This book focuses on the comprehensive prevention and control methods for short-circuit faults in power systems. Based o

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*Table of contents : PrefaceContents1 Risk Evaluation of Short-Circuit Fault in Power System 1.1 Descriptions of Power Systems and Their Risk Issues 1.2 Evaluation Techniques of Short-Circuit Fault Probability 1.2.1 Data-Driven Techniques 1.2.2 Analytical Techniques 1.3 Evaluation Techniques of Short-Circuit Fault Consequences 1.4 Challenges of Short-Circuit Risk Prevention and Control 1.5 Organization of This Book for Resilience Enhancement in Power System Short-Circuit Faults References2 Risk-Based Optimal Configuration of Fault Current Limiter in Power System 2.1 Introduction 2.2 Modeling of FCL and SCC Calculation 2.2.1 Dynamic Response Modeling of FCL 2.2.2 Modeling of SCC Calculation 2.3 Many-Objective FCL Configuration Modeling 2.3.1 Decision Variables 2.3.2 Objective Function 2.4 Solution Method 2.4.1 Coding Strategy 2.4.2 NSGA-III 2.5 Case Studies 2.5.1 Test System and Parameters 2.5.2 Pareto Solutions Analysis 2.5.3 Comparison Between the Proposed Risk-Based Method and the Traditional Deterministic Method 2.5.4 Solution Performance Analysis 2.6 Conclusion References3 5G-Based Optimal Configuration of Centralized Fault Current Limiter in Power System 3.1 Introduction 3.2 Framework of the 5G-Based CSF for FCLs 3.2.1 Applying 5G for CSF of FCLs 3.2.2 Backup Strategy in Case of 5G Communication Failure 3.3 Formulation of the FD Model 3.3.1 Fault SCC Magnitude Constraint 3.3.2 Voltage Sag Constraint 3.3.3 Offline Fault Scanning Scheme for Online Decision 3.4 Description of the FCL Allocation Approach 3.4.1 Description of the FCL Allocation Approach 3.4.2 Formulation of the Bi-level FCL Allocation Model 3.4.3 Solution Method 3.4.4 Case Study 3.5 Conclusion References4 A Multi-state Model for Power System Resilience Enhancement Against Short-Circuit Faults 4.1 Introduction 4.2 Extreme Weather Event Response Schema 4.2.1 Short-Circuit Current Limiting 4.2.2 Transient Stability Maintaining 4.2.3 Formulation of the MINLP in the Proposed Schema 4.3 Multi-state Modeling of Transmission System Resilience Enhancement 4.3.1 Multi-state Resilience Enhancement Against SCFs 4.3.2 State Generating Model 4.3.3 Traversal Procedure 4.4 Solution Method 4.4.1 Problem Reformulation 4.4.2 Searching Space Reduction 4.4.3 Heuristics Solution 4.5 Case Study 4.5.1 Scenario Generating 4.5.2 Sensitivity Analysis 4.5.3 Suggested Resilience Enhancement Scheme 4.5.4 Comparison Studies 4.6 Conclusions References5 Voltage Violations Assessment Considering Reactive Power Compensation Provided by Smart Inverters 5.1 Introduction 5.2 Basic Models 5.2.1 Reactive Power Compensation Mechanism of Smart Inverters 5.2.2 Models of On-Load Tap Changers and Switching Capacitors 5.2.3 Comparisons of Different Compensation Strategies 5.2.4 Voltage Deviation Indexes 5.3 Simulation Methods for Voltage Violation Assessment 5.3.1 Kernel Density Estimation of Electric Vehicle Loads and Photovoltaics Outputs 5.3.2 Slice Sampling for Voltage Violation Assessment 5.3.3 Automated Step Width Selection for Slice Sampling 5.4 Risk Assessment for Voltage Violation 5.4.1 Voltage Regulation Modes 5.4.2 Voltage Regulation Constraints 5.4.3 Risk Assessment Process 5.5 Case Studies 5.5.1 Kernel Density Estimation Results 5.5.2 Performance of Slice Sampling 5.5.3 Sensitivity Analysis 5.6 Conclusion References6 A Distributed MPC to Exploit Reactive Power V2G for Real-time Voltage Regulation of Post-fault Power Systems 6.1 Introduction 6.2 Basic Models of EV Chargers and the Grid 6.2.1 Operation Range of EV Chargers 6.2.2 Modeling of EV Chargers 6.2.3 Modeling of the Grid 6.3 The Framework of Proposed DMPC 6.4 DMPC Formulation 6.4.1 Prediction Models 6.4.2 Models Constraints 6.4.3 Objective Function 6.4.4 Mechanisms for DMPC 6.4.5 Pre-calculation and Communication Process 6.5 Case Study 6.5.1 Test System Description for Balanced DNs 6.5.2 Case 1: Effectiveness of DMPC Under Balanced DNs 6.5.3 Case 2: Impact of Communication Latency in Balanced DNs 6.5.4 Case 3: Effectiveness of DMPC Under Unbalanced DNs 6.6 Conclusion References7 A Stochastic Unit Commitment to Enhance Frequency Security of Post-fault Power Systems 7.1 Introduction 7.2 The Proposed Integration-Based Frequency Security Criterion 7.2.1 Frequency Deviation During the PFR Process 7.2.2 Frequency Deviation During the SFR Process 7.3 Stochastic Frequency Security-Constrained Unit Commitment Model 7.3.1 First-Stage Problem: Coordination of Cost and Risk 7.3.2 Second Stage Problem: Risk Assessment 7.4 Solution Methodology 7.4.1 Problem Linearization 7.4.2 Regularized L-shape Algorithm 7.5 Case Study 7.5.1 Parameter Setting 7.5.2 Reserve Dispatch Under Different Security Requirements 7.5.3 Comparative Analysis Between the Proposed Frequency Security Criterion and the Conventional Criteria 7.5.4 Validation of the Proposed Frequency Security Criterion 7.5.5 Impacts of Operation Life and Hurricane Intensities on Operation Cost 7.6 Conclusion References8 A Data-Driven Reserve Allocation Method with Frequency Security Constraint of Post-fault Power System Considering Inverter Air Conditioners 8.1 Introduction 8.2 Modeling of Power System Frequency Response Integrated with IACs 8.2.1 Power System Frequency Response Model 8.2.2 Equivalent Frequency Response Model of IACs 8.2.3 Power System Frequency Response Model with Aggregated IACs 8.3 Data-Driven Reserve Allocation with the Frequency Security Constraint 8.3.1 Problem Description 8.3.2 Data-Driven Approximation of Frequency Security Constraint 8.3.3 Suggest-and-Improve Method for QCQP 8.4 Case Studies 8.4.1 Aggregation of IACs 8.4.2 Reserve Allocation Results 8.5 Conclusion References9 Iterative Online Fault Identification Scheme for High Voltage Circuit Breaker 9.1 Introduction 9.2 HVCB Condition Monitoring System 9.2.1 Current in the Coil 9.2.2 Vibration Signal 9.2.3 Framework of the HVCB Condition Monitoring System 9.3 Missing Data Repair Method 9.3.1 KNN-Based Clustering 9.3.2 ELM for Data Estimation 9.3.3 K-D Tree-Based Fast Scanning Technique 9.4 Softmax Classifier for HVCB Status Identification 9.5 Procedure of Iterative HVCB Diagnosis Utilizing Repaired Data 9.6 Realistic Case Studies 9.6.1 Case Description 9.6.2 Accuracy Validation 9.6.3 Searching Efficiency Validation 9.7 Conclusion References*

Chengjin Ye Chao Guo Yi Ding

Risk-Based Planning and Operation Strategy Towards Short Circuit Resilient Power Systems

Risk-Based Planning and Operation Strategy Towards Short Circuit Resilient Power Systems

Chengjin Ye · Chao Guo · Yi Ding

Risk-Based Planning and Operation Strategy Towards Short Circuit Resilient Power Systems

Chengjin Ye Zhejiang University Hangzhou, Zhejiang, China

Chao Guo Zhejiang University City College Hangzhou, Zhejiang, China

Yi Ding Zhejiang University Hangzhou, Zhejiang, China

ISBN 978-981-19-9724-2 ISBN 978-981-19-9725-9 (eBook) https://doi.org/10.1007/978-981-19-9725-9 © 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

Preface

The authors shared their work in writing this book. It was a pleasure with Springer Associate Editor. Hangzhou, China

Chengjin Ye Chao Guo Yi Ding

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Contents

1 Risk Evaluation of Short-Circuit Fault in Power System . . . . . . . . . . . 1.1 Descriptions of Power Systems and Their Risk Issues . . . . . . . . . . . . 1.2 Evaluation Techniques of Short-Circuit Fault Probability . . . . . . . . . 1.2.1 Data-Driven Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Analytical Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Evaluation Techniques of Short-Circuit Fault Consequences . . . . . . 1.4 Challenges of Short-Circuit Risk Prevention and Control . . . . . . . . . 1.5 Organization of This Book for Resilience Enhancement in Power System Short-Circuit Faults . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Risk-Based Optimal Configuration of Fault Current Limiter in Power System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Modeling of FCL and SCC Calculation . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Dynamic Response Modeling of FCL . . . . . . . . . . . . . . . . . . . 2.2.2 Modeling of SCC Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Many-Objective FCL Configuration Modeling . . . . . . . . . . . . . . . . . . 2.3.1 Decision Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Coding Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 NSGA-III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Test System and Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Pareto Solutions Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Comparison Between the Proposed Risk-Based Method and the Traditional Deterministic Method . . . . . . . . 2.5.4 Solution Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 5G-Based Optimal Configuration of Centralized Fault Current Limiter in Power System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Framework of the 5G-Based CSF for FCLs . . . . . . . . . . . . . . . . . . . . . 3.2.1 Applying 5G for CSF of FCLs . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Backup Strategy in Case of 5G Communication Failure . . . . 3.3 Formulation of the FD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Fault SCC Magnitude Constraint . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Voltage Sag Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Offline Fault Scanning Scheme for Online Decision . . . . . . . 3.4 Description of the FCL Allocation Approach . . . . . . . . . . . . . . . . . . . 3.4.1 Description of the FCL Allocation Approach . . . . . . . . . . . . . 3.4.2 Formulation of the Bi-level FCL Allocation Model . . . . . . . . 3.4.3 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 A Multi-state Model for Power System Resilience Enhancement Against Short-Circuit Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Extreme Weather Event Response Schema . . . . . . . . . . . . . . . . . . . . . 4.2.1 Short-Circuit Current Limiting . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Transient Stability Maintaining . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Formulation of the MINLP in the Proposed Schema . . . . . . . 4.3 Multi-state Modeling of Transmission System Resilience Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Multi-state Resilience Enhancement Against SCFs . . . . . . . . 4.3.2 State Generating Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Traversal Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Problem Reformulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Searching Space Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Heuristics Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Scenario Generating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Suggested Resilience Enhancement Scheme . . . . . . . . . . . . . 4.5.4 Comparison Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Voltage Violations Assessment Considering Reactive Power Compensation Provided by Smart Inverters . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Basic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Reactive Power Compensation Mechanism of Smart Inverters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Models of On-Load Tap Changers and Switching Capacitors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Comparisons of Different Compensation Strategies . . . . . . . 5.2.4 Voltage Deviation Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Simulation Methods for Voltage Violation Assessment . . . . . . . . . . . 5.3.1 Kernel Density Estimation of Electric Vehicle Loads and Photovoltaics Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Slice Sampling for Voltage Violation Assessment . . . . . . . . . 5.3.3 Automated Step Width Selection for Slice Sampling . . . . . . 5.4 Risk Assessment for Voltage Violation . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Voltage Regulation Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Voltage Regulation Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Risk Assessment Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Kernel Density Estimation Results . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Performance of Slice Sampling . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 A Distributed MPC to Exploit Reactive Power V2G for Real-time Voltage Regulation of Post-fault Power Systems . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Basic Models of EV Chargers and the Grid . . . . . . . . . . . . . . . . . . . . . 6.2.1 Operation Range of EV Chargers . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Modeling of EV Chargers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Modeling of the Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Framework of Proposed DMPC . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 DMPC Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Prediction Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Models Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Mechanisms for DMPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.5 Pre-calculation and Communication Process . . . . . . . . . . . . . 6.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Test System Description for Balanced DNs . . . . . . . . . . . . . . 6.5.2 Case 1: Effectiveness of DMPC Under Balanced DNs . . . . . 6.5.3 Case 2: Impact of Communication Latency in Balanced DNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.5.4 Case 3: Effectiveness of DMPC Under Unbalanced DNs . . . 117 6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7 A Stochastic Unit Commitment to Enhance Frequency Security of Post-fault Power Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Proposed Integration-Based Frequency Security Criterion . . . . 7.2.1 Frequency Deviation During the PFR Process . . . . . . . . . . . . 7.2.2 Frequency Deviation During the SFR Process . . . . . . . . . . . . 7.3 Stochastic Frequency Security-Constrained Unit Commitment Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 First-Stage Problem: Coordination of Cost and Risk . . . . . . . 7.3.2 Second Stage Problem: Risk Assessment . . . . . . . . . . . . . . . . 7.4 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Problem Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Regularized L-shape Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Parameter Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Reserve Dispatch Under Different Security Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Comparative Analysis Between the Proposed Frequency Security Criterion and the Conventional Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.4 Validation of the Proposed Frequency Security Criterion . . . 7.5.5 Impacts of Operation Life and Hurricane Intensities on Operation Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 A Data-Driven Reserve Allocation Method with Frequency Security Constraint of Post-fault Power System Considering Inverter Air Conditioners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Modeling of Power System Frequency Response Integrated with IACs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Power System Frequency Response Model . . . . . . . . . . . . . . 8.2.2 Equivalent Frequency Response Model of IACs . . . . . . . . . . 8.2.3 Power System Frequency Response Model with Aggregated IACs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Data-Driven Reserve Allocation with the Frequency Security Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Data-Driven Approximation of Frequency Security Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Suggest-and-Improve Method for QCQP . . . . . . . . . . . . . . . .

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Contents

8.4 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Aggregation of IACs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Reserve Allocation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Iterative Online Fault Identification Scheme for High Voltage Circuit Breaker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 HVCB Condition Monitoring System . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Current in the Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Vibration Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Framework of the HVCB Condition Monitoring System . . . 9.3 Missing Data Repair Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 KNN-Based Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 ELM for Data Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 K-D Tree-Based Fast Scanning Technique . . . . . . . . . . . . . . . 9.4 Softmax Classifier for HVCB Status Identification . . . . . . . . . . . . . . 9.5 Procedure of Iterative HVCB Diagnosis Utilizing Repaired Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Realistic Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Case Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2 Accuracy Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.3 Searching Efficiency Validation . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

156 158 159 162 162 165 165 166 166 167 169 170 171 171 173 174 176 178 178 178 181 182 182

Chapter 1

Risk Evaluation of Short-Circuit Fault in Power System

1.1 Descriptions of Power Systems and Their Risk Issues Globally, on the one hand, with the expansion of the scale of the power system, the level of short-circuit current gradually increases. On the other hand, with the frequent occurrence of extreme weather, short-circuit faults occur more frequently. Taking China as an example, the power supply and load demand have obvious inconsistencies in spatial distribution. Constrained by resource endowments, the vast majority of China’s coal, the hydro, wind, and solar resources are distributed in the western, southwestern, and northern regions. However, more than 70% of the energy demand is concentrated in the east-central region. The supply and demand centers of electricity are geographically thousands of kilometers apart. To ensure the efficient transmission and consumption of large energy bases, China has planned and built a large number of AC and DC ultra-high voltage transmission and transformation projects [1] and formed a “three vertical and three horizontal” ultra-high voltage backbone network. From a global perspective, driven by the demand for a wide area allocation of electric energy, several transnational power grids have been developed [2, 3]. For example, the U.S. and Canada power grids, the European power grid and the Russian-Baltic power grid, etc. In a significant speech delivered on September 22, 2020, at the 75th session of the United Nations General Assembly, General Secretary Xi Jinping noted that China would increase its national contribution, adopt more aggressive policies and measures, work to reach its peak CO2 emissions by 2030, and work to achieve carbon neutrality by 2060 [4]. In the context of “carbon peaking and carbon neutral”, the inverse distribution of resource endowment and energy demand in China determines that a “large power supply and huge grid” is still the inevitable trend of modern power grid development. It is well known that excessive short-circuit current is one of the prominent problems of large grid operation. Short circuits are the most common type of fault in power systems and can be caused by insulation aging, lightning flashover, and bird and animal cross-connection. In recent years, various extreme weather events,

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Ye et al., Risk-Based Planning and Operation Strategy Towards Short Circuit Resilient Power Systems, https://doi.org/10.1007/978-981-19-9725-9_1

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including heavy rainfall, hurricanes, thunderstorms, and floods, are striking the world in an increasingly frequent and destructive manner [5]. Extreme weather hazards can exacerbate the probability and serious consequences of short-circuit faults in power systems, which is bound to bring great challenges to the safe and stable operation of power systems. For example, the major blackout that occurred in Brazil on November 10, 2009 [6] was mainly caused by a thunderstorm that triggered short-circuit faults in several lines of the power system one after another, resulting in a voltage collapse in the southeastern grid of Brazil, especially in the São Paulo area. The high-voltage DC transmission system of the Itaipu hydroelectric plant was bipolar blocked due to the activation of the minimum DC voltage protection on the inverter side. At the same time, the national interconnection system of Brazil was disconnected from Paraguay’s 50 Hz AC grid. Ultimately, a massive blackout occurred in Brazil, resulting in a load loss of 24.436 GW, or approximately 40% of Brazil’s total load. Similarly, when Super Typhoon Morathi struck Fujian on September 15, 2016, the mechanical load on line equipment in the horizontal direction against the wind increased significantly. As a result, there were up to 2830 short-circuit trips on lines above 10 kV, which increased the financial losses for power-using businesses and the pressure on grid companies to respond to emergencies and disasters. Numerous severe power outages both domestically and internationally have demonstrated that China’s strategic energy security would be constrained by its inability to efficiently address short-circuit faults brought on by significant meteorological disasters. At present, China’s power grid has entered the post-development phase of the “new normal”, and the average annual growth rate of electricity consumption during the 13th Five-Year Plan period has dropped from 8.8% in the 12th Five-Year Plan to 3.6–4.8%. In the context of the slowdown, the concept of precise investment in power grids has received increasing attention. Since the release of “No. 9”, the rapid advancement of the power market construction further requires the power system to change the original relatively sloppy development mode and improve the operation economy [7]. Considering that the existing grid current-limiting measures are based on deterministic safety criteria without risk-awareness [8], and the short-circuit faults of power systems in the context of extreme meteorological disasters are mostly episodic in nature. If the existing decision method focuses only on the consequences of faults and ignores the probability of faults, it will easily lead to overly adventurous current-limiting schemes and affect the overall economy of grid planning and operation [8]. Therefore, the emphasis on accident risk is the overall current trend of the power system. The North American Electric Reliability Council (NERC) assists grid dispatch through risk assessment, empirical learning, and event root cause analysis. The PJM grid in the United States has introduced risk management methods into system and market operations [9]. In China, risk-based power system planning and operation has received increasingly widespread attention, e.g., the Regulations on Emergency Response and Investigation of Electricity Safety Accidents (Decree 599 of the State Council) has also directly proposed requirements for power system

1.2 Evaluation Techniques of Short-Circuit Fault Probability

3

accident classification. In order to realize a comprehensive and optimal configuration of risk-based current-limiting measures and optimal control of short-circuit faults, it is urgent to take into account the binary attributes of the probability of occurrence of short-circuit faults and the consequences of faults in the context of growing attention to the economics of power systems. In addition to leading to faults that cannot be isolated by switchgear, oversized short-circuit currents also have the complex chain and derivative consequences. On the one hand, fault currents trigger significant temperature rise and electrodynamic forces within the transmission and substation equipment, which can easily damage the equipment under the influence of both thermal and dynamic stability, causing loss of load and affecting power supply reliability [8]. On the other hand, a short circuit is equivalent to an increase in branch circuits, and problems such as tripping due to short circuits can cause significant changes in the grid topology, which can lead to a shift in the stability boundary of the power system. If the operating point breaks through the stability boundary, serious consequences such as unit disconnection will occur. Numerous short-circuitinduced blackouts have happened all over the world, and these accidents share the following evolutionary characteristics: equipment short-circuit → faulty equipment decommissioning → normal equipment N − 1 overload → stability problems → fault expansion. Existing short-circuit current limiting schemes generally consider only circuit breaker blocking capacity boundary conditions. Because of the complex secondary consequences of short-circuit faults such as disconnection and machine cutting, it is necessary to consider the reliability and stability problems caused by short-circuit faults in addition to the fault current magnitude when deciding on the short-circuit current limitation scheme and optimizing the control of short-circuit risks in large grids to ensure safe grid operation.

1.2 Evaluation Techniques of Short-Circuit Fault Probability In this section, the intrinsic correlation between the service age and working condition of typical power equipment and the deterioration and aging of insulation is analyzed. A meteorological information-driven proportional risk model is established to assess the probability of short-circuit faults at grid nodes, taking into account internal factors such as equipment insulation aging and external factors such as meteorological statistics and equipment environmental conditions on the probability of short-circuit faults. To realize the probability assessment of short-circuit faults at grid nodes, the grid vulnerability analysis model is established for typhoon meteorological disaster scenarios.

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1.2.1 Data-Driven Techniques Many significant power outage situations, including the 2003 North American Blackout, evolved from short-circuit faults [10]. Because of their proximity to intricate environments, overhead lines are generally more likely to meet shortcircuit faults. Therefore, this paper introduces a data-driven PHM to evaluate the short-circuit fault rate of overhead lines. The framework of the data-driven PHM is illustrated in Fig. 1.1, which includes the reference short-circuit fault rate modeling and covariate connection function formulating. Specifically, the reference short-circuit fault rate is calculated based on the operation data of overhead lines. The covariate connection function considers the climate and surroundings data. The impact of covariates on the reference shortcircuit fault rate is formulated utilizing techniques such as the Levenberg–Marquardt parameter estimator.

1.2.1.1

Basic Proportional Hazard Model

The PHM [11] was applied for the short-circuit fault rate modeling of electrical equipment in this paper, which is illustrated as: h(t) = h 0 (t)ψ(F(t))

(1.1)

where h0 (t) is the reference short-circuit fault rate function; ψ(F(t)) is a connection function used to quantify the effects of different factors on the short-circuit fault rate. The reference short-circuit fault rate without consideration of extreme external conditions is mainly determined by insulation material damage caused by various Fig. 1.1 Short-circuit failure rate modeling of overhead line

1.2 Evaluation Techniques of Short-Circuit Fault Probability Fig. 1.2 Bathtub curve for reference SCF rate

h0(t)

Infant Mortality Stage I

5

Useful Life

We ar-out

Stage II

Stage III

α1 0

TU

TV

TW

defects, aging, or accidental factors [12], which can be well described with the Bathtub curve. The reference short-circuit fault rate typically goes through three stages [12], including infant mortality, useful life, and wear out, as shown in Fig. 1.2. Specifically, the short-circuit fault in Stage I is relatively frequent and is caused by defects in design, materials, and production process, or improper use. The shortcircuit fault in stage II is caused by some random factors whose occurrence rates are relatively constant. Stage III is associated with the aging of insulation materials, where the SCF rate grows rapidly with the increasing accumulated service time. It is important to note that there is a long period of rigorous testing and quality checks prior to the deployment of certain types of electrical equipment. The real infant mortality stage is usually skipped or compressed to a very short duration. In this paper, stages II and III are considered in the reference SCF rate modeling. For the reference SCF rate modeling, the Weibull distribution was selected because it is the most often used mathematical model to characterize the Bathtub curve [13]. [ h 0 (t) =

1.2.1.2

TU ≤ t < TV α1 ; α2 eβ2 t ; TV ≤ t < TW

(1.2)

Covariate Modeling

The climate and surrounding conditions are defined as covariates in connection function modeling. The detailed indices of each covariate are shown in Fig. 1.3. The indices of climate covariates include three climate conditions, i.e., rainstorm, hurricane, and wildfire. The values of the covariates were obtained by calculating the composite score of the correlation index. In this paper, we aim to build a comprehensive evaluation framework to map from the considered indices to the required SCF rate while taking into account the various external circumstances. In particular, a data-driven approach is used that entails three steps: weight assignment, hypothesis testing, and correlation coefficient calculation. First, the correlation between a particular influencing indicator and the SCF rate is measured using the Pearson product-moment correlation coefficient in Eq. (1.3).

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Fig. 1.3 Covariates and indices for short-circuit fault

Σ r = /Σ

(xa − x)(ya − y) Σ (xa − x)2 (ya − y)2

(1.3)

where x a is the standardized value of the influencing factor obtained from statistic data and ya is the statistic value of the SCF rate. Second, the t-test is used for hypothesis testing to determine whether there is a link between the SCF rate and the relevant index. Assuming H 0 : ρ = 0 (No correlation exists); H 1 : ρ /= 0 (Correlation does exist), the test statistics are as follows: √ (r − ρ) N D t= ∼ t(N D − 1) σr

(1.4)

where σ r is the standard deviation of the observed samples. Thirdly, the probability of H 0 being rejected is determined by the testing probability p−a . As shown in Fig. 1.4, p−a can be calculated from the cumulative probability density under the corresponding t-distribution. When p−a is small enough, the original hypothesis H 0 should be rejected. In other words, the correlation of interest is higher. Therefore, the weight of the influencing index ζ a is calculated with Eq. (1.5) based on the obtained testing probability p−a . The assigned principle also meets the requirement that the sum of weights is 1. 1 − p−a ζa = Σ (1 − p−a )

(1.5)

f(t)

Fig. 1.4 t-distribution Pr(τ < − t) = Pr(τ > t)

-t

Testing Probability p− a = 2 Pr(τ > t)

t

1.2 Evaluation Techniques of Short-Circuit Fault Probability

7

Finally, the score of covariate F co in a standard Hundred Score system with Na influencing indices can be obtained as: Fco = 100

( Na Σ

) ζa xa

(1.6)

a=1

1.2.1.3

Final Integrated PHM Model

Exponential function, as the most commonly used connection function, is applied to the covariate connection function modeling in this paper. The SCF rate function for overhead lines is developed by integrating the reference fault rate function and the covariate model: ( NZ ) Σ ( ) ( ) h teq , F; γ = h 0 teq exp γco Fco (1.7) co=1

where t eq represents equivalent service time of lines; F co represents the related covariates, here refers to the scores of climate and surrounding condition. γ co can be estimated by the following parameter estimation method.

1.2.1.4

Model Parameter Estimation

The Levenberg–Marquardt method [14] is utilized to estimate the parameters, i.e., γ 1 , γ 2 in this case. Specifically, a set of initial parameters are assigned as γ (0) = (γ 1 (0) , γ 2 (0) ). Ωw = (t eq,w , F 1,w , F 2,w ) represents the w-th observed historical data. Firstly, Ωw is substituted into Eq. (1.8). Then, the Taylor expansion of Eq. (1.8) at c(0) is obtained and high order terms are omitted as: 2 ( ) ( ) Σ ) ∂h(Ωw ; γ ) ( T E teq , F; γ = h Ωw ; γ (0) + γv − γv(0) ∂γv v=1

(1.8)

Finally, the overall variance is as follows based on the least squares principle: σ=

ND Σ w=1

{h w − T h(Ωw ; γ )} + d

2 Σ ( )2 γv − γv(0)

(1.9)

v=1

where d is a damping coefficient used to prevent the occurrence of a singular matrix. Set the first partial derivatives of Eq. (1.9) for all estimated parameters equal to zero. A set of two-parameter non-linear equations can be obtained, and then γ v is repeatedly calculated using the Levenberg–Marquardt method until the difference

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between the two consecutive results can be ignored. Finally, a numerical solution of the connecting coefficients is obtained.

1.2.2 Analytical Techniques This subsection introduces a fragile model that uses a hurricane instance to explain how to relate weather parameters to SCF rates of overhead lines. The determination of the overall SCF rate, weather-induced covariant SCF rate, and aging-based reference SCF rate are the three steps in the fragile model.

1.2.2.1

Aging-Based Reference SCF Rate Modeling

The reference SCF rate without consideration of extreme external conditions is mainly determined by insulation material damage caused by various defects, aging, or accidental factors [12], which can be well described with the Bathtub curve. Weibull distribution, as the most deployed mathematical model to describe the Bathtub curve [13], was used for the reference SCF rate modeling. [ λ A (t) =

TU ≤ t < TV α1 ; α2 eβ2 t ; TV ≤ t < TW

(1.10)

The parameters in Eq. (1.10) can be fitted through long-term statistics of a large number of samples or obtained through modeling the physical aging mechanism of materials.

1.2.2.2

Covariant SCF Rate Modeling Under the Extreme Weather

Considering the extreme weather, the described SCF rates in Fig. 1.2 are greatly magnified. Under a hurricane, most short-circuit faults are caused by falling towers or trees, thus the SCF rate of overhead lines is mainly influenced by wind direction and speed. The wind load function L W for a transmission line with coordinates (x, y) can be expressed as follows [15]: [ ) )] ( ( R2 R2 L W (x, y, t) = ω (t) ε1 exp − 2 − ε2 exp − 2 2γ1 2γ2 / [ ]2 R= [(x − κx (t)]2 + y − κ y (t)

(1.11) (1.12)

where ε1 and ε2 are hurricane intensity parameters; γ 1 and γ 2 denote influence scopes; R is the distance between hurricane center and transmission line; (x, y) and

1.2 Evaluation Techniques of Short-Circuit Fault Probability

9

G

G Wind Direction

α(t)

(x, y) R

Typhoon Center

(κx(t), κy(t))

1m ηd Transmission corridor

G

Fig. 1.5 A moving hurricane influencing a transmission line

(κ x (t), κ y (t)) are coordinates of the line and the wind center, respectively. ω (t) denotes the influence factor of wind direction on wind load, which is defined as follows: ω (t) = sin α(t)

(1.13)

α(t) is the angle by which the wind hits the line, which is shown in Fig. 1.5. The relationship between wind load and covariant SCF rate of the transmission line can be described below: ] [ ηd η h L W (x, y, t) L (1.14) + a2 λW (t) = exp a1 Vd H where a1 and a2 are parameters obtained from historical statistics; V d is the design wind speed of the transmission line; ηd and ηh are the vegetation density (1/m) and average height (m) within the specified 35 m wide transmission corridor; H and L are the height (m) and length (km) of the transmission line.

1.2.2.3

Overall SCF Rate Calculation

In order to highlight the correlation between SCF rate, equipment aging degree, and extreme weather conditions, this book gathers some realistic operation and SCF data from East China Power Grid, which is depicted in Fig. 1.6. The trend of SCF rate is consistent with that in [16]. The SCF rate of new equipment is nearly identical to that of equipment that is 20 years old under various wind speeds, as shown in Fig. 1.6. In other words, the impact of aging on the SCF rate during some extreme weather events can be disregarded if equipment aging is not a serious issue. Then, one of the main factors contributing to short-circuit problems is the weather parameters. However, beyond

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Fig. 1.6 Fragility curves of overhead lines in different years

a certain point, age gradually starts to have an amplifying effect on the SCF rate during extreme weather events. Additionally, at a specific level of age, the covariant short-circuit fault rate, which is roughly S-type in form, is similar to the fluctuating trend of the short-circuit fault rate. This demonstrates that age scales with time but does not significantly alter the trend of failure rate as a function of wind speed. In order to express the overall SCF rate under the combined effect of aging and extreme weather variables, the following linear function is used: λ(t) = λW (t)ψ1 [λ A (t), λ0 ] [ ψ1 [λ A (t), λ0 ] =

λ A (t)/λ0 , λ A (t) > λ0 1, λ A (t) ≤ λ0

(1.15)

(1.16)

where ψ 1 represents the penalty operator for reference SCF rate λA (t) exceeding a given failure rate λ0 . Both extreme weather and aging equipment require additional data collection efforts in order to more accurately assess SCF rates [17]. The parameters of the SCF rate calculation can then be specified using some parameter estimation or machine learning techniques after the accumulation of short-circuit fault records, equipment aging, and extreme weather data acquisition. The SCF probability of line l in [0, t m ] can be obtained via the integral of corresponding instantaneous rates [15]: ⎡ ρl = 1 − exp⎣−

{tm 0

⎤ λ(t)dt ⎦

(1.17)

1.3 Evaluation Techniques of Short-Circuit Fault Consequences

11

1.3 Evaluation Techniques of Short-Circuit Fault Consequences In huge interconnected grids, short-circuit current overruns can result in failures that cannot be isolated by switches, as well as complex chain and expansion repercussions. On the basis of the development mechanism of typical short-circuit accidents, it is suggested to study the complicated impacts of short-circuit faults on many aspects of the reliability and stability of the power system in stages. A perfect index system for the short-circuit safety level of large power grids is established from several aspects to assess the effects of short-circuit faults in a graded and classified manner. This index system is based on the definition of short-circuit consequence severity functions such as tidal overload, load shedding, unit destabilization, cut-off, and chain fault. As seen in Fig. 1.7, the severity function quantifies the severity of short-circuit current overrun in relation to the original circuit breaker opening capacity. For the direct impact of short-circuit fault, the decay process of fault current after the occurrence of short-circuit fault can be simulated by symmetric component, standardized method, simulation, etc. For short-circuit fault secondary effects, the quantification process is more complex, using the following methods for analysis and calculation. 1. For the voltage stability problem, the sensitivity analysis method or the Jacobi matrix singularity method [18] is used for the static analysis of voltage stability, and the severity of node voltage overruns is quantified by the voltage overrun severity function. 2. For the transient stability problem, the power angle curves of the main units are differenced [19], and the power angle dynamics are simulated by implicit trapezoidal integration with the fault state as the initial point, and the severity of the loss of synchronization of the unit’s power angle is quantified by the power angle divergence severity function.

