎥⎢ ⎢ 0 0 ⎥⎢ 0 0 ⎥ ⎢ 0 0 ⎥⎢ 0 0 ⎥ ⎥ ⎥ ⎢ ⎢ ⎥⎢ ⎥⎢ ⎣ A −CA ⎦⎣ A −CA ⎦ ⎣ 0 0 ⎦⎣ 0 0 ⎦ 0 0 B −CB 0 0 B −CB ⎡ ⎡ ⎤⎡ ⎤T ⎤⎡ ⎤T P1 E TA 0 P1 E BT 0 P1 E TA 0 P1 E BT 0 ⎢ 0 P E T ⎥⎢ 0 P E T ⎥ ⎢ 0 P E T ⎥⎢ 0 P E T ⎥ 2 A ⎥⎢ 2 A⎥ 2 B ⎥⎢ 2 B⎥ ⎢ ⎢ ⎢ ⎢ ⎥⎢ ⎥ ⎥⎢ ⎥ 0 ⎥⎢ 0 0 ⎥ 0 ⎥⎢ 0 0 ⎥ ⎢ 0 ⎢ 0 + ε1i ⎢ ⎥⎢ ⎥ + ε2i ⎢ ⎥⎢ ⎥ ⎢ 0 ⎢ 0 0 ⎥⎢ 0 0 ⎥ 0 ⎥⎢ 0 0 ⎥ ⎢ ⎢ ⎥⎢ ⎥ ⎥⎢ ⎥ ⎣ 0 ⎣ 0 0 ⎦⎣ 0 0 ⎦ 0 ⎦⎣ 0 0 ⎦ 0 0 0 0 0 0 0 0 Simplifying and using Schur complement we obtain:

Sensor Fault Detection for Uncertain T-S DC Model …

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

sym(Ai P1 ) + sym(Z 1 C) + I ∗ ∗ ∗ ∗ Z 1T − P2 C Ai − Z 2 C + I −Z 2 − Z 2T + I ∗ −C Bd P2 −γ 2 I ∗ Bd P1 −C P2 0 −γ 2 I P1 0 0 0 0 0 0 0 0 E A P2 0 0 E A P1 E B P2 0 0 E B P1

851

⎤ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⎥ ⎥ ∗ ∗ ∗ ∗ ⎥ ⎥ ⎥ ∗ ∗ ∗ ∗ ⎥   ⎥