Optimal Coordination of Power Protective Devices with Illustrative Examples
IEEE Press Series on Power Engineering
1. Edition January 2022
528 Pages, Hardcover
Wiley & Sons Ltd
ISBN:
978-1-119-79485-1
John Wiley & Sons
Further versions
Optimal Coordination of Power Protective Devices with Illustrative Examples
Provides practical guidance on the coordination issue of power protective relays and fuses
Protecting electrical power systems requires devices that isolate the components that are under fault while keeping the rest of the system stable. Optimal Coordination of Power Protective Devices with Illustrative Examples provides a thorough introduction to the optimal coordination of power systems protection using fuses and protective relays.
Integrating fundamental theory and real-world practice, the text begins with an overview of power system protection and optimization, followed by a systematic description of the essential steps in designing optimal coordinators using only directional overcurrent relays. Subsequent chapters present mathematical formulations for solving many standard test systems, and cover a variety of popular hybrid optimization schemes and their mechanisms. The author also discusses a selection of advanced topics and extended applications including adaptive optimal coordination, optimal coordination with multiple time-current curves, and optimally coordinating multiple types of protective devices. Optimal Coordination of Power Protective Devices:
* Covers fuses and overcurrent, directional overcurrent, and distance relays
* Explains the relation between fault current and operating time of protective relays
* Discusses performance and design criteria such as sensitivity, speed, and simplicity
* Includes an up-to-date literature review and a detailed overview of the fundamentals of power system protection
* Features numerous illustrative examples, practical case studies, and programs coded in MATLAB(r) programming language
Optimal Coordination of Power Protective Devices with Illustrative Examples is the perfect textbook for instructors in electric power system protection courses, and a must-have reference for protection engineers in power electric companies, and for researchers and industry professionals specializing in power system protection.
Author Biography xvi
Preface xvii
Acknowledgments xviii
Acronyms xix
About The Companion Website xxiii
Introduction xxv
1 Fundamental Steps in Optimization Algorithms 1
1.1 Overview 1
1.1.1 Design Variables 4
1.1.2 Design Parameters 4
1.1.3 Design Function 5
1.1.4 Objective Function(s) 5
1.1.5 Design Constraints 7
1.1.5.1 Mathematical Constraints 8
1.1.5.2 Inequality Constraints 8
1.1.5.3 Side Constraints 9
1.1.6 General Principles 10
1.1.6.1 Feasible Space vs. Search Space 10
1.1.6.2 Global Optimum vs. Local Optimum 11
1.1.6.3 Types of Problem 12
1.1.7 Standard Format 12
1.1.8 Constraint-Handling Techniques 13
1.1.8.1 Random Search Method 17
1.1.8.2 Constant Penalty Function 17
1.1.8.3 Binary Static Penalty Function 18
1.1.8.4 Superiority of Feasible Points (SFPs) - Type I 18
1.1.8.5 Superiority of Feasible Points (SFP) - Type II 18
1.1.8.6 Eclectic Evolutionary Algorithm 18
1.1.8.7 Typical Dynamic Penalty Function 19
1.1.8.8 Exponential Dynamic Penalty Function 19
1.1.8.9 Adaptive Multiplication Penalty Function 19
1.1.8.10 Self-Adaptive Penalty Function (SAPF) 20
1.1.9 Performance Criteria Used to Evaluate Algorithms 21
1.1.10 Types of Optimization Techniques 23
1.2 Classical Optimization Algorithms 23
1.2.1 Linear Programming 25
1.2.1.1 Historical Time-Line 25
1.2.1.2 Mathematical Formulation of LP Problems 26
1.2.1.3 Linear Programming Solvers 26
1.2.2 Global-Local Optimization Strategy 28
