Reactive Power Support Using Photovoltaic Systems: Techno-Economic Analysis and Implementation Algorithms [1st ed.] 9783030612504, 9783030612511

With the widespread adoption of photovoltaic (PV) systems across the world, many researchers, industry players, and regu

983 242 8MB

English Pages XXV, 156 [177] Year 2021

Report DMCA / Copyright

DOWNLOAD FILE

Polecaj historie

Reactive Power Support Using Photovoltaic Systems: Techno-Economic Analysis and Implementation Algorithms [1st ed.]
 9783030612504, 9783030612511

Table of contents :
Front Matter ....Pages i-xxv
Introduction (Oktoviano Gandhi)....Pages 1-20
Analysis of Local Reactive Power Provision Using PV in Distribution Systems (Oktoviano Gandhi)....Pages 21-51
Analytical Approach to Power Dispatch in Distribution Systems (Oktoviano Gandhi)....Pages 53-81
Inverter Degradation Consideration in Reactive Power Dispatch (Oktoviano Gandhi)....Pages 83-108
Reactive Power Dispatch for Large Number of PV Installations (Oktoviano Gandhi)....Pages 109-143
Conclusions and Future Works (Oktoviano Gandhi)....Pages 145-152
Back Matter ....Pages 153-156

Citation preview

Springer Theses Recognizing Outstanding Ph.D. Research

Oktoviano Gandhi

Reactive Power Support Using Photovoltaic Systems Techno-Economic Analysis and Implementation Algorithms

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

Theses are accepted into the series by invited nomination only and must fulfill all of the following criteria • They must be written in good English. • The topic should fall within the confines of Chemistry, Physics, Earth Sciences, Engineering and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder. • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the significance of its content. • The theses should have a clearly defined structure including an introduction accessible to scientists not expert in that particular field.

More information about this series at http://www.springer.com/series/8790

Oktoviano Gandhi

Reactive Power Support Using Photovoltaic Systems Techno-Economic Analysis and Implementation Algorithms Doctoral Thesis accepted by National University of Singapore, Singapore, Singapore

123

Author Dr. Oktoviano Gandhi Solar Energy Research Institute of Singapore National University of Singapore Singapore, Singapore

Supervisor Prof. Dipti Srinivasan Electrical and Computer Engineering National University of Singapore Singapore, Singapore

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-3-030-61250-4 ISBN 978-3-030-61251-1 (eBook) https://doi.org/10.1007/978-3-030-61251-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Supervisor’s Foreword

Electricity generation using Solar Photovoltaic (PV) system is now one of the most promising sources of energy and is expected to become a major component in most, if not all, electricity networks across the world. Due to the rapidly falling cost of PV panels in recent years, solar energy is becoming an attractive source of renewable energy. However, due to this increasing penetration of solar PV systems in the utility grid, many utilities are experiencing technical challenges due to their intermittency and lack of system inertia. One of the potential enablers to solve some of these challenges and to help us transition towards smart grid is PV reactive power management. That is why this thesis, which gives insights into the cost-effectiveness and practical implementations of PV reactive power support, is very timely and relevant. Through this thesis, Okto has analysed the costs and benefits of reactive power support using PV, both to the PV owner and to the power system operator. Although many before him have investigated the use of PV reactive power, none has formulated and quantified the costs as extensively. The reactive power cost formulations developed in this thesis will allow system engineers to calculate the appropriate remunerations that should be given for reactive power support from PV, and to compare the performance of PV inverter as a reactive power compensator with other sources of reactive power currently in use. The literature review provided in the thesis covers the full spectrum of advantages and technical aspects that act as building blocks in the development of models, as well as the essential background to help acquire a thorough understanding of the issues and challenges. The thesis makes extensive use of mathematical modelling techniques to formulate approximations of power lines, losses and impact of reactive power on inverters. The use of real weather, load and electricity market data makes this thesis relevant not only to academics but also to power system regulators, operators and PV owners. In addition to the theoretical contributions, this thesis also proposes practical methods and algorithms to optimise the reactive power dispatch from distributed PV systems.

v

vi

Supervisor’s Foreword

This thesis will make an ideal reference for those working in the field of solar energy, and a valuable learning resource for advanced students undertaking courses and projects as part of a broader sustainable energy programme. Singapore, Singapore July 2020

Prof. Dipti Srinivasan

Abstract

As photovoltaic (PV) systems are becoming more widespread as a supplement, or even replacement, to conventional generators, many researchers have begun utilising reactive power from PV inverters to provide benefits to the power system. However, the economic costs and benefits of the reactive power support, as well as its practical implementation, have not been analysed in detail. Nevertheless, it is of utmost importance to evaluate the impacts that PV reactive power support brings to the PV owners as well as to the power system, so as to prevent costly retrofitting in the future. Hence, this work aims to quantify the economic and technical impacts of reactive power provision using PV and to explore practical and effective ways to optimise the PV reactive power dispatch in the power system. Consequently, this thesis is divided into two main aspects: (1) the formulation of reactive power cost from PV and (2) the development of algorithms to optimise reactive power dispatch efficiently and effectively. The cost of reactive power from PV has been divided into two components, the inverter loss and the inverter degradation costs. This work has formulated both components mathematically, as well as quantified and compared them to other sources of reactive power using numerous case studies. Through comprehensive analysis, the feasible range for monetary incentives for PV reactive power support has been identified. It was noted that although reactive power compensation using PV is not as economical as switched capacitors (SCs) yet, it will be increasingly so with higher PV penetration and inverter efficiency in the future. Nevertheless, it was found that combinations of PV and SC for reactive power provision can be more technically and economically beneficial than just PV or SC alone. Two algorithms have also been developed in this thesis. The first one is an analytical approach to solving optimal reactive power dispatch. The proposed approach overcomes the weaknesses of mathematical and metaheuristic programming, as it is able to incorporate non-convex cost functions, and is up to a 100 times faster compared with metaheuristic approaches commonly used in the literature. The second one is a data-driven local optimisation of global objectives. The proposed method is able to control the reactive power dispatch from PV and other distributed energy resources and was shown to perform almost as well as vii

viii

Abstract

centralised optimisation, without any communication and without any information regarding the grid topology. Therefore, the power system operator can analyse the benefits of PV reactive power, determine the appropriate remuneration for the reactive power support and choose from the two proposed algorithms to optimise the reactive power dispatch based on its situation. All in all, this research has for the first time comprehensively quantified and analysed the techno-economic cost and benefits of reactive power support using PV. Practical methods to implement reactive power dispatch in distribution systems have also been proposed. The findings and approaches in this work would then be able to help power system planners and operators in making sound regulations with regard to reactive power support from PV.

Publications Related to This Thesis Journal Publications [1] O. Gandhi, C. D. Rodríguez-Gallegos, W. Zhang, D. Srinivasan, and T. Reindl, “Economic and technical analysis of reactive power provision from distributed energy resources in microgrids,” Applied Energy, vol. 210, pp. 827–841, 2018. [2] O. Gandhi, W. Zhang, C. D. Rodríguez-Gallegos, D. Srinivasan, M. Bieri, and T. Reindl, “Analytical approach to optimal reactive power dispatch and energy arbitrage in distribution systems with DERs,” IEEE Transactions on Power Systems, vol. 33, no.6, pp. 6522–6533, 2018. [3] O. Gandhi, C. D. Rodríguez-Gallegos, N. B. Y. Gorla, M. Bieri, T. Reindl, and D. Srinivasan, “Reactive Power Cost from PV Inverters Considering Inverter Lifetime Assessment,” IEEE Transactions on Sustainable Energy, vol. 10, no. 2, pp. 738–747, 2019. [4] O. Gandhi, C. D. Rodríguez-Gallegos, T. Reindl, and D. Srinivasan, “Competitiveness of PV Inverter as a Reactive Power Compensator considering Inverter Lifetime Reduction,” Energy Procedia, vol. 150, pp. 74–82, 2018. [5] O. Gandhi, W. Zhang, C. D. Rodríguez-Gallegos, H. Verbois, H. Sun, T. Reindl, and D. Srinivasan, “Local Reactive Power Dispatch Optimisation Minimizing Global Objectives,” Applied Energy, vol. 262, 2020. Conference Publications [1] O. Gandhi, W. Zhang, C. D. Rodríguez-Gallegos, D. Srinivasan, and T. Reindl, “Continuous optimisation of reactive power from PV and EV in distribution system,” in 2016 IEEE Innovative Smart Grid Technologies—Asia (ISGT-Asia). Melbourne: IEEE, nov 2016, pp. 281–287. [2] O. Gandhi, C. D. Rodríguez-Gallegos, and D. Srinivasan, “Review of optimisation of power dispatch in renewable energy system,” in 2016 IEEE Innovative Smart Grid Technologies—Asia (ISGT-Asia). Melbourne: IEEE, nov 2016, pp. 250–257. [3] O. Gandhi, D. Srinivasan, C. D. Rodríguez-Gallegos, and T. Reindl, “Competitiveness of Reactive Power Provision using PV Inverter in Distribution System,” in 2017 IEEE Innovative Smart Grid Technologies— Europe (ISGT-Europe). Torino: IEEE, sep 2017. [4] O. Gandhi, C. D. Rodríguez-Gallegos, T. Reindl, and D. Srinivasan, “Locally-determined Voltage Droop Control for Distribution Systems,” in 2018 IEEE Innovative Smart Grid Technologies—Asia (ISGT-Asia). Singapore: IEEE, may 2018, pp. 425–429.

ix

Acknowledgements

Despite what people say about Ph.D. being an independent research, the journey that I went through will never be the same without the following people, whom I would like to profess my sincere gratitude to. First and foremost, I would like to thank my supervisor, Prof. Dipti Srinivasan, who was willing to take me as her student despite my having no prior background in power systems. I thank her for her constant guidance, and for giving me the opportunities and freedom to expand my knowledge and pursue my interests throughout the 4 years. I am extremely lucky to have Prof. Dipti as my supervisor and academic mentor. I would also like to express my gratitude to my co-supervisor, Dr. Thomas Reindl, who supported me and my work, gave many excellent ideas, as well as technical and business perspectives to my research. It was very kind of him to accept me to the SES cluster. Although I eventually chose a different research field, it was because of Dr. Thomas Mueller and his research on heterojunction solar cells that I applied for Ph.D. at NUS in the first place. Therefore, I am thankful to him, and also to Prof. Armin Aberle, for welcoming me at SERIS. I am grateful to have had the late Prof. Xu Jianxin as the Chair of my Thesis Advisory Committee, who inspired me to never give up and to always face any problem, no matter how dire, with curiosity; Prof. William Haskell, as a member of my TAC who has given me numerous insights into optimisation methods; and Prof. Zhang Rui who readily accepted my request to be the new TAC Chair. During my exchange at Tsinghua University, I had the fortune to be guided by Prof. Hongbin Sun. I am forever grateful to him for hosting me and letting me join the EMS family. His guidance has contributed to this thesis substantially. Among my peers, I am most thankful to Carlos, who is not only a co-author, lab mate, project mate, partner to play tennis and squash with but also a very good friend. Thanks for always willing to listen and bounce off ideas, whenever I have any problem or doubt. Truly, this thesis, and my life, wouldn’t have been the same without him.

xi

xii

Acknowledgements

Wenjie is another important lab mate and co-author whom I am very grateful to. Wenjie is the person who first taught me about optimisation (a major part of this thesis), and is always willing to use his resources to help with teaching, IEEE duties and much more. Hadrien has been extremely helpful in providing weather forecast data on very short notice even when he was busy with his own thesis and other works. And then there is Moni, who always answered my queries regarding economics and financing, and never failed to brighten my day with her glittering smile. I am also indebted to my colleagues at SERIS who have helped expand my knowledge about solar PV, and shared with me the solar irradiance data: Christoph (a co-author and a very good friend whom I have learned tremendously from, regarding PV, storage systems, their practical implementation and much more), Lu (a wonderful group head whom I have also learnt a lot from), Marek, Soe, Haohui, Jason, Mridul, Franco, Vijay, Stephen, Yanqin, Abishek, Darryl, Congyi, Thway, Sagnik, Zibo, Xin Zheng, Ge Jia, Ankit, Vinodh, Naomi, Muzhi, Janet, Ann, Cecilia and others. And of course, Marinel, who has always helped me with all my requests and forms. Without her, my Ph.D. life would be so much more complicated. Furthermore, I am thankful to the lunch boys: Jaffar, who taught me the basics of power systems back when I didn’t even know about the existence of the field, and Amit who has shared his MATLAB knowledge with me. Mr. Seow and Mr. Looi at the EMML have also been very helpful throughout my Ph.D., for which I am grateful. Additionally, I wish to thank all the EMML and EMDL lab members (past and present), especially Dr. Anurag and Dr. Dhivya for gracing me with their wisdom and benevolence, Anupam for his numerous advice, Shada and Binita, who often provided me with frequent sugar boost to go through the working days, Rahul, Utkarsh, Mert, Monika, Salish, Kawsar, Dongdong, Cikai, Naga, Sandeep and many more. My time in China has been very fruitful thanks to the colleagues and friends at Tsinghua University, especially Jiazhen. Although the time I spent there was brief, but their influence on my life will be long-lasting. I also would like to thank NUS and NGS who have given me the opportunity to pursue the Ph.D. degree and have supported me throughout the programme through their dedicated staff, especially Irene, Jenny, Wei Min, Ivy, Vivien and Anu, as well as many more whom I may not have communicated directly with, but who have been instrumental nevertheless. My Ph.D. life became much more interesting thanks to my “group of friends”, Ali, Christoph (again), Carlos (again), Edward and Windrich, whom I have gone through so much with, learning about the real-life implementation of PV systems. I will never forget when we had to arrange ourselves like sardine in the tiny room in Geranting. Moreover, I wish to acknowledge my fellow Indonesian graduate students: Jeffry, Benedict, Cenna, Ci Aci, Miya, as well as the students from PINUS and PPIS, who have made my stay in Singapore more colourful.

Acknowledgements

xiii

Many icons used in the thesis has been provided free of charge under the creative commons license by the designers from the Noun Project: Adrian Coquet, Brennan Novak, Dinosoft Labs, Made, rivercon, Shashank Singh, Sitchko Igor and Stepan Voevodin. Last but certainly not the least, I owe an infinite debt of gratitude to my parents, my brother and my sister, who have always tolerated all my selfishness and have supported me in all my decisions. It is because of my family (especially my parents), that I have been able to come this far. I really hope that I have made them proud and brought them joy through this degree.

Contents

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

1 1 3 6 10 11 12 13 14 15

2 Analysis of Local Reactive Power Provision Using PV in Distribution Systems . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Optimisation of Power Dispatch . . . . . . . . . . . . . . 2.2.1 Proposed Objective Function . . . . . . . . . . . 2.2.2 Power System Constraints . . . . . . . . . . . . . 2.2.3 PV System Constraints . . . . . . . . . . . . . . . . 2.2.4 Tap-Changing Transformer . . . . . . . . . . . . . 2.3 Proposed Reactive Power Costs . . . . . . . . . . . . . . 2.4 Implementation Setup . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Test System . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Cost Parameters . . . . . . . . . . . . . . . . . . . . . 2.4.3 Weather Parameters . . . . . . . . . . . . . . . . . . 2.5 Analysis of Economic and Technical Impact . . . . . 2.5.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . 2.5.2 Optimisation Algorithm . . . . . . . . . . . . . . . 2.5.3 Results and Discussions . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

21 21 23 23 24 25 26 27 28 28 29 30 31 31 32 34

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . 1.2 Reactive Power . . . . . . . . . . . . . . . . . 1.3 Reactive Power Support Using PV . . . 1.4 Optimisation Methods . . . . . . . . . . . . 1.5 Research Objectives . . . . . . . . . . . . . 1.6 Research Methodology and Parameters 1.7 Research Contributions . . . . . . . . . . . 1.8 Thesis Outline . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

xv

xvi

Contents

2.6 Competitiveness of Local Reactive Power Provision Using PV 2.6.1 Objective Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Optimisation Algorithm . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . 2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

38 38 40 40 41 48 48

Systems . . .

. . . . . . .

. . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

53 53 54 54 55 57 57 59 59 64 66 68 68 69 71 72 72 73 75 77 78 78

4 Inverter Degradation Consideration in Reactive Power Dispatch 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 PV System Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Reactive Power Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Power Loss Component . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Inverter Lifetime Reduction Component . . . . . . . . . . . 4.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Implementation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 PV-Inverter Configuration . . . . . . . . . . . . . . . . . . . . . 4.5.2 System Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Total Operating Costs . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Effects on Inverter Lifetime . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

83 83 85 86 86 86 90 91 91 92 93 93 94

3 Analytical Approach to Power Dispatch in Distribution 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Objective Function . . . . . . . . . . . . . . . . . . . . 3.2.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Decision Variables . . . . . . . . . . . . . . . . . . . . 3.2.4 Cost Functions . . . . . . . . . . . . . . . . . . . . . . 3.3 Proposed Analytical Approach . . . . . . . . . . . . . . . . 3.3.1 Optimal Reactive Power Dispatch . . . . . . . . . 3.3.2 Energy Arbitrage Considering Line Losses . . 3.3.3 Voltage and Current Constraints Handling . . 3.4 Implementation Setup . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Benchmark Optimisation Algorithms . . . . . . 3.4.2 Test Systems . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Cost and Weather Parameters . . . . . . . . . . . . 3.5 Results and Discussions . . . . . . . . . . . . . . . . . . . . . 3.5.1 Performance Comparison . . . . . . . . . . . . . . . 3.5.2 Line Loss Consideration in Energy Arbitrage 3.5.3 Power Profiles . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Uncertainty Treatment . . . . . . . . . . . . . . . . . 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

Contents

4.6.3 Economic Balance of Local Reactive Power Provision . 4.6.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Competitiveness of Local Reactive Power Provision Using PV After Considering Inverter Lifetime Reduction . . . . . . . . . . . . 4.7.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Comparison of PV and SC for Reactive Power Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Reactive Power Dispatch for Large Number of PV Installations 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Objective Function and Constraints . . . . . . . . . . . . . . . . . . . . 5.3 Local Optimisation of Global Objectives (LOGO) . . . . . . . . . 5.3.1 Relations Between Global and Local Variables . . . . . . 5.3.2 Relations Between Node Voltage and PV Reactive Power Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Expressing Cost in Terms of QPV x;t and Vx;t . . . . . . . . . . 5.3.4 Inclusion of Reactive Power Cost . . . . . . . . . . . . . . . . 5.3.5 Local Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6 Constraints Handling . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Implementation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Test Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 System Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Weather Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Benchmark Algorithms . . . . . . . . . . . . . . . . . . . . . . . 5.5 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Stability of LOGO . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Performance of LOGO . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Cost Compositions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Reactive Power Profiles . . . . . . . . . . . . . . . . . . . . . . . 5.5.5 Limitations and Further Development of LOGO . . . . . 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xvii

... ...

97 99

. . . 100 . . . 100 . . . 101 . . . 106 . . . 106 . . . . .

. . . . .

. . . . .

109 109 111 112 112

. . . 115 . . . 121 . . . . . . . . . . . . . . . .

6 Conclusions and Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Summary of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Proposed Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 More Comprehensive Reactive Power Cost Formulation for Transmission System . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Impact of PV and Energy Storage in Distribution System on the Transmission System . . . . . . . . . . . . . . . 6.2.3 Enhancement of the Inverter Lifetime Reduction Component of Reactive Power Cost . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

122 123 124 125 125 125 128 128 131 131 131 134 134 139 140 141

. . 145 . . 145 . . 146 . . 146 . . 147 . . 148

xviii

6.2.4 Incorporation of More Variables into Local Optimisation of Global Objectives (LOGO) . . . . . . . . . . . . . . . . . . . 6.2.5 Analysis of PV Reactive Power Support at Shorter Time Scales and in Non-Steady State . . . . . . . . . . . . . . 6.2.6 Comprehensive Formulation of Reactive Power Cost for Other DERs . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents

. . 149 . . 149 . . 150 . . 151

Appendix: Convergence of the Proposed Analytical Approach . . . . . . . . 153

Symbols

a b DPloss nadir DPloss peak DPX;oppcost x;t DPBESS;invloss x;t DPEV;invloss x;t DPPV;invloss x;t DPPV x;t DQloss nadir DQloss peak dCH x;t dDCH x;t gLI;A0 gLI;A1 gLI;B0

temperature coefficient of I PV;SC t temperature coefficient of V PV;OC t difference in active power loss in a system because of the charging of BESS/EV at price nadir difference in active power loss in a system because of the discharging of BESS/EV at price peak active power of the xth DER that has to be curtailed to generate reactive power more than QXx;t;lim additional power loss in the inverter of the xth BESS due to its reactive power injection/absorption total additional power loss in the inverters of the xth EV parking lot due to its reactive power injection/absorption additional power loss in the inverter of the xth PV system due to its reactive power injection/absorption maximum error prediction for PPV x;t difference in reactive power loss in a system because of the charging of BESS/EV at price nadir difference in reactive power loss in a system because of the discharging of BESS/EV at price peak binary variable that takes the value of 1 when PBESS [ 0, and 0 x;t otherwise binary variable that takes the value of 1 when PBESS x;t \ 0, and 0 otherwise numerator coefficient of the zeroth-order LR from fitting the values of LI to those of LR numerator coefficient of the first-order LR from fitting the values of LI to those of LR denominator coefficient of the zeroth-order LR from fitting the values of LI to those of LR

xix

xx

gLI;B1 gLR;0 gLR;1 gLR;2 gLR;3 gP;0

Symbols

denominator coefficient of the first-order LR from fitting the values of LI to those of LR coefficient of the zeroth-order QPV from fitting the values of ILR t PV cost per hour with Qt coefficient of the first-order QPV from fitting the values of ILR cost t per hour with QPV t coefficient of the second-order QPV from fitting the values of ILR t cost per hour with QPV t coefficient of the third-order QPV t from fitting the values of ILR cost PV per hour with Qt ~ loss from fitting the values of P ~ loss to coefficient of the zeroth-order P those of Ploss

g

P;1

gQ;0

~ loss from fitting the values of P ~ loss to coefficient of the first-order P those of Ploss ~ loss from fitting the values of Q ~ loss to coefficient of the zeroth-order Q those of Qloss

g

Q;1

gQLim gT gBESS;DOD x gBESS;P x gBESS;SOC x gBx gCH x gDCH x gPLoss;0 x gPLoss;Q1 x gPLoss;Q2 x gPLoss;V1 x

~ loss from fitting the values of Q ~ loss to coefficient of the first-order Q those of Qloss constant limiting the change in QPV x;t power temperature coefficient of solar cells battery degradation coefficient related to depth of discharge of the xth BESS battery degradation coefficient related to charging power of the xth BESS battery degradation coefficient related to SOC of the xth BESS round-trip efficiency of the xth BESS. It is the multiplication of gCH x and gDCH x charging efficiency of the xth BESS discharging efficiency of the xth BESS coefficient of the zeroth-order from fitting the values of Ploss to V x;t t PV and Qx;t loss coefficient of the first-order QPV to x;t from fitting the values of Pt PV V x;t and Qx;t loss coefficient of the second-order QPV to x;t from fitting the values of Pt PV V x;t and Qx;t coefficient of the first-order V x;t from fitting the values of Ploss to t V x;t and QPV x;t

Symbols

gPLoss;V2 x gxQLoss;0 gxQLoss;Q1 gxQLoss;Q2 gxQLoss;V1 gxQLoss;V2 gV;P1 x gV;Q1 x c / SOCx;max SOCx;min SOCx;t ~ loss;init P ~ loss P ~ loss P i ~ Q

loss;init

~ loss Q ~ loss Q i B cBESS;deg x;t cPgrid t cPPV x cQBESS x;t cQEV x;t cQgrid cQPV x;t

xxi

coefficient of the second-order V x;t from fitting the values of Ploss to t PV V x;t and Qx;t coefficient of the zeroth-order from fitting the values of Qloss to V x;t t PV and Qx;t loss coefficient of the first-order QPV to x;t from fitting the values of Qt PV V x;t and Qx;t loss coefficient of the second-order QPV to x;t from fitting the values of Qt PV V x;t and Qx;t coefficient of the first-order V x;t from fitting the values of Qloss to t V x;t and QPV x;t coefficient of the second-order V x;t from fitting the values of Qloss to t V x;t and QPV x;t PV coefficient of the first-order PPV x;t relating V x;t to Px;t PV coefficient of the first-order Qx;t relating V x;t to QPV x;t irradiance correction factor of V PV;OC t phase angle between voltage and current in an AC circuit or system maximum SOC of the xth BESS minimum SOC of the xth BESS SOC of the xth BESS approximate total initial active power loss in a system before adding any DER into the system approximate total active power loss in a system approximate active power loss on the line from sending node i approximate total initial reactive power loss in a system before adding any DER into the system approximate total reactive power loss in a system approximate reactive power loss on the line from sending node i number of BESS in a system battery degradation cost of the xth BESS cost of active power from the grid cost of active power from the xth PV system. It is equivalent to LCOE of the PV system, and is assumed to be fixed for the lifetime of the PV system cost of reactive power from the xth BESS cost of reactive power from the xth EV parking lot cost of reactive power from the grid cost of reactive power from the xth PV system

xxii

cQPV;invloss x;t cQPV;LR x;t cQSC x;t cBESS;R x cBESS;self x cBESS;V x cPV;R x cPV;self x cPV;V x E E BESS x G70 x;t Gclear t Gx;t

H i I CAP;ACrms t I CAP;ripple t I PV;SC t I PV t I max I i;t j k LCAP;P LCAP;Q LCAP LCAP ref

Symbols

inverter power loss component of the reactive power cost from the xth PV system inverter lifetime reduction component of the reactive power cost from the xth PV system cost of reactive power from the xth SC current-dependent loss coefficient of the inverter of the xth BESS standby loss coefficient of the inverter of the xth BESS voltage-dependent loss coefficient of the inverter of the xth BESS current-dependent loss coefficient of the inverter of the xth PV system standby loss coefficient of the inverter of the xth PV system voltage-dependent loss coefficient of the inverter of the xth PV system number of EV parking lots in a system energy capacity of the xth BESS upper limit of forecasted irradiance falling on the xth PV system in a distribution system at 70% confidence level clear sky irradiance irradiance falling on the xth PV system in a distribution system. In this book, the irradiance is assumed to be uniform within a particular distribution system for simplicity, such that the value of Gx;t is the same for all x length of time period in [hour] index for bus/node in a system RMS AC current flowing through the DC-link capacitors in a PV inverter ripple current flowing through the DC-link capacitors in a PV inverter PV module short circuit current PV module current upper bound current magnitude across the line in a system current flowing from sending node i index for bus/node in a system index for bus/node in a system lifetime of the DC link capacitors, and hence the inverter, when it is only used for PPV lifetime of the DC link capacitors, and hence the inverter, when it is used for both PPV and QPV estimated lifetime of the DC link capacitors, and hence the inverter of a particular PV system reference lifetime of the DC-link capacitors in a PV inverter

Symbols

LCAP t LPV M Mt N PV;par N PV;ser P'i PBESS x;t PEV x;t Pgrid t Pload i;t Ploss i;t Ploss t PPV70 x;t PPV x;t PPV ave Pi PBESS x;max PEV x;max PPV;invloss x;max PBESS x;min PPV x;rated PBESS;invloss x;t PPV;invloss x;t PR0 Q0 i QBESS x;t QEV x;t Qgrid t Qload i;t

xxiii

expected lifetime of the DC-link capacitors in a PV inverter for a particular operating condition. The lifetime of all the DC-link capacitors in an inverter is assumed to be the same for simplicity lifetime of a PV system number of PV systems in a system modulation index of a PV inverter number of PV strings connected in parallel for a PV system number of PV modules connected in series in a string active power flowing across the line from sending node i before the addition of any DER into the system charging (positive)/discharging (negative) power of the xth BESS total charging (positive)/discharging (negative) power of the xth EV parking lot active power from the grid active power load at node i active power loss on the line from sending node i total active power loss in the system upper limit of forecasted active power from the xth PV system at 70% confidence level active power from the xth PV system average active power generated by a PV system at each period t over the lifetime of the system, taking into account the reduction of PR in later years active power flowing across the line from sending node i maximum charging power of the xth BESS maximum charging power of the xth EV parking lot maximum power loss in the inverter of the xth PV system maximum discharging power of the xth BESS power rating at STC of the xth PV system power loss in the inverter of the xth BESS power loss in the inverter of the xth PV system performance ratio of a PV system before taking into account the inverter losses reactive power flowing across the line from sending node i before the addition of any DER into the system reactive power from the xth BESS total reactive power from the xth EV parking lot reactive power from the grid reactive power load at node i

xxiv

Qloss i;t Qloss t QPV x;t Qi QPV x;t;lim QX x;t;lim RCAP;s RCAP;th RPV;s Rij ri Sloss Sloss i;t SPV x;max SX x;max SBESS x;t SPV x;t T t T a70 x;t T ax;t

T CAP;P t T CAP ref T CAP t T PV x;t V CAP ref V CAP t V INV;AC V INV;DC V PV;OC t V PV t

Symbols

reactive power loss on the line from sending node i total reactive power loss in the system reactive power from the xth PV system reactive power flowing across the line from sending node i the maximum reactive power that can be generated by PV without reducing its active power output beyond the additional inverter loss the maximum reactive power that can be generated by the DERs without reducing their active power output series resistance of the DC-link capacitors in a PV inverter thermal resistance of the DC-link capacitors in a PV inverter PV module series resistance effective resistance connecting node i and node j resistance of the line from sending node i total apparent power loss in a system apparent power loss on the line from sending node i inverter power rating of the xth PV system inverter power rating of the xth DER apparent power from the xth BESS apparent power from the xth PV system number of time periods in a simulation index for time period upper limit of ambient temperature experienced by the xth PV system in a distribution system at 70% confidence level ambient temperature experienced by the xth PV system in a distribution system. In this book, the ambient temperature is assumed to be uniform within a particular distribution system for simplicity, such that the value of T ax;t is the same for all x operating temperature of the DC link capacitors in a PV inverter, when it is only used for PPV temperature rating of the DC-link capacitors in a PV inverter operating temperature of the DC-link capacitors in a PV inverter solar cell temperature of the xth PV system. All the solar cells in a PV system is assumed to have the same temperature for simplicity voltage rating of the DC-link capacitors in a PV inverter operating voltage of the DC-link capacitors in a PV inverter line-to-line RMS AC output voltage of a PV inverte rated DC voltage of a PV inverter PV module open circuit voltage PV module voltage

Symbols

V max V min V ref V i;t V meas x;t V pred x;t X x X ij xi

xxv

upper bound voltage magnitude of the nodes in a system lower bound voltage magnitude of the nodes in a system reference voltage magnitude in a system, usually taken as 1.0 p.u. voltage at node i measured voltage at the node where the xth PV system is located predicted voltage at the node where the xth PV system is located superscript denoting that the quantity is associated to DERs (either BESS, EV parking lot, PV system, or SC) index for PV and other DERs in a system effective reactance connecting node i and node j reactance of the line from sending node i

Chapter 1

Introduction

1.1 Overview Climate change is one of the world’s most pressing issues, and most climate scientists agree that the Earth temperature is on track to increase by more than two degrees above pre-industrial levels—thereby inducing flooding of low-lying countries and extreme weather conditions throughout the world, among other severe and irreversible effects—if countries do not reduce their fossil fuels consumption [1]. That is why in the 2015 United Nations Climate Change Conference in Paris (also known as the Paris Agreement), the world leaders have agreed to accelerate the “reduction of global greenhouse gas emissions” through increased adoption of renewable energies [2]. With the rising energy demand and the need to cut down greenhouse gas emissions at the same time, solar energy—especially in the form of photovoltaic (PV) systems— is becoming more popular as it has lower carbon footprint than fossil fuels and can operate at a smaller scale, forming what we call distributed energy resources (DERs). Its low carbon footprint, its modularity, and perhaps most importantly, its continuously decreasing cost, have prompted the large scale adoption of PV. In fact, PV is the type of electricity generation with the largest annual installed capacity since 2016 [3], at 73 GWp of net new PV capacity installed in 2016. From Fig. 1.1, we can see that the PV annual installed capacity has kept on increasing [4] until 2019. The trend will most likely continue because the cost of PV is still rapidly declining and because of the development of many promising applications and technologies, such as building-integrated PV (BIPV) [5], floating PV [6], PV-powered cooling [7, 8], bifacial solar panels [9, 10], tandem solar cells [11], as well as PV trackers [12, 13]. Not to mention that PV also has the highest potential among the available energy sources to provide electricity access to those who do not currently have it [14]. As PV penetration rises, increasing number of researchers have begun utilising PV to provide ancillary services. Ancillary services are “services which make up the basic components needed for open access operation” of power systems, which include © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 O. Gandhi, Reactive Power Support Using Photovoltaic Systems, Springer Theses, https://doi.org/10.1007/978-3-030-61251-1_1

1

2

1 Introduction

Fig. 1.1 Accumulated installed PV capacity, annual installed PV capacity, and PV penetration by electricity generation from 2000 to 2019. Data source [15–18]. Adapted from [19]

regulation, load following, spinning reserve, as well as reactive power and voltage control [20]. While the variability of solar irradiance prevents PV from providing many of the services, many works have shown that the inverters that connect the PV into the power system are capable of providing reactive power for voltage control [21–23]. Compared with traditional power factor correction devices used in distribution systems, i.e. capacitor banks, PV inverters have faster response time and therefore can regulate voltage more accurately [21]. Many researchers have proposed the use of reactive power to minimise losses in the system [24–27] and to maintain voltage within acceptable levels [26–29]. Moreover, line loading and system operational costs can also be reduced through the local reactive power provision using PV [28]. The utilisation of reactive power can even increase the maximum PV penetration in a system [21, 30]. Even though many researchers assume that PV inverters are able to inject reactive power at no cost [25, 26, 28, 29], there are tradeoffs involved in producing reactive power through the inverter which have not been taken into account before. The algorithms used to optimise the reactive power dispatch are still not practical either, requiring high computational and time resources [25, 31–34], extensive communication infrastructure [35, 36], and/or failing to achieve the most economic solution [37]. Therefore, the main aim of this thesis is to advance the field of PV reactive power support, through extensively analysing the technical and economic aspects of reactive power provision using PV for both the power system and the PV owners, and by proposing practical algorithms for PV reactive power dispatch. To achieve that goal, this thesis focuses on the following two main aspects: [1] Theoretical research on the costs and benefits of reactive power support using PV, and [2] The development of algorithms to optimise PV reactive power dispatch efficiently and effectively.

1.1 Overview

3

Through the works and contributions on both the theoretical and implementation aspects of PV reactive power provision, this thesis would be useful to the academic community, industry players, as well as policy makers concerned with PV deployment in power systems. This chapter is organised as follows. Section 1.2 first gives a basic introduction to what reactive power is and why it is important for power system operations. Subsequently, Sect. 1.3 explores how PV can provide reactive power support and briefly reviews the existing literature on the topic. Section 1.4 then gives an overview of optimisation methods and how they are applied in the field of power systems. This thesis’ objectives, methodology, and contributions are outlined in Sects. 1.5, 1.6, and 1.7, respectively. Finally, the outline of the thesis is explained in Sect. 1.8.

1.2 Reactive Power As this thesis focuses on the reactive power support using PV systems, there is a need to first establish what reactive power is, its importance in power system operations, and how PV can play a role in reactive power support. Active and reactive power are properties of alternating current (AC). In an AC system, voltage and current typically have waveforms that vary sinusoidally. The waveforms of the voltage V and current I may either be in phase or out of phase with each other. Active power is then the multiplication of voltage with the current component that is in phase with the voltage waveform, while reactive power is the multiplication of voltage with the remaining current component, as shown in Fig. 1.2 [38]. The vector addition of the active power P and reactive power Q is then termed the apparent power S, given by the following equations. The relation of P, Q, and S is also illustrated in Fig. 1.3. S = V I∗ = P + jQ  |S| = P 2 + Q 2

(1.1) (1.2)

In a purely resistive circuit, i.e. the load only has resistance r and no inductance or capacitance, the voltage and current will always be in phase, such that the magnitude of reactive power Q will be zero, and the apparent power S will be equal to the active power P. The presence of an inductance or capacitance,1 combined to a term called reactance x, will cause V and I to be out of phase with each other. If a circuit has a purely inductive load, then I is said to lag V by 90◦ , whereas a purely capacitive load causes I to lead the voltage by 90◦ . In AC systems, only active power is being 1 Inductive

loads are usually found in any type of wound coil, or induction motors, which are common load for industrial customers, whereas capacitive loads are typically capacitor banks.

4

1 Introduction

Fig. 1.2 Active and reactive power (single phase)

Fig. 1.3 Relation between P, Q, S, and φ

consumed, which is why active power is also called real power. Reactive power is just energy moving back and forth across the circuit and does no useful/real work at the load. Nevertheless, the current that accounts for this reactive power, together with the line resistance, causes energy losses. To quantify the ratio of P to S, generally the term power factor (PF) is used, which is defined as the following: P F = cos φ =

P |S|

(1.3)

1.2 Reactive Power

5

where φ is the phase angle in Figs. 1.2 and 1.3. PF is dimensionless number between −1 to 1. As can be seen from Eq. (1.3), a PF of 1 means that Q = 0. Depending on the sign of φ (and Q), PF can be said to be lagging or leading. On the one hand, a lagging power factor means that the load is inductive, that the load “consumes” reactive power (Q load is positive). On the other hand, leading power factor indicates a capacitive load, which “supplies” reactive power (Q load is negative). Therefore, capacitor banks can be thought as a “source” of reactive power, that they inject or provide reactive power to the system, whereas induction motors can be thought as reactive power “sinks”, which absorb reactive power from the system. As Q increases S (given it has the same P), a load with lower PF draws more current than a load with higher PF, and therefore increases the energy loss in the circuit or system. Loads with lower PF may also necessitates larger wires to support the transfer of higher current. For fundamental concepts of electricity and power systems, readers are referred to [38–40]. In power systems, reactive power consumption is generally less than active power consumption, but it is usually not negligible, especially in industrial areas. As reactive power is traditionally generated by fossil fuel generators located far away from the load centres, the presence of reactive power loads consequently increases the current across the transmission and distribution lines from the generators to the loads, and hence the power losses across these lines. Because of the increased cost from the losses or additional infrastructures required, utilities usually charge large customers (mainly industrial customers) for the reactive power consumption if their loads have a PF below a certain value (e.g. 0.95) [41]. To avoid paying the reactive power charge and to maintain their voltage at an acceptable level, some large customers, especially the ones that have their own electricity network, install capacitor banks or switched capacitors (SC) to locally provide the reactive power required by the inductive loads [42]. On top of its influence on the line losses in a system, reactive power is also important because it affects the voltage of the system. By reducing the line losses and the current flowing through the system, the voltage drop across the system can be reduced significantly. Injecting reactive power into the system will cause the voltage to rise while absorbing reactive power will cause the voltage to drop. Because of the large influence of reactive power on the voltage magnitude, some power flow algorithms, such as Decoupled Load Flow algorithm, assume that P is coupled with δ (voltage angle), while Q is coupled with |V |, the voltage magnitude [43]. In a particular system, the voltage of each node needs to be maintained at a certain acceptable level (e.g. ±10% of the nominal voltage). This is very important as when the voltage deviates too far from the nominal level, some generators will disconnect automatically to prevent damage to themselves, potentially causing cascading failures. In fact, many world major blackouts have been caused by voltage collapse due to insufficient reactive power support [44, 45], further highlighting the importance of reactive power management in power system. While traditionally the role of regulating the system voltage is in the hands of large-scale generators, tap-changing transformers, and flexible AC transmission

6

1 Introduction

system (FACTS) devices in the transmission system, the emergence of local reactive power support using PV may benefit power systems as it has the potential to reduce transmission and distribution losses, as well as maintain voltage stability of the system locally [46].

1.3 Reactive Power Support Using PV PV system converts sunlight to produce direct current (DC) through photovoltaic effect. Since most grids are run on AC, in grid-connected systems, the current produced from PV will pass through an inverter, which converts the DC power into AC. The AC power is subsequently used to fulfil the electrical demand of AC loads in the system. If the PV system is installed nearby an electrical load (e.g. on a roof of a building), then the power produced by PV may first be consumed locally before flowing to the larger grid. A typical non-utility-scale2 on-grid PV schematic is shown in Fig. 1.4. As PV penetration keeps rising, high penetration of PV may introduce major challenges to power system operations. These challenges are caused by four characteristics of solar power as illustrated in Fig. 1.5 [19]: [1] Distributed: Many PV systems are located behind the meters in distribution systems, causing them to be invisible and non-dispatchable to the grid operators. The PV systems might also be installed in different phases and cause unbalance. [2] Diurnal: Solar power is only available during the day. Although it can generate power when it is sunny, PV is unable to contribute to fulfilling the peak load in the evening, when people return to their home from work and start turning on their appliances. This causes strain in the power system with high PV penetration, as the net load during the day is decreased significantly, but the peak load remains unchanged. As a result of growing PV installation, the ramp up necessary to account for the loss of power generation from PV keeps on increasing year after year. This phenomenon is named the “duck curve”, because of its distinct net load curve shape, as can be seen for the case California Independent System Operator (CAISO) in Fig. 1.6. [3] Converter-based: PV systems are connected to the grid via inverters. These inverters do not have inertia—unlike synchronous generators—which is essential in maintaining power system stability during disturbances. Furthermore, the nonlinear switching in the power electronic components may introduce harmonics into the power system.

2 On-grid

PV system, based on its capacity and location where it is installed, can be divided into residential PV (1–10s of kWp, typically installed on residential rooftops); commercial/industrial PV (10s of kWp to approximately 1 MWp, typically installed on rooftop of commercial/industrial buildings); and utility-scale PV (≥5 MWp, typically ground-mounted PV installed away from the load centres).

