Adaptive Switching Control of Large-Scale Complex Power Systems: Theory and Applications [1st ed. 2023] 9819910382, 9789819910380

Table of contents :
Preface
Contents
Acronyms
Part I Switching Control Based on Bang-Bang Funnel Controller and Stability Analysis Methods
1 Switching Control of Impulsively Disturbed Nonlinear Systems Based on Bang-Bang Constant Funnel Controller
1.1 Impulsively Disturbed Nonlinear System and Feasibility Assumptions of the Switching Controller
1.1.1 Bang-Bang Constant Funnel Controller
1.1.2 Feasibility Assumptions of the Switching Controller
1.1.3 Description of Impulsive Disturbance
1.2 IDRA of the Nonlinear System Controlled by SC
1.2.1 Main Results on IDRA of Impulsively Disturbed Nonlinear Systems
1.2.2 State-Dependent Switching Strategy mathscrT of SC
1.2.3 Control Law of BCFC
1.2.4 Solution Framework of the Closed-Loop System Consisting of the Nonlinear System and Its SC
1.3 Proof of Main Results
1.3.1 Boundary Values of e(i)(t) (i=0, 1, …, r-1) and Their Recovery Time of the Impulsively Disturbed Nonlinear System Controlled by BCFC
1.3.2 Preliminaries for the Proof of Main Results
1.3.3 Proof of Theorem 1.1
1.3.4 Proof of Theorem 1.2
1.4 Simulation Studies: Application of BCFC and SC for Frequency Control of a SMIB Power System
1.4.1 Excitation Controller Designed Based on BBFC
1.4.2 Excitation Controller Designed Based on SC
1.5 Summary
References
2 Observer-Based Robust Bang-Bang Funnel Controller and Its Stability in Closed-Loop Systems
2.1 Introduction to Bang-Bang Funnel Control and Main Results on Closed-Loop Stability
2.1.1 System Description
2.1.2 Bang-Bang Funnel Control
2.1.3 High-Gain State Observer
2.1.4 Assumptions of RBC
2.1.5 Main Results on Convergence of Estimation Error and Closed-Loop Stability
2.2 Proof of Convergence of Estimation Errors of High-Gain Observer
2.3 Proof of Stability of Closed-Loop System
2.3.1 Dynamics of RBC Outside Error Funnels
2.4 Simulation Studies: Application of RBC for Frequency Control of a SMIB Test System
2.5 Summary
References
Part II Transient Stability Control of Large-Scale Complex Power Systems Based On Adaptive Switching Controllers
3 Switching Control of Synchronous Generators for Transient Stability Enhancement
3.1 Design of Switching Power System Stabilizer
3.1.1 Power System Model Used for Designing SPSS
3.1.2 Design of SPSS
3.1.3 Closed-Loop Stability
3.2 Design of Switching Excitation Controller and Switching Governor
3.2.1 Power System Model Used for Designing SEC and SG
3.2.2 Design of SEC and SG
3.3 Simulation Studies
3.3.1 Test Results of SPSS
3.3.2 Test Results of SEC and SG
3.4 Summary
References
4 Switching Control of Modular Multi-level Converters in High-Voltage-Direct-Current Transmission Systems Via BBFC-Based SCU
4.1 Model of a MMC-HVDC Transmission System
4.1.1 Model of a MMC
4.1.2 Outer-Loop Controllers of a VC for Rectifier-Side MMC
4.1.3 Outer-Loop Controllers of a VC for Inverter-Side MMC
4.1.4 Inner-Loop Current Controllers of a VC
4.2 Bang-Bang Funnel Controller and Its Applications in Control of MMCs
4.2.1 Bang-Bang Funnel Controller with Neutral Output
4.2.2 BBFCs of SCUs for Rectifier-Side MMC and Inverter-Side MMC
4.3 Switching Laws for Switching Control Units of FRTHCs
4.4 Simulation Studies
4.4.1 Fault Ride-Through Performance of a Two-Machine Test System in the Case of a Three-Phase-to-Ground Fault on the Rectifier-Side AC Grid
4.4.2 Fault Ride-Through Performance in a Two-Machine Test System in the Case that a Line-to-Line Fault Occurred on Inverter-Side AC Grid
4.4.3 Fault Ride-Through Performance in the Case of a Four-Machine Test Power System
4.4.4 Modal Analysis of the Four-Machine Thirteen Bus Test System Controlled by VC
4.5 Summary
References
5 Switching Control of Doubly-Fed Induction Generator-Based Wind Turbines for Transient Stability Enhancement of Wind Power Penetrated Power Systems
5.1 Multi-loop Switching Control of DFIG-Based Wind Turbine Systems
5.1.1 Model Linearization of DFIG-Based Wind Turbine System
5.1.2 Four-Loop Switching Controller Designed for DFIG-Based Wind Turbine System
5.2 Switching Angle Controller and AGC for Frequency Control of DFIG Based WPPS
5.2.1 Internal Voltage and Virtual Rotor Angle of DFIGs
5.2.2 Design of SAC and AGC
5.2.3 Modal Analysis of DFIGs with SAC and AGC
5.3 Simulation Studies
5.3.1 Test Results of the Four-Loop Switching Controller
5.3.2 Test Results of the Switching Angle Controller and AGC
5.4 Summary
References
6 Adaptive Switching Control of Power Electronic Converters
6.1 Switching Fault Ride-Through of GSCs via …
6.1.1 Design of SFRTC
6.2 Bang-Bang Funnel Control of Three-phase …
6.2.1 Introduction to Dual-Buck Scheme
6.2.2 Bang-Bang Funnel Control of the Inverter
6.3 Experiment and Simulation Results
6.3.1 Test Results of the Switching Fault Ride-Through Controller for GSCs
6.3.2 Test Results of the Bang-Bang Funnel Controller of the Three-Phase Full-Bridge Inverter
6.4 Summary
References

Citation preview

Power Systems

Yang Liu Qing-Hua Wu

Adaptive Switching Control of Large-Scale Complex Power Systems Theory and Applications

Power Systems

Electrical power has been the technological foundation of industrial societies for many years. Although the systems designed to provide and apply electrical energy have reached a high degree of maturity, unforeseen problems are constantly encountered, necessitating the design of more efficient and reliable systems based on novel technologies. The book series Power Systems is aimed at providing detailed, accurate and sound technical information about these new developments in electrical power engineering. It includes topics on power generation, storage and transmission as well as electrical machines. The monographs and advanced textbooks in this series address researchers, lecturers, industrial engineers and senior students in electrical engineering. **Power Systems is indexed in Scopus**

Yang Liu · Qing-Hua Wu

Adaptive Switching Control of Large-Scale Complex Power Systems Theory and Applications

Yang Liu School of Electric Power Engineering South China University of Technology Guangzhou, Guangdong, China

Qing-Hua Wu School of Electric Power Engineering South China University of Technology Guangzhou, Guangdong, China

ISSN 1612-1287 ISSN 1860-4676 (electronic) Power Systems ISBN 978-981-99-1038-0 ISBN 978-981-99-1039-7 (eBook) https://doi.org/10.1007/978-981-99-1039-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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

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Preface

With the overwhelming development of renewable power generation and application of flexible power electronics converters, various stability problems are emerging in large-scale complex power systems. Cascaded tripping of generators, transmission line outages caused by natural disasters, failure of renewable power sources, and overloading of transmission networks have resulted in lots of large-scale power failures and humorous financial losses all over the world. One of the common characteristics of these events is that they can damage the network structure and introduce a large amount of imbalance between power supply and consumption. Therefore, control methods that are able to reshape the energy distribution and accelerate the process of power re-balance during post-fault recovery are greatly desired. Over the last century, various linear and nonlinear controller design methods were developed for power systems. They are implemented on exciters of synchronous generators, high-voltage direct-current transmission stations, wind power generators, solar cells, electric energy storage units, and reactive power compensation devices. Dynamic stability of the power system in small neighborhoods of the operation point has been significantly improved. Oscillation stability problems, such as low-frequency oscillation, sub-synchronous oscillation, and super-synchronous oscillation, were widely studied, and mature damping controllers were developed. Nevertheless, the existing control methods fail to offer satisfactory performance for power systems in the events of large disturbances. The core reasons are twofold: (1) Parameters of the controllers are tuned considering the tradeoff between response speed and overshooting. There are still much untapped control effort and control energy hidden in the distributed controllers. (2) Coordination mechanism is lacking among the distributed controllers, and counteraction of control energy exists in the system with a large number of controllers. In order to fulfill the great gap between the need in coordinately utilizing the utmost effort of distributed devices in the events of large disturbances and the compromised design of continuous controllers, this monograph is dedicated to research on the theory of adaptive switching control method and the core technologies of applying the method for stability control of large-scale complex power systems. Overall, this monograph is divided into two parts. Part I presents the theoretical foundation for vii

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the switching controller and the observer-based robust bang–bang controller. Part II includes several applications of the switching controllers and robust bang–bang controllers in the transient stability control of power systems. Specifically, Part I is organized as follows: Chapter 1 proposes a switching controller for the impulsively disturbed nonlinear systems with arbitrary relative degree based on the bang–bang constant funnel controller. Weaker feasibility assumptions are proposed for the switching controller in contrast to the conventional bang–bang funnel controller. The switching controller is designed to switch between a bang–bang constant funnel controller and a continuous controller according to a state-dependent switching strategy. Moreover, the impulsive disturbance recovery ability, which refers to the ability of returning to the pre-disturbed operation point following an impulsive disturbance, of the nonlinear system controlled by the switching controller is analyzed. The boundary values and the recovery time of the system output tracking error and its derivatives of an impulsively disturbed nonlinear system controlled by the bang–bang constant funnel controller are discussed as well. Furthermore, the bang–bang constant funnel controller and the switching controller are applied respectively in the excitation control of a synchronous generator for the frequency supervision of a single-machineinfinite-bus power system, through which the control performance of the switching controller and the bang–bang constant funnel controller is verified with simulation studies. Chapter 2 proposes a robust bang–bang controller based on the high-gain nonlinear observer for a nonlinear system with intermittent disturbances. The intermittent disturbances are able to model the system faults and sudden changes of operating conditions of practical nonlinear systems. A bang–bang constant funnel controller is employed such that the largest control capabilities of the control system are explored to stabilize the nonlinear system against the disturbances. The bang–bang constant funnel controller functions with the estimates of tracking error of control target and its derivatives, obtained with a high-gain state observer. This eliminates the requirement on the derivative calculations of the tracking error signal of system output. Convergence of the observation error of the high-gain observer is verified. Closed-loop stability of the robust bang–bang controller is proved in a bounded-input bounded-state sense. Part II consists of the following contents: Chapter 3 presents the applications of the switching controller in the control of synchronous generators. Firstly, a coordinated switching power system stabilizer (PSS) is designed to enhance the small-signal stability of multi-machine power systems. The switching PSS (SPSS) switches between a bang–bang PSS and a conventional PSS based on a state-dependent switching strategy. The bang–bang PSS is designed as a bang–bang constant funnel controller (BCFC). It is able to provide fast damping of rotor speed oscillations in a bang–bang manner. The closedloop stability of the power system controlled by the switching PSS and the conventional PSS is analyzed. To verify the control performance of the switching PSS, simulation studies were carried out in a 4-generator 11-bus power system and the

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IEEE 16-generator 68-bus power system. The damping ability of the SPSS is evaluated in aspects of small-signal oscillation damping and transient stability enhancement, respectively. Meanwhile, the coordination between different SPSSs and the coordination between the SPSS and the conventional PSS (CPSS) are investigated. Secondly, two switching controllers working in a coordinated manner are designed in Chap. 3 to enhance the transient stability of multi-machine power systems. One is switching excitation controller (SEC) and the other is switching governor (SG). The SEC switches between a bang–bang excitation controller (BEC) and a conventional excitation controller (CEC), and the SG switches between a bang–bang governor (BG) and a conventional governor (CG), via a state-dependent switching strategy. The BEC and the BG are designed as BCFCs, which are able to provide fast switching of excitation voltage and valve position between their upper and lower limits. A detailed steam turbine model including boiler pressure effect is considered in the controller design process. Coordination between the excitation loop and speed governing loop is studied. Two system resilience indexes are introduced, and the short-term resilience is investigated for the power systems with and without switching controllers installed, respectively. Chapter 4 presents an application of the switching controller in the modular multilevel converter (MMC)-based high-voltage direct-current (MMC-HVDC) transmission systems. A fault ride-through hybrid controller (FRTHC) is designed to improve the fault ride-through capability of the MMC-HVDC transmission system. The FRTHC is composed of four loops of cascading switching control units (SCUs). Each switching controller switches between a bang–bang funnel controller (BBFC) and a proportional–integral (PI) control loop according to a state-dependent switching law. The BBFC is able to utilize the full control capability of each control loop through three-value control signals with the maximum available magnitude. A statedependent switching law is designed for each SCU to guarantee the structural stability of SCUs. Simulation studies are undertaken to verify the superior fault ridethrough capability of the MMC-HVDC transmission system controlled by FRTHC, in comparison to that controlled by a vector controller (VC) and a VC with DC voltage droop, respectively. Chapter 5 illustrates the application of switching controllers in the control of doubly-fed induction generator (DFIG)-based wind turbines (DFIGWTs). In the first place, a coordinated four-loop switching controller (SC) is designed for the DFIGWT to improve the transient stability of wind power-penetrated power systems. A shortterm resilience index is defined, and it reflects the dynamics of both frequency and bus voltage. A four-loop SC is driven by the four outputs of the DFIG. Referring to a state-dependent switching strategy, the four-loop SC switches between a logicbased BBFC and a VC theory-based conventional controller (CC) in each control loop. The logic-based BBFC is robust to system nonlinearities, uncertainties, and external disturbances. The control signal of the logic-based bang–bang constant funnel controller is bang–bang with the upper and lower limits of control variables. Simulation studies were undertaken in a modified IEEE 16-generator 68-bus power system, in which four DFIG-based wind farms are penetrated to provide 9.94% power supply. The four-loop switching controller was evaluated in aspects of the

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integral control of the DFIG and the resilience enhancement of the multi-machine power system, respectively. In the second place, a switching angle controller and an automatic generation controller (AGC) are designed for the DFIG to control the frequency of DFIG-based wind power-penetrated power systems. The concept of virtual rotor angle of the DFIG is defined. The virtual rotor angle is controlled by the switching angle controller (SAC) in a bang–bang manner such that the active power of the DFIG is regulated to provide frequency support to the external power network. The output of the SAC is also used for the pitch angle control to offer a short-term regulation of the mechanical power input to the DFIG, and the longterm control of the mechanical power input is achieved with the AGC. Small-signal analysis was undertaken to verify the effectiveness of the SC and the AGC. Simulation studies were carried out in a two-machine power system and a modified Kundur fourmachine two-area power system, respectively. The frequency support performance of DFIGs having different control configurations is investigated. Modal analysis was undertaken to evaluate the effect in providing additional damping to the rotor oscillation of the test systems. Chapter 6 proposes a switching fault ride-through controller for the grid-side converter of permanent magnetic synchronous generator-based wind turbines. The switching fault ride-through controller switches between an observer-based bang– bang funnel controller (OBFC) and a VC via a state-dependent switching strategy. The design of observer-based bang–bang funnel controller does not rely on accurate system model, and it works using the estimates of high-order dynamics of system outputs obtained with a high-gain observer. The fault ride-through control performance of switching fault ride-through controller was studied with hardware-in-theloop experiments. Moreover, a bang–bang funnel control strategy is designed for the three-phase full-bridge inverter under dual-buck scheme. By adopting the dualbuck scheme, only two switches on the different phase legs are controlled at high frequency. The full-bridge topology can be decoupled into two parallel-connected buck units, which allows the inverter to be formulated and analyzed by the model of buck chopper. The BBFC works with only two values, and it restrains the tracking error between the system’s output and the reference within the pre-specified funnel boundaries. The nature of BBFC makes it simpler, faster, and more robust, suitable for the inverter operating in dual-buck scheme. The proposed strategy not only guarantees accurate static output performance, but also promotes the inverter’s dynamic response against rapid variations of the load, the reference, and the input voltage, as well as its robustness to the unexpected parameter changes of the elements. The effectiveness of the proposed strategy is validated by the simulation and experimental results. We wrote this monograph in the belief that adaptive switching control method is a promising way of enhancing transient stability of large-scale complex power system. Adaptive switching control will be a flexible compensation for the emergency control framework of power systems. Especially due to the increasing penetration of power electronics converters, switching of power network components and switching of different control loops will be a common fact and leads to the need of novel stability control and analysis tools. On one hand, this monograph presents bang–bang funnel

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control-based methods to explore and utilize the largest control energy of distributed devices. This enables the reshaping of energy distribution in power networks and takes full advantage of the utmost control potential of flexible controllers in a coordinated manner. On the other hand, this monograph offers instances for stability analysis of switched systems in a bounded-input bounded-state sense. We hope that this monograph will be useful for those postgraduates, academic researchers, and engineers working in the area of stability control and analysis of power systems. We would like to thank Dr. Xindi Chen, Dr. Tianhao Wen, Dr. Yuqing Lin, Mr. Zehui Lin, and all colleagues in the Smart Grid and Its Automation Group, for all assistance provided. Special thanks also goes to Zisheng Wang and Gowtham Chakravarthy V, the editors of Springer, for their professional and efficient editorial work on this monograph. This book is based on the work partly funded by the National Natural Science Foundation of China (Grant No. U1866210, 51807067). Guangzhou, China January 2023

Dr. Yang Liu Prof. Qing-Hua Wu

Contents

Part I

Switching Control Based on Bang-Bang Funnel Controller and Stability Analysis Methods

1 Switching Control of Impulsively Disturbed Nonlinear Systems Based on Bang-Bang Constant Funnel Controller . . . . . . . . . . . . . . . . . 1.1 Impulsively Disturbed Nonlinear System and Feasibility Assumptions of the Switching Controller . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Bang-Bang Constant Funnel Controller . . . . . . . . . . . . . . . . . 1.1.2 Feasibility Assumptions of the Switching Controller . . . . . . 1.1.3 Description of Impulsive Disturbance . . . . . . . . . . . . . . . . . . . 1.2 IDRA of the Nonlinear System Controlled by SC . . . . . . . . . . . . . . . 1.2.1 Main Results on IDRA of Impulsively Disturbed Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 State-Dependent Switching Strategy T of SC . . . . . . . . . . . . 1.2.3 Control Law of BCFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Solution Framework of the Closed-Loop System Consisting of the Nonlinear System and Its SC . . . . . . . . . . . 1.3 Proof of Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Boundary Values of e(i) (t)(i = 0, 1, . . . , r − 1) and Their Recovery Time of the Impulsively Disturbed Nonlinear System Controlled by BCFC . . . . . . . . 1.3.2 Preliminaries for the Proof of Main Results . . . . . . . . . . . . . . 1.3.3 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Simulation Studies: Application of BCFC and SC for Frequency Control of a SMIB Power System . . . . . . . . . . . . . . . . 1.4.1 Excitation Controller Designed Based on BBFC . . . . . . . . . . 1.4.2 Excitation Controller Designed Based on SC . . . . . . . . . . . . . 1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 5 7 8 8 9 10 11 11 16

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Contents

2 Observer-Based Robust Bang-Bang Funnel Controller and Its Stability in Closed-Loop Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction to Bang-Bang Funnel Control and Main Results on Closed-Loop Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Bang-Bang Funnel Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 High-Gain State Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Assumptions of RBC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Main Results on Convergence of Estimation Error and Closed-Loop Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Proof of Convergence of Estimation Errors of High-Gain Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Proof of Stability of Closed-Loop System . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Dynamics of RBC Outside Error Funnels . . . . . . . . . . . . . . . . 2.4 Simulation Studies: Application of RBC for Frequency Control of a SMIB Test System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II

31 31 31 32 33 34 35 36 38 38 42 47 47

Transient Stability Control of Large-Scale Complex Power Systems Based On Adaptive Switching Controllers

3 Switching Control of Synchronous Generators for Transient Stability Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1 Design of Switching Power System Stabilizer . . . . . . . . . . . . . . . . . . 63 3.1.1 Power System Model Used for Designing SPSS . . . . . . . . . . 63 3.1.2 Design of SPSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.1.3 Closed-Loop Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2 Design of Switching Excitation Controller and Switching Governor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.2.1 Power System Model Used for Designing SEC and SG . . . . 72 3.2.2 Design of SEC and SG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.3 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.3.1 Test Results of SPSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.3.2 Test Results of SEC and SG . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4 Switching Control of Modular Multi-level Converters in High-Voltage-Direct-Current Transmission Systems Via BBFC-Based SCU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Model of a MMC-HVDC Transmission System . . . . . . . . . . . . . . . . . 4.1.1 Model of a MMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Outer-Loop Controllers of a VC for Rectifier-Side MMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

4.1.3 Outer-Loop Controllers of a VC for Inverter-Side MMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Inner-Loop Current Controllers of a VC . . . . . . . . . . . . . . . . . 4.2 Bang-Bang Funnel Controller and Its Applications in Control of MMCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Bang-Bang Funnel Controller with Neutral Output . . . . . . . . 4.2.2 BBFCs of SCUs for Rectifier-Side MMC and Inverter-Side MMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Switching Laws for Switching Control Units of FRTHCs . . . . . . . . . 4.4 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Fault Ride-Through Performance of a Two-Machine Test System in the Case of a Three-Phase-to-Ground Fault on the Rectifier-Side AC Grid . . . . . . . . . . . . . . . . . . . . . 4.4.2 Fault Ride-Through Performance in a Two-Machine Test System in the Case that a Line-to-Line Fault Occurred on Inverter-Side AC Grid . . . . . . . . . . . . . . . . . . . . . 4.4.3 Fault Ride-Through Performance in the Case of a Four-Machine Test Power System . . . . . . . . . . . . . . . . . . 4.4.4 Modal Analysis of the Four-Machine Thirteen Bus Test System Controlled by VC . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Switching Control of Doubly-Fed Induction Generator-Based Wind Turbines for Transient Stability Enhancement of Wind Power Penetrated Power Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Multi-loop Switching Control of DFIG-Based Wind Turbine Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Model Linearization of DFIG-Based Wind Turbine System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Four-Loop Switching Controller Designed for DFIG-Based Wind Turbine System . . . . . . . . . . . . . . . . . . 5.2 Switching Angle Controller and AGC for Frequency Control of DFIG Based WPPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Internal Voltage and Virtual Rotor Angle of DFIGs . . . . . . . 5.2.2 Design of SAC and AGC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Modal Analysis of DFIGs with SAC and AGC . . . . . . . . . . . 5.3 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Test Results of the Four-Loop Switching Controller . . . . . . . 5.3.2 Test Results of the Switching Angle Controller and AGC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

106 106 108 108 109 111 113

113

115 119 121 123 123

125 125 125 129 131 131 133 134 144 144 153 163 165

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Contents

6 Adaptive Switching Control of Power Electronic Converters . . . . . . . . 6.1 Switching Fault Ride-Through of GSCs via Observer-Based Bang-Bang Funnel Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Design of SFRTC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Bang-Bang Funnel Control of Three-phase Full-Bridge Inverter Under Dual-Buck Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Introduction to Dual-Buck Scheme . . . . . . . . . . . . . . . . . . . . . 6.2.2 Bang-Bang Funnel Control of the Inverter . . . . . . . . . . . . . . . 6.3 Experiment and Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Test Results of the Switching Fault Ride-Through Controller for GSCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Test Results of the Bang-Bang Funnel Controller of the Three-Phase Full-Bridge Inverter . . . . . . . . . . . . . . . . . 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167 167 167 170 170 175 177 177 181 190 190

Acronyms

CPSS AGC APF AVR BBFC BCFC BEC BG BPSS CC CEC CG DFIG FRT FRTHC GSC HVDC IDRA LBCFC LCC-HVDC MMC OBFC PFC PID PLL PMSGWT PR PSS PWM RBC

Conventional power system stabilizer Automatic generation controller Active power filter Automatic voltage controller Bang–bang funnel controller Bang–bang constant funnel controller Bang–bang excitation controller Bang–bang governor Bang–bang power system stabilizer Conventional controller Conventional excitation controller Conventional governor Doubly-fed induction generator Fault ride through Fault ride-through hybrid controller Grid-side converter High-voltage direct-current transmission Impulsively disturbance recovery ability Logic-based bang–bang constant funnel controller Line-commutated converter-based high-voltage direct-current transmission Modular multi-level converter Observer-based bang–bang funnel controller Power factor correction Proportional–integral–derivative controller Phase-locked loop Permanent magnetic synchronous generator-based wind turbine Proportional resonant Power system stabilizer Pulse width modulation Robust bang–bang controller xvii

xviii

RSC SAC SC SCR SCU SdSEF SEC SFRTC SG SMIB SPSS SVPWM VC VDRC VIC VSI WPG WPGS WPPS

Acronyms

Rotor side converter Switching angle controller Switching controller Short-circuit ratio Switching control unit State-dependent switched energy function Switching excitation controller Switching fault ride-through controller Switching governor Single-machine-infinite bus Switching power system stabilizer Space vector pulse width modulation Vector control Vector controller with DC voltage droop Virtual inertia control Voltage source inverter Wind power generator Wind power generation system Wind power-penetrated power system

Part I

Switching Control Based on Bang-Bang Funnel Controller and Stability Analysis Methods

For large-scale power systems, stability of the system is mostly threatened by events with low possibility and high impact [1], which introduces disturbances into a system in an impulsive manner. To control the effect of such events, novel methodologies are needed to enhance the impulsive disturbance rejection ability (IDRA) [2]. Considering a specific set of system modes, linear controllers cannot provide satisfactory performance in such events since system modes will change, if the system is disturbed in a far distance from its original equilibrium point [3]. System nonlinearity and uncertainties are not considered in the design process as well. Catering to these, adaptive tuning techniques are proposed for parameters of linear controllers [4], and robust linear controllers [5] are designed. However, the design of these controllers achieves the stability of the nonlinear system under a wide range of operation conditions at the cost of compromising performance in each region of operation. In the cases of impulsive disturbances, they may be unable to utilize the utmost control effort to ensure the stability of the nonlinear system. Regarding the nonlinear control methods, nonlinearities and uncertainties are considered in the controller design procedure [6]. Moreover, nonlinear control methods have stronger robustness. Nevertheless, few of them have ever employed the maximum control effort in the control laws, and there stillis untapped potential in the control system. However, the unemployed control energy can possibly be the last straw to save an impulsively disturbed nonlinear system from going unstable. Among nonlinear controllers, the time-optimal control or so-called bang-bang control scheme is the only one that has utilized the maximum control effort in its control law [7]. This provides a solution to improving the IDRA of nonlinear systems. Referring to the traditional bang-bang control [8], the control law is obtained through solving the canonical equations of the Hamiltonian of the system. It requires the accurate parameters of the system, which cannot be obtained in practice. Moreover, the Hamiltonian of a high-order nonlinear system can be very complex, which has restricted the application of bang-bang control in industry. Inspired by constructing a funnel controller with input saturation [9], the conventional BBFC was proposed in [10]. The BBFC uses bang-bang control in highorder nonlinear systems without calculating the canonical equations. The BBFC was

2

Part I: Switching Control Based on Bang-Bang Funnel Controller …

designed to regulate the output tracking error so as to drive it into pre-specified time-varying error bounds. Robustness of BBFC with respect to time delays was investigated in [11]. Afterward, the output tracking control of nonlinear systems having arbitrary known relative degree was studied in [12]. The main result therein ensured that the tracking error of system output and its derivatives can be controlled into the pre-designed error funnels. However, the feasibility assumptions of the BBFC are too conservative for practical applications. Regarding the error funnel design showed in [12], it is too restrictive to require that the error funnels must contain all the high-order derivatives of the tracking error within them. For power systems, a three-phase-to-ground fault on a transmission line can result in sudden changes of the high-order derivatives of system states [13, 14]. Meanwhile, the reference of outputs usually shows step-like changes, such as the torque reference in [15]. It is of great challenge to choose appropriate funnels for the BBFC. Moreover, it can be intuitively seen that BBFC provides superior performance than continuous controllers, in the cases where system states are away from steady-state values. Thus, the requirement that initial values of the error funnels should be large enough to include the initial system output tracking error and its derivatives significantly complicates the application of the BBFC. To improve the applicability of the BBFC, this chapter presents a bang-bang constant funnel controller (BCFC) based on the BBFC. The BCFC is able to drive the system output and its derivatives back into the pre-designed error bounds when they are perturbed outside the error funnels due to the impulsive disturbance. Moreover, the requirement concerning the initial values of error funnels is not needed any more. Despite of the above, the output tracking control of nonlinear systems requires that the output tracking error can stabilize at zero. The BCFC can only ensure that the output tracking error oscillates within the corresponding error funnel rather than stabilizing at zero. Therefore, a continuous controller (CC) should be employed. A SC, which switches between a CC and a BCFC according to a state-dependent switching strategy, should be designed. In Chap. 1, the closed-loop stability of the switching controller composed of a CC and a BCFC is proved for impulsively disturbed nonlinear systems. The BCFC requires the tracking error of control target and its high-order derivatives to be the inputs. The derivative operations are not desirable in practical control systems since they significantly magnify the measurement noise and may result in instability of the system. A robust bang-bang funnel controller (RBC) is proposed in Chap. 2, and it uses constant funnel boundaries instead of time-varying ones, which simplifies its design and enhances its practical feasibility. Weaker feasibility assumptions are proposed and theoretically verified for RBC, and it does not require the funnel boundaries to enclose the system outputs and their derivatives. A high-gain observer is designed for RBC to obtain the estimates of the tracking error of control target and its derivatives, which eliminates the derivative calculations in conventional BBFC. The convergence of the error dynamics of the high-gain observer and the closed-loop stability of RBC are proved. Part I lays the theoretical foundations for the various applications presented in Part II of this monograph.

Part I: Switching Control Based on Bang-Bang Funnel Controller …

3

References 1. Panteli M, Mancarella P (2015) The grid: stronger, bigger, smarter?: Presenting a conceptual framework of power system resilience. IEEE Power Energy Mag 13(3):58–66 2. Wang Y, Chen C, Wang J, Baldick R (2016) Research on resilience of power systems under natural disastersła review. IEEE Trans Power Syst 31(2):1604–1613. 10.1109/TPWRS.2015. 2429656 3. Abido M (2000) Robust design of multimachine power system stabilizers using simulated annealing. IEEE Trans Energy Convers 15(3):297–304. 10.1109/60.875496 4. Precup RE, David RC, Petriu E, Radac MB, Preitl S (2014) Adaptive gsa-based optimal tuning of pi controlled servo systems with reduced process parametric sensitivity, robust stability and controller robustness. IEEE Trans Cybern 44(11):1997–2009. 10.1109/TCYB.2014.2307257 5. Pearson J, Shields R, Staats PJ (1974) Robust solutions to linear multivariable control problems. IEEE Trans Autom Control 19(5):508–517. 10.1109/TAC.1974.1100679 6. Kravaris C, Niemiec M, Berber R, Brosilow CB (1998) Nonlinear Model-Based control of nonminimum-phase processes. Springer, Netherlands 7. Consolini L, Piazzi A (2009) Generalized bang-bang control for feedforward constrained regulation. Automatica 45(10):2234–2243. http://dx.doi.org/10.1016/j.automatica.2009.06.030 8. Athanassiades M, Smith O (1961) Theory and design of high-order bang-bang control systems. IEEE Trans Autom Control 6(2):125–134. 10.1109/TAC.1961.1105182 9. Hackl CM, Hopfe N, Ilchmann A, Mueller M, Trenn S (2013) Funnel control for systems with relative degree two. SIAM J Control Optim 51:965–995 10. Liberzon D, Trenn S (2010) The bang-bang funnel controller. In: 2010 49th IEEE CDC, pp 690–695. 10.1109/CDC.2010.5717742 11. Liberzon D, Trenn S (2013) The bang-bang funnel controller: time delays and case study. In: 2013 ECC, pp 1669–1674 12. Liberzon D, Trenn S (2013) The bang-bang funnel controller for uncertain nonlinear systems with arbitrary relative degree. IEEE Trans Autom Control 58(12):3126–3141. 10.1109/TAC. 2013.2277631 13. Du P, Yuri M (2014) Using disturbance data to monitor primary frequency response for power system interconnections. IEEE Trans Power Syst 29(3):1431–1432 14. Pillay P, Bhattacharjee A (1996) Application of wavelets to model short-term power system disturbances. IEEE Trans Power Syst 11(4):2031–2037 (1996) 15. Lukic S, Emadi A (2010) State-switching control technique for switched reluctance motor drives: theory and implementation. IEEE Trans Ind Electron 57(9):2932–2938. 10.1109/TIE. 2009.2038942

Chapter 1

Switching Control of Impulsively Disturbed Nonlinear Systems Based on Bang-Bang Constant Funnel Controller

1.1 Impulsively Disturbed Nonlinear System and Feasibility Assumptions of the Switching Controller The nonlinear system considered here can be described by the following single-input single-output system  x˙ = F(x) + G(x)u (1.1) y = H (x) where x ∈ Rn , u ∈ R, y ∈ R, F(x) : Rn → Rn , G(x) : Rn → Rn , H (x) : Rn → R, F(x) and G(x) are smooth vector fields on Rn and H (x) is a smooth function. The relative degree of output H (x) with respect to input u is r , which is known and defined as follows. Differentiate output H (x) until input u appears. The relative degree r is the smallest integer such that input u appears explicitly in H (r ) (x), namely, H (r ) (x) = L Fr H (x) + LG L Fr −1 H (x)u, on condition that LG L Fr −1 H (x) = 0 and LG L Fk H (x) = 0 hold in a neighborhood of x 0 for all k = 0, . . . , r − 2, where x 0 is the equilibrium of the nonlinear system. The problem here is to design a SC, which switches between a BCFC and a CC, to regulate the output tracking error and its derivatives of nonlinear system (1.1) following an impulsive disturbance back to zero. For this purpose, an introduction to the basic knowledge of the BCFC is presented first.

1.1.1 Bang-Bang Constant Funnel Controller The BCFC is designed to ensure approximate tracking of a reference signal yref . Precisely, we want to ensure that tracking error e := y − yref meets pre-specified error bounds which are given by the funnel F0 :={e ∈ R|ϕ0− ≤ e ≤ ϕ0+ }, where ϕ0− © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Liu and Q.-H. Wu, Adaptive Switching Control of Large-Scale Complex Power Systems, Power Systems, https://doi.org/10.1007/978-981-99-1039-7_1

5

6

1 Switching Control of Impulsively Disturbed Nonlinear …

and ϕ0+ are the pre-specified error bounds. The index 0 is used for the funnel because funnels Fi , i = 1, 2, . . . , r − 1 for the ith derivative e(i) (t) of e(t) will be considered as well. Switching logic S : e(t) → q is defined with r logic blocks B0 , B1 , . . . , Br −1 recursively as follows. ˙ B0 ) = S1 (e(t), ˙ q1 ) S : e(t) → q := S1 (e(t),

(1.2)

where, for i = 1, 2, . . . , r − 2 

   d (i) e (t), Bi e(i) (t), qi dt  (i+1)  (t), qi+1 = Si+1 e

Si : (e(i) (t), qi ) → q : = Si+1

and

    Sr −1 : e(r −1) (t), qr −1 → q := Br −1 e(r −1) (t), qr −1 .

(1.3)

(1.4)

Moreover, the r logic blocks B0 , B1 , . . . , Br −1 are defined as follows. B0 : e(t) → q1 with q1 (t) = G (e(t), ϕ0+ − ε0+ , ϕ0− + ε0− , q1 (t−)) q1 (0−) = q10 ∈ {true, false}

(1.5)

Bi : (e(i) (t), → qi+1 , i = 1, 2, . . . , r − 2, with ⎧ qi ) (i) G (e (t), −λi− − εi+ , ϕi− + εi− , qi+1 (t−)), ⎪ ⎪ ⎨ if qi (t) = true qi+1 (t) = G (e(i) (t), ϕi+ − εi+ , λi+ + εi− , qi+1 (t−)), ⎪ ⎪ ⎩ if qi (t) = false 0 ∈ {true, false} qi+1 (0−) = qi+1

(1.6)

Br −1 : (e(r −1) (t), qr −1 ) → q given as above with i = r − 1 and q = qr , q(0−) = q 0 ∈ {true, false}

(1.7)

and Bi is

and Br −1 is

where G (e, e, e, qold ) := [e ≥ e ∨ (e > e ∧ qold )], e(·) is the upper switch trigger , e(·) is the lower switch trigger, e(t) is the tracking error of system output which drives the BCFC, qold ∈ {true, false}, ϕi± ∈ R are the constant funnel boundary values used to define funnels Fi := {e(i) (t) ∈ R|ϕi− ≤ e(i) (t) ≤ ϕi+ }, λi± ∈ R≥0 represent the desired increase or decrease rate of e(i−1) (t), εi± ∈ R≥0 are designed as the safety distances, which trigger a switch when the error or its derivatives get close to the funnel boundaries, q(t−) := limε→0+ q(t − ε). q(t) is the output of the switching logic, which maps the tracking error e(t) to the switching signal q(t).

1.1 Impulsively Disturbed Nonlinear System and Feasibility Assumptions …

7

1.1.2 Feasibility Assumptions of the Switching Controller The SC is designed to switch between a CC and a BCFC. The relative degree of system (1.1) is assumed to be known and following feasibility assumptions are made for the BCFC and the CC. Specifically, F1 − F7 are the feasibility assumptions of the BCFC, and F8 is the feasibility assumption of the CC. F1

F2 F3 F4

F5 F6

System (1.1) can be transformed to its equivalent system written in ByrnesIsidori normal form  (r ) y (t) = f (Y r −1 , z) + g(Y r −1 , z)u (1.8) z˙ = h(Y r −1 , z) where Y r −1 := (y, y˙ , . . . , y (r −1) ), Y r −1 (0) ∈ Rr , z(0) = z 0 ∈ Rn−r is the zero state vector of system (1.1), f, g, h are locally Lipschitz continuous, g is positive. Moreover, we assume that the z-system does not have a finite escape time for any bounded “input” vector Y r −1 , namely, ∀Y r −1 ∈ L ∞ (R≥0 → R)r and ∀z 0 ∈ Rn−r , there exists a global solution z : R≥0 → Rn−r for z˙ (t) = h(Y r −1 , z), z(0) = z 0 , where L ∞ (X → Y ) denotes the set of all measurable and essentially bounded functions f : X → Y with the supremum norm f ∞ . yref ∈ C r −1 (R≥0 → R), where C r −1 (X → Y ) or short C r −1 denotes the set of all (r − 1)-times continuously differentiable functions f : X → Y . ϕ0+ − ε0+ > ϕ0− + ε0− and for ∀i ∈ {1, . . . , r − 1}: ϕi+ − εi+ > εi− + λi+ , ϕi− + εi− < −λi− − εi+ with εi± > 0 and λi± > 0. There exist numbers Δi+ > 0, Δi− > 0 for i = 0, 1, . . . , r − 1 and Δr+ ≥ 0, + + − + (|ϕi+ | + |ϕi− |)/λi+1 and Δi− ≥ Δi+1 + (|ϕi+ | + Δr− ≥ 0 such that Δi+ ≥ Δi+1 − − |ϕi |)/λi+1 for i = 0, . . . , r − 1. + + + 2 − − − − 2 + εi+ > Δi+2 |ϕi+1 | + |ϕi+1 | /(2λi+2 ) and εi− > Δi+2 |ϕi+1 | + |ϕi+1 | /(2λi+2 ) hold for i = 0, 1, . . . , r − 2. ⎧ (r ) + f (yt0 ,yt1 ,...,ytr −1 ,z t ) ⎪ ⎨ U + ≥ λr +yref (t)− g(yt0 ,yt1 ,...,ytr −1 ,z t ) (1.9) (r ) − f (yt0 ,yt1 ,...,ytr −1 ,z t ) ⎪ ⎩ U − ≤ −λr +yref (t)− g(y 0 ,y 1 ,...,y r −1 ,z ) t

t

t

t

holds for all t ≥ 0, (yt0 , yt1 , . . . , ytr −1 ) ∈ Φt ref , z t ∈ Z t ref , where U ± are the y values of the control variable and Rr ⊇ Φt ref := y

y

  ∀i ∈ 0, 1, . . . , r − 1 :  (y0 , . . . , yr −1 ) (i) yi (t) − yref (t) ∈ [−χi , χi ]

8

1 Switching Control of Impulsively Disturbed Nonlinear … (i) where χi is the boundary value of yi (t) − yref (t) of system (1.8) controlled by the SC following an impulsive disturbance occurring in the dynamics of y y (r −1) (t), which will be discussed in Sect. 1.3.1. Moreover, Z t ref :=

⎧ ⎪ ⎪ ⎨

⎫ . , y (r −1) , z), z(0) ⎪  z solves z˙ = h(y, y˙ , . .n−r ⎪ ⎬  = z 0 , for some z 0 ∈ R and some  z(t) r −1 (r −1) y∈C with (y(τ ), . . . , y (τ )) ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ y ∈ Φτ ref , τ ∈ [0, t]. F7

F8

The maximum control value Umax of system (1.1) satisfies ∞ > Umax > U ± to guarantee the operation of the BCFC within the error funnels [1], where U ± y is obtained by (1.9) with (yt0 , yt1 , . . . , ytr −1 ) ∈ Φt ref := {(y0 , . . . , yr −1 )|∀i ∈ (i) − + 0, 1, . . . , r − 1 : yi (t) − yref (t) ∈ [ϕi , ϕi ]}. The closed-loop system consisting of (1.1) and the CC is asymptotically stable in F , which denotes a domain of x 0 defined as F := {e(i) (t)|e(t) ∈ F0 , . . . , e( j) (t) ∈ F j , . . . , e(r −1) (t) ∈ Fr −1 } and x 0 is the initial state of system (1.1).

1.1.3 Description of Impulsive Disturbance Concerning that an impulsive disturbance occurs in the dynamics of y (r −1) (t) at t = t0 , we assume that (1.8) holds on [0, t0 ) and (t0 , ∞), and at the time t0 there is a jump in y (r −1) (t) in the following form: y (r −1) (t0 +) = y (r −1) (t0 −) + Δy (r −1)

(1.10)

for some Δy (r −1) ∈ R. In practice, the control signal cannot be arbitrary large, namely, |U ± | ≤ Umax should be satisfied. Meanwhile, F7 ensures that Umax is large enough to guarantee the operation of the BCFC. In order to find the maximum impulsive disturbance that system (1.1) can endue, a searching procedure is given in Fig. 1.1, where h > 0 is the search step length. (r −1) is the maximum impulsive disturbance β that the nonlinear The obtained Δym+1 system can endure with the maximum control input Umax . If the impulsive disturbance |Δy (r −1) | is larger than β, then neither of the SC or the CC can maintain the nonlinear system stable.

1.2 IDRA of the Nonlinear System Controlled by SC Assume that the output of the BCFC is u BCFC (t), and the output of the CC is u CC (t), and u SC (t) denotes the output of the SC. Then the main results of this chapter can be stated as follows.

1.2 IDRA of the Nonlinear System Controlled by SC

9

Fig. 1.1 The searching procedure of the maximum impulsive disturbance that the nonlinear system can endure

1.2.1 Main Results on IDRA of Impulsively Disturbed Nonlinear Systems Theorem 1.1 (IDRA of the impulsively disturbed nonlinear system controlled by the SC) Consider the nonlinear system (1.1) satisfying F1 with known relative degree r > 0, an impulsive disturbance Δy (r −1) occurring in the dynamics of y (r −1) (t) as presented in (1.10) at t = t0 , a reference signal yref (t) satisfying F2 , system (1.1) controlled by a SC given by the following control law: u SC (t) = γ (t) · u BCFC (t) + (1 − γ (t)) · u CC (t) where γ (t) is obtained according to a state-dependent switching strategy T , the error funnels Fi (i = 0, 1, . . . , r − 1) of the BCFC defined via funnel boundaries ϕi± satisfying F3 –F5 , and the BCFC given by the switching logic defined in (1.5)– (1.7) driven by the tracking error e = y − yref , the CC satisfying F8 , if the input values U + and U − of the BCFC are large enough in the sense of F6 , the available maximum control Umax of (1.1) satisfies F7 , and the impulsive disturbance satisfies |Δy (r −1) | ≤ β, then the closed-loop system consisting of (1.1) and its SC

10

1 Switching Control of Impulsively Disturbed Nonlinear …

has for all initial values, x 0 ∈ Rn , γ 0 ∈ {true, false}, q0 ∈ {true, false}r , a unique maximally extended solution:(x, γ , q) : [0, ∞) → Rn × {0, 1} × {true, false}r , and γ and q have in each compact interval within [0, ∞) only finitely many jumps, respectively, and the system output tracking error and its derivatives are ensured to converge to zero after an impulsive disturbance occurs in the dynamics of y (r −1) (t) at t = t0 . Theorem 1.2 (Convergence of the impulsively disturbed nonlinear system controlled by the BCFC) Consider the nonlinear system (1.1) satisfying F1 with known relative degree r > 0, an impulsive disturbance Δy (r −1) occurring in the dynamics of y (r −1) (t) as presented in (1.10) at t = t0 , nonlinear system (1.1) controlled by a SC switching between a CC and a BCFC via a state-dependent switching strategy T , a reference signal yref (t) satisfying F2 , error funnels Fi (i = 0, 1, . . . , r − 1) of the BCFC defined via funnel boundaries ϕi± satisfying F3 –F5 and the BCFC given by the switching logic defined in (1.5)–(1.7) driven by the output tracking error e(t) = y(t) − yref (t), and the BCFC switched on operation on interval [t1 , t2 ) ⊆ R(t2 ∈ (t1 , ∞]), if the input values U + , U − of the BCFC are large enough in the sense of F6 , the available maximum control Umax of (1.1) satisfies F7 , and the impulsive disturbance satisfies |Δy (r −1) | ≤ β, then when the SC switches to the BCFC, the closed-loop system consisting of (1.1) and the BCFC has for all initial values, x t1 ∈ Rn , qt1 ∈ {true, false}r , a unique maximally extended solution (x, q) : [t1 , t2 ) → (Rn × {true, false}) such that q has only locally finitely many switches in each finite interval within [t1 , t2 ) (i.e. Zeno behavior does not occur) and the system output tracking error and its derivatives are ensured to converge into the error funnels within a time length of ζr (ζr ∈ (0, ∞)) after the impulsive disturbance occurs in the dynamics of y (r −1) (t) at t = t0 , i.e. e(i) (t) ∈ [ϕi− , ϕi+ ](i ∈ {0, 1, . . . , r − 1}) holds for all t ≥ t0 + ζr , and it follows that |e(i) (t)|max ≤ χi (i = 0, 1, . . . , r − 1), where χi is the boundary value of |e(i) (t)| and its detailed expression is presented in (1.13).

1.2.2 State-Dependent Switching Strategy T of SC Let sequence Γ := {Γ1 , Γ2 , . . . , Γ j } denote the maximums and the minimums of the system output tracking error e(t), and let |Γ | := {|Γ1 |, |Γ2 |, . . . , |Γ j |}. |Γs |(s ∈ {1, 2, . . . , j}) is the maximum of sequence |Γ |. Then γ (t) is determined by the following state-dependent switching strategy.  γ (t) =

1, if T1 is true 0, if T2 is true

where T1 :

{e(r −1) (t) > ϕr+−1 }∨{e(r −1) (t) < ϕr−−1 },

(1.11)

1.2 IDRA of the Nonlinear System Controlled by SC

11

T2 : {{(|Γs | − |Γ j |)/|Γs | ≥ τ > 0} ∨ {Γ j−1 ∈ [ϕ0− , ϕ0− + ε0− ] and Γ j ∈ [ϕ0+ − ε0+ , ϕ0+ ]} ∨ {Γ j−1 ∈ [ϕ0+ − ε0+ , ϕ0+ ] and Γ j ∈ [ϕ0− , ϕ0− + ε0− ]} } ∧ {e(i) (t) ∈ [ϕi− , ϕi+ ] (i ∈ {1, . . . , r − 2})} ∧ {e(r −1) (t) ∈ [(ϕr−−1 + εr−−1 ), (ϕr+−1 − εr+−1 )]}. Remark 1.1 Two scenarios are included in T2 . Precisely, {(|Γs | − |Γ j |)/|Γs | ≥ τ > 0} ∧ {e(i) (t) ∈ [ϕi− , ϕi+ ] (i ∈ {1, . . . , r − 2})} ∧ {e(r −1) (t) ∈ [(ϕr−−1 + εr−−1 ), (ϕr+−1 − εr+−1 )]} functions in the case that impulsive disturbance is so large that all of e(i) (t)(i ∈ {0, 1, . . . , r − 1}) are perturbed outside the error funnels. {{Γ j−1 ∈ [ϕ0− , ϕ0− + ε0− ] and Γ j ∈ [ϕ0+ − ε0+ , ϕ0+ ]} ∨ {Γ j−1 ∈ [ϕ0+ − ε0+ , ϕ0+ ] and Γ j ∈ [ϕ0− , ϕ0− + ε0− ]}} ∧ {e(i) (t) ∈ [ϕi− , ϕi+ ] (i ∈ {1, . . . , r − 2})} ∧ {e(r −1) (t) ∈ [(ϕr−−1 + εr−−1 ), (ϕr+−1 − εr+−1 )]} functions in the case that e(0) (t) is within the error funnel F0 despite of the impulsive disturbance, and only some of e(i) (t)(i ∈ {1, 2, . . . , r − 1}) are perturbed outside the corresponding error funnels.

1.2.3 Control Law of BCFC On the basis of g(Y r −1 , z) > 0, the control law of the BCFC is given by  u BCFC (t) =

U − , if q(t) = true U + , if q(t) = false.