1.2.2.1 Multi-Start Linear Programming 29
1.2.2.2 Hybridizing LP with Meta-Heuristic Optimization Algorithms as a Fine-Tuning Unit 31
1.3 Meta-Heuristic Algorithms 33
1.3.1 Biogeography-Based Optimization 34
1.3.1.1 Migration Stage 40
1.3.1.2 Mutation Stage 41
1.3.1.3 Clear Duplication Stage 43
1.3.1.4 Elitism Stage 44
1.3.1.5 The Overall BBO Algorithm 45
1.3.2 Differential Evolution 45
1.4 Hybrid Optimization Algorithms 46
1.4.1 BBO-LP 48
1.4.2 BBO/DE 51
Problems 51
Written Exercises 51
Computer Exercises 53
2 Fundamentals of Power System Protection 57
2.1 Faults Classification 57
2.2 Protection System 61
2.3 Zones of Protection 65
2.4 Primary and Backup Protection 66
2.5 Performance and Design Criteria 66
2.5.1 Reliability 66
2.5.1.1 Dependability 66
2.5.1.2 Security 66
2.5.2 Sensitivity 67
2.5.3 Speed 67
2.5.4 Selectivity 67
2.5.5 Performance versus Economics 67
2.5.6 Adequateness 67
2.5.7 Simplicity 67
2.6 Overcurrent Protective Devices 67
2.6.1 Fuses 68
2.6.2 Bimetallic Relays 69
2.6.3 Overcurrent Protective Relays 69
2.6.4 Instantaneous OCR (IOCR) 70
2.6.5 Definite Time OCR (DTOCR) 71
2.6.6 Inverse Time OCR (ITOCR) 72
2.6.7 Mixed Characteristic Curves 73
2.6.7.1 Definite-Time Plus Instantaneous 73
2.6.7.2 Inverse-Time Plus Instantaneous 74
2.6.7.3 Inverse-Time Plus Definite-Time Plus Instantaneous 74
2.6.7.4 Inverse-Time Plus Definite-Time 75
2.6.7.5 Inverse Definite Minimum Time (IDMT) 76
Problems 76
Written Exercises 76
Computer Exercises 77
3 Mathematical Modeling of Inverse-Time Overcurrent Relay Characteristics 79
3.1 Computer Representation of Inverse-Time Overcurrent Relay Characteristics 79
3.1.1 Direct Data Storage 79
3.1.2 Curve Fitting Formulas 82
3.1.2.1 Polynomial Equations 82
3.1.2.2 Exponential Equations 89
3.1.2.3 Artificial Intelligence 93
3.1.3 Special Models 94
3.1.3.1 RI-Type Characteristic 94
3.1.3.2 RD-Type Characteristic 95
3.1.3.3 FR Short Time Inverse 95
3.1.3.4 UK Rectifier Protection 95
3.1.3.5 BNP-Type Characteristic 95
3.1.3.6 Standard CO Series Characteristics 95
3.1.3.7 IAC and ANSI Special Equations 96
3.1.4 User-Defined Curves 98
3.2 Dealing with All the Standard Characteristic Curves Together 99
3.2.1 Differentiating Between Time Dial Setting and Time Multiplier Setting 99
3.2.2 Dealing with Time Dial Setting and Time Multiplier Setting as One Variable 104
3.2.2.1 Fixed Divisor 106
3.2.2.2 Linear Interpolation 108
3.2.3 General Guidelines Before Conducting Researches and Studies 111
Problems 113
Written Exercises 113
Computer Exercises 114
4 Upper Limit of Relay Operating Time 117
4.1 Do We Need to Define T¯max ? 117
4.2 How to Define T¯max ? 118
4.2.1 Thermal Equations 118
4.2.1.1 Thermal Overload Protection for 3Phi Overhead Lines and Cables 118
4.2.1.2 Thermal Overload Protection for Motors 122
4.2.1.3 Thermal Overload Protection for Transformers 124
4.2.2 Stability Analysis 126
Problems 136
Written Exercises 136
Computer Exercises 138
5 Directional Overcurrent Relays and the Importance of Relay Coordination 139
5.1 Relay Grading in Radial Systems 139
5.1.1 Time Grading 140
5.1.2 Current Grading 140
5.1.3 Inverse-Time Grading 143
5.2 Directional Overcurrent Relays 146
5.3 Coordination of DOCRs 148
5.4 Is the Coordination of DOCRs an Iterative Problem? 148
5.5 Minimum Break-Point Set 161
5.6 Summary 163
Problems 164
Written Exercises 164
Computer Exercises 166
6 General Mechanism to Optimally Coordinate Directional Overcurrent Relays 169
6.1 Constructing Power Network 169
6.2 Power Flow Analysis 170
6.2.1 Per-Unit System and Three-to-One-Phase Conversion 172
6.2.2 Power Flow Solvers 173
6.2.3 How to Apply the Newton-Raphson Method 175
6.2.4 Sparsity Effect 179
6.3 P/B Pairs Identification 186