1.3 Reactive Power Support Using PV

7

Fig. 1.4 Schematic of a typical non-utility-scale on-grid PV system

Fig. 1.5 Overview of power system issues caused by high PV penetration. Adopted from [19]

[4] Intermittent/variable: The irradiance that reaches the solar panels forming the PV system is not constant across time. The power generated by PV can change significantly, due to cloud movements or other weather phenomena. While the output of utility-scale PV can be curtailed, it cannot be increased like conventional generators can. Given the potential issues that PV brings about, especially the lack of inertia and the non-dispatchable nature, many utilities have started imposing reserve or balancing charges on PV owners or operators [49–51], which makes PV less competitive. One of the ways for PV to benefit the power system and also improve its competitiveness is through the provision of ancillary services. While the variability of PV prevents it from providing many of these services, such as regulation and load fol-

8

1 Introduction

Fig. 1.6 Historical net load curve of California ISO, also known as the “California duck curve”. Data source [47, 48]. Adopted from [19]

lowing, many works have shown that PV inverters are capable of providing reactive power for voltage control [21–23]. Compared with traditional power factor correction devices at the distribution level, i.e. capacitor banks, inverters have faster response time and therefore can regulate voltage more accurately [21]. Historically, PV systems have mainly been deployed in distribution systems, and the inverters are designed to fulfil that purpose. PV was only allowed to operate at unity power factor—it can only inject active power to the grid—and was forbidden to actively regulate the grid voltage by the Institute of Electrical and Electronics Engineers (IEEE) Standard 15473 [52]. This was probably because at the time there were many unknowns regarding reactive power injection in distribution system, including the impacts that it may have. Nonetheless, this did not stop researchers from exploring the benefits of PV reactive power support. The local reactive power provision has been shown to reduce losses [24–27], decrease line loading [28], maintain the systems’ voltage within limits [26–29, 31], and reduce the cost of running the system [28]. These benefits are enhanced for the distributed PV connected at the low voltage distribution grids, as the local reactive power support can tremendously reduce the flow of reactive power from the traditional generators all the way to the load centres. PV reactive power capability is particularly useful for systems with very high penetration of PV, in which there is a reverse power flow from the distribution system to the transmission system. In such distribution systems, the voltage of the nodes in

3 The

full name of the standard is IEEE Standard for Interconnection and Interoperability of Distributed Energy Resources with Associated Electric Power Systems Interfaces, but it will be referred to IEEE Standard 1547 in the text for brevity.

1.3 Reactive Power Support Using PV

9

the system in the middle of the day—when PV power generation is the highest—may go beyond the allowable range, causing overvoltage problems. In those scenarios, it has been shown that by having the PV systems absorb reactive power, the voltage of the system can be kept within the safety limit [30, 53]. Since voltage violations have been cited as one of the factors limiting PV penetration [54, 55], reactive power from PV can also enable higher penetration of PV in the system [21, 30]. Indeed, a recent review on the measures to mitigate power system impacts of high PV penetration highlighted reactive power support from PV as one of the ways forward [56]. Recognising the capability and the benefits of reactive power injection and absorption using PV and other inverter-based DERs for voltage regulation, in 2014 and 2017 respectively, IEEE Standard 1547a and California’s Electric Tariff Rule 21 have allowed smart inverters to provide dynamic reactive power compensation to keep the voltage of the distribution system within the acceptable range [57, 58]. Many commercial inverter manufacturers have also displayed the inverters’ reactive power capability in their data sheets [59, 60]. [59] also suggested that generating reactive power when PV is unable to produce much active power can be the most economical option to stabilise the grid voltage. In 2018, IEEE Standards Coordinating Committee 21 has revised the Standard 1547 to require DERs to at least be able to inject and absorb reactive power 44% of their nameplate kVA rating, and to provide voltage regulation support demanded by the specific electric power system operator within the aforementioned range [61]. However, even after realising the importance of reactive power support from DERs, most people assume that the reactive power can be provided for free. There is no remuneration for PV that injects or absorbs reactive power to or from the system. Perhaps this is partly because the monetary value of reactive power is much lower than active power. For example, in many countries, only the industrial customers whose power factor is below 0.95 is required to pay for reactive power [41]. Even then, the reactive power charge is about 20 times smaller than the price of active power [62, 63]. It is therefore not surprising that most of the research works utilising PV or other DERs for reactive power have ignored its cost [28, 31, 64, 65], while most of the works which considered the reactive power cost, have assigned arbitrary values [24, 66]. Nevertheless, reactive power from PV does not come for free. As will be explained in detail in Chaps. 2 and 4, there are costs associated with injecting and absorbing reactive power using inverters. When not accounted for, these costs will be borne by the PV system owners/operators unknowingly. Quantifying this cost is imperative to determine the economic viability of reactive power support using PV, and to be able to set appropriate regulations and prices for it. Therefore, one of the main focuses of this thesis is formulating the cost of PV reactive power provision.

10

1 Introduction

1.4 Optimisation Methods In the past, power dispatch optimisation usually only took place at the transmission level, where there are generators whose power generation can be controlled to fulfil the electricity demand in the most cost-effective way possible [67]. Power dispatch problem at the transmission level, also known as generation scheduling problem, consists of a unit commitment [68] and economic dispatch. Unit commitment problem seeks to find the optimum generators’ up and down time to fulfil the power demand in the near future, whereas economic dispatch is about finding the optimum operating point of these generators to minimise the fuel cost without changing the on/off setting. Generation scheduling was usually conducted one day ahead while assuming perfect prediction of future electricity demand. However, with the introduction and the increasing importance of variable DERs—including PV, electric vehicles (EV), and battery energy storage system (BESS)—in power system, the power dispatch problem has become even more complex. There is a need to predict and take into account the availability of the intermittent renewable resources in the future to optimise the system meaningfully. Without such optimisation, demand outage may occur because of the mismatch in electricity demand and supply. Consequently, optimisation methods have become increasingly important in power and energy sector. They need to be able to control increasing number of operating points and power system devices in a cost-effective and efficient manner. With the advent of local reactive power support from the DERs, the generation scheduling problem not only gets more complex, but is also starting to happen at the distribution level. Hence, this thesis aims to help facilitate the practical implementation of reactive power dispatch by proposing optimisation algorithms that are fast and scalable, which can be used for day-ahead generation scheduling and real-time operations alike. Optimisation is simply the process of choosing the values of decision variables, i.e. the variables that can be controlled by the entity carrying out the optimisation, to achieve the best possible outcome. The best possible outcome in an optimisation problem is either the minimisation or maximisation of an objective function, such as minimising the cost of operating a distribution system, minimising voltage deviation in the system, maximising PV penetration in the system, and so on. For the case of reactive power dispatch optimisation that this thesis focuses on, the decision variables are then the reactive power output of PV and other DERs. There are, of course, constraints associated with the optimisation, such as the acceptable voltage limits, the limit of reactive power outputs, etc. These objective functions and constraints will be described mathematically in the later chapters. Optimisation techniques can be broadly classified into exact and metaheuristic optimisations [69]. On the one hand, exact/mathematical optimisation, also known as mathematical programming, guarantees that an optimum solution will be reached in a finite time. However, when the problem is NP-hard, the algorithm may need an exponential time—which might be infeasible for practical purposes—to find the

1.4 Optimisation Methods

11

optimum solution [70]. And when the problem is non-convex, meaning that the objective function has more than one minima/maxima with respect to the decision variables, then exact optimisation algorithms may be trapped in one of the local minima/maxima [71]. [72] and [73], widely used textbooks for exact optimisation, gave an introduction to the field. On the other hand, metaheuristic optimisation gives no guarantee that an optimum solution would be found. Nevertheless, it will give “good enough” solutions within a “reasonable” time through trial and error. Metaheuristic optimisation is categorised into population-based (e.g. Genetic Algorithm (GA), Particle Swarm Optimisation (PSO)) and non-population-based (e.g. Tabu Search, Simulated Annealing). GA [74] and PSO [75] are some of the most widely used optimisation algorithms in power dispatch problems because of their flexibility and scalability [76]. As such, they will be used extensively, either as the optimisation algorithm of choice, or as benchmarks for the algorithms proposed in this thesis. Readers are referred to [77, 78] for introduction to the different metaheuristic optimisation algorithms. Most optimisation problems deal with one objective function, however there are also cases whereby two conflicting objective functions need to be optimised simultaneously, e.g. minimising power system operational cost while minimising the system voltage deviation as done in Chaps. 2 and 4. In this case, a multi-objective optimisation algorithm, which will be explained in more detail in Chap. 2, need to be employed. As will be explained in more detail in the literature review of Chaps. 3 and 5, both the exact and metaheuristic optimisation algorithms found in the literature have many weaknesses that make them unsuitable to optimise reactive power dispatch in real life. In particular, both the mathematical programming and metaheuristic algorithms are not fast enough to be implemented in real time for non-convex objective functions. Moreover, nearly all of them require extensive two-way communication infrastructure [79–86], which may not be feasible with increasing number of PV and DERs in the power system because of cost, privacy, and latency issues. As such, another focus of this thesis is to formulate optimisation algorithms that are suitable to optimise reactive power dispatch in the presence of numerous PV and other DERs in the system, both for day-ahead scheduling, and for real-time implementation.

1.5 Research Objectives As the cost of PV keeps decreasing, and shifting away from fossil fuels becomes more urgent, the adoption of PV across the world will continue to rise. And as discussed in Sect. 1.3, reactive power support from PV is also becoming more widespread, especially at the distribution level, because of its many advantages for the power system. Should a PV system be rewarded appropriately for the support that it provides, the business case of installing PV will become even more compelling, and may further increase the rate of adoption of PV, hence accelerating the shift to cleaner energy sources.

12

1 Introduction

Nevertheless, there are still gaps in the knowledge to be able to determine the suitable remuneration for PV systems which participate in reactive power support, and to implement the reactive power dispatch effectively and efficiently in real life. In particular, the costs and benefits of reactive power from PV have not been previously quantified in detail, making it difficult to determine the economic viability and the appropriate remuneration for the reactive power support. Additionally, the currently available optimisation algorithms are not very suitable to schedule the reactive power dispatch in the presence of large number of PV in real time either. Thus, this thesis aims to tackle these two important problems to facilitate the utilisation of PV for reactive power support. First and foremost, this thesis explores and formulates the cost of reactive power from PV thoroughly, subsequently enabling the comprehensive techno-economic analysis of the impact of PV reactive power provision on the distribution system. Next, this thesis also proposes effective algorithms to implement the PV reactive power dispatch in a fast and scalable manner.

1.6 Research Methodology and Parameters To measure the economic and technical impacts of the PV reactive power support in power system, a few performance indices have been used regularly throughout the work. Total operational costs of running the power system—which consist of the cost of providing active and reactive power to fulfil the demand and loss in the system—have been adopted as the performance index to measure economic impact. The impact of PV and its reactive power provision on the system cost indicates its economic benefit and viability. Other derivative indicators have also been used when necessary, such as the net monetary benefit of reactive power provision to the distribution system in Chaps. 2 and 4. Most of the works are conducted in half-hour resolution, suitable for the economic and market analysis,4 therefore steady state quantities such as power losses, voltage deviations, and minimum/maximum voltage magnitudes have been adopted as the technical performance indices. These indices can reveal if and to what extent the PV and its reactive power support are technically beneficial or detrimental to the power system. As far as possible, the different chapters in this thesis have utilised the same test systems, load profile, weather data,5 electricity price, and cost parameters so that the results from across the thesis can be easily compared. In particular, the 69-bus [87] and 119-bus [88] radial distribution systems have been extensively employed as the test systems due to their wide adoption in the field. Nevertheless, due to the different requirements and nature of some of the works, some variations in the implementation

4 Singapore

electricity market is operating on a half-hour basis [63].

5 The weather forecast data employed in this thesis have been mainly generated by Hadrien Verbois

from Solar Energy Research Institute of Singapore (SERIS), and by Wenjie Zhang from Electrical and Computer Engineering Department, National University of Singapore (NUS).

1.6 Research Methodology and Parameters

13

parameters are present. The implementation parameters are always displayed in every chapter for ease of reading. Although the motivation in proposing the algorithms are to conduct reactive power dispatch from PV efficiently, these algorithms can also include active power dispatch from other sources, as noted and shown in Chaps. 3 and 5. In the analyses, the presence of other technologies such as EV and BESS have also been considered. Both EV and BESS, which are also increasingly becoming more important, will be inseparable aspects of future power systems.

1.7 Research Contributions This thesis contributes to the PV and power system fields in both the theoretical and implementation aspects. On the one hand, theoretically, this thesis offers the following contributions: [1] Formulates the cost of reactive power support from PV inverter and other inverterbased DERs. These costs are divided into two components, namely the power loss component, and the inverter lifetime reduction component. This work is the first to propose such cost formulation and to quantify it at varying PV penetration and power output. [2] Comprehensively analyses the cost and benefits of PV reactive power support to determine the conditions in which PV reactive power support are beneficial to both the PV owners and the power system. Its economic and technical viability have also been compared with switched capacitors, representing traditional reactive power devices. [3] Determines the feasible range of remuneration for PV owners participating in reactive power support. On the other hand, the thesis also contributes towards practical implementation of PV reactive power support by proposing two optimisation approaches, namely: [1] A centralised analytical approach to solving reactive power dispatch. The proposed approach removes the dependence on initial solutions and parameter tuning, shows superior results compared to benchmark algorithms, and is up to 100 times faster compared to existing algorithms commonly used. [2] A local approach which is able to optimise global objectives, such as minimising operational cost of power system and the line losses, without any communication and without any information regarding the grid topology. The approach has been shown to perform better than other local and distributed algorithms in test systems with large number of PV. It is the author’s hope that the above-mentioned contributions will be used as a foundation for future research on PV reactive power provision, and will allow the development of suitable regulations to avoid costly retrofitting in the future [89].

14

1 Introduction

1.8 Thesis Outline The rest of this thesis has been arranged in chronological order of the research conducted to allow the readers to understand the evolution of the research work, and also because the later chapters are built on the findings elaborated in the earlier chapters. Each chapter has its own introduction, outlining the previous works in the literature that have led to the respective work, and highlighting the main contributions of the work done in the chapter. The relationship among the different chapters are illustrated in Fig. 1.7. The publications resulting from the works in the respective chapters are mentioned below as references, and are also reiterated at the end of each chapter’s introduction. Firstly, the first component of reactive power cost from PV, called the inverter loss component, was formulated in Chap. 2. Subsequently, the technical and economic benefits of local reactive power provision were quantified using optimisation of power dispatch in distribution systems [90, 91]. To determine the competitiveness of PV inverters in providing reactive power, the benefits of reactivecapable PVs have also been compared against those arising from traditional reactive power devices [92]. Extensive sensitivity analyses were performed to determine the scenarios in which one technology is more competitive than the other. Next, recognising the benefits of local reactive power provision and the need to optimise the power dispatch in the distribution system quickly and reliably, Chap. 3 proposes an analytical approach to solving optimal reactive power dispatch [93]. Based on the insights in Chap. 2 regarding reactive power provision using PV, other DERs were also incorporated to study their interactions, which will be imperative in future power system.

Fig. 1.7 Thesis Outline

1.8 Thesis Outline

15

Another cost component for reactive power support from PV, the inverter lifetime reduction (ILR) component, was then identified in Chap. 4, by combining the findings from power electronics research and power system economics [94]. Utilising the analytical approach developed in Chap. 3, the benefits of local reactive power provision to the system and its effect on the lifetime of PV inverters were analysed. It was found that the cost of inverter lifetime reduction is a significant part of the reactive power cost (more than 50% at lower PV penetration), but decreases at higher PV penetration when the reactive power support is distributed over more PV systems. The competitiveness of PV as a reactive power compensator was thus revisited, incorporating the newly formulated ILR component [95]. It was noted that the analytical approach developed in Chap. 3 requires comprehensive communication infrastructures in power system, which are not yet present. Therefore, to handle increasing number of PV in distribution systems in the absence of communication infrastructure, a data-driven local optimisation of global objectives algorithm was proposed in Chap. 5 [96, 97]. The algorithm is able to satisfy the voltage constraints using only locally available information, and has been shown to be more stable and have better performance compared to other local and distributed algorithms. Finally, this thesis is concluded in Chap. 6, where possible research works to further the field of PV reactive power support are also elaborated.

References 1. CBC News (2015) How a 2C temperature increase could change the planet. https://www.cbc. ca/news2/interactives/2degrees/. Visited on 09/08/2019 2. United Nations (2015) Adoption of the Paris agreement. Tech Rep, p 32 3. Blakers A (2017) Solar is now the most popular form of new electricity generation worldwide. http://theconversation.com/solaris-now-the-most-popular-form-of-newelectricity-generation-worldwide-81678. Visited on 03/29/2019 4. Hill JS (2018) Global solar market installed 98.9 gigawatts in 2017. https://cleantechnica. com/2018/03/19/global-solar-market-installed-98-9-gigawatts-in-2017/7B7D0A. Visited on 03/29/2019 5. Saretta E, Caputo P, Frontini F (2019) A review study about energy renovation of building facades with BIPV in urban environment. Sustain Cities Soc 44:343–355. ISSN 22106707. https://doi.org/10.1016/j.scs.2018.10.002 6. World Bank Group (2019) ESMAP, and SERIS, Where sun meets water: floating solar market report. World Bank. Washington, Tech Rep, DC 7. Luerssen C, Gandhi O, Reindl T, Sekhar C, Cheong D (2019) Levelised cost of storage (LCOS) for solar-pv-powered cooling in the tropics, Appl Energy 242(February):640–654, ISSN 03062619. https://doi.org/10.1016/j.apenergy.2019.03.133 8. Luerssen C, Gandhi O, Reindl T, Sekhar C, Cheong D (2020) Life cycle cost analysis (LCCA) of PV-powered cooling systems with thermal energy and battery storage for off-grid applications, Appl Energy 273(May):115–145. ISSN 0306-2619. https://doi.org/10.1016/j.apenergy.2020. 115145

16

1 Introduction

9. Rodríguez-Gallegos CD, Gandhi O, Yacob Ali JM, Shanmugam V, Reindl T, Panda SK (2018) On the grid metallization optimization design for monofacial and bifacial Si-based PV modules for real-world conditions, IEEE J Photovolt 1–7. https://doi.org/10.1109/JPHOTOV. 20182882188 10. Rodríguez-Gallegos CD, Bieri M, Gandhi O, Singh JP, Reindl T, Panda S (2018) Monofacial vs bifacial Si-based PV modules: which one is more cost effective. Sol Energy 176(October):412– 438, ISSN 0038-092X. https://doi.org/10.1016/j.solener.2018.10.012 11. Liu H, Rodríguez-Gallegos CD, Liu Z, Buonassisi T, Reindl T, Peters IM (2020) A worldwide theoretical comparison of outdoor potential for various silicon-based tandem module architecture. Cell Rep Phys Sci 1(4):100 037. ISSN 2666-3864. https://doi.org/10.1016/j.xcrp.2020. 100037 12. Rodríguez-Gallegos CD, Liu H, Gandhi O, Singh JP, Krishnamurthy V, Kumar A, Stein JS, Wang S, Li L, Reindl T, Peters IM (2020) Global technoeconomic performance of bifacial and tracking photovoltaic systems. Joule 1–28. ISSN 2542-4351. https://doi.org/10.1016/j.joule. 2020.05.005 13. Rodríguez-Gallegos CD, Gandhi O, Panda SK, Reindl T (2020) On the PV tracker performance: tracking the sun versus tracking the best orientation. IEEE J Photovolt 1–7. ISSN 2156-3381. https://doi.org/10.1109/JPHOTOV.2020.3006994 14. Gandhi O, Srinivasan D (eds) (2020) Sustainable energy solutions for remote areas in the tropics. Series green energy and technology. Cham: Springer International Publishing, ISBN 978-3-030-41951-6. https://doi.org/10.1007/978-3-030-41952-3 15. PVPS IEA (2016) Trends 2016 in photovoltaic applications. International Energy Agency Photovoltaic Power Systems Programme (IEA PVPS), pp 1–30 Tech. Rep 16. PVPS IEA (2018) 2018 snapshot of global photovoltaic markets, International Energy Agency Photovoltaic Power Systems Programme (IEA PVPS), pp 1–33. Tech. Rep 17. PVPS IEA (2019) 2019 snapshot of global photovoltaic markets. International Energy Agency Photovoltaic Power Systems Programme (IEA PVPS), pp 1–35. Tech. Rep 18. PVPS IEA (2020) Snapshot of global PV markets 2020, International Energy Agency Photovoltaic Power Systems Programme (IEA PVPS), pp 33–47. Tech. Rep 19. Gandhi O, Kumar DS, Rodríguez-Gallegos CD, Srinivasan D (2020) Review of power system impacts at high PV penetration part I: factors limiting PV penetration. Solar Energy 20. Power Systems Engineering Research Center (PSERC) (2001) Reactive power support services in electricity markets. Tech Rep 21. Wandhare RG, Agarwal V (2014) Reactive power capacity enhancement of a PV-grid system to increase PV penetration level in smart grid scenario. IEEE Trans Smart Grid 5(4):1845–1854, ISSN 1949-3053. https://doi.org/10.1109/TSG.2014.2298532 22. Wu L, Zhao Z, Liu J (2007) A single-stage three-phase grid-connected photovoltaic system with modified MPPT method and reactive power compensation. IEEE Trans Energy Convers 22(4):881–886, ISSN 0885-8969. https://doi.org/10.1109/TEC.2007.895461 23. Cagnano A, De Tuglie E, Liserre M, Mastromauro RA (2011) Online optimal reactive power control strategy of PV inverters. IEEE Trans Ind Electron 58(10):4549–4558. ISSN 0278-0046. https://doi.org/10.1109/TIE.2011.2116757 24. Kekatos V, Wang G, Conejo AJ, Giannakis GB (2014) Stochastic reactive power management in microgrids with renewables. IEEE Trans Power Syst PP (99):1–10. ISSN 0885-8950. https:// doi.org/10.1109/TPWRS.2014.2369452. arXiv: 1409.6758 25. Zhang L, Tang W, Liang J, Cong P, Cai Y (2016) Coordinated day-ahead reactive power dispatch in distribution network based on real power forecast errors. IEEE Trans Power Syst 31(3):2472–2480. ISSN 0885-8950. https://doi.org/10.1109/TPWRS.2015.2466435 26. Yang HT, Liao JT MF-APSO-based multiobjective optimization for PV system reactive power regulation. IEEE Trans Sustain Energy 6(4):1346–1355. ISSN 19493029. https://doi.org/10. 1109/TSTE.2015.2433957 27. Su X, Masoum MA, Wolfs PJ (2014) Optimal PV inverter reactive power control and real power curtailment to improve performance of unbalanced fourwire LV distribution networks. IEEE Trans Sustain Energy 5(3):967–977. ISSN 1949-3029. https://doi.org/10.1109/TSTE. 2014.2313862

References

17

28. Sousa T, Morais H, Vale Z, Castro R (2015) A multi-objective optimization of the active and reactive resource scheduling at a distribution level in a smart grid context. Energy 85:236–250. ISSN 0360-5442. https://doi.org/10.1016/j.energy.2015.03.077 29. Ding T, Li C, Yang Y, Jiang J, Bie Z, Blaabjerg F, A two-stage robust optimization for centralized-optimal dispatch of photovoltaic inverters in active distribution networks. IEEE Trans Sustain Energy 8(2):744–754. ISSN 1949-3029. https://doi.org/10.1109/TSTE.2016. 2605926 30. Ding T, Liu S, Yuan W, Bie Z, Zeng B (2016) A two-stage robust reactive power optimization considering uncertain wind power integration in active distribution networks. IEEE Trans Sustain Energy 7(1):301–311. ISSN 1949-3029. https://doi.org/10.1109/TSTE.2015.2494587 31. Ziadi Z, Taira S, Oshiro M, Funabashi T (2014) Optimal power scheduling for smart grids considering controllable loads and high penetration of photovoltaic generation. IEEE Trans Smart Grid 5(5):2350–2359. ISSN 1949-3053. https://doi.org/10.1109/TSG.2014.2323969 32. Kim YJ, Kirtley JL, Norford LK (2015) Reactive power ancillary service of synchronous DGs in coordination with voltage control devices. IEEE Trans Smart Grid 8(2):515–527. ISSN 1949-3053. https://doi.org/10.1109/TSG.2015.2472967 33. Chen S, Hu W, Chen Z (2016) Comprehensive cost minimization in distribution networks using segmented-time feeder reconfiguration and reactive power control of distributed generators. IEEE Trans Power Syst 31(2):983–993. ISSN 0885-8950. https://doi.org/10.1109/TPWRS. 2015.2419716 34. Mohseni-Bonab SM, Rabiee A (2016) Optimal reactive power dispatch: a review, and a new stochastic voltage stability constrained multi-objective model at the presence of uncertain wind power generation. IET Gener Transm Distrib 11:815–829. ISSN 1751-8687. https://doi.org/ 10.1049/iet-gtd.2016.1545 35. Yang D, Sharma V, Ye Z, Lim LI, Zhao L, Aryaputera AW (2015) Forecasting of global horizontal irradiance by exponential smoothing, using decompositions. Energy 81:111–119. ISSN 0360-5442. https://doi.org/10.1016/j.energy.2014.11.082 36. Yang Y, Wu W (2018) A distributionally robust optimization model for realtime power dispatch in distribution networks. IEEE Trans Smart Grid. https://doi.org/10.1109/TSG.2018.2834564 37. Zhu T, Luo W, Bu C, Yue L (2016) Accelerate population-based stochastic search algorithms with memory for optima tracking on dynamic power systems. IEEE Trans Power Syst 31(1):268–277. ISSN 0885-8950. https://doi.org/10.1109/TPWRS.2015 38. Weedy BM, Cory B, Jenkins N, Ekanayake J, Strbac G (2012) Electric Power Systems, 5th edn. Wiley, p 513. ISBN 9780470682685 39. Grainger JJ, Stevenson WD (1994) Power system analysis. McGraw-Hill. ISBN 0070612935. http://medcontent.metapress.com/index/A65RM03P4874243N.pdf 40. Kundur P (1994) Power system stability and control, series. The EPRI Power System Engineering. McGraw-Hill. ISBN 978-0070359581 41. UK Power Networks (2016) Use of system charging statement. Tech. Rep. UK Power Networks 42. Pires DF, Antunes CH, Martins AG (2012) NSGA-II with local search for a multi-objective reactive power compensation problem. Int J Electr Power Energy Syst 43(1):313–324. ISSN 01420615. https://doi.org/10.1016/j.ijepes.2012.05.024 43. Stott B, Alsac O (1974) Fast decoupled load flow. IEEE Trans Power Apparatus Syst PAS93(3):859–869. https://doi.org/10.1109/TPAS.1974.293985 44. Hanger J (2003) Reactive power and blackout. https://www.energycentral.com/c/tr/reactivepower-and-blackout. Visited on 04/07/2019 45. Parmar J (2011) How reactive power is helpful to maintain a system healthy. https://electricalengineering-portal.com/howreactive-power-is-helpful-to-maintain-a-system-healthy. Visited on 04/07/2019 46. Bhattacharya K, Zhong J (2001) Reactive power as an ancillary service. IEEE Trans Power Syst 16(2):294–300 47. IEA (2019) The California duck curve, Paris. https://www.iea.org/data-and-statistics/charts/ the-california-duck-curve. Visited on 04/28/2020 48. CAISO (2016) Fast facts: CAISO duck curve

18

1 Introduction

49. Porter K, Fink S, Rogers J, Mudd C, Buckley M, Clark C, Hinkle G (2012) PJM renewable integration study: review of industry practice and experience in the integration of wind and solar generation. GE Energy Tech Rep 50. Riesz J, Macgill I (2013) Frequency control ancillary services: is Australia a model market for renewable integration? In: Proceedings of the 12th international workshop on large-scale integration of wind power into power systems, London 51. Energy Market Authority (2018) Intermittency pricing mechanism for intermittent generation sources in the National Electricity Market of Singapore: final determination paper. Energy Market Authority, Singapore, Tech Rep 52. Ellis A, Nelson R, Engeln EV, Walling R, Mcdowell J, Casey L, Seymour E, Peter W, Barker C, Kirby B (2012) Reactive power interconnection requirements for PV and wind plants—recommendations to NERC, Sandia National Laboratories, Albuquerque. Tech Rep SAND2012-1098 53. National Renewable Energy Laboratory (2014) Advanced inverter functions to support high levels of distributed solar policy and regulatory the need for advanced, NREL Technical Report NREL/TP-5200-57991, pp 1–8 54. Bernards R, Morren J, Slootweg JG (2014) Maximum PV-penetration in lowvoltage cable networks. In: 7th IEEE young researchers symposium. IEEE, Ghent, Belgium, pp 1–5 55. Hoke A, Butler R, Hambrick J, Kroposki B (2013) Steady-state analysis of maximum photovoltaic penetration levels on typical distribution feeders. IEEE Trans Sustain Energy 4(2):350– 357. ISSN 1949-3029. https://doi.org/10.1109/TSTE.2012.2225115 56. Kumar DS, Gandhi O, Rodríguez-Gallegos CD, Srinivasan D (2020) Review of power system impacts at high PV penetration part II: potential solutions and the way forward. Sol Energy Under Second Rev 57. IEEE standards Coordinating Committee (2014) 21, 1547 IEEE Standard for Interconnecting Distributed Resources with Electric Power Systems Amendment 1. New York 58. Southern California Edison (2017) Electric Rule 21 59. SMA, SMA shifts the phase. https://www.sma-australia.com.au/partners/knowledgebase/smashifts-the-phase.html. Visited on 04/05/2019 60. Huawei (2018) Smart string inverter sun2000-105ktl-h1, Shenzhen. http://solar.huawei.com/ en-GB/download?p=7B 61. IEEE Standards Coordinating Committee (2018) IEEE standard for interconnection and interoperability of distributed energy resources with associated electric power systems interfaces, New York. IEEE. ISBN 9781504446396 62. EMC (2016) Use of system charges. https://www.mypower.com.sg/documents/ts-usc.pdf. Visited on 04/01/2018 63. EMC (2018) Energy market price information. https://www.emcsg.com/marketdata/ priceinformation. Visited on 04/01/2018 64. Gabash A, Li P (2012) Active-reactive optimal power flow in distribution networks with embedded generation and battery storage. IEEE Trans Power Syst 27(4):2026–2035. ISSN 08858950. https://doi.org/10.1109/TPWRS.2012.2187315.arXiv: 9605103 [cs] 65. Liang RH, Wang JC, Chen YT, Tseng WT (2015) An enhanced firefly algorithm to multiobjective optimal active/reactive power dispatch with uncertainties consideration, Int J Electr Power Energy Syst 64 1088–1097. ISSN 01420615. https://doi.org/10.1016/j.ijepes.2014.09. 008 66. Samimi A, Kazemi A, Siano P (2015) Economic-environmental active and reactive power scheduling of modern distribution systems in presence of wind generations: a distribution market-based approach. Energy Convers Manage 106 495–509. ISSN 01968904. https://doi. org/10.1016/j.enconman.2015.09.070 67. Lee KY, Park YM, Ortiz JL (1985) A united approach to optimal real and reactive power dispatch. IEEE Power Eng Rev PER-5(5):42–43. ISSN 02721724. https://doi.org/10.1109/ MPER.1985.5526580 68. Saravanan B, Das S, Sikri S, Kothari DP (2013) A solution to the unit commitment problema review. Front Energ 7(2):223–236. ISSN 20951701. https://doi.org/10.1007/s11708-0130240-3

References

19

69. Gandhi O, Rodríguez-Gallegos CD, Srinivasan D (2016) Review of optimization of power dispatch in renewable energy system. In: 2016 IEEE Innovative Smart Grid Technologies— Asia (ISGT-Asia), Melbourne. IEEE, pp 250–257. ISBN 978-1-5090-4303-3. https://doi.org/ 10.1109/ISGT-Asia.2016.7796394 70. Baños R, Manzano-Agugliaro F, Montoya FG, Gil C, Alcayde A, Gómez J (2011) Optimization methods applied to renewable and sustainable energy: a review. Renew Sustain Energy Rev 15(4):1753–1766. ISSN 13640321. https://doi.org/10.1016/j.rser.2010.12.008 71. Wu W, Hu Z, Song Y (2016) A new method for opf combining interior point method and filled function method. In: 2016 IEEE Power and Energy Society General Meeting (PESGM), pp 1–5. IEEE. ISBN 978-1-5090-4168-8. https://doi.org/10.1109/PESGM.2016.7741321 72. Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press. ISBN 9780521833783 73. Bazaraa MS, Sherali HD, Shetty CM (2006) Nonlinear programming: theory and algorithms, 3rd edn. Wiley. ISBN 978-0-471-78777-8 74. Sivanandam S, Deepa S (2008) Introduction to genetic algorithms, Springer, p 453. ISBN 9783540731894. https://doi.org/10.1007/978-3-540-73190-0 75. Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of IEEE international conference on neural networks, vol 4, pp 1942–1948. ISSN 19353812. https://doi.org/ 10.1109/ICNN.1995.488968 76. Logenthiran T, Srinivasan D, Shun TZ (2012) Demand side management in smart grid using heuristic optimization. IEEE Trans Smart Grid 3(3):1244–1252. ISSN 19493053. https://doi. org/10.1109/TSG.2012.2195686 77. Talbi EG (2009) Metaheuristics: From design to implementation. Wiley. ISBN 978-0-47027858-1 78. Yang XS (2010) Nature-inspired metaheuristic algorithms. Luniver Press 79. Ahn C, Peng H (2013) Decentralized voltage control to minimize distribution power loss of microgrids. IEEE Trans Smart Grid 4(3):1297–1304. ISBN 1949-3053. https://doi.org/10. 1109/TSG.2013.2248174 80. Arnold DB, Negrete-Pincetic M, Sankur MD, Auslander DM, Callaway DS (2016) Model-free optimal control of var resources in distribution systems: an extremum seeking approach. IEEE Trans Power Syst 31(5):3583–3593. ISSN 08858950. https://doi.org/10.1109/TPWRS.2015. 2502554 81. Bolognani S, Carli R, Cavraro G, Zampieri S (2015) Distributed reactive power feedback control for voltage regulation and loss minimization, IEEE Trans Autom Control 60(4):966–981. ISSN 00189286. https://doi.org/10.1109/TAC.2014.2363931 arXiv 1303.7173 82. Šulc P, Backhaus S, Chertkov M (2014) Optimal distributed control of reactive power via the alternating direction method of multipliers. IEEE Trans Energy Convers 29(4):968–977. ISSN 08858969. https://doi.org/10.1109/TEC.2014.2363196. arXiv 1310.5748 83. Dall’Anese E, Dhople SV, Giannakis GB (2016) Photovoltaic inverter controllers seeking AC optimal power flow solutions. IEEE Trans Power Syst 31(4):2809–2823, ISSN 08858950. https://doi.org/10.1109/TPWRS.2015.2454856. arXiv: 1501.0188 84. Zhang W, Liu W, Wang X, Liu L, Ferrese F (2014) Distributed multiple agent system based online optimal reactive power control for smart grids. IEEE Trans Smart Grid 5(5):2421–2431, ISSN 19493053. https://doi.org/10.1109/TSG.2014.2327478 85. Erseghe T (2014) Distributed optimal power flow using ADMM. IEEE Trans Power Syst 29(5):2370–2380. ISSN 08858950. https://doi.org/10.1109/TPWRS.2014.2306495 86. Zheng W, Wu W, Zhang B, Sun H, Liu Y (2016) A fully distributed reactive power optimization and control method for active distribution networks. IEEE Trans Smart Grid 7(2):1021–1033. ISSN 19493053. https://doi.org/10.1109/TSG.2015.2396493 87. Savier JS, Das D (2007) Impact of network reconfiguration on loss allocation of radial distribution systems. IEEE Trans Power Delivery 22(4):2473–2480. ISSN 08858977. https://doi. org/10.1109/TPWRD.2007.905370 88. Zhang D, Fu Z, Zhang L (2007) An improved TS algorithm for loss-minimum reconfiguration in large-scale distribution systems. Electr Power Syst Res 77(5-6):685–694. ISSN 03787796. https://doi.org/10.1016/j.epsr.2006.06.005

20

1 Introduction

89. Stetz T, Rekinger M, Theologitis I (2014) Transition from uni-directional to bi-directional distribution grids. International Energy Agency, Kassel, Tech Rep, p 154 90. Gandhi O, Zhang W, Rodríguez-Gallegos CD, Srinivasan D, Reindl T (2016) Continuous optimization of reactive power from PV and EV in distribution system, In: 2016 IEEE innovative smart grid technologies—Asia (ISGT-Asia), Melbourne: IEEE, pp 281–287. ISBN 978-1-50904303-3. https://doi.org/10.1109/ISGT-Asia.2016.7796399 91. Gandhi O, Rodríguez-Gallegos CD, Zhang W, Srinivasan D, Reindl T (2018) Economic and technical analysis of reactive power provision from distributed energy resources in microgrids. Appl Energy 210:827–841. ISSN 03062619. https://doi.org/10.1016/j.apenergy.2017.08.154 92. Gandhi O, Srinivasan D, Rodríguez-Gallegos CD, Reindl T, Competitiveness of reactive power compensation using PV inverter in distribution system. In: 2017 IEEE PES Innovative Smart Grid Technologies Conference Europe (ISGT-Europe), Torino, Italy, pp 1–6. IEEE. ISBN 9781-5386-1953-7. https://doi.org/10.1109/ISGTEurope.2017.8260238 93. Gandhi O, Zhang W, Rodríguez-Gallegos CD, Bieri M, Reindl T, Srinivasan D (2018) Analytical approach to reactive power dispatch and energy arbitrage in distribution systems with DERs. IEEE Trans Power Syst 33(6):6522–6533. ISSN 0885-8950. https://doi.org/10.1109/ TPWRS.2018.2829527 94. Gandhi O, Rodríguez-Gallegos CD, Gorla NBY, Bieri M, Reindl T, Srinivasan D (2019) Reactive power cost from PV inverters considering inverter lifetime assessment. IEEE Trans Sustain Energy 10(2):738–747. ISSN 1949-3029. https://doi.org/10.1109/TSTE.2018.2846544 95. Gandhi O, Rodríguez-Gallegos CD, Reindl T, Srinivasan D (2018) Competitiveness of PV inverter as a reactive power compensator considering inverter lifetime reduction. Energy Procedia 150:74–82. ISSN 18766102. https://doi.org/10.1016/j.egypro.2018.09.005 96. Gandhi O, Rodriguez-Gallegos CD, Reindl T, Srinivasan D (2018) Locally determined voltage droop control for distribution systems. In: 2018 IEEE Innovative Smart Grid Technologies— Asia (ISGT Asia), Singapore, pp 425–429. IEEE. ISBN 978-1-5386-4291-7. https://doi.org/ 10.1109/ISGT-Asia.2018.8467784 97. Gandhi O, Zhang W, Rodríguez-Gallegos CD, Verbois H, Sun H, Reindl T, Srinivasan D (2020) Local reactive power dispatch optimisation minimising global objectives. Appl Energy 262. ISSN 03062619. https://doi.org/10.1016/j.apenergy.2020.114529

Chapter 2

Analysis of Local Reactive Power Provision Using PV in Distribution Systems

2.1 Introduction Approximately 50% of global PV installations have occurred at the distribution level by the end of 2015 [1]. And in many countries, the share of distributed PV systems can go much higher, e.g. 80% of the PV capacity in Germany and nearly all of PV capacity in Italy and Australia [2, 3]. By having the solar power generation close to where the load is consumed, transmission losses can be significantly reduced [4], voltage improved, and congestion of the lines avoided [5]. On top of their active power capability, PVs (and other inverter-based DERs) have been shown to be capable of providing reactive power to the grid through inverters [6–9]. Compared to traditional power factor correction devices used in distribution systems, i.e. capacitor banks, inverters have faster response time and therefore can regulate voltage more accurately [6]. Recognising the capability and the benefits of DERs’ reactive power provision for voltage regulation, IEEE Standard 1547 and California’s Electric Tariff Rule 21 have allowed and even required smart inverters to provide dynamic reactive power compensation to keep the voltage of the distribution system within the acceptable range [10–12]. Yet, despite the increasing implementation of PV reactive power support, the cost of procuring the reactive power is not clear. In the literature, reactive power payment functions have been formulated for traditional generators [13, 14] and reactive power devices [15]. Using the payment functions, the reactive power dispatch (RPD) optimisation has been explored using market-based approaches, with objective functions ranging from minimisation of expected payment function [13, 14, 16], societal advantage function [17], as well as other objectives including maximising voltage stability and minimising transmission losses [18]. However, the market-based approaches were mainly explored in transmission systems with limited number of DERs. In addition, the coefficients in the payment functions were generated randomly and therefore it is not clear how

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 O. Gandhi, Reactive Power Support Using Photovoltaic Systems, Springer Theses, https://doi.org/10.1007/978-3-030-61251-1_2

21

22

2 Analysis of Local Reactive Power Provision Using PV in Distribution Systems

economical it is to produce the reactive power. Furthermore, the optimisation generally occurs for only one time step, which is not suitable for time-dependent DERs such as PV. Recently, there is more research considering the integrated optimisation of active and reactive power [5, 19–23]. The approaches have either been an integrated activereactive power market-based approach [19, 20] or day-ahead generation scheduling [5, 21–23]. The market-based approach still suffers from most of the aforementioned drawbacks. Although [20] considered active and reactive power markets in distribution systems with wind generation for 24 h, the authors did not take the line losses into account and used arbitrary values for the reactive power prices. Meanwhile, authors who employed the generation scheduling approach have optimised the integrated active and reactive power dispatch in distribution systems with DERs, but have not taken into account the costs of reactive power produced [5, 21–23]. Nevertheless, reactive power can be a significant part of the costs and by ignoring them entirely, the optimisation of power dispatch will not yield optimum economic results. It is also of utmost importance to explicitly calculate the cost of reactive power provision from DERs and assess their competitiveness before power system planners and operators can determine whether a shift from traditional reactive power devices towards inverters with reactive power capability is cost effective. Even though reactive power costs are much smaller than the active power costs, the investment involved is significant. The work in this chapter focuses on PV in a small test system, to allow direct and clear understanding of the effects of local reactive power provision. However, the methods and insights from this chapter can be extended to other inverter-based DERs in larger test systems, as shown in Chap. 3 later. This chapter has two main studies. Firstly, the economic and technical impacts of reactive power provision using PV were analysed using proposed reactive power cost formulation. Secondly, the cost competitiveness of PV in providing local reactive power provision was compared with switched capacitors (SCs), representing conventional reactive power devices. Thorough sensitivity analyses were conducted to determine the conditions in which the inverter-based reactive capable PV is preferred for reactive compensation or otherwise. The sensitivity analyses also allow planners to choose the parameters and results presented in this work which are applicable to them based on their market situation. The two studies have been published as conference papers [24, 25] and subsequently in Applied Energy journal [26]. Similar figures and tables as in those publications have been recreated to suit the data and test systems used in this chapter and thesis. The main contributions of this chapter are therefore: [1] Formulates explicitly the cost of reactive power support using PV and analyses its competitiveness to grid reactive power charge and SC [2] Establishes the economic and technical benefits of local reactive power provision [3] Analyses extensively the impacts of changes in parameters on the technoeconomic benefits of reactive power compensation from PV and SC.

