Optimal Coordination of Power Protective Devices with Illustrative Examples (IEEE Press Series on Power and Energy Systems)

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 Tmax ? 117 4.2 How to Define Tmax ? 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 PSyi i and TMSyi 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