(1.12)

1.2.4 Solution Framework of the Closed-Loop System Consisting of the Nonlinear System and Its SC In the first place, we will show that a maximally extended solution γ (t) can be obtained for (1.11) in the close-loop system consisting of (1.1) and its SC on some interval [t1 , t2 ) ⊆ R with any initial value γ t1 ∈ {0, 1}. Lemma 1.1 Consider the nonlinear system (1.1) satisfying F1 , an impulsive disturbance occurring in the dynamics of y (r −1) (t) as presented in (1.10) at t = t0 , system (1.1) controlled by the SC, which switches between a BCFC and a CC via a state-dependent switching strategy T on some interval [t1 , t2 ) ⊆ R(t2 ∈ (t1 , ∞]) with any γ t1 ∈ {0, 1}, if δ := min{εr−−1 , εr+−1 } > 0 holds, then (1.11) has a unique maximally extended solution γ : [t1 , t2 ) → {0, 1}, which is right continuous, i.e. for all t ∈ [t1 , t2 ), there exists ε > 0 such that γ is constant on [t, t + ε). Furthermore, the jumps of γ cannot accumulate within any compact subset of [t1 , t2 ), in particular, γ (t−) := limε→0+ γ (t − ε) is for all t ∈ (t1 , ω) well defined, where ω ∈ (t1 , ∞]. Proof Let us discuss the case that {e(r −1) (t0 ) > ϕr+−1 } ∨ {e(r −1) (t0 ) < ϕr−−1 } = true first. For any fixed t ∈ [t1 , t2 ), γold = γ (t−) = 1 and γnew = γ (t) = 0 if and only if T2 is satisfied at the time t, which also excludes the case that t = t0 . For t0 ∈ (t, t2 ], e(r −1) (t) is continuous on the interval [t, t0 ). By the continuity of

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e(t) and e(r −1) (t), it follows that γnew = 0 and γold = 1 implies γ (t + θ ) = 0 for all small enough θ > 0. Thus there exists ε > 0 such that γ |[t,t+ε) ≡ 0 solves (1.11). Choose the maximal ε > 0 such that the constant γ |[t,t+ε) solves (1.11). This implies that at the time t + ε, T1 is satisfied, i.e. {e(r −1) (t + ε) > ϕr+−1 } ∨ {e(r −1) (t + ε) < ϕr−−1 }=true. In a similar way, it can be verified that there exists ε > 0 such that γ |[t+ε,t+ε+ε ) ≡ 1 solves (1.11). Let us discuss the case t0 ∈ (t + ε + ε , t2 ) first. Assume the contradiction, i.e. T2 is satisfied at the time (t + ε + ε ) with ε → 0. This implies that e(r −1) (t + ε + ε ) = ϕr+−1 − εr+−1 and e(r −1) (t + ε) = ϕr+−1 (or e(r −1) (t + ε + ε ) = ϕr−−1 + εr−−1 and e(r −1) (t + ε) = ϕr−−1 ). Concern t+ε+ε (r )  t+ε+ε (r )  e (t), it has | t+ε e (t)dt| = ing e(r −1) (t + ε + ε ) = e(r −1) (t + ε) + t+ε  t+ε+ε + − + | f + g · u(t) − yref |dt = εr −1 (or εr −1 ). It follows that limε →0 εr −1 (or εr−−1 ) t+ε  t+ε+ε  t+ε+ε  t+ε+ε ≤ limε →0 t+ε | f − yref |dt + limε →0 t+ε |g|dt · t+ε |u(t)|dt. By F1 and F2 , f, g and yref are continuous functions, and thus they are locally bounded. Then  t+ε+ε  t+ε+ε it has limε →0 t+ε | f − yref |dt = 0 and limε →0 t+ε |g|dt = 0, which implies  t+ε+ε that limε →0 t+ε |u(t)|dt → ∞. This conflicts with F7 , whence the contradiction is obtained. Then choose the maximal ε > 0 such that the constant γ |[t+ε,t+ε+ε ) solves (1.11). This implies that at the time t + ε + ε , it has γ (t + ε + ε ) = 0, which excludes the case that t0 = t + ε + ε . For the case that t + ε = t0 , the same results can be obtained as presented above. The only difference is that e(r −1) (t + ε) < ϕr−−1 (or e(r −1) (t + ε) > ϕr+−1 ). For the case that t0 ∈ (t + ε, t + ε + ε ), it can be obviously seen that the appearance of Δy (r −1) does not influence that γ (t) = 1 holds on interval [t + ε, t + ε + ε ). Hence we have shown that we can extend the solution onto a maximal interval [t1 , ω). It remains to show that ω = t2 . Seeking a contradiction, assume ω < t2 , which implies ω < ∞. Considering that e(r −1) (t) has a jump at the time t0 caused by the impulsive disturbance Δy (r −1) , e(r −1) (t) is continuous on the interval [t1 , t0 ) and [t0 , t2 ), respectively. Then there exists ε > 0 such that ∀t ∈ (ω − ε, ω] ⊆ [t1 , t0 ) and ∀t ∈ (ω − ε, ω] ⊆ [t0 , t2 ): |e(r −1) (t) − er −1 (ω)| < δ/2. This implies that either e(r −1) (t) > ϕr+−1 − εr+−1 or e(r −1) (t) < ϕr−−1 + εr−−1 or ϕr−−1 < e(r −1) (t) < ϕr+−1 hold for all t ∈ (ω − ε, ω]. Hence there can be at most one jump of γ (t) on (ω − ε, ω) in this case. For the case that t0 ∈ (ω − ε, ω] ⊆ [t1 , t2 ), there exists ε > 0 such that ∀t ∈ (ω − ε, t0 ): |e(r −1) (t) − e(r −1) (t0 −)| < δ/2 and ∀t ∈ [t0 , ω]: |e(r −1) (t) − e(r −1) (ω)| < δ/2 since e(r −1) (t) is continuous on the interval (ω − ε, t0 ) and [t0 , ω], respectively. If γ (t0 ) = γ (t0 −), then it must be γ (t0 ) = 1 and γ (t0 −) = 0 since we are discussing the case that {e(r −1) (t0 ) > ϕr+−1 } ∨ {e(r −1) (t0 ) < ϕr−−1 }. Similarly, by ∀t ∈ (ω − ε, t0 ): |e(r −1) (t) − e(r −1) (t0 −)| < δ/2, it follows that there can be at most one jump of γ (t) on (ω − ε, t0 ). Concerning ∀t ∈ [t0 , ω]: |e(r −1) (t) − e(r −1) (ω)| < δ/2 and T2 , it follows that γ (t) ≡ 1 holds on the interval [t0 , ω]. Hence, there can be at most two jumps of γ (t) on the interval (ω − ε, ω] in the case that t0 ∈ (ω − ε, ω]. Therefore, γ (ω−) is well defined and yields a unique γ (ω) which can be extended as above. This contradicts the maximality of ω. In particular, this shows that the jumps of γ (t) cannot accumulate in any compact subset of [t1 , t2 ).

1.2 IDRA of the Nonlinear System Controlled by SC

13

With respect to the case that e(r −1) (t0 ) ∈ [ϕr−−1 , ϕr+−1 ], if γ t1 = 0, then system (1.1) is only controlled by the CC during t ∈ [t1 , t2 ), and constant γ (t) ≡ 0 solves (1.11) on the interval t ∈ [t1 , t2 ), and then there is nothing to show. If γ t1 = 1, for the case that e(i) (t1 ) ∈ Fi (i = 0, 1, . . . , r − 1), then there exists ε > 0 such that γ (t) = 1 solves (1.11) on the interval [t1 , t1 + ε). Assume the contradiction, i.e. T2 is satisfied at the time t1 + ε with ε → 0. Referring to T2 , it only needs to show {(|Γs | − |Γ j |)/|Γs | ≥ τ > 0} ∨ {Γ j−1 ∈ [ϕ0− , ϕ0− + ε0− ] and Γ j ∈ [ϕ0+ − ε0+ , ϕ0+ ]} ∨ {Γ j−1 ∈ [ϕ0+ − ε0+ , ϕ0+ ] and Γ j ∈ [ϕ0− , ϕ0− + ε0− ]} = true holds at the time t1 + ε. By the continuity of e(t) on the interval [t1 , t2 ), limε→0 (|Γs | − |Γ j |)/|Γs | = τ = 0, which contradicts τ > 0. Moreover, limε→0 Γ j−1 = limε→0 Γ j and the continuity of e(t) on the interval [t1 , t2 ) imply either ϕ0− + ε0− ≥ ϕ0+ − ε0+ or e(t) is constant on the interval [t1 , t1 + ε). The previous case contradicts F3 . The latter case implies that e(t) ˙ ≡ 0 holds on the interval [t1 , t1 + ε). Then it follows that e(i) (t) ≡ 0(i = 2, 3, . . . , r ) on the interval [t1 , t1 + ε). By F6 , it follows that (r ) > λr+ (or < −λr− ). This implies that λr± < 0, which cone(r ) (t) = f + g · u − yref tradicts F3 . Whence the contradiction results in. Choose the maximal ε > 0 such that the constant γ |[t1 ,t1 +ε) = 1 solves (1.11). This implies that at the time t1 + ε, T2 is satisfied. Since e(r −1) (t0 ) ∈ [ϕr−−1 , ϕr+−1 ], the constant γ (t)|[t1 +ε,t2 ) = 0 with t2 ∈ (t1 + ε, ∞] solves (1.11). To this end, Lemma 1.1 is verified. Lemma 1.1 presents the solution framework of the switching logic of the SC. The solution framework of the closed-loop system consisting of (1.1) and the BCFC is concerned in the following. Assume that the BCFC is switched on operation on the interval [t1 , t2 ), which means the SC switches from the BCFC to the CC at the time t2 . Let us consider the cases that t0 ∈ [0, t1 ] and t0 ∈ (t2 , ∞] first. The closed-loop system, consisting of the nonlinear system (1.1) and its BCFC, is a hybrid system. Lemma 5.1 of [1] shows the right-continuity and well-posedness of the switching signal q of the BCFC, and the right continuity of q is determined by the switching logic S : e → q itself. Concerning the switching logic of the BCFC is the same as that of the BBCF despite of its constant funnel values, Lemma 5.1 of [1] still holds in this chapter. Moreover, Lemma 5.2 of [1] shows that the dynamic logic system (DLS) induced by q(t) = G (e(t), e(t), ¯ e(t), q(t−)) produces right-continuous outputs provided that switching triggers e(t) ¯ and e(t) are continuous and do not intersect. It can be shown that Lemma 5.2 of [1] still holds in the cases that e(t) is out of the error funnels, i.e. e(t) > e(t) ¯ or e(t) < e(t) on the interval [t1 , ω) with ω ∈ (t1 , t2 ). Referring to G (e, e, ¯ e, qold ) = [e ≥ e¯ ∨ (e > e ∧ qold )], it follows that q(t) ≡ true holds on t ∈ ¯ and q(t) ≡ false holds on t ∈ [t1 , ω) in the case [t1 , ω) in the case that e(t) > e(t), that e(t) < e(t). Therefore, there exists a maximal ε > 0 such that the constant q|[t1 ,t1 +ε) solves the DLS. This implies that at the time t1 + ε a trigger was hit, i.e. q(t1 + ε) = q((t1 + ε)−) when the upper trigger was hit in the case that e(t) < e(t) or the lower trigger was hit in the case that e(t) > e(t). ¯ After that, e(t) enters the error funnel, and the solution q(t) can be extended as presented in Lemma 5.2 of [1]. And Zeno behavior is excluded in the working process of the DLS.

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Let us consider the case that t0 ∈ (t1 , t2 ). In the same way, Lemma 5.1 of [1] can be verified. It only needs to show the Lemma 5.2 of [1]. Since e(i) (t)(i = 0, 1, . . . , r − 2) are continuous on [t1 , t2 ) in this case as well. Thus we just consider the DLS of Br −1 with input e(r −1) (t). Similar with Lemma 5.2 of [1], it can be shown that the solution of the DLS of Br −1 can be extended onto a maximal interval [t1 , ω). Then it remains to show that ω = t2 . Seeking a contradiction, assume ω < ∞, which implies ω < ∞. There exists ε > 0 such that t0 ∈ (ω − ε, ω]. The case that t0 being out of interval (ω − ε, ω] is the same as the case discussed in Lemma 5.2 of [1], and thus it is not concerned here. Let δ := e(t) ¯ − e(t) > 0. By continuity of e(r −1) (t) on the interval (ω − ε, t0 ), in can be satisfied that ∀t ∈ (ω − ε, t0 ) : |er −1 (t) − e(r −1) (t0 −)| < δ/2. This implies that ¯ on (ω − ε, t0 ). Hence, there either e(r −1) (t) > e(t) on (ω − ε, t0 ) or e(r −1) (t) < e(t) can be at most one jump of q on interval (ω − ε, t0 ). At the time t0 , the impulsive disturbance occurs and the following two cases should be considered.

1.2.4.1

Case 1 {{e(r −1) (t0 ) < ϕr−−1 } ∨ {e(r −1) (t0 ) > ϕr+−1 }} ∧ {|Δy (r −1) | ≤ β}

In this case, at most one jump of q may happen at t = t0 , i.e. q(t0 −) = true and e(r −1) (t0 ) < ϕr−−1 or q(t0 −) = false and e(r −1) (t0 ) > ϕr+−1 . By choosing a small enough ε, it can be satisfied that ∀t ∈ [t0 , ω] : |e(r −1) (t) − e(ω)| < δ/2. This implies ¯ on [t0 , ω], hence there that either e(r −1) (t) > e(t) for all t ∈ [t0 , ω] or e(r −1) (t) < e(t) can be at most one jump of q on [t0 , ω].

1.2.4.2

Case 2 e(r −1) (t0 ) ∈ [ϕr−−1 , ϕr+−1 ]

In this case, at most one jump of q may happen at the time t0 , namely, q(t0 −) = true and e(r −1) (t0 ) ∈ [ϕr−−1 , ϕr−−1 + εr−−1 ] (or q(t0 −) = false and e(r −1) (t0 ) ∈ [ϕr+−1 − εr+−1 , ϕr+−1 ]). With a small enough ε, it follows that ∀t ∈ [t0 , ω] : |e(r −1) (t) − e(ω)| < ¯ for all t ∈ [t0 , ω] in the case that δ/2. This implies that either e(r −1) (t) < e(t) e(r −1) (t0 ) ∈ [ϕr−−1 , ϕr−−1 + εr−−1 ] or e(r −1) (t) > e(t) for all t ∈ [t0 , ω] in the case that e(r −1) (t0 ) ∈ [ϕr+−1 − εr+−1 , ϕr+−1 ]. Then there can be at one jump of q on [t0 , ω]. Hence, in the case that t0 ∈ (ω − ε, ω], there can be at most two jumps of q on the interval (ω − ε, ω], and q(ω−) is well defined, and it yields a unique q(ω), which can be extended as above. This conflicts with the maximality of ω, whence Lemma 5.2 of [1] is verified for the case that t0 ∈ (t1 , t2 ).

1.2 IDRA of the Nonlinear System Controlled by SC

15

Theorem 1.3 (Well-posedness of the closed-loop system consisting of (1.1) and its BCFC (Theorem 5.3 of [1])) Consider the nonlinear system (1.1) satisfying F1 with known relative degree r > 0, an impulsive disturbance occurring in the dynamics of y (r −1) (t) at the time t0 as presented in (1.10), system (1.1) controlled by a SC switching between a CC and a BCFC via a state-dependent switching strategy T , a reference signal yref (t) satisfying F2 , error funnels Fi (i = 0, 1, . . . , r − 1) defined via ϕi± satisfying F3 –F5 and the BCFC given by the switching logic defined in (1.5)– (1.7) driven by the tracking error e = y − yref , and the BCFC switched on operation on the interval [t1 , t2 ) ⊆ R, then the closed-loop system consisting of (1.1) and its BCFC has for all initial values x t1 ∈ Rn , qt1 ∈ {true, false}r , a unique maximally extended solution (x, q) : [t1 , t2 ) → Rn × {true, false}r . Furthermore, q has in each compact interval within [t1 , t2 ) only finitely many jumps. Proof For the cases that t0 ∈ [0, t1 ] and t0 ∈ (t2 , ∞], e(i) (t)(i = 0, 1, . . . , r − 1) is continuous on the interval [t1 , t2 ) during which the BCFC is switched on operation. Then the necessary conditions for the verification of Theorem 5.3 of [1] are met. By Lemmas 5.1 and 5.2 of [1] and the supplementary statements presented above, the well-poseness of the closed-loop system consisting of (1.1) and its BCFC on the interval [t1 , t2 ) can be just verified as presented in Sect. 5 of [1]. Moreover, the constant funnel values of the BCFC makes the verification of Theorem 1.3 here much easier than that in [1]. Whence Theorem 1.3 can be verified. With respect to the case that t0 ∈ (t1 , t2 ), e(r −1) (t) is piece-wise continuous on the interval (t1 , t0 ) and [t0 , t2 ), respectively. Meanwhile, e(i) (t) is continuous for i = 0, 1, . . . , r − 2. By comparison, it can be seen that the inductive derivations presented in the proof of Theorem 5.3 also suits for the case here. The only difference lies in the solution of the DLS of Br −1 , whose input is e(r −1) (t). Nevertheless, we have shown in the above that there exists a right-continuous unique solution q : [0, ω) → {true, false} of the DLS of Br −1 , and q has in each compact interval within [t1 , ω) only finitely many jumps. Whence Theorem 1.3 is verified for the case that an impulsive disturbance occurs in the dynamics of y (t−1) (t). Combining the results of Lemma 1.1 and Theorem 1.3, the well-poseness of the closed-loop system consisting of (1.1) and its SC can be stated as follows. Theorem 1.4 (Well-posedness of the closed-loop system consisting of (1.1) and its SC) Consider the nonlinear system (1.1) satisfying F1 with known relative degree r > 0, an impulsive disturbance Δy (r −1) occurring in the dynamics of y (r −1) (t) at t = t0 satisfying (1.10), a reference signal yref (t) satisfying F2 , system (1.1) controlled by a SC switching between a CC and a BCFC according to switching strategy T , error funnels Fi (i = 0, 1, . . . , r − 1) of the BCFC defined via ϕi± satisfying F3 –F5 ,and the BCFC given by the switching logic defined in (1.5)–(1.7) driven by the tracking error e = y − yref , the CC satisfying F8 , then the closed-loop system consisting of (1.1) and its SC has for all initial values, x 0 ∈ Rn , γ 0 ∈ {true, false}, q0 ∈ {true, false}r , a unique maximally extended solution:(x, γ , q) : [0, ∞) → Rn × {0, 1} × {true, false}r , and γ and q have in each compact interval within [0, ∞) only finitely many jumps, respectively.

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1 Switching Control of Impulsively Disturbed Nonlinear …

Proof Lemma 1.1 shows that the switching strategy (1.11) has for all initial values γ 0 ∈ {true, false} a unique maximally extended solution γ (t) : [0, ∞) → {0, 1}. Assume that the BCFC is switched on operation on [t1k , t2k )(k ∈ N), and let q(t) ≡ q(t2k ) on the interval [t2k , t1(k+1) ), then q(t1k −) is always well defined. With respect to the closed-loop system consisting of (1.1) and the CC on interval [t2k , t1(k+1) ), it is a continuous system and it only works in the domain e(i) (t) ∈ F (i = 0, 1, . . . , r − 1). Referring to F8 , the closed-loop system consisting of (1.1) and the CC is stable and has for all initial values x t2k ∈ Rn a unique maximally extended solution x : [t2k , t1(k+1) ) → Rn . Therefore, combined with Theorem 1.3, the solution of the closed-loop system as follows. q :=  consisting of (1.1) and the SC can be denoted   q| + q| , x := x| + x| [t ,t ) [t ,t ) [t ,t ) [t2k ,t1(k+1) ) , where 1k 2k 2k 1(k+1) 1k 2k k∈N k∈N k∈N k∈N t11 = 0 for the case γ 0 = 1 and t21 = 0 for the case that γ 0 = 0. Whence Theorem 1.4 is verified.

1.3 Proof of Main Results 1.3.1 Boundary Values of e(i) (t)(i = 0, 1, . . . , r − 1) and Their Recovery Time of the Impulsively Disturbed Nonlinear System Controlled by BCFC As concerned in Theorems 1.1 and 1.2, an impulsive disturbance Δy (r −1) occurs in the dynamics of y (r −1) (t) at t = t0 , and the case that {|Δy (r −1) | ≤ β} ∧ {{e(r −1) (t0 ) < ϕr−−1 } ∨ {e(r −1) (t0 ) > ϕr+−1 }} is considered here. Moreover, the boundary values of e(i) (t)(i = 0, 1, . . . , r − 1) and their recovery time are only discussed on the interval [t0 , tc ), during which the BCFC is switched on operation due to the impulsive disturbance. And tc denotes the time when the SC switches from the BCFC to the CC. In terms of the other cases, in which there is no external disturbance or all of e(i) (t)(i = 0, 1, . . . , r − 1) are contained within the error funnels Fi for all t ∈ [0, ∞), such as the case that e(r −1) (t0 ) ∈ [ϕr−−1 , ϕr+−1 ] and γ 0 = 1, the operation process of the BCFC is the same as that in [1], and there is nothing to show. Furthermore, we just consider the case that Δy (r −1) < 0 and the case of Δy (r −1) > 0 follows analogously. Let ζi ∈ (0, tc − t0 )(i = 1, 2, . . . , r − 1) denotes the shortest time length that e(r −i) (t)(i = 1, 2, . . . , r − 1) takes to converge into [λr+−i + εr−−i , ϕr+−i − εr+−i ] since the impulsive disturbance occurs at the time t0 . And let ζr ∈ (0, tc − t0 ) represent the shortest time length that e(0) (t) takes to converge back to zero since t = t0 . Before the impulsive disturbance occurs, system (1.1) operates at its equilibrium point and thus e(i) (t0 −) = 0(i = 0, 1, . . . , r − 1). Considering the impulsive disturbance Δy (r −1) occurring in the dynamics of y (r −1) (t) at the time t0 , we have e(r −1) (t0 ) = Δy (r −1) , e(i) (t0 ) = 0(i = 0, 1, . . . , r − 2). According to the switching logic of Br −1 , it can be obtained that q(t) = false and u(t) = U + hold on the interval (r ) (t) + f (y(t), y˙ (t), . . . , y (r −1) (t), z(t)) t ∈ [t0 , t0 + ζ1 ). Thus we have e(r ) (t) = −yref

1.3 Proof of Main Results

17

+ g(y(t), y˙ (t), . . . , y (r −1) (t), z(t))U + on interval t ∈ [t0 , t0 + ζ1 ). Considering F6 , Then we have e(r −1) (t) = e(r −1) (t0 ) it yields e(r ) (t) ≥ λr+ .  t (r ) (r −1) + t0 e (t)dt ≥ e(r −1) (t0 ) + λr+ (t − t0 ). Let e(r −1) (t) = 0 and it has t ≤ − e λ+(t0 ) . Then let α1 = − e

(r −1)

(t0 )

and we have ζ1 ≤ α1 +     boundary value of e(r −1) (t) is χr −1 = Δy (r −1) . λr+

λr+−1 +εr−−1 λr+

r

= σ1 . Apparently, the

With respect to the case of e(r −2) (t), e(r −2) (t) keeps decreasing on the interas e(r −2) (t) = val [t0 , t0 + α1 ). When t ∈ [t0 , t0 + α1 ), e(r −2) (t) can bedenoted t t t t The e(r −2) (t0 ) + t0 e(r −1) (t)dt ≥ e(r −2) (t0 ) + t0 e(r −1) (t0 )dt + t0 t0 e(r ) (t)d2 t.    t +α t +α minimum of e(r −2) (t) is e(r −2) (t0 + α1 ) and thus χr −2 =  t00 1 Δy (r −1) dt + t00 1   t0 +α1 + 2  λr d t . Concerning that e(r −2) (t) is increasing during the interval [t0 + t0 t α , t + σ ] , thus e(r −2) (t) can be denoted as e(r −2) (t) = e(r −2) (t0 + α1 ) + t0 +α1  1τ 0 (r ) 1 that e(r ) (t) ≥ λr+ t0 +α1 e (τ )dτ dt on the interval t ∈ [t0 + α1 , t0 + σ1 ). Considering   t +σ holds on this interval, we have e(r −2) (t0 + σ1 ) ≥ e(r −2) (t0 + α1 ) + t00+α11 tτ0 +α1 λr+ dτ dt. There exists an interval t ∈ [t0 + σ1 , t0 + α1 + ε) with 0 < ε < Δr+−1 such that on this interval, and e(r −2) (t) can be denoted as e(r −2) (t) = e(r −2) (t) keeps increasing t (r −2) (r −1) (t0 + σ1 ) + t0 +σ1 e (t)dt. Since e(r −1) (t) ∈ [λr+−1 , ϕr+−1 ] holds on this intere val, let e(r −2) (t) = 0 and it can be obtained that t ≤ − e

ζ2 ≤ σ1 + α2 +

λr+−2 +εr−−2 λr+−1

(r −2)

(t0 +σ1 ) λr+−1

= α2 . Then it has

= σ1 + σ2 .

As for the situation of e(r −i) (t), it keeps decreasing on the interval [t0 , t0 + σ1 + · · · + σi−2 + αi−1 ). When t ∈ [t0 + σ1 + · · · + σi−2 , t0 + σ1 + · · · + σi−2 + αi−1 ), the dynamics of e(r −i) (t) can be written as e(r −i) (t) = e(r −i) (t0 ) +

t0 +σ1 +···+σ  i−3 +αi−2

e(r −i+1) (t0 )dt +

t0 +σ1 +···+σ  i−3 +αi−2

t0

t0

t0 +σ1 +···+σ  i−4 +αi−3

e(r −i+2) (t0 )d2 t + · · · +

t0 +σ1 +···+σ  i−3 +αi−2 t0 +σ1 +···+σ  i−4 +αi−3

t0

t0

t0 +σ  1 +α2 t0+α1 t

... t0

t0

e(r −1) (t0 )di−1 t +

t0

t0 +σ  1 +α2 t0+α1 t  t

... t0

t0

t0 t0

t0

e(r ) (t)di t +

t0

t0 +σ1 +···+σ  i−3 +αi−2 t0 +σ1 +···+σ  i−4 +αi−3 t0

t0 +σ1 +···+σ  i−3 +σi−2

t0 +σ1 +···+σi−3 +αi−2

e(r −i+1) (t0 + σ1 + · · · + σi−3 + αi−2 )dt +

t0 +σ1 +···+σ  i−3 +αi−2 t0 +σ1 +···+σ  i−4 +σi−3 t0

t0 +σ1 +···+σi−4 +αi−3

18

e

1 Switching Control of Impulsively Disturbed Nonlinear …

(r −i+2)

t0 +σ1 +···+σ  i−3 +αi−2 t0 +σ1 +···+σ  i−4 +αi−3

(t0 + σ1 + · · · + σi−4 + αi−3 )d t + · · · + 2

t0 t0 +σ  1 +α2 t0 +σ1

...

e(r −2) (t0 + α1 )di−2 t +

t0 +α1

t0

t

t0

t0 +σ1 +···+σ  i−3 +σi−2

t

t0 +σ1 +···+σi−3 +αi−2 t0 +σ1 +···+σi−3 +αi−2 t0 +σ1 +···+σ  i−3 +αi−2 t0 +σ1 +···+σ  i−4 +σi−3

e(r −i+3) (t)d3 t +

t0 +σ1 +···+σi−3 +αi−2

t0 +σ1 +···+σi−4 +αi−3

t0

t

t e

(r −i+4)

t0 +σ1 +···+σ  i−3 +αi−2

(t)d t + · · · +

...

4

t0 +σ1 +···+σi−4 +αi−3 t0 +σ1 +···+σi−4 +αi−3 t0 +σ  1 +α2 t0 +σ1

t

t

t0

t

(r )

e (t)d t + t0 +α1 t0 +α1 t0 +α1

t0

e

i

(r −i+1)

t0 +σ1 +···+σi−3 +σi−2 t0 +σ1 +···+σ  i−3 +αi−2

(t0 + σ1 + · · · + σi−3 + σi−2 )dt + t0

e

(r −i+2)

t

t0 +σ1 +···+σi−3

t0 +σ1 +···+σ  i−3 +αi−2

(t0 + σ1 + · · · + σi−3 )d t + · · · + t0

t

e(r −2) (t0 + σ1 )di−2 t +

t0 +σ  1 +α2

t

...

2

t

t0

t0 +σ1

e(r −i+2) (t)d2 t

t0 +σ1 +···+σi−3 +σi−2 t0 +σ1 +···+σi−3 +σi−2 t0 +σ1 +···+σ  i−3 +αi−2

t

t

+ t0 +σ1 +···+σi−3 t0 +σ1 +···+σi−3

t0 t0 +σ1 +···+σ  i−3 +αi−2

t0 +σ  1 +α2

t

t

... t0

e(r −i+3) (t)d3 t + · · · +

t0

e(r −1) (t)di−1 t

t0 +σ1 t0 +σ1

Therefore, the minimum of e(r −i) (t) is e(r −i) (t0 + σ1 + · · · + σi−2 + αi−1 ) and we   (r −i)  have e (t0 + σ1 + · · · + σi−2 + αi−1 ) < χr −i and χr −i is given in (1.13). Then e(r −i) (t) starts to increase on the interval [t0 + σ1 + · · · + σi−2 + αi−1 , t0 + σ1 + · · · + σi−2 written as e(r −i) (t) = e(r −i) (t0 + σ1 + · · · + σi−2 +  τ be(r −i+2)  t + σi−1 ) and it can (t)dτ dt. After that, ∃ε : 0 < ε < Δr+−i+1 such αi−1 ) + t0 +σ1 +···+σi−2 +αi−1 0 e that e(r −i) (t) keeps increasing on the interval [t0 + σ1 + · · · + σi−1 , t0 + σ1 + · · · + can be denoted as e(r −i) (t) = e(r −i) (t0 + σ1 + · · · + σi−1 ) σi−1  t + ε) and (rit−i+1) (t)dt. Concerning that e(r −i+1) (t) ≥ λr+−i+1 holds on this + t0 +σ1 +···+σi−1 e

1.3 Proof of Main Results

19

interval and let e(r −i+1) (t) = 0, we have t ≤ − e σ1 + · · · + σi−1 + αi +

λr+−i +εr−−i λr+−i+1

r −i

(t0 +σ1 +···+σi−2 +σi−1 ) λr+−i+1

= αi and ζi ≤

= σ1 + · · · + σi .

(0)

With respect to the case of e (t), it keeps decreasing on the interval [t0 , t0 + σ1 + · · · + σr −2 + αr −1 ). When t ∈ [t0 + σ1 + · · · + σr −2 , t0 + σ1 + · · · + σr −2 + αr −1 ), the dynamics of e(0) (t) can be denoted as in Appendix A.1, in which let i = r . There fore, the minimum of e(0) (t) is e(0) (t0 + σ1 + · · · + σr −2 + αr −1 ) and it has e(0) (t0 +   σ1 + · · · + σr −2 + αr −1 ) < χ0 . Then e(0) (t) starts to increase on the interval [t0 + (0) σ1 + · · · + σr −2 + αr −1 , t0 + σ1 + · ·· + σr −1 ) and it can  τ be denoted as e (t) = t e(0) (t0 + σ1 + · · · + σr −2 + αr −1 ) + t0 +σ1 +···+σr −2 +αr −1 0 e(2) (t)dτ dt. After that, ∃ε (0) : 0 < ε < Δ+ 1 such that e (t) keeps increasing on the interval [t0 + σ1 + · · · + σr −1 , t0 + σ1 + · · · + σr −1 + ε) and it can be written as e(0) (t) = e(0) (t0 + σ1 + t · · · + σr −1 ) + e(1) (t)dt. Considering that e(1) (t) ≥ λ+ 1 holds on this t0 +σ1 +···+σr −1

(0)

r −2 +σr −1 ) = αr interval and let e(0) (t) = 0, it can be obtained that t ≤ − e (t0 +σ1 +···+σ λ+ 1 and ζr ≤ σ1 + · · · + σr −1 + αr . The detailed expressions of χi (i = 0, 1, . . . , r − 2, r − 1) are as follows.

χr −1 = |Δy (r −1) | t 0 +α1

χr −2 = |

Δy

(r −1)

t 0 +α1 t

λr+ d2 t|

dt +

t0

t0

t0

χr −i =   

t0 +σ1 +···+σ  i−3 +αi−2 t0 +σ1 +···+σ  i−4 +αi−3

t0 +σ  1 +α2 t0 +α1 t

... t0

t0

t0

t0 +σ1 +···+σ  i−3 +αi−2 t0 +σ1 +···+σ  i−4 +αi−3

t0

e(r −1) (t0 )di−1 t

t0

t0 +σ  1 +α2 t0 +α1 t

t

...

+ t0

t0

t0 +σ1 +···+σ  i−3 +σi−2

+

t0

t0

t0

λr+ di t

t0

e(r −i+1) (t0 + σ1 + · · · + σi−3 + αi−2 )dt+

t0 +σ1 +···+σi−3 +αi−2 t0 +σ1 +···+σ  i−3 +αi−2 t0 +σ1 +···+σ  i−4 +σi−3

t0

e(r −i+2) (t0 + σ1 + · · · + σi−4 + αi−3 )d2 t

t0 +σ1 +···+σi−4 +αi−3 t0 +σ1 +···+σ  i−3 +αi−2 t0 +σ1 +···+σ  i−4 +αi−3

+ ··· +

t0 +σ  1 +α2 t0 +σ1

... t0

t0

t0

t0 +α1

20

1 Switching Control of Impulsively Disturbed Nonlinear …

e

(r −2)

t0 +σ1 +···+σ  i−3 +σi−2

(t0 + α1 )d

i−2

t

t

t+ t0 +σ1 +···+σi−3 +αi−2 t0 +σ1 +···+σi−3 +αi−2 t0 +σ1 +···+σi−3 +αi−2

λr+−i+3 d3 t

t0 +σ1 +···+σ  i−3 +αi−2 t0 +σ1 +···+σ  i−4 +σi−3

+ t0 +σ1 +···+σi−4 +αi−3 t0 +σ1 +···+σi−4 +αi−3

t0

t

λr+−i+4 d4 t

t0 +σ1 +···+σ  i−3 +αi−2

+ ··· +

t

t0 +σ  1 +α2 t0 +σ1

...

t0 +σ1 +···+σi−4 +αi−3

t

t

t0

t

λr+ di t +

t0 +α1 t0 +α1

t0

t0 +α1

e(r −i+1) (t0 + σ1 + · · · + σi−3 + σi−2 )dt

t0 +σ1 +···+σi−3 +σi−2

t0 +σ1 +···+σ  i−3 +αi−2

t

e(r −i+2) (t0 + σ1 + · · · + σi−3 )d2 t + · · · +

+ t0 +σ1 +···+σi−3

t0 t0 +σ1 +···+σ  i−3 +αi−2

t0 +σ  1 +α2

t

e(r −2) (t0 + σ1 )di−2 t+

... t0

t0 +σ1

t0

t0 +σ1 +···+σ i−3 +σi−2 +αi−1 t0 +σ1 +···+σi−3 +σi−2 +αi−1

λr+−i+2 d2 t

t0 +σ1 +···+σi−3 +σi−2

t0 +σ1 +···+σi−3 +σi−2

t0 +σ1 +···+σ  i−3 +αi−2

t

t

λr+−i+3 d3 t

+ t0 +σ1 +···+σi−3 t0 +σ1 +···+σi−3

t0

t0 +σ1 +···+σ  i−3 +αi−2

+ ··· +

t0 +σ  1 +α2

t

t

... t0

t0

  λr+−1 di−1 t 

t0 +σ1 t0 +σ1

(1.13) where it is assumed that Δy (r −1) < 0. The case that Δy (r −1) > 0 follows analogously by replacing λ+j with λ−j ( j = 2, 3, . . . , r ).

1.3.2 Preliminaries for the Proof of Main Results For the purpose of showing Theorems 1.1 and 1.2 should be verified first. Then the following preliminary results are illustrated for the verification of Theorem 1.2. In order to define the feasibility of Si , the following logical abbreviations for some interval [t0 , t1 ) are introduced.

1.3 Proof of Main Results

21

Eqi := [qi is true on [t0 , t1 )], E¬qi := [qi is false on [t0 , t1 )] Fi+,ε (t0 ) := {e(i) (t0 ) ∈ [λi+ + εi− , ϕi+ − εi+ ]}, Fi−,ε (t0 ) := {e(i) (t0 ) ∈ [ϕi− + εi− , −λi− − εi+ ]}. Then the feasibility of Si implies the following implications [1], Eqi ∧ Fi−,ε (t0 ) ⇒ ∀t ∈ [t0 , t1 ) : e(i) (t) ∈ [ϕi− , −λi− ]

(1.14)

E¬qi ∧ Fi+,ε (t0 ) ⇒ ∀t ∈ [t0 , t1 ) : e(i) (t) ∈ [λi+ , ϕi+ ]

(1.15)

Eqi ∧ t1 − t0 > i− ⇒ ∃t ∈ [t0 , t0 + i− ] : e(i) (t) ≤ −λi− − εi+

(1.16)

E¬qi ∧ t1 − t0 > i+ ⇒ ∃t ∈ [t0 , t0 + i+ ] : e(i) (t) ≥ λi+ + εi− .

(1.17)

Lemma 1.2 Consider the nonlinear system (1.1) satisfying F1 , a reference signal yref (t) satisfying F2 , error funnels Fi (i = 0, 1, . . . , r − 1) satisfying F3 –F5 and a BCFC given by the switching logic defined in (1.5)–(1.7) and input values satisfying F6 and F7 , if ∀t ∈ [t0 , t1 ), qi (t) = false, e(i) (t0 ) ≤ ϕi− + εi− , ∃t2 < t1 and ∀ts ∈ [t0 , t2 ] such that qi+1 (ts ) = false and Si+1 , . . . , Sr −1 are feasible, then {qi+1 (t) = false} ∧ + + + − } = true implies that ∃t ∈ [t0 , t0 + Δi+1 ] : e(i+1) (t) ≥ λi+1 + εi+1 . {t2 − t0 > Δi+1 Proof The proof of Lemma 1.2 is similar with that of Step 2c of Lemma 6.3 presented in Sect. 6 of [1]. Lemma 1.3 Consider the nonlinear system (1.1) satisfying F1 , a reference signal yref (t) satisfying F2 , error funnels Fi (i = 0, 1, . . . , r − 1) satisfying F3 –F5 and a BCFC given by the switching logic defined in (1.5)–(1.7) and input values U ± satisfying F6 and F7 , if ∀t ∈ [t0 , t1 ), qi (t) = false, e(i) (t0 ) ≥ ϕi+ − εi+ , ∃t2 < t1 and ∀ts ∈ [t0 , t2 ) such that qi+1 (ts ) = true and Si+1 , . . . , Sr −1 are feasible, − − } = true implies that ∃t ∈ [t0 , t0 + Δi+1 ]: then {qi+1 (t) = true} ∧ {t2 − t0 > Δi+1 − + e(i+1) (t) ≤ −λi+1 − εi+1 . Proof It can be verified analogously as Lemma 1.2. Lemma 1.4 (Corollary 6.2 of [2]) Assume η : [t0 , t2 ] → R is twice differentiable ˙ ≤ ψ d (t) for all t ∈ [t0 , t1 ) ⊆ and let a continuous ψ d : [t0 , t1 ] → R be such that η(t) ¨ ≤ −λ is satisfied for all [t0 , t2 ]. Furthermore, assume there exists λ > 0 such that η(t) t ∈ [t1 , t2 ). Then for every absolutely continuous ψ : [t0 , t2 ] → R with essentially bounded derivative and ε := ψ(t0 ) − η(t0 ) > 0, it holds that η(t) < ψ(t) for all t ∈ [t0 , t2 ] if ε := ψ(t0 ) − η(t0 ) > (t1 − t0 ) ψ˙ − ψ d ∞ +

˙ ∞) ( ψ d ∞ + ψ . 2λ

22

1 Switching Control of Impulsively Disturbed Nonlinear …

Lemma 1.5 Consider the nonlinear system (1.1) satisfying F1 , a reference signal yref (t) satisfying F2 , error funnels Fi (i = 0, 1, . . . , r − 1) satisfying F3 –F5 and a BCFC given by the switching logic defined in (1.5)–(1.7) and input values U ± satisfying F6 and F7 , if ∀t ∈ [t0 , t1 ), ei (t0 ) ∈ [λi+ + εi+ , ϕi+ − εi+ ], qi (t) = false and Si+1 , . . . , Sr −1 are feasible, then {qi (t) = false} ∧ {ei (t0 ) ∈ [λi+ + εi+ , ϕi+ − εi+ ]} = true implies that ∀t ∈ [t0 , t1 ) : e(i) (t) ∈ (λi+ , ϕi+ ). Proof The proof of Lemma 1.5 is similar with that of Step 2b of Lemma 6.3 presented in Sect. 6 of [1].

1.3.3 Proof of Theorem 1.1 Due to the impulsive disturbance occurs in the dynamics of y (r −1) (t) at t = t0 as illustrated in (1.10), the BCFC is switched on operation on the interval [t0 , tc ). In order to verify that the system output tracking error and its derivatives can be regulated back into the pre-specified error funnels, we assume the worst case occurs, i.e. the output tracking error e(t) and its derivatives (e(t), ˙ e(2) (t), . . . , e(r −1) (t)) all jump out of the (r −1) . Referring to the analysis presented funnel boundaries due to the impact of Δy in Sect. 1.3.1, it follows that the convergence of e(i+1) (t) priors to that of e(i) (t) for i = 0, 1, . . . , r − 2. Therefore, if the BCFC can regulate e(i) (t)(i = 0, 1, . . . , r − 1), all of which jump out of the corresponding error funnels, back into the error funnels, then it is definitely able to deal with the case that only part of e(i) (t)(i = 1, 2, . . . , r − / 1) jump out of the error funnels. Thus we just consider here that e(r −1) (t0 ) ∈ / [ϕr−−2 , ϕr+−2 ], . . ., e(r −i) (t0 + σ1 + σ2 + · · · + σi−2 + [ϕr−−1 , ϕr+−1 ], e(r −2) (t0 + α1 ) ∈ / [ϕr−−i , ϕr+−i ], . . ., e(0) (t0 + σ1 + σ2 + · · · + σr −2 + αr −1 ) ∈ / [ϕ0− , ϕ0+ ], where αi−1 ) ∈ (r −i) (t0 + σ1 + · · · + σi−2 + αi−1 )(i = 1, 2, . . . , r ) denotes the minimum of e e(r −i) (t) on the interval t ∈ [t0 , ∞) in the case that Δy (r −1) < 0, and it represents the maximum of e(r −i) (t) on the interval t ∈ [t0 , ∞) in the case that Δy (r −1) > 0. Then Theorem 1.1 can be verified as follows. Proof Assume Δy (r −1) < 0 and the situation of Δy (r −1) > 0 follows analogously. With the assumption of Δy (r −1) < 0, it can be obtained that e(r −1) (t0 ) < ϕr−−1 , . . ., e(0) (t0 + σ1 + σ2 + · · · + e(r −i) (t0 + σ1 + σ2 + · · · + σi−2 + αi−1 ) < ϕr−−i , . . ., − σr −2 + αr −1 ) < ϕ0 . (a) The Feasibility of Sr −1 : Assume that there exists a minimal tˆ ∈ [t0 , t1 ) such that e(r −1) (tˆ) ∈ [ϕr−−1 + εr−−1 , ϕr+−1 − εr+−1 ]. The existence of tˆ can be verified as follows. Concerning e(r −1) (t0 ) < ϕr−−1 and the switching logic of Sr −1 , q(t) = false holds for all t ∈ [t0 , tˆ). Hence, u = (r ) (r )

∞ + f )/g, thus e(r ) (t) = −yref (t) + U + . Invoking F6 , it has U + > (λr+ + yref

(r ) (r ) f (Y r −1 , z) + g(Y r −1 , z)U + ≥ −yref (t) + f (Y r −1 , z) + λr+ + yref

∞ + f . Due to (r ) (r ) (r ) r −1 +

yref ∞ ≥ yref (t), we have e (t) ≥ f (Y , z) + f + λr , and further e(r ) (t) ≥

1.3 Proof of Main Results

23

λr+ . Concerning e(r ) (t) ≥ λr+ > ϕ˙r−−1 implies dtd (e(r −1) (t) − ϕr−−1 ) > 0, ∃Δt = tˆ − t0 such that e(r −1) (tˆ) ∈ [ϕr−−1 + εr−−1 , ϕr+−1 − εr+−1 ] holds for all t ∈ [t0 + Δt, ∞). Concerning e(r −2) (t0 + α1 ) < ϕr−−2 and the switching logic of Sr −2 , we have qr −1 (t0 + α1 ) = false. Since e(r −2) (t) keeps decreasing until e(r −1) (t) ≥ 0 holds, we have tˆ < t0 + α1 < t1 and e(r −1) (t0 + α1 ) = 0. Therefore, e(r −2) (t) < ϕr−−2 and qr −1 = false hold on the interval t ∈ [t0 , t0 + α1 ]. In the following, we will show that ∃t ∈ [tˆ + Δr+−1 , t1 ) such that e(r −1) (t) ∈ [λr+−1 + εr−−1 , ϕr+−1 − εr+−1 ]. Seeking a contradiction, assume ∀t ∈ [tˆ + Δr+−1 , t1 ) such that e(r −1) (t) < λr+−1 + εr−−1 and thus it can be obtained that λr+−1 + εr−−1 > e(r −1) (t) ≥ e(r −1) (tˆ) + e(r ) (t)(t − tˆ) ≥ e(r −1) (tˆ) + e(r ) (t)Δr+−1 ≥ e(r −1) (tˆ) + λr+ Δr+−1 ≥ ϕr−−1 + λr+ Δr+−1 . Invoking F4 , we + + + (|ϕi+ | + |ϕi− |)/λi+1 and it shows that ϕr−−1 + λr+ Δr+−1 ≥ ϕr−−1 + have Δi+ ≥ Δi+1 + − + (|ϕr −1 | + |ϕr −1 |) ≥ |ϕr −1 |. Invoking F3 , it can be obtained that |ϕr+−1 | > λr+−1 + εr−−1 , whence the sought contradiction, λr+−1 + εr−−1 > |ϕr+−1 | > λr+−1 + εr−−1 , is obtained. Therefore, it shows that ∃t ∈ [tˆ + Δr+−1 , t1 ) such that e(r −1) (t) ∈ [λr+−1 + εr−−1 , ϕr+−1 − εr+−1 ] holds. (b) We Show That the Feasibility of Si+1 Implies That of Si : + − + + Consider that e(i+1) (t) ∈ [λi+1 + εi+1 , ϕi+1 − εi+1 ] holds on the interval [t4 , t7 ). Meanwhile, Si+2 , . . ., Sr −1 are feasible. The problem here is to show that ∃ts ∈ [t0 + tη , t7 ), where tη = σ1 + σ2 + · · · + σi−1 + αi and e(i+1) (t) deceases to its minimum at t = t0 + tη , such that e(i) (t) ∈ [λi+ + εi− , ϕi+ − εi+ ] holds for all t ∈ [ts , t7 ). Assuming the contradiction, e(i) (t) < λi+ + εi− holds for all t ∈ [t4 , t7 ). Meanwhile, it is assumed that e(i+1) (t3 ) = 0 holds, where t3 = t0 + σ1 + σ2 + · · · + σi−2 + αi−1 . We consider that Si+1 , Si+2 , . . . , Sr −1 have recovered from the impulsive disturbance and e(i+1) (t), e(i+2) (t), . . . , e(r −1) (t) have converged back into the pre-specified error funnels when t ≥ t2 . Furthermore, it is assumed that e(i+1) (t) < 0 + − + εi+1 . holds when t < t3 and that there exists t = t4 such that e(i+1) (t4 ) = λi+1 − Meanwhile, it has e(i) (t3 ) < ϕi and e(i) (t) drops to its minimum at t = t3 . With respect to the switching logic Si , it can be obtained that qi+1 (t) = false holds on the interval t ∈ [t3 , t7 ). Invoking F3 , we have εi− > e(i) (t) − λi+ for all t ∈ [t3 , t7 ). + + − εi+1 holds. Then it can be Moreover, ∃t5 ∈ [t4 , t7 ) such that e(i+1) (t5 ) = ϕi+1 + + obtained that e(i+1) (t) ≥ ϕi+1 − εi+1 and qi+2 (t) = true hold on some interval t ∈ [t5 , t6 ) with t6 < t7 . Invoking Lemma 1.3, (1.16) of Si+2 ensures e(i+2) (t) ≤ − + < ϕ˙i+1 = 0. Then e(i+1) (t) will decrease in a short period and Lemma 1.5 −λi+2 + guarantees that e(i+1) (t) cannot increase beyond ϕi+1 . As the switching logic of + + + − (i+1) (t), ϕi+1 − εi+1 , λi+1 + εi+1 , qi+2 (t−)) if qi+1 (t) = false, Si+1 is qi+2 = G (e + − + εi+1 . Consequently, there exists a minimal t6 ∈ (t5 , t7 ) such that e(i+1) (t6 ) = λi+1 + − (i+1) (t) ≤ λi+1 + εi+1 hold on a short interval after t = t6 . Furqi+2 (t6 ) = false and e + > 0. thermore, the feasibility of Si+2 and (1.17) of Si+2 imply that e(i+2) (t) ≥ λi+2 (i+1) (t) will start to increase in a small period of time after t = t6 . MeanHence, e + . According to the while, Lemma 1.3 ensures that e(i+1) (t) cannot decrease below λi+1 + + (i+1) (t) will stay within [λi+1 , ϕi+1 ] when t ≥ t4 , above analysis, we have shown that e which means t7 → ∞. Recalling that qi+1 (t) = false on interval t ∈ [t4 , t7 ) and + > λi+ , it has e(i+1) (t) > λi+ > 0. Thereafter, we have e(i) (t7 ) − λi+ ≤ e(i) (t4 ) − λi+1 t λi+ + t47 e(i+1) (t)dt. Thus it can be shown that e(i) (t7 ) − λi+ → ∞ as t7 → ∞, which

24

1 Switching Control of Impulsively Disturbed Nonlinear …

leads to the contradiction that e(i) (t) − λi+ < εi− . Therefore, ∃ts ∈ [t0 + tη , t7 ) such that e(i) (t) ≥ λi+ + εi− holds on all t ∈ [ts , t7 ), whence the feasibility of Si+1 implies that of Si . Inductively, it can be shown that e(0) (t), e(1) (t), . . . , e(i) (t), . . . , e(r −1) (t) will converge back into the error funnels after the impulsive disturbance occurring in the dynamics of y (r −1) (t). Whence Theorem 1.1 is verified.

1.3.4 Proof of Theorem 1.2 Proof In the case that {|Δy (r −1) | ≤ β} ∧ {{e(r −1) (t0 ) > ϕr+−1 } ∨ {e(r −1) (t0 ) < ϕr−−1 }}, the SC switches from the CC to the BCFC at t = t0 . According to Theorem 1.1, the system output tracking error and its derivatives can be regulated back into the error funnels by the BCFC. Then it remains to show that {(|Γs | − |Γ j |)/|Γs | ≥ τ > 0} ∨ {Γ j−1 ∈ [ϕ0− , ϕ0− + ε0− ] and Γ j ∈ [ϕ0+ − ε0+ , ϕ0+ ]} ∨ {Γ j−1 ∈ [ϕ0+ − ε0+ , ϕ0+ ] and Γ j ∈ [ϕ0− , ϕ0− + ε0− ]} can always be satisfied in the case that e(i) (t)(i = 0, 1, . . . , r − 1) are all within Fi . For the case that e(t) is perturbed outside F0 by the impulsive disturbance, it follows that |Γs | > max{ϕ0+ , |ϕ0− |} and Γ j ∈ [ϕ0− , ϕ0+ ]. Hence, there exists τ > 0 such that {(|Γs | − |Γ j |)/|Γs | ≥ τ > 0} = true. For the case that e(t) ∈ [ϕ0− , ϕ0+ ] for all t ∈ [0, ∞) despite of the impulsive disturbance, {Γ j−1 ∈ [ϕ0− , ϕ0− + ε0− ] and Γ j ∈ [ϕ0+ − ε0+ , ϕ0+ ]} ∨ {Γ j−1 ∈ [ϕ0+ − ε0+ , ϕ0+ ] and Γ j ∈ [ϕ0− , ϕ0− + ε0− ]} is always true due to the mechanism of the BCFC. Then the SC switches from the BCFC to the CC, thus F8 is satisfied. Nonlinear system (1.1) is stabilized to its original operation point by the CC. In the case that e(r −1) (t0 ) ∈ [ϕr−−1 , ϕr+−1 ], let us consider the case that γ 0 = 0 first, in which the BCFC will not be triggered. And system (1.1) operates under the control of the CC only. Referring to F8 , the system output tracking error and its derivatives can be stabilized to zero by the CC. In the case that γ 0 = 1, as presented above, {(|Γs | − |Γ j |)/|Γs | ≥ τ > 0} ∨ {Γ j−1 ∈ [ϕ0− , ϕ0− + ε0− ] and Γ j ∈ [ϕ0+ − ε0+ , ϕ0+ ]} ∨ {Γ j−1 ∈ [ϕ0+ − ε0+ , ϕ0+ ] and Γ j ∈ [ϕ0− , ϕ0− + ε0− ]} can always be satisfied in the case that e(i) (t)(i = 0, 1, . . . , r − 1) are all within Fi . Then the CC is switched on and stabilize the system to its original operation point. Whence Theorem 1.2 is verified.

1.4 Simulation Studies: Application of BCFC and SC for Frequency Control of a SMIB Power System The BCFC and the SC is applied respectively for the excitation control of a synchronous generator to supervise the frequency of a SMIB power system in this section. The excitation controller design is carried out in a SMIB power system, which is shown in Fig. 1.2. The synchronous generator is connected to an infinite

1.4 Simulation Studies: Application of BCFC and SC …

25

Fig. 1.2 The layout of a SMIB power system

bus through a transformer and two parallel transmission lines. A third-order model is adopted for the generator, and the generator is equipped with an AC1A type excitor and a PSS, respectively. The dynamics of the entire system are described as follows. ⎧ ˙ ⎪  ⎨ δ = ω −ω0 ω0 Pm − ωD0 (ω − ω0 ) − Pe ω˙ = 2H (1.18) ⎪ ⎩ E˙  = 1 (E − E ) f q q T d0

where

Pe = E q Iq + (X q − X d )Id Iq   Id = X1 E q − Vs cos(δ) ds E q = E q + (X d − X d )Id  X ds = X d + X T + X s

Pe =

E q Vs sin(δ) X ds Vs sin(δ) X qs

Iq = X qs = X q + X T + X s X ds = X d + X T + X s

and δ represents the relative rotor angle, in rad, ω is the rotor speed of generator, in rad/s, ω0 denotes the synchronous speed of the system, in rad/s, E q and E q are the transient voltage and voltage behind the quadrature-axis, respectively, Pm is the mechanical power input of the generator and assumed to be constant, in p.u., Pe represents the electrical power output of the generator, in p.u., H is the inertia  denotes the direct axis transient short circuit time constant of rotor, in seconds, Td0 constant, in seconds, D is the damping constant of the generator, in p.u., X d and X d are the d-axis synchronous and transient impedances of the generator, respectively, X q is the q-axis synchronous impedance, X T and X s are the impedance of transformer and transmission line, respectively, E f is the excitation voltage, Id and Iq are the daxis and q-axis generator current, respectively, and Vs is the voltage of the infinite bus. By defining state variables as x = [x1 x2 x3 ] = [δ − δ0 ω − ω0 E q ] and the input variable as u = E f , the state equation of system (1.18) can be rewritten in the following matrix form  x˙ = f (x) + g(x)u (1.19) y = h(x) = x2 where

⎡ ⎢ω f (x) = ⎣ 2H0



⎤

⎡

⎤ 0 Pm − ωD0 x2 − Pe ⎥ ⎦ , g(x) = ⎣ 0 ⎦ . 1 1  (−E q ) Td0 T x2

d0



26

1 Switching Control of Impulsively Disturbed Nonlinear …

For system (1.19), we have   ω0 x2 x3 Vs cos(δ) (X q − X d )Vs2 cos2 (2δ) =− −   2H X ds X ds X qs   ω0 Vs sin(δ)E q Dω0 D − Pm − x2 − Pe +   4H 2 ω0 2H Td0 X ds ω0 Vs sin(δ) Lg L f h(x) = −   2H Td0 X ds L f2 h(x)

As Lg L f h(x) = 0 holds for ∀δ = kπ(k = 0, 1, 2, . . .), system (1.19) has relative degree r = 2. The second-order derivative of y with respect to time can be written as y¨ = L f2 h(x) + Lg L f h(x)u.

