Control and Optimization Methods for Complex System Resilience 9819930529, 9789819930524

Citation preview

Studies in Systems, Decision and Control 478

Chao Zhai

Control and Optimization Methods for Complex System Resilience

Studies in Systems, Decision and Control Volume 478

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the worldwide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

Chao Zhai

Control and Optimization Methods for Complex System Resilience

Chao Zhai School of Automation China University of Geosciences Wuhan, Hubei, China

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-981-99-3052-4 ISBN 978-981-99-3053-1 (eBook) https://doi.org/10.1007/978-981-99-3053-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

To my beloved families.

Preface

In contrast to the simple system, complex systems have some unique and interesting properties such as nonlinearity, emergence, self-organization and adaptation, cascading evolution and so on, which can spontaneously arise due to the interaction and interdependence among different components. Such extraordinary phenomena have attracted many researchers from various fields in the past decades. Therein, as one of the most complicated artificial systems in history, power systems have been widely investigated because of its great impact on industrial production and human life. As a result, it is vital to ensure its security and stable operation in case of abnormal conditions or disruptive contingencies, which largely relies on novel methodologies to improve system resilience. Nevertheless, the uncertainties of intrinsic dynamics and external disturbances have made it a challenging task for researchers to develop an effective control and optimization approach to strengthen the resilience of complex systems, while maintaining a desired level of system functionality. To deal with the above issues, this monograph aims to propose a systematic framework and effective control and optimization methods for enhancing the resilience of complex dynamical systems at different evolution phases. On the whole, this monograph can be divided into four parts. The first part of this monograph presents the research background as well as the state-of-the-art literature review on resilience design of complex system and its applications to infrastructure systems. According to the evolution process of complex systems, it is feasible to enhance system resilience at three sequential phases: prior to contingencies or disruptive disturbances, degradation of system performance, and system recovery. At each phase, a large number of relevant control and optimization methods have been proposed to accomplish the identification of adverse risks or disturbances, online protection to absorb the damages and efficient system restoration. The second part centers on the theoretical formulation and technical analysis of control and optimization methods for risk identification and security monitoring of complex dynamical systems. First of all, an optimal control approach is proposed to identify the initial disruptive disturbances that trigger the worst-case cascading failure of power systems, by treating the disruptive disturbances as control inputs of complex vii

viii

Preface

cascade system. An iterative search algorithm is designed to look for optimal solutions to the worst-case cascading failures with theoretical guarantee on its asymptotic convergence. Then the problem of contingency identification in complex cascade system is formulated in the framework of hybrid differential-algebraic system, and it can be solved by the Jacobian-Free Newton–Krylov method to circumvent the Jacobian matrix and reduce computation costs. Moreover, we investigate how to construct the region of attraction of dynamical system centered around a stable equilibrium by using stable state trajectories of system dynamics and converse Lyapunov theorem in control theory. Finally, an online approach for security assessment is proposed to explore the region of attraction of differential-algebraic system. The third part provides reliable control and optimization methods for online protection and efficient recovery of complex systems. First of all, a model predictive approach is proposed to achieve the online protection against cascading failures, which allows to eliminate the propagation of cascading outages in its infancy. Then, a Markov chain model is developed to predict the cascading failure paths, and a robust optimization formulation is proposed to prevent cascading processes. Next, cooperative control and distributed optimization methods are proposed to relieve the stress of complex systems caused by disruptive disturbances. Finally, a reinforcement learning approach is developed to restore generator units after cascading blackouts. The last part summarizes main contributions of this monograph and discusses some potential research directions for future work. Wuhan, China August 2022

Chao Zhai

Acknowledgements

There have been many people who have walked alongside me during my academic journey. They have guided, supported and accompanied me. I would like to, hereby, thank each of them sincerely. First and foremost, I would like to express my deepest gratitude to my respectable supervisors Prof. Yiguang Hong at Institute of Systems Science, Chinese Academy of Science, Beijing, China, and Prof. Hai-Tao Zhang at School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan, China, for their unwavering support, encouragement and constructive guidance. They, upon whose shoulders I stand, explored and paved the path before me. Without their inspiring guidance, this monograph would simply not have been possible. Besides, I would like to thank my postdoctoral supervisors Prof. Gaoxi Xiao and Prof. TsoChien Pan at Nanyang Technological University, Singapore, and Prof. Mario Di Bernardo at University of Bristol, United Kingdom and University of Naples Federico II, Italy. They are always willing to take time to listen and usually provide insightful questions and suggestions, as well as clear instructions as feedback. Their unstinting support and encouragement have driven me to strive for excellence. Secondly, special thanks are given to many of my friends and colleagues including Prof. Hehong Zhang at Fuzhou University, China, Prof. Hung D. Nguyen at Nanyang Technological University, Singapore, Prof. Hans Heinimann at ETH Zurich, and Dr. Xiaozhou Yang at Singapore-ETH Center, ETH Zurich for sharing a wonderful and memorable life with me. In addition, my wholehearted thanks are given to Dr. Wenqi Du, Dr. Chaolie Ning, Dr. Beibei Li and Dr. Min Meng at Nanyang Technological University for their generous support and understanding given in many moments of crisis over the years in Singapore. I cannot list all the names here, but you guys hold a special place in my heart. Thirdly, some thanks should be given to my colleagues at School of Automation, China University of Geosciences (Wuhan) and Graduate students in my team including Mr. Zhaoxu Wang and Miss Huimiao Li. Finally, and most importantly, my most heartfelt and forever gratitude goes to my parents Mr. Shengli Zhai and Mrs. Xiaohui Zhou and other family members, who have always been a constant source of support and encouragement. Thanks ix

x

Acknowledgements

to my parents for putting me through the best education possible and giving me the strength to reach for the stars and chase my academic dream. I appreciate their sacrifices and unending support, and I would not have been able to get to this stage without them. This work is supported by the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) and the 111 Project, China under Grant B17040. Wuhan, China August 2022

Chao Zhai

Contents

1

Introduction to Complex System Resilience . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Risk Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 System Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 System Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Outline of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 5 6 8 9

2

Optimal Control Approach to Identifying Cascading Failures . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Cascading Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 DC Power Flow Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Optimization Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Simulation and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Cascading Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Proof of Lemma 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Proof of Lemma 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Proof of Lemma 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 15 15 18 20 22 25 26 27 29 29 30 30 30 31 32 34

3

Jacobian-Free Newton-Krylov Method for Risk Identification . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Cascading Failure Process . . . . . . . . . . . . . . . . . . . . . . . . .

37 37 39 39 xi

xii

Contents

3.2.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Optimization Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Numeric Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Cascade Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Parameter Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Simulation and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5 Applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 FACTS Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 HVDC Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Protective Relay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40 41 42 46 47 47 48 52 53 53 55 55 55 56 56

4

Security Monitoring Using Converse Lyapunov Function . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 ROA of General Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . 4.3 GP for Learning Unknown Dynamics . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Gaussian Process and RKHS Norm . . . . . . . . . . . . . . . . . . 4.3.2 GP-UCB Based Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Power System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 SMIB System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 IEEE 39 Bus System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 The Class  Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Upper Bound of Discretizing Error . . . . . . . . . . . . . . . . . . 4.7.3 Information Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.4 Computation of RKHS Norm . . . . . . . . . . . . . . . . . . . . . . . 4.7.5 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 59 60 62 63 64 66 67 68 69 70 71 72 73 73 73 76 77 78 79

5

Online Gaussian Process Learning for Security Assessment . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The ROA of DAE System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Windowed Online GP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 GP Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Windowed Online GP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Security Assessment Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 83 85 85 86 88 90

Contents

xiii

5.5.1 Validations with PMU Data . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 The Operator R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.3 Proof of Theorem 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90 93 94 94 94 95 95 96

6

Risk Identification of Cascading Process Under Protection . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 State Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Protective Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Cost Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Theoretical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Simulation and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Parameter Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Validation and Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99 99 100 101 103 104 105 108 108 108 110 111

7

Model Predictive Approach to Preventing Cascading Proces . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Protection Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Nonrecurring Protection Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Recurring Protection Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Parameter Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Validation and Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Effect of Tuning Parameters . . . . . . . . . . . . . . . . . . . . . . . . 7.5.4 Other Test Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.1 Definition of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.2 Proof of Equation 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.3 Proof of Proposition 7.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 113 115 119 122 127 128 128 131 133 136 136 137 137 138 138 139

8

Robust Optimization Approach to Uncertain Cascading Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Prediction of Cascading Failure Paths . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Markov Chain Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Dimensionality Reduction . . . . . . . . . . . . . . . . . . . . . . . . .

141 141 144 144 146

xiv

Contents

8.3 8.4 8.5

Robust Optimization Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Solver Using Dykstra’s Algorithm . . . . . . . . . . . . . . . . Simulation and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Parameter Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Validation and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

148 151 151 152 152 154 154

Cooperative Control Methods for Relieving System Stress . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Coordination Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Generation of Control Signals . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Construction of Jacobian Estimator . . . . . . . . . . . . . . . . . . 9.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157 157 158 161 161 162 166 167 168 168

10 Distributed Optimization Approach to System Protection . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Hybrid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Communication Topology . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Control Design and Theoretical Analysis . . . . . . . . . . . . . . . . . . . . 10.4.1 Control Law of TCPST . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Distributed Optimization Algorithm . . . . . . . . . . . . . . . . . 10.4.3 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Reduction of Branch Capacity . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Bus Overloads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.3 Effect of Tuning Parameters . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171 171 173 173 174 175 176 176 177 180 185 185 186 187 188 189

11 Reinforcement Learning Approach to System Recovery . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Restoration Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Static Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 Dynamic Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

191 191 193 195 197 198 199 201 201

9

Contents

12 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Summary of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

203 203 204 205

Acronyms

AC ADMM BAP DAE DC DSA FACTS GP GP-UCB HVDC JFNK KKT MPC NPS ODE OPF PDC PMU RKHS ROA RPS SIP SMIB SPS SVC TCPST TCSC UPFC WAMS WAPCS

Alternating current Alternating direction method of multipliers Best approximation problem Differential-algebraic equation Direct current Dynamic security assessment Flexible alternating current transmission system Gaussian process Gaussian process upper confidence bound High-voltage direct current Jacobian-free Newton–Krylov Karush–Kuhn–Tucker Model predictive control Nonrecurring protection scheme Ordinary differential equation Optimal power flow Phasor data concentrator Phasor measurement unit Reproducing kernel Hilbert space Region of attraction Recurring protection scheme Set intersection problem Single-machine infinite bus Special protection scheme Static var compensator Thyristor-controlled phase shifting transformer Thyristor-controlled series capacitor Unified power flow controller Wide-area monitoring system Wide-area protection and control system

xvii

Nomenclature

Pi j ci j Y pk y kp , i A Ybk uk θk Si Vi ei Pk E ik m n h α(z) V (x) Vˆ (x) V  (x) φ(x,  t) k x, x S c N t δ AN μi (x)

Power flow on the branch that connects Bus i to Bus j Power flow threshold of the branch that connects Bus i to Bus j Branch admittance vector at the k-th cascading step Admittance on the i-th branch at the k-th cascading step Branch-bus incidence matrix Nodal admittance matrix at the k-th cascading step Control input or disturbance at the k-th cascading step Vector of voltage phase angle on buses at the k-th cascading step The i-th subnetwork in power system Set of Bus IDs in the i-th subnetwork Unit vector with the i-th element being 1 and 0 otherwise Vector of power injected on buses at the k-th cascading step Diagonal matrix to select the ik -th branch at the k-th cascading step Number of buses in power system Number of transmission lines in power system Number of cascading steps in power system A class  function Lyapunov function Estimation of Lyapunov function An existing Lyapunov function State trajectory of nonlinear dynamical system Covariance function of GP or kernel function in the reproducing kernel Hilbert space Real region of attraction for nonlinear system Level set of Lyapunov function with the upper bound c Number of stable sampling points Time interval of numerical method for solving the differential equation Parameter to specify the confidence level Set of N stable sampling points Mean value of a Gaussian process at the i-th iteration

xix

xx

σi (x) ·  · k P pi j xk sk slk λl h Pbk Pek In pover phidden pcont PX (i) (d)

Nomenclature

Standard deviation of a Gaussian process at the i-th iteration 2-norm in Euclidean space   Induced RKHS norm with the kernel k x, x Transition matrix of Markov chain Transition probability from state i to state j Probability distribution vector over states at the k-th cascading step Random vector describing the power system state at the k-th cascading step Random variable describing the connection state of the l-th branch at the k-th cascading step Outage probability of the l-th branch Number of predicted cascading steps during power system cascades Net active power vector injected to buses at the k-th cascading step Power flow vector at the k-th cascading step Set of positive integers, i.e., {1, 2, …n} Probability vector of branch outage due to overloads Probability vector of branch outage due to hidden failures Probability vector of branch outage due to contingencies Projection of the point d onto the set X (i)

Chapter 1

Introduction to Complex System Resilience

Abstract Complex systems appear in a variety of fields such as global climate system, biological organisms, infrastructure systems (e.g., power grid, transportation or communication systems), social and economic organizations, and ecosystems. Undoubtedly, understanding and taming complex systems is of great significance to the advances of human society. Thus, it is crucial to develop a systematic control and optimization approach to enhancing the robustness, reliability, adaptation and stability of concerned systems. To this end, three correlated steps are proposed in order to achieve the resilience of complex systems as follows: identify the initial contingency that causes worst-case cascading failures, develop the protective schemes to prevent cascading outages, and design the recovery strategy and integrate it with disturbance identification and system protection.

1.1 Background Compared to the simple system, it is notoriously difficult to predict and tame the dynamical behaviors of complex systems because of complicated interdependence and coupling among different components [1]. This makes it a challenging task to protect the concerned system from disruptions or huge loss of functionality once it suffers from external perturbations or attacks. Such examples range from cascading blackouts in power grids [2], the outbreak of pandemic [3], population collapse of planktonic organisms [4], and social behavior change in socioeconomic systems [5] to budding yeast due to dilution [6]. For instance, the US-Canada blackout in 2003 has interrupted power to 55 million population, and the economic loss is up to 6 billion US dollars, and it is viewed as the worst-case blackout in terms of financial loss. In comparisons, India blackout in 2012 affected 620 million people, which accounts for almost 50% population of the nation. Other severe blackouts include Italy blackout in 2003 and Brazil blackout in 1999 (see Fig. 1.1). The latest blackout happened on May 21, 2022 in parts of Ontario and Quebec, Canada, where the summer storm has caused around one million citizens to lose electric power supply, and it cost about 70 million US dollars to repair the damaged properties [7]. Besides infrastructure systems, it is suggested that the loss of resilience in biological systems © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Zhai, Control and Optimization Methods for Complex System Resilience, Studies in Systems, Decision and Control 478, https://doi.org/10.1007/978-981-99-3053-1_1

1

2

1 Introduction to Complex System Resilience

Fig. 1.1 Major blackouts in the world. It includes US-Canada blackout in 2003, India blackout in 2012, Italy blackout in 2003, and Brazil blackout in 1999

or ecosystem may result in catastrophic consequences and usually pave the way for a new stable state, as shown in Fig. 1.2 [8]. For examples, the degradation of coral reefs around the world mainly results from their overgrowth by fleshy macro-algae, and the reef ecosystems may shift between alternative stable states, instead of responding to external conditions smoothly [9, 10]. In practice, such shifts can make the system vulnerable to the events that trigger a drastic change of stable states. In addition, it is indicated that woodlands and a grassy open landscape can be alternative stable states with well analysed examples as the woodland in Botswana, African [11]. Therefore, it is of great importance to develop control and optimization strategies for maintaining resilience for the sustainability of complex artificial or biological systems.

1.2 Literature Review In terms of cascading blackouts, it basically goes through 4 stages: precondition, steady-state progression, high-speed cascade, and restoration (see Fig. 1.3). During the process, the initiating event leads to the steady-state progression, while the triggering event gives rise to the high-speed cascade. In the lower panel, a real example of cascading blackouts is also presented by analyzing the number of tripped components with time. In the stage of precondition, it is desirable to identify potential initiating events that may cause the cascading failures in advance (i.e., risk identi-

1.2 Literature Review

3

Fig. 1.2 Resilience dependence of multi-stable ecosystems on external conditions [8]. The stability landscapes depict the evolution of equilibrium points and their basins of attraction due to the change of intrinsic or external conditions. Stable equilibrium points correspond to valleys, while the unstable ones correspond to the mountaintop. If the size of the attraction basin is small, a moderate perturbation may bring the system into the new basin of attraction

fication and security monitoring). In the stage of cascade events, protective actions should be taken to prevent the propagation of line outages as much as possible (i.e., system protection). In the stage of restoration, an efficient recovery strategy has to be developed so that the concerned system is able to go back to normal states as soon as possible (i.e., system recovery). In what follows, existing studies in complex system resilience are reviewed according to the above three stages.

1.2.1 Risk Identification As the first step to improve system resilience, risk identification and security assessment of complex dynamic systems plays a crucial role in taking precautions to avoid the occurrence of catastrophe. By treating the unknown disturbances or attacks as

4

1 Introduction to Complex System Resilience

Fig. 1.3 Stages of cascading process during blackouts

control inputs, it is expected to investigate the problem of risk identification in the framework of control theory. Based on the above strategy, [2] develops an optimal control approach to identifying the worst-case cascading failures of power systems and obtains the feasible solutions by solving a system of algebraic equations with a numerical search algorithm. Nevertheless, the above control approach can only identify the disruptive disturbances on one branch and fails to work for the case of multiple branches. To handle this issue, [12] redesigns the cost function of optimal control problem to take into account the automatic seeking of critical branches and risk identification on the corresponding branches. Actually, the main difficulties lie in the creation of mathematical model to describe the cascading process of line outages as well as the conversion of optimal control problem into a system of algebraic equations. The proposed optimal control approaches can be adopted to identify the risk of power systems with protection and the penetration of renewable energy [13, 14]. Besides optimal control approaches, other optimization and control methods are adopted to identify the risk as well. For instance, nonlinear programming is employed to formulate the problem of determining the disruptive disturbances, and two different optimization formulations are presented to analyze the vulnerability of power grids. Specifically, nonlinear programming enables to cope with the voltage disturbance, while nonlinear two-level optimization deals with the power adjustment. The complicated coupling relation among different components and dynamic evolution of power grids make it a challenging task to identify the malicious disturbances or attacks. By formulating the problem of contingency identification in the mathematical framework of hybrid differential-algebraic system, [15] adopts Jacobian-Free Newton-Krylov (JFNK) method for identifying the initial disruptive contingencies

1.2 Literature Review

5

that can result in the catastrophic cascading failures of power systems. When the complex system suffers from external disturbances, it may be driven away from its stable state. Nevertheless, as long as the system states are still contained in a region of attraction (ROA), the system will finally return to the operating equilibrium. Thus, the ROA can be used for the stability assessment of dynamical power systems under uncertainties. By leveraging the converse Lyapunov theorem in control theory, [16] proposes a Gaussian process (GP) approach to learn unknown Lyapunov functions for the construction of ROA, which allows to eliminate the need for an analytic Lyapunov function. In addition, [17] develops an online approach for security assessment of small-signal stability of power grids using windowed GP.

1.2.2 System Protection Distributed optimization methods have been effectively applied to power system protection, where each bus can be treated as an agent. The agents can communicate with their neighbors to form a multi-agent network. As a result, distributed optimization algorithm is developed to control each agent for improving the overall performance of complex systems (i.e. energy systems, infrastructure, economic systems, ecological systems, etc.) against disruptive disturbances or baleful attacks. For a specific engineering problem, the actual physical constraints are usually quite complicated and have to be approximated before formulating it in a proper mathematical framework. In terms of power systems, distributed optimization algorithms can be used to solve the problems of optimal power flow (OPF), voltage/frequency optimization and disturbance rejection. Due to the practical demand of network partition, alternating direction method of multipliers (ADMM) has been adopted to address the OPF problem. For instance, a fully distributed and robust algorithm is proposed to solve the OPF problem by dividing power network into subnetworks and communicating among neighbors [18]. By integrating ADMM with model predictive control (MPC), a distributed MPC approach is developed to coordinate the demand response and AC-OPF in the condition of renewable reduction [19]. A two-level distributed algorithm is designed to deal with the OPF problem with advantages of convergence, extendability, and robustness over existing variants of ADMM [20]. In addition, [21] compares the loss of power systems on the condition of asynchronous and synchronous communication and analyzes the effect of time delay on system performance. In practice, the voltage and frequency are key indexes to describe the power quality and greatly affects the safety and stability of power networks. With the aid of reactive power generated by renewable, a control method is proposed to handle the voltage regulation in the high penetration of sustainable energy [22]. In order to achieve the restoration of voltage and frequency, [23] proposes a control strategy based on distributed consensus algorithms by integrating improved droop control with secondary power optimization and demonstrates its effectiveness in the numerical simulations. In addition, [24] discusses how to solve the voltage control problem with the injection of distributed generation and seeks the optimal solutions with the

6

1 Introduction to Complex System Resilience

convex programming. By using the photovoltaic inverters, [25] comes up with a distributed mixed time-scale strategy for voltage regulation and solves the optimization problem with ADMM. The distributed frequency regulation also attracts plenty of researchers. To be specific, [26] investigates the effect of communication delays on the distributed regulation of power system frequency. For the islanded microgrids, [27] developes distributed frequency control strategy and hierarchic scheme for voltage regulation. When the communication among microgrids is event-dependent and subject to stochastic noise, [28] designs a control strategy to minimize the cost of generation. Distributed control and optimization approaches can readily cope with the external disturbances or attacks to effectively protect system from severe damages. In case of catastrophes or malicious attacks, a large-scale power network can be divided into multiple subnetworks, and each subnetwork works independently to survive and reduce the power loss. By optimizing the design and management of various generation devices, [29] creates a multi-objective model to deploy the distributed generators in the microgrids. On the basis of average consensus algorithm, [30] proposes a distributed resilient control strategy to protect DC microgrids against denial of service attacks with the consideration of communication faults. For random contingencies, [31] makes a distributed model based on robust optimization to improve the network topology and meet load demands. By realizing the distributed generation and demand response, [32] proposes a control scheme to maximize the system resilience under malicious attacks.

1.2.3 System Recovery Conventional optimization algorithms usually resort to mathematical modeling to optimize the recovery of faulted power system. Considering the uncertainty caused by the microgrid connected to the power system, [33] proposes a new black-start recovery method by coordinating the operation of distributed generators through droop control, which is modeled as a mixed integer linear programming problem. Considering the distribution network with distributed generation and various constraints, the problem is modeled as a mixed integer second-order cone programming problem to minimize the power losses of distribution network [34]. Sekhavatmanesh and Cherkaoui [35] constructs a self-healing framework for service recovery by an agent interactive mechanism to minimize switching operation and maximize restorable load, which formulates the problem as a mixed integer convex second-order cone problem. Chen and Wang [36] presents a new sequential service restoration framework for restoring unbalanced distribution systems by formulating the problem as a mixed integer linear programming problem. Arif et al. [37] proposes a sequential reconfiguration optimization framework using switching device classification, which decomposes the switching operation problem into a determined optimal network topology and two mixed integer linear programs that generate switching operation sequences question. Poudel and Dubey [38] proposes a feeder restoration method for critical load restoration of distributed generation. By optimizing the available

1.2 Literature Review

7

resources of the microgrid, the restoration problem is described as a mixed integer linear programming problem. In [39], a new operation method is proposed to form a microgrid powered by multiple distributed generation through the real-time operation of distribution network to restore critical loads after large-scale outages. In order to realize an optimal coordinated recovery scheme with distributed generators and remote control devices, [40] proposes a restoration framework, which includes remote control switches, manually operated switches and dispatchable distributed generators. In summary, the power system recovery problem is modeled as a mathematical programming model, and then solved based on the optimization theory. Intelligent optimization algorithms are generally based on biological intelligence or physical phenomena. They have specific search rules and are universally applicable to specific problems. During the restoration period, the uncertainty of lines and loads has brought great challenges to the restoration. Considering the uncertainty of load restoration, [41] introduces fuzzy entropy to quantify the uncertainty of the load, establishes a load restoration model, and uses the clear equivalence class of fuzzy chance constraints to convert the uncertainty model into the deterministic model. Genetic algorithm has the characteristics of overall search strategy, which does not rely on gradient information during optimization. Due to the complexity of power system topology, a large number of infeasible solutions will be generated during the genetic operation, which reduces the search efficiency. The local minimum tree problem in graph theory is used to optimize the shortest transmission path, and it is used to solve the shortest path for units and loads [42]. Simulated annealing algorithm allows to jump out of the local optimal solution and eventually converge to the global optimal solution. In order to reasonably plan the sub-areas after largescale outages, a black-start network reconfiguration strategy based on the improved simulated annealing algorithm is proposed in [43]. Particle swarm optimization has a fast convergence speed, which is suitable for solving complex nonlinear system optimization problems [44]. In order to improve the reliability of fault recovery, [45] proposes a recovery strategy based on the mutation particle swarm algorithm, which takes into account the characteristics of load time variability and demand time variability. Ant colony algorithm is a kind of parallel and distributed computing, which has strong global optimization ability. [46] applied ant colony algorithm in the optimization and reconfiguration of the distribution system, which minimizes the loss of the distribution network under the condition of satisfying the relevant operational constraints. The applicability of artificial intelligence methods for solving semi-structured and unstructured problems has been effectively applied to power system restoration, which has promoted the research of system restoration. As the typical representative of artificial intelligence, machine learning has rapidly become a research hotspot in recent years. As an important branch of machine learning, reinforcement learning (RL) has been widely used in science, engineering and so on. It is a learning control method based on Markov model. To resolve the slow calculation time in power system recovery, [47] proposes a fast solution algorithm by combining offline training and real-time fast optimization. In order to search for the optimal control strategy of critical loads, [48] presents a dynamic critical load recovery method for distri-

8

1 Introduction to Complex System Resilience

bution systems. To reduce network loss and improve voltage profile, [49] presents a dynamic distribution network reconfiguration problem based on the offline RL algorithm, which is described as a Markov decision process. In order to quicken the system network reconfiguration, [50] proposes a non-response recovery method based on model-free RL. Considering the uncertainty of load demand and renewable distributed generator units, a distributed multi-agent approach is proposed to solve service restoration problem, which is described as a multi-objective optimization problem to minimize the number of switch actions [51]. However, load requirements and availability of historical data should be available. By taking power flow distribution into account, [52] proposes a service restoration framework with distributed generation. In order to fully coordinate the functions of each component, [53] presents a multi-agent method for power system restoration, which consists of multiple bus agents and a negotiating agent. The interaction among bus agents determines the optimization objective. But this method is only suitable for local information to determine the target configuration and switching order. Considering that the impact of infrastructure interdependence is not considered in detail in previous literature, and only pre-determined weights are assigned to determine priorities, [54] establishes a hybrid multi-agent system for load power supply after partial outages.

1.3 Outline of the Book The remainder of this book is outlined as follows: Chap. 2 proposes an optimal control approach to identify the worst-case cascading failures by regarding the initial disturbances as control inputs of complex cascade systems. Then the Jacobian-free Newton-Krylov method is adopted to detect the disruptive contingencies of complex cascade systems with strong coupling of different components in Chap. 3. To deal with the problem of security assessment, the GP approach is developed to learn the state trajectories of dynamical systems to explore the region of attraction with the aid of converse Lyapunov function in Chaps. 4 and 5. In addition, an optimal control algorithm is proposed to identify the risks of protected systems in Chap. 6. Besides algorithms for risk identification and security assessment, a model predictive approach is designed to prevent cascading outages in Chap. 7. Afterwards, a robust optimization approach is proposed to deal with the uncertain cascading paths in Chap. 8. Moreover, Chap. 9 comes up with a cooperative controller for relieving the system stress using FACTS devices. In Chap. 10, a distributed compensating strategy is developed for system protection against disruptive contingencies. Chapter 11 focuses on the system recovery using the RL approach. Finally, Chap. 12 summarizes the whole book and discusses some potential research topics in the future.

References

9

References 1. Zhai, C., Zhang, H., Xiao, G.: Cooperative Coverage Control of Multi-agent Systems and Its Applications. Springer (2021). ISBN 978-981-16-7625-3 2. Zhai, C., Zhang, H., Xiao, G., Pan, T.: An optimal control approach to identify the worst case cascading failures in power systems. IEEE Trans. Control Netw. Syst. 7(2), 956–966 (2020) 3. Singer, B., Thompson, R., Bonsall, M.: The effect of the definition of ‘pandemic’ on quantitative assessments of infectious disease outbreak risk. Sci. Rep. 11(1) (2021) 4. Veraart, A., Faassen, E., Dakos, V., van Nes, E., Lürling, M., Scheffer, M.: Recovery rates reflect distance to a tipping point in a living system. Nature 481, 357–359 (2012) 5. Prakash, B., Chakrabarti, D., Valler, N., Faloutsos, M., Faloutsos, C.: Threshold conditions for arbitrary cascade models on arbitrary networks. Knowl. Inf. Syst. 33(3), 549–575 (2012) 6. Dai, L., Korolev, K., Gore, J.: Slower recovery in space before collapse of connected populations. Nature 496, 355 (2013) 7. Hydro-Quebec: A look at the outages caused by the May 21 derecho. www.newswire.ca, 19 June 2022 8. Scheffer, M., Carpenter, S., Foley, J., Folke, C., Walker, B.: Catastrophic shifts in ecosystems. Nature 413, 591–596 (2001) 9. Knowlton, N.: Thresholds and multiple stable states in coral reef community dynamics. Am. Zool. 32, 674–682 (1992) 10. McCook, L.: Nutrients and phase shifts on coral reefs: scientific issues and management consequences for the Great Barrier Reef. Coral Reefs 18, 357–367 (1999) 11. Walker, B.: Conservation Biology for the Twenty-first Century, pp. 121–130, Oxford Univ. Press, Oxford (1989) 12. Zhang, H., Zhai, C., Xiao, G., Pan, T.: Identifying critical risks of cascading failures in power systems. IET Gener. Transm. Distrib. 13(12), 2438–2445 (2019) 13. Zhai, C., Xiao, G., Zhang, H.: Identification and analysis of cascading failures in power grids with protective actions. In: Proceedings of ISGT Europe, Bucharest, Romania, 29 Sept.–1 Oct. 2019 14. Zhai, C., Xiao, G., Zhang, H., Pan, T.: Risk identification of power transmission system with renewable energy. In: Proceedings of the 31st Chinese Control and Decision Conference, Nanchang, China, 3–5 June 2019 15. Zhai, C., Xiao, G., Zhang, H., Wang, P., Pan, T.: Identifying disruptive contingencies for catastrophic cascading failures in power systems. Int. J. Electr. Power Energ. Syst. 123 (2020) 16. Zhai, C., Nguyen, H.: Estimating the region of attraction for power systems using Gaussian process and converse Lyapunov function. IEEE Trans. Control Syst. Technol. 30(3), 1328–1335 (2022) 17. Zhai, C., Nguyen, H., Zong, X.: Dynamic security assessment of small-signal stability for power grids using windowed online Gaussian process. IEEE Trans. Autom. Sci. Eng. (2022). https://doi.org/10.1109/TASE.2022.3173368 18. Erseghe, T.: Distributed optimal power flow using ADMM. IEEE Trans. Power Syst. 29(5), 2370–2380 (2014) 19. Shi, Y., Tuan, H., Savkin, A., Lin, C., Poor, H.: Distributed model predictive control for joint coordination of demand response and optimal power flow with renewables in smart grid. Appl. Energ. 290, 116701 (2021) 20. Sun, K., Sun, X.: A two-level ADMM algorithm for AC OPF with global convergence guarantees. IEEE Trans. Power Syst. 36(6), 5271–5281 (2021) 21. Xu, J., Sun, H., Dent, C.: ADMM-based distributed OPF problem meets stochastic communication delay. IEEE Trans. Smart Grid 10(5), 5046–5056 (2019) 22. Kryonidis, G., Malamaki, K., Gkavanoudis, S., Oureilidis, K., Demoulias, C.: Distributed reactive power control scheme for the voltage regulation of unbalanced LV grids. IEEE Trans. Sustain. Energ. 12(2), 1301–1310 (2021)

10

1 Introduction to Complex System Resilience

23. Li, D., Zhao, B., Wu, Z., Zhang, X., Zhang, L.: An improved droop control strategy for lowvoltage microgrids based on distributed secondary power optimization control. Energies 10(9), 1347 (2017) 24. Deshmukh, S., Natarajan, B., Pahwa, A.: Voltage/VAR control in distribution networks via reactive power injection through distributed generators. IEEE Trans. Smart Grid 3(3), 1226– 1234 (2012) 25. Li, P., Wu, Z., Zhang, C., Hu, M., Li, S., Wang, F.: Distributed hybrid-timescale voltage/VAR control in active distribution networks. Autom. Electr. Power Syst. 45(16), 160–168 (2021) 26. Ramachandran, T., Nazari, M., Grijalva, S., Egerstedt, M.: Overcoming communication delays in distributed frequency regulation. IEEE Trans. Power Syst. 31(4), 2965–2973 (2016) 27. Xu, Y., Guo, Q., Sun, H., Fei, Z.: Distributed discrete robust secondary cooperative control for islanded microgrids. IEEE Trans. Smart Grid 10(4), 3620–3629 (2019) 28. Lai, J., Lu, X., Yu, X., Monti, A.: Stochastic distributed secondary control for AC microgrids via event-triggered communication. IEEE Trans. Smart Grid 11(04), 2746–2759 (2020) 29. Shirazi, H., Ghiasi, M., Dehghani, M., Niknam, T., Ramezani, A.: Cost-emission control based physical-resilience oriented strategy for optimal allocation of distributed generation in smart microgrid. In: 7th International Conference on Control, Instrumentation and Automation, pp. 1–6 (2021) 30. Chen, X., Zhou, J., Shi, M., Chen, Y., Wen, J.: Distributed resilient control against denial of service attacks in DC microgrids with constant power load. Renew. Sustain. Energ. Rev. 153, 111792 (2022) 31. Babaei, S., Jiang, R., Zhao, C.: Distributionally robust distribution network configuration under random contingency. IEEE Trans. Power Syst. 35(5), 3332–3341 (2020) 32. Mosquera, D., Trujillo, E., Lopez-Lezama, J.: Vulnerability analysis to maximize the resilience of power systems considering demand response and distributed generation. Electronics 10(12), 1498 (2021) 33. Jiang, Y., Chen, S., Liu, C., et al.: Blackstart capability planning for power system restoration. Int. J. Electr. Power Energ. Syst. 86, 127–137 (2017) 34. Jabr, R.A., Singh, R., Pal, B.C.: Minimum loss network reconfiguration using mixed-integer convex programming. IEEE Trans. Power Syst. 27(2), 1106–1115 (2012) 35. Sekhavatmanesh, H., Cherkaoui, R.: Distribution network restoration in a multiagent framework using a convex OPF model. IEEE Trans. Smart Grid 10(3), 2618–2628 (2019) 36. Chen, B., Wang, J.H.: Sequential service restoration for unbalanced distribution systems and microgrids. IEEE Trans. Power Syst. 33(2), 1507–1520 (2018) 37. Arif, A., Cui, B., Wang, Z.: Switching device-cognizant sequential distribution system restoration. IEEE Trans. Power Syst. 37(1), 317–329 (2022) 38. Poudel, S., Dubey, A.: Critical load restoration using distributed energy resources for resilient power distribution system. IEEE Trans. Power Syst. 34(1), 52–63 (2019) 39. Chen, C., Wang, J.H., Qiu, F., et al.: Resilient distribution system by microgrids formation after natural disasters. IEEE Trans. Smart Grid 7(2), 958–966 (2016) 40. Chen, B., Chen, C., et al.: Toward a MILP modeling framework for distribution system restoration. IEEE Trans. Power Syst. 34(3), 1749–1760 (2019) 41. Li, Z.K., Wei, Y.J., et al.: Research on market mechanism of black start restoration auxilliary service in distribution network. In: Proceedings of the CSEE, pp. 1–16 (2022) 42. Xie, Y.Y., Song, K.L., et al.: Orthogonal genetic algorithm based power system restoration path optimization. Int. Trans. Electr. Energ. Syst. 28(12), 1–17 (2018) 43. Abbaszadeh, A., Doustmohammadi, A.: Optimal islands determination in power system restoration applying multi-objective populated simulated annealing. Int. Trans. Electr. Energ. Syst. 29(3), 1–25 (2019) 44. Yang, M.S., Li, J.Q., et al.: Reconfiguration strategy for DC distribution network fault recovery based on hybrid particle swarm optimization. Energies 14(21), 1–15 (2021) 45. Zhu, H.N., Yang, N., Guo, P.H., et al.: Optimization of generating unit restoration considering restoration path operation success rate. In: 2018 China International Conference on Electricity Distribution (CICED), pp. 1231–1235 (2018)

References

11

46. Carpaneto, E., Chicco, G.: Distribution system minimum loss reconfiguration in the hyper-cube ant colony optimization framework. Electr. Power Syst. Res. 78(12), 2037–2045 (2008) 47. Yang, Z., Marti, J.R.: Real-time resilience optimization combining an AI agent with online hard optimization. IEEE Trans. Power Syst. 37(1), 508–517 (2022) 48. Huang, Y.X., Li, G.F., Li, L.Y.: Deep reinforcement learning method for dynamic load restoration of resilient distribution systems. Autom. Electr. Power Syst. 46(8), 1–15 (2022) 49. Gao, Y., Shi, J., Wang, W., et al.: Dynamic distribution network reconfiguration using reinforcement learning. In: 2019 IEEE International Conference on Communications, Control, and Computing Technologies for Smart Grids (2019) 50. Wang, B.B., Zhu, H., Xu, H.H., et al.: Distribution network reconfiguration based on noisy net deep q-learning network. IEEE Access 9, 90358–90365 (2021) 51. Khederzadeh, M., Zandi, S.: Enhancement of distribution system restoration capability in single multiple faults by using microgrids as a resiliency resource. IEEE Syst. J. 13(2), 1796–1803 (2019) 52. Li, W.G., Chen, C., et al.: A full decentralized multi-agent service restoration for distribution network with DGs. IEEE Trans. Smart Grid 11(2), 1100–1111 (2020) 53. Nagata, T., Sasaki, H.: A multi-agent approach to power system restoration. IEEE Trans. Power Syst. 17(2), 457–461 (2002) 54. Ye, D.Y., Zhang, M.J., Sutanto, D.: A hybrid multi-agent framework with q-learning for power grid systems restoration. IEEE Trans. Power Syst. 26(4), 2434–2441 (2011)

Chapter 2

Optimal Control Approach to Identifying Cascading Failures

Abstract Cascading failures in power systems normally occur as a result of initial disturbances or faults on electrical elements, closely followed by situational awareness errors of human operators. It remains a great challenge to systematically trace the source of cascading failures in power systems. In this chapter, we develop a mathematical model to describe the cascading outages of transmission lines in power networks. In particular, the direct current (DC) power flow equation is employed to calculate the power flow on the branches. By regarding the disturbances on branches as the control inputs, the problem of identifying the initial disruptive disturbances is formulated with optimal control theory, which provides a systematic approach to exploring the most disruptive disturbances that give rise to changes of branch admittance in addition to direct branch outages. Moreover, an iterative search algorithm is proposed to look for the optimal solution leading to the worst-case cascading failures. Theoretical analysis guarantees the asymptotic convergence of the iterative search algorithm. Finally, numerical simulations are carried out on the IEEE test systems to validate the proposed approach.

2.1 Introduction The stability and secure operation of power grids have a great impact on other interdependent critical infrastructure systems such as energy systems, transportation systems, finance systems and communication systems. Nevertheless, contingencies on vulnerable components of power systems and situational awareness errors of human operators1 could trigger a chain reaction of circuit breakers, leading to a large blackout of power networks. For instance, the North America cascading blackout on August 14, 2003 resulted in a power outage affecting 50 million people [2]. The misoperation of a German operator in November 2006 triggered a chain reaction of power grids that caused 15 million Europeans to lose access to power [3]. Recently, a relay fault near the Taj Mahal in India gave rise to a severe cascading blackout on July 31, 2012 1

The errors of human operators due to the inadequate situational awareness, which includes three components: perception of the environment elements, comprehension of the situation, and projection of future status [1].

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Zhai, Control and Optimization Methods for Complex System Resilience, Studies in Systems, Decision and Control 478, https://doi.org/10.1007/978-981-99-3053-1_2

13

14

2 Optimal Control Approach to Identifying Cascading Failures

affecting 600 million people. It is vital to identify the worst possible attacks or initial disturbances on the critical electrical elements in advance and develop effective protection strategies to alleviate cascading blackouts in power systems. A cascading blackout in power systems is defined as a sequence of component outages that include at least one triggering component outage caused by initial contingencies [4, 5] and subsequent tripping component outages due to the overloading of transmission lines and situational awareness errors of human operators [6–8]. Besides line failures, multiple factors contribute to the occurrence of cascading blackouts, such as weather conditions, human factors (e.g., misoperations of operators), equipment aging, and so on. Note that a cascading failure does not necessarily lead to a cascading blackout or load shedding. The existing cascading models basically fall into 3 categories [6]. The first type of model only reveals the topological properties but ignores the physics of power grids. As a result, these models are unable to accurately describe the cascading evolution of power networks in practice [9–11]. The second type of model focuses on the quasi-steady state of power systems and computes the power flow on branches by solving the DC or alternating current (AC) power flow Eqs. [5, 12, 13]. The third one resorts to the dynamic modeling of cascading failures in order to allow for the effects of component dynamics [14–16]. For example, a dynamic model of cascading failure was presented to deal with the interdependencies of different mechanisms with the transient dynamics of generators and protective relays [15]. It is suggested the transient dynamical behaviors in power systems play a crucial role in the emergence of cascading failures [16]. Actually, many factors (e.g., temperature, frequency, short circuit, poor contactor, etc) may contribute to the changes in branch impedance other than line outages. Some disturbances or faults on the transmission lines of power grids can be described by impedance or admittance changes [17, 18]. As a special case, the outage of a transmission line can lead to infinite impedance or zero admittance between two relevant buses. Linear or nonlinear programming is normally employed to formulate the problem of determining disruptive disturbances. [17] presents two different optimization formulations to analyze the vulnerability of power grids. Specifically, nonlinear programming is adopted to address the voltage disturbance, and nonlinear bilevel optimization is employed to deal with the power adjustment. Nevertheless, there is still a lack of mathematical framework and theoretical results for investigating the effects of initial disturbances on the ongoing dynamics of cascading failures. Previous optimization formulations are not sufficient to describe the outage sequence of transmission lines in practice, because the final configuration of a power network strongly depends on the evolution process of transmission lines (i.e., the line outage sequence) in addition to initial conditions. In this chapter, we will develop a cascading model of power networks to describe the outage sequence of transmission lines. Moreover, the problem of identifying initial disturbances causing the worst disruption of cascades is formulated and solved in the framework of optimal control theory by treating the disruptive disturbances as control inputs in the optimal control system. The proposed approach provides new insight into tracing disruptive disturbances on vulnerable components of power grids. It helps to find the theoretical results of the most disruptive disturbances on

2.2 Problem Formulation

15

high-voltage transmission lines that give rise to changes in branch admittance in addition to direct branch outages. Since it is difficult to directly identify the initial disturbances that cause the worst-case cascades of real power systems, this work provides a theoretical approach to search for the worst-case cascades in a simplified model. Moreover, it is expected to determine the most disruptive disturbances at the early stage of cascades in practice. The remainder of this chapter is organized as follows. Section 2.2 presents the cascading model of power systems and the optimal control approach. Section 2.3 provides theoretical results for the problem of identifying disruptive disturbances, followed by simulations and validation on the IEEE test systems in Sect. 2.4. Finally, we conclude the chapter and discuss future work in Sect. 2.5.

2.2 Problem Formulation The power system is basically composed of power stations, transformers, power transmission networks, distribution stations, and consumers (see Fig. 2.1). In this work, we are interested in identifying disruptive disturbances (e.g., lightning, storm, temperature fluctuation [19], etc) on transmission lines that trigger chain reactions and cause cascading blackouts in power grids. The disturbances give rise to the admittance changes of transmission lines, which results in the redistribution of power flow in power grids. The focus of this work is on the identification of initial disruptive disturbances that cause the worst-case cascades. In practice, most initial disturbances can be modeled as the changes in branch admittance, as can be observed in major blackouts in history [2, 20, 21]. Theoretically, it is reasonable to model the branch disturbance as the change of branch admittance [17–19, 22]. The dynamics of the cascade process include the sequence of line outages and the redistribution of power flow on branches. The overloading of transmission lines causes certain circuit breakers to sever the corresponding branches and readjust the power network topology. The above process does not stop until the power grid reaches a new steady state and transmission lines are not severed anymore. In this section, we propose a cascading model to describe the cascading process of transmission lines, where the DC power flow equation is solved to obtain the power flow on each branch. More significantly, the mathematical formulation based on optimal control is presented by treating the disruptive disturbances of power grids as the control inputs in the optimal control system.

2.2.1 Cascading Model The cascading model describes the evolution of branch admittance as a result of overloading on transmission lines and subsequent branch outages. To characterize the connection state of the transmission line, we introduce the state function of the transmission line that connects Bus i and Bus j as follows:

16

2 Optimal Control Approach to Identifying Cascading Failures

Fig. 2.1 Schematic diagram of power systems suffering from the lightning on transmission lines

g(Pi j , ci j ) =

⎧ ⎪ 0, ⎪ ⎨

|Pi j | ≥

 

ci2j +

1, |Pi j | ≤ ⎪ ⎪ ⎩ 1−sin σ (Pi2j −ci2j ) , otherwise. 2

ci2j

−

π 2σ

;

π 2σ

;

(2.1)

where i, j ∈ Im = {1, 2, . . . , m}, i = j and m is the total number of buses in the power system. σ is a tunable positive parameter. Pi j refers to the steady-state power flow on the transmission line that links Bus i and Bus j, and ci j denotes its power flow threshold. The state function g(Pi j , ci j ) is differentiable with respect to Pi j , and more closely resembles a step function as σ increases (see Fig. 2.2). The transmission line is in good condition when g(Pi j , ci j ) = 1, while g(Pi j , ci j ) = 0 implies that the transmission line has been severed by the circuit breaker. Essentially, σ quantifies the approximation level of g(Pi j , ci j ) to a step function. By properly tuning the parameter σ , the function g(Pi j , ci j ) is able to reflect the system characteristic of branch outage while guaranteeing the differentiability with respect to Pi j , which is indispensable to derive the necessary condition for the optimality using optimal control theory. Remark 2.1 The value of g(Pi j , ci j ) is fractional when Pi2j is in the interval (ci2j − π/2σ , ci2j + π/2σ ). Clearly, the interval length is π/σ , and it is sufficiently small when σ is large enough. As a result, we can increase σ to eliminate the fractional state when it occurs in numerical simulations. Alternatively, the post-processing method is adopted to replace the fractional state with values 0 and 1 respectively, and then opt for the one with the smaller cost function value.

2.2 Problem Formulation

17

Fig. 2.2 The state function g(Pi j , ci j ) of the transmission line connecting Bus i to Bus j with the power threshold ci j = 5

1 σ=0.1 σ=0.3 σ=0.5

0.9 0.8 0.7

g

0.6 0.5 0.4 0.3 0.2 0.1 0

3

3.5

4

4.5

5

5.5

6

6.5

7

Pij

The cascading model of power networks at the k-th cascading step can be presented as: Y pk+1 = G(Pikj , ci j ) · Y pk + E ik u k , k = 0, 1, 2, . . . , h − 1

(2.2)

where Y pk = (y kp,1 , y kp,2 , . . . , y kp,n )T is the admittance vector for the n transmission lines or branches at the k-th step, and u k = (u k,1 , u k,2 , . . . , u k,n )T denotes the control inputs or disturbances on transmission lines. h is the total number of cascading steps in power networks. G(Pikj , ci j ) and E ik are the diagonal matrices defined as G(Pikj , ci j ) = diag(gk1 , gk2 , . . . , gkn ) with gks = g(Piks js , cis js ), s ∈ In = {1, 2, . . . , n} and E ik = diag(0, . . . , 0, 1, 0, . . . , 0) ∈ R n×n ,  

ik

respectively. Here E ik is a predetermined matrix. By assigning 1 to the i k -th diagonal element in E ik , we can add the control input onto the i k -th branch and thus change the admittance of the selected branch. The cascading model (2.2) is formulated based on the steady-state power flow on branches, and it mainly allows for the branch outage caused by persistent and steady branch overloads. Thus, the branch admittance becomes zero and remains unchanged once the branch is severed. In fact, Eq. (2.2) can k k T be rewritten in the element-wise form (i.e., y k+1 p,s = g(Pi s js , ci s js ) · y p,s + es E i k u k , s ∈ In ).