6.3.1 Inspection Method 186
6.3.2 Graph Theory Methods 186
6.3.3 Special Software 188
6.4 Short-Circuit Analysis 189
6.4.1 Short-Circuit Calculations 189
6.4.2 Electric Power Engineering Software Tools 190
6.4.2.1 Minimum Short-Circuit Current 190
6.4.2.2 Maximum Short-Circuit Current 192
6.4.3 Most Popular Standards 193
6.4.3.1 ANSI/IEEE Standards C37 & UL 489 193
6.4.3.2 IEC 61363 Standard 194
6.4.3.3 IEC 60909 Standard 194
6.5 Applying Optimization Techniques 201
Problems 202
Written Exercises 202
Computer Exercises 205
7 Optimal Coordination of Inverse-Time DOCRs with Unified TCCC 207
7.1 Mathematical Problem Formulation 207
7.1.1 Objective Function 208
7.1.1.1 Other Possible Objective Functions 210
7.1.2 Inequality Constraints on Relay Operating Times 211
7.1.3 Side Constraints on Relay Time Multiplier Settings 211
7.1.4 Side Constraints on Relay Plug Settings 211
7.1.5 Selectivity Constraint Among Primary and Backup Relay Pairs 212
7.1.5.1 Transient Selectivity Constraint 213
7.1.6 Standard Optimization Model 216
7.2 Optimal Coordination of DOCRs Using Meta-Heuristic Optimization Algorithms 217
7.2.1 Algorithm Implementation 217
7.2.2 Constraint-Handling Techniques 218
7.2.3 Solving the Infeasibility Condition 222
7.3 Results Tester 228
Problems 229
Written Exercises 229
Computer Exercises 231
8 Incorporating LP and Hybridizing It with Meta-heuristic Algorithms 235
8.1 Model Linearization 235
8.1.1 Classical Linearization Approach 236
8.1.1.1 IEC Curves: Fixing Plug Settings and Varying Time Multiplier Settings 236
8.1.1.2 IEEE Curves: Fixing Current Tap Settings and Varying Time Dial Settings 237
8.1.2 Transformation-Based Linearization Approach 237
8.1.2.1 IEC Curves: Fixing Time Multiplier Settings and Varying Plug Settings 238
8.1.2.2 IEEE Curves: Fixing Time Dial Settings and Varying Current Tap Settings 238
8.2 Multi-start Linear Programming 242
8.3 Hybridizing Linear Programming with Population-Based Meta-heuristic Optimization Algorithms 245
8.3.1 Classical Linearization Approach: Fixing PS/CTS and Varying TMS/TDS 245
8.3.2 Transformation-Based Linearization Approach: Fixing TMS/TDS and Varying PS/CTS 245
8.3.3 Innovative Linearization Approach: Fixing/Varying TMS/TDS and PS/CTS 250
Problems 250
Written Exercises 250
Computer Exercises 251
9 Optimal Coordination of DOCRs With OCRs and Fuses 253
9.1 Simple Networks 253
9.1.1 Protecting Radial Networks by Just OCRs 253
9.1.2 Protecting Double-Line Networks by OCRs and DOCRs 255
9.2 Little Harder Networks 257
9.2.1 Combination of OCRs and DOCRs 258
9.2.2 Combination of Fuses, OCRs, and DOCRs 261
9.3 Complex Networks 264
Problems 265
Written Exercises 265
Computer Exercises 266
10 Optimal Coordination with Considering Multiple Characteristic Curves 271
10.1 Introduction 271
10.2 Optimal Coordination of DOCRs with Multiple TCCCs 273
10.3 Optimal Coordination of OCRs/DOCRs with Multiple TCCCs 278
10.4 Inherent Weaknesses of the Multi-TCCCs Approach 279
Problems 280
Written Exercises 280
Computer Exercises 281
11 Optimal Coordination with Considering the Best TCCC 283
11.1 Introduction 283
11.2 Possible Structures of the Optimizer 284
11.3 Technical Issue 287
Problems 290
Written Exercises 290
Computer Exercises 291
12 Considering the Actual Settings of Different Relay Technologies in the Same Network 293
12.1 Introduction 293
12.2 Mathematical Formulation 294
12.2.1 Objective Function 294
12.2.2 Selectivity Constraint Among Primary and Backup Relay Pairs 295
12.2.3 Inequality Constraints on Relay Operating Times 296
12.2.4 Side Constraints on Relay Time Multiplier Settings 296
12.2.5 Side Constraints on Relay Plug Settings 296
12.3 Biogeography-Based Optimization Algorithm 297