1.4.1 Excitation Controller Designed Based on BBFC According to the above analysis, a second-order BCFC can be applied here for a bang-bang excitation controller (BEC) of the synchronous generator. Moreover, it can be seen that a phase-to-ground fault, resulting in sudden changes of Vs , can lead to dramatic variations of Pe . The dramatic variations of Pe further result in the jumps in the dynamics of ω, ˙ which is the (r − 1)th order derivative of the output of this system. This justifies the rationality of the description of the external impulsive disturbance presented in (1.10). Parameters of the SMIB power system are: X d = 1.0 p.u., X d = 0.4 p.u., X q = 0.6 p.u., X T = 0.12 p.u., X L = 1.0 p.u., D = 0.008,  = 5.0 s, and H = 4.34 s. The normal operation conditions of the system are δ0 = Td0 0.6981 rad, Pe = 0.4732 p.u., and ω0 = 314.15 rad/s. Thus we have Lg L f h(x) < 0. Considering F1 and (1.12), the control law of the BEC can be given as  + E f , if q(t) = true u(t) = E f− , if q(t) = false. The parameters of the second-order BCFC are chosen as ϕ0+ = −ϕ0− = 2, ϕ1+ = − + − −ϕ1− = 2, ε0+ = ε0− = 1.95, ε1+ = ε1− = 0.8, λ+ 1 = λ1 = 0.35, λ2 = λ2 = 1.026, + − + − + − −4 Δ0 = Δ0 = 15.34, Δ1 = Δ1 = 3.91, Δ2 = Δ2 = 1 × 10 . E V Δδ¨ + ωD0 Δδ˙ + Xqdss cos(δ0 ) The linearization of (1.18) around its equilibrium is 2H ω0 Δδ = 0, then the eigenvalues of this equation are p = ±4.5205j. Thus we have ˙ = Δω(t) = 4.5205Ae4.5205jt holds. Under the Δδ(t) = Ae4.5205jt . Then δ(t) assumption that the output tracking error e(x) = Δω and its derivative can be regulated back into the error funnels after the impulsive disturbance, it has Δω ∈ [−2, 2]. Therefore, it has |A| ≤ 2/4.5205 = 0.4424 and thus Δδ(t) ∈ [−0.4424, 0.4424]. In the cases that system (1.18) is disturbed and the output h(x) and its derivative are within the error funnels, we have δ(t) ∈ [0.2557, 1.1405] with δ0 = 0.6981.

1.4 Simulation Studies: Application of BCFC and SC … (a) Rotor speed deviation

(d) Rotor speed deviation CEC SEC

5 Δ ω (rad/s)

Δω (rad/s)

4 2 0 BEC CEC

−2 0

5

0 −5

(e) Active power output 1

d Δω / dt Pe (p.u.)

10 0 −10

0.5 0

−0.5 CEC

10

5 (c) Excitation voltage BEC

0

10

5 0

5

5

10 SEC

CEC

0

−5

−5 0

SEC

(f) Excitation voltage CEC E (p.u.) f

0

Ef

10

5

0

10

(b) Derivative of rotor speed deviation

dΔω / dt

27

5 Time (s)

10

0

5 Time (s)

10

Fig. 1.3 The dynamics of the generator controlled by the CEC, BEC and SEC respectively (within a, the dash dot line represents ϕ0± and the dot line denotes ϕ0+ − ε0+ and ϕ0− + ε0− , within b, the dash dot line represents ϕ1± and the dot line denotes λ± 1)

Concerning the most serious fault that may happen in the power system, the ω0 Pe caused by a three-phase-todynamics of Δω˙ could see a largest jump of − 2H ground fault on the infinite bus (Vs = 0). Therefore, we have χ1 = 17.13. Referring to (1.13), it has χ0 = 143. On condition that Δω ∈ [−143, 143] and Δω˙ ∈ [−17.13, 17.13], we have L f2 h(x) ∈ [0.9934, 8.7494] and Lg L f h(x) ∈ [−4.3290, −1.2058]. Invoking F6 , we have E f− < 2 −λ+ 2 −L f h(x)|max

2 λ+ 2 +L f h(x)|max L g L f h(x)|max

= −8.107 and E f+ >

= 8.107. Hence, we choose E f+ = −E f− = 8.2 p.u., which is realizable in practice [3]. A three-phase-to-ground fault occurs on a transmission line between t = 1 s and t = 1.28 s. The BEC is switched on at t = 1.02 s. The dynamics of the rotor speed deviation and its derivative and the excitation voltage are illustrated in Fig. 1.3a–c, respectively. In the process of the fault, the voltage of infinite bus drops to Vs = E V sinδ 0.01 p.u. This results in a decline of the active power output Pe as Pe = q Xsds . L g L f h(x)|max

28

1 Switching Control of Impulsively Disturbed Nonlinear …

Consequently, Δω and Δω˙ can leave the error funnels as depicted in Fig. 1.3a and b, respectively. As can be noticed, Δω˙ is outside the error funnel at the end of the fault, while Δω is within the error funnel. With the effort of the BEC depicted in Fig. 1.3c, Δω˙ is able to converge back into the error funnel. Then Δω and Δω˙ can be regulated within the error funnels by the BEC. By contrast, the generator controlled by the CEC goes unstable as depicted in Fig. 1.3a.

1.4.2 Excitation Controller Designed Based on SC In practice, the rotor speed is desired to operate on a stable equilibrium. Then the rotor speed deviation of the synchronous generator is favored to converge to zero in the cases that external disturbances occur in the power system. Considering the same fault in the SMIB system, the SC is applied as a switching excitation controller (SEC) for the frequency control of the SMIB power system. The switching strategy T described with (1.11) is adopted, and the control parameter of T2 is selected as τ = 0.5. As depicted in Fig. 1.3d and e, the rotor speed deviation and the active power output of the generator controlled by the SEC can be stabilized to their original equilibriums. By contrast, the generator controlled by the CEC goes unstable. This is due to that the SEC makes use of the largest damping energy of the excitor and it is able to provide proper switching time of the excitation voltage as depicted in Fig. 1.3f. It can be noticed that the BEC has been switched on two times. After the first time that the SEC switches from the BEC to the CEC, the terminal voltage deviation of the generator is still large. Since the terminal bus voltage is also a part of the input of the CEC, the excitation voltage generated by the CEC will drive the terminal voltage deviation to zero. The side-effect of this action is that the rotor angle is driven away from its equilibrium as well. Therefore, the BEC is switched on again. After the second time that SC switches from the BEC to the CEC, both of the terminal voltage and rotor angle deviation of the synchronous generator are near their equilibriums. Then the CEC can stabilize the generator to the pre-fault operation point. Moreover, it should be mentioned that the same set of parameters are used for the BCFCs in the two cases. Therefore, it can be verified that the BCFC is able to fully explore the potential of the excitation system and provides the utmost damping to the oscillations of system output.

1.5 Summary This chapter has proposed a SC based on the BCFC for the IDRA enhancement of impulsively disturbed nonlinear systems. The SC is designed to switching between a BCFC and a CC based upon a statedependent switching strategy. The BCFC is able to utilize the maximum control

References

29

energy of the nonlinear system, which has greatly enhanced the IDRA of the nonlinear system controlled by the SC. Meanwhile, the SC is able to stabilize the nonlinear system to its pre-fault operation point with hybrid control signals. The BCFC does not require the system output tracking error and its derivatives, especially their initial values, to be contained within the pre-specified error funnels. Moreover, the funnel values of the BCFC are constant, in contrast to the time-varying funnels required in the BBFC, which makes the BCFC design easier than the design of the BBFC. A searching method has been given to find the largest impulsive disturbance β that the nonlinear system can endure. In the cases that the magnitude of the impulsive disturbance is smaller than β, the boundary values and the recovery time of system output tracking error and its derivatives of an impulsively disturbed nonlinear system controlled by the BCFC are analyzed. In this way, the feasibility assumption F6 can be checked. The BCFC and the SC have been applied in the excitation control of a synchronous generator to supervise the frequency of a SMIB power system, respectively. The simulation results meet the results stated by Theorems 1.1 and 1.2. Owning to the effort of the BEC, the synchronous generator controlled by the SEC can be stabilized to its pre-fault operation point in the case that a three-phase-to-ground fault occurs in the power system. In contrast, the power system only controlled with the CEC goes unstable. This reveals that the BCFC is able to explore the utmost potential of the control system and makes use of the largest control energy for the IDRA enhancement of nonlinear systems suffering from impulsive disturbances.

References 1. Liberzon D, Trenn S (2013) The bang-bang funnel controller for uncertain nonlinear systems with arbitrary relative degree. IEEE Trans Autom Control 58(12):3126–3141. https://doi.org/ 10.1109/TAC.2013.2277631 2. Liberzon D, Trenn S (2013) The bang-bang funnel controller: time delays and case study. In: 2013 ECC, pp 1669–1674 3. Lee D, Baker D, Bess K et al (1992) IEEE recommended practice for excitation system models for power system stability studies. Energy Development and Power Generation Committee of Power Engineering Society

Chapter 2

Observer-Based Robust Bang-Bang Funnel Controller and Its Stability in Closed-Loop Systems

2.1 Introduction to Bang-Bang Funnel Control and Main Results on Closed-Loop Stability 2.1.1 System Description Consider the following single-input single-output nonlinear system 

x˙ = F(x) + G(x)u y = H (x)

(2.1)

where x ∈ Rn , u ∈ R, y ∈ R, F(x) : Rn → Rn , G(x) : Rn → Rn , H (x) : Rn → R, F(x) and G(x) are smooth on Rn and H (x) is smooth as well. Using the following map, x → (Y, z  ) , Y := (y, y˙ , . . . , y (r −1) ), (2.1) is transformed into its controllable canonical form  Y˙ = AY + B[ f (Y, z) + g(Y, z)u], Y  (0) = y 0 ∈ Rr (2.2) z˙ = h(Y, z), z(0) = z 0 ∈ Z 0 ⊆ Rn−r where

⎡

0 ⎢ 0 ⎢ A=⎢ ⎢··· ⎣ 0 0

⎤ ⎡ ⎤ 1 0 ··· 0 0 ⎢ 0 ⎥ 0 1 ··· 0 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ···⎥ ⎥ , B = ⎢···⎥ , ⎦ ⎣ 0 ⎦ 0 0 ··· 1 0 0 · · · 0 r ×r 1 r ×1

f (Y, z) = L Fr H (x), g(Y, z) = LG L Fr −1 H (x), r is the relative degree of y with respect to u, and it has, y (r ) = L Fr y + LG L Fr −1 H (x)u = f (Y, z) + g(Y, z)u, © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Liu and Q.-H. Wu, Adaptive Switching Control of Large-Scale Complex Power Systems, Power Systems, https://doi.org/10.1007/978-981-99-1039-7_2

31

32

2 Observer-Based Robust Bang-Bang Funnel Controller …

with LG L Fr −1 y = 0 and LG L Fk y = 0 holding in a neighborhood of x 0 for all k = 0, . . . , r − 2, and x 0 represents the equilibrium of (2.1). Moreover, it is assumed that g(Y, z) > 0 and the zero dynamics of the system is stable. Stability of the nonlinear system confronting external disturbances is the subject of concern, a piece-wise continuous function is considered in y (r ) (t) so as to model the intermittent disturbance. Then (2.2) is rewritten as 

Y˙ = AY + BΨ (Y, u, z, t) z˙ = h(Y, z)

(2.3)

where Ψ (Y, u, z, t) = f (Y, z) + g(Y, z)u + d(t), d(t) : R≥0 → R represents a piece-wise continuous external disturbance, i.e.,  d(t) =

0, t2ζ −1 ≤ t < t2ζ θ (t), t2ζ ≤ t < t2ζ +1

(2.4)

where |θ (t)| ≤ β, β ∈ R+ denotes the largest magnitude of the disturbance concerned, ζ ∈ N+ and {t1 , t2 , t3 , . . .} is the set of discrete time points.

2.1.2 Bang-Bang Funnel Control A BCFC is designed based on [1], and the output tracking error of (2.1), i.e. e := y − yref , meets the pre-specified error bounds, given by F0 := {(t, e) ∈ R≥0 × R| − ϕ0 ≤ e ≤ ϕ0 }(ϕ0 ∈ R represents a constant error bound). The largest effort of the control system is utilized to enhance the robustness of system (2.1) to the external disturbance (2.4). Constant error funnels are designed for the ease of application. The switching logic S : e(t) → q of the BCFC is defined with r switching blocks B0 , B1 , . . . , Br −1 in a recursive manner as: ˙ B0 ) = S1 (e(t), ˙ q1 ) S : e(t) → q := S1 (e(t),

(2.5)

where, for i = 1, 2, . . . , r − 2



(i)  d (i) e (t), Bi e (t), qi Si : (e (t), qi ) → q : = Si+1 dt

(i+1)  (t), qi+1  = Si+1 e

(r −1) 

(r −1) Sr −1 : e (t), qr −1 → q := Br −1 e (t), qr −1 (i)

The r logic blocks B0 , B1 , . . . , Br −1 are defined as

(2.6)

2.1 Introduction to Bang-Bang Funnel Control and Main Results on Closed-Loop Stability

33

B0 : e(t) → q1 with q1 (t) = G (e(t), ϕ0 − ε0 , −ϕ0 + ε0 , q1 (t−)) q1 (0−) = q10 ∈ {true, false} Bi : (e(i) (t), → qi+1 , i = 1, 2, . . . , r − 2, with ⎧ qi ) (i) G (e (t), −λi − εi , −ϕi + εi , qi+1 (t−)), ⎪ ⎪ ⎨ if qi (t) = true qi+1 (t) = G (e(i) (t), ϕi − εi , λi + εi , qi+1 (t−)), ⎪ ⎪ ⎩ if qi (t) = false 0 ∈ {true, false} qi+1 (0−) = qi+1

(2.7)

Br −1 : (e(r −1) (t), qr −1 ) → q given as above with i = r − 1 and q = qr , q(0−) = q 0 ∈ {true, false} where G (e, e, e, qold ) := [e ≥ e ∨ (e > e ∧ qold )], e(·) is the upper error bound, e(·) is the lower error bound, e(t) is the error of the control target, qold ∈ {true, false}, ϕi ∈ R≥0 are the constant funnel values that define funnels Fi := {e(i) (t) ∈ R| − ϕi ≤ e(i) (t) ≤ ϕi }, λi ∈ R≥0 denote the desired increase and decrease rate of e(i−1) (t), εi ∈ R≥0 are designed as the safety distance, which trigger a switch event when the error or its derivatives reach the error bounds, q(t−) := limε→0+ q(t − ε). q(t) maps the tracking error e(t) to the switching control logic and is the control command. Consider g(Y, z) > 0, the output of the BCFC is  u(t) =

−U, if q(t) = true U, if q(t) = false

(2.8)

2.1.3 High-Gain State Observer Referring to (2.7), inputs of the BCFC are the tracking error of outputs and its 1st-order to (r − 1)th-order derivatives. However, calculating the derivatives of the measured signals will amplify the measurement noise, which may result in the instability of the control system. To get rid of the derivative calculation, a high-gain state observer is employed to obtain the estimates of the derivatives of tracking errors. The estimates are transferred to the BCFC. Let Wˆ = [wˆ 1 wˆ 2 . . . wˆ r ] be the estimates of Ye = [e e˙ . . . e(r −1) ] and w˜ 1 = e − wˆ 1 , an r th-order high-gain state observer is designed as ⎧ ˙ α1 ⎪ wˆ 1 = wˆ 2 + ε w˜ 1 ⎪ ⎪ ⎪ ⎨ w˙ˆ 2 = wˆ 3 + αε22 w˜ 1 ··· (2.9) ⎪ αr −1 ⎪ ˙ ⎪ wˆ r −1 = wˆ r + εr −1 w˜ 1 ⎪ ⎩ ˙ wˆ r = αεrr w˜ 1 where ε ∈ (0, 1), and αi (i = 1, 2, . . . , r ) are selected and the roots of

34

2 Observer-Based Robust Bang-Bang Funnel Controller …

Y z

AY

B (Y , u, z, t ) y h(Y , z )

e

yref

BCFC

q

U

High-gain state observer

U

wˆ 1 , wˆ 2 ,

, wˆ r

Funnel parameters

Fig. 2.1 Structure of the closed-loop system [2].

s r + α1 s r −1 + · · · + αr −1 s + αr = 0

(2.10)

are in the open left-half complex plain, where s denotes the Laplace operator. Using the high-gain state observer, the entire system can be illustrated by Fig. 2.1. The estimates are used to drive the switching logic of the BCFC.

2.1.4 Assumptions of RBC F1

Ψ (Y, u, z, t) : Rr × R × Rn−r × R → R is globally bounded in Y : |Ψ (Y, u, z, t)| ≤ γ1

where γ1 > 0 satisfies γ1 > β, and it has Ψ (0, 0, 0, 0) = 0 for Ψ (Y, u, z, t). yref ∈ C r −1 (R≥0 → R), where C r −1 (X → Y ) or short C r −1 is the set of all (r − 1)-times continuously differentiable f : X → Y . ∗ ), F3 Gains of the high-gain state observer should be chosen such that ε ∈ (0, εspo where  −1 2λ λ (P ) (P )γ max 10 max 10 1 ∗ = +α εspo λmin (P10 ) δspo

F2

where δspo > 0 represents the maximum estimation error of Ye obtained by the high-gain state observer, and it satisfies δspo εi , and P10 is a positive definite solution of P10 A10 + A 10 P10 = −I , where ⎡ A10

−α1 −α2 .. .

⎢ ⎢ ⎢ =⎢ ⎢ ⎣ −αr −1 −αr

⎤ 0 0⎥ ⎥ .. ⎥ .⎥ ⎥ 0 0 ··· 1⎦ 0 0 · · · 0 r ×r 1 0 .. .

0 1 .. .

··· ··· .. .

2.1 Introduction to Bang-Bang Funnel Control and Main Results on Closed-Loop Stability

35

ϕ0 − ε0 > −ϕ0 + ε0 and for ∀i ∈ {1, . . . , r − 1}: ϕi − εi > εi + λi , −ϕi + εi < −λi − εi with εi > 0, λi > 0, and λr β. There exist numbers i > 0 for i = 0, 1, . . . , r such that i ≥ i+1 + 2ϕi /λi+1 for i = 0, . . . , r − 1. εi > 2 δspo + i+2 ϕi+1 + ϕi+1 /(2λi+2 ) holds for i = 0, 1, . . . , r − 2. (i) F5 e (0) ∈ [−ϕi + εi , ϕi − εi ], i = 0, 1, . . . , r − 1. F6 (r ) (t) − f (Y, z) λr + yref U≥ g(Y, z)

F4

y

y

y

holds for all t ≥ 0, Y ∈ t ref , z ∈ Z t ref , and Rr ⊇ t ref := {Y |∀i ∈ 0, 1, . . . , r − 1 (i) (t) ∈ [−μi , μi ]}, where μi ∈ R>0 is the boundary value of yi (t) − : y (i) (t) − yref (i) yref (t) considering the most severe external disturbance d(t), i.e. θ (t) ≡ β or y θ (t) ≡ −β. Moreover, Z t ref := ⎧ ⎨

⎫  z solves z˙ = h(Y, z), z(0) = z 0 , for some ⎬  z(t) z 0 ∈ Rn−r and some Y ∈ C r −1 with ⎩ ⎭ y (y(τ ), . . . , y (r −1) (τ )) ∈ τ ref , τ ∈ [0, t]. F7

The time interval between two consecutive disturbances should not be smaller than a positive number κ ∈ R>0 , i.e., t2ζ − t2ζ −1 ≥ κ.

2.1.5 Main Results on Convergence of Estimation Error and Closed-Loop Stability Define W˜ = Ye − Wˆ . Gains of the observer are selected to ensure the estimation error W˜ converges exponentially to a region which is arbitrarily close to the origin. Hence, we need to prove the following result in the first place. Theorem 2.1 (Convergence of the estimation error) Consider system (2.3) with bang-bang control input (2.8) generated by the BCFC, design a high-gain state observer (2.9) and choose observer gains αi as described in (2.10). If assumption F1 holds, then given any positive constant δspo > 0, there exists a positive constant ∗ ∗ , such that ∀ε, 0 < ε < εspo , the estimation error W˜ , from any initial value W˜ (0), εspo converges exponentially to the neighbourhood W˜  ≤ δspo . According to Theorem 2.1, the error of the estimates of is smaller than δspo on condition that the gains of the high-gain observer are appropriately selected. With Wˆ (the estimates of Ye ) being the inputs of the BCFC, the closed-loop system consisting of (2.3) and the RBC is stable. The above results can be summarized as follows. Theorem 2.2 (Closed-loop stability) Consider the nonlinear system (2.1) with a controllable canonical form representation of (2.3) satisfying F1 , a reference signal

36

2 Observer-Based Robust Bang-Bang Funnel Controller …

satisfying F2 , error funnels Fi satisfying F4 , external disturbances satisfying F7 , and the BCFC driven by Wˆ generated by a high-gain state observer satisfying F3 . If the initial tracking error of control target e(0) and its derivatives e(i) (0) are within the error boundaries, namely, F5 is satisfied, and U is large enough to satisfy F6 , then the system in Fig. 2.1 owns a global solution (x, q) : [0, ∞) → Rn × {true, false} and q has locally finite times of switching behavior and the tracking error of control target and its derivatives are able to converge into the error funnels between two consecutive disturbances, namely, ∃τ ∈ (t2ζ −1 , t2ζ ) such that e(i) (t) ∈ [−ϕi , ϕi ] holds ∀t ∈ [τ, t2ζ ) and i = 0, 1, . . . , r − 1.

2.2 Proof of Convergence of Estimation Errors of High-Gain Observer The error dynamics of the high-gain state observer is ⎧ ˙ w˜ 1 = w˜ 2 − αε1 w˜ 1 ⎪ ⎪ ⎪ ⎪ ⎨ w˙˜ 2 = w˜ 3 − αε22 w˜ 1 ··· ⎪ −1 ⎪ ⎪ w˙˜ = w˜ r − αεrr−1 w˜ 1 ⎪ ⎩ ˙ r −1 w˜ r = Ψ (Y, z, t) − αεrr w˜ 1 which can be rewritten in the matrix form as W˙˜ = (A − H C)W˜ + BΨ (Y, u, z, t)

(2.11)

where H = [ αε1 αε22 . . . αεrr ] , C = [1 0 . . . 0]. Substituting the observer error dynamics with the following scaled estimating error, i.e. ηi =

w˜ i , 1≤i ≤r εr +1−i

(2.12)

Let η = [η1 η2 . . . ηr ] , and D(ε) = diag[εr εr −1 . . . ε]r ×r . Then we have W˜ = D(ε)η

(2.13)

Substituting (2.13) into (2.11), we have η˙ = D −1 (ε)(A − H C)D(ε)η + D −1 (ε)BΨ (Y, u, z, t) which is rearranged as η˙ =

1 [A10 η + BΨ (Y, u, z, t)] ε

(2.14)

2.2 Proof of Convergence of Estimation Errors of High-Gain Observer

37

where A10 is Hurwitzian. Proof of Theorem 2.1 For system (2.14), A10 is a Hurwitz matrix, thus a Lyapunov function can be defined as (2.15) V10 (η) = η P10 η where P10 is a positive definite solution of the Lyapunov equation P10 A10 + A 10 P10 = −I . (2.15) satisfies λmin (P10 )η2 ≤ V10 (η) ≤ λmax (P10 )η2 ∂ V10 A10 η ≤ −η2 ∂η    ∂ V10     ∂η  ≤ 2λmax (P10 )η

(2.16)

Differentiating V10 along (2.14), we have 1 2 V˙10 = − η2 + Ψ (Y, u, z, t)B  P10 η ε ε

(2.17)

Referring to F1 , (2.17) can be rearranged as 1 2 V˙10 ≤ − η2 + γ1 λmax (P10 )η ε ε

(2.18)

Taking α with 0 < α < 1, it has V˙10 ≤ −αη2

(2.19)

max (P10 ) > 0. According to (2.16), it can be obtained holds, if η ≥ δspo . δspo = 2γ1 λ1−αε 2 that λmin (P10 )η ≤ V10 (η) ≤ λmax (P10 )η2 . Applying Corollary 5.3 of Theorem 5.1 in [3], it has if η(0) ≥ δspo1 , ∃t1 > 0, such that 



 η ≤

λmax (P10 ) η(0)e−(α/(2λmax (P10 )))t , ∀t < t1 λmin (P10 ) 

and η ≤ where t1 ≤

2λmax (P10 ) log α



η δspo1

λmax (P10 )  δ , ∀t ≥ t1 λmin (P10 ) spo

 .

∗ Based on (2.12) and 0 < ε < 1, it has W˜  ≤ η. Therefore, ∀δspo , ∃spo ∈ (0, 1) ∗ given by F3 such that, ∀ε ∈ (0, εspo ),

38

2 Observer-Based Robust Bang-Bang Funnel Controller …

W˜ (t) ≤ δspo , ∀t ≥ t1 whence Theorem 2.1 is proved.

2.3 Proof of Stability of Closed-Loop System Theorem 2.3 indicates that the estimation errors of tracking error of control target and its derivatives can be arbitrary small, through selecting the gains of the high-gain state observer properly. In order to guarantee the proper function of the BCFC, the high-gain state observer and the BCFC are designed such that δspo ε. The proof of Theorem 2.2 consists of two parts. The first part is to prove that the solution of the closed-loop system is well-defined. The second part is to show that ∃τ ∈ (t2ζ −1 , t2ζ ) such that e(i) (t) ∈ [−ϕi , ϕi ] holds for all t ∈ [τ, t2ζ ). Input of the BCFC is Wˆ , and Wˆ is continuous according to (2.9). Referring to [1, 4, 5], closed-loop system shown in Fig. 2.1 has a single solution (Y, q) : [0, ω) → Rr × {true, false}, ω ∈ (0, ∞] for all Y 0 ∈ Rr , q0 ∈ {true, false}r , and q has finite switching frequency in each compact region within [0, ω). For investigating the closed-loop dynamics, it needs to be figured out the impact of the tracking errors of the observer on the dynamics of the BCFC within and outside the error funnels Fi , respectively.

2.3.1 Dynamics of RBC Outside Error Funnels According to [6, 7], the original BCFC functions normally on condition that the safety distance εi is enlarged by δspo , which is obtained in Sect. 2.2. The feasibility of the robust bang-bang funnel controller within the error funnels can be verified in the same manner with that in [1] concerning the revised feasibility assumption on controller parameters (F4 ). With respect to the dynamics of the RBC outside the error funnels, the worst case, in which all components of Ye are outside the error funnels, is concerned here. Due to this, it is assumed that θ (t) ≡ −β holds on t ∈ [t2ξ , t2ξ +1 ), where ξ ∈ N+ , and the case of θ (t) ≡ β follows analogously. Theorem 2.2 should be proved in a recursive manner. Hence, let ξ = 1 firstly, and it has e(i) (t2ξ ) ∈ (−ϕi , ϕi ) according to F5 and the feasibility of the BCFC within the error funnels. Moreover, it is assumed that wˆ r (t2ξ ) ≤ −ϕr −1 + εr −1 . According to the switching logic of Br −1 , it has q(t2ξ ) =false and u(t2ξ ) = U . Hence, on the time interval [t2ξ , t2ξ +1 ) it has

2.3 Proof of Stability of Closed-Loop System

39

e(r −1) (t2ξ +1 ) = e(r −1) (t2ξ ) + ≥ −ϕr −1 +

t2ξ +1  t2ξ

t2ξ +1

e(r ) (t)dt

t2ξ

 (r ) f (Y, z) + g(Y, z)U − β − yref dτ

(r ) Referring to F7 , it has g(Y, z)U + f (Y, z) − yref ≥ λr . Let t = t2ξ +1 − t2ξ , then it has (2.20) e(r −1) (t2ξ +1 ) ≥ −ϕr −1 + (λr − β) t = χ(re −1),0

Considering F4 , it has χ(re −1),0 0 since λr β. On the interval [t2ξ , t2ξ +1 ), the dynamics of e(r −2) (t) satisfies e(r −2) (t2ξ +1 ) = e(r −2) (t2ξ ) + ≥ −ϕr −2 +

t2ξ +1

e(r −1) (τ )dτ

t2ξ

t2ξ +1

[−ϕr −1 + (λr − β)(τ − t2ξ )]dτ

t2ξ

= −ϕr −2 + (−ϕr −1 ) t + 21 (λr − β) t 2 = χ(re −2),0 Similarly, the dynamics of e(r −i) (t) on [t2ξ , t2ξ +1 ) satisfies e

(r −i)

(t2ξ +1 ) ≥ −ϕr −i −

i−1  ϕr −i+ j t j j!

j=1

=

+

(λr − β) t i i!

χ(re −i),0

According to the description of the external disturbance, d(t) turns to zero from t2ξ +1 . Therefore, there exists tb(r −1) ∈ (0, t2ξ +2 − t2ξ +1 ) such that e(r −1) (t2ξ +1 + tb(r −1) ) = 0 and e(r −1) (t) keeps increasing on [t2ξ +1 , t2ξ +1 + tb(r −1) ) and

e

(r −1)

t (t)

≥χ(re −1),0

+

e(r ) (t)dt

t2ξ +1

≥χ(re −1),0

+ λr (t − t2ξ +1 )

Referring to (2.20), we have tb(r −1) ≤ −χ(re −1),0 /λr The switching logic of Sr −1 leads to that q(t) =false until wˆ r hits the upper switching trigger. The upper switching trigger of Sr −1 is ϕr −1 − εr −1 + δspo . Concerning the maximum increasing rate of e(r −1) (t) is λr and letting e(r −1) (t2ξ +1 + tb(r −1) +

40

2 Observer-Based Robust Bang-Bang Funnel Controller …

te(r −1) ) = ϕr −1 − εr −1 + δspo , it has te(r −1) ≤ (ϕr −1 − εr −1 + δspo )/λr With respect to the dynamics of e(r −2) (t), it decreases until t = t2ξ +1 + tb(r −1) and it has on [t2ξ +1 , t2ξ +1 + tb(r −1) ) that

e

(r −2)

t (t)

≥χ(re −2),0

e(r −1) (t)dτ

+ t2ξ +1

t ≥χ(re −2),0

+

[χ(re −1),0 + λr (t − t2ξ +1 )]dt

t2ξ +1

=χ(re −2),0 + χ(re −1),0 (t − t2ξ +1 ) +

1 λr (t − t2ξ +1 )2 2!

e(r −2) (t) starts to increase from t = t2ξ +1 + tb(r −1) , and it satisfies 1 λr t 2 + 2! b(r −1) (χ(re −1),0 + λr tb(r −1) )(t − tb(r −1) − t2ξ +1 ) 1 + λr (t − tb(r −1) − t2ξ +1 )2 = ψr −2 2!

e(r −2) (t) ≥χ(re −2),0 + χ(re −1),0 tb(r −1) +

on [t2ξ +1 + tb(r −1) , t2ξ +1 + tb(r −1) + te(r −1) ). Assume e(r −2) (t) reaches zero at t = t2ξ +1 + tb(r −1) + te(r −1) + tb(r −2) , then we have tb(r −2) ≤ −ψr −2 /λr −1 Referring to the switching logic Sr −2 , qr −1 =false until e(r −2) (t) hits the upper switching trigger, i.e. ϕr −2 − εr −2 + δspo under the assumption delayed switching. Let the time point at which e(r −2) (t) reaches the upper switching trigger for the first time be t = t2ξ +1 + tb(r −1) + te(r −1) + tb(r −2) + te(r −2) , then it has te(r −2) ≤ (ϕr −2 − εr −2 + δspo )/λr −1 In the same manner, it can be obtained the minimum of e(r −i) (t) on the interval t ∈ [t2ξ +1 , t2ξ +2 ). Let τ ∈ {e, b} and τ χi,k =

i  1 τ¯ j χ t j! s, f τ,(r −k) j=0

(2.21)

2.3 Proof of Stability of Closed-Loop System

41

where i ∈ {1, . . . , r }, k ∈ {0, 1, . . . , r − 1}, τ¯ denotes the complement of τ in the set {e, b},  r − i + j, for k = 1 and τ = b s= i − j, otherwise  f =  e χm,n

= 

b χm,n =

k − 1, for τ = b k, for τ = e λr −m , when m = n 0, when m < n

λr −n+1 , when n − m = 1 0, when n − m > 1

Then the minimum of e(r −i) (t) is min{e(r −i) (t)}|[t2ξ +1 ,t2ξ +2 ) =e(r −i) (t2ξ +1 + ςr −i+1 ) b ≥χi,(i−1) = −μr −i

(2.22)

where ςr −i+1 = tb(r −1) + te(r −1) + . . . + te(r −i+2) + tb(r −i+1) . e(r −i) (t) keeps increasing on t ∈ [t2ξ +1 + ςr −i+1 , t2ξ +1 + ςr −i + te(r −i) ), and we have e(r −i) (t2ξ +1 + ςr −i+1 + te(r −i+1) ) ≥ b b 2 χi,(i−1) + χ(i−1),(i−2) te(r −i+1) + 2!1 λr −i+2 te(r −i+1) = ψr −i Therefore, it can be obtained that  tb(r −i) ≤ −ψr −i /λr −i+1 te(r −i) ≤ (ϕr −i − εr −i + δspo )/λr −i+1

(2.23)

Then the shortest interval σ between two consecutive disturbances, which ensures that each element of Ye has converged back to the error funnels, satisfies r −1  −ψ0 − ϕ0 + δspo [tb(r −i) + te(r −i) ] + λ1 i=1 ! r −1  −ψ0 − ϕ0 + δspo ϕr −i − εr −i + δspo − ψr −i + ≤ =κ λr −i+1 λ1 i=1

σ =

(2.24)

Therefore, it has been verified that the tracking error of the output and its derivatives can be regulated back into the error boundaries subjected to the most severe disturbance that concerned, on condition that the interval between two consecutive disturbances is larger than κ, i.e., t2ζ − t2ζ −1 ≥ κ. With respect to the less severe cases, the performance of the tracking error of control target and its derivatives can

42

2 Observer-Based Robust Bang-Bang Funnel Controller …

be guaranteed in the same manner as above. For the cases of ξ = 2, 3, . . ., it can be recursived obtained that e(i) (t2ξ ) ∈ (−ϕi , ϕi ). Then Theorem 2.2 can be verified similarly as above, then Theorem 2.2 is proved.

2.4 Simulation Studies: Application of RBC for Frequency Control of a SMIB Test System The RBC is applied in the design of the exciter of a synchronous generator to regulate the frequency of a SMIB test system. The structure of the SMIB system is shown in Fig. 2.2. A third-order model is used for the dynamics of the synchronous generator, and the entire system is described as [8] ⎧ ˙ ⎪  ⎨ δ = ω −ω0 ω0 Pm − ωD0 (ω − ω0 ) − Pe ω˙ = 2H (2.25) ⎪ ⎩ E˙  = 1 (E − E ) f q q T d0

where

Pe = E q Iq + (X q − X d )Id Iq   Id = X1 E q − Vs cos(δ) ds E q = E q + (X d − X d )Id  X ds = X d + X T + X s

Pe =

E q Vs sin(δ) X ds Vs sin(δ) X qs

Iq = X qs = X q + X T + X s X ds = X d + X T + X s

and δ represents the rotor angle, in rad, ω is the rotor speed of generator, in rad/s, ω0 denotes the synchronous speed of the system, in rad/s, E q and E q are voltage behind  the quadrature-axis and its transient value, respectively, Td0 denotes the d-axis opencircuit transient time constant, E f is the excitation voltage, Id and Iq are the d-axis and q-axis generator current, respectively, and Vs is the voltage of the infinite bus. The parameters of the SMIB system are given as follows: X d = 1.00 p.u., X d = 0.40  = 5.00 s, and p.u., X q = 0.60 p.u., X T = 0.12 p.u., X L = 1.00 p.u., D = 0.008, Td0 H = 4.34 s. Letting x = [x1 x2 x3 ] = [δ − δ0 ω − ω0 E q ] and the input variable as u = E f , the state equation of system (2.25) can be rewritten as 

Fig. 2.2 The layout of a SMIB power system [2].

x˙ = f (x) + g(x)u y = h(x) = x2

(2.26)

2.4 Simulation Studies: Application of RBC for Frequency Control of a SMIB Test System

where

⎡

43

⎤

⎡ ⎤ 0 ⎢ ω0 P − D x − P ⎥ ⎣ 0 ⎦. f (x) = ⎣ 2H m e ⎦ , g(x) = ω0 2 1 1  (−E q ) Td0 T x2





d0

For system (2.26), we have ω0 x2 x3 Vs cos(δ) (X q − X d )Vs2 cos2 (2δ) =− −   2H X ds X ds X qs ω0 Vs sin(δ)E q Dω0 D − Pm − x2 − Pe +   4H 2 ω0 2H Td0 X ds ω0 Vs sin(δ) Lg L f h(x) = −   2H Td0 X ds

!

L f2 h(x)

As Lg L f h(x) = 0 holds for ∀δ = kπ(k = 0, 1, 2, . . .), system (2.26) has relative degree of r = 2, then it can be obtained that y¨ = L f2 h(x) + Lg L f h(x)u According to the above results, a second-order high-gain state observer and a second-order BCFC is employed for the RBC. The RBC functions as a bang-bang excitation controller (BEC) for the synchronous generator. According to Theorem 2.1 and (2.10), the high-gain state observer is configured as: ε = 0.01, α1 = 300, α2 = 9000. In steady state, the operation point of the SMIB power system is at δ0 = 0.6981 rad, Pe = 0.4732 p.u., and ω0 = 314.15 rad/s. Thus it has Lg L f h(x) < 0. Referring to (2.8), the BEC can be given as  u(t) =

E f , if q(t) = true −E f , if q(t) = false

The parameters of BCFC are chosen as ϕ0+ = −ϕ0− = 2, ϕ1+ = −ϕ1− = 2, ε0+ = − + − + − 0.8, λ+ 1 = λ1 = 0.35, λ2 = λ2 = 1.026, 0 = 0 = 15.34, − −4 = 2 = 1 × 10 . E V δ¨ + ωD0 δ˙ + Xqds s cos(δ0 ) δ = The linearization of (2.25) at equilibrium is 2H ω0 0, then the eigenvalues of this equation are p = ±4.5205j. Thus it has δ(t) = ˙ = ω(t) = 4.5205Ae4.5205jt holds. With the assumption that Ae4.5205jt . Then δ(t) ω and its derivative are able to go back to the error funnels after the impulsive disturbance, we have ω ∈ [−2, 2]. Therefore, it has |A| ≤ 2/4.5205 = 0.4424 and then δ(t) ∈ [−0.4424, 0.4424]. Considering (2.25) is disturbed and the output h(x) and its derivative are within the error funnels, it has δ(t) ∈ [0.2557, 1.1405] with δ0 = 0.6981. ε0− = 1.95, ε1+ = ε1− = − + + 1 = 1 = 3.91, 2

44

2 Observer-Based Robust Bang-Bang Funnel Controller …

r

(p.u.)

KSTAB

Ef

Power system stabilizer sTw 1 sT1 1 sTw 1 sT2

1 sTA KA 1 sTB Exciter

1 sT3 1 sT4

Vs

Vs _ ref

Fig. 2.3 The schematic of the AC1A-type exciter and the power system stabilizer [2]

With respect to the most serious fault, a metallic three-phase fault on the infinite ω0 Pe . Therefore, it has μ1 = 17.13 bus (Vs = 0), ω˙ could see a largest jump of − 2H and μ0 = 143 according to (2.22). If ω ∈ [−143, 143] and ω˙ ∈ [−17.13, 17.13], we have L f2 h(x) ∈ [0.9934, 8.7494] and Lg L f h(x) ∈ [−4.3290, −1.2058]. Refer−λ+ −L 2 h(x)|max

f ring to F6 , it has E f > L2g L f h(x)| = 8.107. Hence, it is chosen that E f = 8.2 p.u., max which is realizable according to [9]. To this end, the RBC-based BEC is designed. To evaluate the BEC, the SMIB system controlled by the BEC is compared with that controlled by a conventional excitation controller (CEC). The CEC is composed of an AC1A-type excitor and a power system stabilizer (PSS). Structure of the system is as illustrated in Fig. 2.3. The PSS is designed as: Tw =0.3, T1 =0.07, T2 =0.03, T3 =0.07, and T4 =0.03. The AC1A-type excitor is set as: K A =150, TA =0.2, and TB =5. The three-phase fault is applied on the line between t = 1s and t = 1.28s as depicted in Fig. 2.2. To start the simulation from the steady state, the BEC switched on after t = 1.02s. Before that, a CEC is utilized for the stability of the synchronous generator. Dynamics of the test system having the BEC and CEC implemented respectively is shown in Figs. 2.4 and 2.6. During the fault, the voltage of the infinite bus drops to Vs = 0.01 p.u. Referring to Fig. 2.4c, the drop of the voltage of the infinite bus results in a decline of Pe as E V sinδ Pe = q Xsds . Consequently, both ω and ω˙ leave the error funnels as depicted in Fig. 2.4a, b, respectively. As is observed, ω˙ goes outside the error funnel at the end of the fault, while ω is within the error funnel. With the BEC depicted in Fig. 2.4c, ω˙ is able to converge back into the error funnel. Both ω and ω˙ are regulated back into the error funnels by the BEC. By contrast, the generator controlled by the CEC shows more oscillations in both rotor speed error and active power as depicted in Fig. 2.4a, c, respectively. The high-gain observer provides accurate estimation for the rotor speed tracking error ω and its derivative ω, ˙ as shown in Fig. 2.5a, b, respectively. We can see that the estimation errors of the high-gain observer is larger when the fault is applied and cleared. Relative smaller errors are observed on the estimates of ω˙ at the switching points of the control input generated by the BEC as illustrated in Fig. 2.5b. Hence,

2.4 Simulation Studies: Application of RBC for Frequency Control of a SMIB Test System

Fig. 2.4 The dynamics of the generator controlled by the CEC and BEC, respectively [2].

45

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2 Observer-Based Robust Bang-Bang Funnel Controller …

Fig. 2.5 The dynamics of the generator controlled by the CEC and BEC, respectively [2].

Terminal voltage of generator

V t (p.u.)

1.2 1 0.8 0.6 BEC CEC

0.4 0.2

1

2

3

4

5

6

7

8

9

10

Time (s) Fig. 2.6 Terminal voltage of the generator controlled by BEC and CEC, respectively [2].

References

47

it shows that the high-gain observer is completely capable of tracking the error of control target and its derivative in the cases where the input to the objective system is discrete. Dynamics of the terminal voltage of the generator is as depicted in Fig. 2.6. The case controlled by CEC presents more fluctuation in the magnitude of terminal voltage. Since the excitation control signals generated by BEC only contain bang-bang values, the terminal voltage of the generator controlled by BEC shows persistent small-magnitude oscillations. This can be solved by switching the BEC with a CEC through a properly designed switching strategy [6–8, 10].

2.5 Summary This chapter has presented a RBC to improve the robustness of disturbed nonlinear systems and explore the control system potential. In comparison with the original BBFC theory, weaker feasibility assumptions have been raised. The convergence of the estimating error of the high-gain observer has been justified, and it is able to offer accurate tracking performance for a system with discrete inputs. The combination of high-gain observer and BCFC effectively eliminates the derivative calculations in the BCFC. It has also been shown that the error of control target and its derivatives can be regulated into the error funnels by the RBC if the time interval between two consecutive disturbances is larger than κ. Simulation results have shown that the RBC is able to fully utilize the effort of the exciter and presents better performance than the conventional controller, which meets Theorem 2.2. The nonlinear observer is able to offer accurate estimation for the rotor speed tracking error and its derivatives, which meets Theorem 2.1. Based on the theoretical results, the RBC can be used in the stability control of various systems.

References 1. Liberzon D, Trenn S (2013) The bang-bang funnel controller for uncertain nonlinear systems with arbitrary relative degree. IEEE Trans Autom Cont 58(12):3126–3141. https://doi.org/10. 1109/TAC.2013.2277631 2. Liu Y, Xiahou K, Wu QH et al (2020) Robust bang-bang control of disturbed nonlinear systems based on nonlinear observers. CSEE J Power Energy Syst 6(1):193–202 (2020). https://doi. org/10.17775/CSEEJPES.2018.01310 3. Khalil HK (1996) Nonlinear systems. Prentice-Hall Inc, London 4. Liberzon D, Trenn S (2010) The bang-bang funnel controller. In: 2010 49th IEEE CDC, pp 690–695 (2010). https://doi.org/10.1109/CDC.2010.5717742 5. Liberzon D, Trenn S (2013) The bang-bang funnel controller: time delays and case study. In: 2013 ECC, pp 1669–1674

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6. Liu Y, Xiahou K, Lin X, Wu QH (2019) Switching fault ride-through of GSCs via observerbased bang-bang funnel control. IEEE Trans Ind Electron 66(9):7442–7446. https://doi.org/ 10.1109/TIE.2018.2864709 7. Liu Y, Xiahou K, Wang L, Wu QH (2018) Switching control of GSC of DFIGWTs for disturbance rejection based on bang-bang control. IEEE Trans Power Del 33(6):3256–3259 8. Liu Y, Wu QH, Zhou XX (2016) Coordinated switching controllers for transient stability of multi-machine power systems. IEEE Trans Power Syst 31(5):3937–3949 9. Lee D, Baker D, Bess K et al (1992) IEEE recommended practice for excitation system models for power system stability studies. Energy Dev Power Gener Committee Power Eng Soc 10. Liu Y, Wu QH, Kang H, Zhou X (2016) Switching power system stabilizer and its coordination for enhancement of multi-machine power system stability. CSEE J Power Energy Syst 2(2):98– 106

Part II

Transient Stability Control of Large-Scale Complex Power Systems Based On Adaptive Switching Controllers

Stability of power systems can be divided into power angle stability, frequency stability, and voltage stability. Power angle stability includes small-signal stability and transient stability, which are the main issues of concern in Part II. Enhancing transient stability of power systems requires the coordinated behavior of various distributed controllers in the power system, and the largest control effort of each device should be utilized within the first 2 s, i.e., the fist-swing, of the fault-on stage. Therefore, time-optimal controllers should be designed for various distributed controllers in power systems. Conventional PSS (CPSS) is the most commonly used distributed controllers for enhancing small-signal stability in power systems. The CPSS has significant impact on the instability modes of the system. If the energy involved within the instability modes is dissipated in an early stage, transient stability and small-signal stability of the entire system can be improved. The mechanism of a CPSS is to add auxiliary damping to rotor-speed oscillations [1] of synchronous generators. However, CPSS has intrinsic shortcomings, which significantly diminish its control performance in the following aspects. First, the CPSS is designed using the linearized model of a power system operating at equilibrium, which is based on the Lyapunov’s first stability theorem. The location setting and parameter selection of a CPSS rely on the equilibrium point of the system, at which constant operating conditions and load levels are assumed [2]. However, these factors are continuously changing in normal operation, and the operating point of the system always varies [3]. Second, the performance of the CPSS is largely impacted by the accuracy of parameters used in the controller design; however, absolutely accurate parameters are hardly available [4]. Third, CPSSs in a large-scale power system have complex interactions, and this leads to the contradiction between controllers [5]. In addition to CPSSs, robust, probabilistic, and adaptive PSSs were designed. A robust PSS was designed in [6], and the PSS was robust to both the transient and small-signal oscillations of the system. As a matter of fact, the same set of parameters cannot be optimal to all oscillatory modes of the power system. The same problem exists in the probabilistic PSS designed in [7]. The expectation and variance of the eigenvalues of a system operating under different operation modes

50

Part II: Transient Stability Control of Large-Scale Complex Power …

are evaluated to determine the parameters of PSS. Moreover, a decentralized adaptive PSS was proposed in [8] based on a reduced-order state observer. The information of the relative rotor speed between the generators was used to generate the estimation of state variables. Moreover, the accuracy of the estimated states is determined by the parameters of the state observer. However, measured noise can be magnified and seriously impacts the performance of the adaptive PSS. Catering to the above, Chap. 3 firstly proposes a bang-bang PSS (BPSS) based on the BBFC to achieve fast damping of small-signal oscillations. With respect to the transient stability of power systems, the excitation control of synchronous generators is a traditional but effective way to improve the transient stability of power systems [9]. The existing literature studying the excitation control can be classified as follows. The first kind is linear excitation control methods, and the PID excitation controller combined with an AVR compensator was designed in [10]. The PID excitation controller in combination with PSS was capable of improving both the transient and the small-signal stabilities of power systems [11]. To obtain the robustness of the PSS for damping the oscillations, the PSS design methods concerning multiple operating points were studied in [12, 13]. Moreover, much effort was dedicated to the linear optimal and suboptimal exciters [14]. The linear output feedback excitation control was proposed in [15], which was used for stabilizing the torsional oscillation of synchronous generators. As for nonlinear excitation controllers, the exact linearization feedback excitation control was investigated in [16]. A partial linearization feedback excitation control was presented in [17]. Moreover, the fuzzy theory, Lyapunov stability, and recursive design methods were applied in the design of the exciter in [18–20], respectively. Nevertheless, the aforementioned nonlinear controllers need accurate parameters and thus lack robustness to uncertainties of power systems. As far as we know, all excitation control needs system information. In addition to the excitation control, the speed control of synchronous generators was expected to improve the transient stability of power systems as well [21]. However, the speed control and the excitation control loop of a synchronous generator were generally decoupled in different timescales [22]. However, it is no longer the situation of the next-generation power systems, in which the rapid switch of valves is allowed by accumulators and hydraulic speed governing systems. Therefore, tight mutual interaction between the excitation loop and speed governing loop exists in the synchronous generator [23]. The coordination between these two loops was studied in [24], which was proved to be able to enlarge the stability margin and achieve better transient stability. Apart from the above continuous control methods, bang-bang control, also known as time-optimal control, was applied to improve the transient stability of smallscale power systems in [25, 26]. On the basis of the minimum principle, the bangbang control law was obtained by solving the canonical equation of the Hamiltonian of the power system. The bang-bang controller makes use of the largest damping power to dissipate the oscillations of the power system in the shortest time, thereby achieving a time-optimal performance. Although it has shown the potential in the transient stability control of the power system, the bang-bang control law requires the

Part II: Transient Stability Control of Large-Scale Complex Power …

51

derivatives of the Hamiltonian of the system. Therefore, the requirement of accurate parameters and the complexity of constructing the Hamiltonian definitely hindered its application in large-scale systems. Utilizing the logic-based control theory, a BBFC was proposed in [27]. It enables the application of bang-bang control in large-scale power systems without complex computation and accurate system information. The BCFC is employed in Chap. 3 for the design of the SEC and the SG for synchronous generators to improve the transient stability of power systems. Besides synchronous generators, the MMC-HVDC is widely used for longdistance power transmission nowadays [28]. In comparison with traditional twolevel or three-level converters, each bridge arm of a MMC consists of a number of submodules connected in series. The MMC is easy to expand and shows low harmonic distortion rate and switching losses [29]. MMC-HVDC transmission systems operate robustly to the disturbance of external AC grids and help AC grids to recover from a severe disturbance. Hence, fault ride-through capability of MMCs has a significant impact on the stability of the entire power system [30, 31]. A large amount of work has been undertaken to improve the control of MMC-HVDC transmission system, for better fault ride-through performance. Chapter 4 focuses on the study of MMCs with half-bridge submodules. Owing to easy implementation, independent control of active and reactive power, vector controller (VC) is widely used for MMCs [32]. A VC is implemented by PI controllers or proportional resonant (PR) controllers. A PI controller is of first order and responds to any frequency range [33]. In comparison, a PR controller is of second order or higher and responds to a specific frequency [34]. To enhance the fault ride-through performance, a DC voltage droop control scheme is implemented for the MMC in [35]. A PR controller is proposed in [36] to suppress the circulating current in the MMC-HVDC transmission system during fault process. A systematic tuning rule is designed for parameters of dual-loop voltage-controlled MMC in [37], which considers voltage stability, transient fault current limitation, as well as stiffness against load current changes. An optimization approach is proposed for VC of MMCs in [38]. With the optimized parameters, small-signal stability and dynamic responses of the hybrid AC/DC power system were enhanced. By injecting additional current components with feed-forward control or optimizing controller parameters, linear controllers [38, 34–36, 37, 32, 33] can improve the robustness of MMCs against grid faults. These methods have been well studied, and mature linear analysis tools can be used for stability analysis [39]. However, limitations of these controllers are obvious. First, the parameter tuning of linear controllers relies on small-signal stability analysis. Mode transfer cannot be taken into account, which is the situation when severe grid faults occur in a power system [40]. Second, the trade-off between response speed and overshot must be considered in the design of linear controllers; thus, only moderate performance can be achieved in most cases [41]. Maximum control effort of the MMC cannot be used for robust stability at all operating regions of the system. Nonlinear controllers have also been designed, so as to improve the fault ridethrough capability of MMC-HVDC transmission systems. Output feedback linearization method was employed in [42] and [43] to realize the decoupled control of the

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Part II: Transient Stability Control of Large-Scale Complex Power …

MMC. Nonlinearities in the MMC can be fully considered in the feedback linearization [43, 42], and it improves the robustness and stability of these controllers to the external disturbances. A nonlinear phase-unsynchronized decoupling control was proposed for the MMC in [44]. By input–output linearization, output of the MMC is decoupled without phase-locked loop (PLL). An optimization-based current reference algorithm has been designed in [45], for real-time control of MMC operating under balanced and unbalanced AC as well as DC network conditions. Mathematical optimization-based optimal controllers were proposed in [46], and the robust control of a MMC was implemented with optimized operational costs. [47] employed the exponentials of matrices to represent the state-space model of the MMC, such that a current control law was realized by pole-zero configuration. A unified numerical method was proposed for calculating voltages and currents at the DC side of a MMC under fault conditions in [48]. [49] proposed a model predictive control method to stabilize the MMC system under various conditions by taking nonlinear properties of MMC into account in control laws. However, these nonlinear methods [47, 46, 49, 43, 45, 42, 44] have a common shortcoming, i.e., the control laws of these methods were much more complex than a vector controller, and parameter tuning is also more difficult. Therefore, these nonlinear controllers are not widely employed in industry. In addition to the above restrictions, the existing control methods [47, 46, 38, 49, 34–37, 43, 32, 33, 45, 42, 44] cannot make full use of the capability of MMC during a fault ride-through event. Along with the development of renewable power generators, exploring the most controllable energy in power systems during transient process is an unprecedented trend. Achieving the re-balance between power generation and consumption rapidly will help to retain the voltage and frequency levels. Therefore, it is of great need to explore the full control effort of the MMC in the early stage when a power system recovers from a severe fault. Such investigations have been carried out in [50, 51], where BBFCs were employed for the maximum energy control of voltage source converters. In Chap. 4, a fault ride-through hybrid controller (FRTHC) is proposed to improve the fault ride-through performance of MMC-HVDC transmission systems. Chapter 4 includes the following work: First, a switching control unit (SCU) in modular structure is proposed, and the SCU can be used individually or with another SCU, in a control loop of the MMC. Inclusion of the BBFC enables the SCU to explore more control power of the MMC, in comparison with continuous controllers. Second, a state-dependent switching law is proposed for each SCU. Based on the law, a BBFC and a PI control loop work in a switched manner. Delay modules are implemented to guarantee the stably switching between the two control loops. Along with the booming development of wind power generation, transient stability of large-scale wind power penetrated power systems (WPPS) is increasingly influenced by wind power plants [52]. In an extreme event, a strong control system of wind power generators can improve the reliability of WPPS and prevent wind farms from tripping. This helps to mitigate the power unbalance and improve the transient stability of large-scale WPPSs. The DFIGWT is the most widely used wind power generators in large-scale wind farms, due to its high efficiency and relatively low

Part II: Transient Stability Control of Large-Scale Complex Power …

53

cost [53]. Therefore, Chap. 5 mainly focuses on enhancing the control of DFIGWT for transient stability enhancement of large-scale WPPSs. During the last decades, various kinds of control techniques have been designed for the DFIG. In terms of linear control methods, the VC is the most mature one that has been widely applied in the control of the DFIG [54]. Using a stator flux-oriented [55] or a stator voltage-oriented [56] reference frame, active and reactive power components of the rotor current are obtained. The decoupled control of stator active and reactive power is realized by regulating the corresponding rotor current components [57]. Based upon the VC scheme, direct torque control [58] and direct power control [59] were investigated for the control of rotor-side converters of the DFIG. To offer frequency support to power grids, a virtual inertial control was implemented in the vector control framework of the DFIG [60]. The VC-based control methods are simple in terms of controller tuning and efficiency near the pre-specified operating point. However, the nonlinearity, parameter uncertainty, and external disturbances are not considered in the design process of these controllers [61]. These parameterdependent methods are unable to offer optimal control in the cases where the WPPS operates far away from its original equilibrium. Moreover, a great number of nonlinear control techniques, such as fuzzy control [62], sliding-mode control [63], and model predictive control [64], have also been used for controller DFIGs. Although these nonlinear controllers have better robustness to the system nonlinearity and uncertainty in comparison with linear controllers, these controllers did not employ the maximum control power of the converter of DFIG. Unused control power still exists in the converter of the DFIG. However, few of them is able to coordinate between each other nor coordinate with external power grid [65]. Combining the advantages of linear and nonlinear controllers, bang-bang control scheme is employed in Chap. 5 for the integral control of the DFIG to enhance the transient stability of WPPSs. Based on the BBFC, a logic-based BCFC (LBCFC) is proposed in Chap. 5 for the design of a four-loop SC of the DFIG. Chapter 5 covers the following contents: First, four LBCFCs are designed, respectively, for the integrated control of the DFIG. These LBCFCs are installed in four loops, i.e., the rotor-speed control loop, the reactive power control loop of the stator winding, the DC-link voltage control loop, and the reactive power control loop of the grid-side converter. The LBCFC regulates the output tracking error of each control loop into a pre-defined error funnel using bang-bang control signals after an external disturbance occurs. The LBCFC uses the largest control power of converters to suppress the oscillations of the DFIG. Due to the model free characteristic of the LBCFC, coordination between different DFIGs and that between wind farms and power grids automatically can be achieved. Moreover, a state-dependent switching strategy is presented for the SCs. The SC switches between a LBCFC and a CC in each control loop. If a severe disturbance occurs in a WPPS, outputs of the DFIG are driven far away from their steady-state values. The LBCFCs are switched, respectively, in the four control loops. Thereafter, the CCs are switched on in the four control loops, respectively, to regulate the output tracking errors asymptotically to zero.

