Modern Optimization Techniques with Applications in Electric Power Systems (eBook)

Preface 8
Acknowledgments 10
Contents 12
Chapter 1: Introduction 20
1.1 Introduction [1-11] 20
1.2 Optimization Techniques 21
1.2.1 Conventional Techniques (Classic Methods) [2, 3] 22
1.2.2 Evolutionary Techniques [4-11] 26
1.2.2.1 Heuristic Search [3] 26
1.2.2.2 Evolutionary Computation [8] 27
What Do You Mean by Pareto Optimal Set? 28
1.2.2.3 Genetic Algorithm [7] 28
1.2.2.4 Evolution Strategies and Evolutionary Programming 30
1.2.2.5 Differential Evolutions 32
1.2.2.6 Particle Swarm [9] 33
1.2.2.7 Tabu Search [8-12] 35
1.2.2.8 Simulated Annealing [8-12] 35
1.2.2.9 Stochastic Approximation 36
1.2.2.10 Fuzzy [13] 37
1.3 Outline of the Book 39
References 40
Chapter 2: Mathematical Optimization Techniques 42
2.1 Introduction 42
2.2 Quadratic Forms [1] 43
2.3 Some Static Optimization Techniques [1-10] 45
2.3.1 Unconstrained Optimization 46
2.3.2 Constrained Optimization 49
2.4 Pontryagin´s Maximum Principle [11-14] 56
2.5 Functional Analytic Optimization Technique [6] 61
2.5.1 Norms 61
2.5.2 Inner Product (Dot Product) 62
2.5.3 Transformations 64
2.5.4 The Minimum Norm Theorem 65
2.6 Simulated Annealing Algorithm (SAA) [16-26] 67
2.6.1 Physical Concepts of Simulated Annealing [79] 68
2.6.2 Combinatorial Optimization Problems 69
2.6.3 A General Simulated Annealing Algorithm [16-26] 69
2.6.4 Cooling Schedules 70
2.6.5 Polynomial-Time Cooling Schedule 70
2.6.5.1 Initial Value of the Control Parameter 70
2.6.5.2 Decrement of the Control Parameter 71
2.6.5.3 Final Value of the Control Parameter 71
2.6.5.4 The Length of Markov Chains 71
2.6.6 Kirk´s Cooling Schedule 72
2.6.6.1 Initial Value of the Control Parameter 72
2.6.6.2 Decrement of the Control Parameter 72
2.6.6.3 Final Value of the Control Parameter 72
2.6.6.4 Length of the Markov Chain 73
2.7 Tabu Search Algorithm 73
2.7.1 Tabu List Restrictions 73
2.7.2 Aspiration Criteria 74
2.7.3 Stopping Criteria 74
2.7.4 General Tabu Search Algorithm 75
2.8 The Genetic Algorithm (GA) 76
2.8.1 Solution Coding 77
2.8.2 Fitness Function 77
2.8.3 Genetic Algorithms Operators 78
2.8.4 Constraint Handling (Repair Mechanism) 78
2.8.5 A General Genetic Algorithm 79
2.9 Fuzzy Systems [78] 79
2.9.1 Basic Terminology and Definition 83
2.9.2 Support of Fuzzy Set 84
2.9.3 Normality 85
2.9.4 Convexity and Concavity 85
2.9.5 Basic Operations [53] 85
2.9.5.1 Inclusion 86
2.9.5.2 Equality 86
2.9.5.3 Complementation 86
2.9.5.4 Intersection 86
2.9.5.5 Union 87
2.9.5.6 Algebraic Product 87
2.9.5.7 Algebraic Sum 87
2.9.5.8 Difference 87
2.9.5.9 Fuzzy Arithmetic 87
2.9.5.10 LR-Type Fuzzy Number 89
2.9.5.11 Interval Arithmetic 90
2.9.5.12 Triangular and Trapezoidal Fuzzy Numbers 90
2.10 Particle Swarm Optimization (PSO) Algorithm 90
2.11 Basic Fundamentals of PSO Algorithm 93
2.11.1 General PSO Algorithm 95
References 97
Chapter 3: Economic Operation of Electric Power Systems 101
3.1 Introduction 101
3.2 A Hydrothermal-Nuclear Power System 102