12.3.1 Clear Duplication Stage 297
12.3.2 Avoiding Facing Infeasible Selectivity Constraints 297
12.3.2.1 Linear Programming Stage 297
12.3.3 Linking PS¯yi ¯i and TMS¯yi ¯i with yi 298
12.4 Further Discussion 299
Problems 300
Written Exercises 300
Computer Exercises 301
13 Considering Double Primary Relay Strategy 303
13.1 Introduction 303
13.2 Mathematical Formulation 306
13.2.1 Objective Function 307
13.2.2 Selectivity Constraint 308
13.2.3 Inequality Constraints on Relay Operating Times 308
13.2.4 Side Constraints on Relay Time Multiplier Settings 308
13.2.5 Side Constraints on Relay Plug Settings 309
13.3 Possible Configurations of Double Primary ORC Problems 309
Problems 315
Written Exercises 315
Computer Exercises 316
14 Adaptive ORC Solver 319
14.1 Introduction 319
14.2 Types of Network Changes 320
14.2.1 Operational Changes 321
14.2.2 Topological Changes 321
14.3 AI-Based Adaptive ORC Solver 322
14.3.1 Generating Datasets 323
14.3.2 Applying ANN to Solve ORC Problems 324
Problems 328
Written Exercises 328
Computer Exercises 329
15 Multi-objective Coordination 333
15.1 Basic Principles 333
15.1.1 Conventional Aggregation Method 334
15.2 Multi-objective Formulation of ORC Problems 335
15.2.1 Operating Time vs. System Reliability 336
15.2.2 Operating Time vs. System Cost 336
15.2.3 Operating Time vs. System Reliability vs. System Cost 342
15.3 Further Discussions 342
Problems 345
Written Exercises 345
Computer Exercises 345
16 Optimal Coordination of Distance and Overcurrent Relays 347
16.1 Introduction 347
16.2 Basic Mathematical Modeling 348
16.3 Mathematical Modeling with Considering Multiple TCCCs 350
16.3.1 Inequality Constraints 351
16.3.2 Objective Function 352
16.4 Mathematical Modeling with Considering Different Fault Locations 353
16.4.1 Objective Function 353
16.4.2 Inequality Constraints 354
16.4.2.1 Near-End Faults 354
16.4.2.2 Middle-Point Faults 354
16.4.2.3 Far-End Faults 355
17 Trending Topics and Existing Issues 357
17.1 New Inverse-Time Characteristics 357
17.1.1 Scaled Standard TCCC Models 357
17.1.2 Stepwise TCCCs 358
17.1.3 New Customized TCCCs 359
17.2 Smart Grid 359
17.2.1 Distributed Generation 359
17.2.2 Series Compensation and Flexible Alternating Current Transmission System 360
17.2.3 Fault Current Limiters 360
17.3 Economic Operation 360
17.4 Power System Realization 361
17.4.1 Power Lines 361
17.4.2 Economic Operation 363
17.4.2.1 Combined-Cycle Power Plants 364
17.4.2.2 Degraded Efficiency Phenomenon 364
17.4.2.3 Unaccounted Losses in Power Stations 365
17.5 Locating Faults in Mesh Networks by DOCRs 367
17.5.1 Mechanism of the Proposed Fault Location Algorithm 370
17.5.1.1 Approach No. 1: Classical Linear Interpolation 373
17.5.1.2 Approach No. 2: Logarithmic/Nonlinear Interpolation 374
17.5.1.3 Approach No. 3: Polynomial Regression 375
17.5.1.4 Approach No. 4: Asymptotic Regression 375
17.5.1.5 Approach No. 5: DTCC-Based Regression 375
17.5.2 Final Structure of the Proposed Fault Locator 377
17.5.3 Overall Accuracy vs. Uncertainty 379
17.5.4 Further Discussion 380
Appendix A Some Important Data Used in Power System Protection 381
A.1 Standard Current Transformer Ratios 381
A.2 Standard Device/Function Number and Function Acronym Descriptions 382
A.2.1 Standard Device/Function Numbers 382
A.2.2 Device/Function Acronyms 383
A.2.3 Suffix Letters 383
A.2.3.1 Auxiliary Devices 383
A.2.3.2 Actuating Quantities 383
A.2.3.3 Main Device 384
A.2.3.4 Main Device Parts 384
A.2.3.5 Other Suffix Letters 384
Appendix B How to Install PowerWorld Simulator (Education Version) 387
Appendix C Single-Machine Infinite Bus 391
Appendix D Linearizing Relay Operating Time Models 393
D.1 Linearizing the IEC/BS Model of DOCRs by Fixing Time Multiplier Settings 393