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Part II: Transient Stability Control of Large-Scale Complex Power …

A short-term resilience index is defined for the WPPS in Chap. 5. The resilience of a WPPS can be evaluated with the unbalanced kinetic energy and load bus voltage. Several recent studies [66, 67] have proposed the concept of power system resilience. Nevertheless, the concept of short-term resilience discussed in a smaller timescale still remains undeveloped. Using the short-term resilience index, it obtains a quantified evaluation of transient stability control performance of controllers. The first category of control methods used the de-loading operation of WPGs to take part in the primary frequency control [71]. The active power was reserved, and WPGs can ramp up active power and down in the frequency deviation events to offer frequency support to the external power grid. In order to de-load WPGs, over-speeding and pitching techniques were employed [72]. An application was presented in [73], in which the de-loading of DFIGs was realized with a power versus rotor-speed lookup table. Similar method was presented in [74]. Nevertheless, the over-speeding technique was only preferable when WPGs operate below rated wind speed [71]. With respect to the pitching technique, de-loading was realized through increasing the pitch angle of WPGs such that the active power margin was obtained. The applications of this scheme can be found in [75, 76], through which primary frequency support of WPGs can be obtained by pitch angle control. The second category of strategies was referred to as virtual inertia control. WPGs were controlled to emulate the synchronous generators in frequency deviation events. For instance, the set point of the torque reference of the speed control loop of a WPG was regulated with the deviation of grid frequency and its changing rate in [77]. In [78], a synchronous generator model was configured in the rotor-side controller such that the active power of the WPG was controlled by a rotor motion equation. As a matter of fact, the virtual inertia control was realized on the basis of the de-loading operation of WPGs as well. The primary frequency control requires WPGs to boost their active power when system frequency drops. In addition to the above, energystorage devices were employed for the frequency control of WPPSs in [79, 80]. Inertial response of WPGs was achieved by the bidirectional active power control of energy storages. However, the main restriction of using energy-storage devices for frequency control was their high cost. On the basis of the de-loading of WPGs realized by pitch control, Chap. 5 proposes a switching angle controller (SAC) and an automatic generation controller (AGC) for the DFIG to control the frequency of DFIGWTs. In comparison with the existing studies [75, 71, 76, 79, 77, 73, 78, 72, 74, 80], the SAC is a discrete controller with only two control values. It regulates the active power of the DFIG by directly controlling the relative angle between the internal voltage and the terminal voltage. This relative angle is defined as the virtual rotor angle, and it enables the DFIG to response to the system frequency deviation in a bang-bang manner. In comparison with the continuous controllers, the SAC has the following advantages. On the one hand, the fast-response capability of the converter is fully explored by the SAC. The DFIG controlled by the SAC can provide faster active power support than that controlled by the continuous controller. On the other hand, continuous controllers respond to frequency deviation of all magnitudes and frequencies, while it can only be triggered when frequency deviation exceeds the pre-defined boundaries. Therefore, the SAC

Part II: Transient Stability Control of Large-Scale Complex Power …

55

has stronger robustness to the small-magnitude oscillations and measurement noise of system frequency. Similar to the AGC of a synchronous generator, an AGC is implemented in the pitch angle control loop of the DFIG. This enables the DFIG to take part in the primary and secondary frequency controls of power systems. Compared with the method proposed in [77], the SAC and the AGC do not result in any derivative items in their control loops. The SAC and AGC act as complementary controllers for the conventional VC. They are easier to be implemented than the method presented in [78], in which completely new control system was employed for the rotor-side converter (RSC), and another VC must be used for the grid-side converter (GSC). In contrast to the work shown in [79, 80], the SAC and AGC do not require additional costs on WPGs. Besides DFIGWTs, fault ride-through capability of GSCs is a key factor that influences the stable operation of permanent magnetic synchronous generator-based wind turbines (PMSGWTs). VC with supplementary control loops [81–83] was used in the control of the GSC of PMSGWTs. However, VC and the supplementary continuous control loops cannot make full use of the largest control power of converters. For a power system with a high penetration level of power electronics devices, it is desirable to make full use of the largest control effort of power electronics devices for transient stability improvement of the entire system. Chapter 6 constitutes an observer-based BBFC (OBFC) for the GSC of the PMSGWT. An OBFC is designed for the d-axis control loop of the GSC. Meanwhile, a BBFC is designed for the q-axis control loop for reactive current regulation. The OBFC and BBFC operate in a switched manner with the d-axis and q-axis control loops of VC, respectively, according to state-dependent switching strategies. The fault ride-through performance of the d-axis and q-axis control loops of the switching fault ride-through controllers is tested through hardware-in-the-loop experiments. Three-phase voltage source inverters (VSIs) have a variety of topologies. The most widely used one is the three-phase full-bridge topology. However, it suffers from the risk of shoot-through [84], since there are two active switches connected in series on the same phase leg. To prevent the shoot-through problem, additional deadtime is required in the switching commands. This introduces more low-frequency harmonics and leads to the output waveform distortion. The harmonics cannot be filtered effectively by the conventional LC filter and will reduce the system reliability. The physical decoupling method was firstly reported in [85] and has been used for the three-phase power-factor correction (PFC) rectifiers [86], grid-connected inverters [87–89], and active power filters (APF) [90, 91]. Moreover, the half-cycle mode of the three-phase dual-buck inverter [92, 93] is originated from it as well. Referring to [89], one AC line cycle of the inverter is divided into six regions by the zero-crossing points of phase voltages. Only two switches on different phase legs are operating at high frequency. The rest kept on or off, and there is no need for dead-time. Moreover, the three-phase VSI can be decoupled into a parallel connected dual-buck circuit in each 60° region. However, most of the existing literature [87, 89] involving this vector operation just analyzed the inverter by using its averaged steadystate model. System’s dynamics on the aspect of the equivalent dual-buck topology

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is omitted. [88] adopted the inverter’s large-signal model and rearranges it as two independent differential equations for the design of controllers, but this modeling is based on the three-phase frame. To focus on the transient behavior of dual-buck system, Chap. 6 presents a new approach to realize the system’s formulation and analysis, with the standard model of the buck chopper. The proposed decoupling method is called “dual-buck scheme”, which distinguishes from the traditional vector operation. Since the dual-buck scheme acts as buck chopper, control methods with desirable performance are the prime candidate. In this way, the inverter is able to track the reference signals rapidly and accurately. Practically, this kind of dual-buck circuit is normally controlled by the conventional PID controller with pulse width modulation (PWM) or space vector pulse width modulation (SVPWM) [92, 94]. However, this method cannot meet the requirement for improving dynamic responses, owing to the delay of the signals in the PID controller. Meanwhile, the controller is not able to track the sinusoidal signals without steady-state error. Hence, the coordinate transformation is required. The one-cycle control has fixed switching frequency and can obtain zero steady-state error and zero transient error [87, 95, 89]. The sliding-mode control is insensitive to the disturbances and has fast-response speed [96, 97, 88]. Nevertheless, it is rather difficult to acquire the ideal sliding surface. The fuzzy-logic control [98] suits for converters due to its simple realization and quick execution. The main shortcoming is the lack of systematic process to choose the appropriate membership functions. Owing to the disadvantages of these existing control methods, Chap. 6 also develops a three-phase full-bridge inverter based on dual-buck scheme with the BBFC. The BBFC is actually a generalization of the PD funnel controller presented by [99], and it has the following superiorities: (1) The control command consists of only two values, and no modulation technique is needed. (2) The maximal control values yield time-optimal performance with fast dynamic response. (3) Through the funnels, the desired tracking accuracy can be regulated arbitrarily small. (4) The controller is more robust to parameter uncertainties, changes in system topology, the external disturbances, as well as time delays and measurement noises. Apart from these, the BBFC allows direct analysis of the binary input signal, especially for the power electronics converters [27]. Up to now, the funnel control has been proved to be appropriate for tracking problems and successfully applied to power electronic converters, such as [100–102, 40, 51, 41, 103].

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Chapter 3

Switching Control of Synchronous Generators for Transient Stability Enhancement

3.1 Design of Switching Power System Stabilizer 3.1.1 Power System Model Used for Designing SPSS A fourth-order synchronous generator model with a first-order excitation control was used for designing the SPSS. For a n machine power system, the nth machine is chosen as the reference machine, and the system can be described with [1] ⎧ Δδ˙i = ωB Δωi ⎪ ⎪ ⎪ ⎨ Δω˙ i = 1 (Pmi − Pei − Di Δωi ) 2Hi  1  [E fi − E qi ] ⎪ E˙ qi = Td0i ⎪ ⎪ ⎩ E˙ = 1 [−E + K (ΔU + ΔU )] fi fi Ai ti PSSi TAi

(3.1)

where i = 1, 2, . . . , n. The above parameters of the ith generator are defined in [1]. Using this model, partial linearization can be realized.

3.1.1.1

Partial Linearization of Model of Multi-machine Power Systems

The state variables are defined as z = [z 1 , . . . , z i , . . . , z n ] , where z i = [z i1 , z i2 ,  z i3 , z i4 ] = [Δδi Δωi E qi E fi ] . The control variables and outputs are defined as u = [u 1 , u 2 , . . . , u n ] and y = [y1 , y2 , . . . , yn ] , respectively, where yi = z i2 and u i = ΔUPSSi . Then the model of ith synchronous generator can be denoted as 

z˙ i = Fi (z) + G i (z)u i yi = Hi (z)

i = 1, 2, . . . , n

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Liu and Q.-H. Wu, Adaptive Switching Control of Large-Scale Complex Power Systems, Power Systems, https://doi.org/10.1007/978-981-99-1039-7_3

(3.2)

63

64

3 Switching Control of Synchronous Generators for Transient …

where

⎤ ⎤ ⎡ ωB z i2 Fi1 1 [Pmi − Pei − Di z i2 ] ⎥ ⎢ Fi2 ⎥ ⎢ 2Hi ⎥ ⎥=⎢  Fi (z) = ⎢ 1 ⎢ ⎣ Fi3 ⎦ ⎣  [z i4 − z i3 − (xdi − xdi )Idi ] ⎥ ⎦ Td0i 1 Fi4 (−z i4 + K Ai ΔUti ) ⎡

TAi

G i (z) = 0 0 0

K Ai TAi



Hi (z) = z i2 .

The entire system can be written as 

where

z˙ = F(z) + G(z)u y = H (z)

(3.3)

F(z) = [F1 (z), F2 (z), . . . , Fn (z)] G(z) = block diag [G 1 (z), G 2 (z), . . . , G n (z)] H (z) = [H1 (z), H2 (z), . . . , Hn (z)] .

A nonlinear coordinate transformation is introduced as x = [x11 , x12 , . . . , x1r , . . . , xi1 , xi2 , . . . , xir , . . . , xn1 , xn2 , . . . , xnr ] = T (z) = [H1 (z), L F H1 (z), . . . , Hi (z), L F Hi (z), . . . , L Fr −1 Hi (z), . . . , Hn (z), L F Hn (z), . . . , L Fr −1 H1 (z), . . . , r −1  L F Hn (z)] , where r is the relative degree of ith generator model. As a result, the multi-machine power system model is decoupled and the model of ith subsystem can be written as ⎧ x˙i1 = xi2 ⎪ ⎪ ⎪ ⎪ x ⎪ ⎨ ˙i2 = xi3 n  (3.4) x ˙ = α (x) + βi j (x)u j i = 1, 2, . . . , n i3 i ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎩ yi = xi1 where

αi (x) = (L Fr Hi (z))|z=T −1 (x) βi j (x) = (LG j L Fr −1 Hi (z))|z=T −1 (x) L F Hi (z) = Δω˙ i

3.1 Design of Switching Power System Stabilizer

65

 n  1  ∂ Pei ∂ Pei Di =− F j1 (z) + F j3 (z) − Fi2 (z) 2Hi j=1 ∂z j1 ∂z j3 2Hi n 1  L F3 Hi (z) = − Fm1 (z) 2Hi m=1 ⎧ ⎫  n  ⎨ ∂ Pei ∂ Pei ∂ Pei ∂ F j3 (z) ⎬ F j1 (z) + F j3 (z) + ⎩ ∂z j1 ∂z m1 ∂z j3 ∂z m1 ∂z j3 ∂z m1 ⎭ j=1     n n ∂ Pei ωB  Di  ∂ Fi2 (z) Fm1 (z) − Fm2 (z) − 2Hi m=1 ∂z m1 2Hi m=1,m=i ∂z m1  2   n Di 1  ωB ∂ Pei + Fm3 (z) Fi2 (z) − − − 2Hi ∂z i1 2Hi 2Hi m=1 ⎧ ⎫ ⎬ n  ⎨ ∂ Pei ∂ Pei ∂ Pei ∂ F j3 (z) F j1 (z) + F j3 (z) + ⎩ ∂z j1 ∂z m3 ∂z j3 ∂z m3 ∂z j3 ∂z m3 ⎭

L F2 Hi (z)

j=1

  n  n  1 ∂ Pei Di  ∂ Fi2 (z) 1  − Fm3 (z) − Fm4 (z)  2Hi m=1 ∂z m3 2Hi m=1 Td0m ∂z m3 LG m L F1 Hi (z) = 0 K Am ∂ Pei 1 . LG m L F2 Hi (z) = −  2Hi Td0m TAm ∂z m3 Considering the derivatives of Pei , the system obtained from the previous nonlinear coordination transformation can be further decoupled. Based on that, the derivatives of Pei can be written as n   ∂ Pei d Pei = F j3 + dt ∂z j3 j=1 n   ∂ Pd d 2 Pei = F j1 + dt 2 ∂z j1 j=1

+

∂ Pei F j1 ∂z j1

 = Pd

∂ Pd ∂ Pd ∂ Pd F j2 + F j3 + F j4 ∂z j2 ∂z j3 ∂z j4

n n   K A j ∂ Pd K A j ∂ Pd u j = L F Pd + u j. T ∂z T ∂z j4 j4 j=1 A j j=1 A j



66

3 Switching Control of Synchronous Generators for Transient …

Meanwhile, the following can be obtained L F3 Hi (z)

  Di 1 =− L F Pd − L F Fi2 (z) 2Hi 2Hi ⎛ ⎞   n Di Fi2 (z) 1 ⎝ d 2 Pei  K A j ∂ Pd ⎠ =− − L − u j F 2Hi dt 2 T ∂z j4 2Hi j=1 A j

L F Fi2 (z) =

n   ∂ Fi2 (z) j=1

∂z j1

F j1 (z) +

∂ Fi2 (z) ∂ Fi2 (z) F j2 (z) + F j3 (z) ∂z j2 ∂z j3



∂ Pei 1 ∂ Pd = .  ∂z m4 ∂z m3 Td0m Therefore, it has L F3 Hi (z) = −

n  1 ¨ Di L F [Fi2 (z)] − [LG m L F2 Hi (z)u m ]. Pei − 2Hi 2Hi m=1

(3.5)

Hence, according to (3.4), it can be obtained that n  1 d 2 Pei Di αi (z) = − − L F [Fi2 (z)] − βim u m 2Hi dt 2 2Hi m=1

then the system model can be represented as ⎧ x˙i1 = xi2 ⎪ ⎪ ⎪ ⎪ ⎨ x˙i2 = xi3  2  d Pei 1 + Di L F [Fi2 (z)] ⎪ ⎪ x˙i3 = − 2H ⎪ dt 2 i ⎪ ⎩ yi = xi1 .

(3.6)

The second-order derivative of Pei with respect to time is 2 d 2 Pei    ˙qi E˙ qi ¨ qi ¨qi + (xqi − xdi ) d (Idi Iqi ) = 2 I + I + E E I qi qi dt 2 dt 2

xi3 can be rewritten as

x˙i3 = f i (x) + bi (x)u i

(3.7)

3.1 Design of Switching Power System Stabilizer

67

where   −1 d2     2 I˙qi E˙ qi + Iqi E¨ qi + E qi I¨qi + (xqi − xdi ) 2 (Idi Iqi ) 2Hi dt Iqi Di − L F [Fi2 (z)] − [Fi4 (z) − E˙ qi ]  2Hi 2Hi Td0i Iqi K Ai . bi (x) = −  2Hi Td0i TAi f i (x) =

3.1.2 Design of SPSS 3.1.2.1

Introduction to Third-Order BCFC

Since the relative degree of (3.1) is r = 3, a third-order BCFC is employed based on [2]. The BCFC should meet the following feasibility assumptions. F1

(3.1) can be transformed to its equivalent system written in Byrnes-Isidori normal form  (r ) y = f (Y r −1 , z) + g(Y r −1 , z)u (3.8) z˙ = h(Y r −1 , z)

where Y r −1 := (y, y˙ , . . . , y (r −1) ), y is the system output, r is the relative degree of y with respect to u, z is the zero state vector of system, f, g, h are locally Lipschitz continuous, and g is positive. The zero state of the system is assumed to be stable. (r −1) is absolutely continuous with right-continuous derivaF2 yref ∈ C r −1 and yref tive. Logic of a third-order bang-bang constant funnel controller can be given as q1 (t) = G (e(t), ϕ0+ − ε0+ , ϕ0− + ε0− , q1 (t−)) q1 (0−) = q10 ∈ {true, false}  + − − G (e(t), ˙ −λ− 1 − ε1 , ϕ1 + ε1 , q2 (t−)), if q1 (t) = true q2 (t) = + + + − G (e(t), ˙ ϕ1 − ε1 , λ1 + ε1 , q2 (t−)), if q1 (t) = false q2 (0−) = q20 ∈ {true, false}  + − − G (¨e(t), −λ− 2 − ε2 , ϕ2 (t) + ε2 , q(t−)), if q2 (t) = true q(t) = + + + − if q2 (t) = false G (¨e(t), ϕ2 − ε2 , λ2 + ε2 , q(t−)),

(3.9)

q(0−) = q 0 ∈ {true, false} where q(t), q1 (t), q2 (t) ∈ {true, false}, q(t) is the output of the switching logic, G (e(t), e, e, q(t−)) := [e(t) ≥ e ∨ (e(t) > e ∧ qold )], e(t) is the tracking error of system output, e(·) is the upper trigger of a switch event, e(·) is the lower trigger of

68

3 Switching Control of Synchronous Generators for Transient …

a switch event, q(t−) := limε→0 q(t − ε), ϕi± , εi± and λ±j are constant values used to define funnel Fi (i = 0, 1, 2; j = 1, 2). The BCFC is given as  u(t) =

3.1.2.2

U − , if q(t) = true, U + , if q(t) = false.

Design of Third-Order BPSS

Based on (3.7) and the BCFC, a third-order BPSS is designed. The switching logic of the third-order BPSS has been given in Sect. 3.1.2.1. The funnel parameters should be chosen such that ϕ0+ − ε0+ > ϕ0− + ε0− and ϕi+ − εi+ > εi− + λi+ , ϕi− + εi− < −λi− − εi+ (i = 1, 2). With respect to F1 , the multi-machine power system model has been transformed into Byrnes-Isidori form and its zero dynamics is stable. With respect to the positive  ness of bi (x), since Pei = E qi Iqi and E qi > 0 and Pei > 0, it has Iqi > 0. Consider ing bi (x) = −(Iqi K Ai )/(2Hi Td0i TAi ), it can be obtained that bi (x) < 0. In order   to satisfy the positiveness of bi (x), (3.7) is rewritten as x˙i3 = f i (x) + bi (x)u i ,    where u i = −u i , bi (x) = −bi (x). Hence, we have bi (x) > 0. Correspondingly,   U + = U − > 0, and U − = U + < 0 hold. The control law of the BPSS is  + u PSSi (if q(t) = true) u i (t) = (3.10) u− PSSi (if q(t) = false). 3.1.2.3

Switching Strategy of SPSS

The SPSS switches between a BPSS and a CPSS to damp the oscillations of rotor angle based on a state-dependent switching strategy T . The SPSS is illustrated in Fig. 3.1. The maxima sequence of the absolute value of the output tracking error |ei (t)| = |yi (t) − yiref (t)| is denoted by Γi (t) = {Γi1 , Γi2 , . . . , Γi j } in the case that ei (t) is oscillating after disturbances occur in the system, and Γis (s ∈ {1, 2, . . . , j}) is the maximum of Γi (t). The SPSS switches from CPSS to BPSS if T1 is true and switches from BPSS to CPSS if T2 is true, where T1 and T2 are illustrated as follows, T1 : {|ei (t)| ≥ }, T2 : {The switching frequency of BPSS reaches its maximum} ∨ {{(Γis − Γi j )/ Γis ≥ τ }∧{ei (t) converges within [(ϕ0− + ε0− ), (ϕ0+ − ε0+ )]}}, where , τ are parameters of the SPSS installed on ith generator. is the value of output tracking error that triggers the BPSS. can be determined according to the desired oscillation magnitude to be concerned. τ is chosen according to the desirable damping rate of the magnitude of the tracking error of the system output.

3.1 Design of Switching Power System Stabilizer

69

Fig. 3.1 Schematic of SPSS [3]

3.1.3 Closed-Loop Stability The PSS is implemented to add additional damping to the rotor speed. The kinetic energy of a synchronous generator is regarded as its energy function, which is used for closed-loop stability analysis of the multi-machine power system. The energy function of the system can be written as 1 1 2 (2Hi Δωi2 ) = (2Hi xi1 ). 2 i=1 2 i=1 n

V =

n

(3.11)

Assume that SPSSs are installed on the first m machines. For the last n − m generators with CPSSs configured only, the energy function can be denoted as Vm−n =

n 1  2 (2Hi xi1 ). 2 i=m+1

(3.12)

The difference of the energy function can be written as d Vm−n =

n 

(2Hi xi1 d xi1 )

(3.13)

i=m+1

where dΘ = [Θ(i + 1) − Θ(i)], Θ(i) is the value of Θ(t) at t = ti and 2Hi d xi1 = −d Pei which is obtained based on the assumption that Di = 0. Since Pei = ωi Tei and ωi ≈ 1 if the operation point of the system is around the equilibrium, it has d Pei = dTei . Moreover, the CPSS is usually used to produce an electromechanical torque which is in the same phase with that of Δωi [4]; namely, dTei = Dei Δωi = Dei xi1 holds, where Dei is determined by the parameters of the CPSS, and this ensures that Dei > 0. Therefore, (3.12) can be rearranged as

70

3 Switching Control of Synchronous Generators for Transient …

d Vm−n = −

n 

2 Dei xi1 .

(3.14)

i=m+1

If the operation point of the power system leaves the normal stable point and |xi1 | > 0, then d Vm−n < 0 holds. Therefore, Vm−n satisfies the necessary condition of an energy function of the synchronous generators installed with CPSS. For generators implemented with the SPSS, it is assumed that a negative disturbance occurs at t = t0 . The BPSS is switched on first to control the error of rotor speed and its derivatives, such that they will converge back into the prespecified error funnels. Figure 3.2 supports the following discussions, in which e(0) (t) = xi1 = Δωi (i = 1, 2, . . . , m). Assume a minimal t4 such that e(1) (t4 ) ≥ 0 and minimal t3 such that e(0) (t3 ) = ϕ0+ − ε0+ . In the following, it will be shown that e(0) (t) is monotonically increasing on [t4 , t3 ). Seeking a contradiction, assume that there exists [t11 , t12 ] ∈ [t4 , t3 ) on which e(0) (t) is decreasing. For switching logic q2 , there is a minimal t9 ∈ (t4 , t11 ) such that e(1) (t9 ) = ϕ1+ − ε1+ . Hence, q2 (t9 ) = true and q2 (t) = true are satisfied on [t9 , t10 ) and it is assumed that e(1) (t) hits ϕ1− + ε1− at t = t10 . However, the switching logic of q1 , q1 = false holds on interval [t4 , t3 ). Concerning q2 (t) = G (e(1) (t), ϕ1+ − − (1) ε1+ , λ+ 1 + ε1 , q2 (t−)) (if q1 (t) = false), q2 (t) will turn false when e (t) hits + − λ1 + ε1 . Hence, q2 (t) = false holds on the interval t ∈ [t13 , t10 ), and the contradiction is obtained in this way. According to the above analysis and Fig. 3.2, it can be known that xi1 will converge into the error funnels monotonically from t4 after the impulsive disturbance occurs in the system. Therefore, the energy function of the generators installed with SPSS is monotonically decreasing, and the rotor speed converges region defined m back the 2 (2Hi xi1 ) will decrease by the error funnels on [t4 , t15 ]. Therefore, V1−m = 21 i=1 into the invariant region defined by the error funnels eventually. This is to say d V1−m =

m  (2Hi xi1 d xi1 ) < 0

(3.15)

i=1

holds on [t4 , t15 ]. Within F0 , the CPSS will be triggered if the condition of switching the BPSS to the CPSS is met. The difference of the energy function of the generators installed with the SPSS can be written as d V1−m = −

m 

2 Dei xi1 ≤ 0.

(3.16)

i=1

Therefore, it has been shown that the energy function of n generators will converge to zero. The trace of the energy function of the generators are presented in Fig. 3.3.

3.1 Design of Switching Power System Stabilizer

71

Fig. 3.2 Illustration showing e(0) (t) is monotonically increasing on interval [t4 , t3 ) [3]

Fig. 3.3 Trajectories of the energy functions of generators installed with the CPSSs and the SPSSs, respectively [3]

72

3 Switching Control of Synchronous Generators for Transient …

3.2 Design of Switching Excitation Controller and Switching Governor 3.2.1 Power System Model Used for Designing SEC and SG 3.2.1.1

Model of Power Systems

The fast switching of valve will be used by the BGs, and the internal dynamics of the boiler and its control system should be modeled. The detailed dynamics of a tandem-compound double-reheat steam turbine and its boiler system is included in the system model [6], which is as illustrated in Fig. 3.4. A n-generator system can be modelled with the following (12 × n)th-order ordinary differential equation (ODE) [1, 6, 7].

Fig. 3.4 Structure of a tandem-compound double-reheat steam turbine and its boiler system [5]

3.2 Design of Switching Excitation Controller and Switching Governor

⎧ Δδ˙i = ωB Δωi ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ Δω˙ i = 2Hi (Pmi − Pei − Di Δωi ) ⎪ ⎪    ⎪ ⎪ E˙ qi = T1 [E fi − E qi − (xdi − xdi )Idi ] ⎪ ⎪ d0i ⎪ ⎪ ⎪ ˙Hi = 1 (−PHi + PTi μCVi ) ⎪ P ⎪ T ⎪ CHi ⎪ ⎪ 1 ⎪ ⎪ P˙RH1i = TRH1i (PHi − PRH1i μIVi ) ⎪ ⎪ ⎪ ⎨ P˙ 1 RH2i = TRH2i (−PRH2i + PRH1i μIVi ) 1 ⎪ ˙ PCi = TCOi (−PCi + PRH2i ) ⎪ ⎪ ⎪ √ ⎪ ⎪ 1 ˙ ⎪ PTi = C (K i PDi − PTi − PTi μCVi ) ⎪ SHi ⎪ ⎪ √ ⎪ ˙ ⎪ PDi = C1Di (m wi − K i PDi − PTi ) ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ m˙ wi = Twi (−m wi + Q i ) ⎪ ⎪ ⎪ ⎪ Q˙ i = 1 [−Q i + CPi + K Pi (PTrefi − PTi )] ⎪ TFi ⎪ ⎪ ⎩ ˙ CPi = K Ii (PTrefi − PTi )

73

(3.17)

     where i = 1, 2, 3, . . . , n, Pei = E qi2 G ii + E qi nj=1, j=i E q j Bi j sinδi j , E qi = E qi −  (xdi − xdi )Idi , Pmi = FVHPi PHi + FHPi PRH1i μIVi + FIPi PRH2i + FLPi PCi , Δδi is the rotor angle error of a generator, Δωi is the rotor speed error, Hi denotes the inertia coefficient of the rotor, Di represents the damping coefficient in pu torque/pu speed   deviation, E qi denotes the transient voltage behind the q-axis, Td0i is the transient time constant of d-axis open-circuit, E fi is the excitation voltage, Pei is the active power output, Pmi is the mechanical power input from the prime mover, xdi denotes  synchronous impedance in d-axis, xdi is the transient impedance in d-axis, Idi is the d-axis stator current, PHi represents the high pressure flow, TCHi represents the time constant of main inlet volumes and steam chest, μCVi is the control valve flow area, PRH1i denotes the pressure of the first reheater flow, PRH2i represents the pressure of the second reheater flow, TRH1i denotes the time constant of the first reheater, TRH2i is the time constant of the second reheater, μIVi denotes the intercept valve flow area, PCi represents crossover pressure, TCOi is the time constant of crossover piping and low pressure inlet volumes, PTi is throttle pressure, CSHi denotes the time constant of superheater, K i is the gain of the square root of the pressure drop between drum and throttle, PDi is drum pressure, CDi denotes the time constant of drum, Twi represents the time constant of water wall lag, m wi is the steam generation of water wall, Q i represents the heat released by fuel, TFi is the time constant of fuel dynamics, K Pi denotes the proportion coefficient of the boiler pressure controller, PTrefi is the throttle pressure set point, CPi represents the boiler pressure control signal, K Ii is the integration coefficient of the boiler pressure controller. We can find that Pmi can be controlled by the intercept valve μIVi directly without time delay. The fast closing and opening of intercept valve enable it to change the mechanical power input of a synchronous generator on the faster time scale. In order to achieve the linearization of the model of multi-machine power systems, (3.17) should be rearranged in a matrix form. State variables of (3.17) are written as X = [X 1 X 2 . . . X n ] , where X i = [x1i x2i . . . x12i ] , x1i = Δδi ,

74

3 Switching Control of Synchronous Generators for Transient … 

x2i = Δωi , x3i = E qi , x4i = PHi , x5i = PRH1i , x6i = PRH2i , x7i = PCi , x8i = PTi , x9i = PDi , x10i = m wi , x11i = Q i , x12i = CPi (i = 1, 2, . . . , n). Control variables of (3.17) are defined as U = [U1 U2 . . . Un ] , where Ui = [u 1i u 2i u 3i ] , u 1i = E fi , u 2i = μCVi , u 3i = μIVi . Referring to [8], the output feedback excitation of a synchronous generator had been applied for the rotor speed deviation successfully. This resulted in a lower-order exciter in contrast to that designed with the rotor angle error being the input. Therefore, the output variables here are chosen as Y = [y1 y2 . . . yn ] , yi = Δωi . (3.17) can be rearranged as  X˙ = F(X ) + G(X )U (3.18) Y = H (X ) where F(X ) = [F1 (X ) . . . Fi (X ) . . . Fn (X )] , Fi (X ) = [F1i (X ) . . . Fmi (X ) . . . F12i (X )] (m = 1, 2, . . . , 12), G(X ) = block diag[G 1 (X ) . . . G i (X ) . . . G n (X )], G i (X ) = [G 1i (X ) G 2i (X ) G 3i (X )], H (X ) = [H1 (X ) . . . Hi (X ) . . . Hn (X )] , Hi (X )=yi =Δωi . Meanwhile, Fi (X ) and G i (X ) can be written in the following form: Fi (X ) = [F1i F2i F3i F4i F5i F6i F7i F8i F9i F10i F11i F12i ] ⎤ ⎡ ωB x2i ⎥ ⎢ 1 (F ⎢ 2Hi VHPi x4i + FIPi x6i + FLPi x7i − Pei − Di x2i ) ⎥ ⎥ ⎢  1 ⎥ ⎢  [−x 3i − (x di − x di )Idi ] ⎥ ⎢ Td0i ⎥ ⎢ 1 x ⎥ ⎢ − 4i ⎥ ⎢ TCHi ⎥ ⎢ 1 x ⎥ ⎢ 4i TRH1i ⎥ ⎢ ⎥ ⎢ 1 x ⎥ ⎢ − 6i =⎢ TRH2i ⎥ ⎥ ⎢ 1 (−x + x ) ⎥ ⎢ 7i 6i T COi ⎥ ⎢ √ ⎥ ⎢ 1 K ⎥ ⎢ i x 9i − x 8i C SHi ⎥ ⎢ √ 1 ⎥ ⎢ (x − K x − x ) i 9i 8i CDi 10i ⎥ ⎢ ⎥ ⎢ 1 (−x + x ) ⎥ ⎢ 10i 11i Twi ⎥ ⎢ 1 ⎦ ⎣ [−x + x + K (P − x )] 11i 12i Pi Trefi 8i TFi K Ii (PTrefi − x8i ) ⎡ 0 0 0 1 F ⎢ 0 0 ⎢ 2Hi HPi x 5i ⎢ 1 ⎢  0 0 ⎢ Td0i ⎢ 1 ⎢ 0 0 TCHi x 8i ⎢ ⎢ 0 0 −T 1 ⎢ RH1i ⎢ 1 G i (X ) = [G 1i (X ) G 2i (X ) G 3i (X )] = ⎢ 0 0 TRH2i ⎢ ⎢ 0 0 0 ⎢ ⎢ 0 − 1 x 0 ⎢ CSHi 8i ⎢ ⎢ 0 0 0 ⎢ .. .. ⎢ .. ⎣ . . . 0 0 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 12×3

3.2 Design of Switching Excitation Controller and Switching Governor

3.2.1.2

75

Linearization of the Model of Multi-machine Power Systems

A n-generator system (3.18) can be decoupled as 3n independent subsystems using the partial linearization technique. For such, the following nonlinear coordinate transformation Z 1 , Z 2 and Z 3 are employed, where Z j = [z 11 , . . . , z 1r j , . . . , z i1 , . . . , z ir j , r −1 . . . , z n1 , . . . , z nr j ] = T j (X ) = [H1 (X ), L F H1 (X ), . . . , L F j H1 (X ), . . . , Hi (X ), r −1 r −1 L F Hi (X ), . . . , L F j Hi (X ), . . . , Hn (X ), L F Hn (X ), . . . , L F j Hn (X )] ( j = 1, 2, 3), r j is the relative degree of Hi (X )(i = 1, 2, . . . , n) with respect to u ji . The relative degree of Hi (X ) with respect to u ji is defined in the following way. Differentiating Hi (X ) until u ji appears explicitly. The relative degree r j is the smallest (r ) integer such that at least one of the inputs appears in the expression of Hi j (X ), i.e., (r j )

Hi

r

(X ) = L F j Hi (X ) +

n  3  r −1 [LG jm L F j Hi (X )u jm ], m=1 j=1

r −1

if LG jm L F j Hi (X ) = 0 holds for at least one j. Subsequently, the subsystem with output Hi (X ) and input u ji is written as: ⎧ z˙ 1i = z 2i ⎪ ⎪ ⎪ ⎪ z ˙ 2i = z 3i ⎪ ⎪ ⎪ ⎨···

n  ⎪ ⎪ = α (Z ) + (βim1 (Z )u 1m + βim2 (Z )u 2m + βim3 (Z )u 3m ) z ˙ ⎪ rji ji ⎪ ⎪ ⎪ m=1 ⎪ ⎩ yi = z i1 r −1

r

where α ji (Z ) = (L F j Hi (X ))| X =T j−1 (Z ) , βim j (Z ) = (LG jm L F j As for u 3i , we have

(3.19)

Hi (X ))| X =T j−1 (Z ) .

LG 1m Hi (X ) = LG 2m Hi (X ) = 0 (m = 1, 2, . . . , n) 1 LG 3m Hi (X ) = FHPi x5i 2Hi L F Hi (X ) = F2i (X ). Therefore, the relative degree of Hi (X ) with respect to u 3i is r3 = 1. The obtained subsystems can be written as z˙ 1i = L F Hi (X ) + LG 3i Hi (X )u 3i . Moreover, for u 1i and u 2i , we have

76

3 Switching Control of Synchronous Generators for Transient …

LG 1m L F Hi (X ) = −  LG 2m L F Hi (X ) = 

∂ Pei 1 (i = 1, 2, . . . , n)  2Hi Td0m ∂ x3m

0 (m = i) K 1 x8i (m = i) 2Hi TCHi

0 (m = i) FIPi x5i i FHPi x 5i − D(2H + 2H (m = i) 2 T i)  n  i RH2i  1  ∂ Pei ∂ Pei 2 L F Hi (X ) = − F1s + F3s 2Hi s=1 ∂ x1s ∂ x3s

LG 3m L F Hi (X ) =

+

1 (FVHPi F4i + FIPi F6i + FLPi F7i − Di F2i ). 2Hi

Hence, the relative degree of Hi (X ) with respect to u 1i and u 2i is r1 = 2 and r2 = 2. Then subsystems can be written as ⎧ ⎪  n  ⎪ z˙ 1i = z 2i  ⎪ ⎪ ⎪ 1  ∂ Pei ∂ Pei ⎪ ⎪ F1s + F3s ⎪ ⎨ z˙ 2i = − 2Hi ∂ x1s ∂ x3s s=1 1 + 2H (FVHPi F4i + FIPi F6i + FLPi F7i − Di F2i ) ⎪ ⎪ i ⎪    n  ⎪  ⎪ ∂ Pei FVHPi x8i FIPi x5i Di FHPi x5i 1 ⎪ ⎪ u 3i . u 1s + u 2i + − ⎪ ⎩ − 2H T  ∂ x3s 2Hi TCHi 2Hi TRH2i (2Hi )2 i d0i s=1 (3.20) According to [9], the time derivative of Pei is  ∂ Pei ∂ Pei 1 ∂ Pei d Pei = ( F1s + F3s +  u 1s ). dt ∂ x ∂z Td0s ∂z 3s 1s 3s s=1 n

(3.21)

Substituting (3.21) into (3.20), it can be obtained that   −1 d Pei − FVHPi F4i − FIPi F6i − FLPi F7i + Di F2i 2Hi dt   FVHPi x8i FIPi x5i Di FHPi x5i u 3i . + u 2i + − 2Hi TCHi 2Hi TRH2i (2Hi )2

z˙ 2i =



(3.22)



Active power can also be written as Pei = E qi Iqi + (xqi − xdi )Iqi Idi , and it has Iqi E qi Iqi d Pei d   = E qi I˙qi + (xqi − xdi ) (Iqi Idi ) − +  u i1 .  dt dt Td0i Td0i

(3.23)

where Iqi denotes the stator current in q-axis, and xqi denotes the q-axis synchronous impedance. Substituting (3.23) into (3.22), it has

3.2 Design of Switching Excitation Controller and Switching Governor

z˙ 2i = f i (X ) + b1i (X )u 1i + b2i (X )u 2i

77

(3.24)

1 [E qi I˙qi +(xqi − xdi ) dtd (Iqi Idi ) − qiT  qi − FVHPi F4i − FIPi F6i − where f i (X ) = − 2H i d0i 

I x5i Di FHPi x5i FVHPi x8i u FLPi F7i + Di F2i ] + 2HFIPi − , b (X ) = − 2H qiT  , b2i (X ) = 2H . 3i 1i (2Hi )2 i TRH2i i TCHi i d0i Therefore, the relative degree of (3.1) is 5n and the last 7nth-order dynamics is internal dynamics. The internal dynamics is zero in this case and it has no effect on the stable operation of the system. 



I E

3.2.2 Design of SEC and SG 3.2.2.1

Preliminaries of Bang-Bang Constant Funnel Controller

Feasibility assumptions of the BCFC can be described as: F1

(3.1) can be written in Byrnes-Isidori form as 

y (r ) = f (Y r −1 , z) + g(Y r −1 , z)u z˙ = h(Y r −1 , z)

(3.25)

where Y r −1 := (y, y˙ , . . . , y (r −1) ), y is the output variable of the system, r represents the relative degree of y with respect to u, z is the zero state vector, f, g, h are locally Lipschitz continuous functions, g(Y (r −1) , z) > 0 and the zero state of the system is stable is assumed to be stable. (r −1) are absolutely continuous with rightF2 The references of yref ∈ C r −1 and yref continuous derivative. With C r −1 (X → Y ), or short C r −1 , the set of all (r − 1)times continuously differentiable functions are written as f : X → Y . The switching logic of a second-order BCFC can be written as q1 (t) = G (e(t), ϕ0+ − ε0+ , ϕ0− + ε0− , q1 (t−)) q1 (0−) = q10 ∈ {true, false}  + − − G (e(t), ˙ −λ− 1 − ε1 , ϕ1 + ε1 , q(t−)), if q1 (t) = true q(t) = + + + − G (e(t), ˙ ϕ1 − ε1 , λ1 + ε1 , q(t−)), if q1 (t) = false

(3.26)

q(0−) = q 0 ∈ {true, false} where q(t), q1 (t) ∈ {true, false}, q(t) is the output of the switching logic, G (e(t), e, e, q(t−)) := [e(t) ≥ e ∨ (e(t) > e ∧ q(t−))], e(t) = yref (t) − y(t), e(·) is the upper trigger of a switch event, e(·) is the lower trigger of a switch event, q(t−) := limε→0 q(t − ε), ∨ presents logic operation “or”, ∧ represents logic operation “and”, ϕi± denotes the upper and lower bound of the ith error funnel, εi± represents the safety distances triggering an event when the error or its derivative get close to the funnel ˙ (if q1 (t) = true, it is expected to boundaries, λ±j denotes the desirable values of e(t)

78

3 Switching Control of Synchronous Generators for Transient …

Fig. 3.5 The working mechanism of the second-order bang-bang constant funnel controller [5]

control e(t) ˙ to be smaller than −λ− 1 and vise versa). The control law of the BCFC can be written as  − U if q(t) = true, u(t) = U + if q(t) = false. Funnels Fi (i = 0, 1) are defined as Fi := {(t, e(t)) ∈ R≥0 × R|ϕi− ≤ e(i) (t) ≤ ϕi+ }. Schematic of Fi , ϕi± , εi± and λi± is shown in Fig. 3.5, and the mechanism of the second-order BCFC is presented therein. Output tracking error and its firstorder derivative of the system can be regulated and operate in the pre-specified error funnels Fi through the bang-bang control signal u(t). The switching logic of the first-order BCFC is the same as the first-order part of the second-order BCFC.

3.2.2.2

Design of Bang-Bang Excitation Controller and Bang-Bang Governor

Since the relative degree of z 1i with respect to u 1i is r1 = 2, a second-order BEC was designed for the excitation control of a synchronous generator. A second-order BG was designed for the regulation of the control valve, since the relative degree of z 1i with respect to u 2i is r2 = 2. Concerning r3 = 1 is the relative degree of z 1i with respect to u 3i , a first-order BG was designed for the control of the intercept valve. Schematic of the BEC and the BG is as shown in Fig. 3.7. F1 of the BCFC is satisfied, and (3.1) has been transformed into its Byrnes-Isidori form (3.19). Zero dynamics of (3.1) is stable, which can be easily verified.   Since Pei = E qi Iqi and E qi > 0 and Pei > 0, we have Iqi > 0. Concerning I

b1i (X ) = − 2H qiT  , b1i (X ) < 0 holds. The control law of the BEC can be written i d0i

3.2 Design of Switching Excitation Controller and Switching Governor

79

Fig. 3.6 The fast closing and opening characteristics of control valve and intercept valve [5]



as u 1i =

E fi+ (if q1i (t) = true) E fi− (if q1i (t) = false)

(3.27)

where q1i (t) is the output of the BEC. The positiveness of b2i (X ) and LG 3m Hi (X ) is obvious. The control laws of the two BGs designed for the control and intercept valve respectively can be written as  u 2i =

μ− CVi (if q2i (t) = true) μ+ CVi (if q2i (t) = false)

 u 3i =

μ− IVi (if q3i (t) = true) μ+ IVi (if q3i (t) = false)

(3.28)

where q2i (t) and q3i (t) are the outputs of the second-order BG and the first-order BG.

3.2.2.3

Fast Closing and Opening Characteristics of Control Valve and Intercept Valve

Fast switching of valve position was employed in the BG, which needs careful validation for its feasibility and rationality. The fast closing and opening characteristics of the control and the intercept valve were as presented in Fig. 3.6. Referring to [10], T1 denotes the delay between the initiation of the valve and the time when the valve starts to close, T2 represents the valve closing time, T3 is the length of time that the valve can be remained closed and T4 represents the valve open time. The fast closing of the valve allows a complete valve closure in 0.08 to 0.4 s. Using accumulators and hydraulic systems [11, 12], the fast open of a valve can be achieved and it allows complete valve open in 0.85 to 3 s. Parameters in [10] are employed here for the control and the intercept valve, which are T1 = 0.1 s, T2 = 0.25 s, T3 = 0.1 s and T4 = 0.85 s.

80

3.2.2.4

3 Switching Control of Synchronous Generators for Transient …

Switching Strategy of Switching Excitation Controller and Switching Governor

Configuration of the SEC and SG in the control system of a synchronous generator is as shown in Fig. 3.7, where TSM represents the speed relay of the servo motor of a governor, L.R. denotes the load reference obtained from the generic turbine controller, which is defined in [6]. According to Fig. 3.7, a SEC is composed of a CEC and a BEC. Based on a state-dependent switching strategy T , the CEC and the BEC operate in a switching manner. Moreover, the SG is consist of a CG and two BGs, i.e., a second-order BG and a first-order BG. Parameters of the CG are cited from [6]. The BGs have fast closing and opening capabilities for the control and intercept valve. Γi (t) = {Γi1 , Γi2 , . . . , Γi j } are the maximum sequence of |ei (t)|, and Γis (s ∈ {1, 2, . . . , j}) is the maximum of the sequence Γi (t). The SEC (SG) switches from the CEC to the BEC (vice versa) if T1 is met and switches from the BEC to the CEC (switches from the BG to the CG) if T2 is met. T1 : {|ei (t)| ≥ i }, T2 : {The switching frequency of the BEC (BG) reaches its maximum} ∨ {{(Γis − Γi j )/Γis ≥ τi } ∧ {|ei (t)| converges within [2(ϕ0− + ε0− ), 2(ϕ0+ − ε0+ )]}}, where i ∈ R, τi ∈ R. The same set of funnel parameters can be used for the BEC and the BG.

3.2.2.5

Parameter Set for Switching Excitation Controller and Switching Governor

The BEC and the BG are set as ⎧ + ϕ − ε0+ ⎪ ⎪ ⎨ 0+ ϕ1 − ε1+ ϕ − + ε1− ⎪ ⎪ ⎩ 1± ϕi > 0,

> ϕ0− + ε0− > ε1− + λ+ 1 + < −λ− 1 − ε1 ± ± εi > 0, λ1 > 0.

(3.29)

Funnel parameters following (3.29) are able to achieve the control objectives of the BEC and the BG. Parameters given in this chapter can be applied for the BCFC in many other applications. Parameters of T , i determines how fast the bang-bang controller can respond to the oscillation of rotor speed. So as to prevent the bang-bang controller from being activated when only small oscillation occurs, i ∈ [1, 1.5] is able to work properly if the input signal is measured in rad/s. The bang-bang controllers are expected to consume 30–50% of the swing energy. With this concern, τi ∈ [0.3, 0.5] is selected.

3.2 Design of Switching Excitation Controller and Switching Governor

81

Fig. 3.7 The configuration of the control system of a synchronous generator having a SEC and a SG installed [5]

82

3 Switching Control of Synchronous Generators for Transient …

Fig. 3.8 Structure of the four-generator eleven-bus power system [3] Table 3.1 Parameters of CPSS and excitation controller of the 4-generator 11-bus power system [3] Parm. Value Parm. Value Parm. Value Tsi T3i u+ PSSi K Ai

0.08 0.015 −0.1 3

T2i K PSSi TAi E f−j

0.015 20 0.01 −3

Table 3.2 Parameters of SPSS of test systems [3] Parm. Value Parm. Value

Parm.