Fig. 1.7 Quantification of short circuit accident consequences based on fault evolution mechanism

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3. For the frequency stability problem, the maximum frequency deviation analysis of the power system is carried out by the time-domain simulation method [108] or the analytical method [20], and the severity of the system frequency crossing limit is quantified by the frequency crossing limit severity function.

1.4 Challenges of Short-Circuit Risk Prevention and Control Compared with traditional short-circuit fault analysis, short-circuit risk prevention and control of power systems in the context of disasters need to consider the binary properties of the severity and probability of short-circuit accidents while taking into account the reliability, stability, and other complex secondary effects. Traditional deterministic accident-based decision-making methods and theories face many challenges, mainly the following difficulties. 1. The physical characteristics of different types of extreme weather hazards vary greatly, and the complex nonlinear mapping model between external causes such as extreme weather hazards and the probability of short-circuit failure of power equipment has not been established, while the age and working condition of equipment and the degradation and aging degree of insulation are intrinsically related, and the mechanism of the influence of internal causes such as aging of equipment insulation on the probability of short-circuit failure has not been established, resulting in multiple internal and external factors. The short-circuit risk of complex power grids under the superposition of multiple internal and external factors lacks methodological support. 2. In power systems, fault current limiters are an effective solution to prevent shortcircuit current crossing. However, the access of fault current limiters affects both grid voltage distribution and transient stability level. The new fault current limiter is characterized by diverse functions, which limit the current amplitude and change the reactive voltage distribution and grid stability boundary, making the optimal configuration of the fault current limiter extremely difficult. In order to establish an optimal configuration model that takes into account the dynamic response process of fault current limiter switching and to provide an effective solution method, it is important to take into account the trade-off between current limiting, voltage, temporary stability, and economic objectives. 3. The local and sudden nature of extreme meteorological disasters makes the power system subject to disaster uncertainty when responding to short-circuit faults. Therefore, how to comprehensively consider the disaster prediction uncertainty, focus on the current crossing limit and transient instability caused by short-circuit faults, establish the optimization model of power grid current-limiting measures, and realize the mutual coordination of short-circuit risk prevention and control and emergency dispatch is one of the difficulties faced by the optimal operation of the system.

1.4 Challenges of Short-Circuit Risk Prevention and Control

13

Fig. 1.8 Organization of this book for comprehensive control of short circuit risks towards resilient power systems

4. Extreme meteorological disasters lead to short-circuit current overrun and transient instability in the power system, while they are also very likely to trigger serious voltage and frequency drops. It is a big challenge to solve the voltage problem, frequency problem, and transient stability problem caused by shortcircuit fault from both planning and operation levels, using resources on both sides of generation and load. In the face of existing challenges, it is essential to develop new theories and methods for evaluating short-circuit risks in power systems. The proposed methods can help system operators and planners to understand accurately the risk level of

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1 Risk Evaluation of Short-Circuit Fault in Power System

short-circuit faults in power systems, and thus guide the development of risk control measures.

1.5 Organization of This Book for Resilience Enhancement in Power System Short-Circuit Faults The organization of this book is shown in Fig. 1.8. First, the challenges of shortcircuit risk assessment and short-circuit risk prevention and control are described in this chapter. In this chapter, the motivation for the study of this book is answered. In addition, short-circuit risk control methods at the planning level are given in Chaps. 2–4. Finally, risk control measures at the operational level are presented in Chaps. 5–9.

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15

14. K. Madsen, H. B. Nielsen, and O. Tingleff, “Methods for non-linear least squares problems,” 2004. 15. E. Broström and L. Söder, “Modelling of ice storms for power transmission reliability calculations,” in 15th Power Systems Computation Conference (PSCC’05), Liège, Belgium, August 22–26, 2005, 2005: PSCC. 16. A. M. Salman, Y. Li, and M. G. Stewart, “Evaluating system reliability and targeted hardening strategies of power distribution systems subjected to hurricanes,” Reliability Engineering & System Safety, vol. 144, pp. 319–333, 2015. 17. M. H. J. Bollen, “Effects of adverse weather and aging on power system reliability,” IEEE Transactions on Industry Applications, vol. 37, no. 2, pp. 452–457, 2001. 18. Xie Q, Hui H, Ding Y, et al. Use of demand response for voltage regulation in power distribution systems with flexible resources [J]. IET Generation, Transmission & Distribution, 2020, 14(5): 883–892. 19. Gan D, Thomas R J, Zimmerman R D. Stability-constrained optimal power flow [J]. IEEE Transactions on Power Systems, 2000, 15(2): 535–540. 20. Restrepo J F, Galiana F D. Unit commitment with primary frequency regulation constraints [J]. IEEE Transactions on Power Systems, 2005, 20(4): 1836–1842.

Chapter 2

Risk-Based Optimal Configuration of Fault Current Limiter in Power System

2.1 Introduction As illustrated in Chap. 1, short-circuit is one of the most prevalent fault types in power systems [1]. Short-circuit currents (SCCs) are rising dramatically as electrical networks are connected and expanded [2]. Out-of-range SCC has been a significant issue in several nations, particularly in China [3]. The out-of-range SCC has already existed in nearly 30% of the 500 kV buses in the Eastern China power grid [4]. To achieve the best decision-making during the planning stage, it is important to develop an automatic allocation method of current limiting measures [5]. As an effective current limiting solution, fault current limiter (FCL) has been widely adopted in transmission systems [6]. In the existing studies [6, 7], FCLs are mostly configured in deterministic expected short-circuit fault (SCF) situations. The SCC limitation techniques were developed with the assumption that nondiscriminatory SCFs could occur on any bus [6]. A deterministic three-phase SCF set was utilized in [7] to select the optimal locations of the superconducting fault current limiters (SFCL). However, the short-circuit rates in power systems are relatively low while the mainstream FCLs are still expensive [8], such as SFCLs [7], saturated core FCL [9], and high-temperature superconducting fault current limiters (HTS-FCLs) [10]. The over-configuration of FCLs and inefficient use of transmission system investments occur when fault situations are modeled as deterministic events. SCF has a risk linked with both severity and probability. The SCF rate is closely related to equipment aging and may demonstrate certain statistical rules. To activate SCFs in power equipment, conductors must come into close contact with one another due to aging and insulation damage [11]. Numerous outside factors, in addition to the equipment’s state of health, also increase SCF rates. The bulk of SCF-related power outages in the Northeastern United States is specifically attributed to fallen trees and debris [12]. Similarly, there is a connection between the occurrence of SCF and severe weather events including hurricanes, wildfires, and snowstorms. For instance, 129 line SCF occurrences were brought on by a snowstorm in South China in 2008

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Ye et al., Risk-Based Planning and Operation Strategy Towards Short Circuit Resilient Power Systems, https://doi.org/10.1007/978-981-19-9725-9_2

17

18

2 Risk-Based Optimal Configuration of Fault Current Limiter in Power …

[13]. In general, little research has been done on how to evaluate or estimate SCF rates while taking various aspects into account. For the FCL siting and sizing, substantial research has been done on the impact of SCFs on power systems. In general, excessive SCCs may make it more challenging to isolate failures using circuit breakers and result in cascading failures [14]. As a result, the SCC amplitude was primarily specified as the constraint in earlier FCL design approaches [15, 16]. In [15], the smallest FCLs’ circuit parameters were obtained to restrain SCCs under circuit breakers’ interrupting ratings. A mixedinteger nonlinear programming approach was suggested in [16] for configuring FCLs with the restriction that the SCCs could be restricted to security scope. Additionally, after the fault current is recognized, the switching in and out of FCL is a dynamic process because of the material properties [17]. The aforementioned FCLs have varying reaction times, and the increased investment in FCLs is due to the necessity for a faster response time [18]. It is critical to coordinate the action time of relay protection with the reaction time of FCL since the development of fault detection techniques improves the response speed of relay protection [19]. Besides the SCC, SCF has multiple impacts on power system operation [17]. The system stability boundary, for instance, alters when a SCF takes place or a specific FCL is installed in the grid [2, 7]. Consequently, the stability influence of SCF has been widely integrated into the FCL configuration model. One of the objectives of the FCL configuration model was to reduce the generators’ rotor angle variations with regard to the center of inertia [10]. Because both the dynamic response process of FCL and the transient stability simulation operate at the millisecond (ms) temporal level, the dynamic process of FCL must be taken into account. Additionally, it has been observed that the SCF creates irregular voltage sags, which have a negative impact on power quality and raise security issues, particularly in hybrid AC/DC power systems. The voltage sags at the location of the AC/DC coupling buses may result in HVDC system commutation failures and severe power shortages for the receiving ends [20]. Recently, novel modified structures have been presented in [21, 22], covering both the FCL and dynamic voltage restorer (DVR) functions. In DVR mode, it has a more complicated effect on the voltage distribution in power systems and can supply a specific fixed voltage. Hence, to optimally allocate FCLs in power systems, it is crucial to take into account the multiple effects of SCFs, whose main difficulty resides in the solution approach. Heuristic approaches have taken over as the go-to solution for FCL configuration issues because of their mixed-integer, nonconvex, and nonlinear characteristics. The overall FCL optimal configuration of interest should be expressed in many-objective (objectives ≥ 4) forms with the further increase of associated factors such as transient instability and voltage sags, while existing multi-objective (objectives ≤ 3) heuristics methods such as the classic NSGA-II tend to fall into local optimums when dealing with many objectives simultaneously. Fortunately, NSGA-III [23] was claimed to provide excellent performance in optimizations with large-scale objectives. The innovations of this paper can be concluded as follows:

2.2 Modeling of FCL and SCC Calculation

19

1. The resistance is modeled as a time-dependent variable in the time-domain dynamic response process that controls the switching in and out of FCL. In order to realize the coordination between fault current limiting and relay protection, it is integrated into the FCL configuration model. 2. A risk-based many-objective framework for FCL configuration is established, taking into account both the total cost of the FCL and the hazards associated with out-of-range SCC, post-fault voltage sag, and rotor angle oscillation. To seek the Pareto optimal FCL configuration alternatives, the most recent many-objective heuristics NSGA-III are introduced.

2.2 Modeling of FCL and SCC Calculation 2.2.1 Dynamic Response Modeling of FCL The FCLs exhibit many dynamic switching in and out properties, including SFCL, saturated core FCL, and HTS-FCLs. Here, high-temperature superconducting materials-based resistive-type FCL (R-FCL) is adopted. Its circuit diagram is shown in Fig. 2.1. The line current follows Path 1 through the bypass switch with nearly zero resistance during steady-state operation. The switch should be opened when the SCF occurs, and the fault current should then go via Path 2 through the variable resistance of R-FCL. As a result, the bypass switch’s opening time and the quenching time of the superconducting materials are both included in the FCL’s total action time. The dynamic response process of R-FCL [24] is modeled as illustrated in Fig. 2.2, and the quenching process is determined by the superconducting materials. The two FCLs undergo various quenching processes despite having the same final current limiting resistance. Due to its greater ramping rate, RFCL(A) performs better in terms of fault current limitation. Here, Rns represents the resistance of the R-FCL; τ signifies the time constant; t qs , t fr , t sr , and t cr are the times of the quench starting, the first-stage recovery, the secondary-stage recovery, and the completed recovery, respectively. The coefficients a1 ~ a2 as well as b1 ~ b2 are to define different slopes and turning points of the resistance lines. Fig. 2.1 R-FCL circuit diagram

Bypass switch Steady state

Path 1

Fault state

Path 2 RFCL(t)

20

2 Risk-Based Optimal Configuration of Fault Current Limiter in Power …

Fig. 2.2 Dynamic response process of R-FCL

2.2.2 Modeling of SCC Calculation In this research, the three-phase SCF is regarded as the preset fault for the FCL configuration. The SCC at bus b can be calculated as follows: b ISC =

E b0 X bb

(2.1)

According to Eq. (2.1), the bus admittance matrix can be changed to achieve the fault current limiting. As the self-impedance of the corresponding buses grows, the SCC drops [2]. If only the impedance of branch l (line i − j) is added by Δx l . The following formula can be used to express the increment of the diagonal element X bb [25]. ( ΔX bb = −

X bi − k T X bj

)2

( ) 1/Δyl + X ii + k T2 X j j − k T X i j + X ji ) ( 2 Δyl = −Δxl / xl0 + Δxl xl0

(2.2) (2.3)

where k T is the standard voltage ratio for the transformer branch imputed to i side (k = 1 for ordinary lines). When the power system protects against SCFs, the SCC amplitude must be within the circuit breaker’s interrupting capacity at the moment of the protection components’ action. The impedance of branches is added when branches are equipped with FCLs. The impedance implemented in the branches is time-dependent when taking into account the FCLs’ dynamic response. Therefore, Eq. (2.3) can be reformulated as follows: ) ( | 2 (2.4) Δyl |t=t pr = −Δxl |t=t pr / xl0 + xl0 Δxl |t=t pr Then the increment of diagonal element X bb can be obtained, and the SCC limitation constraint is formulated as follows:

2.3 Many-Objective FCL Configuration Modeling

| b | ISC = t=t pr

21

E b0 ≤ Ibc X bb +ΔX bb |t=t pr

(2.5)

2.3 Many-Objective FCL Configuration Modeling 2.3.1 Decision Variables The decision variables of the FCL configuration model include integer and continuous types, which are defined as: 1. Location status (Integer variable) u l ∈ {0, 1, 2, . . .}

(2.6)

If the FCL is not installed at transmission branch l, the status ul should be 0. The non-zero integers represent FCLs with different dynamic response processes. 2. Sizing status (Continuous variable) 0 ≤ zlFCL ≤ 1

(2.7)

max The upper limit of FCL impedance to be installed is set to be z FCL . Then, the FC L max actual configured FCL can be expressed as zl z FCL .

2.3.2 Objective Function There are numerous effects of FCL configuration on power system operation. The FCL configuration model’s objective set is simultaneously optimized and is formulated as follows: {min C I, min R1 , min R2 , min R3 , . . .}

(2.8)

where CI represents the FCLs investment cost, R1 , R2 , and R3 , etc. are the corresponding risks associated with power system operation. The specific risk item is defined as: Risk =

NF Σ f =1

pf Sf

(2.9)

22

2 Risk-Based Optimal Configuration of Fault Current Limiter in Power …

where pf is the occurrence probability of SCF f ; S f indicates the severity of the corresponding SCF imposed on the power system. The detailed objectives are illustrated as follows. 1. FCL investment The total investment of FCLs, including fixed costs of FCL installation and incremental costs of impedance, is formulated as follows: CI =

NL Σ [ F ] Cl (u l ) + ClV (u l )zlFC L

(2.10)

l=1

For the FCLs installed on line l, C F l (ul ) and C V l (ul ) indicate the fixed and variable cost coefficients, respectively. Note that the FCLs with higher response speeds have larger variable cost coefficients. 2. Risk due to out-of-range SCCs Relay protection is used to cope with the SCF in the power system, therefore when the relay protection kicks in, the SCC needs to be within the circuit breaker’s breaking rating. Therefore, the risk of SCCs in the action time of protection components exceeding the breaking rating of the circuit breaker is formulated: R1 =

NF Σ f =1

[ ( | ) ] b | λ1 p f ψ ISC , I bc t=t pr

(2.11)

where I bc is the permissible SCC threshold. The penalty operator ψ is formulated as follows: { m1 − m2, m1 > m2, (2.12) ψ(m 1 , m 2 ) = 0 m1 ≤ m2. 3. Risk due to post-fault transient instability With the installation of FCLs, the transient stability boundary shifts. To generate precise rotor angle curves, the dynamic process of FCL is additionally taken into account in the transient stability simulation. As a result, the following formula is used to express the probability that the generators’ rotor angle departure from the center of inertia (COI) exceeds the threshold: R2 =

NF Σ f =1

pf

N G TΣ s /Δt Σ | [ (| ) ] ψ |δn,k ( f ) − δCOI,k ( f )|, δthr e λ2

(2.13)

n=1 k=1

of generator n at integration step k in contingency | δ n,k (f ) is the rotor angle | f.|δn,k ( f ) − δC O I,k ( f )| corresponds to the rotor angle deviation of generator n with respect to COI.

2.3 Many-Objective FCL Configuration Modeling

23

The curve of the relative angle between the generators over time is calculated using the swing equation. δ˙˜g = ω˜ g ) 1 1 ( Pmg − Peg − PCOI − Dg ω˜ g ω˙˜ g = Mg MT

(2.14)

PCOI is the power associated with the COI reference frame; ωN is the synchronous rotor speed; δ g and ωg are rotor angle and speed; δ˜ = δg − δ0 , ω˜ = ωg − ω0 ; δ 0 and ω0 are the weighted average rotor angle and speed, respectively; M T is the total inertial constant of generators; Pmg is the mechanical input power; Peg is the active electromagnetic power, which is: Peg = E g

NG Σ

( ( [ ) )] E n G gn cos δg − δn + Bgn sin δg − δn

(2.15)

n=1

where Ggn and Bgn represent the reduced admittance between generator g and n. Note that the reduced admittance changes dynamically with the dynamic response process of FCL. The COI can be calculated through a weighted average of the generators as follows [2]: ΣN G

g=1 δg M g δC O I = Σ N G g=1 M g

(2.16)

4. Risk due to abnormal voltage sag The FCL configuration alters the system’s topology impedance, which has an impact on the voltage distribution. The newly designed FCL, meanwhile, has the capability of voltage compensation, which can maintain the post-fault voltage amplitude of crucial nodes. To demonstrate the total voltage stability, the following out-of-range voltage drop is used: R3 =

NF Σ f =1

pf

NB Σ | [ (| ) ] ψ |−I f X f b |, ΔUthr e λ3

(2.17)

b=1

where ΔU thre means the voltage sag threshold, I f is the SCC after the occurrence of SCF f , X fb denotes the impedance value.

24

2 Risk-Based Optimal Configuration of Fault Current Limiter in Power … Initial choromosome

Fig. 2.3 The chromosome structures

Location 1 Location 2 Location 3

x x

1

x

1

x

3

x

2

3

x

4

x . . . x 5

2M-3

x

5

x

Binary choromosome Integer choromosome

Selection

2M-1

Location M-1 Location M

x . . . x 6

x

2

x

2M-3

4

x

2M-2

x

x

2M-1

2M

x . . . x 6

2M-2

x

2M

Continuous choromosome

GA operator Crossover Mutation

Combination

2.4 Solution Method The heuristic algorithm is the most common approach for solving the mixed-integer, non-convex, and nonlinear programming issues represented by the suggested FCL configuration model. Heuristic approaches must contend with the aforementioned many-objective optimization. The search effect is significantly diminished by the exponential growth of non-dominated solutions with the rising objective number. In this section, NSGA-III [23] is used to optimize the many objectives at once.

2.4.1 Coding Strategy The chromosome structure for NSGA-III is illustrated in Fig. 2.3. M branches are selected as the candidate locations to configure FCLs. Thus, the length of the chromosome is 2M. Each candidate location is represented with an integer number and a continuous number. The integer number C int indicates the configuration situation of the FCL, in which 0 indicates that the FCL is not used on the branch while nonzero integers indicate the types of the installed FCLs. The continuous number C con max represents the ratio of the deployed FCL impedance to the reference value z FCL .

2.4.2 NSGA-III NSGA-II [26] utilized crowded distances to select individuals with the same nondominated level. Individuals with greater crowded distances are preferred. While a novel reference point-based approach is utilized in NSGA-III to select individuals. The basic process of NSGA-III can be described in Table 2.1. Firstly, the technique in [27] is used to select the placement of reference points. The number of reference points N r is determined by the number of objectives N 0 and the number of divisions N d , which is given by:

2.4 Solution Method

25

Table 2.1 Procedure of NSGA-III Generation t of NSGA-III procedure Input: Define a set of reference points Z s and parent population Pt Output: Pt+1 1:

S t = ϕ, i = 1

2:

Qt = Recombination + Mutation (Pt )

3:

Rt = P t ∪ Qt

4:

(F 1 , F 2 , …) = Non-dominated-sort (Rt )

5:

Repeat

6:

S t = S t ∪ F i and i = i + 1

7:

until |St| ≥ N

8:

The last front to be included: F l = F i

9:

if |S t | = N then

10:

Pt+1 = S t , break

11:

else

12:

Pt+1 = F 1 ∪ F 2 … ∪ F l−1

13:

Points to be chosen from F l : K = N − |Pt+1 |

14:

Choose K members one at a time from F l based on the reference point-based method to construct Pt+1

15:

end if

(

N0 + Nd − 1 Nr = Nd

) (2.18)

Then, the perpendicular distance d (S p , W r ) between the population member (Set of population S p ) and the reference lines (Set of reference lines W r ) is calculated as follows: ( ) ||( )|| d ⊥ S p , W r = || S p − W rT S p W r /||W r ||2 ||

(2.19)

Finally, based on the perpendicular distance, K members from F l are chosen to fill the vacant population slots of Pt+1 . The NSGA-II method falls into local optimums when dealing with many-objective optimization because the solutions it produces are unevenly distributed on the nondominated layer. The reference points (Grey circles) for NSGA-III are equally inclined to all objective axes, as seen in Fig. 2.4. Individuals (Black circles) that are non-dominated and close to the predetermined reference lines are highlighted by NSGA-III (Dotted line). The reference point-based strategy utilized by NSGA-III can effectively address the many-objective problems of poor convergence and high diversity requirements since individuals are widely distributed in the global feasible space.

26

2 Risk-Based Optimal Configuration of Fault Current Limiter in Power …

Fig. 2.4 The reference points and reference lines

Reference line

f1

Reference point

Perpendicular distance Chromosome

Normalized hyperplane

f3

f2

2.5 Case Studies 2.5.1 Test System and Parameters The proposed technique is tested using the New England 39-bus system. In addition, the data-driven SCF rate model is fed with climate statistics and system operation data from a real power grid in eastern China. There are basic network parameters in [28]. In order to maintain security, SCC magnitudes at the critical buses must be limited below 25 p.u. Additionally, the nodal voltage’s minimum and maximum values are 0.95 and 1.1 p.u., respectively. FCL cost coefficients are available in [29]. We select two types of FCL with distinct time constants in order to achieve coordination between the FCL dynamic response process and the action time of relay protection components. Note that the time constant τ of the first type of FCL (Type 1) is 5 ms, while the second type of FCL (Type 2) has a slow response speed and its time constant τ is 8 ms. The parameters of the FCL configuration model are shown in Table 2.2. The SCF probabilities obtained with the model proposed in subsection 1.2.1 are shown in Fig. 2.5. The lines with largest SCF rates are 20–34, 29–38, 23–36, 19–33, Table 2.2 Parameters of the configuration model

Parameter

Value

Parameter

Value

I bc

25 p.u

Ts

1s

Δt

0.01 s

λ1

10,000

λ2

10,000

λ3

10,000

M

10

δ thre

100°

N0

4

Nd

10

8

max z FCL

60 Ω

NF

2.5 Case Studies

27

Fig. 2.5 SCF probability gradient and candidate FCL lines

19–20, 6–31, 22–35 and 10–32, respectively. The candidate lines 16–17, 2–30, 17– 27, 16–27, 28–29, 26–28, 26–29, 15–16, 22–23 and 14–15, numbered L1 to L10 , are chosen for FCL deployment based on the SCF defending sensitivity.

2.5.2 Pareto Solutions Analysis To visualize the results, three objectives are optimized including Eqs. (2.10), (2.11), and (2.17). The population distribution of the raw generation and the 50th generation are depicted in Figs. 2.6 and 2.7, respectively. As can be seen from Fig. 2.7, the population presents an obvious assembling effect with a non-dominated front appearing after the evolution of NSGA-III. The considered objective functions conflict with each other and the proposed heuristic solution method is demonstrated to provide a set of Pareto solutions for decision-makers to select according to different preferences. For example, to reduce the cost, lines with the best current limiting effects and 4

x 10

3rd Objective

Fig. 2.6 Population distribution of the raw generation

2 1.5 1 0.5 10

8 6 4 4 x 10 2nd Objective

2

2

4

6

8

10

1st Objective

12 4

x 10

28

2 Risk-Based Optimal Configuration of Fault Current Limiter in Power …

lowest cost coefficients are preferred, which will inevitably increase the electrical distance to generators and undermine the overall stability of the system. To further analyze the Pareto solutions, four objectives Eqs. (2.10), (2.11), (2.13), and (2.17). are considered simultaneously. The schemes selected from the Pareto optimal solutions are shown in Table 2.3. As can be seen, if there is no FCL placed in the system, the investment will be the lowest (Solution 6). However, the SCC amplitude, voltage quality, and transient stability are relatedly inferior. The FCLs placed at L1 , L3 , L5 , L6 , L7 , L8, and L10 can deal with out-of-range SCCs, L2 , L6, and L8 have a positive effect on the voltage stability. Moreover, when 103 Ω type 1 (installed at L1 , L6 , L7 , and L8 , respectively) and 15 Ω type 2 FCLs (installed at L10 ) are installed, the rotor speed oscillations are the smallest (Solution 2). Thus, L8 is relatively a critical line to defend against SCF, which is effective in terms of current, voltage, and power angles. 4

Fig. 2.7 Population distribution of the 50th generation

x 10

3rd Objective

1.8 1.6 1.4 1.2 1 0.8 0.6 8 6 4 4 x 10 2nd Objective

2

0

8

6

4

2

4

1st Objective

x 10

Table 2.3 Pareto solutions located on the non-dominated edge S/N

Current limiting strategies (Ω) NFCL

1

Objective functions 2

J1

J2

J3

J4

1

7

218

55

90,706

0

16,666

7322

2

5

103

15

42,754

59,829

8396

0

3

3

58

65

35,668

87,458

0

10,771

4

4

59

51

34,163

66,003

8481

9652

5

6

30

121

40,303

55,725

10,232

8350

6

0

0

0

0

77,523

6302

13,266

7

4

0

118

27,793

84,075

7400

6337

8

7

181

34

74,431

39,680

12,359

7405

9

4

12

57

19,778

78,772

6497

7333

10

7

218

51

89,938

27,990

16,326

7400

2.5 Case Studies

29

2.5.3 Comparison Between the Proposed Risk-Based Method and the Traditional Deterministic Method The scheme costs obtained by the deterministic and the proposed risk-based models are compared in Fig. 2.8, where the deterministic refers to the method in which SCF probability is excluded. An average cost index AVC is utilized to evaluate the overall economy of the obtained solutions as follows: Σ N pop AV C =

i=1

Ji1

(2.20)

N pop

The average cost of the risk-based schemes is 50,192.34 $. Meanwhile, a higher average cost of 54,259.19$ is given by the deterministic model. The proposed riskbased method can deploy FCLs more economically. As can be seen from Fig. 2.8, the 30th solution of the deterministic and risk-based methods are the same in terms of overall FCL investment. The FCL configuration schemes obtained with the riskbased method are shown in Table 2.4, while the deterministic configuration schemes are shown in Table 2.5. Fig. 2.8 Cost of deterministic and risk-based methods

Table 2.4 Configuration schemes with the risk-based method Lines

16–17

17–27

28–29

26–28

15–16

FCL type

1

2

2

2

1

FCL value (Ω)

47

4

43

7

48

Table 2.5 Configuration schemes with the deterministic method Lines

16–17

2–30

26–28

26–29

15–16

14–15

FCL type

2

1

1

2

1

1

FCL value (Ω)

44

14

5

22

22

36

30

2 Risk-Based Optimal Configuration of Fault Current Limiter in Power …

In the deterministic configuration model, the probability of each scenario is the same and the pf in Eqs. (2.11), (2.13), and (2.17) is set to 1, which indicates that all the scenarios are treated indiscriminately. The proposed risk-based method takes the SCF probabilities of the scenario as the weight. Therefore, the risk-based model emphasizes the scenario with higher SCF probability than the deterministic model. Then, the most probable SCF on line 20–34 was selected to compare the risk-based scheme with the deterministic scheme. Specifically, the corresponding out-of-range SCC, voltage instability, and transient instability penalty values of the risk-based scheme are 1,455,111.12, 63,477.32, and 7288.32, approximate 95, 97, and 99% of those obtained by the deterministic scheme, which can be clearly illustrated from Figs. 2.9 and 2.10. Overall, the risk-based scheme is capable to provide a betterdefending effect than the deterministic one. The above defending effect difference can be explained as the fact that fault probability and SCC magnitude are inconsistent attributes for nodes. For example, the SCC amplitudes of buses 34 and 38 are higher than buses 20 and 35 while their SCF probabilities are lower. If the SCF rates are not considered in the decision procedure, the out-of-range SCC buses would be treated indiscriminately. In contrast, the proposed risk-based method gives priority to limiting the SCCs of buses with higher fault possibilities, where the probability acts as a certain kind of weight and helps to avoid investments against rare SCFs. Fig. 2.9 SCC penalty on critical buses

Fig. 2.10 Voltage sag penalty on critical buses

2.5 Case Studies

31

2.5.4 Solution Performance Analysis A comparison between NSGA-II [26] and NSGA-III [23] is implemented in this section. The population sizes of the two heuristics method are both set at 50. In each generation, the non-dominated individuals of NSGA-II and III are combined into a mixed one. Then, the non-dominated sorting is implemented to select the better ones. For many-objective minimization problems, the vector of objective components is given by Eq. (2.21). f (X) = ( f 1 (X), f 2 (X), . . . f n (X)), X ∈ U

(2.21)

The definition of non-domination is given by Eqs. (2.22)–(2.23). In other words, when Eqs. (2.22)–(2.23) are satisfied, the individual X u is dominated by X v ∀i ∈ {1, . . . n}, f i (Xv ) ≤ f i (Xu )

(2.22)

∃i ∈ {1, . . . n}, f i (Xv ) < f i (Xu )

(2.23)

The non-dominated-sorting results of the four-objective problem are illustrated in Fig. 2.11. N pop and N non denote the sizes of the mixed population set and the nondominated subset, respectively. The algorithm which achieves more non-dominated survivors in the competition should be identified as the superior solution method. As can be seen, from the fourth generation, the number of non-dominated individuals is approximate 70 in the 100 mixed individuals, of which about 50 are provided by the NSGA-III. Note that NSGA-III constantly provides more non-dominated solutions than NSGA-II in each generation. In the 50th mixed generation, 29% of the nondominated individuals are from NSGA-II, and as high as 71% are from NSGA-III. The impact of problem size on the optimization performance of heuristics is illustrated in Fig. 2.12, where the percentage of non-dominated individuals provided by NSGA-III increases with the increasing objective dimension. NSGA-III is demonstrated to achieve a more effective global searching ability compared with the Fig. 2.11 Non-dominated solutions of different generations in the mixed NSGA-II and NSGA-III population

32

2 Risk-Based Optimal Configuration of Fault Current Limiter in Power …

Fig. 2.12 Non-dominated solutions under different objective numbers in the mixed NSGA-II and NSGA-III population

benchmark NSGA-II, especially for many-objective problems. The performance gap between NSGA-III and II grows with the increase of objectives.