D.2 Linearizing the ANSI/IEEE Model of DOCRs by Fixing Time Multiplier Settings 394
Appendix E Derivation of the First Order Thermal Differential Equation 397
Appendix F List of ORC Test Systems 399
F.1 Three-Bus Test Systems 399
F.1.1 System No. 1 399
F.1.2 System No. 2 399
F.2 Four-Bus Test Systems 403
F.2.1 System No. 1 403
F.2.2 System No. 2 403
F.3 Five-Bus Test System 408
F.4 Six-Bus Test Systems 410
F.4.1 System No. 1 410
F.4.2 System No. 2 410
F.4.3 System No. 3 411
F.4.4 System No. 4 413
F.5 Eight-Bus Test Systems 418
F.5.1 System No. 1 418
F.5.2 System No. 2 422
F.5.3 System No. 3 423
F.5.4 System No. 4 424
F.5.5 System No. 5 425
F.6 Nine-Bus Test System 427
F.7 14-Bus Test Systems 430
F.7.1 System No. 1 431
F.7.2 System No. 2 433
F.8 15-Bus Test System 437
F.9 30-Bus Test Systems 441
F.9.1 System No. 1 441
F.9.2 System No. 2 444
F.10 42-Bus Test System 448
F.11 118-Bus Test System 453
References 457
Index 479
Preface xvii
Acknowledgments xviii
Acronyms xix
About The Companion Website xxiii
Introduction xxv
1 Fundamental Steps in Optimization Algorithms 1
1.1 Overview 1
1.1.1 Design Variables 4
1.1.2 Design Parameters 4
1.1.3 Design Function 5
1.1.4 Objective Function(s) 5
1.1.5 Design Constraints 7
1.1.5.1 Mathematical Constraints 8
1.1.5.2 Inequality Constraints 8
1.1.5.3 Side Constraints 9
1.1.6 General Principles 10
1.1.6.1 Feasible Space vs. Search Space 10
1.1.6.2 Global Optimum vs. Local Optimum 11
1.1.6.3 Types of Problem 12
1.1.7 Standard Format 12
1.1.8 Constraint-Handling Techniques 13
1.1.8.1 Random Search Method 17
1.1.8.2 Constant Penalty Function 17
1.1.8.3 Binary Static Penalty Function 18
1.1.8.4 Superiority of Feasible Points (SFPs) - Type I 18
1.1.8.5 Superiority of Feasible Points (SFP) - Type II 18
1.1.8.6 Eclectic Evolutionary Algorithm 18
1.1.8.7 Typical Dynamic Penalty Function 19
1.1.8.8 Exponential Dynamic Penalty Function 19
1.1.8.9 Adaptive Multiplication Penalty Function 19
1.1.8.10 Self-Adaptive Penalty Function (SAPF) 20
1.1.9 Performance Criteria Used to Evaluate Algorithms 21
1.1.10 Types of Optimization Techniques 23
1.2 Classical Optimization Algorithms 23
1.2.1 Linear Programming 25
1.2.1.1 Historical Time-Line 25
1.2.1.2 Mathematical Formulation of LP Problems 26
1.2.1.3 Linear Programming Solvers 26
1.2.2 Global-Local Optimization Strategy 28
1.2.2.1 Multi-Start Linear Programming 29
1.2.2.2 Hybridizing LP with Meta-Heuristic Optimization Algorithms as a Fine-Tuning Unit 31
1.3 Meta-Heuristic Algorithms 33
1.3.1 Biogeography-Based Optimization 34
1.3.1.1 Migration Stage 40
1.3.1.2 Mutation Stage 41
1.3.1.3 Clear Duplication Stage 43
1.3.1.4 Elitism Stage 44
1.3.1.5 The Overall BBO Algorithm 45
1.3.2 Differential Evolution 45
1.4 Hybrid Optimization Algorithms 46
1.4.1 BBO-LP 48
1.4.2 BBO/DE 51
Problems 51
Written Exercises 51
Computer Exercises 53
2 Fundamentals of Power System Protection 57
2.1 Faults Classification 57
2.2 Protection System 61
2.3 Zones of Protection 65
2.4 Primary and Backup Protection 66
2.5 Performance and Design Criteria 66
2.5.1 Reliability 66
2.5.1.1 Dependability 66
2.5.1.2 Security 66
2.5.2 Sensitivity 67
2.5.3 Speed 67
2.5.4 Selectivity 67
2.5.5 Performance versus Economics 67
2.5.6 Adequateness 67
2.5.7 Simplicity 67
2.6 Overcurrent Protective Devices 67
2.6.1 Fuses 68
2.6.2 Bimetallic Relays 69
2.6.3 Overcurrent Protective Relays 69
2.6.4 Instantaneous OCR (IOCR) 70
2.6.5 Definite Time OCR (DTOCR) 71
2.6.6 Inverse Time OCR (ITOCR) 72
2.6.7 Mixed Characteristic Curves 73
2.6.7.1 Definite-Time Plus Instantaneous 73
2.6.7.2 Inverse-Time Plus Instantaneous 74
2.6.7.3 Inverse-Time Plus Definite-Time Plus Instantaneous 74