Value

ϕ1+ ϕ2− ε1+ ε2− λ+ 2 τ

8 −100 0.5 5 16.95 0.8

ϕ0+ ϕ1− ε0+ ε1− λ+ 1 λ− 2 U+

7.5 0.08 0.1 20

3 −8 2.95 0.5 6.95 16.95 0.1

T1i T4i u− PSSi E f+j

ϕ0− ϕ2+ ε0− ε2+ λ− 1 U−

−3 100 2.95 5 6.95 2 −0.1

3.3 Simulation Studies 3.3.1 Test Results of SPSS To validate the damping performance of the SPSS, simulation studies were undertaken in a four-generator eleven-bus system and the IEEE 16-generator 68-bus system, respectively. The CPSSs studied were designed based on [13, 14]. These techniques based on the information of multiple operation points to ensure robustness of the PSS. They are effective and, with minor variations, they are still the state-of-theart techniques.

Rotor Angle 2 (rad)

3.4 3.2 3 CPSS 5

SPSS

379

10

15

CPSS 5

(b)

CPSS

376

PSS Output 3 (p.u.)

SPSS

1.8 0

(a)

377

5

10

15

SPSS

379

5

10

15

15

376 0

5

(d)

0

10

377

(c) CPSS

15

378

Time (s)

SPSS

10

CPSS

Time (s)

0.1

−0.1 0

2

Time (s)

378

0

2.2

Time (s) Rotor Speed 4 (rad/s)

SPSS 0

Rotor Speed 3 (rad/s)

83

PSS Output 4 (p.u.)

Rotor Angle 1 (rad)

3.3 Simulation Studies

SPSS

0.1

CPSS

0 −0.1 0

5

10

Time (s)

Time (s)

(e)

(f)

15

Fig. 3.9 Mechanical power change applied on generator 1 in the 4-generator 11-bus power system. a Rotor angle deviation between generator 1 and generator 3. b Rotor angle deviation between generator 2 and generator 3. c Rotor speed of generator 3. d Rotor speed of generator 4. e Control input of the PSS of generator 3. f Control input of the PSS of generator 4 [3]

3.3.1.1

Test Results of Mechanical Power Changes in Four-Generator Eleven-Bus Power System

A four-generator eleven-bus power system was employed, and the system structure is as presented in Fig. 3.8. The simplified first-order exciter was installed on each generator, and the parameters of the exciters are given in Table 3.1. In order to evaluate the performance of the SPSS, generator 3 and generator 4 were configured with the SPSS, and the parameters of the SPSS were as given in Table 3.2. The other generators did not have any PSS. Parameters of the CPSS of the SPSS installed on generator 3 and generator 4 were as shown in Table 3.1. The simulation step length was h = 0.01 s.

84

3 Switching Control of Synchronous Generators for Transient …

Table 3.3 Parameters of CPSS and exciter of synchronous generators in IEEE 16-generator 68-bus test system [3] Parm.a Value Parm. Value Parm. Value Tsi T3i u+ PSSi K Ai

7.5 0.08 0.1 30

T1i T4i u− PSSi E fi+

0.08 0.015 −0.1 3

T2i K PSSi TAi E fi−

0.015 50 0.01 −3

Mechanical power input to generator 1 had a 0.1 p.u. step at t = 1 s. Dynamics of the system was as presented in Fig. 3.9. According to Fig. 3.9a and b, less intermachine oscillations were found in the system with SPSS. The system with the CPSS implemented only went unstable. According to Fig. 3.9c and d, generator 3 and generator 4 controlled by the SPSS showed less oscillations in their rotor speeds. This was attributed by the control signal generated by the SPSS, presented in Fig. 3.9e, f. It can be seen that the SPSS employed the maximum magnitude of the output of the PSS and was able to offer appropriate switching of control inputs. As a result, the SPSS made use of the largest control power of the PSS and enhanced the small-signal stability of the test system. Moreover, the inputs of the SPSS were generated by magnifying the rotor speed error of generators in rad/s, i.e., 200Δω. Consequently, the same set of parameters of the SPSS were applied, and this did not impact the performance of the SPSS.

3.3.1.2

Test Results of Load Changes in the IEEE 16-Generator 68-Bus Power System

To validate the small-signal oscillation damping capability of the SPSS in a largescale power system, the IEEE 16-generator 68-bus power system was simulated, and the layout was as illustrated in Fig. 3.10. The setting of synchronous generators, transmission lines, and loads were as presented in [15]. A simplified first-order excitation controller, was installed on each synchronous generator. Generator 1, 10, 11, 12, 13 and 16 were installed with a PSS, respectively, and the SPSS is implemented on generator 1, 10, and 11, respectively. The parameters of the SPSS were as given in Table 3.2. The parameters of the CPSS and excitation controller were as given in Table 3.3. The simulation step-length was h = 0.01 s. In this case, a 0.5 p.u. load decrease occurred on bus 20 at t = 1 s in the IEEE 16-generator 68-bus power system, and it lasted for 0.2 s. The load decrease lead to a 0.03 Hz frequency increase in the system with only the CPSS installed, while it only lead to a 0.01 Hz frequency increase in the system having the SPSS installed as shown in Fig. 3.11j. From Fig. 3.11a–f, it can be seen that the rotor angle deviation and the rotor speed of the generators having the SPSS installed showed less oscillations than those of the generators having only the CPSS implemented. Although the BPSS of

3.3 Simulation Studies Area 3

G

85

1

14 48

40

41

47

G

8

54

G 26

25

29

28

53 27

31

55

30

G

10

38

G

61

35

15

45

44

50

12

16

G

19

G

4

65

63

20 5

59

36

43

58

60

G

37 39

66 64

57

52

G

56

34

42

Area 5

6 67

32

49

51

22

Area 1

33 Area 4

21

68

G

9

17

18

Area 2

11 46

G

24

G

13

G

62

2

G

G

3

23

G

7

Fig. 3.10 The layout of the IEEE 16-generator 68-bus power system [3]

generator 11 was not on during this stage as presented in Fig. 3.12c, effort of the SPSS of generator 1 and generator 10 was able to control the power system to its original operating point. As shown in Fig. 3.12a and b, the SPSSs of generator 1 and generator 10 could offer appropriate switching of control signal and they took use of the maximum control power to suppress the oscillations of rotor speed. Due to the above, the frequency of the power system having the SPSS installed presented less deviation and less fluctuations than that of the system with CPSS installed only as shown in Fig. 3.12d.

3.3.1.3

Test Results of Load Trip in the IEEE 16-Generator 68-Bus Power System

The influence of the SPSS with respect to the transient stability of power systems was studied in a a new case, where load trip on bus 20 was applied in the IEEE 16-generator 68-bus power system. The load of bus was tripped at t = 1 s and the event lasted 0.05 s. This lead to a frequency increase of about 0.14 Hz in the power system with the CPSS installed merely. In comparison, the frequency increase of the power system with the SPSS installed only was 0.04 Hz, as depicted in Fig. 3.13j.

3 Switching Control of Synchronous Generators for Transient …

−7

SPSS

CPSS

−7.2 −7.4 −7.6 −7.8 0

2

4 Time (s)

6

Rotor Speed 1 (rad/s)

Rotor Angle 1 (rad)

86

377.2

SPSS

377 376.8 0

2

CPSS

−5.1 −5.2 2

4 Time (s)

6

Rotor Speed 10 (rad/s)

Rotor Angle 10 (rad)

SPSS

0

377.05

SPSS

CPSS

−2.65 4 Time (s)

(e)

6

Rotor Speed 11 (rad/s)

Rotor Angle 11 (rad)

SPSS

2

CPSS

376.95 376.9 0

2

4 Time (s)

6

(d)

−2.6

0

6

377

(c) −2.55

4 Time (s)

(b)

(a) −5

CPSS

377.02

SPSS

CPSS

377 376.98 376.96 0

2

4 Time (s)

6

(f)

Fig. 3.11 Load change applied on the IEEE 16-generator 68-bus power system. a Rotor angle of generator 1. b Rotor speed of generator 1. c Rotor angle of generator 10. d Rotor speed of generator 10. e Rotor angle of generator 11. f Rotor speed of generator 11. g Control input of the PSS of generator 1. h Control input of the PSS of generator 10. i Control input of the PSS of generator 11. j Frequency response of the power system [3]

The rotor angle errors and rotor speeds of generator 1, generator 10 and generator 11 were as shown in Fig. 3.13a–f. We can see that the generators having the SPSS installed presented better damping to the oscillations than the generators with only the CPSS installed. Control commands generated by the PSSs were as shown in Fig. 3.14a–d. The BPSS of generator 1 was not switched on at this time, and the damping of the power system having the SPSS installed was mainly obtained by the effort of the SPSS of generator 10 and generator 11, respectively. Less frequency errors can be found in the system with the SPSS installed as presented in Fig. 3.14d.

87

PSS Ouput 10 (p.u.)

PSS Output 1 (p.u.)

3.3 Simulation Studies

0.1

0

−0.1 0

2

4 Time (s)

6

0.1

SPSS

0

−0.1 0

2

CPSS

0 −0.01 2

4 Time (s)

6

(c)

System Frequency (Hz)

PSS Output 11 (p.u.)

SPSS

0.01

0

4 Time (s)

6

(b)

(a) 0.02

CPSS

CPSS

60.02

SPSS

60 59.98 0

2

4 Time (s)

6

(d)

Fig. 3.12 Load change applied on the IEEE 16-generator 68-bus power system. a Control input of the PSS of generator 1. b Control input of the PSS of generator 10. c Control input of the PSS of generator 11. d Frequency response of the power system [3]

3.3.2 Test Results of SEC and SG In order to evaluate the performance of the SEC and SG, studies were undertaken using the IEEE 16-generator 68-bus test system, whose structure is as illustrated in Fig. 3.15. Parameters of the power system can be found in [15]. All generators are equipped with excitation controllers and PSSs. G10 , G11 , G12 and G13 have a SEC installed, respectively. Transfer function of the PSS can be written as G PSSi (s) =  2 1+T1i s Twi s K PSSi 1+T , where i = 1, 2, . . . , 16 and Twi = 7.5. PSSs are as configured s 1+T s wi 2i in Table 3.4. Moreover, G1−15 are implemented with a governor, respectively. A SG is implemented on G1 , G10 , G12 , G13 , G14 and G15 respectively. Other generators are equipped with a CG respectively except for G16 . Table 3.6 gives the configuration of the BEC, the BG and their switching strategies. The limits of control commands are denoted by the error with respect to their steady-state values. Moreover, the CEC equipped were IEEE AC4A exciters, whose parameters are shown in [16]. Tandem-compound double-reheat steam turbine were configured based on [7] and given by Table 3.5 (i = 1, 2, . . . , 15). Boiler model, electro-hydraulic speed governing system and the generic boiler turbine control system can be found in [6].

3 Switching Control of Synchronous Generators for Transient …

SPSS

−7

CPSS

−7.2 −7.4 −7.6 −7.8 0

4

2

6

Rotor Speed 1 (rad/s)

Rotor Angle 1 (rad)

88

SPSS

377.2 377 376.8 376.6 0

Time (s)

CPSS

−5.1

4

6

Rotor Speed 10 (rad/s)

Rotor Angle 10 (rad)

SPSS −5

2

−2.65 6

Rotor Speed 11 (rad/s)

Rotor Angle 11 (rad)

−2.6

2 4 Time (s)

CPSS

376.95 376.9 0

2 4 Time (s)

6

(d) SPSS CPSS

(e)

6

377

(c)

−2.55

SPSS

377.05

Time (s)

0

4 2 Time (s)

(b)

(a)

−5.2 0

CPSS

377.05

SPSS

CPSS

377 376.95 0

2 4 Time (s)

6

(f)

Fig. 3.13 Load trip applied on the IEEE 16-generator 68-bus power system. a Rotor angle of generator 1. b Rotor speed of generator 1. c Rotor angle of generator 10. d Rotor speed of generator 10. e Rotor angle of generator 11. f Rotor speed of generator 11. g Control input of the PSS of generator 1. h Control input of the PSS of generator 10. i Control input of the PSS of generator 11. j Frequency response of the power system [3]

3.3.2.1

Test Results of Three-Phase-to-Ground Fault on Transmission Line

The SEC and SG were also tested in a case where a three-phase-to-ground fault was applied on the transmission line between bus 53 and bus 54 at t = 0.5 s and the fault line lasted 0.1 s. Rotor angle error, active power output and terminal voltage of G1 , were as shown in Fig. 3.16a–c. We can see that the G1 having a SG installed had presented less oscillations than that having a CG installed. Owing to the fast switching of valve in Fig. 3.16e and f, the mechanical power of G1 with a SG implemented responded

89

0.1

SPSS

CPSS

0

−0.1 0

2

PSS Output 10 (p.u.)

PSS Output 1 (p.u.)

3.3 Simulation Studies

6

4 Time (s)

0.1

0

−0.1 0

2

CPSS

0

2

6

4 Time (s)

System Frequency (Hz)

PSS Output 11 (p.u.)

SPSS

−0.1 0

6

(b)

(a) 0.1

4 Time (s)

60.15

CPSS

SPSS

60.1 60.05 60 0

2

4 Time (s)

6

(d)

(c)

Fig. 3.14 Load trip applied on the IEEE 16-generator 68-bus power system. a Control input of the PSS of generator 1. b Control input of the PSS of generator 10. c Control input of the PSS of generator 11. d Frequency response of the power system [3] Table 3.4 Parameters of PSSs [5] No. K PSS T1 1 3 5 7 9 11 13 15

372 508 182 458 228 108 121 200

0.075 0.097 0.091 0.063 0.092 0.074 0.090 0.094

T2

No.

K PSS

T1

T2

0.008 0.011 0.042 0.011 0.019 0.017 0.020 0.041

2 4 6 8 10 12 14 16

169 172 169 422 188 100 506 100

0.060 0.090 0.092 0.061 0.089 0.089 0.093 0.081

0.009 0.026 0.035 0.022 0.018 0.014 0.051 0.015

Table 3.5 Parameters of steam turbine system [5] Parameter Value Parameter Value TCHi TCOi FIPi CDi TFi

0.2 s 0.3 s 0.3 300 5s

TRH1i FVHPi FLPi Ki

8s 0.22 0.26 3.5

Parameter

Value

TRH2i FHPi CSHi Twi

8s 0.22 15 7s

90

3 Switching Control of Synchronous Generators for Transient …

Fig. 3.15 Schematic of the IEEE 16-generator 68-bus power system [5] Table 3.6 Parameters of the BEC, the BG and switching strategies [5] Parameter Value Parameter Value Parameter ϕ0+ (ϕ0− ) ε1+ (ε1− ) τ1 11 τ12 14 τ15 ΔE fi+ Δμ− CV j

4 0.5 0.4 1.5 0.5 1 0.3 2 −0.1

ϕ1+ (ϕ1− ) − λ+ 1 (λ1 ) 10 τ11 13 τ14 16 ΔE fi−

Δμ+ IV j

10.5 9.4 2 0.5 1 0.3 1 −2 0.1

ε0+ (ε0− ) 1 τ10 12 τ13 15 τ16 Δμ+ CV j Δμ− IV j

Value 3.9 1.5 0.5 1 0.5 1 0.5 0.1 −0.1

3.3 Simulation Studies

91

(a) Rotor angle deviation of generator 1 −0.7

SG

(b) Active power output of generator 1 4

CG Pe1 (p.u.)

Δδ 1 (rad)

−0.8 −0.9 −1

SG

CG

3 2 1

−1.1 0

1

2

3

4

5

6

0

7

(d) Mechanical power of generator 1

Pm1(p.u.)

Vt1(p.u.)

SG

2.6

1.05 1 0.95 0.9 0

1

2

3

4

5

2.5 2.45 2.4

SG CG 6 7

(f) Intercept valve of generator 1

SG

0.9

CG

IV1

0.8 0.75

SG

CG

0.85

(p.u.)

0.85

μ

(p.u.) CV1

μ

7

6

5

4

3

2

1

0

0.8 0.75 0.7

0.7

0

7

6

5

4

3

2

1

0

SEC+SG

2

3

4

5

SEC+SG

CEC+CG

6

7

10

−0.6

0

Δω

−0.8

CEC+CG

2

(rad/s)

−0.4

1

(h) Rotor speed deviation of generator 10

(g) Rotor angle deviation of generator 10

Δδ 10 (rad)

CG

2.55

(e) Control valve of generator 1 0.9

7

6

5

4

3

2

1

0

(c) Terminal voltage of generator 1

−2 −1 7

6

5

4

3

2

1

0

0

SEC+SG

CEC+CG

2

3

t10

(p.u.)

6 4

1

2

3

4

5

6

5

6

7

0.9 0.8 0.7 0.6

2 0

4

1

V

Pe10(p.u.)

8

1

(j) Terminal voltage of generator 10

(i) Active power output of generator 10

7

0

SEC+SG 1

2

3

4

5

CEC+CG 6

7

Fig. 3.16 The dynamics of G1 , G10 and G11 respectively obtained in the case that a three-phaseto-ground fault was applied on a transmission line [5]

92

3 Switching Control of Synchronous Generators for Transient … (k) Mechanical power of generator 10 SEC+SG

5.1 5 4.9

2 1

0

1

2

3

4

5

6

0

7

0.9

SEC+SG

2

0.9

CEC+CG μ IV10 (p.u.)

0.85

μ CV10 (p.u.)

1

3

4

7

6

5

(n) Intercept valve of generator 10

(m) Control valve of generator 10

0.8

0.75

SEC+SG

CEC+CG

0.85 0.8 0.75 0.7

0.7 0

1

2

3

4

5

6

0

7

(o) Rotor angle deviation of generator 11 SEC

−0.1

1

2

3

4

5

6

7

(p) Active power output of generator 11

CEC

SEC

CEC

14 (p.u.)

Δδ11 (rad)

−0.2

12 10

P

e11

−0.3 −0.4

8

−0.5

6 0

1

2

3

4

5

6

7

0

1

2

3

4

1

SEC

(p.u.)

2 1

f11

0.9

E

3

0.8 0.7

SEC 0

1

2

3 4 Time(s)

5

CEC

0

CEC 6

7

6

5

(r) Excitation voltage of generator 11

(q) Terminal voltage of generator 11

Vt11(p.u.)

CEC+CG

0

4.8 4.7

SEC+SG

3 Ef10(p.u.)

Pm10(p.u.)

5.2

(l) Excitation voltage of generator 10

CEC+CG

7

0

1

2

3 4 Time(s)

5

6

7

Fig. 3.16 (continued)

appropriately to the active power fluctuation in the first swing as shown in Fig. 3.16d. This helped to re-balance the mechanical power input and the active power output of G1 . Moreover, the rotor speed deviation, active power output and terminal voltage of the G11 having a SEC installed showed less error than the one having a CEC implemented, as shown in Fig. 3.16o–q, respectively. The BEC utilized the maximum magnitude of its excitation voltage to suppress the rotor speed oscillation in the first stabilization stage as shown in Fig. 3.16r.

3.3 Simulation Studies

93

With respect to the generators with a SEC and a SG installed, dynamics of G10 was as depicted in Figs. 3.16g to 3.16n. According to the above results, the errors of rotor angle, rotor speed, active power output and terminal voltage of the G10 with a SEC and a SG were able to obtain faster convergence than the one having conventional controllers installed. The fast valve switching, as presented in Fig. 3.16m, n respectively, was able to coordinate with the excitation voltage shown in Fig. 3.16l. In the first place, the mechanical power presented in Fig. 3.16k was anti-phase with the rotor speed deviation in the first stabilization stage. In the second place, the excitation voltage generated by the BEC was in a bang-bang manner to offer an anti-oscillation control signal as shown in Fig. 3.16l. Therefore, the SEC and the SG were able to provide damping to the oscillation of rotor speed in a coordinated manner. Since the expression of mechanical power is Pmi = FVHPi PHi + FHPi PRH1i μIVi + FIPi PRH2i + FLPi PCi , we can see that it was directly controlled by the intercept valve. The fast switching of intercept valve enabled the fast step of mechanical power as shown in Figs. 3.16d and 3.16k. The active power of G12 , G13 , G14 and G15 , with a SG installed respectively, were as illustrated in in Fig. 3.17a–d. Attributed to the contribution of BGs, the mechanical power input of these generators was immediately decreased at t = 0.5 s when their active power reduced as shown in Fig. 3.17e, f. The mechanical power controlled by the SG can coordinate with the active power to reach a novel balanced state. Hence, the generators with a SG installed respectively were able to suppress the oscillation of active power faster than the ones with a CG installed respectively. Figure 3.18 shows the inter-machine oscillations of the power system. According to the above results, the inter-area dynamics of the system with conventional controllers installed only showed more severe oscillation than that of the system with switching controllers equipped, as shown in Fig. 3.18a–c and f, respectively. Meanwhile, the relative values of rotor angles between different generators in the same area showed less oscillations in the system with switching controllers installed than those obtained in the system with conventional controllers only. This is illustrated in Fig. 3.18d and e, respectively. In order to assess the resilience of a power system,we defined two short-term n Hi Δωi2 (t j )] was introresilience indexes. In the first place, Rk (t j ) = 1/[1 + i=1 duced to measure the system resilience with its kinetic energy, where t j is the sampling time of jth step. In the second place, Rp (t j+1 ) = ! n [Δδi (t j+1 )−Δδi (t j )][Pei (t j+1 )−Pmi (t j+1 )] " 1/ 1 + | i=1 | was defined to measure the sysh tem resilience with its potential energy, where h is the sampling time step length. At steady-state, we have Rk = 1 and Rp = 1. Both Rk and Rp tend to decrease when the system is disturbed. After the perturbation is cleared, they will increase to 1 respectively. The restorative time of the system is defined as the length of time that this recovery process takes. From Fig. 3.19a and b, we can see that the system with switching controllers installed showed shorter restorative time than the one controlled by conventional controllers only.

94

3 Switching Control of Synchronous Generators for Transient … (a) Active power output of generator 12 SEC+SG

(b) Active power output of generator 13

CEC+CG

CG

35

e13

(p.u.)

14

SG

40

12

P

P

e12

(p.u.)

16

30 10 0

1

2

3

4

5

0

7

6

(c) Active power output of generator 14 SG

3

4

5

7

6

(p.u.) e15

SG

CG

11 10

P

(p.u.)

16

P

e14

2

12

CG

18

9

14 0

1

2

3

4

5

6

8 0

7

SG

2

3

4

5

7

6 SG

(p.u.)

CG

10

P

35

CG

10.2

m15

36

34 0

1

(f) Mechanical power of generator 15

(e) Mechanical power of generator 13 37 Pm13 (p.u.)

1

(d) Active power output of generator 15

1

2

3 4 Time(s)

5

6

7

9.8 0

1

2

4 3 Time(s)

5

6

7

Fig. 3.17 The active power outputs of G12 , G13 , G14 and G15 and the mechanical power inputs of G13 and G15 obtained in the case that a three-phase-to-ground fault occurred on a transmission line [5]

3.3.2.2

Test Results of Transmission Line Outage in Power System

In this situation, the transmission line between bus 36 and bus 37 was tripped at t = 0.5 s due to the fault of the protection system. The line was turned on again at t = 0.7 s. SS, SC and CC showed in the following figures represent the results of three power systems having different control system. SS denotes the results in the case where the SECs, the CECs, the SGs and the CGs were installed in the power system and we call this system has SS configuration. SC denotes the results obtained in the case where the SECs, the CECs and the CGs were applied in the system and we call this system has SC configuration. Then CC denotes the results obtained in the case where only the CECs and the CGs were installed in the system and we call this power system has CC configuration. As shown in Fig. 3.20a–c, G1 of the system with CC configuration cannot be stabilized within the simulation time. In comparison, G1 of the system with SC

3.3 Simulation Studies

95 (b)Relative rotor angle between generator 12 and 14

(rad)

−0.15 −0.2 −0.25 −0.3

1

2

3

4

5

6

0

7

SS

CC

1

9

0.95 0.9 1

2

3

4

−1 −1.1 −1.2 0

7

6

5

Δδ14−Δδ15 (rad)

CC

0.6

11

Δδ −Δδ13 (rad)

SS

3

4

5

6

7

1

2

3

4

5

6

7

(f)Relative rotor angle between generator 14 and 15 0.3 SS CC

(e)Relative rotor angle between generator 11 and 13

0.8

2

−0.9

9

1.05

1

Δδ −Δδ (rad)

Δδ −Δδ13 (rad)

1.1

1

(d)Relative rotor angle between generator 1 and 9 −0.8 SS CC

(c)Relative rotor angle between generator 9 and 13

0.4 0

CC

12

0.4

0.85 0

SS

−0.1

Δδ −Δδ

0.6

0.2 0

−0.05

14

0.8

1

Δδ11−Δδ (rad)

(a)Relative rotor angle between generator 11 and 1 1 SS CC

0.25

0.2 1

2

4 3 Time(s)

7

6

5

0

1

2

3 4 Time(s)

5

6

7

Fig. 3.18 The inter-machine oscillations obtained in the case that a three-phase-to-ground fault occurred on a transmission line (SS: the power system having the SECs and the SGs installed, CC: the power system only having the CECs and the CGs equipped) [5] 1

Rp

Rk

1

0.5

0 0

SS 1

2

CC

3 4 5 6 7 Time(s) (a) System resilience measured with its kinetic energy

0.5

0 0

SS 1

CC

2

3 4 5 6 7 Time(s) (b) System resilience measured with its potential energy

Fig. 3.19 The short-term resilience measured with the kinetic energy and the potential energy of the system, respectively (SS: the power system having the SECs and the SGs installed, CC: the power system only having the CECs and the CGs equipped) [5]

96

3 Switching Control of Synchronous Generators for Transient … (b) Active power output of generator 1

(a) Rotor angle deviation of generator 1 SS

−1

SC

CC

SS

SC

5

6

CC

3.5 Pe1(p.u.)

Δδ (rad)

−1.1

1

−1.2 −1.3

3 2.5 2

−1.4 −1.5 0

1

2

3

4

5

6

1.5

7

0

(c) Terminal voltage of generator 1 SS

SC

CC

2

3

4

2.5

1.06

Pm1(p.u.)

Vt1 (p.u.)

1.08

1

7

(d) Mechanical power of generator 1

1.04

SS

SC

CC

5

6

2.45

1.02

2.4

0

1

2

3

4

5

0

7

6

−0.2

SS

SC

1

2

3

4

7

(f) Rotor speed deviation of generator 10

(e) Rotor angle deviation of generator 10 4

CC

SS

SC

5

6

CC

Δ ω 10 (rad/s)

(rad)

−0.4

10

−0.6

Δδ

−0.8 −1 −1.2

2 0 −2 −4

0

1

2

3

4

5

6

0

7

1

SS

SC

CC

4

1.05

t10

(p.u.)

6 4

7

SS

SC

5

6

CC

1

0.95

2 5

4

3

2

1

0

6

0

7

SS

5.2

SC

CC

2

3

4

7

3 Ef10(p.u.)

5.1 5 4.9

2 1 0

4.8 4.7

1

(j) Excitation voltage of generator 10

(i) Mechanical power of generator 10

Pm10(p.u.)

3

V

P

e10

(p.u.)

8

2

(h) Terminal voltage of generator 10

(g) Active power output of generator 10

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

Fig. 3.20 The dynamics of G1 , G10 and G11 respectively obtained in the case that transmission line outage occurred in the system [5]

3.3 Simulation Studies

97

(k) Control valve of generator 10 0.9

SS

SC

(l) Intercept valve of generator 10

μ

IV10

0.8 0.75 1

2

3

4

5

6

SS

0

SC

0

1

2

3

4

7

(n) Active power output of generator 11 SS

SC

5

6

CC

14 (p.u.)

12

e11

−0.4

P

(rad) 11

16

CC

−0.2

Δδ

6

0.75

7

(m) Rotor angle deviation of generator 11

−0.6

10 8

−0.8 0

1 2 3 4 5 6 (o) Terminal voltage of generator 11 SS

SC

6 0

7

1

2

3

4

7

(p) Excitation voltage of generator 11 CC

3 (p.u.)

2 1

t11

f11

1

E

1.05 (p.u.)

5

CC

0.7 0

V

SC

0.8

0.7

0.95 0.9

SS

0.85

(p.u.)

0.85

μ CV10 (p.u.)

0.9

CC

SS

SC

5

6

CC

0 0

1

2

4 3 Time(s)

5

6

7

0

1

2

3 4 Time(s)

7

Fig. 3.20 (continued)

configuration can be stabilized before t = 5 s. G1 of the system with SS configuration presented the least oscillations among the three configuration plan. Because of the transmission line outage, the mechanical power of the G1 showed the largest error in the system with CC configuration as presented in Fig. 3.20d. It did not reach its original operating point until at the end of the simulation. Concerning G10 , the one with SS configuration was stabilized to the original operating point in the shortest time as shown in Fig. 3.20e–h. Referring to the results in Fig. 3.20j, we can see that the SEC was able to offer stronger damping to the rotor speed oscillation with less power than the CEC. The control and intercept valve of the G10 presented continuous oscillation after the fault occurred in the system with CC configuration as presented in Fig. 3.20k and l, respectively. This lead to the sustained oscillation of the mechanical power of G10 in the system with CC configuration as shown in Fig. 3.20i. By contrast, the valve position and the mechanical power input of G10 presented less oscillation with SC and SS configuration respectively. Especially, we can see that the BG for the control valve of the G10 in the system with SS configuration did not participate in the control at the beginning of the first swing as shown in Fig. 3.20k. This is attributed to the inherent initiation delay of the fast

98

3 Switching Control of Synchronous Generators for Transient …

valving module, i.e., T1 introduced in Sect. 3.2.2.3. However, the BG for the intercept valve was switched on since the first swing as illustrated in Fig. 3.20l. The reason was that the relative degree of the rotor speed with respect to the intercept valve is r3 = 1, thus the intercept valve was able to respond to the rotor speed deviation faster than the control valve. As a result, the mechanical power of the G10 provided appropriate damping to the oscillation of the generator in the systems with SS and SC configuration respectively as shown in Fig. 3.20i. Moreover, from the results, the largest variation of the intercept valve of G10 was 0.12 p.u. in the system with CC configuration. In comparison, 10% of the valve flow area was employed in the BG. The BG avoided the full closure and full reopening of control and intercept valves. In this way, the wear and tear of the SG should be revaluated in practice concerning the trade-off between the additional cost and stability improvement. Similarly, G11 showed sustained fluctuation in the system with CC configuration. In comparison, G11 converged back to its original operating point within t = 4.5 s in the system with SC configuration as showed in Fig. 3.20m–p. Meanwhile, G11 of the system with SS configuration showed the strongest robustness to the transmission line outage and the event can be stabilized within t = 4 s. The active power of G12 , G13 , G15 , and G16 in the system with CC configuration have presented the largest oscillations as shown in Fig. 3.21a–d, respectively. By contrast, results observed in the system with SS configuration converged to their original operating points in the shortest time. The system with SC configuration presented a moderate performance.

(a) Active power output of generator 12 SS

SC

60 (p.u.)

20

e13

15 10

(b) Active power output of generator 13

70

CC

P

P

e12

(p.u.)

25

SS

SC

5

6

CC

50 40 30 20

5

10 0 0

1

2

3

4

5

6

0

7

(c) Active power output of generator 15 SC

2

3

4

7

CC

44 (p.u.)

11

e16

10

SS

SC

5

6

CC

42 40

P

Pe15 (p.u.)

SS

1

(d) Active power output of generator 16

38

9

36 0

1

2

3 4 Time(s)

5

6

7

0

1

2

3 4 Time(s)

7

Fig. 3.21 The active power output of G12 , G13 , G15 , and G16 respectively obtained in the case that transmission line outage occurred in the system [5]

3.3 Simulation Studies

99

The system with CC configuration cannot be stabilized to its original operating point as presented in Fig. 3.22a–f. In area 4 and 5, the relative rotor angle between G14 and G15 even went unstable due to the line outage as shown in Fig. 3.22f. At the same time, the system with SC configuration presented more inter-area and inner-area fluctuation compared with the system with SS configuration. Referring to Fig. 3.23b, systems with CC, SC and SS configuration respectively could recover from the event. However, the system with SS configuration showed shorter restorative time than the other two systems. By contrast, the three systems showed extinct resilience performance as depicted in Fig. 3.23a. According to the resilience evaluated by its kinetic energy, the systems with SC and CC configuration respectively cannot go back to the original operating points. In comparison, the system with SS configuration presented the shortest restorative time and was able to go to its original operating point. The dynamics of Rk and Rp revealed that the system with SS configuration was more resilient than the other two systems.

(b) Relative rotor angle between generator 12 and 14 0.5 SS SC CC

(a) Relative rotor angle between generator 11 and 1 SC

CC

0.8

0.4 1

2

3

4

5

6

SS

0

SC

2

3

4

CC

5

6

7

SS

SC

5

6

CC

−0.9

9

1

−1

1

0.5

−1.1 0 0

1

2

3

4

5

0

7

6

(e) Relative rotor angle between generator 11 and 13 SS

1.5

SC

CC

1 0.5 0 0

1

2

4 3 Time(s)

5

6

7

1

2

3

4

7

(f) Relative rotor angle between generator 14 and 15 Δδ14−Δδ15 (rad)

11

1

(d) Relative rotor angle between generator 1 and 9

Δδ −Δδ (rad)

Δδ9−Δδ13 (rad)

1.5

−1

7

(c) Relative rotor angle between generator 9 and 13

Δδ −Δδ (rad) 13

−0.5

12

0.6

0

0

14

(rad)

1

Δδ −Δδ

11

Δδ −Δδ1 (rad)

SS

0.32

SS

SC

5

6

CC

0.3 0.28 0.26 0.24 0.22 0

1

2

4 3 Time(s)

7

Fig. 3.22 The inter-machine oscillations obtained in the case that transmission line outage occurred in the system [5]

100

3 Switching Control of Synchronous Generators for Transient … SS

SC

CC

1

Rp

Rk

1

0.5

0

0.5

0 0

1

2

3 4 5 6 7 Time(s) (a) System resilience measured with its kinetic energy

0

1

2

3 4 5 6 7 Time(s) (b) System resilience measured with its potential energy

Fig. 3.23 The short-term resilience measured with the kinetic energy and the potential energy of the power system, respectively [5]

3.4 Summary In the first place, this chapter has proposed a coordinated SPSS to enhance the stability in the small of the power system. Design of the BPSS does not rely on accurate system information and only the knowledge of relative degree of outputs is needed. The switching strategy utilizes the time-optimum characteristic of the BPSS and makes full use of the damping ability of the CPSS. According to the simulation results, the SPSS is able to provide stronger damping ability to the small-signal oscillations than the CPSS. Moreover, the BPSS can utilize the largest control energy of the PSS to offer the utmost damping with respect to the oscillations of rotor speed. In the case of mechanical power change in the 4generator 11-bus system, the SPSS has prevented the system from going unstable. The SPSS improves the transient stability of multi-machine power systems as well. Referring to the case of IEEE 16-generator 68-bus power system, the system having the SPSS installed has shown stronger transient stability than that having the CPSS implemented only. Although SPSSs are installed locally, they are able to coordinate with the SPSSs on different generators. This can be seen from the results of the relative rotor angle between different generators. The interacted dynamics of generators can be cancelled with the switching of control inputs. Moreover, the same set of SPSS parameters are used in the three cases. Thus the robustness of the SPSS, with respect to the change of power system operation conditions, is verified. In the second place, this chapter has proposed a SEC and a SG working in a coordinated manner to improve the transient stability of multi-machine power systems. The BEC and BG have further explored the potential of the excitation system and the governor of a synchronous generator for the damping of unbalanced energy of the power system. The same set of parameters are used for BEC and BG in the two cases. The SEC and the SG have shown their superior robustness than the CEC and the CG respectively in the cases that operation point change and external disturbance occur respectively in the power system. In the process of the first stabilization stage, the BEC utilizes the maximum  amplitude of the excitation voltage to regulate E q . The unbalanced power between the mechanical input and the electrical output is reduced then. Compared with the

References

101

CG, the SG makes fully use of the control effort of the control and intercept valve in the process of the first swing of the power system. The unbalance between the mechanical power input and the active power output of a synchronous generator is directly controlled through the fast switching of valve position. The SG provides a potential solution to enhancing the transient stability of severely perturbed power systems through the fast switching of control and intercept valves. This would also help to reduce other investments for transient stability improvement in power systems. The application of the SG has further enhanced the stability of the power system having the SECs installed. The simulation studies have shown that the SEC and the SG are able to offer coordinated control performance. Less inter-area and innerarea oscillations are observed in the power system having the SECs and the SGs equipped. The power system only having conventional controllers installed even cannot be stabilized to its original operation point. Attributed to the fast switching of valve positions, the speed governing loop of the synchronous generator can respond almost in the same time-scale with its excitation loop. With Rk and Rp , the shortterm resilience of a power system is evaluated. It has been verified that switching controllers can reduce the restorative time and improve the short-term resilience of multi-machine power systems.

References 1. Jiang L, Wu QH, Wen JY (2004) Decentralized nonlinear adaptive control for multimachine power systems via high-gain perturbation observer. IEEE Trans Circ Syst I Regul Pap 51(10):2052–2059 2. Liberzon D, Trenn S (2013) The bang-bang funnel controller for uncertain nonlinear systems with arbitrary relative degree. IEEE Trans Autom Control 58(12):3126–3141. https://doi.org/ 10.1109/TAC.2013.2277631 3. Liu Y, Wu QH, Kang H, Zhou X (2016) Switching power system stabilizer and its coordination for enhancement of multi-machine power system stability. CSEE J Power Energy Syst 2(2):98– 106 4. Lu Q, Sun Y, Mei S (2001) Nonlinear control systems and power system dynamics. Springer, Berlin 5. Liu Y, Wu QH, Zhou XX (2016) Coordinated switching controllers for transient stability of multi-machine power systems. IEEE Trans Power Syst 31(5):3937–3949 6. (1991) Dynamic models for fossil fueled steam units in power system studies. IEEE Trans Power Syst 6(2):753–761. https://doi.org/10.1109/59.76722 7. Report IC (1973) Dynamic models for steam and hydro turbines in power system studies. IEEE Trans Power Apparatus Syst PAS-92(6):1904–1915. https://doi.org/10.1109/TPAS.1973. 293570 8. Mahmud M, Pota H, Aldeen M, Hossain M (2014) Partial feedback linearizing excitation controller for multimachine power systems to improve transient stability. IEEE Trans Power Syst 29(2):561–571. https://doi.org/10.1109/TPWRS.2013.2283867 9. Guo G, Wang Y, Hill DJ (2000) Nonlinear output stabilization control for multimachine power systems. IEEE Trans Circ Syst I Regul Pap 47(1):46–53 10. Kundur P, Balu NJ, Lauby MG (1994) Power system stability and control. McGraw-Hill, New York 11. Park RH (1973) Fast turbine valving. IEEE Trans Power Apparatus Syst 3:1065–1073

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3 Switching Control of Synchronous Generators for Transient …

12. Cushing E Jr, Drechsler G, Killgoar W, Marshall H, Stewart H (1972) Fast valving as an aid to power system transient stability and prompt resynchronization and rapid reload after full load rejection. IEEE Trans Power Apparatus Syst 4:1624–1636 13. Larsen E, Swann D (1981) Applying power system stabilizers. Part I: General concepts. IEEE Trans Power Apparatus Syst PAS-100(6):3017–3024. https://doi.org/10.1109/TPAS. 1981.316355 14. Larsen E, Swann D (1981) Applying power system stabilizers. Part II: Performance objectives and tuning concepts. IEEE Trans Power Apparatus Syst 6:3025–3033 15. Sadikovic R (2006) Use of facts devices for power flow control and damping of oscillations in power systems. Ph.D. thesis, Swiss Federal Institute of Technology Zurich 16. (2006) IEEE recommended practice for excitation system models for power system stability studies. IEEE Std 421.5-2005 (Revision of IEEE Std 421.5-1992), pp 1–85. https://doi.org/10. 1109/IEEESTD.2006.99499

Chapter 4

Switching Control of Modular Multi-level Converters in High-Voltage-Direct-Current Transmission Systems Via BBFC-Based SCU

4.1 Model of a MMC-HVDC Transmission System A MMC-HVDC transmission system includes a rectifier, an inverter, and their control systems. The per phase schematic of a MMC converter is as depicted in Fig. 4.1a, in which a symmetric monopole configuration with DC bus midpoint grounded is considered. Each phase leg of the MMC includes an upper and a lower arm. Each arm has N submodules in series connection, which are half-bridge converters as shown in Fig. 4.1b. Each half-bridge submodule is composed of two IGBTs and one DC i represents the capacitor voltage of the i-th submodule storage capacitor, where vcu,l in the upper or lower arm. By the on-off of IGBT switches, each submodule works in one of the following three states: inserted, bypassed, and blocked. The sum of voltage levels of submodules in the arm are provided by each bridge arm. Capacitor voltage of submodules in a bridge arm should be kept close to each other, which is realized by the capacitor voltage balancing module in MMC control. Structure of a MMC-HVDC transmission system is as illustrated in Fig. 4.2. The most common controller of the MMC is VC. Four parallel control loops are used for the decoupled control of active power and reactive power of the MMC. Within each control loop, a two-layer structure is employed, namely, an outer-loop PI controller and an inner-loop PI controller.

4.1.1 Model of a MMC The average model is employed for a MMC converter connected on node i, which can be written as [2]

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Liu and Q.-H. Wu, Adaptive Switching Control of Large-Scale Complex Power Systems, Power Systems, https://doi.org/10.1007/978-981-99-1039-7_4

103

104

4 Switching Control of Modular Multi-level Converters in High …

Fig. 4.1 a Per phase schematic of a MMC. b Submodule structure [1]

  ⎧ ξ ξ ⎪ ⎪ d Vdi = 1 I ξ − Pei Srate ⎪ ⎪ di ξ ⎪ ⎪ dt Cdi Vdi ⎪ ⎪  ⎨ ξ d Isdi 2 1 1 ξ ξ ξ ξ E = R ω − V − I + L I pui sdi s pui sqi sdi adi ⎪ dt L pui 2 2 ⎪ ⎪ ⎪  ξ ⎪ d Isqi ⎪ 2 1 1 ⎪ ξ ξ ξ ξ ⎪ E sqi − Vaqi − Rpui Isqi − ωs L pui Isdi = ⎩ dt L pui 2 2

(4.1)

where ξ ∈ {r, i} represents variables describing the rectifier-side MMC or inverterξ side MMC, Vdi is pole-to-pole DC bus voltage of the DC side of the converter station in Volt, C represents submodule capacitance in Farad, N is the number of ξ submodules in each arm, Idi represents the DC current of the transmission line on the DC side in Ampere, Idir = (Vdi j − Vdir )/Rdc , Idi j = (Vdir − Vdi j )/Rdc , Rdc represents the resistance of DC transmission line in , Cdi = Cdi + 2MC/N , Cdi is the DC ξ bus capacitance in Farad, M denotes the number of phases, Pei denotes the active

4.1 Model of a MMC-HVDC Transmission System

105

Fig. 4.2 Schematic of a MMC-HVDC transmission system controlled by VC [1]

ξ power output in p.u., Srate represents the rated MVA for the system, Isdi is the d-axis ξ output current in p.u., Isqi is the q-axis output current in p.u., L pui represents the ξ arm inductance of the MMC in p.u., E sdi is the d-axis controller output voltage in ξ ξ p.u., E sqi denotes the q-axis controller output voltage in p.u., Vadi represents the ξ d-axis component of AC terminal bus voltage in p.u., Vaqi is the q-axis component of AC terminal bus voltage in p.u., Rpui is the parasitic arm resistance in p.u., and ωs denotes the rated angular speed in p.u. With the phase-locked loop (PLL), d-axis of − →ξ the dq-rotational reference frame can be aligned with the node voltage vector V ai . − →ξ ξ ξ ξ In this way, we have Vadi = Vai = | V ai | and Vaqi = 0. Active and reactive power the MMC can be described with



ξ

ξ

ξ

ξ

ξ

Pei = Vadi Isdi + Vaqi Isqi ξ

ξ

ξ

ξ

ξ

Q ei = Vaqi Isdi − Vadi Isqi

(4.2)

The dynamics of circulation current, sum capacitor voltages, and circulating current controller can be found in [2], and they are not showed here.

4.1.2 Outer-Loop Controllers of a VC for Rectifier-Side MMC The DC bus voltage and AC bus voltage on the rectifier-side are controlled by the outer-loop controllers of the VC. The outer-loop controllers are described with ⎧ dxr 1 ⎪ αidi αdi Cdi (Vdir2 − Vdir∗2 ) ⎨ odi = dt 2Srate r ⎪ ⎩ d xoqi = K (V r − V r∗ ) Ioqi ai ai dt

(4.3)

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4 Switching Control of Modular Multi-level Converters in High …

r where xodi represents the state variable of d-axis outer-loop controller of the rectifierr represents the state variable of q-axis outer-loop controller of the side MMC and xoqi rectifier-side MMC, αidi is the bandwidth of the DC bus voltage integrator and αdi is the bandwidth of the DC bus voltage control loop, Vdir∗ represents the reference of Vdir in Volt, and K Ioqi represents the integral coefficient of the AC bus voltage controller. Outputs of the outer-loop controllers can be written as

⎧ ⎨

 1 αdi Cdi Vdir2 − Vdir∗2 2Srate + K Poqi (Vair − Vair∗ )

r∗ r = xodi + Isdi

⎩ I r∗ = x r sqi oqi

(4.4)

where K Poqi represents the proportional coefficient of the AC bus voltage controller, r∗ r∗ r r and Isqi denote the references for Isdi and Isqi in p.u., respectively, and Vair∗ Isdi r denotes the reference for Vai in p.u.

4.1.3 Outer-Loop Controllers of a VC for Inverter-Side MMC The active power output and the AC bus voltage on the inverter side are controlled by the outer-loop controllers, and the state equations are described with ⎧ i d xod j ⎪ ⎪ ⎨ = K Iod j (Pei∗j − Pei j ) dt ⎪ dxi ⎪ ⎩ oq j = K Ioq j (V i − V i∗ ) aj aj dt

(4.5)

i where xod j represents the state variable of d-axis outer-loop controller of the inverteri side MMC and xoq j denotes state variable of q-axis outer-loop controller of the inverter-side, K Iod j and K Ioq j represent the integral coefficients of the active power and AC bus voltage controllers, and Pei∗j is the reference of Pei j in p.u. Outer-loop controllers generate

i∗ i i∗ i Isd j = x od j + K Pod j (Pe j − Pe j ) i∗ i i i∗ Isq j = xoq j + K Poq j (Va j − Va j )

(4.6)

where K Pod j and K Poq j denote the proportional coefficients of the active power controller and AC bus voltage controller of the inverter-side MMC.

4.1.4 Inner-Loop Current Controllers of a VC State equations of the inner-loop current controllers of the rectifier-side and inverterside MMC connected on node i are

4.1 Model of a MMC-HVDC Transmission System

⎧ ξ dx ⎪ ξ∗ ξ ⎪ ⎨ idi = K Iidi (Isdi − Isdi ) dt ξ ⎪ ⎪ ⎩ d xiqi = K (I ξ ∗ − I ξ ) Iiqi sqi sqi dt ξ

107

(4.7)

ξ

where xidi denotes the state variable of d-axis inner-loop current controllers and xiqi represents the state variable of q-axis inner-loop current controllers, and K Iidi and K Iiqi denote integral coefficients. Inner-loop current controllers generate the following control commands. ⎧ 1 ξ ξ ξ ξ∗ ξ ξ ⎪ ⎨ E sdi = xidi + Vadi + K Pidi (Isdi − Isdi ) − ωs L pui Isqi 2 ⎪ ⎩ E ξ = x ξ + V ξ + K Piqi (I ξ ∗ − I ξ ) + 1 ωs L pui I ξ sqi iqi aqi sqi sqi sdi 2

(4.8)

where K Pidi and K Piqi denote proportional coefficients of inner-loop current controllers. State equations and outputs of the inner-loop current controllers of a VC on the inverter side are as given in (4.7) and (4.8). Designed with the same four-loop structure of the VC, a FRTHC was applied for the control of rectifier-side and inverter-side MMCs. Layout of a FRTHC is as shown in Fig. 4.3, in which a four-loop structure is used for the decoupled control of active and reactive power of rectifier-side MMC and inverter-side MMC. In each of the four control loops, two cascading SCUs are configured. Each SCU switches between a BBFC and a PI control loop according to a switching law. The operating principle of a BBFC and its applications for the control of different variables of MMCs are presented in the following.

Fig. 4.3 A MMC-HVDC transmission system controlled by FRTHC [1].

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4 Switching Control of Modular Multi-level Converters in High …

4.2 Bang-Bang Funnel Controller and Its Applications in Control of MMCs In each control loop of the FRTHC, two cascading SCUs are implemented, i.e., an outer-loop SCU and an inner-loop SCU. A BBFC with neutral output is used in each SCU.

4.2.1 Bang-Bang Funnel Controller with Neutral Output The schematic of the four-loop FRTHC is as shown in Fig. 4.3. With the d-axis of dq rotational reference frame aligned to AC terminal voltage vector of the MMC by ξ ξ ξ the PLL, we have Vadi = Vai and Vaqi = 0. (4.2) can be rearranged as ξ

ξ ξ

Pei = Vai Isdi , ξ

ξ

ξ

ξ

ξ ξ

Q ei = −Vai Isqi

(4.9)

ξ

Therefore, Pei , Isdi , Q ei , and Isqi are related with each other through algebraic equations. Meanwhile, referring to the last two differential equations of (4.1), we can see ξ ξ ξ ξ and E sdi , as well as Isqi and E sq , are described by that the relationship between Isdi first-order ordinary differential equations. Hence, first-order BBFCs can be employed in SCUs presented in Fig. 4.3. The switching logic of a BBFC with neutral output is ⎧ ⎨ −1, if e(t) ≥ e+ ∨ (q old = −1 ∧ e(t) > 0) q(t) = 0, if e(t) = 0 ∨ (q old = 0 ∧ e− < e(t) < e+ ) ⎩ +1, if e(t) ≤ e− ∨ (q old = +1 ∧ e(t) < 0)

(4.10)

where q(t) denotes the output of the first-order BBFC, e(t) = y(t) − y ∗ (t) represents the tracking error of y(t), y ∗ (t) is the reference of y(t), q old is of the value of q(t) at the previous sampling interval. The control signal generated by the BBFC is written as ⎧ + ⎪ ⎨ U + u 0 , if q(t) = −1 u0, if q(t) = 0 u(t) = ⎪ ⎩ − U + u 0 , if q(t) = +1 If e(t) and u(t) are positively correlated, then BBFC is ⎧ + ⎪ ⎨ U + u 0 , if u0, if u(t) = ⎪ ⎩ − U + u 0 , if

(4.11)

the control command given by the q(t) = +1 q(t) = 0 q(t) = −1

(4.12)

4.2 Bang-Bang Funnel Controller and Its Applications in Control of MMCs

109

such that a large positive u(t) will drive a positive e(t) to be larger. In (4.11) and (4.12), U + > 0 and U − < 0 are the positive and negative maximal outputs of the BBFC, and u 0 is the steady-state value of u(t) at the equilibrium. For first-order nonlinear system in the following y˙ (t) = f (x) + g(x)u(t)

(4.13)

where x denotes the state vector, f (x) and g(x) are piecewise right-continuous functions. If

+ U + u 0 > max[− f (x)/g(x)] (4.14) U − + u 0 < min[− f (x)/g(x)] are satisfied, then e(t) is ensured to be driven into [e− , e+ ] by (4.10)–(4.12) [3].

4.2.2 BBFCs of SCUs for Rectifier-Side MMC and Inverter-Side MMC With respect to the DC bus voltage control module of rectifier-side MMC in the FRTHC, an outer-loop SCU and an inner-loop SCU are implemented in a cascaded manner. The logic of the BBFC in the outer-loop SCU is written as qvr di (t) = ⎧ r r r+ r_old ⎪ ⎪ −1, Vdi ≥ evdi ∨ (qvdi = −1 ∧ Vdi ≥ 0) ⎨ 0, Vdir = 0 ∨ (qvr_old = 0 ∧ evr−di < Vdir < evr+di ) di ⎪ ⎪ ⎩ +1, Vdir ≤ evr−di ∨ (qvr_old = +1 ∧ Vdir ≤ 0) di

(4.15)

where Vdir = Vdir − Vdir∗ . For the inverter-side MMC, two cascading SCUs for the control of the active power output are composed of an outer-loop SCU and an inner-loop SCU. The BBFC in the outer-loop SCU is configured as q ipe j (t) = ⎧ i i_old −1, Pei j ≥ ei+ ⎪ pe j ∨ (q pe j = −1 ∧ Pe j ≥ 0) ⎪ ⎨ i i− i+ 0, Pei j = 0 ∨ (q i_old pe j = 0 ∧ e pe j < Pe j < e pe j ) ⎪ ⎪ ⎩ +1, P i ≤ ei− ∨ (q i_old = +1 ∧ P i ≤ 0) ej

pe j

pe j

(4.16)

ej

where Pei j = Pei j − Pei∗j . r i is negatively correlated with Vdir and Isd Since Isdi j is negatively correlated with i Pe j , outputs of the BBFCs for rectifier-side and inverter-side MMC can be written as

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4 Switching Control of Modular Multi-level Converters in High …

ξ∗

Isdi, j

⎧ ξ ∗+ ξ ∗0 I + Isdi, j , if qvξdi, j (t) = −1 ⎪ ⎪ ⎨ sdi, j ξ ∗0 if qvξdi, j (t) = 0 Isdi, = j, ⎪ ⎪ ⎩ ξ ∗− ξ ∗0 Isdi, j + Isdi, j , if qvξdi, j (t) = +1

(4.17)

ξ ∗0 ξ∗ where Isdi, j is the steady-state value of Isdi, j , obtained at the equilibrium.