18

2 Optimal Control Approach to Identifying Cascading Failures

2.2.2 DC Power Flow Equation In this work, we care about the outage sequence of transmission lines in power systems and thus compute the DC power flow to deal with the overloading problem [23, 24]. Specifically, the DC power flow equation is given by Pi =

m 

Bi j θi j =

j=1

m 

Bi j (θi − θ j )

(2.3)

j=1

where Pi and θi refer to the injection power and voltage phase angle of Bus i, respectively. Bi j represents the mutual susceptance between Bus i and Bus j, where i, j ∈ Im . Equation (2.3) can be rewritten in matrix form [25] P = Bθ, where P = (P1 , P2 , . . . , Pm )T , θ = (θ1 , θ2 , . . . , θm )T and ⎛ m i=2

⎞ . B1m ⎟ B2m i=1,i=2 B2i . ⎟ ⎠ . . . m−1 −Bm2 . i=1 Bmi

B1i  −B12 m

⎜ −B21 B=⎜ ⎝ . −Bm1

Actually, B is the nodal admittance matrix of power networks while employing the DC power flow. The nodal admittance matrix Ybk at the k-th cascading step can be obtained as Ybk = A T diag(Y pk )A, where A denotes the branch-bus incidence matrix [26]. Therefore, the matrix B at the k-th step of cascading process can be calculated as B k = Ybk = A T diag(Y pk )A, where Y pk = (y kp,1 , y kp,2 , . . . , y kp,n )T and y kp,i = −

1 , i ∈ In I m(z kp,i )

(2.4)

z kp,i denotes the impedance of the i-th branch at the k-th step. Then the DC power flow equation at the k-th step is given by P k = B k θ k = Ybk θ k ,

(2.5)

where P k = (P1k , P2k , . . . , Pmk )T and θ k = (θ1k , θ2k , . . . , θmk )T . During the cascading process, the power network may be divided into several subnetworks (i.e., islands), which can be identified by analyzing the nodal admittance matrix Ybk . Suppose

2.2 Problem Formulation

19

Ybk is composed of q isolated components or subnetworks denoted by Si , i ∈ Iq = {1, 2, . . . , q} and each subnetwork Si includes ki buses. Let Vi = {v1 , v2 , . . . , vki } denote the of bus IDs in the subnetwork Si , where v1 , v2 ,…, vki denote the bus set q IDs and i=1 ki = m. Notice that Bus v1 in Subnetwork Si is designated as the reference bus, which is normally a generator bus connected to the largest generating station in practice. Thus, the power variation of the reference bus accounts for a small percentage of its generating capacity. When there are no generator buses in the subnetwork, the power flow on each branch of this subnetwork is zero. In addition, the injected power on the reference bus can be adjusted to balance the power supply and consumption in the subnetwork. Moreover, the nodal admittance matrix of the i-th subnetwork can be computed as k = EiT Ybk Ei , i ∈ Iq , Yb,i

where Ei = (ei1 , ei2 , . . . , eiki ). For simplicity, we introduce two operators ∗ and −1∗ to facilitate the analytical expression and theoretical analysis of solving the DC power flow equation. Definition 2.1 Given the nodal admittance matrix Ybk , the operators ∗ and −1∗ are defined by 

∗ Ybk

=

q 

 Ei

0 0ki −1

i=1

0kTi −1 Iki −1



 k Yb,i

0 0ki −1

0kTi −1 Iki −1

 EiT

and 

−1∗ Ybk

=

q  i=1

respectively, where

 Ei

0kTi −1 Iki −1



 k −1   0ki −1 Iki −1 EiT , Yb,i

 k  k = 0ki −1 Iki −1 Yb,i Yb,i



0kTi −1 Iki −1

 .

Iki −1 is the (ki − 1) × (ki − 1) identity matrix and 0ki −1 denotes the (ki − 1) dimensional column vector with zero elements. In this work, the DC power flow equation is solved with the operator −1∗ directly without using any power system simulators. Remark 2.2 The power network represented by the nodal admittance matrix Ybk can be decomposed into q isolated subnetworks, and each subnetwork is described by a k , i ∈ Iq . The operators ∗ and −1∗ replace all the elements in the 1st submatrix Yb,i k with 0. Moreover, the operator −1∗ also replaces the row and the 1st column of Yb,i k with its inverse matrix. According to algebraic graph theory, remaining part of Yb,i

20

2 Optimal Control Approach to Identifying Cascading Failures

k the rank of the nodal admittance matrix Yb,i is ki − 1 since each subnetwork Si , i ∈ Iq k is connected [27]. Thus it is guaranteed that the matrix Yb,i has full rank ki − 1, and hence it is invertible.

Remark 2.3 For the branch impedance with the resistance R and the reactance X , the dependence of branch impedance on the temperature can be described by the formulas [28]: R(Tc ) = R(T0 ) · [1 + α(Tc − T0 )] and X (Tc , ω) = X (T0 , ω) · [1 + β(Tc − T0 )], where α and β refer to the temperature coefficients of resistivity and reactance, respectively. T0 denotes the reference temperature (usually 20 ◦ C), and Tc represents the conductor temperature. R(T0 ) and X (T0 , ω) are determined at the reference temperature T0 , and ω is the operating angular frequency.

2.2.3 Optimization Formulation The cascading dynamics of power system are composed of the cascading model defined by Eq. (2.2) and the DC power flow equation described by Eq. (2.5). These two components, coupled with each other, characterize the cascading process of power grids after suffering from disruptive disturbances. The optimal control algorithm allows us to obtain the disruptive disturbances by treating the disturbances as the control inputs of the optimal control system (see Fig. 2.3). Specifically, the cascading model describes the outage of overloading branches and updates the admittance on transmission lines with the latest power flow, which is provided by the DC power flow equation. Meanwhile, the DC power flow equation is solved with the up-to-date admittance of branches from the cascading model. The above two processes occur iteratively in describing the evolution of admittance and power flow on transmission lines. Moreover, the cascading dynamics of power system exactly function as the state equation of the optimal control system. In this way, optimal control theory enables us to gain the optimal control inputs that lead to the worst-case cascading failures in power systems. The identification of disruptive disturbances in power systems can be formulated as the following optimal control problem: min J (Y ph , u k ) uk

(2.6)

with the cost function J (Y ph , u k ) = T(Y ph ) + 

h−1  k=0

u k 2 max{0, ι − k}

(2.7)

where  is a positive weight, and  ·  represents the 2-norm. As mentioned before, the state equation of the optimal control system consists of Eqs. (2.2) and (2.5). The above cost function includes two terms. Specifically, the first term T(Y ph ) denotes the

2.2 Problem Formulation

21

Fig. 2.3 Optimal control approach to identify disruptive disturbances

endpoint cost of cascading failures and it is differentiable with respect to Y ph . The second term characterizes the control energy at the first ι time steps with the constraint 1 ≤ ι ≤ (h − 1). In practice, T(Y ph ) is designed according to the specific concerns about the worst-case scenario of power systems. In particular, the parameter  is set small enough that the first term dominates the cost function. In brief, the idea behind Eq. (2.7) is that the cost function is dominated by its endpoint cost T(Y ph ) that quantifies the final disruption of cascades triggered by initial disturbances (i.e., control inputs). Essentially, the minimization of cost function is done over the first ι control inputs (i.e., u k , k = 0, 1, . . . , ι − 1), and the design of the term u k 2 /max{0, ι − k} enables us to add the control input u k at the specified time steps by setting ι. The objective of this work is to minimize T(Y ph ) (e.g., to minimize the connectivity of power networks or the power consumption) by adding the appropriate control inputs on the selected branch at the beginning of cascades. Optimal control theory enables us to obtain the optimal control inputs (i.e., the most disruptive disturbances) and the minimum T(Y ph ) (i.e., the worst-case cascading failures). Remark 2.4 The proposed approach can be extended to allow for the physical characteristics of other relays. For example, the time-inverse characteristics of overcurrent relays are described by comparing the integral of the branch current over a time interval with the threshold [15]. In addition, the time delay in distance relays can be addressed with the aid of adaptive dynamic programming in optimal control [29]. Remark 2.5 Protection actions of power systems have impacts on the evolution of cascading blackouts [30, 31]. By updating P k at certain cascading steps, the protection actions such as load shedding and generator tripping can be taken into

22

2 Optimal Control Approach to Identifying Cascading Failures

account without affecting the applicability of the proposed approach, though it shall increase the computation burden and complexity of the optimal control problem (2.6). By minimizing the change of injected power on buses, a nonlinear programming problem can be formulated to allow for protective actions (e.g., load shedding and generation dispatch) against the cascades. It follows from the Karush-Kuhn-Tucker (KKT) conditions that the nonlinear programming problem can be converted into a system of (7m + 6n) algebraic equations. By combining it with the system (2.8), we obtain an extended system of 7m + (6 + h)n algebraic equations with 7m + (6 + h)n unknowns. The solutions to the extended system of algebraic equations enable us to compute the disruptive disturbances that cause the worst-case cascading failures of power grids with protective actions. Compared to the system (2.8), both the number of algebraic equations and the number of unknowns in the extended system increase by (7m + 6n). Remark 2.6 The uncertainties in power systems and hidden failures of the relays are important factors in cascading failures. Optimal control of uncertain systems can be adopted to deal with the uncertainties (e.g., the variations of power generation, consumption, and branch impedance, etc) in power systems [32]. For hidden failures of the relays, the probabilistic model based on the Markov chain is more suitable to allow for the stochastic factors [33], and statistics of cascading line outages from utility data can be used as the benchmark to validate the model [34–36]. For this case, optimal control of Markov decision processes provides a useful framework to identify the disruptive disturbance by regarding the disturbances in power systems as the policy or actions in Markov decision processes [37, 38]. Remark 2.7 Other than line failures, there are multiple contributors to cascading failures such as weather conditions, human factors, protection actions, voltage instability, device aging, and so on. Considering that the triggering events (i.e., the initial disruptive disturbances) of most major blackouts in history can be modeled by changes in branch admittance [2, 20, 21], this work focuses on the identification of initial disruptive disturbances or faults that change the branch admittance and cause the worst-case cascades. Actually, the proposed approach is also applied to identify the fluctuation of injected power on buses caused by load variation and generation control. In addition, the probabilistic model based on the Markov chain is more suitable to allow for the stochastic contributing factors such as device aging and human factors, as explained in Remark 2.6.

2.3 Theoretical Analysis In this section, we present some theoretical results on the proposed optimal control problem. The main theoretical results are presented in Theorem 2.1, which provides the necessary conditions of optimal solutions to the proposed optimal control problem (i.e., a system of algebraic equations). Specifically, Lemma 2.1 contributes to the proof of Lemmas 2.3 and 2.2 allows us to compute the power flow on each branch.

2.3 Theoretical Analysis

23

Both Lemmas 2.2 and 2.3 are used to derive the system of algebraic equations in Theorem 2.1. First of all, the properties of operators ∗ and −1∗ are given by the following lemmas. Lemma 2.1 For the nodal admittance matrix Ybk ∈ R m×m , the equations 

∗ Ybk



−1∗ Ybk

=



−1∗ Ybk



∗ Ybk

=

q 

Ei diag(0, 1kTi −1 )EiT

i=1

hold, where 1ki −1 = (1, 1, . . . , 1)T ∈ R (ki −1) . 

Proof See Appendix 2.6.1.

Lemma 2.1 indicates that the two operators ∗ and −1∗ are commutative for the same square matrix. Given the injection power for each bus P k = (P1k , P2k , . . . , Pmk )T at the k-th time step, the quantitative relationship between Y pk and power flow on each branch is presented as follows. Lemma 2.2

∗

Pikj = eiT Ybk e j (ei − e j )T (Ybk )−1 P k , i, j ∈ Im

where ei = (0, . . . , 0, 1, 0, . . . , 0)T ∈ R m .  

i

Proof See Appendix 2.6.2.



Similar to the matrix inversion, the operators ∗ and −1∗ satisfy the following equation in terms of the derivative operation. Lemma 2.3

∗

∂(Ybk )−1 ∗ ∗ = −(Ybk )−1 (A T diag(ei )A)∗ (Ybk )−1 . k ∂ y p,i

Proof See Appendix 2.6.3.



Next, we present theoretical results about the optimal control problem (2.6), which is actually a special case (i.e., the time-invariant case) of the optimal control for the time-varying discrete-time nonlinear system in [39]. By applying the approach in [39] to the optimal control problem (2.6), we obtain the necessary conditions to identify the disruptive disturbance of power systems with the cascading model (6.1) and the DC power flow Eq. (2.5). Theorem 2.1 The necessary conditions for the optimal control problem (2.6) are the solutions of the following system of algebraic equations. Y pk+1 − G(Pikj , ci j )Y pk − E ik u k = 0n ,

(2.8)

24

2 Optimal Control Approach to Identifying Cascading Failures

where the control input u k is given by h−k−1  ∂Y ph−s ∂T(Y ph ) max{0, ι − k} E ik uk = − · h−s−1 2 ∂Y ph ∂Y p s=0

(2.9)

with k = 0, 1, . . . , h − 1. Proof See Appendix 2.6.4.



Remark 2.8 Substituting (2.9) into (2.8) yields the system of h × n algebraic equations with h × n unknowns (i.e., y kp,i , i ∈ In , k ∈ Ih ). Thus, the solution to the above system of algebraic equations enables us to obtain the branch admittances at each cascading step. The optimal control input u k is also available by replacing the unknowns in (2.9) with the computed branch admittances. It is necessary to winnow the solutions to Eq. (2.8) since they just satisfy the necessary conditions for optimal control problem (2.6). Thus, we introduce a search algorithm to explore the optimal control input or initial disturbances. Table 2.1 presents the implementation process of the Iterative Search Algorithm (ISA) in detail. First of all, we set the maximum iteration steps i max of the ISA and the initial value of cost function J ∗ , which is a sufficiently large number Jmax and is larger than the maximum value of the cost function. The solution to the system of algebraic equations (2.8) allows us to obtain the control input u i from (2.9). Then we compute the cost function J i from (2.7) by adding the control input u i in power systems. Then J ∗ and u ∗ are replaced with J i and u i if J i is less than J ∗ . Finally, the algorithm goes to the next iteration and solves the system of algebraic equations (2.8) once again. Regarding the Iterative Search Algorithm in Table 2.1, we have the following theoretical result. Theorem 2.2 The Iterative Search Algorithm in Table 2.1 ensures that the cost function J ∗ and control input u ∗ converge to the optima as the iteration steps i max go to infinity.

Table 2.1 Iterative search algorithm 1: Set the maximum iteration steps i max , i = 0 and J ∗ = Jmax 2: while (i h(t, x, y, θ )

(3.1)

where x denotes a vector of continuous state variables subject to differential relationships, and y represents a vector of continuous state variables under the constraints of algebraic equations. In addition, θ refers to a vector of discrete binary state variables (i.e., θi ∈ {0, 1}). The set of differential equations in the system (3.1) characterize the dynamic response of machines, governors, exciters and loads in power grids. The algebraic components mainly describe the AC power flow equations, and the inequality terms reflect the discrete events (e.g., the automatic line tripping by protective relays, manual operations, lightning, etc.) during cascading failures. In practice, the structure of power grids (e.g., network topology, component parameters) is affected once a discrete event occurs. Thus, the discrete events directly influence the dynamic response of relevant devices and power flow distribution. To incorporate the effect of discrete events at time instants tk , the time axis is divided into a series of time intervals [tk−1 , tk ), k ∈ Im = {1, 2, . . . , m}. At each time interval, the set of differential-algebraic equations is solved using the updated parameters and initial conditions of power system model due to discrete events. By solving the system (3.1) in each time interval, the vectors of state variables x and y at the terminal of each time interval can be obtained as follows

3.2 Problem Formulation

41



xk = F(tk , xk−1 , yk−1 , θk−1 ) yk = G(tk , xk−1 , yk−1 , θk−1 )

(3.2)

where xk = x(tk ), yk = y(tk ) and θk = θ (tk ), k ∈ Im . And the iterated functions F and G characterize the discrete-time evolution of state variables xk and yk , respectively.

3.2.3 Optimization Formulation The cascading blackouts result in severe damages to power networks and may paralyze the service of power supply. Our goal is to search for the initial contingencies that cause the worst disruptions of power grids at the end of cascading blackouts. Therefore, the problem of identifying initial contingencies is formulated as min J (δ, xm , ym ) δ∈

s. t. xk = F(tk , xk−1 , yk−1 , θk−1 ) yk = G(tk , xk−1 , yk−1 , θk−1 ), k ∈ Im

(3.3)

where δ denotes the initial contingencies in power grids that change the state variables x, y, or θ in the initial time interval [t0 , t1 ). It can describe both single disturbance and multiple disturbances as the initial contingency. And  represents the of n physical set n {δ | vi (δ) ≤ restrictions on initial contingencies, which can be described as i=1 0} with the inequality constraints vi (δ) ≤ 0. For simplicity, it is assumed that the triggering event or initial contingency occurs at time τ ∈ [t0 , t1 ). Then we have (x(τ+ ), y(τ+ ), θ (τ+ )) = (x(τ ), y(τ ), θ (τ ), δ), and the function  characterizes the effect of the contingency δ on the state variables at time τ . The objective function J (δ, xm , ym ) quantifies the disruptive level of power grids at the end of cascading failures. A smaller value of J (δ, xm , ym ) indicates a worse disruption of power grids due to cascading blackouts. Then it follows from the KarushKuhn-Tucker (KKT) conditions that the necessary conditions for optimal solutions to Optimization Problem (3.3) is presented as follows [21]. Theorem 3.1 The optimal solution δ ∗ to the Optimization Problem (3.3) with the multipliers μi , i ∈ In satisfies the KKT conditions ∗

∇ J (δ , xm , ym ) +

n 

μi ∇vi (δ ∗ ) = 0

i=1

vi (δ ∗ ) + ωi2 = 0 ∗

μi · vi (δ ) = 0 μi − σi2 = 0, i ∈ In where ωi and σi , i ∈ In are the unknown variables.

(3.4)

42

3 Jacobian-Free Newton-Krylov Method for Risk Identification

Proof The KKT conditions for Optimization Problem (3.3) are composed of four components: stationary, primal feasibility, dual feasibility and complementary slackness. Specifically, the stationary condition allows us to obtain ∇ J (δ ∗ , xm , ym ) +

n 

μi ∇vi (δ ∗ ) = 0,

i=1

where =

n  {δ | vi (δ) ≤ 0}. i=1

Moreover, the primal feasibility leads to gi (δ) ≤ 0, i ∈ In , which can be converted into equality constraints vi (δ ∗ ) + ωi2 = 0, i ∈ In with the unknown variables ωi ∈ R. Further, the dual feasibility corresponds to μi ≥ 0, which can be replaced by μi − σi2 = 0, i ∈ In with the unknown variables σi ∈ R. Finally, the complementary slackness gives μi · vi (δ ∗ ) = 0, i ∈ In 

This completes the proof.

Remark 3.1 To reduce the computational burden, the gradient ∇ J (δ ∗ , xm , ym ) can be approximated by

∂ J (δ, xm , ym ) |δ=δ∗ ∈ R dim(δ) ∂δi

∗ J (δ + ξ ei , xm , ym ) − J (δ ∗ , xm , ym ) ≈ ξ

∇ J (δ, xm , ym )|δ=δ∗ =

with a sufficiently small ξ and the unit vector ei with 1 in the i-th position and 0 elsewhere. And the symbol dim(δ) denotes the dimension of the variable δ.

3.3 Numeric Solver The branch outages result in the discontinuity of cascading failure process, which makes it infeasible to compute partial derivatives for identifying the initial contingency by the Jacobian matrix-based methods [7]. Moreover, other existing methods

3.3 Numeric Solver

43

focus on the identification of direct branch outages as the initial disturbances [13– 15]. For this reason, the Jacobian-Free Newton-Krylov method is employed to solve the system of nonlinear algebraic equations (3.4) without computing the Jacobian matrix. It also enables us to identify the catastrophic cascading failures caused by the continuous changes of branch admittance, in addition to the direct branch outages. Essentially, the JFNK methods are synergistic combinations of Newton methods for solving nonlinear equations and Krylov subspace methods for solving linear equations [22]. To facilitate the analysis, the system (3.4) is rewritten in matrix form S(z) = 0,

(3.5)

where the unknown vector z is composed of δ ∗ , μi , ωi , σi , i ∈ In . And 0 refers to a zero vector with the proper dimension. To obtain the iterative formula for solving (3.5), the Taylor series of S(z) at zs+1 is computed as follows S(zs+1 ) = S(zs ) + J(zs )(zs+1 − zs ) + O(zs )

(3.6)

with zs = zs+1 − zs . By neglecting the high-order term O(zs ) and setting S(zs+1 ) = 0, we obtain J(zs ) · zs = −S(zs ),

s ∈ Z+

(3.7)

where J(zs ) represents the Jacobian matrix and s denotes the iteration index. Thus, solutions to Eq. (3.5) can be approximated by implementing Newton iterations zs+1 = zs + zs , where zs is obtained by Krylov methods. First of all, the Krylov subspace is constructed as follows  K i = span rs , J(zs )rs , J(zs )2 rs , . . . , J(zs )i−1 rs

(3.8)

with rs = −S(zs ) − J(zs ) · z0s , where z0s is the initial guess for the Newton correction and is typically zero [22]. Actually, the optimal solution to zs is the linear combination of elements in the Krylov subspace K i . zs = z0s +

i−1 

λ j · J(zs ) j rs ,

(3.9)

j=1

where λ j , j ∈ {1, 2, . . . , i − 1} is obtained by minimizing the residual rs with the Generalized Minimal RESidual (GMRES) method subject to the constraint of step size zs  ≤ c [23]. In particular, the matrix-vector products in (3.9) can be approximated by S(zs + rs ) − S(zs ) , (3.10) J(zs )rs ≈

44

3 Jacobian-Free Newton-Krylov Method for Risk Identification

where is a sufficiently small value [24]. In this way, the computation of Jacobian matrix is avoided via matrix-vector products in (3.10) while solving Eq. (3.5). Actually, the accuracy of the forward difference scheme (3.10) can be estimated as follows. Theorem 3.2

S(zs + rs ) − S(zs )

rs 2 s s

− J(z sup S (2) (zs + t rs ) )r

≤

2 t∈[0,1] where S (2) (z) denotes the second order derivative of S(z) with respect to the unknown vector z. Proof It follows from NR 3.3-3 in [25] that S(zs + rs ) − S(zs ) − J(zs )rs =

1

(1 − t)S (2) (zs + t rs )rs rs dt

0

which implies



1

S(zs + rs ) − S(zs )

s )rs = (1 − t)S (2) (zs + t rs )rs rs dt

− J(z



0

1 ≤



(1 − t) S (2) (zs + t rs )rs rs dt

0

1 ≤

(1 − t)S (2) (zs + t rs ) · rs 2 dt

0

≤ sup S (2) (zs + t · rs ) · rs 2 t∈[0,1]

1 (1 − t)dt 0

rs 2 sup S (2) (zs + t rs ) = 2 t∈[0,1]

The proof is thus completed.



Remark 3.2 The choice of greatly affects the accuracy and robustness of the JFNK method. For the forward difference scheme (3.10), can be set equal to a value larger than the square root of machine epsilon to minimize the approximation error [26]. Table 3.1 presents the implementation process of Contingency Identification Algorithm (CIA) with the aid of the JFNK method. First of all, the initial values for some variables are specified as follows: δ = l = 0, min , 0 with the condition min < 0 ,

3.3 Numeric Solver

45

Table 3.1 Contingency identification algorithm Initialize: lmax , min , 0 , and l = δ = 0 Goal: δ ∗ and J (δ ∗ , xm , ym ) 1: while (l < lmax ) 2: s = 0 3: while ( s > min ) 4: Calculate the residual rs = −S(zs ) − J(zs ) · z0s 5: Construct the Krylov subspace K i in (3.8) 6: Approximate J(zs ) j rs in (3.9) using (3.10) 7: Compute λ j in (3.9) with the GMRES method 8: Compute zs with (3.9) 9: zs+1 = zs + zs 10:

s+1 = zs /zs  11: s =s+1 12: end while 13: Update δ ∗ and J (δ ∗ , xm , ym ) 14: if (J (δ ∗ , xm , ym ) < J (δ, xm , ym )) 15: δ = δ∗ 16: end if 17: l = l + 1 18: end while

and the maximum iterative step lmax . Then the JFNK method is employed to obtain the optimal disturbance δ ∗ and the cost J (δ ∗ , xm , ym ) from Step 4 to Step 16. Specifically, the residual rs is calculated in each iteration in order to construct the Krylov subspace K i . For elements in K i , the matrix-vector products are approximated by Eq. (3.10) without forming the Jacobian. Next, the term zs for Newton iterations is obtained via the GMRES method [23]. The tolerance s and step number s are updated after implementing the Newton iteration for zs . Afterwards, a new iteration loop is launched if the termination condition s ≤ min fails. After adopting the JFNK method, a disturbance value δ ∗ in (3.4) is saved if it results in a worse cascading failure (i.e., J (δ ∗ , xm , ym ) < J (δ, xm , ym )). The above algorithm does not terminate until the maximum iterative step lmax is reached. The following theoretical results allow us to roughly estimate the convergence accuracy of the optimal disturbances or contingencies δ ∗ before running the CIA. Theorem 3.3 With the CIA in Table 3.1, the increment δ is upper bounded by  δ ≤ min · z 0  + c · smax where z 0 denotes the initial value for the unknown vector z in the numerical algorithm, and smax refers to the maximum iteration steps.

46

3 Jacobian-Free Newton-Krylov Method for Risk Identification

Proof According to the CIA, we have the following inequality zs  ≤ min zs  s+1 after adopting the JFNK method. = s−1 Ini addition, it follows from the updating law z s s s 0 z + z that z = z + i=0 z , which allows us to obtain

zs  ≤ min · zs 

s−1

0  i z = min · z +

i=0   s−1  0 i ≤ min · z  + z  i=0

 ≤ min · z 0  + c · smax , due to zs  ≤ c and s ≤ smax . Moreover, it follows from δ ≤ zs  that we have  δ ≤ min · z 0  + c · smax , which completes the proof.



Remark 3.3 According to the CIA in Table 3.1, the value of cost function J (δ ∗ , xm , ym ) decreases monotonically as the iteration step lmax increases. Considering that J (δ ∗ , xm , ym ) is normally designed to have a lower bound (i.e., J (δ ∗ , xm , ym ) ≥ 0), J (δ ∗ , xm , ym ) converges to a local minimum. This enables us to identify the corresponding initial disturbances δ ∗ .

3.4 Case Study In this section, a cascading failure model based on the DC power flow is introduced to characterize the branch outage sequence of power grids with FACTS devices, HVDC links and protective relays. Then the CIA in Table 3.1 is implemented to search for the disruptive contingencies on selected branches of IEEE 118 Bus System [27]. Moreover, statistical analysis is conducted to investigate the effect of random factors on the identification of initial contingencies. Finally, the CIA is implemented to identify the initial disruptive disturbances on the branch of the IEEE 39 Bus System with the cascade model that includes the AC power flow and transient process.

3.4 Case Study

47

3.4.1 Cascade Model In the simulations, a cascade model is taken into account, which includes FACTS devices, HVDC links and protective relays. The mathematical descriptions of these components are presented in the Appendix. In addition, the DC power flow equation is employed to ensure computational efficiency and avoid the numerical nonconvergence [5]. When power grids are subject to malicious contingencies, the FACTS devices take effect to adjust the branch admittance and balance the power flow for relieving the stress of power networks. If the stress is not eliminated, protective relays will be activated to sever the overloading branches on the condition that the timer of circuit breakers runs out of the preset time [5]. The outage of overloading branches may result in severer stress on power transmission networks and end up with a cascading blackout. The evolution time of cascading failure is introduced to allow for the time factor of cascading blackouts. Essentially, the time interval between two consecutive cascading steps basically depends on the preset time of the timer in protective relays [5]. Thus, the evolution time of cascading failure is roughly estimated by t = kT at the k-th cascading step, and T denotes the preset time of the timer in protective relays.

3.4.2 Parameter Setting The per-unit system is adopted with the base value of 100 MVA in numerical simulations, and the power flow threshold for each branch is 5% larger than the normal power flow on each branch without any contingencies. The power flow on each branch is close to saturation, although it does not exceed their respective thresholds. In this way, the power system is vulnerable to initial contingencies and thus is likely to suffer from cascading blackouts. The cost function in (3.3) is designed as Pe (δ, P m , Y pm )2 to minimize the total power flow on branches by identifying the initial contingency δ. Here Pe represents the vector of power flow on branches. P m and Y pm denote the vector of injected power on buses and that of branch admittance at the end of cascading failure, respectively. The maximum iterative step lmax is equal to 10 in the CIA. Other parameters are given as follows: = 10−2 in Eq. (3.10), min = 10−8 in the JFNK method. Branch 8 (i.e., the red link connecting Bus 5 to Bus 8 in Fig. 3.2) is randomly selected as the disturbed element of power networks. The lower and upper bounds of initial disturbances on Branch 8 are given by δ = 0 and δ¯ = 37.45, respectively. Actually, the upper bound of initial disturbances directly leads to the branch outage. And the total number of cascading steps is m = 12. For simplicity, we specify the same values for the parameters of three HVDC links as follows: Rci = Rcr = R L = 0.1, α = π/15 and γ = π/4. Regarding the FACTS devices, we set X min,i = 0, X max,i = 10 and X i∗ = 0 for the TCSC, and

48

3 Jacobian-Free Newton-Krylov Method for Risk Identification

K P = 4, K I = 3 and K D = 2 for the PID controller via the trial-and-error method. ∗ accounts for 80% with respect to the power In addition, the reference power flow Pe,i flow capacity of relevant branches.

3.4.3 Simulation and Validation Figure 3.2 shows the initial state of IEEE 118 Bus System in the normal condition, and this power system includes 53 generator buses, 64 load buses, 1 reference bus (i.e., Bus 69) and 186 branches. And the HVDC links are denoted by blue lines, which include Branch 4 connecting Bus 3 to Bus 5, Branch 16 connecting Bus 11 to Bus 13 and Branch 38 connecting Bus 26 to Bus 30. In practice, the time delay of circuit breaker ranges from 0.3 to 1 s with the consideration of reclosing time of circuit breakers [28]. In the simulations, two preset values of the timer are taken into consideration for protective relays, i.e., T = 0.5 s, and T = 1 s. The CIA is carried out to search for the initial contingency that results in the catastrophic cascading failures of power systems (i.e., relatively small values of cost function

1

2

117

3

39

4

11

12

14 33 35

5

6

7

13

16

15

17

18 19 34 38

8 9

40 41 54 56 43 44 45

42

37 36

53 58 57 46

48

52

47 49

51 50

20

10

63 59

31 113 30 21 29 28

116 66

22 72 71 32

23

60

73

67

70 74 75 69 68 65

27

115

24 25

118

99

83 84 87 86 85

106 107

79 97 98

76 82 78 96 94 100

26

62 64 61

77 81 80 114

55

104 105 108

95 93 89

88 90 91

103 109

92 101 102

111 110 112

Fig. 3.2 Initial state of IEEE 118 Bus System. Red balls denote the generator buses, while green ones stand for the load buses. Cyan lines represent the branches of power systems. In addition, the red line is selected as the disturbed branch, and three blue lines are the HVDC links, including Branch 4, Branch 16 and Branch 38

3.4 Case Study

49

1

2

117

4

11

12

14 33 35

5

6

7

13

16

15

17

18 19 34 38

3

39

8 9

40 41 54 56 43 44 45

37 36

53 58 57 46

48

52

47 49

51 50

20

10

63 59

31 113 30 21 29 28

116 66

22 72 71 32

23

60

73

67

70 74 75 69 68 65

27

115

24 25

118

99

83 84 87 86 85

106 107

79 97 98

76 82 78 96 94 100

26

62 64 61

77 81 80 114

55

42

104 105 108

95 93 89

88 90 91

103 109

92 101 102

111 110 112

Fig. 3.3 Final configuration of IEEE 118 Bus System without FACTS devices

Pe (δ, P m , Y pm )2 ). For the IEEE 118 Bus System without the FACTS devices, the computed magnitude of disturbance on Branch 8 is 37.45, which exactly leads to the outage of Branch 8. To be exact, the disturbance magnitude refers to the change of branch admittance caused by the disturbance. For the power system with the FACTS devices and the preset time of the timer T = 0.5 s, the disturbance magnitude identified by the CIA is 36.77, while it is 35.98 for T = 1 s. Next, we validate the proposed identification approach by adding the computed disturbances on Branch 8 of IEEE 118 Bus Systems. Specifically, Fig. 3.3 demonstrates the final state of IEEE 118 Bus System with no FACTS devices and with the preset time of circuit breaker T = 1 s. The cascading process terminates with 95 outage branches and the value of cost function is 53.28 after 16 s, and the system collapses with 42 islands in the end. These 42 islands include 24 isolated buses and 18 subnetworks. In contrast, Fig. 3.4 presents the final configuration of IEEE 118 Bus Systems with the protection of the FACTS devices and with the preset time T = 0.5 s. The cascading process ends up with 40 outage branches and the value of cost function is 102.56 after 10 s, and the power system is separated into 17 islands, which include 6 subnetworks and 11 isolated buses. Figure 3.5 presents the final state of power systems with FACTS devices and T = 1 s. It is observed that the power network is eventually split into 3 islands (Bus 14, Bus 16 and a subnetwork composed of all other buses) with only 6 outage branches and the cost function of 153.69. Note that the initial contingencies identified by the CIA fail to cause the outage of

50

3 Jacobian-Free Newton-Krylov Method for Risk Identification 1

2

117

4

11

12

14 33 35

5

6

7

13

16

15

17

18 19 34 38

3

39

8 9

40 41 54 56 43 44 45

37 36

53 58 57 46

48

52

47 49

51 50

20

10

63 59

31 113 30 21 29 28

116 66

22 72 71 32

23

60

73

67

70 74 75 69 68 65

27

115

24 25

118

99

83 84 87 86 85

106 107

79 97 98

76 82 78 96 94 100

26

62 64 61

77 81 80 114

55

42

104 105 108

95 93 89

88 90 91

103 109

92 101 102

111 110 112

Fig. 3.4 Final configuration of IEEE 118 Bus System with FACTS devices and T = 0.5 s

Branch 8 in the end for both T = 0.5 s and T = 1 s. The above simulation results demonstrate the advantage of the FACTS devices in preventing the propagation of cascading outages. A larger preset time of timer enables the FACTS devices to sufficiently adjust the branch admittance in response to the overload stress. As a result, the less severe damages are caused by the contingency for the larger preset time of the timer. Figure 3.6 presents the time evolution of branch outages in the IEEE 118 Bus System as a result of disturbing Branch 8 in three different scenarios. The cyan squares denote the number of outage branches with no FACTS devices and T = 1 s, while the green and blue ones refer to the numbers of outage branches with FACTS devices and with T = 0.5 s and T = 1 s, respectively. The contingencies identified by the CIA are added to change the admittance of Branch 8 at t = 0 s. With no FACTS devices, the cascading outage of branches propagates quickly from t = 2 s to t = 10 s and terminates at t = 16 s. When the FACTS devices are adopted and the preset time of timer is T = 0.5 s, the cascading failure starts at t = 2 s and speeds up till t = 8 s and stops at t = 10 s. For T = 1 s, the cascading outage propagates slowly due to the larger preset time of timer and comes to an end with only 6 outage branches at t = 8 s. Together with protective relays and HVDC links, the FACTS devices succeed in protecting power systems against blackouts by adjusting the branch impedance in real time. More precisely, the number of outage branches decreases by 57.9% with FACTS devices and T = 0.5 s and decreases by 93.7% with FACTS devices and T = 1 s.

3.4 Case Study

51

1

2

117

4

11

12

14 33 35

5

6

7

13

16

15

17

18 19 34 38

3

39

8

40 41 54 56 43 44 45

37 36

9

53 58 57 46

48

52

47 49

51 50

20

10

63 59

31 113 30 21 29 28

116 66

22 72 71 32

23

60

73

67

70 74 75 69 68 65

27

115

24

118

25

99

83 84 87 86 85

106 107

79 97 98

76 82 78 96 94 100

26

62 64 61

77 81 80 114

55

42

104 105 108

95 93 89

103 109

92 101

88 90 91

102

111 110 112

Fig. 3.5 Final configuration of IEEE 118 Bus System with FACTS devices and T = 1 s Fig. 3.6 Time evolution of outage branches during cascades

100 NO FACTS FACTS, T=0.5s FACTS, T=1.0s

90

Number of outage branches

80 70 60 50 40 30 20 10 0 0

5

10

Time [s]

15

20

52

3 Jacobian-Free Newton-Krylov Method for Risk Identification

3.4.4 Statistical Analysis To further validate the proposed approach, statistical analysis is conducted by taking into account random factors such as hidden failures of branches. If one branch is exposed to line tripping through a common bus, it is more likely to be tripped incorrectly due to hidden failures [29]. Suppose the probability of an exposed line tripping incorrectly is 0.05, and the time delay of protective relay is T = 0.5 s. Then multiple cascading failure paths could be generated by adding an initial disturbance on the selected branch. In the simulations, 5 branches (i.e., Branch 5, Branch 26, Branch 45, Branch 64, and Branch 108) are randomly selected on IEEE 118 Bus System, and the CIA is implemented for 100 times on each selected branch to identify the initial contingency. Table 3.2 presents the statistics on the cascading failures identified by the CIA in terms of initial disturbances and the values of cost function. In this table, J ∗ denotes the minimum value of cost function in the 100 simulation trials, and δ ∗ refers to the initial disturbance that leads to the minimum value of cost function J ∗ . In addition, μδ and σδ represent the mean value and standard deviation of the initial disturbances δ identified by the CIA, respectively. Similarly, μ J and σ J denote the mean value and standard deviation of values of cost function at the end of cascades by adding the identified initial disturbances, respectively. Note that the identified initial disturbance δ ∗ = 25.38 on Branch 26 results in the direct branch outage (i.e., sever Branch 26). In contrast, the identified disturbances δ ∗ on other four branches (Branches 5, 45, 64 and 108) actually give rise to the change of their respective branch admittances instead of the direct branch outages, which subsequently causes the overload and outage of other branches in power systems. It is observed that the disturbance δ ∗ = 3.66 on Branch 64 results in the worst-case cascading failures of the five identified disturbances δ ∗ in terms of the values of cost function J ∗ . In addition, the identified disturbances on Branch 64 have a relatively small standard deviation σδ = 1.61, which implies that Branch 64 is vulnerable to catastrophic cascading failures within a relative small range of initial disturbances. On the whole, the identified disturbances on Branch 108 can lead to the worst-case cascading failures in terms of mean values μ J with the smallest standard deviation σ J = 93.21.

Table 3.2 Statistics on cascading failures identified by the CIA Branch ID δ ∗ J∗ μδ σδ 5 26 45 64 108

14.93 25.38 3.41 3.66 3.67

91.68 69.51 89.84 63.82 86.35

8.95 13.36 2.09 3.32 4.18

6.46 9.35 1.64 1.61 2.43

μJ

σJ

190.83 206.52 237.52 192.38 176.75

94.25 100.22 107.21 96.35 93.21

3.5 Conclusions

53

3.4.5 Applicability The proposed CIA is also applied to the cascade models with the AC power flow, various protective relays and transient process. To demonstrate its efficacy, the CIA is implemented to identify the initial disturbance on the branch of IEEE 39 Bus System by using the Cascading Outage Simulator with Multiprocess Integration Capabilities (COSMIC) [5]. The COSMIC is able to describe various mechanisms of cascading outages, which include protective relays, the AC power flow, transient process, load shedding and so on. In the simulation, Branch 2 is randomly selected to identify the initial disturbances that cause the catastrophic cascading failures by the CIA, and the parameter setting remains unchanged [5]. In addition, the cost function J is the same as that in IEEE 118 Bus System and it is computed based on the real power flow. The initial disturbance can give rise to the continuous change of branch admittance, in addition to the direct line outage. The cascades last for 10 s and Branch 35 is tripped by accident (e.g., due to human errors) at the 3rd second. The CIA allows to obtain the initial disruptive disturbance δ ∗ = 3.3 with the cost function J = 575.72 after 10 iterations (i.e., lmax = 10). It is demonstrated that the identified initial disturbance on Branch 2 initially results in the tripping of Branch 3 at t = 0.001 s and subsequently a sequence of under voltage load shedding (UVLS) at Buses 3, 4, 7, 8, 12, 15, 16, 18, and 27. Then Branch 2 is tripped at t = 0.86 s, which leads to the UVLS at Bus 26 and Bus 31 in succession. Afterwards, Branch 35 is tripped at t = 3 s, which causes the failure of numeric solver in COSMIC and thus the termination of simulations. Figures 3.7 and 3.8 show the time responses of bus voltage magnitudes and angles to the identified disturbance on Branch 2 by using the COSMIC, respectively. It is observed that several discontinuous changes of both voltage magnitudes and angles occur during the cascades due to the UVLS and line tripping. The numerical simulation lasts for 3 s and fails to proceed because it suffers from the problem of numerical non-convergence. This is a major issue of cascade models of power grids that take into account the AC power flow and transient process.

3.5 Conclusions In this chapter, we investigated the problem of identifying the initial contingencies that lead to cascading blackouts of power systems equipped with FACTS devices, HVDC links and protective relays. A general optimization formulation was proposed to identify the initial disruptive contingencies, and an efficient numerical method was presented to solve the optimization problem. Numerical simulations were carried out on IEEE test systems to validate the proposed identification approach. Significantly, the proposed contingency identification algorithm allows to detect some nontrivial contingencies that result in the catastrophic cascading failure of power grids, in addition to the direct branch outage. It is demonstrated that the coordination of FACTS devices and protective relays can enhance the capability of power grids

54

3 Jacobian-Free Newton-Krylov Method for Risk Identification

Fig. 3.7 Time responses of bus voltage magnitudes to the initial disruptive disturbance on Branch 2 of IEEE 39 Bus System using the COSMIC

1.5 1.4 1.3

|V|

1.2 1.1 1 0.9 0.8 0.7 0

0.5

1

1.5

2

2.5

3

2

2.5

3

Time[s]

Fig. 3.8 Time responses of bus phase angles to the initial disruptive disturbance on Branch 2 of IEEE 39 Bus System using the COSMIC

1.5

1

0.5

0

-0.5

-1

-1.5 0

0.5

1

1.5

Time[s]

against cascading failures. Future work may include the effect of model uncertainty on the identification of disruptive contingencies that can initiate catastrophic cascading failures. Moreover, considering that the error of human operators is one of the key factors leading to power system blackouts, it is desirable to incorporate human factors for the contingency identification in the next step.

3.6 Appendix

3.6 3.6.1

55

Appendix FACTS Devices

FACTS devices can greatly enhance the stability and transmission capability of power systems. As an effective FACTS device, TCSC has been widely installed to control the branch impedance and relieve system stresses. The dynamics of TCSC is described by a first order dynamical model [30] TC,i

dX C,i = −X C,i + X i∗ + u i , dt

X min,i ≤ X C,i ≤ X max,i

(3.11)

where X i∗ refers to its reference reactance of Branch i for the steady power flow. X min,i and X max,i are the lower and upper bounds of the branch reactance X C,i respectively and u i represents the supplementary control input, which is designed to stabilize the disturbed power system [31]. For simplicity, PID controller is adopted to regulate the power flow on each branch t u i (t) = K P · ei (t) + K I ·

ei (τ )dτ + K D ·

dei (t) dt

(3.12)

0

where K P , K I and K D are tunable coefficients, and the error ei (t) is given by  ei (t) =

∗ ∗ − |Pe,i (t)|, |Pe,i (t)| ≥ Pe,i ; Pe,i 0, otherwise.

∗ and Pe,i (t) denote the reference power flow and the actual power flow Here, Pe,i on Branch i, respectively. Note that TCSC fails to function when the transmission line is severed.

3.6.2

HVDC Links

HVDC links work as a protective barrier to prevent the propagation of cascading outages in practice, and it is normally composed of a transformer, a rectifier, a DC line and an inverter. Actually, the rectifier terminal can be regarded as a bus with real power consumption Pr , while the inverter terminal can be treated as a bus with real power generation Pi . The direct current from the rectifier to the inverter is computed as follows [32] √ 3 3(cos α − cos γ ) Id = , π(Rcr + R L − Rci )

56

3 Jacobian-Free Newton-Krylov Method for Risk Identification

where α ∈ [π/30, π/2] denotes the ignition delay angle of the rectifier, and γ ∈ [π/12, π/9] represents the extinction advance angle of the inverter. Rcr and Rci refer to the equivalent communicating resistances for the rectifier and inverter, respectively. Additionally, R L denotes the resistance of the DC transmission line. Thus the power consumption at the rectifier terminal is √ 3 3 Id cos α − Rcr Id2 , Pr = π and at the inverter terminal is √ 3 3 Id cos γ − Rci Id2 = Pr − R L Id2 . Pi = π

(3.13)

(3.14)

Note that Pr and Pi keep unchanged when α and γ are fixed.

3.6.3

Protective Relay

The protective relays are indispensable components in power systems protection and control. When the power flow exceeds the given threshold of the branch, the timer of circuit breaker starts to count down from the preset time [5]. Once the timer runs out of the preset time, the transmission line is severed by circuit breakers and its branch admittance becomes zero. Specifically, a step function is designed to reflect the physical characteristics of branch outage as follows  g(Pe,i , bi ) =

0, |Pe,i | > bi and tc > T ; 1, otherwise.

where T is the preset time of the timer in protective relays, and tc denotes the counting time of the timer. In addition, Pe,i denotes the power flow on Branch i with the threshold bi .