2.6 Conclusion The out-of-range and ever-increasing SCC has become a significant concern for the secure operation of power systems. The out-of-range and ever-increasing SCC has become a significant concern for the secure operation of power systems. A data-driven PHM is developed to evaluate the long-term nodal SCF rate and generate the expected SCF scenarios. Based on these scenarios, this paper proposes a risk-based method to configure FCLs in an innovative many-objective optimization framework, integrating related effects of SCCs amplitude, voltage sag, transient stability, and investment cost. The time-domain dynamic response process of FCL is integrated into the proposed configuration model to realize the coordination between the FCL response time with the relay protection action time in terms of SCC limiting. Moreover, the timevarying resistance of FCL is also considered in the transient stability simulation to obtain more accurate rotor angle curves. The many-objective Pareto optimal FCL configuration schemes are effectively searched with the advanced NSGA-III, which is insensitive to the number of objectives and makes it free to integrate more factors in FCL configuration. The simulation results verified that the risk-based method can configure FCLs more economically. Besides, at the same investment cost, the risk-based method achieves a better defending effect against typical short-circuit faults. The enhanced NSGA-III, which is insensitive to the number of objectives and free to incorporate more factors in FCL configuration, successfully searches the many-objective Pareto optimal FCL configuration schemes. The simulation results demonstrated that FCLs can be configured more affordably using the risk-based approach. Additionally, the risk-based strategy delivers a greater protection impact against ordinary short-circuit problems at the same investment cost.

References

33

References 1. Z. Yang, H. Zhong, Q. Xia, and C. Kang, “Optimal transmission switching with short-circuit current limitation constraints,” IEEE Transactions on Power Systems, vol. 31, no. 2, pp. 1278– 1288, 2015. 2. C. Ye and M. Huang, “Multi-objective optimal configuration of current limiting strategies,” Science China Technological Sciences, vol. 57, no. 9, pp. 1738–1749, 2014. 3. Y. Dong, Z. Kang, and L. Yutian, “Coordinated optimization for controlling short circuit current and multi-infeed DC interaction,” Journal of Modern Power Systems and Clean Energy, vol. 2, no. 4, pp. 374–384, 2014. 4. J. Zhou, H. Hu, K. Zhuang, and R. zhu, “Short-circuit current of 500 kV East China power grid and its limitation,” East China Electric Power, vol. 34, no. 7, pp. 55–59, 2006. 5. G.-H. Moon, Y.-M. Wi, K. Lee, and S.-K. Joo, “Fault current constrained decentralized optimal power flow incorporating superconducting fault current limiter (SFCL),” IEEE Transactions on Applied Superconductivity, vol. 21, no. 3, pp. 2157–2160, 2010. 6. H.-T. Yang, W.-J. Tang, and P. R. Lubicki, “Placement of fault current limiters in a power system through a two-stage optimization approach,” IEEE Transactions on Power Systems, vol. 33, no. 1, pp. 131–140, 2017. 7. G. Didier, J. Leveque, and A. Rezzoug, “A novel approach to determine the optimal location of SFCL in electric power grid to improve power system stability,” IEEE Transactions on Power Systems, vol. 28, no. 2, pp. 978–984, 2012. 8. M. A. Baferani, M. R. Chalaki, N. Fahimi, A. A. Shayegani, and K. Niayesh, “A novel arrangement for improving three phase saturated-core fault current limiter (SCFCL),” in 2018 IEEE Texas Power and Energy Conference (TPEC), 2018, pp. 1–6: IEEE. 9. L. Wei, B. Chen, and J. Yuan, “Limiting mechanism and equivalent analytical model of a bridge-type saturated core fault current limiter,” Electric Power Systems Research, vol. 180, p. 106134, 2020. 10. S. Alaraifi, M. S. El Moursi, and H. H. Zeineldin, “Optimal Allocation of HTS-FCL for Power System Security and Stability Enhancement,” IEEE Transactions on Power Systems, vol. 28, no. 4, pp. 4701–4711, 2013. 11. W. Li, E. Vaahedi, and P. Choudhury, “Power system equipment aging,” IEEE Power and Energy Magazine, vol. 4, no. 3, pp. 52–58, 2006. 12. G. Li et al., “Risk analysis for distribution systems in the northeast US under wind storms,” IEEE Transactions on Power Systems, vol. 29, no. 2, pp. 889–898, 2013. 13. Z. Bie, Y. Lin, G. Li, and F. Li, “Battling the Extreme: A Study on the Power System Resilience,” Proceedings of the IEEE, vol. 105, no. 7, pp. 1253–1266, 2017. 14. M. Noe, M. Steurer, S. Eckroad, and R. Adapa, “Progress on the R&D of fault current limiters for utility applications,” in 2008 IEEE Power and Energy Society General Meeting-Conversion and Delivery of Electrical Energy in the 21st Century, 2008, pp. 1–4: IEEE. 15. J.-H. Teng and C.-N. Lu, “Optimum fault current limiter placement with search space reduction technique,” IET generation, transmission & distribution, vol. 4, no. 4, pp. 485–494, 2010. 16. P. Yu, B. Venkatesh, A. Yazdani, and B. N. Singh, “Optimal location and sizing of fault current limiters in mesh networks using iterative mixed integer nonlinear programming,” IEEE Transactions on Power Systems, vol. 31, no. 6, pp. 4776–4783, 2016. 17. M. El Moursi and R. Hegazy, “Novel technique for reducing the high fault currents and enhancing the security of ADWEA power system,” IEEE Transactions on Power Systems, vol. 28, no. 1, pp. 140–148, 2012. 18. A. Safaei, M. Zolfaghari, M. Gilvanejad, and G. B. Gharehpetian, “A survey on fault current limiters: Development and technical aspects,” International Journal of Electrical Power & Energy Systems, vol. 118, p. 105729, 2020. 19. Z. Bo, X. Lin, Q. Wang, Y. Yi, and F. Zhou, “Developments of power system protection and control,” Protection and Control of Modern Power Systems, vol. 1, no. 1, pp. 1–8, 2016. 20. H. Xie, Z. Bie, and G. Li, “Reliability-oriented networking planning for meshed VSC-HVDC grids,” IEEE Transactions on Power Systems, vol. 34, no. 2, pp. 1342–1351, 2018.

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21. F. Jiang, C. Tu, Q. Guo, Z. Shuai, X. He, and J. He, “Dual-functional dynamic voltage restorer to limit fault current,” IEEE Transactions on Industrial Electronics, vol. 66, no. 7, pp. 5300–5309, 2018. 22. H. Nourmohamadi, M. Sabahi, P. T. Balsara, E. Babaei, S. H. Hosseini, and A. FakhimBabaei, “New Concept for Fault Current Limiter with Voltage Restoration Capability,” IEEE Transactions on Industrial Electronics, 2020. 23. K. Deb and H. Jain, “An evolutionary many-objective optimization algorithm using referencepoint-based nondominated sorting approach, part I: solving problems with box constraints,” IEEE Transactions on Evolutionary Computation, vol. 18, no. 4, pp. 577–601, 2013. 24. L. Chen et al., “Pareto optimal allocation of resistive-type fault current limiters in active distribution networks with inverter-interfaced and synchronous distributed generators,” Energy Science & Engineering, vol. 7, no. 6, pp. 2554–2571, 2019. 25. J. H. Teng and C. N. Lu, “Optimum fault current limiter placement with search space reduction technique,” IET Gene. Transm. Distrib., vol. 4, no. 4, pp. 485–494, 2010. 26. K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGA-II,” IEEE transactions on evolutionary computation, vol. 6, no. 2, pp. 182– 197, 2002. 27. I. Das and J. E. Dennis, “Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems,” SIAM journal on optimization, vol. 8, no. 3, pp. 631–657, 1998. 28. R. D. Zimmerman, C. E. Murillo-Sánchez, and D. Gan, “A MATLAB power system simulation package,” ed, 2005. 29. L. Chen, M. Huang, J. Wu, D. Wang, and D. Gan, “An Optimal Strategy for Short Circuit Current Limiter Deployment,” in Asia-Pacific Power and Energy Engineering Conference, 2010, pp. 1–4.

Chapter 3

5G-Based Optimal Configuration of Centralized Fault Current Limiter in Power System

3.1 Introduction The reduction of SCCs has become an urgent task for grid planning divisions [1–3]. The studies in Chap. 2 focus on the planning configuration of the installed limiter, which is dispersed and autonomously regulated, taking into account the various installation impacts. However, with the development of 5G technology, centralized control of the fault limiter becomes possible, which is the focus of this chapter. Particularly, the fault current limiters (FCLs), one of the most appealing methods to handle short-circuit faults (SCFs), ensure that the SCC is switched off by CBs to prevent cascade failures. In particular, an FCL has a low impedance when it is not active but changes to a very large impedance when it is. Additionally, the stability of post-fault power systems can be increased and voltage sags can be mitigated by appropriately switching FCLs [4]. Transmission systems have largely embraced FCLs, particularly in China [5]. The mainstream FCLs include superconductors, solid-state, hybrid types, and so on [6]. Compared with other protective devices, the prices of FCL are relatively high [7]. To create both efficient and cost-effective schemes, the candidate allocation positions of FCLs are thoroughly researched [7–10]. These studies focused on the location and size of the FCL by minimizing installation costs while limiting the size of the SCC. In particular, a two-stage approach is proposed in [7] for FCL placement, which includes stage I sorting feasible solutions and stage II optimizing the FCL parameters. To select candidates for FCL installations, a sensitivity factor is established in [8]. To locate the superconductor FCL in the optimal place and increase system transient stability, the generator rotor angular factor is introduced in [9]. A multi-objective optimal FCL allocation model is developed in [10] by taking into account the impact of FCL on power system reliability. The studies mentioned above, meanwhile, use heuristics like particle swarm optimization and genetic algorithms (GA) to find the ideal FCL sites and sizes. An iterative mixed integer nonlinear programming method to allocate FCLs is proposed in [11], taking into account the restricted optimality and efficiency of heuristics while dealing with © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Ye et al., Risk-Based Planning and Operation Strategy Towards Short Circuit Resilient Power Systems, https://doi.org/10.1007/978-981-19-9725-9_3

35

36

3 5G-Based Optimal Configuration of Centralized Fault Current Limiter …

large-scale systems. The worldwide optimal configuration of FCLs is still difficult in terms of both economics and SCF defending effect, despite the tremendous efforts made in the prior-art studies. Deterministic fault scenarios are used in the economic aspect without taking the probability of SCF into account in earlier studies. Since the direct contact of conductors brought on by aging is the most direct cause of equipment SCF, there are often some statistical laws that can be seen about the occurrence of SCF, which are closely related to internal factors like aging [3]. Additionally, outside variables also raise SCF rates. For instance, the bulk of SCF-related power outages in the Northeastern United States is attributed to fallen trees [12]. Similarly, there is a connection between SCF and severe weather phenomena like hurricanes and wildfires. Many SCFs are caused by the SCC that results from lightning strikes into the phase-conductors of overhead power lines and cable damage from hurricanes [13]. It is crucial to take into account the stochastic properties of SCFs in the FCL allocation studies to prevent over-configuring FCLs to protect against a few uncommon candidate short-circuit events. In terms of technology, current research assumes that configured FCLs can only be executed on a local switch framework (LSF). The FCLs switch independently under the LSF without communicating to one another or a center; instead, they monitor the current variation near where they are mounted and raise impedances to lessen the excessive current surges [14]. Therefore, in addition to the essential FCLs, many more are likely to switch when a certain SCF happens. Furthermore, without taking into account their impact on nodal voltages, the LSF solely activates the FCLs based on the local limitation of SCCs. The voltage sags at some buses may not be efficiently reduced as a result of the LSF; instead, they may be made worse by the improper switching of FCLs [15, 16]. The impact of voltage sags on power quality and even the safe operation of the power system has been thoroughly investigated. Wind turbines without low voltage ride-through (LVRT) capability will be offline as a result of the heavy voltage sags [17]. Even worse, irregular voltage sags at the point of common coupling (PCC) buses may result in converter commutation failures with the integration of the multi-infeed High Voltage Direct Current system (MI-HVDC), which may temporarily stop the transfer of power [18]. In order to ensure the efficient and secure operation of the power system, the FCLs need typically be dispatched from a global perspective in particular situations. The centralized control of FCLs was thought to be impractical due to the old communication infrastructure’s significant delays and low dependability issues, which would interfere with the sequential coordination between FCLs and other security measures. Recent advancements in information and communication technology (ICT) [19] have led to the widespread adoption of terminals with communication modules in power systems, including PMUs and smart meters [19–21]. As a result, it is possible to use 5G to centrally update the switching logic of FCLs while retaining their sequential coordination with other security measures. In light of the foregoing, this research seeks to present a framework for centralized switching logic of flexible FCLs based on 5G as well as a technique for optimally

3.2 Framework of the 5G-Based CSF for FCLs

37

allocating such flexible FCLs in transmission networks. The following are this paper’s main contributions: 1. A centralized switch framework (CSF) of FCLs based on 5G is utilized from the standpoint of economic and security issues worldwide. In the event of a communication breakdown, a backup method is created to automatically swap FCLs after a certain interval, ensuring the protection’s dependability. 2. A FCL dispatching (FD) model is suggested, taking security limitations of fault current and voltage into account. The FD can provide online FCL switching schemes to fulfill the urgent need for power system safety by utilizing the quick communication capabilities of 5G and an offline fault detection technique.

3.2 Framework of the 5G-Based CSF for FCLs 3.2.1 Applying 5G for CSF of FCLs Considering the fact that FCLs are usually located around the substations, a central hub substation (CHS) is selected as the center for the 5G-based CSF for FCLs. The proposed framework has two main modules embedded in the CHS named Fault Location Determination (FLD) and FCL Switching Determination (FSD). As illustrated in Fig. 3.1, the 5G network is the communication carrier for signal transmission among various terminals during the real-time FCL operation. Considering the time for FCL dispatching globally in the whole system may not meet the requirement of fast SCC limitation, the system should be firstly divided into several subzones based on historical SCF records and the topology characteristics (e.g. the zones with MI-HVDC as shown in Fig. 3.1). Then a CHS will be selected to carry out the centralized FCL switching within the zone of interest. Specifically, the real-time voltage and current of each bus are measured by various sensors and then uploaded to the CHS, where the FLD program is implemented to precisely find the suspected fault locations. Take the three-phase SCF as an example, for bus g, the FLD logic can be given by the following equation. Ig =

Σ

Ii > 0

(3.1)

i∈NB

Vg = V0 → 0

(3.2)

where V 0 is the voltage of the ground and NB is the set of the branches linked to bus g. I g is the ground current of bus g and satisfies Eq. (3.1) if the fault occurred at bus g. V g is the voltage of bus g and satisfies Eq. (3.2) when the fault is detected. As indicated by the equations above, the FLD module is simple and only needs local

38

3 5G-Based Optimal Configuration of Centralized Fault Current Limiter …

Fig. 3.1 Diagram of the 5G-based CSF for FCLs

measurements within the zone, consequently, the communication and computing tasks can be finished online. After the SCF locations are confirmed, the FSD module is performed by the CHS. The FSD determines the optimal FCL switching schemes based on the FD model, which will be detailed and formulated in the next section. The optimal switching schemes are then broadcast to the terminal FCLs in the zone to perform the SCC limitation. As shown in Fig. 3.2, the total required time for FCL switching t S can be represented by: t S = 2tC + t M +t F +t F L D +t F S D

(3.3)

where t C is the time for communication, which only takes a few milliseconds with the deployment of smart sensors and 5G network [21]. t M and t F are the measuring Zone 1 Distance-based Protection( 3 cycles ) Relay Action Time (0.5 cycles)

CB Action Time(2.5 cycles)

Time for Commercial FCL Operation (tS )

CB Cut Off

Time of CSF Operation

2tC+tM+tFLD+tFSD

SCF Indentification

tF CHS Signal arrival

FCL Switching

Fig. 3.2 Time requirements for FCL switching and CSF operation

3.2 Framework of the 5G-Based CSF for FCLs

39

time of sensors and switching time of FCLs, respectively. t FLD and t FSD are the time for running the FLD and FSD modules, respectively. It is worth noting that t M and t F are consistent with the time for commercial FCLs under the prior-art LSF. As a result, the additional time required under the proposed CSF is the sum of 2t C , t FLD and t FSD . The time requirements of FCL and proposed CSF operation are given in Fig. 3.2. In practice, the FCLs should be switched before the cut-off of the corresponding CB conducted by the zone-1 distance-based protection [22]. Since the zone-1 distance-based protection is implemented after the action of the corresponding relay and CB, the time for the FCL operation (t S ) is expected to be within the sum of the time for the relay and CB actions, which is around three cycles [23]. Meanwhile, considering the FCL switching will have an influence on the activated CB’s selection made by the corresponding relay, the FCL switching results made by the CSF operation need to be transmitted to the relay before its implementation. Therefore, the time of CSF operation (t M + 2t C + t FLD + t FSD ) is expected to be within the time of relay action, which is around one-half cycle [23].

3.2.2 Backup Strategy in Case of 5G Communication Failure As seen from the operation process of the CSF, the role of the 5G network is to exchange lots of information among sensors, CHS, and FCLs. Therefore, the FCL switching is highly dependent on the working status of the 5G network. To improve the reliability, the prior-art LSF is retained as a backup protection strategy to conduct the FCL switching in case of 5G failure or other conditions causing the CSF paralysis. Specifically, when the FCL detects the SCC, it reserves a pre-set delay Δt to implement the autonomous switching. As shown in Fig. 3.3a, the delay Δt should cover the average CSF signal arrival duration in normal circumstances so that the FCLs can be switched by the CSF first. Meanwhile, under the 5G failure circumstance, as shown in Fig. 3.3b, the Δt should ensure that the FCL can be timely switched by the backup strategy before CB is cut off, which is expressed by: Δt+t M +t F ≤ t Z

(3.4)

where t Z is the time for zone-1 distance-based protection, which is usually around three cycles. As a result, if the FCL cannot receive the CSF signal from the CHS within Δt (Δt < t M + 2t C + t FLD + t FSD ), the FCL will automatically switch its status with the prior-art LSF logic after Δt, which can be expressed in the form as: [ v=

1, i f : I 1 ≥ λ · I 0 0, i f : I 1 ≤ λ · I 0

(3.5)

40

3 5G-Based Optimal Configuration of Centralized Fault Current Limiter … Zone 1 Distance-based Protection Relay Action Time (0.5 cycles)

tZ =3cycles

CB Action Time(2.5 cycles)

Δt 2tC+tM+tFLD+tFSD < Δt

SCF Indentification

CB Cut Off

tF CHS Signal arrival

FCL Switched by CSF

(a) Normal circumstance with the proposed CSF Zone 1 Distance-based Protection Relay Action Time (0.5 cycles)

tZ =3cycles

CB Action Time(2.5 cycles) Time of LSF operation

Δt 2tC+tM+tFLD+tFSD > Δt

tM+tF

CB Cut Off

FCL Back up Strategy Switched by LSF Initiation

SCF Indentification

(b) 5G failure circumstance with the backup protection strategy

Fig. 3.3 Diagram of the backup strategy under different circumstances

where v is a 0–1 decision variable. V = 1 means the FCL is used and v = 0 means the opposite. I 1 and I 0 are the pre- and post-fault currents measured by the FCL, respectively. λ is the pre-set coefficient of excessive current on the FCL. With the above backup strategy enabled by a reasonably pre-set delay, the FCLs can be switched to limit SCC effectively and timely even if the 5G network fails. In this way, the proposed CSF can be feasible to be applied in power system protection with the high-reliability requirement.

3.3 Formulation of the FD Model The main purpose of FCL switching is to ensure the fault current within a limit and suppress the voltage sags with fewer switched impedance. As a result, the proposed FD model is a multi-objective problem aiming at minimizing the operation costs of FCL, which are denoted as the penalties for the FCL switching amounts and the out-of-range voltage sags, shown as: Min : α1

Σ f ∈F

x f v f +α2

Σ

{ } max ΔV j − ΔV jT , 0

(3.6)

j∈NC

where α 1 and α 2 refer to the penalty coefficients of FCL switching and voltage sags, respectively. V f is the switching decision of the f -th FCL. X f is the installed impedance of the f -th FCL, which is a constant in the FD model. F is the set of the installed FCLs.

3.3 Formulation of the FD Model

41

Meanwhile, the max function max{ΔV j − ΔV j T , 0} is used to describe the voltage sag magnitude at bus j. NC is the set of buses. ΔV j is the post-fault voltage sag of bus j and ΔV j T is the corresponding voltage sag threshold. The above max function can be linearized by: ⎧ h= max{ y, 0} ⎪ ⎪ ⎨ y − (1 − δ)M ≤ h ≤ y + (1 − δ)M ⎪ −(1 − δ)M ≤ y ≤ δ M ⎪ ⎩ 0 ≤ h ≤ δM

(3.7)

where δ is a 0–1 variable and M is a large value. In addition to the objective function, constraints of SCC magnitudes and voltage sags should also be considered in the FD model.

3.3.1 Fault SCC Magnitude Constraint As the most severe case, the three-phase SCC I S,i can be obtained by [24]: I S,i = Vi0 /(Z ii0 + ΔZ ii )

(3.8)

where Vi0 is the pre-fault voltage of bus i, which is assumed to be 1 p.u. Z ii0 is the pre-fault self-impedance of bus i, which can be obtained from the diagonal entries of the pre-fault bus impedance matrix Z. ΔZ ii is the incremental self-impedance after FCL switching, through the Thevenin equivalence [24], it can be formulated by: ΔZ ii =

Σ f ∈F

vf

xlk0

+

0 2 (Z il0 − Z ik ) 0 2 0 [(xlk ) /x f ] − Z ll −

0 0 Z kk + 2Z lk

, ∀l, k ∈ NC

(3.9)

where the f -th FCL is assumed to be installed on the branch between bus l and k 0 0 (branch l − k). xlk0 is the original impedance of branch l − k. Z il0 , Z ik , Z ll0 , Z kk , and 0 Z lk are corresponding elements in matrix Z. The SCC constraint in the FD model can be expressed as: I S,i ≤ I S,i ≤ I S,i

(3.10)

where I S,i is the upper bound of SCC magnitude, which is usually set as the cut-off limit of the deployed CBs. I S,i is the lower bound of SCC magnitude, which is set as the ratio between measured voltage and setting impedance value of the corresponding relay in order to ensure the reduced SCC by FCL switching can stimulate the CB action.

42

3 5G-Based Optimal Configuration of Centralized Fault Current Limiter …

3.3.2 Voltage Sag Constraint For a three-phase SCF at the bus i, the voltage sag ΔV j can be calculated by [25]: ΔV j =

Z i0j +ΔZ i j Z ii0 +ΔZ ii

Vi0 : ∀ j ∈ NC

(3.11)

where Vi0 are the pre-fault voltage of SCF bus i. Z i0j is the pre-fault mutual-impedance between bus i and j. ΔZ ij is the incremental mutual-impedance between bus i and j after the f -th FCL switching, through the Thevenin equivalence, it can be formulated by: ΔZ i j =

Σ f ∈F

vf

0 (Z il0 − Z ik )(Z 0jl − Z 0jk ) 0 0 xlk0 + [(xlk0 )2 /x f ] − Z ll0 − Z kk + 2Z lk

, ∀ j, l, k ∈ NC

(3.12)

where Z 0jl and Z 0jk are corresponding elements of Z as well. Typically, the voltage sag threshold ΔV j T is determined based on the nodal voltage requirement. For buses linked with wind turbines, their voltage sag thresholds ΔV j T should be set as the critical offline voltage sag V w C . Moreover, for zones with MIHVDC as shown in Fig. 3.1, the voltage sags at the PCC buses may cause HVDC commutation failure, specifically, they may cut down the extinction angle γ of the converters below this minimum allowable value γ m min [26]. As a result, if bus j is a PCC bus linked with converter m, the voltage sag threshold ΔV j T can be obtained by: ΔV jT

=

V j0

√ 2n m Im,DC X m − cos γmmin + cos θm

(3.13)

where I m,DC is the rated DC current of converter m. nm is the transformer ratio of converter m. X m and θ m are the equivalent commutating reactance and the firing angle of converter m, respectively.

3.3.3 Offline Fault Scanning Scheme for Online Decision As seen from Eqs. (3.6)–(3.13), the FD model is a mixed integer quadratic constrained problem (MIQCP) which contains the product of the binary variable vf ∈ [0,1] and the continuous variable h, and the product of two binary variables vf ∈ [0,1] and δ ∈ [0,1]. A linear equivalent method for product vf h and vf δ can be found through the method introduced in the appendix of [27]. Then the proposed MIQCP can be transferred into a MILP to obtain a global optimal solution. However, the state-of-art commercial solvers such as CPLEX may not be capable to solve the above MILP

3.4 Description of the FCL Allocation Approach

43

Offline All Potential SCF Scenarios

FD Model [Eq.(3.6)(3.13)]

FCL Switching Set

Online SCF Locations

SCF Scenario Identification

Real Time SCF Scenario

Optimal FCL Switching Schemes

Fig. 3.4 The offline fault scanning scheme to enable online FCL dispatching

online. Therefore, an offline fault scanning scheme is proposed to facilitate the FCL dispatching to meet the time requirement of power system protection. As shown in Fig. 3.4, the FD model is solved offline and an optimal FCL switching set based on all the potential SCF scenarios is formulated. During the real-time operation, the CHS firstly identifies the SCF scenario using the FLD and then directly calls the corresponding optimal FCL switching scheme by searching in the offline built FCL switching set. It is worth noting that the potential SCF scenarios should be renewed routinely according to the future expansions or reconfiguration of the transmission system. Besides, the offline fault scanning algorithm should be continually running to find the FCL switching schemes for the new potential SCF scenarios. Meanwhile, if the SCFs that are not considered in the offline optimization occur, the FD model has to be solved online to find the FCL switching scheme. Besides, if the FD model cannot be solved and transmit the switching scheme to FCLs within the pre-set delay Δt introduced, the backup strategy will also be initiated and then the FCLs switch autonomously based on Eq. (3.5).

3.4 Description of the FCL Allocation Approach As aforementioned, with the CSF enabled by 5G, the number of switched FCLs can be reduced compared with the prior-art LSF. Thus, if the centralized switching characteristics are involved in the planning stage, the FCLs that rarely operate under specific scenarios are prone to be excluded from the suggested scheme. In this section, a scenario-based bi-level FCL allocation approach is developed. As shown in Fig. 3.5, the proposed bi-level FCL allocation model is scenariobased, in which the assessed nodal SCF probabilities and candidate FCL locations are utilized as inputs. Meanwhile, the upper-level problem minimizes the sum of FCL installation and expected operating costs while the lower-level problem minimizes the expected operating costs of FCLs in each SCF scenario. Specifically, the upperlevel problem sends the temporarily optimal FCL allocation scheme to the lower level and receives the minimum expected FCL operation cost, the interaction continues

44

3 5G-Based Optimal Configuration of Centralized Fault Current Limiter …

SCF scenario probabilities

Minimize the installation costs of FCL Placement

SCF scenarios and candidate FCL placement locations

Upper-level problem

Installation results of FCLs Grid Planner

Lower-level problem

System operator

Optimal installed locations and impedances of FCLs

Minimize the expected costs of FCL Operation Expected penalty costs of FCL switching and voltage sag out of range considering all SCF scenarios

Optimal switching of installed FCLs in each scenario

Fig. 3.5 Framework of the scenario-based bi-level FCL allocation approach

until an equilibrium result is reached, which contains the optimal siting and sizing scheme of FCLs as well as the optimal switching plan of the installed FCLs.

3.4.1 Description of the FCL Allocation Approach In this subsection, a scenario generation approach is utilized to consider nodal SCF possibilities in the power system. According to the calculated three-phase SCCs, the SCF buses are selected by: 0 I S,i = Vi0 /Z ii0 > I Bmax , ∀i ∈ NS

(3.14)

0 where I S,i is the SCC of bus i before FCL installation. I Bmax is the maximum CB cut-off current. NS is the set of the SCF buses. In this paper, the covariates affecting the SCF possibility include insulation level, aging condition, weather, and surrounding environment, which can be measured by the indices shown in Fig. 3.6. Specifically, the insulation level is assessed by equipment electric strength E s equaling to the ratio between the breakdown voltage V b and insulation thickness d; the aging condition is evaluated by equipment utilization rate U R which equals the ratio between operating time t o and designed working lifespan S; the weather condition is described by extreme weather frequency F E which equals to the ratio between the number of extreme weather occurrence N E and the number of operation

3.4 Description of the FCL Allocation Approach Insulation Level

Equipment Electric Strength

Aging Condition

Equipment Utilization Rate

Weather Condition

Extreme Weather Frequency

Surrounding Environment

45

SCF Scenario Probabilities

Vegetation Coverage Rate

Fig. 3.6 Indexes used to formulate factors affecting the SCF probabilities

cycle N T ; the surrounding environment is evaluated by vegetation coverage rate V C , which equals to the ratio between the vegetation projected area S V and total area S B occupied by the corresponding bus. The PHM is used to describe SCF probabilities subject to the above factors [28]. The SCF probability pi of bus i can be obtained by: [

pi = p0 eγ2 U R +γ3 FE +γ4 VC ) p0 =γ1 E s

(3.15)

where p0 is the base SCF probability, which is assumed to be only affected by the insulation level. γ 1 , γ 2 , γ 3 , and γ 4 are the parameters that can be obtained by regression analysis based on historical statistics of SCFs and the measured values of the factor within a certain period.

3.4.2 Formulation of the Bi-level FCL Allocation Model The objective function of the scenario-based bi-level FCL allocation model is introduced as follows: [ ] min C1 (u f , x f ) + min C2 (v f , h j )

(3.16)

where the upper-level objective function is to minimize the sum of the FCL installation costs and the corresponding FCL expected operating costs given from the lower-level problem. The lower-level problem C 2 (vf , hj ) is to minimize the expected FCL operating costs including the expected penalty costs for both the switched FCLs and out-of-range voltage sags. They can be specifically written as: C1 (u f , x f ) =

Σ f ∈F

(u f c F, f + cV , f x f )

(3.17)

46

3 5G-Based Optimal Configuration of Centralized Fault Current Limiter …

C2 (v f , h j ) =

Σ i∈NS

⎛ pi ⎝α1

Σ

x f v f +α2

f ∈F

Σ

⎞ h j⎠

(3.18)

j∈NC

where the 0–1 variable uf determines the installation status of the f -th FCL. X f is the installed impedance variable of the f -th FCL. C F, f , and cV, f is the fixed and variable installation cost coefficients of the f -th FCL. Besides, the variables and parameters in C 2 (vf , hj ) have already been introduced in the FD model. The upper-level objective function is subject to constraints formulated in Eqs. (3.19) and (3.20). 0 ≤ x f ≤ u f x max f , ∀f ∈F

(3.19)

u f ∈ [0, 1], ∀ f ∈ F

(3.20)

where Eq. (3.19) limits the installed impedance of the FCL. 0–1 variable uf = 1 means the f -th FCL is installed and uf = 0 means the opposite. After solving the upper-level model, the installed FCL impedance x f will be transmitted to the lower-level model and treated as input parameters. Meanwhile, since the FD model is integrated into the lower-level model, the lower-level objective function is subject to the SCC magnitude constraints (Eq. 3.10), the voltage sag constraints of each bus (Eq. 3.11). Then the minimum FCL operating costs will be transmitted from the lower level to the upper level and treated as a constant.

3.4.3 Solution Method As aforementioned, the lower-level problem is the FD model that can be equivalently transferred as a MILP, which renders the proposed model becomes a bi-level MILP problem. In this paper, a hybrid method integrating the GA and the MILP solver is adopted. The detailed solving procedure can be divided into the following steps. Step 1: Formulate the SCF scenarios based on Eq. (3.14) and SCF probabilities based on Eq. (3.15). Then select the candidate FCL placement locations by a two-stage method given in Appendix B. Step 2: Generate the initial installation results through the given input parameters including the SCF scenarios, SCF probabilities, and candidate FCL placement locations. Step 3: Obtain the optimal switching scheme of FCL under each SCF scenario based on the FCL installation results by solving the FD model through the MILP solver.