Logics of BBFC of the inner-loop SCU of DC bus voltage control loop of rectifierside MMC and active power control loop of inverter-side MMC are qiξsdi, j (t) = ⎧ ξ ξ+ ξ _old ξ ⎪ ⎪ −1, Isdi, j ≥ eisdi, j ∨ (qisdi, j = −1 ∧ Isdi, j ≥ 0) ⎪ ⎨ ξ ξ _old ξ− ξ ξ+ 0, Isdi, j = 0 ∨ (qisdi, j = 0 ∧ eisdi, j < Isdi, j < eisdi, j ) ⎪ ⎪ ⎪ ⎩ +1, I ξ ≤ eξ − ∨ (q ξ _old = +1 ∧ I ξ ≤ 0) sdi, j i sdi, j i sdi, j sdi, j ξ

ξ

ξ∗

(4.18)

ξ

Since Isdi, j = Isdi, j − Isdi, j and E sdi, j are positively correlated, the BBFC generates

ξ∗

E sdi, j

⎧ ξ ∗+ ξ ∗0 ξ E + E sdi, j , if qisdi, j (t) = +1 ⎪ ⎪ ⎨ sdi, j ξ ∗0 ξ E sdi, j , if qisdi, j (t) = 0 = ⎪ ⎪ ⎩ ξ ∗− ξ ∗0 ξ E sdi, j + E sdi, j , if qisdi, j (t) = −1

(4.19)

ξ ∗0 ξ∗ where E sdi, j is the steady-state value of E sdi, j at the equilibrium. Concerning AC terminal voltage control loop of rectifier-side and inverter-side MMC in FRTHC, an outer-loop SCU and an inner-loop SCU are employed as well. The BBFC in the outer-loop SCU is configured as

qvξai, j (t) = ⎧ ξ ξ ξ+ ξ _old −1, Vai, ⎪ j ≥ evai, j ∨ (qvai, j = −1 ∧ Vai, j ≥ 0) ⎪ ⎨ ξ ξ − + = 0 ∧ evξai, < Vai, j < evξai, ) 0, Vai, j = 0 ∨ (qvξai,_old j j j ⎪ ⎪ ⎩ ξ ξ − ∨ (qvξai,_old = +1 ∧ Vai, j ≤ 0) +1, Vai, j ≤ evξai, j j ξ

ξ

ξ∗

ξ

(4.20)

Since Vai, j = Vai, j − Vai, j and Isqi, j are negatively correlated, the BBFC is configured as ⎧ ξ ∗+ ξ ∗0 ξ I + Isqi, ⎪ j , if qvai, j (t) = −1 ⎪ ⎨ sqi, j ξ ∗0 ξ∗ if qvξai, j (t) = 0 Isqi, j , Isqi, j = (4.21) ⎪ ⎪ ⎩ ξ ∗− ξ ∗0 Isqi, j + Isqi, j , if qvξai, j (t) = +1 ξ ∗0

ξ∗

where Isqi, j is the steady-state value of Isqi, j at the equilibrium.

4.3 Switching Laws for Switching Control Units of FRTHCs

111

Logics of the BBFC of the inner-loop SCU of the terminal voltage control loop of the rectifier-side and inverter-side MMC are given by ξ

qisqi, j (t) = ⎧ ξ ξ+ ξ _old ξ ⎪ −1, Isqi, ⎪ j ≥ ei sqi, j ∨ (qi sqi, j = −1 ∧ Isqi, j ≥ 0) ⎪ ⎨ ξ ξ _old ξ− ξ ξ+ 0, Isqi, j = 0 ∨ (qisqi, j = 0 ∧ eisqi, j < Isqi, j < eisqi, j ) ⎪ ⎪ ⎪ ⎩ +1, I ξ ≤ eξ − ∨ (q ξ _old = +1 ∧ I ξ ≤ 0) sqi, j i sqi, j i sqi, j sqi, j ξ

ξ

ξ∗

(4.22)

ξ

Since Isqi, j = Isqi, j − Isqi, j and E sqi, j are positively correlated, the BBFC is configured as ⎧ ξ ∗+ ξ ∗0 ξ E + E sqi, j , if qisqi, j (t) = +1 ⎪ ⎪ ⎨ sqi, j ξ ∗0 ξ ξ∗ if qisqi, j (t) = 0 E sqi, j , E sqi, j = (4.23) ⎪ ⎪ ⎩ ξ ∗− ξ ∗0 ξ E sqi, j + E sqi, j , if qisqi, j (t) = −1 ξ ∗0

ξ∗

where E sqi, j is the steady-state value of E sqi, j obtained at the equilibrium.

4.3 Switching Laws for Switching Control Units of FRTHCs The following performance is achieved by the switching law implemented in each of the SCU. When the magnitude of the tracking error of an output variable overrides a threshold value for a specified length of time, the BBFC is switched on and generates bang-bang control signals. When the magnitude of the tracking error of an output variable remains within a pre-designed interval for a selected length of time, PI control loop is switched on and asymptotical convergence of the tracking error is obtained. A state-dependent switching law as shown in Fig. 4.4 is proposed for each SCU in the FRTHC. The switching law is composed of two modules, namely, a disturbance indicator and a switching signal generator. In the disturbance indicator, |e| denotes the absolute value of the tracking error of y, i.e., Vdir , Vair , Pei j , Vai j for outer-loop controllers, r r i i , Isqi , Isd and Isdi j , Isq j for inner-loop controllers. τ1y represents the threshold value, overrides which it indicates a disturbance occurs. τ1y is selected as half of the width of the desired error interval, i.e., τ1y = (e+ − e− )/2. K y is the gain to set the triggering speed of the BBFC of a SCU. It is usually selected within 1 to 100 according to the practice. τ2y is generally set as half of τ1y , which denotes a valve value below which the clearing port of the integrator generates C = 1 and the integrator is reset to be zero. The comparator generates 1 when A > B. DBlk represents the blocking signal of the MMC, and the MMC is blocked when DBlk=0. χ = 1 indicates that |e| > τ1y lasts longer than γ1y seconds, γ1y coordinates with K y .

112

4 Switching Control of Modular Multi-level Converters in High …

Fig. 4.4 Schematic of the switching law designed for SCUs of the FRTHC [1].

With respect to the switching signal generator, JK flip flop module outputs Q = 1 when a step-up signal of χ is sensed by the clearing port C, and the initial value of Q is 0. When χ = 1, the integrator of the switching signal generator is reset to 0 and T = 1. When χ decreases from 1 to 0, the output of the integrator then increases. Until the output of the integrator is greater than γ2y , it has T = 0. According to T , control signals generated for the corresponding SCU is u scu = T ∗ u bbfc (t) + (1 − T ) ∗ u pi (t)

(4.24)

where u pi (t) denotes the output of the corresponding PI control loop. According to the above, outputs of the SCUs of the FRTHC are written as ⎧ r∗_pi r∗_scu r∗_bbfc ⎪ = Tir ∗ Isd,qi + (1 − Tir ) ∗ Isd,qi Isd,qi ⎪ ⎪ sd,qi sd,qi ⎪ ⎪ ⎪ r∗_pi ⎪ r∗_bbfc ⎪ E r∗_scu = Tvrdi ∗ E sdi + (1 − Tvrdi ) ∗ E sdi ⎪ ⎪ ⎪ sdi ⎪ ⎪ ⎪ r∗_pi r∗_scu r∗_bbfc ⎪ ⎪ E sqi = Tvr ∗ E sqi + (1 − Tvr ) ∗ E sqi ⎨ ai

ai

i∗_pi i∗_scu i∗_bbfc i i ⎪ Isd,q ⎪ ⎪ j = Ti sd,q j ∗ Isd,q j + (1 − Ti sd,q j ) ∗ Isd,q j ⎪ ⎪ ⎪ i∗_scu ⎪ i∗_pi i∗_bbfc ⎪ ⎪ E sd j = T pi e j ∗ E sd + (1 − T pi e j ) ∗ E sd j ⎪ j ⎪ ⎪ ⎪ ⎪ i∗_scu i∗_pi ⎪ i∗_bbfc ⎪ + (1 − Tvia j ) ∗ E sq j ⎩ E sq j = Tvia j ∗ E sq j

(4.25)

4.4 Simulation Studies

113

where superscripts “_scu”, “_bbfc”, and “_pi” denote the control signals generated by SCUs, BBFCs, and PI controllers, respectively.

4.4 Simulation Studies In order to evaluate the performance of the FRTHC, simulation studies were carried out in a two-machine test power system in PSCAD and a four-machine test power system in Matlab, respectively.

4.4.1 Fault Ride-Through Performance of a Two-Machine Test System in the Case of a Three-Phase-to-Ground Fault on the Rectifier-Side AC Grid Structure of a two-machine test system is as shown in Fig. 4.5. A 1000 MVA/640 kV half-bridge-monopolar MMC-HVDC transmission system is employed to connect two AC systems. Short circuit ratio (SCR) of both the rectifier-side and inverterside AC grid is selected as SCR=2.5. The MMC-HVDC transmission system is configured as: f n = 60 Hz, M = 3, N = 76, Srate = 1000 MVA, Vdc∗ = 640 kV, Carm = 2800 µF, R = 0.005 , L = 50 mH. In order to assess the FRTHC, simulation results of the test system implemented by a single VC and a VDRC respectively are illustrated for comparison. The droop controller can adjust reference value according to the DC voltage sag in real time to stabilize DC side voltage of the MMC [4]. Tracking error of the active power output of the MMC ξ ξ (Vdiξ ∗ − Vdiξ ), where K droop is the droop is described with: e = Peiξ ∗ − Peiξ + K droop i i i = 1, coefficient. The VC is set as: αd = 5, αid = 4, K Pod = 0.25, K Iod = 5, K Poq i i i i i r r K Ioq = 10, K Pid = 0.6, K Iid = 10, K Piq = 0.6, K Iiq = 10, K Pod = 6, K Iod = 20, r r r r r r = 1, K Ioq = 10, K Pid = 0.65, K Iid = 100, K Piq = 0.65, and K Iiq = 100. The K Poq PI control loops of SCUs in the FRTHC also use the above parameters. The droop i r = 0.32, K droop = 0.01. control are set as: K droop

Fig. 4.5 A two-machine test power system with MMC-HVDC transmission [1]

114

4 Switching Control of Modular Multi-level Converters in High …

Table 4.1 Parameters of the BBFCs and switching laws of the SCUs in the FRTHC of the rectifierside MMC [1] Parameter

Vdir loop

e+

evr+ =0.1(0.02)p.u. eir+ =0.02(0.2)p.u. di sdi r− evr− =-0.1(-0.02)p.u. e i sdi =-0.02(-0.2)p.u. di K vr di =70(1) K ir =100(1) sdi r =0.1(0.02)p.u. r τ1v τ1i =0.02(0.2)p.u. di sdi r r τ2v =0.05(0.02)p.u. τ2i =0.01(0.2)p.u. di sdi r =0.18(0)s γ1v γ1ir =0.02(0)s di sdi r =0.54(0)s γ2v γ2ir =0.24(0)s di sdi r∗+ r∗+ Isdi =0.5(1.0)p.u. E sdi =1.0(0.5)p.u. r∗− r∗− Isdi =-0.5(-1.0)p.u. E sdi =-1.0(-0.5)p.u.

e− Ky τ1y τ2y γ1y γ2y U+ U−

r loop Isdi

Vair loop

r loop Isqi

evr+ =0.05(0.2)p.u. ai

eir+ =0.1(0.02)p.u.

evr− =-0.05(-0.2)p.u. ai

eir− =-0.1(-0.02)p.u.

K vr ai =70(1)

K ir =100(1)

r =0.05(0.2)p.u. τ1v

r τ1i

r =0.03(0.2)p.u. τ2v ai r =0.54(0)s γ1v ai r =0.54(0)s γ2v ai r∗+ Isqi =0.5(1.0)p.u. r∗− Isqi =-0.5(-1.0)p.u.

r τ2i =0.05(0.02)p.u. sqi γ1ir =0.06(0)s sqi γ2ir =0.24(0)s sqi r∗+ E sqi =1.0(2.0)p.u. r∗− E sqi =-1.0(-2.0)p.u.

ai

sqi sqi

sq

sqi

=0.1(0.02)p.u.

Table 4.2 Parameters of the BBFCs and switching laws of the SCUs in the FRTHC of the inverterside MMC [1] Parameter

Pei j

i Isd j

Vai j

i Isq j

e+

ei+ pe j =0.2(0.04)p.u. ei− pe j =-0.2(-0.04)p.u. K ip =70(1) ej τ1i p =0.2(0.04)p.u. ej τ2i p =0.1(0.04)p.u. ej γ1i p =0.54(0.001)s ej γ2i p =0.72(0.001)s ej i∗+ =-0.1(2.0)p.u. Isd j i∗− =-1.5(-1.5)p.u. Isd j

eii+ =0.04(0.02)p.u. sd j eii− =-0.04(-0.02)p.u. sd j K ii =100(1) sd j i τ1i =0.04(0.02)p.u. sd j i τ2i =0.02(0.02)p.u. sd j i γ1i =0.12(0.0001)s sd j i γ2i =0.24(0)s sd j i∗+ E sd j =1.0(0.7)p.u. i∗− =-1.0(-0.7)p.u. E sd j

evi+ =0.05(0.2)p.u. aj evi− =-0.05(-0.2)p.u. aj i K v =70(1) aj i τ1v =0.05(0.2)p.u. aj i τ2v =0.03(0.2)p.u. aj i γ1v =0.18(0.001)s aj i γ2v =0.18(0)s aj i∗+ Isq j =0.5(1.0)p.u. i∗− =-0.5(-1.0)p.u. Isq j

eii+ =0.1(0.04)p.u.

e− Ky τ1y τ2y γ1y γ2y U+ U−

sq j

eii− =-0.1(-0.04)p.u. sq j

K ii

=100(1)

i τ1i

=0.1(0.04)p.u.

sq j

sq j i τ2i =0.05(0.04)p.u. sq j i γ1i =0.18(0.001)s sq j i γ2i =0.24(0.005)s sq j i∗+ E sq j =1.5(2.0)p.u. i∗− =-1.5(-2.0)p.u. E sq j

Parameters of the BBFCs and switching laws of SCUs in the FRTHC of the rectifier-side MMC are configured as presented by the non-bracketed values in Table 4.1. BBFCs and switching laws of SCUs in the FRTHC of the inverterside MMC were configured as presented by the non-bracketed values in Table 4.2. Figure 4.6 presents the results obtained in the case where a three-phase-to-ground fault happened on node 1. The fault was applied at t = 2.0s and it lasted for 0.1 s. With respect to the rectifier station, both BBFCs in the outer-loop SCUs were switched on and generated bang-bang current reference signals for the inner-loop controllers, as shown by Fig. 4.6c, d. The current tracking errors were as presented in Fig. 4.6e, f. After the d-axis current tracking error went greater than the predefined level, the inner-loop BBFC was triggered in d-axis based on the switching law of r∗ as the inner-loop SCU. Bang-bang voltage control signals were generated for E sd presented in Fig. 4.6a. In the q-axis inner-loop SCU, the BBFC was not triggered and

4.4 Simulation Studies

115

the q-axis voltage references were as shown in Fig. 4.6b. Attributed to the combined effort of the d-axis and q-axis control voltages, smaller oscillation and less tracking error were observed on the DC side voltage of the rectifier-side MMC controlled by the FRTHC compared with those controlled by VC and VDRC as presented in Fig. 4.6g. The voltage of node 1 in the three test systems were similar, which are presented in Fig. 4.6h. Concerning the inverter-side MMC, BBFCs of SCUs in the d-axis control loop were switched on as illustrated by the d-axis current reference in Fig. 4.6k and the daxis voltage in Fig. 4.6i. Bang-bang signals were provided by BBFCs, and the control power of the MMC was fully explored. Consequently, less tracking errors were found in the d-axis output current and the active power of the inverter-side MMC controlled by the FRTHC than those controlled by VC and VDRC, as illustrated in Fig. 4.6m, o, respectively. The BBFCs of the SCUs in the q-axis control loop were not switched on, and the references of q-axis output current and control voltage were offered by the PI loops in SCUs, as depicted by Fig. 4.6l, j. Attributed to the combined effort of both the d-axis and q-axis controllers, the tracking error and oscillation in the output current and AC terminal voltage of the inverter-side MMC regulated by the FRTHC were smaller than those controlled by VC and VDRC, as presented by Fig. 4.6n, p. The control performance of the VDRC was better than the VC. Nevertheless, the VDRC was essentially PI control, and the oscillating phenomenon still existed.

4.4.2 Fault Ride-Through Performance in a Two-Machine Test System in the Case that a Line-to-Line Fault Occurred on Inverter-Side AC Grid To evaluate the FRTHC in asymmetrical fault disturbances, a line-to-line fault was applied on node 2 at t = 2.0s in the test system of Fig. 4.5. The fault lasted for 0.2 s. In this case, the SCR of the inverter-side AC grid was reduced to SCR = 1.7, and the SCR of the rectifier-side AC grid was kept at SCR = 2.5. Due to the asymmetrical fault, MMCs controlled by FRTHC, VC, and VDRC respectively showed large tracking errors in the active power and AC bus voltage, as presented in Fig. 4.7o, p. Unstable oscillatory modes were excited by the fault in the system controlled by VC and VDRC, as can be found in Fig. 4.7o, p, in which unstable oscillations occurred after the fault was cleared at t = 2.2 s. With respect to the inverter station, both of the two BBFCs in the outer-loop SCUs were switched on. Bang-bang current reference signals were generated for the inner-loop controllers, as presented in Fig. 4.7k, l. Tracking errors of output currents were as shown in Fig. 4.7m, n. We can see that both d-axis and q-axis currents presented unstable fluctuation in the system controlled by VC and VDRC, respectively. In comparison, the system controlled by the FRTHC did not show any unstable oscillations. This was mainly attributed to that both BBFCs in the inner-loop SCUs of the FRTHC were switched on after the fault occurred. During the initial

116

4 Switching Control of Modular Multi-level Converters in High … 0.5

2 1 0 0

FRTHC VC VDRC

1

2

3

4

5

0 -0.5 0

1

(a) Time (s) 1.5 1 0.5 0 0

1

2

3

4

0

5

1

1

2

3

4

0

5

2

3

4

5

1

0

-0.5 1

2

3

4

5

-1 0

1

2 1 2

3

3

4

5

2

3

4

5

4

5

4

5

(h) Time (s)

(g) Time (s)

1

2

(f) Time (s) 0

4

5

0.2 0 -0.2 -0.4 0

1

2

3

(j) Time (s)

(i) Time (s) 0

0 -0.05 -0.1

-1 -2 0

5

0.2 0 -0.2

0.2

0 0

4

(d) Time (s)

(e) Time (s)

-0.2 0

3

0 -0.2 -0.4

(c) Time (s) 0.5 0 -0.5 -1 0

2

(b) Time (s)

1

2

3

(k) Time (s)

4

5

0

1

2

3

(l) Time (s)

Fig. 4.6 Fault ride-through performance of the MMC-HVDC transmission system controlled by FRTHC, VC and VDRC respectively obtained in the case where a 0.1 s three-phase-to-ground fault occurred on node 1 [1] (The black solid line represents the system dynamics controlled by FRTHC, the red dotted line represents the system dynamics controlled by VC, and the blue dotted dash line represents the system dynamics controlled by VDRC)

4.4 Simulation Studies

117

1

0.2

0

0

-1

-0.2

0

1

2

3

4

5

0

1

(m) Time (s)

2

3

4

5

4

5

(n) Time (s) 0.2

0

0.1

-0.5 -1 0

0 1

2

3

(o) Time (s)

4

5

0

1

2

3

(p) Time (s)

Fig. 4.6 (continued)

stage of the fault and the fault recovery process, large control power was employed as shown by Fig. 4.7i, j. Oscillating power of the system then can be consumed in the early stage of the post-fault process. As a result, the MMC controlled by the FRTHC was stable after the PI control loops in SCUs were triggered again. With respect to the rectifier-side MMC, neither of the BBFC in the d-axis outerloop SCU nor q-axis outer-loop SCU was triggered in the fault process. As depicted by Fig. 4.7c, d, reference currents were offered by PI control loops in the outer-loop SCUs during the simulation interval. The BBFC of the inner-loop SCU in the d-axis control loop was switched on after the fault was applied as shown in Fig. 4.7a. The BBFC of the inner-loop SCU in the q-axis control loop was not switched on, and the reference of q-axis control voltage was provided by the PI loop in SCU, as shown r∗ r r∗ r − Isd and Isq − Isq , were by Fig. 4.7b. The tracking error of currents, written as Isd as shown in Fig. 4.7e, f, respectively. The above current dynamics were unstable in the system controlled by VC and VDRC respectively, owing to the impact of the inverter-side MMC. Instability of the active power output of the inverter-side MMC resulted in the fluctuation of Vdci of DC terminal. Current flowing through the DC transmission line oscillated, and the oscillation was transferred to the rectifier-side MMC. Consequently, unstable oscillations were found in the voltage of DC bus of the rectifier-side MMC controlled by VC and VDRC, as shown in Fig. 4.7g. Oscillations of the DC bus voltage lead to the oscillations in output currents and the power of the MMC. We can see from Fig. 4.7h that the AC terminal voltage was unstable as well in the system controlled by VC and VDRC. The test system was unstable controlled by VC after the fault was cleared, and it oscillated and finally went unstable.

118

4 Switching Control of Modular Multi-level Converters in High … 2 1 0 0

FRTHC VC VDRC

1

2

3

4

5

0.4 0.2 0 -0.2 0

1

1 0.5 0 0

1

2

3

4

5

0

1

2

3

4

5

4

5

4

5

(d) Time (s) 0.2 0 -0.2

1

2

3

4

5

0

1

1

2

3

2

3

(f) Time (s)

0 4

5

0.2 0.1 0 -0.1 0

1

(g) Time (s)

2

3

(h) Time (s) 0.5

2 1

0

0

-0.5 1

2

3

4

5

0

1

(i) Time (s)

-1 -2 1

2

3

(k) Time (s)

2

3

4

5

4

5

(j) Time (s)

0

0

5

0

0.2

0

4

-0.1

(e) Time (s)

-0.2 0

3

0.1

(c) Time (s) 0.5 0 -0.5 -1 0

2

(b) Time (s)

(a) Time (s)

4

5

0 -0.2 -0.4 -0.6 0

1

2

3

(l) Time (s)

Fig. 4.7 Fault ride-through performance of the MMC-HVDC transmission system controlled by FRTHC, VC, VDRC respectively obtained in the case where a 0.2 s line-to-line fault occurred on node 2 [1] (The black solid line represents the system dynamics controlled by FRTHC, the red dotted line represents the system dynamics controlled by VC, and the blue dotted dash line represents the system dynamics controlled by VDRC)

4.4 Simulation Studies

0.5 0 -0.5 -1 0

1

119

2

3

4

5

0.8 0.6 0.4 0.2 0 -0.2 -0.4 0

1

(m) Time (s)

2

3

4

5

4

5

(n) Time (s) 0.4

0 -0.5 -1 0

-0.25 1

2

3

(o) Time (s)

4

5

-0.9 0

1

2

3

(p) Time (s)

Fig. 4.7 (continued)

Fig. 4.8 A four-machine test power system with MMC-HVDC transmission [1].

4.4.3 Fault Ride-Through Performance in the Case of a Four-Machine Test Power System The FRTHC was also tested in a four-machine thirteen-bus test power system in Matlab, and the schematic of test system was as shown in Fig. 4.8. Parameters of SGs and the network are given in [5]. The MMC-HVDC transmission system was configured as: M=3, N =180, Srate =900MVA, Vd∗ = 800 kV, Vsmax = 400 kV, Ismax = 1 kA, L pu = 0.08 p.u., Rpu = 0.008 p.u., Carm = 9.375μF, Cd =100μF, C =0.0017F, Rdc =3.058. The VC was set as: αd = 50, αid = 25, K Pod =1, K Iod = 5, K Poq = 1, K Ioq = 5, K Pid = 4, K Iid = 80, K Piq = 4, K Iiq = 80, Ra = 20, and αc = 200. The above parameters were also employed for the PI loops in the SCUs of the FRTHC. i r = 0.26, K droop = 0.062. Droop control were configured as: K droop The BBFCs and switching laws of the FRTHC of the rectifier-side MMC were configured as presented by the bracketed values in Table 4.1. Results obtained in the case where a 0.1 s three-phase-to-ground fault occurred on node 2 at t =0.1 s were

120

4 Switching Control of Modular Multi-level Converters in High … 2

2 1 VC VDRC FRTHC

0 0

0.1

0.2

0.3

0.4

0 -2 0

0.1

(a) Time (s)

-1 0.1

0.2

0.3

0

0.4

0.1

105

7.5 0

0.5

1

1 0.8 0.6 0.4 0.2 0

0.1

(e) Time (s)

0.95 0.2

0.3

0.3

0.4

0.2

0.3

0.4

(f) Time (s)

1

0.1

0.2

(d) Time (s)

8

0.9 0

0.4

-0.5

(c) Time (s) 8.5

0.3

0

0.5 0 -0.5 -1 0

0.2

(b) Time (s)

0.4

0.4 0.2 0 -0.2 -0.4 0

0.1

(g) Time (s)

0.2

0.3

0.4

(h) Time (s) 0

0.32 0.3

-0.1

0.28 0.26 0

0.1

0.2

0.3

0.4

0

0.1

(i) Time (s)

0.2

0.3

0.4

(j) Time (s)

0.3

1 0.95

0.25 0

0.1

0.2

0.3

(k) Time (s)

0.4

0.9 0

0.1

0.2

0.3

0.4

(l) Time (s)

Fig. 4.9 Dynamics of the MMC-HVDC transmission system controlled by FRTHC, VC, and VDRC respectively obtained in the case where a 0.1 s three-phase-to-ground fault occurred on node 2 in the four-machine test power system [1] (The black solid line represents the system dynamics controlled by FRTHC, the red dotted line represents the system dynamics controlled by VC, and the blue dotted dash line represents the system dynamics controlled by VDRC)

4.4 Simulation Studies

121

as shown in Fig. 4.9. The performance of the FRTHC was compared with that of a VC and a VDRC, respectively. Owing to the rectifier-side fault, both BBFCs of the outer-loop SCUs in the d-axis and q-axis loop of the FRTHC of the rectifier-side MMC were triggered. As depicted in Fig. 4.9c, d, bang-bang current reference signals were offered to the inner-loop controllers. Similarly, both BBFCs of the inner-loop SCUs in the d-axis and q-axis loop were switched on. Voltage references were generated in a bang-bang manner for the rectifier-side MMC as shown in Fig. 4.9a, b. Because of the bang-bang signals generated by the FRTHC, the DC bus voltage and AC terminal voltage of the rectifierside MMC presented less fluctuation and less error than those controlled by VC and VDRC, as can be seen in Fig. 4.9e, f. In regards to the inverter-side MMC, BBFCs in SCUs were not switched on in the d-axis nor q-axis loop, and current and voltage references were offered by the PI controllers, as shown in Fig. 4.9i, j, g, h. However, oscillatory modes were found in the current and voltage references generated by the VC and VDRC, respectively. In comparison, the MMC controlled by the FRTHC did not show any oscillation. As a result, the active power and AC terminal voltage of the inverter-side MMC controlled by the FRTHC showed less fluctuation and magnitude error than those controlled by VC and VDRC respectively, as can be seen from Fig. 4.9k, l. According to the above results, we can see that the inverter-side MMC was closely related with the rectifier-side MMC. Although BBFCs on the inverter side were not triggered, the rectifier-side MMC controlled by FRTHC suppressed the imbalance between the input and output energy of the MMC-HVDC transmission system. Therefore, the outputs of MMCs controlled by the FRTHC presented better than those controlled by VC and VDRC respectively. Meanwhile, the test power system did not go into the oscillatory region and no oscillation was excited.

4.4.4 Modal Analysis of the Four-Machine Thirteen Bus Test System Controlled by VC Modal analysis was carried out for the four-machine thirteen-bus test power system with a MMC-HVDC transmission system controlled by VC, and results were as shown in Table 4.3. ωri denotes the speed error of i-th synchronous generator in p.u., δi is the power angle deviation of i-th SG in rad, fdi , 1di , 1qi and 2qi are flux linkages of the field winding, d-axis amortisseur winding, and two q-axis amortisseur windings of i-th SG in p.u. There were 42 modes in the system, and three of them are unstable. Unstable modes are marked red in Table 4.3. Referring to the participation factor of state modes, state variables with unstable modes were δ2 and δ3 . Therefore, we can find that the MMC-HVDC controlled by VC did not provide enough damping to the unstable models in δ2 and δ3 . These unstable models lead to the divergent fluctuation in the MMCs controlled by VC. Hence, modal analysis agrees with the simulation given in Sect. 4.4.3.

122

4 Switching Control of Modular Multi-level Converters in High …

Table 4.3 Oscillatory modes of the four-machine thirteen-bus test power system with MMC-HVDC controlled by VC [1] Number

Eigenvalue

Frequency

Damping ratio

Participation factor

Relevant variable

1

−4185.2

0 Hz

1.0

0.5003

r Vd5

2

−188.8

0 Hz

1.0

1.0712

i Isd6

3

−109.6

0 Hz

1.0

1.3418

i Isq6

4

−99.6

0 Hz

1.0

0.9929

v14

5

−99.0

0 Hz

1.0

0.5166

v12

6

−99.5

0 Hz

1.0

0.5175

v11

7

−83.9

0 Hz

1.0

2.5411

r Isq5

8

−99.4

0 Hz

1.0

0.9788

v13

9

−50.3

0 Hz

1.0

1.7514

r Isd5

10

−15.1+29.8j

4.7425 Hz

0.4529

0.7029

r xod5

11

−15.1−29.8j

4.7425 Hz

0.4529

0.7029

r xod5

12

−29.1+4.1j

0.657 Hz

0.9901

2.4076

r xid5

13

−29.1−4.1j

0.657 Hz

0.9901

2.4076

r xid5

14

−35.9

0 Hz

1.0

1.2782

1d2

15

−36.3+0.1j

0.0086 Hz

1.0

0.3897

2q2

16

−36.3−0.1j

0.0086 Hz

1.0

0.3897

2q2

17

−35.5

0 Hz

1.0

0.5889

1d4

18

−34.2

0 Hz

1.0

0.8650

1d1

19

−33.6

0 Hz

1.0

0.5787

1d3

20

−30.7

0 Hz

1.0

0.5844

r xid5

21

−31.2

0 Hz

1.0

0.5606

2q1

22

−24.1

0 Hz

1.0

1.1068

i xiq6

23

−14.4

0 Hz

1.0

0.5320

r xoq5

24

3.3

0 Hz

−1.0

0.1819

δ2

25

−8.0+7.0j

1.1118 Hz

0.7512

0.5378

i xid6

26

−8.0−7.0j

1.1118 Hz

0.7512

0.5378

i xid6

27

−0.5+6.8j

1.0827 Hz

0.0759

0.2598

ωr1

28

−0.5−6.8j

1.0827 Hz

0.0759

0.2598

ωr1

29

−0.5+7.1j

1.1292 Hz

0.0681

0.3042

ωr4

30

−0.5−7.1j

1.1292 Hz

0.0681

0.3042

ωr4

31

0.4 + 3.8 j

0.6116 Hz

−0.1141

0.2387

δ3

32

0.4 − 3.8 j

0.6116 Hz

−0.1141

0.2387

δ3

33

−4.9

0 Hz

1.0

0.3354

1q2

34

−5.0

0 Hz

1.0

0.3627

1q3

35

−3.7+1.4j

0.2274 Hz

0.9335

0.1661

1q3

36

−3.7−1.4j

0.2274 Hz

0.9335

0.1661

1q3

37

−3.7

0 Hz

1.0

0.2956

1q1

38

−2.5

0 Hz

1.0

0.3399

i xoq6

39

−1.3

0 Hz

1.0

0.8318

fd1

40

−1.0

0 Hz

1.0

0.5583

fd3

41

−0.8

0 Hz

1.0

0.8127

fd2

42

−0.7

0 Hz

1.0

0.6458

fd4

References

123

4.5 Summary This chapter has proposed a FRTHC to enhance the fault ride-through performance of MMC-HVDC transmission system. Following conclusions are obtained: First, the FRTHC is structurally stable. The switching laws designed are effective, and the BBFCs cannot be triggered by measurement noise and impulsive disturbances. Second, tracking errors of outputs are smaller in the system controlled by the FRTHC. The BBFCs of a SCU can utilize more control power of the MMC than a PI control loop. This helps to dissipate the unbalanced power in DC capacitors and submodule capacitors of MMCs. After achieving the re-balance between the input and output power of MMCs, stability of the MMC-HVDC system is ensured and better performance is achieved by the FRTHC. Third, FRTHC ensures the stability of the MMC-HVDC system. The piecewise constant bang-bang control signals generated by BBFCs do not inspire any oscillating modes. We can see that BBFCs have resisted the MMC-HVDC to consume the oscillatory power of synchronous generators and circumvent the poor damping of PI control loops in SCUs.

References 1. Liu Y, Lin Z, Xu C, Wang L (2022) Fault ride-through hybrid controller for mmchvdc transmission system via switching control units based on bang-bang funnel controller. J Modern Power Syst Clean Energy 1–12. https://doi.org/10.35833/MPCE.2021.000470 2. Kamran S, Lennart H, Hans-Peter N et al (2016) Design, control, and application of modular multilevel converters for HVDC transmission systems. Wiley 3. Liu Y, Xiahou K, Wu QH et al (2020) Robust bang-bang control of disturbed nonlinear systems based on nonlinear observers. CSEE J Power Energy Syst 6(1):193–202. https://doi.org/10. 17775/CSEEJPES.2018.01310 4. Huang K, Li Y, Zhang X et al (2021) Research on power control strategy of household-level electric power router based on hybrid energy storage droop control. Protection Control Modern Power Syst 6(1):1–13 5. Kundur P, Balu NJ, Lauby MG (1994) Power system stability and control. McGraw-hill, New York

Chapter 5

Switching Control of Doubly-Fed Induction Generator-Based Wind Turbines for Transient Stability Enhancement of Wind Power Penetrated Power Systems

5.1 Multi-loop Switching Control of DFIG-Based Wind Turbine Systems 5.1.1 Model Linearization of DFIG-Based Wind Turbine System 5.1.1.1

Model of DFIG-Based Wind Turbine System

3 The mechanical power extracted from the wind is described with Pm = 0.5ρ Ar vwind 2 gCp (λ, β) [1], where Ar = π R . The model of GE 3.6 MW wind turbine is investigated here. The mechanical power extraction coefficient Cp is Cp (λ, β) = 4  4 ωt R i j i=0 j=0 αi j β λ [1], where αi j are given in [1], λ = vwind . The optimal rotor speed ωt_opt can ensure that Cp reaches its maximum value Cp_ max , which is referred to as maximum power extraction. A two-mass model is employed to describe the shaft system of the wind turbine [2], and the mode is written as

⎧ P ⎨ 2Ht ω˙ t = ωmt − Dt ωt − Dtg (ωt − ωr ) − Ttg 2H ω˙ = Ttg + Dtg (ωt − ωr ) − Dg ωr − ωPer ⎩ g r T˙tg = K tg (ωt − ωr )

(5.1)

where ωt denotes the rotor speed of wind turbine, ωr represents the rotor speed of DFIG, Ttg denotes the internal torque of the connecting resilient shaft, Ht represents the inertia constant of wind turbine, Pm denotes the mechanical power input, Dt represents the mechanical damping coefficient, Dtg denotes the damping coefficient of connecting resilience shaft between the rotor of turbine and the rotor of induction generator, Dg represents the mechanical damping coefficient of the DFIG, K tg denotes the stiffness of the connecting resilient shaft between the rotor of turbine and the rotor of the induction generator, Pe represents the active power output of the DFIG. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Liu and Q.-H. Wu, Adaptive Switching Control of Large-Scale Complex Power Systems, Power Systems, https://doi.org/10.1007/978-981-99-1039-7_5

125

126

5 Switching Control of Doubly-Fed …

The stator-flux oriented frame is employed and d-axis of the d-q reference frame is aligned with the stator-flux vector. In this way, we have vqs = vs and vds = 0. Model of the DFIG is described with [3] i˙dr = i˙qr =

1 (−Rr i dr + sωs σ L r i qr + vdr ) σ Lr    σ L r i dr +L 2m i ms 1 −R i − sω r qr s σ Lr Ls v −R i

L2

+ vqr



(5.2)

−i dr ) , L s = L ls + L m , where σ = 1 − L s mL r , i ms = sωs Lsm qs , i qs = − mL sqr , i ds = L m (ims Ls L r = L lr + L m , s = (ωs − ωr )/ωs , i dr ] denotes the d-axis rotor current of rotor-side circuit, i qr represents the q-axis rotor current of rotor-side circuit, vdr denotes the d-axis voltage of rotor-side converter, vqr is the q-axis voltage of rotor-side converter, L m represents the mutual inductance of rotor-side circuit, vs is the voltage of the terminal bus of DFIG, L s represents the inductance of stator winding, L r is the inductance of rotor-side circuit, Rr denotes the resistance of rotor-side circuit, ωs is the synchronous rotational speed, i ms represents the magnetizing current of induction generator, L ls denotes the stator leakage inductance, L lr is the rotor leakage inductance. For the grid-side converter, the d-axis is aligned with the stator voltage, and we have vds = vs and vqs = 0. The grid side converter is described with [4]

i˙dg = i˙qg =

L i

1 (−Rg i dg Lg 1 (−Rg i qg Lg

+ ωs L g i qg + vs − vdg ) − ωs L g i dg − vqg ).

(5.3)

where i dg denotes the d-axis current of grid-side converter circuit, i qg is the q-axis current of grid-side converter circuit, vdg represents the d-axis voltage of grid-side filter, vqg is the q-axis voltage of grid-side filter, L g denotes the inductance of gridfilter circuit, Rg is the resistance of gird-filter circuit. Neglecting the harmonics introduced by the switch, the dynamics of the DC-link of the back-back converter is [3, 5] 1 V˙dc = C



3m 1 Pr √ i dg − V 4 2 dc

 (5.4)

where Pr = 1.5(vdr Idr + vqr Iqr ), Vdc represents the voltage of DC-link capacitor, Pr is the active power from the rotor side, m 1 denotes the modulation depth.

5.1.1.2

Output Feedback Linearization of DFIG-Based Wind Turbine Model

Using the above model, the output feedback linearization was undertaken for the DFIG-based wind turbine system. Define the state vector as X = [x1 x2 . . . x8 ] = [ωt ωr Ttg i dr i qr i dg i qg Vdc ] . The vector of control variables is U = [u 1 u 2 u 3 u 4 ] =

5.1 Multi-loop Switching Control of DFIG-Based Wind Turbine Systems

127

[vdr vqr vdg vqg ] . The vector of output variables is written as Y = [y1 y2 y3 y4 ] = 1.5ωs L 2m i ms (i ms −i dr ) is the [ωr − ωref Q s − Q sref Q g − Q gref Vdc − Vdcref ] , where Q s = Ls stator reactive power, and Q g = −1.5vs i qg is the reactive power of the grid-side converter. The DFIG based wind turbine system is described with

where

⎡

1 2H t

X˙ = F(X ) + G(X )U Y = H (X )



Pm x1

− Dt x1 − Dtg (x1 − x2 ) − x3

(5.5)

⎤

⎢ ⎥ ⎢ 1 ⎥ ⎢ 2Hg x3 + Dtg (x1 − x2 ) − Dg x2 + 1.5Lx2mLvss x5 ⎥ ⎢ ⎥ ⎢ ⎥ K tg (x1 − x2 ) ⎢ ⎥ 1 ⎢ ⎥ (−Rr x4 + sωs ρ L r x5 ) ⎢ ⎥ ρ Lr   2 F(X ) = ⎢ ⎥ sωs L m i ms sωs ρ L r x4 1 ⎢ ⎥ −R x − − r 5 ρ Lr Ls Ls ⎢ ⎥ ⎢ ⎥ 1 (−R x + ω L x + v ) g 6 s g 7 s ⎢ ⎥ Lg ⎢ ⎥ 1 ⎢ ⎥ (−R x − ω L x ) s g 6 Lg ⎣ ⎦ g 7 Pr 1 3m√1 x6 − C x8 4 2 = [F1 (X ) F2 (X ) . . . ⎡

0 0 0

⎢ ⎢ ⎢ ⎢ 1 ⎢ ⎢ ρ Lr G(X ) = ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ ⎣ 0 0 and

0 0 0 0

F8 (X )] ,

0 0 0 0 0 0 0 0 1 0 0 ρ Lr 0 − L1g 0 0 0 − L1g 0 0 0

⎤ ⎥ ⎥ ⎡ ⎤ ⎥ G 1 (X ) ⎥ ⎥ ⎢ ⎥ ⎢ G 2 (X ) ⎥ ⎥ , ⎥=⎣ G 3 (X ) ⎦ ⎥ ⎥ G 4 (X ) ⎥ ⎥ ⎦

H (X ) = [H1 (X ) H2 (X ) H3 (X ) H4 (X )] = [y1 y2 y3 y4 ] .

The following nonlinear coordinate transformation is employed Z = [z 11 z 12 . . . z 1r1 z 21 z 22 . . . z 2r2 z 31 z 32 . . . z 3r3 z 41 z 42 . . . z 4r4 ] = T (X ) = [H1 (X ) L F H1 (X ) L Fr2 −1 H2 (X ) H3 (X ) L F H3 (X ) . . . . . . L Fr1 −1 H1 (X ) H2 (X ) L F H2 (X ) . . . r3 −1 r4 −1 L F H3 (X ) H4 (X ) L F H4 (X ) . . . L F H4 (X )], where r1 , r2 , r3 and r4 denote the relative degree of Hi (X ) (i = 1, 2, 3, 4) with respect to u j ( j = 1, 2, 3, 4), respectively. The relative degree r j is the smallest integer such that at least one (r ) of the inputs appears explicitly in Hi j (X ), i.e.,

128

5 Switching Control of Doubly-Fed … (r j )

Hi

r

(X ) = L F j Hi (X ) +

4  r −1 [LG j L F j Hi (X )u j ], j=1

r −1

if LG j L F j Hi (X ) = 0 holds for at least one j. (5.5) is decoupled into four independent subsystems, and the ith (i = 1, 2, 3, 4) subsystem can be described with ⎧ z˙ i1 = z i2 ⎪ ⎪ ⎪ z˙ = z ⎪ i2 i3 ⎪ ⎪ ⎨···

4  ⎪ ⎪ βi j (Z )u j z˙ iri = αi (Z ) + ⎪ ⎪ ⎪ j=1 ⎪ ⎩ yi = z i1

(5.6)

where αi (Z ) = (L Fri Hi (X ))| X =T −1 (Z ) , βi j (Z ) = (LG j (X ) L Fri −1 Hi (X ))| X =T −1 (Z ) ( j = 1, 2, 3, 4), which are written as follows. ⎧ m vs LG 2 L F H1 (X ) = 2H1.5L = 0 ⎪ g σ L r x2 L s ⎪ ⎪ ⎪ ⎨ LG 1 L F H1 (X ) = LG 3 LF H1 (X ) = LG 4 L F H1 (X ) = 0 D F (X )

D +D

L F2 H1 (X ) = tg2H1g − tg2Hg g + 1.5LL mxv2s x5 F2 (X )+ ⎪ s 2 ⎪ ⎪ ⎪ ⎩ F3 (X ) + 1.5L m vs F5 (X ) 2Hg L s x2 ⎧ 2Hg 1.5ωs L 2m i ms ⎪ ⎪ ⎨ LG 1 H2 (X ) = − σ L r L s = 0 LG 2 H2 (X ) = LG 3 H2 (X ) = LG 4 H2 (X ) = 0 ⎪ ⎪ ⎩ L F H2 (X ) = − 1.5ωs L 2m ims F4 (X ) Ls ⎧ 1.5vs L H (X ) = = 0 ⎪ G 3 4 Lg ⎨ LG 1 H3 (X ) = LG 2 H3 (X ) = LG 3 H3 (X ) = 0 ⎪ ⎩ L H (X ) = −1.5v F (X ) s 7 ⎧ F 3 √3m 1 ⎪ L L H (X ) = − = 0 G F 4 ⎪ 3 4 2C L g ⎨ LG 1 L F H4 (X ) = LG 2 L F H4 (X ) = LG 4 L F H4 (X ) = 0 ⎪ ⎪ ⎩ L 2 H4 (X ) = 3m √ 1 F (X ) + Pr 2 F8 (X ) F Cx 4 2C 6 8

Therefore, we have r1 = 2, r2 = r3 = 1, r4 = 2 and (5.5) has internal dynamics of second-order.

5.1.1.3

Internal Dynamics of DFIG-Based Wind Turbine System

The following two equations are added to the four decoupled subsystems (5.6) to describe the internal dynamics η1 (X ), η2 (X ).

5.1 Multi-loop Switching Control of DFIG-Based Wind Turbine Systems



η˙ 1 (X ) = L F η1 (X ) + LG 1 η1 (X )u 1 + . . . + LG 4 η1 (X )u 4 η˙ 2 (X ) = L F η2 (X ) + LG 1 η2 (X )u 1 + . . . + LG 4 η2 (X )u 4

129

(5.7)

where η1 (X ), η2 (X ) satisfy LG 1 ηi (X ) = 0, LG 2 ηi (X ) = 0, LG 3 ηi (X ) = 0 and LG 4 ηi (X ) = 0(i = 1, 2). As a result (5.7) is rearranged as

η˙ 1 (X ) = L F η1 (X ) η˙ 2 (X ) = L F η2 (X )

(5.8)

The four-loop SC of (5.5) will ensure the steady-state tracking error to be zero, i.e., limt→∞ Hi (X ) = 0 (i = 1, 2, 3, 4) hold. If t → ∞ holds, we have x˙2 (t) = 0, x˙4 (t) = 0, x˙5 (t) = 0, x˙6 (t) = 0, x˙7 (t) = 0 and x˙8 (t) = 0. The zero states can be selected as η1 (X ) = x1 and η2 (X ) = x3 . Since x˙2 = 0, x3 + Dtg (x1 − x2 ) − Dg x2 + 1.5L m vs x5 = 0 holds if t → ∞. Therefore, we have x3 = Dtg (x2 − x1 ) + Dg x2 − x2 L s 1.5L m vs x5 . Then x˙3 = −Dtg x1 holds. Considering x˙3 = K tg (x1 − x2∗ ) if t → ∞, it can x2 L s be obtained that −Dtg x˙1 − K tg x1 + K tg x2∗ = 0. Therefore, we have x1 (t) is stable if Dtg > 0 and K tg > 0 hold. Owing to x˙1 (t) = 0 if t → ∞, it has x1∗ = x2∗ . Moreover, we have x˙3 = 0 if t → ∞. Therefore, the internal dynamics of (5.6) is zero, which has no impact on the stability of (5.5).

5.1.2 Four-Loop Switching Controller Designed for DFIG-Based Wind Turbine System A four-loop SC is designed for the DFIG based wind turbine system. The SC switches between a LBCFC and a CC in four control loops respectively according to a statedependent switching strategy. The schematic of the four-loop SC is shown in Fig. 5.1. The CC is designed based on the VC given by Pena et al. [3, 6, 7].

5.1.2.1

Four LBCFCs Designed for Four-loop SC of DFIG

(1) The LBCFC for the Rotor Speed Control Loop According to the analysis presented in 5.1.1.2, the relative degree of ωr with respect to vqr is r1 = 2. A second-order LBCFC was employed for the rotor speed control m vs > 0, the control law of the loop. Since g(y1r1 −1 , z) = LG 2 L F H1 (X ) = 2H1.5L g σ L r x2 L s LBCFC for the rotor speed control loop is written as vqr (t) =

vqr_ min , if q(t) = true vqr_ max , if q(t) = false.

130

5 Switching Control of Doubly-Fed …

(2) The LBCFC for the Stator Reactive Power Control Loop The relative degree of Q s with respect to vdr is r2 = 1. A first-order LBCFC is designed for the stator reactive power control loop. Since g(y2r2 −1 , z) = LG 1 H2 (X ) = 1.5ω L 2 i − σ Ls r Lms ms < 0, the control law of the LBCFC for the stator reactive power control loop is written as vdr_ max , if q(t) = true vdr (t) = vdr_ min , if q(t) = false. (3) The LBCFC Designed for the Grid-side Reactive Power Control Loop The relative degree of Q g with respect to vqg is r3 = 1. A first-order LBCFC is designed for the grid-side reactive power control loop. Since g(y3r3 −1 , z) = s > 0, the control law of the LBCFC of the grid-side reactive power LG 4 H3 (X ) = 1.5v Lg control loop is written as vqg (t) =

vqg_ min , if q(t) = true vqg_ max , if q(t) = false.

(4) The LBCFC Designed for the DC-link Voltage Control Loop The relative degree of Vdc with respect to vdg is r4 = 2. A second-order LBCFC is designed for the DC-link voltage control loop. Since g(y4r4 −1 , z) = LG 3 L F H4 (X ) = 1 < 0, the control law of the LBCFC for the DC-link voltage control loop is − 4√3m 2C L g described as vdg_ max , if q(t) = true vdg (t) = vdg_ min , if q(t) = false. 5.1.2.2

State-Dependent Switching Strategy for the Switching Controller

The four-loop SC of the DFIG works based on a state-dependent strategy T . The switching strategy is stated as that the SC switches from the CC to the LBCFC in a control loop if T1 is satisfied and switches from the LBCFC to the CC in a control loop if T2 is met. T1 and T2 are given as: T1 : {|e(t)| ≥ }, T2 : {The switching frequency of the control signal generated by a LBCFC reaches its maximum} ∨ {{(s −  j )/ s ≥ τ } ∧ {e(t) converges within [2(ϕ0− + ε0− ), 2(ϕ0+ − ε0+ )]}}, The structure of the four-loop SC of the DFIG is as shown in Fig. 5.1.

5.2 Switching Angle Controller and AGC for Frequency Control of DFIG Based WPPS

131

Fig. 5.1 The schematic of the four-loop SC of the DFIG [8]

5.1.2.3

Short-Term Resilience Index of a Power System

The resilience of a power system is used to describe its capability to recover from an external, high-impact, low-probability event. A conceptual definition for the resilience is given by Panteli and Mancarella [9, 10]. A short-term resilience index is defined as R(t) =

1+

n

2 i=1 [2Hi ωi (t) ] +

1 m

i=1 (|Vli (t)

− Vlrefi |2 · |Sli |)

(5.9)

where Vli is the voltage of ith load bus, Vlrefi denotes the reference voltage of ith load bus, Sli is the apparent power of ith load bus,  denote the relative error of the variable. In the steady state, we have R(t) = 1. If a disturbance occurs, R(t) will decrease. After the event, R(t) will return to its steady-state value 1 gradually. The length of the time that R(t) takes to recover to its steady-state value is called as the restorative time.