3.2.1 Problem Formulation [10-15] 102
3.2.2 The Optimization Procedure 105
3.2.3 The Optimal Solution Using Minimum Norm Technique 109
3.2.4 A Feasible Multilevel Approach 112
3.2.5 Conclusions and Comments 114
3.3 All-Thermal Power Systems 114
3.3.1 Conventional All-Thermal Power Systems Problem Formulation [17, 19, 20, 22, 27, 37-38]
3.3.2 Fuzzy All-Thermal Power Systems Problem Formulation [39]
3.3.3 Solution Algorithm 123
3.3.4 Examples 123
3.3.5 Conclusion 129
3.4 All-Thermal Power Systems with Fuzzy Load and Cost Function Parameters 130
3.4.1 Problem Formulation 131
3.4.2 Fuzzy Interval Arithmetic Representation on Triangular Fuzzy Numbers 141
3.4.3 Fuzzy Arithmetic on Triangular L-R Representation of Fuzzy Numbers 146
3.4.4 Example 147
3.5 Fuzzy Economical Dispatch Including Losses 163
3.5.1 Problem Formulation 164
3.5.2 Solution Algorithm 182
3.5.3 Simulated Example 183
3.5.4 Conclusion 185
Appendix A.1 200
Appendix A.2 200
References 201
Chapter 4: Economic Dispatch (ED) and Unit Commitment Problems (UCP): Formulation and Solution Algorithms 203
4.1 Introduction 203
4.2 Problem Statement 204
4.3 Rules for Generating Trial Solutions 204
4.4 The Economic Dispatch Problem 204
4.5 The Objective Function 205
4.5.1 The Production Cost 205
4.5.2 The Start-Up Cost 205
4.6 The Constraints 206
4.6.1 System Constraints 206
4.6.1.1 Load Demand Constraints 206
4.6.1.2 Spinning Reserve Constraint 206
4.6.2 Unit Constraints 207
4.6.2.1 Generation Limits 207
4.6.2.2 Minimum Up/Down Time 207
4.6.2.3 Units Initial Status Constraint 208
4.6.2.4 Crew Constraints 208
4.6.2.5 Unit Availability Constraint 208
4.6.2.6 Units Derating Constraint 208
4.7 Rules for Generating Trial Solutions 209
4.8 Generating an Initial Solution 211
4.9 An Algorithm for the Economic Dispatch Problem 211
4.9.1 The Economic Dispatch Problem in a Linear Complementary Form 212
4.9.2 Tableau Size for the Economic Dispatch Problem 214
4.10 The Simulated Annealing Algorithm (SAA) for Solving UCP 214
4.10.1 Comparison with Other SAA in the Literature 215
4.10.2 Numerical Examples 216
4.11 Summary and Conclusions 225
4.12 Tabu Search (TS) Algorithm 226
4.12.1 Tabu List (TL) Restrictions 227
4.12.2 Aspiration Level Criteria 230
4.12.3 Stopping Criteria 231
4.12.4 General Tabu Search Algorithm 231
4.12.5 Tabu Search Algorithm for Unit Commitment 233
4.12.6 Tabu List Types for UCP 234
4.12.7 Tabu List Approach for UCP 234
4.12.7.1 Approach (1) 234
4.12.7.2 Approach (2) 234
4.12.7.3 Approach (3) 235
4.12.7.4 Approach (4) 235
4.12.7.5 Approach (5) 235
4.12.8 Comparison Among the Different Tabu Lists Approaches 235
4.12.9 Tabu List Size for UCP 236
4.12.10 Numerical Results of the STSA 236
4.13 Advanced Tabu Search (ATS) Techniques 238
4.13.1 Intermediate-Term Memory 239
4.13.2 Long-Term Memory 240
4.13.3 Strategic Oscillation 240
4.13.4 ATSA for UCP 241
4.13.5 Intermediate-Term Memory Implementation 241
4.13.5.1 Approach (1) 242
4.13.5.2 Approach (2) 242