References 1. McLinn, J.: Major power outages in the US, and around the world. In: Annual Technology Report of IEEE Reliability Society (2009) 2. Günther, B., Povh, D., Retzmann, D., Teltsch, E.: Global blackouts-lessons learned. In: PowerGen Europe, vol. 28(30) (2005) 3. Fang, Y., Pedroni, N., Zio, E.: Resilience-based component importance measures for critical infrastructure network systems. IEEE Trans. Reliab. 65(2), 502–512 (2016) 4. Jovcic, D., Pillai, G.N.: Analytical modeling of TCSC dynamics. IEEE Trans. Power Deliv. 20(2), 1097–1104 (2005)

References

57

5. Song, J., et al.: Dynamic modeling of cascading failure in power systems. IEEE Trans. Power Syst. 31(3), 2085–2095 (2016) 6. Almassalkhi, M.R., Hiskens, I.A.: Model-predictive cascade mitigation in electric power systems with storage and renewables-Part I: theory and implementation. IEEE Trans. Power Syst. 30(1), 67–77 (2015) 7. Taedong, K., Wright, S.J., Bienstock, D., Harnett, S.: Analyzing vulnerability of power systems with continuous optimization formulations. IEEE Trans. Netw. Sci. Eng. 3(3), 132–146 (2016) 8. Yan, J., Tang, Y., He, H., Sun, Y.: Cascading failure analysis with DC power flow model and transient stability analysis. IEEE Trans. Power Syst. 30(1), 285–297 (2015) 9. Zhang, H., Zhai, C., Xiao, G., Pan, T.: Identifying critical risks of cascading failures in power systems. IET Gener. Transm. Distrib. (2019). https://doi.org/10.1049/iet-gtd.2018.5667 10. Chen, Q., McCalley, J.: Identifying high risk n−k contingencies for online security assessment. IEEE Trans. Power Syst. 20(2), 823–834 (2005) 11. Davis, C.M., Overbye, T.J.: Multiple element contingency screening. IEEE Trans. Power Syst. 26(3), 1294–1301 (2011) 12. Donde, V., Lopez, V., Lesieutre, B., Pinar, A., Yang, C., Meza, J.: Severe multiple contingency screening in electric power systems. IEEE Trans. Power Syst. 23(2), 406–417 (2008) 13. Bienstock, D., Verma, A.: The n−k problem in power grids: new models, formulations, and numerical experiments. SIAM J. Optimiz. 20(5), 2352–2380 (2010) 14. Rocco, C., Ramirez-Marquez, J., Salazar, D., Yajure, C.: Assessing the vulnerability of a power system through a multiple objective contingency screening approach. IEEE Trans. Reliab. 60(2), 394–403 (2011) 15. Eppstein, M.J., Hines, P.D.: A “random chemistry” algorithm for identifying collections of multiple contingencies that initiate cascading failure. IEEE Trans. Power Syst. 27(3), 1698– 1705 (2012) 16. Zhai, C., Zhang, H., Xiao, G., Pan, T.: An optimal control approach to identify the worst-case cascading failures in power systems. IEEE Trans. Control Netw. Syst. (2019). https://doi.org/ 10.1109/TCNS.2019.2930871 17. Babalola, A.A., Belkacemi, R., Zarrabian, S.: Real-time cascading failures prevention for multiple contingencies in smart grids through a multi-agent system. IEEE Trans. Smart Grid 9(1), 373–385 (2018) 18. Cai, Y., Cao, Y., Li, Y., Huang, T., Zhou, B.: Cascading failure analysis considering interaction between power grids and communication networks. IEEE Trans. Smart Grid 7(1), 530–538 (2016) 19. Zhai, C., Zhang, H., Xiao, G., Pan, T.: A model predictive approach to preventing cascading failures of power systems. Int. J. Electr. Power Energ. Syst. 113, 310–321 (2019) 20. Hines, P.D., Rezaei, P.: Cascading failures in power systems. In: Smart Grid Handbook, pp. 1–20 (2016) 21. Mangasarian, O.L.: Nonlinear Programming. SIAM (1994) 22. Knoll, D.A., Keyes, D.E.: Jacobian-free Newton-Krylov methods: a survey of approaches and applications. J. Comput. Phys. 193(2), 357–397 (2004) 23. Saad, Y., Martin, H.S.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986) 24. Brown, P.N., Saad, Y.: Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Stat. Comput. 11(3), 450–481 (1990) 25. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. SIAM, Philadelphia (2000) 26. Chan, T.F., Jackson, K.R.: Nonlinearly preconditioned Krylov subspace methods for discrete Newton algorithms. SIAM J. Sci. Stat. Comput. 5, 533–542 (1984) 27. Zimmerman, R.D., Murillo-Sánchez, C.E., Thomas, R.J.: Matpower: steady-state operations, planning and analysis tools for power systems research and education. IEEE Trans. Power Syst. 26(1), 12–19 (2011)

58

3 Jacobian-Free Newton-Krylov Method for Risk Identification

28. PJM Relay Subcommittee, Protective Relaying Philosophy and Design Guidelines, 12 July 2018 29. Thorp, J., Phadke, A., Horowitz, S., Tamronglak, C.: Anatomy of power system disturbances: importance sampling. Int. J. Electr. Power Energ. Syst. 20(2), 147–152 (1998) 30. Paserba, J., et al.: A thyristor controlled series compensation model for power system stability analysis. IEEE Trans. Power Deliv. 10(3), 1471–1478 (1995) 31. Son, K.M., Park, J.K.: On the robust LQG control of TCSC for damping power system oscillations. IEEE Trans. Power Syst. 15(4), 1306–1312 (2000) 32. Kundur, P., Balu, N.J., Lauby, M.G.: Power System Stability and Control, vol. 7. McGraw-hill, New York (1994)

Chapter 4

Security Monitoring Using Converse Lyapunov Function

Abstract This chapter introduces a novel framework to construct the region of attraction (ROA) of a power system centered around a stable equilibrium by using stable state trajectories of system dynamics. Most existing works on estimating ROA rely on analytical Lyapunov functions, which are subject to two limitations: the analytic Lyapunov functions may not be always readily available, and the resulting ROA may be overly conservative. This work overcomes these two limitations by leveraging the converse Lyapunov theorem in control theory to eliminate the need for an analytic Lyapunov function and learning the unknown Lyapunov function with the Gaussian Process (GP) approach. In addition, a Gaussian Process Upper Confidence Bound (GP-UCB) based sampling algorithm is designed to reconcile the trade-off between the exploitation for enlarging the ROA and the exploration for reducing the uncertainty of sampling region. Within the constructed ROA, it is guaranteed in the probability that the system state will converge to the stable equilibrium with a confidence level. Numerical simulations are also conducted to validate the assessment approach for the ROA of the single-machine infinite bus system and the New England 39-bus system. Numerical results demonstrate that our approach can significantly enlarge the estimated ROA compared to that of the analytic Lyapunov counterpart.

4.1 Introduction The region of attraction (ROA) for complex dynamical systems provides a useful measure of stability level and robustness against external disturbances. Thus, it is of great importance to safety-critical systems (e.g., power systems [1, 2], nuclear reactor control systems, engine control systems, etc.), where the system stability has to be guaranteed before they are implemented in practice. In terms of power systems, the ROA refers to a subspace of operating states that can converge to a steady-state equilibrium. There are various approaches for estimating the ROA of a general nonlinear system such as contraction analysis, level sets of Lyapunov function [3], sum of square technique [4, 5], sampling-based method [6], and so on. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Zhai, Control and Optimization Methods for Complex System Resilience, Studies in Systems, Decision and Control 478, https://doi.org/10.1007/978-981-99-3053-1_4

59

60

4 Security Monitoring Using Converse Lyapunov Function

Nevertheless, these approaches largely rely on deterministic models and may not be applicable to deal with uncertainties in more realistic systems. As a non-parametric method, GP is flexible to incorporate the prior information as well as to quantify the uncertainty [7]. By regarding the unknown function or dynamics as a GP, Bayesian machine learning provides a powerful tool for both regression and inference using the prior belief and sample data, and it can generate the posterior distribution for the unknown function [8]. Therefore, the GP approach has found wide applications in various fields such as bandit setting [9], robotics [10], and classifier design [11], to name just a few. As is well known, it is always a challenging problem for determining a suitable Lyapunov function for a complex nonlinear dynamical system. For a stable equilibrium point of the dynamical system, the converse Lyapunov theorem ensures the existence of Lyapunov functions and enables us to compute the values using stable state trajectories [3]. In practice, the accurate values of Lyapunov function are not available due to numerical error and time restrictions. For this reason, it would be desirable to regard the unknown Lyapunov function as a GP. In this way, the posterior distribution of Lyapunov function can be obtained by learning the sampling data (i.e., estimated values of Lyapunov function), which makes it possible to construct and evaluate the ROA for nonlinear dynamical systems. Thus, this chapter centers on the quantitative evaluation of ROA for power systems from a GP perspective. Compared with existing work [10, 12, 13], the key contributions of this work lie in 1. Propose a theoretical framework based on the converse Lyapunov theorem for estimating the ROA of general nonlinear systems around an equilibrium without constructing an analytic Lyapunov function. 2. Develop a GP-UCB based sampling algorithm for creating a sampling set and learning the unknown Lyapunov function using stable state trajectories. 3. For an existing Lyapunov function, our approach allows for extending the certified ROA with a guaranteed confidence level. The remainder of this chapter is organized as follows. Section 4.2 introduces the region of attraction for a general dynamical system and the estimation of the Lyapunov function. Section 4.3 presents the GP approach for learning the known dynamics, followed by the main results on the GP-UCB based sampling algorithm in Sect. 4.4. Numerical simulations are conducted to validate the proposed approach on the IEEE test systems in Sect. 4.5. Finally, Sect. 4.6 draws a conclusion and discusses future work.

4.2 ROA of General Dynamical Systems The ROA of a general dynamical system normally refers to a region where each state can converge to the stable equilibrium point as time goes to infinity. Likewise, the ROA of a power system is viewed as a set of operating states such as rotor angles

4.2 ROA of General Dynamical Systems

61

Fig. 4.1 Illustration on the level set of Lyapunov function. The red ellipse describes the level set {x ∈ R 2 | V (x) ≤ C1 }, while the blue one denotes the level set {x ∈ R 2 | V (x) ≤ C2 } with C1 < C2 and x = (x1 , x2 ). The black dot represents a stable equilibrium point of dynamic system. Each state in the level set can converge to the equilibrium point on condition of V˙ (x) < 0

and frequencies able to converge to the stable equilibrium, which corresponds to the solution of power flow problem [14, 15], after being subject to a disturbance. The corresponding convergent trajectory is regarded as a stable state trajectory. The estimation of ROA is basically dependent on the construction of Lyapunov function and its level set (see Fig. 4.1). In practice, constructing an analytic Lyapunov function for a nonlinear system is a challenging task. The converse Lyapunov theorem [3] allows for estimating the value of Lyapunov function without using its analytic form. For a power system that has a stable state x(t), t ≥ 0, a commonly used converse Lyapunov function is V (x) = trajectory ∞ 2 x(t) dt [13]. We generalize the converse Lyapunov function by introducing 0 a more general function α(·) (see the definition in Appendix 4.7.1) and a solution trajectory φ(·) in Lemma 4.1. The existence of such generalized Lyapunov function is guaranteed in theory (i.e., Theorem 4.17 in [3]) as follows. Lemma 4.1 Without loss of generality, let x = 0 be an asymptotically stable equilibrium point for the nonlinear system x˙ = f(x), where f : X → R n is locally Lipschitz, and S is the region of attraction, then there is a continuous positive definite function W (x) such that ∞ V (x) = α(φ(x, t))dt, V (0) = 0 0

and

with

dV (x) ∂ V (x) = f(x) ≤ −W (x), ∀ x ∈ S dt ∂x

(4.1)

dφ(x, t) = f(φ(x, t)), φ(x, 0) = x. dt

(4.2)

62

4 Security Monitoring Using Converse Lyapunov Function

α(z) is a class  function (see Appendix 4.7.1), and the level set c = {x ∈ R n | V (x) ≤ c, ∀ c > 0} is a compact subset of S = {x ∈ R n | limt→+∞ φ(x, t) = 0}. In Lemma 4.1, the dynamics x˙ = f(x) of a power system normally represent that of generators which are known as swing equations [16]. Here we explicitly assume that the system dynamics are available, although our framework can be generally extended to incorporate uncertainties with a predefined complexity level, i.e., the smoothness of the unknown components, in the system’s dynamical model. Property (4.1) implies that the function V (x) decays over time. Equation (4.2) defines a stable trajectory φ(x, t) with the initial state x. More importantly, the Lyapunov function V (x) in Lemma 4.1 has a nice property that, by increasing the level set of V (x), we can approach the real region of attraction (see Remark 4.7 for a formal explanation). Note that this property may not hold for any analytical Lyapunov function. Thus, we can leverage the converse Lyapunov function approach to obtain a better ROA by collecting more sampling points in order to enlarge the level sets. Due to the discrete nature of sampling data, the converse Lyapunov function V (x) in Lemma 4.1 can be discretized and approximated as follows Vˆ (x) =

n 

α(φ(x, ti ))t,

(4.3)

i=1

where t denotes the sampling time interval and ti = (i − 1)t, i ∈ {1, 2, . . . , n}. The error caused by the above discretization can be further estimated (see Lemma 4.2 in Appendix 4.7.2). While the approximated Vˆ (x) can be calculated directly based on samples, the explicit Lyapunov function V (x) is unknown. This work therefore learns this unknown Lyapunov function V (x) from the discretized counterpart. By treating V (x) and Vˆ (x) as a GP and its measurement, respectively, the estimation error V (x) − Vˆ (x) can be regarded as the measurement noise. This enables us to learn the unknown Lyapunov function V (x) using the GP approach.

4.3 GP for Learning Unknown Dynamics The above section introduces the discretized, approximated Lyapunov function and the respective unknown Lyapunov function. In this section, we propose the GP approach for learning the unknown Lyapunov function (i.e., GP regression). Normally, a general GP regression requires a prior distribution of unknown functions specified by a mean function, a covariance function, and the probability of the observations and sampling data to obtain the posterior distribution. Without loss of generality, we consider an unknown function h(x) as a GP, which can be sequentially measured by y (i) = h(x(i) ) + , i ∈ Z +

4.3 GP for Learning Unknown Dynamics

63

where y (i) refers to the observed function value for the input x(i) at the i-th sampling step, and the measurement noise  is zero-mean, independent and bounded by σ . With the GP approach, we can obtain the posterior distribution over h(x) by using sampling data in the training set {(x(1) , y (1) ), (x(2) , y (2) ), . . . , (x(i) , y (i) )}. In this work, it is assumed that the unknown Lyapunov function is a GP prior or its “complexity” can be measured by the RKHS norm. The sampling data are obtained by implementing the GP-UCB based sampling algorithm. Note that, for an existing Lyapunov function, the GP approach allows to extend the certified ROA with a given confidence level.

4.3.1 Gaussian Process and RKHS Norm By regarding the values of V (x) as random variables, any finite collection of them is multivariate distributed in an overall consistent way. The unknown Lyapunov function V (x) can be approximated by a GP. Note that the covariance or kernel function k(x, x ) encodes the smoothness property of V (x) from the GP (see Fig. 4.2). For a sample from a known GP distribution Vˆ N = [Vˆ (x(1) ), . . . , Vˆ (x(N ) )]T at points A N = {x(1) , . . . , x(N ) }, Vˆ (x(i) ) = V (x(i) ) +  with  ∼ N (0, σ 2 ), there are the analytic formulas for mean μ N (x), covariance k N (x, x ) and variance σ N2 (x) of the posterior distribution as follows [9] μ N (x) = k N (x)T (K N + σ 2 I )−1 Vˆ N k N (x, x ) = k(x, x ) − k N (x)T (K N + σ 2 I )−1 k N (x ) σ N2 (x)

(4.4)

= k N (x, x)

Fig. 4.2 Examples of basic kernel functions: a Squared Exponential Kernel k(x, x ) = 2 2 e−x−x  /2l with a length scale parameter l and b Linear Kernel k(x, x ) = x T x with x = 1. Each kernel allows to approximate the unknown function with a certain “complexity”

64

4 Security Monitoring Using Converse Lyapunov Function

where k N (x) = [k(x(1) , x), . . . , k(x(N ) , x)]T and K N is the positive definite kernel matrix [k(x, x )]x,x ∈A N . If the prior distribution over V (x) is unknown, V (x) can be approximated by the  linear combination of kernel functions in the RKHS, i.e. V (x) = i ci k(x, x(i) ). The RKHS Hk (X ) with the kernel k(x, x ) is a complete subspace of L 2 (X ), and its inner product ·, · k is endowed with the reproducing property: V (x), k(x, ·) k = V (x), ∀ V (x) ∈ Hk (X ). Moreover, there exists a uniquely associated √ RKHS Hk (X ) for each kernel k(x, x ) [7]. The induced RKHS norm V k = V, V k is used to quantify the smoothness of the function V with respect to the kernel k(x, x ). In brief, the function V (x) gets smoother as V k decreases. In this work, we consider that Lyapunov functions V (x) have relatively low “complexity” or high smoothness, which can be measured by the RKHS norm. Remark 4.1 There is an interesting connection between the kernel function in RKHS and the covariance function of a GP. If the kernel function in RKHS is the same as the covariance function of a GP, the posterior mean of a GP is equivalent to the estimator of kernel ridge regression in RKHS [17]. In addition, the posterior variance of a GP can be regarded as a worst-case error in RKHS. Within the GP approach, how one samples plays an important role because the sampling rule affects the confidence level of the estimated ROA, which is the probability at least with which a trajectory initiated from an inner state of such estimated ROA will converge to the corresponding stable equilibrium. In other words, that is the probability that an estimated ROA is valid. For this work, we use the GP-UCB based sampling rule. Compared with other heuristics in GP optimization [18–20], the GP-UCB based sampling rule is able to deal with the trade-off between exploitation for optimizing the objective function and exploration for reducing the uncertainty with the guaranteed theoretical performance.

4.3.2 GP-UCB Based Algorithm For a given value of δ ∈ (0, 1) and the sampling domain X ∈ R n , which is a subset of the state space, our goal is to maximize the region of attraction, wherein each point converges to the origin with probability at least the confident level of 1 − δ. Thus, a GP-UCB based algorithm is developed in Table 4.1 to select the sampling points for enlarging the ROA with a guaranteed confidence level. Specifically, a sampling 1/2 point x(i) is selected in X by searching for the maxima of μi−1 (x) + βi σi−1 (x), where the term μi−1 (x) contributes to enlarge the level set of Lyapunov function and the term σi−1 (x) allows to reduce the uncertainty of sampling region. Essentially, the sampling rule aims to reconcile the trade-off between the exploitation for enlarging the ROA and the exploration for reducing the uncertainty of sampling region. Let the sampling point x(i) serve as the initial condition of nonlinear system x˙ = f(x), and this allows to generate a state trajectory φ(x(i) , t), t ≥ 0. If this

4.3 GP for Learning Unknown Dynamics

65

Table 4.1 GP-UCB based algorithm Input: X ∈ R n , δ, ξ , tn , μ0 , σ0 , k(x, x ), i = 1 Output: x(i) , φ(x(i) , t), Vˆ (x(i) ), μi (x), σi (x), i ∈ {1, 2, . . . , N } 1: while (i ≤ N ) 2:

  1/2 Choose x(i) = argmaxx∈X μi−1 (x) + βi σi−1 (x)

3: 4: 5: 6: 7: 8: 9: 10: 11:

Generate a state trajectory φ(x(i) , t), t ≥ 0 if (φ(x(i) , tn ) < ξ ) Sample Vˆ (x(i) ) = V (x(i) ) +  with (4.3) Update μi (x) and σi (x) with (4.4) Update i = i + 1 else Update X = X/{x(i) }. end if end while

state trajectory can converge to the origin, the point x(i) is called as a stable sampling point. Then the value of Lyapunov function at x(i) is estimated by Vˆ (x(i) ) with (4.3). By choosing {(x(1) , Vˆ (x(1) )), . . . , (x(i) , Vˆ (x(i) ))} as the training set, μi (x) and σi (x) for the unknown Lyapunov function V (x) can be updated according to (4.4). On the other hand, if the state trajectory φ(x(i) , t), t ≥ 0 fails to converge to the origin, the sampling point x(i) is removed from the sampling region X . The above process does not terminate until it achieves the specified number of stable sampling points. Figure 4.3 illustrates three different sampling schemes, i.e., Scheme A, Scheme B, and Scheme C which correspond to sampling points A, B, and C, respectively. Mathematically, Scheme A can be described as x(i) = argmaxx∈X μi−1 (x) and Scheme B is expressed as x(i) = argmaxx∈X σi−1 (x). In practice, Scheme A is too greedy and tends to get local optima, while Scheme B provides a good rule for exploring V (x) globally [9]. As a result, Scheme C is designed to integrate A with B by adopting x(i) = argmaxx∈X Ji (x), where the reward function Ji (x) is given by 1/2

Ji (x) = μi−1 (x) + βi σi−1 (x). In addition, the parameter βi depends on the “complexity” of Lyapunov function, the sample size and information gain (see Appendix 4.7.3). Remark 4.2 To obtain the global maximum of Ji (x) in the sampling region X is generally infeasible due to the non-convexity of Ji (x). In practice, there are multiple ways to heuristically search for its local maxima. For instance, the local maxima can be readily identified from the finite historic dataset of sampling points instead of the region X . In addition, it is expected to approach the local maxima along the gradient ascent of Ji (x), and the gradient can be approximated with the finite difference method [21].

66

4 Security Monitoring Using Converse Lyapunov Function 4

3 A 2

C

V(x)

B 1

0

−1

−2 −1

0

−0.5

0.5

1

x Fig. 4.3 Illustration on three different sampling rules. The blue line denotes mean values of the unknown function V (x), and the gray shade describes the 95% confidence interval. The three red circles A, B and C represent three different sampling rules, respectively. A prefers the point with the maximum posterior mean value in order to achieve the large V (x), and B selects the point with the maximum posterior variance in order to reduce the uncertainty. C aims to allow for both the exploitation for large V (x) and the exploration for eliminating the uncertainty

Remark 4.3 For a certified ROA, it is sufficient to judge a stable sampling point if the corresponding state trajectory can overpass the boundary of this ROA without checking its convergence to the stable equilibrium point. This will reduce the computation time for selecting stable sampling points.

4.4 Main Results This section presents main theoretical results on the construction and evaluation of ROA with a given confidence level by using the GP-UCB based algorithm. Theorem 4.1 Let δ ∈ (0, 1), and the measurement noise is bounded by σ . Then it holds with probability at least 1 − δ that |V (x) − μ N −1 (x)| ≤ β N σ N −1 (x), ∀x ∈ X, ∀N ∈ Z + 1/2

where β N = 2V 2k + 300γ N ln3 (N /δ). Proof The results follow directly from Theorem 6 in [9].



Remark 4.4 The RKHS norm  · k characterizes the “complexity” or “smoothness” of the function in RKHS. If the bound of V 2k is unknown a prior, it can be computed using kernel ridge regression as V 2k = Vˆ NT P T diag(λi (λi + N θ )−2 )P Vˆ N ,

4.5 Numerical Simulations

67

where K N = P T diag(λi )P and diag(λi ) ∈ R N ×N denotes a diagonal matrix with the diagonal elements λi , i = 1, 2, . . . , N . Moreover, θ is a positive tunable parameter for controlling the smoothness of V (x) to avoid overfitting. A larger value of θ leads to the smoother function V (x) (see Appendix 4.7.4). For a stable equilibrium point, the Lyapunov function V (x) is viewed as a GP, and Theorem 4.1 allows to estimate the ROA of nonlinear system with a given confidence level. If the ROA  is already established according to an existing Lyapunov function V  (x), the proposed approach is applied to evaluate the stability of the region outside the ROA  by treating the mismatch between V (x) and V  (x) as a GP. Thus, theoretical results are summarized as follows. Theorem 4.2 Let δ ∈ (0, 1) and V  (x) be an existing Lyapunov function for nonlinear system x˙ = f(x). With GP-UCB based Algorithm in Table 4.1, it holds that Prob(x ∈ S) ≥ 1 − δ, ∀x ∈ Sδ,N where Sδ,N is given by 

x ∈ X | V  (x) + μ N −1 (x) + β¯N σ N −1 (x) ≤ Cmax,N 1/2



with Cmax,N = maxx(i) ∈A N Vˆ (x(i) ) and

2 β¯N = 2 V (x) − V  (x) k + 300γ N ln3 (N /δ). For a stable equilibrium point, the above conclusion also holds with V  (x) ≡ 0. Proof See Appendix 4.7.5.



Remark 4.5 There are alternative schemes to characterize the mismatch between the converse Lyapunov function V (x) and the existing Lyapunov function V  (x) other than the difference scheme V (x) = V (x) − V  (x). For example, it is also feasible to adopt the proportion scheme V (x)/V  (x) as an unknown function for the GP learning. Essentially, a stable equilibrium point can be thought of as a special case of the ROA when the ROA shrinks into a point and thus the existing Lyapunov function becomes V  (x) ≡ 0.

4.5 Numerical Simulations This section presents simulation results using the GP-UCB based algorithm for both single-machine infinite bus (SMIB) system and IEEE 39 bus system. First of all, the swing dynamics of a power system are introduced as follows.

68

4 Security Monitoring Using Converse Lyapunov Function

4.5.1 Power System Model Consider a M-bus power system described by a set of swing equations as follows [12, 23]  1 m i ψ¨ i + di ψ˙ i = pi − sin(θij + ψi j ), (4.5) B i j j∈N i

where ψi = θi − θi , ψi j = ψi − ψ j and θij = θi − θ j . Note that θi denotes the angle of the generator at bus i, and the superscript  represents the steady state condition or the power flow solution. The swing equation (4.5) describes the evolution of the phase angle due to the power mismatch between the mechanical power pi and the electrical power. In addition, Ni is the neighborhood set of bus i, and Bi j denotes the susceptance of branch i j. The parameters m i and di refer to inertia and damping coefficients for the machine, respectively. For a load bus i, we normally assume that m i = di = 0. In  the steady state condition, the mechanical power pi can be expressed as pi = j∈Ni 1/Bi j sin θij . This allows us to obtain the following perturbed dynamic model m i ψ¨ i + di ψ˙ i =

 1  sin θij − sin(θij + ψi j ) . Bi j j∈N i

A number of approaches to assess the stability of these systems are presented, and we use an energy-like function for the above system of nonlinear differential equations as follows ˙ ψ) = V (ψ, 

1 1 2 i=1 j∈N Bi j M

i

+

1 2

M 

ψi j

 sin θij − sin(θij + τ ) dτ

0

m i ψ˙ i2

i=1

and the origin (i.e., ψ = ψ˙ = 0) is locally asymptotically stable, and the certified ROA can be estimated by [12] 

˙ | ψ T Lψ + ψ˙ T ψ˙ ≤ Cλ ,  = (ψ, ψ)

(4.6)

where ψ = (ψ1 , . . . , ψ M )T and ψ˙ = (ψ˙ 1 , . . . , ψ˙ M )T . In addition, L denotes the Laplacian matrix of power network with weights 1/Bi j , and  is a diagonal matrix that satisfies  = diag(m) with m = (m 1 , . . . , m M )T . Moreover, Cλ is given by Cλ = mini j 1/Bi j (2 cos λ − (π − 2λ) sin λ) with λ = maxi j |θij |. For simplicity, the class  function α(z) in Definition 4.1 is chosen as α(z) = z 2 in order to estimate the value of converse Lyapunov function.

4.5 Numerical Simulations

69

4.5.2 SMIB System In this SMIB system, bus 1 is the generator bus which is connected to the infinity bus 2 where θ2 = ψ2 = 0. The swing equation for this SMIB system is given by m 1 ψ¨ 1 + d1 ψ˙ 1 =

1  sin θ1 − sin(θ1 + ψ1 ) B12

  where m 1 = 12, d1 = 20, p1 = 0.5 and B12 = 0.1, and thus we have sin θ12 = 0.05  and θ12 = arcsin(0.05). All values are in p.u. Then the ROA can be estimated by {(ψ1 , ψ˙ 1 )|10ψ12 + 12ψ˙ 12 ≤ 18.45} according to (4.6) and it is described by the red dashed ellipse in Fig. 4.4. The GP-UCB based Algorithm in Table 4.1 is adopted to assess the confidence level of operating states around the certified ROA. The parameter setting is given as follows: N = 100, δ = 0.05, ξ = 0.01, t = 0.01 and tn = 100. In addition, the squared exponential kernel or covariance function is employed to learn the unknown Lyapunov function with the unit characteristic length-scale and μ0 = 0. Figure 4.5 presents the simulation result for the SMIB system. The sampling points are denoted by small green dots, and the region of operating states with confidence level at least 95% has been marked in yellow. This implies that each state point in the yellow region converges to the origin with the probability that is not less than 95%. It is observed that the red dashed ellipse is closely surrounded by the yellow region. Notably, the yellow region covers most sampling points outside the certified ROA except for four sampling points that are far away from the origin.

5 4 3 2 1

ψ˙ 1

Fig. 4.4 Sampling points and state trajectories of SMIB system. Red dots represent the unstable sampling points that fail to converge to the origin, while green dots refer to stable sampling points that converge to the origin. Blue lines indicate their state trajectories, and the arrows point in the direction of state trajectories. The red dashed ellipse denotes the certified ROA according to (4.6)

0 −1 −2 −3 −4 −5 −5

0

ψ1

5

Fig. 4.5 Confidence evaluation for the ROA of SMIB system with 100 sampling points. The green dots denote the stable sampling points, and the red ellipse region refers to the certified ROA with an existing Lyapunov function. The yellow region indicates the state that converges to the origin with the probability at least 95%

4 Security Monitoring Using Converse Lyapunov Function 2 1.5 1 0.5

ψ˙ 1

70

0 −0.5 −1 −1.5 −2 −2

−1.5

−1

−0.5

0

ψ1

0.5

1

1.5

2

4.5.3 IEEE 39 Bus System In order to validate the scalability of the proposed approach, numerical simulations are also conducted on the IEEE 39-bus 10-machine system (see Fig. 4.6), and the parameters are the same as those in [24]. Bus 31 with a generator is assigned as the swing bus. We then consider the dynamics for the remaining 9 machines. The detailed parameter setting and the algorithm can be found in our MATLAB code made available in [25]. Figure 4.7 shows the assessment result for the ROA of IEEE 39 bus system with 9 machines. To facilitate the visualization, we project the stable sampling points, the certified ROA and the confidence region onto 9 distinct two-dimensional planes, respectively. Essentially, each plane acts as a cross section to showcase the profile of confidence region and certified ROA with respect to a different machine. In each panel of Fig. 4.7, the state points inside the red dashed ellipse is guaranteed to converge to the origin, while those in the yellow region are asymptotically convergent with the probability at least 95%. By Monte Carlo-like estimate, the volume of yellow region is about 2.3 × 104 times larger than that of the certified ROA. This achievement is due to the large number of state dimensions or the size of the considered dynamical system. Note that this comparison is not entirely fair as the certified ROA can ensure that the system state will always converge to the stable equilibrium if it starts from the inside, while our estimated yellow region is only 95% confident. Similar to the case of the SMIB system, the yellow region does not cover the state points that are relatively far away from the origin. Actually, the profile of yellow regions largely depends on the distribution and size of sampling points as well as the choice of kernel functions for GP learning.

4.5 Numerical Simulations

71

Fig. 4.6 IEEE 39 Bus System

4.5.4 Discussions The computational burden associated with the GP-UCB algorithm is mostly due to two processes: solving the dynamical equation or swing equation using ODE solvers, and GP optimization presented in (4.4). While the former depends heavily on the system size, the dimension of the differential equation for power systems does not result in the visible increase of computational cost in GP optimization. Indeed, the computational cost is essentially immune to the dimension of power system dynamics. This is because the computation burden largely results from the operation of matrix inverse in (4.4), where the size of kernel matrix K N mainly depends on the number of sampling points N rather than the dimension of power system dynamics. With Matlab 2017b in the desktop (Intel i7-3770 CPU 3.40 GHz and installed RAM 8GB), it takes around 10 min for IEEE 39-bus system and 50 s for the SMIB system in our simulations. For the large-size sampling data, many efficient approaches for training the GP with the desirable performance [26] are available. For instance, the sparse representation of GP model is developed to overcome the limitations for large data sets via the sequential construction of a sub-sample of the entire sampling data [27].

72

4 Security Monitoring Using Converse Lyapunov Function

Fig. 4.7 Confidence evaluation for the ROA of IEEE 39 bus system with 400 sampling points. In each panel, the green dots denote the stable sampling points, and the red dashed ellipse represents the boundary of a certified ROA. The yellow region indicates the state that converges to the origin with the probability at least 95%

4.6 Conclusions and Future Work In this chapter, we investigated the problem of estimating the ROA for power systems. By treating the unknown Lyapunov function as a Gaussian Process, we assessed the state stability of power systems with the aid of the converse Lyapunov function. For an existing Lyapunov function, our approach allows for extending the pre-existing ROA with a provable confidence level. In addition, a GP-UCB based algorithm was developed to deal with the trade-off between exploration and exploitation in selecting stable sampling points. Numerical simulations are conducted to validate the proposed approach on the IEEE test cases. In the next step, we will consider the online learning applications of unknown dynamics in practical power systems and the real-time prediction and stability assessment. This requires the creation of an efficient numerical algorithm with the aid of Gaussian Process and the converse Lyapunov function, which introduces the second part of this work. Another improvement is to optimally rescale state variables to be aligned to the shape of the real ROA which is not equal in all dimensions.

4.7 Appendix

4.7

73

Appendix

This section provides a mathematical definition of the class  function and theoretical proofs for Lemmas 4.2, 4.3 and Theorem 4.2, respectively. First of all, the definition is presented as follows.

4.7.1

The Class  Function

Definition 4.1 The class  function consists of all continuous functions α : [0, a) → [0, ∞] which satisfy the following conditions: 1. ∀z > 0, α(z) ∈ C 2 . 2. ∀z > y ≥ 0, α(z) > α(y) and α(0) = 0. 3. ∀z ≥ 0, ∃ m > 0, such that α(z) ≤ z m . Remark 4.6 Condition 1 in Definition 4.1 indicates that the first two derivatives of the class  function α(z) exist and are continuous. Condition 2 implies that α(z) is a strictly increasing function. In addition, Condition 3 aims to impose the restriction on the rate of α(z). Remark 4.7 The construction of V (x) in Lemma 4.1 allows us to obtain an important property on the region of attraction S (i.e., limc→+∞ c = S). Specifically, ∀x ∈ ∞ c , we have 0 α(φ(x, t))dt ≤ c, which implies limt→+∞ α(φ(x, t)) = 0 and limt→+∞ φ(x, t) = 0. Thus, we have x ∈ S and c ⊆ S. In addition, it is guaranteed that V (x) ≤ ∞, ∀x ∈ S. This indicates x ∈ ∞ and thus S ⊆ ∞ . It follows that c ⊆ S ⊆ ∞ . Considering that the constant c can be sufficiently large, we have limc→+∞ c = S.

4.7.2

Upper Bound of Discretizing Error

An upper bound of the estimation error due to discretizing converse Lyapunov function V (x) is given below. Lemma 4.2 Let [∂f/∂x](0) be Hurwitz. There are positive constants κ, η, λ and m such that the following inequality holds  κn(t)3  ηm φ(x, tn )m   + V (x) − Vˆ (x) ≤ 12 mλ

(4.7)

74

4 Security Monitoring Using Converse Lyapunov Function

The proof is given below. V (x) − Vˆ (x) = V (x) −

tn α(φ(x, t))dt 0

tn +

α(φ(x, t))dt −

n 

α(φ(x, ti ))t,

i=1

0

that      tn n         α(φ(x, ti ))t  ≤ V (x) − α(φ(x, t))dt  V (x) −     i=1 0  t   n  n     +  α(φ(x, t))dt − α(φ(x, ti ))t    i=1 0

∞ =

α(φ(x, t))dt tn

 t   n  n    +  α(φ(x, t))dt − α(φ(x, ti ))t    i=1 0

By the trapezoidal rule for numerical integration in the interval [0, tn ] [28], the error bound is given by   t   n n   κn(t)3  ≤  α(φ(x, t))dt − α(φ(x, t ))t i   12   i=1 0

where |∂tt α| ≤ κ, t ∈ [0, tn ]. Considering that [∂f/∂x](0) is Hurwitz, x = 0 is an exponentially stable equilibrium point from Corollary 4.3 in [3]. This implies that there are positive constants η and λ, such that φ(x, t) ≤ ηφ(x, tn )e−λ(t−tn ) , ∀t ≥ tn . In terms of Condition 3 of the  class function in Definition 4.1, there is the positive constant m such that α(φ(x, t)) ≤ α(ηφ(x, tn )e−λ(t−tn ) ) ≤ ηm φ(x, tn )m e−mλ(t−tn ) , ∀t ≥ tn

4.7 Appendix

75

Therefore, we obtain ∞

∞ α(φ(x, t))dt ≤

tn

ηm φ(x, tn )m e−mλ(t−tn ) dt

tn

∞ = η φ(x, tn ) m

m

e−mλ(t−tn ) dt

tn

ηm φ(x, tn )m = mλ This completes the proof. We discuss some notes on the Inequality (4.7) in the following remark. Remark 4.8 For the fixed total sampling time tn , the first term on the right hand side of Inequality (4.7) converges to 0 as the sampling time interval t goes to 0. Since limn→∞ φ(x, tn ) = 0, the second term on the right hand side of Inequality (4.7) converges to 0 as n goes to the positive infinity. Remark 4.9 The first partial derivative of α(φ(x, t)) with respect to time t is given by ∂t α =

∂α(φ(x, t)) ∂t

φ(x, t)T dφ(x, t) · φ(x, t) dt T f(φ(x, t)) φ(x, t) = α (φ(x, t)) · φ(x, t)

= α (φ(x, t)) ·

where α (z) = dα(z)/dz. And the second partial derivative of α(φ(x, t)) with respect to time t is given by ∂ 2 α(φ(x, t)) ∂t 2   ∂ φ(x, t)T f(φ(x, t)) α (φ(x, t)) · = ∂t φ(x, t)   [φ(x, t)T f(φ(x, t))]2 = α (φ(x, t)) − 1 · φ(x, t)3 f(φ(x, t))2 + φ(x, t)T ∂t f , + α (φ(x, t)) · φ(x, t)

∂tt α =

(4.8)

76

4 Security Monitoring Using Converse Lyapunov Function

where α (z) = d2 α(z)/dz 2 and ∂t f =

∂f(φ(x, t)) = f (φ(x, t))f(φ(x, t)). ∂t

Equation (4.8) enables us to estimate the upper bound of |∂tt α| in [0, tn ] and obtain the constant κ.

4.7.3

Information Gain

To quantify the reduction in uncertainty on V (x) from observations Vˆ (x), the information gain is introduced for a Gaussian process as follows I (VˆA ; VA ) =

1 log |I + σ −2 K A |, 2

where K A = [k(x, x )]x,x ∈A is the covariance matrix of VA = [V (x)]x∈A with the sample set A. Let γ N denote the upper bound of I (VˆA ; VA ) for the sample set A with |A| = N . That is γ N = max I (VˆA ; VA ) A⊂X :|A|=N

Actually, γ N is related to the choice of kernel functions k(x, x ). Lemma 4.3 For A ⊆ X ⊂ R n , it holds that γN ≤

N 2σ 2

when k(x, x ) ≤ 1. Proof Since K A is a positive definite kernel matrix, it follows that |I + σ −2 K A | =

N 

(1 + σ −2 λi )

i=1

where λi , i ∈ I N = {1, . . . , N } are all positive eigenvalues of K A . Considering that log(1 + x) ≤ x, ∀x ≥ 0, we have

4.7 Appendix

77

log |I + σ −2 K A | = log

N 

 (1 + σ −2 λi )

i=1

=

N 

log(1 + σ −2 λi )

i=1

≤ σ −2

N 

λi

i=1

Because of

N

λi = tr (K A ) and k(x, x ) ≤ 1, we obtain

i=1

N 

λi = tr (K A ) =

i=1

N 

k(xi , xi ) ≤ N

i=1

Therefore, we get log |I + σ −2 K A | ≤ σ −2 N , ∀x ∈ A, |A| = N and γN ≤

N 2σ 2

This completes the proof.



Remark 4.10 The upper bound for γ N in Lemma 4.3 is applied to all kernel functions satisfying k(x, x ) ≤ 1. For a specific kernel (e.g., finite dimensional linear kernel, squared exponential kernel and Matérn kernel, etc.), the tighter upper bound is available [9].

4.7.4

Computation of RKHS Norm

Kernel ridge regression can be formulated as a regularized empirical risk minimization problem over the RKHS Hk (X ) as follows min

V ∈Hk (X )

N 1  ˆ (i) (V (x ) − V (x(i) ))2 + θ V 2k . N i=1

It follows from Theorem 3.4 in [7] that there is a unique solution to the above N ci k(x, x(i) ) with minimization problem, and the solution is given by V (x) = i=1 T −1 ˆ the coefficient vector c = (c1 , c2 , . . . , c N ) = (K N + N θ I ) VN . Thus, V 2k can be computed by

78

4 Security Monitoring Using Converse Lyapunov Function

V 2k = V, V k =

N  N 

ci c j k(x(i) , x( j) ) = cT K N c.

i=1 j=1

By substituting the vector c = (K N + N θ I )−1 Vˆ N and the positive semidefinite matrix K N = P T diag(λi )P, we have V 2k = c T K N c = Vˆ NT (K N + N θ I )−1 K N (K N + N θ I )−1 Vˆ N = Vˆ NT (K N + N θ I )−1 P T diag(λi )P(K N + N θ I )−1 Vˆ N = Vˆ NT P T diag(λi (λi + N θ )−2 )P Vˆ N .

4.7.5

Proof of Theorem 4.2

The proof consists of two parts. The first part aims to estimate the ROA for a stable equilibrium point, and the second part centers on the ROA for the case of an existing Lyapunov function V  (x). (1) Since x = 0 is an asymptotically stable equilibrium point for the nonlinear system x = f(x), Lemma 4.1 guarantees the existence of Lyapunov function V (x). Considering a sequence of sampling points x(1) , x(2) ,…, x(N ) in A ⊂ X that can generate the stable state trajectory, the values of V (x(i) ), i ∈ {1, . . . , N } can be estimated as Vˆ (x(i) ) according to Eq. (4.3). By treating Vˆ (x(i) ) as the observations of a GP, the measurement noise  is uniformly bounded by σ , which satisfies ηm φ(x, tn )m κn(t)3 + σ ≤ 12 mλ from Lemma 4.2. In addition, it follows from Theorem 4.1 that with probability at least 1 − δ, the inequality |V (x) − μ N −1 (x)| ≤ β N1/2 σ N −1 (x) holds, where β N = 2V 2k + 300γ N ln3 (N /δ) and γ N ≤ allows us to deduce that the inequality

N 2σ 2

from Lemma 4.3. This

1/2

V (x) ≤ μ N −1 (x) + β N σ N −1 (x), ∀x ∈ X holds with probability at least 1 − δ. Then we define a level set for the converse Lyapunov function V (x) as W N = {x ∈ X | V (x) ≤ Cmax,N }, which includes all the stable state trajectories of sampling points x(i) , i ∈ {1, . . . , N } and is a compact subset of the region of attraction S (i.e., W N ⊆ S). Thus, if we have μ N −1 (x) +

References

79

1/2

β N σ N −1 (x) ≤ Cmax,N , the inequality V (x) ≤ Cmax,N holds with probability at least 1 − δ, which implies Prob(x ∈ W N ) ≥ 1 − δ, ∀x ∈ δ,N with  1/2 δ,N = x ∈ X | μ N −1 (x) + β N σ N −1 (x) ≤ Cmax,N . Considering that W N ⊆ S, we obtain Prob(x ∈ S) ≥ 1 − δ, ∀x ∈ δ,N . (2) Considering that V  (x) is an existing Lyapunov function for the nonlinear system x = f(x) with the certified ROA  , it is suggested that the origin is an asymptotically stable equilibrium point. This ensures the existence of a converse Lyapunov function V (x) according to Lemma 4.1. Define V (x) = V (x) − V  (x) as an unknown function for the GP learning. By observing the measurements Vˆ (x(i) ) = Vˆ (x(i) ) − V  (x(i) ) as a GP at the sampling points x(i) , i ∈ {1, . . . , N } according to GP-ROA based algorithm in Table 4.1, it follows from Theorem 4.1 that the inequality |V (x) − μ N −1 (x)| ≤ β¯N1/2 σ N −1 (x) holds with probability at least 1 − δ. By replacing V (x) with V (x) − V  (x), we obtain   V (x) − V  (x) − μ N −1 (x) ≤ β¯ 1/2 σ N −1 (x). N In light of the proof for a stable equilibrium point, it is concluded that for any x ∈ Sδ,N , the inequality Prob(x ∈ S) ≥ 1 − δ holds. This completes the proof.

References 1. Zhai, C., Zhang, H., Xiao, G., Pan, T.: A model predictive approach to protect power systems against cascading blackouts. Int. J. Electr. Power Energ. Syst. 113, 310–321 (2019) 2. Zhai, C., Nguyen, H., Xiao, G.: A robust optimization approach for terminating the cascading failure of power systems. Electr. Power Syst. Res. 189, 106794 (2020) 3. Khalil, H.K., Grizzle, J.W.: Nonlinear Systems, vol. 3. Prentice Hall, Upper Saddle River, NJ (2002) 4. Izumi, S., Somekawa, H., Xin, X., Yamasaki, T.: Estimation of regions of attraction of power systems by using sum of squares programming. Electr. Eng. 100(4), 2205–2216 (2018) 5. Chesi, G.: Domain of Attraction: Analysis and Control Via SOS Programming, vol. 415. Springer Science & Business Media (2011) 6. Bobiti, R., Lazar, M.: Automated-sampling-based stability verification and DOA estimation for nonlinear systems. IEEE Trans. Autom. Control 63(11), 3659–3674 (2018) 7. Kanagawa, M., Hennig, P., Sejdinovic, D., Sriperumbudur, B.K.: Gaussian processes and kernel methods: a review on connections and equivalences. arXiv preprint arXiv:1807.02582 (2018) 8. Rasmussen, C.E., Nickisch, H.: Gaussian processes for machine learning (GPML) toolbox. J. Mach. Learn. Res. 11, 3011–3015 (2010) 9. Srinivas, N., Krause, A., Kakade, S.M., Seeger, M.W.: Information-theoretic regret bounds for Gaussian process optimization in the bandit setting. IEEE Trans. Inform. Theor. 58(5), 3250–3265 (2012)

80

4 Security Monitoring Using Converse Lyapunov Function

10. Berkenkamp, F., Krause, A., Schoellig, A.P.: Bayesian optimization with safety constraints: safe and automatic parameter tuning in robotics. arXiv preprint arXiv:1602.04450 (2016) 11. Gibbs, M.N., MacKay, D.J.: Variational Gaussian process classifiers. IEEE Trans. Neural Netw. 11(6), 1458–1464 (2000) 12. Munz, U., Romeres, D.: Region of attraction of power systems. IFAC Proc. Volumes 46(27), 49–54 (2013) 13. Jones, M., Mohammadi, H., Peet, M.M.: Estimating the region of attraction using polynomial optimization: a converse Lyapunov result. In: 2017 IEEE 56th Annual Conference on Decision and Control (CDC), pp. 1796–1802. IEEE (2017, December) 14. Lesieutre, B., Wu, D.: An efficient method to locate all the load flow solutions-revisited. In: 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 381–388 (2015) 15. Liu, J., Cui, B., Molzahn, D.K., Chen, C., Lu, X.: Optimal power flow for DC networks with robust feasibility and stability guarantees (2019) 16. Kundur, P., Balu, N.J., Lauby, M.G.: Power System Stability and Control, vol. 7. McGraw-Hill, New York (1994) 17. Kimeldorf, G.S., Wahba, G.: A correspondence between Bayesian estimation on stochastic processes and smoothing by splines. Annals Math. Stat. 41(2), 495–502 (1970) 18. Mockus, J.: On Bayesian methods for seeking the extremum. In: Optimization Techniques IFIP Technical Conference, pp. 400–404. Springer, Berlin, Heidelberg (1975) 19. Mockus, J.: Bayesian Approach to Global Optimization: Theory and Applications, vol. 37. Springer Science & Business Media (2012) 20. Brochu, E., Cora, V.M., De Freitas, N.: A tutorial on Bayesian optimization of expensive cost functions, with application to active user modeling and hierarchical reinforcement learning. arXiv preprint arXiv: 1012.2599 (2010) 21. Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, vol. 12. Springer Science & Business Media (2013) 22. Zimmerman, R.D., Murillo-Sanchez, C.E., Thomas, R.J.: MATPOWER: steady-state operations, planning and analysis tools for power systems research and education. IEEE Trans. Power Syst. 26(1), 12–19 (2011) 23. Sauer, P.W., Pai, M.A.: Power System Dynamics and Stability, vol. 101. Prentice hall, Upper Saddle River, NJ (1998) 24. Dorfler, F., Jovanovic, M.R., Chertkov, M., Bullo, F.: Sparsity-promoting optimal wide-area control of power networks. IEEE Trans. Power Syst. 29(5), 2281–2291 (2014) 25. https://github.com/Chaocas/ROA-for-Power-Systems 26. Liu, H., Ong, Y.S., Shen, X., Cai, J.: When Gaussian process meets big data: a review of scalable GPs. arXiv preprint arXiv:1807.01065 (2018) 27. Csato, L., Opper, M.: Sparse on-line Gaussian processes. Neural Comput. 14(3), 641–668 (2002) 28. Atkinson, K.E.: An introduction to Numerical Analysis. Wiley (2008)

Chapter 5

Online Gaussian Process Learning for Security Assessment

Abstract The online small-signal stability assessment of complex power systems is typically a challenging problem due to uncertainties and parameter variations of system dynamics as well as the incurred high computational complexity. This chapter proposes a novel theoretical framework for dynamic small-signal stability assessment of power systems by estimating the region of attraction for operating states in real time. By analyzing the latest sampling data of power grids in a fixed time window, an up-to-date training set is constructed with the aid of converse Lyapunov function, which enables us to develop an online learning approach based on Gaussian Process (GP) to assess the stability level of power grids. As a result, an iteration algorithm is designed to update the assessment parameters by learning the input-output pairs in the training set. Theoretical analysis is conducted to ensure the existence of converse Lyapunov function for differential-algebraic system that serves to describe power system dynamics, as well as to estimate the region of attraction for operating states with a given confidence level. In particular, a practical method is proposed to leverage phasor measurement unit (PMU) data of real power grids for validating the online GP approach. Moreover, experimental validations are taken to substantiate the proposed online assessment approach by using PMU data of smart-grid infrastructure on EPFL campus. The proposed assessment approach contributes to situational awareness of human operators in the control station, thereby taking proactive remedial actions prior to emergencies.