5.2 Switching Angle Controller and AGC for Frequency Control of DFIG Based WPPS 5.2.1 Internal Voltage and Virtual Rotor Angle of DFIGs Voltage equations of the DFIG are written as [11]. − → − → − → − → − → V s = Rs I s + jωs L ls I s + jωs L m ( I s + I r ) − → − → − → − → − → Vr = Rsr I r + jωs L lr I r + jωs L m ( I s + I r ) s

(5.10)

132

5 Switching Control of Doubly-Fed …

Fig. 5.2 Steady-state circuit of the DFIG and its equivalent circuit [12]

− → where V s denotes the terminal voltage (stator voltage) of the DFIG, Rs is the resis− → tance of stator windings, I s denotes the stator current, ωs represents the synchronous − → rotational speed (system frequency measured by PLL), I r denotes the rotor current, L m represents the mutual inductance of rotor-side circuit, L ls is the leakage induc− → tance of stator windings, V r denotes the rotor voltage of RSC, s = (ωs − ωr )/ωs is the slip speed, Rr denotes the rotor winding resistance, L lr is the leakage inductance of − → − → − → − → rotor windings, I ms denotes the excitation current and defined as I ms = I s + I r , Ims is obtained with Ims = (Vs − Rs Iqs )/(ωs L m ). − → The equivalent circuit of the DFIG is as depicted in Fig. 5.2a, in which I ms − → − → − → denotes the excitation current and defined as I ms = I s + I r . According to Thevenin’s theorem, the circuit can be transformed to the one shown in Fig. 5.2b, ω2 L 2m Rr /s L m [ωs2 L lr L r +(Rr /s)2 ] . The internal where Req = Rs + (Rr /s)s 2 +(ω 2 and L eq = L ls + (Rr /s)2 +(ωs L m )2 s Lr) voltage is written as sωs2 L m L r + jωs L m Rr − → − → Vr (5.11) E ms = Rr2 + (sωs L r )2 − → − → Let V s = Vs ∠θs , and E ms = E ms ∠(θ + θs ), where θ is defined to be the vir− → − → tual rotor angle of the DFIG. The active power transferred from E ms to V s is V s Pe = √ 2 2 [Req (E ms cosθ − Vs ) + X eq E ms sinθ ]. Since X eq is ten times larger Req +X eq

than Req , the active power of the DFIG is denoted by Pe = EXmseqVs sinθ . The DFIG concerned has the following configuration: Nominal active power Pn = 1.5MW, nominal voltage of stator windings Vs_nom = 575V, nominal voltage of rotor windings Vr_nom = 1975V, nominal frequency f = 60Hz, Rs =0.023 p.u., L ls =0.18 p.u., Rr =0.016 p.u., L lr =0.16 p.u., L m = 2.9 p.u., inertia constant of induction generator H1 =0.685s, pairs of poles p = 3, grid-side converter maximum current (pu of generator nominal current) Ig_ max = 0.8p.u., grid-side coupling resistance Rg = 0.003 p.u., grid-side coupling inductance L g = 0.3 p.u., nominal DC bus voltage Vdc_nom = 1150V, DC bus capacitor C = 1e-2 F, wind speed at Cp_ max is 11 m/s, nominal mechanical output power is 1.5e6 W, inertia constant of wind turbine H2 = 4.32s, turbine initial speed (pu of nominal speed) ωm = 1.2 p.u., turbine initial speed ωm0 = 1.2 p.u., initial output torque Tm0 = 0.83 p.u.

5.2 Switching Angle Controller and AGC for Frequency Control of DFIG Based WPPS

133

5.2.2 Design of SAC and AGC The virtual rotor angle is controlled by the SAC through regulating the phase angle measured by the PLL. The SAC is bang-bang with two control values [13–15]. To ensure the stability of the DFIG, the virtual rotor angle ia always be positive. The input of the SAC is the load frequency error f load − f 0 , where f 0 denotes the rated frequency of the system. Control logic of the SAC is q(t) = S (e(t), ϕ0+ − ε0+ , ϕ0− + ε0− , q(t−))

= [e(t) ≥ ϕ0+ − ε0+ ∨ (e(t) > ϕ0− + ε0− ∧ q(t−))] q(0−) ∈ {true, false}

(5.12)

where q(t) ∈ {true, false} denotes the output of S , e(t) = f load − f 0 represents the load frequency error, ϕ0± and ε0± are constants to define the error funnel F0 := {(t, e(t)) ∈ R≥0 × R|ϕ0− ≤ e(t) ≤ ϕ0+ }, q(t−) := limε→0+ q(t − ε). The switching of q(t) only happens at e(t) = ϕ0+ − ε0+ or e(t) = ϕ0− + ε0− . The sliding mode does not exist in the switching process of the SAC. The control law of the SAC is described with q(t) as u(t) =

−δ, if q(t) = true δ, if q(t) = false.

(5.13)

According to (5.12) and (5.13), the SAC does not have a neutral value, such as u(t) =0. θ is expected to be constant in steady-state. In other words, the output of the SAC should be constant in steady-state. Hence, we have introduced a conditional delay module for the SAC: u(t) =

u(t − 1), if e(t) ∈ (ϕ0− + ε0− , ϕ0+ − ε0+ ) u(t), if e(t) ≤ ϕ0− + ε0− ∨ e(t) ≥ ϕ0+ − ε0+

(5.14)

Output of the SAC is added on the phase angle θPLL . The obtained phase angle is then applied to generate the three phase voltage reference for the PWM module as shown in Fig. 5.3. Therefore, the active power of the DFIG provides frequency support to the external network. To keep the rotor speed within an expected range, the mechanical power input of the DFIG should be controlled properly. It is realized by adding K δ u(t) to the reference power in the pitch angle controller as shown in Fig. 5.3, where Pref_0 denotes the reference power. The DFIG is then able to offer inertia support to the external network and keep the stability. In primary frequency control, the balance of power generation and load is obtained by the total effort of generators and loads. Active power of generators is controlled through governors, while the power consumption of loads vary based on the speed-droop characteristics. The DFIG takes part in the primary and even secondary frequency control, and an

134

5 Switching Control of Doubly-Fed …

Fig. 5.3 The schematic of the control system of the DFIG

AGC is added on the pitch angle control loop. The integration of frequency error is added to the active power reference in the pitch angle control loop as well, as shown in Fig. 5.3.

5.2.3 Modal Analysis of DFIGs with SAC and AGC Figure 5.4 presents the small-signal model of the DFIG. In the model, the aerodynamic character of wind, a single-mass rotor model, an induction generator model [8], a pitch angle controller, and a VC are included. According to (5.11) and the

5.2 Switching Angle Controller and AGC for Frequency Control of DFIG Based WPPS

135

Fig. 5.4 The small-signal model of the DFIG

control system showed in Fig. 5.3, we have θPLL − θr + θad = θ + θs , where θPLL denotes the phase angle measured by PLL, θs represents the real phase angle of the stator terminal voltage, θr denotes the rotor angle of the DFIG, and θad = atan sωRsrL r . √ (ωs L m Rr )2 +(sωs2 L m L r )2 Vr Then it has θ = θPLL + θad − θr − θs . Since E ms = , Pe is Rr2 +(sωs L r )2 rearranged as  2  2 E ms Vs Rsr + ωs L m sinθ Pe =  2   2 + ωs L m [ωs2 L lr L r + Rsr ωs L ls Rsr )2 + ωs L m

(5.15)

Linearizing (5.15), we have Pe =

∂ Pe ∂ Pe ∂ Pe ∂ Pe θ E ms + ωs + ωr + ∂ E ms ∂ωs ∂ωr ∂θ

(5.16)

136

5 Switching Control of Doubly-Fed …

where θ = θPLL + θad − θr − θs , and K1 =

sωs σ L r L 2s

L s (sωs σ L r )2 + (Rr L s )2 + s L 2m Rs Rr  K 2 = − Rr σ L r [L s (sωs σ L r )2 + (Rr L s )2 + s L 2m Rs Rr ]  ωr + 2(sωs )2 (σ L r )3 Rr L s − s σ Rr2 Rs L r L 2m ωs (s Rr L s L m Vs − Rr L 2s vqr + sωs σ L s L r vdr )/[Rr L s (sωs σ L r ) + Rr3 L 2s ωr + s Rs Rr L 2m ]2 + sωs σ L r [Rr 2 L s L m Vs − σ L s L r vdr ]/[Rr L s (sωs σ L r )2 ωs + Rr3 L 2s + s Rs Rr2 L 2m ]  K 3 = Rr σ L r [L s (sωs σ L r )2 + (Rr L s )2 + s Rs Rr L 2m ]  − 2(sωs )2 Rr (σ L r )2 L s − s Rs Rr2 σ L r L 2m (s Rr L s L m Vs − Rr L 2s vqr + sωs σ L s L r vdr )/[Rs L s (sωs σ L r )2 + Rr3 L 2s + s Rs Rr2 L 2m ]2 + sωs σ L r [Rr L s L m Vs /ωs + L s L r σ vdr ]/[Rr L s (sωs σ L r )2 + Rr3 L 2s + s Rs Rr2 L 2m ] sωs L s L r K4 = − L s (sωs σ L r )2 + (Rr L s )2 + s Rs Rr L 2m

K 5 = [−2sωs L s (σ L r )2 − Rs Rr L 2m /ωs ](s Rr L s L m Vs − Rr L 2s vqr + sωs L s σ L r vdr )/[L s (sωs σ L r )2 + (Rr L s )2 + s L 2m Rs Rr ]2 + (Rr L s L m Vs /ωs + L s L r σ vdr )/[L s (sωs σ L r )2 + (Rr L s )2 + s Rs Rr L 2m ] K 6 = [2sωs L s (σ L r )2 − Rs Rr L 2m ωr /ωs2 ](s Rr L s L m Vs − Rr L 2s vqr + sωs L s σ L r vdr )/[L s (sωs σ L r )2 + (Rr L s )2 + s L 2m Rs Rr ]2 + (Rr L s L m Vs ωr /ωs2 − L s L r σ vdr )/[L s (sωs σ L r )2 + (Rr L s )2 + s Rs Rr L 2m ] sωs σ L r K 7 = sωs σ L r , K 8 = σ L r i qr , K 9 = −σ L r i qr , K 10 = Ls 2 2 K 11 = (ωs σ L r Idr + ωr L m Vs )/(ωs L s ), K 12 = −(ωs σ L r Idr + L m Vs )/(ωs L s ) K 13 =

∂ Pm k2 vw3 λ ∂Cp (λ, β) , = ∂ωr ωr ∂λ

K 14 =

∂Cp (λ, β) ∂ Pm = k2 vw3 ∂β ∂β

5.2 Switching Angle Controller and AGC for Frequency Control of DFIG Based WPPS

137



    Rr 2 Rr2 ωr 2 K 15 = E ms Vs sinθ − 3 2 + 2L m ωs ωs L ls s ωs s   2    Rr + (ωs L m )2 + ωs L m ωs2 L lr L r + ] − E ms Vs sinθ s       2 2  Rr Rr 2 2 L ls + (ωs L m ) + (ωs L m ) + ωs L ls s s



 R 2  Rr2 ωr r 2 2 − 3 2 + 2L m ωs + L m ωs L lr L r + + ωs L m s ωs s

   R2ω  2  Rr 2  r − 3r 2 + 2ωs L lr L r ωs L ls + ωs L m s ωs s

   2 Rr 2 + ωs L m + ωs2 L lr L r s

   Rr 2 2E ms Vs Rr sinθ  2 + ωs L m ω K 55 = L + (ω L ) s ls s m ωs s 3 s

   2 Rr 2 2Rr L s E ms Vs sinθ  Rr 2  − + ωs2 L lr L r + ω L s m s s3 s



2  R 2  R 2  r r 2 2 ωs L ls + (ωs L m ) + ωs L m + ωs L lr L r s s ∂ Pe ∂ Pe ∂ Pe ∂ Pe = K 17 , = K 15 , = K 16 , = K 55 ∂ωs ∂ E ms ∂θ ∂ωr Linearizing (5.11), we have E ms =

∂ E ms ∂ E ms ∂ E ms ∂ E ms vdr + vqr + ωs + ωr ∂vdr ∂vqr ∂ωs ∂ωr

where

  2

K 17



2 

+ ωs L m

    2 Rr Rr 2 + ωs L m ωs L ls + ωs L m + ωs L lr L r s s

   2  2 cosθ E ms Vs Rsr + ωs L m

  

 =  2 2  2 Rr 2L L + ωs L m ωs L ls Rsr + ωs L m + ω s ls r s Vs sinθ

K 16 =

  2

Rr s

2 

∂ E ms ∂ E ms ∂ E ms ∂ E ms = K 20 K 21 , = K 20 K 22 , = K 18 , = K 19 ∂vdr ∂vqr ∂ωs ∂ωr

(5.17)

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5 Switching Control of Doubly-Fed …

Defining state variables as [ωr ωs ], vdr and vqr can be expressed with the state variables [8]. Under the following assumptions, A.1 The reference for reactive power is constant, thus we have Idr_ref = 0. This assumption is used in linearizing of the reactive power regulator of RSC. A.2 The reference for rotor speed tracks the active power slowly through the maximum power point tracking (MPPT) with a large time constant, and we have ωr_ref = 0. This assumption is used in the linearization of the speed regulator of RSC. A.3 The external network is strong enough and the magnitude of the terminal bus voltage is constant, i.e., Vs = 0. This assumption is used in the linearization of the current control loops of RSC. we have

M2 (s) ωr + M1 (s) M6 (s) ωr + vdr = M4 (s)

vqr =

where

M3 (s) ωs M2 (s) M5 (s) ωs M4 (s)

 P1 s + I1 P3 s + I3 M1 (s) = 1 + K 24 + K 4 K 7 ][1 − K 1 K 10 + K 23 s s  P3 s + I3 P1 s + I1 ][K 1 + K 7 K 23 + K 10 K 24 − K 4 s s  P3 s + I3 (P3 s + I3 )(P2 s + I2 ) · − K5 M2 (s) = K 3 K 10 + K 12 + s s2  P1 s + I1 1 + K 4 K 7 + K 24 s  P3 s + I3  P1 s + I1 K3 + K5 K7 + K9 − K 10 K 24 − K 4 s s   P1 s + I1 P3 s + I3 M3 (s) = 1 + K 24 + K 4 K 7 K 2 K 10 + K 11 − K 6 s s  P3 s + I3  P1 s + I1 − K 10 K 24 − K 4 K2 + K8 + K6 K7 s s   P1 s + I1 P3 s + I3 M4 (s) = 1 + K 24 + K 4 K 7 K 10 K 24 − K 4 s s  (P1 s + I1 )  M5 (s) = 1 + K 4 K 7 + K 24 1 − K 1 K 10 + K 23 · s  P1 s + I1 (P3 s + I3 ) M3 (s) − 1 + K 4K 7 + K 24 · s M1 (s) s  P3 s + I3 K 2 K 10 + K 11 − K 6 s  P1 s + I1  P3 s + I3 M6 (s) = 1 + K 4 K 7 + K 24 1 − K 1 K 10 + K 23 s s M2 (s)  (P3 s + I3 )(P2 s + I2 ) − K 3 K 10 + K 12 + M1 (s) s2 P3 s + I3  I1 − K5 ) 1 + K 4 K 7 + K 24 (P1 + ) s s

(5.18)

5.2 Switching Angle Controller and AGC for Frequency Control of DFIG Based WPPS

139

The structure of the conventional orthogonal-signal-generation based PLL is as illustrated in [16]. The small-signal model of the PLL is as shown in Fig. 5.4. We can see that s2 + P4 s + I4 ωs , θPLL = P4 s2 + I4 s −Rr L r = (ωs − ωr ), (sωs L r )2 + Rr2

θs = θad

1 ωs , s 2π f 0 θr = ωr s

(5.19)

Substituting (5.17)–(5.19) into (5.16), (5.16) can be rearranged as Pe = M12 (s)ωr + M15 (s)ωs + K 17 SAC(ωs )

(5.20)

where M6 (s) M2 (s) + K 20 K 22 + K 19 M4 (s) M1 (s) M5 (s) M3 (s) + K 20 K 22 + K 18 M8 (s) = K 20 K 21 M4 (s) M1 (s) P4 s + I4 M9 (s) = 2 , M10 (s) = sM9 (s) s + P4 s + I4 M11 (s) = K 17 M9 (s) − K 17 + [K 16 M8 (s) + K 15 + K 17 K 25 ]M10 (s) 2π f 0 K 17 + K 16 M7 (s) − K 17 K 25 M12 (s) = − s M11 (s) M15 (s) = M12 (s) M7 (s) = K 20 K 21

According to (5.12), (5.13) of the SAC can be written as SAC(ωs ) ≈ −δ · sgn(ωs ) if ϕ0+ − ε0+ and ϕ0− + ε0− are small enough. So as to get a transfer function for the SAC, η is introduced and η = δ · sgn(ωs )/ωs = δ/|ωs |. Then (5.13) can be further rearranged as SAC(ωs ) ≈ −ηωs . We can see that η is not a constant, and it varies according to the magnitude of ωs . The SAC is configured as follows: ϕ0+ = 2, ϕ0− = −2, ε0+ = 1.95, ε0− = 1.95, u(0) = 0 rad, K δ = 0.4, K f = 2, and we have η ∈ (0, 6] when the error of system frequency varies within ωs ∈ [0.05, ∞). Using the above approximation, (5.20) can be rearranged as Pe = M12 (s)ωr + (M15 (s) − ηK 17 )ωs where the active power response of the DFIG to frequency error ωs is determined by M15 (s) − ηK 17 , whose bode diagram is as depicted in Fig. 5.5. Six samples of η are selected and used for analysis. Decrease of η denotes the increase of ωs . We can find that the SAC strengthens the active power response of the DFIG to system frequency deviation. For the ease of analysis, the drive train and rotor of the induction generator are represented with a single-mass model. Rotor speed dynamics can be written as

140

5 Switching Control of Doubly-Fed …

Bode Diagram

100 Without SAC

η=1

η=2

η=3

η=4

η=5

η=6

Gain (dB)

0 −100 −200 −300 −6

10

10

−4

−2

10

0

10 −1 Frequency (s )

10

2

10

4

Fig. 5.5 The bode diagram of M15 (s) − ηK 17 [12]

1 (Pm − Pe ) = ω˙ r 2H ωr

(5.21)

The mechanical power extracted from the wind is [17] Pm = k2 vw3 Cp (λ, β)

(5.22)

where Cp (λ, β) = 0.73[(151/λi ) − 0.58β − 0.002β 2.14 − 13.2]e(−18.4/λi ) , 1/λi = k 1 ωr 1 − β0.003 , and k1 = λCp_max vw_Cpmax /rated . Linearizing (5.22) and 3 +1 , λ = v λ−0.02β w considering the dynamics of the pitch angle controller, the AGC, and the SAC, we have (5.23) Pm = M13 (s)ωr + M16 (s)ωs + M14 (s)SAC(ωs ) where

 I5  M13 (s) = K 13 + K 14 K pitch + K 14 M12 (s) P5 + s   I5  I5  − K 14 K δ P5 + M14 (s) = K 14 K 17 P5 + s s   Kf I5 M11 (s) − M16 (s) = K 14 P5 + s M10 (s) s

Linearizing (5.21) with (5.20) and (5.23), we have ωr M15 (s) − M16 (s) + η(K 56 − K 17 ) = ωs M13 (s) − M12 (s) − 2H ωr s

(5.24)

ωr Figure 5.6 shows the bode diagram of ω with different η. We can find that the SAC s enhances the response of rotor speed to the error of system frequency. The magnitude ωr decreases as ωs increases. Under the control of SAC, the rotating masses are of ω s able to release or restore kinetic energy to support the system frequency. Figure 5.7

5.2 Switching Angle Controller and AGC for Frequency Control of DFIG Based WPPS

141

Bode Diagram

Gain (dB)

20 0 −20 −40 η=1

Without SAC

−60 −2

10

η=3

η=4

η=5 4

2

0

η=6

10

10

Phase (degree)

10 0

η=2

−50

−100 −3 10

10

ωr ωs

considering different η [12]

Bode Diagram

15 K =0

K =−1

Gain (dB)

f

f

K =−2 f

K =−3

K =−4

f

f

K =−5 f

K =−6 f

10

5

0 −3 10 −10 Phase (degree)

10

10

10 10 Frequency (s−1)

Fig. 5.6 The bode diagrams of

2

1

0

−1

−2

10

−2

10

−1

0

10

−20 −30 −40 −50 −60

0

10 −1 Frequency (s ) Fig. 5.7 The bode diagrams of

ωr ωs

considering different K f [12]

1

10

2

10

142

5 Switching Control of Doubly-Fed …

ωr denotes the bode diagram of ω with different K f . We can see that the AGC mainly s influence the low frequency dynamics, and it helps to reduce the low frequency and steady-state system frequency error.

L 2s Rr L s (sωs σ L r )2 + (Rr L s )2 + s Rs Rr L 2m 1 L s (sωs σ L r )2 = − Rr Rr [L s (sωs σ L r )2 + (Rr L s )2 + s Rs Rr L 2m ] −Rr L r = (sωs L r )2 + Rr2

K 23 = K 24 K 25

K 53 = (K 14 − 1)(K 15 + K 17 K 25 ) + η[K 14 (K 17 − K δ ) − K 17 ] + (K 14 − 1)K 16 K 18 K 54 = −K 4 K 24 I1 I3 [(K 14 − 1)K 17 − K 14 K f ] K 26 = K 53 (1 + K 4 K 7 + K 24 P1 )(K 10 K 24 − K 4 P3 ) −

1 · 2π f 0

(K 14 − 1)K 16 K 20 K 21 (1 + K 4 K 7 + K 24 P1 )(K 2 K 10 + K 11 − K 6 P3 ) K 27 = K 53 [K 24 I1 (K 10 K 24 − K 4 P3 ) − K 4 I3 (1 + K 4 K 7 + K 24 P1 )] + [(K 14 − 1)K 17 − K 14 K f ](1 + K 4 K 7 + K 24 P1 )(K 10 K 24 1 − K 4 P3 ) − (K 14 − 1)K 16 K 20 K 21 [K 24 I1 (K 2 K 10 2π f 0 + K 11 − K 6 P3 ) − K 6 I3 (1 + K 4 K 7 + K 24 P1 )] K 28 = −K 4 K 24 K 53 I1 I3 + [(K 14 − 1)K 17 − K 14 K f ][K 24 I1 1 (K 10 K 24 − K 4 P3 ) − K 4 I3 (1 + K 4 K 7 + K 24 P1 )] + 2π f 0 · (K 14 − 1)K 6 K 16 K 20 K 21 K 24 I1 I3 K 29 = (1 + K 4 K 7 + K 24 P1 )(1 − K 1 K 10 + K 23 P3 ) + (K 10 K 24 − K 4 P3 )(K 1 P1 + K 7 K 23 ) K 30 = K 24 I1 (1 − K 1 K 10 + K 23 P3 ) + K 23 I3 (1 + K 4 K 7 + K 24 P1 ) + K 1 I1 (K 10 K 24 − K 4 P3 ) − K 4 I3 (K 1 P1 + K 7 K 23 )

5.2 Switching Angle Controller and AGC for Frequency Control of DFIG Based WPPS

K 31 = K 23 K 24 I1 I3 − K 1 K 4 I1 I3 1 K 32 = (K 14 − 1)K 16 K 20 K 21 (1 + K 4 K 7 + K 24 P1 )(1− 2π f 0 1 K 1 K 10 + K 23 P3 ) + (K 14 − 1)K 16 K 20 K 22 (1 + K 4 K 7 2π f 0 + K 24 P1 )(K 10 K 24 − K 4 P3 ) 1 (K 14 − 1)K 16 K 20 K 21 [K 23 I3 (1 + K 4 K 7 + K 24 P1 )+ 2π f 0 1 K 24 I1 (1 − K 1 K 10 + K 23 P3 )] + (K 14 − 1)K 16 K 20 K 22 2π f 0 [K 24 I1 (K 10 K 24 − K 4 P3 ) − K 4 I3 (1 + K 4 K 7 + K 24 P1 )] 1 = (K 14 − 1)K 16 K 20 (K 21 K 23 K 24 I1 I3 − K 4 K 22 K 24 I1 I3 ) 2π f 0 = (1 + K 4 K 7 + K 24 P1 )(K 2 K 10 + K 11 − K 6 P3 )−

K 33 =

K 34 K 35

(K 10 K 24 − K 4 P3 )(K 8 + K 6 K 7 + K 2 P1 ) K 36 = K 24 I1 (K 2 K 10 + K 11 − K 6 P3 ) − K 6 I3 (1 + K 4 K 7 + K 24 P1 ) K 37

− K 2 I1 (K 10 K 24 − K 4 P3 ) + K 4 I3 (K 8 + K 6 K 7 + K 2 P1 ) = K 2 K 4 I1 I3 − K 6 K 24 I1 I3

K 38 = 2H ωr (1 + K 4 K 7 + K 24 P1 )(K 10 K 24 − K 4 P3 ) K 39 = 2H ωr [K 24 I1 (K 10 K 24 − K 4 P3 ) − K 4 I3 (1 + K 4 K 7 + K 24 P1 )] + [−K 13 − K 14 K pitch + (1 − K 14 )K 16 K 19 ](1 + K 4 K 7 + K 24 P1 )(K 10 K 24 − K 4 P3 ) − (1 − K 14 )K 16 K 20 K 21 · (1 + K 4 K 7 + K 24 P1 )(K 3 K 10 + K 12 + P2 P3 − K 5 P3 ) K 40 = −2H ωr K 4 K 24 I1 I3 + [−K 13 − K 14 K pitch + (1 − K 14 )K 16 K 19 ][K 24 I1 (K 10 K 24 − K 4 P3 ) − K 4 I3 (1 + K 4 K 7 + K 24 P1 )] + 2π f 0 K 17 (K 14 − 1)(1 + K 4 K 7 + K 24 P1 )(K 10 K 24 − K 4 P3 ) − (1 − K 14 )K 16 K 20 K 21 [(1 + K 4 K 7 + K 24 P1 )· (P3 I2 + P2 I3 − K 5 I3 ) + K 24 I1 (K 3 K 10 + K 12 + P2 P3 − K 5 P3 )]

143

144

5 Switching Control of Doubly-Fed …

K 41 = −K 4 K 24 I1 I3 [−K 13 − K 14 K pitch + (1 − K 14 )K 16 K 19 ]+ 2π f 0 K 17 (K 14 − 1)[K 24 I1 (K 10 K 24 − K 4 P3 ) − K 4 I3 (1+ K 4 K 7 + K 24 P1 )] − (1 − K 14 )K 16 K 20 K 21 [I2 I3 (1 + K 4 K 7 + K 24 P1 ) + K 24 I1 (P3 I2 + P2 I3 − K 5 I3 )] K 42 = −2π f 0 K 4 K 17 (K 14 − 1)K 24 I1 I3 − (1 − K 14 )K 16 K 20 K 21 K 24 I1 I2 I3 K 43 = (1 + K 4 K 7 + K 24 P1 )(1 − K 1 K 10 + K 23 P3 ) + (K 10 K 24 − K 4 P3 )(K 7 K 2 3 + K 1 P1 ) K 44 = I3 K 23 (1 + K 4 K 7 + K 24 P1 ) + K 24 I1 (1 − K 1 K 10 + K 23 P3 ) + K 1 I1 (K 10 K 24 − K 4 P3 ) − K 4 I3 (K 7 K 23 + K 1 P1 ) K 45 = K 23 K 24 I1 I3 − K 1 K 4 I1 I3 K 46 = K 16 K 20 K 21 (1 − K 14 )(1 + K 4 K 7 + K 24 P1 )(1 − K 1 K 10 + K 23 P3 ) K 47 = K 16 K 20 K 21 (1 − K 14 )[K 23 I3 (1 + K 4 K 7 + K 24 P1 ) + K 24 I1 (1 − K 1 K 10 + K 23 P3 )] K 48 = K 16 K 20 K 21 K 23 K 24 (1 − K 14 )I1 I3 K 49 = (1 + K 4 K 7 + K 24 P1 )(K 3 K 10 + K 12 + P2 P3 − K 5 P3 ) K 50

− (K 10 K 24 − K 4 P3 )(K 5 K 7 + K 9 + K 3 P1 ) = (1 + K 4 K 7 + K 24 P1 )(P3 I2 + P2 I3 − K 5 I3 ) + K 24 I1 (K 3 K 10 + K 12 + P2 P3 − K 5 P3 ) − K 3 I1 (K 10 K 24 − K 4 P3 ) + K 4 I3

(K 5 K 7 + K 9 + K 3 P1 ) K 51 = I2 I3 (1 + K 4 K 7 + K 24 P1 ) + K 24 I1 (P3 I2 + P2 I3 − K 5 I3 ) K 52

+ K 3 K 4 I1 I3 = K 24 I1 I2 I3

5.3 Simulation Studies 5.3.1 Test Results of the Four-Loop Switching Controller Simulation studies were carried out in a modified IEEE 16-generator 68-bus power system, the structure of which was as presented in Fig. 5.8. Four DFIG-based wind farms (WF) were connected to bus 17, 18, 19 and 20, respectively. The active power of WF17 , WF19 and WF20 was set as 305 MW, respectively. The active power of WF18 was set as 915 MW. The nominal power of WF17 , WF19 and WF20 was 360 MW, respectively. The nominal power of WF18 was 1080 MW. In total, wind power took up 9.94% of the total power in the system. All DFIGs of the four wind farms were

5.3 Simulation Studies

145

Fig. 5.8 The layout of a modified IEEE 16-generator 68-bus power system (20-generator 72-bus system) [8]

controlled by the four-loop SC. The CC of the four-loop SC was designed according to [3, 6, 7]. The pitch angle controller as shown in [5], and the parameters of the DFIG based wind turbine and those of the step-up transformer of wind farms were configured as: Wind turbine: rated capacity = 3.6 MW, rotor diameter = 104 m, swept area = 8495 m2 , wind speed = 16 m/s, air density = 1.296 kg/m3 , initial blade pitch angle β0 = 1◦ , λ v ωref = optRwind , λopt = 8.68. Mechanical shaft system (on 3.6 MW base): Ht = 4.29 s, Hg = 0.9 s, Dt = Dg = 0, Dtg = 1.5 p.u., K tg = 296.7 p.u. DFIG (on 3.6 MW, 4.16 kV bases): rated power = 3.6 MW, rated stator voltage = 4.16 kV, number of poles p=4, Rs = 0.0079 p.u., Rr = 0.025 p.u., L ls = 0.07937 p.u., L lr = 0.4 p.u., L m = 4.4 p.u., Rg = 0.01, L g = 5 mH, C = 20 mF. Step-up transformer: rated power = 360 MVA, turn ratio = 4.16/34.5 kV, leakage reactance = 6%. The LBCFC and switching strategy of the SC were configured as. First-order LBCFC: ϕ0+ = −ϕ0− = 4, ε0+ = ε0− = 3.9. Second-order LBCFC: + − ϕ0 = −ϕ0− = 4, ε0+ = ε0− = 3.9, ϕ1+ = −ϕ1− = 10.5, ε1+ = ε1− = 0.5, λ+ 1 = λ1 = 9.4. Switching strategy: ωr = 1, τωr = 0.5, Q s = 0.5, τ Q s = 0.5, Q g = 2, τ Q g = 0.5, Vdc = 3, τVdc = 0.3. Control variables: vdr_ max = −vdr_ min = 1.5, vqr_ max = −vqr_ min = 5, vdg_ max = −vdg_ min = 2, vqg_ max = −vqg_ min = 1.5 p.u.

146

5 Switching Control of Doubly-Fed …

Table 5.1 The parameters of the CCs [8] PI1 PI2 PI3 P I

27.778 19.380

22.309 21.883

6.159 49.949

PI4

PI5

PI6

PI7

0.100 49.406

44.645 35.930

39.338 45.488

3.561 8.830

Each synchronous generator was configured with a PSS and an IEEE AC4A exciter. Parameters of the original IEEE 16-generator 68-bus system were given in [18]. Constant impedance load model was employed in the system. The WPPS was simulated with a self-developed power system simulation toolbox in Matlab/Simulink. Parameters of the PI controller of the CCs were as given in Table 5.1, in which P denotes the proportional parameter and I represents the integral parameter of PIi (i = 1, 2, . . . , 7). The structure of these PI controllers was as shown in Fig. 5.1. The same set of parameters were used in the PI controllers of the CC and those of the four-loop SC in the following two cases.

5.3.1.1

A Three-Phase-to-Ground Fault Occurs in Power System

A three-phase-to-ground fault was simulated on one transmission line between bus 53 and 54 at t = 1.5s. The fault was tripped at t = 1.6s. The tripped line was switched on operation again at t = 2.1s. Dynamics of WF18 and WF20 was as shown in Fig. 5.9. Referring to Fig. 5.9a, the rotor speed of the DFIG controlled by the CC showed continuous unstable fluctuation in WF18 . The WF18 controlled by the CC was tripped at t = 6.409s to keep the system stable. In the following, the results of the DFIG were set to zero as shown in Fig. 5.9a–g, i and k–p. In comparison, as presented in Fig. 5.9a, c, the rotor speed and the DC-link voltage of WF18 controlled by the four-loop SC were stabilized to their equilibriums. This was due to the bang-bang control effort Vqr and Vdg offered by the LBCFC as presented in Fig. 5.10l, m, respectively. The LBCFCs were triggered respectively after the fault occurred. They utilized the largest control power of the converters to offer the largest damping to the oscillation of output tracking errors. We can see that the control signals provided by the CC did not exceed their limits and cannot respond as fast as the LBCFCs as presented in Fig. 5.10l, m, respectively. As a result, the active power of WF18 controlled by the four-loop SC could go back to its steady-state value, which was as shown in Fig. 5.9e. According to the reactive power of WF18 depicted in Fig. 5.9g, the four-loop SC presented better performance than the CC. As shown in Fig. 5.10k, n, the proper switching of Vdr and Vqg ensured the fast convergence of the reactive power controlled by the four-loop SC. Consequently, less fluctuation was observed in the terminal bus voltage of the WF18 controlled by the four-loop SC as shown in Fig. 5.10i. As presented in Fig. 5.10o, p, the fault currents of the rotor-side circuit of WF18 with the four-loop SC presented smaller oscillation.

5.3 Simulation Studies

147

(b) Rotor speed of a DFIG of WF

(a) Rotor speed of a DFIG of WF

20

18

CC

SC ω (p.u.)

1

CC

1.01

0.5

1 0.99 0.98

0

6 8 10 4 2 (d) DC−link voltage of a DFIG of WF

6 8 10 4 2 (c) DC−link voltage of a DFIG of WF 18

CC

20

SC

4470 Vdc (V)

Vdc (V)

4480 4460

4465

4440

4460

CC

4420 18

CC

CC

SC

15

P20 (p.u.)

P

18

(p.u.)

20

10

SC

3.5 3

5 0

2.5

2 4 6 8 10 (g) Reactive power output of WF (p.u.)

SC

20

2

CC

0.01

SC

0

Q

1

2 4 6 8 10 (h) Reactive power output of WF 20

18

CC (p.u.)

SC

2 4 6 8 10 (f) Active power output of WF20

10

8 6 4 2 (e) Active power output of WF

−0.01

Q

18

SC

r

ωr (p.u.)

1.5

0

−0.02 2

4

6

8

10

2

4

6

8

10

Fig. 5.9 The dynamics of a DFIG of WF18 and WF20 respectively obtained in the case that a three-phase-to-ground fault occurred in the system [8]

t Moreover, the efficiency of controllers were evaluated with index η = 1/( 0 |y(t) − t yref (t)|dt · 0 |u(t)|dt). The index synthesizes the error of system outputs and the effort of control commands. Four indexes ηωr , η Q s , η Q g and ηVdc were calculated for the four-loop SC and the CC respectively, with the output variables ωr , Q s , Q g , Vdc and control inputs. As presented in Table 5.2, the four-loop SC was able to obtain higher control efficiency than the CC. With respect to WF20 , the rotor speed and the DC-link voltage controlled by the four-loop SC showed better dynamic performance as depiced in Fig. 5.9b, d, respectively. The active power and reactive power outputs of WF20 were as illustrated

148

5 Switching Control of Doubly-Fed …

(j) Voltage of bus 20

0.8

0.9

(p.u.)

0.95 0.85

s20

0.6

0.8

V

V

s18

(p.u.)

(i) Voltage of bus 18 1

0.4 CC

0.2

0.75

SC

CC

SC

Vqr (p.u.)

dr

V (p.u.)

2 4 6 8 10 6 8 10 4 2 (l) q−axis voltage of RSC of a DFIG of WF (k) d−axis voltage of RSC of a DFIG of WF 18 18 5 CC SC CC SC 1 0

0

−1

Vqg (p.u.)

V

dg

(p.u.)

−5 2 4 6 8 10 2 4 6 8 10 (m) d−axis voltage of GSC of a DFIG of WF (n) q−axis voltage of GSC of a DFIG of WF18 18 2 CC SC CC SC 1 1 0 −1

−1

−2

2 4 6 8 10 (o) d−axis rotor current of a DFIG of WF 18

2 4 6 8 10 (p) q−axis rotor current of a DFIG of WF 18 0 Iqr (p.u.)

0.2 Idr (p.u.)

0

0 −0.2 −0.4 2

4

6 Time (s)

CC 8

SC 10

−2 −4 2

4

6 Time (s)

CC 8

SC 10

Fig. 5.10 The dynamics of a DFIG of WF18 and WF20 respectively obtained in the case that a three-phase-to-ground fault occurred in the system [8] Table 5.2 Control efficiency of the four-loop SC and the CC of the DFIG of WF18 [8] ηωr η Qs η Qg ηVdc CC Four-loop SC

0.3251 10.1613

1.0354 4.0670

212.0556 329.3408

5.7474 8.9530

5.3 Simulation Studies

149

(b) Relative rotor angle between G11 and G14 0.8 CC SC 0.6 0.4 0.4 0.2 0.2 0 0 −0.2 CC SC −0.2 −0.4 10 10 8 8 6 6 4 4 2 2 (c) Relative rotor angle between G14 and G15 (d)Relative rotor angle between G11 and G13 0.4 CC SC 1.2 (a) Relative rotor angle between G

and G

1

(rad)

14 11

Δδ −Δδ (rad)

13 11

0.35

Δδ −Δδ

Δδ14−Δδ15 (rad)

Δδ11−Δδ1 (rad)

11

0.6

0.3 2

4

6 Time(s)

CC 8

SC 10

1 0.8 0.6 0.4 4

2

6 Time(s)

8

10

Fig. 5.11 The inter-area and inner-area oscillation obtained in the case that a three-phase-to-ground fault occurred in the power system [8]

60.5

(b) Voltage of load buses 1.1

Vn

f (Hz)

(a) Frequency response of the power system 61 CC SC

60

1

0.9 2

4

6 Time (s)

8

10

2

4

6 Time(s)

8

10

Fig. 5.12 The frequency response and load bus voltage of the power system obtained in the case that a three-phase-to-ground fault occurred in the power system [8]

in Fig. 5.9f, h. We can find that the WF20 controlled by the CC required more reactive power after WF18 was tripped at t = 6.409s. The terminal voltage of WF20 droped after WF18 was switched off as shown in Fig. 5.10j. The inter-area oscillation was found in the relative rotor angle between G11 and G1 , G11 and G14 and G14 and G15 respectively, which were as shown in Fig. 5.11a–c, respectively. We can see that the system with the four-loop SCs presented less interarea oscillations and smaller change in the operating point. Moreover, the inner-area dynamics of area 2 was as shown in Fig. 5.11d. The relative rotor angle between G11 and G13 showed less oscillation when the four-loop SCs were configured. System frequency was as presented in Fig. 5.12a, based on  which was calculated n n   1 (Pmi ) − Pl , where Meq = 2Hi . According [19] with  f˙ = ω˙ r = Meq i=1

i=1

150

5 Switching Control of Doubly-Fed …

(a) Short−term resilience of the system

0.5

0

1

R

R

1

2

4

6 Time(s)

CC 4 SCs 8 10

(b) Short−term resilience of the system

0.5

0

2

4

6 Time(s)

CC 1 SC 8 10

Fig. 5.13 The short-term resilience of the system obtained in the case that a three-phase-to-ground fault occurred in the power system (“4 SCs” indicates that the DFIGs of WF17 , WF18 , WF19 and WF20 all have the four-loop SC installed, and “1 SC” denotes that only the DFIGs of WF18 have the four-loop SC equipped) [8] Table 5.3 The mean value steady state tracking error of R(t) [8] Name VC 1 SC The mean value of R(t) Steady state tracking error of R(t)

4 SCs

0.5376

0.6563

0.6584

0.1464

0.0092

0.0087

to Fig. 5.12a, the four-loop SCs performed better during the first swing process of the system. Figure 5.12b shows the load bus voltage, and the one of the system with CCs presented larger drop. WF18 was tripped after t = 6.409s in the system with CCs and the power supply of the system declines. Drop of load bus voltages enabled the balance between power generation and consumption in the system. The short-term resilience of the system is as shown in Fig. 5.13, where “4 SCs” indicates that the DFIGs of WF17 , WF18 , WF19 and WF20 were installed with fourloop SC, and “1 SC” denotes that only WF18 installed the four-loop SC. The resilience curve of the system with four-loop SCs installed presented less oscillation and less steady-state error. The short-term resilience was determined by the overwhelming kinetic energy of the system and the voltage deviation of load buses. As shown in Fig. 5.12a, b, the system with the four-loop SCs showed less error in the system frequency and load bus voltage. Hence, the system with four-loop SCs had stronger resilience as presented in Fig. 5.13a, b. Moreover, the mean value and steady-state error of R(t) were as shown in Table 5.3, such that a quantified evaluation of the short-term resilience of the power system is obtained. R(t) of the system with the four-loop SC installed presented the largest mean value and the smallest steady-state error. Therefore, the four-loop SC showed better coordination between each other than the CC.

5.3 Simulation Studies

5.3.1.2

151

Results Obtained in the Case Where Transmission Line Outage Occurred

The transmission line between bus 36 and bus 37 was tripped at t = 1.5s due to the fault on the protection system. This fault was corrected and the tripped line was switched on again at t = 1.65s. The line outage fault introduced a severe disturbance to the WPPS. The following simulation results were presented on a 100 MVA base.

1.02

(a) Rotor speed of a DFIG of WF

18

CC

ωr (p.u.)

ωr (p.u.)

1

5 10 15 (d) DC−link voltage of a DFIG of WF

5 10 15 (c) DC−link voltage of a DFIG of WF18

V

19

SC

4465

4400

15 10 5 (e) Active power output of WF18 10

CC

5 10 15 (f) Active power output of WF19

SC

CC P19 (p.u.)

9.5

(p.u.)

SC

4450

4470

9

18

CC

4500 Vdc (V)

4475

dc

(V)

CC

P

1

0.98

0.98

8.5

SC

3.2 3 2.8

8

5 10 15 (h) Reactive power output of WF19 −3 x 10

5 10 15 (g) Reactive power output of WF18 CC

SC

0.02

5

0

0

Q

Q (p.u.)

(p.u.)

SC

0.99

0.99

18

CC

1.01

1.01

0.04

(b) Rotor speed of a DFIG of WF19

1.02

SC

s

−0.02 −0.04

−5 −10 CC

−15 5

10

15

5

10

SC 15

Fig. 5.14 The dynamics of a DFIG of WF18 and WF19 respectively obtained in the case that transmission line outage occurs in the system [8]

152

5 Switching Control of Doubly-Fed …

(i) Voltage of bus 18 CC

SC (p.u.)

1.1

0.9

(j) Voltage of bus 19 CC

SC

1

s19

1

0.9

V

V

s18

(p.u.)

1.1

0.8 0.8

5 10 (k) Ds−ω of a DFIG of WF r

15

5 10 (l) Ds−Qs of a DFIG of WF18

18

0.5 s

1

Ds−Q

Ds−ωr

2

0 −1 10 5 (m) Ds−Qg of a DFIG of WF18

15

0

−0.5

5 (n) Ds−V

0

10 of a DFIG of WF

15

Time (s) 10

15

dc

4 Ds−Vdc

Ds−Qg

2

15

18

2 0 −2

−2

5

Time (s)10

15

−4

5

Fig. 5.15 The dynamics of a DFIG of WF18 and WF19 respectively obtained in the case that transmission line outage occurred in the system [8]

The dynamics of WF18 and WF19 was as depicted in Fig. 5.14. The rotor speed, DC-link voltage controlled by the four-loop SC showed less fluctuation as illustrated in Fig. 5.14a–d, respectively. The rotor speed was controlled to its optimal value for maximum power point tracking. As a result, the rotor cannot participate in frequency regulation, and less fluctuation in rotor speed was observed. This lead to the oscillation of the active power and reactive power of WF18 and WF19 controlled by the LBCFCs as presented in Fig. 5.14e–h, respectively. Nevertheless, the four loop SC consumed more unbalanced power. This was attributed to the LBCFCs, and the active power and reactive power of WF18 and WF19 controlled by the four-loop SC converged to their steady-state values faster than those controlled by the CC as presented in Fig. 5.14e–h, respectively. This further resulted in less fluctuation and faster stabilization speed in the terminal voltage of WF18 and WF19 as shown in Fig. 5.15i–j, respectively. The real control commands Ds-ωr , Ds-Q s , Ds-Q g and Ds-Vdc of the four-loop SC of WF18 were as presented in Fig. 5.15k–n. The SC switched between the CC and the LBCFC in four control loops according to the switching strategy T . These

61 60.5 60 59.5

153

(a) Frequency response CC SC

1 R

f (Hz)

5.3 Simulation Studies

5 Time (s)10

15

(b) Short−term resilience CC 4 SCs

0.5 0

5 Time(s)10

15

Fig. 5.16 The frequency response and short-term resilience of the WPPSs obtained in the case that transmission line outage occurred in the system [8]

control commands were obtained with the magnified outputs. A coordinated damping performance was obtained. The dotted lines denote parameters ωr , Q s , Q g and

Vdc . If the trigger signal hit the dotted line, then the corresponding LBCFC would be triggered. The frequency response of the WPPS was as shown in Fig. 5.16a. We can see that the frequency of the system with the four-loop SC presented less fluctuation and faster stabilization speed. Moreover, the short-term resilience of the WPPS was as presented in Fig. 5.16b. We can see that the system with the four-loop SC installed presented shorter restorative time and less steady-state error in R(t) compared with the system with the CC implemented only.

5.3.2 Test Results of the Switching Angle Controller and AGC To validate the performance of the SAC and AGC, simulation studies were undertaken with a two-machine power system and a modified Kundur four-machine two-area power system, respectively. The SAC and the AGC were configured as: ϕ0+ = 2, ϕ0− = −2, ε0+ = 1.95, ε0− = 1.95, u(0) = 0, K δ = 0.4, K f = 2, K p = 3, K i = 0.6, K p = 0.6, K i = 8, K p = 0.83, K i = 5, K p = 8, K i = 400, K p = 3, K i = 30. ϕ0± and ε0± were set such that ϕ0+ − ε0+ and ϕ0− + ε0− were small values. K δ and K f were selected to eliminate secondary active power drop of the DFIG, after the active power of the DFIG has been elevated by the SAC when system frequency drops. Parameters of the RSC, GSC, SAC and AGC of the DFIG were set to offer enough damping to the oscillation modes of the external power network.

5.3.2.1

Results of the Case Where Dynamic Load Changes Occurred in a Two-machine Power System

Structure of the two-machine power system was as shown in Fig. 5.17. A 9 MW DFIG-based wind farm was consist of 6 × 1.5 MW DFIG-based wind turbines. The

154

5 Switching Control of Doubly-Fed …

Fig. 5.17 The layout of the two-machine power system [12]

wind farm was connected to a 80 MW power system through a 25 kV transmission line and two transformers. A 20 MW load was connected on the 120 kV bus at t = 0.3s, and the load was tripped at t = 2s. The wind speed of the wind farm was set to be 15 m/s. Without AGC and the K δ loop given in Fig. 5.3, the frequency support performance of the SAC was investigated. Due to the load change, frequency of the power grid dropped at t = 0.3s and stepped up at t = 2s as shown in Fig. 5.18a. We can find that the systems with SAC showed less frequency error. Moreover, the frequency of was improved as δ of the SAC was increased. By Fig. 5.18b, we can see that the wind farms with larger δ were able to offer more active power support to grid frequency error. The rotor speed of the DFIG changed along with its active power, as shown in Fig. 5.18c. The rotor decelerated and released its kinetic energy such that excessive active power was provided by the wind farm since t = 0.3s. The rotor accelerated and stored more kinetic energy when the active power of the wind farm was reduced at t = 2s. To prevent over-speed of wind turbines, the pitch angle was ramped up since the rotor speed was greater than 1.2 p.u. as presented in Fig. 5.18d. The control commands of the SAC was as presented in Fig. 5.18e.

5.3.2.2

Results of the Case Where Load Increase Occurred Under Variable Wind Speed Conditions

The frequency control performance of the SAC and AGC was studied with a modified Kunder four-machine two-area power system, the structure of which was as shown in Fig. 5.19. Four 75 MW wind farms was simulated by an aggregated model of 50 × 1.5 MW DFIGs. The wind farms were connected to bus 12, 13, 14, and 15, respectively. The switching dynamics of the IGBTs were considered. The active power of the four wind farms were reserved and set as 0.63 p.u., respectively. The active power of G1 and G4 was decreased by a 94.5 MW, respectively. The other configurations of the system were the same as the Kundur four-machine two-area system given in [19]. The Hydro-Québec grid codes was employed, namely, the error of grid frequency should be within ±0.2 Hz in normal operation. The four 75 MW wind farms supplied 10.64% of the system load in peak generation. Bus 5 and bus 11 were set as wind power generation buses such that the

5.3 Simulation Studies

155 (a)

f (Hz)

60.2 60 59.8 0

0.5

3

2.5

2

1.5

1

3.5

4

(b)

Pe (p.u.)

1 0.9 Without SAC Δδ=0.1 rad Δδ=0.2 rad Δδ=0.3 rad

0.8 0.7 0.6 0

1

2

3

(c)

4

5

6

7

1.2 Without SAC Δδ=0.1 rad Δδ=0.2 rad Δδ=0.3 rad

r

ω (p.u.)

1.22

1.18 1.16 0

1

2

3

(d)

4

5

6

7

2

3

(e) 4

5

6

7

β (degree)

12 10 8 6 4 0

Without SAC Δδ=0.1 rad Δδ=0.2 rad Δδ=0.3 rad

1

Δδ=0.1 rad Δδ=0.2 rad Δδ=0.3 rad

Δδ (rad)

0.2 0 −0.2 0

1

2

3

4 Time (s)

5

6

7

Fig. 5.18 Dynamics of the wind farm obtained in the case where dynamic load changes occurred in a two-machine power system (a System frequency b The active power output of the wind farm c The rotor speed of the DFIG d The pitch angle of the DFIG e The output of the SAC) [12]

inter-area and local oscillation modes were investigated. Variable wind speed was applied on four wind farms [20], and the wind speed was as shown in Fig. 5.20a–d. A 125 MW load was connected on bus 11 at t = 0.5s. Simulation results of the DFIG without any frequency controller are denoted by“original", those of the DFIGs with only the SAC, only the AGC, and both the SAC and AGC are indicated by “SAC”, “AGC”, and “AGC+SAC”, respectively. “VIC” represents the results controlled by the virtual inertia controller (VIC) [21].

156

5 Switching Control of Doubly-Fed …

Fig. 5.19 The layout of the modified Kundur 4-machine 11-bus power system with four wind farms connected [12] (m/s)

(a) 15.5 15

V

wind12

16

(m/s)

0

2

4

6

10

12

14

16

18

10

12

14

16

18

10

12

14

16

18

8 10 Time (s)

12

14

16

18

8 (b)

15.5 15

V

wind13

16

V

4

6

8

15.5 15

14.5

wind15

(m/s)

0

V

2

(c)

wind14

(m/s)

0

2

4

6

8 (d)

15.5 15 14.5 0

2

4

6

Fig. 5.20 The wind speed subjected to wind farm 12, 13, 14, and 15, respectively [12]

System frequency measured with a PLL at load bus 7 was as presented in Fig. 5.21a. Detailed data of system frequency as given in Table 5.4. RoCoF represents the rate of change of frequency, which is the absolute value of the changing rate of frequency after a disturbance occurs. Since the frequency was measured with the PLL, the voltage distortion lead to a sharp drop of the frequency after the load was increased as presented in Fig. 5.21a. Therefore, the RoCoF is calculated with two frequency points, i.e., the pre-disturbance frequency point and the first maximum frequency point after the load increase occurs. The RoCoF is denoted with the change rate of AB and AC shown in Fig. 5.21a. Frequency nadir denotes the lowest

5.3 Simulation Studies

157 (a)

60

B C

59.96

f

load

(Hz)

59.98

A

59.94 Original

59.92 0

2

4

6

AGC

8

AGC+SAC

10

12

VIC

14

SAC

16

18

(b) Original

AGC

AGC+SAC

VIC

SAC

0.8

e

P (p.u.)

0.9

0.7 0.6 0

2

4

6

8

10

12

14

16

18

(c)

ωr (p.u.)

1.2 1.18 1.16 1.14 0

Original

2

4

AGC

6

8

AGC+SAC

10

12

VIC

14

SAC

16

18

Fig. 5.21 The dynamics of the DFIG of wind farm 12 obtained in the case where a load increase occurred in the modified Kundur four-machine two-area power system (a System frequency measured by a phase-locked loop at load bus 7. b The active power output of the DFIG. c The rotor speed of the DFIG) [12]

system frequency during the post-disturbance stage. Settling frequency represents the average value of frequency calculated in the last cycle of oscillation. According to the results in Table 5.4, the frequency controllers reduced the RoCoF. The system with both the AGC and the SAC had the lowest RoCoF. Moreover, the system also presented the highest frequency nadir, and the system with SAC only presented the lowest frequency nadir. Frequency of the systems with only AGC or both the SAC and AGC showed higher settling frequency.