4.13.6 Long-Term Memory Implementation 243
4.13.7 Strategic Oscillation Implementation 244
4.13.8 Numerical Results of the ATSA 244
4.14 Conclusions 248
4.15 Genetic Algorithms for Unit Commitment 249
4.15.1 Solution Coding 250
4.15.2 Fitness Function 250
4.15.3 Genetic Algorithms Operators 251
4.15.4 Constraint Handling (Repair Mechanism) 251
4.15.5 A General Genetic Algorithm 252
4.15.6 Implementation of a Genetic Algorithm to the UCP 252
4.15.7 Solution Coding 253
4.15.8 Fitness Function 254
4.15.9 Selection of Chromosomes 255
4.15.10 Crossover 255
4.15.11 Mutation 255
4.15.11.1 Mutation Operator (1) 256
4.15.11.2 Mutation Operator (2) 256
4.15.12 Adaptive GA Operators 257
4.15.13 Numerical Examples 257
4.15.14 Summary 262
4.16 Hybrid Algorithms for Unit Commitment 264
4.17 Hybrid of Simulated Annealing and Tabu Search (ST) 264
4.17.1 Tabu Search Part in the ST Algorithm 265
4.17.2 Simulated Annealing Part in the ST Algorithm 266
4.18 Numerical Results of the ST Algorithm 266
4.19 Hybrid of Genetic Algorithms and Tabu Search 269
4.19.1 The Proposed Genetic Tabu (GT) Algorithm 269
4.19.2 Genetic Algorithm as a Part of the GT Algorithm 269
4.19.3 Tabu Search as a Part of the GT Algorithm 271
4.20 Numerical Results of the GT Algorithm 273
4.21 Hybrid of Genetic Algorithms, Simulated Annealing, and Tabu Search 277
4.21.1 Genetic Algorithm as a Part of the GST Algorithm 279
4.21.2 Tabu Search Part of the GST Algorithm 279
4.21.3 Simulated Annealing as a Part of the GST Algorithm 281
4.22 Numerical Results of the GST Algorithm 281
4.23 Summary 286
4.24 Comparisons of the Algorithms for the Unit Commitment Problem 287
4.24.1 Results of Example 1 287
4.24.2 Results of Example 2 289
4.24.3 Results of Example 3 290
4.24.4 Summary 292
References 292
Chapter 5: Optimal Power Flow 298
5.1 Introduction 298
5.2 Power Flow Equations 304
5.2.1 Load Buses 305
5.2.2 Voltage Controlled Buses 305
5.2.3 Slack Bus 305
5.3 General OPF Problem Formulations 308
5.3.1 The Objective Functions 309
5.3.1.1 Minimization of Active Power Transmission Loss 309
5.3.1.2 Minimization of Generation Fuel Cost 310
5.3.1.3 Maximization of Reactive Power Reserve Margin 310
5.3.1.4 Minimization of Reactive Power Transmission Loss 310
5.3.1.5 Minimization of Emission Index 311
5.3.1.6 Maximization of Security Margin Index 311
5.3.2 The Constraints 312
5.3.2.1 Equality Constraints 312
5.3.2.2 Inequality Constraints 312
5.3.3 Optimization Algorithms for OPF 314
5.4 Optimal Power Flow Algorithms for Single Objective Cases [90-102] 316
5.4.1 Particle Swarm Optimization (PSO) Algorithm for the OPF Problem 317
5.4.2 The IEEE-30 Bus Power System 318
5.4.3 Active Power Loss Minimization 318
5.4.4 Minimization of Generation Fuel Cost 324
5.4.5 Reactive Power Reserve Maximization 326
5.4.6 Reactive Power Loss Minimization 327
5.4.7 Emission Index Minimization 329
5.4.8 Security Margin Maximization 334
5.5 Comparisons of Different Single Objective Functions 336
5.6 Multiobjective OPF Algorithm 344