3.4 Description of the FCL Allocation Approach

47

Step 4: Calculate the sum of the FCL installation costs and the expected penalty costs of switched FCLs and out-of-range voltage sags obtained from the FD model as the fitness value. Step 5: Generate new installation results of FCLs by genetic manipulation including selection, crossover, and mutation. Step 6: Repeat Step 3–5 until the predefined maximum generation is reached.

3.4.4 Case Study In this section, the IEEE 39-bus system is modified as a representative test zone with MI-HVDC to validate the effectiveness of the proposed model and methods. As shown in Fig. 3.7, a three-infeed HVDC system is linked with the 39-bus system, and buses 7, 8, and 9 are the PCC buses. The detailed network parameters are available in [29]. The operating parameters of the three-infeed HVDC system are given in [26]. The threshold voltage sags at the AC load buses and PCC buses are set to be 0.7 and 0.6 p.u., respectively. The penalty coefficients for the switched FCL and out-of-range voltage sags are set to be 10,000 and 1000/p.u., respectively. Considering the events with second-order or simultaneous SCFs among different buses are quite rare, they are ignored and thus only the events with SCF that occur

=

~

G

AC line DC line Converter

G

Generator

2

G

30

37 25

18

3

38

17

G

39

21 15 4

16 24

14

G

=

~

5

13

9

36

23

6

19 12

=

~

Threeinfeed HVDC System

29

27

1

G

28

26

22

11 7

20 10

31

8

G

=

~

Fig. 3.7 Modified New England 39-bus testing system

32

G

34

G

33

G

35

G

48

3 5G-Based Optimal Configuration of Centralized Fault Current Limiter …

Table 3.1 SCF scenarios and corresponding probabilities Scenario

Bus

SCC without FCL (p.u.)

SCF probability (p)

1

20

22.3836

0.04

2

31

21.8847

0.05

3

32

20.6055

0.06

4

33

23.1535

0.02

5

34

38.3615

0.04

6

36

25.1352

0.08

7

38

26.7188

0.03

Table 3.2 Candidate branches and parameters for FCL installation No

Candidate branches (from bus-to bus)

cF, f ($million)

cV, f ($million/p.u.)

x max (p.u.) f

1

10–32

0.5

1

1

2

16–19

0.6

1

0.5

3

19–20

0.8

1

1.5

4

19–33

0.3

1

0.8

5

20–34

0.4

1

1

at one single bus location are considered in this paper. If the maximum CB cut-off current I Bmax is set to be 20 p.u., then 7 buses can be selected for SCF consideration by Eq. (3.14) and the events of SCF that occur at these buses can be classified into 7 independent SCF scenarios in the proposed FCL allocation method. The probability of each SCF scenario can be obtained through the PHM model expressed by Eq. (3.15), which is shown in Table 3.1. The candidate branches and parameters for FCL installation are given in Table 3.2.

3.4.4.1

Results of the Proposed FCL Allocation Method

Through the proposed bi-level method, both the optimal FCL installation results and the switching scheme under each SCF scenario can be obtained. The impedance of the installed FCLs on each candidate branch is given in Table 3.3. Apparently, with the minimal installation costs and the maximum installable impedance, branch 19–33 is deployed with the largest impedance FCL, reaching 0.792 p.u. Meanwhile, with the highest investment costs, the FCL installed on branches 19–20 has the lowest installed impedance. The corresponding overall FCL installation cost is $2.613 million. The switching schemes of the installed FCLs under expected SCF scenarios are shown in Fig. 3.8.

3.4 Description of the FCL Allocation Approach Table 3.3 Impedance of installed FCL in each candidate branch

49

Branches

Installed impedance of FCL x f (p.u.)

1

10–32

0.753

2

16–19

0.231

3

19–20

0.103

4

19–33

0.792

5

20–34

0.234

Inactivated FCL

Activated FCL Scenario 1 and 5

Scenario 2

Scenario 3

Switched FCL Impedance

Switched FCL Impedance

Switched FCL Impedance

0.103 p.u.

0.895 p.u.

0.987 p.u.

Scenario 4

Scenario 6

Scenario 7

Switched FCL Impedance

Switched FCL Impedance

Switched FCL Impedance

0.231 p.u.

1.648 p.u.

1.545 p.u.

Fig. 3.8 Switching scheme of the installed FCLs under different SCF scenarios

As can be seen, the switching schemes of FCLs are different under the expected scenarios, indicating the necessity of the proposed CSF. Specifically, the switching schemes are consistent under scenarios 1 and 5, in which only FCL 3 is activated and the impedance is the lowest, reaching 0.103 p.u. Under scenario 6, three FCLs are switched and the corresponding impedance is the highest, reaching 1.648 p.u. The fault current reduction effect under each scenario is illustrated in Fig. 3.9. It can be seen that the SCC can be limited between the given range of 20 p.u. ~ 10 p.u. under all SCF scenarios by switching the installed FCLs. Meanwhile, under scenario 5, the switching of FCLs achieves the most significant SCC reduction effect, from 38.36 to 12.17 p.u., specifically. It can be demonstrated by the above analysis that the proposed bi-level method is capable to ensure the secure operation of the power system considering multiple scenarios.

3 5G-Based Optimal Configuration of Centralized Fault Current Limiter …

Fault Current (p.u.)

50 50 45 40 35 30 25 20 15 10 5 0

With FCL

No FCL

SCC Upper Limits SCC Lower Limits

1

2

3

4

5

6

7

Scenario Fig. 3.9 SCC reduction effect under each SCF scenario

3.4.4.2

Comparison with Traditional Allocation Methods

In this section, the FCL allocation results of the proposed method are compared with several benchmark methods to illustrate the advantages of probabilistic modeling as well as the CSF. Four cases are considered as follows: • Case 1: FCLs are configured by the proposed bi-level allocation model with the proposed CSF based on the probabilistic SCF scenarios. • Case 2: FCLs are configured by the same bi-level allocation model as Case 1 based on the deterministic SCF scenarios (SCF probabilities in Table 3.1 are all set at 1.0). • Case 3: FCLs are configured by the traditional allocation model in [7] based on the probabilistic SCF scenarios. The traditional allocation model assumes the FCL switching follows the LSF in the form of Eq. (3.5). • Case 4: FCLs are configured by the same allocation model as Case 3 based on the deterministic SCF scenarios. A. Comparison of FCL Allocation Results If the coefficient λ is assumed to be 5, the FCL allocation schemes of the above four cases are shown in Fig. 3.10. Comparing the allocation results of Case 1 and 2, the installed impedance has a relatively notable change in FCL 2 and 5. Specifically, when the probabilistic SCFs are considered, the installed impedance in FCL 2 decreases from 0.494 p.u. in Case 2 to 0.234 p.u. in Case 1. Meanwhile, the variation of installed impedances of FCL 1 and 4 is relatively less obvious, which seems that the SCF probabilities have little effect on them. Moreover, compared to the allocation results between Case 1 and Case 3, the variation of installed impedance is quite significant in FCL 3, from 0.103 p.u. in Case 1–1.220 p.u. in Case 3. It may due to the fact that excessive current surges are relatively high on the branch where FCL 3 is installed. Consequently, FCL 3 acts more frequently in the LSF and thus larger impedance value is assigned to this FCL.

3.4 Description of the FCL Allocation Approach

51

Fig. 3.10 Allocation results of FCLs in different cases

The FCL investment costs in different cases are given in Table 3.4. With the consideration of both the FCL centralized switching and the probabilistic SCFs, the FCL installation cost is significantly reduced, which drops from $4.482 million in Case 4 to $2.613 million in Case 1. Moreover, comparing the investment costs between Cases 1 and 2, it can be found that by introducing probabilistic SCFs, the FCL investment cost effectively decreases by nearly 20%. Meanwhile, comparing the investment costs between Cases 1 and 3, the investment cost decreases by over 34% from LSF to the proposed CSF. These results indicate that the proposed method is feasible to improve the economy of FCL allocation. If wind power is involved in the studying case, the voltage sag constraints for buses should be considered to prevent the off-line accident as aforementioned. If the generators in buses 33 and 34 in Fig. 3.7 are connected with wind turbines without LVRT capability. And the critical offline voltage sag Vw C equals 0.7 p.u. The corresponding FCL investment costs are compared in Table 3.5. It can be seen that the investment costs of both cases are increased with the wind turbine considered. It is because the added constraints in voltage sag caused by wind integration lift the installed FCL impedances needed for switching during SCF. Besides, the increased investment cost in Case 2 is larger than in Case 1. It seems Table 3.4 FCL investment costs in different cases Cases

Case 1

Case 2

Case 3

Case 4

Investment cost ($million)

2.613

3.236

3.960

4.482

Table 3.5 FCL investment costs with different wind turbine considerations Cases Investment cost ($million)

Wind power

Thermal power

Case 1

Case 2

Case 1

Case 2

2.613

3.236

3.014

3.862

52

3 5G-Based Optimal Configuration of Centralized Fault Current Limiter …

Table 3.6 Switched impedance of FCLs in Case 1 and Case 3 Switched FCL impedance (p.u.) Scenarios

1

2

3

4

5

6

7

Case 1

0.103

0.895

0.987

0.231

0.103

1.648

1.545

Case 3

0.476

1.192

1.372

1.276

0.572

2.020

2.020

that by involving the probabilistic SCF scenarios, the increased FCL investment cost can be reduced. B. Comparison of FCL Switching Results In order to compare the influences of different FCL switch frameworks, the switched FCL impedance under each SCF scenario in Cases 1 and 3 are given in Table 3.6, where λ is still assumed to be 5. It can be seen that the switched FCL impedance decreases under all SCF scenarios when changed from the LSF in Case 3 to the proposed CSF in Case 1. Specifically, the switched impedance in Case 3 is consistent under scenarios 6 and 7, which means the switched FCLs are the same under these two scenarios with the LSF. However, with the proposed CSF, the switched FCL impedance under both scenarios is reduced and inconsistent. It can be concluded that the proposed centralized framework can better utilize the allocated FCLs to limit the SCCs and reduce the redundant FCL switching. C. Comparison of Voltage Sag Mitigation Effect To compare the effects of different FCL switch frameworks in terms of voltage sag mitigation, the number of buses with abnormal voltage sags in Cases 1 and 3 are given in Table 3.7. As can be seen, the number of load buses with out-of-range voltage sag is effectively reduced under all SCF scenarios when changed from the LSF in Case 3 to the proposed CSF in Case 1. Meanwhile, the detailed voltage sags at load buses in Case 1 and 3 under scenario 6 are given in Fig. 3.11, in which the voltage sags with no FCL installation are added for comparison. It can be seen that the LSF in Case 3 cannot effectively reduce the voltage sags for load buses 21, 23, and 24 which locate quite close to the SCF bus 36. Even worse, the voltage sags at bus 29 increase from 0.73 to 0.75 p.u. after the installation of FCLs. However, when changed from the LSF to the proposed CSF in Case 1, the Table 3.7 Numbers of load buses with voltage sag out of range Number of load buses with voltage sag out of range Scenarios

1

2

3

4

5

Case 1

5

6

6

7

5

7

7

Case 3

7

8

8

9

7

11

10

6

7

3.4 Description of the FCL Allocation Approach

53

Voltage Sag Amplitude (p.u.)

1.00

Case 1

Case 3

No FCL

0.95 0.90 0.85 0.80 0.75 0.70

Voltage Sag Limit

0.65 0.60 0.55 3

4

7

8

12

15

16

18

20

21

23

24

25

26

27

28

29

31

39

Load Bus Fig. 3.11 Voltage sag at different load buses indifferent Cases under scenario 6

voltage sag amplitudes of the above load buses are improved substantially. Besides, the voltage sags of AC load buses are suppressed around or under the AC threshold (0.7 p.u.) while the voltage sag amplitudes of PCC buses 7 and 8 are restricted within the PCC threshold (0.6 p.u.). These results indicate that the proposed CSF can better mitigate the voltage sags during the SCFs compared to the LSF, which further helps to improve the power quality for loads and prevent HVDC commutation failure for PCC buses.

3.4.4.3

Extended Test Systems

A. Calculation Time To compare the total calculation time for getting the FCL switching results in different test systems, we simulate the FD model in the modified IEEE 118-bus system [30] and 300-bus system [31] receptively. It’s assumed that buses 2, 5, and 8 are the PCC buses for the 118-bus system, and buses 17, 20, and 25 are the PCC buses for the 300-bus system. The constraints for scenario generation and candidate FCL location selection are the same as those in the 39-bus system introduced above. Considering all the SCF scenarios, the average calculation time for the FD model in each test system is given in Table 3.8. It can be seen that the calculation time increases with the expansion of the test system scale. Specifically, calculation time under the 39-bus system meets the time requirement of CSF operation (one-half circle), which means the FCL switching Table 3.8 Average calculation time for getting FCL switching results

Test system Modified IEEE Modified IEEE Modified IEEE 39 bus 118 bus 300 bus Time (ms)

6.12

37.46

120.34

54

3 5G-Based Optimal Configuration of Centralized Fault Current Limiter …

can be conducted by the FD model online. However, the calculation time under the 118-bus and 300-bus systems is beyond the time requirement, which reflects the significance of the proposed offline fault scanning scheme. B. Communication Time In this section, the communication times for the proposed CSF with different communication platforms are compared. The communication time t C is assumed to be made up by: tC = t I L + tT R

(3.21)

where t IL is the inherent latency of the network, which equals to the time for data access and decoding between terminals and network base station. For the 5G and 4G networks, t IL is 1 ms and 10 ms, respectively [32]. t TR is the transmission time of the network, which equals to: tT R =

NC Σ

Bi /v N

(3.22)

i=1

where Bi is the bytes of data at bus i to be communicated with the CHS, which usually contains the sampled amplitudes and phase angles of voltage and current for each linked feeder as well as the switching instruction of the installed FCLs. The size of Bi is similar to the data required for PMU communication which is around hundreds of bytes [33]. vN is the average data rate of the network, which equals to 1 Gbps and 50 Mbps for 5G and 4G networks, respectively [32]. It can be seen from Eq. (3.22) that the t TR will increase with the expansion scale of the buses communicating with the CHS. Assuming that Bi for each bus equals to 512 bytes, the communication times for the 5G and 4G networks in the different test systems are given in Table 3.9. It can be seen from the results that with the increasing number of buses from the 39-bus system to the 300-bus system, the scale of data communication between CHS and buses are expanded, which lifts the communication times. However, by using the 5G network the communication times under each test system are all below 2.5 ms, which leaves enough time for the CHS to run the FLD and FSD module to get the FCL switching decision. Meanwhile, by using the 4G network, the communication times increases significantly from 3.04 to 23.36 ms, which obviously exceeds the Table 3.9 Communication times for 4G and 5G networks Network

4G network t IL (ms)

5G network

t TR (ms)

t C (ms)

t IL (ms)

t TR (ms)

t C (ms)

Modified 39-bus system

10

3.04

13.04

1

0.15

1.15

Modified 118-bus system

10

9.19

19.19

1

0.45

1.45

Modified 300-bus system

10

23.36

33.36

1

1.15

2.15

References

55

time required for CSF operation (one-half cycle) The results verify the 5G network’s advantage in enabling the signal transmission to meet the millisecond-level time requirement for communication under the proposed CSF.

3.5 Conclusion In this paper, a bi-level model is proposed to find the optimal allocation scheme of centrally switched FCLs in a 5G-based framework, in which the probabilistic characteristics of SCFs are considered. Security constraints of SCC magnitudes and voltage sags are integrated into the proposed model. A hybrid GA and MILP solution method is used to solve the proposed bi-level problem. Numerical results demonstrate that the proposed scenario-based bi-level method is capable to defend against out-of-range SCCs. Meanwhile, the proposed method effectively reduces allocation investment costs and promotes the effect of FCL switching in voltage sag mitigation. With the continuous deployment of sensors and improving ICT infrastructure in power systems, the real-time precisely sensing of SCF locations and the online switching of FCLs are becoming increasingly realistic, which is expected to be deployed in future power dispatching centers.

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Chapter 4

A Multi-state Model for Power System Resilience Enhancement Against Short-Circuit Faults

4.1 Introduction A resilient power system should be able to withstand extreme occurrences and recover quickly [1, 2]. One of the main goals of smart grid development has been to increase grid resilience to extreme weather events [3]. The prior-art research mostly focused on expected line damages caused by hurricanes or wildfires [4–6]. However, many contingencies triggered by extreme weather events evolved with short-circuit faults (SCFs), e.g. the massive power blackout in Brazil on 10th November 2009 [7]. In particular, during hurricanes, heavy rain frequently results in SCFs at cable connections with compromised insulation, and powerful winds may uproot trees and result in SCFs for overhead lines [8]. SCFs exhibit a typical cascading impact on power systems. The response architecture against extreme weather occurrences for transmission systems should take the SCC constraint into account. Common SCC limiting strategies include the deployment of fault current limiters (FCLs) [9, 10], switch shifting of lines [11], or bus splitting [12]. The feasibility of genetic algorithms (GA) for SCC limiting measure configuration has been reported [13]. The maintenance of transient stability is another essential function for the power system to resist SCFs. A combined numerical discretization with the Interior Point Method (CNDIPM) [14] has been adopted as a mainstream solution for the transient stability constrained optimal power flow (TSCOPF) problems. In distinct spatial and temporal situations, the prior-art SCC restricting procedures and TSCOPF operations are often implemented individually. The power system is therefore unable to withstand the cascading impacts of SCF brought on by the extreme weather event under the obtained solutions. Furthermore, as aforementioned, the disturbance of extreme weather events on the power system is indeed probable [15]. Robust optimization techniques in a min-maxmin framework have been extensively used in grid resilience studies of the past to provide the best hardening or emergency power supply measures for power systems

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Ye et al., Risk-Based Planning and Operation Strategy Towards Short Circuit Resilient Power Systems, https://doi.org/10.1007/978-981-19-9725-9_4

59

60

4 A Multi-state Model for Power System Resilience Enhancement …

against the worst fault scenario [16–18]. Consequently, the low probability or rarity of the most severe fault leads to a relatively excessive cost. In this study, the post-fault instability of units and excessive SCC magnitudes are used to examine the cascade effects of SCFs on power systems. In order to give an ideal resilience enhancement method for the electricity system taking the unpredictability of weather forecasts and cost-effectiveness into account, a multistate model is constructed. The main contributions of this paper are: 1. In order to ensure system resilience against the cascading effects of triggered SCFs, an integrated extreme weather response framework for transmission systems is proposed. This framework makes use of a MINLP model to provide the best line switch shifting, generation rescheduling, and load shedding schemes. 2. With the NAS forecasts being unclear, a multi-state model is created to find the most cost-effective grid resilience upgrade plan, taking into account the anticipated urgent adaptations of the initial scheme due to weather state transition.

4.2 Extreme Weather Event Response Schema In this section, the evolution or spreading of a SCF in a transmission system is investigated, including the relay protection and system stability control stages, to conduct the contingency management strategy pertinently. Specifically, the circuit breaker is utilized in the relay protection stage to remove the fault. Generally, the transmission line fault can cause huge SCCs, which cannot be cut through circuit breakers when the current exceeds the limited value. Restraining SCC within the limit is the first task of extreme weather event management. Besides, SCF is considered to be a severe disturbance for the power system operation, which might cause transient instability. Thus, post-fault transient stability maintenance is another task of SCF management. The response framework of a transmission system for extreme weather events is shown in Fig. 4.1. The activities deployed by grid operators include non-networked schemes without adding equipment and networked ones associated with grid planning. As for the out-of-rage SCC limitation, the deployment of FCLs is a kind of networked scheme, while the switch shifting and bus splitting belong to nonnetworked schemes. As the NAS is in the short-term temporal scale, which is mainly performed one or two weeks ahead. Networked schemes cannot be installed so timely. Therefore, non-network measures like switch shifting are utilized to limit the SCC in the proposed schema. For transient stability concerns which are in operation scale, non-networked schemes such as generation rescheduling and load shedding are deployed. The aforementioned non-networked strategies against SCFs caused by extreme weather events are modeled as follows.

4.2 Extreme Weather Event Response Schema

61

Receiving-end

Sending-end Transmission lines t0

Fault occurrence Relay Protection

Exceeding current

Short-circuit fault Fault detected t1 Circuit breaker Stability control Fault Evolution System States

Transient instability

Extreme weather event response schema Generation Rescheduling

Switch shifting

Load shedding

Fig. 4.1 Response framework for extreme weather events

4.2.1 Short-Circuit Current Limiting As mentioned before, the line switch shifting is utilized to limit the exceeding SCCs. The detailed scheme cost and the current limiting effect are described as follows.

4.2.1.1

Cost of Line Switch Shifting

Switch shifting on lines may result in direct load loss and the change in system operating conditions. The overall switch shifting scheme cost C 1 is formulated as follows: Σ C1 = χi j ci j (4.1) (i, j)∈Ω B

where χ ij represents the switch shifting status on line i − j. ΩB is the set of switches to be shifted. cij is the cost of switch shifting on line i − j, which include the direct load loss cost, and the difference of system operation cost (F B − F A ). F B and F A are determined with the optimal power flow model before and after deploying the switch shifting scheme.

62

4 A Multi-state Model for Power System Resilience Enhancement …

4.2.1.2

The Current Limiting with Switch Shifting

Generally, the SCCs of three-phase faults are higher than single-phase ones [11]. Here, the effect of switch shifting on SCC limiting is illustrated with a three-phase SCF. The three-phase SCC [19] should be limited to the given threshold I M : Ib =

Vb0 ≤ IM X bb

(4.2)

where V b 0 and X bb are the pre-fault voltage and the self-impedance of bus b. An effective method for SCC limiting is to modify the node admittance matrix and thus increase the self-impedance of the corresponding buses [11]. If the impedance of line l is added by Δx l , the incremental of the diagonal element X bb is [20]: )2 ( X bi − k X bj ) ) ( ΔX bb = − ( 2 − xl0 + Δxl xl0 /Δxl + X ii + k 2 X j j − k X i j + X ji

(4.3)

where X bi , X bj , X ii , X jj , X ij, and X ji are all elements of impedance matrix X; k is the standard voltage ratio imputed to the i side; x l0 is the initial impedance of line l from bus i to j. The switch shifting on line l is equivalent to Δx l → ∞, whose effect is: ( ΔX bb = −

4.2.1.3

X bi − k X bj

)2

) ( −xl0 + X ii + k 2 X j j − k X i j + X ji

(4.4)

Islanding Detection

The switch shifting on lines may cause islanding, which should be excluded from the feasible solutions. An islanding detection technique based on the incidence matrix [21] is introduced to screen the candidate switches to be shifted. The bus-branch incidence matrix can be expressed as an N b × N l matrix A = (ai,l )Nb ×Nl through the bus set Ωb , the line set Ωl , and the line status Ωs . The detailed elements of A are: [ ail =

1i ⊂l 0 i /⊂ l

(4.5)

where i = 1, 2, …N b , l = 1, 2, …N l , and i ⊂ l represents that bus i is connected to line l, otherwise i /⊂ l. When the switch on line l breaks, the elements of column l in A are all set to 0. Then the first level bus-bus incidence matrix can be expressed as C (1) = (c(1) i,j )Nb × Nb , in which the elements are:

4.2 Extreme Weather Event Response Schema

ci(1) j =

63

Nl | | ( ) aik ∩ a jk

(4.6)

k=1

Moreover, the final level bus-bus incidence matrix C (Nl) is calculated based on C : (1)

C

(Nl )

=

Nl −1) 2(Π

C (1)

(4.7)

l=1

If the final level bus-bus incidence matrix C (Nl) contains 0 elements, the initial switch shifting scheme can cause islanding and should be removed.

4.2.2 Transient Stability Maintaining The generation rescheduling and load shedding are adopted for transient stability maintenance. The corresponding cost and constraints are described as follows.

4.2.2.1

Transient Stability Cost

The total operating costs of the power system considering the generation rescheduling and load shedding are as follows: Co =

Σ Σ

( ) Σ Σ CigP Pig + L pis Ds (do )

i∈NG g∈N G i

i∈N L s∈N L i

(4.8)

where Pig and C ig P are the active power output and the generation cost function of generator g on bus i; L pis represents the load curtailment of the user s on the bus i; N G and N L are the set of generator buses and load buses; NGi and NL i are the set of generators and loads on the bus i; d o represents the outage time, Ds (d o ) is the cost damage function.

4.2.2.2

Steady Constraints and Dynamic Equations

First, the power flow equations are given as follows: Σ g∈N G i

Pig −

Σ ( s∈N L i

Nb ) Σ ( ) Pis − L pis = Vi V j G i j cos θi j + Bi j sin θi j j=1

64

4 A Multi-state Model for Power System Resilience Enhancement …

Σ

Q ig −

g∈N G i

Σ (

Nb ) Σ ( ) Q is − L qis = Vi V j G i j sin θi j − Bi j cos θi j

s∈N L i

(4.9)

j=1

where N b is the number of buses; Pis and Qis are the active and reactive load of user s on the bus i; Qig is the reactive power of generator g on the bus i; V i and θ i are the voltage magnitude and angle of bus i; Gij and Bij are the admittance of line i − j. Besides, the limits of the generator outputs and nodal voltages are also considered in the TSCOPF. Then, a classical generator model in [22, 29] is adopted: E g ∠δg0

·

= Vi +

) ( j Pig − j Q ig ∗ Vi

, X dg

(4.10)

where E g , δ g 0 , and X, dg represent the internal voltage magnitude, initial rotor angle, and direct-axis transient reactance of generator g. The initial value equation is transformed into the following real and imaginary parts: {

( ) , E g Vi cos δg0 − θi − Vi2 − Q ig X dg =0 ( 0 ) , E g Vi sin δg − θi − Pig X dg = 0

(4.11)

Finally, two groups of discretized swing equations are given to describe the dynamics: ) Δt ( t ωg + ωgt−1 − 2 ω N 2 ) ] ( ) [ ( t t−1 ω N / 2Mg ωgt = ωgt−1 + Δt 2Pig − Dg ωgt − Dg ωgt−1 − Peg − Peg δgt = δgt−1 +

(4.12)

where ωN is the synchronous rotor speed; Δt is the step length; δ g t and ωg t are rotor angle and rotor speed of generator g at time step t; M g and Dg are the inertial constant and damping constant of generator g. Peg t is the active electromagnetic power of generator g at time step t, which is given by: Peg = E g

NG Σ

[ ) )] ( ( E n G gn cos δg − δn + Bgn sin δg − δn π

(4.13)

n=1

where NG is the number of generators; Ggn and Bgn represent the reduced admittance between generator g and n. Transient stability constraints are as follows: Σ −δmax ≤

δgt

− δC O I = t

δgt

g∈N G

− Σ

δgt Mg

g∈N G

Mg

≤ δmax

(4.14)

4.3 Multi-state Modeling of Transmission System Resilience Enhancement

65

where δ COI t is the center of inertia (COI) at time t and δ max represents the angle deviation thresholds with respect to the center of inertia.

4.2.3 Formulation of the MINLP in the Proposed Schema Based on the previous SCC limitation and stability maintenance models, the optimal integrated response strategy of a transmission system against an extreme weather event of interest can be obtained by solving the following combined MINLP. The objective function is to minimize the overall costs of switch shifting, generation rescheduling, and load shedding. min

Σ (i, j )∈Ω B

χi j ci j +

Σ Σ

( ) Σ Σ L( ) CigP Pig + Cis L pis

i∈NG g∈N G i

(4.15)

i∈N L s∈N L i

Subject to the following constraints: SCC limitation constraints: ∀Ib ≤ I M

(4.16)

−δmax ≤ δgt − δCt O I ≤ δmax

(4.17)

Transient stability constraints:

Other operational and dynamic security constraints: (4.2)–(4.7), (4.9)–(4.14). The decision variables include 0–1 variable of switch shifting status χ ij , continuous variables of nodal load curtailment L pis, and generator active power output Pig . χ ij indicates the status of the switch. χ ij = 1 means that the switch on line i − j is opened and χ ij = 0 means it is closed. Constraint (4.16) denotes that nodal SCCs should not exceed the threshold I M . Constraint (4.17) means the generator rotor angle oscillation should be limited within a range at each time step. Note that both the objective function and SCC constraint contain 0–1 variables χ ij . Thus, the proposed MINLP is nonconvex.

4.3 Multi-state Modeling of Transmission System Resilience Enhancement The weather forecast is of complexity. As illustrated in Fig. 4.2, for a newly generated extreme hurricane, usually several possible moving paths [23], and intensities [24] will be announced by the meteorological bureaus or groups but not a deterministic

66

4 A Multi-state Model for Power System Resilience Enhancement …

Fig. 4.2 Uncertain moving paths of extreme weather events

one [25]. Consequently, power systems are faced with great uncertainties in preevent dispatching. In this section, a comprehensive multi-state model is established to describe the uncertain influences of extreme weather events on the power system in terms of triggered SCFs. Then a two-stage decision approach is proposed to achieve a balance between economy and system resilience.

4.3.1 Multi-state Resilience Enhancement Against SCFs In the established model, states represent certain sets of SCFs due to different predicted hurricane situations. Each state corresponds to a specific initial defending strategy and their performance can be described in terms of the following initial defending strategy cost and state probability: ( ) Ck = f k Ω Tk , Ω Sk =Ck1 + Ck2 Pk = p˜ s

Π l∈Sk

ρl

Π

(1 − ρo )

(4.18) (4.19)

o∈S / k

where Ck and Pk represent the prevention cost of state k and its corresponding probability; Ωk T and Ωk S represent the generation rescheduling/load shedding, and switch shifting strategies; C k 1 and C k 2 are the costs of system transient stability maintenance and SCC limitation, respectively; p˜ s is the standardized probability of the s-th hurricane, which is obtained based on hurricane forecast; S k is the set of the

4.3 Multi-state Modeling of Transmission System Resilience Enhancement

67

short-circuit lines in state k under the s-th hurricane scenario, ρ l is SCF probability of line l obtained with the fragile model. Especially, Eq. (4.18) represents the initial prevention cost of state k, which is a function of the generation rescheduling/load shedding Ωk T , and switch shifting strategies Ωk S . These optimal integrated response strategies of a transmission system against an extreme weather event of interest can be obtained by solving the proposed MINLP in Sect. 4.2. Besides, the prevention cost can also be represented as the summation of the costs of system transient stability maintaining C k 1 and SCC limiting C k 2 . With regard to Eq. (4.19), it represents the probability of state k, which is obtained by multiplying the standardized probability p˜ s of the s-th weather scenario and the calculated SCF or normal operation event probabilities given by the fragile model in the corresponding scenario. For instance, there is a set of short-circuit lines in state k, the probability of the expected fault in the s-th weather scenario is the product of the SCF probabilities of the short-circuit lines and the normal operation probabilities of the non-fault lines. System state transition may occur due to errors in NAS result of the extreme weather event. In other words, when the hurricane lands, the situation may not be the same as previously expected. Consequently, an urgent adaption of the initial defending strategy is required. The transition probability is a conditional probability associated with both the current and future states. If the correlation of states is ignored, then the transition formula from state k to h can be expressed as: ( T ) S 1 2 = Ck→h Ck→h = f k→h Ω k→h , Ω k→h + Ck→h Pk→h = Ph = p˜ m

Π

ρl

l∈Sh

Π

(1 − ρo )

(4.20) (4.21)

o∈S / h

where Ck→h represents the adaption cost of state k to h; Pk→h represents the transition 1 2 and Ck→h denote the system stability maintenance probability from state k to h; Ck→h S T and Ωk→h represent and SCC limitation costs from state k to h, separately; Ωk→h the generation rescheduling/load shedding and switch shifting strategies from state k to h. p˜ m is the standardized probability of the m-th hurricane; S h is the set of the short-circuit lines in state h under the m-th hurricane scenario. Due to the higher requirement of response speed, the cost parameters of urgent adaption are higher than those of the initial defending, such as the climbing cost of units and shedding cost of loads. The urgent adaption in the second stage is to expand the initial defending strategy to an updating state. As shown in Fig. 4.3, considering the probabilistic state transition, the overall cost of resilience enhancement strategy under state k can be modeled as the sum of the initial defending cost and the expected urgent adaption cost of the initial plan being adapted to other future states, which is expressed as follows: min E k = Ck + pr e

k∈S

Σ h∈S pr e ,h/=k

Ck→h Pk→h

(4.22)

68

4 A Multi-state Model for Power System Resilience Enhancement …

Fig. 4.3 Multi-state resilience enhancement model against short-circuits

where S pre represents the set of expected faults.