2.1 Introduction

23

This chapter continues as follows. Section 2.2 describes the optimisation problem, followed by Sect. 2.3, where the reactive power cost for PV is formulated. The test system and parameters employed in the simulations are outlined in Sect. 2.4. Section 2.5 analyses the economic and technical impact of reactive power provision from PV, whereas Sect. 2.6 thoroughly assesses its competitiveness under different conditions when compared to SC. This chapter is finally summarised in Sect. 2.7.

2.2 Optimisation of Power Dispatch 2.2.1 Proposed Objective Function In this chapter (as well as in the rest of the thesis, unless otherwise stated), the perspective of an independent system operator (ISO) has been assumed, whose task is to ensure the electricity demand in the system is fulfilled in the most cost-efficient way. Here, the goal of the ISO is to minimise the total costs of running a grid-connected distribution system containing M number of PV systems with reactive power capability. The ISO is assumed to have been authorised to utilise unused inverters’ capacity of the PV systems for reactive power provision. The PV systems are paid for the active power they generate to recover the investment, and are compensated for the reactive power produced as explained in Sect. 2.3. The objective function is therefore expressed as Eq. (2.1):

Cost =

T   t=1 M  x=1



Pgrid

ct 

grid

cost of P from grid PV cPPV Px,t + x





cost of P from PV

Pgrid

grid

P + cQgrid Q t +  t     M  x=1



(2.1)

cost of Q from grid

 QPV cx,t Q PV x,t H 



cost of Q from PV

QPV where ct (cQgrid ) and cxPPV (cx,t ) are the cost of active (reactive) power from the PPV grid and PV, respectively. cx is constant throughout the lifetime of the PV, as the levelised cost of electricity (LCOE) has taken into account the energy that will be grid grid generated by the PV throughout its lifetime. Pt and Q t are the active and reactive PV PV power purchased from the grid at period t. Px,t and Q x,t are the active and reactive power generated by the x th PV system at period t. Finally, H is the length of the period in [hour]. For this chapter and most of the thesis, one-day simulations have been carried out with half-hour time steps, i.e. H = 0.5 and T = 48. On top of Cost as the objective function, there are other performance indices that are used to quantify the results of reactive power dispatch using PV. The technical indicators used in this chapter, and in the subsequent ones, are total apparent voltage deviation index (WVDI), and minimum voltage power loss, S loss , weighted

magnitude, min Vi,t .

24

2 Analysis of Local Reactive Power Provision Using PV in Distribution Systems

S loss is calculated based on the following formula: Sloss [%] =

Total energy loss [kVAh] × 100% Total energy input [kVAh]

(2.2)

WVDI, which penalises larger magnitude of voltage deviation more heavily, is therefore favoured over absolute voltage deviation, is obtained through the following: WVDI =

T  N 

Vi,t − Vref

2

(2.3)

t=1 i=1

where Vref is the reference voltage, generally taken to be the voltage magnitude at the substation and is fixed at 1 p.u.

2.2.2 Power System Constraints The constraints of the power system are: grid

Pt



N 

Pi,tloss −

i=1 grid

Qt



N 

N 

Pi,tload +

i=1

P j,t = Pi,t − ri j

N 

load Q i,t +

2 Pi,t2 + Q i,t

Q j,t = Q i,t − xi j

(2.4)

M 

Q PV x,t = 0, t = 1, ..., T

(2.5)

x=1

i=1

Vi,t2

PV Px,t = 0, t = 1, ..., T

x=1

i=1 loss Q i,t −

M 

load PV − P j,t + P j,t , ∀(i, j) ∈ L

2 Pi,t2 + Q i,t

PV − Q load j,t + Q j,t , ∀(i, j) ∈ L Vi,t2 2 loss 2 ∗ = = Vi,t − V j,t Ii,t , ∀(i, j) ∈ L Pi,tloss + Q i,t

loss Si,t



Ii,t ≤ Imax , i = 1, ..., N , t = 1, ..., T Vmin ≤ |Vi,t | ≤ Vmax , i = 1, ..., N , t = 1, ..., T

(2.6) (2.7) (2.8) (2.9) (2.10)

where index i and j indicates quantities for the ith and jth node in a system composed loss loss , and Si,t are the active, reactive, and apparent power losses of N nodes. Pi,tloss , Q i,t on the line from sending node i. The active and reactive load are represented by Pi,tload load and Q i,t respectively. ri j and xi j are the resistance and reactance of the line (from PV the set of lines L) connecting node i and j. P j,t (Q PV j,t ) are the active (reactive) PV 1 generation at node j, respectively. Vi,t is the voltage of node i, whereas Vmin and 1 P PV i,t

PV are different from P PV , in the sense that the former two quantities are concerned and P j,t x,t with the PV generation at some particular nodes i and j (which may or may not have PV), whereas

2.2 Optimisation of Power Dispatch

25

Vmax are its lower and upper bound. Ii,t is the current flowing from sending node i, limited by Imax as its upper bound. Equation (2.4) is the active power balance constraint. The reactive power balance constraint is expressed in Eq. (2.5) where positive Q PV x,t represents reactive power provision from PV. Equations (2.6) and (2.7) are the active and reactive power flow from sending node i to receiving node j. Equation (2.8) illustrates the relation between active and reactive power, as well as the line loss calculation from voltage and current. The line losses are therefore linked to the power of the PV via the voltage and current quantities. The current flowing through the distribution lines are limited according to Eq. (2.9). Lastly, the voltage at each node needs to lie within a certain range (Eq. (2.10)). The current and voltage for all the lines and nodes are obtained through the backward forward sweep (BFS) algorithm, an accurate load flow algorithm for radial distribution systems [27]. To declutter the subsequent equations, the qualifiers are removed. However, the equations are still valid for all i = 1, ..., N , x = 1, ..., M, and t = 1, ..., T when the respective indices appear in the equations, unless otherwise stated.

2.2.3 PV System Constraints PV The forecasted power of a PV system, Px,t , can be calculated using the following Eqs. [28, 29]: PV = Px,t

PV G x,t PV PV,invloss Px,rated × 1 − η T Tx,t − 25[◦ C] − Px,t 2 1000[W/m ]

(2.11)

PV is its DC where G x,t is the irradiance received by the x th PV system and Px,rated PV,invloss T is the power loss in the inverter, while η is the power temperature rating. Px,t coefficient of the solar cells/panels in the PV system. PV , the temperature of the solar cells, is then calculated through: Tx,t PV a = Tx,t + Tx,t

G x,t × (NOCT − 20[◦ C]) 800[W/m2 ]

(2.12)

a where Tx,t is the ambient temperature experienced by the x th PV system and NOCT is the nominal operating temperature of solar cells. a PV , and Tx,t also vary within a particular PV Although quantities like G x,t , Tx,t system, they are assumed to be the same in this thesis for simplicity. This is a reasonable assumption because we are only concerned with the PV power at halfhourly resolution and with the potential of PV for reactive power support.

the latter is concerned with the generation of the x th PV. Similar notations are also used for Q PV . As is the case throughout the thesis, i and j are indices reserved for nodes in a system, whereas x is the index for PV and other DERs.

26

2 Analysis of Local Reactive Power Provision Using PV in Distribution Systems

a a70 PV70 By replacing G x,t and Tx,t with G 70 x,t and Tx,t in Eq. (2.11) and (2.12), Pt 70 a70 PV70 can be obtained. G x,t , Tx,t , and Px,t are the upper limit of forecasted irradiance, ambient temperature, and PV power at 70% confidence level respectively. The values PV70 is subsequently used to find the reactive power constraint for PV: of Px,t

 2

PV

PV,invloss PV70 2

Q ≤ Q PV = S PV − Px,max − Px,t (2.13) x,t

x,t,lim

x,max

where Q PV x,t,lim is the maximum reactive power that can be generated by the PV system PV is without reducing its active power output beyond the additional inverter loss. Sx,max PV,invloss is the maximum inverter loss (when PV is operating the inverter rating and Px,max at its maximum power). From Eq. (2.13), it can be seen that the maximum PV reactive power output is limited by the upper limit of the PV active power forecast, rather than the forecasted value. This ensures that the owner of the PV system is not curtailing the PV active power generation because of the reactive power provision since P PV in general has higher monetary value than Q PV . The 70% confidence level has also been utilised by Pennsylvania New Jersey Maryland (PJM) Interconnection for unit commitment [2]. The optimisations implemented in this work represent the day-ahead scheduling and the values of the forecast can be updated accordingly.

2.2.4 Tap-Changing Transformer Traditionally, many distribution systems use tap-changing transformers to control the voltage in the system, which are usually located at the first bus in a distribution system [5, 30]. The turns ratio inside a tap-changing transformer can be adjusted, hence enabling the reference voltage of the distribution system to be modified to prevent voltage violations. A type of tap-changing transformer, called on-load tap changer (OLTC), is commonly used as it allows changing the turns ratio while being connected to the load. As such, an OLTC has been included in this study to see the impact of PV reactive power support with and without OLTC in the system. OLTC generally has discrete taps and can be operated according to the following constraints: τmin ≤ τt ≤ τmax |τt − τt−1 | ≤ τmax

(2.14) (2.15)

where τt is the tap position of the OLTC, which must be within the minimum (τmin ) and maximum (τmax ) tap position. τmax is the maximum tap change within a single time step. τt determines the voltage of the node where the OLTC is located (usually the first node) and hence the voltage of the subsequent nodes. By modifying the voltage, it also changes the current values and therefore the power losses. As such, OLTC operations can be optimised to minimise Eq. (2.1). The parameters for the OLTC and the impact that OLTC has on the system voltage profile are elaborated in Sects. 2.4 and 2.5 respectively.

2.3 Proposed Reactive Power Costs

27

2.3 Proposed Reactive Power Costs The inverters that connect PV to the system are assumed to have reactive power capability. The reactive power injection or absorption are limited by Eq. (2.13) where PV PV is taken to be the same as Px,rated . Therefore, no additional investment is Sx,max necessary to provide reactive power using the PV system [7]. Nevertheless, it is PV PV can be larger than Sx,max by up to noted that for many commercial systems, Px,rated 30%. To minimise the objective function in Eq. (2.1), we first need to formulate the costs of producing reactive power using PV. In [31], Braun showed that inverters suffer from power losses which are dependent on the apparent power output of the inverters. When the inverters inject reactive power, the apparent power flowing through them increases, inducing additional power loss in the inverters. The power flow through the inverter is illustrated in Figure 2.1. The cost of reactive power is therefore the additional power loss multiplied by the cost of the electricity to fulfil the additional loss. The power loss in the inverter is approximated by [31]: P invloss = cself + cV S + cR S 2

(2.16)

where cself , cV , and cR are standby loss coefficient, voltage-dependent loss coefficient, and current-dependent loss coefficient of an inverter, respectively. They are constants determined experimentally that fit the efficiency curve of the inverter. When there is no reactive power involved, the power loss in the inverter can be expressed as: PV,invloss Px,t

 PV PV 2 cxPV,self + cxPV,V Px,t + cxPV,R Px,t = 0

PV Px,t = 0 PV Px,t = 0

(2.17)

And when there is reactive power usage in the inverter, the power loss is now increased to:  PV PV 2 PV cxPV,self + cxPV,V Sx,t + cxPV,R Sx,t Px,t = 0 PV,invloss (2.18) = PV,self Px,t 2 PV,V PV PV,R PV PV cx Q x,t + cx Q x,t + cx Px,t = 0

Fig. 2.1 PV power flow through an inverter

28

2 Analysis of Local Reactive Power Provision Using PV in Distribution Systems

PV 2 PV 2 PV 2 PV where Sx,t is the apparent power output from the x th PV, Sx,t = Px,t + Q x,t . Therefore, the additional losses due to the reactive power injection can be expressed by subtracting Eq. (2.17) from (2.18) [31]:  PV PV 2  PV PV 2 cxPV,V Sx,t + cxPV,R Sx,t − Px,t − Px,t = PV 2 PV,R cxPV,self + cxPV,V Q PV Q x,t x,t + cx 

PV,invloss Px,t

PV Px,t = 0 PV Px,t =0 (2.19)

The calculation of the costs can be different because different sources of active power may be used to compensate for the additional losses. For PV, P PV,invloss is compensated by P PV during the daytime and by P grid in the dawn and at night. However, from the perspective of the ISO, any active power loss must be compensated by taking more power from the grid (because the PV penetration2 is always less than 100% in the current study). Therefore, the cost of reactive power for PV is: Pgrid

QPV PV Q x,t = ct cx,t

PV,invloss × Px,t

(2.20)

2.4 Implementation Setup 2.4.1 Test System A 69-bus radial distribution system [32] has been employed for the case study and PV systems have been added at six nodes for the base cases and at additional ten nodes for higher penetration case (see Sect. 2.6.4.6), as indicated in Fig. 2.2. The total active and reactive energy demand of the system are 96748 kWh and 66690 kvarh respectively over 24 h with normalised Singapore load profile from 27 May 2016 [33]. The locations of the PV have been chosen arbitrarily and the rated power of each PV system is 300 kWp. The OLTC is located at node 1 with τmin and τmax of 0.9 and 1.1 p.u. respectively. There are 15 tap positions from τmin to τmax . τmax is set to 14, since the time step considered is half hour and the time delay in tap changing is much smaller [30]. The OLTC tap position determines the voltage of node 1. Imax , Vmin , Vref , and Vmax are set at 1.0, 0.9, 1.0, and 1.05 p.u. respectively.

2 PV

penetration is defined here as the ratio of the total active power generated by PV systems to the total active power demand in the same period.

2.4 Implementation Setup

29

Fig. 2.2 69-bus radial distribution system with PV systems

2.4.2 Cost Parameters Pgrid

ct is the wholesale electricity price in Singapore on 27 May 2016 plus the associQgrid , is the ated grid charges [34] as shown in Fig. 2.3. The grid reactive power cost, ct reactive power charge in Singapore for consumers at 22 kV or 6.6 kV [35]. The LCOE of PV, cPPV , is taken to be 10.28 cents SGD/kWh, as calculated using Eq. (4.16) for the case of Singapore [36], assuming PV system lifetime and inverter lifetime of 20 and 14 years (see Sect. 4.5 for details). This value is similar to the industrial PV LCOE in Singapore in 2016 [37]. The reactive power cost from switched capacitors (SC), cQSC , is calculated based on the installation and maintenance cost from [38] (adopted due to its suitability for short term case study), and financial parameters, such as debt ratio and discount rate, from [36], resulting in value of 0.00248 SGD/kvarh. The cost parameters used are listed in Table 2.1. Figure 2.4 shows the unit reactive power cost (both provision and absorption), QPV , assuming electricity price of 10.28 cents SGD/kWh. The area at the top right cx,t

Fig. 2.3 Singapore wholesale electricity price plus grid charges on 27 May 2016

30

2 Analysis of Local Reactive Power Provision Using PV in Distribution Systems

Table 2.1 Cost Parameters Parameters [Units] cPPV [SGD/kWh] cQGrid [SGD/kvarh] cQSC [SGD/kvarh]

Values 0.1028 0.00630 0.00248

Fig. 2.4 Unit reactive power cost for inverter-based PV as a function of active and reactive power flowing through the inverter

of the figure is unfeasible as it lies outside of the inverters’ capacity. It can be seen QPV PV is lower at higher Px,t and lower Q PV that cx,t x,t .

2.4.3 Weather Parameters The irradiance forecasting was done by Hadrien Verbois from the Solar Energy Research Institute of Singapore (SERIS), using numerical weather prediction model and quantum gradient boosting as in [39], and post-processing method from [40] to

2.4 Implementation Setup

31

Fig. 2.5 Forecasted (day-ahead) and real solar irradiance data for typical irradiance scenarios

generate the solar irradiance data in Fig. 2.5. Going forward, all the cases in this chapter use the weather data on 27 May 2016, except for Sect. 2.6.4.4 which uses all four weather scenarios shown in Fig. 2.5 to analyse the effects of difference irradiance profile on the techno-economic benefits of local reactive power support using PV. Real irradiance and temperature data are obtained by SERIS at 1.2491◦ , 103.8414◦ . The irradiance forecast are used in Eq. (2.11) while the upper limits of 70% confidence level are used in Eq. (2.13). The real irradiance data are not used in the simulations, but serve as an illustration that the even for day-ahead forecast, the real data would broadly be within the 70% confidence level, except for when there is a washout as in Fig. 2.5c, in which case the values would need to be updated within the day for hour-ahead scheduling.

2.5 Analysis of Economic and Technical Impact 2.5.1 Simulation Setup To analyse the effects of PV reactive power provision on the system, the following cases are considered: Case 1: the distribution system only contains load. This is the original system from [32] with load profile from [33]; Case 2: PV systems are placed in the distribution system according to Fig. 2.2, but the PV systems are not able/allowed to provide reactve power support;

32

2 Analysis of Local Reactive Power Provision Using PV in Distribution Systems

Case 3: the PVs can inject both active and reactive power. Case 3 is optimised using genetic algorithm (GA) explained in Sect. 2.5.2; Case 4: this case has the same setup as Case 3, but instead of optimising the RPD, the PV systems provide the maximum possible reactive power allowed by Eq. (2.13) at all time periods (the total reactive power injection is always below the total reactive power demand in the system, so no cut-off limits are applied). Many distribution systems also have OLTC installed, as mentioned in Sect. 2.2.4. Therefore it is important to see the impact of PV reactive power provision in the presence of OLTC. Similarly, 4 cases with OLTC are considered, namely Case OLTC 1, OLTC 2, OLTC 3, and OLTC 4. These cases are the same as the previous 4 cases, except that now OLTC is also considered and its operations are optimised using GA with constraints (2.14) and (2.15), on top of the rest of the power system constraints.

2.5.2 Optimisation Algorithm The objective function and constraints outlined in Eqs. (2.1)–(2.10) are known to be NP-hard [41–44]. The complexity of the problems and the vastness of the search space, on top of the recursive algorithm—backward forward sweep (BFS)—used to calculate the nodes’ current and voltage values for radial distribution systems, have prompted the use of metaheuristic optimisation. BFS is widely used to calculate load flow in distribution systems due to their solution accuracy and computational efficiency [45]. A possible way to implement BFS is shown in Algorithm 2.1. Other stopping or convergence criteria can be set for step 3. For our case studies, the BFS algorithm is found to converge after a maximum of four iterations, in line with [27, 45]. It is noted that Step 9 and 13 require identification of the loads and nodes beyond a particular line. Any method for the identification, such as that suggested by the original authors in [27], can be used. Genetic Algorithm (GA) [46] has been employed to optimise Q PV x,t and τt according to Eq. (2.1). GA is an optimisation algorithm inspired by the evolution of organisms [47, 48], and is therefore categorised as an evolutionary algorithm. The algorithm starts by generating a population of chromosomes, randomly and/or through educated estimates. Each of the chromosomes is a set of values for all the decision variables, i.e. a possible solution to the optimisation problem. For this chapter (and most of the thesis), the chromosomes are a set of the PV reactive power output values. All the chromosomes’ fitness values are evaluated based on the objective function(s) (e.g. Eq. (2.1)). The chromosomes with higher fitness values (in this case, lower operating cost) are favoured and are more likely to be selected as parents; they subsequently produce the next generation of chromosomes through crossover operation. The process is repeated until the maximum number of generations or other stopping criteria are reached. To increase the diversity of the solutions and prevent them from being trapped in local optima, mutation operator is applied to portions of the population. At the end of the algorithm run, the values of the best chromosome are recorded (the ones displayed in Tables 2.2 and 2.3 as Case 3) The properties of GA, such as the

2.5 Analysis of Economic and Technical Impact

33

Algorithm 2.1 Backward Forward Sweep Algorithm 1: Vi,t = 1, ∀i = 1, . . . , N 2: Ii,t = 0, ∀i = 1, . . . , N − 1 3: for iter ≤ itermax do ∗ load = S load 4: Ii,t /Vi,t , ∀i = 1, . . . , N i,t 5: for i = N : −1 : 1 do  Backward Sweep 6: if there’s only 1 bus connected to bus i (i is a last bus in the lateral) then load 7: Ii,t ← Ii,t 8: else load + I 9: Ii,t ← Ii,t i+1,t 10: end if 11: end for 12: for i = 1 : 1 : N − 1 do  Forward Sweep 13: Vi+1,t ← Vi,t − Ii,t (ri + j xi ) 14: end for 15: iter = iter + 1 16: end for Table 2.2 Simulation Results Parameters [Units] Total Costs [SGD] Reduction in costs w.r.t. Case 1 [%] S loss [%] S loss [kVA] WVDI [p.u.] min Vi,t [p.u.]

Case 1

Case 2

Case 3

Case 4

11436 0 5.22 6188 2.5724 0.8898

11106 2.89 4.54 5344 1.9996 0.8981

10861 5.03 3.32 3865 1.4647 0.9126

10862 5.02 3.35 3905 1.4588 0.9129

Case 2

Case 3

Case 4

11053 2.81 4.13 4841 1.7567 0.9461

10820 4.85 3.01 3489 1.8139 0.9597

10824 4.82 3.04 3522 1.8856 0.9600

Table 2.3 Simulation Results for OLTC Cases Parameters [Units] Case 1 Total Costs [SGD] Reduction in costs w.r.t. Case 1 [%] S loss [%] S loss [kVA] WVDI [p.u.] min Vi,t [p.u.]

11372 0 4.74 5595 1.7361 0.9384

mutation rate, crossover operator, selection operator, number of chromosomes, and number of generations can be modified to suit the optimisation problem at hand. GA is widely used in optimising power system economic dispatch as it has the advantage of being flexible and scalable [49]. In this work, a population of 120 chromosomes, 50 generations, initial mutation probability of 0.9, and crossover probability of 0.85, have been adopted. The number of chromosomes and generations are increased to 200 and 70 respectively for the high penetration cases so as to handle more decision variables.

34

2 Analysis of Local Reactive Power Provision Using PV in Distribution Systems

Fig. 2.6 Active power profile of the system

Fig. 2.7 Reactive power profile of the system for Case 3

2.5.3 Results and Discussions Table 2.2 shows the economic and technical results obtained from the four different cases. Values in bold are the best results among the four cases. The active and reactive power profiles of the system for Case 3 is shown in Figs. 2.6 and 2.7. From the figure, it can be noted that even with low penetration of PV, it can already fulfil a significant amount of reactive power demand in the system. It is also notable that the reactive power can be provided at a comparable cost with switched capacitors (as will be explained in Sect. 2.6) and reactive power charge from the grid with the exception of the evening peak price (Fig. 2.8).

2.5.3.1

Economic Impact

As seen on Table 2.2, having PV systems that have reactive power capability reduces the operational costs of the test system. Case 1 has the highest cost, followed by Case 2, Case 4 and Case 3. Comparing only Case 1 and Case 2, most of the savings

2.5 Analysis of Economic and Technical Impact

35

Fig. 2.8 Reactive power cost

come from the reduction in distribution losses due to the distributed active power generation, followed by more economical active power generation using PV – cPPV is constant and PV is producing electricity during the 2:00 to 4:00 p.m. peak electricity price. Although not considered in the problem formulation, the addition of PV may also benefit the system as it reduces the peak demand from the grid. A notable improvement in costs is also seen from Case 2 to Case 3 as the PV systems are allowed to produce reactive power. Again, this cost reduction mainly comes from the decrease in active power losses as the reactive power demand is fulfilled locally in Case 3. Note that the PV systems are not sized or sited optimally and therefore the cost reductions are the lower bound benefits for optimally sized and sited systems. When compared to Case 4, where the PV systems are always producing the maximum reactive power possible, Case 3 still performs better economically, albeit only slightly. The reduction in costs happens despite the observation that the costs Qgrid . This is because of producing reactive power using PV (ctQPV ) are lower than ct the higher active power losses in Case 4 outweighs the savings from the cheaper reactive power generation. This result underscores the importance of optimising the RPD instead of just injecting maximum reactive power possible. The importance of the optimisation will only become more apparent as the system’s size and complexity grow as is the case in later chapters. The power loss detailed in Table 2.2 is the power lost within the distribution system and does not take into account the power lost during transmission from the conventional generators to the distribution system. This means that the savings due to distributed and continuous reactive power provision from PV are still underestimated. Overall, the PV reactive power support has been shown to reduce the total operational costs of running the distribution system significantly compared to the other cases presented. The cost reduction is largely attributed to substantial curtailment in active power loss.

36

2 Analysis of Local Reactive Power Provision Using PV in Distribution Systems

Fig. 2.9 Voltage profiles of the four cases. Each line represents a voltage magnitude of a bus throughout the day

2.5.3.2

Technical Impact

The PV systems and their reactive power capability have reduced the distribution losses in the system significantly (29.7%, comparing Case 3 and Case 1) as can be seen in Table 2.2. The lower losses not only translate to economic benefits, but they also decrease the line loading in the system, which may delay the need to upgrade the distribution and transmission lines. The voltage profiles of the buses in the distribution system are also benefited significantly. The voltage magnitudes of the Case 1 to 4 are detailed in Fig. 2.9, while the voltage deviation (represented by WVDI) and minimum voltage across time and nodes are listed on Table 2.2. Starting with Case 1 (Fig. 2.9a), the variation of voltage magnitudes throughout the day follows the load profile inversely. When the loads increase, the voltage magnitudes of the buses decrease. The further away the buses are from the substation/point of common coupling (PCC) (node 1 in Fig. 2.2), the lower the voltage magnitudes are. Current needs to flow from the PCC, where power flows from the grid to the distribution system, all the way to the last bus to fulfil the load there. Without any generation or voltage regulators in the system, there are undervoltage problems at the nodes located at the end of the system. As shown in Fig. 2.9b, the voltage magnitudes of the whole distribution system are improved significantly from 8:00 a.m. until 6:00 p.m. for Case 2. This is due to the active power generated by the PV systems at the different locations in the distribution

2.5 Analysis of Economic and Technical Impact

37

system. When the power generation from bus 7, 19, 24, 61, 64, and 66 exceeds the load at the respective buses, current flows from these buses to other neighbouring buses. Less power is taken from the grid and therefore there is less current flowing from the PCC to other buses in the system. As the voltage magnitude of the PCC is kept constant by the grid (at 1 p.u.), the voltage magnitudes of the other buses are pushed up in comparison to Case 1. However, there are still voltage violations at 7:00 and 7:30 p.m. since the PV are no longer generating any active power. The voltage profile of the buses in Case 3 (Fig. 2.9c) is further improved with reactive power provision from PV. As the reactive power generation from the PV partially fulfils the reactive power demand from the load, less current is flowing across most of the distribution lines. Thus, the power losses across them are reduced and the voltage profile is improved. Thanks to the RPD from PV, there is no more voltage violation in the system as PV can provide reactive power support even in the absence of sunshine. By providing maximum reactive power possible (Case 4), the voltage profile is further improved (Fig. 2.9d). However, it has been shown that Case 4 is less economical than Case 3. This highlights the complexity of the optimisation problem and that minimisation of costs is not necessarily aligned with improving the voltage stability of the system. Therefore in the next section, the benefits of reactive power provision using multi-objective optimisation will be analysed, with minimisation of costs and of voltage deviation as the objectives. Through the case studies, distributed power generation and reactive power injection by PV have been proven to be beneficial. They improve the voltage profile of the system and reduce the line current. As a result, less power is lost along the distribution lines and overloading of these lines is reduced. Should the PV penetration be increased, the minimum voltage in the systems will also be raised. Moreover, if each PV is to be sized and sited optimally for the system, the technical benefits to the system will become even larger.

2.5.3.3

Impact in the Presence of OLTC

The results when there is OLTC in the system whose operations are optimised to minimise Eq. (2.1) are shown in Table 2.3 while the voltage profiles are shown in Fig. 2.10. The optimisation algorithm set the OLTC tap setting to always be 1.429 p.u., the closest tap setting to Vmax (there is no tap change at all throughout the day). This is most likely because at higher voltage, the current is lower and therefore the line losses are lower, as can be seen by comparing Tables 2.3 and 2.2. There was no need to lower the tap setting because there was no overvoltage problems. The trends elaborated in Sects. 2.5.3.1 and 2.5.3.2 remain the same, and optimising the reactive power support from PV (Case 3) still yield the best results for most of the indicators. One obvious change is the values of WVDI. WVDI of Case 1 is markedly improved in the presence of OLTC as now the node voltages are closer to 1 p.u. However, the WVDI of the OLTC cases becomes worse with the addition of PV and its reactive power support. This is because most of the node voltages are already

38

2 Analysis of Local Reactive Power Provision Using PV in Distribution Systems

Fig. 2.10 Voltage profiles of the four cases with OLTC. Each line represents a voltage magnitude of a bus throughout the day

above 1 p.u. (the value of Vref used in calculating WVDI) and the addition of PV and its reactive power support increases the voltage further. As can be seen from Fig. 2.10, the voltage profiles of the nodes are identical with the cases without OLTC, except that the magnitudes are pushed up by the OLTC closer to 1.05 p.u, solving the undervoltage problems in Case 1 and 2. From these results, we can be confident that as long as there is no overvoltage problem, i.e. when the PV generation does not exceed the load, PV reactive power support will not interfere with OLTC operation and will still yield benefits for the system. Hence, in the subsequent sections and chapters, we will focus our attention on PV reactive power support without OLTC in the distribution system.

2.6 Competitiveness of Local Reactive Power Provision Using PV 2.6.1 Objective Functions After quantifying the benefits of PV reactive power support, it is then necessary to QPV in compare the competitiveness of local reactive power provision using PV (cx,t QSC Eq. (2.1)) and SCs (cx,t ) before power system planners and operators can deter-

2.6 Competitiveness of Local Reactive Power Provision Using PV

39

mine whether a move from traditional reactive power devices towards inverters with reactive power capability is beneficial. Therefore, the reactive power compensation (RPC) from PV and SC are assessed through economic and technical objectives. To evaluate the benefits of local reactive power provision with changing parameters, the improvement in voltage stability and net monetary benefits are adopted as the technical and economic objectives, respectively.

2.6.1.1

Economic Objective

The economic objective function is the net monetary benefits (NMB) of having either PV inverters or capacitors for RPC. It can be calculated by subtracting the cost of running the system when there is RPC by PV or SC from the cost of running the system when there is not: NMB = Cost NoQ − Cost SC/PV

(2.21)

where the total cost is calculated from Eq. (2.1). NoQ refers to the system without local reactive power compensation (Case 2 in Sect. 2.5.1) while SC/PV refers to cases where the reactive power are provided by SCs or PVs respectively, as will be explained in greater detail in Sects. 2.6.2 and 2.6.4. A positive NMB means that the RPC is economically beneficial.

2.6.1.2

Technical Objective

Voltage deviation has been used as a technical objective because voltage stability is one of the most important aspects in power system with high PV penetration [16, 50, 51]. Large deviations in voltage may cause voltage collapse and power system failure [52]. Therefore, one of the objectives is to minimise the voltage deviation in the system. Weighted voltage deviation index (WVDI)—calculated using Eq. (2.3)—penalises larger magnitude of voltage deviation more heavily and is therefore favoured over absolute voltage deviation. The technical benefits from RPC can be represented through the reduction in the voltage deviation, or equivalent to the WVDI improvement: WVDI Improv. = WVDINoQ − WVDISC/PV

(2.22)

Similar to NMB, positive WVDI Improvement means that RPC is technically beneficial for the system.

40

2 Analysis of Local Reactive Power Provision Using PV in Distribution Systems

2.6.2 Simulation Setup Two base cases are considered to assess the competitiveness of reactive power compensation using PV. PV Base Case: The PV inverters in the systems are able to generate reactive power following Constraint (2.13). SC Base Case: The PV inverters are not able to generate reactive power. In their place, a switched capacitor of 300 kvar (consisting of 3 capacitor banks of 100 kvar) each is available for RPC—a total of 6 SCs in the system. Each SC has four operating points and can change its operating point from one time period to the next without any restriction. Based on the two cases, thorough sensitivity analyses were conducted by varying electricity price, levelised cost of electricity (LCOE) of PV, SC cost, solar irradiance profile, as well as inverter efficiency. The two objective functions, WVDI Improvement and NMB are also calculated based on the NoQ cases with the respective parameters changed (e.g. NMB of PV case with 120% electricity price is calculated by subtracting Cost of the PV NoQ case with 120% electricity price). The total cost (Eq. (2.1)) and WVDI (Eq. (2.3)) of the NoQ cases are calculated with the PV systems producing active power, but no reactive power is generated by the PV or SC.

2.6.3 Optimisation Algorithm Non-dominated Sorting Genetic Algorithm-II (NSGA-II) has been adopted to analyse the possible schedules for RPD. As the total costs and voltage stability of the system are not always aligned, there will be pareto optimal solutions where the solutions are non-dominated with respect to other solutions. GA described in Sect. 2.5.2 is only capable of optimising one objective function. As such, to optimise both the economic and technical objectives simultaneously, multi-objective version of GA, called Non-dominated Sorting Genetic Algorithm II (NSGA-II) [53], has been employed in this section. Unlike GA, NSGA-II does not only produce one best solution, but rather a set of pareto optimal solutions. None of these solutions are better in both the technical and economic objectives than each other. In other words, they are non-dominated with respect to each other. NSGA-II utilises similar mechanisms as GA to generate the chromosomes. The algorithm has been found to be superior to other multi-objective algorithms [54, 55] and thus was employed in this work. Continuous [46] and binary values were adopted for PV and SC cases respectively. The same parameters as in Sect. 2.5.2 (i.e. initial mutation probability of 0.9, crossover probability of 0.85, 120 (200) chromosomes, and 50 (70) generations) have been adopted for the base (high penetration) PV and SC cases.

2.6 Competitiveness of Local Reactive Power Provision Using PV

41

2.6.4 Results and Discussions Results of the base cases are illustrated in Fig. 2.11. There are a few things to be noted from this graph. Firstly, the blue crosses and orange circles comprise the pareto fronts for PV and SC, where each point is a different reactive power strategy and none of the points are better than another in both the technical and economic objectives. Secondly, RPC from SC yield higher economic and technical benefits compared with using PV. This is not because reactive power from SC is cheaper, but because SCs fulfil more reactive power demand in the system since PV reactive power is constrained by the active power usage (Eq. (2.13)). On average, the solutions in the SC base case’s pareto front generate 38000 kvarh locally, compared to 33000 kvarh for the PV base case. To put it into context, 57.0% and 49.5% of the reactive power demand in the system are fulfilled locally in the SC and PV base case respectively. Therefore, SC can improve the voltage profile and reduce line losses better. That being said, the NMB of the RPC using PV is still significant— at about 14 cents SGD/kWp per day or 50 SGD/kWp annually— approximately 3.9% of the PV system price in 2016 (1300 SGD/kWp [37]).

Fig. 2.11 Base PV and SC cases. Y-axis represents the technical objective while x-axis the economic objective

42

2 Analysis of Local Reactive Power Provision Using PV in Distribution Systems

Lastly, there are far fewer pareto solutions for SC. This is because SC can only generate discrete values of reactive power, whereas the PV inverters can generate continuous values.

2.6.4.1

Effect of Electricity Price

As can be seen from Fig. 2.12, higher cPGrid increases the NMB of both SC and PV cases linearly because the value of the power loss increases as electricity price rises. The increase in NMB for PV cases is not as considerable (1.5% for every 10% increase in cPGrid , compared to 5% for SC cases) as the PV cases did not reduce power loss as much as SC cases as previously explained. ctQPV also depends on electricity price (Eq. (2.20)), and therefore increases when cPGrid rises. The pareto front is largest when the electricity price is the lowest, and becomes smaller until electricity price is 120% of the original values. Interestingly, the pareto front starts to enlarge again as the electricity price continues to increase. The likely explanation is that, when ctPGrid is lower, ctQPV is also lower, and consequently, there are more operating points where ctQPV < cQGrid . As such, the PV systems may generate Q PV x,t more than necessary, even when it would increase the line losses or not

Fig. 2.12 Impact of changes in electricity price on a SC and b PV cases

2.6 Competitiveness of Local Reactive Power Provision Using PV

43

improve the WVDI. And as ctPGrid increases, while ctQPV is more likely to be larger than cQGrid , there are also more technical and economic tradeoffs involved in proQPV . These two opposing trends are likely the cause of the ducing Q PV x,t at higher ct behaviour of the pareto fronts observed in Fig. 2.12b. The same behaviour is also observed in Sect. 2.6.4.5, which elaborates the impact of inverter efficiency on the competitiveness of RPC.

2.6.4.2

Effect of PV LCOE

Since ctQPV and ctQSC formulation is not affected by the LCOE of PV, varying PV LCOE has no observable impact on the PV and SC cases (Figs. 2.13 and 2.14). The slight variation in the shape and size of the pareto fronts are purely due to randomness inherent in the metaheuristic algorithm used. Nevertheless, should the formulation of ctQPV in Eq. 2.20 include cPPV , then the impact of changes in PV LCOE on the PV cases will be similar to that of changes in electricity price, albeit smaller as there are more Q PV x,t generated at night than those during the day.

Fig. 2.13 Impact of changes in PV LCOE on PV cases

Fig. 2.14 Impact of changes in PV LCOE on SC cases

44

2 Analysis of Local Reactive Power Provision Using PV in Distribution Systems

Fig. 2.15 Impact of changes in switched capacitor’s price on SC cases

Fig. 2.16 Impact of changes in irradiance profile on SC cases

2.6.4.3

Effect of Capacitor Price

Figure 2.15 shows that for every 10% change in ctQSC , the NMB of SCs changes by approximately 3% in the opposite direction. There is no change in WVDI improvement as there is no change in SC capacity.

2.6.4.4

Effect of Solar Irradiance

The more power generated during the day by PV, the more reduction in losses in the system. Hence, RPC from PV/SC is not able to reduce as much losses anymore, bringing less NMB and lower WVDI improvement. SC under variable irradiance has slightly higher NMB and WVDI improvement compared to SC base case, corresponding to lower forecast energy generated by PV PV ) on 18 November 2016 (variable irradiance case) compared to that on 27 May (Px,t 2016 (base case) (8027 and 8618 kWh respectively). This, and the trend in Fig. 2.16 suggest that the benefits brought about by SC’s RPC depends more on the energy

2.6 Competitiveness of Local Reactive Power Provision Using PV

45

Fig. 2.17 Impact of changes in irradiance profile on PV cases

generated by PV, or that the variability of solar irradiance does not have significant effect at half-hour resolution. For PV cases, the trend is further strengthened as when there is low irradiance PV70 ) , PV is able to generate more reactive power according during the day (lower Px,t to Eq. (2.13), giving higher monetary and technical benefits to the system for its reactive power provision. Therefore, the reactive power capability boosts the overall benefits of PV to the distribution system, as even when the PV is not able to generate active power during rainy days, its reactive power capability can compensate and reduce the losses in the system, as well as improve the voltage profile of the system. The results for PV variable irradiance and PV base case are similar (Fig. 2.17) because the total P PV70 of variable irradiance case and base case is approximately the same (10960 kWh for the full day) and therefore the reactive power that can be generated are also similar.

2.6.4.5

Effect of Inverter Efficiency

Higher inverter efficiency means that the PV systems can generate more P PV (Eq. (2.11)). Higher local PV generation, in turn, increases the voltage magnitudes within the distribution system closer to Vref , leading to lower WVDI. The WVDI improvement and NMB of SC cases therefore decreases with increasing PV inverter efficiency, albeit only very slightly, as seen in Fig. 2.18. This is the same as the trend observed for the impact of irradiance in Fig. 2.16. The effects of change in inverter efficiency are much more pronounced for the PV cases, as can be seen from Fig. 2.19. Higher inverter efficiency means that there is lower P PV,invloss , and therefore the PV reactive power capability is improved (Eq. (2.13)) while the reactive power cost is reduced (Eq. (2.20)). With only 2% increase in inverter efficiency, ctQPV drops by 41.9% and the total reactive power cost from PV decreases by SGD 52. The 2% increase in efficiency translates to an inverter with peak efficiency of 98.2%, an achievable standard even with current

46

2 Analysis of Local Reactive Power Provision Using PV in Distribution Systems

Fig. 2.18 Impact of changes in inverter efficiency on SC cases. The InvEff 97%, 98%, 99%, Base, 101%, 102%, and 103% refer to peak inverter efficiency of 93.4%, 94.3%, 95.3%, 96.3%, 97.2%, 98.2% and 99.1%, respectively

Fig. 2.19 Impact of changes in inverter efficiency on PV cases. The InvEff 97%, 98%, 99%, Base, 101%, 102%, and 103% refer to peak inverter efficiency of 93.4%, 94.3%, 95.3%, 96.3%, 97.2%, 98.2% and 99.1%, respectively

inverter topologies [56]. In fact, many inverters in 2019 already have peak efficiency of 99% [57], nearing the InvEff103% case (equivalent to peak efficiency of 99.1%). Comparing Figs. 2.15 and 2.19, it is important to note that the impact of a 2% increase of inverter efficiency in PV case is larger than a 20% decrease of SC cost in SC case. Nevertheless, the voltage deviations in the PV cases are still larger than in the SC cases because of the aforementioned reasons.