5.1 Introduction The online security assessment of power system states is crucial to situational awareness and emergency decisions of operators in the control station, as it enables to take responsive regulation and corrections against disruptive contingencies [1]. False assessment may lead to major system blackouts such as August 10, 1996 blackout of the WECC that induces tremendous economic losses. Among the other things, power system security level can be assessed based on the stability margin of operating states. Nevertheless, frequent external disturbances (e.g., load variations and fluctuation of power generation) and internal parameter variations (e.g., changes © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Zhai, Control and Optimization Methods for Complex System Resilience, Studies in Systems, Decision and Control 478, https://doi.org/10.1007/978-981-99-3053-1_5

81

82

5 Online Gaussian Process Learning for Security Assessment

of branch impedance and device aging) make quantifying the stability margin and possible risks of power girds challenging. Small-signal stability of operating states is an important class of stability in power systems that characterizes the local behavior of the system associated with a particular equilibrium [2, 3]. Dynamic security assessment (DSA) of small-signal stability aims to evaluate the stability margin according to the dynamic response of power grids subject to contingencies. In practice, the DSA focuses on the stability of a given operating state determined either from measurement tools (e.g., supervisory control and data acquisition, PMU) or constructed in the framework of scenario analysis for planning or operation purposes [4]. By using PMU data, various machine learning based approaches have been proposed to investigate the DSA of power grids, such as decision trees [5, 6], generative adversarial network [7], extreme learning machine [8], semi-supervised learning and data editing [9]. Specifically, [5] proposes an online DSA scheme for large-scale power systems using phasor measurements and decision trees, which can identify stability problems for operating conditions and provide guidelines for preventive control. In addition, [7] designs a fully data-driven approach for pre-fault DSA of power grids in spite of incomplete data measurements. Moreover, [8] develops a transient stability assessment model with extreme learning machine to improve the learning speed and on-line updating. On the basis of semisupervised learning and data editing, [9] presents a new DSA framework to reduce the number of labeled samples, which can relieve the computation burden of training neural networks. Nevertheless, the online performance of existing approaches might not be sufficient for practical operation. The computational cost and using machine learning as a black box are the major hindrances to effective and reliable estimation of the security level of power systems under uncertainties online. Thus, the region of attraction around the equilibrium of power systems is estimated in this work to quantify the security margin of operating states. In addition, the GP approach is employed to deal with uncertainties of power grids by learning the sampling data with a priori assumption [10]. One major obstacle in the online application of GP approach is the non-Gaussianity of posterior process, which can be resolved by approximating this non-Gaussian posterior process using a Gaussian one [11]. The other obstacle results from the size of kernel matrix, which can be handled by the development of sparse approximation techniques [12]. By integrating the region of attraction in stability theory with the online GP approach, this work aims to propose an online DSA scheme for power grids. Compared with existing studies [5–9, 13, 14], key contributions of this work lie in 1. Construct the region of attraction (ROA) for a general differential-algebraic equation system with the aid of converse Lyapunov function. 2. Develop an online GP approach for the dynamic security assessment of power systems with limited sampling data in a time window. 3. Propose a practical method to validate the online GP approach by using PMU data of real power grids.

5.2 The ROA of DAE System

83

The remainder of this chapter is organized as follows. Section 5.2 introduces the region of attraction for a general differential-algebraic system with the aid of the converse Lyapunov theorem. Section 5.3 presents an online GP approach for learning the unknown Lyapunov function by using sampling data in a time window. Section 5.4 elaborates on the online DSA scheme and presents theoretical results. Experimental validations are conducted with real data of smart-grid infrastructure to substantiate the proposed approach in Sect. 5.5. Finally, Sect. 5.6 draws a conclusion and discusses future work.

5.2 The ROA of DAE System Consider the differential-algebraic equation (DAE) system as follows x˙ = f(x, y) 0 = g(x, y)

(5.1)

with x ∈ Rn and y ∈ Rm . And the functions f : Rn × Rm → Rn and g : Rn × Rm → Rm are twice continuously differentiable in an open connected set . Suppose that there exists an equilibrium point (x∗ , y∗ ) in the DAE system (5.1), and the partial derivative of g with respect to y has full rank on an open connected set that contains this equilibrium point [15]. This guarantees the existence and uniqueness of solutions to the DAE system (5.1) for any initial points (x0 , y0 ) that satisfies the algebraic equation 0 = g(x0 , y0 ) in the connected set [16]. Contraction analysis can be employed to construct the ROA of nonlinear DAE systems [17]. The regularity of the algebraic equation 0 = g(x, y) is defined in order to ensure the existence and uniqueness of solutions to the system (5.1) in a connected set [15]. Definition 5.1 The algebraic equation 0 = g(x, y) is regular if the Jacobian of g(x, y) with  respect to y has the full rank on the connected set , that is rank ∇y g(x, y) = m, ∀(x, y) ∈ . The regularity of the algebraic equation 0 = g(x, y) allows us to convert the DAE system (5.1) into an ordinary differential equation (ODE) system. Then the converse Lyapunov theorem can be employed to estimate the value of Lyapunov function without its analytic form [18]. For the DAE system (5.1) that has a stable state trajectory φ(x, t), t ≥ 0, a converse Lyapunov function can be constructed [19]. Here the stable state trajectory is defined as the trajectory that converges to a stable equilibrium point as time goes to infinity. The existence and construction of such converse Lyapunov function are presented as follows. Theorem 5.1 Without loss of generality, let the origin be an asymptotically stable equilibrium point for the DAE system (5.1), where f(x, y) is locally Lipschitz with respect to x, and 0 = g(x, y) is regular in the connected set  that contains the

84

5 Online Gaussian Process Learning for Security Assessment

origin. S is the region of attraction. Then there exist a continuous positive definite function W (x) and a smooth, positive definite function V (x) such that ∞ V (x) =

α( φ(x, t) )dt, V (0) = 0 0

and

with

dV (x) dt

=

∂ V (x) f ∂x

(x, Y(x)) ≤ −W (x), ∀ x ∈ S

dφ(x, t) = f (φ(x, t), Y(φ(x, t))) dt

(5.2)

(5.3)

and φ(x, 0) = x, where α(z) is a class  function (see Appendix 4.7.1). Y(x) is the implicit function determined by the equation 0 = g(x, y). The level set with c > 0 is given by c = {x ∈ Rn | V (x) ≤ c}, which is a compact subset of S = {x ∈ R n | limt→+∞ φ(x, t) = 0}. 

Proof See Appendix 5.7.1.

In Theorem 5.1, the differential equation x˙ = f(x, y) normally represents swing equation of generators in power systems [20]. The algebraic equation 0 = g(x, y) characterizes the power flow distribution. Inequality (5.2) indicates that the Lyapunov function V (x) decays over time, while Equation (5.3) defines a stable trajectory φ(x, t) with the initial state x. For the Lyapunov function V (x) constructed in Theorem 5.1, it is feasible to approach the real ROA by enlarging the level set. This property may not hold for any analytical Lyapunov function. This advantage enables us to obtain a better ROA by collecting more sampling points in order to enlarge the level sets. In practice, the converse Lyapunov function V (x) proposed in Theorem 5.1 can be estimated by Vˆ (x) =

n 

α( φ(x, ti ) )t,

(5.4)

i=1

where t denotes the sampling time interval and ti = (i − 1)t, i ∈ {1, 2, . . . , n}. While Vˆ (x) can be calculated directly using (5.4) and the sampling data, the analytical Lyapunov function V (x) is unknown. This work aims to learn this unknown Lyapunov function V (x) online by capitalizing on the discrete sampling data and values of converse Lyapunov function in a time window. By treating V (x) and Vˆ (x) as a GP and its measurement, respectively, the estimation error V (x) − Vˆ (x) can be regarded as the measurement noise. This enables us to learn the unknown Lyapunov function V (x) using the online GP approach. In comparison, the eigenvalue-based approaches are effective for a fixed operating point but degrade quickly when the system moves

5.3 The Windowed Online GP

85

Fig. 5.1 The time window with h sampling points at the N -th sampling step. The red dashed rectangle refers to the time window, and it includes h sampling points, which are denoted by the squares with blue boundaries. At each sampling step, the newest sampling point enters the time window, while the oldest one is removed from the window. For example, x(N ) enters the time window WhN and x(N −h) goes out of it at the N -th sampling step

away from such an operating point [21]. Most importantly, the dependence of critical eigenvalues on system parameters is highly nonlinear and lacks of patterns. The critical and noncritical eigenvalues can switch while scaling system parameters. This makes the assessment of small-signal stability via tracking critical eigenvalues cumbersome and thus becomes a major challenge for online applications.

5.3 The Windowed Online GP The above section introduces the approximated Lyapunov function Vˆ (x) and the unknown Lyapunov function V (x). In this section, we propose the windowed online GP approach for learning the unknown Lyapunov function.

5.3.1 GP Regression Normally, a general GP regression requires a prior distribution of unknown functions specified by a mean function, a covariance function, and the probability of the observations and sampling data to obtain the posterior distribution. Without loss of generality, we consider the unknown Lyapunov function V (x) as a GP, which can be sequentially measured by y (i) = V (x(i) ) + , i ∈ Z + , where y (i) refers to the observed function value for the input x(i) at the i-th sampling step, and the measurement noise  is zero-mean, independent and bounded by σ . With the GP approach, we can obtain the posterior distribution over V (x) by using sampling data in the training set. By regarding the values of V (x) as random variables, any finite collection of them is multivariate distributed in an overall consistent way. The unknown Lyapunov function V (x) can be approximated by a GP. Note that the covariance or kernel function k(x, x ) encodes the smoothness property of V (x) from the GP. Essentially, the estimated value of Lyapunov function at a sampling point can be regarded as one observation of GP

86

5 Online Gaussian Process Learning for Security Assessment

Vˆ (x(i) ) = V (x(i) ) + , i ∈ Z + with  ∼ N (0, σ 2 ). Then the first N observations form the vector ˆ N = [Vˆ (x(1) ), . . . , Vˆ (x(N ) )]T . V There are the analytic formulas for mean μ N (x), covariance k N (x, x ) and standard deviation σ N (x) of the posterior distribution as follows [22] ˆN μ N (x) = k N (x)T (K N + σ 2 I N )−1 V k N (x, x ) = k(x, x ) − k N (x)T (K N + σ 2 I N )−1 k N (x )  1 σ N (x) = k N (x, x) 2

(5.5)

where k N (x) = [k(x(1) , x), . . . , k(x(N ) , x)]T and K N is the positive definite kerN N N . And the set W∞ is given by W∞ = {x(1) , nel matrix [k(x, x )] with x, x ∈ W∞ (2) (N ) x , . . . , x }. Although it is convenient to adopt the formula (5.5) for the inference, non-Gaussian of the posterior distribution and the size of matrix K N preclude its direct application [10]. In practice, the proper selection of the width of time window can resolve the problem of matrix size due to large datasets. The non-Gaussian of the posterior distribution can be handled by the approximation of Gaussian ones. Thus, an online algorithm is developed to integrate the windowed GP with the approximation of non-Gaussian posterior process.

5.3.2 Windowed Online GP By approximating the posterior in the sense of the Kullback-Leibler divergence, the online GP is given by [10] μ N (x) = μ N −1 (x) + q N · k N −1 (x, x(N ) ) k N (x, x ) = k N −1 (x, x ) + r N · k N −1 (x, x(N ) )k N −1 (x(N ) , x )  1 σ N (x) = k N (x, x) 2

(5.6)

where q N and r N are updated as follows qN = rN =

∂ ∂ E[V N ] N −1 ∂2 ∂ E[V N ]2N −1

ln E[ p(Vˆ N |V N )] N −1 ln E[ p(Vˆ N |V N )] N −1

(5.7)

with V N = V (x(N ) ) and Vˆ N = Vˆ (x(N ) ). Due to limited resources of computation and the evolution of power systems, it is necessary to remove the old sampling points and add the latest ones into the sampling set when the number of sampling points is larger than a given threshold. Thus, a time window is introduced to include the

5.3 The Windowed Online GP

87

latest h sampling points for GP learning. In order to allow for the evolution of power systems and relieve the computational burdens, a time window is constructed to take into account the latest h sampling points for the GP learning (see Fig. 5.1). Thus, a set of sampling data is defined as  WhN = x(N −h+1) , x(N −h+2) , . . . , x(N −1) , x(N ) to include the latest h sampling points at the N -th sampling step. By unfolding the recursion steps in (5.6), the parametrization of approximate posterior GP can be obtained as follow μhN (x) = (α N )T khN (x)

khN (x, x ) = k(x, x ) +

x(i) ,x( j) ∈WhN

σhN (x)

CiNj · k(x, x(i) )k(x( j) , x )

 1 = khN (x, x) 2

(5.8)

where khN (x) = [k(x(N −h+1) , x), . . . , k(x(N ) , x)]T . Here WhN denotes the set of sampling points in the time window at the N -th iteration. Let KhN = {k(x, x )} ∈ R h×h represent the kernel matrix with x, x ∈ WhN . Then an operator R is introduced to update KhN as follows KhN = R(KhN −1 ) + [0h×(h−1) , T (khN )] + [0h×(h−1) , T (khN )]T with

(5.9)

khN = [k(x(N −h+1) , x(N ) ), . . . , k(x(N −1) , x(N ) )]T .

The definition of the operator R is presented in Appendix 5.7.2, and the coefficients α N and C N = {CiNj } with x(i) , x( j) ∈ WhN are obtained according to the updating rule: khi si αi Ci

= (KhN ehi ) = T (Ci−1 khi ) + ei+h−N = T (α i−1 ) + q i · si = U (Ci−1 ) + r i · si (si )T ,

(5.10)

where khi = [k(x(i−h+1) , x(i) ), . . . , k(x(i−1) , x(i) )]T and the iteration number i increases sequentially from N − h + 1 to N . Moreover, ehi represents a h dimensional unit vector with the i-th element being 1. And is an operator that can construct a vector by extracting the first (i − 1) elements from a given vector. T and U are two operators that extend the vector and matrix by one dimension, respectively. Specifically, T adds zero at the end of the vector, and U appends zeros to the last row and column of the matrix. In addition, ei+h−N refers to the (i + h − N )-th unit vector.

88

5 Online Gaussian Process Learning for Security Assessment

Fig. 5.2 Information flow of the online DSA scheme for a smart grid

5.4 Security Assessment Scheme For a given value of δ ∈ (0, 1) and the sampling domain X ∈ R n , which is a subset of the operating state space, our goal is to estimate the region of attraction online, wherein each point converges to the origin with the probability of δ at least. Thus, a security assessment scheme is developed in Table 5.1 to select the sampling points for enlarging the ROA with a guaranteed confidence level. Specifically, a sampling point x(N ) is selected in X at the N -th sampling step by searching for the maxima of μhN −1 (x) + βδ · σhN −1 (x), where the term μhN −1 (x) helps to enlarge the level set of Lyapunov function and the term σhN −1 (x) allows to reduce the uncertainty of sampling region. Essentially, the sampling rule aims to reconcile the trade-off between the exploitation for enlarging the ROA and the exploration for reducing the uncertainty of sampling region. Let x(N ) serve as the sampling point of the DAE system (5.1), and it enables us to obtain a state trajectory φ(x(N ) , t), t ≥ 0. For a real power grid, φ(x(N ) , t) can be reconstructed from its PMU data. If this state trajectory can converge to the origin, the point x(N ) is called as a stable sampling point. Then the value of Lyapunov function at x(N ) is estimated by Vˆ (x(N ) ) with (5.4). The time window WhN is updated by removing an old sampling point x (N −h) and including a new one x N . By choosing {(x(N −h+1) , Vˆ (x(N −h+1) )), . . . , (x(N ) , Vˆ (x(N ) ))} as the training set, μhN (x) and σhN (x) for the unknown Lyapunov function V (x) can be updated according to (5.8). Note that the initial training set is assigned as {(0, 0), (0, 0), . . . , (0, 0)}. If the state trajectory φ(x(N ) , t), t ≥ 0 fails to converge to the origin, the sampling point x(N ) will be reselected according to the sampling rule. Finally, the region of attraction can be estimated online by constructing δN with (5.11).

5.4 Security Assessment Scheme

89

Table 5.1 Online DSA scheme Input: X ∈ Rn , δ, ξ , tn , h, μ0 , σ0 , k(x, x ), N = 1 Output: WhN , μhN (x), σhN (x), δN 1: while (true) 2:

 Choose x(N ) = arg maxx∈X μhN −1 (x) + βδ · σhN −1 (x)

3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14:

Obtain φ(x(N ) , t) from (5.1) or PMU data if ( φ(x(N ) , tn ) < ξ ) Sample Vˆ (x(N ) ) = V (x(N ) ) +  with (5.4) Update WhN = WhN −1 ∪ {x(N ) } \ {x(N −h) } Compute α N and C N with (5.9) and (5.10) Update μhN (x) and σhN (x) with (5.8) else Go to Step 2 end if Construct δN with (5.11) Update N = N + 1 end while

If a certified ROA is available, it is sufficient to judge a stable sampling point if the corresponding state trajectory can enter this ROA. Each state in the ROA is guaranteed to approach the stable equilibrium point as time goes to the infinity. This can reduce the computation time for determining the stable sampling points. Next, we present theoretical results on the construction and evaluation of ROA with a given confidence level by using the Security Assessment Scheme in Table 5.1. N = maxx(i) ∈WhN Vˆ (x(i) ). Then the region of Theorem 5.2 Let δ ∈ (0, 1) and Vˆmax attraction of DAE system (5.1) at the N -th sampling step is given by



N δN = x ∈ R n |μhN (x) + βδ · σhN (x) ≤ Vˆmax

(5.11)

with the probability of δ and βδ = −1 ( 1+δ ), where −1 is the inverse cumulative 2 distribution function of the standard normal distribution. Proof See Appendix 5.7.3.



Theorem 5.2 allows to estimate the ROA of the DAE system (5.1) with a certain probability. If a certified ROA can be obtained, the proposed scheme can be employed to assess the stability of sampling region outside the certified ROA. Besides the ROA, the proposed scheme is applied to a more general concept of security for power systems. For example, it is also feasible to estimate the invariant set of the system (5.1) for the online security assessment.

90

5 Online Gaussian Process Learning for Security Assessment

Fig. 5.3 Configuration of smart-grid infrastructure on EPFL campus [24]

5.5 Case Study This section presents the case study of a real smart grid on EPFL campus by implementing the online DSA Scheme in Table 5.1. Figure 5.2 illustrates the information flow and data processing for the online DSA of a smart grid. First of all, the measurement data are collected from a smart grid using measurement devices (e.g., PMU). Then the sampling data are sent to a time window WhN that can accommodate the latest h sampling points, and the out-of-date sampling points are deleted from the time window in order to reflect the recent system evolution and reduce the computational burden. For each sampling point x N , the state trajectory φ(x N , t) is used to estimate the value of Lyapunov function Vˆ N . The h sampling points in the time window are used to train the GP model and implement the online DSA scheme, which can provide the real-time estimation of ROA for smart grids. Matlab codes for the online DSA scheme are available at GitHub [23].

5.5.1 Validations with PMU Data The online DSA scheme in Table 5.1 can be directly applied to the PMU data from a real power grid on EPFL campus as shown in Fig. 5.3. The active power injections of this smart-grid infrastructure with variable load demands include 2MW of photovoltaic panels and 6MW of combined heat and power generation units. In addition, PMUs are deployed at 5 locations of smart grid with the refresh rate of 20 ms

5.5 Case Study

91 10 4

U [V]

1.2195 1.219 1.2185 0

5

10

15

20

25

30

20

25

30

20

25

30

[rad]

Time [s] 2 1 0

0

5

10

15

[rad/s]

Time [s] 0 -0.1 -0.2 0

5

10

15

Time [s]

Fig. 5.4 Time series of voltage magnitude, phase and angular velocity from one PMU in EPFL smart-grid infrastructure

[24]. Different from numerical simulators, it is difficult to determine the equilibrium of real power grids due to uncertainties of power system dynamics and frequent external disturbances. To address this issue, a new method is proposed in this work to determine the quasi-equilibrium of real power grids by processing PMU data. Figure 5.4 presents time series of power system states (i.e., voltage magnitude, phase and angular speed) from one PMU of EPFL smart-grid infrastructure in 30 s. In theory, power system states keep unchanged at the equilibrium, which implies that their time derivatives are equal to zero at the equilibrium. As a result, power system states approach the equilibrium as their time derivatives get close to zero. In order to quantify the distance between the current operating state and the equilibrium of smart grids, a metric d(t) is introduced as follows  ˙  d(t) =  sup U (t) |U˙ (τ )| τ ∈I

θ˙ (t) ω(t) ˙ ˙ )| supτ ∈I |ω(τ ˙ )| supτ ∈I |θ(τ

  

2

with power system states U (t), θ (t), ω(t) and a prespecified time interval I . By setting the threshold  and solving t ∗ = inf{t : d(t) ≤ }, the quasi-equilibrium of power grid can be determined as (U (t ∗ ), θ (t ∗ ), ω(t ∗ )). By choosing  = 0.05 and I = [0, 30 s], the quasi-equilibrium of smart grid can be identified as (12.19 kV, 2.46 rad, 0.002 rad/s) and t ∗ = 4.38 s (see the red dot on the green line in Fig. 5.5). After the coordinate transformation and normalization, time response of power system states can be reconstructed as

92

5 Online Gaussian Process Learning for Security Assessment

Fig. 5.5 Identification of quasi-equilibrium of smart grid using PMU data

1.5 Distance to equilibrium Threshold =0.05

1

0.5

0 0

5

10

15

20

25

30

Time [s]

⎞ U (t)−U (t ∗ ) ⎞ ⎛ supτ ∈[0, t ∗ ] |U (τ )−U (t ∗ )| U¯ (t) ∗ ⎜ ⎟ θ(t)−θ(t ) x(t) = ⎝ θ¯ (t) ⎠ = ⎝ supτ ∈[0, t ∗ ] |θ(τ )−θ(t ∗ )| ⎠ . ∗ ω(t)−ω(t ) ω(t) ¯ ∗ ⎛

supτ ∈[0,

t∗]

(5.12)

|ω(τ )−ω(t )|

By selecting h different time points ti ∈ [0, t ∗ ], i ∈ {1, 2, . . . , h}, the training set of a time window can be created as

 (x(t1 ), Vˆ (x(t1 )), (x(t2 ), Vˆ (x(t2 )), . . . , (x(th ), Vˆ (x(th )) according to (5.4) and (5.12). With the above training set, the ROA of power system states can be estimated online via the scheme in Table 5.1. Once a new quasiequilibrium of power grid is identified, the training set will be updated as well for the online ROA estimation. By projecting the estimated ROA of power system states ¯ θ¯ − ω¯ and ω¯ − U¯ ), Fig. 5.6 presents onto three different planes (i.e., Planes U¯ − θ, the ROA snapshots in four time instants by using PMU data from EPFL smart-grid infrastructure. In each subfigure, the red star denotes the quasi-equilibrium, and the blue circle indicates the operating state of smart grid. The green area represents the ROA of power system states with the confidence level of 95%, and it updates online according to training data in a time window with the width h = 20 and the sampling period 0.02 s. As is observed, the green area gradually increases and covers the quasi-equilibrium of smart grid as more latest training data goes into the time window. The blue circle is always located in the green area, which is in accordance with the fact that the operating states converge to the quasi-equilibrium. In practice, the green area enables us to assess the domain of security for power grids in real time, as well as to provide the awareness of abnormal operating states timely. In addition, the actual power system states U (t), θ (t) and ω(t) can be obtained by solving (5.12) with U¯ (t), θ¯ (t) and ω(t), ¯ respectively.

5.5 Case Study

93

Fig. 5.6 Snapshots of estimated ROA on three different projection planes for EPFL smart-grid infrastructure

5.5.2 Discussions The computational cost associated with online DSA Scheme in Table 5.1 mainly results from two factors: the acquisition of state trajectories in power grids and the implementation of online GP algorithm as presented in (5.8), (5.9) and (5.10). For power system model, the former is related to the structure and dimension of DAE system. The computation burden depends on the number of training points h in a time window rather than the dimension of power system dynamics. With Matlab 2019b in the desktop (Intel i5-8500 CPU 3.00 GHz and installed RAM 8 GB), it takes around 1s for the experimental validation to update the estimated ROA in real time. For large-size sampling data, the sparse representation of GP model can be employed to overcome the size limitation by reconstructing a sub-sample of the whole sampling data [25]. Essentially, the introduction of time window allows to reconstruct sampling data for online GP learning. When PMU data are available from real power grids, the computational cost mainly results from the width of time window h. In practice, it is feasible to relieve the computational burdens of online DSA schemes with the guaranteed confidence level by tuning h properly.

94

5 Online Gaussian Process Learning for Security Assessment

5.6 Conclusions and Future Work In this chapter, an online GP approach was developed to assess the security of operating states of power systems by learning the sampling data in a finite time window. Theoretical analysis was conducted to estimate the real-time domain of stability for power system states with a guaranteed confidence level. Experimental validations were carried out to substantiate the proposed approach by using the PMU data from a real smart grid. The proposed approach enables us to quantify the security level of operating states in real time, as well as to provide timely situational awareness for operational decision making. For the DSA of power systems, it is of great significance to enlarge the domain of security by capitalizing on historical sampling data. This requires the acquisition of sampling data from power grids subject to “large” disturbances and the effective integration of GP approaches with online learning strategies. As a result, future work may include the filtering of redundant sampling data in the time window, the identification of sampling data with “large” disturbances and the integration of real-time protection schemes for enhancing the resilience of power systems [26].

5.7 5.7.1

Appendix Proof of Theorem 5.1

It follows from the implicit function theorem and the regularity of g(x, y) with respect to y that there is a neighborhood U ∈ R n and a unique twice differentiable function Y : Rn → Rm such that 0 = g(x, Y(x)), x ∈ U [16]. Thus, the DAE system (5.1) reduces to the ODE system x˙ = f(x, Y(x)), x ∈ U . Since the origin is a stable equilibrium point of the DAE system (5.1), 0 is a stable equilibrium point of the ˙ = f (x, Y(x)). It follows from Theorem 4.17 in [18] that there exist a ODE x smooth, positive definite function V (x) and a continuous, positive definite function W (x) such that ∂ V (x) f (x, Y(x)) ≤ −W (x), ∀ x ∈ S ∂x with the converse Lyapunov function given by ∞ α( φ(x, t) )dt, V (0) = 0

V (x) = 0

according to Lemma 1 in [27]. This completes the proof.

5.7 Appendix

5.7.2

95

The Operator R

For any matrix D ∈ Rh×h , the operator R allows to move all elements of D up along its main diagonal by one slot. Mathematically, it is described as   T  0h−1 0 0h−1 Ih−1 D R(D) = T 0 0h−1 Ih−1 0h−1 

where Ih−1 denotes the (h − 1) dimensional unit matrix and 0h−1 refers to the (h − 1) dimensional zero vector.

5.7.3

Proof of Theorem 5.2

For the sampling points in WhN and the fixed x ∈ X , it follows from (5.8) that V (x) ∼  N N N μh (x), σh (x) , which leads to V (x) − μhN (x) ∼ N (0, 1). σhN (x) Thus, for a positive constant c, it holds that  

  V (x)−μ N (x)  Prob  σ N (x)h  ≤ c = h

√1 2π

c

τ2

e− 2 dτ

−c

= 2(c) − 1 where  denotes the cumulative distribution function (CDF) of the standard normal distribution. Since σhN (x) > 0, this indicates that   Prob V (x) − μhN (x) ≤ c · σhN (x) = 2(c) − 1, which is equivalent to   Prob V (x) − μhN (x) ≤ βδ · σhN (x) = δ with δ = 2(c) − 1 and βδ = −1 ( 1+δ ). Therefore, x is in the level set {x ∈ 2 n N ˆ R |V(x) ≤ Vmax } with the probability of δ when it satisfies the inequality N μhN (x) + βδ · σhN (x) ≤ Vˆmax

96

5 Online Gaussian Process Learning for Security Assessment

N with Vˆmax = maxx(i) ∈WhN Vˆ (x(i) ). Considering that the level set is a compact subset of region of attraction S, the estimated region of attraction with the probability of δ is given by 

N . δN = x ∈ R n |μhN (x) + βδ · σhN (x) ≤ Vˆmax

This completes the proof.

References 1. Zhai, C., Zhang, H., Xiao, G., Pan, T.: A model predictive approach to protect power systems against cascading blackouts. Int. J. Electr. Power Energ. Syst. 113, 310–321 (2019) 2. Nguyen, H., Turitsyn, K.: Robust stability assessment in the presence of load dynamics uncertainty. IEEE Trans. Power Syst. 31(2), 1579–1594 (2016) 3. Qin, B., Ma, J., Li, W., Ding, T., Sun, H., Zomaya, A.: Decomposition-based stability analysis for isolated power systems with reduced conservativeness. IEEE Trans. Autom. Sci. Eng. 17(3), 1623–1632 (2020) 4. Huang, J., Valette, A., Beaudoin, M., Morison, K., Moshref, A., Provencher, M., Sun, J.: An intelligent system for advanced dynamic security assessment. In: Proceedings of International Conference on Power System Technology, vol. 1, pp. 220–224 (2002) 5. Sun, K., Likhate, S., Vittal, V., Kolluri, V., Mandal, S.: An online dynamic security assessment scheme using phasor measurements and decision trees. IEEE Trans. Power Syst. 22(4), 1935– 1943 (2007) 6. He, M., Zhang, J., Vittal, V.: Robust online dynamic security assessment using adaptive ensemble decision-tree learning. IEEE Trans. Power Syst. 28(4), 4089–4098 (2013) 7. Ren, C., Xu, Y.: A fully data-driven method based on generative adversarial networks for power system dynamic security assessment with missing data. IEEE Trans. Power Syst. 34(6), 5044–5052 (2019) 8. Xu, Y., Dong, Z., Meng, K., Zhang, R., Wong, K.: Real-time transient stability assessment model using extreme learning machine. IET Gener. Transm. Distrib. 5(3), 314–322 (2011) 9. Liu, R., Verbic, G., Ma, J.: A new dynamic security assessment framework based on semisupervised learning and data editing. Electr. Power Syst. Res. 172, 221–229 (2019) 10. Csato, L., Opper, M.: Sparse on-line Gaussian processes. Neural Comput. 14(3), 641–668 (2002) 11. Seeger, M.: Bayesian model selection for support vector machines, Gaussian processes and other kernel classifiers. In: Advances in Neural Information Processing Systems, pp. 603–609 (2000) 12. Smola, A., Schölkopf, B.: Sparse greedy matrix approximation for machine learning. In: International Conference on Machine Learning (2000) 13. Berkenkamp, F., Krause, A., Schoellig, A.P.: Bayesian optimization with safety constraints: safe and automatic parameter tuning in robotics. arXiv preprint arXiv:1602.04450 (2016) 14. Münz, U., Romeres, D.: Region of attraction of power systems. IFAC Proc. Volumes 46(27), 49–54 (2013) 15. Persis, C., Monshizadeh, N., Schiffer, J., Dorfler, F.: A Lyapunov approach to control of microgrids with a network-preserved differential-algebraic model. In: Proceedings of IEEE Conference on Decision and Control, December 2016 16. Hill, D., Mareels, I.: Stability theory for differential/algebraic systems with application to power systems. IEEE Trans. Circuits Syst. 37(11), 1416–1423 (1990) 17. Nguyen, H., Vu, T., Slotine, J., Turitsyn, K.: Contraction analysis of nonlinear DAE systems. arXiv preprint arXiv:1702.07421 (2017)

References

97

18. Khalil, H., Grizzle, J.: Nonlinear Systems, vol. 3. Prentice Hall, Upper Saddle River, NJ (2002) 19. Jones, M., Mohammadi, H., Peet, M.M.: Estimating the region of attraction using polynomial optimization: a converse Lyapunov result. In: Proceedings of the 56th Conference on Decision and Control, Dec. 2017, Melbourne, Australia, pp. 1796-1802 (2017) 20. Kundur, P., Balu, N., Lauby, M.: Power System Stability and Control, vol. 7. McGraw-Hill, New York (1994) 21. Pareek, P., Turitsyn, K., Dvijotham, K., Nguyen, H.D.: A sufficient condition for small-signal stability and construction of robust stability region. In: IEEE Power and Energy Society General Meeting, pp. 1–5 (2019) 22. Srinivas, N., Krause, A., Kakade, S., Seeger, M.: Information-theoretic regret bounds for Gaussian process optimization in the bandit setting. IEEE Trans. Inform. Theor. 58(5), 3250– 3265 (2012) 23. https://github.com/Chaocas/DSA 24. Pignati, M., et al.: Real-time state estimation of the EPFL-campus medium-voltage grid by using PMUs. In: 2015 IEEE Power and Energy Society Innovative Smart Grid Technologies Conference, pp. 1–5, 18 Feb. 2015 25. Liu, H., Ong, Y., Shen, X., Cai, J.: When Gaussian process meets big data: a review of scalable GPs. arXiv preprint arXiv:1807.01065 (2018) 26. Wu, D., Yang, T., Stoorvogel, A., Stoustrup, J.: Distributed optimal coordination for distributed energy resources in power systems. IEEE Trans. Autom. Sci. Eng. 14(2), 414–424 (2017) 27. Zhai, C., Nguyen, H.D.: Region of attraction for power systems using Gaussian process and converse Lyapunov function—Part I: theoretical framework and off-line study. arXiv preprint arXiv:1906.03590 (2019)

Chapter 6

Risk Identification of Cascading Process Under Protection

Abstract This chapter aims to identify and analyze the initial contingencies or disturbances that could lead to the worst-case cascading failures of power grids. An optimal control approach is proposed to determine the most disruptive disturbances on the branch of power transmission system by regarding the disturbances as the control inputs. Moreover, protective actions such as load shedding and generation dispatch are taken into account in a convex optimization framework to prevent the cascading outages of power grids. In theory, the necessary conditions for identifying the most disruptive disturbances are obtained by solving an integrated system of algebraic equations. Finally, numerical simulations are carried out to validate the proposed approach on the IEEE 24 Bus System.

6.1 Introduction In the past decades, the world has suffered from several major blackouts such as US-Canada Blackout in 2003 [1], European Blackout in 2006 [2], India Blackout in 2012 and Brazil Blackout in 1999 [3]. All the major blackouts have caused huge economic losses and affected millions of people. Due to the complexity of electrical power systems, it has been a great challenge to understand, analyze and identify the cascading blackouts in practical power grids. According to the analysis of technological reports, power system blackouts normally go through five stages: precondition, initiating event, cascade events, final state, and restoration [3]. The precondition usually happens in the winter or summer peak time due to the excessive power demand. The initiating event (e.g., short-circuit, overload, protection hidden failure, etc.) triggers the chain reaction of branch outages, which starts the cascade events. During the cascade event, power systems may take protective actions such as load shedding and generation dispatch to prevent the cascading failure. If protective actions fail to prevent further cascades, power systems may end up with the final state of cascading failures, which tends to result in the blackout. As a result, the recovery strategy has to be taken in order to restore the normal state of power grids. To avoid the occurrence of cascading blackouts, it would be desirable to identify the initial malicious disturbances or contingencies © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Zhai, Control and Optimization Methods for Complex System Resilience, Studies in Systems, Decision and Control 478, https://doi.org/10.1007/978-981-99-3053-1_6

99

100

6 Risk Identification of Cascading Process Under Protection

before cascading failures, so that the precautions can be taken in advance to eliminate the risk of power system blackouts. Some identification approaches have been developed to search for the critical branches or initial malicious disturbances that can cause large-scale disruptions [4– 11]. For instance, some methods are proposed to identify the collections of n − k contingencies based on the event trees [4], line outage distribution factor [5] and other optimization techniques [6–8]. Nevertheless, these optimization approaches are not efficient to identify the large collections of n − k contingencies that result in cascading blackouts. To address this problem, a “random chemistry” algorithm is designed with the relatively low computational complexity [9]. In addition, an optimal control approach is proposed to identify the initial contingencies by treating these contingencies as the control inputs [10, 11]. The above optimal control approach is able to identify the continuous changes of branch impedance other than direct branch outages as the initial contingencies. Moreover, it can be extended to identify the dangerous fluctuation of injected power on buses caused by the generation of renewable energies or load variations [12]. It is demonstrated that the optimal control approach can effectively determine the worst-case cascades of power grids without protective actions. As is well known, protective actions such as load shedding and generation dispatch play a key role in preventing cascading blackouts of power grids. This work aims to extend the optimal control approach to identify the malicious contingencies by taking into account protective actions during the cascades. The main challenge in theory is to incorporate protective actions in the framework of optimal control theory, since protective actions are taken according to solutions of a different optimization problem (linear or nonlinear programming). By converting both the optimal control problem and the optimization problem for protective actions into an integrated system of algebraic equations, we manage to formulate and solve the problem of identifying initial contingencies that could lead to the worst-case cascading failures in power grids endowed with protective actions. The remainder of this chapter is organized as follows. Section 6.2 formulates the contingency identification problem of power grids involving protective actions in the framework of optimal control theory. Section 6.3 presents the theoretical analysis of the optimal control problem. Section 6.4 provides simulation results to validate the proposed approach. Finally, the conclusion is drawn in Sect. 6.5.

6.2 Problem Formulation This section presents the problem formulation of identifying the most disruptive disturbances on transmission lines of power grids with n branches and m buses. Figure 6.1 presents an illustration of identifying the worst-case cascades of power grids with protective actions in the framework of optimal control theory. When the initial contingency or disturbance is added to the branch of power grids, the branch impendence is changed. This leads to the redistribution of power flow on the branches.

6.2 Problem Formulation

101

Fig. 6.1 Schematic diagram on the identification of power system cascades with protective actions

If the power flow on the branch exceeds its threshold, the branch is severed with the change of network topology. The above branch outage could result in the overloads of other branches and give rise to further cascades. At the given cascading step (e.g., the l-th cascading step in Fig. 6.1), protective actions (e.g., load shedding and generation dispatch) are taken to prevent the cascading failure. The above cascade process can be described by the state equation in optimal control theory. By designing a cost function to quantify the final disruption level of cascading failures, an optimal control algorithm can be developed to obtain the optimal control inputs, which are exactly the initial contingencies or disturbances that can cause the worst-case cascading failures. By treating the branch impedance or admittance as the state variable, a state equation can be established to characterize the line outage sequence during the power system cascades. The power flow on each branch is obtained by solving the power flow equation. The protective actions during cascades are implemented at the given cascading step by adjusting the injected power on buses. Finally, an optimal control problem is formulated to search for the initiating event of cascading blackouts with the consideration of load shedding and generation dispatch.

6.2.1 State Equation The state equation is used to describe the line outage sequence of power grids during cascades. Suppose the status of branch or transmission line depends on both the power flow on the branch and its threshold. Thus, an approximate function on the

102

6 Risk Identification of Cascading Process Under Protection

branch status is designed to characterize the operation of circuit breakers in protective relays as follows  ⎧ ⎪ 0, |Pi j | ≥ ci2j + ⎪ ⎨  g(Pi j , ci j ) = 1, | ≤ ci2j − |P i j ⎪ ⎪ ⎩ 1−sin σ (Pi2j −ci2j ) , otherwise. 2

π 2σ

;

π 2σ

;

where σ is a tunable positive parameter. Pi j refers to the power flow on the branch between Bus i and Bus j, and ci j denotes its threshold of power flow. Pi j can be obtained by solving the power flow equation. Notably, the function g is differentiable with respect to Pi j , and it approximates to the step function as the parameter σ increases. It can reflect the physical characteristic of protective relays while contributing to theoretical analysis on the identification of initial contingencies using optimal control theory. The line outage sequence can be described by the following state equation (i.e., branch outage model in Fig. 6.1) Y pk+1 = G(Pikj ) · Y pk + E ik u k

(6.1)

where Y pk = (y kp,1 , y kp,2 , ..., y kp,n )T ∈ R n refers to the vector of branch admittance, and the control input or disturbance is denoted by u k = (u k,1 , u k,2 , ..., u k,n )T . In addition, G(Pikj ) is a diagonal matrix ⎞ 0 . 0 g(Pi1 j1 , ci1 j1 ) ⎟ ⎜ 0 0 g(Pi2 j2 , ci2 j2 ) . ⎟ G(Pikj ) = ⎜ ⎠ ⎝ . . . . 0 0 . g(Pin jn , cin jn ) ⎛

and it describes the status of each branch at the k-th cascading step. The matrix E ik is constructed as

 E ik = diag eiTk = diag(0, .., 0, 1, 0, ..., 0) ∈ R n×n    ik

and it enables us to select the i k -th branch for adding the initial disturbances. For high-voltage transmission systems, the DC power flow is a good substitute for the AC power flow [13], and it is computationally efficient and immune to numerical non-convergence. Thus, the DC power flow equation is employed in this work. The classic DC power flow equation is given by Pi =

m  j=1

Bi j θi j =

m  j=1

Bi j (θi − θ j )

(6.2)

6.2 Problem Formulation

103

where Pi and θi refer to the injection power and voltage phase angle of Bus i, respectively. Bi j represents the mutual susceptance between Bus i and Bus j. The symbol m denotes the number of buses in power networks. Equation (6.2) can be rewritten in matrix form [14] P = Bθ, where P = (P1 , P2 , ..., Pm ), θ = (θ1 , θ2 , ..., θm ) and ⎛ m i=2

⎞ . B1m ⎟ B2m i=1,i=2 B2i . ⎟ ⎠ . . . m−1 −Bm2 . i=1 Bmi

B1i  −B12 m

⎜ −B21 B=⎜ ⎝ . −Bm1

It is worth pointing out that B can be obtained by removing the real part of Yb . In fact, the nodal admittance matrix Ybk at the k-th cascading step can be obtained as Ybk = A T diag(Y pk )A, where A denotes the branch-bus incidence matrix [15]. Thus, the power flow on the branch between Bus i and Bus j can be computed as ∗

Pikj = eiT Ybk e j (ei − e j )T (Ybk )−1 P k , i, j ∈ Im = {1, 2, ..., m}, where ei is the m-dimensional unit vector with 1 at the i-th position and 0 elsewhere. P k refers to the vector of injected power on buses at the k-th cascading step. The symbol −1∗ represents a generalized inverse for solving DC power flow equation [10]. Remark 6.1 During the cascading blackout, power network may be divided into several subnetworks (i.e., islands), which can be identified by analyzing the nodal admittance matrix Ybk . To solve the DC power flow equation, the generator bus connected to the largest generating station is selected as the new slack bus in the subnetwork. And thus the power variation of slack bus accounts for a small percentage of its generating capacity. If there is no generator bus in the subnetwork, the power flow is zero on each branch of this subnetworks.

6.2.2 Protective Actions If generation dispatch and load shedding are taken into account in the formulation, P k has to be updated at certain steps of cascading failure. For simplicity, suppose that load shedding and generation control are implemented at the l-th cascading step (1 < l < h). This implies that P k = P l for k ≥ l. Thus, a nonlinear programming problem can be proposed to allow for load shedding and generation dispatch as

104

6 Risk Identification of Cascading Process Under Protection

follows min P l − P 0 2 Pl

P i ≤ Pil ≤ P¯i

s. t.

(6.3)

− ci j ≤ Pilj ≤ ci j where P l = (P1l , P2l , ..., Pml )T , and P 0 denotes the vector of original injected power on buses. The symbols P i and P¯i denote the upper and lower bounds of injected power on Bus i, respectively. The cost function in (6.3) quantifies the changes of injected power on buses due to load shedding and generation control. Essentially, the objective of Optimization Problem (6.3) is to achieve the minimum adjustment of injected power on buses while preventing further branch outages of power grids. Remark 6.2 The linear programming formulation can also be adopted to allow for protection actions in power systems [16]. The proposed approach is also applied to the linear programming formulation. In addition, it also has a chance to be extended for dealing with protection actions at multiple cascading steps.

6.2.3 Cost Function Next, a cost function of optimal control problem is presented to quantify the disruption level of power grids at the end of cascading failures. Suppose the cascades come to an end at the h-th cascading step, and the cost function is designed as min J (Y ph , u k ) uk

with J (Y ph , u k )

=

(Y ph )

+

h−1  k=0

(6.4)

u k 2 max{0, 1 − k}

and the adjustment of injected power on buses P k for the system protection is implemented according to the solutions to Optimization Problem (6.3) at the l-th cascading step. The first term in the cost function (i.e., (Y ph )) describes the final status of cascading failures (e.g., network connectivity or power flow), and the second term characterizes the accumulated control energy. In addition,  is a positive weight, and the symbol  ·  represents the 2-norm. Normally,  is small enough so that more efforts are taken to minimize the first term in the cost function. Remark 6.3 The second term in the cost function ensures that only initial disturbances can be added on the branch of power transmission system. This is due to max{0, 1 − k} = 1 for k = 0 and max{0, 1 − k} = 0 for k ≥ 1. When max{0, 1 −

6.3 Theoretical Results

105

k} = 0, the second term goes to the infinity. This implies that it is impossible to minimize the cost function when k ≥ 1.

6.3 Theoretical Results This section provides theoretical results on the identification of initial contingencies on branches that can cause catastrophic cascading failures of power grids. First of all, an equivalent condition is presented for Problem (6.3). Lemma 6.1 The optimal solutions of Optimization Problem (6.3) are equivalent to the solutions of the following system of algebraic equations 

2(P l − P 0 ) + μ¯ − μ +

(λ¯ i j − λi j )eiT Ybl e j (Ybl )−1 ei j = 0 ∗

(i, j)∈

− P¯i + x¯i2 = 0,

μ¯ i (Pil − P¯i ) = 0,

μ¯ i − z¯ i2 = 0

P i − Pil + x i2 = 0,

μi (Pil − P i ) = 0,

μi − z i2 = 0

Pil

Pilj − ci j + y¯i2j = 0, (Pilj − ci j )λ¯ i j = 0, λ¯ i j − w¯ i2j = 0 Pilj + ci j − y i2j = 0, (Pilj + ci j )λi j = 0, λi j − wi2j = 0

(6.5)

where Ybl = A T diag(Y pl )A, ei j = ei − e j , i ∈ Im , (i, j) ∈ , μ¯ = (μ¯ 1 , μ¯ 2 , ..., μ¯ m )T and μ = (μ1 , μ2 , ..., μm )T . And the symbol  denotes the set of branches in power systems with the cardinality || = n (i.e., the number of branches). Proof The inequality constraints in the optimization problem (6.3) can be converted into the equality constraints by introducing the unknown variables x¯i , x i , y¯i j and y i j as follows. min P l − P 0 2 Pl

s. t. Pil − P¯i + x¯i2 = 0 P i − Pil + x i2 = 0 Pilj − ci j + y¯i2j = 0 Pilj + ci j − y i2j = 0

(6.6)

According to the Karush-Kuhn-Tucker (KKT) conditions, the necessary condition for the solutions to Optimization Problem (6.6) can be obtained as follows. Specifically, we have ∇P l − P 0 2 + μ¯ − μ +

 (i, j)∈

(λ¯ i j − λi j )eiT Ybl e j (Ybl )−1 ei j = 0 ∗

106

6 Risk Identification of Cascading Process Under Protection

with ∇P l − P 0 2 = 2(P l − P 0 ) for the stationarity condition and the constraints Pil − P¯i + x¯i2 = 0,

Pilj − ci j + y¯i2j = 0

P i − Pil + x i2 = 0,

Pilj + ci j − y i2j = 0

The conditions for complementary slackness are given by μ¯ i (Pil − P¯i ) = 0, (Pilj − ci j )λ¯ i j = 0 μi (Pil − P i ) = 0, (Pilj + ci j )λi j = 0 Moreover, the dual feasibility can be described by μ¯ i − z¯ i2 = 0, λ¯ i j − w¯ i2j = 0 μi − z i2 = 0, λi j − wi2j = 0 where ei j = ei − e j , i ∈ Im , (i, j) ∈ , μ¯ = (μ¯ 1 , μ¯ 2 , ..., μ¯ m )T and μ = (μ1 , μ2 , ..., μm )T . Since the cost function in Optimization Problem (6.3) is a convex function and the inequality constraints are affine, the above necessary conditions are also sufficient for optimality. This implies that the solutions to Optimization Problem (6.3) are equivalent to solutions to the system of algebraic equations (6.6). The proof of this lemma is thus completed.  Remark 6.4 As we can see, the system (6.5) is composed of (7m + 6n) equations and (7m + 6n) additional unknown variables (i.e., P l , wi j , w¯ i j , λi j , λ¯ i j , y i j , y¯i j , x i ,

x¯i , z i , z¯ i , μi , μ¯ i ). Note that Ybl contains the existing unknown variables Y pl in the state Eq. (6.1).