5.7 Basic Concept of Multiobjective Analysis 344
5.8 The Proposed Multiobjective OPF Algorithm 346
5.8.1 Multiobjective OPF Formulation 346
5.8.2 General Steps for Solving Multi-Objective OPF Problem 347
5.9 Generating Nondominated Set 347
5.9.1 Generating techniques 347
5.9.2 Weighting method 349
5.10 Hierarchical Cluster Technique 350
5.11 Conclusions 355
Appendix 356
References 359
Chapter 6: Long-Term Operation of Hydroelectric Power Systems 364
6.1 Introduction 364
6.2 Problem Formulation 366
6.3 Problem Solution: A Minimum Norm Approach 367
6.3.1 System Modeling 367
6.3.2 Formulation 368
6.3.3 Optimal Solution 371
6.3.4 Practical Application 373
6.3.5 Comments 373
6.3.6 A Nonlinear Model 374
6.3.6.1 Formulation 375
6.3.6.2 The Optimal Solution 380
6.3.6.3 Practical Application 381
6.3.6.4 Comments 381
6.4 Simulated Annealing Algorithm (SAA) [28-32] 383
6.4.1 Generating Trial Solution (Neighbor) 384
6.4.2 Details of the SAA for the LTHSP 385
6.4.2.1 Generating an Initial Feasible Solution 385
6.4.2.2 Testing the Simulated Annealing 386
6.4.2.3 Step Size Vector Adjustment 386
6.4.2.4 Cooling Schedule 386
6.4.2.5 Equilibrium Test 387
6.4.3 Practical Applications 387
6.4.4 Conclusion 388
6.5 Tabu Search Algorithm [34-41] 388
6.5.1 Problem Statement 389
6.5.2 TS Method 390
6.5.3 Details of the TSA 390
6.5.3.1 Tabu List in the LTHSP 390
6.5.3.2 TS Test 390
6.5.3.3 Generating Trial Solution (Neighbor) 392
6.5.4 Step-Size Vector Adjustment 393
6.5.5 Stopping Criteria 393
6.5.6 Numerical Examples 393
6.5.7 Conclusions 395
References 395
Chapter 7: Electric Power Quality Analysis 398
7.1 Introduction 398
7.2 Simulated Annealing Algorithm (SAA) 401
7.2.1 Testing Simulated Annealing Algorithm 402
7.2.2 Step-Size Vector Adjustment 402
7.2.3 Cooling Schedule 403
7.3 Flicker Voltage Simulation [2] 403
7.3.1 Problem Formulation 403
7.3.2 Testing the Algorithm for Voltage Flicker 404
7.3.3 Effect of Number of Samples 405
7.3.4 Effects of Sampling Frequency 405
7.4 Harmonics Problem Formulation [60-80] 405
7.5 Testing the Algorithm for Harmonics 406
7.5.1 Signal with Known Frequency 406
7.5.2 Signal with Unknown Frequency 407
7.6 Conclusions 410
7.7 Steady-State Frequency Estimation [48-59] 411
7.7.1 A Constant Frequency Model, Problem Formulation [34-37] 413
7.7.2 Computer Simulation 414
7.7.2.1 Noise-free signal 414
7.7.2.2 Effects of Number of Samples 414
7.7.2.3 Effects of Sampling Frequency 415
7.7.3 Harmonic-contaminated Signal 415
7.7.3.1 Effects of Number of Samples 415
7.7.3.2 Effects of Sampling Frequency 416
7.7.4 Actual Recorded Data 417
7.8 Conclusions 418
7.8.1 A Variable Frequency Model 418
7.8.1.1 Problem Formulation 419
7.8.1.2 The Algorithm of Solution 419
7.8.2 Simulated Example 419
7.8.2.1 Effects of Number of Samples 420
7.8.2.2 Effects of Sampling Frequency 420
7.8.2.3 Effects of Harmonics 421
7.8.3 Exponential Decaying Frequency 422
7.8.3.1 Actual Recorded Data 422
7.9 Conclusions 424
References 424
Index 427