2.6.4.6

Future Scenarios

The main reason for higher WVDI improvement and NMB of SC cases is because an SC of the same rating has higher capacity to produce reactive power than a PV inverter and there is still enormous reactive power demand to be fulfilled. Therefore

2.6 Competitiveness of Local Reactive Power Provision Using PV

47

Fig. 2.20 Future scenarios of PV and SC cases

it is interesting to look at scenarios with higher PV penetration such that most of the reactive power demand can be fulfilled locally in both the PV and SC cases. Figure 2.20a demonstrates the results when the number of PVs in the system is increased from 6 to 16, giving PV penetration of 25% by produced energy or 106% by installed capacity over peak demand of the day. Results in Fig. 2.20b represents a scenario for a distribution system with high PV penetration (same as (a)) in the future. The electricity price, SC cost, and inverter efficiency are 110%, 80%, and 103% of the values in base case respectively. The PV LCOE is taken to be the same for a conservative case. At higher PV penetration, PV inverters are almost as economically competitive as SCs in providing reactive power, and that in the future PV would be more economically competitive. Nevertheless, RPC from PV are still not as technically beneficial as that from SC even at higher penetration of PV, particularly when the forecast PV generation is high (from 10 a.m. to 5 p.m. as shown in Fig. 2.21), due to the limited reactive power capability of PV during these periods. For the rest of the day, it can be seen that the WVDI improvement of PV matches that of SC. With improvement in PV forecasting (hour-ahead instead of day-ahead, and development of more accurate PV70 PV is closer to that of Px,t , PV would be even methods) such that the value of Px,t more technically beneficial.

48

2 Analysis of Local Reactive Power Provision Using PV in Distribution Systems

Fig. 2.21 WVDI improvement comparison between future PV and SC cases

2.7 Summary In this chapter, the cost of reactive power provision using PV in grid-connected distribution system has been explicitly formulated and incorporated into a practical objective function of minimising the total cost of operating the system. Through various case studies, the economic and technical benefits of local reactive power provision for the system have been quantified and analysed. Reactive power provision from PV is also shown to be cost competitive with the reactive power charge from the grid, but not with producing reactive power using switched capacitors (SCs) yet. Among the considered parameters, inverter efficiency has been identified as the most important factor affecting the benefits derived from reactive power provision using the PV inverters, whereas electricity price is the most important factor for SCs. Higher PV penetration is also shown to increase the competitiveness of PV reactive power support. Therefore, as PV penetration and inverter efficiencies are expected to increase in the future, investment in reactive power capable PV will be even more beneficial. The sensitivity analyses have also shown us that regardless of the change in parameters, reactive power support using distributed PV are beneficial for distribution systems. In the next section, the applicability of implementing the metaheuristic optimisation approach that has been discussed will be analysed in greater detail.

References 1. International Energy Agency (IEA) (2018) Trends in photovoltaic applications 2018. ISBN 9783906042794 2. Porter K, Fink S, Rogers J, Mudd C, Buckley M, Clark C, Hinkle G (2012) PJM renewable integration study: review of industry practice and experience in the integration of wind and solar generation. GE Energy Tech Rep

References

49

3. Ogimoto K (2014) Power system operation and augmentation planning with PV integration. International Energy Agency Photovoltaic Power Systems Programme (IEA PVPS), Tokyo. Tech Rep T14–04:2014 4. Atwa YM, El-Saadany EF, Salama MMA, Seethapathy R (2010) Optimal renewable resources mix for distribution system energy loss minimization. IEEE Trans Power Syst 25(1):360–370, ISSN 08858950. https://doi.org/10.1109/TPWRS.2009.2030276 5. Ziadi Z, Taira S, Oshiro M, Funabashi T (2014) Optimal power scheduling for smart grids considering controllable loads and high penetration of photovoltaic generation. IEEE Trans Smart Grid 5(5):2350–2359, ISSN 19493053. https://doi.org/10.1109/TSG.2014.2323969 6. Wandhare RG, Agarwal V (2014) Reactive power capacity enhancement of a PV grid system to increase pv penetration level in smart grid scenario. IEEE Trans Smart Grid 5(4):1845–1854, ISSN 19493053. https://doi.org/10.1109/TSG.2014.2298532 7. Wu L, Zhao Z, Liu J (2007) A single-stage three-phase grid-connected photovoltaic system with modified MPPT method and reactive power compensation. IEEE Trans Energy Convers 22(4):881–886, ISSN 08858969. https://doi.org/10.1109/TEC.2007.895461 arXiv: z0024 8. Cagnano A, De Tuglie E, Liserre M, Mastromauro RA (2011) Online optimal reactive power control strategy of PV inverters. IEEE Trans Ind Electron 58(10):4549–4558, ISSN 0278-0046. https://doi.org/10.1109/TIE.2011.2116757 9. Kisacikoglu MC, Ozpineci B, Tolbert LM (2010) Examination of a PHEV bidirectional charger system for v2g reactive power compensation. In: Proceedings of conference on–IEEE applied power electronics conference and exposition–APEC, pp 458–465, ISSN 1048-2334. https:// doi.org/10.1109/APEC.2010.5433629 10. Southern California Edison. (2017) Electric rule 21 11. IEEE Standards Coordinating Committee 21 (2018) IEEE standard for interconnection and interoperability of distributed energy resources with associated electric power systems interfaces, New York. IEEE. ISBN 9781504446396 12. Kumar DS, Gandhi O, Rodríguez-Gallegos CD, Srinivasan D (2020) Review of power system impacts at high PV penetration part ii: potential solutions and the way forward. Solar Energy Under Second Rev 13. Bhattacharya K, Zhong J (2001) Reactive power as an ancillary service. IEEE Trans Power Syst 16(2):294–300 14. Zhong J, Nobile E, Bose A, Bhattacharya K (2004) Localized reactive power markets using the concept of voltage control areas. IEEE Trans Power Syst 19(3):1555–1561 15. Zhong J (2005) A pricing mechanism for reactive power devices in competitive market. In: 2006 IEEE power india conference, vol 2005, IEEE, pp 67–72, ISBN 0-7803-9525-5. https:// doi.org/10.1109/POWERI.2006.1632493 16. Kargarian A, Raoofat M, Mohammadi M (2011) Reactive power market management considering voltage control area reserve and system security. Appl Energy 88(11):3832–3840, ISSN 03062619. https://doi.org/10.1016/j.apenergy.2011.04.024 17. El-samahy I, Bhattacharya K, Cañizares C, Anjos MF, Pan J (2007) A procurement market model for reactive power services considering system security. IEEE Trans Power Syst 1–13 18. Reddy S, Abhyankar AR, Bijwe PR (2011) Reactive power price clearing using multiobjective optimization. Energy 36(5):3579–3589, ISSN 03605442. https://doi.org/10.1016/ j.energy.2011.03.070 19. Rabiee A, Shayanfar H, Amjady N (2009) Coupled energy and reactive power market clearing considering power system security. Energy Convers Manag 50(4):907–915, ISSN 01968904. https://doi.org/10.1016/j.enconman.2008.12.026 20. Samimi A, Kazemi A, Siano P (2015) Economic-environmental active and reactive power scheduling of modern distribution systems in presence of wind generations: a distribution market-based approach. Energy Convers Manage 106:495–509, ISSN 01968904. https://doi. org/10.1016/j.enconman.2015.09.070 21. Gabash A, Li P (2012) Active-reactive optimal power flow in distribution networks with embedded generation and battery storage. IEEE Trans Power Syst 27(4):2026–2035, ISSN 08858950. https://doi.org/10.1109/TPWRS.2012.2187315. arXiv: 9605103 [cs]

50

2 Analysis of Local Reactive Power Provision Using PV in Distribution Systems

22. Liang RH, Wang JC, Chen YT, Tseng WT (2015) An enhanced firefly algorithm to multiobjective optimal active/reactive power dispatch with uncertainties consideration. Int J Electr Power Energy Syst 64:1088–1097, ISSN 01420615. https://doi.org/10.1016/j.ijepes.2014.09. 008 23. Sousa T, Morais H, Vale Z, Castro R (2015) A multi-objective optimization of the active and reactive resource scheduling at a distribution level in a smart grid context. Energy 85:236–250, ISSN 03605442. https://doi.org/10.1016/j.energy.2015.03.077 24. Gandhi O, Zhang W, Rodríguez-Gallegos CD, Srinivasan D, Reindl T (2016) Continuous optimization of reactive power from PV and EV in distribution system. In: 2016 IEEE Innovative Smart Grid Technologies–Asia (ISGT-Asia), Melbourne, Nov. 2016. IEEE, pp 281–287. ISBN 978-1-5090-4303-3. https://doi.org/10.1109/ISGT-Asia.2016.7796399 25. Gandhi O, Srinivasan D, Rodríguez-Gallegos CD, Reindl T (2017) Competitiveness of reactive power compensation using PV inverter in distribution system. In: 2017 IEEE PES Innovative Smart Grid Technologies Conference Europe (ISGT-Europe), Torino, Italy. Sep. 2017. IEEE, pp 1–6. ISBN 978-1-5386-1953-7. https://doi.org/10.1109/ISGTEurope.2017.8260238 26. Gandhi O, Rodríguez-Gallegos CD, Zhang W, Srinivasan D, Reindl T (2018) Economic and technical analysis of reactive power provision from distributed energy resources in microgrids. Appl Energy 210:827–841, ISSN 03062619. https://doi.org/10.1016/j.apenergy.2017.08.154 27. Ghosh S, Das D (1999) Method for load-flow solution of radial distribution networks. IEE Proc Gener Transm Distrib 146(6):641–648, ISSN 13502360. https://doi.org/10.1049/ip-gtd: 19990464 28. Skoplaki E, Palyvos JA (2009) On the temperature dependence of photovoltaic module electrical performance: a review of efficiency/power correlations. Solar Energy 83(5):614–624, ISSN 0038092X. https://doi.org/10.1016/j.solener.2008.10.008 29. Rodríguez-Gallegos CD, Gandhi O, Yang D, Alvarez-Alvarado MS, Zhang W, Reindl T, Panda SK (2018) A siting and sizing optimization approach for pv-battery-diesel hybrid systems. IEEE Trans Ind Appl 54(3):2637–2645, ISSN 0093-9994. https://doi.org/10.1109/TIA.2017. 2787680 30. Li C, Disfani VR, Pecenak ZK, Mohajeryami S, Kleissl J (2018) Optimal OLTC voltage control scheme to enable high solar penetrations. Electr Power Syst Res 160:318–326. https://doi.org/ 10.1016/J.EPSR.2018.02.016. arXiv: arXiv:1804.06025v1 31. Braun M (2008) Provision of ancillary services by distributed generators, Ph.D Thesis, Kassel University, p 273, ISBN 9783899586381 32. Savier JS, Das D (2007) Impact of network reconfiguration on loss allocation of radial distribution systems. IEEE Trans Power Delivery 22(4):2473–2480, ISSN 08858977. https://doi. org/10.1109/TPWRD.2007.905370 33. Energy Market Authority (2018) Singapore half-hourly system demand data. https:// www.ema.gov.sg/statistic.aspx?sta%7B%5C_%7Dsid=20140826Y84sgBebjwKV Visited on 07/10/2017 34. EMC, Energy market price information. https://www.emcsg.com/marketdata/priceinformation Visited on 04/01/2018 35. EMC (2016) Use of system charges. https://www.mypower.com.sg/documents/ts-usc.pdf Visited on 04/01/2018 36. Bieri M, Winter K, Tay S, Chua A, Reindl T (2017) An irradiance-neutral view on the competitiveness of life-cycle cost of PV rooftop systems across cities. Energy Procedia 37. National Solar Repository, Solar economics handbook. http://www.solar-repository.sg/pvadoption-in-singapore Visited on 05/02/2017 38. Chen SX, Eddy YSF, Gooi HB, Wang MQ, Lu SF (2015) A centralized reactive power compensation system for LV distribution networks. IEEE Trans Power Syst 30(1):274–284, ISSN 08858950. https://doi.org/10.1109/TPWRS.2014.2326520 39. Verbois H, Rusydi A, Thiery A (2018) Probabilistic forecasting of day-ahead solar irradiance using quantile gradient boosting. Solar Energy 173(March):313–327, ISSN 0038-092X. https:// doi.org/10.1016/j.solener.2018.07.071

References

51

40. Verbois H, Huva R, Rusydi A, Walsh W (2018) Solar irradiance forecasting in the tropics using numerical weather prediction and statistical learning. Solar Energy 162 no. December 2017:265–277, ISSN 0038-092X. https://doi.org/10.1016/j.solener.2018.01.007 41. Zheng W, Wu W, Zhang B, Sun H, Liu Y (2016) A fully distributed reactive power optimization and control method for active distribution networks. IEEE Trans Smart Grid 7(2):1021–1033, ISSN 19493053. https://doi.org/10.1109/TSG.2015.2396493 42. Lavaei J, Member S, Low SH (2012) Zero duality gap in optimal power flow problem. 27(1):92– 107 43. Takeuchi A, Hayashi T, Nozaki Y, Shimakage T (2012) Optimal scheduling using metaheuristics for energy networks. IEEE Trans Smart Grid 3(2):968–974, ISSN 19493053. https://doi. org/10.1109/TSG.2012.2191580 44. Low SH (2014) Convex relaxation of optimal power flow part ii: exactness. IEEE Trans Control Network Syst 1(2):177–189, ISSN 2325-5870. https:// doi.org/10.1109/TCNS.2014.2323634. arXiv: 1405.0814. [Online]. Available: http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6815671 45. Chang GW, Chu SY, Wang HL (2007) An improved backward/forward sweep load flow algorithm for radial distribution systems. IEEE Trans Power Syst 22(2):882–884, ISSN 08858950. https://doi.org/10.1109/TPWRS.2007.894848 46. Chelouah R, Siarry P (2000) A continuous genetic algorithm designed for the global optimization of multimodal functions. J Heuristics 6:191–213 47. Sivanandam S, Deepa S (2008) Introduction to genetic algorithms. Springer, p 453, ISBN 9783540731894. https://doi.org/10.1007/978-3-540-73190-0 48. Gandhi O, Rodríguez-Gallegos CD, Srinivasan D (2016) Review of optimization of power dispatch in renewable energy system. In: 2016 IEEE Innovative Smart Grid Technologies–Asia (ISGT-Asia), Melbourne. Nov. 2016, IEEE. pp 250–257. ISBN 978-1-5090-4303-3. https://doi. org/10.1109/ISGT-Asia.2016.7796394 49. Logenthiran T, Srinivasan D, Shun TZ (2012) Demand side management in smart grid using heuristic optimization. IEEE Trans Smart Grid 3(3):1244–1252, ISSN 19493053. https://doi. org/10.1109/TSG.2012.2195686 50. Savic A, Durišic Ž (2014) Optimal sizing and location of SVC devices for improvement of voltage profile in distribution network with dispersed photovoltaic and wind power plants. Appl Energy 134:114–124, ISSN 03062619. https://doi.org/10.1016/j.apenergy.2014.08.014 51. Gandhi O, Kumar DS, Rodríguez-Gallegos CD, Srinivasan D (2020) Review of power system impacts at high PV penetration part i: factors limiting pv penetration. Solar Energy 52. Sode-Yome A, Mithulananthan N (2004) Comparison of shunt capacitor, SVC and STATCOM in static voltage stability margin enhancement. Int J Electr Eng Educ 41(2):158–171, ISSN 0020-7209. https://doi.org/10.7227/IJEEE.41.2.7 53. Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197, ISSN 1089778X. https://doi. org/10.1109/4235.996017 54. Pires DF, Antunes CH, Martins AG (2012) NSGA-II with local search for a multi-objective reactive power compensation problem. Int J Electr Power Energy Syst 43(1):313–324, ISSN 01420615. https://doi.org/10.1016/j.ijepes.2012.05.024 55. Rodríguez-Gallegos CD, Singh JP, Yacob Ali JM, Gandhi O, Nalluri S, Kumar A, Shanmugam V, Aguilar ML, Bieri M, Reindl T, Panda SK (2019) PV-GO: a multiobjective and robust optimization approach for the grid metallization design of Si-based solar cells and modules. Prog Photovoltaics Res Appl 27(2):113–135, ISSN 10627995. https://doi.org/10.1002/pip. 3036 56. Araújo SV, Zacharias P, Mallwitz R (2010) Highly efficient single-phase transformerless inverters for grid-connected photovoltaic systems. IEEE Trans Ind Electr 57(9):3118–3128, ISSN 02780046. https://doi.org/10.1109/TIE.2009.2037654 57. Growatt New Energy Technology Co. Ltd (2019) 3 phase inverters datasheet, Shenzhen. http://www.ginverter.com/upload/file/contents/2019/06/5d0344b5ef461.pdf

Chapter 3

Analytical Approach to Power Dispatch in Distribution Systems

3.1 Introduction In Chap. 2, the benefits of local reactive power provision using PV inverters have been demonstrated. The next logical step would be to assess the implementation of the power dispatch optimisation in the distribution system in the presence of other DERs that are becoming inseparable aspects of future power system. This chapter expands the work based on the insights from the previous chapter on reactive power compensation (RPC) using PV to include other inverter-based DERs and proposes a practical way to optimise the power dispatch. To solve the non-convex reactive power dispatch (RPD) elaborated in Chap. 2, most researchers employ metaheuristic algorithms [1–7], which require high computational and time resources, rendering them not yet suitable to be used in real applications. Some researchers have also utilised exact/mathematical optimisation [8–10]. Even though exact optimisation is typically fast, it relies on convexification of the problem and has stringent restrictions on the objective functions and constraints to achieve globally optimal solutions. For non-convex problems, exact optimisation algorithms often converge to local optima and fail to achieve the most economic solution [11]. The field would therefore benefit from an analytical approach, which can efficiently and consistently find the solutions to an optimal power dispatch. Analytical approaches have been proposed in the literature to determine the optimal size and location for DERs in grid-connected radial distribution systems [12, 13]. Nevertheless, the accuracy and the performance of the methods were not benchmarked with other algorithms. Moreover, the approaches have not incorporated the active and reactive power cost/price even though they can affect the optimisation results. As mentioned in Sect. 2.1, most works have utilised the reactive power support from DERs without taking into account the reactive power cost [3, 14], or by assigning arbitrary values as the cost [15]. Energy storage devices have also been optimised © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 O. Gandhi, Reactive Power Support Using Photovoltaic Systems, Springer Theses, https://doi.org/10.1007/978-3-030-61251-1_3

53

54

3 Analytical Approach to Power Dispatch in Distribution Systems

for energy arbitrage (EA) without considering the effect of charging and discharging on the line losses [16–19]. To overcome the shortcomings in the literature, the work in this chapter proposes a comprehensive analytical approach, incorporating price of electricity, cost of reactive power, reactive power charge, line losses, and energy arbitrage from battery energy storage system (BESS). To the best of the author’s knowledge, this work is the first to integrate reactive power costs and line losses into an analytical approach. The results were subsequently published in IEEE Transactions on Power Systems journal [20]. The main contributions of the work in this chapter are therefore: [1] Proposes an analytical approach to RPD that is up to 100 times faster, and can achieve better solutions compared to the widely used exact optimisation and metaheuristic algorithms, and [2] Proposes a rule-based analytical approach for energy arbitrage considering line losses. The approach performs better economically compared with existing approaches which do not take line losses into consideration. These two contributions build on the findings from the previous chapter to allow a practical implementation of RPD in distribution systems with PV and other DERs. Subsequently, Sect. 3.2 describes the optimisation problem and the associated cost functions. The analytical approach for RPD and EA is elaborated in Sect. 3.3. Section 3.4 describes the test systems and parameters used. Next, the results are presented and discussed in Sect. 3.5 and the chapter is concluded in Sect. 3.6.

3.2 Problem Formulation 3.2.1 Objective Function Maintaining the perspective of an ISO in Chap. 2, the objective function is to minimise the total cost. The case studies have been extended by adding B number of battery energy storage systems (BESS) and E electric vehicle (EV) parking lots to the distribution system, such that the objective function proposed in Eq. (2.1) becomes Eq. (3.1). The ISO is assumed to have control over the BESS and has been authorised to utilise the spare capacity of the PV inverters and the unused EV off-board chargers for RPC. The cost of active power from the grid in Eq. (3.1) already includes the net cost of charging of BESS and EV. The latter is subtracted from the objective function as the cost of charging the EVs is paid privately by the owners. Cost =

T   t=1

Pgrid grid ct Pt







cost of P from grid

+ c 

Qgrid

grid Qt





cost of Q from grid

+

M  x=1



PV cxPPV Px,t





cost of P from PV

3.2 Problem Formulation

+

M 

55

QPV PV cx,t Q x,t +

x=1





B  x=1





cost of Q from PV Pgrid

− ct

E  x=1







E 

 QEV EV cx,t Q x,t H

x=1





net cost of charging EV



BESS,deg

cx,t

x=1



cost of Q from BESS

EV Px,t +

B 

QBESS BESS cx,t Q x,t +





(3.1)



cost of BESS degradation



cost of Q from EV

EV where Px,t is the charging/discharging power of the x th EV parking lot. Q BESS and x,t EV Q x,t are the reactive power generated by the x th BESS and EV parking lot at the cost BESS,deg QBESS QEV of cx,t and cx,t respectively. cx,t is the battery degradation cost of the x th BESS, calculated using Eq. (3.18). Similar to Chap. 2, S loss , and min Vi,t have also been used as the technical indicators to compare the performance of the proposed algorithm.

3.2.2 Constraints The power balance and the power flow constraints (2.4–2.7) are therefore extended to (3.2–3.5), while the voltage and current constraints remain the same as in Sect. 2.2.2. grid Pt



N 



Pi,tloss

i=1 grid

Qt



N 

N 

Pi,tload

+

i=1

P j,t = Pi,t − ri j

N 

load Q i,t +



B 

2 Pi,t2 + Q i,t

Vi,t2

Q PV x,t +

B  x=1



E 

EV Px,t =0

(3.2)

x=1

Q BESS + x,t

E 

Q EV x,t = 0

(3.3)

x=1

load PV BESS EV − P j,t + P j,t − P j,t − P j,t , ∀(i, j) ∈ L (3.4)

2 Pi,t2 + Q i,t

Vi,t2

M 

BESS Px,t

x=1

x=1

i=1

Q j,t = Q i,t − xi j

PV Px,t

x=1

i=1 loss Q i,t −

M 

PV BESS − Q load + Q EV j,t + Q j,t + Q j,t j,t , ∀(i, j) ∈ L



Ii,t ≤ Imax , i = 1, . . . , N Vmin ≤ |Vi,t | ≤ Vmax , i = 1, . . . , N

(3.5) (3.6) (3.7)

BESS is the charging/discharging power of the x th BESS. Similar to Sect. 2.2.2, where Px,t the subscript j for PV, BESS, and EV quantities indicate their respective values at node j, which may or may not have any PV, BESS, or EV. BESS EV and Px,t represent charging of BESS and EV, and are considered Positive Px,t as load, while positive Q Xx,t (where the superscript X can refer to PV, BESS or EV)

56

3 Analytical Approach to Power Dispatch in Distribution Systems

represents reactive power injection from the DERs. The reactive power constraints for PV follow the inequality constraint (2.13) from Chap. 2:

PV

2 PV70 2 PV

Q ≤ Q PV = Sx,max − Px,t (3.8) x,t x,t,lim whereas the reactive power for BESS and EV are limited by the following equation:

X

2 X 2 X

Q ≤ QX Sx,max = − Px,t (3.9) x,t x,t,lim where Q Xx,t,lim is the maximum reactive power that can be generated by the DERs PV PV without reducing their active power output. Sx,max is taken to be equal to Px,rated , X BESS EV is the same as Px,max and Px,max for BESS and EV respectively. The while Sx,max inverters are not oversized for reactive power and hence no additional investment is required for the RPC from the DERs [21]. The forecast values of active power of EV and BESS are not used because when the EV charger is operating, Q EV x,t is set to zero, and because the ISO is assumed to BESS , respectively. be in control of Px,t The charging and discharging of the BESS are limited by the maximum charging BESS BESS ) and discharging power (Px,min ). The BESS’ state of charge (SOCx,t ) must (Px,max also lie within its minimum (SOCx,min ) and maximum (SOCx,max ) limit. The SOC of the BESS is updated according to the following equations: BESS BESS BESS ≤ Px,t ≤ Px,max Px,min

SOCx,min ≤ SOCx,t ≤ SOCx,max

CH CH BESS SOCx,t+1 = SOCx,t + δx,t ηx Px,t  DCH −1 BESS DCH BESS,invloss ηx +δx,t Px,t − Px,t

(3.10) (3.11) (3.12) H E xBESS

CH DCH BESS where δx,t (δx,t ) is a binary variable that takes the value of 1 when Px,t >0 BESS < 0) and 0 otherwise. ηxCH and ηxDCH are the charging and discharging (Px,t efficiency of the BESS. Energy capacity of the BESS is represented by E xBESS . BESS,invloss is the power loss in the BESS inverter, calculated using Eq. (2.16) and Px,t repeated here for convenience:

BESS 2 BESS,invloss BESS = cxBESS,self + cxBESS,V Sx,t + cxBESS,R Sx,t Px,t

(3.13)

BESS is the apparent power output of the x th BESS. cxBESS,self , cxBESS,V , and where Sx,t BESS,R are the standby loss coefficient, voltage-dependent loss coefficient, and cx current-dependent loss coefficient of the x th BESS inverter, respectively. The constraints for the charging and discharging of EVs are already accounted for in [18], from which the EV charging schedule is adopted.

3.2 Problem Formulation

57

3.2.3 Decision Variables BESS EV The decision variables considered in the problem are Q PV , Q BESS x,t , Px,t x,t , and Q x,t . The PV systems are assumed to always operate at maximum power point and produce PV EV possible (Eq. (2.11)). Px,t follows the fast charging strategy that the maximum Px,t minimises total charging and battery degradation costs in [18] and is not within the control of the ISO.

3.2.4 Cost Functions 3.2.4.1

Reactive Power Cost

The PV, BESS, and EV inverters are assumed to be three-phase inverters with reactive power capability [21]. The reactive power support of PV is limited by Eq. (3.8), while that of BESS and EV by (3.9). The reactive power cost of the DERs in this chapter are derived following the same foundation laid out in Sect. 2.3, where the calculation of the additional losses in the inverter follows Eq. (2.19) and is repeated below for the case of generic DER X:  X X 2  X X 2 cxX,V Sx,t + cxX,R Sx,t − Px,t − Px,t = 2 cxX,self + cxX,V Q Xx,t + cxX,R Q Xx,t 

X,invloss Px,t

X Px,t = 0 X Px,t =0

(3.14)

Even though the calculation of the additional losses in the inverter are the same for PV, BESS, and EV, the calculation of their costs can be different as different sources of active power may be used to compensate for the additional losses. For PV, the reactive power cost is the same as the one formulated in Chap. 2 [22], and is repeated here for convenience: Pgrid

QPV PV Q x,t = ct cx,t

PV,invloss × Px,t

(3.15)

For BESS, when P BESS = 0, the additional power loss reduces the BESS’ charging or discharging power, i.e. P BESS,invloss reduces the power bought from the grid or QBESS is associated decreases the power that can be sold to the grid. Therefore cx,t Pgrid with ct . When P BESS = 0, P BESS,invloss decreases the SOC of the BESS. The associated costs are then the average cost of electricity that has been purchased to charge the battery. Hence, they take the form of: QBESS BESS cx,t Q x,t =

⎧ Pgrid BESS,invloss ⎨ct × Px,t ⎩

t

Pgrid CH BESS t  =1 δx,t  Px,t  ×ct  t CH BESS t  =1 δx,t  Px,t 

BESS,invloss × Px,t

BESS Px,t = 0 BESS Px,t =0

where t  indicates the periods from the beginning of the day until period t.

(3.16)

58

3 Analytical Approach to Power Dispatch in Distribution Systems

The PV systems can technically also charge the BESS, but since the PV penetration considered in this work is always below 100%, any charging of the BESS (even using QBESS PV power) will cause the ISO to buy more power from the grid. Therefore cx,t Pgrid is always associated with ct . If PV penetration at period t goes above 100% and QBESS will be associated with PV is used to charge the BESS at that period, then cx,t cxPPV . When the BESS’ SOC is at SOCmin and is not initially charging, the BESS will QBESS will also be charge just enough to compensate for the losses, and therefore cx,t Pgrid proportional to ct . In the case of EV, the RPC comes from the stationary charger. The EV chargers are only allowed to produce/absorb reactive power when the EV is not charging or discharging to prioritise EV owners’ charging strategy. Consequently, the power loss in the stationary chargers’ inverters is compensated by the grid. The cost of reactive power by EV chargers are therefore: Pgrid

QEV EV Q x,t = ct cx,t

EV,invloss × Px,t

(3.17)

As can be seen from Eqs. (3.15) to (3.17), the reactive power cost depends on Q Xx,t , Pgrid X,invloss ct (known by the ISO), and Px,t , which is available from the inverters’ data sheet or can be obtained directly from the manufacturer [23]. As such, the ISO can QX to determined the appropriate remuneration for DERs’ RPC. easily calculate cx,t This topic is further explored in Sect. 4.6.3.

3.2.4.2

Battery Degradation Cost

Batteries have a finite cycle life and experience degradation with each cycle. Different types of batteries produced by different manufacturers have different degradation parameters. To be accurate and consistent, the BESS’ type, size, and degradation parameters are taken from a single source [24]. The cost function of the battery degradation is [24]: BESS,deg

cx,t

2 = ηxBESS,DOD SOCx,t+1 − SOCx,t BESS 2 2 + ηxBESS,P Px,t + ηxBESS,SOC SOCx,t+1 − SOCx,ref

(3.18)

where ηxBESS,DOD , ηxBESS,P , and ηxBESS,SOC are battery degradation constants related to depth of discharge, charging power, and SOC respectively [24]. SOCx,ref is the SOC at which there is minimal potential induced battery degradation [24]. The battery degradation in EVs have been taken into account in the charging strategy implemented [18].

3.3 Proposed Analytical Approach

59

Fig. 3.1 Line section diagram to illustrate the power flow across the distribution lines

3.3 Proposed Analytical Approach 3.3.1 Optimal Reactive Power Dispatch To calculate the impact of DERs’ power injection on line losses, it is necessary to first modify the power flow across the lines (Pi and Q i shown in Fig. 3.1), and to calculate line losses using BFS [25] (see Algorithm 2.1 on page 33), an algorithm suitable for radial distribution systems [26]. Nevertheless, it is difficult to analytically solve the relations among the variables due to the iterative nature of BFS. Therefore, BFS algorithm is replaced with a non-iterative approximation of power losses. Subsequently, an analytical solution to RPD is derived using the following line loss approximations [12, 27]: 2 2 iloss = ri Pi + Q i , P 2 |Vi |

2 2 iloss = xi Pi + Q i Q 2 |Vi |

(3.19)

where the tilde serves to indicate approximate quantities. ri and xi are the resistance and reactance of the line connecting sending node i. The index t has been removed loss are: loss and Q to simplify the notations. The total P loss = P

N −1  i=1

P 2 + Q i2 ri i , |Vi |2

loss = Q

N −1 

xi

i=1

Pi2 + Q i2 |Vi |2

(3.20)

Equations (3.19) and (3.20) are approximate as Pi and Q i are only the power flowing through line i to fulfil the net load of the receiving nodes (all the nodes downstream of i), while the second order terms are ignored [27]. Although they loss largely follow a linear relation with the exact loss loss and Q are approximate, P calculated using BFS, even when Vi is kept constant at the original value in the system (Figs. 3.2 and 3.3). The approximate power losses are calculated through the following: loss = P

N −1  i=1

P 2 + Q2 r i i 2 i ;

Vi

loss = Q

N −1  i=1

xi

Pi2 + Q i2

2

Vi

(3.21)

60

3 Analytical Approach to Power Dispatch in Distribution Systems

loss , and from BFS, P loss , for Fig. 3.2 Relation between Approximate PLoss from Eq. (3.21), P the 69-bus system

loss , and from BFS, Q loss , for Fig. 3.3 Relation between Approximate QLoss from Eq. (3.21), Q the 69-bus system

3.3 Proposed Analytical Approach

61

Fig. 3.4 Errors between P loss obtained from Eq. (3.22) and from BFS, for a 69-bus system and b 119-bus system



where Vi refers to the voltage magnitude of node i in the system when no PV has



been added. Vi has been used instead of |Vi | to illustrate that the losses can be estimated accurately even if up-to-date information regarding the node voltages are loss , as shown in Fig. 3.3. not available. Similar relationships were also obtained for Q The accurate losses across the line are then: loss ; P loss = η P,0 + η P,1 P

loss Q loss = η Q,0 + η Q,1 Q

(3.22)

loss to P loss where η P,0 , η P,1 , η Q,0 , and η Q,1 are the constants obtained from fitting P loss loss  to Q (Fig. 3.2) and Q (Fig. 3.3). The maximum error encountered between P loss from Eq. (3.22) and from BFS is less than 5%, as can be seen in Fig. 3.4, validating the use of Eq. (3.22) in the formulation. The method to express the operational costs in terms of the decision variables will now be outlined. To illustrate the concept with the simplest notations possible, a system with one PV at node k is used (Fig. 3.1). And the objective function in Eq. (3.1) is simplified to: Cost =

 T   Pgrid grid grid ct Pt + cQgrid Q t + cPPV PtPV + ctQPV Q PV H t

(3.23)

t=1

Expanding the loss in power balance equation (Eqs. (3.2) and (3.3)) using grid grid Eq. (3.22), Pt and Q t can be expressed as :

62

3 Analytical Approach to Power Dispatch in Distribution Systems

P grid =

N 

loss Piload − P PV + η P,0 + η P,1 P

(3.24)

loss Q iload − Q PV + η Q,0 + η Q,1 Q

(3.25)

i=1

Q grid =

N  i=1

loss and Q loss expressions now include the power injection from the PV: where P

loss = P

k−1 

ri

P  i − P PV

2 N −1  P 2 + Q2 + Q  i − Q PV + ri i 2 i

2

Vi

Vi

i=k

2

i=1

(3.26)

where Pi and Q i are the original active and reactive power flow from sending node i before the addition of PV at node k. By expanding the square terms, we get: 2 2 k−1 P  i2 + Q  i2  P PV − 2P  i P PV + Q PV − 2Q  i Q PV r i 2 + ri

2

Vi

Vi

i=1 i=1 (3.27) Substituting Eq. (3.20) to Eq. (3.27), the following equation is obtained: loss = P

N −1 

loss = P loss,init + P

k−1 

ri

P PV

2

i=1

2 − 2Pi P PV + Q PV − 2Q i Q PV

2

Vi

(3.28)

loss,init is the active power loss before adding the DERs and is constant for a where P loss can be expanded, with ri changed to xi . particular time step. Similarly, Q

loss = Q loss,init + Q

k−1 

xi

P PV

i=1

2

2 − 2Pi P PV + Q PV − 2Q i Q PV

2

Vi

(3.29)

Now Cost can be explicitly expressed in terms of Q PV : Cost = cPgrid

 N 

 loss Piload − PtPV + η P,0 + η P,1 P

(3.30)

i=1

+c

Qgrid

 N 

+ cPPV P

i=1 PV

 Q iload



Q PV t

+ cQPV Q PV



Q,0



Q,1 loss

Q

3.3 Proposed Analytical Approach

63

loss and Q loss are described in Eqs. (3.28) and (3.29). By differentiating Cost where P with respect to Q PV and having its derivative equal to zero, the optimal value of Q PV to minimise the cost of running the system can be obtained. There are two cases, one when P PV = 0 and another when P PV = 0 since the ctQPV for the two cases are different (Eqs. (2.19) and (2.20)). For P PV = 0, PV 2 2  k−1  P − 2Pi P PV + Q PV − 2Q i Q PV ∂Cost ∂ Pgrid P,1 = η ri (3.31) c

2 ∂ Q PV ∂ Q PV

Vi

i=1   PV 2 2 k−1  P − 2Pi P PV + Q PV − 2Q i Q PV Qgrid Q,1 PV +c xi −Q η

2

Vi

i=1   PV 2  Pgrid PV,self PV,V PV PV,R c Q +c Q +c +c ∂Cost cQgrid PV PV PV =0 =A Q − B + C Q − D + E + F Q − k k k k ∂ Q PV 2

(3.32)

where Ak =

k−1  ri Pgrid P,1 η ;

2 c



i=1 Vi

Ck =

k−1  xi Qgrid Q,1 η ;

2 c



i=1 Vi

Bk =

k−1  ri Q i Pgrid P,1 η ;

2 c



i=1 Vi

Dk =

k−1  xi Q i Qgrid P,1 η ;

2 c



i=1 Vi

E=

cPgrid cPV,V 2

F = cPgrid cPV,R

Ak , Bk , Ck , and Dk depend on the location of the DERs in the system, but not on the type of DERs, whereas E and F depend on the inverter characteristics of the DERs. The optimal value of Q PV is therefore: Q PV = When P PV = 0, by setting

Bk + Dk − E + cQgrid /2 Ak + C k + F

∂Cost ∂ Q PV

= 0, we obtain:

⎛ Q PV ⎝ Ak + Ck + F +

(3.33)

⎞ E

P PV

2



+ Q PV

2

Qgrid ⎠ = Bk + Dk + c 2

(3.34)

Although the root of Eq. (3.34) is difficult to be obtained manually, the value of Q PV can be easily calculated using software such as MATLAB. Therefore, the optimal RPD by the PV at node k at each time slot can be obtained using Eqs. (3.33) and (3.34), subject to Eq. (3.8). When there are more than one DERs, the steps required to schedule the RPD are:

64

3 Analytical Approach to Power Dispatch in Distribution Systems

[1] Determine the constants η P,0 , η P,1 , η Q,0 , η Q,1 for the system [2] Determine Ak , Bk , Ck , Dk , E, F for each DER. Calculate the optimal reactive power output of each DER Q X separately, taking other decision variables as constant [3] Incorporate the optimal reactive power from all the DERs into the power flow (Q i ) and update the constants Bk and Dk for each DER [4] Start the subsequent iteration to calculate Q X until the values of Q X converge Steps 2 to 4 are repeated for all time periods. In Appendix A, the convergence of the analytical approach for two variables is proven. Although the convergence has not been proven for any n variables, through rigorous testing it was found that the proposed approach always converge within eight iterations even for more than 100 variables. Although it is possible to calculate the values of the decision variables simultaneously, it is not practical to do so, as the time requirement increases exponentially with the number of decision variables. The constants in Eqs. (3.33) and (3.34), except for cQgrid and Ck , vary with time, depending on the power flowing through the lines and the electricity price. The analytical approach should generate optimal solutions as long as the approximation in Eq. (61) and (63) remain valid. The approach can also be extended to include other/existing reactive power equipment or DERs, such as static VAR compensator (SVC) or wind turbine generators, by replacing Eq. (3.8) with the rating of the reactive power device/generator, and incorporating the cost of the reactive power [SGD/kvarh] from the device/generator to the objective function, Eq. (3.1). For reactive power equipment with continuous reactive power output, the optimal output of the equipment can be obtained similar to Eq. (3.33), whereas for equipment with discrete values, other methods such as GA can also be incorporated into the approach.

3.3.2 Energy Arbitrage Considering Line Losses BESS enables energy arbitrage (EA) as it allows the owner/operator to buy electricity from the grid when price is the lowest (price nadir), store the electrical energy in the BESS, and then sell it back to the grid when price is the highest (price peak). Figure 3.5 indicates the price peaks and nadirs in 2015 average wholesale electricity price (including grid charge) in Singapore [28, 29]. When the BESS is charging (buying electricity from the grid), power losses in the system are increased, and when it is discharging (selling electricity to the grid), the power losses are reduced.1 As such, these losses have to be considered to make sure that either (1) EA is not only beneficial for the BESS owner/operator, but also to the ISO, or (2) the ISO charges (remunerates) the BESS owner/operator should the EA disadvantages (benefits) the system. 1 This

is true as long as the distribution system is a net importer of electricity at that period.

3.3 Proposed Analytical Approach

65

Fig. 3.5 Price peaks and nadirs for 2015 Singapore annual wholesale electricity price plus grid charge

To determine whether EA is profitable for a particular pair of peak and nadir, the following condition has to be satisfied: Pgrid

Pgrid

Pgrid

Pgrid

BESS BESS loss loss cpeak ηxB Pnadir > cnadir Pnadir + cnadir Pnadir + cpeak Ppeak   loss + cQgrid Q loss nadir + Q peak

(3.35)

loss where ηxB = ηxCH ηxDCH is the round-trip efficiency of the x th BESS. Pnadir (Q loss nadir ) loss loss and Ppeak (Q peak ) are the difference in active (reactive) power loss because of the charging at price nadir and discharging at the price peak, respectively. The left hand side of Eq. (3.35) is the price that the system has to pay by taking power BESS ) from the grid at the price peak without using the BESS, while the right (ηxB Pnadir hand side is the cost of delivering that power by charging the BESS at price nadir BESS BESS BESS ) and then discharging it at price peak (Ppeak = ηxB Pnadir ). Equation 3.35 (Pnadir does not include battery degradation terms from Eq. 3.18 because they are not as substantial as the power losses (demonstrated in Sect. 3.5.2 on EA results) but add disproportionate complexity. Nevertheless, they are included when calculating the final cost (Eq. (3.1)). Assuming the BESS is located at node k, the power losses can be calculated using Eq. (3.22) to yield:

BESS 2 loss BESS = αk Pnadir + 2βk Pnadir Pnadir loss BESS 2 BESS Ppeak = αk ηxB Pnadir − 2γk ηxB Pnadir BESS 2 BESS Q loss + 2ζk Pnadir nadir = k Pnadir B BESS 2 BESS Q loss − 2λk ηxB Pnadir peak = k ηx Pnadir where

66

3 Analytical Approach to Power Dispatch in Distribution Systems

αk =

k−1  ri P,1

2 η ;



i=1 Vi

βk =

k−1  ri Pi,nadir P,1

2 η ;

Vi

i=1

γk =

k−1  ri Pi,peak P,1

2 η

Vi

i=1

k =

k−1  xi Q,1

2 η ;



i=1 Vi

ζk =

k−1  xi Pi,nadir Q,1

2 η ;

Vi

i=1

λk =

k−1  xi Pi,peak Q,1

2 η

Vi

i=1

The net revenue of EA, Rev, is the left hand side of Eq. (3.35) minus the right BESS for a particular pair of hand side. Taking ∂∂PRev BESS = 0, we can find the optimal P nadir and peak: BESS Pnadir

Pgrid Pgrid cpeak ηxB (1 + 2γk ) − cnadir (1 + 2βk ) + 2cQgrid λk η B − ζk   = 2  2 Pgrid  Pgrid 2αk cnadir + ηxB cpeak + 2k cQgrid 1 + ηxB

(3.36)

To schedule for the most profitable EA for the system, an analytical rule-based approach is proposed (Fig. 3.62 ). When power losses are not considered, the condition Pgrid Pgrid in Eq. (3.35) becomes [17]: cpeak ηxB > cnadir , which suggests that as long as the condition is fulfilled, the BESS should take in as much power as possible during the price nadir and discharge the power at the price peak. As shown in Sect. 3.5.2, this will not yield the most economical result for the system.