The equivalent conditions in Lemma 6.1 allows to obtain the necessary conditions for the optimal control problem (6.4) as follows Theorem 6.1 The necessary conditions for the optimal control problem (6.4) with protective actions according to (6.3) are given by solving the system of algebraic equations as follows 

Y pk+1 − G(Pikj )Y pk − F(Y pl ,

h−k−2

∂Y ph−s 1 ∂Y ph−s−1 n

=0 ¯ P , wi j , w¯ i j , λi j , λi j , y i j , y¯i j , x i , x¯i , z i , z¯ i , μi , μ¯ i ) = 0 l

max{0,1−k} E ik 2

s=0

(6.7)

where the second equation represents System (6.5), and the optimal adjustment of injected power on buses for protective actions satisfies P k = P l for k ≥ l, and P l is the solution to Problem (6.3). In addition, the optimal control input is given by uk =

h−k−2  ∂Y ph−s max{0, 1 − k} E ik 1n , k ∈ Ih−1 2 ∂Y ph−s−1 s=0

(6.8)

6.3 Theoretical Results

107

Proof With Pontryagin’s maximum principle in optimal control theory for the discrete-time system [17], the necessary conditions for the optimal control problem (6.4) can be determined as Y pk+1 = G(Pikj ) · Y pk + E ik u k 

∂Y pk+1

T λk+1 +

∂u k  λk =

∂Y pk+1

∂u k 2  · =0 max{0, 1 − k} ∂u k

T

∂Y pk

λk+1 + ∂T(Y ph ) ∂Y ph

 ∂u k 2 · max{0, 1 − k} ∂Y pk

− λh = 0

where 0 = (0, 0, ..., 0)T ∈ R n . By reorganizing the above equations, we can obtain the optimal control input uk =

h−k−2  ∂Y ph−s max{0, 1 − k} E ik 1n , k ∈ Ih−1 2 ∂Y ph−s−1 s=0

and the system of algebraic equations Y pk+1 − G(Pikj )Y pk −

h−k−2  ∂Y ph−s max{0, 1 − k} E ik 1n = 0 2 ∂Y ph−s−1 s=0

(6.9)

where the vector P l is determined by the solutions to System (6.5), which can be rewritten as F(Y pl , P l , wi j , w¯ i j , λi j , λ¯ i j , y i j , y¯i j , x i , x¯i , z i , z¯ i , μi , μ¯ i ) = 0

(6.10)

for ease of notation according to Lemma 6.1. By combining Eqs. (6.9) and (6.10), an extended system of algebraic equations is obtained with 7m + (6 + h)n equations and 7m + (6 + h)n unknown variables. By substituting the solutions of this extended system into (6.8), we can identify the most disruptive disturbances for the cascades of power grids with protective actions at the given cascading step. This completes the proof.  Remark 6.5 The extended system of algebraic equations (6.7) can be solved using the numerical solver in numerical-analysis software. In this way, load shedding and generation dispatch can be taken into account in the proposed optimal control formulation of identifying the worst-case cascading failures.

108

6 Risk Identification of Cascading Process Under Protection

Remark 6.6 Since the solutions to the system (6.7) can only provide the necessary conditions for the optimal control problem (6.4), extensive numerical simulations have to be conducted to search for feasible solutions in practice. By comparing the values of cost function with the identified initial disturbances (i.e., control inputs), it is expected to obtain the optimal or suboptimal solutions to Problem (6.4) if numerical simulations are repeated for a sufficiently large number of times.

6.4 Simulation and Validation To demonstrate the effectiveness of the proposed identification approach based on optimal control, numerical simulations are conducted on the IEEE RTS 24 Bus System to determine the initial disruptive disturbances on each branch (see Fig. 6.2). In addition, we make a comparison of simulation results between the cases with and without protective actions (i.e., load shedding and generation dispatch), respectively.

6.4.1 Parameter Setting Per unit values are adopted with the base value of power 100 MVA. Moreover, the solver “fsolve" in Matlab is employed to solve the integrated system of algebraic equations (6.7). Other parameters are given as follows: σ = 5 × 104 ,  = 10−4 , h = 10, and (Y ph ) = Y ph 2 /2 in the cost function (6.4). For each branch, numerical simulations are carried out to solve System (6.7) for 10 times, and the worst-case disturbances are selected by comparing the values of cost function. A performance index γ is defined to quantify the disruption level of cascades as follows γ =

J (Y ph , u) J (Y ph , 0)

.

Intuitively, the index γ is the ratio between the final cost of power grids with the control input u and that without any control inputs, and a smaller γ indicates a worse cascade of power systems. The power flow threshold on each branch is 10% larger than the normal power flow on the corresponding branch of power systems without any disturbances.

6.4.2 Validation and Comparison In the simulations, generation dispatch and load shedding are implemented at the 4-th cascading step (i.e., l = 4) according to solutions of Optimization Problem (6.3). Figure 6.3 presents the initial disturbances (i.e., control inputs) on each branch identified by the proposed optimal control approach and the resulting normalized costs (i.e., the index γ ). Specifically, the blue bars denote the control inputs and

6.4 Simulation and Validation

109

Fig. 6.2 IEEE RTS 24 bus system with initial disturbances on branches (red star)

normalized costs without generation dispatch and load shedding, while the green bars represent those with generation dispatch and load shedding. As we can observe in the upper panel of Fig. 6.3, the height of green bar is not smaller than that of blue bar for each branch. This indicates that the larger initial disturbances (i.e., the magnitude of control input) are required to trigger the worst-case cascades of power grids with generation dispatch and load shedding compared to those without generation dispatch

110

6 Risk Identification of Cascading Process Under Protection

Control input

80 No dispatich & shedding With dispatich & shedding

60 40 20 0 5

10

15

20

25

30

35

25

30

35

Branch ID Normalized cost

1

0.5

0 5

10

15

20

Branch ID

Fig. 6.3 Control inputs and normalized costs of the cascades with and without generation dispatch and load shedding on the IEEE RTS 24 Bus System.

and load shedding. This demonstrates that protection actions are able to effectively enhance the robustness of power systems and relieve the final disruption level after malicious disturbances. It is worth pointing out that the branches with equal heights of blue bar and green bar (e.g., Branch 1, Branch 2, Branch 3, Branch 4, Branch 5, Branch 6, etc.) are directly severed by the initial disturbances (i.e., control inputs). The lower panel of Fig. 6.3 demonstrates that the cascades with generation dispatch and load shedding are less disruptive on the whole except for the cascades triggered by the initial disturbances on Branch 28, Branch 32, and Branch 33. This is because the optimal solutions for generation dispatch and load shedding (i.e., solutions to the optimization problem (6.3)) are not obtained by the numeric solver at the specified cascading step. Thus, the adjustment of injected power on buses (i.e., generation dispatch and load shedding) actually deteriorates branch overloads and results in worse disruptions of cascades in the end.

6.5 Conclusions In this chapter, we investigated the problem of identifying the initial contingencies that result in the worst-case cascading failures of power grids. In particular, power grids are equipped with protection devices to prevent the cascading blackouts by load shedding or generation dispatch in time. Moreover, a theoretical framework was proposed to allow for both the identification of the most disruptive disturbances and the

References

111

optimal adjustment of injected power buses for the protection of power girds. Numerical simulations were conducted to better understand the effect of protection actions on the identification of initial malicious contingencies and the final disruptions of cascading failures. In this work, the deterministic cascading failure paths are taken into account for the identification of initial disruptive disturbances and the implementation of protective actions. In practice, power grids are subject to uncertainties (e.g., hidden failure, device aging, human errors, etc.). Therefore, we will consider the contingency identification of power grids with uncertain cascading failure paths in the next step.

References 1. Report, Final, on the August 14: Blackout in the United States and Canada, p. 2004. Electricity Markets and Policy Group Technical report, US-Canada Power System Outage Task Force (2003) 2. Final Report System Disturbance on 4 November 2006, Technical Report, Union for the Coordination of Transmission of Electricity (2007) 3. Lu, W., Besanger, Y., Zamai, E., Radu, D.: Blackouts: description, analysis and classification. In: WSEAS International Conference on Power Systems, Sept (2006) 4. Chen, Q., McCalley, J.: Identifying high risk n-k contingencies for online security assessment. IEEE Trans. Power Syst. 20(2), 823–834 (2005) 5. Davis, C.M., Overbye, T.J.: Multiple element contingency screening. IEEE Trans. Power Syst. 26(3), 1294–1301 (2011) 6. Donde, V., Lopez, V., Lesieutre, B., Pinar, A., Yang, C., Meza, J.: Severe multiple contingency screening in electric power systems. IEEE Trans. Power Syst. 23(2), 406–417 (2008) 7. Bienstock, D., Verma, A.: The n-k problem in power grids: New models, formulations, and numerical experiments. SIAM J. Optimiz. 20(5), 2352–2380 (2010) 8. Rocco, C., Ramirez-Marquez, J., Salazar, D., Yajure, C.: Assessing the vulnerability of a power system through a multiple objective contingency screening approach. IEEE Trans. Reliab. 60(2), 394–403 (2011) 9. Eppstein, M.J., Hines, P.D.: A “random chemistry" algorithm for identifying collections of multiple contingencies that initiate cascading failure. IEEE Trans. Power Syst. 27(3), 1698– 1705 (2012) 10. Zhai, C., Zhang, H., Xiao, G., Pan, T.: Modeling and identification of worst-case cascading failures in power systems (2017). arXiv preprint arXiv:1703.05232 11. Zhai, C., Zhang, H., Xiao, G., Pan, T.: Comparing different models for investigating cascading failures in power systems. In: International Workshop on Complex Systems and Networks (IWCSN) pp. 230–236. IEEE (2017) 12. Zhai, C., Zhang, H., Xiao, G., Pan, T.: Risk identification of power transmission system with renewable energy (2018). arXiv preprint arXiv:1811.08984 13. Yan, J., Tang, Y., He, H., Sun, Y.: Cascading failure analysis with DC power flow model and ransient stability analysis. IEEE Trans. Power Syst. 30(1), 285–297 (2015) 14. Stott, B., Jardim, J., Alsaç, O.: DC power flow revisited. IEEE Trans. Power Syst. 24(3), 1290–1300 (2009) 15. Stagg, W., Ahmed, H.: Computer Methods in Power System Analysis. McGraw-Hill (1968) 16. Carreras, B.A., Lynch, V.E., Dobson, I., Newman, D.E.: Critical points and transitions in an electric power transmission model for cascading failure blackouts. Chaos Interdisc J Nonlinear Sci 12(4), 985–994 (2002) 17. Frank, L., Vassilis, L.S.: Optimal Control, 2nd Edition. Wiley-Interscience, 2nd edition, Oct (1995)

Chapter 7

Model Predictive Approach to Preventing Cascading Proces

Abstract Large-scale blackouts normally go through several sequential phases according to the propagation speed of branch outages and the disruption level of power grids. It is crucial to eliminate the propagation of cascading outages in its infancy. In this chapter, a model predictive approach is proposed to protect power grids against cascading blackouts. First of all, the cascading dynamics of power grids is described by the outage model of transmission lines and the DC power flow equation, which allows us to predict the cascading failure path. Then a nonlinear convex optimization formulation is established to terminate the cascading outages by adjusting the injected power on buses. Afterwards two protection schemes are designed according to the optimization formulation: one scheme carries out protective actions to terminate cascading outages once for all, while the other takes protection measures in two consecutive steps. Saddle point dynamics is employed to provide a numerical solution to the proposed optimization problem, and its global convergence is guaranteed in theory. Finally, numerical simulations on IEEE test systems are implemented to validate the proposed approach.

7.1 Introduction The protection of power grids against cascading blackouts has always been a great challenge to both the power industry and academia due to unexpected contingencies and the evolving nature of power grids [1, 2]. In the last decades, the technologies of phasor measurement, communication and data processing have made great progress, which allows us to detect the real-time state of power systems, transmit the data and generate control signals for emergency control. The developments of power system protection basically go through three stages [3]. Specifically, the conventional protection of power systems mainly resorts to electro-mechanical protective relays for tripping overloading branches [4]. In spite of high reliability and simplicity in construction, these relays need to be calibrated periodically, and they are unable to determine the direction of a fault with respect to the relay’s location. The introduction of computers features the second stage of power system protection as advanced control algorithms can be applied to protect power © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Zhai, Control and Optimization Methods for Complex System Resilience, Studies in Systems, Decision and Control 478, https://doi.org/10.1007/978-981-99-3053-1_7

113

114

7 Model Predictive Approach to Preventing Cascading Proces

grids [5, 6]. The third stage is characterized by the utilization of global positioning system (GPS), which enables engineers to synchronize time precisely and obtain the global phase information for the wide-area protection [7]. The availability of global information on power systems allows to establish a systematic approach to cope with catastrophic scenarios in wide-area power networks. Thus, the special protection scheme (SPS) is proposed to mitigate global stresses by separating power systems into several islands and isolating the faulted areas according to predetermined actions [8]. It is demonstrated that the installation of SPSs is economically profitable [9]. Nevertheless, the SPSs are designed for particular power systems suffering from limited unusual stresses (e.g., frequency instability, voltage instability and transient angle instability), which inevitably restricts their compatibility and the universality for contingencies. As a result, some effective protection algorithms are designed to deal with the disruptive contingencies by means of adaptive relaying [10], network of phasor measurements [11] and multi-agent approach [12]. In practice, the signalling latency makes up the critical barrier against the online implementation of protection algorithms [13]. Power system blackouts normally go through five phases: precondition, initiating events, cascade events, final state, and restoration [14]. As classified in existing studies [14] and evidenced by previous large blackouts (e.g., [15, 16]), cascade events can be divided into two stages: steady-state progression where the cascades propagate slowly while keeping the balance between the power generation and consumption, and high-speed cascade where the cascades propagate quickly and may end up with the system collapse in a short time. The steady-state progression may give rise to a triggering event (e.g., the tripping of a certain line), which leads to the occurrence of high-speed cascade. In the period of steady-state progression, the cascade overloads are the major incidents, and thus it is feasible to predict the cascading failure path according to the overloading branches during this period. In this chapter, we propose a protection architecture to prevent cascading blackouts by predicting the cascading outages and adjusting the injected power on buses in the period of steady-state progression. Since the DC power flow is a desirable substitute for the AC power flow in high-voltage transmission networks [17, 18], for simplicity, the DC power flow equation is employed to compute the power flow. The main contributions of this work are listed as follows. 1. Propose a disturbance-related real-time protection architecture of power systems by predicting the cascading evolution in the period of steady-state progression. 2. Integrate time delays of faults detection, signal transmission and processing into the protection architecture to achieve reliable protections. 3. Develop two protection schemes to prevent the propagation of cascading outages and succeed in implementing the optimal adjustment of injected power on buses with saddle point dynamics. The outline of this chapter is organized as follows. Section 7.2 presents the protection architecture of power systems against cascading blackouts. Section 7.3 provides the optimization formulation of nonrecurring protection scheme and theoretical analysis, followed by the scheme of recurring protection in Sect. 7.4. Simulations and

7.2 Protection Architecture

115

validation on IEEE test systems are given in Sect. 7.5. Finally, we discuss the extensions of the proposed protection architecture in Sect. 7.6 and conclude the chapter in Sect. 7.7.

7.2 Protection Architecture Thanks to phasor measurement units (PMUs), the operating state of modern power grids can be monitored in real time [19, 20]. This enables the wide-area protection and control system (WAPCS) to identify the disturbances or faults as soon as possible [21] and then take remedial actions. In this work, remedial actions mainly refer to the adjustment of injected power on buses (e.g., load shedding, generation ramping/tripping) for protecting power system against blackouts. Specifically, the disturbance-related signals are detected by PMUs and then transmitted to the phasor data concentrator (PDC) [22]. Through the dedicated communication networks (e.g., virtual private network [20]), PDCs send the disturbance-related data to data server and WAPCS, where the disturbances are identified and the cascading process of transmission lines is predicted via the outage model of branches. Finally, the protection architecture produces corrective control signals for local actuators in order to terminate cascading outages. In practice, the local actuators include different relays (e.g., voltage relays and frequency relays) and circuit breakers for load shedding, and supplementary controllers and turbine valves for generation control. Time delays of the above sequential operations in WAPCS are summarized in Table 7.1. The overall time delays from the disturbance detection to the implementation of control command can be roughly estimated. During this period, the power system probably has gone through some cascading failures due to initial contingencies and subsequent branch outages. Thus, proper control and remedial actions should be taken in time by incorporating the above time delays. In this work, our goal is to develop a protection architecture that integrates widearea monitoring, prediction and control of power networks so that WAPCS is able to make disturbance-related remedial actions in time to achieve the least power loss during emergency. First of all, we clarify the concept of cascading step in order to describe the evolution of branch outages during cascading blackouts. A cascading

Table 7.1 Time delays of sequential operations in WAPCS [23, 24] Sequence Operation Time delay 1 2 3 4 5

Signal detection using PMUs Transmission of PMU data Generation of control signals Transmission of control signals Operation of local actuators

≤ 150 ms ≤ 700 ms ≈ 100 ms ≈ 10 ms ≈ 50 ms

116

7 Model Predictive Approach to Preventing Cascading Proces

step is defined as one topological change (e.g., one branch outage) of power networks due to contingencies, human factors or the overloading of transmission lines during the cascading process. Once the transmission line is overloaded, the timer of circuit breaker will be triggered to count down from the preset time T p . And the transmission line is tripped when the timer runs out. Thus, the time interval between two consecutive cascading steps basically depends on the preset time of the timer in protective relays [25]. The evolution time of cascading failure is defined as the time duration from the first activation of the timer of circuit breaker to the current time during the cascades. Then the evolution time of cascading failure is tc ≈ kT p at the k-th cascading step. Remedial/protective actions are taken at a specified cascading step, which ensures that the estimated evolution time of cascades exceeds the total time delay of sequential operations in WAPCS. This work focuses on the evolution of branch outages in the early stage of cascading failure (i.e., the steady-state progression), during which the branch outage basically occurs as a result of branch overloads and running out of preset time in the timer. This enables us to roughly estimate the evolution time of cascading failure based on the cascading steps. Remark 7.1 It takes a fixed time delay T p for the relay to trip the branch once it is overloaded. The timer in the relay starts to count down from the preset time T p , and the circuit breaker trips the overloaded branch when the timer runs out. The topology of power networks is updated after the overloaded branches are tripped. Then the power flow is recomputed based on the updated topology of power networks. Again the branch is overloaded if the recomputed power flow exceeds the given threshold. In this way, the cascading outage of branches proceeds until the power flow on each branch is less than the threshold. Actually, the proposed protection architecture is also compatible with other types of relays with different temporal operational characteristics, which will introduce the variable time delays of branch outages according to the overloading level. For the tripping delay under different time-inverse characteristics, we can choose the minimum value of TP (i.e., the minimum reclosing time of circuit breaker) as a fixed time delay to compute the prediction horizon m for the timely implementation of protection actions at the end of prediction horizon. To predict the cascading failure path, it is necessary to obtain the cascading dynamics of power networks, which includes the DC power flow equation and the outage model of branches. Consider a power network with n branches and n b buses, and the initial disturbances δ (e.g., lightning, storm, poor contactor and collapsed vegetation, etc.) affect the branch admittance as follows Y p1 = Y p0 + δ where Y p0 ∈ R n refers to the n-dimensional vector of the original branch admittance and Y p1 denotes the branch admittance at the first cascading step. For simplicity, the DC power flow equation is employed to compute the power flow on each transmission line. (7.1) P = A T diag(Y pk )Aθ k , k ∈ N

7.2 Protection Architecture

117

where P denotes the n b -dimensional vector of injected power on each bus and A ∈ R n×n b refers to the branch-bus incidence matrix [26]. θ k represents the n b dimensional vector of voltage angle on each bus at the k-th cascading step. In addition, the operation diag(x) obtains a square diagonal matrix with the elements of vector x on the main diagonal. The solution to Equation (7.1) is expressed as ∗

θ k = (A T diag(Y pk )A)−1 P where the operator −1∗ is used to compute the inverse of a square matrix, as defined in [27] (see Sect. 7.8.1 in Appendix). Thus, the vector of power flow on branches is ∗ diag(Y pk )A(A T diag(Y pk )A)−1 P. According to Lemma 3.2 in [27] (see Sect. 7.8.2 in Appendix), the power flow from Bus i to Bus j at the k-th cascading step is given by ∗ (7.2) Pikj = eiT A T diag(Y pk )Ae j (ei − e j )T (A T diag(Y pk )A)−1 P, with i, j ∈ In b = {1, 2, . . . , n b }, where ei represents the n b dimensional unit vector with the i-th element being 1 and other elements being 0. The evolution of branch admittance (i.e., the outage model of branches) is described by the following formula  Y pk+1

=

k = 0; Y pk + δ, G(Pikj ) ◦ Y pk , k ≥ 1.

(7.3)

where the operator ◦ represents the Hadamard product and G(Pikj ) is given by G(Pikj ) = (g(Pik1 j1 , ci1 j1 ), g(Pik2 j2 , ci2 j2 ), ..., g(Pikn jn , cin jn ))T . And therein is the approximation function [27]  ⎧ k ⎪ 0, |P | ≥ c2 + ⎪ i j ⎪ ⎨  ij |Pikj | ≤ ci2j − g(Pikj , ci j ) = 1,  ⎪ ⎪ ⎪ ⎩ 1−sin σ (Pikj )2 −ci2j , otherwise. 2

π 2σ

;

π 2σ

;

where ci j denotes the threshold of power flow on the transmission line connecting Bus i to Bus j. The approximation function is introduced to describe the change of branch admittance when the power flow exceeds the threshold and the transmission line is tripped by the circuit breaker. It gets close to the step function that reflects the real system characteristic of branch outage when the parameter σ increases. Moreover, it is differentiable with respect to Pikj , which ensures the feasibility of optimal adjustment of injected power on buses in protection schemes. Essentially, the protection architecture is composed of three building blocks including contingency identification, prediction of cascading failure and protection schemes (see Fig. 7.1). It is in line with the typical architecture of wide-area monitoring, protection and control system [20]. The disturbance is identified by WAPCS

118

7 Model Predictive Approach to Preventing Cascading Proces

Fig. 7.1 Protection architecture of power systems

[21], then it triggers the evolution of cascading dynamics, which is composed of the DC power flow Eq. (7.1) and the outage model of branches (7.3). When the power network is split into isolated subnetworks, the generator bus connected to the largest generating station is selected as the new slack bus in the subnetwork. And thus the power variation of slack bus accounts for a small percentage of its generating capacity. Then the DC power flow is solved for each subnetwork to redistribute the power flow. Significantly, the cascading dynamics allows us to preplan remedial actions by taking into account the time delays of sequential operations and solving the optimization problem in protection schemes. After power blackouts are prevented by implementing the control commands from protection schemes, the protection architecture starts the detection and identification of disturbances once again. Thus, the key task is to design the effective protection schemes to terminate the propagation of cascading outages. The current practices for power system emergencies largely resort to the predetermined protection actions in the SPS. Compared to the SPS, our approach requires the real-time computation of mitigating actions for the specific power system state. It can effectively reduce the cost of power system protection and increase the flexibility of power grids to deal with diverse emergencies. Remark 7.2 Mathematically, the cascading failure process of power grids can be regarded as a series of operations/transformations on a mathematical model (i.e., the outage model of branches) that characterizes the redistribution of power flow and protective relays in power systems. And protection schemes ensure that the cascading

7.3 Nonrecurring Protection Scheme

119

failure process converges towards a desired fixed point, which stabilizes power grids with the least cost. In the subsequent two sections, we propose two different protection schemes (i.e., nonrecurring protection scheme and recurring protection scheme).

7.3 Nonrecurring Protection Scheme This section presents the first scheme, i.e. Nonrecurring Protection Scheme (NPS). NPS implies that protective actions are taken only once at a given cascading step. Due to time delays in WAPCS, we will only be able to carry out the optimal adjustment of injected power on buses against the propagation of cascading failures at the m-th cascading step. And thus the evolution time of cascading failure is tc ≈ mT p , which should be larger than the total time delays in WAPCS so that protective actions are available at the m-th cascading step. Specifically, the value of prediction horizon m should satisfy m · T p > Tdelay + Tdetection + Tr un + Tactuation with time delay of data transmission Tdelay , the detection time of disruptive disturbances Tdetection , the runtime of numerical algorithm Tr un , and the actuation time of relays Tactuation . In this way, the protection architecture is ready to take protective actions (i.e., optimal adjustment of injected power on buses) when the cascading failure reaches the m-th cascading step. The optimization problem can be formulated as min J (P, W )

P s.t.Y pm

= G(Pim−1 ) ◦ G(Pim−2 ) ◦ · · · ◦ G(Pi1j ) ◦ Y p1 j j ∗

Pimj = eiT A T diag(Y pm )Ae j (ei − e j )T (A T diag(Y pm )A)−1 P Pimj )2 ≤ ci2j , (i, j) ∈ E P i ≤ Pi ≤ P¯i , i ∈ In b

(7.4)

where the objective function J (P, W ) is given by J (P, W ) = W ◦ (P − P 0 )2

(7.5)

P 0 represents the vector of original injected power on each bus, and P = (P1 , P2 , . . . Pn b )T refers to the injected power vector after the adjustment. P i and P¯i denote the lower and upper bounds of injected power on Bus i, respectively. The weight vector W = (W1 , W2 , . . . , Wn b )T characterizes the bus significance in power systems, and Pimj denotes the power flow on the branch connecting Bus i and Bus j at the m-th cascading step. The cost function of Problem (7.4) characterizes the mismatch of injected power on each bus between the original distribution of load and power generation and the reassigned one from the optimization algorithm. The first term in the constraint conditions predicts the cascading process of power grids before taking remedial

120

7 Model Predictive Approach to Preventing Cascading Proces

measures. The second term calculates the power flow on each branch at the m-th cascading step, and the third one imposes the restriction on the upper bound of power flow on each branch. The final term stipulates the adjustment range of injected power on each bus. Moreover, it can be observed that Optimization Problem (7.4) is convex. Remark 7.3 TP largely depends on the parameter setting of protective relays (normally larger than 0.3s). In addition, the parameters Tdelay , Tdetection , Tr un and Tactuation can be roughly estimated for the actual power grid and the WAPCS (see Table 7.1). Thus, the prediction horizon m can be chosen by

m=

 Tdelay + Tdetection + Tr un + Tactuation , Tp

where the symbol x gives the least integer that is greater than the real number x. Proposition 7.3.1 Optimization Problem (7.4) is convex. Proof The cost function J (P, W ) and the constraint function (Pimj )2 −ci2j are convex, and Pimj is affine with respect to P. Thus, (7.4) is a convex optimization problem.  Then we present the necessary and sufficient condition for the optimal solution to Optimization Problem (7.4). Proposition 7.3.2 Suppose Slater’s condition holds (nonempty feasible region) for Convex Optimization Problem (7.4). Then P ∗ is the optimal solution if and only if there exist Lagrangian multipliers λi∗j , τ¯i∗ and τ i∗ satisfying the KKT conditions: nb nb nb 

m∗ 2 ∗ ∗ 2 ∇ J (P , W ) + λi j ∇ (Pi j ) − ci j + (τ¯i∗ − τ i∗ )ei = 0 i=1 j=1

i=1

and 2 2 (Pim∗ j ) ≤ ci j ∗

T T m T T m −1 Pim∗ P∗ j = ei A diag(Y p )Ae j (ei − e j ) (A diag(Y p )A) P i ≤ Pi∗ ≤ P¯i 2 2 λi∗j [(Pim∗ j ) − ci j ] = 0 τ¯i∗ (Pi∗ − P¯i ) = 0

τ i∗ (P i − Pi∗ ) = 0 Proof The result directly follows from Theorems 3.25–3.27 in [28].



Actually, many numerical methods are available to solve Convex Optimization Problem (7.4). Here, Saddle Point Dynamics is employed due to its great success in designing distributed network control protocols [29, 30], which helps to enhance the resilience of power systems. Design the following Lagrangian function L(P, λ, τ ) = J (P, W ) +

(i, j)∈E

nb nb  λi j (Pimj )2 − ci2j + τ¯i (Pi − P¯i ) + τ i (P i − Pi ) i=1

i=1

7.3 Nonrecurring Protection Scheme

121

where (i, j) is an element of set E if Bus i and Bus j are connected in power networks. According to Saddle Point Theorem in [28], the optimal solution P ∗ to Optimization Problem (7.4) satisfies the KKT condition in Proposition 7.3.2 with Lagrangian multipliers λ∗ and τ ∗ if and only if (P ∗ , λ∗ , τ ∗ ) is a saddle point of the Lagrangian function L(P, λ, τ ). Next, we present the saddle point dynamics to search for the saddle point of Lagrangian function L(P, λ, τ ) [31]. P˙ = −∇ P L(P, λ, τ ) = −2 W ◦ (P − P 0 ) − 2



λi j Pimj Rimj − (τ¯ − τ )

(i, j)∈E m 2 2 + λ˙ i j = [∇λi j L(P, λ, τ )]+ λi j = [(Pi j ) − ci j ]λi j

¯ + τ˙¯i = [∇τ¯i L(P, λ, τ )]+ τ¯i = [Pi − Pi ]τ¯i + τ˙ i = [∇τ i L(P, λ, τ )]+ τ i = [P i − Pi ]τ i

where

(7.6)

∗ T Rimj = eiT A T diag(Y pm )Ae j · (A T diag(Y pm )A)−1 (ei − e j )

and the operator [ ]+ is defined as [x]+ y =



x, y > 0; max{x, 0}, y = 0.

(7.7)

Significantly, it is guaranteed in theory that Saddle Point Dynamics (7.6) approaches the optimal solution of Optimization Problem (7.4) as time goes to infinity. Proposition 7.3.3 Saddle Point Dynamics (7.6) globally asymptotically converges to the optimal solution to Optimization Problem (7.4). Proof See Sect. 7.8.3 in Appendix.



Finally, we summarize the Nonrecurring Protection Scheme in Table 7.2. First of all, we set the number of cascading steps m and estimate the evolution time of cascading failure tc ≈ mT p for protective actions. NPS detects electrical signals of power network in real time with the aid of PMUs and then estimates the current branch admittance vector Y p (t) [21]. This allows WAPCS to locate the disturbed branches by comparing the current branch admittance vector Y p (t) with the original one Y p0 . Then NPS computes the admittance changes on the branches to identify the disturbance δ. The above disturbance initiates the cascading process to predict the state of power grids at the m-th cascading step, which enables us to solve Saddle Point Dynamics (7.6) and work out remedial actions. When the cascading failure evolves to the m-th step, WAPCS will take the remedial actions of adjusting injected power on buses. Finally, NPS computes the power flow on each branch to validate the remedial actions.

122

7 Model Predictive Approach to Preventing Cascading Proces

Table 7.2 Nonrecurring protection scheme 1: Detect electrical signals using PMUs 2: Estimate branch admittance vector Y p (t) 3: if (Y p (t) = Y p0 ) 4: Identify the disturbance δ = Y p (t) − Y p0 5: Initiate cascading dynamics with (7.2) and (7.3) 6: Save system state at the m-th cascading step 7: Solve Saddle Point Dynamics (7.6) 8: if (k = m) 9: Take protective actions with solutions to (7.6) 10: end if 11: Compute the power flow on each branch 12: else 13: Go back to Step 1 14: end if

7.4 Recurring Protection Scheme In this section, we propose the second scheme, i.e. Recurring Protection Scheme (RPS), with which WAPCS takes remedial actions and implements the corrective control by adjusting the injected power on buses at two consecutive cascading steps (i.e., m − 1 and m). Compared with NPS, more control variables are available in RPS to optimize the objective function. Essentially, RPS increases the flexibility of preventing cascading outages. Theoretically, the optimization formulation of RPS can be presented as min C(P m−1 , P m , W ) Pk

∗ s.t. Pikj = eiT A T diag(Y pk )Ae j (ei − e j )T (A T diag(Y pk )A)−1 P˜ k

Y pk+1 = G(Pikj ) ◦ Y pk Pimj )2 ≤ ci2j , (i, j) ∈ E P i ≤ P˜ik ≤ P¯i , i ∈ In b , k ∈ {m − 1, m}

(7.8)

where C(P m−1 , P m , W ) = W ◦ (P m−1 − P 0 )2 + W ◦ (P m − P 0 )2 and P˜ k =



P 0 , k ≤ m − 2; P k , otherwise.

Remark 7.4 It is assumed that the fluctuation of P k is negligible prior to load shedding and generator ramping/tripping. Thus, the value of P k keeps constant (i.e.,

7.4 Recurring Protection Scheme

123

P 0 ) in the optimization formulations (7.4) and (7.8) before implementing protection schemes. Actually, the above optimization formulations are applied to the case with the time-varying values of P k when the dynamics of P k (1 ≤ k ≤ m − 1) is available. Proposition 7.4.1 Solutions to Optimization Problem (7.8) do not underperform those to the Convex Optimization Problem (7.4) in terms of minimizing the changes of injected power on buses. Proof Let P k∗ , k ∈ {m − 1, m} denote the solution to Optimization Problem (7.8), and P ∗ represents the solution to Convex Optimization Problem (7.4). Then it follows from W ◦ (P m∗ − P 0 )2 ≤ C (P (m−1)∗ , P m∗ , W ) ≤ C (P 0 , P ∗ , W ) = W ◦ (P ∗ − P 0 )2

that

W ◦ (P m∗ − P 0 )2 ≤ W ◦ (P ∗ − P 0 )2 , 

which completes the proof.

Actually, it is difficult to obtain the optimal solution to Optimization Problem (7.8) due to its non-convexity. The linearized method can be applied to approximate the non-convex constraint for achieving the global optima. To simplify mathematical expressions, we define ∗

F(Y pk ) = eiT A T diag(Y pk )Ae j (ei − e j )T (A T diag(Y pk )A)−1 Then we obtain Pikj = F(Y pk ) P˜ k and Pimj = F(Y pm )P m  m−1 Pm = F G(Pim−1 ) ◦ Y p j 

= F G(F(Y pm−1 )P m−1 ) ◦ Y pm−1 P m The gradient of Pimj with respect to [P m−1 , P m ] is given by

T ⎞ G (F(Y pm−1 )P m−1 ) ◦ Y pm−1 ∇Y pm F(Y pm )P m ⎟ ⎜ ·F(Y pm−1 )T ∇ Pimj = ⎝ ⎠ 

T F G(F(Y pm−1 )P m−1 ) ◦ Y pm−1 ⎛

Therefore, Pimj = F(Y pm )P m can be approximated by the following linear equation in the neighborhood of [P 0 , P 0 ].

124

7 Model Predictive Approach to Preventing Cascading Proces  T  G (F (Y pm−1 )P 0 ) ◦ Y pm−1 F (Y pm−1 )(P m−1 − P 0 ) Pˆimj |[P 0 ,P 0 ] = ∇Y pm F (Y pm | P 0 )P 0  + F G(F (Y pm−1 )P 0 ) ◦ Y pm−1 P m (7.9)

where Y pm | P 0 = G(F(Y pm−1 )P 0 ) ◦ Y pm−1 . In this way, Optimization Problem (7.8) can be approximated by the following problem. min C(P m−1 , P m , W ) Pk

T  G (F(Y pm−1 )P 0 ) ◦ Y pm−1 F(Y pm−1 )(P m−1 − P 0 ) s.t. Pˆimj = ∇Y pm F(Y pm | P 0 )P 0

 + F G(F(Y pm−1 )P 0 ) ◦ Y pm−1 P m ( Pˆimj )2 ≤ ci2j , (i, j) ∈ E P i ≤ Pik ≤ P¯i , i ∈ In b , k ∈ {m − 1, m}

(7.10)

Proposition 7.4.2 Optimization Problem (7.10) is convex. Proof The objective function C(P m−1 , P m , W ) is convex, and Pˆimj is an affine function of variables P m−1 and P m . Also, ( Pˆimj )2 − ci2j is convex. This indicates that Optimization Problem (7.10) is convex.  Next, we discuss the numerical solution to Optimization Problem (7.10). Design the Lagrangian function for Optimization Problem (7.10) as follows L(P m , P m−1 , λ, τ ) = C(P m−1 , P m , W ) +

nb

m m  τ¯i (Pi − P¯i ) + τ¯im−1 (Pim−1 − P¯i ) i=1

nb 

m + τ i (P i − Pim ) + τ im−1 (P i − Pim−1 ) i=1

+



 λi j ( Pˆimj )2 − ci2j

(i, j)∈E

Saddle point dynamics to search for the saddle point of Lagrangian function L(P m , P m−1 , λ, τ ) is given by P˙k = −∇ P k L(P m , P m−1 λ, τ ) = −2W ◦ (P k − P 0 ) − 2



λi j Pˆimj Rikj − (τ¯ k − τ k )

(i, j)∈E 2 + ˆm 2 λ˙ i j = [∇λi j L(P, λ, τ )]+ λi j = [( Pi j ) − ci j ]λi j

= [Pik − P¯i ]+ τ¯˙ik = [∇τ¯ik L(P, λ, τ )]+ τ¯ k τ¯ k i

i

τ˙ ik = [∇τ ik L(P, λ, τ )]+ = [P i − Pik ]+ τk τk i

i

(7.11)

7.4 Recurring Protection Scheme

125

where k ∈ {m − 1, m} and ⎧

  m 0 T ⎪ G (F(Y pm−1 )P 0 ) ◦ Y pm−1 ⎨ ∇Y pm F(Y p | P 0 )P Rikj = ·F(Y pm−1 )T , k = m − 1 ⎪ ⎩ F G(F(Y m−1 )P 0 ) ◦ Y m−1 T , k = m p p and the operator [ ]+ is defined in Eq (7.7). Remark 7.5 The solution to Optimization Problem (7.10) merely guarantees | Pˆimj | ≤ ci j instead of |Pimj | ≤ ci j , ∀(i, j) ∈ E. Therefore, it is necessary to check whether the constraints |Pimj | ≤ ci j , ∀(i, j) ∈ E hold after adjusting the injected power on buses at the (m − 1)-th step and the m-th step according to the solution to (7.10). The solution to (7.6) can be adopted as a remedy if there exists (i, j) ∈ E such that |Pimj | > ci j . Remark 7.6 For the RPS, only two consecutive steps are taken into consideration for protective actions, because protective actions on more cascading steps will greatly increase the computational complexity of the optimization problem and thus require more time for its numerical solutions. This may delay the real-time implementation of protection schemes. In theory, we can guarantee that Saddle Point Dynamics (7.11) achieves the asymptotic convergence of the global optimal solution to Optimization Problem (7.10). Proposition 7.4.3 Saddle Point Dynamics (7.11) globally asymptotically converges to the optimal solution to Convex Optimization Problem (7.10). Proof The proof follows by the same argument as that of Proposition 7.3.3, and is thus omitted.  Table 7.3 presents the procedure of RPS, which resembles NPS except for controllable cascading steps and Saddle Point Dynamics (7.11). Compared with NPS, RPS adjusts the injected power on buses to prevent cascading outages at two consecutive cascading steps (k ∈ {m − 1, m}). If RPS fails to prevent further cascading outages through the model validation of two consecutive protective actions, the one-off protection from the solution to (7.6) takes effect as a remedy. In practice, NPS and RPS are two independent mechanisms that cannot be executed subsequently in one potential cascade. In other words, if RPS is executed, NPS will not be executed, and vice versa. Essentially, NPS can be regarded as a special case of RPS by specifying protection actions at a given cascading step. Normally, more branches can be protected if the prediction horizon m is shorter. This implies that NPS with a shorter prediction horizon probably outperforms RPS with a relatively long prediction horizon. In addition, NPS and RPS are designed based on the steady-state assumption. If the assumption fails to hold, the SPS is initiated as a remedy to protect power grids against blackouts. Remark 7.7 Different types of relays introduce variable line-tripping delays based on the severity of overloading. The outage model of branches can be improved to reflect the temporal operational characteristics of relays, and the variable time delays

126

7 Model Predictive Approach to Preventing Cascading Proces

Table 7.3 Recurring protection scheme 1: Detect electrical signals using PMUs 2: Estimate branch admittance vector Y p (t) 3: if (Y p (t) = Y p0 ) 4: Identify the disturbance δ = Y p (t) − Y p0 5: Initiate cascading dynamics with (7.2) and (7.3) 6: Get system states at the (m − 1)-th cascading step 7: Solve Saddle Point Dynamics (7.11) 8: Calculate Pimj , ∀(i, j) ∈ E with solutions to (7.11) 9: if (|Pimj | ≤ ci j , ∀(i, j) ∈ E ) 10: Solution to (7.11) ⇒ Psl 11: else 12: Solution to (7.6) ⇒ Psl 13: end if 14: if (k ∈ {m − 1, m}) 15: Take protective actions according to Psl 16: end if 17: Compute the power flow on each branch 18: else 19: Go back to Step 1 20: end if

of line tripping can be roughly estimated based on the overloading level. Thus, this does not affect the applicability and performance of protection schemes (i.e., NPS and RPS). Remark 7.8 NPS and RPS allow for both load shedding and generation ramping/tripping while implementing protection actions against cascading blackouts. The generation ramping/tripping can be achieved by adjusting the injected power on generator buses, and load shedding can be achieved by adjusting the injected power on load buses. Moreover, the generator is tripped when the injected power on the generator bus is adjusted to its lower bound. In practice, the generation ramping/tripping shall be applied first before any load shedding is called. Actually, the proposed optimization formulation is able to allow for the order of protection actions (i.e., generation ramping/tripping and load shedding). Specifically, the optimization problems (7.4) and (7.10) can be solved by specifying the constant load and the variable power generation on buses, which can be achieved by adjusting the lower and upper bounds of Pi in (7.4) and Pik in (7.10). If the solutions to the above optimization problems are available, the generation ramping/tripping can be applied to prevent the cascading failure without load shedding. If there are no feasible solutions, the original optimization problems (7.4) and (7.10) should be solved to obtain the optimal injected power on both load buses and generator buses. In this way, the generation ramping/tripping can be applied first, immediately followed by load shedding in order to prevent the cascades.

7.5 Numerical Simulations

127

7.5 Numerical Simulations In this section, we validate and compare two protection schemes (i.e., NPS and RPS) on IEEE test Systems. Two flow charts are presented to illustrate the simulations of NPS and RPS, respectively (see Fig. 7.2). Specifically, the disturbance is added on the selected branch with the initial cascading step k = 0. Then the power flow on each branch is computed by solving the DC power flow equation. The overloading branch will be tripped once its timer runs out of the preset time in the outage model

Fig. 7.2 Flow chart of simulations for two protection schemes a NPS and b RPS

128

7 Model Predictive Approach to Preventing Cascading Proces

of branches. When the cascading process arrives at the specified steps (i.e., k = m for NPS, k = m − 1 and k = m for RPS), the protection schemes (NPS or RPS) are initiated to prevent further cascading outage of power grids.

7.5.1 Parameter Setting The parameter setting for both NPS and RPS is given in detail as follows. For simplicity, we set W = (1, 1, . . . , 1)T to ensure that all buses are equally treated in protection schemes. Per unit values are adopted with the base value of power 100 MV in the simulations. The preset time of the timer is T p = 1s in protective relays. The minimum injected power on each bus is equal to the negative value of its total load, and the maximum injected power is provided by the generator connected to this bus. For load buses, their maximum injected power is 0. Consequently, for load buses, we set P¯i = 0 and P i is the total injected power. For generator buses, P¯i is the total power from the generator, while P i is the injected power from the load. The threshold of power flow on each branch is 1 pu with σ = 103 in the approximation function. In addition, Euler method is employed to implement the numerical algorithms (7.6) and (7.11) with the step size 0.01.

7.5.2 Validation and Comparison If the branch connecting Bus 9 and Bus 11 (i.e., Branch 10) is tripped as the initial disturbance of IEEE 57 Bus Systems, the cascading process comes to an end after 6 cascading steps (tc ≈ 6s) without protection schemes. Branch 10 is selected to add the initial contingency because it can result in a relatively severe cascading failure. Actually, other branches can also be chosen to trigger the cascades. Figure 7.3 shows the final configuration of IEEE 57 Bus Systems after suffering from the above disturbance. Red balls represent the generator buses, while green ones denote the load buses. Bus ID is also marked on each ball, and the cyan links refer to branches in power systems. The power system ends up with 43 connected branches (5 branches with the power flow) and the total power flow is 1.004pu. Notably, the power network is separated into 5 islands encircled with the dashed curves and 13 isolated buses. Two islands survive due to the power supply from their respective generator buses (Bus 6 and Bus 12), while the other three islands without generator buses go through blackouts. In Fig. 7.3, two islands subject to blackouts include a lot of load buses, which implies that the initial contingency results in large disruptions of power grids in terms of power supply. Thus, it is crucial to initiate real-time protection schemes during the emergency in order to terminate the cascading blackout. In contrast, Fig. 7.4 presents the final configuration of IEEE 57 Bus Systems with protection schemes (NPS or RPS) after adding the same disturbance (i.e., tripping Branch 10) as that in Fig. 7.3. It is observed that the power system is well protected since most branches in the network are in a good state of transmitting power flow

7.5 Numerical Simulations

129

Fig. 7.3 Final configuration of the IEEE 57 bus system without protection schemes

among buses. The power network is decomposed into 3 islands and 9 isolated buses with 53 connected branches. Two islands are composed of generator buses (Bus 6, Bus 9, and Bus 12), while blackouts occur on the island that only includes load buses (Bus 32 and Bus 33). Thanks to NPS and RPS, a lot of load buses can still obtain the power supply in spite of the separation of the whole power network. This indicates the effectiveness of protection schemes in protecting power grids against blackouts. Specifically, for NPS, the cascading process stops after implementing the optimal adjustment of injected power on buses at the 4th cascading step (tc ≈ 4s), and the system remains unchanged since then, with the total power flow of 9.156pu. Moreover, the objective function is minimized with the value of 0.1068. Let Pk = P k − P 0 , k ∈ {3, 4} denote the vector of load change on each bus at the k-th cascading step. Figure 7.5 shows the distribution of changes of injected power on each bus according to NPS. There are no negative values for the changes of injected power on buses, which implies the absence of generator tripping during power systems protection. In Fig. 7.5, the largest change of injected power occurs on Bus 51 using NPS, and its variation magnitude exceeds 0.1pu. By adjusting injected power on buses at the 3rd and the 4th cascading steps, RPS succeeds in preventing the cascading outage at the 4th cascading step (tc ≈ 4s). The total power flow is 12.496pu, and the value of objective function is 0.0979 in the

130

7 Model Predictive Approach to Preventing Cascading Proces

Fig. 7.4 Final configuration of the IEEE 57 bus system with protection schemes (NPS or RPS)

end, less than 0.1068 in NPS. This demonstrates the better performance of RPS to maintain the power transmission and prevent cascading outages in spite of high computational cost. Figure 7.6 presents the distribution of changes of injected power on each bus using RPS. The green bars denote the changes of injected power on buses at the 3rd cascading step (tc ≈ 3s), while the blue ones refer to those at the 4th cascading step (tc ≈ 4s). Note that the negative values at Bus 3 and Bus 12 indicate the decrease of power supply to these generator buses. In Fig. 7.6, four buses (i.e., Bus 1, Bus 3, Bus 8, and Bus 12) adjust their respective injected power at the 3rd and 4th cascading steps, while other buses only adjust the injected power at the 4th cascading step. Moreover, the total change of injected power on Bus 8 is the largest of all buses and it exceeds 0.2 pu. To compare NPS and RPS in depth, we consider the effect of different operating points (i.e., prediction horizon m) on the performance of two schemes. As we can observe in Fig. 7.7, NPS and RPS lead to the same performance in terms of Ncb and Nab (as green balls and red balls are all overlapping) for each different prediction horizon m. In terms of the total power flow Pt , RPS performs better than NPS in the power transmission because it can contribute to having more total power flow when m < 5. As for the changes of injected power on buses Pm , RPS can protect power

7.5 Numerical Simulations

131

0.15 P4

P

0.1

0.05

0 0

10

20

30

40

50

Bus ID

Fig. 7.5 Distribution of changes of injected power on each bus using NPS 0.25 P3 P4

0.2

P

0.15 0.1 0.05 0 -0.05 0

10

20

30

40

50

Bus ID

Fig. 7.6 Distribution of changes of injected power on each bus using RPS

grids with a lower cost (i.e., fewer changes of injected power on buses) when m ≤ 5. This partially validates theoretical results in Proposition 7.1. It is worth pointing out that NPS and RPS perform the same for all four indexes when m ≥ 6. This is because the cascading failure process comes to an end at m = 6 and there is no chance that protection schemes can prevent power grids from blackouts.