2.6.7.4 Inverse-Time Plus Definite-Time 75
2.6.7.5 Inverse Definite Minimum Time (IDMT) 76
Problems 76
Written Exercises 76
Computer Exercises 77
3 Mathematical Modeling of Inverse-Time Overcurrent Relay Characteristics 79
3.1 Computer Representation of Inverse-Time Overcurrent Relay Characteristics 79
3.1.1 Direct Data Storage 79
3.1.2 Curve Fitting Formulas 82
3.1.2.1 Polynomial Equations 82
3.1.2.2 Exponential Equations 89
3.1.2.3 Artificial Intelligence 93
3.1.3 Special Models 94
3.1.3.1 RI-Type Characteristic 94
3.1.3.2 RD-Type Characteristic 95
3.1.3.3 FR Short Time Inverse 95
3.1.3.4 UK Rectifier Protection 95
3.1.3.5 BNP-Type Characteristic 95
3.1.3.6 Standard CO Series Characteristics 95
3.1.3.7 IAC and ANSI Special Equations 96
3.1.4 User-Defined Curves 98
3.2 Dealing with All the Standard Characteristic Curves Together 99
3.2.1 Differentiating Between Time Dial Setting and Time Multiplier Setting 99
3.2.2 Dealing with Time Dial Setting and Time Multiplier Setting as One Variable 104
3.2.2.1 Fixed Divisor 106
3.2.2.2 Linear Interpolation 108
3.2.3 General Guidelines Before Conducting Researches and Studies 111
Problems 113
Written Exercises 113
Computer Exercises 114
4 Upper Limit of Relay Operating Time 117
4.1 Do We Need to Define T¯max ? 117
4.2 How to Define T¯max ? 118
4.2.1 Thermal Equations 118
4.2.1.1 Thermal Overload Protection for 3Phi Overhead Lines and Cables 118
4.2.1.2 Thermal Overload Protection for Motors 122
4.2.1.3 Thermal Overload Protection for Transformers 124
4.2.2 Stability Analysis 126
Problems 136
Written Exercises 136
Computer Exercises 138
5 Directional Overcurrent Relays and the Importance of Relay Coordination 139
5.1 Relay Grading in Radial Systems 139
5.1.1 Time Grading 140
5.1.2 Current Grading 140
5.1.3 Inverse-Time Grading 143
5.2 Directional Overcurrent Relays 146
5.3 Coordination of DOCRs 148
5.4 Is the Coordination of DOCRs an Iterative Problem? 148
5.5 Minimum Break-Point Set 161
5.6 Summary 163
Problems 164
Written Exercises 164
Computer Exercises 166
6 General Mechanism to Optimally Coordinate Directional Overcurrent Relays 169
6.1 Constructing Power Network 169
6.2 Power Flow Analysis 170
6.2.1 Per-Unit System and Three-to-One-Phase Conversion 172
6.2.2 Power Flow Solvers 173
6.2.3 How to Apply the Newton-Raphson Method 175
6.2.4 Sparsity Effect 179
6.3 P/B Pairs Identification 186
6.3.1 Inspection Method 186
6.3.2 Graph Theory Methods 186
6.3.3 Special Software 188
6.4 Short-Circuit Analysis 189
6.4.1 Short-Circuit Calculations 189
6.4.2 Electric Power Engineering Software Tools 190
6.4.2.1 Minimum Short-Circuit Current 190
6.4.2.2 Maximum Short-Circuit Current 192
6.4.3 Most Popular Standards 193
6.4.3.1 ANSI/IEEE Standards C37 & UL 489 193
6.4.3.2 IEC 61363 Standard 194
6.4.3.3 IEC 60909 Standard 194
6.5 Applying Optimization Techniques 201
Problems 202
Written Exercises 202
Computer Exercises 205
7 Optimal Coordination of Inverse-Time DOCRs with Unified TCCC 207
7.1 Mathematical Problem Formulation 207
7.1.1 Objective Function 208
7.1.1.1 Other Possible Objective Functions 210
7.1.2 Inequality Constraints on Relay Operating Times 211
7.1.3 Side Constraints on Relay Time Multiplier Settings 211
7.1.4 Side Constraints on Relay Plug Settings 211
7.1.5 Selectivity Constraint Among Primary and Backup Relay Pairs 212
7.1.5.1 Transient Selectivity Constraint 213
7.1.6 Standard Optimization Model 216
7.2 Optimal Coordination of DOCRs Using Meta-Heuristic Optimization Algorithms 217
7.2.1 Algorithm Implementation 217
7.2.2 Constraint-Handling Techniques 218
7.2.3 Solving the Infeasibility Condition 222
7.3 Results Tester 228
Problems 229
Written Exercises 229
Computer Exercises 231