3.3.3 Voltage and Current Constraints Handling So far, the proposed analytical approach has not limited the current and voltage values in the system as required by constraints (3.6) and (3.7). To do so, their values need to be calculated, which can be done using the following equations from Simplified Distflow method [27]: 2 = Vi,t2 − 2 ri Pi,t + xi Q i,t Vi+1,t



Si,t

Ii,t =

Vi,t

(3.37) (3.38)

After determining the BESS schedule according to Fig. 3.6 and calculating the reactive power outputs for each time step using Eqs. (3.33) and (3.34), the current and voltage values in the system are calculated using Eqs. (3.37) and (3.38). For each bus and line in the system, a priority list is created, indicating the nearest DER (highest priority) to the furthest (lowest priority). If either constraint (3.6) or (3.7) 2 In

Fig. 3.6, a sub-nadir refers to a period between the previous peak and the peak that a particular nadir is paired with, that has the next lowest cPgrid . Similarly, a sub-peak is a period between the nadir that a particular peak is paired with and the subsequent nadir, that has the next highest cPgrid . For example, in Fig. 3.5, if nadir 1 (04:30 a.m.) is paired with peak 3 (10:30 a.m.), the sub-nadir is at 04:00 a.m. and the sub-peak is at 11:00 a.m.

3.3 Proposed Analytical Approach Fig. 3.6 Proposed analytical rule-based approach to conduct the most profitable EA and low-cost RPD. Appropriate constraint handling algorithm refers to under-voltage (Algorithm 3.1), overvoltage, and overcurrent constraint handling algorithm, depending on the constraint violated

67

68

3 Analytical Approach to Power Dispatch in Distribution Systems

Algorithm 3.1 Undervoltage Constraint Handling

1: if min V i,t < Vmin , ∀i = 1, . . . , N then 2: while Vi,t < Vmin do 3: for x = 1 to si ze(Priorit y List)

do X



4: while Q X x,t < Q x,t,lim && Vi,t < Vmin do X X X 5:

x,t

Q x,t ← Q

x,t + Q

Vi,t ← newVi,t

6: 7: end while 8: end for 9: for x = 1 to B do



BESS > P BESS && V < V 10: while Px,t i,t min do x,min BESS ← P BESS − P BESS 11: P x,t

x,t

x,t

Vi,t ← newVi,t

12: ← new Q BESS 13: Q BESS x,t,lim x,t,lim



14: while Vi,t < Vmin && Q BESS < Q BESS x,t x,t,lim do BESS BESS BESS 15: Q ← Q + Q x,t x,t x,t





Vi,t ← newVi,t

16: 17: end while 18: end while 19: end for 20: end while 21: end if

 adjusting RPD



 recalculate Vi,t

 adjusting EA

is violated for any bus, then the power outputs of the DERs are adjusted (subject to constraints (3.8) to (3.10)) according the the order in the priority list until all the constraints are satisfied. Algorithm 3.1 outlines the undervoltage constraint handling algorithm. BESS is only modified if the constraints are still vioQ Xx,t is adjusted first while Px,t BESS lated. When Px,t is changed, then the optimal EA strategy of the x th BESS and the RPD at the affected time periods are recalculated. Figure 3.10 (in Sect. 3.5) illusBESS has to be adjusted to fulfil the undervoltage constraint. The trates a case when Px,t changes to the DERs’ output in line 5, 11, and 15 of Algorithm 3.1 are proportional to

Vmin − Vi,t . Overvoltage, and overcurrent constraints are implemented similarly.

3.4 Implementation Setup 3.4.1 Benchmark Optimisation Algorithms The proposed analytical approaches are benchmarked with both exact and metaheuristic optimisation algorithms. Interior Point Method (IPM) is a common exact optimisation method to solve non-convex optimal power flow [30–32] and is therefore employed in this work as a benchmark. The IPM was run using MATLAB

3.4 Implementation Setup

69

built-in IPM algorithm (fmincon) using the data structures and perturbed KarushKuhn-Tucker (KKT) conditions employed in [30, 32]. Meanwhile, Particle Swarm Optimisation (PSO) [3, 33] and Genetic Algorithm (GA) [34], are chosen to represent the metaheuristic optimisation algorithms because they are widely used in the field [6, 35]. The metaheuristic algorithms are run with 120 particles/chromosomes and 60 iterations. It is noted that there are increasingly many exact and metaheuristic optimisation algorithms developed to solve complex, non-convex problems, such as surrogate optimisation [36, 37], unified differential evolution [38, 39], salp swarm algorithm [40], and many others. These algorithms have shown promising results in other fields and may also be useful for power dispatch optimisation. Nevertheless, the proposed approach is benchmarked against the optimisation algorithms that have been tried and tested within the field of power systems to enable easy replication by other researchers. All algorithms are implemented in MATLAB R2016a on Intel Core i5-4690 CPU @ 3.50GHz, with 12GB RAM. All approaches utilised the same following parameters.

3.4.2 Test Systems In Chap. 2, a smaller test system was used so that the effects of reactive power compensation from PV can be analysed and quantified more easily. Building on those insights, a larger test system with a variety of DERs can now be adopted to validate the applicability and the robustness of the proposed approach. The proposed analytical approach was first validated using the 69-bus [41] radial distribution system used previously. Subsequently, a 119-bus system [42] was also employed to demonstrate the scalability of the approach. The locations of the DERs in the systems are shown in Fig. 3.7. PV of each PV system is 300 kWp and 500 kWp for 69-bus and 119-bus system Px,rated respectively, while the BESS are Lithium ion batteries with 500 kWh capacity and BESS of 500 kVA [24] for both systems. Each EV charger is assumed to have 4 kVA Sx,max rating. Node 3, 17, 48 of 69-bus system have 44, 100, 25 EVs respectively. In 119-bus system, node 6, 13, 38, 52, 61, 65, 81, 96, 100 have 150, 50, 100, 35, 75, 80, 40, 20, 90 EVs respectively. The first vehicle arrives at 6 a.m. and the last vehicle leaves at 8 p.m (typical industrial area). The placement of the DERs in the systems is completely arbitrary. Penetration levels of PV, EV, and BESS power are 30.0%, 15.6%,3 and 34.5% of peak load respectively, in 69-bus system. The respective penetration levels in 119-bus system are 32.4%, 10.1%, and 19.6% of peak load. The load data are varied to form a more realistic case where the load variation of each bus over 24 h follows an industrial load profile [18], but the variation is not 3 Assuming

energy consumption of 6000 kWh per employee, there are about 3 EVs per 100 people in the industrial area.

70

3 Analytical Approach to Power Dispatch in Distribution Systems

Fig. 3.7 Test distribution systems with (a) 69, and (b) 119 buses. DERs are connected to the indicated nodes

identical to each other. The power factor of the load is constant throughout the day, assuming that the resistance and inductance of the load do not change with voltage or current. Vmin and Vmax are 0.9 p.u. and 1.05 p.u. respectively, while Imax is set at 1 p.u.

3.4 Implementation Setup

71

3.4.3 Cost and Weather Parameters Pgrid

ct is taken from the 2015 average of half-hourly wholesale electricity price plus grid charge in Singapore [29]. As in Chap. 2, cQgrid is the reactive power charge in Singapore for consumers taking supplies at 22kV or 6.6kV [43]. cxPPV value follows the value from Chap. 2 (the same throughout the thesis), which is 10.28 cents SGD/kWh. cxPPV is constant throughout the lifetime of the PV, conservatively assumed to be 20 years throughout the thesis. To obtain the values of cself , cV , cR of the different inverters, empirical values of self c , cV , cR for a 208kVA inverter are first obtained from [44]. Next, the efficiency curve for the whole range of apparent power rating of the 208 kVA is obtained. The 4, 300, 500 kVA inverters are assumed to have the same efficiency as the 208 kVA inverter. From the efficiency curves, the values of cself , cV , cR for the different inverters are obtained. Wenjie Zhang from Electrical and Computer Engineering (ECE) department of the National University of Singapore (NUS) has forecasted G x,t and G 70 x,t using a method adapted from [45]. Solar irradiance and ambient temperature data from 1st January 2012 to 25th February 2015 have been used as training data to generate the day-ahead forecast for 26th February 2015 (Fig. 3.8). Only the forecast and upper limit of the 70% confidence level data are employed in the optimisation. The data were measured every second in Singapore at latitude 1.3026◦ and longitude 103.7729◦ . They were averaged to half-hour periods to suit the electricity price information data. All PV systems in this work use the same weather data, provided by SERIS.

Fig. 3.8 Forecast and real solar irradiance data

72

3 Analytical Approach to Power Dispatch in Distribution Systems

Table 3.1 Results Comparison for Five Runs of the Algorithms 69-bus system Mean (Time) [s]

119-bus system

PSO

GA

IPM

AS

ASEA PSO

5445

10579 194

138

153

GA

IPM

AS

ASEA

13780 28626 2240

286

346

Mean (Obj) [SGD] 11554 11556 11672 11549 11507 69342 69271 69358 68950 68811 Std (Obj) [SGD] 1.86   Mean S loss [%] 2.96

0.25

9.09

0

0

3.83

12.57

1.58

0

0

2.98

3.37

2.95

3.03

3.78

3.67

3.59

3.41

3.45

Min (|V i,t |) [p.u.]

0.924

0.909

0.925

0.909

0.895

0.901

0.903

0.906

0.900

0.922

3.5 Results and Discussions 3.5.1 Performance Comparison Table 3.1 shows the results of five runs of the proposed analytical solution (AS), AS with energy arbitrage (ASEA), and the benchmark algorithms. It is clear that AS performs better than the exact and metaheuristic algorithms for both the 69-bus and loss and P loss the 119-bus system. This confirms that although the relation between P in Fig. 3.2 is not perfectly linear, the proposed approach works well. Notably, for the 119-bus system, AS yields a 24%, 19%, and 25% greater reduction in costs through RPD compared to PSO, GA, and IPM, respectively. Notably, the performance improvement of AS over the other methods is even more superior for the 119-bus system than that for the 69-bus system, proving that it can handle a large number of variables better than other widely used methods. AS with energy arbitrage (ASEA) further reduces the cost. The other approaches discharge the BESS to SOCx,min at the first period and do not charge them again because that is the most economic choice for every individual period. The better performance of AS and ASEA is due to the lower losses in the system and lower reactive power cost, compared with the results from GA, PSO, and IPM.

3.5.1.1

Comparison with PSO and GA

For the five runs, PSO performed better than GA on the 69-bus system, but worse on the larger system. Nevertheless, GA took about twice the time compared to PSO. It is true that parameter tuning is important to improve the performance of metaheuristic algorithms, and increasing the population size and number of iterations will allow PSO and GA to reach better solutions. However, even with the current number of population and iteration, it is already impractical to implement the metaheuristic algorithms for real-time control. Additionally, removing the need for parameter tuning and dependence on initial solution estimate is one of the main advantages of the proposed analytical approach.

3.5 Results and Discussions

73

More importantly, both AS and ASEA are decisively faster than the metaheuristic methods. The bulk of the time is spent on evaluating the solution using BFS, which has to be done for each particle/chromosome in each iteration for PSO and GA, but only has to be done once for the proposed approaches. The benefits in terms of costs and time savings are expected to further increase with larger systems and more variables.

3.5.1.2

Comparison with IPM

Although IPM is faster than PSO and GA, it is still slower than AS and ASEA. Moreover, the time taken to generate the Jacobian and Hessian to be employed in IPM is actually longer than to optimise the problem. The memory requirement to handle the Jacobian and Hessian matrices are also much larger. The poor performance of IPM compared to the other methods is likely due to the multiple local minima that exist in the problem, which IPM is prone to be trapped in [46]. Faster convergence has been reported in the literature using nonlinear commercial solvers such as BARON, DICOPT, and others. However, because of lack of access, no comparison has been made with the commercial solvers. It is also noted that GA has been combined with IPM [47] to employ the global perspective of GA and local convergence of IPM. Nevertheless, as the computational time of GA and IPM are both higher than AS and ASEA, the hybrid algorithm would most probably be slower than the analytical approaches, and it will still require parameter tuning. Therefore, compared to mathematical programming and metaheuristic algorithms, the proposed analytical approaches have four distinct advantages: [1] Up to 100 times faster, scalable for larger systems with more variables, and suitable for use in real time [2] Give consistent and repeatable results [3] Remove dependence on initial solutions and parameters [4] Display explicit relations among the variables, allowing analysis on how the parameters affect each other (Eqs. (3.33), (3.34), and (3.36)).

3.5.2 Line Loss Consideration in Energy Arbitrage Next, in this subsection, the benefits of considering line loss in EA are investigated. After carrying out the steps in Fig. 3.6, the profitable pairs are nadir 1 and peak 3, as well as nadir 5 and peak 7 (Fig. 3.5). Sub-nadir 1 is also profitable with sub-peak 3, but sub-nadir 5 is only profitable with peak 7, not sub-peak 7. The charging and discharging power, as well as the SOCs of the BESS for the 69 and the 119-bus system are shown in Figs. 3.9 and 3.10, respectively. When the line losses are not considered, all the BESS in the 69-bus system behave in the same manner as BESS 1

74

3 Analytical Approach to Power Dispatch in Distribution Systems

Fig. 3.9 (a) Power and (b) SOC of the BESS using proposed ASEA (69-bus). BESS 1, 2, 3 are located at node 6, 11, 63, respectively

Fig. 3.10 (a) Power and (b) SOC of representative BESS using proposed ASEA (119-bus). BESS 1, 3, 7 are located at node 3, 25, 73, respectively

(Fig. 3.9). Nevertheless, as BESS 3 is located further inside the system, the charging of the BESS 3 increases the power flow in more lines and therefore induces more losses. Since ηxB is always less than 1, the loss reduction during discharging is always lower than the loss increase during charging. It can therefore be discerned from Eq. (3.35), that for EA, the best location for BESS is at the substation, unless the BESS . DERs near the BESS can fulfil Pnadir

3.5 Results and Discussions

75

Table 3.2 Comparison of BESS Revenue ASEA 69-bus 119-bus BESS Revenue [SGD] Cost of line loss [SGD] Battery degradation [SGD] Net Revenue [SGD]

46.65 (4.58) (1.68) 40.39

155.31 (8.72) (5.49) 141.10

Without considering line loss and voltage 69-bus 119-bus 47.05 (6.69) (1.82) 38.54

156.82 (13.48) (6.05) 137.29

Additionally, for the 119-bus system, the undervoltage constraint handling was triggered and the EA of BESS 7 was modified (Fig. 3.10) to satisfy the minimum voltage constraint, validating the efficacy of the proposed algorithm. The difference in economics of EA with and without considering line losses can be seen in Table 3.2. While the differences in revenue and battery degradation are not very significant, the lower cost of line losses makes the proposed ASEA more profitable. The Net Revenue shown in Table 3.2 has excluded the revenue from the BESS discharge at period 1 (Fig. 3.6). By utilising the BESS for EA, the reactive power capability of the BESS is also more restricted (Eq. (3.9)) and the BESS may not be able to provide as much reactive power support. These findings are not limited to BESS, and can also be extended to other storage devices. Therefore, any scenario considering the economics of energy arbitrage using BESS or EV should account not only for the battery degradation but also for the losses in the system due to the arbitrage. It may also be more economical for the battery to provide spinning reserve or peak-shaving service compared to arbitraging but its investigation is outside the scope of this work. Nevertheless, the proposed analytical approach will also be able to calculate the optimal dispatch for the different purposes, given the cost functions.

3.5.3 Power Profiles The active and reactive power outputs of ASEA for the 69-bus system are shown in Figs. 3.11 and 3.12, respectively. The power outputs of each type of DER have been aggregated for easier visualisation. The following observations can be made from the two figures: [1] The EA from BESS and EV increases the peak load of the system while reducing the minimum load. This may introduce difficulty to the grid to ramp up and ramp down its generation, especially at higher penetration of DERs. While the losses resulting from the BESS’ EA have been considered, the impact of energy storage scheduling at the distribution level on the transmission level is also worth investigating in future research.

76

3 Analytical Approach to Power Dispatch in Distribution Systems

Fig. 3.11 Active power profile for Case ASEA in the 69-bus test system

Fig. 3.12 Reactive power profile for Case ASEA in the 69-bus test system

[2] Even at relatively low penetration of DERs, most of the reactive power demand in the system can be fulfilled locally. This has the advantage of reducing the line losses and maintaining voltage within the acceptable range. [3] Having many different types of DERs is beneficial for the system because when some of them are not able to produce reactive power because of their active power usage, the other DERs are able to provide the reactive power support. It is also observed from Fig. 3.13 that RPC using DERs is cost competitive with the reactive power charge from the grid, as also observed for PV in Chap. 2. And Pgrid QX is affected proportionally by ct as the grid compensates for the additional that cx,t losses due to reactive power. Reactive power costs are lower when they are generating

3.5 Results and Discussions

77

Fig. 3.13 Active power price and reactive power costs for Case ASEA in the 69-bus test system. The reactive power costs of the DERs are the average among the DERs of the same type. The missing points are when the DERs are not producing any reactive power

(or consuming) active power as the power losses in the inverters are assigned to both X and Q Xx,t . Px,t

3.5.4 Uncertainty Treatment In this chapter, the analytical approaches have been applied for day-ahead scheduling using day-ahead forecast of solar irradiance and deterministic load, EV charging, and electricity price. A conservative approach has been taken in setting the limits of the RPD. For example, the reactive power output from PV has been limited by the upper bound of the solar irradiance forecast (Eq. (3.8)) so as not to limit the PV active power output. For EV, the EV charger will not produce or absorb reactive power when an EV is being charged or discharged in case the owner would like to change its charging strategy. This conservative approach can be modified depending on the preference of the ISO. When more accurate forecast is available, e.g. for hour-ahead scheduling, it can be incorporated into the analytical approaches. The proposed approaches should also be able to be implemented in real time when there is a sudden change in the load or EV charging as they are much faster than the benchmark algorithms (approximately 2.9 s and 6.0 s per time step for 69-bus and 119-bus systems, respectively, as shown in Table 3.1). Like any other optimisation methods, the results from the proposed AS and ASEA also depend on the accuracy of the parameters’ forecast. Should erroneous forecast be used as inputs, then the results obtained through the algorithms may not be economically or technically satisfactory.

78

3 Analytical Approach to Power Dispatch in Distribution Systems

3.6 Summary In this chapter, an analytical approach for reactive power dispatch and energy arbitrage, incorporating reactive power costs and line losses has been proposed. The proposed approach is able to efficiently handle large number of variables, giving near-optimal solution up to 100 times faster compared to existing approaches. It is therefore suitable for both day-ahead and real-time power dispatch applications in large systems. The same analytical approach can also be used for general power dispatch with different parameters and different cost functions. The formulations and findings from this work can be used to create a pricing scheme for energy arbitrage and EV charging, taking into account the resulting line losses. Nevertheless, implementing the analytical approach requires extensive communication and monitoring infrastructure not available in most of today’s distribution systems. Consequently, Chap. 5 will explore ways to decentralise the proposed analytical approach to further reduce the computational burden and communication requirements.

References 1. Liang RH, Wang JC, Chen YT, Tseng WT (2015) An enhanced firefly algorithm to multiobjective optimal active/reactive power dispatch with uncertainties consideration. Int J Electr Power Energy Syst 64:1088–1097, ISSN: 01420615. https://doi.org/10.1016/j.ijepes.2014.09. 008 2. Sousa T, Morais H, Vale Z, Castro R (2015) A multi-objective optimization of the active and reactive resource scheduling at a distribution level in a smart grid context. Energy 85:236–250, ISSN: 03605442. https://doi.org/10.1016/j.energy.2015.03.077 3. Ziadi Z, Taira S, Oshiro M, Funabashi T (2014) Optimal power scheduling for smart grids considering controllable loads and high penetration of photovoltaic generation. IEEE Trans Smart Grid 5(5):2350–2359, ISSN: 19493053. https://doi.org/10.1109/TSG.2014.2323969 4. Kim YJ, Kirtley JL, Norford LK (2015) Reactive power ancillary service of synchronous DGS in coordination with voltage control devices. IEEE Trans Smart Grid 8(2):515–527, ISSN: 1949-3053. https://doi.org/10.1109/TSG.2015.2472967 5. Chen S, Hu W, Chen Z (2016) Comprehensive cost minimization in distribution networks using segmented-time feeder reconfiguration and reactive power control of distributed generators. IEEE Trans Power Syst 31(2):983–993, ISSN: 08858950. https://doi.org/10.1109/TPWRS. 2015.2419716 6. Mohseni-Bonab SM, Rabiee A (2016) Optimal reactive power dispatch: a review, and a new stochastic voltage stability constrained multi-objective model at the presence of uncertain wind power generation. IET Gener Transm Distrib 11:815–829, ISSN: 1751-8687. https://doi.org/ 10.1049/iet-gtd.2016.1545 7. Zhao B, Guo C, Cao Y (2005) A multiagent-based particle swarm optimization approach for optimal reactive power dispatch. IEEE Trans Power Syst 20(2):1070–1078, ISSN: 0885-8950. https://doi.org/10.1109/TPWRS.2005.846064 8. Kekatos V, Wang G, Conejo AJ, Giannakis GB (2014) Stochastic reactive power management in microgrids with renewables. IEEE Trans Power Syst PP (99):1–10, ISSN: 0885-8950. https:// doi.org/10.1109/TPWRS.2014.2369452 arXiv: 1409.6758

References

79

9. Robbins BA, Domínguez-garcía AD (2016) Optimal reactive power dispatch for voltage regulation in unbalanced distribution systems. IEEE Trans Power Syst 31(4):1–11, ISSN: 08858950. https://doi.org/10.1109/TPWRS.2015.2451519 10. Gabash A, Li P (2012) Active-reactive optimal power flow in distribution networks with embedded generation and battery storage. IEEE Trans Power Syst 27(4):2026–2035, ISSN: 08858950. https://doi.org/10.1109/TPWRS.2012.2187315. arXiv: 9605103 [cs] 11. Zhu T, Luo W, Bu C, Yue L (2016) Accelerate population-based stochastic search algorithms with memory for optima tracking on dynamic power systems. IEEE Trans Power Syst 31(1):268–277, ISSN: 08858950. https://doi.org/10.1109/TPWRS.2015.2407899 12. Hung DQ, Mithulananthan N, Lee KY (2014) Determining PV penetration for distribution systems with time-varying load models. IEEE Trans Power Syst 29(6):3048–3057, ISSN: 0885-8950. https://doi.org/10.1109/TPWRS.2014.2314133 13. Hung DQ, Mithulananthan N, Bansal R (2014) Integration of PV and BES units in commercial distribution systems considering energy loss and voltage stability. Appl Energy 113:1162–1170, ISSN: 03062619. https://doi.org/10.1016/j.apenergy.2013.08.069 14. Zhang L, Tang W, Liang J, Cong P, Cai Y (2016) Coordinated day-ahead reactive power dispatch in distribution network based on real power forecast errors. IEEE Trans Power Syst 31(3):2472–2480, ISSN: 0885-8950. https://doi.org/10.1109/TPWRS.2015.2466435 15. Rabiee A, Feshki Farahani H, Khalili M, Aghaei J, Muttaqi KM (2016) Integration of plug-in electric vehicles into microgrids as energy and reactive power providers in market environment. IEEE Trans Ind Informatics 12(4):1312–1320, ISSN: 15513203. https://doi.org/10.1109/TII. 2016.2569438 16. Cheng B, Powell W (2016) Co-optimizing battery storage for the frequency regulation and energy arbitrage using multi-scale dynamic programming. IEEE Trans Smart Grid PP (99):1– 10, ISSN: 19493053. https://doi.org/10.1109/TSG.2016.2605141 17. Sanjari M, Karami H, Gooi HB (2017) Analytical rule-based approach to online optimal control of smart residential energy system. IEEE Trans Ind Informatics ISSN: 1551-3203. https://doi. org/10.1109/TII.2017.2651879 18. Mehta R, Srinivasan D, Khambadkone AM, Yang J, Trivedi A (2016) Smart charging strategies for optimal integration of plug-in electric vehicles within existing distribution system infrastructure. IEEE Trans Smart Grid 3053 no. c:1–1, ISSN: 1949-3053. https://doi.org/10.1109/ TSG.2016.2550559 19. Sarker MR, Olsen DJ, Ortega-Vazquez MA (2016) Co-optimization of distribution transformer aging and energy arbitrage using electric vehicles IEEE Trans Smart Grid 1–11, ISSN: 19493053. https://doi.org/10.1109/TSG.2016.2535354 20. Gandhi O, Zhang W, Rodríguez-Gallegos CD, Bieri M, Reindl T, Srinivasan D (2018) Analytical approach to reactive power dispatch and energy arbitrage in distribution systems with ders. IEEE Trans Power Syst 33(6):6522–6533, ISSN: 0885-8950. https://doi.org/10.1109/TPWRS. 2018.2829527 21. Wu L, Zhao Z, Liu J (2007) A single-stage three-phase grid-connected photovoltaic system with modified mppt method and reactive power compensation. IEEE Trans Energy Convers 22(4):881–886, ISSN: 08858969. https://doi.org/10.1109/TEC.2007.895461 arXiv: z0024 22. Gandhi O, Rodríguez-Gallegos CD, Zhang W, Srinivasan D, Reindl T (2018) Economic and technical analysis of reactive power provision from distributed energy resources in microgrids. Appl Energy 210:827–841, ISSN: 03062619. https://doi.org/10.1016/j.apenergy.2017.08.154 23. SMA, Sunny tripower inverter. https://usa.krannichsolar.com/fileadmin/content/dataSTPTLUS12-24EXP-DUS132533W.pdf 24. Koller M, Borsche T, Ulbig A, Andersson G (2013) Defining a degradation cost function for optimal control of a battery energy storage system. In: 2013 IEEE Grenoble Conference, IEEE, Jun. 2013, pp 1–6, ISBN: 978-1-4673-5669-5. https://doi.org/10.1109/PTC.2013.6652329 25. Ghosh S, Das D (1999) Method for load-flow solution of radial distribution networks. IEE Proc Gener Transm Distrib 146(6):641–648, ISSN: 13502360. https://doi.org/10.1049/ip-gtd: 19990464

80

3 Analytical Approach to Power Dispatch in Distribution Systems

26. Teng JH (2003) A direct approach for distribution system load flow solutions. IEEE Trans Power Delivery 18(3):882–887, ISSN: 0885-8977. https://doi.org/10.1109/TPWRD.2003.813818 27. Baran ME, Wu FF (1989) Network reconfiguration in distribution systems for loss reduction and load balancing. IEEE Trans Power Delivery 4(2):1401–1407, ISSN: 0885-8977. https:// doi.org/10.1109/61.25627 28. Bieri M (2017) Methodology description: SERIS future power price scenarios 29. EMC, Energy market price information. https://www.emcsg.com/marketdata/priceinformation Visited on 04/01/2018 30. Wei H, Sasaki H, Kubokawa J, Yokoyama R (1998) An interior point nonlinear programming for optimal power flow problems with a novel data structure. IEEE Trans Power Syst 13(3):870– 877, ISSN: 08858950. https://doi.org/10.1109/59.708745 31. Liu M, Tso S, Cheng Y (2002) An extended nonlinear primal-dual interior-point algorithm for reactive-power optimization of large-scale power systems with discrete control variables. IEEE Trans Power Syst 17(4):982–991, ISSN: 0885-8950. https://doi.org/10.1109/TPWRS. 2002.804922 32. Li YW, Kao CN (2009) An accurate power control strategy for powerelectronics- interfaced distributed generation units operating in a low-voltage multibus microgrid. IEEE Trans Power Electron 24(12):2977–2988, ISSN: 08858993. https://doi.org/10.1109/TPEL.2009.2022828 33. Gandhi O, Zhang W, Rodríguez-Gallegos CD, Srinivasan D, Reindl T (2016) Continuous optimization of reactive power from PV and EV in distribution system. In: 2016 IEEE Innovative Smart Grid Technologies–Asia (ISGT-Asia), Melbourne. Nov. 2016, IEEE. pp 281–287. ISBN: 978-1-5090-4303-3. https://doi.org/10.1109/ISGT-Asia.2016.7796399 34. Chelouah R, Siarry P (2000) A continuous genetic algorithm designed for the global optimization of multimodal functions. J Heuristics 6:191–213 35. Gandhi O, Rodríguez-Gallegos CD, Srinivasan D (2016) Review of optimization of power dispatch in renewable energy system. In: 2016 IEEE Innovative Smart Grid Technologies–Asia (ISGT-Asia), Melbourne. Nov. 2016. IEEE. pp 250–257. ISBN: 978-1-5090-4303-3. https:// doi.org/10.1109/ISGT-Asia.2016.7796394 36. Bhosekar A, Ierapetritou M (2018) Advances in surrogate based modeling, feasibility analysis, and optimization: a review. Comput Chem Eng 108:250–267, ISSN: 00981354. https://doi. org/10.1016/j.compchemeng.2017.09.017 37. Queipo NV, Haftka RT, Shyy W, Goel T, Vaidyanathan R, Tucker PK (2005) Surrogate-based analysis and optimization. Prog Aerosp Sci 41(1):1–28. https://doi.org/10.1016/j.paerosci. 2005.02.001 38. Trivedi A, Sanyal K, Verma P, Srinivasan D (2017) A unified differential evolution algorithm for constrained optimization problems. 2017 IEEE Congress on Evolutionary Computation CEC 2017 - Proceedings, pp 1231–1238. https://doi.org/10.1109/CEC.2017.7969446 39. Trivedi A, Biswas N, Chakroborty S, Srinivasan D (2017) Extending unified differential evolution with a new ensemble of constraint handling techniques. In: 2017 IEEE Symposium Series on Computational Intelligence (SSCI). Nov. 2017, IEEE. pp 1–8, ISBN: 978-1-5386-2726-6. https://doi.org/10.1109/SSCI.2017.8285446 40. Mirjalili S, Gandomi AH, Mirjalili SZ, Saremi S, Faris H, Mirjalili SM (2017) Salp swarm algorithm: a bio-inspired optimizer for engineering design problems. Adv Eng Softw 114:163– 191, ISSN: 18735339. https://doi.org/10.1016/j.advengsoft.2017.07.002 41. Savier JS, Das D (2007) Impact of network reconfiguration on loss allocation of radial distribution systems. IEEE Trans Power Delivery 22(4):2473–2480, ISSN: 08858977. https://doi. org/10.1109/TPWRD.2007.905370 42. Zhang D, Fu Z, Zhang L (2007) An improved TS algorithm for loss-minimum reconfiguration in large-scale distribution systems. Electr Power Syst Res 77(5–6):685–694, ISSN: 03787796. https://doi.org/10.1016/j.epsr.2006.06.005 43. EMC (2016) Use of system charges https://www.mypower.com.sg/documents/ts-usc.pdf Visited on 04/01/2018 44. Braun M (2008) Provision of ancillary services by distributed generators, Ph.D Thesis, Kassel University, p 273, ISBN: 9783899586381

References

81

45. Zhang W, Quan H, Gandhi O, Rodríguez-Gallegos CD, Sharma A, Srinivasan D (2017) An ensemble machine learning based approach for constructing probabilistic PV generation forecasting. In: 2017 IEEE PES Asia-Pacific Power and Energy Engineering Conference (APPEEC). Nov. 2017, IEEE. pp 1–6. ISBN: 978-1-5386-1379-5. https://doi.org/10.1109/ APPEEC.2017.8308947 46. Wu W, Hu Z, Song Y (2016) A new method for OPF combining interior point method and filled function method. In: 2016 IEEE Power and Energy Society General Meeting (PESGM). Jul. 2016, IEEE. pp 1–5. ISBN: 978-1-5090-4168-8. https://doi.org/10.1109/PESGM.2016. 7741321 47. Kelner V, Capitanescu F, Léonard O, Wehenkel L (2008) A hybrid optimization technique coupling an evolutionary and a local search algorithm. J Comput Appl Math 215(2):448–456, ISSN: 03770427. https://doi.org/10.1016/j.cam.2006.03.048

Chapter 4

Inverter Degradation Consideration in Reactive Power Dispatch

4.1 Introduction Even though many researchers assume that PV inverters are able to provide reactive power compensation (RPC) at no cost [1–4], there are tradeoffs involved in injecting/absorbing reactive power through the inverter. The first tradeoff is the additional losses in the inverter because of the RPC [5, 6], which either reduces the active power output of the PV or has to be compensated by the grid [5, 7]. This reactive power cost is the cost introduced in Chap. 2 [8] and then incorporated into the analytical reactive power dispatch in Chap. 3 [9]. However, there is another trade-off in providing reactive power using inverters, which has not been taken into account by power system or optimisation researchers, namely the increase in wear and tear of the inverter components. The increased wear and tear has been analysed by power electronics researchers, and shown to reduce the lifetime of the inverter. The additional losses from RPC cause temperature rise in the inverter components, especially during the time when PV is not generating any active power [10, 11]. Consequently, the increase in thermal stress has adverse impacts on the lifetime of inverter components [10, 12–15]. Therefore, the detrimental effects of reactive power usage on the PV inverter lifetime can be analysed in depth at the component level. Figure 4.1 shows the schematic of how the two components of the reactive power cost arise. Most of the components in a PV system, especially the solar panels, are designed to last for 25 years or more. Among these components, inverter is the most prone to failure, which increases the maintenance and/or replacement cost. Should the lifetime of PV inverters be further reduced by the reactive power support, the cost incurred by the PV owner/operator would increase, reducing the economic viability of the PV system. Nevertheless, no work to date has translated the lifetime reduction (LR) due to the RPC to a cost borne by the PV owners or the power system operators. However, this is crucial for both the PV owners and power system operators to determine the viability of PV to provide reactive power support as ancillary services. The © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 O. Gandhi, Reactive Power Support Using Photovoltaic Systems, Springer Theses, https://doi.org/10.1007/978-3-030-61251-1_4

83

84

4 Inverter Degradation Consideration in Reactive Power Dispatch

Fig. 4.1 Schematic of the source of PV reactive power cost

main aim of PV owners is generally to receive as much compensation as possible— subject to satisfying grid requirements—either through the active power generation, or provision of ancillary services. Therefore, PV reactive power provision needs to be accompanied by suitable monetary incentives. With the rapid increase of PV penetration, timely investigation is imperative so that appropriate regulations can be implemented and costly retrofitting of the PV systems or change in the regulations can be avoided [16]. In this chapter, effects of reactive power injection from PV inverters on the inverter LR have been quantified into an increase in the levelised cost of electricity (LCOE) of the PV system, which is subsequently translated into reactive power cost. Next, this work expands the existing literature and the work in the previous chapters by quantifying and analysing the inverter LR (ILR) cost of PV RPC at different penetration levels. The net benefits for the system are also evaluated such that the feasible payment range for the RPC can be determined. Finally, in the light of the newly formulated PV reactive power cost, the competitiveness of local RPC using PV is reassessed using the same methodology as in Chap. 2. Although in this chapter only reactive power injection is being considered, the insights can be adapted for the case of reactive power absorption. In this chapter, only the inverter LR cost of PV reactive power provision has been formulated. Formulation of the cost for other DERs, such as EV and BESS, requires life cycle analysis of the respective technologies, and is outside the scope of this thesis. The formulation of the ILR component of the reactive power cost has been published in IEEE Transactions on Sustainable Energy [17], while the reevaluation of the competitiveness of PV for RPC was presented in SNEC PV Power Expo 2018 and then published in Energy Procedia [18].

4.1 Introduction

85

Hence, the contributions of this chapter are: [1] Translating the reduction in inverter lifetime due to RPC into reactive power cost for PV, [2] Establishing a relationship between PV reactive power output and the inverter lifetime, which can be incorporated into any cost function and optimisation problem, [3] Identifying the feasible range for monetary incentives for PV owners to provide RPC, and [4] Reassessing the competitiveness of PV as reactive power compensator, considering the inverter lifetime reduction. The rest of the chapter is laid out as follows: The PV system model and reactive power constraint are described in Sect. 4.2. Section 4.3The rest of the chapter is outlines the derivation of the ILR component of reactive power cost. Section 4.4 describes the case studies to analyse the significance of ILR cost while Sect. 4.5 elaborates the parameters. In Sect. 4.6, the results are presented and the impacts of inverter lifetime consideration on the cost and benefits of reactive power from PV are discussed. Based on the newly formulated cost and findings, the competitiveness of PV for local reactive power support is compared with that of switched capacitors (SC) in Sect. 4.7, and finally Sect. 4.8 summarises the chapter.

4.2 PV System Modeling The PV model used in the previous chapters (Eq. (2.11)) has to be replaced with a more detailed model involving the values of PV module current (ItPV ) and voltage (VtPV ) at any particular period t, as these values are required to calculate the lifetime of PV inverters, as explained in Sect. 4.3. Therefore, an experimentally validated PV module model has been adopted from [19], which is described by the following equations:   G t PV,SC  PV 1 + α TtPV − Tref ItPV,SC = Iref G ref PV,SC PV It ItPV = Iref PV,SC Iref     PV  Gt PV,OC PV,OC PV 1 + β Tt − Tref + γln = Vref Vt G ref

  PV,OC PV PV VtPV = Vref + R PV,s Iref + VtPV,OC − Vref − ItPV

(4.1) (4.2) (4.3) (4.4)

where ItPV,SC and VtPV,OC represent the short circuit current and the open circuit voltage of the PV module respectively. The subscript “ref” indicates a reference value which can be obtained from the data sheet of a particular PV module. α, β, and

86

4 Inverter Degradation Consideration in Reactive Power Dispatch

γ are the temperature coefficient of ItPV,SC , temperature coefficient of VtPV,OC , and the irradiance correction factor of VtPV,OC respectively. The series resistance, R PV,s , is calculated using the formula in [19]. PV systems comprise many modules arranged in series and parallel. The active power output of such system, PtPV , can be obtained using: PtPV = N PV,par ItPV × N PV,ser VtPV × P R  − PtPV,invloss

(4.5)

where N PV,par is the number of PV strings connected in parallel, while N PV,ser is the number of PV modules connected in series for each string. P R  is defined as the performance ratio of the PV system before taking into account the inverter losses (PtPV,invloss ). The reactive power output of the PV, Q PV t , is limited by the PV inverter rating, the forecast active power generation, and the maximum error prediction for active power generation, PtPV : PV     PV 2 − P PV + P PV 2 Q ≤ Smax (4.6) t t t PtPV has been used in Eq. (4.6), rather than PtPV70 as in the previous two chapters to avoid the need to forecast the PV active power production for a full year, the length of the simulation employed in this chapter.

4.3 Reactive Power Costs 4.3.1 Power Loss Component RPC using PV inverters has been shown to induce additional inverter losses [5] and the resulting reactive power cost has been explicitly formulated in the previous chapters: Pgrid

ctQPV,invloss Q PV t = ct

× PtPV,invloss

(4.7)

Pgrid

is the active power price purchased by the distribution system from the where ct main grid, and PtPV,invloss is the additional power loss in the inverter because of the RPC, calculated using Eq. (2.19). ctQPV,invloss is the inverter loss component of PV reactive power cost formulated in Chap. 2.

4.3.2 Inverter Lifetime Reduction Component On top of the additional losses, reactive power usage also causes higher thermal stress on the inverter components [10, 11]. In particular, DC-link capacitors are determined

4.3 Reactive Power Costs

87

to be the components most likely to age faster from thermal and power cycling [12, 13, 20, 21]. Therefore, in this work, the lifetime of the DC-link capacitor has been assumed to be the same as the inverter lifetime.

4.3.2.1

Capacitor Degradation

First, the lifetime of the capacitor for a particular operating condition (L CAP ) can be t calculated from the following formula [14]:  = L CAP L CAP t ref

VtCAP CAP Vref

−n 2

CAP −T CAP Tref t 10

(4.8)

CAP CAP where L CAP are reference lifetime, voltage rating, and temperaref , Vref , and Tref ture rating of the capacitor. n is the voltage stress exponent of the lifetime model. voltage VtCAP is the operating  of the DC-link capacitor, which can be taken to be:  VtCAP = max VtPV , V INV,DC where V INV,DC is the rated DC voltage of the inverter, that is required to produce the nominal inverter line-to-line RMS AC output voltage, V INV,AC . Typically, for sinusoidal pulse-width modulation (SPWM) inverters, the DC voltage of the inverter is calculated using [22]:

√ √ V INV,DC ≥ V INV,AC × 2 2/ 3

(4.9)

The temperature of the capacitor, TtCAP , can be obtained using the following [15]:

2 CAP,ripple TtCAP = Tta + R CAP,th R CAP,s It

(4.10) CAP,ripple

where R CAP,th and R CAP,s are the capacitor’s thermal and series resistances. It is the ripple current flowing through the capacitor, which can be calculated using [23]:  √ √    9 3 3 CAP,ripple CAP,ACrms  + cos2 φ − Mt It = It 2Mt (4.11) 4π π 16

2 ( PtPV )2 +( Q PV t ) where = is the RMS AC current flowing through the V INV,AC capacitor, cosφ is the power factor of the inverter, and Mt 1 is the modulation index given by the following [23]:

ItCAP,ACrms

is always the same in the simulations done, i.e. VtCAP = V INV,DC , and therefore can be taken as constant, and the t index can be removed from Mt .

1 V CAP t

88

4 Inverter Degradation Consideration in Reactive Power Dispatch

√ 2 2V INV,AC Mt = √ CAP 3Vt

(4.12)

is different at every t, due to different TtCAP The expected capacitor lifetime L CAP t (Eq. (4.8)). Therefore, to obtain the lifetime in years, the following calculation is carried out: Degradation =

T  H L CAP t t=1

L CAP = 1/Degradation

(4.13) (4.14)

where Degradation is the total degradation of the capacitor in the first year. In this work, the analysis are done over a year (2016) at half-hour resolution, such that H = 0.5 and T = 17568. L CAP is the operating lifetime of the capacitor, assuming that the weather condition of the subsequent years is the same as the first year. The lifetime reduction (L R) due to RPC is then: L R = L CAP,P − L CAP,Q

2 = ν θt − 2λt −μQ

(4.15)

where L CAP,P and L CAP,Q are the operating lifetime of the capacitor—and hence the inverter—when it is only used for P PV , and for both P PV and Q PV , respectively. The other symbols are defined as:  2   R CAP,th R CAP,s (ζ + κ) PtPV 1 CAP a − Tt − λt = T  2 10 0 V INV,AC

θt = 2

CAP, P T0CAP −Tt 10

 ν=

L CAP 0 √

κ=

VtCAP V0CAP

ζ R CAP,th R CAP,s 2  10 V INV,AC √ 2M 3 ζ= 4π

μ= −n

9 3 − M π 16

where TtCAP,P is the temperature of the capacitors when the PV inverter is only producing active power.