7.5.3 Effect of Tuning Parameters The parameter W describes the bus weight in the optimal adjustment of injected power on buses. To be specific, the buses with larger weights are deemed less significant in power networks, thus WAPCS prefers to adjust the injected power on these buses during emergency. This section aims to investigate the effects of W on the

132

7 Model Predictive Approach to Preventing Cascading Proces

Control input

80 No dispatich & shedding With dispatich & shedding

60 40 20 0 5

10

15

20

25

30

35

25

30

35

Branch ID Normalized cost

1

0.5

0 5

10

15

20

Branch ID

Fig. 7.7 Comparison between NPS and RPS with different prediction horizons m on IEEE 57 bus system. Ncb and Nab refer to the number of connected branches and the number of active branches with power flow at the end of cascading failures, respectively. Pt and Pm represent the total power flow and the total amount of changes of injected power on buses, respectively

performance of protection schemes. For simplicity, we assign the same weight Wl to load buses and the same weight Wg to generator buses and take into account the weight distribution and proportion γ = Wg /Wl on the two types of buses. Numerical simulations are conducted to implement NPS and RPS on the IEEE 57 Bus Systems with the same initial disturbance, respectively (i.e., tripping Branch 10). For NPS, the protective action is implemented at the 4th cascading step, while remedial actions are taken at the 3rd and 4th cascading steps for RPS. Then we look into the power transmission and intact branches in the final cascading step by tuning the parameter γ from 0.1 to 10 gradually. Figure 7.8 presents the dependence of connected branches Ncb , active branches Nab , the power flow Pt and the change of injected power on buses Pm on the parameter γ in the final configuration. It is observed that NPS is insensitive to the variation of γ , and all the four measures keep stable. In contrast, RPS behaves differently as γ varies in the range of [0.1, 1]. Specifically, Ncb and Nab jump from 51 to 53 as γ increases from 0.3 to 0.4. In addition, Pm declines greatly as γ varies from 0.1 to 0.3. It is worth pointing out that RPS outperforms NPS in terms of total power transmission after terminating the cascading outages. Compared with NPS, RPS ensures more power flow with fewer changes of injected power on buses when γ is larger than 1. In both NPS and RPS, we can observe that Ncb is always equal to Nab for the same parameter γ .

7.5 Numerical Simulations

133

55 54

NPS RPS

54

ab

53

53

N

Ncb

55

NPS RPS

52

52

51

51

50 −1 10

0

10

50 −1 10

1

10

γ 16

0

10

1

10

γ 5

NPS RPS

NPS RPS

4

14 m

P

Pt

3 12

2 10 8 −1 10

1 0

10

γ

1

10

0 −1 10

0

10

1

10

γ

Fig. 7.8 The effect of weight proportion γ on the final configuration of power systems. Ncb and Nab refer to the number of connected branches and the number of active branches with power flow at the end of cascading failures, respectively. Pt and Pm represent the total power flow and the total amount of changes of injected power on buses, respectively

7.5.4 Other Test Systems Two protection schemes are also implemented on IEEE 118 Bus System and IEEE 300 Bus System to validate the proposed approach. Specifically, Branch 3 is tripped as the initial disturbance to trigger the cascading failure of 118-bus system. NPS is implemented at 6th cascading step to protect power grids against blackouts, and RPS is taken at the 5th and 6th cascading steps. Both NPS and RPS achieve the same performance in terms of Ncb and Nab (i.e., Ncb = 59 and Nab = 58 for both NPS and RPS). RPS outperforms NPS in terms of the total power flow on branches because of Pt = 20.4 for RPS and Pt = 18.4 for NPS. In addition, it follows from Pm = 0.06 for RPS and Pm = 0.29 for NPS that RPS can protect power grids against blackouts with a lower cost (i.e., the fewer changes of injected power on buses) in comparison to NPS. Figures 7.9 and 7.10 present the changes of injected power on each bus of the IEEE 118 Bus System by taking NPS and RPS, respectively. As we can see in Fig. 7.9, the largest change of injected power occurs on Bus 59 with NPS. Since most changes of injected power are positive, the protection action of load shedding is taken on most buses. Similarly, the largest change of injected power also occurs on Bus

134

7 Model Predictive Approach to Preventing Cascading Proces

Fig. 7.9 Distribution of changes of injected power on buses of IEEE 118 Bus System with NPS

0.3 P6 0.25 0.2

P

0.15 0.1 0.05 0 -0.05 -0.1 0

20

40

60

80

100

Bus ID

Fig. 7.10 Distribution of changes of injected power on buses of IEEE 118 bus system with RPS

0.4 P5 P6

0.3 0.2

P

0.1 0 -0.1 -0.2 -0.3 -0.4 0

20

40

60

80

100

Bus ID

59 with RPS in Fig. 7.10. In terms of changes of the injected power on buses, it is suggested that more efforts are taken to prevent the cascades at the 6th cascading step. As for the 300-bus system, Branch 6 is tripped as the initial disturbance to start cascading failures of power grids. NPS is implemented at the 4th cascading step, and RPS is taken at the 3rd and the 4th cascading steps. Compared to RPS, NPS results in more connected branches and active branches (i.e., Ncb = 124 and Nab = 66 for NPS, Ncb = 116 and Nab = 46 for RPS) after protective actions. In addition, NPS outperforms RPS in terms of total power flow on branches due to Pt = 13.6 for NPS and Pt = 8.8 for RPS. Nevertheless, RPS can protect power grids against blackouts with much fewer changes of injected power on buses compared to NPS (i.e., Pm = 0.87 for RPS and Pm = 48.97 for NPS), which is consistent with the conclusion in Proposition 7.4.1. Figures 7.11 and 7.12 show the distribution of changes of injected power on buses of the IEEE 300 Bus System by taking NPS and RPS, respectively. As we can observe in Fig. 7.11, Bus 171 adjusts the largest amount of injected power

7.5 Numerical Simulations

135

Fig. 7.11 Distribution of changes of injected power on buses of IEEE 300 bus system with NPS

7 P4 6 5

P

4 3 2 1 0 -1 0

50

100

150

200

250

300

Bus ID

Fig. 7.12 Distribution of changes of injected power on buses of IEEE 300 bus system with RPS

P3

0.3

P4

0.2 0.1 0

P

-0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 0

50

100

150

200

250

300

Bus ID

with NPS. In comparison, few changes of injected power are made on the buses from Bus 250 to Bus 300. In Fig. 7.12, it is demonstrated that both load shedding and generation ramping/tripping are taken at the 3rd and the 4th cascading steps with RPS. In terms of changes of the injected power on buses, the amount of changes at the 3rd step is comparable to that at the 4th step on the whole. On the whole, NPS is superior to RPS in terms of computation complexity, and it takes less time for NPS to work out Optimization Problem (7.4) according to Saddle Point Dynamics (7.6). This enables NPS to efficiently implement protective actions at the early stage of cascading blackouts. Considering that more control variables in RPS are available, RPS is able to provide more flexible solutions of adjusting injected power on buses to control and protect power systems against cascading outages. In addition, RPS is able to prevent cascading outages of power grids with fewer changes of injected power on buses.

136

7 Model Predictive Approach to Preventing Cascading Proces

7.6 Discussions The proposed protection architecture has the potential to be extended to deal with the uncertainty of power systems and coordinate different types of protection measures for more reliable protection. In practice, the initial contingency triggers chain reactions, which may evolve into multiple cascading outage paths due to various uncertainties (e.g., load variation, temperature variation, breaker faults, and other unexpected events) in the period of steady-state progression. The proposed protection architecture is able to prevent the cascading failure for each specific cascading failure path. By taking into account the uncertainties in the protection architecture, multiple possible cascading failure paths and the remedial scheme for each specific path can be generated and recorded in advance. During the emergency, the power system can implement the prerecorded remedial action for the path that matches the real cascading outage path. In case of emergency, the proposed protection architecture goes through three sequential stages: real-time detection and identification of disruptive contingencies, prediction of cascading outage paths and protection schemes based on optimal adjustment of injected power on buses. Essentially, this architecture is also compatible to other protection measures such as proactive line tripping, the online adjustment of FACTS devices and the isolation of faulted elements. More effective protection of power grids can be achieved with the coordination of different types of protection schemes. Specifically, multiple possible cascading failure paths can be described as the corresponding trajectories in the state space, and the objective is to terminate the evolution of each state trajectory by cooperatively implementing multiple protection schemes with the least cost (which may have different definitions e.g., power loss, network connectivity, fluctuation of voltage and frequency, etc., in different applications).

7.7 Conclusions In this chapter, we aim to propose a model predictive approach for the real-time protection of power grids against blackouts. A protection architecture was adopted to identify the initial contingency, predict the cascading failure paths and compute the optimal protection strategies. The proposed protection architecture overcomes the shortcoming of the classic SPSs and opens up the opportunity to develop disturbancerelated protection schemes. Based on the protection architecture, two types of protection schemes (i.e., NPS and RPS) were designed to prevent the cascading outage of power systems. The first scheme NPS takes remedial actions at a given cascading step, while the second one implements the corrective control at two consecutive cascading steps. Moreover, It is proved that RPS does not underperform NPS in terms of minimizing the changes of injected power on buses, and these two schemes are able to achieve the optimal adjustment of injected power on buses against blackouts.

7.8 Appendix

137

Finally, extensive simulations were conducted to validate the proposed approach and theoretical results. It is demonstrated that the protection schemes perform well on different IEEE test systems. Regarding our future research work, other than the possible extensions as mentioned in Sect. 7.6, we may also study on distributed protection scheme, where challenging issues such as synchronization, convergence rate and fault tolerance will be carefully investigated.

7.8

Appendix

This section presents the mathematical definition of the operators ∗ and −1∗ and the proofs of Eq. (7.2) and Proposition 7.3.3 in detail.

7.8.1

Definition of Operators

Suppose the nodal admittance matrix Ybk = A T diag(Y pk )A is composed of q isolated subnetworks denoted by Si , i ∈ Iq = {1, 2, . . . , q} and each subnetwork Si includes ki buses. Let Vi = {i 1 , i 2 , . . . , i ki } denote the set of bus IDs in the subnetwork Si , where i 1 , i 2 ,..., i ki denote the bus IDs. Notice that Bus i 1 in Subnetwork Si is designated as the reference bus. Moreover, the nodal admittance matrix of the i-th subk = EiT Ybk Ei , i ∈ Iq , where Ei = (ei1 , ei2 , . . . , eiki ). network can be computed as Yb,i ∗ Then the operators ∗ and −1 are defined as follows [26]. Definition 7.1 Given the nodal admittance matrix Ybk , the operators ∗ and −1∗ are defined by     q  k ∗ 0 0kTi −1 0 0kTi −1 k Yb,i EiT Ei Yb = 0ki −1 Iki −1 0ki −1 Iki −1 i=1

and



−1∗ Ybk

=

q 

 Ei

i=1

respectively, where k Yb,i



0kTi −1 Iki −1





k Yb,i

= 0ki −1 Iki −1



−1 

 k Yb,i

 0ki −1 Iki −1 EiT ,

0kTi −1 Iki −1

 .

Iki −1 is the (ki − 1) × (ki − 1) identity matrix, and 0ki −1 denotes the (ki − 1) dimensional column vector with zero elements.

138

7 Model Predictive Approach to Preventing Cascading Proces

7.8.2

Proof of Equation 7.2 ∗

According to the DC power flow equation, we have θ k = (B k )−1 P, where B k = A T diag(Y pk )A and θ k = (θ1k , θ2k , . . . , θnkb )T . Let Bikj denote the element in the i-th row and j-th column of the matrix B k . Then the power flow from Bus i to Bus j can be computed by Pikj = Bikj (θik − θ kj ) = eiT B k e j (ei − e j )T θ k ∗

= eiT B k e j (ei − e j )T (B k )−1 P ∗

= eiT A T diag(Y pk )Ae j (ei − e j )T (A T diag(Y pk )A)−1 P.

7.8.3

Proof of Proposition 7.3.3

Design Lyapunov function as follows V (P, λ, τ ) =

1 (P − P ∗ 2 + λ − λ∗ 2 + τ − τ ∗ 2 ) 2

The time derivative of V (P, λ, τ ) along Saddle Point Dynamics (7.6) gives V˙ (P, λ, τ ) = (P ∗ − P)T ∇ P L(P, λ, τ ) ∗ T + + (λ − λ∗ )T [∇λ L(P, λ, τ )]+ λ + (τ − τ ) [∇τ L(P, λ, τ )]τ

Equation (7.7) leads to ∗ T (λ − λ∗ )T [∇λ L(P, λ, τ )]+ λ ≤ (λ − λ ) ∇λ L(P, λ, τ )

and

∗ T (τ − τ ∗ )T [∇τ L(P, λ, τ )]+ τ ≤ (τ − τ ) ∇τ L(P, λ, τ ).

Since L(P, λ, τ ) is convex in P and concave in λ and τ , we have (P ∗ − P)T ∇ P L(P, λ, τ ) ≤ L(P ∗ , λ, τ ) − L(P, λ, τ ) and (λ − λ∗ )T ∇λ L(P, λ, τ ) + (τ − τ ∗ )T ∇τ L(P, λ, τ ) ≤ L(P, λ, τ ) − L(P, λ∗ , τ ∗ ),

References

139

which leads to V˙ (P, λ, τ ) ≤ L(P ∗ , λ, τ ) − L(P, λ, τ ) + L(P, λ, τ ) − L(P, λ∗ , τ ∗ )

  = L(P ∗ , λ, τ ) − L(P ∗ , λ∗ , τ ∗ ) + L(P ∗ , λ∗ , τ ∗ ) − L(P, λ∗ , τ ∗ ) Moreover, it follows from L(P ∗ , λ, τ ) − L(P ∗ , λ∗ , τ ∗ ) ≤ 0 and L(P ∗ , λ∗ , τ ∗ ) − L(P, λ∗ , τ ∗ ) < 0 that V˙ (P, λ, τ ) < 0. This indicates V (P, λ, τ ) converges to 0 (and P also converges to the optimal value P ∗ ) as time goes to infinity. The proof is thus completed.

References 1. Begovic, M., Novosel, D., Karlsson, D., Henville, C., Michel, G.: Wide-area protection and emergency control. Proc. IEEE 93(5), 876–891 (2005) 2. Zhai, C., Nguyen, H., Xiao, G.: A robust optimization approach for terminating the cascading failure of power systems. Electr. Power Syst. Res. 189, 106794 (2020) 3. Bo, Z., Lin, X., Wang, Q., Yi, H., Zhou, F.: Development of power system protection and control. Prot. Control Mod. Power Syst. 1–7 (2016) 4. Hewitson, L., Mark, B., Ramesh, B.: Practical power system protection. Elsevier (2004) 5. Molina-Cabrera, A., Rios, M.A., Besanger, Y., HadjSaid, N.: A latencies tolerant model predictive control approach to damp Inter-area oscillations in delayed power systems. Int. J. Electr. Power Energy Syst. 98, 199–208 (2018) 6. Xyngi, I., Marjan, P.: An intelligent algorithm for the protection of smart power systems. IEEE Trans. Smart Grid 4(3), 1541–1548 (2013) 7. Heng, L., et al.: Reliable GPS-based timing for power systems: a multi-layered multi-receiver architecture. In: Power and Energy Conference at Illinois (PECI). IEEE (2014) 8. Madami, V., Adamiak, M., Thakur, M.: Design and implementation of wide area special protection schemes. In: 2004 57th Annual Conference for Protective Relay Engineers. IEEE (2004) 9. Zima, M.: Special Protection Schemes in Electric Power Systems. Technical Report, Chicago (2002) 10. Centeno, V., Phadke, A.G., Edris, A., Benton, J., Gaudi, M., Michel, G.: An adaptive out-of-step relay for power system protection. IEEE Trans. Power Del. 12(1), 61–71 (1997) 11. Milosevic, B., Begovic, M.: Voltage-stability protection and control using a wide-area network of phasor measurements. IEEE Trans. Power Syst. 18(1), 121–127 (2003) 12. Zhu, Y., Song, S., Wang, D.: Multiagents-based wide area protection with best-effort adaptive strategy. Int. J. Electr. Power Energy Syst. 31(2–3), 94–99 (2009) 13. Aminifar, F., et al.: Synchrophasor measurement technology in power systems: panorama and state-of-the-art. IEEE Access 2, 1607–1628 (2014) 14. Lu, W., et al.: Blackouts: Description, analysis and classification. In: Proceedings of the 6th WSEAS International Conference on Power Systems. Lisbon, Portugal, 22–24 Sept (2006) 15. UCTE Investigation Committee: Interim Report of the Investigation Committee on the 28 September 2003 Blackout in Italy. UCTE Report, 27 Oct (2003) 16. NERC Steering Group: Technical Analysis of the August 14, 2003, Blackout: What Happened, Why, and What Did We Learn. Report to the NERC Board of Trustees (2004) 17. Yan, J., Tang, Y., He, H., Sun, Y.: Cascading failure analysis with DC power flow model and transient stability analysis. IEEE Trans. Power Syst. 30(1), 285–297 (2015) 18. Mousavi, O.A., Sanjari, M.J., Gharehpetian, G.B., Naghizadeh, R.A.: A simple and unified method to model HVDC links and FACTS devices in DC load flow. Simulation 85(2), 101– 109 (2009)

140

7 Model Predictive Approach to Preventing Cascading Proces

19. Phadke, A.G., Thorp, J.S.: History and applications of phasor measurements. In: Power Systems Conference and Exposition, pp. 331–335, PSCE’06. IEEE PES (2006) 20. Terzija, V., Valverde, G., Cai, D., Regulski, P., Madani, V., Fitch, J., Phadke, A.: Wide-area monitoring, protection, and control of future electric power networks. Proc. IEEE 99(1), 80–93 (2011) 21. Omid, A., Yuan, Y., Dobbe, R., Meier, A., Low, S., Tomlin, C.: Event detection and localization in distribution grids with phasor measurement units. arXiv preprint at https://arxiv.org/abs/ 1611.04653 22. Armenia, A., Chow, J.H.: A flexible phasor data concentrator design leveraging existing software technologies. IEEE Trans. Smart Grid 1(1), 73–81 (2010) 23. Naduvathuparambil, B., Valenti, M.C., Feliachi, A.: Communication delays in wide area measurement systems. In: Proceedings of the IEEE 34th Southeastern Symposium on System Theory, pp. 118-122 (2002) 24. Leelaruji, R., Vanfretti, L.: Power System Protective Relaying: Basic Concepts, IndustrialGrade Devices, and Communication Mechanisms. Technical Report, KTH (2011) 25. Song, J., et al.: Dynamic modeling of cascading failure in power systems. IEEE Trans. Power Syst. 31(3), 2085–2095 (2016) 26. Stagg, G.W., El-Abiad, A.H.: Computer Methods in Power System Analysis. McGraw-Hill (1968) 27. Zhai, C., Zhang, H., Xiao, G., Pan, T.: Modeling and identification of worst cast cascading failures in power systems, arXiv preprint at https://arxiv.org/abs/1703.05232 28. Ruszczynski, A.P.: Nonlinear Optimization. Princeton University Press, Princeton, NJ (2006) 29. Bai, L., et al.: Distributed control for economic dispatch via saddle point dynamics and consensus algorithms. In: Proceedings of the IEEE 55th Conference on Decision and Control, pp. 6934–6939 (2016) 30. Nedi´c, A., Ozdaglar, A.: Subgradient methods for saddle-point problems. J. Optim. Theor. Appl. 142(1), 205–228 (2009) 31. Ashish, C., Gharesifard, B., Cortes, J.: Saddle-point dynamics: conditions for asymptotic stability of saddle points. SIAM J. Control Optim. 55(1), 486–511 (2017)

Chapter 8

Robust Optimization Approach to Uncertain Cascading Process

Abstract Due to uncertainties and the complicated intrinsic dynamics of power systems, it is difficult to predict the cascading failure paths once the cascades occur. This makes it challenging to achieve the effective power system protection against cascading blackouts. By incorporating uncertainties and stochastic factors of the cascades, a Markov chain model is developed in this chapter to predict the cascading failure paths of power systems. The transition matrix of Markov chain is dependent on the probability of branch outage caused by overloads or stochastic factors. Moreover, a robust optimization formulation is proposed to prevent cascading blackouts by optimal load shedding and generation control for multiple cascading failure paths with relatively high probabilities. Since each state on the cascading failure paths can be described by one convex set, the proposed robust optimization problem is equivalent to the best approximation problem in Euclidean space. Thus, an efficient numerical solver based on Dykstra’s algorithm is employed to deal with the robust optimization problem. In theory, we provide a lower bound for the probability of being able to prevent the cascading blackouts of power systems. Finally, the proposed approach for power system protection is verified by a case study of IEEE 118 bus system.

8.1 Introduction The past several decades have witnessed major blackouts of power systems in the world, such as the 1999 Southern Brazil Blackout, the 2003 US-Canada Blackout and the 2012 India Blackout, which have affected millions of people and caused huge economic losses [1]. According to technical reports on major power outages, power system blackouts normally go through five stages: precondition, initiating event, cascading events, final state, and restoration [2]. The successful prevention of cascading blackouts relies on the effective identification of disruptive initiating events and the accurate prediction of cascading failure paths. Here, a cascading failure path refers to the sequence of branch outages during the cascades, and it describes how the failure propagates throughout power systems. However, the uncertainties and interdependencies of diverse components make it difficult to predict the cascading © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Zhai, Control and Optimization Methods for Complex System Resilience, Studies in Systems, Decision and Control 478, https://doi.org/10.1007/978-981-99-3053-1_8

141

142

8 Robust Optimization Approach to Uncertain Cascading Process

failure paths once the cascade is triggered. As a result, it is a great challenge for protecting power systems against cascading blackouts. The insights into the cascading sequences of power grids play an important role in designing an effective protection scheme to prevent cascading blackouts. The conventional protection of power systems largely relies on the protective relays for directly severing overloaded branches or the predetermined protection schemes to isolate the faulted components in power systems during emergencies, such as under-voltage load shedding [3], under-frequency load shedding [4, 5] and the special protection scheme (SPS) [6, 7]. According to the definition from the North American Electric Reliability Council (NERC), an SPS is designed to detect abnormal system conditions and take preplanned, corrective actions to provide acceptable system performance. It normally results in the decomposition of the whole system into several islands in order to isolate the faulty components [8]. Actually, each power system has its own emergency control practices and operating procedures, which are dependent on the different operating conditions and characteristics of the system [9]. This implies that the operating procedure for each power system is unique. Although the effectiveness of conventional protection methods has been demonstrated in practice, they are actually subject to multiple limitations in terms of compatibility and universality for various contingencies that could trigger the cascades. For instance, the SPS is designed based on a number of specific scenarios, characterized by certain abnormal stresses, following which the system will collapse. A fault that is not contained in such a list of scenarios, however, may not be covered by the SPS. For this reason, it would be desirable to develop a disturbance-related real-time protection scheme that is able to prevent the degradation of power systems during cascades. The introduction of phasor measurement units (PMUs) in power systems allows to develop the real-time protection scheme by estimating the system states for emergency control [10]. For example, [11] proposes a model predictive approach to prevent the cascading blackout of power systems by predicting the cascading failure path and taking the corresponding remedial actions in time. Nevertheless, the remedial actions are computed based on a deterministic cascading failure path without allowing for uncertainties and stochastic factors such as hidden failures, contingencies, and malfunction of protective relays. To address this problem, we propose a Markov chain model to describe the effect of uncertainties and stochastic factors on the cascading failure paths in this chapter. The transition matrix of Markov chain is dependent on the probability of branch outage that is caused by overloads or other relevant stochastic factors. This Markov chain model allows to predict multiple cascading failure paths of relatively high probabilities. Compared with existing probabilistic models of power system cascades [12–15], our proposed model contributes to the effective selection of the most possible cascading failure paths with relatively low computational burdens. Given predicted failure paths, further blackouts can be prevented by executing an effective loadshedding scheme. Collating multiple cascading failure paths that are most likely to occur, a robust optimization problem is formulated to compute the minimum amount of loads to shed. Geometrically, each considered state in cascading failure paths is associated with a convex set, and the intersection of such convex domains constitutes

8.1 Introduction

143

the search space to find the optimal solution to the robust optimization problem. In practice, the proposed Markov chain model and protection scheme can be adopted in the wide-area measurement and protection system to enhance the capability of preventing the cascading blackout (see Fig. 8.1). The main contributions of this work are summarized as follows: 1. Propose a Markov chain model to predict the cascading failure paths with the consideration of uncertainties and stochastic factors. 2. Develop a robust optimization formulation for emergency control and protection to prevent cascading blackouts. 3. Design an efficient numerical solver for the robust optimization problem using Dykstra’s algorithm. The remainder of this chapter is organized as follows: Sect. 8.2 presents a Markov chain model for the prediction of cascading failure paths. Section 8.3 provides the robust optimization formulation for the prevention of cascading blackouts, followed by an efficient numerical solver in Sect. 8.4. Numerical simulations are conducted to validate the protection approach in Sect. 8.5. Finally, we draw a conclusion and discuss the future work in Sect. 8.6.

Fig. 8.1 Information flow in the wide-area measurement, protection and control system. The block of risk identification is used to detect the contingency (e.g. branch outage). And the block of model prediction produces the possible cascading failure paths. The block of protection scheme implements the protective actions (e.g. load shedding, generation control)

144

8 Robust Optimization Approach to Uncertain Cascading Process

8.2 Prediction of Cascading Failure Paths This section presents a Markov chain model for the prediction of cascading failure paths and an approach for handling the curse of dimensionality. Without loss of generality, we consider a power system with n branches, and each branch has two connection states: “on” and “off”, represented by the binary bits 1 and 0, respectively. Thus, the state of power system on branch connection can be described by a string of n binary bits (see Fig. 8.2). To avoid vagueness, it is necessary to clarify the concept of cascading step. To be precise, a cascading step refers to one topological change (e.g. one branch outage) of power networks due to branch overloads or stochastic factors such as hidden failures of relays, human errors and the weather [11].

8.2.1 Markov Chain Model Suppose that the probability of moving to the next power system state only depends on the present state during the cascades. Given the probability of initial states, a sequence of random variables can be generated to describe the cascading process. Then a n n state-dependent transition matrix P = ( pi j ) ∈ R 2 ×2 can be created to describe the random process, and each element in P is given by pi j = Prob(s k+1 = j|s k = i), i, j ∈ B n , B = {0, 1} where B n denotes the state set of Markov chain and s k = (s1k , s2k , ..., snk ) ∈ B n is a vector of n binary elements characterizing the branch connection (i.e. “on” or “off”) at the k-th cascading step. Specifically, pi j is the transition probability of moving from the state i to the state j in one cascading step. Let phidden and pcont denote the probabilities of branch outage due to hidden failures and the contingency, respectively. In addition, pover refers to the outage probability due to branch overloads. Suppose the above factors that cause branch outages are independent. Thus the outage probability of the l-th branch is given by       λl = 1 − 1 − pover,l · 1 − phidden,l · 1 − pcont,l where pover,l , l ∈ In = {1, 2, ..., n} represents the probability of branch outage, which depends on the current or power flow on the l-th branch. Actually, the power flow or current on each branch is dependent on the connectivity of power networks. The following theoretical results reveal the relationship between the elements of transition matrix and the probability of branch outage. Proposition 8.1 The elements in the transition matrix P satisfy  pi j =

0, 

l∈(i, j) λl ·

i ∨ j = i; (1 − λ ), i ∨ j = i, l l∈(i, j)



(8.1)

8.2 Prediction of Cascading Failure Paths

145

Fig. 8.2 Branch ID and its state in a string of n binary bits. The head bit on the left side stores the state of Branch n, and the tail bit on the right side records the state of Branch 1

where the symbol ∨ denotes the bitwise logical disjunction. In addition, (i, j) and (i, j) are two sets of branch ID as follows: (i, j) = {l ∈ In |s k = i, s k+1 = j, slk = 1, slk+1 = 0} and (i, j) = {l ∈ In |s k = i, s k+1 = j, slk = 1, slk+1 = 1}. Proof The condition i ∨ j = i implies that there exist branches, whose connection status turns to “1” at the (k + 1)-th cascading step from “0” at the k-th cascading step. This is in contradiction with our assumption that the branch cannot be reconnected any longer once it is severed. And thus we have pi j = 0. The condition i ∨ j = i characterizes the normal cascading process of power system. The set (i, j) includes the ID numbers of severed branches in the state shift from i to j, while the set (i, j) keeps those of connected branches. Since the events of branch outage are independent, the probability of severing the branches in (i, j) is l∈(i, j) λl and the probability of keeping the connected branches in (i, j) is 

(1 − λl ).

l∈(i, j)

is

It follows that the transition probability of moving from the state i to the state j   λl · (1 − λl ), l∈(i, j)

l∈(i, j)



which completes the proof.

Once the transition matrix P is computed according to (8.1), it becomes feasible to predict the cascading failure paths of power systems (see Fig. 8.3). Note that the elements in P are related to the state shift (e.g. from state i to state j) instead of the iteration number. The probability distribution over states of power system on branch connection evolves as follows x k+1 = x k P, k ∈ N,

146

8 Robust Optimization Approach to Uncertain Cascading Process

Fig. 8.3 The predicted and actual cascading failure paths. The blue balls with solid boundaries denote the actual power system state during the cascades, while the dashed ones refer to the possible states predicted by the Markov chain model. The arrows indicate the shift of power system states with those in yellow shade implying the most probable paths. The solutions to robust optimization problem are implemented to safeguard power system states in the green ellipse against further cascades

where N denotes the set of nonnegative integers, and x k refers to the 2n -dimensional probability vector of power system states. This allows us to obtain the probability distribution over power system states at the k-th cascading step x k = x 0 P · P · · · P = x 0 P k , k

where x 0 denotes the initial probability distribution over the states of power systems, and it can be provided by the identification algorithm of branch outage in the widearea monitoring system (WAMS) [16]. In practice, the transition matrix P and its k-th power P k can be computed and recorded offline in order to save time and reduce the computational burden for online protection.

8.2.2 Dimensionality Reduction As is known, the dimension of transition matrix P increases exponentially with the size of power networks (i.e. number of branches). For a power system with n branches, the size of transition matrix P goes up to 2n × 2n . In practice, when n is larger than 20, the cost of computing P becomes extremely high. Therefore, it is necessary to develop an efficient approach for estimating the probability distribution over states of power system, while avoiding the curse of dimensionality. To solve

8.2 Prediction of Cascading Failure Paths

147

the problem, a simplified model of power system cascades is proposed by ruling out the states with extremely low probabilities. With this simplified model, it becomes workable to predict the cascading failure paths and compute the probability of power system topologies at each cascading step. Since plenty of states occur with extremely low probabilities during the cascades, we only consider the states in the uncertainty set D , where the elements or states satisfy a certain condition (i.e. their probabilities are larger than a certain threshold ). The mathematical expression of D is given by k > , ∀k ∈ Ih }, D = {s ∈ B n | xν(s)

(8.2)

k where Ih = {1, 2, ..., h}, and xν(s) denotes the probability that the cascades evolve to the state s at the k-th cascading step. The state s is described by a string of binary bits and ν(s) transforms s to a decimal sequence number that ranks the state s in the state space B n . For example, ν(s) can be defined as ν(s) = B2D(s) + 1, where B2D(s) denotes the conversion from a binary number to an equivalent decimal number. In this way, the computational burden can be significantly relieved with the guaranteed probability of evolving into some states at a certain cascading step. For the 2n -dimensional row vector x, a vector function  (x) is introduced to reset x as follows (8.3)  (x) = x · diag (1 (x1 ), 1 (x2 ), ..., 1 (x2n )) ,

where the operation diag(x) obtains a square diagonal matrix with the elements of vector x being on the main diagonal, and 1 (xi ) is the indicator function defined by  1 (xi ) =

1, xi > ; 0, xi ≤ .

where 1 ≤ i ≤ 2n . Thus, an iteration equation is established to estimate the probability distribution with relatively low computation burdens as follows 

xˆ 0 = x 0 , xˆ k+1 =  (xˆ k P), k ∈ N.

(8.4)

By using the above iteration equation, it is feasible to estimate the probability of power system states at the k-th cascading step. Proposition 8.2 With the iteration equation (8.4), it holds that Prob(s k ∈ D ) ≥ xˆ k 1

(8.5)

where  · 1 denotes the l1 norm, and s k refers to the power system state at the k-th cascading step.

148

8 Robust Optimization Approach to Uncertain Cascading Process

Proof It follows from the iteration equation (8.4) that xˆ k =  (xˆ k−1 P). Considering that xˆik ≤ xik , ∀i ∈ I2n , we have Prob(s k ∈ D ) =  (x k )1 =  (x k−1 P)1 ≥  (xˆ k−1 P)1 = xˆ k 1 , which completes the proof.



8.3 Robust Optimization Formulation In this section, we present the formulation of robust optimization for the online protection of power systems against the cascades. By incorporating the uncertainties of cascading failure paths into the change of branch admittance, a robust optimization problem is formulated as follows min Pb − Pb0  Pb

∗ s.t. Pe = diag(Yˆ p )A(A T diag(Yˆ p )A)−1 Pb

Yˆ p = diag(s) · Y p , s ∈ D Pb,i ∈ [P b,i , P¯b,i ], i ∈ Im

(8.6)

Pe,2 j ≤ σ j2 , j ∈ In where Pb0 refers to the original vector of injected power on buses before load shedding or generation control, and Pb = (Pb,1 , Pb,2 , ..., Pb,m )T denotes the vector of injected power on buses after load shedding or generation control. P b,i and P¯b,i characterize the lower and upper bounds of the injected power on the i-th bus, respectively. In theory, the minimum injected power on each bus is equal to the negative value of its total load, while the maximum injected power depends on the power generation from the generator connected to this bus [11]. For load buses, we have P¯b,i = 0 and P b,i is the total injected power. For generator buses, P¯b,i is the total power from the generator and P b,i the injected power from the load. Pe = (Pe,1 , Pe,2 , ..., Pe,n )T represents the vector of power flow on branches, and σ j specifies the threshold of power flow on the j-th branch. Y p denotes the vector of branch susceptance. In addition, the matrix A refers to the incidence matrix from branch to bus in power networks [17]. The set D contains all the predicted power system states during the cascades, and the proposed optimization formulation can allow for the online protection of all predicted states in D . If protective actions are scheduled to be taken at the k-th cascading step,

8.3 Robust Optimization Formulation

149

the vector s in the constraints of Problem (8.6) can be replaced by the vector s k to reduce the computation burden. It is worth pointing out that the symbol −1∗ denotes a pseudo-inverse to solve the DC power flow equation [18]. The power system state s determines the topology of power networks and thus affects the power flow distribution on branches. The iteration Eq. (8.4) allows us to predict power system states and estimate their respective probabilities at a given cascading step. By implementing the solution to Problem (8.6), it is expected to prevent the cascades for the predicted power system states with a guaranteed probability. Proposition 8.3 Robust Optimization Problem (8.6) is equivalent to a convex optimization problem as follows (8.7) min Pb − Pb0  Pb ∈X

with X =

 s∈D

X s and ⎧ ⎨

⎫ ∗ Pe = diag(Yˆ p )A(A T diag(Yˆ p )A)−1 Pb ⎬ X s = Pb Yˆ p = diag(s) · Y p , Pb,i ∈ [P b,i , P¯b,i ] . ⎩ ⎭ Pe,2 j ≤ σ j2 , i ∈ Im , j ∈ In Proof The constraints of (8.6) can be described by the intersection of finite sets X=



Xs

s∈D

where X s is given by ⎧ ⎨

⎫ ∗ Pe = diag(Yˆ p )A(A T diag(Yˆ p )A)−1 Pb ⎬ X s = Pb Yˆ p = diag(s) · Y p , Pb,i ∈ [P b,i , P¯b,i ] . ⎩ ⎭ Pe,2 j ≤ σ j2 , i ∈ Im , j ∈ In For any j ∈ In , the constraint function Pe,2 j − σ j2 is convex. For any s ∈ D , Pe is affine with respect to Pb , and thus X s is a convex set. Since the intersection of finite convex sets is a convex set, X is a convex set as well. Thus, Robust Optimization Problem (8.6) is equivalent to the following convex optimization problem min Pb − Pb0 

Pb ∈X

This completes the proof.



The solutions to Robust Convex Optimization Problem (8.7) can be interpreted as the projection of a point onto the intersection of convex sets (see Fig. 8.4). To determine the nonempty X s , we introduce the following two convex sets on the vector of power flow α = {Pe | Pe,2 j ≤ σ j2 , j ∈ In }

150

8 Robust Optimization Approach to Uncertain Cascading Process

Fig. 8.4 Geometric interpretation of optimal solutions to the problem (8.6). For instance, there are three possible cascading failure paths, which allows us to obtain three convex sets X 1 , X 2 and X 3 . The set X denotes the intersection of X 1 , X 2 and X 3 , and the optimal solution Pb∗ to the problem (8.6) is obtained by the projection of Pb0 in the set X

and

⎧ ⎨

⎫ Yˆ p = diag(s) · Y p ⎬ . βs = Pe Pb,i ∈ [P b,i , P¯b,i ], i ∈ Im ⎩ ⎭ ∗ Pe = diag(Yˆ p )A(A T diag(Yˆ p )A)−1 Pb

It follows that X s = ∅ if and only if dist(α, βs ) = 0. Actually, the distance between two convex sets dist(α, βs ) can be obtained by solving the following convex optimization problem: min x − y subject to x ∈ α and y ∈ βs . Efficient numerical algorithms are available to compute the distance between two convex sets and check the non-null of their intersection [19]. To verify that X is a non-null set, we need to find a point in X . Actually, it is associated with the set intersection problem (SIP), and the method of alternating projections can be adopted to solve the SIP [20]. Thus, we have the following theorem. Theorem 8.1 For the non-null set X , the solutions to Problem (8.6) can prevent further cascades with the probability as least xˆ k 1 at the k-th cascading step. Proof Proposition 8.2 provides the lower bound of the probability that power system states are in the set D at the k-th cascading step. Since X is a non-null set that is the intersection of convex sets X s , s ∈ D , the optimal solutions to Problem (8.6) can  allow for all the states in D . This completes the proof.

8.5 Simulation and Validation

151

Table 8.1 Numerical solver using Dykstra’s algorithm (0) Initialize: N , Pb,[i] ∈ [P b,1 , P¯b,1 ] × · · ·[P b,m , P¯b,m ], ∀i ∈ I|D | Goal: argmin Pb ∈X Pb − Pb0  |D | (0) 0 1: E |(0) D | = Pb − i=1 Pb,[i] 2: for k = 1 to N 3: E 0(k) = E |(k−1) D | 4: for i = 1 to |D | (k) (k) (k−1) 5: Z i = E i−1 + Pb,[i] (k) (k) 6: E i = PX (i) (Z i ) (k) 7: Pb,[i] = Z i(k) − E i(k) 8: end for 9: end for

8.4 Numerical Solver Using Dykstra’s Algorithm Essentially, Robust Optimization Problem (8.6) belongs to the best approximation problem (BAP), which considers the projection of a point onto the intersection of a number of convex sets [21]. Note that Dykstra’s algorithm ensures the convergence of optimal solutions to the BAP [22]. Thus, a numerical solver based on Dykstra’s algorithm is developed in Table 8.1 in order to find the optimal solution to Problem (8.6). Before implementing the numerical solver, it is required to specify the iteration number N and initial values in each convex set X (i) , i ∈ I|D | . The superscript i of the set X (i) denotes the sequence number of X s in the set D . Here |D | denotes the cardinality of the set D . And the goal is to find a point in the intersection set X (0) is computed for the that is closest to the point Pb0 . Then the initial deviation E |D | (k) (i) correction of projection in each set X . Note that Pb,[i] , i ∈ I|D | is a vector, which is different from the scalar Pb,i , i ∈ Im in (8.6). An iteration loop is established to implement a sequence of projections onto each convex set and the corrections, and the loop terminates once the iteration number is reached.

8.5 Simulation and Validation In this section, we validate the proposed robust optimization approach for power system protection on IEEE 118 bus system. Suppose that the initial malicious contingencies can be identified by power systems, and it immediately triggers the power system cascades. In the simulation, the Markov chain model enables us to determine the most probable cascading failure paths. By implementing the numerical solver in Table 8.1, the optimal solutions for load shedding and generation control are obtained to terminate the cascades at a specified cascading step. Note that our focus is on the

152

8 Robust Optimization Approach to Uncertain Cascading Process

power system protection at the steady-state progression of cascades, where the progression of the cascade events is slow, and the balance between the power generation and consummation is not lost [23].

8.5.1 Parameter Setting For simplicity, the probability of branch outage due to contingencies is identical for each branch with pcont = 10−4 . The probability of branch outage due to hidden failure is dependent on the location of outage branches. It is suggested that one branch is more likely to fail if it is connected to the outage branches through a common bus [24]. Thus, we specify phidden = 10−2 for the branches connected to one outage branch and phidden = 10−4 for those without connecting to any outage branches. As for the probability of branch outage due to overloads pover , we introduce a ratio ri between the actual power flow on Branch i and its threshold, and pover for Branch i is computed as follows:

pover

⎧ ri < 0.8; ⎨ 0, ri > 1.05; = 1, ⎩ 4ri − 3.2, otherwise.

In addition, the DC power flow equation is employed to compute the power flow on branches. Per unit values are adopted with the base value of power 100 MVA. It is assumed that the threshold of power flow on each branch is two times larger than the normal power flow (i.e. power flow without any branch outages) on the corresponding branch, and the minimum threshold of power flow is 2 pu for each branch. The value of  is equal to 0.1 in order to select the most probable cascading failure paths. The iteration number N is equal to 50 for implementing the numerical solver in Table 8.1.

8.5.2 Validation and Discussion With the proposed Markov chain model and the above parameter setting, we can identify the most probable cascading paths at the first several cascading steps by severing Branch 8 with the probability 0.6 or severing Branch 4 with the probability 0.4 as the initial contingency, respectively. Thus, we can obtain two most probable cascading failure paths as follows. Path A: 8 → (21, 36, 37) → 32 → (38, 50, 54) and Path B: 4 → 12 → 5 → 37, where each number denotes the ID of outage branch at the corresponding cascading step, and the arrow indicates the shift of cascading steps. Our goal is to terminate the further cascades at the 3rd cascading step by implementing the solutions of numerical solver in Table 8.1. This can be achieved by taking protective actions to terminate the power system cascades at two states:

8.5 Simulation and Validation

153

2

Pb

0 -2 -4 0

20

40

60

80

100

120

Bus ID 6

4

2

0 5

10

15

20

25

30

35

40

45

50

Iteration Number

Fig. 8.5 Distribution of changes of injected power on each bus after implementing the numerical solver in Table 8.1 (the upper panel) and the convergence of the numerical solver (the lower panel)

(8, 21, 36, 37, 32) and (4, 12, 5) simultaneously. Let Pb = Pb∗ − Pb0 denote the changes of injected power on buses after taking protective actions according to the solutions of numerical solver. At each iteration step, the index δ = Pb,[1] − Pb,[2]  is computed to describe the distance between the projected points with corrections in the two convex sets. The upper panel in Fig. 8.5 shows the distribution of Pb for each bus after implementing the solver in Table 8.1, and the lower panel presents the time evolution of δ while implementing the solver. It is observed that δ quickly goes to a steady-state value smaller than 0.5 after 5 iteration steps. Moreover, simulation results demonstrate that the two most probable cascading failure paths can be terminated at the 3rd cascading step by adjusting the injected power on buses with numerical results from the solver in Table 8.1. According to Theorem 8.1, the cascades can be prevented with the probability of 0.92 at least. Actually, there is a tradeoff between the optimal adjustment of injected power on buses and the effective termination of multiple cascading failure paths. In other words, the larger changes of injected power on buses have to be made for terminating more cascading failure paths. In the worst cases, it might be infeasible to prevent further cascades by only adjusting the injected power on buses (e.g. load shedding and generation control) at a given cascading step in practice. Thus, other remedial actions should be taken in a cooperative manner to protect power systems against blackouts (e.g. proactive line tripping, adjustment of branch impedance using FACTS devices, etc.).

154

8 Robust Optimization Approach to Uncertain Cascading Process

8.6 Conclusions This chapter investigated the problem of preventing cascading blackouts of power systems with uncertainties. A Markov-chain model was proposed to deal with the uncertainties and predict the cascading failure paths. In particular, a robust convex optimization problem was formulated to terminate the cascades by optimally shedding loads at the possible power system states. Moreover, an efficient numerical solver based on Dykstra’s algorithm was adopted to solve the proposed robust optimization problem. In theory, we provide the lower bound for the probability of preventing the cascading blackouts of power systems. Numerical simulations were conducted to validate the proposed robust optimization approach and obtain the theoretical lower bound on the probability of terminating the cascades. Future work may include the calibration of the proposed Markov chain model by using the statistical data on real power system cascades as well as the validation of the proposed protection scheme [25]. In addition, more efforts will be taken on the development of efficient numerical algorithms to solve the robust optimization problem so that the cascades can be terminated in short time.