4.3.2 State Generating Model As introduced above, the probability threshold ρ thre is mainly determined in consideration of the risk preference of system operators. Specifically, the smaller the threshold is set, the more SCFs are considered in the fault-defending process, compromising the economy of system operation while benefiting the security. The appropriate probability threshold can be determined by coordinating the fault-defending cost and system risk. For one state, the optimization model to obtain the probability threshold is given: min C T = C p + λ R ρthre

NC Σ

P(n)Ce (n)

(4.23)

n=1

where λR is the risk reference coefficient. C p is the total cost of defending against all the expected SCFs whose probabilities are larger than ρ thre . Nc is the number of unexpected SCFs whose probabilities are smaller than ρ thre . C e (n) is the severity of the power system when the n-th unexpected SCF happens. C e (n) can be measured by the cost of urgent generation rescheduling and load shedding. P(n) represents the probability of the n-th unexpected SCF. C p and C e (n) can be obtained by solving Eqs. (4.18) and (4.20), respectively. It should be noted that C p has some monotone decreasing relationship with ρ thre , as illustrated in Fig. 4.4. Since the feasible region of the expected fault-defending

4.3 Multi-state Modeling of Transmission System Resilience Enhancement

69

Fig. 4.4 Relationship between the control cost/system risk and ρ thre

optimization problem will enlarge as ρ thre increases. Moreover, the system risk has some monotone increasing relationship with ρ thre . When ρ thre is very small or large, the value of C T will be large. Thus, C T has a minimum value, which is mainly determined by ρ thre under the given risk coordination coefficient [26]. After ρ thre is determined, the state of the system is determined accordingly.

4.3.3 Traversal Procedure As illustrated in Fig. 4.5, the process of the multi-state modeling for the transmission system resilience enhancement can be divided into four steps:

Fig. 4.5 The flowchart of the multi-state modeling for resilience enhancement

70

4 A Multi-state Model for Power System Resilience Enhancement …

Step 1 State generating. Based on released forecasts of extreme weather events, the SCF rates of transmission lines are calculated with the fragile model in Sect. 1.2. The faults whose probabilities are greater than ρ thre are added to the expected fault set. Lines with higher fault rates and their combinations are utilized to generate system states as well as the assessed probabilities. Step 2 Initial prevention. According to the obtained states, the initial prevention strategy and its cost are determined by solving the MINLP defending model. Shifted switches are selected to limit SCCs and load curtailment, as well as generation adjustments, are optimized to guarantee the transient stability under each initial system state. Step 3 Urgent adaption. All possible state transitions from initial states to others are evaluated. The urgent adaption strategies are obtained by solving the MINLP defending model to handle possible abrupt weather changes. Step 4 Strategy selection. The initial prevention cost and expected urgent adaption cost are summed for each state. The initial preventive measure with the lowest overall cost is selected as the final scheme to enhance the resilience of the power system before the arrival of the extreme weather event.

4.4 Solution Method As mentioned above, the proposed MINLP is non-convex and high dimensional. To solve the problem, we reformulated the proposed MINLP model. Then, a searching space reduction technique and a combined heuristics and CNDIPM solution method are developed in this section.

4.4.1 Problem Reformulation In regard to the proposed problem, conducting the switch-shifting strategy is not necessarily capable to guarantee that the out-of-range SCCs can be limited below the threshold. The load shedding and generation rescheduling cannot definitely guarantee the transient stability of the post-fault system. Moreover, the conducted solutions may be contradictory in terms of the effects of stability maintenance and current limiting. Thus, there may be no solution to this MINLP. In this paper, a reformulation strategy is adopted to deal with this problem. Specifically, a penalty function for the critical constraints [22] is introduced, and the original constrained problem is transformed into the following form: ⎧⎛ ⎞ ⎨ Σ Σ Σ Σ Σ ( ) ( ) min ⎝ χi j ci j + CigP Pig + CisL L pis ⎠ ⎩ (i, j)∈Ω B

i∈NG g∈N G i

i∈N L s∈N L i

4.4 Solution Method

+

Nb Σ

71

(ψo (Ib , I M )λc )

b=1

⎫ N G TΣ s /Δt ⎬ Σ | ( (| t ) ) ψo |δg − δCt O I |, δmax λs + ⎭

(4.24)

g=1 t=1

Subject to the following operational constraints: (4.2)–(4.7), (4.9)–(4.14). The penalty operator ψ o is formulated as follows: [ ψo (m 1 , m 2 ) =

m1 − m2, m1 > m2, 0 m1 ≤ m2.

(4.25)

In (4.24), the first part denotes the original objective function, while the second and the third parts denote the penalties of the exceeding fault currents and the transient instability, respectively. It should be noted that the penalty coefficients λc and λs are set to very large numbers (such as 10e6) to ensure no violation of the solutions in terms of fault current limitation and post-fault transient power angle convergence in most cases (these two penalty items are zero).

4.4.2 Searching Space Reduction In the proposed problem, the discrete and continuous variables are coupled, which increases the difficulty of optimization. Therefore, we try to divide it into discrete and continuous sub-problems and deal with them separately and interactively. For the discrete or current limiting sub-problem, it will be very time-consuming to restart the searching of switch shifting schemes without any prior knowledge for each iteration. Considering the generality of switch shifting schemes, a sensitivity-based searching space reduction technique [27] is introduced to choose some switching candidates or line switching patterns so as to avoid the curse of dimensionality and speed up the solution. Considering the excessive level of SCCs and impedance incremental sensitivity factor of all buses, the comprehensive sensitivity of the switch shifting on line l to the whole system is expressed as follows: [ ] )2 ( Σ ( Ib )τ − X bi − k X bj ξs (l) = Ithr e X ii + k 2 X j j − 2k X i j − xl0 b∈N

(4.26)

I

where τ is an adjustment coefficient; N I is the set of the buses whose fault currents exceed the threshold. ξ s (l) is used to choose the candidate shifted switches to limit SCCs.

72

4 A Multi-state Model for Power System Resilience Enhancement …

4.4.3 Heuristics Solution For the continuous or stability control sub-problem, as a widely-used analytic tool, the TSCOPF solver is utilized to calculate the stability-oriented scheme considering the fact that power systems are transient stability in most cases. Based on the above content, an efficient heuristic based on GA and CNDIPM is developed to solve the nonconvex reformulated problem considering an interaction strategy, which is illustrated in Fig. 4.6. The framework of the method can be decomposed into two levels, the top level is the decision sub-problem of switch shifting, the bottom level is the sub-problem of TSCOPF optimization. Firstly, within the candidate shifted switches indicated by ξ s (l), GA is utilized at the top level to generate a parent population, in which each chromosome denotes a raw configuration scheme of switch shifting. For each chromosome, its cost of switch shifting C 1 is calculated, and then the two following calculations are performed: 1. The nodal SCCs are calculated and the corresponding current penalty F 1 is quantified. 2. The CNDIPM [14] is utilized at the bottom level to obtain the optimal load curtailments and generation adjustments. The corresponding cost is denoted as C 2 and the transient stability instability penalty F 2 is evaluated. For each switch shifting scheme, its fitness is defined as the reciprocal of the summation of the current limiting cost C 1 , the SCC penalty F 1 , the transient stability maintenance cost C 2 , and the transient stability instability penalty F 2 , which is written as Fitness = (C 1 + F 1 + C 2 + F 2 )−1 . Indicated by the obtained fitness values, elite switch shifting schemes are selected with higher fitness values. Then, crossover and mutation are performed to generate an

Fig. 4.6 The procedure of the proposed solution method

4.5 Case Study

73

offspring population and the iterative evolution continues until the maximum iteration is reached. The shifting scheme with the highest fitness in the last population is selected as the final suggested scheme, as well as its corresponding load shedding and generation rescheduling solution. It should be noted that the cases with unsolvable TSCOPF can be ignored as the nonzero penalty term restricts the fitness of the corresponding solutions.

4.5 Case Study The New England 39-bus system is adopted to verify the proposed method. The simulation time is 1 s with a step of 0.05 s. All the SCFs are uniformly set to be three-phase-to-ground ones occurring at 0 s and cleared by tripping the fault lines at 0.1 s. The parameters of hurricanes and lines are listed in Table 4.1. The topology diagram of the system is shown in Fig. 4.7. Moreover, there are two aging conditions of lines considered. Note that the red lines in Fig. 4.7 are assumed to be in the wear-out stage of the bathtub curve, while the other transmission lines are at the stage of useful life. Specifically, line 25–26, 3–4, 6–31, 10–13, 10–32, 20–34, 19–33, 22–35, and 23–36 are in stage III, and their failure rate is 0.06 (1/year). The failure rate of the other lines (In stage II) is 0.008 (1/year).

4.5.1 Scenario Generating In this paper, both the moving path and hurricane strength are considered to generate the SCF scenarios. Referring to a historical Pacific Ocean hurricane available from the China hurricane weather website [28], four possible landings or moving paths and two possible strengths are revealed to the public. Thus, eight hurricane scenarios are considered to guide power system resilience enhancement. The detailed shortcircuit probabilities of lines subject to hurricane forecasts are determined by the fragile model, which is shown in Fig. 4.8. Table 4.1 The parameters of hurricane and transmission lines

Parameters

Values

Parameters

Values

ε1 (G1)

50 m/s

ε2 (G1)

20 m/s

ε1 (G2)

45 m/s

ε2 (G1)

18 m/s

γ1

120 km

γ2

15 km

Vd

30 m/s

Δt

0.05 s

a1

11

a2

−8

74

4 A Multi-state Model for Power System Resilience Enhancement … G

Fig. 4.7 Topology diagram of the New England system

G

30

37

2 1 G

26

25 3

18

17

28

29 38

27

G

39

21 16

15 4

14 13

5 9

6

19

11

8

10 32

31 G

G

36

23

12

7

G

24

22

20 34 G

33 G

35 G

The expected SCF event’s probabilities are determined by the standardized probabilities of hurricanes and the corresponding theoretical SCF probabilities of transmission lines. According to the risk preference of the system operator, the number of considered expected faults is determined. Then, the expected SCF states are selected in the order of ultimate probability from high to low. The probability threshold for a SCF event to be integrated into the expected SCF set is set to be 0.1 according to the risk preference of system operators. Here, 8 SCF states comply with this threshold, and the corresponding expected faults are shown in Table 4.2. Note that the probability of the second-order SCF denoted as a product of isolated failure rates is much lower than that of the first-order SCF. Thus, the case of simultaneous faults is ignored and there is one line encountering SCF for each state.

4.5.2 Sensitivity Analysis The sensitivity factors of transmission lines are shown in Fig. 4.9. The top 10 transmission lines with the largest comprehensive sensitivity factors are lines 16–17, 2–30, 17–27, 16–27, 28–29, 26–28, 26–29, 15–16, 22–23 and 14–15. Some switch shifting options like the switches on lines 22–35 and line 25–37 are excluded, which caused islanding with Eq. (4.7). The optimal switch shifting scheme is searched among the above 10 candidate lines.

4.5 Case Study

75

Probability

(a) Path 1: Grade 1

(b) Path 2: Grade 1

(c) Path 3: Grade 1

(d) Path 4: Grade 1

(e) Path 1: Grade 2

(f) Path 2: Grade 2

(g) Path 3: Grade 2

(h) Path 4: Grade 2

Fig. 4.8 Failure rates of transmission lines under different hurricanes

76

4 A Multi-state Model for Power System Resilience Enhancement …

Table 4.2 The probability of different scenarios Scenarios

Lines

Probability

Scenarios

Lines

Probability

1

19–20

0.1651

5

20–34

0.1247

2

6–31

0.1529

6

22–35

0.1478

3

10–13

0.1013

7

23–36

0.1454

4

19–33

0.1306

8

25–37

0.1124

Fig. 4.9 Candidate branches with the largest sensitivity factors

4.5.3 Suggested Resilience Enhancement Scheme Based on the proposed MINLP model, the optimal defending strategies of 8 states are calculated. The urgent adaption strategies are illustrated in Fig. 4.10 and the prevention strategies are listed in Table 4.3. For instance, the initial strategy for state 3 does not involve switch shifting, while 163 MW load curtailment and a generation rescheduling cost of $5796 are needed to guarantee transient stability. The urgent strategy of initial state 3 being adapted to future state 4 includes the switch shifting on line 14–15 to limit SCCs, 74 MW load curtailment, and a generation rescheduling cost of $3529 to guarantee transient stability. Figure 4.11 further illustrates the initial prevention cost, expected urgent adaptation cost, as well as the overall cost of each state. As can be seen, the initial prevention cost of state 8 is the lowest while its adaption cost is the highest ($133,318). Comparing the costs of states 1 and 3, it can be concluded that if fewer initial preventions are adopted, more urgent adaptions are demanded when the weather suddenly changes. The overall cost is utilized to select the optimal preventive action. The preventive action which is specially designed for state 7 achieves the lowest overall cost of $76,210. Therefore, the most cost-effective strategy to defend against the uncertain hurricane is to adopt the prevention strategy under state 7 with the expected SCF on line 23–36.

4.5 Case Study L10,143,335 L1,154,4802 0,87,568

S1

L6,100,23541

77 L1/L5,92,548

S2

0,166,3180

S3

L9,80,606

S4 S5

S2

L10,0,4522

L8,0,4343

0,157,-208 0,169,394

S3 S4 S6

L6/L8,147, 3855

0,0,1940

S2

0,0,15.57

S7

L8,0,8112

S8

S4

L3/L7,0, -3145

S5

S6

L9,80,416 L6/L10,0, 4647 L7/L8,0, 4546 L3/L9/L10,0, 3477

S4

L6,195,-2686

S1

0,140,4834

L5/L8,381, -3534

S2

0,171,-19314

S7

L1/L3/L6,18, 3061

S8

0,68,5240

S3

L4/L7/L10,0, 214

S7

L3/L5,0, -3567

S2

S2 S3 S5 S6

S6

S1

S4

0,0,1405

S1

L7,0,3284

L7/L8,105, 4645

S8

S1

L3/L8,268,12 75

0,125,-977

L4,0,244

S8

S5

S3

S7

S7

0,80,4671

S5

0,94,3649

L4,0,4061

0,212,5238

S4

0,156,-618

S2

L10,74,3529

S6

S6

0,175,3552

S3

0,0,2352

S1

0,156,212

0,123,186

L3,100,3541

L8,0,8112

L5,0,-1339

S1

L7,0,3507

S3

0,0,8365

S4

S8

L10,0,8365

L9,446,-3701

S7 S8 S1 S2 S3 S4

0,0,175

S5

S5

S5 0,0,454

L7,0,3834

S6

S6

S7 L4,0,3235

0,0,77

S8

S8

S8

Fig. 4.10 Urgent adaption strategies of 8 states

Table 4.3 The initial prevention strategies

Scenario

Strategies

Scenario

Strategies

1

L5, 160 MW, 1381$

5

L3/L7, 0, 3189$

2

L6, 157 MW, 1273$

6

124 MW, 1445$

3

163 MW, 5796$

7

L6/L10, 0, 3208$

4

L9, 81 MW, 1862$

8

L4, 0, 1539$

4.5.4 Comparison Studies To verify the validity of the proposed multi-state resilience enhancement method and the necessity of considering SCFs, three benchmark cases are designed as follows: Benchmark 1: The most likely scenario with the largest occurrence probability is considered (S1 in Table 4.2) while other possible scenarios are excluded. In this case, the suggested scheme should survive from the most probable fault.

78

4 A Multi-state Model for Power System Resilience Enhancement …

Fig. 4.11 Solution costs of different states

Benchmark 2: All possible fault scenarios (S1-8 in Table 4.2) are considered simultaneously. In this case, the suggested scheme should survive the whole set of expected faults. Benchmark 3: S7 is considered while its expected fault type is set as direct damage of line 23–36 but not a SCF. The deterministic initial preventions and the urgent adaptions of benchmark 1 can be found in Table 4.3 and Fig. 4.10. Table 4.4 shows the prevention actions of benchmark 2. The costs of benchmarks 1 and 2 are framed in Fig. 4.11. The costs of benchmark 1 include $73,931.70 for prevention and $53,237.82 for adaption, and the total cost is $127,169.52. Although the urgent adaption cost of benchmark 1 is relatively low, its overall cost is higher than that of the suggested scheme (state 7). All possible scenarios are considered simultaneously in benchmark 2, therefore, there is no need for urgent adaption. In other words, the cost of preventive action is indeed the total cost with a value of $270,334.24, which is the highest compared with the other benchmarks. For the proposed method, initial prevention and expected urgent adaption costs are considered as the objective, which is the most economical compared with the above two benchmarks. In benchmark 3, the rescheduling of generators with the cost of $1312 is required to ensure the security of the system in terms of the breakage of line 23–36, and no load shedding is needed. If the actual fault is a SCF in line 23–36 but not its line damage, simulations are performed to check the consequences. Firstly, the SCC is within the threshold. Meanwhile, the post-fault rotor angle of unit 6 deviates from the COI and causes transient instability, which is illustrated in Fig. 4.12. To guarantee transient stability, 147 MW load curtailment and a generation rescheduling cost of $4926 are needed. In other words, the preventive actions considering transmission Table 4.4 The prevention action of benchmark 2

Contingencies

Lines with shifted switch

Load curtailment

Scenario 1–8

L3, L6, L7, L10

537 MW

References

79

Fig. 4.12 Rotor angles with respect to COI in benchmark 3

line breakages are inadequate. Therefore, it is of great necessity to consider SCFs caused by extreme weather in the resilience enhancement of power systems.

4.6 Conclusions Faced with the severe impacts of weather disasters, this paper proposed a multi-state model to determine the transmission system resilience enhancement strategy against SCFs caused by extreme weather events. Specifically, the weather-induced nodal SCF probabilities as well as grid defending measures against the out-of-range SCC magnitude and system transient instability are all integrated. The results show that the proposed multi-state model for the transmission system resilience enhancement achieves both increased scheme flexibility and decreased investment cost, which is easily extended to other extreme disasters, such as wildfires, line icing, etc.

References 1. I. A. R. Berkeley, and M. Wallace. A Framework for Establishing Critical Infrastructure Resilience Goals: Final Goals and Recommendations [Online]. 2. D. N. Trakas and N. D. Hatziargyriou, “Optimal Distribution System Operation for Enhancing Resilience Against Wildfires,” IEEE Trans. Power Syst., vol. 33, no. 2, pp. 2260–2271, 2018. 3. T. T. Dan and W. T. P. Wang, “A More Resilient Grid: The U.S. Department of Energy Joins with Stakeholders in an R&D Plan,” IEEE Power Energy Mag., vol. 13, no. 3, pp. 26–34, 2015. 4. A. Gholami, T. Shekari, F. Aminifar, and M. Shahidehpour, “Microgrid Scheduling With Uncertainty: The Quest for Resilience,” IEEE Trans. Smart Grid, vol. 7, no. 6, pp. 2849–2858, 2016. 5. A. Khodaei, “Resiliency-Oriented Microgrid Optimal Scheduling,” IEEE Trans. Smart Grid, vol. 5, no. 4, pp. 1584–1591, 2014.

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6. M. Ouyang and L. Dueñas-Osorio, “Multi-dimensional hurricane resilience assessment of electric power systems,” Structural Safety, vol. 48, pp. 15–24, 2014. 7. J. P. Conti, “The day the samba stopped,” Engineering & Technology, vol. 5, no. 4, pp. 46–47, 2010. 8. G. Li et al., “Risk Analysis for Distribution Systems in the Northeast U.S. Under Wind Storms,” IEEE Transactions on Power Systems, vol. 29, no. 2, pp. 889–898, 2014. 9. P. Yu, B. Venkatesh, A. Yazdani, and B. N. Singh, “Optimal location and sizing of fault current limiters in mesh networks using iterative mixed integer nonlinear programming,” IEEE Trans. Power Syst., vol. 31, no. 6, pp. 4776–4783, 2016. 10. C. Guo, C. Ye, Y. Ding, Z. Lin, and P. Wang, “Risk-based many-objective configuration of power system fault current limiters utilising NSGA-III,” IET Gene. Transm. Distrib., 2020. 11. C. Ye and M. Huang, “Multi-objective optimal configuration of current limiting strategies,” Science China Technological Sciences, vol. 57, no. 9, pp. 1738–1749, 2014. 12. Y. Dong, Z. Kang, and L. Yutian, “Coordinated optimization for controlling short circuit current and multi-infeed DC interaction,” Journal of Modern Power Systems and Clean Energy, vol. 2, no. 4, pp. 374–384, 2014. 13. K. Hongesombut, Y. Mitani, and K. Tsuji, “Optimal location assignment and design of superconducting fault current limiters applied to loop power systems,” IEEE Trans. Appl. Supercond., vol. 13, no. 2, pp. 1828–1831, 2003. 14. Q. Jiang and G. Geng, “A Reduced-Space Interior Point Method for Transient Stability Constrained Optimal Power Flow,” IEEE Trans. Power Syst., vol. 25, no. 3, pp. 1232–1240, 2010. 15. Y. Wang, J. Min, and Y. Chen, “Impact of the hybrid gain ensemble data assimilation on meso-scale numerical weather prediction over east China,” Atmospheric Research, vol. 206, pp. 30–45, 2018. 16. W. Yuan, J. Wang, F. Qiu, C. Chen, C. Kang, and B. Zeng, “Robust Optimization-Based Resilient Distribution Network Planning Against Natural Disasters,” IEEE Trans. Smart Grid, vol. 7, no. 6, pp. 2817–2826, 2016. 17. C. Lee, C. Liu, S. Mehrotra, and Z. Bie, “Robust distribution network reconfiguration,” IEEE Trans. Smart Grid, vol. 6, no. 2, pp. 836–842, 2015. 18. Z. Wang, B. Chen, J. Wang, J. Kim, and M. M. Begovic, “Robust optimization based optimal DG placement in microgrids,” IEEE Trans. Smart Grid, vol. 5, no. 5, pp. 2173–2182, 2014. 19. D. P. Kothari and I. J. Nagrath, “Modern Power System Analysis, Third Edition,” 2003. 20. J. H. Teng and C. N. Lu, “Optimum fault current limiter placement with search space reduction technique,” IET Gene. Transm. Distrib., vol. 4, no. 4, pp. 485–494, 2010. 21. T. L, T. W, C. X, M. S, W. L, and Y. X, “Network Connectivity Identification Method based on Incidence Matrix and Branch Pointer Vector,” in 2019 IEEE Innovative Smart Grid Technologies - Asia (ISGT Asia), 2019, pp. 429–433. 22. C.-J. Ye and M.-X. Huang, “Multi-objective optimal power flow considering transient stability based on parallel NSGA-II,” IEEE Transactions on Power Systems, vol. 30, no. 2, pp. 857–866, 2014. 23. G. J. Alaka Jr, X. Zhang, S. G. Gopalakrishnan, Z. Zhang, F. D. Marks, and R. Atlas, “Track Uncertainty in High-Resolution HWRF Ensemble Forecasts of Hurricane Joaquin,” Weather and Forecasting, vol. 34, no. 6, pp. 1889–1908, 2019. 24. A. T. Hazelton, M. Bender, M. Morin, L. Harris, and S.-J. Lin, “2017 Atlantic hurricane forecasts from a high-resolution version of the GFDL fvGFS model: Evaluation of track, intensity, and structure,” Weather and Forecasting, vol. 33, no. 5, pp. 1317–1337, 2018. 25. Burma cyclone was forecast four days in advance. Available: https://www.newscientist.com/ article/dn13868-burma-cyclone-was-forecast-four-days-in-advance/ 26. Z. Wang, X. Song, H. Xin, D. Gan, and K. P. Wong, “Risk-based coordination of generation rescheduling and load shedding for transient stability enhancement,” IEEE Transactions on Power Systems, vol. 28, no. 4, pp. 4674–4682, 2013. 27. L. Chen, M. Huang, J. Wu, and D. Wang, “An Optimal Strategy for Short Circuit Current Limiter Deployment,” in Asia-pacific Power & Energy Engineering Conference, 2010.

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28. China weather hurricane network, China climate network, 2020. [online]. Available: http://typ hoon.weather.com.cn/ 29. D. Gan, R. J. Thomas, and R. D. Zimmerman, “Stability-constrained optimal power flow,” IEEE Trans. Power Syst., vol. 15, no. 2, pp. 535–540, 2000.

Chapter 5

Voltage Violations Assessment Considering Reactive Power Compensation Provided by Smart Inverters

5.1 Introduction Power electronics advancements in distribution networks benefit distributed photovoltaics, EV chargers, and other AC-DC inverter-equipped devices. However, because of the erratic nature of photovoltaics outputs and the load impact of EV charging, distribution networks are now more susceptible to voltage violation issues [1]. On-load tap changers (OLTCs) and switching capacitors, which are common reactive power compensation facilities, can only give step-wise and high-delay reactive power at the feeder head, which restricts the regulatory effect [1]. Distribution Static Synchronous Compensators (D-STATCOMs) have limited use in distribution networks due to their high cost [2]. Therefore, step-less compensating facilities must be implemented at the feeder terminal. Meanwhile, the terminal end power electronics in distribution networks can take part in voltage control thanks to the two-way reactive power capability of smart inverters (EVs and photovoltaics). To deliver reactive electricity via smart inverters, some research has previously been done. Reactive power exchange for photovoltaic inverters is extended by Sharma and Das [3], Feng et al. [4], which also contribute to balancing the active and reactive power transmission of each phase. In [5], the theoretical and experimental analysis and validation of the reactive power compensation capabilities of EV chargers are conducted. The usage of EV chargers is greatly expanded by the observations made in [6–8] that the reactive power correction mechanism used by photovoltaics and EV chargers does not interfere with the active power delivery or harm EV batteries. The revolutionary smart inverter photovoltaic-STATCOM, which operates photovoltaic inverters as a dynamic reactive power compensator, is presented by Varma and Siavashi [6] as a means of integrating these flexible power electronics. Furthermore, In [9], voltage management in both the global and local domains using smart inverters for photovoltaics and EV charging stations has been done. Reactive power compensation has been thoroughly studied and implemented using smart inverters [10, 11]. Therefore, switching to

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Ye et al., Risk-Based Planning and Operation Strategy Towards Short Circuit Resilient Power Systems, https://doi.org/10.1007/978-981-19-9725-9_5

83

84

5 Voltage Violations Assessment Considering Reactive Power …

smart inverters for precise and quick voltage management in distribution networks is conceivable. Research has been done on using intelligent inverters to control voltage quality. In [12], the distributed voltage regulation system that makes use of the photovoltaics inverters’ abilities to handle reactive power has been developed. Reactive power control in distribution networks is accomplished by Singh et al. [9] and Quirós-Tortós et al. [13] with the aid of EV charges and photovoltaics. The use of intelligent inverters for voltage regulation, however, is still a research area that requires application. Nonparametric estimate strategies are needed for irregular photovoltaics outputs and EV loads to derive their probability density functions (PDFs), and more versatile sampling procedures are needed for each irregular PDF [14]. The conventional Monte Carlo sampling methods, like [15–17], are not appropriate. Piecewise sampling is a sophisticated Markov chain monte Carlo simulation technique [18]. Because it can effectively sample from irregular PDFs, it is feasible for photovoltaic output and EV loads. The model for assessing voltage violations is proposed in this paper with the involvement of smart inverters. Samples of EV loads and photovoltaic outputs are produced. The voltage violation assessment is done to gauge how well smart inverters work. The contributions of this chapter are as follows: This paper proposes a voltage violation assessment model considering the participation of smart inverters. Samples of photovoltaic outputs and EV loads are generated. The voltage violation assessment is carried out to evaluate the performance of smart inverters. The main contributions of this paper can be summarized as two-fold: 1. It is suggested that smart inverters’ quantitative compensatory ability be evaluated using the available reactive power capacities under active power limitations. The voltage control optimization model is developed taking into account both conventional reactive power compensation facilities and smart inverters. 2. The proposed voltage deviation indices, non-parametric kernel density estimation (KDE), and slice sampling are used to assess voltage violations under the uncertainty of photovoltaic outputs and EV behaviors. KDE precisely fits the non-standard PDFs of photovoltaics and EVs. Additionally, to effectively create samples from the received PDFs, the automated step width selection for slice sampling is used.

5.2 Basic Models 5.2.1 Reactive Power Compensation Mechanism of Smart Inverters Figure 5.1 shows the equivalent smart inverter models. These intelligent inverters can run their output voltages with flexibility thanks to their structure. The inverter can modify the transmission of both active and reactive electricity by altering the

5.2 Basic Models

85

amplitude and phase angle of the output voltage. Smart inverters’ reactive power adjustment technique is depicted in Fig. 5.2. Two typical operating modes, shown in Fig. 5.2, are shown to supply the grid with pure capacitive or inductive reactive power. The voltage of smart inverters affects the reactive power compensation modes, as shown in Fig. 5.2. The rated current limit, which guards against over-loading the inverter, is used to draw the yellow circle range in Fig. 5.2. Within this yellow circle, as the inverter voltage varies, so does the current. The inverter then outputs active power P and reactive power Q in accordance with the grid voltage and current. As shown in Fig. 5.1, the operation range of P and Q of smart inverters is subsequently calculated [19, 20]. However, the operating ranges of EV chargers and solar panels are distinct. EV chargers can function in four quadrants, however, photovoltaics cannot absorb active power. Therefore, the operation range can be expressed as: Q

Q

Fig. 5.1 Topology and operation range of smart inverters

Qmax

Qmax

P

P Do not consume active power

Operation range with 0.1Pmax

vInverter Inverter

Grid

vGrid

Pmax

Pmax

DC-DC

i

EV Charger or PV

Filter L R

jω Li

vInverter

i

vGrid vGrid vInverter

(a) capacitive mode

Ri

jω Li Ri

i

(b) inductive mode

Fig. 5.2 Voltage vector diagrams of two typical operation modes

86

5 Voltage Violations Assessment Considering Reactive Power …

[

2 P 2 + Q 2 ≤ Smax

(5.1)

EV EV ≥ P E V ≥ Pmin Pmax PV P P V = Pout put

(5.2)

where Smax is the maximum value of the inverter capacity; P E V and P P V are the EV EV and Pmin denote the active power of EV chargers and photovoltaics respectively; Pmax maximum and minimum allowed active charging power for EV chargers respectively; PV Pout put is the active power generated by photovoltaics. Equation (5.1) indicates that when chargers and photovoltaics are idle, these smart inverters are capable of providing considerable reactive power compensation. This advantage is illustrated by the orange rectangle in Fig. 5.1, where the inverter has a vast operation range under a low active power level.

5.2.2 Models of On-Load Tap Changers and Switching Capacitors OLTCs and switching capacitors are fundamental voltage regulation resources in distribution networks, which are usually equipped at the transformer substation. The reactive power compensation mechanism for OLTCs is to change the turn ratio by adjusting the tap position. The model of the OLTCs can be expressed as [21]: ki j = kimin j + tapi j · Δki j , 0 ≤ tapi j ≤ tapi j

(5.3)

where k ij is the turns ratio of the transformer between node i and node j; kimin j is the minimum turns ratio; tapi j is the tap position of the OLTC; tapi j tapij is the maximum value of the tap position; Δki j is the ratio change per tap. Switching capacitors provide capacitive reactive power to maintain the voltage level. According to the capacity of each switching capacitor connected to the grid, the model of switching capacitors can be described as Q SC =

N SC Σ

(Q iSC · n iSC ), 0 ≤ n iSC ≤ n iSC

(5.4)

i=1

where Q SC is the total reactive power provided by switching capacitors; N SC is the total number of the switching capacitor types; Q iSC and n iSC are the capacity and the number of switching capacitors that belong to type i; n iSC is the maximum value of n iSC .