8 Incorporating LP and Hybridizing It with Meta-heuristic Algorithms 235
8.1 Model Linearization 235
8.1.1 Classical Linearization Approach 236
8.1.1.1 IEC Curves: Fixing Plug Settings and Varying Time Multiplier Settings 236
8.1.1.2 IEEE Curves: Fixing Current Tap Settings and Varying Time Dial Settings 237
8.1.2 Transformation-Based Linearization Approach 237
8.1.2.1 IEC Curves: Fixing Time Multiplier Settings and Varying Plug Settings 238
8.1.2.2 IEEE Curves: Fixing Time Dial Settings and Varying Current Tap Settings 238
8.2 Multi-start Linear Programming 242
8.3 Hybridizing Linear Programming with Population-Based Meta-heuristic Optimization Algorithms 245
8.3.1 Classical Linearization Approach: Fixing PS/CTS and Varying TMS/TDS 245
8.3.2 Transformation-Based Linearization Approach: Fixing TMS/TDS and Varying PS/CTS 245
8.3.3 Innovative Linearization Approach: Fixing/Varying TMS/TDS and PS/CTS 250
Problems 250
Written Exercises 250
Computer Exercises 251
9 Optimal Coordination of DOCRs With OCRs and Fuses 253
9.1 Simple Networks 253
9.1.1 Protecting Radial Networks by Just OCRs 253
9.1.2 Protecting Double-Line Networks by OCRs and DOCRs 255
9.2 Little Harder Networks 257
9.2.1 Combination of OCRs and DOCRs 258
9.2.2 Combination of Fuses, OCRs, and DOCRs 261
9.3 Complex Networks 264
Problems 265
Written Exercises 265
Computer Exercises 266
10 Optimal Coordination with Considering Multiple Characteristic Curves 271
10.1 Introduction 271
10.2 Optimal Coordination of DOCRs with Multiple TCCCs 273
10.3 Optimal Coordination of OCRs/DOCRs with Multiple TCCCs 278
10.4 Inherent Weaknesses of the Multi-TCCCs Approach 279
Problems 280
Written Exercises 280
Computer Exercises 281
11 Optimal Coordination with Considering the Best TCCC 283
11.1 Introduction 283
11.2 Possible Structures of the Optimizer 284
11.3 Technical Issue 287
Problems 290
Written Exercises 290
Computer Exercises 291
12 Considering the Actual Settings of Different Relay Technologies in the Same Network 293
12.1 Introduction 293
12.2 Mathematical Formulation 294
12.2.1 Objective Function 294
12.2.2 Selectivity Constraint Among Primary and Backup Relay Pairs 295
12.2.3 Inequality Constraints on Relay Operating Times 296
12.2.4 Side Constraints on Relay Time Multiplier Settings 296
12.2.5 Side Constraints on Relay Plug Settings 296
12.3 Biogeography-Based Optimization Algorithm 297
12.3.1 Clear Duplication Stage 297
12.3.2 Avoiding Facing Infeasible Selectivity Constraints 297
12.3.2.1 Linear Programming Stage 297
12.3.3 Linking PS¯yi ¯i and TMS¯yi ¯i with yi 298
12.4 Further Discussion 299
Problems 300
Written Exercises 300
Computer Exercises 301
13 Considering Double Primary Relay Strategy 303
13.1 Introduction 303
13.2 Mathematical Formulation 306
13.2.1 Objective Function 307
13.2.2 Selectivity Constraint 308
13.2.3 Inequality Constraints on Relay Operating Times 308
13.2.4 Side Constraints on Relay Time Multiplier Settings 308
13.2.5 Side Constraints on Relay Plug Settings 309
13.3 Possible Configurations of Double Primary ORC Problems 309
Problems 315
Written Exercises 315
Computer Exercises 316
14 Adaptive ORC Solver 319
14.1 Introduction 319
14.2 Types of Network Changes 320
14.2.1 Operational Changes 321
14.2.2 Topological Changes 321
14.3 AI-Based Adaptive ORC Solver 322
14.3.1 Generating Datasets 323
14.3.2 Applying ANN to Solve ORC Problems 324
Problems 328
Written Exercises 328
Computer Exercises 329
15 Multi-objective Coordination 333
15.1 Basic Principles 333
15.1.1 Conventional Aggregation Method 334
15.2 Multi-objective Formulation of ORC Problems 335
15.2.1 Operating Time vs. System Reliability 336
15.2.2 Operating Time vs. System Cost 336
15.2.3 Operating Time vs. System Reliability vs. System Cost 342