References 1. McLinn, J.: Major power outages in the US, and around the world. Ann. Technol. Rep. IEEE Reliab. Soc. (2009) 2. Lu, W., Besanger, Y., Zamai, E., Radu, D.: Blackouts: description, analysis and classification. In: Proceedings of the 6th WSEAS International Conference on Power Systems, pp. 429–434. Lisbon, Portugal, Sept (2006) 3. Arnborg, S., Andersson, G., Hill, D., Hiskens, I.: On undervoltage load shedding in power systems. Int. J. Electr. Power Energy Syst. 19(2), 141–149 (1997) 4. “Survey of underfrequncy relay tripping of load under emergency conditions—IEEE committee report," presented at the Summer Power Meeting, Portland, OR (1967) Paper 31 TP 67-402 5. “A status report on methods used for system preservation during underfrequency conditions," In: Summer Meeting Energy Resources Conference. Anaheim, CA (1974). Paper 74 310-9 6. Hewitson, L., Mark, B., Ramesh, B.: Practical Power System Protection, Elsevier (2004) 7. Anderson, P., LeReverend, B.: Industry experience with special protection schemes. IEEE Trans. Power Syst. 11(3), 1166–1179 (1996) 8. (1998, Sep.) Maintaining reliability in a competitive U.S. electricity industry: Final report of the North American Electric Reliability Council (NERC) Task Force on Electric System Reliability. [Online]. Available: http://www.nerc.com/~filez/reports.html 9. Begovic, M., Novosel, D., Karlsson, D., Henville, C., Michel, G.: Wide-area protection and emergency control. Proc. IEEE 93(5), 876–891 (2005) 10. Nuqui, R., Phadke, A.: Phasor measurement unit placement techniques for complete and incomplete observability. IEEE Trans. Power Delivery 20(4), 2381–2388 (2005) 11. Zhai, C., Zhang, H., Xiao, G., Pan, T.: A model predictive approach to protect power systems against cascading blackouts. Int. J. Electr. Power Energ. Syst. 113, 310–321 (2019) 12. Wang, Z., Scaglione, A., Thomas, R.J.: A Markov-transition model for cascading failures in power grids. In: The 45th Hawaii International Conference on System Science, pp 2115–2124 (2012) 13. Rahnamay-Naeini, M., Wang, Z., Ghani, N., Mammoli, A., Hayat, M.M.: Stochastic analysis of cascading-failure dynamics in power grids. IEEE Trans. Power Syst. 29(4), 1767–1779 (2014)

References

155

14. Dobson, I., Carreras, B., Newman, D.: A loading-dependent model of probabilistic cascading failure. Probabil. Eng. Inf. Sci. 19(1), 15–32 (2005) 15. Yao, R., Huang, S., Sun, K., Liu, F., Zhang, X., Mei, S., Wei, W., Ding, L.: Risk assessment of multi-timescale cascading outages based on Markovian tree search. IEEE Trans. Power Syst. 32(4), 2887–2900 (2016) 16. Zhao, Y., Chen, J., Goldsmith, A., Poor, H.: Identification of outages in power systems with uncertain states and optimal sensor locations. IEEE J. Sel. Top. Sign. Process. 8(6), 1140–1153 (2014) 17. Stagg, G., Ahmed, H.: Computer Methods in Power System Analysis. McGraw-Hill (1968) 18. Zhai, C., Zhang, H., Xiao, G., Pan, T.: An optimal control approach to identify the worst-case cascading failures in power systems. IEEE Trans. Control Netw. Syst. July (2019) https://doi. org/10.1109/TCNS.2019.2930871 19. Llanas, B., De Sevilla, M., Feliu, V.: An iterative algorithm for finding a nearest pair of points in two convex subsets of Rn. Comput. Math. Appl. 40(8–9), 971–983 (2000) 20. Bauschke, H., Combettes, P.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011) 21. Jeffrey Pang, C.: The supporting halfspace-quadratic programming strategy for the dual of the best approximation problem. SIAM J. Optim. 26(4), 2591–2619 (2016) 22. Boyle, J., Dykstra, R.: A method for finding projections onto the intersection of convex sets in Hilbert spaces. Lect. Notes Stat. 37, 28–47 (1986) 23. Lu, W., Besanger, Y., Zama, E., Radu, D.: Blackouts: description, analysis and classification. Network 2, 14 (1996) 24. Thorp, J., Phadke, A., Horowitz, S., Tamronglak, C.: Anatomy of Power System Disturbances: Importance Sampling. PSCC, Dresden (1996) 25. Dobson, I., Carreras, B.A., Newman, D.E., Reynolds-Barredo, J.M.: Obtaining statistics of cascading line outages spreading in an electric transmission network from standard utility data. IEEE Trans. Power Syst. 31(6), 4831–4841 (2016)

Chapter 9

Cooperative Control Methods for Relieving System Stress

Abstract In this chapter, we study coordination control and effective deployment of Thyristor-Controlled Series Compensation (TCSC) to protect power grids against disruptive disturbances. The power grid consists of flexible alternate current transmission systems (FACTS) devices for regulating power flow, phasor measurement units for detecting system states, and control station for generating the regulation signals. We propose a novel coordination control approach of TCSC devices to change branch impedance and regulate the power flow against unexpected disturbances on buses or branches. More significantly, a numerical method is developed to estimate a gradient vector for generating regulation signals of TCSC devices and reducing computational costs. To describe the degree of power system stress, a performance index is designed based on the error between the desired power flow and actual values. Moreover, technical analysis is presented to ensure the convergence of the proposed coordination control algorithm. Numerical simulations are implemented to substantiate that the coordination control approach can effectively alleviate the stress caused by different types of contingencies on IEEE test systems, as compared to the classic PID control. It is also demonstrated that the deployment of TCSCs can alleviate the system stress greatly by considering both impedance magnitude and active power on branches.

9.1 Introduction Due to huge economic loss, power system blackouts have become an issue of great concern to both electrical power industries and governments in the past several decades. Thus, a lot of efforts are taken to develop various protection schemes against the blackout. Some researchers are dedicated to the prevention of cascading failures when the cascade propagates in the early stage [1, 2], while others focus on the identification of initial contingencies [3, 4]. Actually, most blackouts are closely related to the precondition of excessive power demand, which results in the stress of power system. For instance, more than 60% blackouts take place in the summer and winter peaks when the power demand is relatively high [5]. Therefore, it is significant to

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Zhai, Control and Optimization Methods for Complex System Resilience, Studies in Systems, Decision and Control 478, https://doi.org/10.1007/978-981-99-3053-1_9

157

158

9 Cooperative Control Methods for Relieving System Stress

eliminate the undesired consequence of preconditions (e.g., power system stress due to branch overloads) and reduce the risk of cascading blackouts. The application of FACTS greatly improves the performance of power system in terms of power oscillation damping and transient stability enhancement. As an important member in the FACTS family, the TCSC plays a major part in the reliable operation of power transmission system to relieve the power system stress, provide the voltage support, schedule the power flow, etc. [6]. As a result, various control schemes of TCSC have been proposed in the past decades, and they include PID control [7], Fuzzy logic control [8], energy function method [9], auto-disturbance rejection control [10], neural network based control [11], to name just a few. Nevertheless, most control schemes focus on the separate operation of TCSC controllers to improve the transient stability and dynamic performance of power system, which ignores the collaboration of TCSC controllers to achieve the better performance. On the other side, cooperative control of multi-agent system has attracted much interest of researchers in the field of control and systems engineering [12–14]. Essentially, cooperative control refers to control actions that aim to achieve the control goal through sharing the information of multiple components in a cooperative way. Actually, cooperative control is also applied to the power system protection by regarding each TCSC as an agent that is able to collaborate with each other for scheduling the power flow. Specifically, cooperative control allows the TCSC agents to work together for a common goal by reconciling the conflict of interest among individual TCSCs, which helps to achieve the desired performance in a shorter time. Moreover, it contributes to strengthening the capability of power system against malicious disturbances by absorbing the stress or damage in a systematic manner. Therefore, a cooperative control scheme is proposed in this chapter to deal with the problem of power system stress. The main contributions of this work are listed as follows: 1. Develop a cooperative control algorithm of TCSC with the guaranteed convergence of performance index in theory. 2. Propose a simple and efficient approach to estimating the Jacobian matrix, which greatly reduces the computation burdens. 3. Validate the cooperative control approach on the standard IEEE bus system with unknown load variations. The remainder of this chapter is arranged as follows. Problem formulation on the coordination control of TCSC devices is provided in Sect. 9.2. The novel control scheme of TCSC is presented in Sect. 9.3. IEEE test systems are employed to validate the proposed approach in Sect. 9.4. The conclusion is drawn in Sect. 9.5.

9.2 Problem Formulation The smart power grid is comprised of transmission networks, phase measurement units, FACTS devices and control station (see Fig. 9.1). More specifically, PMU is used to detect the state information of buses and send it to the control station in real

9.2 Problem Formulation

159

Fig. 9.1 Cyber-physical power systems

time. The control station generates proper control signals to drive FACTS devices by using the state information. Finally, TCSC devices update the impedance of branches to regulate the power flow. Without loss of generality, consider a power grid with n branches and m buses. When the power flow on the branch changes, TCSC devices adjust the branch impedance, so that the actual power flow Pe is restored to the desired values σ = (σ1 , σ2 , ..., σn ) ∈ C n . To eliminate power system stresses, we need to design a control input U for TCSC devices, and the optimization problem is formulated as follows. (9.1) min H (Z) U

with the objective function H (Z) = PeR − σ R 2 + PeI − σ I 2 , where the vector of branch impedance is denoted by Z and  ∈ [0, 1] is a tuning parameter. Note that PeR and PeI refer to the real part and imaginary part of Pe , respectively. The superscripts R and I apply to other complex variables as well. To solve Problem (9.1), the Lyapunov function candidate V (Z) is constructed below V (Z) = H (Z) − H (Z∗ ), where H (Z∗ ) is the minimum value of Problem (9.1). Then the derivative of V (Z) with respect to the time t gives

160

9 Cooperative Control Methods for Relieving System Stress d V (Z) dt

with

and

=

d H (Z) dt

T  = 2 PeR − σ R

dPeR dt

T  + 2 PeI − σ I

dPeI dt

(9.2)

∂PeR dZ R ∂P R dZ I dPeR = + eI · · R dt ∂Z dt ∂Z dt

(9.3)

dPeI ∂PeI dZ R ∂PeI dZ I = + . · · dt ∂Z R dt ∂Z I dt

(9.4)

By substituting (9.3) and (9.4) into (9.2), we obtain d V (Z) dt

T ∂P R T ∂P I   R R = 2 PeR − σ R ∂ZeR · dZ + 2 PeI − σ I ∂ZeR · dZ dt dt T ∂P R T ∂P I   I I +2 PeR − σ R ∂ZeI · dZ + 2 PeI − σ I ∂ZeI · dZ dt dt  R  R T dZ Pe − σ R dt I =2 J (Z) dZ . I I (Pe − σ )

(9.5)

dt

The J (Z) in (9.5) is described by  J (Z) =

∂PeR ∂Z RI ∂Pe ∂Z R

∂PeR ∂Z II ∂Pe ∂Z I



  = Ji, j (Z) ∈ R 2n×2n .

(9.6)

The control input U is designed as 

UR U= UI



 =

dZ R dt dZ I dt

 .

(9.7)

Because of the complexity of system model, it is not easy to directly obtain the accurate value of J (Z) . As a result, a numerical approach is proposed to estimate J (Z). Moreover, we do not convert the minimization problem (9.1) into KKT condition, but directly design a coordination control method to minimize the value of H (Z). The convergence of objective function H (Z) is guaranteed by the proposed coordination control algorithm, instead of KKT condition. In Fig. 9.2, PMUs collect time series of Pb , V and I. By injecting a small disturbance λ on the branch, the Jacobian matrix can be estimated. The controller generates command signals for TCSC devices to regulate the power flow accordingly. The TCSC device consists of two main control blocks, and the function of external control block is to improve the transmission capacity or stability of power grid. According to different control objectives, external control can be designed using different methods. The traditional PI controller is a slow automatic control for power flow regulation, and the coordination control proposed in the present work can be regarded as a type of external control. The function of internal control block is to provide appropriate gate drive signals for thyristors to generate compensation

9.3 Coordination Controller

161

Fig. 9.2 Regulation signals in smart power grids equipped with TCSC devices

reactance. The relation between TCSC impedance and active power of branch is given in the appendix. Remark 9.1 The goal of this work is to propose a novel coordination control method to regulate the impedance of branch by TCSC devices so as to alleviate the system stress. In fact, the proposed method is universal. It not only can be applied to TCSC devices, but also to other FACTS devices.

9.3 Coordination Controller This section focuses on the design of coordination control law and discusses how to calculate J (Z).

9.3.1 Generation of Control Signals The coordination control law for TCSC devices is given by  U = −κ(Z) ◦ J (Z)

T

 PeR − σ R , (PeI − σ I )

(9.8)

162

9 Cooperative Control Methods for Relieving System Stress

where ◦ is the Hadamard product. Each element in κ(Z) = (κ1 (Z), κ2 (Z), ..., κ2n (Z))T is designed as  c, Z i ∈ [Z i , Z i ]; κi (Z) = 0, otherwise. with the constant c > 0 and the upper limit Z i and lower limit Z i . The coordination controller (9.8) is composed of three terms: κ(Z), J (Z) and the error vector  PeR − σ R . Pe = (PeI − σ I ) 

Specifically, κ(Z ) allows to adjust the branch impedance in the given interval with upper and lower bounds. J (Z) indicates the incremental direction of objective function with respect to branch impedance. Pe can be obtained by comparing the desired values with the actual ones. Proposition 9.1 The control input (9.8) for TCSC devices can guarantee the convergence of H (Z). Proof Note that both Z R and Z I are adjusted according to (9.8). By substituting (9.7) into (9.5) , one obtains T   R  d V (Z) Pe − σ R UR . =2 J (Z) (PeI − σ I ) UI dt

(9.9)

By substituting (9.8) into (9.9), one has d V (Z) dt

 = −2

PeR − σ R (PeI − σ I )

T

J (Z) · κ(Z) ◦ J (Z)T  R  2 Pe − σ R T ≤0 ¯ ◦ J (Z) = −2 κ(Z) (PeI − σ I )



PeR − σ R (PeI − σ I )



√ √ √ where κ(Z) ¯ = ( κ1 (Z), κ2 (Z), ..., κ2n (Z))T . This indicates that the objective function decreases monotonously as t → +∞. Thus, H (Z) is convergent because of H (Z) ≥ 0. 

9.3.2 Construction of Jacobian Estimator The implementation of the proposed coordination control algorithm calls for the estimation of J (Z). The linear total least-square can be employed to calculate J (Z). Although approximations and relaxations exist in the modeling of relevant problems, the process is still complicated and calls for a large amount of calculations. Thus, it is indispensable to come up with a numerical method to estimate J (Z) with low computation burdens. The approximation of Jacobian matrix includes four steps

9.3 Coordination Controller

163

Fig. 9.3 A small perturbation approach for estimating Jacobian matrix

(see Fig. 9.3). In the 1st step, Pe (Z) is calculated. Then a small disturbance λ is injected onto each branch, and Pe (Z R + λei ) is obtained, where ei denotes the unit vector with the i-th entry being 1. In light of Taylor’s theorem, Pe,R j (Z R + λei ) and Pe,I j (Z R + λei ) can be rewritten as ∂PR

Pe,R j (Z R + λei ) = Pe,R j (Z) + λ ∂Ze,Rj ei + O(λ) = Pe,R j (Z) + λJ j,i (Z) + O(λ) and ∂PI

Pe,I j (Z R + λei ) = Pe,I j (Z) + λ ∂Ze,Rj ei + O(λ) = Pe,I j (Z) + λJ j+n,i (Z) + O(λ), respectively. After removing O(λ), elements in the i-th column of J (Z) are given by Pe,R j (Z R + λei ) − Pe,R j (Z) J j,i (Z) ≈ , (9.10) λ J j+n,i (Z) ≈

Pe,I j (Z R + λei ) − Pe,I j (Z) λ

.

(9.11)

Similarly, elements in the (i + n)-th column of J (Z) are presented as J j,i+n (Z) ≈

Pe,R j (Z I + λei ) − Pe,R j (Z)

J j+n,i+n (Z) ≈

λ

,

Pe,I j (Z I + λei ) − Pe,I j (Z) λ

(9.12)

.

(9.13)

164

9 Cooperative Control Methods for Relieving System Stress

Table 9.1 Jacobian estimation algorithm Input: λ and Pb Output: J (Z ) 1: Set λ and detect Pb 2: Compute Pe (Z ) 3: for i = 1 to n do 4: Re(Z ) = Re(Z ) + λei 5: Compute Pe (Z ) 6: Estimate J j,i (Z ) with (9.10) and (9.11) 7: Re(Z ) = Re(Z ) − λei 8: Im(Z ) = Im(Z ) + λei 9: Compute Pe (Z ) 10: Estimate J j,i+n (Z ) with (9.12) and (9.13) 11: Im(Z ) = Im(Z ) − λei 12: end for

Table 9.2 Coordination control algorithm Input: s = 0, k = 0, T = 100 and S0 = H0 (Z ) Output: Z and Pe 1: while (Hs (Z ) = 0) 2: Detect Pe 3: if (mod(s, T ) = 0) 4: Update k = k + 1 5: Compute Sk 6: if (Sk ≥ Sk−1 ) or (s = 0) 7: Sk ← Sk−1 8: Detect Pb , V and I 9: Run the JEA for J (Z ) 10: end if 11: end if 12: Update Z with (9.8) 13: Update s = s + 1 14: Compute Hs (Z ) 15: end while

Table 9.1 summaries how to estimate J (Z) in the numerical method. To alleviate computation burdens, a performance index is designed as Sk = maxi∈I (k) Hi (Z), where I (k) = [(k − 1)T + 1, kT ], ∀k ∈ Z+ , and Hi (Z) denotes the value of H (Z) at the i-th step. The implementation of proposed coordination controller is described in Table 9.2, and it allows to decrease the performance index Sk .

9.3 Coordination Controller

165

Fig. 9.4 IEEE test system equipped with TCSC devices and phase measurement units

Proposition 9.2 Coordination control algorithm for TCSC devices in Table 9.2 guarantees the monotonous convergence of Sk . Proof Coordination control algorithm enables to produce a sequence {Sk }∞ k=1 . It follows from Sk+1 ≤ Sk and Sk ≥ 0, ∀k ∈ Z+ that Sk converges to a constant value  inf k∈Z+ Sk monotonously as k approaches the positive infinity.

166

9 Cooperative Control Methods for Relieving System Stress

Table 9.3 The location and parameters of contingencies Test No. Branch ID Fault magnitude 1 2 3

29 5 5, 6, 29, 36

0.06 0.6 0.6, 0.36, 0.06, 0.06

Disruption 1 10 15

Fig. 9.5 Performance comparison of the coordination control and the PID control in terms of Sk

9.4 Numerical Simulations Simulation results are presented to demonstrate the performance of the proposed control approach (see Fig. 9.4). Some key parameters are given below: c = 0.02,  = 0.2, λ = 106 and T = 100. The parameters of PID control are presented as C P = 2 × 10−5 , C I = 10−2 , C D = 3 × 10−6 . Euler method is employed to implement the control algorithm with the step size of 0.01 and the total time steps 104 . In addition, the adjustment range of branch impedance is 0.5 to 4 times larger than the magnitude of the steady-state value. Per unit systems are employed using base value of 100 MVA. In this part, we will analyze how to use the coordination control algorithm to relieve the stress of power system caused by disruptive disturbances. Three types of emergencies are tested, with small change of branch impedance on one transmission line, large change of branch impedance on one transmission line and changes of branch impedance on multiple transmission lines (i.e., the red lightning in Fig. 9.4), respectively. The details of three cases are summarized in Table 9.3. Specifically, the first column in Table 9.3 lists the test ID, the second column presents the branch

9.5 Conclusions

167

Table 9.4 The initial and final values of Sk Test No. 1 2 Control Initial Final Initial strategies Proposed method PID control

Final

3 Initial

Final

0.1518

0.0007

0.2256

0.0044

0.7083

0.0335

0.1519

0.1504

0.2280

0.2261

0.7106

0.7082

ID where disruptive disturbances are added, the third column provides the value of branch impedance after contingencies, and the last column provides the disruption caused by faults to the whole buses. For simplicity, it is assumed that the injected bus power is subject to the disturbances, which satisfy the normal distribution with zero mean value and standard deviation of 0.01 in Test 1. We define the disturbances in Test 1 as the unit value, and the disturbances in subsequent tests are compounded by the unit value. The desired power flow Pσ is specified as power flow in the normal condition before the initial contingency. The green trajectories in Fig. 9.5 show the monotonous decrease of Sk from the initial value to the final value, which partially confirms the conclusion of Proposition 9.2. The total number of k is 100 since Sk is computed for every T = 100 steps with the total steps of 104 . In addition, the Jacobian matrix J (Z) is updated 30 times in the simulation. By contrast, the red trajectories display the evolution of Sk with the PID control for each TCSC. In Table 9.4, it is obvious that Sk decreases rapidly by using coordination control scheme, while Sk only changes slightly with the PID control algorithm. Finally, we evaluate the stability of coordination control algorithm by changing the mean value and standard deviation of the disturbances. In each group, simulations are carried out for 10 times, and the average values of the objective function H (Z) in the final step are analyzed. Figure 9.6 shows the mean value and standard deviation of H (Z) in the final step with the intensity of disturbances. Again, it is shown that the proposed coordination control algorithm can effectively eliminate the stress of power system.

9.5 Conclusions A novel coordination control algorithm was developed to alleviate the system stress by regulating power flow with the assistance of TCSC devices. Simulation results for different types of disturbances indicated the excellent stability of the proposed coordination control approach, as compared to the traditional control methods. Future work may include optimal deployment of limited TCSC agents on the entire power network, and the estimation of Jacobian elements by analyzing the real PMU data without disturbing the branch impedance.

168

9 Cooperative Control Methods for Relieving System Stress

Fig. 9.6 The changes of the average value of H (Z) in the final step with the increase of mean value μ and standard deviation σ of Gaussian noise disturbance

9.6

Appendix

This section provides the relation between TCSC impedance and active power of branch. The TCSC model between Bus i and Bus j and the reactance of TCSC can be calculated with the power flow equations. X T C SC



|Vi | V j sin θi j = − XL, Pi j

where X T C SC and X L represent the reactance of TCSC and transmission line between Bus i and Bus j, respectively.

References 1. Begovic, M., Novosel, D., Karlsson, D., Henville, C., Michel, G.: Wide-area protection and emergency control. Proc. IEEE 93(5), 876–891 (2005) 2. Zhai, C., Zhang, H., Xiao, G., Pan, T.C.: A model predictive approach to preventing cascading failures of power systems (2017). arXiv preprint arXiv:1710.05184 3. Kim, T., Wright, S.J., Bienstock, D., Harnett, S.: Analyzing vulnerability of power systems with continuous optimization formulations. IEEE Trans. Netw. Sci. Eng. 3(3), 132–146 (2016) 4. Zhai, C., Zhang, H., Xiao, G., Pan, T.C.: Modeling and identification of worst-case cascading failures in power systems (2017). arXiv preprint arXiv:1703.05232 5. Lu, W., et al.: Blackouts: description, analysis and classification. In: Proceedings of the 6th WSEAS International Conference on Power Systems. Lisbon, Portugal, 22–24 Sept 2006 6. Zhou, X., Liang, J.: Overview of control schemes for TCSC to enhance the stability of power systems. IEE Proc.-Gener., Transm. Distrib. 146(2), 125–130 (1999)

References

169

7. Panda, S.: Multi-objective PID controller tuning for a FACTS-based damping stabilizer using non-dominated sorting genetic algorithm-II. Int. J. Electr. Power Energy Syst. 33(7), 1296– 1308 (2011) 8. Hiyama, T., et al.: Coordinated fuzzy logic control for series capacitor modules and PSS to enhance stability of power system. IEEE Trans. Power Delivery 10(2), 1098–1104 (1995) 9. Gronquist, J.F., et al.: Power oscillation damping control strategies for FACTS devices using locally measurable quantities. IEEE Trans. Power Syst. 10(3), 1598–1605 (1995) 10. Zhang, C., Zhou, X.: The auto-disturbances rejection control of TCSC. Control Eng. Pract. 7(2), 195–199 (1999) 11. Dai, X., et al.: Neural network αth-order inverse control of thyristor controlled series compensator. Electr. Power Syst. Res. 45(1), 19–27 (1998) 12. Zhai, C., Hong, Y.: Decentralized sweep coverage algorithm for uncertain region of multi-agent systems. Proc. Am. Control Conf. Montréal, Can. 4522–4527(June), 27–29 (2012) 13. Zhai, C., Hong, Y.: Decentralized sweep coverage algorithm for multi-agent systems with workload uncertainties. Automatica 49(7), 2154–2159 (2013) 14. Zhai, C., He, F., Hong, Y., Wang, L., Yao, Y.: Coverage-based interception algorithm of multiple interceptors against the target involving decoys. J. Guidance Control Dyn. 39(7), 1647–1653 (2016) 15. Zimmerman, R.D., Murillo-Scnchez, C.E., Thomas, R.J.: MATPOWER: steady-state operations, planning and analysis tools for power systems research and education. IEEE Trans. Power Syst. 26(1), 12–19, Feb 2011

Chapter 10

Distributed Optimization Approach to System Protection

Abstract The responsive and effective protective actions are vital to the resilience of electric power systems. This chapter addresses the online protection problem of power systems subject to disruptive disturbances caused by contingencies. A novel distributed optimization algorithm for flexible alternate current transmission systems devices is proposed to adjust the branch impedance and voltage phase angle with the aid of thyristor-controlled series compensation and thyristor-controlled phase shifting transformer (TCPST), which enables to regulate power flow and eliminate the disruptions caused by bus overloads and transmission line fault. Moreover, theoretical analysis is conducted to guarantee the convergence of numerical algorithm for solving the distributed optimization problem. Finally, numerical simulations are carried out to validate the proposed optimization algorithm on IEEE bus system and compare with existing methods.

10.1 Introduction The rapid development of the economy and human society has brought new challenges to the power industry. Although the construction and update of electric power equipment can relieve the stress of power grids, cascading blackouts still occur occasionally [1]. The latest blackout in Texas, US caused by freeze in February 2021 has left more than 10 million people without electricity at its worst [2]. According to technical reports, most blackouts are triggered by the misoperation of human operators, extreme weather, the aging of power system equipment, etc. [3, 4]. To avoid large-scale blackouts, some research focuses on the massive power blackout caused by the cascading failure. Compared with other factors of blackouts, it is more difficult to cope with the uncertainties of triggering events or contingencies [5]. Since the transmission line plays an important role in power system reliability, it has a significant impact on the security of power grids subject to attacks or external disturbances [6, 7]. Therefore, it is necessary to ensure the safe and reliable operation of power transmission system. In practice, it has been widely studied to enhance the efficiency of branch and transient stability of transmission systems using FACTS devices. In [8], the unified power © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Zhai, Control and Optimization Methods for Complex System Resilience, Studies in Systems, Decision and Control 478, https://doi.org/10.1007/978-981-99-3053-1_10

171

172

10 Distributed Optimization Approach to System Protection

flow controller (UPFC) is equipped to enhance power quality and regulate power flow in a power system. In [9], a remedial action scheme is proposed to relieve tie-line congestions in transmission systems based on TCSCs. [10] proposes a novel control scheme that uses wide area measurement system and TCSC to suppress interregional oscillation in large power systems. The location of static var compensator (SVC) and thyristor-controlled phase shifting transformer (TCPST) are presented based on the optimal power flow [11]. As one type of special FACTS device that can change the voltage angle of bus nodes, the influence of TCPST on angle stability is analyzed in [12]. In fact, one type of FACTS device may fail to effectively mitigate the overload and handle serious blackout contingencies or demand increase [13]. On the other hand, various optimization algorithms are applied to the control of TCSC or TCPST, which include the improved harmony search algorithm for TCSC [14], and cross-entropy approach, genetic algorithm for TCPST [11, 15]. In particular, a cooperative control algorithm is proposed to reduce the stress of power systems and improve the resilience of power systems through the cooperation of TCSC installed on branches [16]. Nevertheless, these studies are mainly based on centralized control methods, which require massive data transmission into a control center via communication networks. When the control center fails, it will damage the control performance of FACTS devices and impair power system security. Thus, it is of great significance to design the distributed algorithm for the online control of FACTS and alleviate the stress of power systems caused by branch or bus faults. Therefore, we develop a novel distributed optimization method by using FACTS devices to relieve the stress of power stress. Specifically, each TCSC is regarded as an agent, through the communication between TCSC agents to exchange information and relieve the stress of power systems. Furthermore, to better enhance the system resilience, TCPST device is installed on each bus to regulate the voltage phase angle. In short, the core contributions of this work are listed as follows: • Create a novel network model of TCSC-TCPST for power grids in the framework of multi-agent system, which can decouple the physical layer and cyber layer, thereby reducing the probability of faults occurrence. • Develop a distributed optimization algorithm by integrating TCSC with TCPST for the real-time protection of power systems against external disturbances. • Provide the theoretical analysis for the distributed optimization algorithm with the guaranteed convergence and analyze the effect of parameters on system performance. The remainder of this chapter is organized as follows. Section 10.2 provides the preliminaries of optimization problem and system model. Section 10.3 presents the problem formulation of power system protection by adjusting the branch impedance with TCSC and bus voltage phase angle with TCPST. Section 10.4 provides the distributed optimization algorithm and theoretical analysis. Numerical simulations are conducted in Sect. 10.5. Finally, we conclude the chapter and discuss future work in Sect. 10.6.

10.2 Preliminaries

173

Fig. 10.1 The proposed TCSC-TCPST model

10.2 Preliminaries In this section, we present the TCSC-TCPST hybrid model and then introduce the direct-current (DC) power flow equation, followed by the communication network of TCSC agents.

10.2.1 Hybrid Model The model of FACTS device is linked to the problem to be solved. For example, the modeling of TCSC in [17] just takes into account the reactance, which helps to achieve the goal of controlling power flow. In [11, 15], to relieve power grid congestion, the modeling of TCPST only considers voltage magnitude, and voltage phase angle is ignored based on optimal power flow. We aim to reduce or eliminate power system stress by using TCSC and TCPST devices. Hence, the model of TCSC-TCPST is designed. Figure 10.1 presents the diagram of TCSC-TCPST based on DC power flow. In the above model, the reactive power is ignored, and the voltage magnitude of each bus is considered to be equal, while the variation of voltage phase angle is very small [18]. In the proposed model, the variation of Bus G’ s voltage magnitude is ignored, which means that the voltage change of TCPST VT is equal to zero, and only the variation of voltage phase angle is considered. Then the DC power flow equation of two buses in Fig. 10.1 can be expressed as: PlG L = BlG L (θG − θ L ), 

(10.1)

where θG = θT + θG is the voltage phase angle of Bus G adjusted by TCPST. BlG L = 1/(X T C SC + X L ) denotes the susceptance of Branch l G L . In practice, the adjustment range of FACTS devices is limited. In [19], the adjustment range of TCSC is set as 0 to 3 times of the normal branch impedance. Considering the inaccuracy of the DC power flow model, the adjustment range of TCSC is 0.2 to 1 times in this work. In [11], the

174

10 Distributed Optimization Approach to System Protection

Fig. 10.2 A simple five Bus System with six TCSC agents

range of angle θT is set as ±35% of normal value. Since the TCPST device only plays an auxiliary role in the proposed coordination control strategy of TCSC-TCPST, the adjustment range of TCPST is set as ±10% of normal value. Consider a power network with M branches and N bus nodes. Let M = {li j |li j = 1, ..., M; i, j ∈ N } denote the set of branches and N = {1, ..., N } denote the set of bus nodes. It is assumed that li j exists if and only if Bus i and Bus j are connected by a branch. Equation (10.1) can be rewritten as Pli j = Bli j (θi − θ j )

∀i, j ∈ N , li j ∈ M.

(10.2)

Then the DC power flow equation of the power network is given in vector form as PM = BM · A · θN

(10.3)

with P M denotes the active power of branches and θ N denotes the bus voltage phase angles, respectively. Incidence matrix A describes the parameters relation between branches and bus nodes. B M is a M×M-diagonal matrix with the line susceptance on the diagonal. The relationship between Eqs. (10.2) and (10.3) is that Pli j and Bli j are the elements of P M and B M , respectively.

10.2.2 Communication Topology Suppose that all branches are installed with TCSC devices and all bus nodes are installed with TCPST devices. If the branches are connected to the same bus, their TCSC agents are neighbors [20]. Furthermore, neighboring agents can communicate with each other. To make this clear, an example is presented in Fig. 10.2, where TCSC agents T12 , T14 and T15 are the neighbors because they all connect to Bus 1. Agent T45 also is the neighbor of Agent T15 on account of Bus 5. In the TCSC-equipped network,

10.3 Problem Formulation

175

each agent can exchange information with neighbor agents. The network topology is an undirected graph G = (T , E) with T = {1, ..., M} representing the set of TCSC agents, and E ∈ T × T provides the branch information. The Laplacian matrix L for the undirected graph G is also a M×M-dimensional matrix, which describes the correlation of TCSC agents.

10.3 Problem Formulation Contingencies on bus nodes or transmission lines may impair the whole power grid and lead to cascading blackouts in the worst scenario, which brings huge losses and inconveniences to consumers. To avoid large-scale blackouts, this work aims to relieve or eliminate the disturbances by simultaneously adjusting voltage phase angles and branch susceptance via the coordination between TCSC and TCPST. Consider a desired power flow on each branch, denoted by the M-dimensional vector PσM , and Plσi j is the element of PσM . When the actual power flow deviates from the desired one, TCSC agents start to update the branch impedance to drive the actual power flow towards the desired value on each branch. Meanwhile, TCPST devices guarantee the stability of voltage phase angles in real time. Hence, the control goal is to minimize the deviation of the actual power flow from the desired power flow. To be more practical, we consider three constraints: (1) active power balance of the power network. (2) branch impedance constraint because of the finite range of branch impedance. (3) active power constraint on the branch. With the above constraints, the stress of power systems can be relieved or eliminated by adjusting the branch impedance using TCSC and the voltage phase angle using TCPST, and the optimization formulation is presented as follows min f (B M )

(10.4)

BM

subject to

PGi −

N 

D Bli j θli j − PLi = 0

∀i, j ∈ N , li j ∈ M,

(10.5)

j=1

B li j ≤ Bli j ≤ B li j

∀li j ∈ M,

(10.6)

P li j ≤ Pli j ≤ P li j

∀li j ∈ M

(10.7)

 2     with f (B M ) = lMi j =1 fli j Bli j = 21 lMi j =1 Bli j θli j − Plσi j . In (10.4), the branch susceptance Bli j on the diagonal of B M is an independent variable of local objective     function fli j Bli j . In addition, f (B M ) is the sum of local objective function fli j Bli j ,   and fli j Bli j is the quadratic term of the error between the actual active power and the desired active power on each branch. TCSC agents need to optimize f (B M )

176

10 Distributed Optimization Approach to System Protection

under all agents’ global constraint (10.5) and local constraints (10.6) and (10.7). In Constraint (10.5), the variable D represents the direction of active power, if active power Pli j goes out of Bus i, D = 1 otherwise D = −1. B li j and B li j are the lower and upper bounds of susceptance Bli j , P li j and P li j are the lower and upper bounds of actual active power Pli j , respectively. In particular, both the local objective function and local constraints of TCSC agent Ti j are only known by itself and cannot be shared with other TCSC agents. Then a distributed optimization algorithm can be proposed to solve the problem. It can be observed that each local objective function is strictly convex and Problem (10.4) has the optimal solution. The Karush-KuhnTucker (KKT) conditions of Problem (10.4) is given as M  li j =1

N M M M M        ∗ ∇ fli j Bl∗i j − αi∗ Dθli j − λl∗i j + μl∗ θli j + λli j − μl∗i j θli j = 0 li j =1

i=1

li j =1

li j =1

ij

li j =1

(10.8) and PGi

−

N 

D Bl∗i j θli j − PLi = 0,

j=1

B li j − Bl∗i j ≤ 0, Bl∗i j − B li j ≤ 0, P li j − Pl∗i j ≤ 0, Pl∗i j − P li j ≤ 0, ∗

λl∗i j , λli j ≥ 0,

  λl∗i j B li j − Bl∗i j = 0,   ∗ λli j Bl∗i j − B li j = 0,   μl∗ P li j − Pl∗i j = 0, ij   μl∗i j Pl∗i j − P li j = 0,

(10.9)

μl∗ , μl∗i j ≥ 0, ij

where αi and αi∗ are the multiplier and optimal multiplier of Constraint (10.5), respectively. λli j , λli j ∈ λli j , μl , μli j ∈ μli j are Lagrangian multipliers of inequality conij straints. In addition, both the sets of equality constraint multipliers and inequality constraint multipliers are closed, convex and bounded. Bl∗i j and Pl∗i j are the optimal solution of Problem (10.4), and they are the optimal susceptance and active power of branch, respectively.

10.4 Control Design and Theoretical Analysis 10.4.1 Control Law of TCPST In this section, we introduce the control laws for TCSC agents and TCPST devices, respectively. The implementation of the distributed control algorithm of TCSC agents

10.4 Control Design and Theoretical Analysis

177

Algorithm 1 Self-Adjustment of Bus Voltage Angle Input: θiσ , θiin , K , k Output: θiout 1: if abs(θiin − θiσ ) ≤ K · abs(θiσ ) then 2: θiout = θiin 3: else 4: while abs(θiin − θiσ ) > K · abs(θiσ ) do 5: if θiin − θiσ ≤ K · θiσ then 6: θiin = θiin + k · abs(θiσ ) 7: else 8: θiin = θiin − k · abs(θiσ ) 9: 10: 11:

end if end while θiout = θiin

12: end if

is based on the stability of voltage phase angles. The control law for TCPST is introduced to maintain voltage phase angles stability. Then the distributed control algorithm of TCSC is presented. Finally, we provide the convergence analysis of the designed control law for TCSC agents. In Eq. (10.3), although the variation range of θi is comparatively small, it cannot be ignored. Denote N-dimensional vector θ Nσ with its element θiσ as the desired voltage phase angles. For each TCPST device, we design the control law to adjust the voltage phase angle in real time. Table 10.1 shows the control algorithm of TCPST installed on Bus i. Specifically, if the deviation between the actual value θi and desired value θiσ meets the requirement, Algorithm 1 does not work. Otherwise, θi is adjusted by Algorithm 1 with k a regulating factor. Then, we need to control TCSC agents to accomplish the optimization goal.

10.4.2 Distributed Optimization Algorithm In this subsection, we solve the Problem (10.4) by designing the distributed primaldual gradient algorithm. As mentioned in Sect. 10.3, the TCSC agents can exchange information with their neighbors via the communication network. To solve the problem in a distributed fashion, suppose that the communication network topology graph G is strongly connected, which implies that Laplacian matrix L satisfies L1 M = 0 M and L1TM = 0TM . Thus a penalty term is considered, which ensures that the error of power flow on each branch tends to be consistent and converges to the optimal solution. The Problem (10.4) can be rewritten as min fˆ (B M ) BM

(10.10)

178

10 Distributed Optimization Approach to System Protection

with M 2 1  T   1  Bli j θli j − Plσi j + κ B M θ M − PσM L ⊗ I1 B M θ M − PσM fˆ (B M ) = 2 l =1 2 ij

(10.11)

PGi −

subject to

N 

D Bli j θli j − PLi = 0

∀i, j ∈ N , li j ∈ M.

j=1

(10.12)

  L ⊗ I1 B M θ M − PσM = 01M

(10.13)

B li j ≤ Bli j ≤ B li j

∀li j ∈ M.

(10.14)

P li j ≤ Pli j ≤ P li j

∀li j ∈ M

(10.15)

with the tuning parameter κ and the M-dimensional column vector θ M . In [21], the parameter κ is not taken into account. In fact, κ may affect the information exchanged between TCSC agents, which changes the stability and optimal performance of the designed control algorithm. Constraint (10.13) implies that the deviation between real active power and desired active power are consensus, that is ((Bl12 θl12 − Plσ12 )=(Bl23 θl23 − Plσ23 )= · · · =(Bli j θli j − Plσi j )). Denote ν M ∈ R M as the multiplier vector of Constraint (10.13), and νli j is the element of ν M . The KKT conditions of Problem (10.10) with Constraints (10.12) − (10.15) are as follows M 

M        ∇ fˆli j Bl∗i j + θli j aTi j _n Bl∗i j θli j − Plσi j − Bl∗i j_n θli j_n − Plσi j_n

li j =1

+

li j_n =1

M 

N M       ∗ ∗ ∗ ∗ λl∗i j − λli j aTi j _n θli j νli j − νli j_n − αi Dθli j −

li j_n =1

− θli j

M  li j =1

i=1

li j =1

  μl∗ − μl∗i j = 0, ij

  L ⊗ I1 B∗M θ M − PσM = 01M ,

(10.16) (10.17)

and Eq .(10.9). Based on the Lemma 3.2 in [21] and KKT condition (10.9), (10.16)−(10.17), TCSC agent Ti j can find the optimal solution Bl∗i j and the opti∗

mal multipliers (vl∗i j , αi∗ , λl∗i j , λli j , μl∗ , μl∗i j ) in a distributed manner, the estimations ij of all agents are synchronized to the optimal solution of Problem (10.10), which also is the optimal solution of Problem (10.4). Then we design the distributed algorithm by allowing for KKT conditions (10.9), (10.16)−(10.17) and the saddle point theorem. The Lagrangian function is defined as

10.4 Control Design and Theoretical Analysis

179

M   1 2  Bli j θli j − Plσi j L B M , v M , αi , λli j , λli j , μl , μli j = ij 2 l =1 ij

T     1  T + κ B M θ M − PσM L ⊗ I1 B M θ M − PσM + ν M L ⊗ I1 B M θ M − PσM 2 ⎛ ⎞ N N M         λl B l − Bli j + λli j Bli j − B li j + αi ⎝ PGi − D Bli j θli j − PLi ⎠ + ij

i=1

+

ij

li j =1

j=1

M       μl P li j − Pli j + μli j Pli j − P li j . ij

li j =1

(10.18) To apply the distributed gradient algorithm, we denote Z M = B M θ M − P M with its element Z li j = Bli j θli j − Plσi j . The Lagrangian function (10.18) can be rewritten as 

Lˆ Z M , v M , αi , λli j , λli j , μl , μli j



ij

+

N 

⎛ αi ⎝ PGi −

i=1

+

M  li j =1

N 

= ⎞

M 1  2 1 T Z li j + κZTM L ⊗ I1 Z M + ν M L ⊗ I1 Z M 2 2 li j =1

D Bli j θli j − PLi ⎠ +

M  

    λli j B li j − Bli j + λli j Bli j − B li j

li j =1

j=1

     μl P li j − Pli j + μli j Pli j − P li j . ij

(10.19) By differentiating variables and multipliers of Lagrangian function (10.19), we get the dynamics of Agent Ti j as follows ⎧ ⎨

M      aTi j _n Z li j − Z li j_n Z˙ li j = − ∇ fli j Z li j + κ ⎩ li j_n =1

+

M 



aTi j _n νli j − νli j_n



li j_n =1

  aTi j _n Z li j − Z li j_n ,

M 

ν˙li j =

⎫ ⎬  1  − Dαi − λli j − λli j − μl + μli j , ij ⎭ θli j

(10.20)

li j_n =1

α˙ i = PGi −

N 

D Bli j θli j − PLi ,

j=1

+ +  λ˙ li j = B li j − Bli j , λ˙ li j = Bli j − B li j , λli j λl ij + +   ˙ l = Pl − P l , μ , μ ˙ l = P li j − Pli j ij ij i j ij μ μ 

li j

li j

180

10 Distributed Optimization Approach to System Protection

Algorithm 2 Coordination Control of TCSC-TCPST. Input: T , s, r Output: B M , f (B M ), fli j (Bli j ) 1: for s = 1 : T do 2: if 0 ≤ r em(s, r ) < 10 then 3: Run Algorithm 1 to compute θ N 4: else 5: Update Z M (B M ), ν M , αi λli j , μli j with Dynamics (10.20) 6: end if 7: end for

where aTi j _n is the element of Laplacian L that matches Agent Ti j and its n-th neighbor, and n = 0 means the diagonal element of L matrix that Agent Ti j matched. Z li j_n , νli j_n are the mismatch power flow of Branch li j_n corresponding to the n-th neighbor of Agent Ti j and its multiplier, respectively. Furthermore, the symbol [ ] is defined as  , if [] > 0 or φ > 0. + []φ = 0, otherwise. Here  denotes the set {B li j − Bli j , Bli j − B li j , P li j − Pli j , Pli j − P li j }, and φ denotes the set {λli j , λli j , μl , μli j }, respectively. ij

Remark 10.1 As aforementioned, both the change of branch impedance Bli j and the change of voltage angle θli j affect active power, and we can control the power flow by changing Bli j and θli j via FACTS devices. Therefore, the varying of Bli j and θli j means Z li j is also changed. Dynamics (10.20) shows the control low of TCSC agent for changing Bli j and the reason why Problem (10.4) can be solved. TCSC agents can obtain the optimal solution of Problem (10.4) with Dynamics (10.20). Nevertheless, the change of voltage phase angle θi is not taken into account during the iteration of Dynamics (10.20), which means that the control law of TCSC agents is independent of the control law of TCPST device. Algorithm 2 illustrates the relationship between those two control laws in detail. In Table 10.2, it is observed that voltage phase angles are calculated in the first 10 steps, and Dynamics (10.20) is carried out in the remaining steps in each time interval. Here r em(x, y) means the remainder of x/y is an integer, and r determines the time interval.

10.4.3 Convergence Analysis The convergence of Dynamics (10.20) is analyzed in this subsection. To prove con(Pli j ) = P li j − Pli j vergence straightforward, the constraints are rearranged. Let gllow ij

10.4 Control Design and Theoretical Analysis

181

up

and gli j (Pli j ) = Pli j − P li j denote the lower and upper bounds constraint of active up (Pli j ), gli j (Pli j )}, μlPi j denotes its power Pli j , and glPi j (Pli j ) refers to the set of {gllow ij B B multiplier set. Similarly, gli j (Bli j ) and λli j are the set of impedance constraints of Bli j and its multipliers, respectively. For Equality Constraint (10.12), which i = PGi − PLi , we obtain needs to be converted to inequality constraints. Donate Pout N i sli j (Bli j ) = Pout − j=1 D Bli j θli j . Here sli j (Bli j ) = 0 when the power grid is normal, i,σ i,σ i,σ ≤ sli j (Bli j ) ≤ 1.3Pout , where Pout and by adding a slack on sli j (Bli j ). We have 0.7Pout up i,σ is desired power output of Bus i. Denote sllow (Bli j ) = 0.7Pout − sli j (Bli j ), sli j (Bli j ) = ij   up i,σ sli j (Bli j ) − 1.3Pout , slsi j (Bli j ) can be referred to the set of sllow (Bli j ), sli j (Bli j ) with ij its multiplier set ζlsi j . In addition, notice that all inequality constraints of Agent Ti j can be expressed as constraints related Z li j (as defined above: Z li j = Bli j θli j − Plσi j ). Here let gli j (Z li j ) denote the inequality constraints set of Agent Ti j , and its multiplier set is denoted by li j . Remark 10.2 Constraint (10.12) is a power balance constraint that is directly related to the TCSC agents, and it is a basic global constraint that all agents must be satisfied. By slacking Constraint (10.12), we make up the deficiency in [21] is that did not take into consideration of equality constraint, which is conducive to the proof of convergence. Here, we assume that the range of active power output of Bus i is ±30% of the normal value, but of course, it can be other ranges. Obviously, Equality Constraint (10.12) is a special case of inequality constraint after relaxation. In what follows, we prove the trajectories of Dynamics (10.20) are bounded. It is ∗ observed that when all agents reach to the equilibrium point (Z∗M , ν M ,  M∗ ), Dynam˙ M = 0, and we can know that the equilibrium ics (10.20) satisfy Z˙ M = 0, ν˙ M = 0,  ∗ ,  M∗ ) is the optimal solution of Problem (10.10). point (Z∗M , ν M Lemma 10.1 The trajectories of Dynamics (10.20) are bounded. Proof Define the function: (Z M , ν M ,  M ) =

M  2  2  2  1  Z li j − Z l∗i j + νli j − νl∗i j + li j − l∗i j , 2 l =1 ij

(10.21) ∗ where (Z l∗i j , νl∗i j , l∗i j ) is the element of equilibrium point (Z∗M , ν M ,  M∗ ). By taking the derivative with respect to (Z M , ν M ,  M ) along the trajectories of the Dynamics (10.20), we obtain

182

10 Distributed Optimization Approach to System Protection M    d (Z M , ν M ,  M ) − Z li j − Z l∗i j ∇ Z li j L (Z M , ν M ,  M ) = dt li j =1   + νli j − νl∗i j ∇νli j L (Z M , ν M ,  M )   + + li j − l∗i j ∇li j L (Z M , ν M ,  M ) li j

≤

M 





(10.22)

{− Z li j − Z l∗i j ∇ Z li j L (Z M , ν M ,  M )

li j =1

  + νli j − νl∗i j ∇νli j L (Z M , ν M ,  M )   + li j − l∗i j ∇li j L (Z M , ν M ,  M )}. In (10.22), the Lagrangian function L (Z M , ν M ,  M ) is convex in decision vari∗ ,  M∗ ) is a saddle able Z li j and concave in Lagrangian multiplier li j , thus (Z∗M , ν M point of L (Z M , ν M ,  M ). Based on saddle point theorem and first order optimal condition [22], we have M 

    − Z li j − Z l∗i j ∇ Z li j L (Z M , ν M ,  M ) ≤ L (Z M , ν M ,  M ) − L Z ∗M , ν M ,  M

li j =1

and T    ∗  ∇ν M L (Z M , ν M ,  M ) , ∇ M L (Z M , v M ,  M ) ,  M∗ (ν M ,  M ) − ν M   ∗ ,  M∗ . ≤ L (Z M , ν M ,  M ) − L Z M , ν M Substituting the above inequalities into (10.22), we have   d (Z M , ν M ,  M ) ≤ L Z ∗M , ν M ,  M − L (Z M , ν M ,  M ) dt   ∗ ,  M∗ + L (Z M , ν M ,  M ) − L Z M , ν M     ∗ ≤ L Z ∗M , ν M ,  M − L Z ∗M , ν M ,  M∗     ∗ ∗ + L Z ∗M , ν M ,  M∗ − L Z M , ν M ,  M∗ ≤ 0.