5.2 Basic Models

87

Fig. 5.3 Voltage diagrams of different compensation strategies

1.10 1.05

Voltage (p.u.) Using OLTC & SCs

Using OLTC, SCs & Smart Inverters

1.00 0.95 0.90

No Compensation

Using Smart Inverters

Node

5.2.3 Comparisons of Different Compensation Strategies Since OLTCs and switching capacitors are commonly located in the transformer substation, traditional compensation methods regulate the voltage levels by injecting reactive power at the head of the feeders, resulting in the difficulty of balancing the voltage of the whole feeder line. Figure 5.3 illustrates the effect of different compensation strategies. When the voltage violation occurs, the voltage profile with no compensation adopted drops seriously at the feeder terminal. Although the OLTC and switching capacitors are able to raise the voltage profile as shown in the blue line, they cause over-voltage at the head of feeders because of too much reactive power injection. Smart inverters can regulate the voltages on the demand side in a local way, as they are distributed close to terminal users in distribution networks. Therefore, smart inverters are capable of compensating reactive power locally as the orange line in Fig. 5.3 depicts. However, in severe voltage violations, smart inverters may not be able to raise all voltages above the safe line due to the limited capacities of demand-side power electronics. The green line in Fig. 5.3 illustrates that by using both traditional compensation resources and smart inverters, a satisfactory compensation effect can be achieved without causing over-compensation or being constrained by capacities, which is feasible for voltage regulation in distribution networks.

5.2.4 Voltage Deviation Indexes To further evaluate the voltage deviation, the voltage violation probability pvio and the expected comprehensive deviation of voltage violations E dev are proposed in this

88

5 Voltage Violations Assessment Considering Reactive Power …

paper. The definition of voltage violations in this paper is the scenario where any nodal voltage exceeds the safe range, which is set as 0.95–1.05 p.u. In Markov chain monte Carlo simulation analysis, pvio can be expressed as the proportion of voltage violation scenarios in all simulated scenarios: sim 1 Σ L vio Nsim i=1 i [ 1, L ivio = 0,

N

Pvio =

(5.5)

(5.6)

where N sim is the number of all simulated scenarios and L i vio marks the simulation result in the ith scenario. E dev describes the deviation degree of the voltage violation. The expression of E dev is E dev = /Σ Di =

Nsim 1 Σ Di Nsim i=1

Nnode j=1

[

V j (i) − V r e f

(5.7) ]2

Nnode

(5.8)

where Di is the comprehensive voltage deviation of the ith scenario; is the number of nodes in the DN; V j (i) is the nodal voltage of node j in the ith scenario; V r e f is the reference nodal voltage, which is set to 1 p.u. in this paper.

5.3 Simulation Methods for Voltage Violation Assessment 5.3.1 Kernel Density Estimation of Electric Vehicle Loads and Photovoltaics Outputs The KDE method is utilized to model the PDFs of EV loads and photovoltaic outputs. Its basic idea is to treat each sample as a kernel function and sum these functions to form an overall PDF. Several load samples and the normal distribution kernel function are selected in Fig. 5.4 to illustrate the mechanism of KDE. Different kernel functions create different PDFs of the sample data. The kernel function defined as K by KDE is [22]: ∽ f (x) =

1

Nsam Σ

Nsam h

i=1

K(

x − Xi ) h

(5.9)

5.3 Simulation Methods for Voltage Violation Assessment

89

Fig. 5.4 The mechanism of kernel density estimation

where N sam is the number of sample data; h is the bandwidth; X 1 , X 2 , …, and X n are the sample data from the target distribution. The standard normal distribution kernel function is adopted in this paper. Hence, Eq. (5.9) can be specified as: ∽ f (x) =

1

Nsam Σ

Nsam h

i=1

(x−X i )2 1 √ e− 2h2 2π

(5.10)

To avoid over-smooth or under-smooth in KDE, the bandwidth h is determined according to the formula provided by Silverman [23]. ( h=

4 3Nsam

)1/5 σ

(5.11)

where σ is the standard deviation of the sample data.

5.3.2 Slice Sampling for Voltage Violation Assessment It is essential to accurately sample from the PDF obtained by KDE to generate adequate data for Markov chain monte Carlo simulation. Slice sampling is an efficient method for handling the continuous PDF with an irregular shape and is mainly composed of two procedures as illustrated in Fig. 5.5 [18]. The first procedure is defined as the step-out process in Fig. 5.5a. Assume x k is the[ previous] sampling result and yk is an auxiliary variable randomly drawn from U 0, ∽ f (xk ) . The step-out process is to extend the slice range from the initial point of x k step by step in a step width ω until the probability densities of both ends are

90

5 Voltage Violations Assessment Considering Reactive Power …

f ( x)

f ( x)

yk ~ U [0, f ( xk )] yk

yk

3rd Time ~ U [ L1max , R1max ] 2nd Time max ~ U [ Lmax ] 0 , R1

xnl xnl −1

step-out

xk

xnr−1 xnr

step-out

(a) step-out process

x

shrinkage Lmax 0 Lmax 0

L1max

xk +1

1st Time max ~ U [ Lmax ] 0 , R0 R1max R1max

shrinkage

x

R0max

(b) shrinkage process

Fig. 5.5 Procedures of slice sampling

all below yk . Therefore, the eventual slice range completely contains the PDF range which is above yk . Then, the shrinkage process is carried out to obtain the next sample xk+1 that and R0max are the initial values of the satisfies ∽ f (xk+1 ) ≥ yk . In Fig. 5.5b, L max 0 left the [ end andmax ] right end of the slice range respectively. The first sample is drawn U L max , R with its corresponding PDF value calculated. If the value is below 0 0 yk , the new slice range boundary will be updated and the next sample will be drawn according to the new range until the sample satisfies ∽ f (xk+1 ) ≥ yk . The step-out and shrinkage procedures continue to provide samples until the simulation converges.

5.3.3 Automated Step Width Selection for Slice Sampling The inappropriate value of the step width ω can decrease the sampling efficiency significantly. Therefore, the automated selection mechanism is introduced in this paper to find the optimal ω [24]. The selection algorithm is shown in Fig. 5.6. This algorithm helps to minimize the effort of step-out and shrinkage operations with several pre-sampling iterations. In each iteration, the numbers of step-out and shrinkage operations are recorded to optimize ω until the tuning iteration converges. After that, the optimal ω is utilized in slice sampling to accelerate the Markov chain monte Carlo simulation.

5.4 Risk Assessment for Voltage Violation In this section, the tap position of the OLTC and the reactive power from smart inverters and switching capacitors are regarded as manipulated variables to regulate voltage violations in distribution networks. As different EV loads and photovoltaic

5.4 Risk Assessment for Voltage Violation

91 Initialize φ= 0.1, t = 0, N0auto= 1, and 0= 1

Fig. 5.6 Flow chart of the step width selection

Draw Ntauto samples with using slice sampling Record the number of expansions Ntexpan during the step-out process

Ntexpan If 0.5-φ< expan reduc t0 + t g { 0 L Id · t−t , t0 ≤ t ≤ t0 + td td ΔPd (t) = L I d , t > t0 + td

(7.3)

(7.4)

where the maximum response time of generators tg and flexible loads td can be obtained from the engineering experience, e.g., 10 and 0.5 s in the UK, respectively [3, 4]. The frequency deviation in the time domain can be obtained by solving Eq. (7.1), as shown in Eq. (7.5) at the bottom of this page, where D, = D · PD. ) (Σ ⎧[ ) )]( (Σ Σ Σ D , )t−t ) ) ) [ ) ⎪ LI − 2H g P F Rg g P F Rg ⎪ d L Id 0 − ⎪ ΔPL + 2H + d , d t − t0 , t ∈ t0 , td + 1 − e ⎪ , , tg tg td ⎪ ,2 D D D t ⎪ D d ⎪ ⎪ ]( [ ) Σ Σ ⎪ ) ) ) ) ,) ,) ⎨ 2H g P F R g, ) ) − D t−t ) [ ) − D t−td g P F Rg ) d + ΔPL −P F R td + Δ f (t) = Δ f td e 2H t − td , t ∈ td , t g 1 − e 2H − ,2 tg ⎪ D, D , tg D ⎪ ⎪ ( ) ⎪ ) ) ) ) ) ) ⎪ , , ⎪ ) ) − D t−tg [ ) ⎪ ΔPL −P F R tg − D t−tg ⎪ ⎪ 1 − e 2H + , t ∈ t g , t1 ⎩ Δ f tg e 2H D,

(7.5)

7.2 The Proposed Integration-Based Frequency Security Criterion

127

Note that the duration of the PFR process is normally much longer than t g [3]. It can be deduced from Eq. (7.5) that when t ≫ t g , the system frequency will gradually arrive at the quasi-steady state and the corresponding frequency deviation Δ f (t1 ) can be approximated as: Σ Δ f (t1 ) ≈

g

Σ

P F Rg +

d

L Id − ΔPL

(7.6)

D,

By integrating Eqs. (7.1) and (7.2) on both sides: ,

{t1

2H · [Δ f (t1 ) − Δ f (t0 )] + D ·

=

{t1 (Σ

ΔPg (t) +

g

t0

Δ f (t)dt t0

Σ

)

ΔPd (t) − ΔPL dt

(7.7)

d

The frequency dead band is generally a small value, e.g., 0.018 Hz in ERCOT [22]. Compared to the frequency deviation under large power loss, this value can be neglected, whence Δ f (t0 ) is assumed to be zero. Let t, = t1 − t0 and take Eqs. (7.3)– (7.6) into Eq. (7.7), the integration of frequency deviation during the PFR process can be calculated as: ) t, 2H · ΔPL − D, (D , )2 t0 ) Σ ( tg − 2t , 2H · + P F Rg + 2D , (D , )2 g ) Σ ( 2H td − 2t , · + L Id + 2D , (D , )2 d (

{t1

F S1 = −

Δ f (t)dt =

(7.8)

7.2.2 Frequency Deviation During the SFR Process After the frequency arrives at the quasi-steady state at time t 1 , the generators start to provide SFR to the power system. Flexible loads are assumed to provide the LI with full capacity during the PFR process, whence they are not considered in the SFR process. With enough SFR, the frequency can be recovered to the nominal value at the end of the SFR process, whose duration t,, = t2 − t1 is normally taken as 10 min [7]. The output increment of generators during the SFR process is governed by the automatic generation control (AGC) based on the area control error (ACE) signal

128

7 A Stochastic Unit Commitment to Enhance Frequency Security …

[28]: AC E(t) = P F R(t1 ) + S F R(t) − ΔPL − 10 · γ · Δ f (t)

(7.9)

where γ is the frequency bias of the power system, SFR(t) is the SFR provided by generators in the receiving-end power system at time t, which can be calculated by: 1 S F R(t) = −α · AC E(t) − β

{t AC E(τ )dτ

(7.10)

t1

where α and β are the proportional and integral control parameters of AGC, respectively. Since frequency is recovered to the nominal value at the end of SFR process t2, both Δf(t2) and ACE(t2) are zero. It thus can be deduced from Eq. (7.9) that: S F R(t2 ) = ΔPL − P F R(t1 )

(7.11)

The SFR is assumed to be provided by generators to the receiving-end power system following a constant rate rr, then SFR(t) can be represented as: S F R(t) = rr · (t − t1 ) =

S F R(t2 ) · (t − t1 ), t ∈ [t1 , t2 ] t ,,

(7.12)

Combining Eqs. (7.9)–(7.12), the integration of frequency deviation during the SFR process can be calculated as: {t2 F S2 = −

Δ f (t)dt = (ΔPL − P F R(t1 )) · t1

( = ΔPL −

Σ g

P F Rg −

Σ d

) L Id

·

t ,, − 2β 20γ

t ,, − 2β 20γ

(7.13)

The proposed frequency security criterion FS can be calculated by Eqs. (7.8) and (7.13) as: F S = F S1 + F S2 ) ( , t 2H t ,, − 2β · ΔPL = − + D, 20γ (D , )2 ) Σ ( tg − 2t , 2H t ,, − 2β · + − P F Rg + 2D , 20γ (D , )2 g ) Σ ( 2H t ,, − 2β td − 2t , · + − L Id + 2D , 20γ (D , )2 d

(7.14)

7.3 Stochastic Frequency Security-Constrained Unit Commitment Model

129

where the coefficients of PFRg and LId reflect how much the value of frequency security criterion FS can be reduced given per unit of additional PFR provided by generators and per unit of additional LI by flexible loads, respectively. Based on the coefficients, the effectiveness of flexible loads and generators to the frequency regulation process can be compared quantitatively by a performance factor Γ, which is defined as: ∂ F S/∂ L Id −1= Γ= ∂ F S/∂ P F Rg

tg −2t , 2D ,

+

td −tg 2D , 2H (D , )2

−

t ,, −2β 20γ

(7.15)

Since t d is generally smaller than tg, it is surely that Γ > 0, which means flexible loads outperform the generators in the frequency regulation process with higher efficiency, as we shall explain further in the case studies. Remark II: as can be seen from Eq. (7.14), the proposed frequency security criterion FS is linear to the power loss ΔPL, the PFRg provided by generators, and the LI d provided by flexible loads. This linearity makes it easy to be implemented into optimization problems.

7.3 Stochastic Frequency Security-Constrained Unit Commitment Model The proposed FCUC model is formulated into a two-stage stochastic optimization model to cope with frequency deviation caused by uncertain transmission line failures under extreme weather conditions.

7.3.1 First-Stage Problem: Coordination of Cost and Risk The first stage problem aims to minimize the total operation cost of generation, PFR, SFR, and the expected LI cost under all transmission line failure events. min

Σ Σ[ ) ) ] Σ s PFR SF R f g Pg,t + C gSU · Yg,t + C g,t · P F Rg,t + C g,t · S F Rg,t + IC t

g

s

(7.16) where f g (·) is the energy cost function of generator g, which can be piece-wise linearized to reduce the computational complexity. The objective function Eq. (7.16) is subject to the following constraints: Σ g∈G i

Pg,t −

Σ d∈Ωi

Pd,t =

Σ j∈Φi

) ) Bi j θi,t − θ j,t

(7.17)

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7 A Stochastic Unit Commitment to Enhance Frequency Security …

) ) −Fimax ≤ Bi j θi,t − θ j,t ≤ Fimax j j

(7.18)

Pgmin · X g,t ≤ Pg,t

(7.19)

Pg,t + P F Rg,t + S F Rg,t ≤ Pgmax · X g,t

(7.20)

0 ≤ P F Rg,t ≤ η · Pgmax

(7.21)

0 ≤ S F Rg,t ≤ X g,t · R Rg · t ,,

(7.22)

2Yg,t ≤ X g,t+1 − X g,t + 1

(7.23)

Constraint (7.17) enforces the power balance on each bus. Constraint (7.18) limits the power flow within the nominal transmission capacity. Constraint (7.19) and (7.20) represent the output limits of generators considering the reserve margin. Constraints (7.21) and (7.22) limit the maximum PFR and SFR that can be provided by generators. Constraint (7.23) links the startup binary variables to the binary commitment variables. Other constraints also include the ramping up speed and ramping down speed constraints, the minimum start-up time and the minimum shutdown time constraints, etc. These constraints are conventional and can be readily found in other literature such as [3].

7.3.2 Second Stage Problem: Risk Assessment Based on the decision variables X g,t , Pg,t , P F Rg,t and S F Rg,t from the first stage problem, the second stage problem dispatches the power system under each transmission line failure event s and aims to minimize the corresponding expected LI cost ICs. The expected LI cost is returned to the first-stage problem to minimize the total operation cost. The objective function of the second stage problem is expressed as: I C s = min

ΣΣ t≥t s

LI s Pr(s) · Cd,t · L Id,t

(7.24)

d

The objective function of the second stage problem is subject to the following constraints: (1) Post-contingency operation constraints: Σ g∈G i

s Pg,t −

Σ) d∈Ωi

) Σ s ) s ) s Pd,t − L Id,t = Ii j,t Bi j θi,t − θ sj,t j∈Φi

(7.25)

7.4 Solution Methodology

131

s Pgmin · X g,t ≤ Pg,t ≤ Pg,t + P F Rg,t + S F Rg,t

(7.26)

) s ) s s max −Fimax j,ctgy ≤ Ii j,t Bi j θi,t − θ j,t ≤ Fi j,ctgy

(7.27)

Constraints (7.25)–(7.27) are the post-contingency power balance constraint, generation constraint, and power flow constraint, respectively. Constraint (7.27) allows the power flow to exceed the nominal transmission limits shortly after contingencies and supports the frequency regulation in the receiving-end system. (2) Post-contingency frequency security constraints: ΔPLs ≤

Σ

P F Rg,t +

g

Σ

S F Rg,t +

g

Σ

s L Id,t

(7.28)

d

s 0 ≤ L Id,t ≤ ξ · Pd,t

(7.29)

) ) s F S ΔPLs , P F Rg,t s , L Id,t ≤ ρ · F S max s

(7.30)

where FS(·) is the proposed integral-based frequency security criterion under failure events, which can be calculated from (7.14) with the amount of PFR provided by generators, the LI of flexible loads, and the power loss ΔPLs under failure events. Constraint (7.28) requires that the overall amount of PFR, SFR, and LI be greater than the power loss due to transmission line failures so that the frequency can be recovered to the nominal value. Constraint (7.29) means that the amount of LI cannot exceed the maximum interruption capacity of flexible loads. Constraint (7.30) ensures that the frequency deviation satisfies the frequency security requirement under all failure events.

7.4 Solution Methodology 7.4.1 Problem Linearization The system inertia H(t) is determined by the online generators at hour t: H (t) =

Σ

X g,t · Hg · Pgmax / f 0

(7.31)

g

This makes constraint (7.30) bi-linear and should be linearized to reduce compuΘg,t to replace the bi-linear items tational Define a new variable ) ( complexity. Σ Σ X g,t · ΔPL − d L Id,t − g P F Rg,t in constraint (7.30). These bi-linear terms

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7 A Stochastic Unit Commitment to Enhance Frequency Security …

can be replaced by the following equivalent linear constraints with the McCormick envelope method [19]: −M · X g,t ≤ Θg,t ≤ M · X g,t ΔPL −

Σ

L Id,t −

Σ

) ) P F Rg,t − M · 1 − X g,t ≤ Θg,t

(7.32) (7.33)

g

d

Θg,t ≤ ΔPL −

Σ

L Id,t −

Σ

) ) P F Rg,t + M · 1 − X g.t

(7.34)

g

d

where M is a large number. As the largest transmission line capacity is greater than ΔPL, it can be taken as the value of M. If X g,t = 0, constraint (7.32) enforces Θg,t = 0, and constraints (7.33) and (7.34) are unbinding. Σ Otherwise, Σ if X g,t = 1, constraints (7.33) and (7.34) enforce Θg,t = ΔPL − d L Id,t − g P F Rg,t and constraint (7.32) become unbinding. Hence, the problem is reformulated into an equivalent two-stage stochastic mixed-integer linear problem (MILP).

7.4.2 Regularized L-shape Algorithm The two-stage stochastic FCUC problem is solved by the regularized L-shape algorithm [29], which decomposes the two-stage problem as a regularized master problem (MP) and multiple sub-problems (SP). The regularized MP corresponds to the first stage problem but with an additional regularized item. The SPs correspond to the second stage problems. By iteratively generating feasibility cuts and optimality cuts based on the SPs, the problem can be solved efficiently. For the convenience of discussion, the MP is reformulated in a compact form: min f T x + gT y +

Σ s

)2 ) σs + κ x − x, , y − y, 2

s.t. Ax + By ≤ C

(7.35)

where the objective function corresponds to Eq. (7.16) and the constraints correspond to Eqs. (7.17)–(7.23), in which x and y are the vectors of the variables in the first stage problem; σs is the auxiliary variable of ICs; x, and y, are the initial feasible solutions of the FCUC problem; κ is the penalty parameter for the regularized term; A and B are constant coefficient matrices. f, g, and C are constant-coefficient vectors. The vector x, which includes Xg,t , Pg,t , PFRg,t , and SFRg,t , is sent to the SPs to determine the expected LI cost ICs. The SPs can be reformulated as: I C s (x) = min qsT zs s.t. Wzs = hs − Ts x, zs ≥ 0

(7.36)

7.5 Case Study

133

where the objective function corresponds to Eq. (7.24) and the constraints correspond to Eqs. (7.25)–(7.30), in which zs is the vector of second-stage variables. Ts is the coefficient matrix, qs and hs are coefficient vectors under failure event s. Whereas, W is a fixed matrix and will not change in different scenarios. Denote Ns as the number of SPs. If (x, y, σs) given by the regularized MP is an optimal solution to the FCUC problem, it must satisfy the following constraints: σs ≥ (πs )T (hs − Ts x) ) , )T ) )T πs Ts x ≥ πs, hs

(7.37)

where πs and π, s represent the extreme points and the extreme rays of the SPs, which can be readily obtained based on the dual problems of SPs [29]. Note that the interruption cost is non-negative, whence instead of starting the iterations from σs = −∞ as the standard algorithm, we start from σs = 0 to reduce iterations. Moreover, the initial feasible solutions x, and y, can be carefully chosen to significantly reduce the iterations of the regularized L-shape algorithm. Specifically, massive historical solutions under different operating conditions, such as load curves, weather conditions, and generator fuel costs, have been accumulated through the long-term scheduling practice. Some typical patterns can be extracted from historical solutions to act as initial feasible solutions for x, and y, under similar operating conditions. In addition, machine learning techniques can be adopted to obtain some good initial solutions based on the historical data set [30]. This is also an important promotion factor for the efficient regularized L-shape algorithm in this data-intensive FCUC problem.

7.5 Case Study In this section, a modified IEEE 118-bus system [31] is used to illustrate the proposed FCUC method. The system is divided into the sending-end system and the receivingend system, which are connected by three HVDC lines with different transmission capacities [32], as shown in Fig. 7.2.

7.5.1 Parameter Setting Assume all three HVDC lines are under wear-out period with failure rate 0.03 (year— 1/100 km) [22]. In this paper, both the moving paths and intensities from the historical Pacific Ocean hurricanes [33] are considered to generate the HVDC failure events. The parameters for calculating the weather condition factor can be found in [17]. The probabilities of different failure events are shown in Fig. 7.3. The duration of PFR process t, and SFR process t,, are set as 60 s and 600 s, respectively. The maximum

134

7 A Stochastic Unit Commitment to Enhance Frequency Security …

Fig. 7.2 Two systems connected by three HVDC lines subject to possible hurricane paths. The red numbers denote the buses connecting HVDC lines, and the blue numbers denote the location buses of flexible loads

response time of generators tg and flexible loads td is set as 10 s and 0.5 s, respectively [3, 4]. The nominal frequency f0 is taken as 50 Hz. For each generator, the inertia Hg is set as 5 s [4]. The AGC parameter α is set as zero, β is set as 50 s, and γ is set as 1% of the peak load demand [34]. The costs of PFR and SFR of generators are set as $600/MW. The maximum percentage of generation capacity for PFR η is set as 20%. 11 flexible loads in the receiving-end system are selected to provide LI. The costs of LI on different buses and the maximum ratio of interruption of flexible loads ξ are given in Table 7.1, where the data is taken from Ref. [3]. The load-damping rate D is set as 2%/Hz [22]. The maximum frequency security criterion FSmax is set as 200. The penalty parameter of the regularized L-shape algorithm κ is set as 100.

7.5.2 Reserve Dispatch Under Different Security Requirements To demonstrate the effectiveness of the proposed FCUC method, three simulations under different frequency security requirements are carried out. The reserve scheduling results including the PFR, the expected LI, and SFR are shown in Fig. 7.4. As can be seen, both the amount of PFR provided by generators and the expected

7.5 Case Study

135

Fig. 7.3 Failure probabilities of three HVDC lines at each specific hour, e.g., the failure probability of HVDC line 2 at 6 am is 0.004

Table 7.1 Load interruption at different buses in the receving-end system

Location bus

Cost ($/MWh)

Maximum ratio of LI (%)

42, 49, 54, 56, 59, 60

30,000

20

62, 78, 80, 90, 116

6000

20

LI provided by flexible loads increases as ρ decreases from 0.75 to 0.25, i.e., the frequency security requirement becomes tighter. Whereas, the amount of scheduled SFR decreases gradually. With more PFR available in the power system, the frequency deviation can be better restrained and both the frequency deviation at the nadir time and the quasi-steady state are reduced. As a result, less SFR is required to recover the frequency to the nominal value during the SFR process. The cause-andeffect relationship between the results and the proposed criterion can be explained by Eqs. (7.14) and (7.30). Furthermore, as can be observed from Fig. 7.4, more LI is scheduled from 9 to 12 a.m. than in the other periods, whereas less PFR and SFR are scheduled for generators. This is because the load demand level is higher during this period and allows more capacity for LI.

7.5.3 Comparative Analysis Between the Proposed Frequency Security Criterion and the Conventional Criteria In this subsection, we will first show that the restraining effect of the proposed criterion on the shape of the system frequency dynamic curve is similar to the conventional criteria. The frequency deviations under all failure events are shown in Fig. 7.5, where the bold lines represent the frequency deviations subject to the HVDC 1 failure at 5 a.m. As can be seen, the integration of frequency deviation decreases when ρ decreases from 0.75 to 0.25. Meanwhile, the time nadir tnadir arrives earlier, and both the frequency nadir fnadir and the quasi-steady-state frequency fqss are elevated.

136

7 A Stochastic Unit Commitment to Enhance Frequency Security …

Fig. 7.4 Scheduling results of the frequency regulation reserve (MW) under three different frequency security requirements

According to the UK practice [3], the frequency nadir and the quasi-steady-state frequency should be above 49.2 Hz and 49.5 Hz, respectively. As is evident from Fig. 7.5, these frequency requirements can be well satisfied when ρ is set as 0.25. Note that ρ is fixed at 0.25 in the subsequent simulations. Overall, the proposed frequency security criterion is not controversial but compatible with the conventional criteria.

Fig. 7.5 Scheduling results of the frequency regulation reserve (MW) under three different frequency security requirements

7.5 Case Study

137

Next, we show that the proposed criterion can quantify the out-performance of flexible loads compared to generators in regulating frequency. In Fig. 7.6, the frequency dynamic curves under the different percentages of PFR provided by flexible loads are presented, where the power loss is 300 MW, the total amount of PFR is 285 MW, and the system inertia H is 500 MW s/Hz. The frequency deviation is better constrained with more PFR provided by flexible loads. However, this benefit of flexible loads cannot be reflected by the ROCOF and quasi-steady-state criteria, since they only depend on the system inertia and the overall amount of PFR [3]. Although the benefit of flexible loads can be reflected by the frequency nadir criterion, the out-performance of flexible loads is intractable to be analytically quantified by the frequency nadir criterion. Whereas, this can be readily achieved by the performance factor Γ in Eq. (7.22). As shown in Fig. 7.7, Γ is negatively correlated to the duration of the PFR process and system inertia. Here, Γ can be as high as 8%, indicating that 1 MW LI can be as effective as 1.08 MW PFR provided by generators. Further, to demonstrate the cost-effectiveness of the fast-responding flexible loads, the FCUC results under two cases are compared: Case A: the response time of flexible loads td is set as 0.5 s; Case B: td is set the same as the response time of generators, i.e., 10 s. In Fig. 7.8, if flexible loads are not distinguished from generators, more frequency reserve will be dispatched by the FCUC model. The total reserve cost of Case A is about 5% less than the reserve cost of Case B. It can also be noticed from Fig. 7.9 that the amount of reserve provided by generators is decreased by 72%, as the maximum ratio of LI increases from 15 to 30%. Moreover, the decrement of generator reserve cost (2.1 M$) is greater than the increment of LI cost (0.6 M$) as shown in Fig. 7.8, leading to a reduced total reserve cost. This result indicates that flexible loads are more cost-effective than generators. In summary, the proposed frequency security criterion can quantify the merit of the fast response rate of flexible loads and enables the system operators to dispatch frequency reserve resources more effectively.

Fig. 7.6 Frequency deviations under different percentages of PFR provided by flexible loads

138

7 A Stochastic Unit Commitment to Enhance Frequency Security …

Fig. 7.7 The performance factor of flexible loads compared to generators

Fig. 7.8 Reserve cost under different maximum ratios of load interruption

Fig. 7.9 PFR and SFR scheduling results of generators under different maximum load interruption ratios

7.5 Case Study

139

7.5.4 Validation of the Proposed Frequency Security Criterion In this subsection, the Power System Analysis Software Package (PSASP) [35] is used to further verify the effectiveness of the proposed criterion. The power loss is assumed to be 300 MW. The system inertia H is set as 500 MW s/Hz and the load damping parameter D, is set as 100 MW/Hz. Three cases are simulated with different combinations of PFR and SFR: Case A: 275 and 25 MW; Case B: 250 and 50 MW; Case C: 225 and 75 MW. The frequency dynamic curves simulated by PSASP are shown in Fig. 7.10. There exist frequency damping oscillations during the realistic PFR process. This is because the frequency response models of generators and flexible loads are within a closedloop process, wherein the reserve delivery functions ΔPg(t) and ΔPd(t) depend on the frequency deviation Δf(t) in PSASP, as shown in Fig. 7.11. In contrast, the reserve delivery functions are linearly approximated by Eqs. (7.3) and (7.4), which are independent of the frequency deviation. Hence, no damping oscillation exists in the estimated frequency dynamic curves, as described by Eq. (7.5).

Fig. 7.10 Simulated frequency dynamic curve and the integration of frequency deviation during the PFR process FS1 as well as the SFR process FS2

Fig. 7.11 Block diagram for the simulation of system frequency dynamics

140

7 A Stochastic Unit Commitment to Enhance Frequency Security …

However, despite the absence of damping oscillations, the proposed criterion is still applicable to estimate the real value of the integration of frequency deviation. Based on Eqs. (7.8) and (7.13), the estimated integration of frequency deviation during the PFR and SFR process are respectively: 26.2 and 10.4 for Case A; 37.5 and 20.8 for Case B; 48.7 and 31.4 for Case C. The simulated values by PSASP are shown in Fig. 7.10. The simulated values are smaller than the estimated values, with relative errors between 2 and 25% amongst the three cases. These results show that the proposed frequency security criterion is a relatively conservative criterion, which may cause the FCUC model to dispatch slightly more frequency reserve. This is because the linear approximations of reserve delivery functions are based on the maximum response time of generators and flexible loads, leading to an underestimation of the realistic reserve response rates and frequency regulation capacity. Consequently, the realistic value of FS is overestimated and more reserve will be dispatched to satisfy the frequency security requirement. Nevertheless, it should be noted that the FCUC model is a day-ahead dispatch model. Scheduling relatively more reserve is acceptable as the real-time condition may vary greatly under extreme weather conditions. Moreover, the day-ahead scheduling results can be adjusted in the realtime market.

7.5.5 Impacts of Operation Life and Hurricane Intensities on Operation Cost A set of simulations are implemented to show the impacts of the HVDC lifetime in the wear-out period and the hurricane intensity on the power system operation cost. As illustrated in Fig. 7.12, the operation cost increases when the operation life becomes longer and the hurricane intensity becomes larger. This is because the failure rates of HVDCs are aggravated by the operation life and hurricane intensity. Consequently, the probability of each failure event and the expected cost of the FCUC model are increased. It can also be observed that the total operation cost of the worst scenario (upper-right) is 40% higher compared to the benchmark scenario (lowerleft). Therefore, the high operation cost due to the failure-prone HVDCs under the wear-out period and their replacement cost should be carefully balanced by the system operators to achieve overall cost-effectiveness in the long-term run.

7.6 Conclusion In order to describe the accumulated frequency deviation during the frequency regulation process, a novel frequency security criterion is suggested in this study. The proposed criterion can measure how flexible loads outperform generators when it comes to frequency management. According to numerical findings, the flexible loads

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Fig. 7.12 Operation costs (M$) of the power system under different operation lifetimes in the wear-out period and hurricane intensities. The lower-left box represents the operation cost of the benchmark scenario

can be up to 8% more effective than the generators in the conditions of this research. Additionally, the suggested criterion’s linearity makes it easily adaptable to optimization issues. Additionally, a stochastic FCUC method is suggested to schedule the generators and flexible loads optimally while meeting the proposed criterion against potential transmission line failures caused by unknown extreme weather. Case studies demonstrate that, in comparison to the adaptive robust approach and the deterministic N−1 method, the suggested stochastic FCUC method can achieve a better trade-off between security and economy.