15.3 Further Discussions 342
Problems 345
Written Exercises 345
Computer Exercises 345
16 Optimal Coordination of Distance and Overcurrent Relays 347
16.1 Introduction 347
16.2 Basic Mathematical Modeling 348
16.3 Mathematical Modeling with Considering Multiple TCCCs 350
16.3.1 Inequality Constraints 351
16.3.2 Objective Function 352
16.4 Mathematical Modeling with Considering Different Fault Locations 353
16.4.1 Objective Function 353
16.4.2 Inequality Constraints 354
16.4.2.1 Near-End Faults 354
16.4.2.2 Middle-Point Faults 354
16.4.2.3 Far-End Faults 355
17 Trending Topics and Existing Issues 357
17.1 New Inverse-Time Characteristics 357
17.1.1 Scaled Standard TCCC Models 357
17.1.2 Stepwise TCCCs 358
17.1.3 New Customized TCCCs 359
17.2 Smart Grid 359
17.2.1 Distributed Generation 359
17.2.2 Series Compensation and Flexible Alternating Current Transmission System 360
17.2.3 Fault Current Limiters 360
17.3 Economic Operation 360
17.4 Power System Realization 361
17.4.1 Power Lines 361
17.4.2 Economic Operation 363
17.4.2.1 Combined-Cycle Power Plants 364
17.4.2.2 Degraded Efficiency Phenomenon 364
17.4.2.3 Unaccounted Losses in Power Stations 365
17.5 Locating Faults in Mesh Networks by DOCRs 367
17.5.1 Mechanism of the Proposed Fault Location Algorithm 370
17.5.1.1 Approach No. 1: Classical Linear Interpolation 373
17.5.1.2 Approach No. 2: Logarithmic/Nonlinear Interpolation 374
17.5.1.3 Approach No. 3: Polynomial Regression 375
17.5.1.4 Approach No. 4: Asymptotic Regression 375
17.5.1.5 Approach No. 5: DTCC-Based Regression 375
17.5.2 Final Structure of the Proposed Fault Locator 377
17.5.3 Overall Accuracy vs. Uncertainty 379
17.5.4 Further Discussion 380
Appendix A Some Important Data Used in Power System Protection 381
A.1 Standard Current Transformer Ratios 381
A.2 Standard Device/Function Number and Function Acronym Descriptions 382
A.2.1 Standard Device/Function Numbers 382
A.2.2 Device/Function Acronyms 383
A.2.3 Suffix Letters 383
A.2.3.1 Auxiliary Devices 383
A.2.3.2 Actuating Quantities 383
A.2.3.3 Main Device 384
A.2.3.4 Main Device Parts 384
A.2.3.5 Other Suffix Letters 384
Appendix B How to Install PowerWorld Simulator (Education Version) 387
Appendix C Single-Machine Infinite Bus 391
Appendix D Linearizing Relay Operating Time Models 393
D.1 Linearizing the IEC/BS Model of DOCRs by Fixing Time Multiplier Settings 393
D.2 Linearizing the ANSI/IEEE Model of DOCRs by Fixing Time Multiplier Settings 394
Appendix E Derivation of the First Order Thermal Differential Equation 397
Appendix F List of ORC Test Systems 399
F.1 Three-Bus Test Systems 399
F.1.1 System No. 1 399
F.1.2 System No. 2 399
F.2 Four-Bus Test Systems 403
F.2.1 System No. 1 403
F.2.2 System No. 2 403
F.3 Five-Bus Test System 408
F.4 Six-Bus Test Systems 410
F.4.1 System No. 1 410
F.4.2 System No. 2 410
F.4.3 System No. 3 411
F.4.4 System No. 4 413
F.5 Eight-Bus Test Systems 418
F.5.1 System No. 1 418
F.5.2 System No. 2 422
F.5.3 System No. 3 423
F.5.4 System No. 4 424
F.5.5 System No. 5 425
F.6 Nine-Bus Test System 427
F.7 14-Bus Test Systems 430
F.7.1 System No. 1 431
F.7.2 System No. 2 433
F.8 15-Bus Test System 437
F.9 30-Bus Test Systems 441
F.9.1 System No. 1 441
F.9.2 System No. 2 444
F.10 42-Bus Test System 448
F.11 118-Bus Test System 453
References 457
Index 479
ALI R. AL-ROOMI, PhD, is a Research Assistant at Dalhousie University, Halifax, Canada. He is an IEEE Member and has served as an academic reviewer for Transactions on Automation Science and Engineering and Electrical Power and Energy Conference. His research interests include power systems operation, control systems, sensors, optimization algorithms, and machine learning computing systems.