4.3 Reactive Power Costs

4.3.2.2

89

Economic Effect

To integrate L R into a reactive power cost value, the effect of the inverter lifetime reduction needs to be quantified in relation to LCOE of the PV system. LCOE of a PV system can be calculated using (Eq. (4.16)) [24].

LC O E =

E PC I + I DC +

 L PV

 L PV I R In  O M ∗ +I C ∗ LP n=1 (1+D R)n + (1+D R)n  + n=1 (1+D R)n

 T PV ×(1−S D R)n  L PV t=1 Pt n=1 (1+D R)n ∗

+

∗ RVn=20 (1+D R)20

(4.16)

* inflation-adjusted

where n  = L CAP , 2L CAP , 3L CAP ... ≤ 20. Since the unit of LCOE is [SGD/kWh], the numerator in Eq. (4.16) consists of all the possible costs for a PV system throughout its lifetime (assumed to be 20 years, i.e. L PV = 20), whereas the denominator is the total energy generated by the PV system in the 20 years. The equity project cost investment (EPCI) and the interest during construction (IDC) comprise the initial investment cost, while the operations and maintenance (OM) as well as the insurance cost (IC) make up the annual operating cost. The inverter replacement investment (IRI) represents the cost to replace inverters every L CAP years (Eq. (4.14)). When part of the upfront investment is debt financed (assumed to be 60% here), the loan payments (LP) need to be made every year, including annual interest and amortisation expense. At the end of the 20 years, a residual value (RV) might still be present. The denominator is the sum of the annual energy productions. Energy generation after the first year is reduced according to the system degradation rate (SDR). Annual values in the numerator and the denominator are adjusted by the discount rate (DR) for net present value calculations [24, 25]. Because of the inverter lifetime reduction, IRI from Eq. (4.16) will increase when the inverter is used for RPC, causing the LCOE of PV to increase. The LCOE increase (L I ) for different values of L R is plotted in Fig. 4.2. It shows that L I can be approximated as: L I [$/kWh] =

η LI,A1 L R + η LI,A0 L R 2 + η LI,B1 L R + η LI,B0

(4.17)

where η LI,A0 , η LI,A1 , η LI,B0 , and η LI,B1 are constants. To translate the LCOE increase (LI) from [SGD/kWh] to cost increase (CI) in PV , the average active power generated by the [SGD], Eq. (4.17) is multiplied by Pave PV at each period t over the lifetime of the system, considering the reduction in PR in later years. Finally, the reactive power cost due to inverter lifetime reduction, ctQPV,LR , is obtained through the following: ctQPV,LR Q PV t = C I [SGD]

(4.18)

90

4 Inverter Degradation Consideration in Reactive Power Dispatch

Fig. 4.2 Relation between the increase in LCOE and inverter lifetime reduction (ILR). The data points are fitted with Eq. (4.17). The original inverter lifetime with only active power generation (Case 1) has been set to 14 years, as was found for the system parameters used in this chapter. r2 value is 0.99908

4.4 Methodology To analyse the importance of the cost of inverter lifetime reduction (ILR), the operation of a distribution system with reactive-capable PV is simulated. The operational cost of running the grid-connected distribution system takes the same form as in Chap. 2: Cost =

T   t=1

+

Pgrid

grid

grid

c P + cQgrid Q t t  t    

cost of P from grid X  x=1



PV cxPPV Px,t +





cost of P from PV

(4.19)

cost of Q from grid X  x=1



QPV PV cx,t Q x,t H 



cost of Q from PV

QPV QPV,invloss QPV,LR The reactive power cost is represented by cx,t = cx,t + cx,t . The rest of the terms are the same as in previous chapters. To quantify the costs and benefits of reactive power, the operational costs of the following two cases are compared: Case 1: The PV systems are only generating active power (Q PV x,t is always zero). Case 2: The PV systems are providing RPC, which is optimised by the analytical approach developed in Chap. 3. The cost-benefit analysis was conducted with 16 penetration levels (represented by the additions of a new PV into the system, as explained in more detail in Sect. 4.5.2).

4.4 Methodology

91

The method for the reactive power dispatch is the analytical approach proposed in Chap. 3, although it has not incorporated the ILR. The analytical approach is employed because it performs better compared with conventional optimisation methods—namely, particle swarm optimisation (PSO), genetic algorithm (GA), and interior point method (IPM). More importantly, the analytical approach is much faster and therefore more suitable to optimise the one-year dispatch.

4.5 Implementation Setup 4.5.1 PV-Inverter Configuration Each PV system capacity is assumed to be 300 kWp, consisting of 40 strings of 25 Trina 300 W modules [26] per string. The system is connected to a 3-phase 300 kW inverter (Fig. 4.3) with peak efficiency of 98.2% and efficiency curve from [5]. Vishay capacitors 500 PGP-ST [27] are assumed to be the inverter DC-link capacitors. Aluminium electrolytic capacitors have been used as they are the most widely utilised in PV inverters due to their favourable size and cost [12, 13, 23]. Although film capacitors have higher reliability, their adoption is still limited because of higher cost and lower capacitance values for the same volume as electrolytic capacitors [12, 28]. The values of the PV module, inverter and DC-link capacitor parameters are provided in Table 4.1. The value of the thermal resistance of the capacitor, R CAP,th , has been calculated based on the formula in [15].

Fig. 4.3 The configuration of the PV system with a central PV inverter

92

4 Inverter Degradation Consideration in Reactive Power Dispatch

Table 4.1 Parameters of the PV System Parameters Values

Parameters

Values

PV module PV [V] Vref PV,OC Vref [V] PV Iref [A] PV,SC Iref [A] Inverter V INV,DC [V] V INV,AC [V] PV [kVA] Smax

Capacitor C [μF] R CAP,s [] R CAP,th [K/W] L CAP ref [h] CAP [V] Vref n No. in series No. in parallel

15000 0.009 1.072 3000 400 5 2 7

815 997.5 367.6 385.6 800 480 300

Fig. 4.4 69-bus radial distribution system with PV systems. The numbers in red are the order of placement of the PV to analyse the increasing PV penetration

4.5.2 System Data Same as in the previous chapters, the 69-bus radial distribution system [29] has been employed as the test system. To simulate increasing PV penetration in the system, one PV system is added for each penetration level according to the order shown in Fig. 4.4. The locations of PV are the same with the ones from Chap. 2. For the current study, the maximum installed capacity considered is composed of 16 PV systems. This represents 18.0% of electricity consumption generated by PV, and also equivalent to 100.6% PV installed capacity compared to peak load (4771 kW). One-year Singapore load data from 2016 [30], scaled to the magnitude of the test system’s load, has been utilised. Pgrid is the Singapore half-hourly wholesale electricity price plus grid charge for ct Pgrid Pgrid is 8.87 SGD cents per kWh. The average ct the year 2016 [31]. The average ct when PV is generating power (from 8 a.m. to 6 p.m.) is 9.55 SGD cents per kWh. LCOE of PV (cxPPV ) is calculated from (4.16) for the case of Singapore [24]. Assuming inverter lifetime of 14 years—as was found for the system parameters

4.5 Implementation Setup

93

used in this chapter—cxPPV is found to be 10.28 SGD cents per kWh, still higher than the average wholesale electricity price for large electricity consumers. The solar irradiance and temperature data are also for the year 2016, taken in Singapore at latitude 1.2491◦ , longitude 103.8414◦ . They are provided by SERIS. PV PV has been taken to be 10% of the Px,t . Px,t

4.6 Results and Discussions In this section, first, results comparing the operational costs for the year 2016 (calculated using Eq. (4.19)) of Case 1 and 2 are presented. Subsequently, the effect of reactive power injection on the inverter lifetime is investigated, and the costs and benefits of PV RPC are assessed. Through the results presented in the following subsections, three salient insights were obtained: [1] Reactive power injection from PV to fulfil local reactive power demand is still economically beneficial for the system even after taking into account the inverter lifetime reduction (ILR). [2] The reactive power cost due to PV ILR can be expressed as a cubic or a piecewise quadratic function of the PV reactive power output. [3] ILR cost is a significant part of the reactive power cost. However, it gradually decreases with increasing PV penetration, when reactive power support is distributed over more PV.

4.6.1 Total Operating Costs When PV systems are added into the system, the operational cost of the system Pgrid decreases even though cxPPV is higher than the average ct , as observed in Fig. 4.5. When the PV systems are only producing active power (Case 1), the cost starts to drop and reach a minimum when 6 PV systems are placed in the system (37.7% penetration by installed capacity). The decrease in cost—despite higher active power cost from PV— comes from the reduced power losses as more PV are installed in the system. However, the diminishing loss reduction2 cannot compensate for the more costly active power generation from PV and the operational cost increases afterwards. Figure 4.6 shows that 100% of the reactive load has been fulfilled from 90% penetration level, yet, as seen in Fig. 4.5, the cost of Case 2 continuously drops even beyond 90% penetration level. The addition of reactive-capable PV is always economically beneficial for the system regardless of the location. Notably, there is a larger decrease in operational costs when PV is added at strategic locations, 2 This

is also partly due to the random allocation of PV systems, instead of installing them at the optimal locations in the system. In fact, the power losses increase after the addition of PV at node 38, 34, and 44.

94

4 Inverter Degradation Consideration in Reactive Power Dispatch

Fig. 4.5 Operating cost of the system as calculated using Eq. (4.19)

Fig. 4.6 Percentage of reactive load and losses fulfilled by PV. In this work, the distribution system is not allowed to inject reactive power to the grid

namely the third (at node 61), fifth (at node 64), and fifteenth addition (at node 60). Unsurprisingly, all three nodes are located at the same lateral, the most heavily loaded in the system [29].

4.6.2 Effects on Inverter Lifetime When PV is only generating P PV , the inverter is operating at approximately 50% utilisation rate (the sun shines from 7 a.m. to 7 p.m. throughout the year in our case study), and the capacitors are mostly at ambient temperature, as depicted by the blue histogram in Fig. 4.7. The PV penetration does not affect the lifetime of the PV from each PV remains the same regardless of the PV penetration inverters as Px,t (PV is always operating at maximum power point). In this case, the lifetime of the PV inverter, L CAP,P , is 14 years.

4.6 Results and Discussions

95

Fig. 4.7 Temperature profile of the DC-link capacitors for PV system at node 66 throughout the year, when the PV is only generating P (blue), generating Q at 12.6% penetration level (red), and at 94.3% penetration level (yellow)

Fig. 4.8 Average yearly reactive power generated by a PV system at different penetration levels

When PV is also generating Q PV , the inverter components operate at full utilisation, causing T CAP to increase significantly. Figure 4.8 shows that, for PV penetration up to 30%, the PV systems are always generating reactive power close to their maximum capacity (calculated using Eq. (4.6)). This means that the inverters are always operating close to their rated apparent power. However, at 100.6% penetration, the PV systems are only generating approximately 60% of the available reactive power generation. That is the reason why the temperatures of the capacitors are on average higher when the PV penetration is lower (Fig. 4.7), as each PV is generating more reactive power at lower penetration levels (Fig. 4.8). For example, the lifetime of the PV inverter at node 66 is 3.9 and 7.9 years, at 12.6% and 94.3% PV penetration, respectively (Fig. 4.9).

96

4 Inverter Degradation Consideration in Reactive Power Dispatch

Fig. 4.9 Lifetime of the PV inverters at different nodes

Fig. 4.10 Relationship between hourly PV reactive power output and hourly inverter LR cost with (a) cubic fitting, (b) quadratic fitting, and (c) piecewise quadratic (PQ) fitting. There are two domains for the PQ function shown in (c), namely Q PV ≤ 150 kvar and Q PV ≥ 150 kvar. At 150 kvar, the two quadratic functions yield approximately the same value. The coefficients of the curves are listed in Table 4.2

At the same penetration level, most of the PV inverters have very similar lifetime, except for the PV inverters at node 19 and node 24, which recorded longer lifetimes. These two nodes are located at a lightly loaded lateral and are relatively far away from the other loads (Fig. 4.4). Hence, they provide less reactive power. By comparing Figs. 4.8 to 4.9, a correlation between the reactive power generated and the inverter lifetime can be noticed. The existence of such correlation means that Eqs. (4.15) and (4.17) can be simplified into a single variable polynomial equation to be incorporated into an optimisation objective function. In fact, by plotting the Q generated per hour with the LR cost per hour, the following cubic function can be obtained (Fig. 4.10):

4.6 Results and Discussions Table 4.2 Coefficients of the Fitting Cubic η LR,0 η LR,1 η LR,2 η LR,3 r2

0 2.133 × 10−4 −1.986 × 10−6 2.726 × 10−8 0.99984

97

Quadratic

Piecewise Quadratic ≤150 kvar ≥150 kvar

0 −6.731 × 10−4 8.551 × 10−6 0 0.99159

0 2.768 × 10−5 4.041 × 10−6 0 0.99976

0.3068 3.967 × 10−3 1.667 × 10−5 0 0.99992

 PV 2  3 LR,0 LR,2 Qt ctQPV,LR Q PV + η LR,1 Q PV + η LR,3 Q PV t =η t +η t

(4.20)

where η LR,0 , η LR,1 , η LR,2 , and η LR,3 are coefficients obtained from fitting the ILR cost per hour with Q PV t . Their values for the different function fittings are listed in Table 4.2. Figure 4.10 and Table 4.2 show that although quadratic ctQPV,LR does not fit the simulated data as accurately as the cubic fitting, it was found that a piecewise quadratic (PQ) function fits the data well. To directly use Eqs. (4.15) and (4.17) to yield ctQPV,LR not only requires involved calculations, but also necessitates the simulation to be run for at least over a year to get an accurate estimation of the inverter lifetime.3 Therefore, through the formulation of Eq. (4.20), ctQPV,LR can be incorporated into an optimisation objective function with any time horizon. While the results shown are only for a particular type of inverter and capacitor (to allow for direct comparison), the trend will be similar for other configurations, even if the exact lifetime and cost vary, as demonstrated in Sect. 4.6.4. Such relationship will be crucial for PV owners to decide whether RPC makes economic sense for their system configuration.

4.6.3 Economic Balance of Local Reactive Power Provision To quantify the benefits of Q PV to the system, the following terms are used: • Benefits from Q: the monetary benefit of having RPC from PV without taking into QPV account the reactive power cost, cx,t . It is mathematically defined as Cost Case 1 −   QPV PV T X Cost Case 2 + t=1 x=1 cx,t Q x,t H . • Net benefits from Q: the monetary benefit of having RPC from PV, after taking into QPV,invloss QPV,LR and cx,t . It is mathematically defined as Cost Case 1 − account both cx,t Case 2 . Cost

3 The

simulation over at least a year is required to take into account the variation of inverter usage due to different weather conditions throughout the year.

98

4 Inverter Degradation Consideration in Reactive Power Dispatch

Fig. 4.11 Average benefits and costs of reactive power provision using PV throughout the year at different penetration levels

Figure 4.11 presents the costs and benefits of reactive power provision from PV at different penetration levels. We can observe that overall, by increasing the PV penetration in a system, the benefits of reactive power injection (in [SGD/kvarh]) decreases. The spikes in the monetary benefits occur when PV is added at strategic locations (at 18.9%, 31.4%, and 94.3% PV penetration by installed capacity), as previously explained in Sect. 4.6.1. However, as the PV penetration increases, the reactive power cost decreases, thanks to lower ILR cost. As a result, the net benefits from Q increase from 56.6% penetration onwards. The gap between the benefits from Q (the blue diamonds in Fig. 4.11) and the reactive power cost (the sum of the two areas in Fig. 4.11) is the possible range of monetary incentives that can be given to PV owners to inject reactive power. According to the information in Figs. 4.9 and 4.11, it is evident that it is more efficient for the PV to inject smaller amount of reactive power, to reduce the losses in the system, but not to reduce the lifetime of the inverter significantly. It is also worth noting that the reduction in the inverter lifetime due to RPC is a major component of the reactive power cost, especially at lower penetration level (cQPV,LR > cQPV,invloss until 31.4% PV penetration). In Chap. 2, it has been found that reactive power provision from PV is competitive with switched capacitors. Nevertheless, this might no longer be true after incorporating cQPV,LR . Therefore the competitiveness of PV as a reactive power compensator is reevaluated in Sect. 4.7.

4.6 Results and Discussions

99

4.6.4 Limitations The ILR component in this chapter has been calculated for a specific model of capacitors with a particular inverter size, topology, efficiency curve, and modulation technique. When even one of these parameters change, the coefficients listed on Table 4.2 would also change. Nevertheless, although the exact lifetime of the inverters and hence the ILR cost would vary, the relation in Eq. (4.20) is expected to still be valid. To illustrate the change in the inverter lifetime when one of the parameters change, the results from Figs. 4.7 and 4.9 are re-simulated using a different modulation technique, namely space vector pulse width modulation (SVPWM), instead of SPWM. The change in modulation technique changes the V INV,DC , which is calculated using the Eq. (4.21), rather than using (4.9). V INV,DC ≥ V INV,AC ×



2

(4.21)

The results of such change to the temperature profile of the inverters, and consequently their lifetime, can be appreciated in Figs. 4.12 and 4.13, respectively. It

Fig. 4.12 Comparison of temperature profile of the DC-link capacitors for PV system at node 66 for (a) SPWM, and (b) SVPWM modulation technique

Fig. 4.13 Comparison of PV inverters’ lifetime at different nodes for (a) SPWM, and (b) SVPWM modulation technique

100

4 Inverter Degradation Consideration in Reactive Power Dispatch

can be seen that although the exact temperature and lifetime of the inverters vary, the observed trend remains the same. Therefore the insights and methods developed in this chapter can be applied to quantify the reactive power cost of other types of inverters in different systems.

4.7 Competitiveness of Local Reactive Power Provision Using PV After Considering Inverter Lifetime Reduction Similar to Sect. 2.6, the competitiveness of PV for local reactive power provision is analysed through comparison with switched capacitors (SC) on economic and technical objectives, namely the net monetary benefits (NMB) and weighted voltage deviation index (WVDI) improvement. Calculation of the two objectives are laid out in Eqs. (2.21) and (2.22), and are repeated here for convenience: NMB = Cost NoQ − Cost SC/PV

(4.22)

WVDI Improv. = WVDINoQ − WVDISC/PV

(4.23)

where NoQ refers to the case when there is no local reactive power compensation (RPC) in the system. Cost and WVDI are calculated through Eqs. (4.19) and (2.3), respectively. Cost has included the ILR component of PV reactive power cost, approximated through the cubic function of Eq. (4.20). As in Sect. 2.6, the two objectives are optimised simultaneously through multiobjective optimisation algorithm called Non-dominated Sorting Genetic AlgorithmII (NSGA-II) [32]. As NSGA-II is not as fast as the analytical approach developed in Chap. 3, the optimisation was only done over one day, rather than over one year as was done in the previous section.

4.7.1 Simulation Setup Following Sect. 2.6, these two base cases are considered to assess the competitiveness of PV as a reactive power compensator. PV Base Case: Six PV systems are present in the distribution system (at node 7, 19, 24, 61, 64, and 66, as indicated in and 4.4). The PV systems are able to generate P PV and Q PV . SC Base Case: The PV systems are only generating P PV , while SCs at the same nodes as the PV systems are providing the local RPC.

4.7 Competitiveness of Local Reactive Power Provision Using ...

101

All the cases employ irradiance and electricity price data from 27 May 2016, the same as the base cases in Chap. 2. They are also compared with the NoQ case where there are PV generating P PV in the system but there is no local RPV by PV or SC. The cost and WVDI of the NoQ case are used as comparison in Eqs. (4.22) and (4.23) for the base cases and for sensitivity analyses. From Sect. 2.6, it was found that two of the most important factors affecting the competitiveness of PV and SC for RPC are the electricity price and the inverter efficiency (only on PV case). Therefore, the impact of the two variables are analysed in detail in this section. For high penetration cases, the number of PV in the system is increased to 16 (as in Fig. 4.4) to allow PV to provide as much RPC as SC. For each change in variable, the comparison is made with the NoQ case with the respective change in variable. On top of the sensitivity analyses, this section also highlights the importance of considering the ILR cost. This was done by optimising the PV base case twice, with different reactive power cost considerations: [1] PV Base Case: Cost in Eq. (4.19) considers both ctQPV,invloss and ctQPV,LR [2] PV Base w/o ILR Case: Cost in Eq. (4.19) only considers ctQPV,invloss The ILR cost is also incorporated to the results found in the PV Base w/o ILR to illustrate its importance. cQSC , the load profile, and the weather parameters are the same as Chap. 2 to enable direct comparison for the cases with and without ILR.

4.7.2 Comparison of PV and SC for Reactive Power Compensation 4.7.2.1

Impact of Considering Inverter Lifetime Reduction

First, the base case for PV and SC are presented in Fig. 4.14. Three optimisation results are presented, namely SC Base Case, PV Base Case, PV Base w/o ILR Case. The ILR cost were also added to the PV Base w/o ILR results, which are then labelled as “PV Base w/o ILR + ILR” in Fig. 4.14. From Fig. 4.14, it can be seen that for the base case, SC is more competitive for RPC compared to PV, regardless of the ILR cost consideration. This is mainly because SC is able to fulfil more reactive load, hence reduces more losses in the system, and push the voltage profile closer to Vref . When the effect of RPC on the inverter lifetime is not considered, the NMB of RPC is overestimated by approximately 24%. By including the ILR cost in Eq. (4.19), the multi-objective optimisation yields larger pareto front, as there are more tradeoffs between the technical and economic objective when cQPV is higher. To analyse the significance of considering the ILR cost, cost composition of the results that maximise the NMB from PV Base and PV Base w/o ILR + ILR is presented in Fig. 4.15. It is shown that the result from PV Base w/o ILR reduces the power losses and Q grid as PV provides higher RPC compared with PV Base.

102

4 Inverter Degradation Consideration in Reactive Power Dispatch

Fig. 4.14 Technical and economic comparison of local reactive power provision. The three panels below the main panel illustrate the shape of the pareto front of each case

Nevertheless, this induces higher cost of reactive power, for both ctQPV,invloss and ctQPV,LR . Because of that, the overall cost of PV Base w/o ILR is higher than that of the PV Base. When ILR is not considered, ctQPV,LR is approximately 40% of ctQPV,invloss . Thus, neglecting ILR cost in calculating the remuneration to the PV owners for local RPC will cause the PV owners to be underpaid.

4.7.2.2

Effect of Change in Electricity Price and Inverter Efficiency

Each percentage change in electricity price causes approximately 0.5% and 0.25% change in NMB for the SC and PV cases respectively (Fig. 4.16). This is because the loss reduction from RPC becomes more valuable at higher electricity prices. Nevertheless, cQPV,invloss also increases with increasing cPgrid . Hence, there is more tradeoff in generating reactive power using PV, resulting in larger pareto front. Inverter efficiency increases the competitiveness of RPC from PV significantly— a 14% change in NMB for every 1% increase in inverter efficiency—as shown in Fig. 4.17. Nevertheless, even when the peak inverter efficiency is 99.1%, the RPC from SC is still more competitive. This finding differs from what was previously

4.7 Competitiveness of Local Reactive Power Provision Using ...

103

Fig. 4.15 Cost composition of the most economical solution in PV Base (left) and PV Base w/o ILR + ILR (right). The cost of P grid and Q grid are not shown because they are the same for the two cases. Both P loss and Q loss are fulfilled by the grid. They are separated to highlight the power losses in the system

Fig. 4.16 Impact of changes in electricity price on a SC and b PV cases

104

4 Inverter Degradation Consideration in Reactive Power Dispatch

Fig. 4.17 Impact of changes in inverter efficiency on PV cases a considering and b without considering ILR cost. The InvEff 97%, 98%, 99%, Base, 101%, 102%, and 103% refer to peak inverter efficiency of 93.4%, 94.3%, 95.3%, 96.3%, 97.2%, 98.2% and 99.1%, respectively

found in Chap. 2 (Fig. 2.19) [33] when the ILR cost is not considered. Figure 2.19 is re-shown as Fig. 4.17b with the axes adjusted for direct comparison with Fig. 4.17a.

4.7.2.3

Conditions Required for PV to Be Competitive

In the previous scenarios, SC cases have higher NMB and WVDI improvement because the SCs are able to fulfil more reactive load compared with PV cases. Therefore, a case with higher penetration of PV is also analysed. The effects of higher PV penetration on the competitiveness of PV for RPC are twofold. Firstly, higher penetration of PV increases its reactive power capability. Consequently, the PV systems have more operating points to reduce the line losses and maintain the voltage nearer to the Vref . Secondly, higher PV penetration also reduces the ILR cost, as the reactive power fulfilment is distributed over more PV and each PV generates lower Q PV , as was found in Sect. 4.6.3. At higher PV penetration and peak inverter efficiency of 99.1%, PV is already competitive in terms of the economic objective as seen in Fig. 4.18. The lower

4.7 Competitiveness of Local Reactive Power Provision Using ...

105

Fig. 4.18 Results of PV and SC cases considering high PV penetration and inverter efficiency 103% of the base value (peak inverter efficiency of 99.1%)

Fig. 4.19 Pareto fronts comparison between the results of PV and SC cases with the results from combinations of PV and SC, a for the base case, and b for cases with higher PV penetration and higher inverter efficiency. The notation ‘X’PV-‘Y’SC indicates that the RPC comes from X number of PV and Y number of SC

technical benefits are due to lower WVDI improvement when PV is generating active power, as illustrated in Fig. 2.21 in Sect. 2.6.4.6. It is also interesting to use a mix of PV and SC for RPC, i.e. SC can be used to fulfil the base reactive load while PV is used to fine tune the RPC and respond to rapid variation in voltage and/or load. Figure 4.19 shows the results of the RPC from combination of sources, for both the base case as well as for the high penetration case. When RPC comes from a combination of PV and SC, the pareto fronts contain more operating points that that of SC.

106

4 Inverter Degradation Consideration in Reactive Power Dispatch

From Fig. 4.19b, it is observed that replacing some of the SC with PV as the reactive power source enhances the NMB of the system without sacrificing the WVDI improvement. The WVDI improvement of the PV-SC combinations is higher than the PV case, whereas their NMB can be higher than that both PV and SC cases. These results show that a combination of both PV and SC for RPC can be better than cases with only PV or SC.

4.8 Summary In this chapter, reactive power cost from PV considering inverter lifetime reduction (ILR) has been formulated and quantified. One-year simulations with real load and weather data for the case of Singapore at different PV penetration levels have been conducted to analyse the benefits of local reactive power compensation (RPC) to the system, and its effect on the lifetime of the PV inverters. The ILR cost has been shown to be a major component of the reactive power cost and should be incorporated into the optimisation of reactive power dispatch. Accordingly, a simple polynomial relation between the ILR cost and the PV reactive power output, which can be incorporated into any cost function, has been formulated. Moreover, having a more distributed RPC from multiple PV in the system has been shown to be beneficial, as the reactive power cost will be reduced due to a lower reduction in the inverter lifetime. Therefore, the system operator can have a larger range for monetary compensation for the local RPC. The chapter has also highlighted the importance of considering the ILR due to RPC and reassessed the competitiveness of PV as reactive power compensator against switched capacitors (SC). While PV is still not as competitive as SC in providing RPC at low PV penetration, it becomes increasingly so at higher penetration. A combination of PV and SC can also be used to provide local RPC, resulting in higher technical and economic benefits for the system compared with those using only PV or only SC. The results from this work can be used as a foundation for future research on market mechanism or formulating strategies for RPC from PV. Additionally, the findings from this chapter will also be useful for PV owners and operators to account for the costs of complying to reactive power requirement set by the ISO.

References 1. Sousa T, Morais H, Vale Z, Castro R (2015) A multi-objective optimization of the active and reactive resource scheduling at a distribution level in a smart grid context. Energy 85:236–250. ISSN: 03605442. https://doi.org/10.1016/j.energy.2015.03.077 2. Zhang L, Tang W, Liang J, Cong P, Cai Y (2016) Coordinated day-ahead reactive power dispatch in distribution network based on real power forecast errors. IEEE Trans Power Syst 31(3):2472–2480. ISSN: 0885-8950. https://doi.org/10.1109/TPWRS.2015.2466435

References

107

3. Ding T, Li C, Yang Y, Jiang J, Bie Z, Blaabjerg F (2017) A two-stage robust optimization for centralized-optimal dispatch of photovoltaic inverters in active distribution networks. IEEE Trans Sustain Energy 8(2):744–754. ISSN: 19493029. https://doi.org/10.1109/TSTE.2016. 2605926 4. Yang HT, Liao JT (2015) MF-apso-based multiobjective optimization for pv system reactive power regulation. IEEE Trans Sustain Energy 6(4):1346–1355. ISSN: 19493029. https://doi. org/10.1109/TSTE.2015.2433957 5. Braun M (2008) Provision of ancillary services by distributed generators. Ph.D thesis, Kassel University, p 273. ISBN: 9783899586381 6. Su X, Masoum MA, Wolfs PJ (2014) Optimal PV inverter reactive power control and real power curtailment to improve performance of unbalanced fourwire LV distribution networks IEEE Trans Sustain Energy 5(3):967–977. ISSN: 19493029. https://doi.org/10.1109/TSTE. 2014.2313862 7. Kekatos V, Wang G, Conejo AJ, Giannakis GB (2014) Stochastic reactive power management in microgrids with renewables. IEEE Trans Power Syst PP (99):1–10. ISSN: 0885-8950. https:// doi.org/10.1109/TPWRS.2014.2369452, arXiv:1409.6758 8. Kekatos V, Wang G, Conejo AJ, Giannakis GB (2014) Stochastic reactive power management in microgrids with renewables. IEEE Trans Power Syst PP (99):1–10. ISSN: 0885-8950. https:// doi.org/10.1109/TPWRS.2014.2369452, arXiv:1409.6758 9. Gandhi O, Zhang W, Rodríguez-Gallegos CD, Bieri M, Reindl T, Srinivasan D (2018) Analytical approach to reactive power dispatch and energy arbitrage in distribution systems with ders. IEEE Trans Power Syst 33(6):6522–6533. ISSN: 0885-8950. https://doi.org/10.1109/TPWRS. 2018.2829527 10. Anurag A, Yang Y, Blaabjerg F (2015) Thermal performance and reliability analysis of singlephase pv inverters with reactive power injection outside feeding operating hours. IEEE J Emerg Sel Top Power Electron 3(4):870–880. ISSN: 21686785. https://doi.org/10.1109/JESTPE. 2015.2428432 11. Sreechithra SM, Jirutitijaroen P, Rathore AK (2013) Impacts of reactive power injections on thermal performances of PV inverters. In: IECON proceedings (industrial electronics conference), pp 7175–7180. https://doi.org/10.1109/IECON.2013.6700325 12. Sreechithra SM, Jirutitijaroen P, Rathore AK (2013) Impacts of reactive power injections on thermal performances of PV inverters. In: IECON proceedings (industrial electronics conference), pp 7175–7180. https://doi.org/10.1109/IECON.2013.6700325 13. Wang H, Liserre M, Blaabjerg F, De Place Rimmen P, Jacobsen JB, Kvisgaard T, Landkildehus J (2014) Transitioning to physics-of-failure as a reliability driver in power electronics. IEEE J Emer Sel Top Power Electron 2(1):97–114. ISSN: 21686785. https://doi.org/10.1109/JESTPE. 2013.2290282 14. Wang H, Blaabjerg F (2014) Reliability of capacitors for DC-link applications in power electronic converters - an overview. IEEE Trans Ind Appl 50(5):3569–3578. ISSN: 0093-9994. https://doi.org/10.1109/TIA.2014.2308357 15. Albertsen A (2010) Electrolytic capacitor lifetime estimation. http://www.powerguru.org/ electrolytic-capacitor-lifetimeestimation/ 16. Stetz T, Rekinger M, Theologitis I (2014) Transition from uni-directional to bi-directional distribution grids. Technical report, International Energy Agency, Kassel, p 154 17. Gandhi O, Rodríguez-Gallegos CD, Gorla NBY, Bieri M, Reindl T, Srinivasan D (2019) Reactive power cost from PV inverters considering inverter lifetime assessment. IEEE Trans Sustain Energy 10(2):738–747. ISSN: 1949-3029. https://doi.org/10.1109/TSTE.2018.2846544 18. Gandhi O, Rodríguez-Gallegos CD, Reindl T, Srinivasan D (2018) Competitiveness of PV inverter as a reactive power compensator considering inverter lifetime reduction. Energy Proc 150:74–82. ISSN: 18766102. https://doi.org/10.1016/j.egypro.2018.09.005 19. Ding K, Bian X, Liu H, Peng T (2012) A matlab-simulink-based pv module model and its application under conditions of nonuniform irradiance. IEEE Trans Energy Convers 27(4):864– 872. ISSN: 08858969. https://doi.org/10.1109/TEC.2012.2216529

108

4 Inverter Degradation Consideration in Reactive Power Dispatch

20. Flicker JD, Kaplar R, Marinella M, Granata J PV inverter performance and reliability: what is the role of the bus capacitor? In: 2012 38th IEEE photovoltaic specialists conference, June 2012. IEEE, Austin, TX, USA, pp 1–3. https://doi.org/10.1109/PVSC-Vol2.2013.6656709 21. Ton D, Bower W (2005) Summary report on the doe high-tech inverter workshop. Technical report, US Department of Energy 22. Holmes DG, Lipo TA (2003) Pulse width modulation for power converters. IEEE. ISBN: 9780470546284. https://doi.org/10.1109/9780470546284, arXiv:1011.1669v3 23. Kolar J, Round S (2006) Analytical calculation of the RMS current stress on the DC-link capacitor of voltage-PWM converter systems. In: IEE proceedings - electric power applications, vol 153, no 4, p 535. ISSN: 13502352. https://doi.org/10.1049/ip-epa:20050458, arXiv:1011.1669v3 24. Bieri M, Winter K, Tay S, Chua A, Reindl T (2017) An irradiance-neutral view on the competitiveness of life-cycle cost of PV rooftop systems across cities. Energy Proc 25. Rodríguez-Gallegos CD, Yang D, Gandhi O, Bieri M, Reindl T, Panda SK (2018) A multiobjective and robust optimization approach for sizing and placement of PV & batteries in off-grid systems fully operated by diesel generators: an Indonesian case study. Energy 160:410– 429. ISSN: 03605442. https://doi.org/10.1016/j.energy.2018.06.185 26. Trina Solar, Allmax m plus framed 60-cell module. http://static.trinasolar.com/sites/default/ files/Datasheet 27. Vishay (2017) Aluminum capacitors, power general purpose screw terminals 28. Messo T, Jokipii J, Puukko J, Suntio T (2014) Determining the value of DC-link capacitance to ensure stable operation of a three-phase photovoltaic inverter. IEEE Trans Power Electron 29(2):665–673. ISSN: 08858993. https://doi.org/10.1109/TPEL.2013.2255068 29. Savier JS, Das D (2007) Impact of network reconfiguration on loss allocation of radial distribution systems. IEEE Trans Power Deliv 22(4):2473–2480. ISSN: 08858977. https://doi.org/ 10.1109/TPWRD.2007.905370 30. Energy Market Authority (2018) Singapore half-hourly system demand data. https://www.ema. gov.sg/statistic.aspx?sta 31. EMC (2018) Energy market price information. https://www.emcsg.com/marketdata/ priceinformation. Accessed 04 Jan 2018 32. Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197. ISSN: 1089778X. https://doi. org/10.1109/4235.996017 33. Gandhi O, Srinivasan D, Rodríguez-Gallegos CD, Reindl T Competitiveness of reactive power compensation using pv inverter in distribution system. In: 2017 IEEE PES innovative smart grid technologies conference Europe (ISGT-Europe), September 2017. IEEE Torino, Italy, pp 1–6. ISBN: 978-1-5386-1953-7. https://doi.org/10.1109/ISGTEurope.2017.8260238

Chapter 5

Reactive Power Dispatch for Large Number of PV Installations

5.1 Introduction The reactive power controls proposed in the literature can be categorised into three types: centralised, distributed, and local control. The classification of decentralised control in [1], defined as “intermediate state between centralised and distributed control” shall be classified as distributed control for the remainder of this work as both distributed and decentralised controls in [1] require some forms of communication. It is also noted that there are other definitions of distributed and decentralised which assume no communication [2], but those controls are defined as local in this work. Thus far, all the works in the previous chapters [3–8] have been relying on centralised control. Indeed, many works have explored centralised control to mitigate the under/overvoltage problems [9–14]. The centralised control is generally carried out by the ISO, assuming perfect knowledge of the distribution system information (load, power flow, DERs’ generation, line impedance, etc.), whose values and/or measurements are sent to the central controller. The reference power outputs are subsequently communicated to each DER in the system. Among the centralised algorithms, mutation fuzzy adaptive particle swarm optimisation was presented in [15] to minimise system losses and voltage deviations. Yang and Yu [16] formulated two-stage distributionally robust optimisation that reduces operational costs and mitigates overvoltage problems. The analytical approaches developed in Chap. 3 (AS and ASEA) are also centralised. Recently there have been more works which optimised power dispatch using distributed approaches, where a DER needs to only communicate with the neighbouring nodes. Power losses have been minimised and voltage regulated using a two-level distributed controller [17], a multiagent system (MAS)-based distributed subgradient method [18], dual subgradient method [19], a distributed feedback algorithm [20], and an extremum seeking approach [21]. In [22], the authors employed MAS to integrate electric vehicle charging with volt-var optimisation. [23] and [24] optimised reactive power dispatch for DERs using alternating direction method of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 O. Gandhi, Reactive Power Support Using Photovoltaic Systems, Springer Theses, https://doi.org/10.1007/978-3-030-61251-1_5

109

110

5 Reactive Power Dispatch for Large Number of PV Installations

multipliers (ADMM)-based method. The distribution system was divided into multiple areas, each solving the optimal dispatch individually with some information exchange among the areas. [25] also used ADMM-based method which rely on local computations with inputs from neighbouring nodes and local measurements to optimally control reactive power output of PV inverters. Lastly, reactive power from the DERs can also be controlled locally, especially to maintain the local voltages within an acceptable range. [26] implemented networkcognizant droop control, but required the DERs’ owners to have access to the grid parameters to determine the droop coefficients. [27] proposed an optimal activepower-dependent reactive power control (Q(P)) to solve overvoltage problems, whereas a local voltage control under limited reactive power was presented in [28]. The centralised controls typically have shown satisfying results, and many guarantee optimality of the dispatch solutions. However, as the number of DERs in the power system continues to grow, the communication structure required for centralised control may be prohibitive. Although more robust than the centralised control with respect to single-point failures, distributed control algorithms still suffer from some issues. Most of the distributed control algorithms in the literature at least require each of the DER to know some information regarding the grid topology or parameters [18–20], and for the DERs to communicate in some ways with the neighbouring DERs or a central controller (be they one-way or two-way communication) [17–21, 23–25]. When the communication is not available, the results tend to be significantly worse [17]. With many DERs installed in a system, the sharing of the data will be difficult because of the infrastructure required, as well as privacy and latency issues, among others. More recently, [29] proposed a hybrid voltage control that can operate locally when there is a communication outage. Nevertheless, the DERs still need to know incident line reactance values to optimise their reactive power output, and the full system topology is still required to reliably determine some constants. Meanwhile, even though the local controls are able to regulate the node voltage where the inverter is located, they are not able to minimise the power losses or operational costs in the system. Most recently, a local control that is able to optimise the system losses and prevent voltage violations was proposed in [30, 31] and [32] employing regression-based and machine learning techniques, respectively. Once the models are trained, the individual DERs are able to use local variables to determine the near-optimal reactive power outputs. However, to train the models, centralised optimal power flow needs to be carried out in the beginning to determine the reference optimal reactive power outputs for each DER [30–32]. As such, extensive communication and metering infrastructure is still required. It is also not clear how the machine learning model or the regression coefficients can be updated without a centralised optimisation when changes in the grid topology or new DERs are introduced. Hence, a local control of reactive power dispatch which does not need extensive— or even any—communication infrastructure nor confidential grid parameters, yet can optimise global objectives, will be beneficial for power system operators. As such, in this chapter, the author proposes a method for local optimisation of global objectives (LOGO), which is capable of optimising global variables based on local measurements (i.e. voltage magnitudes and reactive power injection) without any communication or knowledge of the grid parameters.

5.1 Introduction

111

Compared to the distributed or centralised approach, the proposed LOGO does not need to wait for response from any other entity, as it does not require any communication. Moreover, there is no need to divide a system into regions or to assign a local coordinator for each region. Last but not least, LOGO can also be implemented in real time without the need to calculate power flow. Therefore, the main contribution of this chapter is to propose a local reactive power control with the following characteristics: [1] is able to optimise global variables based on local measurements without any communication [2] is scalable for large systems and large number of DERs [3] is stable under changing load and solar irradiance [4] does not require the DERs to have clock synchronisation or the same length of time step [5] can be set up without extensive communication or metering infrastructure. The rest of the chapter is organised as follows: Sect. 5.2 lays out the objective and constraints of the system. Section 5.3 presents LOGO formulation, while the case studies to validate the approach are explained in Sect. 5.4. In Sect. 5.5, the results and future development of the approach are discussed. Finally, the chapter is concluded in Sect. 5.6. The derivation of the relation between node voltage and reactive power output (Sect. 5.3.2) has been presented in IEEE conference [33] and subsequently the overall LOGO algorithm was published in Applied Energy journal [34].