158

5 Switching Control of Doubly-Fed …

Table 5.4 System frequency performance indexes obtained in the case where load increase occurred [12] Configuration RoCoF (Hz/s) Frequency nadir (Hz) Settling frequency (Hz) Original AGC SAC AGC+SAC VIC

0.1947 0.1917 0.1040 0.1034 0.1919

59.9291 59.9293 59.9181 59.9358 59.9305

59.9476 59.9836 59.9491 59.9969 59.9478

The frequency of the system with both SAC and AGC dropped due to the load increase at t = 0.5s. When the frequency error was larger than 0.05 Hz, the SAC was switched on and presented a positive step in δ as shown in Fig. 5.22e. The virtual rotor angle θ was increased as presented in Fig. 5.22d. The active power presented a positive step as well as depicted in Fig. 5.21b. Attributed to K δ and K f loops in the pitch angle controller, the pitch angle β decreased continuously as shown in Fig. 5.22f. The mechanical power of the DFIG was elevated such that input-output power of the DFIG was balanced again. The active power an the rotor speed of the DFIG were increased and stabilized, as presented in Fig. 5.21b, c. Dynamics of the active power of the DFIG with only AGC was similar to that of the DFIG with both SAC and AGC, except for the active power jump at t = 0.5s as illustrated in Fig. 5.21b. This lead to that the RoCoF of the system with only AGC was nearly equal to that of the system with original configuration. Owning to the negative frequency, the AGC pitch angle of the DFIG was continuously decreased such that the mechanical power was increased. Then the active power output was improved to support the recovery of frequency in external power grid as presented in Fig. 5.21b. Due to the jump of active power at t = 0.5s, the lowest frequency point in the system with only the SAC was elevated during the first swing of the system as shown in Fig. 5.21a. The increased active power was released by the rotor of the DFIG as presented in Fig. 5.21c. Attributed to the active power feed-forward loop in the pitch angle controller shown in Fig. 5.3, the pitch angle increased as depicted in Fig. 5.21f. As a result, the active power of the DFIG reduced, and a secondary frequency drop in the external power grid was observed, as shown in Fig. 5.21a, b, respectively. Then the system showed the lowest frequency nadir as presented in Table 5.4. For the system with VIC installed, the RoCoF and frequency nadir illustrated in Table 5.4 were satisfactory. Nevertheless, the VIC did not take part in the primary and secondary frequency control of the system, and the settling frequency of the was lower than the cases with other configurations.

5.3 Simulation Studies

159 (d)

0.4

Original

AGC

AGC+SAC

VIC

SAC

θ (rad)

0.35 0.3 0.25 0.2 0

2

4

6

8

10

12

14

16

18

(e)

u(t) (rad)

0.2 0.15 0.1 0.05 AGC+SAC 0 0

2

4

6

8

10

12

14

SAC 16

18

(f)

β (degree)

14 12 10 8 0

Original

2

4

AGC

6

AGC+SAC

8 10 Time (s)

VIC

12

SAC

14

16

18

Fig. 5.22 The dynamics of the DFIG of wind farm 12 obtained in the case where a load increase occurred in the modified Kundur four-machine two-area power system (d The virtual rotor angle of the DFIG e The output of the SAC installed on the DFIG f The pitch angle of the DFIG) [12]

5.3.2.3

Modal Analysis of the Modified Kundur Four-Machine Two-Area Power System

A thyristor excitor and a PSS were modeled in the synchronous generator of the multimachine power system, and structures of the exciter and PSS were as presented in Figs. 5.23 and 5.24. The exciter was configured by: K A = 400, TR = 0.01, TA = 1.2, TB = 5.0. The PSS was configured by: K PSS = 25.0, Ts = 7.5, T1 = 0.055, T2 = 0.02, T3 = 3.0, T4 = 5.5.

160

5 Switching Control of Doubly-Fed …

Table 5.5 Oscillation modes of the modified Kundur four-machine two-area power system [12] Configuration Original

AGC

SAC

η=1 η=6

AGC

η=1

+SAC

η=6

Eigenvalue/(Frequency in Hz, damping ratio) Inter-area mode Area 1 local mode Area 2 local mode −0.173± j3.967 −1.240± j5.438 −0.682± j5.848 ( f = 0.6313, ( f = 0.8655, ( f = 0.9307, ξ = 0.0437) ξ = 0.2223) ξ = 0.1159) −0.175± j3.972 −1.242± j5.441 −0.685± j5.851 ( f = 0.6321, ( f = 0.8660, ( f = 0.9312, ξ = 0.0440) ξ = 0.2225) ξ = 0.1163) −0.177± j3.968 −1.244± j5.440 −0.686± j5.849 ( f = 0.6315, ( f = 0.8658, ( f = 0.9310, ξ = 0.0447) ξ = 0.2229) ξ = 0.1165) −0.197± j3.972 −1.263± j5.449 −0.705± j5.858 ( f = 0.6322, ( f = 0.8672, ( f = 0.9323, ξ = 0.0496) ξ = 0.2258) ξ = 0.1194) −0.178± j3.972 −1.245± j5.443 −0.687± j5.852 ( f = 0.6322, ( f = 0.8663, ( f = 0.9314, ξ = 0.0448) ξ = 0.2230) ξ = 0.1166) −0.198± j3.977 −1.264± j5.452 −0.706± j5.861 ( f = 0.6329, ( f = 0.8677, ( f = 0.9327, ξ = 0.0497) ξ = 0.2259) ξ = 0.1196)

Fig. 5.23 The layout of the exciter of synchronous generators

Fig. 5.24 The layout of the PSS of synchronous generators

The oscillation modes were as presented in Table 5.5. We can see that the AGC elevated the rotor oscillation frequency, and the SAC improved the damping ratio of inter-area and local oscillation modes. The SAC also showed more damping in the case of η = 6 than in the case of η = 1. Therefore, the damping performance of the SAC was more obvious during the initial post-disturbance stage.

5.3 Simulation Studies

161

Table 5.6 System frequency performance indexes obtained in the case where generator trip occurred [12] Configuration RoCoF (Hz/s) Frequency nadir (Hz) Settling frequency (Hz) Original AGC SAC AGC+SAC VIC

5.3.2.4

0.5172 0.5163 0.2132 0.2033 0.4804

59.5191 59.5362 59.5459 59.6283 59.6095

59.7371 59.8066 59.7330 59.8089 59.7429

Results in the Case Where Generator Trip Occurred in a System with Wind Power Penetration Under Constant Wind Speed Conditions

The frequency control performance of the SAC and AGC was investigated in a case where generator trip occurred in the modified Kundur four-machine two-area system as presented in Fig. 5.19. Configuration of the system was the same as that studied in 5.3.2.2. The wind speed applied on the wind farms was set as Vwind = 15 m/s. Due to the fault on protection devices, generator 1 and its step-up transformer were tripped at t = 0.5s. Figure 5.25a presents the dynamics of system frequency, and Table 5.6 shows the three frequency indexes. According to Table 5.6, the system with both SAC and AGC showed the lowest RoCoF, as presented by the active power decrease shown in Fig. 5.25b. Along with the error of system frequency was larger than 0.05 Hz, the SAC was switched on and generated a positive step control signal as shown in Fig. 5.26e. The virtual rotor angle and the active power of the DFIG were elevated to compensate for the power shortage as presented in Figs. 5.26d and 5.25b, respectively. Due to the AGC, the pitch angle of the DFIG showed a continuous decrease after a short increase since generate 1 was tripped as presented in Fig. 5.26f. In this way, the system having both SAC and AGC possessed the highest frequency nadir. The short increase of the pitch angle was introduced by the active power feed-forward loop in the pitch angle controller given in Fig. 5.3. To prevent the rotor from over-speed, the pitch angle was increased after t = 4.2s to a different value. The active power and rotor speed of the DFIG then settled as shown in Figs. 5.25b and 5.26d, respectively. Moreover, Fig. 5.25b shows that the DFIGs with both SAC and AGC or only the AGC were able to offer more active power support to the external grid. These two systems had higher settling frequency as shown in Table 5.6. According to Fig. 5.25b, the systems with SAC provided more active power support to the external grid. This resulted in a lower RoCoF as presented in Table 5.6. The DFIG with only SAC presented more severe secondary active power drop, which lead to that the system with VIC had higher frequency nadir than that with only SAC as presented in Table 5.6. Nevertheless, both SAC and VIC could release the kinetic

162

5 Switching Control of Doubly-Fed … 60

(a) Original

AGC

2

4

AGC+SAC

VIC

SAC

f

load

(Hz)

59.9 59.8 59.7 59.6 0

6

10

8

(b) 1

Original

AGC

2

4

AGC+SAC

VIC

SAC

Pe (p.u.)

0.9 0.8 0.7 0.6 0.5 0

6

8

AGC+SAC

VIC

6

8

10

(c)

1.3 Original

AGC

SAC

1.2

r

ω (p.u.)

1.25

1.15 1.1 0

2

4

10

Fig. 5.25 The dynamics of the DFIG of wind farm 12 obtained in the case where generator trip occurred in the modified Kundur four-machine two-area power system (a System frequency measured by phase-locked loop at load bus 7. b The active power output of the DFIG. c The rotor speed of the DFIG) [12]

energy stored in the rotor of the DFIG and offered inertial response to the external power network. Generator 3 was set as the reference bus of the system, and the relative rotor angles of generator 2 and generator 4 were as presented in Fig. 5.27a, b, respectively. We can see that the systems with SAC installed presented stronger damping to the interarea and local rotor angle oscillations. Moreover, the systems with AGC implemented showed higher oscillation frequency and stronger damping to rotor angle oscillations. The results given in Fig. 5.27a, b showed that the SAC and AGC improved the first swing and transient stability of WPPSs.

5.4 Summary

163 (d)

0.5 0.45

Original

AGC

AGC+SAC

VIC

6

8

SAC

θ (rad)

0.4 0.35 0.3 0.25 0.2 0

2

4

10

(e)

u(t) (rad)

0.3

0.2

0.1 AGC+SAC 0 0

2

4

6

SAC

8

10

(f) Original

AGC

AGC+SAC

VIC

SAC

β (degree)

20 15 10 5 0 0

2

4

Time (s)

6

8

10

Fig. 5.26 The dynamics of the DFIG of wind farm 12 obtained in the case where generator trip occurred in the modified Kundur four-machine two-area power system (d The virtual rotor angle of the DFIG. e The output of the SAC installed on the DFIG. f The pitch angle of the DFIG) [12]

5.4 Summary In the first place, this chapter has proposed a four-loop SC for the resilience enhancement of WPPSs. The four-loop SC presents stronger robustness to the external disturbances in comparison to the CC. With the bang-bang control commands and the proper switching of the LBCFC, the four-loop SC employs the largest damping power of the converters. The four-loop SC presents stronger robustness than the CC. The four-loop SC has improved the low-voltage-ride-through capability of the DFIG-based wind turbine.

164

5 Switching Control of Doubly-Fed … (a) 0.4

Original

AGC

AGC+SAC

VIC

SAC

δ −δ (rad) 3

0.2

2

0 −0.2 −0.4 0 −0.26

2 Original

4 AGC

6 (b) AGC+SAC

8 VIC

10 SAC

δ4−δ3 (rad)

−0.28 −0.3

−0.32 −0.34 0

2

4

Time (s)

6

8

10

Fig. 5.27 The dynamics of the relative rotor angles of synchronous generators obtained in the case where a generator trip occurred in the modified Kundur four-machine two-area power system (a Relative rotor angle between generator 2 and generator 3. b Relative rotor angle between generator 4 and generator 3) [12]

Referring to ηωr , η Q s , η Q g and ηVdc , the four-loop SC achieves higher control efficiency than the CC with less control effort. The four-loop SC achieves better performance in damping the output oscillation. The four-loop SC is capable of maintaining the WPPS stable without tripping WF18 in the case that a three-phase-to-ground fault occurred. Meanwhile, less inter- and inner-area oscillations can be found in the WPSS with the four-loop SC configured. Moreover, the four-loop SC effectively improved the short-term resilience of large-scale WPPSs. On one hand, the frequency of the WPPSs with the four-loop SC installed shows less oscillation in the process of the first swing. On the other hand, less steady-state error is observed in the voltage of load buses in the WPPSs with the four-loop SC configured. The DFIGs controlled by the four-loop SC can coordinate with each other, and the four-loop SC has improved the transient stability of the large-scale WPPSs. In the second place, this chapter has proposed a SAC and an AGC for the DFIG to regulate the frequency of DFIG-based WPPSs. The SAC is a bang-bang funnel controller and it is robust to parameters, oscillations and measurement noise. The small-signal analysis indicates that the SAC and the AGC can strengthen the responses of the active power and rotor speed of DFIGs to system frequency error. The SAC utilizes the rotating masses of DFIGs to release or restore kinetic energy to offer frequency support to the external network. Modal analysis reveals that the SAC

References

165

and the AGC are able to offer additional damping to the rotor oscillations. The AGC improves mode oscillation frequency, and the SAC mainly increases the damping ratio with respect to the oscillation modes of the system. Due to the SAC, the virtual rotor angle of the DFIG offers fast active power support to the external network when the frequency error is larger than 0.05 Hz. The SAC with larger δ provides larger step change of the active power. For the system with the SAC and the AGC, the active power reference of the pitch angle controller is controlled by the two controllers. The SAC provides small rapid variation of the pitch angle, and the AGC provides slow but sustained control. The systems with the SAC and the AGC installed have presented the highest frequency nadir and settling frequency, and the lowest RoCoF. According to the RoCoF, both the SAC and VIC can enhance system inertia. However, the VIC cannot offer sustained active power support to the external network. The settling frequency is always lower in the system with the VIC than that with the AGC installed.

References 1. Miller NW, Price WW, Sanchez-Gasca JJ (2003) Dynamic modeling of GE 1.5 and 3.6 wind turbine-generators. GE-Power Syst Energy Consult (2003) 2. Qiao W, Zhou W, Aller J, Harley R (2008) Wind speed estimation based sensorless output maximization control for a wind turbine driving a DFIG. IEEE Trans Power Electron 23(3):1156– 1169. https://doi.org/10.1109/TPEL.2008.921185 3. Pena R, Clare J, Asher G (1996) Doubly fed induction generator using back-to-back PWM converters and its application to variable-speed wind-energy generation. IEE Proc Electric Power Appl 143(3):231–241 4. Liu Y, Wu Q, Zhou X, Jiang L (2014) Perturbation observer based multiloop control for the DFIG-WT in multimachine power system. IEEE Trans Power Syst 29(6):2905–2915 5. Yang L, Xu Z, Ostergaard J, Dong ZY, Wong KP (2012) Advanced control strategy of DFIG wind turbines for power system fault ride through. IEEE Trans Power Syst 27(2):713–722. https://doi.org/10.1109/TPWRS.2011.2174387 6. González L, Figueres E, Garcerá G, Carranza O (2010) Maximum-power-point tracking with reduced mechanical stress applied to wind-energy-conversion-systems. Appl Energy 87(7):2304–2312 7. Li S, Haskew T, Williams K, Swatloski R (2012) Control of DFIG wind turbine with directcurrent vector control configuration. IEEE Trans Sustainable Energy 3(1):1–11. https://doi. org/10.1109/TSTE.2011.2167001 8. Liu Y, Wu QH, Zhou XX (2016) Co-ordinated multiloop switching control of DFIG for resilience enhancement of wind power penetrated power systems. IEEE Trans Sustainable Energy PP(99):1–11. https://doi.org/10.1109/TSTE.2016.2524683 9. Panteli M, Mancarella P (2015) The grid: stronger, bigger, smarter?: presenting a conceptual framework of power system resilience. IEEE Power Energy Mag 13(3):58–66 10. Panteli M, Mancarella P (2015) Modeling and evaluating the resilience of critical electrical power infrastructure to extreme weather events. IEEE Syst J (in press, 2015) 11. Li S, Challoo R, Nemmers MJ (2009) Comparative study of DFIG power control using statorvoltage and stator-flux oriented frames. In: Power and energy society general meeting, PES’09. IEEE. IEEE, pp 1–8 12. Liu Y, Jiang L, Wu QH, Zhou X (2017) Frequency control of DFIG-based wind power penetrated power systems using switching angle controller and AGC. IEEE Trans Power Syst 32(2):1553–1567. https://doi.org/10.1109/TPWRS.2016.2587938

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5 Switching Control of Doubly-Fed …

13. Liberzon D, Trenn S (2013) The bang-bang funnel controller for uncertain nonlinear systems with arbitrary relative degree. IEEE Trans Autom Cont 58(12):3126–3141. https://doi.org/10. 1109/TAC.2013.2277631 14. Liu Y, Wu QH, Zhou XX (2016) Coordinated switching controllers for transient stability of multi-machine power systems. IEEE Trans Power Syst 31(5):3937–3949 15. Liu Y, Wu QH, Kang H, Zhou X (2016) Switching power system stabilizer and its coordination for enhancement of multi-machine power system stability. CSEE J Power Energy Syst 2(2):98– 106 16. Han Y, Luo M, Zhao X, Guerrero JM, Xu L (2016) Comparative performance evaluation of orthogonal-signal-generators-based single-phase pll algorithmsł survey. IEEE Trans Power Electron 31(5):3932–3944 17. Wang S, Hu J, Yuan X, Sun L (2015) On inertial dynamics of virtual-synchronous-controlled DFIG-based wind turbines. IEEE Trans Energy Convers 30(4):1691–1702 18. Pal B, Chaudhuri B (2006) Robust control in power systems. Springer Science and Business Media (2006) 19. Kundur P, Balu NJ, Lauby MG (1994) Power system stability and control. McGraw-hill New York (1994) 20. Liu Y, Wu QH, Zhou XX, Jiang L (2014) Perturbation observer based multiloop control for the DFIG-WT in multimachine power system. IEEE Trans Power Syst 29(6):2905–2915 21. Morren J, De Haan SW, Kling WL, Ferreira J (2006) Wind turbines emulating inertia and supporting primary frequency control. IEEE Trans Power Syst 21(1):433–434

Chapter 6

Adaptive Switching Control of Power Electronic Converters

6.1 Switching Fault Ride-Through of GSCs via Observer-Based Bang-Bang Funnel Control 6.1.1 Design of SFRTC 6.1.1.1

Dynamics of GSC and Capacitor Voltage

The GSC and capacitor voltage of a PMSGWT can be described by [1] ⎧˙ i = L1g (−Rg i dg + ωs L g i qg + vs − vdg ) ⎪ ⎪ dg ⎨ i˙qg = L1g (−Rg i qg − ωs L g i dg − vqg )   ⎪ ⎪ ⎩ V˙ = 1 3m √1 i dg − Ps dc C 4 2 Vdc

(6.1)

where the grid-voltage oriented reference frame is employed in the model of GSC, i.e., vds = vs and vqs = 0, i dg and i qg represent the d- and q-axis currents flowing through GSC, vdg and vqg represent the d- and q-axis voltages generated by GSC, Rg and L g denote the resistance and inductance through which the GSC is connected to the step-up transformer, ωs is the frequency of grid voltage, Vdc and C represent the voltage and capacity of the dc-link capacitor, Ps denotes the active power generated by the PMSG, and m 1 denotes the modulation depth of the GSC. Layout of the PMSGWT are shown in Fig. 6.1. ∗ , where Vdc∗ is the Output variables are e1 (t) = Vdc − Vdc∗ and e2 (t) = i qg − i qg ∗ reference of Vdc and i qg denotes the reference of i qg . According to (6.1), e1 (t) can be described with d 2 e1 (t) = f (x) + g(x)vdg (6.2) dt 2

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Y. Liu and Q.-H. Wu, Adaptive Switching Control of Large-Scale Complex Power Systems, Power Systems, https://doi.org/10.1007/978-981-99-1039-7_6

167

168

6 Adaptive Switching Control of Power Electronic Converters

Ps

RSC

PMSG

PI2

* iqg

Vs

Vdc

vqg2

(3) and (6) vqg1 First-order BBFC s idg

PI4

PI3 e1 (t ) High-gain observer (5)

Vdc

Li

s g qg

Control law (3) and (4)

vdg2

Switching scheme

Vdc* Vdc*

iqg

vs

* * vdg vqg

s g dg

e2 (t ) Control law Vs*

idg iqg

Li

iqg

Filters

* iqg

Rg Lg

C

Switching scheme

PI1

GSC

Vdc

vdg1

Second-order OBFC

SFRTC

Fig. 6.1 The schematic of a PMSGWT having SFRTC implemented [2]

3m 1 where f (x) = −Vdc∗(2) + C1 [ 4√ (−Rg i dg + ωs L g i qg + vs ) − 2L g

P˙s Vdc0

+

Ps0 ˙ 2 Vdc ], g(x) Vdc0

1 = − 4√3m , Vdc0 and Ps0 are the steady-state values of capacitor voltage and the 2C L g active power of the PMSG, respectively. Therefore, the relative degree of e1 (t) with respect to the vdg is r1 = 2. Referring to the second equation of (6.1), the relative degree of e2 (t) with respect to vqg is r2 = 1.

6.1.1.2

Design of OBFC and BBFC for Vdc and i qg

According to the above, a second-order BBFC is employed in the OBFC for controlling e1 (t) and a first-order BBFC is utilized for controlling e2 (t). A second-order BBFC can be written as [3]: ⎧ q1 (t) = G (e(t), ˆ ϕ0 − ε0 , −ϕ0 + ε0 , q1 (t−)) ⎪ ⎪  ⎪ ⎪ G (e(t), ˙ˆ −λ1 − ε1 , −ϕ1 + ε1 , q(t−)) (q1 (t) = 1) ⎨ q(t) = G (e(t), ˙ˆ ϕ1 − ε1 , λ1 + ε1 , q(t−)) (q1 (t) = 0) ⎪ ⎪ ⎪ (0−) ∈ {1, 0}, q(0−) ∈ {1, 0} q ⎪ ⎩ 1 G (x(t), x, x, q(t−)) := [x(t) ≥ x ∨ (x(t) > x ∧ q(t−))]

(6.3)

A second-order high-gain observer is utilized to constitute an OBFC with the BBFC. d eˆ1 (t) α1 d eˆ˙1 (t) α2 (6.4) = eˆ˙1 (t) + e˜1 (t), = 2 e˜1 (t) dt  dt 

6.1 Switching Fault Ride-Through of GSCs via …

169

where e˜1 (t) = e1 (t) − eˆ1 (t) is the estimation error of e1 (t). The OBFC for controlling e1 (t) is given by  vdg1 =

Ud+ + vdg0 , if q(t) = 1 Ud− + vdg0 , if q(t) = 0

(6.5)

where vdg0 is the steady-state value of vdg , Ud+ and Ud− are the upper and lower bounds of the output of the OBFC. For the control of i qg , the switching logic of the first-order BBFC is described with the first equation in (6.3), and the control law is written as  vqg1 =

Uq+ + vqg0 , if q1 (t) = 1 Uq− + vqg0 , if q1 (t) = 0

(6.6)

where vqg0 is the steady-state value of vqg .

6.1.1.3

Switching Scheme of SFRTC

The OBFC and BBFC are designed to switch with the d-axis and q-axis control loops of VC respectively according to a state-based switching scheme: vg∗ = γi vg1 + (1 − γi )vg2

(6.7)

∗ ∗ , vg1 = vdg1 for the d-axis control loop, vg∗ = vqg , vg1 = vqg1 for the where vg∗ = vdg q-axis control loop, vg2 denotes the d-axis or q-axis control voltage generated by VC, i ∈ {1, 2}, and

 γi =

1, if |ei (t)| is larger than e¯ i 0, if |ei (t)| keeps being smaller than ei for over τi s

where e¯ i > ei . Due to the restriction of modulation in converters, the OBFC in d-axis loop and the BBFC in q-axis loop cannot be triggered at the same time. Hence, the modulation

∗2 ∗2 /V ≤ 1. Large + vqg signals given by the controllers of GSC should meet vdg dc ∗ ∗ vdg and vqg will weaken the control performance of each other. Therefore, only one of the bang-bang controllers in the parallel loops can be activated. The layout of a PMSGWT with SFRTC is as presented in Fig. 6.1.

6.1.1.4

Feasibility of OBFC

For a nonlinear system controlled by OBFC, the convergence of the high-gain observer has been validated in Chap. 1 of this book. Due to the estimation errors

170

6 Adaptive Switching Control of Power Electronic Converters

ˆ˙ ˆ˙ of [e(t) ˆ e(t)], (6.3) of OBFC driven by [e(t) ˆ e(t)] will not switch when [e(t) e(t)] ˙ hit the corresponding switching triggers. The switching of [q(t) q1 (t)] will present some delay or advance compared with the observer-independent BBFC in [3]. The advancement of the switching of [q(t) q1 (t)] does not violate the feasibility of OBFC. Therefore, we just need to consider the feasibility of the OBFC in delayed switching, i.e., e(t) ˜ > 0 when e(t) > 0 and e(t) ˜ < 0 when e(t) < 0. Under the assumption of delayed switching, the feasibility of the OBFC can be maintained by enlarging the safety distance εi (i = 0, 1) by δspo , which is the largest estimation error of the ˆ˙ ˙ − e(t)| ≤ δspo . Therefore, the feasiobserver (6.4), i.e. |e(t) − e(t)| ˆ ≤ δspo and |e(t) bility of OBFC can be proved in the same way with that in [3]. Simulations results of a fourth-order nonlinear system regulated by a fourth-order OBFC are presented to validate the feasibility of OBFC. The nonlinear system is given by ⎧ ⎨ x˙1 = x2 , x˙2 = x3 , x˙3 = x4 , x˙4 = zx32 + ez u z˙ = z(0.09 − z)(z + 0.05) − 0.008x4 (6.8) ⎩ y = x1 , yref = 5sint, z(0) = 0 The fourth-order high-gain observer of OBFC is written as 

wˆ˙ 1 = wˆ 2 + w˙ˆ 3 = wˆ 4 +

10 100 (w1 − wˆ 1 ), w˙ˆ 2 = wˆ 3 + 0.05 2 (w1 − 0.05 300 1000 ˙ (w1 − wˆ 1 ), wˆ 4 = 0.054 (w1 − wˆ 1 ) 0.053

wˆ 1 )

(6.9)

where w1 = e(t) = y − yref . The fourth-order BBFC of the OBFC is configured by: ϕ0 = 1, ε0 = 0.9, ϕ1 = 0.5, ε1 = 0.1, λ1 = 0, ϕ2 = 0.5, ε2 = 0.1, λ2 = 0.2, ϕ3 = 4.5, ε3 = 0.1, and λ3 = 4. The performance of the OBFC is as shown in Fig. 6.2, where the output variable, control signal generated by the OBFC, and output tracking error and its high-order derivatives are presented. The estimation errors of the observer, i.e., e(t) = e(t) − 2 2 3 3 = de − wˆ 2 ,  ddt 2e = ddt 2e − wˆ 3 , and  ddt 3e = ddt 3e − wˆ 4 , are shown in Fig. 6.2. wˆ 1 ,  de dt dt We can see that the high-gain observer provides accurate estimates of system states. The OBFC enables the system output tracking error and its high-order derivatives to be regulated within the pre-defined error funnels.

6.2 Bang-Bang Funnel Control of Three-phase Full-Bridge Inverter Under Dual-Buck Scheme 6.2.1 Introduction to Dual-Buck Scheme Figure 6.3 presents the topology of the three-phase full-bridge inverter, and the variables are defined as follows:

6.2 Bang-Bang Funnel Control of Three-phase …

171 500

10

-500 0

5

10

15

20

0.5

de/dt

e=y-y

ref

-10 0.5

0

u

y

0

0 -0.5 5

10

15

0

0

5

10

15

20

0 -3 ×10

5

10

15

20

0

5

10

15

20

0

5

10

15

20

20 0

0 -6 ×10

5

10

15

20

Δde/dt

Δe(t)

20

-20

0 -5 0

5

10

15

20

0.2

Δd3e/dt3

2

15

20

-1

2

10

0

d3e/dt3

d2e/dt2

0

Δd e/dt

5

-0.5

1

5

0

0 -0.2 0

5

10

15

20

1 0 -1 10 0

-10

Time (s)

Time (s)

Fig. 6.2 Simulation results of a nonlinear system controlled by OBFC [2]

P D1 S1

D3 S3

A

D5 S5

B

U dc

C

L1 L2 L3

ua

ub

ia ib uc ic

O Load

D2

D4 S4

S2

D6

C1

S6

N

Fig. 6.3 Topology of the three-phase full-bridge inverter [4]

Udc ua , ub, uc ia , ib , ic S1 − S6 D1 − D6 L 1, L 2, L 3 C1 , C2 , C3 P, N O

DC link voltage; three-phase output voltages; three-phase output currents; active switches; anti-parallel diodes; inductors of the LC filter; capacitors of the LC filter; positive and negative points of DC link; neutral point of AC side.

C2

C3

172

6 Adaptive Switching Control of Power Electronic Converters

Fig. 6.4 Waveforms of the three-phase output voltages and the inductor currents in one AC line cycle [4]

1

ua

ub

iLa

iLb

uc

iLc

Voltage (p.u.)

0.5

0

-0.5

-1 0

Region I

Region II Region III Region IV Region V Region VI 0.005

0.01

0.015

0.02

Time (s)

For a symmetrical three-phase system, it has L 1 = L 2 = L 3 = L and C1 = C2 = C3 = C. The sum of the output current and the filter capacitor current is defined as inductor current i L , and the reference directions of voltages and currents have been labeled in the figure. The three-phase inverter is normally modelled in natural frame by ⎧ ⎨ L i˙L j = Udc s j + u ON − u C j , C u˙ = i L j − i j , ⎩ Cj u ON = − U3dc (sa + sb + sc ),

(6.10)

where u ON denotes the voltage between O and N , s j ∈ {0,1} describes the state of the switches on phase leg J , i L j , i j and u C j represent the inductor current, output current and capacitor voltage of phase J respectively (J = A,B,C, j = a,b,c). u C j is equivalent with the output voltage u j . Being an extension of conventional vector operation [5], the operating region of the active switches in dual-buck scheme is determined by the filter inductor currents. The normalized waveforms are as shown in Fig. 6.4, where an AC cycle is equally divided into six regions based on the zero-crossing points of the inductor current. Region I is taken as an example to illustrate the operating principle: In region I, i La and i Lc are positive, while i Lb is negative. Accordingly, S4 should be ON (S3 is OFF), and S1 , S5 are modulated at high frequency (S2 , S6 are OFF). In other words, the state of switches in phase B is fully definite and constant during this interval. Meanwhile, phase A and C can be viewed as two buck circuits in parallel. The equivalent circuit of this region is as depicted in Fig. 6.5. The above equivalence method applies to the other five regions, with which yields two different patterns of generalized dual-buck topology, which are as shown in Fig. 6.6. The cross reference of symbols between the three-phase full-bridge topology and the parallel-connected dual-buck topology as well as the output mechanism are listed in Table 6.1. With respect to both the equivalent dual-buck circuits, four switching states are used for two active switches Tk (k = m,n). Fig. 6.7 shows all switching states in region I.

6.2 Bang-Bang Funnel Control of Three-phase …

173

P D1

D5

S1

S5

A U dc

C

ua

L1 L2 L3

ub

O

uc

Load D2

D6

C1

C2

C3

S6

S2

N Buck unit-m

S1

Buck unit-n

Leq

U dc

Req

S5

Leq

U dc

Ceq

Req Ceq

D6

D2

Fig. 6.5 Equivalent circuit for region I [4] Fig. 6.6 Equivalent dual-buck topology [4]

iLeqm

Tm

iLeqn

Tn

uCeqm uCeqn

U in

(a) For region I, III, and V iLeqm

Tm

iLeqn

Tn U in

uCeqm uCeqn

(b) For region II, IV, and VI

174

6 Adaptive Switching Control of Power Electronic Converters iLa

iLa S1

S1

iLc

S5

ucb uab

U dc

iLc

S5

ucb uab

U dc

(a) State1: S1 ON, S5 ON

(b) State2: S1 ON, S5 OFF iLa

iLa S1

S1

iLc

S5

ucb uab

U dc

iLc

S5

ucb uab

U dc

(c) State3: S1 OFF, S5 ON

(d) State4: S1 OFF, S5 OFF

Fig. 6.7 Switching states of the equivalent circuit for region I [4] Table 6.1 Cross reference of the symbols and output mechanism [4] Region

Input voltage

|Equivalent voltage

|Equivalent current

Output mechanism

Uin

u Ceqm

u Ceqn

i Leqm

i Leqn

S1

S2

S5

S6

I

Udc

u ab

u cb

i La

i Lc

Tm

OFF OFF ON

Tn

OFF

II

−Udc

u ba

u ca

i Lb

i Lc

ON

OFF OFF Tm

OFF Tn

III

Udc

u bc

u ac

i Lb

i La

Tn

OFF Tm

S3

S4

OFF OFF ON

IV

−Udc

u cb

u ab

i Lc

i La

OFF Tn

ON

OFF OFF Tm

V

Udc

u ca

u ba

i Lc

i Lb

OFF ON

Tn

OFF Tm

VI

−Udc

u ac

u bc

i La

i Lb

OFF Tm

OFF Tn

ON

OFF OFF

Since only two switches are operating at high frequency on different phase legs, the chance of shoot-through is quite small with the rest either kept ON or OFF. Hence, we do not need the dead-time in the switching signals. In this way, waveform distortion caused by dead-time is avoided, and the switching loss is much lower [6], compared with inverters modulated using PWM. The three-phase full-bridge inverter is decoupled into two parallel buck units in each 60◦ region in dual-buck scheme. The three-phase output voltages follow the reference signals by regulating the equivalent dual-buck circuit. Hence, the inverter can be analyzed with the dynamic model of the buck chopper. Dynamics of the equivalent inductor current i Leqk and capacitor voltage u Ceqk in the corresponding buck unit-k (k = m, n) can be written as:



 Uin 0 − L1eq i Leqk i˙Leqk L eq = 1 + u k (t), 1 u˙ Ceqk − u 0 Ceqk Ceq Req Ceq

(6.11)

6.2 Bang-Bang Funnel Control of Three-phase …

175

where Uin represents the input voltage of the buck circuit, L eq and Ceq are the equivalent filter inductance and capacitance, Req is the equivalent load resistance. u k (t) ∈ {0, 1} denotes the switching signal of Tk , which is the pulse signal that triggers Tk (k = m, n).

6.2.2 Bang-Bang Funnel Control of the Inverter In the first place, (6.11) should be written in the single-input, single-output (SISO) form [7]. u k (t) is the control input, and u Ceqk is chosen as the output yk (t). State variables are defined as xk = [xk1 ,xk2 ] = [i Leqk ,u Ceqk ] . The buck unit-k (k = m,n) can be described with x˙k (t) = f (xk )+g(xk )u k (t) = Axk (t)+bu k (t), xk (0) = xk0 yk (t) = h(xk ) = cxk (t),

(6.12)

where A=

0

1 Ceq

Uin − L1eq   , b = L eq , c = 0 1 . − Req1Ceq 0

The relative degree of (6.12) is calculated as follows. Lg h(xk ) = cb = 0, L f h(xk ) = c Axk , Lg L f h(xk ) = c Ab =

Uin L eq Ceq

= 0,

(6.13)

where L f and Lg are Lie derivatives. The relative degree of yk (t) with respect to u k (t) is r = 2, and the “high frequency” gain is g = Uin /L eq Ceq (g = Udc /L eq Ceq > 0 in region I, III and V, while g = −Udc /L eq Ceq < 0 in region II, IV and VI), which means buck unit-k (k = m,n) has a relative degree of two with sign-definite gain in each 60◦ region. The positive g ensures that the output of BBFC acts in the same “direction” with its input [8]. As for the three-phase full-bridge inverter under dual-buck scheme, g is positive and the output of BBFC u k (t) can be directly used as the pulse signal S1 , S3 or S5 . However, when g is negative, the two output values of the BBFC should be interchanged. Through a diffeomorphism xk → (yk , y˙k ) , the system of buck unit-k (k = m,n) is transformed into Byrnes-Isidori normal form, i.e.,  y¨k = − L eq1Ceq yk

−

1 y˙ Req Ceq k

+

Uin u , L eq Ceq k

yk (0) y˙k (0)

 = yk0

(6.14)

the above system has stable zero dynamics (equivalently called minimum phase, cf. [7]) since

176

6 Adaptive Switching Control of Power Electronic Converters Region Selection

iL abc g ( S1 ) g (S2 ) Inverter g ( S3 ) x(t ) Ax(t ) bu (t ) g (S4 ) y (t ) cx(t ) g ( S5 ) g ( S6 )

Output Mechanism

S2 , S4 , S6

uC abc (uabc )

S1 , S3 , S5

NOT

A

B

u (t )

C

uC abc

q (t ) Switching Logic

U

e(t )

U

uCref abc (uref abc )

Control Law

Second-order BBFC Fig. 6.8 Control diagram of the proposed method [4]

 Uin s In − A b =− det = 0 c 0 L eq Ceq

is independent of s ∈ C. Based on dual-buck scheme, the three-phase inverter can be completely decoupled into two independent SISO systems with relative degree r = 2 and sign-definite gain in each 60◦ region. We found that the buck models of switches on the same phase leg have identical form, but the sign of Uin is different. As a result, these two switches can be controlled by one BBFC. Therefore, three second-order BBFCs are employed for the control of a three-phase full-bridge inverter. The schematic of control is shown in Fig. 6.8, and the law of output is shown in Table 6.1. The control law of the BBFC is  − U , if q(t) = true , (6.15) u(t) = U + , if q(t) = false , The switching logic of the BBFC is written as

6.3 Experiment and Simulation Results

177

  q1 (t) = S e(t), ϕ0+ (t) − ε0+ , ϕ0− (t) + ε0− , q1 (t−) , q1 (0−) = q10 ∈ { true, false },  St (t), if q1 (t) = true , q(t) = S f (t), if q1 (t) = false ,

(6.16)

q(0−) = q 0 ∈ { true, false }, where S : R × R × R × { true, false } → { true, false } is a switching predicate given by   (6.17) S(e, e, ¯ e, qold ) := e ≥ e¯ ∨ (e > e ∧ qold ) , and

 + −    − ˙ min ϕ˙0+ (t),−λ− St (t) = S e(t), 1 −ε1 ,ϕ1 (t)+ε1 ,q(t−)  −   −  + + S f (t) = S e(t),ϕ ˙ ˙0 (t),λ+ 1 (t)−ε1 , max ϕ 1 +ε1 ,q(t−) .

(6.18)

In the symmetrical three-phase circuit, these controllers are configured with the same set of parameters, i.e., ϕ0+ = −ϕ0− ≡ 0.1, ε0± ≡ 0.0999, ϕ1+ = −ϕ1− ≡ 177, ε1± ≡ 88,

λ± 1 ≡ 0.99,

and U + = 1, U − = 0. U ± are the pulse signals for switches (1 means ON, and 0 reflects OFF). The references yref for outputs are written as ⎧ ∗ ⎨ u refa = Um sin(ωt), u = Um∗ sin(ωt − ⎩ refb u refc = Um∗ sin(ωt +

2π ), 3 2π ). 3

(6.19)

where Um∗ is the expected amplitude of the phase output voltages, and ω indicates its angular frequency.

6.3 Experiment and Simulation Results 6.3.1 Test Results of the Switching Fault Ride-Through Controller for GSCs Setup of the experiment was as shown in Fig. 6.3, where a PMSGWT connected to an external power grid and the controller for RSC were simulated in RTDS and the control system for the GSC was implemented in dSPACE. The GSC was configured by: Sn = 2.5 MVA (rated MVA), ωs = 314.16 rad/s, Rg = 0.001p.u., L g =0.15p.u., C = 5F, Vdc∗ = 1.4p.u., Vdc_norm = 1000 V . The VC was configured by: P1 =10, I1 =1000,

178

6 Adaptive Switching Control of Power Electronic Converters

GSC controller Grid PMSGWT RSC controller

Fig. 6.9 The configuration of the experiment platform [2]

P2 = 0.4, I2 =8, P3 = 0.5, I3 =50, P4 =0.4, I4 =8. The parameters of the OBFC in were selected as: α1 = 30, α2 = 300,  = 0.1, ϕ0 = 0.6, ϕ1 = 10 , ε0 =0.58, ε1 = 0.5, λ1 = 8.5, Ud+ = −Ud− = 0.5p.u., e¯ 1 = 0.1p.u., e1 =0.06p.u., and τ1 =0.01 s. The parameters of BBFC in the controller of i qg were chosen as: ϕ0 = 0.6, ε0 = 0.5, Uq+ = −Uq− = 1p.u., e¯ 2 = 0.5p.u., e2 =0.3p.u., and τ2 =0.01 s (Figs. 6.9 and 6.10). The SFRTC was compared with VC and dc-chopper method [9] in testing the controller for the dc-link voltage. The brake resistance of the dc-chopper was chosen as 1.8 . The dc-chopper was triggered when the error of dc-link voltage was larger than 0.6p.u., and it was switched off in the case where the error decreased below 0.4p.u. Experiment results of the case in which a 0.2 s three-phase-to-ground fault was applied on the transformer bus (labeled in Fig. 6.1) were as presented in Fig. 6.4. The peak value of capacitor voltage controlled by VC reached 2.78p.u. as presented in Fig. 6.4a. This may damage the switch of the converter and result in a generator trip event. A generator trip under a 0.2 s’ grid fault violates the grid code requirements of countries such as Australia and New Zealand [10]. The recovery time of the dc-link voltage was 1.0 s in the system controlled by VC only as shown in Fig. 6.4a. By contrast, the capacitor voltage controlled by VC and dc-chopper presented smaller increase at 0.75p.u. and shorter settling time of 0.72s, as presented in Fig. 6.4b. The power from n the PMSG was supplied to the brake resistance of the dc-chopper. The capacitor voltage controlled by SFRTC presented the least magnitude increase at 0.56p.u. and the shortest settling time of 0.6s, which were as shown in Fig. 6.4c. The active power of the PMSGWT was as presented in Fig. 6.4d. Due to the OBFC, the active power controlled by SFRTC showed the least magnitude drop and fastest recovery time. Consequently, the excess power supplied to the capacitor was sent to the external system after the fault was cleared, and over-voltage was less severe as presented in Fig. 6.4c. In the tests for reactive current control loop, SFRTC was compared with VC. ∗ in VC and SFRTC were provided by a PI controller as shown in Fig. 6.1 (PI1 ), i qg ∗ was restricted to [−2, 2] p.u. The PI controller enabled the GSC to inject where i qg reactive power into the external network when the RMS value of Vs was lower than Vs∗ . The phase-to-ground fault applied in the reactive current tests was the same as the previous case. The OBFC in d-axis control loop was not triggered, and only the

6.3 Experiment and Simulation Results

Fig. 6.10 Dc-link voltage and active power output of the PMSGWT [2]

179

180

6 Adaptive Switching Control of Power Electronic Converters

Fig. 6.11 Reactive current control tests of the PMSGWT [2]

BBFC in q-axis loop was implemented. The DC bus voltage controlled by SFRTC increased by 1.25p.u. and its settling time was 0.88s as presented in Fig. 6.5a, which was smaller than those controlled by VC only depicted in Fig. 6.4a. Hence, the BBFC in q-axis helped to restrict the capacitor voltage. The reactive current controlled by VC presented more oscillations in the fault and fault-recovery processes as shown in Fig. 6.5b. As a result, the terminal voltage controlled by SFRTC recovered to 0.95p.u., which was 0.56 ms faster than that controlled by VC only as presented in Fig. 6.5c. Therefore, the GSC controlled by SFRTC provided more reactive power support to the external power network. Both GSCs controlled by SFRTC and VC realized reactive current injection in the fault process (Fig. 6.11).

6.3 Experiment and Simulation Results

181

Table 6.2 Configuration and parameters of the simulation models [4] Three-phase full-bridge inverter Nominal parameters Model-1 Model-2

Model-3

The proposed BBFC & Dual-buck scheme Voltage-mode PI controller & Traditional PWM One-cycle controller & Dual-buck scheme

Udc Um∗ f

720 V 310 V 50 Hz

L QC PLoad

3 mH 3 kvar 15 kW

6.3.2 Test Results of the Bang-Bang Funnel Controller of the Three-Phase Full-Bridge Inverter 6.3.2.1

Simulation Results and Comparisons

A MATLAB/Simulink-based simulation (sampling time: h = 0.01 ms, solver: ode4) was carried out to test the bang-bang funnel control, the conventional PI control [11] with kp = 4, ki = 498, f sw2 = 15 kHz and the one-cycle control [5] with f sw3 = 6 kHz. The configuration and parameters of the simulation model were as presented in Table 6.2. Outputs are expressed as per-unit values in the simulation results. Output voltages are represented by the solid line, while output currents are shown by the dotted line. At t0 = 0 s, Um∗ gradually raised up to 1 p.u. from 0 within 3 ms for safety purposes. The inverter started to operate with the nominal parameters, which was the initial status of the following tests (Fig. 6.12; Table 6.3).

6.3.2.2

Steady-state Performance

Since t0 , all three models started up smoothly. We can find that Model-1 reached the reference amplitude using the least settling time ts , and it presented the lowest total harmonic distortion (THD) and voltage unbalanced factor (VUF) in output voltages. The steady-state error of Model-1 was 0.27 %, which was slightly larger than those of Model-2 and Model-3 (0.17 % and 0.1 %, resp.). Both Model-1 and Model-2 ensured non-overshoot start-up, by contrast, Model-3 showed the peak voltage overshoot of about 0.024 p.u. (2.4 %). VUF was defined to describe the extent of voltage imbalance among the three phases, which is written as VUF = max {|Ua1 −Ub1 | , |Ub1 −Uc1 | , |Uc1 −Ua1 |} ×100%, Ua1(nominal)

(6.20)

0.44 0.39 0.45 0.50 0.48 0.42 0.47 0.44

0.9973 0.9972 0.9970 1.1430 0.8493

0.9984 0.9954 0.9975

Nominal PLoad : 15 kW → 18 kW PLoad : 18 kW → 12 kW Um∗ : 310 V → 310 × 1.15 V Um∗ : 310 × 1.15 V → 310 × 0.85 V Udc : 720 V → 780 V Udc : 780 V → 660 V Udc : 720 + 20sin(30π t) + 10sin(3000π t) (V)

THD(%)

Model-1 Ua1 (p.u.)

Variations (Load or Reference or Input ) 0.9983 0.9991 0.9991 1.1490 0.8503 0.9995 1.0010 0.9998

< 0.1 < 0.1 < 0.1

Model-2 Ua1 (p.u.)

2 0.1 1 1 2

ts (ms)

1.69 1.32 1.23

1.40 0.90 1.10 1.34 1.05

THD(%)

< 0.1 3 < 0.1

18 1 2 3 8

ts (ms)

Table 6.3 Simulation results under the cases of the three-phase load, reference and input voltage variations [4]

1.0050 1.0100 1.0110

1.0010 0.9652 1.0470 1.1600 0.8515

Model-3 Ua1 (p.u.)

THD(%)

1.04 1.09 1.18

0.68 0.89 1.14 0.98 2.44

ts (ms)

< 0.1 2 < 0.1

5 4 4 3 9

182 6 Adaptive Switching Control of Power Electronic Converters

6.3 Experiment and Simulation Results

183

Fig. 6.12 Simulation results of the three-phase instantaneous power, the phase output voltages and currents response to the three-phase load, reference and input voltage variations [4]

where Ua1 , Ub1 and Uc1 are the fundamental magnitude of phase A, B, and C. The evolution of tracking error and its derivatives was as shown in Fig. 6.13.

6.3.2.3

Dynamic Performance

The three-phase load PLoad was stepped up to 18 kW at t1 and then stepped down to 12 kW at t2 . Model-1 reached the new steady state with the smallest peak undershoot/overshoot and the shortest settling time, while preserving favorable output quality. Model-2 ensured its static output performance, but it was inferior to Model-1 in the transient response. Model-3 performed the worst for its large peak undershoot/overshoot, and steady-state tracking error cannot be eliminated (the deviations were 3.48 % and 4.7 %). The reference amplitude Um∗ was changed from 1 p.u. to 1.15 p.u. at t3 and 0.85 p.u. at t4 . According to the simulation results, Model-1 provided good tracking performance and smooth transition from one amplitude to another, and it achieved admissible steady-state tracking error and low THD. Model-2 presented accurate steady-state tracking (0.1 % and 0.03 %, resp.), but the voltage fluctuations in the transient process was noticeable. Although Model-3 achieved smooth transition, it presented the longest settling time and the lowest output quality (when the reference amplitude was 85 % of the nominal value, the output THD reached 2.44 %).

Region Pulse signals de/dt (p.u.)

e (p.u.)

Voltage (p.u.)

184

6 Adaptive Switching Control of Power Electronic Converters 1

ua urefa

0 -1

+ 0

0.1 0 -0.1 200

0 + 1

0 -200 1 0.5 0 -0.5

1

g ( S1 ) g ( S2 )

iLa

5 0

0

0.005

0.01

0.015

0.02

0.025 Time (s)

0.03

0.035

0.04

0.045

0.05

Fig. 6.13 Simulation results of Model-1 showing the tracking performance with how the tracking error and its derivative evolve in the funnels, as well as the control actions and the transition of different regions [4]

With respect to the input voltage variations, Udc was stepped up to 780 V at t5 and then declined to 660 V at t6 . According to the simulation results, Model-1 outperformed the other two, and no transient behaviors and no deterioration in its steady-state output were found. The impact of input steps was suppressed by the  = Udc + v˜ was timebang-bang controller. Considering that the DC link voltage Udc varying in practice, a low frequency noise 20 sin(30π t) V and a high frequency noise 10 sin(3000π t) V were added to the DC supply after t7 . In this case, instability and severe output distortions were not found in the test results of three models.

6.3.2.4

Test Results of the Robustness to the Parameter Uncertainties

THD of outputs under the three-phase LC parameter variations were as presented in Fig. 6.14. The inductance and the three-phase capacitive power of the LC filter, denoted by L and Q C , varied from −25 % to 25 % of the nominal value in the step-size of (25/6) %. We can find that Model-2 was sensitive to the changes of both L and Q C . By contrast, Model-3 was robust to different values of Q C , but its performance was impacted by L. The THD of the output of Model-1 was the same despite of the parameter variation.

6.3 Experiment and Simulation Results

185

1.75

Fig. 6.14 Simulation results of the output THD under the LC parameter variations (−25 % → 25 %) [4]

Model-2

1.5

THD (%)

1.25

1

0.75

Model-3

0.5 Model-1

0.25 Inductance variation Capacitance variation

0 -25%

0

25%

Parameter Variation (%)

The parameter disturbances of the load and the LC filter in single phase (e.g. PLoad1 , L 1 and Q C1 of phase A) were concerned as well. Results of tests were as given in Table 6.4. Model-1 offered lower VUF than the others when the singlephase load varied. VUFs of Model-1 and Model-2 were affected by the variation of L 1 , and they were magnified to around 2.7 % when Q C1 changed. It was the change of L 1 that resulted in severe voltage imbalance among the three phases (the VUF reached 6.14 %) in Model-3. The above simulation results reveal that Model-1 presented an overall better performance than Model-2 and Model-3. The proposed strategy can not only ensure predominant static performance, but also improve the dynamic performance of the system with respect to the sudden changes of the load, reference and input voltage.

6.3.2.5

Test Results of Hardware-in-the-Loop Experiments

The experimental setup was as shown in Fig. 6.15. The control algorithm was realized by dSPACE (DS1104). Parameters of the circuit and controllers were iden = tical to those in the above simulation models. The nonideal DC supply Udc 720 + 20sin(30π t) + 10sin(3000π t) (V) was considered in the experiment. Figure 6.16 shows the results of the voltage harmonic spectra and output THD in steady-state. We can see that Prototype-1 ensured the output quality with the lowest THD. In the presence of the same rapid changes as those in IV, performances were compared as shown in Fig. 6.17 and Table 6.5. It was found that the experimental results were analogous to the simulation ones. Prototype-1 was able to restore the

186

6 Adaptive Switching Control of Power Electronic Converters

Table 6.4 Simulation results under the cases of the load and LC parameter disturbances in single phase [4] Disturbances in phase A

Model-1

Model-2

Model-3

(Load or LC parameter s )

VUF(%)

THDmax (%) VUF(%)

THDmax (%) VUF(%)

Nominal

0.05

0.45

0.10

1.48

0.19

THDmax (%)

0.74

PLoad1 : +10 % kW (resistive) 4.16

0.52

5.29

1.49

6.96

0.74

PLoad1 : −10 % kW (resistive) 4.42

0.47

5.44

1.64

7.36

0.74

PLoad1 : +10 % kvar (inductive)

4.57

0.51

5.75

1.51

4.90

0.75

PLoad1 : −10 % kvar (capacitive)

4.42

0.47

5.38

1.54

4.31

0.73

L 1 : +25 %

0.08

0.45

0.19

1.49

6.14

0.75

L 1 : −25 %

0.11

0.63

0.12

1.79

5.78

0.77

Q C1 : +25 %

2.73

0.48

2.74

1.62

0.29

0.74

Q C1 : −25 %

2.76

0.52

2.76

1.45

0.40

0.74

Power Circuit Model