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Chapter 8

A Data-Driven Reserve Allocation Method with Frequency Security Constraint of Post-fault Power System Considering Inverter Air Conditioners

8.1 Introduction To preserve the stability of power systems, frequency regulation, including primary frequency regulation (PFR) and secondary frequency regulation (SFR), is commonly used [1–3]. One of the key problems with frequency regulation is reserve allocation. In earlier studies of reserve allocation for PFR and SFR, which place a greater emphasis on the power balance of the systems, the frequency dynamics are seldom ever taken into account [4]. The increasing penetration of renewable energies has recently posed a serious danger to the frequency stability of the power system [5], and more studies have followed reserve allocation taking the frequency dynamics into consideration with interest. The frequency response model is typically essential to understanding frequency dynamics. The PFR and SFR models are constructed using the swing equations in [6]. Shekari et al. [7] proposes an adaptive frequency response model that integrates with the load shedding strategy. An adaptive multiple-machine frequency response model with the governor response is provided in [8]. The frequency security restriction associated with frequency dynamics cannot be easily addressed in the power system reserve allocation problem due to the high nonlinearity and complexity of frequency response models. The relatively imprecise generating frequency dynamics are linearized in [9] for simplicity. The unit commitment problem’s affine constraint of the sufficient condition for frequency security is presented in [10], which is a conservative estimate and might result in an inflated reserve demand. The frequency limit is converted into a linear arithmetic equation in [11] using the piece-wise linearization technique, and as the number of system components increases, so does the complexity of the equation. Demand side resources, among which inverter air conditioners (IACs) are the most concerned, have recently been widely believed to have the capacity for power system frequency management [12, 13]. The following are the causes: (1) One of the top energy-guzzlers is air conditioning (AC), and the majority of recently installed ACs are IACs [15]. (2) IACs can swiftly change the input power [16]. Furthermore, © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Ye et al., Risk-Based Planning and Operation Strategy Towards Short Circuit Resilient Power Systems, https://doi.org/10.1007/978-981-19-9725-9_8

145

146

8 A Data-Driven Reserve Allocation Method with Frequency Security …

consumers are relatively unaffected by short-term adjustments to IAC power [17]. IACs should be taken into consideration while regulating the frequency of the power system because they are regarded as the perfect demand response resource. IACs are modeled as a thermal batteries in [18]. Hui et al. [19] presents the equivalent transfer functions, control parameters, and assessment criteria for the equivalent frequency response model of IACs. For frequency regulation service in [20], the IACs are integrated with the traditional generator concept. However, due to the complexity of the IACs’ model, simulation is mostly used in this current research to support the response characteristics of IACs, which has not been incorporated into the reserve allocation optimization problem. Due to its alluring model-free benefit, a machine learning classifier (MLC) is frequently used [21] and has been successfully employed in power system studies such as component failure diagnosis, load forecasting, power quality evaluation, and so on [22–24]. Whether the frequency security limitation is satisfied might be seen as a binary problem from the standpoint of classification. MLC can therefore be used to address the power system frequency security constraint that is implicit but has not yet been reported. This chapter proposes a data-driven reserve allocation approach with frequency security constraints taking IACs into consideration. First, the power system model is combined with the equivalent frequency response model of IACs. The reserve allocation problem between IACs and conventional generators taking into account frequency security constraints is then converted to a general quadratically constrained quadratic program (QCQP) by using a machine learning frequency security classifier (MLFSC) based on support vector machine (SVM) [25]. The general QCQP is solved by the heuristic Suggest-and-Improve (SI) method. Figure 8.1 presents the paper’s organizational structure.

8.2 Modeling of Power System Frequency Response Integrated with IACs The model for the power system frequency response is initially introduced in this section. After that, the IAC equivalent frequency response model is shown. Finally, the power system frequency response framework incorporates the aggregated model of IACs. It should be noted that this section’s models were all built in the frequency domain.

8.2.1 Power System Frequency Response Model The classical frequency response model participated by m conventional generators is illustrated in Fig. 8.2 [6].

8.2 Modeling of Power System Frequency Response Integrated with IACs

147

Fig. 8.1 Framework of the proposed data-driven power system reserve allocation

Fig. 8.2 Classical power system frequency response model

In Fig. 8.2, the input ΔP is the power imbalance of the power system; the output Δ f is the corresponding frequency deviation considering the frequency regulation. H is the equivalent system inertia and D is the load-damping rate. The frequency regulation provided by the conventional generator i is illustrated in the dashed line box. R i and TGi are the governor speed regulation and time constant of the generator i,

148

8 A Data-Driven Reserve Allocation Method with Frequency Security …

respectively. TCi H and TRi H are the steam chest time and reheat turbine time constants of the generator i, respectively. FHi P is the high-pressure power fraction of the reheat turbine of the generator i. SatGi (x) is a saturation block representing the reserve capacity RGi of the generator i. It is defined as: { SatGi (x)

=

x, 0 < x < RGi RGi , x ≥ RGi

(8.1)

8.2.2 Equivalent Frequency Response Model of IACs The IACs can be viewed as a valuable resource to maintain the frequency stability of the power system due to their considerable regulatory capability [20]. Based on its thermal and electrical models, an IAC’s equivalent frequency response model can be created. Based on the modeling of the room temperature variation, the thermal model of an IAC is constructed [26]: Cr oom Vr oom ρ A Tr oom (s) = Q r oom (s) − Q AC (s)

(8.2)

Q r oom (s) = Hr [Tout (s) − Tr oom (s)]

(8.3)

where Cr oom and Vr oom are the room thermal mass and room volume, respectively; ρ A is the density of the air; Q r oom and Q AC are the room heat gain and the refrigerating capacity of the IAC, respectively; Tout and Tr oom are the outdoor and room temperature, respectively; Hr is the equivalent thermal conductance between the room and the outdoor air. IACs can adjust the operating frequency in order to control their operating power. And the electrical model of an IAC is constructed based on this feature [17]: PAC (s) =

θP f AC (s) + Pc Tc s + 1

(8.4)

Q AC (s) =

θQ f AC (s) + Q c Tc s + 1

(8.5)

where θ P and θ Q are the control parameters of the IAC; f AC and PAC are the operating frequency and operating power, respectively; Tc is the compressor time constant of the IAC; Pc and Q c are the baseline operating power and refrigerating capacity, respectively. An IAC can participate in the power system frequency response by changing its operating frequency in accordance with the system frequency deviation, which will

8.2 Modeling of Power System Frequency Response Integrated with IACs

149

affect its operating power. It is possible to determine the link between the system frequency and the IAC operating frequency [19]: Δ f AC (s) = AΔ f (s) + C(s)(ΔTr oom (s) − ΔTset (s)) C(s) = α +

β s

(8.6) (8.7)

where A is the control coefficient that is analogous to the 1/R of generators; C(s) is the temperature controller of IACs, which is a proportional-integral (PI) controller as illustrated in Eq. (8.7); Tset is the IAC set temperature. The operating frequency of an IAC is determined jointly by the system frequency and the deviation of room and set temperature. Based on the thermal and electrical models represented by Eqs. (8.1)–(8.7), the response of the operating power of IACs to the power system frequency deviation can be obtained by: ΔPAC (s) = F1 (s) + F2 (s) F1 (s) =

(8.8)

θ p (Cr oom Vr oom ρ A s + Hr )( AΔ f (s) + C(s)ΔTset (s)) (Tc s + 1)(Cr oom Vr oom ρ A s + Hr ) + θ Q C(s)

(8.9)

θ p C(s)(Hr ΔTout (s)) (Tc s + 1)(Cr oom Vr oom ρ A s + Hr ) + θ Q C(s)

(8.10)

F2 (s) =

It can be seen from Eqs. (8.8)–(8.10) that the IAC operating power is affected by the system frequency, the room temperature, and the set temperature. As the primary frequency response is implemented within seconds [6], it is reasonable to assume that the room temperature and set temperature remain constant during this process. Consequently, Eqs. (8.8)–(8.10) can be simplified as follows: ΔPAC (s) =

θ p A(Ta s + 1)Δ f (s) (Tc s + 1)(Ta s + 1) + Φ AC C(s) Ta =

Cr oom Vr oom ρ A Hr

Φ AC =

θQ Hr

(8.11) (8.12) (8.13)

It can be seen that Eqs. (8.11)–(8.13) exactly constitute the equivalent frequency response model of a single IAC. As demand response has spread, more IACs, particularly central types for industrial or commercial buildings, have been involved in power system operation control. Such IACs are set up with certain control programs and are wirelessly connectable,

150

8 A Data-Driven Reserve Allocation Method with Frequency Security …

allowing for remote monitoring and control. Only by accepting or actively promoting such programs and agreeing to disclose the required IAC criteria are the manufacturers able to raise the added values and popularity of their products and improve sales volumes [29, 30]. Some manufacturers directly participate in the demand response initiatives and split the profits with the load aggregators in specific situations. This essay is based on the aforementioned scenario, in which the IAC manufacturers are involved in the demand response initiatives and have access to the relevant equipment data.

8.2.3 Power System Frequency Response Model with Aggregated IACs The operating power of a single IAC has no significance for a realistic power system. The response of IACs only makes sense with the simultaneous participation of multiple ones. Considering the large quantity of existing IACs (like tens of thousands) and their non-uniform parameters, the clustering method is utilized in this section to avoid the curse of dimensionality. Specifically, the k-means clustering algorithm [27] is utilized to cluster massive IACs. Assume that the IACs are of different Tc and A. Then the dataset to be clustered is {(Tc1 , A1 ), (Tc2 , A2 ), . . . , (Tcd , Ad )} where Tci and Ai are the corresponding parameters of IAC i and d is the total IACs number. C = {(T˜c1 , A˜ 1 ), (T˜c2 , A˜ 2 ), . . . , (T˜cn , A˜ n )} is the cluster centers, which can be calculated by: C = arg min

d Σ

(Tci − Tc(i ) )2 + (Ai − A(i) )2

i=1

(Tc(i) , A(i) ) =

arg ( j) (Tc , A( j ) )∈C

min[(Tci − Tc( j ) )2 + ( Ai − A( j) )2 ]

(8.14)

where n is the number of clusters and (Tc(i) , A(i ) ) represents the cluster centroid that (Tci , Ai ) belongs. The IACs are divided into n aggregated ones with the centroid parameters C. Then the power system frequency response model considering the demand response of IACs can be shown in Fig. 8.3, where the aggregated IAC can be considered as a kind of Virtual Power Plants (VPPs) [28].

8.3 Data-Driven Reserve Allocation with the Frequency Security Constraint

151

Fig. 8.3 Power system frequency response model integrated with IACs and conventional generators

8.3 Data-Driven Reserve Allocation with the Frequency Security Constraint 8.3.1 Problem Description The distribution of power system reserves is the primary concern with primary frequency regulation. Because of the smaller time constant, IACs respond more quickly than conventional generators [19]. IAC reserve usage can naturally reduce frequency variation by a large amount, although in many cases IAC reserves are more expensive than those of conventional generators. The distribution of reserves offered by IACs and generators is thus in fact an optimization problem, where the frequency security constraint and the total cost of the reserves should be taken into account simultaneously. The objective of the allocation problem fitting the established frequency regulation framework in Fig. 8.3 is formulated as follows: T min C GT RG + C AC R AC

RG, R AC

(8.15)

where C G and C AC are the reserve price vectors of generators and aggregated IACs, respectively; RG and R AC are the reserve capacity of generators and aggregated IACs, respectively. The maximum allowed frequency deviation Δ f max is utilized to ensure the frequency security as illustrated in Fig. 8.4. The system reserve allocation scheme should meet the security requirement that the maximum frequency deviation caused by a specific power imbalance ΔP0 is within the threshold Δ f max . Therefore the

152

8 A Data-Driven Reserve Allocation Method with Frequency Security …

Fig. 8.4 Security constraint in terms of the maximum allowed frequency deviation

frequency security constraint can be written as follows: − min[F R(ΔP0 , RG , R AC )] ≤ Δ f max

(8.16)

where F R(·) is the system frequency response model. However, the frequency security constraint in Eq. (8.16) is implicit due to the nonlinear and complex frequency response models as illustrated in Fig. 8.3, which deters it from being integrated into a solvable optimization problem. In order to deal with the difficulty, the MLFSC is utilized in the next section to reconstruct the frequency security constraint in a data-driven way. In addition, the allowed range of the reserve capacity of generators and IACs, as well as the system power flow, should be considered in the reserve allocation problem: RGmin ≤ RG ≤ RGmax

(8.17)

max R min AC ≤ R AC ≤ R AC

(8.18)

8.3 Data-Driven Reserve Allocation with the Frequency Security Constraint

| [ ]| | RG || | | ≤ Fmax |T · A | R AC |

153

(8.19)

where RGmin and RGmax are the minimum and maximum allowed reserves of generators, max respectively; R min AC and R AC are the minimum and maximum allowed reserves of IACs, respectively; T is the power transmission distribution factor (PTDF) and A is the adjacency matrix; Fmax is the power flow threshold.

8.3.2 Data-Driven Approximation of Frequency Security Constraint In this paper, the MLFSC is trained to judge whether the system frequency deviation violates the Δ f max . Firstly, the feature vector x for the MLFSC is: x = (ΔP, RG1 , RG2 , . . . , RGm , R 1AC , R 2AC , . . . , R nAC )

(8.20)

j

where RGi and R AC are the reserve capacities of generator i and aggregated IAC j, respectively; m and n are the numbers of generators and aggregated IACs, respectively. The training dataset {x, y} is composed of multiple feature vectors along with their labels:

{ yi =

x = {x1 , x2 , . . . , xk }

(8.21)

y = {y1 , y2 , . . . , yk }

(8.22)

1, − min(F R(xi )) > Δ f max −1, − min(F R(xi )) ≤ Δ f max

(8.23)

As illustrated in Fig. 8.5, the feature vectors are generated randomly, and the corresponding labels are provided by massive simulation results performed on the power system frequency response model in Fig. 8.3. Later the MLFSC is established with SVM. The main idea is to find a hyperplane represented by parameters {w, b} to separate the data with different labels. With the obtained hyperplane, we can get the label of any new feature vector by judging which side of the hyperplane it is located in. With the training dataset {x, y}, the SVM can be trained by solving the following optimization problem [21]: Σ 1 ||w||2 + C δi 2 i=1 k

min

w,b,δ

(8.24)

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8 A Data-Driven Reserve Allocation Method with Frequency Security …

Fig. 8.5 Generation of the training dataset

s.t. ∀i, yi [w T ϕ(xi ) + b] ≥ 1 − δi

(8.25)

δi ≥ 0

(8.26)

where ϕ(·) is a mapping from the feature space to a new space, which is of great significance to improve the performance of SVM [21]. δ and C are slack variables and the corresponding cost parameters utilized to penalize feature vectors that are misclassified or too close to the hyperplane. Assume that λ∗ is the optimal Lagrange multipliers associated with Eq. (8.25). It can be obtained by solving the Lagrange dual problem of the original problem Eqs. (8.24)–(8.26) [31]. Then the parameters of the optimal hyperplane can be calculated by: ∗

w =

k Σ

λi∗ yi xi

(8.27)

yi λi∗ (xi · x j )

(8.28)

i=1

b∗ = y j −

k Σ i=1

where y j is the label of any point that satisfies 0 < λ∗j < C. With the optimal hyperplane, the frequency security constraint (8.16) can be converted by the trained MLFSC into the following form: k Σ

λi∗ yi K (xi , x) + b∗ +α ≤ 0

(8.29)

i=1

x(1) = ΔP0

(8.30)

where x(1) represents the first element of x; α is the shift coefficient; K (xi , x) is the kernel function of the mapping ϕ(·) that satisfies:

8.3 Data-Driven Reserve Allocation with the Frequency Security Constraint

K (xi , x) = ϕ(xi ) · ϕ(x)

155

(8.31)

It’s clear that the kernel function represents the mapping ϕ(·) implicitly in the form of vector inner products. The polynomial kernel function is a widely used kernel function [21] which is adopted in this paper: K (xi , x) = (a0 xi · x + a1 )2

(8.32)

where a0 and a1 are the polynomial kernel coefficients. It can be seen from Eqs. (8.29) and (8.32) that the original frequency security constraint Eq. (8.16) is converted into a quadratic constraint. As can be seen, the adoption of polynomial kernel function is a necessary step to enable the general QCQP formulation. Therefore, it is utilized in the SVM model instead of the radial basis function (RBF) [21] or other kernel functions.

8.3.3 Suggest-and-Improve Method for QCQP The reserve allocation QCQP is nonconvex because yi in Eq. (8.29) is not ensured to be positive. Consequently, the global optima problem cannot be solved by the general polynomial-time method for convex QCQP [32]. In this paper, the heuristic SI method [33] is utilized to deal with the nonconvex QCQP of interest. The main idea of the SI method is to find a candidate which is the lower bound of the original nonconvex QCQP (Suggest procedure) and shift the candidate point to the approximated optima (Improve procedure). (1) Suggest procedure by semidefinite relaxation In the Suggest procedure, semidefinite relaxation (SDR) is utilized to relax the problem and solve it to get the lower bound of the original problem. We first introduce a new variable S and rewrite Eq. (8.29) as: tr(P S) + q T x + r ≤ 0

(8.33)

S = xxT

(8.34)

P=

k Σ

λi∗ yi a02 xiT xi

(8.35)

2λi∗ yi a0 a1 xi

(8.36)

i=1

q=

k Σ i=1

156

8 A Data-Driven Reserve Allocation Method with Frequency Security …

r=

k Σ

λi∗ yi a12 + b∗ + α

(8.37)

i=1

where tr() represents the trace of a matrix. Note that with the reformulated constraints above, all the objectives and constraints are convex except Eq. (8.34). In order to accomplish a convex relaxation, Eq. (8.34) is substituted with: S ≥ xxT

(8.38)

S(r 1) = ΔP0 x T S(c1) = ΔP0 x

(8.39)

where S(r 1) and S(c1) are the first row and first column of S, respectively. Due to the convexity, the relaxed problem can be solved easily. Here, the candidate ˜ in the Suggest procedure is denoted as the solution x. (2) Improve procedure by coordinating descent The first step of the Improve procedure is to find a feasible point of the original nonconvex QCQP based on the candidate x˜ obtained in Suggest procedure. We iterate each element x˜( j) of x˜ to solve the following problem: min t t,x˜( j )

∀i, f i (x) ≤ t

(8.40)

where f i (x) ≤ 0 represents all the constraints of the reserve allocation problem. It is obvious that when t ≤ 0, the feasible point is obtained and the iteration is terminated. As all elements are fixed except one, the problem (8.40) can be easily solved by the bisection method proposed in [34]. The second step of Improve procedure is to improve the solution. We iterate each element of the feasible point to solve the nonconvex QCQP until no more improvements can be achieved. Similarly, as only one element is variable during the iteration, the nonconvex QCQP can be solved by the bisection method [34]. The pseudocodes of the SI method are given in Table 8.1.

8.4 Case Studies In this section, a realistic 12-bus system in Haining, China [35] is utilized to verify the proposed data-driven reserve allocation method, whose topology is illustrated in Fig. 8.6. The total load of the system is 405 MW. Besides 4 conventional generators, 5104 commercial IACs in the testing system are considered to participate in the

8.4 Case Studies Table 8.1 Pseudocodes of the suggest-improve method

157 Suggest procedure Step 1: Relax the QCQP with a new variable S to a convex problem Step 2: Output candidate x˜ ← Solution of the above convex problem Improve procedure Step 1: Find a feasible point based on x: ˜ Input x˜ Repeat for j = 1, 2, . . . , m + n x˜( j) ← Solution of Eq. (8.40) Until t ≤ 0 Output x˜ Step 2: Improve the feasible point: Input x˜ k ← x, ˜ e Repeat for j = 1, 2, . . . , m + n x˜(k+1 j) ← Solution of the original QCQP k ←k+1 || || Until ||x˜ k+1 − x˜ k || ≤ e Output x˜ k+1 as the approximated optimal solution

frequency regulation. TC H of the four generators are 0.43 s, 0.52 s, 0.51 s, and 0.59 s, respectively. TR H of the four generators are 4.9 s, 4.8 s, 5.1 s, and 4.8 s, respectively. Other parameters of the generators are listed in Fig. 8.6.

Fig. 8.6 Topology of the testing system

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8 A Data-Driven Reserve Allocation Method with Frequency Security …

8.4.1 Aggregation of IACs Considering the fact that most of the thermal characteristics of the commercial IACs in the same testing area are roughly homogeneous, two parameters named Tc and A are utilized to identify the IACs. The widely utilized elbow method is introduced to determine the cluster number in the k-means algorithm [26], which coordinates the clustering accuracy with the computation efficiency. As indicated in Fig. 8.7, the objective value shown in Eq. (8.14) which indicates the error of the clustering, drops with the increase of the number of clusters while the dropping speed gradually slows down. Specifically, the value of the objective has become already quite low (around 50) since the number of clusters reaches 4. And the further significant decrease in the objective value cannot be achieved by adopting more clusters. Therefore, the number of clusters is selected as 4 to ensure a satisfactory accuracy and a considerable computation efficiency of the k-means simultaneously. The aggregation result of the IACs with n = 4 is presented in Fig. 8.8, where the black dots represent the four cluster centroids. Note that the parameters of the IACs in Fig. 8.8 are normalized to [0, 1] for better clustering performance in advance by the following min–max normalization: TCi N =

TCi − min(Tc ) max(Tc ) − min(Tc )

(8.41)

AiN =

Ai − min(A) max(A) − min(A)

(8.42)

where TCi N and AiN are the normalized TCi and Ai , respectively; Tc and A are the sets of Tc and A to be clustered, respectively.

Fig. 8.7 Objective value of the k-means clustering with number of clusters

8.4 Case Studies

159

Fig. 8.8 IAC aggregation result by k-means clustering

8.4.2 Reserve Allocation Results In this section, the numerical results of reserve allocation are studied. The maximum allowed frequency deviation Δ f max is set as 0.4 Hz. The power imbalance ΔP0 is 81 MW (20% of the total load). In order to build the training dataset for the MLFSC, 12,000 feature vectors are randomly generated subject to Gaussian distributions. The labels of feature vectors are obtained from frequency dynamic simulations. The result from cross-validation [36] indicates that the adoption of the polynomial kernel function can achieve higher accuracy (98.91%) than that of RBF (94.17%). The performance of the utilized SI method is compared with the commercial Gurobi solver in terms of the QCQP presented in Sect. 8.3. Table 8.2 illustrates the obtained objectives and CPU time of the two methods. It is indicated that the reserve cost obtained from the SI method are slightly lower than that of the Gurobi solver. Besides, the CPU calculation time by the SI method is shorter than that by Gurobi solver, which proves that the SI method are more efficient than the latter. In order to verify the proposed data-driven method better, the numerical comparison with the state-of-the-art method presented in [9] is also implemented. Specifically, the state-of-the-art or the benchmark method linearizes the frequency response Table 8.2 Performance of the SI and Grobi solver

Obtained objective ($)

CPU (ms)

SI method

633.48

78.5

Gurobi solver

633.76

99.2

160

8 A Data-Driven Reserve Allocation Method with Frequency Security …

of generators and ensures the frequency security constraint by proposing its sufficient condition, which may lead to rather conservative results. The reserve allocation results are illustrated in Figs. 8.9 and 8.10. As can be seen, the proposed data-driven method is superior to the benchmark method, because the benchmark method is obviously too conservative. Specifically, it is shown in Fig. 8.9 that the maximum frequency deviation obtained by the benchmark

Fig. 8.9 Maximum frequency deviations and the overall costs achieved by the two methods

Fig. 8.10 Reserve allocation results obtained from the two methods

8.4 Case Studies

161

method is more deviated to Δ f max than that of the data-driven method. It indicates that the benchmark method tends to overestimate the impact of the power imbalance on the frequency deviation. The total cost obtained from the data-driven method is only 75% of that of the benchmark method. Figure 8.10 illustrates how the suggested data-driven approach tends to allot less reserve to IACs. The reason is that, in this testing environment, IACs have a greater reserve cost but a faster response time, making them more likely to be used in conservative systems. The suggested data-driven strategy enables power systems to operate closer to the frequency security boundary and hence achieves lower cost thanks to the more accurate estimation of frequency response results. From the comparisons, it can be inferred that the reserve allocation recommended by the data-driven strategy is superior to that of the benchmark method. In order to demonstrate the adaptability of the proposed method, the maximum frequency deviations under different power imbalances are shown in Fig. 8.11. It is clear that with the proposed method, the frequency security constraints can be better satisfied. Specifically, the maximum frequency nadir can be kept close to but always within the threshold of 0.4 Hz with the increase of ΔP0 . In addition, to illustrate the impact of IACs on the frequency security, the system in which the IACs are excluded from the frequency regulation is also studied and acts as the benchmark. Its maximum frequency deviation values are also shown in Fig. 8.10. It can be seen that there are no solutions satisfying the frequency security constraint in the investigating horizon of ΔP0 . With the inadequate reserve completely provided by conventional generators, the system frequency nadir deviates greatly from Δ f max and rises significantly with the increase of ΔP0 .

Fig. 8.11 Maximum frequency deviation with different power imbalances

162

8 A Data-Driven Reserve Allocation Method with Frequency Security …

8.5 Conclusion This research proposes an IAC-based data-driven reserve allocation strategy to address the implicit frequency security limitation. The framework for power system frequency regulation includes the equivalent frequency model of aggregated IACs. The MLFSC then transforms the reserve allocation into a solvable nonconvex QCQP, which is dealt with by a heuristic SI approach. The suggested approach is data-driven since the labeled dataset of frequency dynamics from the frequency simulations is used to train the MLFSC. The data-driven method estimates the system frequency dynamics more precisely than the state-of-the-art method and yields lower reserve allocation costs. Furthermore, the frequency security of the power system can be considerably enhanced by the reserves offered by IACs. The suggested method is extremely important to enhance model-free frequency regulation and manage massive data in smart grids, especially in light of the engagement of rising demand-side resources in power system operation. However, it is also possible for other flexible resources, such as electric vehicles (EV), to contribute to power system frequency management [37, 38].

References 1. Atputharajah, and T. Saha, “Power system blackouts-literature review,” in ICIIS., Sri Lanka, Sri Lanka, 2009, pp. 460–465. 2. U. Rudez and R. Mihalic, “WAMS-based under frequency load shedding with short-term frequency prediction,” IEEE Trans. Power Deliver., vol. 31 no. 4, pp. 1912–1920, Nov. 2015. 3. H. Chvez, R. Baldick and J. Matevosyan, “The joint adequacy of AGC and primary frequency response in single balancing authority systems,” IEEE Trans. Sustain Enery., vol. 6, no. 3, pp. 959–966, Jul. 2015. 4. Y. G. Rebours, D. S. Kirschen, M. Trotignon and S. Rossignol, “A survey of frequency and voltage control ancillary services—Part I: Technical features,” IEEE Trans. Power Syst., vol. 22, no. 1, pp. 350–357, Feb. 2007. 5. R. Doherty, A. Mullane, G. Nolan, D. J. Burke, A. Bryson and M. O’Malley, “An assessment of the impact of wind generation on system frequency control,” IEEE Trans. Power Syst., vol. 25, no. 1, pp. 452–460, Oct. 2009. 6. P. Kundur, N. J. Balu and M. G. Lauby, “Power system stability and control,” New York, USA: McGraw-Hill, 1994. 7. T. Shekari, F. Aminifar and M. Sanaye-Pasand, “An analytical adaptive load shedding scheme against severe combinational disturbances,” IEEE Trans. Power Syst., vol. 31, no. 5, pp. 4135– 4143, Dec. 2015. 8. Y. Lu, W. Kao, and Y. Chen, “Study of applying load shedding scheme with dynamic D-factor values of various dynamic load models to Taiwan power system,” IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1976–1984, 2005. 9. Liang, P. Wang, X. Han, W. Qin, R. Billinton and W. Li, “Operational reliability and economics of power systems with considering frequency control processes,” IEEE Trans. Power Syst., vol. 32, no. 4, pp. 2570–2580, Nov. 2016. 10. Y. Wen, W. Li, G. Huang and X. Liu, “Frequency dynamics constrained unit commitment with battery energy storage,” IEEE Trans. Power Syst., vol. 31, no. 6, pp. 5115–5125, Nov. 2016. 11. Y. Bao, Y. Li, B. Wang, M. Hu and Y. Zhou, “Day-Ahead Scheduling Considering Demand Response as a Frequency Control Resource,” Energies., vol. 10, no. 1, pp. 82, Jan. 2017.

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12. Xie, H. Hui, Y. Ding and Z. Lin, “Operating reserve capacity evaluation of aggregated heterogeneous TCLs with price signals,” Appl. Energy., vol. 216, pp. 338–347, Apr. 2018. 13. F. Ruelens, B. J. Claessens, S. Vandael, B. De Schutter, R. Babuška and R. Belmans, “Residential demand response of thermostatically controlled loads using batch reinforcement learning,” IEEE Trans. Smart Grid, vol. 8, no. 5, pp. 2149–2159, Feb. 2016. 14. M. Isaac and D. P. Van, “Modeling global residential sector energy demand for heating and air conditioning in the context of climate change,” Energy policy, vol. 37, no. 2, pp. 507–521, Feb. 2009. 15. BusinessLine, Tech. Rep, “AC makers betting on consumers’ shift to inverter models,” 2017. Available: http://www.thehindubusinessline.com. 16. J. Hwang, “Assessment of air condition load management by load survey in Taipower,” IEEE Trans. Power Syst., vol. 16, no. 4, pp. 910–915, Nov. 2001. 17. Bijli Bachao, Tech. Rep, “What is Inverter Technology AC,” 2017. Available: https://www.bij libachao.com. 18. M. Song, C. Gao, H. Yan and J. Yang, “Thermal battery modeling of inverter air conditioning for demand response,” IEEE Trans. Smart Grid., vol. 9, no. 6), pp. 5522–5534, Mar. 2018. 19. H. Hui, Y. Ding and M. Zheng, “Equivalent modeling of inverter air conditioners for providing frequency regulation service,” IEEE Trans. Ind Electron., vol. 66, no. 2, pp. 1413–1423, Apr. 2019. 20. Y. Ding, Y. Song, H. Hui and C. Shao, “Integration of Air Conditioning and Heating into Modern Power Systems,” Singapore: Springer, 2019. 21. B. Scholkopf and A. J. Smola, “Learning with kernels: support vector machines, regularization, optimization, and beyond,” Cambridge, USA: MIT press, 2001. 22. N. Huang, H. Chen, G. Cai, L. Fang and Y. Wang, “Mechanical fault diagnosis of high voltage circuit breakers based on variational mode decomposition and multi-layer classifier,” Sensors., vol. 16, no. 11, pp. 1887, Nov. 2016. 23. P. Chan, W. Chen, W. Ng and D. Yeung, “Multiple classifier system for short term load forecast of microgrid,” in ICMLC, Guilin, China, 2011, pp. 1268–1273. 24. P. Mahela, A. G. Shaik and N. Gupta, “A critical review of detection and classification of power quality events,” Renew Sust Energ Rev., vol. 41, pp. 495–505, Jan. 2015. 25. Ma, W. K. K, “Semidefinite relaxation of quadratic optimization problems and applications,” IEEE Signal Process Mag., vol. 1053, no. 5888, May. 2010. 26. S. Ihara and F. C. Schweppe., “Physically based modeling of cold load pickup,” IEEE Trans. Power App Syst., vol. 9, pp.4142–4150, Sep. 1981. 27. J. A. Hartigan and M. Wong, “Algorithm AS 136: A k-means clustering algorithm,” J R Stat Soc Ser C Appl Stat., vol. 28, no. 1, pp. 100–108, Jan. 1979. 28. N. Ruiz, I. Cobelo and J. Oyarzabal, “A direct load control model for virtual power plant management,” IEEE Trans. Power Syst., vol. 24, no. 2, pp. 959–966, May. 2009. 29. “Midea and State Grid Jiangsu Corporation reached strategic cooperation, piloting Internet + smart power consuming project,” China Household Appliance Standard and Technology Industry Alliance, [Online]. Available: http://www.chasa.com.cn/2016/0826/208.shtml, 2016. 30. “Raising the curtain of the popularization of smart air conditioners: Haier became the exporter of innovative models,” Ifeng, [Online]. Available: http://home.ifeng.com/a/20170216/44544226_ 0.shtml, 2017. 31. S. Boyd and L. Vandenberghe, “Convex optimization,” Cambridge, UK: Cambridge university press, 2004. 32. R. Misener and C. A. Floudas, “GloMIQO: Global mixed-integer quadratic optimizer,” J Glob Optim., vol. 57, no. 1, pp. 3–50, Sep. 2013. 33. J. Parkand S. Boyd, “General heuristics for nonconvex quadratically constrained quadratic programming,” arXiv preprint arXiv:1703.07870, Mar. 2017. 34. J. Park, “Approximation techniques for mixed-integer quadratic programs,” Ph.D. dissertation. Dept. Math., Univ. Stanford., Palo Alto, USA, 2017. 35. W. Wang, L. Yang, L. Wang, P. Zhang, J. Huang and K. Wang, “Optimal dispatch of integrated electricity-heat energy system considering heat storage characteristics of heating network,” Automation of Electric Power Systems, vol. 21, pp. 45–52, Nov, 2018.