5.2 Objective Function and Constraints As in the previous chapters, the perspective of an ISO, who aims to minimise the cost of running the system through power dispatch optimisation, has been adopted. Similarly, without the loss of generality, we have assumed a grid-connected distribution system, with PV systems with reactive power capability as the DERs. The cost of running such a system is expressed through: Cost =

T   t=1

+

Pgrid

grid

grid

c P + cQgrid Q t t  t    

cost of P from grid M  x=1



PV cxPPV Px,t +





cost of P from PV

(5.1)

cost of Q from grid M  x=1



 QPV PV cx,t Q x,t H 



cost of Q from PV

As done previously, S loss and min Vi,t have been used to measure the technical performance of the

proposed algorithm. In addition, the maximum voltage magnitude values, max Vi,t , are also shown for each algorithm, since a case with more than

112

5 Reactive Power Dispatch for Large Number of PV Installations

100% PV penetration is also presented in this chapter, to measure the algorithm’s efficacy in preventing overvoltage situations. Same as in the previous chapters, the ISO has been assumed to be able to utilise the unused capacity of the PV inverters in the system—the owners are assumed to QPV . have given ISO permission to do so in exchange for monetary compensation cx,t The optimisation is subject to the following constraints to ensure safe and reliable performance of the system: grid

Pt



N 

Pi,tloss −

i=1

grid Qt −

N  i=1

N 

Pi,tload +

N  i=1

PV Px,t = 0, t = 1, ..., T

(5.2)

Q PV x,t = 0, t = 1, ..., T

(5.3)

x=1

i=1

loss Q i,t −

M 

load Q i,t +

M  x=1



Ii,t ≤ Imax , i = 1, ..., N

(5.4)



Vmin ≤ Vi,t ≤ Vmax , i = 1, ..., N

(5.5)



PV

2 PV90 2 PV

Q ≤ Q PV = − Px,t , x = 1, ..., M Sx,max x,t x,t,lim

(5.6)

PV90 is the upper limit of forecasted active power from the x th PV system where Px,t PV70 used in Chaps. 2 and 3 at 90% confidence level and has been used instead of Px,t because the forecast utilised in this chapter is 5-minute-ahead forecast instead of the day-ahead in the previous chapters. Equations (5.2) and (5.3) are the active and reactive power balance constraints. The current flowing through the distribution lines are limited according to Eq. (5.4), while the voltage at each node needs to lie within a certain range (Eq. (5.5)). Lastly, the PV reactive power output is limited by Eq. (5.6).

5.3 Local Optimisation of Global Objectives (LOGO) 5.3.1 Relations Between Global and Local Variables To allow the PV with access only to local variables, Vx 1 and Q PV x , to optimise global variables, P loss and Q loss , we need to find the relations between the global and the local variables. To obtain the relations between these quantities, a data-driven approach was employed, where power dispatch optimisation was run over a period of time (one 1V x

is the voltage at the node where the x th PV is located. Perhaps the notation Vi x is more suitable, but Vx has been adopted for simplicity.

5.3 Local Optimisation of Global Objectives (LOGO)

113

month in this work, with half-hour time step). For each system, a reactive power dispatch optimisation was run for the one month using the analytical approach developed in Chap. 3, employing real Singapore load and irradiance data. Optimal power flow algorithm is run at each time step such that for every value of Q PV x,t , corresponding are obtained. The values are then plotted in a threevalues of Vx,t , Ptloss , and Q loss t as the y-axis, as well as Ptloss and dimensional graphs with Vx,t as the x-axis, Q PV x,t loss Q t as the z-axis. Such graphs for one node each in a 24-bus, 69-bus [35], and 119-bus [36] radial distribution systems are shown in Figs. 5.1 and 5.2. These relations are obtained for all the buses with PV in all the systems described in Sect. 5.4. Figure 5.1 shows that Ptloss can be expressed as:

Fig. 5.1 Relation between P loss , |Vx |, and Q PV x for a bus 17 in 24-bus system with 24 PV, b bus 61 in 69-bus system with 12 PV, and c bus 29 in 119-bus system with 30 PV in the system. The surface fitting follows the form in Eq. (5.7). r2 values for a, b, and c are 0.9991, 0.9875, and 0.9831 respectively. Other buses in all the distribution systems analysed have similar relations and r2 values

114

5 Reactive Power Dispatch for Large Number of PV Installations



Fig. 5.2 Relation between Q loss , Vx,t , and Q PV x for a bus 17 in 24-bus system with 24 PV, b bus 61 in 69-bus system with 12 PV, and c bus 29 in 119-bus system with 30 PV in the system. The surface fitting follows the form in Eq. (5.8). r2 values for a, b, and c are 0.9992, 0.9855, and 0.9893 respectively. Other buses in all the distribution systems analysed have similar relations and r2 values



2 Ptloss =ηxPLoss,0 + ηxPLoss,V1 Vx,t + ηxPLoss,V2 Vx,t PV 2 PLoss,Q2 + ηxPLoss,Q1 Q PV Q x,t x,t + ηx where Ptloss =

N

i=1

(5.7)

Pi,tloss . The symbols ηxPLoss,0 , ηxPLoss,V1 , ηxPLoss,V2 , ηxPLoss,Q1 , and

ηxPLoss,Q2 are constants relating the active power losses in the system to the voltage magnitude and reactive power output of the x th PV system. These constants can change when there is a change in the network topology or when new generators or DERs are installed. Therefore, the data needed for the approach can be shared during

5.3 Local Optimisation of Global Objectives (LOGO)

115

the grid interconnection process, and the updating of the coefficients for the existing system can be automated. Although not considered in this work, the value of the coefficients can also be updated continuously through the use of machine learning, supervised learning, or any other methods, should the computing power be available. As the data are not time-specific, high sampling rate is not necessary to generate the data in Fig. 5.1 and to get the coefficients in Eq. (5.7). Figure 5.1 also shows that the relations from (5.7) are valid for both undervoltage and overvoltage scenarios. can also be expressed as the following: Similarly, from Fig. 5.2, we see that Q loss t



2 Q loss =ηxQLoss,0 + ηxQLoss,V1 Vx,t + ηxQLoss,V2 Vx,t t PV 2 QLoss,Q2 + ηxQLoss,Q1 Q PV Q x,t x,t + ηx

(5.8)

where ηxQLoss,0 , ηxQLoss,V1 , ηxQLoss,V2 , ηxQLoss,Q1 , and ηxQLoss,Q2 are constants relating the reactive power losses in the system to the voltage magnitude and reactive power output of the x th PV system. Now we can express Ptloss and Q loss t , and therefore Cost (Eq. (5.1)) in terms of PV Q x,t and Vx,t . However, to obtain the optimal values of Q PV x,t minimising Cost, Vx,t still needs to be expressed in terms of Q PV , which is derived in the next subsection. x,t

5.3.2 Relations Between Node Voltage and PV Reactive Power Output To know how the DERs’ power output will affect the voltage of the bus it is connected to, power flow analysis needs to be conducted. The equations considered in most works on the distribution system have the following form [37, 38]: P j = Pi − ri j

Pi2 + Q i2 − P jload + P jPV , ∀(i, j) ∈ L Vi2

Pi2 + Q i2 − Q load + Q PV j j , ∀(i, j) ∈ L Vi2 P 2 + Q2 V j2 = Vi2 − 2 ri j Pi + xi j Q i + ri2j + xi2j i 2 i , ∀(i, j) ∈ L Vi

Q j = Q i − xi j

(5.9) (5.10) (5.11)

where Pi and Q i are the active and reactive power flowing from node i to node j. ri j and xi j are the resistance and reactance of the line (from the set of lines L) PV PV connecting node i and j. P jload (Q load j ) and P j (Q j ) are the active (reactive) load 2 and PV generation at node j, respectively. Vi is the voltage of node i. 2 P PV i

and P jPV are different from PxPV , in the sense that the former two quantities are concerned with the PV generation at some particular nodes i and j (which may or may not have PV), whereas the latter is concerned with the generation of the x th PV. Similar notations are also used for Q PV .

116

5 Reactive Power Dispatch for Large Number of PV Installations

There are several disadvantages in using Eqs. (5.9) to (5.11) to obtain the relation between the reactive power and voltage (Q-V relation) of the same bus. Firstly, they are recursive: knowledge of the state and power flow of the neighbouring buses is necessary. Secondly, the equations rely on the grid parameters’ values that may not be known by the PV operators. The non-convexity of the equations also poses challenges for optimisation methods. To simplify the Q-V relation in Eq. (5.11), the following assumptions can be made: line losses smaller than power flows Pi and Q i ; |Vi | ≈ 1, such that are much V j2 − Vi2 = 2 V j − Vi . These approximations introduce a maximum error of 0.25% (1%) for a voltage deviation of 5% (10%) [39]. Therefore |V |, an N -dimensional vector consisting of |Vi | , ∀i = 1, . . . , N , can be approximated as: |V | = |V0 | − R P load − P PV − X Q load − Q PV

(5.12)

where R, X ∈ R N ×N with elements Ri j and X i j , which are the effective resistance and reactance connecting node i and node j (i.e. the sum of all the line resistance and reactance connecting node i with node j), respectively. P load , P PV , Q load , Q PV ∈ R N . N is the number of nodes in the system. Assuming that the active power output and the load are for time period

constants t, the voltage magnitude of node i at time period t, Vi,t can be expressed as [39, 40]: N





Vi,t = Vi,t−1 + X i j Q PV (5.13) j,t j=1

By controlling the value of Q PV j,t , one can then control the value of Vi,t , which is the basis of the voltage droop control, where 1/ X i j is the ideal droop coefficient. Nevertheless, despite the approximation, Eq. (5.13) still requires the knowledge of the effective reactance at all the PV nodes and all the PV reactive power outputs, which might be impractical. PV by controlling Q i,t without knowing To allow the optimisation of Ptloss and Q loss t PV the grid parameters, there is a need to establish a direct relation between Q i,t and



Vi,t . The relation can be found through the following procedure: [1] Assume that there is a PV with adjustable reactive power output at node i whose Q-V relation is to be obtained [2] While keeping all the other variables constant (e.g. load, PV active power genPV PV eration), vary Q iPV from −Q i,t,lim to Q i,t,lim and run an optimal power flow algorithm to obtain |Vi | for every value of Q iPV [3] Vary a variable in the system (e.g. active and reactive load, as well as active and reactive power outputs of PV at other nodes) and repeat step 2 until the desired number of scenarios is obtained. As is the case throughout the thesis, i and j are indices reserved for nodes in a system, whereas x is the index for PV and other DERs.

5.3 Local Optimisation of Global Objectives (LOGO)

117

Fig. 5.3 Relationship between voltage magnitudes and reactive power output for IEEE 13-bus system

In this work, eight scenarios were obtained by varying P load , Q load , P PV , and Q PV at other nodes. To validate its efficacy, the proposed procedure is applied in the most commonly used distribution test systems, namely IEEE 13-bus system [41], IEEE 37-bus system [41], 33-bus system [38], PGCE 69-bus system [35], 119-bus system [36], as well as a real 24-bus system topology based in Anhui, China. The details of the 24-bus system is given in Sect. 5.4.1. For simplicity, single-phase equivalent of the test systems have been utilised. Nevertheless, the proposed procedure can also be applied on three-phase unbalanced distribution networks. Each line in Figs. 5.3, 5.4, 5.5, 5.6, 5.7 and 5.8 was obtained by varying the Q PV at a particular node while keeping all the other variables (P load , Q load , P PV ) constant. It can be seen that for a particular node in all the test systems, reactive power affects the voltage level linearly. In fact, the slopes of the line for node i, ki , are approximately constant regardless of the scenario (within the errors mentioned previously). This means that almost for any system condition, the voltage can be expressed as:



Vi,t = Vi,t−1 + ki Q PV i,t

(5.14)



where Vi,t−1 is the voltage magnitude before the absorption/injection of reactive power (Q iPV ) at node i. Equation (5.14) has the same form as Eq. (5.13), except that Eq. (5.14) only considers the diagonal terms of X . The values of ki obtained from the proposed method are verified with the values of X ii , as shown in Fig. 5.9.

118

5 Reactive Power Dispatch for Large Number of PV Installations

Fig. 5.4 Relationship between voltage magnitudes and reactive power output for IEEE 37-bus system

Fig. 5.5 Relationship between voltage magnitudes and reactive power output for 33-bus system

5.3 Local Optimisation of Global Objectives (LOGO)

119

Fig. 5.6 Relationship between voltage magnitudes and reactive power output for 69-bus system

Fig. 5.7 Relationship between voltage magnitudes and reactive power output for 119-bus system

120

5 Reactive Power Dispatch for Large Number of PV Installations

Fig. 5.8 Relationship between voltage magnitudes and reactive power output for Anhui 24-bus system

Fig. 5.9 Relationship between proposed coefficient ki with effective reactance X ii for the a 69-bus and b IEEE 37-bus system

PV PV In real systems, the value of X ii can be obtained by varying Q i,t from −Q i,t,lim



PV to Q i,t,lim , and recording the corresponding Vi,t in a short period of time (e.g. one minute), while the load and active power output are assumed as constant (for PV, X ii can be determined during clear sky condition or at night time, so that Pi,tPV can be assumed as constant during the process. The relation is robust to change as illustrated in Figs. 5.3 to 5.8, but might need to be updated in case the network topology changes. In a real-time application, the PV owners are likely to only have access to the values of voltage measured by the PV systems and their previous power outputs. Therefore,

5.3 Local Optimisation of Global Objectives (LOGO)

121

utilising only the locally available information, the PV owners can control the node voltage through the following equation:



PV

pred

meas

PV V,P1 PV Px,t − Px,t−1

Vx,t = Vx,t−1 + ηxV,Q1 Q PV x,t − Q x,t−1 + ηx

(5.15)

meas

is the measured voltage magnitude at time step t − 1, at the node where Vx,t−1



pred

where the x th PV is located, while Vx,t is the predicted voltage magnitude at the





pred

same node in the subsequent time step. Vx,t is used instead of Vx,t because



(5.15) is only exact for Vx,t when the other parameters in the grid (e.g. active and reactive loads) are constant from time step t − 1 to t. ηxV,Q1 (previously termed ki in PV Eq. (5.14)) and ηxV,P1 are the coefficients relating Vx,t to Q PV x,t and Px,t respectively. V,Q1 V,P1 The values of ηx and ηx can be obtained locally without any communication nor information from the grid as discussed previously in this subsection.

5.3.3 Expressing Cost in Terms of Q PV x,t and Vx,t To be able to benchmark LOGO with other approaches or algorithms in the QPV —optimisation of Cost in literature—most of which do not take into account cx,t Eq. (5.1) without incorporating the reactive power costs first needs to be developed. Therefore, Eq. (5.1) becomes the following:  T  M   Pgrid grid Qgrid grid PPV PV ct Pt + c Qt + cx Px,t H Cost = 

t=1

(5.16)

x=1

Substituting Eqs. (5.2) and (5.3) into (5.16), expression of Cost  in terms of P loss and Q loss can be obtained. Subsequently, differentiating Cost  with respect to Q PV x,t yields:  N   M   ∂Cost  ∂ Pgrid PV ct = Pi,tload − Px,t + Ptloss ∂ Q PV ∂ Q PV x,t x,t x=1 i=1  N  M   load loss Q i,t − Q PV + cQgrid x,t + Q t i=1

+

M 



x=1

PV cxPPV Px,t

x=1 Pgrid PLoss,V1 V,Q1 = ct ηx ηx + ηxPLoss,Q1 + cQgrid ηxQLoss,V1 ηxV,Q1 + ηxQLoss,Q1

− cQgrid

(5.17)

122

5 Reactive Power Dispatch for Large Number of PV Installations

  Pgrid meas + 2 ct ηxPLoss,V2 ηxV,Q1 + cQgrid ηxQLoss,V2 ηxV,Q1 Vx,t−1   2 Pgrid ηxPLoss,V2 ηxV,Q1 + ηxPLoss,Q2 Q PV + 2ct x,t   V,Q1 2 Qgrid QLoss,V2 QLoss,Q2 ηx Q PV ηx + 2c + ηx x,t Setting ∂Cost  /∂ Q PV x,t = 0, we obtain Q PV x,t =

meas  −Ax,t − Bx,t Vx,t−1  C x,t

(5.18)

where PLoss,V1 V,Q1 ηx ηx + ηxPLoss,Q1 + cQgrid ηxQLoss,V1 ηxV,Q1 + ηxQLoss,Q1 − 1   Pgrid =2 ct ηxPLoss,V2 ηxV,Q1 + cQgrid ηxQLoss,V2 ηxV,Q1   2 Pgrid ηxPLoss,V2 ηxV,Q1 + ηxPLoss,Q2 =2ct   2 + 2cQgrid ηxQLoss,V2 ηxV,Q1 + ηxQLoss,Q2

Ax,t =ct

Pgrid

 Bx,t  C x,t

(5.19)

This means that for each PV in the system, the reactive power output that minimises Eq. (5.16) can be found using (5.18) without any communication or information from outside of the respective PV nodes.

5.3.4 Inclusion of Reactive Power Cost QPV,invloss The cost of RPC using an inverter can be divided into cost of power loss (cx,t ) QPV,LR ) as elaborated in Chap. 4, where and cost of inverter lifetime reduction (cx,t QPV QPV,invloss QPV,LR cx,t = cx,t + cx,t . For the readers’ convenience, the equations for reactive power cost components are rewritten in this section. The cost of power loss can be expressed as: Pgrid

QPV,invloss PV cx,t Q x,t = ct

PV,invloss Px,t

(5.20)

where PV,invloss Px,t   PV 2  PV PV PV 2 cxPV,V Sx,t − Px,t − Px,t + cxPV,R Sx,t = PV 2 PV,R cxPV,self + cxPV,V Q PV Q x,t x,t + cx

(5.21) PV Px,t = 0 PV Px,t =0

5.3 Local Optimisation of Global Objectives (LOGO)

123

Meanwhile, the quadratic version of the inverter lifetime reduction cost because of the reactive power usage can be expressed as: PV 2 QPV,LR PV LI,2 Q x,t = ηxLI,0 + ηxLI,1 Q PV Q x,t cx,t x,t + ηx

(5.22)

Incorporating the reactive power cost into the optimisation, the constants in Eq. (5.19) will become the following: ηxPLoss,V1 ηxV,Q1 + ηxPLoss,Q1 + cxV + cQgrid ηxQLoss,V1 ηxV,Q1 + ηxQLoss,Q1 − 1 + ηxLI,1   Pgrid =2 ct ηxPLoss,V2 ηxV,Q1 + cQgrid ηxQLoss,V2 ηxV,Q1   2 Pgrid ηxPLoss,V2 ηxV,Q1 + ηxPLoss,Q2 + cxR =2ct   2 + 2cQgrid ηxQLoss,V2 ηxV,Q1 + ηxQLoss,Q2 + 2ηxLI,2 Pgrid

A x,t =ct Bx,t C x,t



(5.23)

As the formula for reactive power cost for a particular PV system is different depending on whether it generates active power (Eq. (5.21)), the optimal value of Q PV x,t needs to be calculated differently depending on the active power usage. For PV = 0, Px,t meas −A x,t − B x,t Vx,t−1 = (5.24) Q PV x,t C x,t PV whereas for Px,t = 0, meas C x,t Q PV x,t + A x,t + B x,t Vx,t−1 ⎛ ⎞ PV Q Pgrid x,t ⎠ + ct cxV ⎝ PV 2 − 1 = 0 PV 2 Px,t + Q x,t

(5.25)

Therefore the reactive power output of each PV minimising Eq. (5.1) can be obtained using Eqs. (5.24) and (5.25).

5.3.5 Local Control Using the proposed LOGO, regardless of the number of DERs in the system (generalised as PV in this chapter), each DER participating in the reactive power support will do the following: [1] Determine the constants for P loss (ηxPLoss,0 , ηxPLoss,V1 , ηxPLoss,V2 , ηxPLoss,Q1 , ηxPLoss,Q2 ), Q loss (ηxQLoss,0 , ηxQLoss,V1 , ηxQLoss,V2 , ηxQLoss,Q1 , ηxQLoss,Q2 ), and |Vx | (ηxV,Q1 , ηxV,P1 )

124

5 Reactive Power Dispatch for Large Number of PV Installations

Algorithm 5.1 LOGO Undervoltage Constraint Handling pred

1: Calculate Vx,t using Eq. (5.15) pred 2: while Vx,t < Vmin do PV PV 3: Q x,t ← Q PV x,t + Q x,t pred

pred

4: Vx,t ← newVx,t PV 5: if Q PV x,t > Q x,t,lim then PV 6: Q x,t ← Q PV x,t,lim 7: break 8: end if 9: end while

adjusting RPD pred

recalculate Vx,t

[2] Calculate the constants A x,t , Bx,t , C x,t

meas

) and calculate [3] Note the voltage magnitude from the previous time step ( Vx,t−1 PV the reactive power output Q x,t using Eq. (5.24) or (5.25) subject to constraints (5.2)–(5.6). Step 3 is conducted every time step (which can be chosen depending on the PV owners’ or ISO’s preference, e.g. 5 s), Step 2 whenever there is a change in electricity price (e.g. every half hour), and Step 1 whenever there is a change in the network configuration or a new installation of grid elements. Step 1–3 are carried out independently by each DER and there is no need for clock synchronisation or for the DERs to have the same length of time step. This means that the proposed reactive power control does not require any comPgrid munication with the ISO nor any of the DERs in system. Even the updating of ct does not require any direct communication since the electricity price is generally published online as soon as it is available. As mentioned previously, determining and updating the constants in Step 1 can be done as part of the commissioning or interconnection procedures of new grid elements or when owners/operators want to register their systems to provide reactive power support. During this period, the grid to the existing and new DERs operator can either provide access to the Ptloss and Q loss t to determine the constants themselves. This should alleviate the or to control Q PV x,t privacy issues immensely as there is no need to share the network topology of the grid, nor any generator and line characteristics, with other parties.

5.3.6 Constraints Handling To ensure that Q PV x,t generated by LOGO always satisfy the power system constraints, algorithms are run to adjust the reactive power outputs generated by Eqs. (5.18), (5.24), and (5.25). For example, Algorithm 5.1 is incorporated into the optimisation to make sure that the undervoltage constraints are fulfilled. These constraint-handling algorithms are implemented by each DER at each run of the optimisation algorithm.

5.3 Local Optimisation of Global Objectives (LOGO)

125

On top of constraints (5.2) to (5.6), the change in Q PV x,t from one time step to the next is also limited by the following equation:

PV

Q

x,t+1

QLim PV

− Q PV Q x,t,lim x,t ≤ η

(5.26)

where ηQLim is a constant, which can be set by the ISO. The purpose of constraint (5.26) is to prevent a sudden voltage spike or sag caused by an abrupt change in Q PV x,t , especially when there are many reactive-capable DERs in the system, which do not know the dispatch scheduling of the other DERs, e.g. when a particular bus is experiencing an under/overvoltage violation, another bus may also be experiencing a similar violation and if the change in Q PV x,t is not limited, the response from multiple DERs may produce an unintended voltage swing or oscillation.

5.4 Implementation Setup 5.4.1 Test Systems The 69-bus test system [35] and 119-bus test system [36] have been employed for verification of the effectiveness of LOGO. 12 and 30 PV systems are present in the 69-bus and 119-bus system respectively. Their locations are determined arbitrarily and are shown in Fig. 5.10. The simulations are also carried out on a real distribution system topology based in Anhui, China. The Anhui test system has 24 buses, each equipped with a PV system, with a base voltage of 10 kV. The topology can be seen in Fig. 5.10 while the network parameters are listed in Table 5.1. It is worth noting that both the 69 and 119-bus system scenarios only have undervoltage problems (as the PV power production never exceeds the total load in the system), whereas the 24-bus system scenario suffers from both overvoltage and undervoltage problems. Vx in the 24-bus system is also much more difficult to manage because the PVs are very close to each other (every bus has a PV system) and their Q PV would affect the Vx of the neighbouring buses immensely, making communication all the more important.

5.4.2 System Parameters To test the effectiveness and the stability of LOGO, a one-day scheduling problem with 30-minute load and one-minute irradiance data are employed. The simulations PV = 0) using data on 6 March are run over 11 h (from 8 a.m. to 7 p.m., when Px,t 2018. The length of time step t is chosen to be 5 s, such that the algorithm is run 12 times for every irradiance value. The load data at each node in the test systems are

126

5 Reactive Power Dispatch for Large Number of PV Installations

Fig. 5.10 Test distribution systems with a 69, b 119, and c 24 buses. PV systems are connected to the indicated nodes

5.4 Implementation Setup

127

Table 5.1 Network Data of 24-bus Anhui Distribution System Branch Sending Receiving Branch No Node Node r [] x[] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0 1 2 3 4 5 6 7 8 2 10 11 10 11 3 4 5 6 18 18 6 21 21 8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0 0.8063 0.5934 0.5618 0.6709 1.4500 0.5079 1.3884 0.5490 1.7171 0.3600 1.1158 0.0921 0.1280 0.8477 0.5950 0.0504 0.0850 0.0521 1.0889 1.4069 0.2887 1.2286 0.1523

0 0.5057 0.3834 0.3630 0.4335 0.9369 0.3282 0.8971 0.3548 0.8080 0.1694 0.5251 0.0434 0.0603 0.4413 0.2800 0.0237 0.0400 0.0245 0.5124 0.7191 0.1358 0.5782 0.0717

Receiving Node P load [kW] Q load [kVar] 0 100 90 120 60 60 200 200 60 60 45 60 60 120 60 60 60 90 90 90 90 90 90 420

0 60 40 80 30 20 100 100 20 20 30 35 35 80 10 20 20 401.553 40 40 40 40 50 200

the aggregate Singapore load data [42], multiplied by a random number within [0.9, 1.1], and normalised to the systems’ load. Pgrid has also been taken to be the Singapore electricity price on 6 March 2018 ct plus the associated grid charge [43], while cQgrid is the Singapore reactive power charge [44]. The PV systems in the test networks all have an installed capacity of 300 kWp and a 300 kVA inverter. cPPV is assumed to be 10.28 SGD cents per kWh, as found in Chap. 4 for the case of PV with a system lifetime of 20 years and 14 years inverter lifetime. Vmax and Vmin are 1.05 and 0.9 p.u. respectively, while Imax is set to 1 p.u. ηQLim is set to be 0.1. To obtain the coefficients in Eq. (5.17), half-hourly Singapore load and irradiance data from 1 to 30 January 2016 have been used as training data.

128

5 Reactive Power Dispatch for Large Number of PV Installations

Fig. 5.11 Irradiance and load data

5.4.3 Weather Parameters Tree-based forecasting methods, and in particular Random Forests, have been shown to produce accurate deterministic [45] and probabilistic [46] forecasts of irradiance in the tropics. In this work, a Quantile Random Forest (QRF) is employed to produce 5-minute-ahead probabilistic forecasts of 1-minute resolution of the irradiance on 6 March 2018. Using the 30 previous values of the clear sky index kt , QRF forecasts the next five 1-minute values of kt , as well as their 90% prediction intervals. The corresponding irradiance values were then calculated using the following formula: , where G clear is the clear sky irradiance, estimated using the Ineichen G t = kt G clear t t and Perez clear sky model [47]. The irradiance and temperature data provided by SERIS from latitude 1.3005◦ and longitude 103.771◦ were then used to generate the forecasts, and subsequently PV PV90 and Px,t based on equations described in Chap. 2 (Eqs. (2.11) and (2.12)). The Px,t irradiance forecast for this chapter was performed by Hadrien Verbois of SERIS. Figure 5.11 shows the irradiance and load data employed in this chapter. A closer look at the data from 15:55 to 16:05 is to allow a clearer understanding of Fig. 5.13 shown in the next section.

5.4.4 Benchmark Algorithms As none of the benchmark algorithms include reactive power cost in the optimisation, LOGO is also run without considering the reactive power cost, i.e. Q PV x,t is

5.4 Implementation Setup

129

Table 5.2 LOGO performance comparison for the 11-hour Period Cost [SGD] S loss [%] Min Vi,t

[p.u.] 69

119

24

LOGO AS [6] MASDS [18] Droop Constant PF No reactive power LOGO AS [6] MASDS [18] Droop Constant PF No reactive power LOGO AS [6] MASDS [18] Droop Constant PF No reactive power

∗ Max Vi,t Q PV [kvarh] [p.u.]

5533 5507 5537

2.09 1.96 1.99

0.918 0.927 0.918

1 1 1

18519 29010 21389

5603 5568 5697

2.53 2.38 3.37

0.918 0.910 0.909

1 1 1

6368 13991 0

37803 37777 37894

3.83 3.67 3.89

0.890 0.893 0.890

1 1 1

68893 81584 54778

37921 37959 38493

3.90 4.13 4.96

0.890 0.886 0.868

1 1 1

39205 44042 0

2028 1940 2173

2.38 2.11 3.44

0.905 0.903 0.881

1.050 1.049 1.055

9016 17271 49245

2088 2150 2141

3.82 4.53 4.68

0.912 0.886 0.858

1.057 1.050 1.035

11172 10585 0

∗ Reactive power injection and absorption are both considered (not netted), i.e. 10000 kvarh may mean 5000 kvarh injected and 5000 kvarh absorbed

calculated using Eq. (5.18). Nevertheless, the reactive power cost is still considered when calculating the total cost shown in Table 5.2. Constraint (5.26) is also imposed on the benchmark algorithms, except for centralised optimisation as it is only run every minute, instead of every 5 s. On top of the results from the benchmark algorithms below, the results of the case without any reactive power injection are also included for comparison.

5.4.4.1

Centralised Optimisation

The centralised control is adopted from the Analytical Solution (AS) developed in Chap. 3, as it has been shown to give superior results in a shorter time when compared with exact and metaheuristic optimisation algorithms. AS requires each of the PV to know the parameters of the grid, such as resistance and reactance of the lines, as well as power flow across all the lines at each time step. The centralised optimisation is run every minute, when the irradiance value changes.

130

5.4.4.2

5 Reactive Power Dispatch for Large Number of PV Installations

Distributed Optimisation

This work also employs a MAS-based distributed subgradient method (MASDS) [18] as a benchmark. MASDS needs local and neighbouring nodes’ voltage magnitudes, voltage phase angles, as well as the bus admittance elements to obtain the DERs’ reactive power output. Moreover, MASDS requires the value of a constant α to be determined [18]. The value of α has been varied from 0.05 to 1 in steps of 0.05 for all the test systems. The best results for the 69-bus, 119-bus, and 24-bus systems are obtained when α has the value of 0.45, 0.35, and 0.05, respectively. Only these best results are shown in Table 5.2.

5.4.4.3

Droop Control

Two common local controls are also implemented to serve as benchmark. They are droop control (Q(V )) and constant power factor (PF) control, (Q(P)). In constant PF control, different values of inverters’ PF are varied from 0.70 to meas < Vref , Q PV > 0 (leading 0.95 in steps of 0.05 for for each system. When Vx,t meas PV PF), whereas when Vx,t > Vref , Q < 0 (lagging PF). The best results for the 69bus, 119-bus, and 24-bus systems are obtained when the PF is 0.70, 0.70, and 0.95, respectively. Likewise, only these best results are shown in Table 5.2. The droop control mechanism is illustrated in Fig. 5.12. The values of Vmin , Vlo , Vref , Vhi , Vmax are set to 0.95, 0.98, 1.00, 1.02, 1.05 p.u., respectively.

Fig. 5.12 Traditional droop control used for benchmark

5.5 Results and Discussions

131

5.5 Results and Discussions This section is divided into three parts, namely: (1) stability of the proposed LOGO; (2) general performance of LOGO and benchmark algorithms across the different test systems; and (3) limitations and possible improvements of the algorithm. The first aspect is explored by employing LOGO to optimise the reactive power dispatch of increasing number of DERs in the 69-bus distribution system (from 1 PV to 12 PV in the system). Subsequently, the performance of LOGO is compared with the benchmark algorithms for the case of 69, 119, and 24-bus distribution systems.

5.5.1 Stability of LOGO Figure 5.13 shows the values of Q PV and |V meas | from 15:55 to 16:05 at different PV penetrations in the 69-bus, 24-bus and 119-bus test systems. Recall that the irradiance changes every minute, while the time step for LOGO is 5 s such that there are 12 runs of the algorithm for every value of irradiance. It can be seen from Fig. 5.13 that the values of Q PV and |V meas | are very stable: the “spikes” in their values are relatively small and only last for the first 3 runs of the algorithm, even for the system with 30 PV systems providing reactive power support in 119-bus network. The undervoltage violation in 119-bus test system is explained in Sect. 5.5.2. It can also be seen from Fig. 5.13e and k that LOGO is able to handle overvoltage situations in 24-bus system without problem—some PV systems whose nodes are prone to overvoltage conditions continuously absorb reactive power. Should there be oscillations in Q PV , they will be limited to the values set in Eq. (5.26).

5.5.2 Performance of LOGO As expected, the centralised algorithm performs the best compared to all the other algorithms across all the test systems (Table 5.2), as it has access to the grid topology, as well as all the line and bus information, including those that do not have PV system. The proposed algorithm, LOGO, performs the next best in all the test systems in terms of Cost, notably better than the distributed algorithm and local controls. In Table 5.2, the best results for each test system are written in bold, whereas the results that are worse than the case with no RPC are highlighted in red. For the 119-bus system, all of the dispatch control algorithms violated the undervoltage constraint. This is because at certain time slots within the simulated study, it is not possible to satisfy the constraint, given the P load , Q load , and P PV . In the case PV to Q PV , of AS, at certain

values of t, even though all the Q x,t are already equal x,t,lim



the min ( Vi,t ) is still 0.893 p.u.. The other algorithms have lower min Vi,t values as some of the PV do not generate the maximum reactive power possible (since their

132

5 Reactive Power Dispatch for Large Number of PV Installations

Fig. 5.13 Values of Q PV in 69-bus a–d, 24-bus e, 119-bus test system f, and values of V meas in 69-bus g–j, 24-bus k, 119-bus test systems l from 15:55 to 16:05. Each line in the figure represents reactive power output of a PV system (a–f), or voltage magnitude of the node where the respective PV system is located (g–l)

voltage magnitudes are within the boundary, and they do not know that other buses may not satisfy the undervoltage constraint, e.g. see Fig. 5.13f and l). For the 24-bus system, AS and LOGO managed to handle both the overvoltage and undervoltage problems. The other algorithms violated the constraints at certain time slots, particularly for the first 1 or 2 runs when the load and/or irradiance changes rapidly. LOGO is able to handle the problem even though the local droop control’s

pred

inability to do so is due to the calculation of Vx,t in Eq. (5.15), which made use of the irradiance forecast to mitigate voltage violation. Even though MASDS yields lower losses than LOGO and the local controls in 69 and 24-bus system respectively, it still yields higher Cost. This is due to higher

5.5 Results and Discussions

133

utilisation of Q PV in MASDS, resulting in higher cQPV . This confirms the importance of considering cQPV , the focus of this thesis. MASDS also violated the undervoltage constraint in the 24-bus system as it was not able to coordinate Q PV of the different buses well enough. The difference in performance between LOGO and MASDS is negligible in the 69-bus system with 12 PV,3 but is much larger in the 119-bus system with 30 PV. This difference in performance becomes largest in the 24-bus system whose every node is equipped with PV. The better performance of LOGO over MASDS in more complex distribution systems highlight LOGO’s effectiveness and scalability in optimising reactive power dispatch without any communication. From these results, LOGO’s superiority in minimising system operating cost is expected to be even more apparent in larger and more complex systems, especially those without sufficient communication infrastructure. When compared with the case with no reactive power, all the reactive power controls in 69 and 119-bus systems are more economical, showing that RPC from PV is beneficial for distribution systems with relatively low penetration of PV. Nevertheless, looking at the results for the 24-bus system, MASDS and constant PF control actually increase the operating cost of the system through the RPC. The maximum voltage magnitude in the system is also increased by reactive power support in all the tested algorithms. The increase in maximum voltage— even to the point of voltage violation—is because when irradiance increases suddenly, the PV systems may still be producing the reactive power and did not decrease its reactive power output fast enough. This underlines the danger of using algorithms which have not been rigorously tested for power dispatch and the importance of accurate forecasting for better integration of high PV penetration in electrical grids. To compare the performance of the five algorithms in detail, the total cost at each time step (5-second-long) for the five algorithms in the three test systems are shown in Fig. 5.14. The time steps covered are t = 4500 to t = 6000 (T = 7920), approximately from 14:15 to 16:20. Not all 7920 time steps are shown so that more details in the figure can be appreciated. It can be seen that LOGO performs the closest to the centralised approach and is also the most stable, while both droop control and constant PF4 control perform significantly worse and have long periods of oscillations. Although LOGO, droop control, and constant PF control only use local information, the results generated by LOGO for all test systems are decisively better than those from the droop and constant PF controls. Thus, it has been shown that LOGO is able to perform better than the distributed and local controls in all the test systems as well as to handle the system constraints.

3 MASDS

even performed better than LOGO at certain periods in 69-bus system, as shown in Fig. 5.14a. 4 It might be counterintuitive for a constant PF control to have any oscillation since the PV reactive power outputs are supposed to constant with constant irradiance. The oscillations result from the

be

meas − V ), which also changes the sign of Q PV . constant shift in the sign of ( Vx,t ref x,t

134

5 Reactive Power Dispatch for Large Number of PV Installations

Fig. 5.14 Total cost at each time step for t = 4500 to t = 6000 (approximately from 14:15 to 16:20) of the simulation for the five algorithms in the three test systems

5.5.3 Cost Compositions To analyse the difference in cost further, we look at the cost compositions of the six cases for the three test systems, shown in Fig. 5.15. The costs of active power from the grid and from PV are not shown in the figure because they are the same for all algorithms in each of the system. As can be seen from the figure, LOGO cut down the cost of Q grid from the case with no reactive power—“NoQ”— by more than half in 69 and 24-bus systems by generating reactive power from PV at lower cost. The cost reduction in 119-bus system is not as significant because the percentage PV penetration is much lower compared with that in the other two systems. The difference between LOGO and the other algorithms is most striking in the 24-bus system where MASDS used too much reactive power support while droop and constant PF controls used too little. AS yields the lowest cost in all the systems by minimising the cost of Q grid and, to some extent, of P loss .

5.5.4 Reactive Power Profiles In this subsection, the reactive power profiles of the different algorithms in 119 and 24-bus systems are shown to illustrate the effectiveness of the algorithms and to explain the cost compositions illustrated in the previous subsection. The reactive

5.5 Results and Discussions

135

Fig. 5.15 Cost composition of the different reactive power dispatch algorithms, including the case with no Q PV , for the three different distribution systems. The costs of P grid and P PV are not shown because they are the same for all algorithms in each of the system. Both P loss and Q loss are fulfilled by P grid and Q grid respectively; they are separated to highlight the power losses in the system

Fig. 5.16 Reactive power outputs of 30 PV systems controlled using LOGO in 119-bus system

power profiles from the 69-bus system are not shown for brevity and because the insights are similar to those from the 119-bus system. Figures 5.16, 5.17, 5.18, 5.19, and 5.20 show the reactive power profiles of LOGO, AS, MASDS, droop, and constant PF, respectively, in 119-bus system. Each line in the figures represents the reactive power profile of a PV system throughout the simulation. Comparing Figs. 5.16 and 5.18, it is obvious that the reactive power outputs of LOGO are much more stable than those of MASDS. The oscillations of reactive power outputs by MASDS suggest that a particular PV system is unable to account for the change in parameters caused by other PV systems that it is not in communication with. There is only one line in Fig. 5.17 because all the PV controlled using AS have the same reactive power output, Q PV x,t,max . This is because of the low PV penetration and that the system gains from local reactive power provision. The reactive power

136

5 Reactive Power Dispatch for Large Number of PV Installations

Fig. 5.17 Reactive power outputs of 30 PV systems controlled using AS in 119-bus system. Only one line can be seen because all the PV systems have the same reactive power output, i.e the maximum reactive power possible

Fig. 5.18 Reactive power outputs of 30 PV systems controlled using MASDS in 119-bus system

Fig. 5.19 Reactive power outputs of 30 PV systems using droop control in 119-bus system

5.5 Results and Discussions

137

Fig. 5.20 Reactive power outputs of 30 PV systems controlled using constant PF method in 119-bus system

Fig. 5.21 Reactive power outputs of 24 PV systems controlled using LOGO in 24-bus system

outputs of the Droop control (Fig. 5.19) follow the voltage profile of the respective nodes inversely; the voltage magnitudes become higher as the day progresses until about 11:30 when the irradiance drops while the total load keeps rising (see Fig. 5.11). Meanwhile for Constant PF control (Fig. 5.20), the reactive power profiles are inverted compared with the rest because the value of Q PV is proportional

to that

Vx,t violates is reached (see Eq. (5.6)) and when of P PV , except for when Q PV lim PV constraint (2.10), in which case Q PV x,t is increased to Q x,t,lim subject to constraint (5.26). The reactive power profiles of LOGO, AS, MASDS, Droop, and Constant PF in 24-bus system are shown in Figs. 5.21, 5.22, 5.23, 5.24, and 5.25 respectively. As can be seen in Fig. 5.21, most of the PV reactive power outputs controlled by LOGO are stable, unlike those controlled by MASDS (Fig. 5.23), droop (Fig. 5.24), and constant PF (Fig. 5.25). For LOGO, only 3 PV systems are consistently absorbing reactive power and 3 more PV that are absorbing reactive power for short periods of time. 8 PV systems experience reactive power oscillations for brief periods.

138

5 Reactive Power Dispatch for Large Number of PV Installations

Fig. 5.22 Reactive power outputs of 30 PV systems controlled using AS in 119-bus system

Fig. 5.23 Reactive power outputs of 24 PV systems controlled using MASDS in 24-bus system

Fig. 5.24 Reactive power outputs of 24 PV systems using droop control in 24-bus system

5.5 Results and Discussions

139

Fig. 5.25 Reactive power outputs of 24 PV systems controlled using constant PF method in 24-bus system

This is in contrast with the other local and distributed methods where virtually all the PV systems have injected and absorbed reactive power, as well as experienced prolonged reactive power oscillations at different periods. In droop control, Q PV 1,t is always 0 because bus number 1 is the point of common coupling to the main grid and V1,t = 1 p.u., ∀t = 1, ..., T . AS is the most efficient in providing reactive power support (evident in Fig. 5.15c) with only 2 PV systems absorbing reactive power at specific periods, as illustrated in Fig. 5.22.