(10.23)

∗ In fact, (Z∗M , ν M ,  M∗ ) is a finite point and it also means that the trajectories of Dynamics (10.20) are bounded. In addition, the set {(Z M , ν M ,  M ) | (Z M , ν M ,  M ) ≤ (Z M (0), ν M (0),  M (0))} can be regarded as a positive invariant set of Dynamics (10.20). 

The next step shows the set of points that decision variable Z M and multipliers ν M ,  M converged, which satisfies Z˙ M = 0 M , ν˙ M = 0 M and  ˙ M = 0 M by using LaSalle invariance principle. First, we define the index set ωli j = {li j |li j = 0, gli j (Z li j ) < 0}

10.4 Control Design and Theoretical Analysis

183

as the local constraint function of Agent Ti j , which implies that the inequality constraints matching the set ωli j are independent of the optimization process of Agent Ti j . Let ω = {ωl12 , ωl23 , ..., ω M } donate different multiplier dynamics that different agent’s constraints function matched. Since Dynamics (10.20) can be treated as a hybrid system, ω indicates which dynamics a multi-agent system performs. Notice that both the agent and constraint numbers of optimization problem are finite, hence ω is a finite index set. Construct the Lyapunov function as follows M    1  Z˙ l2i j + ν˙l2i j +  ˙ M, ω = ˙ l2i j , V Z˙ M , ν˙ M ,  2 l =1

(10.24)

ij

here since state (Z M , ν M ,  M ) and  ˙ li j = 0 of each constraint determine the index set ω, the Lyapunov function (10.24) only depends on the state (Z M , ν M ,  M ). However, the change of state (Z  lead to the variation of ω, which results  M , ν M ,  M ) may ˙ M , ω discontinuous. Fortunately, the followin Lyapunov function V Z˙ M , ν˙ M ,    ˙ M , ω is non-increasing no matter ω is fixed or ing proof shows that V Z˙ M , ν˙ M ,  time-varying. Lemma 10.2 Lyapunov Function (10.24) is non-increasing along Dynamics (10.20). Proof For fixed value of ω, the time derivative of Lyapunov function (10.20) gives M  dV = ( Z˙ li j Z¨ li j + ν˙li j ν¨li j +  ˙ li j  ¨ li j ) dt li j =1 ⎧ M ⎨ M   = aTi j_n ( Z˙ li j − Z˙ li j_n ) Z˙ li j [−∇ 2 fli j (Z li j ) Z˙ li j − ⎩ li j =1

−

li j_n =1

M 

aTi j_n (˙νli j − ν˙li j_n ) − ∇gli j (Z li j ) ˙ li j − li j ∇ 2 gli j (Z li j ) Z˙ li j ]

li j_n =1

+˙νli j

M 

aTi j_n ( Z˙ li j − Z˙ li j_n )

li j_n =1

⎫ ⎬ ⎭

+

M 

[ ˙ li j ∇gli j (Z li j ) Z˙ li j ]

li j =1

M    − Z˙ li j ∇ 2 fli j ( Z˙ li j ) Z˙ li j −li j ∇ 2 gli j (Z li j ) Z˙ li j = li j =1

−

⎧ M ⎨  li j =1

− ν˙li j

⎩

Z˙ li j

M  li j_n =1

M 

aTi j_n ( Z˙ li j − Z˙ li j_n ) + Z˙ li j

li j_n =1

aTi j_n ( Z˙ li j − Z˙ li j_n )

⎫ ⎬ ⎭

M  li j_n =1

aTi j_n (˙νli j − ν˙li j_n )

(10.25)

184

10 Distributed Optimization Approach to System Protection

=

M    − Z˙ li j ∇ 2 fli j (Z li j ) Z˙ li j −li j Z˙ li j ∇ 2 gli j (Z li j ) Z˙ li j li j =1



T T T − Z˙ M L Z˙ M + ν˙ M L Z˙ M − Z˙ M L ν˙ M



M    T = − Z˙ M diag[∇ 2 f M (Z M )] + L Z˙ M − {li j Z˙ li j ∇ 2 gli j (Z li j ) Z˙ li j } li j =1

≤ 0. Notice that both the objective function and constraint function are convex in their feasible set. Hence ∇ 2 f M (Z M ) and ∇ 2 gli j (li j ) are positive semi-definite, and Laplacian matrix L is also positive semi-definite. Therefore (10.25) holds with Slater’s condition and li j ≥ 0. The changing of ω may be caused by a change in state (Z M , ν M ,  M ), we need to consider the state of the Lyapunov function ˙ M , ω when ω changes. There are two scenarios that need to be V Z˙ M , ν˙ M ,  considered: variation caused by gli j (Z li j ) goes through zero from negative to positive when ωli j reduces from t− to t+ , and multiplier li j decreases from positive to zero. As for the first scenarios, it is observed that the term  ˙ l2i j is related to     ˙ M , ω from state V Z˙ M , ν˙ M ,  ˙ M , ω(t− ) to state the changing of V Z˙ M , ν˙ M ,    ˙ M , ω(t+ ) , whereas ωl2i j (t− ) = ωl2i j (t+ ) = 0. One can obtain that the V Z˙ M , ν˙ M ,      Lyapunov function V Z˙ M , ν˙ M ,  ˙ M , ω is continuous and V Z˙ M , ν˙ M ,  ˙ M , ω (t− ) =   ˙ M , ω (t+ ) for arbitrary time t. As for the second scenarios, when mulV Z˙ M , ν˙ M ,    ˙ M , ω(t− ) tiplier li j decreases from positive to zero, it causes that V Z˙ M , ν˙ M ,    ˙ M , ω(t+ ) , which also means has one more nonnegative term than V Z˙ M , ν˙ M ,      ˙ M , ω (t+ ) ≤ V Z˙ M , ν˙ M ,  ˙ M , ω (t− ). With the discussed above, we V Z˙ M , ν˙ M ,  obtain ddtV ≤ 0 both for t ≤ t− and t ≥ t+ .  The main theoretical result is presented as follows. Theorem 10.1 All agents converge to the same optimal solution of Problem (10.10), which also is the optimal solution of Problem (10.4) with Dynamics (10.20). Proof To apply the LaSalle invariance principle to the hybrid system as discussed above, we need to ensure there is a compact, positively invariant set . All of the trajectories of Dynamics (10.20) start in  and will stay in  all the time. Then each trajectory in  converges to its infimum , the maximal positively invariant set within d V Z˙ ,˙ν , ˙ ,ω  with trajectories satisfying [23]. (1) ( M dtM M ) ≡ 0 in intervals of fixed ;     (2) V Z˙ M , ν˙ M ,  ˙ M , ω− = V Z˙ M , ν˙ M ,  ˙ M , ω+ if ω switches between ω− and ω+ .   ˙ M, ω = According to Theorem 10.1, for a fixed value of ω, one has V˙ Z˙ M , ν˙ M ,  0. It is observed that all local objective functions have positive definite Hessian ∇ 2 fli j (Z li j ), and the matrix diag[∇ 2 f M (Z M )] + L is positive semidefinite. Thus, Z˙ M = 0 on the basis of (10.25). When Z˙ M = 0, there is no doubt that ν˙ M = 0, because Z˙ M = 0 means that all power flow mismatch of branches must reach consensus due to KKT condition (10.17). According to Lemma 10.1, we know that the trajectories

10.5 Numerical Simulations

185

of ν M are bounded. If Constraint (10.17) does not hold, there exists at least one element of ν M that goes to infinity. Hence ν˙ M = 0. If gli j (Z l∗i j ) > 0, it automatically causes the matched element of li j to goes to infinity according to KKT condition. However, Lemma 10.1 also indicates li j is bounded. Inversely gli j (Z l∗i j ) < 0 means l∗i j = 0. Otherwise, li j will decrease to zero, which is conflict with the continuity of Lyapunov function (10.24) according to Lemma 10.2. Hence  ˙ M = 0. Consequently, the trajectories of Dynamics (10.20) converge to an optimal point Z˙ M = 0, ν˙ M = 0,  ˙ M = 0, which also satisfies the KKT conditions (10.16) − (10.17). Thus, all TCSC agents converge to the same optimal solution of Problem (10.4). 

10.5 Numerical Simulations In this section, numerical simulations are conducted to validate the proposed distributed optimization algorithm of TCSC agents and TCPST devices on IEEE 9 bus system. The parameters and adjustment ranges of variables are set as follows. First, Bus 1 is the reference bus, and the Matlab function “rundcpf" is used for solving nine bus system to obtain the steady-state power flow data, which was set as the desired value (θ Nσ for the bus voltage angle and PσM for the desired power flow) [24]. The lower and upper bounds of branch active power are set as ±30% of steady-state value. Parameters K , k in Algorithm 1, r in Algorithm 2, and the penalty parameter κ are set as 0.01, 0.2, 100 and 1, respectively. The total simulation step is given by T = 5000. To better compare the simulation results and the control effect, performance index Sopt is defined as follows Sopt =

f (B M )max − f (B M )sta × 100%, f (B M )max

(10.26)

where f (B M )max , f (B M )sta are the maximum value and stable value of global objective function f (B M ), respectively. Note that Sopt clearly demonstrates the optimization effect of the control algorithm.

10.5.1 Reduction of Branch Capacity In this subsection, contingency is set as the delivery capacity of Branch l82 suddenly reduced by 20% compared to normal status. As shown in Table 10.1, the value of global objective function f (B M ) converges to 0.0093 with the number of steps increases, Sopt is 82.59%. In comparison, the centralized algorithm in [16] demonstrates better convergence precision, the performance index Sopt is 99.09%. However, the proposed distributed algorithm is superior to the centralized algorithm in computation costs and system resilience. for the centralized algorithm, when Agents T82 and T89 cannot send impedance data to the control center in real time due to commu-

186

10 Distributed Optimization Approach to System Protection

Table 10.1 Comparison between centralized algorithm and distributed algorithm Parameters Centralized algorithm [16] Proposed algorithm Communication network Normal Fault Normal Fault Iterations when f (B M ) reaches stable point Simulation time when f (B M ) reaches stable point (s) Maximum value of f (B M ) Stable value of f (B M ) Sopt

2319

–

996

1934

82.28

–

8.90

20.80

0.0110 0.0001 99.09%

– – –

0.0539 0.0093 82.59%

0.0533 0.0400 24.92%

Fig. 10.3 a Value of voltage phase angle θi for each bus before and after contingency. b Value of branch susceptance Bli j before and after contingency

nication fault, the control center is out of action, and the algorithm failure. For the proposed distributed algorithm, we can see that it still works when the communication between Agents T82 and T89 fails, but the control effect is impaired. In Fig. 10.3a, bus voltage angles hold steady under the control algorithm of TCPST. From Fig. 10.3b, we can see that all TCSC agents participate in the regulation and the susceptance Bl82 changes remarkably, which demonstrates that the cooperation between TCSC agents and TCPST devices can effectively relieve the power system stress caused by branch faults.

10.5.2 Bus Overloads In the above case, it is demonstrated that the proposed distributed control method of TCSC agents and TCPST devices can effectively relieve the power system stress caused by branch faults. In this subsection, bus overload as a common contingency is taken into account. It is assumed that load Buses 5, 7, 9 overload by 10%, which undoubtedly causes the stress of power system due to grid interconnection.

10.5 Numerical Simulations

187

Table 10.2 Comparison between centralized algorithm and distributed algorithm Parameters Centralized algorithm [16] Proposed algorithm Communication network Normal Fault Normal Fault Iterations when f (B M ) reaches stable point Simulation time when f (B M ) reaches stable point (s) Maximum value of f (B M ) Stable value of f (B M ) Sopt

700

–

1750

5000

17.22

–

13.15

19.36

0.0791 0.0693 12.37%

– – –

0.4783 0.1737 63.68%

0.4455 0.1756 60.58%

Fig. 10.4 a Value of voltage phase angle θi for each bus before and after contingency. b Value of branch susceptance Bli j before and after contingency

Table 10.2 shows that the index Sopt of distributed algorithm exceeds the centralized algorithm under this contingency. Moreover, the proposed distributed control algorithm of TCSCs is still better than the centralized algorithm in computation costs. In Fig. 10.4a, the final value of the bus voltage angle is almost the same as the initial value, which demonstrates the effectiveness of control law for TCPST devices. In Fig. 10.4b, it is also observed that all TCSC agents participate in the regulation process of resisting the disturbances.

10.5.3 Effect of Tuning Parameters In the above cases, we show that the proposed distributed control method of TCSC agents and TCPST devices can relieve the power system stress both caused by branch faults and bus overloads. To reduce the computation burden and improve the control effect while ensuring control algorithm stability, the influence of parameter κ is studied in this subsection.

188

10 Distributed Optimization Approach to System Protection

Fig. 10.5 Simulation results for the influence of κ. a Trajectories of global objective function f (B M ) for different κ based on Case 1. b Trajectories of global objective function f (B M ) for different κ based on Case 2. Table 10.3 Comparison with different value of κ. Parameters Iterations

k=0.01

k=0.1

k=0.5

k=1

k=2

Case1

Case2

Case1

Case2

Case1

Case2

Case1

Csae2

Csae1

Csae2

160

505

213

679

597

883

996

1750

–

– –

Time

3.32

6.42

4.15

7.91

7.63

10.99

8.90

13.15

–

f max (B M )

0.0531

0.4105

0.0531

0.4139

0.0532

0.4110

0.0539

0.4783

1.6703

1.2487

f sta (B M )

0.0007

0.1861

0.0007

0.1842

0.0017

0.1781

0.0093

0.1737

–

–

Sopt

98.68%

54.67%

98.68%

55.50%

96.80%

56.67%

82.59%

63.68%

–

–

Based on Case 1 and Case 2, the parameter κ is set as 0.01, 0.1 0.5, 1 and 2, respectively. The trajectories of global objective function f (B M ) are shown in Fig. 10.5. It is observed that the proposed distributed control algorithm loses stability when κ = 2. Table 10.3 shows the performance index of the proposed control algorithm with different values of κ. Specifically, the iteration times decrease with the decreasing of κ. Meanwhile, Sopt increases significantly for contingency is set as branch fault and Sopt decreases slightly for contingency is set as bus overloads. It is worth looking for a suitable κ to balance the control effect in future work. In addition, it is noted that the time spent by the distributed control algorithm to handle contingencies decreases with the decrease of κ, which effectively improves the speed of fault processing and improves the protection performance.

10.6 Conclusions This chapter addressed the online protection problem of power systems by adjusting branch impedance in a distributed manner to redistribute the power flow on the branches, which contributes to the resilience of power grids against disruptive distur-

References

189

bances or external contingencies. A distributed optimization algorithm was proposed to control the TCSC for relieving the stress of power systems subject to contingencies. Simulation results were presented to demonstrate the effectiveness of the proposed algorithm on IEEE test bus system. In addition, the influence of parameters on system performance and control effect was analyzed. Future work includes the consideration of alternating current (AC) power flow so that physical implementations are more practical. To reduce the construction cost, we will also consider the optional location and the number of TCSC and TCPST to achieve the desired goal and improve the optimization of effect based on the sub-area division of the power systems.

References 1. Shield, S., Quiring, M., Pino, V., Buckstaff, K.: Major impacts of weather events on the 270 electrical power delivery system in the united states. Energy 218, 119434 (2021) 2. Busby, W., Baker, K., Bazilian, D., Gilbert, Q., Grubert, E., Rai, V., Rhodes, D., Shidore, S., Smith, A., Webber, E.: Cascading risks: understanding the 2021 winter blackout in texas. Energy Res. Soc. Sci. 77, 102106 (2021) 3. Mathaios, P., Pierluigi, M.: Influence of extreme weather and climate change on the resilience of power systems: impacts and possible mitigation strategies. Electr. Power Syst. Res. 127, 259–270 (2015) 4. Ali, E., Enshaee, P.: A viable controlled splitting strategy to improve transmission systems resilience 280 against blackouts. Electr. Power Syst. Res. 175, 105913 (2019) 5. Zhai, C., Xiao, G., Meng, M., Zhang, H., Li, B.: Identification of catastrophic cascading failures in protected power grids using optimal control. J. Energy Eng. 1, 06020001 (2021) 6. Aliyan, E., Aghamohammadi, M., Kia, M., Heidari, A., Shafie-khah, M., Catalao, J.: Decision tree analysis to identify harmful contingencies and estimate blackout indices for predicting system vulnerability. Electr. Power Syst. Res. 178, 106036 (2020) 7. Zhai, C., Nguyen, H.D., Xiao, G.: A robust optimization approach for protecting power systems 290 against cascading blackouts. Electr. Power Syst. Res. 189, 106794 (2020) 8. Qader, M.: Design and simulation of a different innovation controller-based UPFC (unified power flow controller) for the enhancement of power quality. Energy 89, 576 (2015) 9. Ehsan, N., Samaneh, P., Arash, A.: A remedial action scheme against false data injection cyberattacks in smart transmission systems: application of thyristor-controlled series capacitor. IEEE Trans. Ind. Inf. 18(4), 2297 (2022) 10. Ranjbar, S., Al-Sumaiti, A.S., Sangrody, R., Byon, Y.J., Marzband, M.: Dynamic clusteringbased 300 model reduction scheme for damping control of large power systems using series compensators from wide area signals. Int. J. Electr. Power Energy Syst. 131, 107082 (2021) 11. Sebaa, K., Bouhedda, M., Tlemcani, A., Henini, N.: Location and tuning of TCPSTS and SVCS based on optimal power flow and an improved cross-entropy approach. Int. J. Electr. Power Energy Syst. 54, 536 (2014) 12. Noroozian, M., Angquist, L., Ghandhari, M., Andersson, G.: Improving power system dynamics by series-connected facts devices. IEEE Trans. Power Delivery 12(4), 1635 (1997) 13. Jordehi, A.: Optimal allocation of facts devices for static security enhancement in power systems via imperialistic competitive algorithm (ica). Appl. Soft Comput. 48, 317 (2016) 14. Naresh, G., Ramalinga Raju, M., Narasimham, S.L.: Coordinated design of power system stabilizers and tcsc employing improved harmony search algorithm. Swarm Evol. Comput. 315(27), 169 (2016) 15. Vilmair, E., Fernandes, S., Tortelli, O.: TCPST allocation using optimal power flow and genetic algorithms. Int. J. Electr. Power Energy Syst. 33(4), 880 (2011)

190

10 Distributed Optimization Approach to System Protection

16. Zhai, C., Xiao, G., Zhang, H., Pan, T.C.: Cooperative control of TCSC to relieve the stress of cyber- 320 physical power system. In: 2018 15th International Conference on Control, Automation, Robotics and Vision (ICARCV) 4849 (2018) 17. Kamel, S., Abokrisha, M., Selim, A., Jurado, F.: Power flow control of power systems based on a simple TCSC model. Ain Shams Eng. J. 12(3), 2781 (2021) 18. Brian, S., Jorge, J., Ongun, A.: Dc power flow revisited. IEEE Trans. Power Syst. 24(3), 1290 (2009) 19. Khederzadeh, M., Sidhu, T.: Impact of TCSC on the protection of transmission lines. IEEE Trans. Power Delivery 21(1), 80 (2006) 20. Li, S., Du, H., Lin, X.: Finite-time consensus algorithm for multi-agent systems with double20330 integrator dynamics. Automatica 47(8), 1706 (2011) 21. Yi, P., Hong, Y., Liu, F.: Distributed gradient algorithm for constrained optimization with application to load sharing in power systems. Syst. Control Lett. 83, 45 (2015) 22. Stephen, B., Lieven, B.: Convex Optimization. Cambridge University Press (2004) 23. Lygeros, J., Johansson, K., Simic, S., Zhang, J., Sastry, S.: Dynamical properties of hybrid automata. IEEE Trans. Automa. Control 48(1), 2 (2003) 24. Daniel, Z., Edmundo, M., John, T., Matpower: Steady-state operations, planning, and analysis tools for power systems research and education. IEEE Trans. Power Syst. 340, 26(1), 12 (2011)

Chapter 11

Reinforcement Learning Approach to System Recovery

Abstract Cascading blackouts have caused serious damage to power systems and affected the normal operation of society, so it is crucial to quickly restore the damaged power system to normal operation. In this chapter, a reinforcement learning (RL) approach is developed for the generator restoration in power system. Based on the proposed restoration approach, the Q-learning algorithm has a stronger trial-and-error ability and finds an optimal strategy to make generator units active power optimum after a fault, which maximizes discount reward. Finally, numerical analysis is carried out on the IEEE-9 bus system. The simulation results demonstrate the effectiveness and feasibility of the proposed approach.

11.1 Introduction Power system is an important infrastructure in modern society, as its safe and reliable operation is significant to social production [1, 2]. In recent years, due to various factors such as natural disasters, cyber-attacks, protection refusal or misoperation, there have been many major power blackouts around the world. For instance, there was a major blackout in Ukraine in 2016 due to cyber attacks, which caused 225,000 customers without power [3]. Large power outages caused by continuous and very low temperatures in Texas resulted in 4.5 million customers without electricity [4]. When a power system is subject to blackout, it is in a fragile state, which affects people’s life. Although large-scale power blackouts are uncommon, they greatly increase the risk of outages and the reliability of electricity [5]. Therefore, it is necessary to carry out timely and effective approaches to restore power system [6]. Power system recovery is a very complex process. For traditional recovery methods, it is usually divided into three stages: identifying the initial disturbance sources, determining the generator start-up strategy according to the black-start unit and the recovery path, and finally recovering the critical loads [2, 7]. To optimize black-start units’ recovery sequences, [8] proposed a stochastic heuristic algorithm that satisfies various constraints of power grid and executes multiple times in a high-performance computing environment to find feasible solutions. To find the network operation structure that minimizes the power loss, [9] proposed a harmony search algorithm to © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Zhai, Control and Optimization Methods for Complex System Resilience, Studies in Systems, Decision and Control 478, https://doi.org/10.1007/978-981-99-3053-1_11

191

192

11 Reinforcement Learning Approach to System Recovery

solve the problem of network reconstruction, using the music process to find the perfect harmony state. [10] proposed the genetic algorithm to change the control action to modify the network topology, finding an optimal network structure to reconfigure the topology path. In order to restore more loads, considering frequency deviation, stability of microgrid and various power flow constraints, [11] proposed a recovery method using microgrid to restore critical load, regarding it as a mixed integer linear programming problem. Generally, generator units’ recovery includes the recovery sequence of the generator units and the corresponding recovery path. In order to maximize the power generation capacity of the system and optimize the complex constraint problems that change with time, a new generator start-up sequence is proposed in [12], which is regarded as a mixed integer linear programming problem. [13] proposes an online decision-making strategy for generator startup based on deep learning and Monte Carlo tree search. The encoder is trained on the samples to determine the recovery lines. In order to reduce the loss of load during power blackouts, [14] proposes an optimization sequence of generator start-up recovery based on the greedy algorithm, establishes different generator and transmission path models, and determines the generator start-up priority. And [15] presents a preference multi-objective model considering the difference of objective importance, and the model is solved by a genetic algorithm. Considering the actual power system topology and generator startup sequence, [16] proposes a recovery strategy composed of the generator start-up sequence model, the transmission line recovery model, and the load start model to maximize the total power generation of the system. Most of them use traditional optimization algorithms and heuristic algorithms, while reinforcement learning is rarely used in this research. In this work, in order to simplify the problem, we use the reinforcement learning method, because it can describe the restoration problem as a sequential decision problem. At present, reinforcement learning has been successfully applied to other fields, such as robotics, computer vision, games, etc. An agent makes trial and error in a dynamic environment and finally finds an optimal action that maximizes the discount reward [17]. In this chapter, we introduce the application of Q-learning algorithm in generators’ restoration in power systems, which helps generators quickly restore to normal operation. The main contributions of this chapter are as follows: • Establish a mathematical model of generator recovery. • Propose that actions in RL are the active power output of generation units. • Find the optimal active power output of generation units and validate the performance of the proposed strategy. The outline of this chapter is as follows. Section 11.2 presents problem formulation of generators restoration model. Section 11.3 introduces the restoration of generators in power system based on RL. Simulation and validation on IEEE-9 bus system are conducted in Sect. 11.4. Finally, Sect. 11.5 gives a conclusion and a discussion about future work.

11.2 Problem Formulation

193

11.2 Problem Formulation In this chapter, we assume that load shedding fault occurs at the user terminal, and the generator units are disturbed. Figure 11.1 shows the schematic diagram of power system occurring at the load end. In this problem, we adopt an off-line algorithm: Q-learning, even if the power system is in a faulty state, the agent still makes the optimal choice from the offline data set, that is, adjusting the output power of the generator units. According to the maximal discount reward finds the optimal output power of generator units maximizing the recovery of the faulted power system. In this recovery decision problem, agent needs to make decisions based on previous experience [18]. The recovery of faulted distribution system is a multi-objective, multi-constraint and nonlinear problem. For the proposed problem, we assume the reward funtion attains the maximum. At the same time, constraints such as safety, stability and low operation cost should be considered. Generally, the objective function of this chapter is mainly obtained by power flow calculation, and the power flow calculation method is the Newton-Raphson method. It expands the power flow equation with the Taylor series and solves it after omitting the second-order and higher-order terms. The essence is successive linearization. The core is to solve the nonlinear equation and becomes a process of repeatedly solving the corresponding linear equation, and the convergence effect is better, but the stability is more sensitive to the initial value. The following are objective functions and constraints related to the proposed problem. Objective function: NG  (11.1) max (c0 + c1 P1 + c2 P2 ) k=1

where NG is number of generators. • Node voltage constraint:

Vi ≤ Vi ≤ Vi

(11.2)

• Generators active power constraints: P i ≤ Pi ≤ P i

(11.3)

where Pi refers to generatora i active power output, P i is the lower limit of generator active power output, P i is the upper limit of generator active power output. • Power flow constraint:

194

11 Reinforcement Learning Approach to System Recovery

Industrial Consumer Generator Unit1

Transformer

Distribution Station

Commercial Consumer

Transmission Network Generator Unit2

Residential Consumer

Fig. 11.1 Schematic diagram of power system with load fault

⎧ N    ⎪ ⎪ ⎪ PGi − PDi − Vi V j G i j cosδi j + Bi j sinδi j = 0 ⎪ ⎪ ⎨ j=1 N ⎪  ⎪   ⎪ ⎪ − V V j G i j sinδi j − Bi j cosδi j = 0 Gi − Q ⎪ Di i ⎩

(11.4)

j=1

where PGi is the active power of generator i, PDi is the reactive power of generator i, G i j is conductance of nodes i and j, Bi j is susceptance of nodes i and j, δi j is the difference in phase angle between the voltages of the two nodes i and j. Figure 11.2 shows the flowchart of generator restoration in power system. • Initial state: Based on the AC power flow model, calculating power flow of power system normal operation. • Faulted power system: It is assumed that the normal operating system is suddenly disturbed, that is, the electrical load suffers a fault and power system is in an unbalanced state at this time. • Select action: Based on Q-learning algorithm, we assume that the active power output of the generators other than the slack generator is adjusted each time, and the active power output of each fine-tuning is as action. • Power flow calculation: Every time agent selects an action, the active power output of the generator units changes subsequently. After recalculating the power flow, the reward function is updated, and observing the changes of generator units.

11.3 Restoration Scheme

195

Fig. 11.2 The flowchart of generators restoration in power system

• Whether obtaining the best recovery states: According to each update action, after recalculating the power flow, whether the corresponding average reward value is the largest, the active power output of generator units is the best state, and it is considered to be the best recovery state. • Steady state: If the average reward is the maximum, and the active power of generator units reaches the optimal. Power system at this time will be restored to what we expect, and optimal Q-table will be obtained.

11.3 Restoration Scheme Reinforcement learning mainly includes the agent and environment, which can interact with each other and be modeled as a Markov model. The environment includes action space A and state space S, state transition rate P(st+1 |st , at ) and reward function R. The agent chooses an action a in the current state s, and this action can be determined according to -greedy strategy. After executing this action, the environment will update states and return a reward to the agent. The agent will continue to change the network to update the Q-table until the discount reward is maximized [19]. Figure 11.3 presents the restoration of generators based on Q-learning algorithm. In this model, the agent refers to power system operator making a restoration decision. The environment is the faulted power system. Action space refers to increasing or decreasing the active power output of the other two generators in addition to the slack generator. State space means that every time the active power output of the

196

11 Reinforcement Learning Approach to System Recovery

Fig. 11.3 The restoration of generators basd on Q-learning algoithm

generators changes, power flow is calculated again. Observation space contains the active power of generator units, node voltage, load power and line power. The reward function directly affects the decision-making behavior of the agent, so the recovery objective function and constraints are considered when designing the reward function. During the recovery period, if the state corresponding to the agent’s decision reaches the set state, it will get a positive reward; if the agent’s decision violates the relevant constraints, such as the upper and lower limits of the active power of the generator units and the upper and lower limits of the node voltage, the negative penalty will be given respectively. If the average reward reaches the biggest value, and the active power output value of the generator units reaches the optimal combination. At this time, the generator units restore to the best state. The reward function is as follows: ⎧ c0 + c1 P1 + c2 P2 if the state corresponding to ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ the agent’s decision reaches the set state. (11.5) R = α if active output power of generator units ⎪ ⎪ ⎪ beyond the limit. ⎪ ⎪ ⎪ ⎩ β if the node voltage is beyond the limit. P1 refers to the active power of generator 1, and P2 refers to the active power of generator 2. c0 , c1 andc2 refers to coefficient. Usually, we define discounted cumulative rewards in RL: n  γ t−1 rt (st , at ) R= (11.6) t=1

where rt (st , at ) refers to the tth, the immediate reward obtained by taking an action a in the state s. And γ is a discount factor. If γ = 0, the discount cumulative reward only considers the current reward. If γ = 1, the discount cumulative reward will consider the long-term reward.

11.4 Numerical Results

197

at∗ = argmaxai ∈A Q (st , at )

(11.7)

Q (st+1 , at+1 ) =(1 − α)Q (st , at ) + α {rt+1 Q (st , at ) +γ max Q (st+1 , at )

(11.8)

a

where α refers to learning rate and 0 < α < 1. If α = 0, the agent can learn prior experience influences subsequent action decisions. While if α = 1, the agent only focuses on the immediate reward and ignores the information learned from the prior trial experience. The agent constantly keeps trial and error to make decisions. According to the obtained state-action pair to update the Q value [20]. During training, the agent may not necessarily take the optimal action to maximize the discounted reward function. When the agent takes an action with the greedy strategy, it finds a balance between exploration and exploitation [21]. However, exploration and exploitation are a pair of contradictory relationships. Exploration is to find out more information about the environment, but some interest may be sacrificed, while exploitation is using current known information to maximize the discount reward. In reinforcement learning, it is important to balance the relationship between exploration and exploitation, so generally using the —greedy strategy. And it can avoid local optimal problems. By setting a small value of , the probability 1- is used greedily to select the optimal action a ∗ , while the probability  used randomly to select the action from the available actions [22]. The -greedy strategy is defined:

π (a|st ) =

⎧ ⎪ ⎨1 −  + ⎪ ⎩

 , if a = a ∗ . A (st )  , if a = a ∗ . A (st )

(11.9)

Appropriate values of  can balance the relationships between exploration and exploitation. The -greedy strategy guarantees the effectiveness of Q-learning algorithm and avoids falling into the local optimum problem.

11.4 Numerical Results In this section, numerical simulations are conducted to validate the proposed algorithm on the IEEE-9 bus system. And the IEEE-9 bus system in Fig. 11.4 is used for testing the Q-learning algorithm. In IEEE-9 bus system, there are 9 buses, 3 generators and 3 loads. All the voltage of generators are controllable and all the active power flow are controllable except the slack generator. In Fig. 11.5, the yellow generator unit is a slack generator, the red generators are generator units 1 and 2. Assume that power system is in the state of outage, the fault happened on load 5. It complies with N-1 contingency. Here, the learning rate is set to 0.1, reward decay is

198

11 Reinforcement Learning Approach to System Recovery

Fig. 11.4 IEEE-9 bus system

0.99 and  is 0.99, c1 = 0.01, c2 =0.02, c3 = 0.03. The simulations are run on a Core-i5 computer, with a 3.10-GHz processor and 8GB RAM, and the agent is programmed with Python. Considering the randomness and uncertainty of load disturbance, we will discuss it in two cases, case A: static load and case B: dynamic load.

11.4.1 Static Load Suppose the given disturbance is fixed, Figure a shows the curves of average reward with different negative reward values of the static load. The x-axis is iteration and the y-axis is the average reward, we replace the average reward with η. The red line represents the situation of α = -200, β = −50, the black line denotes the situation of α = -300, β = −100, and the blue line is the situation of α = −200, β = −50. According to Fig. 11.5a, during the third iteration of the red line, the average reward value gradually stabilizes. However, the blue and black lines are different, the average reward attains the max value with the second iteration. In Fig. 11.5b, the upper picture shows the active power of generation unit 1 and the lower picture shows the active power of generation unit 2. The X-axis is iteration and the y-axis is the active power output of generator units 1 and 2 respectively. The active power output of generator 1 and generator 2 are 165.03 MW and 85.82 MW respectively in the third iteration. And the active power output of generator 1 and generator 2 are 163.08 MW, 167.10 MW and 85.32 MW, 86.18 MW respectively in the second iteration. All of them attain a similar max reward value.

11.4 Numerical Results

199

Fig. 11.5 Average reward and active power output of generator units under static load

11.4.2 Dynamic Load Assume the load is subjected to different levels of disturbance. Figure 11.6a shows the curves of average reward with different negative reward values of dynamic load. In Fig a, the red line is the situation of α = −200, β = −100, the blue line is the situation of α = −300, β = −100 and the green line is the situation of α = -200, β = −50. The X-axis is disturbance strength, the y-axis is the average reward. We can see when the load is under ± 10% random fluctuations with α = −200, β = -100 and α = −300, β = −100 is max. The average reward is −30.13 and −53.65 respectively. Although the load is in ± 40% with α = −200, β = −50, which its value is −49.67. As the random fluctuations in load increase, the average reward is in a gradual decrease trend. In Fig. 11.6b, the upper and lower picture shows the active power of generation unit 1 and 2 respectively. The X-axis is disturbance strength, the

200

11 Reinforcement Learning Approach to System Recovery

Fig. 11.6 Average reward and active power output of generator units under dynamic load

y-axis is the active power of generator units. We can see the load is in ± 10%, the active power of generator unit 1 is 170.92 MW, 170.36 MW. And the active power of generator unit 2 is 76.691 MW, 76.16 MW. When the load is in ± 40%, the active power of generator units 1 and 2 is 163.95 MW, and 81.37 MW respectively. Comparing Fig. 11.5 with Fig. 11.6, we can find the average reward is different under static load and dynamic load. Under static load, the average reward value reached the maximum and tended to a stable state. While the average reward value varies with the degree of disturbance to the load, the smaller the disturbance to the load, the larger the average reward value. However, the active power output of generator units corresponding to the maximum value of the average reward value is the best combination.

References

201

11.5 Conclusion The rapid restoration of the generator units is important for the fast, reliable, and safe operation of power system. In this chapter, a reinforcement learning algorithm was proposed to restore the generator units. The agent learns from interaction with power system and makes a time-saving exploration in the case of insufficient knowledge. The reinforcement learning algorithm enabled us to find the optimal output power of the generator units corresponding to the maximum reward even under offline conditions. By setting different negative reward values, it is observed that the reward value can reach the maximum value under different iterations, and we can get different combinations of generator units. However, this method does not apply to the more complex power systems, because more action-state pairs result in dimension explosion. Future efforts will be devoted to adding the neural network to the proposed algorithm to realize the recovery of complex power systems, considering dimension explosion caused by large action-state space.

References 1. Panteli, M., Mancarella, P.: Influence of extreme weather and climate change on the resilience of power systems: Impacts and possible mitigation strategies. Electr. Power Syst. Res. 127, 259–270 (2015) 2. Wang, Y., Chen, C., Wang, J., Baldick, R.: Research on resilience of power systems under natural disasters: a review. IEEE Trans. Power Sys. 31(2), 1604–1613 (2016) 3. Lee, R., Assante, M.: Analysis of the cyber attack on the Ukrainian power grid. SANS Ind. Control Syst. 388, 1–29 (2016) 4. Busby, J., Baker, k., et al.: Cascading risks: Understanding the 2021 winter blackout in Texas. Energy Res. Soc. Sci. 77, 1–10 (2021) 5. Vaiman, M., Bell, K., Chen, Y., et al.: Risk assessment of cascading outages: methodologies and challenges. IEEE Trans. Power Syst. 27(2), 631–641 (2012) 6. Zhai, C., Nguyen, H.D., Xiao, G.: A robust optimization approach for protecting power systems against cascading blackouts. Electr. Power Syst. Res. 189, 106794 (2020) 7. Zhai, C., Zhang, H., Xiao, G., Pan, T.C.: An optimal control approach to identify the worst-case cascading failures in power systems. IEEE Trans. Control Netw. Syst. 7(2), 956–966 (2020) 8. Patsakis, G., Rajan, D., Aravena, I., et al.: Optimal black start allocation for power system restoration. IEEE Trans. Power Syst. vpl. 33(6), 6766–6776 (2018) 9. Rao, R., Narasimham, S., Raju, R., et al.: Optimal network reconfiguration of large-scale distribution system using harmony search algorithm. IEEE Trans. Power Syst. 26(3), 1080– 1088 (2011) 10. Granelli, G., Montagna, G., Zanellini, F., et al.: Optimal network reconfiguration for congestion management by deterministic and genetic algorithms. Electr. Power Syst. Res. 76, 549–556 (2006) 11. Xu, Y., Liu, C., Schneider, K., et al.: Microgrids for service restoration to critical load in a resilient distribution system. IEEE Trans. Smart Grid 9(1), 426–437 (2018) 12. Sun, W., Liu, C., Zhang, L.: Optimal generator start-up strategy for bulk power system restoration. IEEE Trans. Power Syst. 26(3), 1357–1366 (2011) 13. Sun, R., Liu, Y.: An on-line generator start-up strategy based on deep learning and tree search. In: 2018 IEEE Power and Energy Society General Meeting (2018)

202

11 Reinforcement Learning Approach to System Recovery

14. Zhang, J., Wang, D.,Xu, Q., et al.: Greedy algorithm for generator start-up sequence optimization in power system restoration considering transmission path. In: 2019: IEEE PES Innovative Smart Grid Technologies Europe (2019) 15. Sun, R., Zhu, H., Liu, Y.: A r-NSGA-II algorithm based generator start-up for network reconfiguration. In: 2015 5th International Conference on Electric Utility Deregulation and Restructuring and Power Technologies (2016) 16. Sun, L., Lin, Z., Xu, Y., et al.: Optimal skeleton-network restoration considering generator start-up sequence and load pickup. IEEE Trans. Smart Grid 10(3), 3174–3185 (2019) 17. Sutton, R., Barto, A.: Reinforcement Learning: An Introduction. Cambridge MIT Press (1998) 18. Ghorbani, J., Choudhry, M., Feliachi, A.: A MAS learning framework for power distribution system restoration. In: IEEE PES T and D Conference and Exposition (2014) 19. Ernst, D., Glavic, M., Wehenkel, L.: Power systems stability control: reinforcement learning framework. IEEE Trans. Power Syst. 19(1), 427–435 (2004) 20. Wang, Y., Li, T., Lin, C.: Backward q-learning: the combination of sarsa algorithm and qlearning. Eng. Appl. Artif. Intell. 26(9), 2184–2193 (2013) 21. Wu, J., Fang, B., Fang, J., Chen, X., Tse, C.: Sequential topology recovery of complex power systems based on reinforcement learning. Stat. Mech. Appl. Phys. (2019) 22. Gomes, R., Kowalczyk, R.: Modelling the dynamics of multiagent q-learning with -greedy exploration. In: Proceedings of the 8th International Conference on Autonomous Agents and Multiagent System (2009) 23. Zhen, H., Zhai, H., Ma, W. et al.: Design and tests of reinforcement-learning-based optimal power flow solution generator. Energy Rep. 43–50 (2022)

Chapter 12

Summary and Future Work

Abstract This chapter gives a brief summary of this book based on the evolution process of complex cascade systems. The relevant control and optimization methods are outlined at each stage. Moreover, some promising research topics are discussed for future work in the field of complex system resilience from the perspective of control and optimization, which include the systematic and rigorous framework, data-driven control and optimization method, distributed design, and validation of cooperative control and optimization strategies, and resilience modeling and control with uncertainties and incomplete information.

12.1 Summary of the Book This book aims to present a systematic framework for the resilience improvement of complex systems. A series of control and optimization methods have been developed to heighten the system at different evolution phases, including prior to disruptive disturbances, the decline of system performance and system restoration. Before disruptive disturbances, it is crucial to identify the potential risks and explore the region of stability for the complex dynamical system. As a result, an optimal control approach is developed to identify the triggering events that may cause the worst-case cascading failures of concerned complex systems, by regarding the initial disturbances or events as control inputs. This allows to investigate the problem of risk identification in the framework of optimal control theory. Moreover, the proposed optimal control approach can be employed to deal with risk identification problem for dynamical system with protective actions. Further, the JFNK-based optimization method has been adopted to deal with the issues caused by the evolution of system parameters and the complicated coupling relationship of different components in complex cascade systems. In order to quantify the stability margin of dynamical systems, it is necessary to estimate the region of attraction around the equilibrium of concerned systems, and an approach based on Gaussian process and converse Lyapunov function has been proposed to determine the stable domain with certain conference level, by leveraging the stable state trajectories of concerned dynamical system. It is worth pointing out that the proposed approach is applied to DAE systems, which can be © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Zhai, Control and Optimization Methods for Complex System Resilience, Studies in Systems, Decision and Control 478, https://doi.org/10.1007/978-981-99-3053-1_12

203

204

12 Summary and Future Work

used to describe dynamics of power systems. In the stages of system protection and restoration, a model predictive approach is proposed to preplan protective actions against the propagation of cascading failures by predicting the cascading failure paths using the model of the dynamical system. With the assistance of cooperative control, the FACTS devices can play a key role in relieving the stress of the cyber-physical power system, and the methods based on robust optimization and distributed optimization are also adopted to enhance the robustness and resilience of concerned system, respectively. Finally, a reinforcement learning approach has been adopted to restore the generation units with uncertain disturbances.

12.2 Future Directions Given the remarkable progress of complex system resilience, the control and optimization methods have been elaborated in this book according to the evolution process of dynamical systems. In what follows, the potential research topics are discussed for future studies. • Systematic and Rigorous Framework for Complex System Resilience Most existing studies focus on the specific methodologies to deal with issues at different phases of complex dynamical systems, and there is lack of a systematic framework that can be employed to analyze and control the resilience of complex systems. It may be feasible to investigate this research topic in the framework of control theory, and the state-space approach can be applied to the analysis of dynamical behaviors of concerned complex systems [1, 2]. The disturbances and attacks at different evolution stages can be regarded as the inputs of complex systems, and it is expected to propose a systematic control strategy to identify the vulnerable components, absorb the disruptive effects, adapt to intrinsic and external changes and restore the functionality in a coordination manner, which contributes to the resilience improvement and technical analysis of concerned systems. In the absence of such systematic framework, the control of the dynamical behaviors of concerned systems may be interdependent and even counteractive. • Data-Driven Control and Optimization Methods for Resilience Enhancement Data-driven control and optimization methods have been applied to various research fields such as movement science [3, 4], industrial production [5], power and energy systems [6, 7], and so on. For complex systems, it is difficult to accurately characterize the behaviors of concerned system due to the complicated intrinsic dynamics and coupling relationship as well as uncertain disturbances. Thus, massive real data from the practical systems can play a key role in the modeling and analysis of complex real-world systems, and it may help to uncover the unknown dynamics that is unable to describe with analytical methods. This inevitably requires the development of technical tools for data extraction, data analysis and data interpretation, which can effectually complement the analytic model of complex systems, thereby leading to achieving more desirable control and optimization performance.

References

205

• Distributed Design and Validation of Cooperative Control and Optimization Methods for System Resilience Compared to centralized control or optimization methods, distributed strategies have significant advantages in maintaining the robustness, stability, and flexibility of complex systems [8, 9]. This is because the control ability still survives even if some local control centers are damaged due to external attacks or other factors. Moreover, the property of self-organization enables distributed control and optimization methods to adapt to sharp changes in external environments and thus protect complex systems against catastrophic consequences. Thus, the communication protocol among multiple decision centers is of great significance to the design of distributed algorithms. This research direction is closely related to the systematic framework for complex resilience, and the design of distributed control and optimization strategies should be taken into account in a systematic manner and integrated with other protective actions. • Resilience Modelling and Control of Complex Systems with Uncertainties and Incomplete Information In order to create a resilient system, it is necessary to investigate the dynamic characteristics of concerned real-world systems. Nevertheless, the uncertainties or incomplete information in complex systems exist universally in practice [10]. Thus, it is a challenging task to handle uncertainties or incomplete information while ensuring the control performance of complex systems. As one potential solution to the above issue, bifurcation theory may provide a powerful theoretical framework for the resilience modeling of concerned uncertain systems by regulating the equilibrium points of dynamical systems. Here, the core difficulty lies in the quantification of uncertainties and incomplete information using the strict mathematical tool.

References 1. Barabasi, A.: Barzel, et al, Universal resilience patterns in complex networks. Nature 530, 307 (2016) 2. Gama Dessavre, D., Ramirez-Marquez, J., Barker, K.: Multidimensional approach to complex system resilience analysis. Reliab. Eng. Syst. Saf. 149, 34–43 (2016) 3. Zhai, C., Alderisio, F., Slowinski, P., Tsaneva-Atanasova, K., di Bernardo, M.: Design and validation of a virtual player for studying interpersonal coordination in the mirror game. IEEE Trans. Cybern. 48(3), 1018–1029 (2018) 4. Zhai, C., He, Y., Zhang, C.: Design and validation of feedback controller for social motor coordination with time-varying delays. Control Eng. Pract. 109, 104756 (2021) 5. Dai, W., Chai, T., Yang, S.: Data-driven optimization control for safety operation of hematite grinding process. IEEE Trans. Ind. Electron. 62(5), 2930–2941 (2015) 6. Ashuri, T., Li, Y., Hosseini, S.: Recovery of energy losses using an online data-driven optimization technique. Energy Convers. Manage. 225, 113339 (2020) 7. Zhai, C., Nguyen, H.: Estimating the region of attraction for power systems using Gaussian process and converse Lyapunov function. IEEE Trans. Control Syst. Technol. 30(3), 1328–1335 (2022)

206

12 Summary and Future Work

8. Hong, Y., Zhai, C.: Dynamic coordination and distributed control design of multi-agent systems. Control Theor. Appl. 28(10), 1506–1512 (2011) 9. Yang, P., Freeman, R., Lynch, K.: Multi-agent coordination by decentralized estimation and control. IEEE Trans. Autom. Control 53(11), 2480–2496 (2008) 10. Moutsinas, G., Zou, M., Guo, W.: Uncertainty of resilience in complex networks with nonlinear dynamics. IEEE Syst. J. 15(3), 4687–4695 (2021)