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36. Goodfellow, Y. Bengio and A. Courville, “Deep learning,” London, England: MIT Press, 2016. 37. H. Hao, B. M. Sanandaji, K. Poolla and T. L. Vincent, “Frequency regulation from flexible loads: Potential, economics, and implementation,” In ACC, Portland, USA, 2014, pp. 65–72. 38. S. Izadkhast, P. Garcia-Gonzalez and P. Frías, (2014). “An aggregate model of plug-in electric vehicles for primary frequency control,” IEEE Trans. Power Syst., vol. 30, no. 3, pp. 1475–1482, Aug. 2014.

Chapter 9

Iterative Online Fault Identification Scheme for High Voltage Circuit Breaker

9.1 Introduction The high-voltage circuit breaker (HVCB), a basic piece of electrical equipment with the ability to break the current under both fault and normal situations, is crucial to the safe operation of the power system. Blackouts and cascading grid outages can both result from HVCB faults [1]. According to statistical calculations, poor mechanical properties are to blame for 80% of real-world HVCB defects [2]. Planned maintenance is the key source of information for traditional HVCB mechanical fault diagnosis [3]. The deployment of cutting-edge sensors and the quick growth of information communication technology (ICT) have substantially aided the advancement of online monitoring and intelligent diagnostic techniques for HVCB. The distinctive characteristics of HVCB diagnostic methods among emerging techniques include vibration signals during the closing or opening operation [4], contact stroke displacements [5], and electromagnetic coil currents [6]. The interaction of mechanical parts and structures causes the vibration signal to typically exhibit nonlinearity and nonstationarity [7], hence signal decomposition procedures like the Wavelet Transform (WT) and Fourier Analysis are frequently used to extract information in the time–frequency domain. Many classifiers for HVCB fault identification have been adopted based on the retrieved characteristics, with the Support Vector Machine (SVM) being the model that has been used the most frequently [10]. Most electrical non-invasive monitoring systems still use distribution carriers, ZigBee, Bluetooth, or other wireless ICT with relatively poor quality due to network topology, geographic constraints, or economic factors. These cheap channels are frequently interrupted by strong electromagnetic interference, such as over-voltage, and large current surge, which results in monitoring anomalies and negatively impacts the accuracy of fault identification. The accuracy of the measurements has thus become a legitimate problem for HVCB monitoring, and it has become a hot topic to use the missing data to accomplish an accurate online diagnosis of HVCB. A clustering method is employed in [13] to detect the aberrant data in the training set of the SVM, and the segregated abnormal data is then eliminated immediately, © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 C. Ye et al., Risk-Based Planning and Operation Strategy Towards Short Circuit Resilient Power Systems, https://doi.org/10.1007/978-981-19-9725-9_9

165

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potentially destroying the continuity and integrity of the data. Reference [14] suggests two ways to deal with measurement anomalies: identifying them and replacing the anomaly with the average value of the nearby data points. In [15], the Artificial Neural Network (ANN) and Autoregressive with Exogenous Input (ARX) models are used to impute the aberrant data. One of the oddities in [16] is the missing data, which the Autoregressive Integrated Moving Average model fits (ARIMA). Generally speaking, measurements with sharp variation are likely to show potential problems and should be recognized in detail for the fault diagnosis of electrical infrastructures, such as HVCB. Unfortunately, the missing sharp variation component cannot be accurately characterized by time series. To deal with data in power systems with low-value density, other intelligent models, such as Rough Sets (RS), have also been researched [17]. In [18], the condition estimation is carried out using the corresponding reduced set after the state information of the power quality sensing device is reduced by the RS theory. In state-of-the-art efforts, the key concern for missing data recovery is accuracy [1–18]. For a genuine power grid, there are tens or hundreds of thousands of electrical components, which produce close to ten or one hundred million monitoring data annually. The total amount of data has already surpassed TBs. The execution efficiency of the missing data recovery and the defect diagnosis program has become a difficult problem that needs to be implemented online in real applications due to the enormous volume of data [19]. This chapter proposes an online HVCB fault diagnosis approach that takes into account missing measurements. The missing data is specifically estimated using an ELM based on the chosen nearest neighbors. After the data have been rectified, the multi-class Softmax classifier is used to find the fault labels that are most likely to be true. The nearest neighbors are updated by loop iterations until their estimated labels match the repaired sample’s estimated labels. Due to a useful K-D tree data structure, the iterative scanning process can be implemented online.

9.2 HVCB Condition Monitoring System The selection of variables that can characterize the infrastructure’s operational status and be measurably under current technical conditions is a crucial first step in condition monitoring. The mechanical and electrical feature signals of the HVCB are investigated in this section.

9.2.1 Current in the Coil In the switch on/off process, the current flowing in the HVCB tripping and closing coil can reflect the status of the HVCB, mainly the electromagnet and its controlled parts [6]. A typical current curve in the coil is illustrated in Fig. 9.1.

9.2 HVCB Condition Monitoring System

167

Fig. 9.1 A typical current curve in the coil

In Fig. 9.1, t 1 and t 2 are the moments when the iron core starts and stops moving, respectively. t 3 is the timing when the coil current rises to the maximum value of steady-state current. t 4 is the cut-off timing of the auxiliary contact of the circuit breaker. The duration of the four stages and the corresponding current magnitudes can reveal the voltage and resistance of the coil as well as the core movement speed. For instance, at t 0 , the coil is energized, the current grows rapidly, and the flux in the coil also rises. Before t 1 , the current is not large enough to drive the core. If this stage lasts too long, it may due to an abnormal low coil voltage or an invalid empty stroke of the iron core, etc.; Besides, if the peak current increases simultaneously, perhaps a jammed iron core exists. When the current rises to t 1 , the current value reaches the threshold and the core acts, the current drops then. Until t 2 , the iron core gradually stops, and then the core contacts with the load of the operating mechanisms. The current curve in this stage can reveal whether a jammed iron core, a deformed pole or a failed trip exists. After t 4 , the decreasing current can reflect the status of the auxiliary switch.

9.2.2 Vibration Signal The mechanical vibration signal generated by the HVCB action contains valuable equipment status information. The speed and strength of the vibration signal are quite high due to the rapid action of HVCB. The velocity increases from the static to meters per second (m/s) in a matter of milliseconds, and the acceleration of the moving contact can reach hundreds of times that of gravity (ms). The obtained vibration signal is complex [4] because of the change in electromagnetic force brought on by the poor contact inside the HVCB, partial discharge, the movement of conductive particles, and the loosening of mechanical parts. The mechanical vibration of the HVCB is brief in comparison to the periodic signal. The primary technique is to use wavelet decomposition or other techniques to extract its pattern properties. A vibration signal obtained from HVCB is shown in Fig. 9.2 as a collection of wavelet decomposition diagrams, having a total of 11 sub-waves. The wavelet function of interest is the Harr function. The original signal is represented by various

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9 Iterative Online Fault Identification Scheme for High Voltage Circuit …

sub-waves at various frequency ranges. The wavelet marginal spectrum energy of five different fault types is compared in Fig. 9.3. The 11 sub-waves respective marginal spectral energies are denoted by the letters E1–E11. The vibration signals of the normal and four fault types are clearly identifiable in terms of marginal spectral energy, as shown in Fig. 9.3 [20]. As a result, another element of the HVCB monitoring is the spectral energy of the vibration.

E11 Wavelet signal

E10 E9 E8 E7 E6

Original signal

E5 E4 E3

Wavelet signal

E2 E1

Fig. 9.2 An illustrative decomposition sample of vibration signal obtained from HVCB

Fig. 9.3 Comparison of wavelet marginal energy of five failure types

9.2 HVCB Condition Monitoring System

169

9.2.3 Framework of the HVCB Condition Monitoring System A data collection system with numerous indicators is used to comprehensively describe the HVCB’s operational state. In particular, three additional indicators— the interrupting current, the energy motor current, and the contact angle-travel—are used in addition to the introduced coil current and vibration signal in the preceding section. The standard measures of current, vibration, and displacement are all available in a wide range of developed sensors, as shown in Table 9.1. Hall sensors come in two varieties: open-loop and closed-loop. The former uses the hall direct release concept, which is appropriate for monitoring high currents like the coil’s current and interrupting currents. The latter, on the other hand, uses the magnetic balance principle, which is appropriate for low current monitoring. Figure 9.4 depicts the overall design of the HVCB condition monitoring system. Distributed sensors, a communication module, and a computing center make up the system. All of the sensors are distributed around the circuit breaker, so the measurements can be transmitted wirelessly utilizing Zigbee, Bluetooth, 4G, or NB-IoT in the substation. Especially in China, where all of the 110/220/500 kV substations have been connected with an optical Ethernet, the bulk of contemporary HV substations is set up using optical fiber. The optical fiber network is used to send the wirelessly obtained HVCB readings from the substation to the cloud data center. In general, communication system concerns like packet loss or delay dislocation, particularly with regard to the wireless component in the substation, are valid. As a result, a missing data repair program and a data-driven HVCB fault diagnosis model are both implemented in the cloud data center. Table 9.1 Utilized features and the appropriate sensors for HVCB operation status monitoring Feature

Significance

Sensors

Tripping and closing coil current

Status of the mechanisms, mainly the electromagnet

Closed-loop Hall sensor

Interrupting current

Worn, aged, or deformed mechanical transmission

Open-loop Hall sensor

Energy motor current

Wear conditions of the contact

Vibration signal

Overall condition, mainly the operating mechanism

Piezoelectric acceleration sensor

Contact angle-travel

Load or working state of the spring system

Axis tilt sensor

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9 Iterative Online Fault Identification Scheme for High Voltage Circuit …

HVCB Mechanical signal

Electrical signal

Vibration of the operating instrument

Angle-travel of the movable contact

Current in the tripping and closing coil

Piezoelectric acceleration sensor

Axis tilt sensor

Closed-loop Hall Sensor

Interrupting current Energy storage motor current

Open-loop Hall Sensor

Zigbee, Bluetooth, 4G, NB-IoT, 5G, Optical fiber Missing data repair

Cloud data center

Condition diagnose

Fig. 9.4 Framework for HVCB operation condition monitoring

9.3 Missing Data Repair Method The K-Nearest Neighbor (kNN) searching algorithm is used to identify similar training samples, and a K-Dimensional (K-D) Tree-based quick scanning technique is also used. The Extreme Learning Machine (ELM) is used to estimate the missing data with an incredibly fast learning capacity.

9.3 Missing Data Repair Method

171

9.3.1 KNN-Based Clustering The kNN algorithm is a kind of supervised learning method, which is used to find neighbor samples with similar patterns [21]. Based on the basic premise that the faults of the same equipment are of similar data mode, this paper adopts the improved kNN algorithm to repair the lost measurement of HVCB condition monitoring. Given a test sample with lost measurements to be repaired, find the closest K samples in the historical database, and then synthesize the information of the K training samples to estimate the lost part. In order to reveal the relevance among different monitoring indicators, the similarities of samples are measured in terms of Manhattan distances weighted by the negative exponential of the related coefficient. If there are N HVCB monitoring sample pairs with m features in each pair xi = [xi1 , xi2 , . . . , xim ]T (1 ≤ i ≤ N ), the correlation degree of the p-th and q-th feature values ρ pq can be obtained by [21]: )( ) xi p − μ p xiq − μq ρ pq = / ( )2 / 1 Σ N ( )2 1 ΣN x − μ ip p i=1 i=1 x iq − μq N −1 N −1 1 N −1

ΣN ( i=1

(9.1)

where μ is the average of the corresponding feature, and the range of the related coefficient ρ pq is from –1 to +1. –1 means a strong negative correlation between the two features. +1 means a strong positive correlation, while 0 indicates no correlation. Assume that the test sample xi lacks the p-th (1 ≤ p ≤ m) feature measurement xi p , the Manhattan distance weighted by the negative exponential of related coefficient between training sample i and test sample j can be expressed as [22]: d i j| p =

m Σ

| | e−|ρ pq | · |xiq − xiq |,

p /= q

(9.2)

q=1

The spatial distances of measurements are weighted by the correlation coefficients. For features highly correlated with the feature of the missing value, smaller weights will be assigned. In this way, the test sample is prone to approach the training samples with similar measurements on strong correlated features.

9.3.2 ELM for Data Estimation ELM is a single-hidden layer feedforward neural network (SLFN) with fast learning speed and excellent generalization. ELM randomly sets the input weights and biases, which are not required to be tuned iteratively in the training procedure, thus greatly saving the overall training workload. N As can be seen in Fig. 9.5, given a dataset with N samples {(xi , yi )}i=1 , where xi ∈ R and yi ∈ R are inputs and targets respectively, A K hidden unit-ELM with

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9 Iterative Online Fault Identification Scheme for High Voltage Circuit …

Hidden layer a, b x1 δ

x2

xn

Input layer

Output layer

Fig. 9.5 Topology of the ELM-based missing data estimator

activation function φ(·) approximates the N samples without error as follows [23]: yi =

K Σ

) ( βi φ ai · x j + bi , ( j = 1, 2, . . . , N )

(9.3)

i=1

where ai = [ai1 , ai2 , . . . , ain ]T is the vector of the weights connecting hidden unit i and the input units of the ELM, δi = [δi1 , δi2 , . . . , δim ]T is the vector of weights connecting hidden unit i and the output units, bi is the bias of hidden unit i, and ) ( φ ai · x j + bi is the output of hidden unit i with respect to xi . Equation (9.3) can be expressed in compact form: Hδ = Y, where H is the output matrix of ELM hidden layers, and ⎤ φ(a1 · x1 + b1 ) . . . φ(a K · x1 + b K ) ⎥ ⎢ .. .. H =⎣ ⎦ . . φ(a1 · x N + b1 ) . . . φ(a K · x N + b K ) ⎡

(9.4)

Column i of the matrix H corresponds to the output vector of hidden unit i with respect to input xi = [xi1 , xi2 , . . . , xin ]T . In addition, δ is the output weight matrix and Y is the target matrix. With the randomly set ai and bi , the output matrix H can be determined uniquely, consequently, the approximated parameters ai , bi and δ can be acquired by [23] || || ( ∗ ) || H a , . . . , a ∗ , b∗ , . . . , b∗ δ − T || = min||H (a1 , . . . , ak , b1 , . . . , bk )δ − T || 1

k

1

k

δ

(9.5)

9.3 Missing Data Repair Method

173

Training an ELM is actually equivalent to searching the unique minimal norm least-square scheme of the linear model in Eq. (9.4), i.e. δ∗ = H + Y

(9.6)

where H+ is the Moore–Penrose Generalized Inverse (MPGI) of H, which can be obtained through algorithms such as the Singular Value Decomposition (SVD) [23]. The conventional gradient-based ANN models like the BP network are involved with iterative time-consuming training. The ELM achieves extremely efficient training due to the simple matrix transformation, and can always guarantee global optimality. In addition, it has many advantages such as avoiding overtraining.

9.3.3 K-D Tree-Based Fast Scanning Technique If the conventional linear scanning technique is utilized to locate the K closest samples, the efficiency is not capable to meet the requirements of online identification. In this paper, an efficient searching structure named the K-D tree is introduced. (1) Building a K-D tree The HVCB monitoring usually covers more than 10 indices, therefore, the HVCB condition monitoring is a high dimensional data space. The K-D tree is an enhanced data structure to segment data space with K dimensions and speeds up its searching efficiency in it [24]. Figure 9.6 illustrates a 3-dimension dataset to explain the formation of a K-D structure including three specific steps. Step 1: Calculate the data variance of each feature or dimension, and select the dimension with the maximum variance value as the Split Dimension (SD). Generally, a larger variance means a better diversity of data in this dimension, dividing samples in the direction indicated by the maximum variance can effectively balance the overall tree.

Fig. 9.6 Structure of a sample 3-dimensional tree

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Step 2: Sort the data samples based on the values of the SD, and find the pivot point with the most intermediate value on the SD. In Fig. 9.6, the pivot point data of the SD are highlighted in red font. Step 3: Divide the whole dataset on the SD to form two branch k-d trees. When the size of the branch tree is less than the preset maximum leaf node which is 2 in Fig. 9.6, stop generating branch k-d trees. Otherwise, return to step 1. Each node of the obtained structure is a k-dimensional binary tree. All nonleaf nodes can be regarded as a hyperplane for space division. Through the nonoverlapping hierarchical division of the search space, an index structure suitable for efficient data searching is established. (2) Neighborhood searching For structured storage, each HVCB monitoring data sample is of k indices and placed on a node of the K-D tree. It has been proved in Ref. [25] that the complexity of the neighborhood searching in the K-D tree structure is O(log n) (n is the dimension of the data space), which significantly surpasses the conventional linear scanning algorithm with O(n) complexity. The K-D tree-based neighborhood searching strategy consists of three steps. Step 1: Locate the leaf nodes of the samples along the k-d tree, obtain the closest data point, and put all nodes on the searching route into a queue. Step 2: Check whether the distance from the data point in the queue to the test sample is closer than that from the test sample to the closest data point. If not, remove the node out of the queue and duplicate step 2 until the queue becomes empty, then terminate the search process. Otherwise, forward to step 3. Step 3: Update the closest data point and add its offspring nodes to the queue. Then, go back to step 2.

9.4 Softmax Classifier for HVCB Status Identification The essence of HVCB condition identification is to identify its operation label based on a series of measurements, which is a multi-class classification problem. First of all, there is no strict one-to-one correspondence between the external measurements with limited dimensions and the operation state, multiple possible states exist under the observations theoretically. The task is to find the most probable state, which is a probability problem. In addition, the rules of classification cannot be given in advance, and some machine learning models are required to train the rules from historical data. This section introduces a Softmax classifier for the HVCB condition diagnosis task. As illustrated in Fig. 9.7, the Softmax classifier is a two-layer ANN, which quantifies the probability value of classifying a certain input sample to each predefined label and selects the label with the maximal probability. Given an HVCB measurement input xi = [xi1 , xi2 , . . . , xin ]T , ti is the label that can be assigned as r values.

9.4 Softmax Classifier for HVCB Status Identification

175

low coil voltage

Normal

deformed pole

jammed iron core

abnormal p ( t i = 1 xi ; θ )

p ( t i = 2 xi ; θ )

z2 1

p ( ti = r x i ; θ )

z2 2

z2 k

z1 1

z1 2

z1 3

z1 n

x1

x2

x3

xn

+1

θ

Fig. 9.7 Topology of a Softmax classifier for HVCB fault diagnose

As shown in Eq. 9.6, the Softmax classifier can give a vector with r dimensions corresponding to r calculated probabilities [26]. ⎡ T ⎤ ⎤ eθ 1 x i p(ti = 1|x i ; θ ) ⎢ T ⎥ ⎢ p(ti = 2|x i ; θ )⎥ ⎢ eθ 2 x i ⎥ ⎥ ⎢ ⎢ ⎥ 1 ⎥ ⎢ ⎢ ⎥ h θ (x i ) = ⎢ .. r ⎥= Σ ⎢ .. ⎥ T θ x ⎢ ⎥ ⎦ ⎣ i j . e ⎣ . ⎦ j=1 T p(ti = r |x i ; θ ) eθ i x i ⎡

(9.7)

Σ T where θ1 , θ2 , . . . , θr ∈ Rr ×1 are the weights to be trained. rj=1 eθ j x i is utilized to normalize the probability vector. The probability of sorting input xi into label j is: eθ j x i r Σ T eθ l x i T

p( ti = j|x i ; θ )=

(9.8)

l=1

For supervised learning with the labeled dataset, the specific label dimension of the Softmax output is assigned as value 1 and other dimensions are set as value 0. The indicator function 1{·} is introduced to express the cost function of Softmax classification: { 1, i f t = j 1{ti = j} = (9.9) 0, i f t /= j

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9 Iterative Online Fault Identification Scheme for High Voltage Circuit …

The process of training the Softmax model is actually to optimize the parameter θ. The better θ is trained, the better the classification effect can be achieved. Generally, the Cross Entropy index is utilized to measure the effect of model classification [27]. The minimizing of the CE value is equivalent to the process of improving Softmax classification accuracy continuously. The loss function is: ⎤ ⎡ T m r 1 ⎣Σ Σ eθ j x i ⎦ J (θ ) = − 1{ti = j} log Σr θ lT x i m i=1 j=1 l=1 e

(9.10)

There is no closed method to minimize J(θ ). Generally, an iterative optimization algorithm such as the gradient descent method is used instead. For each parameter of Eq. 9.9, the partial derivative is obtained: m δ J (θ ) 1 Σ {xi [1{ti = j} − p(ti = j|xi ; θ )]} =− δθ j m i=1

(9.11)

By substituting the above partial derivative into the gradient descent method, the J(θ ) can be minimized. While the obtained θ is not unique due to theΣ non-convexity Σ of Eq. (9.9) [27]. To address the non-convexity, a weight decay term λ2 ri=1 nj=0 θi2j is added to the cost function, then the training procedure of a Softmax classifier can be abstracted as [28]: ⎛

⎞ m Σ r Σ r n θ Tj x i Σ Σ e λ 1 1{ti = j} log Σr + θi2j ⎠ θ ∗ = arg min⎝− θ lT x i m i=1 j=1 2 e θ l=1 i=1 j=0

(9.12)

In a gradient descent framework, the unique optimal Softmax weight θ can be calculated by iterative BP algorithms.

9.5 Procedure of Iterative HVCB Diagnosis Utilizing Repaired Data In the HVCB fault identification framework integrated with the lost data repair technique, the k-NN algorithm is adopted to gather a set of similar samples, and then the ELM is utilized to establish a map from the sample data to the measurement in terms of the dimension or feature with lost data. The ELM is capable to calculate or repair a value for the lost feature with a satisfactory speed. The Softmax network is exploited to calculate the probability of the sample of interest being classified to each of the preset labels. In this way, preliminary or raw estimated fault labels subject to the repaired dataset are obtained, on the basis of which, the sample space is reconfigured and the k-NN and ELM are used to estimate the lost data again. In this way, the

9.5 Procedure of Iterative HVCB Diagnosis Utilizing Repaired Data

177

Start Read in measurements k-NN algorithm Similar samples ELM lost data repair

Filled data

Check whether the identification is consistent with the k closest samples

Fill the training set using k-NN

NO Remove the inconsistent training samples

Softmax classifier

YES

Diagnosis result Train the Softmax classifier regularly Update the historical dataset

END

Fig. 9.8 Procedure of the iterative HVCB fault identification framework utilizing lost data repair

fault labels of the neighbor samples are iterated until they are in accordance with the detected fault labels of the test samples. The flow chart of the proposed HVCB fault identification method based on the repaired dataset is illustrated in Fig. 9.8, including the following five steps: Step 1: Read in the HVCB condition monitoring data including coil current, interrupting the current, motor current, vibration signal, and angle-travel of the contact, etc. via Internet-of-Things from the sensors deployed on HVCBs, and run the data check program to scan the measurements, if the data is complete, call the trained Softmax classifier to identify the most probable condition label. If certain data is lost, run to Step 2. Step 2: Call the k-NN algorithm to select N training samples with the most similar measurements on strong correlated features and run the ELM to repair the lost data based on the sample data. Then, call the trained Softmax classifier to identify the fault labels. Step 3: Check whether the identification result is in accordance with the fault labels of the k closest training samples. If not, forward to step (4), otherwise, jump to step (5). Step 4: Remove the samples which are inconsistent with the identification label of the repaired data, and use the k-NN algorithm to supplement the required samples and establish an updated training set with N size again. Then, go back to step (2).

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Step 5: After the above implementations, store the sample with final repaired data into the historical sample dataset along with its identification result, and the Softmax classifier is trained regularly with the latest version of the historical labeled sample dataset. Through the above iterative cleaning and maintenance mode, the quality of fault identification database of HVCB can be continuously improved against metering anomalies.

9.6 Realistic Case Studies 9.6.1 Case Description To support the online HVCB fault identification, a labeled training set is established based on sample data obtained from both the laboratory and real operation. Specifically, defective or failure HVCBs are detected during intended fatigue tests or condition-based maintenance, the corresponding fault types are confirmed and their historical condition measurements are called. In general, 3759 pairs of complete HVCB condition monitoring data are obtained from the State Grid Corporation of China (SGCC). Note that the HVCB measurements include point-by-point mechanical vibration energy spectrum and coil current signals and the input feature is an 8-dimension vector. All the fault labels include loose joint pin (F1), loose connecting bolt (F2), jammed closing release (F3), lower coil voltage (F4), stuck auxiliary switch contact (F5), as well as normal (N). The samples with lost data are generated by random data deletion.

9.6.2 Accuracy Validation In order to fully exploit the training data and illustrate the effects, fault identification precisions of different methods are compared under a Leave-one-out Cross Validation (LOM) framework [28]. The competition classifiers include the Softmax, multi-class SVM [10, 13], as well as RS [17, 18]. The studied missing data repair techniques are the proposed k-NN combined with ELM, k-NN combined with simple average computing, and ARIMA [16]. The fault identification precisions of methods with different configurations are shown in Table 9.2. As can be seen from Table 9.2, the precision of the Softmax classifier is 85.4– 92.6%, which is obviously higher than the RS method (77.2–85.8%) based on complete measurements. After the data are randomly deleted, the precisions of the Softmax and RS methods are reduced to 68.4%–75.8% and 63.2%–71.8%, respectively. It can be concluded obviously that the diagnosis models are highly sensitive to data integrity. As the utilized vibration energy spectrum and the current indices

623 656 614 641 689 536

F1 F2 F3 F4 F5 N

Remarks

Training sample

Fault type

The best results

86.3 87.6 86.3 90.8 85.4 92.6

Softmax (complete) (%)

75.8 70.6 68.4 69.8 70.2 76.9

Soft max (lost) (%)

Table 9.2 Fault identification precisions of different methods

The proposed

84.2 88.6 86.5 87.2 82.4 90.2

k-NN (ELM)+ softmax (lost) (%)

k-NN (ELM)+ multiclass SVM (lost) (%) 81.3 84.6 83.7 82.7 78.9 85.2 79.5 82.3 80.2 78.4 72.6 83.8

k-NN (average) +softmax (lost) (%) 80.3 83.9 79.5 83.4 74.6 82.9

ARIMA +softmax (lost) (%) 81.8 77.2 82.6 79.7 81.9 85.8

RS (complete) (%)

The worst results

71.8 68.6 65.4 66.8 63.2 70.9

RS (lost) (%)

9.6 Realistic Case Studies 179

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are all key features for operation status characterizing of HVBC, the absence of any attribute affects the accuracy greatly. By utilizing the proposed lost data repair technique (kNN + ELM), the precision of the Softmax classifier can be restored to 82.4–90.2%, which is only a small precision decline with subject to data missing. It is validated that the proposed data-driven iterative framework is capable to meet the application requirements which minimizes the impact of one or several data defects on the identification precision. Adopting the k-NN integrated ELM data repair technique and multi-class SVM classifier, the obtained precision is 78.9–85.2%. Compared with the precision of the proposed Softmax classifier, there is an about 5% decrease in accuracy. Because the SVM is a kind of binary classifier, multiple SVM networks need to be cascaded or stacked to fit multi-class classification problems, such as this HVCB fault identification case, which is prone to enlarge the error to some extent [10, 13]. If the lost data are approximated by computing the average of samples searched by k-NN in the dimension of interest, the Softmax accuracy decreases to 72.6– 83.8%. As can be seen, though a certain correlation indeed exists in the sample data, modeling it in linear form results in significant errors. The nonlinear ELM adopted in this paper achieves a better data recovery effect compared with replacing the anomaly with the average value of the data points lying before and after it [14]. Figure 9.9 illustrates the relationship of lost data estimation error and number of selected neighborhood samples in the k-NN framework. For the ELM estimator, when insufficient or excessive samples are selected, the prediction error will increase, and the most appropriate number of neighborhood samples is 25 (average error = 2.43%). For the average computing estimator, the obtained average error has been fluctuating around 10%. Moreover, if the ARIMA time series analysis method is used for missing data estimation, i.e., extrapolation estimation based on some measurements before data missing timing, and then Softmax is still used for fault diagnosis. The corresponding 16.00%

14.22% 13.06%

Average error

14.00% 12.85%

10.81%

9.81% 9.45%

10.00% 8.00%

12.35%

11.15%

12.00%

11.86%

8.69%

6.00%

6.89% 5.68%

4.00%

4.68%

0.00%

4.21%

3.53%

2.00%

2.43% 1

2

3

4

The proposed k-NN(ELM)

Fig. 9.9 Lost data estimation errors of different methods

5

6

7

8

kNN+average computing [14]

9.6 Realistic Case Studies

181

Current A Real current His torical current ARIMA fitted current curve [16]

4

ARIMA estimated current [16] Observation timing

2

10

20

30

40

50

60

70

80

Time/ms

Fig. 9.10 Sample of data repair using ARIMA in terms of energy storage motor current

precision is 74.6–83.4%, obviously lower than that of the proposed k-NN (ELM) repair technique. Figure 9.10 illustrates a sample of data repair using ARIMA [16]. As can be seen, due to a mechanical jam fault, the spring load torque increases, and the energy storage motor current increases sharply. If the historical current values are used for extrapolation prediction, the prediction curve of ARIMA fits the historical data in the normal state too much, resulting in an over-conservative final result [16]. Compared with the time series method, k-NN is a global data searching and analysis method, which breaks the mandatory temporal causal relationship, thus achieving better abnormal detecting ability.

9.6.3 Searching Efficiency Validation Figure 9.11 compares the relationship between the neighborhood sample size (k value in the k-NN framework) and the CPU time of the K-D tree-based and conventional linear scanning algorithms. The comparison is implemented on a dataset with an 8dimension HVCB condition monitoring dataset in terms of the Manhattan distance. It can be illustrated that for the K-D tree-based structure, the search time of neighborhood samples increases slowly with the growing sample size. While for the benchmark linear scanning, the time increases rapidly with the growing dataset size. Therefore, the proposed K-D tree scanning has a significant advantage when applied for online fault identification of HVCB with large-scale measurements.

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Fig. 9.11 Comparison of algorithm performances for different search strategies

9.7 Conclusion In this paper, a framework for HVCB defect identification that takes into account lost data is given. Similar samples are collected using the k-NN approach, and the lost data is then recreated using the ELM method. The Softmax classifier is used to determine the repaired data’s most likely condition status. The fault kinds of the neighboring samples are iterated in a loop until they agree with the fault labels of the samples. In the suggested framework, employing a K-D tree scanning technique considerably improves sample searching efficiency, and the lost data may be repaired in a nonlinear manner by ELM without iterative BP computations. Additionally, the time-consuming classifier training is carried out in advance of the line, allowing for the efficient online condition monitoring of the HVCB with anomaly measurements. The suggested method has a wide range of possible applications as the Internet of Things develops since it is frequently necessary for data-driven application scenarios to utilize the potential value of defective data due to inadequate collection or transmission infrastructure.

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