5.5.5 Limitations and Further Development of LOGO It has been shown that LOGO provides a very viable alternative for reactive power dispatch optimisation, especially considering that the necessary infrastructure to facilitate reliable communication among increasing number of DERs may not be able to keep up with their rapid deployment. Nevertheless, as a local optimisation algorithm, LOGO does suffer from some limitations. For example, by only using local variables, it will not be able to enforce voltage/current constraints on buses that do not have DERs/reactive power control. However, should information from other neighbouring buses be available, it would be possible to enforce the constraints using the constraint handling algorithms (e.g. Algorithm 5.1). Estimation of the voltage levels can also be carried out as in [48], should the network topology be available to the DERs. Another limitation of LOGO is that it has only been shown to work in distribution systems under steady state conditions (as the training data are all from steady-state situations). Should a transient state occur, the correlations used for the method may no longer be valid, and a separate transient control will need to be implemented.

140

5 Reactive Power Dispatch for Large Number of PV Installations

While the process of updating the constants in Eq. (5.23) can become part of the DER interconnection procedures and do not require subsequent communication, if there is at least a one-way communication from the ISO to the DERs, then the DERs can constantly update their coefficients. This work has shown that it is possible to optimise global objectives using only local variables. That being said, information from other neighbouring buses can also be incorporated. This will only increase the dimension of the regression analysis and the complexity of Eq. (5.7). Common machine learning or other algorithms can be used to obtain the equations to optimise the reactive power dispatch based on more information. In addition, when more types of data, such as load forecast, are available locally, they can also be incorporated into the approach.

5.6 Conclusion In this chapter, an approach to locally optimise reactive power dispatch of DERs while minimising a global objective, has been proposed. The data-driven approach has utilised correlations between global variables such as power losses with locally available information such as voltage magnitudes and the reactive power output of the DERs. The use of training data removes the need for communication in real time. The proposed approach has yielded better results and are much more stable compared with other distributed and local algorithms; the improvements of the proposed approach, LOGO, over other algorithms increase with system size and complexity. Across all the simulated distribution systems, LOGO performs almost as well as a centralised approach. Therefore, this work has provided a powerful alternative for distributed optimisation of global variables without the need for communication. It will be a cost-effective solution that provides reliability and robustness against communication errors and delays, bad or malicious data, as well as privacy issues. The proposed approach can also be applied to optimise other variables depending on the perspective adopted or ownership of the DERs. The importance of accurate solar forecasting and of validating the control algorithms before implementation have also been demonstrated in this work. Future research can explore continuous adjustment of values of the constants relating system losses and nodes’ voltage when consistent streams of data are available. In this work, only local information has been used in optimising the reactive power dispatch. However, information from other neighbouring buses can also be incorporated, should they be available.

References

141

References 1. Antoniadou-Plytaria KE, Kouveliotis-Lysikatos IN, Georgilakis PS, Hatziargyriou ND (2017) Distributed and decentralized voltage control of smart distribution networks : models, methods, and future research. IEEE Trans Smart Grid 8(6):2999–3008. ISSN: 1949-3053. https://doi. org/10.1109/TSG.2017.2679238 2. Jahangiri P, Aliprantis DC (2013) Distributed volt/var control by pv inverters. IEEE Trans Power Syst 28(3):3429–3439. ISSN: 08858950. https://doi.org/10.1109/TPWRS.2013.2256375 3. Gandhi O, Rodríguez-Gallegos CD, Zhang W, Srinivasan D, Reindl T (2018) Economic and technical analysis of reactive power provision from distributed energy resources in microgrids, Appl Energy 210:827–841. ISSN: 03062619. https://doi.org/10.1016/j.apenergy.2017.08.154 4. Gandhi O, Zhang W, Rodríguez-Gallegos CD, Srinivasan D, Reindl T (2016) Continuous optimization of reactive power from pv and ev in distribution system. In: 2016 IEEE innovative smart grid technologies—Asia (ISGT-Asia). IEEE, Melbourne, November 2016, pp 281–287. ISBN: 978-1-5090-4303-3. https://doi.org/10.1109/ISGT-Asia.2016.7796399 5. Gandhi O, Srinivasan D, Rodríguez-Gallegos CD, Reindl T (2017) Competitiveness of reactive power compensation using pv inverter in distribution system. In: 2017 IEEE PES innovative smart grid technologies conference Europe (ISGT-Europe), Torino. IEEE, Italy, September 2017, pp 1–6. ISBN: 978-1- 5386-1953-7. https://doi.org/10.1109/ISGTEurope.2017.8260238 6. Gandhi O, Zhang W, Rodríguez-Gallegos CD, Bieri M, Reindl T, Srinivasan D (2018) Analytical approach to reactive power dispatch and energy arbitrage in distribution systems with ders. In: IEEE Trans Power Syst 178 Bibliography 33(6):6522–6533. ISSN: 0885-8950. https://doi. org/10.1109/TPWRS.2018.2829527 7. Gandhi O, Rodríguez-Gallegos CD, Gorla NBY, Bieri M, Reindl T, Srinivasan D (2019) Reactive power cost from pv inverters considering inverter lifetime assessment. IEEE Trans Sustain Energy 10(2):738–747. ISSN: 1949-3029. https://doi.org/10.1109/TSTE.2018.2846544 8. Gandhi O, Rodríguez-Gallegos CD, Reindl T, Srinivasan D (2018) Competitiveness of pv inverter as a reactive power compensator considering inverter lifetime reduction. Energy Proc 150:74–82. ISSN: 18766102. https://doi.org/10.1016/j.egypro.2018.09.005 9. Ziadi Z, Taira S, Oshiro M, Funabashi T (2014) Optimal power scheduling for smart grids considering controllable loads and high penetration of photovoltaic generation. IEEE Trans Smart Grid 5(5):2350–2359. ISSN: 19493053. https://doi.org/10.1109/TSG.2014.2323969 10. Ziadi Z, Taira S, Oshiro M, Funabashi T (2014) Optimal power scheduling for smart grids considering controllable loads and high penetration of photovoltaic generation. IEEE Trans Smart Grid 5(5):2350–2359. ISSN: 19493053. https://doi.org/10.1109/TSG.2014.2323969 11. Zare M, Niknam T, Azizipanah-Abarghooee R, Amiri B (2014) Multi-objective probabilistic reactive power and voltage control with wind site correlations. Energy 66:810–822. ISSN: 03605442. https://doi.org/10.1016/j.energy2014.01.034 12. Rahbar K, Xu J, Zhang R (2015) Real-time energy storage management for renewable integration in microgrid: an off-line optimization approach. IEEE Trans Smart Grid 6(1):124–134. ISSN: 1949-3053. https://doi.org/10.1109/TSG.2014.2359004 13. Wang R, Wang P, Xiao G (2015) A robust optimization approach for energy generation scheduling in microgrids. Energy Convers Manag 106:597–607. ISSN: 01968904. https://doi.org/10. 1016/j.enconman.2015.09.066 14. Elsied M, Oukaour A, Gualous H, Hassan R Energy management and optimization in microgrid system based on green energy. Energy 84:139–151 (2015). ISSN: 03605442. https://doi.org/ 10.1016/j.energy.2015.02.108 15. Yang HT, Liao JT (2015) MF-apso-based multiobjective optimization for pv system reactive power regulation. IEEE Trans Sustain Energy 6(4):1346–1355. ISSN: 19493029. https://doi. org/10.1109/TSTE.20152433957 16. Yang Y, Wu W (2018) A distributionally robust optimization model for realtime power dispatch in distribution networks. IEEE Trans Smart Grid. https://doi.org/10.1109/TSG.2018.2834564

142

5 Reactive Power Dispatch for Large Number of PV Installations

17. Ahn C, Peng H (2013) Decentralized voltage control to minimize distribution power loss of microgrids. IEEE Trans Smart Grid 4(3):1297–1304. ISSN: 1949-3053. https://doi.org/10. 1109/TSG.2013.2248174 18. Zhang W, Liu W, Wang X, Liu L, Ferrese F (2014) Distributed multiple agent system based online optimal reactive power control for smart grids. IEEE Trans Smart Grid 5(5):2421–2431. ISSN: 19493053. https://doi.org/10.1109/TSG.2014.2327478 19. Dall’Anese E, Dhople SV, Giannakis GB (2016) Photovoltaic inverter controllers seeking AC optimal power flow solutions. IEEE Trans Power Syst 31(4):2809–2823. ISSN: 08858950. https://doi.org/10.1109/TPWRS.2015.2454856, arXiv: 1501.0188 20. Bolognani S, Carli R, Cavraro G, Zampieri S (2015) Distributed reactive power feedback control for voltage regulation and loss minimization. IEEE Trans Autom Control 60(4):966– 981. ISSN: 00189286. https://doi.org/10.1109/TAC.2014.2363931, arXiv: 1303.7173 21. Arnold DB, Negrete-Pincetic M, Sankur MD, Auslander DM, Callaway DS (2016) Model-free optimal control of var resources in distribution systems: an extremum seeking approach. IEEE Trans Power Syst 31(5):3583–3593. ISSN: 08858950. https://doi.org/10.1109/TPWRS.2015. 2502554 22. Zhang W, Gandhi O, Quan H, Rodríguez-Gallegos CD, Srinivasan D (2018) A multi-agent based integrated volt-var optimization engine for fast vehicle-togrid reactive power dispatch and electric vehicle coordination. Appl Energy 229:96–110. ISSN: 03062619. https://doi.org/ 10.1016/j.apenergy.2018.07.092 23. Erseghe T (2014) Distributed optimal power flow using ADMM. IEEE Trans Power Syst 29(5):2370–2380. ISSN: 08858950. https://doi.org/10.1109/TPWRS.2014.2306495 24. Zheng W, Wu W, Zhang B, Sun H, Liu Y (2016) A fully distributed reactive power optimization and control method for active distribution networks. IEEE Trans Smart Grid 7(2):1021–1033. ISSN: 19493053. https://doi.org/10.1109/TSG.2015.2396493 25. Šulc P, Backhaus S, Chertkov M (2014) Optimal distributed control of reactive power via the alternating direction method of multipliers. IEEE Trans Energy Convers 29(4):968–977. ISSN: 08858969. https://doi.org/10.1109/TEC.2014.2363196, arXiv: 1310.5748 26. Baker K, Bernstein A, Dall’Anese E, Zhao C (2018) Network-cognizant voltage droop control for distribution grids. IEEE Trans Power Syst 33(2):2098–2108. ISSN: 0885-8950. https://doi. org/10.1109/TPWRS2017.2735379, arXiv: 1702.02969 27. Weckx S, Driesen J (2016) Optimal local reactive power control by pv inverters. IEEE Trans Sustain Energy 7(4):1624–1633. ISSN: 19493029. https://doi.org/10.1109/TSTE.2016. 2572162 28. Zhu H, Liu HJ (2016) Fast local voltage control under limited reactive power: optimality and stability analysis. IEEE Trans Power Syst 31(5):3794–3803 29. Liu HJ, Shi W, Zhu H (2018) Hybrid voltage control in distribution networks under limited communication rates. IEEE Trans Smart Grid 1. ISSN: 1949-3053. https://doi.org/10.1109/ TSG.2018.2797692 30. Sondermeijer O, Dobbe R, Arnold D, Tomlin C (2019) Regression-based inverter control for decentralized optimal power flow and voltage regulation. arXiv:1902.08594v1 31. Dobbe R, Sondermeijer O, Fridovich-keil D, Arnold D, Callaway D, Tomlin C (2018) Datadriven decentralized optimal power flow 1–10. arXiv:1806.06790v1 32. Bellizio F, Karagiannopoulos S, Aristidou P, Hug G (2018) Optimized local control for active distribution grids using machine learning techniques. In: IEEE power & energy society general meeting, Portland, Oregon. IEEE, USA. ISBN: 9781538677032 33. Gandhi O, Rodriguez-Gallegos CD, Reindl T, Srinivasan D (2018) Locally determined voltage droop control for distribution systems. In: 2018 IEEE innovative smart grid technologies Asia (ISGT Asia). IEEE, Singapore, pp 425–429. ISBN: 978-1-5386-4291-7. https://doi.org/ 10.1109/ISGT-Asia.20188467784 34. Gandhi O, Zhang W, Rodríguez-Gallegos CD, Verbois H, Sun H, Reindl T, Srinivasan D (2020) Local reactive power dispatch optimisation minimising global objectives. Appl Energy 262. ISSN: 03062619. https://doi.org/10.1016/j.apenergy.2020.114529

References

143

35. Savier JS, Das D (2007) Impact of network reconfiguration on loss allocation of radial distribution systems. IEEE Trans Power Deliv 22(4):2473–2480. ISSN: 08858977. https://doi.org/ 10.1109/TPWRD.2007.905370 36. Zhang D, Fu Z, Zhang L (2007) An improved TS algorithm for loss-minimum reconfiguration in large-scale distribution systems. Electr Power Syst Res 77(5–6):685–694. ISSN: 03787796. https://doi.org/10.1016/j.epsr.2006.06.005 37. Eminoglu U, Hocaoglu MH (2009) Distribution systems forward/backward sweep-based power flow algorithms: a review and comparison study. Electr Power Compon Syst 37(1):91–110. ISSN: 15325008. https://doi.org/10.1080/15325000802322046 38. Baran ME, Wu FF (1989) Network reconfiguration in distribution systems for loss reduction and load balancing. IEEE Trans Power Deliv 4(2):1401–1407. ISSN: 0885-8977. https://doi. org/10.1109/61.25627 39. Farivar M, Zhou X, Chen L (2015) Local voltage control in distribution systems: an incremental control algorithm. In: 2015 IEEE international conference on smart grid communications (SmartGridComm), pp 732–737 40. Li N, Qu G, Dahleh M (2014) Real-time decentralized voltage control in distribution networks. In: Fifty-second annual allerton conference, pp 582–588 41. Kersting WH (2001) Radial distribution test feeders. In: Proceedings of the IEEE power engineering society transmission and distribution conference, winter meeting, vol 2, pp 908–912. ISSN: 21608563. https://doi.org/10.1109/PESW.2001.916993 42. Energy Market Authority (2018) Singapore half-hourly system demand data. https://www.ema. gov.sg/statistic.aspx?sta 43. EMC (2018) Energy market price information. https://www.emcsg.com/marketdata/ priceinformation. Accessed 04 Jan 2018 44. Use of system charges (2016). https://www.mypower.com.sg/documents/ts-usc.pdf. Accessed 04 Jan 2018 45. Dong Z, Zhao L, Yang D, Reindl T (2017) Combine deep neural network and tree based machine learning models using stacked generalization to forecast hourly solar irradiance in tropical regions. In: 33rd European photovoltaic solar energy conference and exhibition, pp 2075–2078 46. Verbois H, Rusydi A, Thiery A (2018) Probabilistic forecasting of day-ahead solar irradiance using quantile gradient boosting. Sol Energy 173:313–327. ISSN: 0038-092X. https://doi.org/ 10.1016/j.solener.201807.071 47. Ineichen P, Perez R (2002) A new airmass independent formulation for the Linke turbidity coefficient. Sol Energy 73(3):151–157 48. Elkhatib ME, Shatshat RE, Salama MM (2012) Decentralized reactive power control for advanced distribution automation systems. IEEE Trans Smart Grid 3(3):1482–1490. ISSN: 19493053. https://doi.org/10.1109/TSG.2012.2197833

Chapter 6

Conclusions and Future Works

6.1 Summary of the Thesis This thesis presents a comprehensive analysis on reactive power compensation (RPC) using PV in distribution systems, which includes detailed quantification of the costs and benefits of the RPC, as well as centralised and local algorithms to optimise the reactive power dispatch of numerous PVs in the system. The reactive power cost from PV has been comprehensively formulated for the first time, and has been divided into two components, namely the inverter loss component [1] and the inverter lifetime reduction component [2]. Subsequently, the technical and economic competitiveness of RPC using PV has been compared with that of switched capacitor (SC) [3, 4]. It was found that while PV reactive power support is not yet competitive with SC given current Singapore market conditions, it will become increasingly so with higher PV penetration and inverter efficiency in the future. Combinations of SC and PV for RPC are also more beneficial for distribution systems, compared wtih SC or PV alone. Additionally, the payment range for PV RPC has also been identified. Both the centralised and local algorithms developed in this thesis have been shown to be capable of optimising reactive power dispatch from large number of PV effectively and efficiently. On the one hand, when there is a central controller with access to the grid topology and the network data (e.g. power flow, load, generation), the proposed analytical approach [5] is shown to perform better than metaheuristic and exact optimisation algorithms, and is up to 100 times faster. On the other hand, in the absence of communication infrastructure, individual PV system or other DER can use the proposed local optimisation [6] to minimise the power losses and operational cost of the distribution system. Since the local optimisation does not require any grid topology or network data, latency and privacy issues can be avoided. The formulation of the reactive power cost, which enables the evaluation of impact and competitiveness of PV RPC, together with the development of both

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 O. Gandhi, Reactive Power Support Using Photovoltaic Systems, Springer Theses, https://doi.org/10.1007/978-3-030-61251-1_6

145

146

6 Conclusions and Future Works

the centralised and local algorithms to implement the reactive power dispatch, are the main contributions of this thesis. The findings from this thesis may therefore be useful to the following parties: • Power system regulators, in determining the regulations governing reactive power dispatch in distribution systems, and in determining the remuneration mechanism and value for reactive power support; • Grid operators, in developing ways to implement reactive power dispatch efficiently and effectively; • PV system owners and operators, in accounting for the cost required for reactive power support (which may be an interconnection requirement [7]); • PV system modeling engineers (e.g. PVSyst), to incorporate aspects of reactive power generation and absorption to their PV system modeling; • Inverter manufacturers, in developing inverters with reactive power support modes and control capabilities outlined in this thesis, and to explore ways to extend the lifetime of the inverters/components when used for reactive power support.

6.2 Proposed Future Works 6.2.1 More Comprehensive Reactive Power Cost Formulation for Transmission System This thesis is mainly concerned with reactive power support in distribution systems, and has developed the reactive power cost from PV and other DERs in the context of distribution systems. In [8], the cost of reactive power from a “generating plant” can be divided into three large components: [1] Capital costs; [2] Energy losses; and [3] Life and operating factor implications. Although energy losses, as well as life and operating factor implications, have been analysed to a great extent in this thesis, the capital costs have not. In the case of PV, capital costs mean the cost of oversizing inverters such that the PV system can still provide reactive power support even when the PV is generating active power at its rated capacity. Throughout this thesis, the inverter size has been assumed to be the rated DC installed capacity of the PV system. Moreover, the opportunity cost of reducing the active power to provide more reactive power support has not been analysed comprehensively either. It has been shown in [1] that when the power system requires inverter-based DERs to curtail its active power generation to provide more reactive power support, there will be lost opportunity cost (LOC) incurred by the DERs’ owners. In such cases, the active X,oppcost , which can be expressed as: power output has to be reduced by Px,t

6.2 Proposed Future Works

147

 X,oppcost Px,t

=

X − Px,t

0



X Sx,max

2

2  X    Q  > QX − Q Xx,t x,t,lim  Xx,t   Q  ≤ QX x,t x,t,lim

(6.1)

  X,oppcost X Px,t = 0 when  Q Xx,t  ≤ Q Xx,t,lim because Px,t is not reduced. Both the inverter oversizing cost and LOC have not been investigated extensively because they are costs that can be avoided, whereas ctQPV,invloss and ctQPV,LR are unavoidable operational costs. As has been shown in Sect. 4.7, PV reactive power support is still not as competitive as SC even when only considering ctQPV,invloss and ctQPV,LR , therefore oversizing the inverter or curtailing active power for reactive power support will not be economical. Nonetheless, while this is the case for PV in distribution systems where the value of active power is always higher than that of reactive power, at the transmission level with very high penetration of renewable energy, the price of active power may become negative during certain hours [9]. At those times, the value of reactive power may then be more valuable and it would make sense to curtail PV active power generation to provide reactive power instead. Therefore, although the cost formulation developed in this thesis is sufficient for developing payment mechanism or formulating strategies for reactive power provision from PV in distribution systems, it might have to be expanded for the case of transmission systems, especially if PV were to participate in reactive power market.1 In that case, the cost of other sources of reactive power in the transmission system, such as traditional generators, Static Synchronous Compensator (STATCOM), and Static VAR Compensator (SVC), have to be analysed in detail too.

6.2.2 Impact of PV and Energy Storage in Distribution System on the Transmission System The works in this thesis has comprehensively analysed the impact of PV (and its RPC) at the distribution system (DS) on the DS itself. In Chap. 3, the impact of energy arbitrage by battery energy storage system (BESS) at the distribution level has also been investigated [5]. It has been assumed that the transmission system (TS), which the DS is connected to, is flexible enough to provide any value of active and reactive power that the DS requires. However, this may not actually be the case, especially when there are many DS with high penetration of PV and BESS, which are “invisible” to the TS operator (TSO) [15, 16]. Although in the literature researchers have analysed the effects of large scale renewable energy on TS [17–20], and distributed PV on DS—as is done in this the1

To the best of the author’s knowledge, such reactive power market has not been implemented in the real world. Additionally, although reactive power market models have been proposed by many works in the literature, e.g. [10–14], the topic is still an ongoing research and there has not been a consensus on the best model yet.

148

6 Conclusions and Future Works

sis and some of the cited works throughout this thesis—few works have analysed the impacts of distributed PV on electricity market operations at the transmission level [21–23]. “Invisible” PV are problematic because they are netted out with load, are non-dispatchable and they change the electricity demand patterns, complicating scheduling and planning. There is also a lack of information about their capacity such that the TSO could underestimate the amount of load if and when the distributed generation become unavailable [15]. Nevertheless, [21] did not analyse the impacts of distributed PV on its own but combine with utility-scale PV and wind. And [22] solved deterministic unit commitment and economic dispatch on hourly basis rather than using stochastic or robust methods at shorter time scale. Furthermore, both [21] and [22] do not consider transmission constraints. [23] advanced the analysis by developing Integrated Grid Modeling System (IGMS) to model integrated transmission-distribution system. The authors investigated the impacts of distributed PV invisibility and reactive power capability on the transmission voltage and power balance. This type of analysis should be conducted for other impacts of distributed PV on the overall grid operations. Additionally, none of the works has analysed the effects of invisible distributed storage combined with PV on the power system operations. Many have analysed the benefits of having utility-scale storage to smooth the variability of renewable energy [24–27] or to perform energy arbitrage [28, 29], but the impacts of the operations of these energy storage systems by the residential, commercial and industrial customers on the TS operations have not been explored. Therefore there will be a need to analyse the impact of distributed PV and BESS (both their active and reactive power dispatch) deployed in DS on TS, and determine if TS indeed can adapt to the variability of the DS. To do so, an integrated model of DS and TS needs to be built.

6.2.3 Enhancement of the Inverter Lifetime Reduction Component of Reactive Power Cost The inverter lifetime reduction (ILR) component of the reactive power cost developed in Chap. 4 [2], cQPV,LR , has assumed that as soon as the DC-link capacitors reach their end of life, the whole inverter will be replaced. When an inverter fails, usually there are two approaches to solve the problem: (a) by replacing the inverter, and (b) by repairing the inverter and replacing the worn components. Approach (a) has been assumed in Chap. 4 and provides a first step towards understanding the reactive power cost due to the associated inverter degradation. It will be very interesting to analyse approach (b)—to replace the capacitors and analyse the associated cost—to form a more comprehensive understanding of PV reactive power cost. In this case, “inverter degradation” is perhaps a more suitable term, rather than “inverter lifetime reduction”. This will require the modeling of other inverter components’ lifetime, such as the switches, as even though the capacitors

6.2 Proposed Future Works

149

have been replaced at, say year 8, the switches may fail within the next 3 years, and either the switches or the whole inverter then need to be replaced. On top of that, detailed costs of each component, both the present and future values, are necessary to make the model accurate. Such tasks are left for future work. Moreover, it has also been assumed that the DC-link capacitors are electrolytic capacitors as the alternative, film capacitors, have much higher cost and lower capacitance values for the same volume [30, 31]. However, it is possible that in the future, either metallised polypropylene film (MPPF) or other type of capacitors will be used as the DC-link capacitors in the inverter. The lifetime of MPPF capacitors is longer (above 20 years, which is the lifetime of the PV system assumed in this thesis), and therefore the MPPF capacitors will not become the limiting factor for the inverter lifetime. Therefore, the calculation of the inverter lifetime, and consequently the inverter degradation cost, will need to be re-evaluated when other technologies are used as the DC-link capacitors in the inverters. This will also be the case when other inverter topologies are considered. Lastly, it will be very beneficial for the scientific community and the PV industry to conduct empirical study of the capacitor and inverter degradation and to compare it with the theoretical models developed in this thesis.

6.2.4 Incorporation of More Variables into Local Optimisation of Global Objectives (LOGO) In Chap. 5, a local control capable of optimising global objectives, using only the local voltage magnitude and its own reactive power output, has been proposed [6]. This approach has also been shown to perform better than other distributed and local control algorithms. That being said, there may still be room for improvement in the performance should other data be available and incorporated into LOGO. Although only local information has been used in optimising the reactive power dispatch so far, it will be interesting to analyse how incorporating information from other neighbouring buses may affect the performance of LOGO. Future research can also explore continuous adjustment of values of the constants relating system losses and nodes’ voltage when consistent streams of data are available in reinforced or deep learning.

6.2.5 Analysis of PV Reactive Power Support at Shorter Time Scales and in Non-Steady State In this thesis, all the reactive power dispatches are optimised at half-hour time scale, with the exception of power dispatch optimised by LOGO in Chap. 5, which has a time step of 5 s and has minutely resolution of irradiance data. While it has been shown

150

6 Conclusions and Future Works

that reactive power support from PV can stabilise the undervoltage and overvoltage problems at these time scales, the potential of reactive power to mitigate the power quality problems (e.g. voltage flickers) that occur in the order of seconds [32], has not been analysed yet. Likewise, no analysis of the impact of PV reactive power support during transient state has been investigated in this thesis. These analyses will require sub-minute resolution of irradiance, PV power generation, and load data, which are not readily available, together with an in-depth research in power electronics and inverter topologies, which is outside the scope of this thesis.

6.2.6 Comprehensive Formulation of Reactive Power Cost for Other DERs This thesis has mainly focused on exploring the reactive power support from PV inverters. Although in Chap. 3, the inverter loss component for the reactive power cost for BESS and electric vehicle (EV) has also been formulated, it was seen in Chap. 4 that reactive power generation/absorption also affects the lifetime of the DERs’ inverters, and the inverter degradation component can be a significant part of the reactive power cost. The formulation of the inverter degradation component has only been done for PV since such formulation requires life cycle analyses of the particular technology. For PV, the life cycle cost has been established in the forms of levelised cost of electricity (LCOE) with a lifetime of 20 years. It will also be important to establish a comprehensive and accurate reactive power cost for other DERs that are also becoming more and more prominent in power systems, such as BESS and EV. Understanding their reactive power cost will enable power system operators to recruit their service in reactive power support and to remunerate the DERs’ owners or operators appropriately. Nevertheless, for BESS and EV, their life cycle cost is not as clear, and depends on their lifetime and usage, among other factors. For example, if we assume that the lifetime of a BESS inverter has been reduced from 15 years to 10 years, but the BESS itself only has a lifetime of 5 years, then the inverter degradation due to the reactive power support may not have any impact on the BESS life cycle cost, and the inverter degradation component of the reactive power cost may be taken to be zero. As such, more work still needs to be done to formulate the reactive power cost for other DERs. Recently, there has been more research in establishing the levelised cost of storage and life cycle cost for BESS [33, 34], which will facilitate the formulation of the BESS’ reactive power cost.

References

151

References 1. Gandhi O, Rodríguez-Gallegos CD, Zhang W, Srinivasan D, Reindl T (2018) Economic and technical analysis of reactive power provision from distributed energy resources in microgrids. Appl Energy, 210:827–841. ISSN 03062619. https://doi.org/10.1016/j.apenergy.2017.08.154 2. Gandhi O, Rodríguez-Gallegos CD, Gorla NBY, M. Bieri, T. Reindl, Srinivasan D (2019) Reactive power cost from PV inverters considering inverter lifetime assessment. IEEE Trans Sustainable Energy, 10(2):738–747. ISSN 1949-3029. https://doi.org/10.1109/TSTE.2018.2846544 3. Gandhi O, Rodríguez-Gallegos CD, Reindl T (2017) Competitiveness of reactive power compensation using PV inverter in distribution system. In: 2017 IEEE PES innovative smart grid technologies conference Europe (ISGT-Europe), Torino, Italy: IEEE, pp 1–6. ISBN 978-15386-1953-7. https://doi.org/10.1109/ISGTEurope.2017.8260238 4. Gandhi O, Rodríguez-Gallegos CD, Reindl T, Srinivasan D (2018) Competitiveness of PV inverter as a reactive power compensator considering inverter lifetime reduction. Energy Procedia 150:74–78. ISSN 18766102. https://doi.org/10.1016/j.egypro.2018.09.005 5. Gandhi O, Rodríguez-Gallegos CD, Reindl T (2018) Analytical approach to reactive power dispatch and energy arbitrage in distribution systems with ders. IEEE Trans Power Syst 33(6):6522–6533. ISSN 0885-8950. https://doi.org/10.1109/TPWRS.2018.2829527 6. Gandhi O, Zhang W, Rodríguez-Gallegos CD, Verbois H, Sun H, Reindl T, Srinivasan D (2020) Local reactive power dispatch optimisation minimising global objectives. Applied Energy 262:03062619. https://doi.org/10.1016/j.apenergy.2020.114529 7. IEEE Standards Coordinating Committee 21 (2018) IEEE Standard for Interconnection and Interoperability of Distributed Energy Resources with Associated Electric Power Systems Interfaces. New York: IEEE, 9781504446396 8. Turner PJ (1996) Provision of reactive power from generating plant. In: IEE colloquium on economic provision of reactive power for system voltage control (Digest No. 1996/190), London: IET, 1996, pp 1–5. https://doi.org/10.1049/ic:19961076 9. EPEX SPOT, Negative prices 10. Bhattacharya K, Zhong J (2001) Reactive power as an ancillary service. IEEE Trans Power Syst 16(2):294–300 11. Zhong J, Nobile E, Bose A, Bhattacharya K (2004) Localized reactive power markets using the concept of voltage control areas. IEEE Trans Power Syst 19(3):1555–1561 12. Zhong J (2005) A pricing mechanism for reactive power devices in competitive market, in 2006 IEEE Power India Conference, vol. 2005, pp 67–72. IEEE. ISBN 0-7803-9525-5. https://doi. org/10.1109/POWERI.2006.1632493 13. Rabiee A, Feshki Farahani H, Khalili M, Aghaei J, Muttaqi KM (2016) Integration of plug-in electric vehicles into microgrids as energy and reactive power providers in market environment. IEEE Trans Industr Informat 12(4):1312–1320. ISSN 15513203. https://doi.org/10.1109/TII. 2016.2569438 14. Reddy S, Abhyankar AR, Bijwe PR (2011) Reactive power price clearing using multiobjective optimization, Energy 36(5):3579–3589. ISSN 03605442. https://doi.org/10.1016/ j.energy.2011.03.070 15. Porter K, Fink S, Rogers J, Mudd C, Buckley M, Clark C, Hinkle S (2012) PJM renewable integration study: review of industry practice and experience in the integration of wind and solar generation, GE Energy, Tech. Rep. November 16. Gandhi O, Kumar DS, Rodríguez-Gallegos CD, Srinivasan D (2020) Review of power system impacts at high PV penetration part i: factors limiting PV penetration. Solar Energy 17. Bouffard F, Galiana FD (2008) Stochastic security for operations planning with significant wind power generation. IEEE Trans Power Syst 23(2):306–316. ISSN 08858950. https://doi. org/10.1109/TPWRS.2008.919318 18. Zheng QP, Wang J, Liu AL (2014) Stochastic optimization for unit commitment—a review, IEEE Trans Power Syst 30(4):1913–1924. ISSN 0885-8950. https://doi.org/10.1109/TPWRS. 2014.2355204

152

6 Conclusions and Future Works

19. Morales JM, Conejo AJ, Liu K, Member S (2012) Pricing electricity in pools with wind producers. IEEE Trans Power Syst 27(3):1366–1376 20. Morales JM, Conejo AJ, Pérez-Ruiz J (2009) Economic valuation of reserves in power systems with high penetration of wind power, IEEE Transactions on Power Systems 24(2):900–910. ISSN 08858950. https://doi.org/10.1109/TPWRS.2009.2016598 21. Mills A, Botterud A, Wu J, Zhou Z, Hodge BM Heaney M (2013) Integrating solar PV in utility system operations, Argonne National Laboratory, Tech. Rep. ANL/DIS-13/18 22. Alqahtani BM, Holt KM, Patiño-Echeverri D, Pratson L (2016) Residential solar PV systems in the Carolinas: opportunities and outcomes. Environ Sci Technol 50(4):2082–2091. ISSN 15205851. https://doi.org/10.1021/acs.est.5b04857 23. Palmintier B, Hale E, Hansen TM, Jones W, Biagioni D, Baker K, Wu H, Giraldez J, Sorensen H, Lunacek M, Merket N, Jorgenson J, Hodge BM, Final technical report : integrated distributiontransmission analysis for very high penetration solar pv, NREL, Tech. Rep. NREL/TP-5D0065550 24. Jiang R, Wang, J, Guan Y, Robust unit commitment with wind power and pumped storage hydro, IEEE Transactions on Power Systems, vol 27, no 2, pp 800–810, 2012, ISSN: 08858950. https:// doi.org/10.1109/TPWRS.2011.2169817 25. Harris C, Meyers JP, Webber ME, A unit commitment study of the application of energy storage toward the integration of renewable generation, Journal of Renewable and Sustainable Energy, vol 4, no 1, 2012, ISSN: 19417012. https://doi.org/10.1063/1.3683529 26. Virasjoki V, Rocha P, Siddiqui AS, Salo A (2016) Market impacts of energy storage in a transmission-constrained power system. IEEE Trans Power Syst 31(5): 4108–4117. ISSN 08858950. https://doi.org/10.1109/TPWRS.2015.2489462 27. Pozo D, Contreras J, Sauma EE (2014) Unit commitment with ideal and generic energy storage units. IEEE Trans Power Syst 29(6):2974–2984. ISSN 0885-8950. https://doi.org/10.1109/ TPWRS.2014.2313513 28. Awad ASA, Fuller JD, El-Fouly THM, Salama MMA (2014) Impact of energy storage systems on electricity market equilibrium. IEEE Trans Sustain Energy 5(3): 875–885. ISSN 19493029. https://doi.org/10.1109/TSTE.2014.2309661 29. Krishnan V, Das T (2015) Optimal allocation of energy storage in a co-optimized electricity market: benefits assessment and deriving indicators for economic storage ventures, Energy 81:175–188. ISSN 03605442. https://doi.org/10.1016/j.energy.2014.12.016 30. Messo T, Jokipii J, Puukko J, Suntio T (2014) Determining the value of DC-link capacitance to ensure stable operation of a three-phase photovoltaic inverter. IEEE Trans Power Electron 29(2):665–673. ISSN 08858993. https://doi.org/10.1109/TPEL.2013.2255068 31. Wang H, Yang Y, Blaabjerg F (2013) Reliability-oriented design and analysis of input capacitors in single-phase transformer-less photovoltaic inverters, Conference Proceedings - IEEE Applied Power Electronics Conference and Exposition—APEC, pp 2929–2933. https://doi. org/10.1109/APEC.2013.6520714 32. Kumar DS, Gandhi O, Rodríguez-Gallegos CD, Srinivasan D (2020) Review of power system impacts at high PV penetration part ii: potential solutions and the way forward. Solar Energy, Under Second Review 33. Luerssen C, Gandhi O, Reindl T, Sekhar D, Cheong (2019) Levelised cost of storage (LCOS) for solar-PV-powered cooling in the tropics. Applied Energy 242: 640–654. ISSN 0306-2619. https://doi.org/10.1016/j.apenergy.2019.03.133 34. Luerssen C, Gandhi O, Reindl T, Sekhar C, Cheong D, Life cycle cost analysis (LCCA) of PVpowered cooling systems with thermal energy and battery storage for off-grid applications, Applied Energy, 273(May):115–145. ISSN 0306-2619. https://doi.org/10.1016/j.apenergy. 2020.115145

Appendix

Convergence of the Proposed Analytical Approach

Although the convergence of the analytical approach developed in Chap. 3 for any n variables cannot be guaranteed, upon rigorous testings with more than 100 variables, we found that the algorithm always converges within eight iterations, suggesting that the algorithm is robust. To prove the convergence of an iterative algorithm, we first need to express the iteration in the following format: Qz+1 = AQz + b

(A.1)

where Qz is a column vector of the variables in the system (reactive power output of the DERs), and z indicates the iteration number. If the maximum absolute value of the eigenvalues (they are all in unit circle) of matrix A is less than 1 [1], then the algorithm is convergent. First, the convergence of the proposed analytical approach will be proven for the case of two variables. For simplicity, consider two PVs located at node k and l in an N -bus system, PV where k < l. Assuming P PV = 0, Q PV k,z and Q l,z can be expressed as: Bk,z + Dk,z − E + cQgrid /2 Ak + C k + F Bl,z + Dl,z − E + cQgrid /2 = Al + C l + F

Q PV k,z =

(A.2)

PV Q l,z

(A.3)

Only values of B and D are updated at each iteration, as they contain the term Q i . Values of E and F are the same for DERs with the same inverter parameters. Updating the values of B and D yield:

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 O. Gandhi, Reactive Power Support Using Photovoltaic Systems, Springer Theses, https://doi.org/10.1007/978-3-030-61251-1

153

154

Appendix: Convergence of the Proposed Analytical Approach

 k−1   PV  ri Q i − Q PV k,z − Q l,z = cPgrid η P,1  2   Vi i=1  PV  PV = Bk,z − Al Q k,z + Q l,z  k−1   l−1   PV PV  ri Q i − Q PV  ri Q i − Q l,z k,z − Q l,z = + cPgrid η P,1  2  2     Vi Vi i=1 i=k

Bk,z+1

Bl,z+1

PV = Bl,z − Ak Q PV k,z − Al Q l,z

Similarly,   PV Dk,z+1 = Dk,z − Cl Q PV k,z + Q l,z PV Dl,z+1 = Dl,z − Ck Q PV k,z − Cl Q l,z

Now we can obtain the expression for Q PV k,z+1 : Bk,z+1 + Dk,z+1 − E + cQgrid /2 Ak + C k + F  Ak + C k Ak + C k PV PV − Q l,z = Q k,z 1 − Ak + C k + F Ak + C k + F

Q PV k,z+1 =

(A.4)

Similarly, PV Q l,z+1

=

PV Q l,z

 1−

Al + C l Al + C l + F

− Q PV k,z

Ak + C k Al + C l + F

(A.5)

k +C k k +C k l +Cl Let Jkk = AkA+C , Jkl = AlA+C , Jll = AlA+C . From Sect. 3.3, looking at the k +F l +F l +F definitions of Ak and Ck , it follows that Ak < Al and Ck < Cl for k < l. Therefore, Jll > Jkk > Jkl . Any Jkk is positive for most, if not all, practical applications, as the values of ri and xi are mostly positive. Even if there is some negative xi in the system (e.g. due to a capacitor), Jkk will most likely still be positive because of the summation of all xi for Ck and ri for Ak . Expressing the variables in the format of Eq. (A.1),

 

Q PV k,z+1 PV Q l,z+1



 =

1 − Jkk −Jkk −Jkl 1 − Jll



Q PV k,z PV Q l,z

(A.6)

1 − Jkk −Jkk where A = and b = 0. The next step is to find λ, the eigenvalues −Jkl 1 − Jll of A, by having det (A − λI) = 0:

Appendix: Convergence of the Proposed Analytical Approach



1 − Jkk − λ −Jkk



=0

−Jkl 1 − Jll − λ

155

(A.7)

which yields λ2 + λ (Jkk + Jll − 2) + (1 − Jkk ) (1 − Jll ) − Jkk Jkl = 0

λ=

1 2 − Jkk − Jll ± (Jkk − Jll )2 + 4Jkk Jkl 2

(A.8)

As Jkk , Jll , and Jkl are all positive, λ is real. The larger λ is then λ=

1 2 − Jkk − Jll + (Jkk − Jll )2 + 4Jkk Jkl 2

(A.9)

which has to be less than one for the algorithm to converge. Therefore we need to prove that: 2 − Jkk − Jll + (Jkk − Jll )2 + 4Jkk Jkl < 2 (A.10) By rearranging the variables, we obtain: Jkk + Jll >

(Jkk − Jll )2 + 4Jkk Jkl

(Jkk + Jll )2 > (Jkk − Jll )2 + 4Jkk Jkl Jll > Jkl

(A.11)

which is always true for k < l 1 . Hence, the convergence of the algorithm for two variables when P X = 0 has been proven. To prove the convergence for n variables, the maximum absolute value of the eigenvalues of n × n matrix A has to be less than one. This means that the absolute value of the sum of each A’s row and column must be less than one. The components of A, akl can be expressed as: ⎧ 1 − Jkl ∀k = l ⎪ ⎪ ⎪ ⎨−J ∀k < l kl akl = ⎪ ∀l < k −Jkk ⎪ ⎪ ⎩ 0 otherwise < l means that the power flowing to node l has to go through node k. If node k and node l are on a different lateral and the power flowing through one node does not have to go through the other, then the “value” of k and l cannot be compared. In this case, Akl , the element of A connecting Q k,z+1 and Q l,z will be 0.

1k

156

Appendix: Convergence of the Proposed Analytical Approach

Even though for small number of DERs in the system, the absolute value of the sum of each row and column of A will be less than one, this is not the case when there is a large number of variables in the system (more than 20 in the test systems used in this thesis). Nevertheless, upon rigorous testing, even with more than 100 variables, the proposed approach always converges by the eighth iteration and provide results that are better than the benchmark approaches. Therefore, although it has not been proven that the proposed approach will always converge for any n variables and for when P X = 0, the author is confident that it will work in practical applications.

Reference 1. Strong DM (2005) Iterative methods for solving [i]ax[/i] = [i]b[/i] - convergence analysis of iterative methods. Convergence. https://www.maa.org/press/periodicals/loci/joma/ iterativemethods-for-solving-iaxi-ibi-convergence-analysis-of-iterativemethods