Preface ...................................................... xvii
Author ...................................................... xxiii
Chapter 1 Introduction .......................................... 1
1.1. Structure of a Generic Electric Power System ............... 1
1.2. Power System Models ........................................ 3
1.3. Power System Control ....................................... 5
1.4. Power System Security Assessment ........................... 8
1.5. Power System Optimization as a Function of Time ........... 11
1.6. Review of Optimization Techniques Applicable to Power
Systems ................................................... 13
References ................................................ 16
Chapter 2 Electric Power System Models ......................... 17
2.1. Introduction .............................................. 17
2.2. Complex Power Concepts .................................... 18
2.3. Three-Phase Systems ....................................... 20
2.3.1. Y-Connected Systems ................................ 21
2.3.2. Delta-Connected Systems ............................ 23
2.3.3. Power Relationships ................................ 25
2.4. Per Unit Representation ................................... 27
2.5. Synchronous Machine Modeling .............................. 28
2.5.1. Classical Representation of the Synchronous
Machine ............................................ 29
2.6. Reactive Capability Limits ................................ 30
2.7. Prime Movers and Governing Systems ........................ 31
2.7.1. Hydraulic Turbines and Governing Models ............ 31
2.7.2. Steam Turbines and Governing System Models ......... 33
2.8. Automatic Gain Control .................................... 34
2.8.1. Power Control in a Multigenerator Environment ...... 34
2.8.2. AGC System Models .................................. 38
2.8.2.1. Case A: Two Generating Units .............. 38
2.9. Transmission Subsystems ................................... 40
2.10.Y-Bus Incorporating the Transformer Effect ................ 42
2.10.1.Fixed Tap-Setting Transformer ...................... 42
2.10.2.TCUL Transformer ................................... 45
2.10.3.Phase-Shifting Transformer ......................... 45
2.11.Load Models ............................................... 50
2.11.1.Static Load Models ................................. 50
2.12.Available Transfer Capability ............................. 52
2.12.1.АТС Definition and Formulation ..................... 52
2.12.2.АТС Calculation .................................... 53
2.13.Illustrative Examples ..................................... 54
2.14.Conclusions ............................................... 58
2.15.Problem Set ............................................... 58
References ................................................ 61
Chapter 3 Power-Flow Computations .............................. 63
3.1. Introduction .............................................. 63
3.2. Types of Buses for PF Studies ............................. 64
3.3. General Form of the PFEs .................................. 66
3.3.1. PF Control by Transformer Regulation ............... 67
3.4. Practical Modeling Considerations ......................... 69
3.4.1. Generation Subsystem ............................... 69
3.4.1.1. Rectangular Formulation ................... 72
3.4.1.2. Polar Formulation ......................... 72
3.5. Iterative Techniques for PF Solution ...................... 72
3.5.1. G-S Iterative Technique ............................ 73
3.5.1.1. G-S Algorithm ............................. 73
3.5.1.2. G-S Method Applied to the PFEs ............ 74
3.5.1.3. G-S Iterative Technique ................... 75
3.5.1.4. Line Flow and Losses ...................... 77
3.5.2. N-R Method ......................................... 78
3.5.2.1. N-R Algorithm in the Scalar Case .......... 78
3.5.2.2. N-R Algorithm in the n-Dimensional
Case ...................................... 83
3.5.2.3. N-R Algorithm Applied to the PFEs ......... 84
3.5.3. Fast-Decoupled PF Method ........................... 97
3.5.4. Linearized (DC) PF Method ......................... 100
3.6. Practical Applications of PF Studies ..................... 102
3.6.1. Case Study Discussion ............................. 103
3.7. Illustrative Examples .................................... 103
3.8. Conclusion ............................................... 108
3.9. Problem Set .............................................. 109
References ............................................... 112
Chapter 4 Constrained Optimization and Applications ........... 113
4.1. Introduction ............................................. 113
4.2. Theorems on the Optimization of Constrained Functions .... 114
4.2.1. Continuity Assumption ............................. 115
4.2.2. Theorems .......................................... 115
4.3. Procedure for Optimizing Constrained Problems
(Functions) .............................................. 116
4.4. Karush-Kuhn-Tucker Condition ............................. 117
4.5. Illustrative Problems .................................... 118
4.5.1. Nonpower Systems Application Examples ............. 119
4.6. Power Systems Application Examples ....................... 120
4.6.1. Optimal Operation of an All-Thermal System:
Equal Incremental Cost-Loading .................... 120
4.6.2. Optimal Operation of an All-Thermal System,
Including Losses .................................. 123
4.7. Illustrative Examples .................................... 127
4.8. Conclusion ............................................... 133
4.9. Problem Set .............................................. 134
References ............................................... 136
Chapter 5 Linear Programming and Applications ................. 137
5.1. Introduction ............................................. 137
5.2. Mathematical Model and Nomenclature in LP ................ 138
5.2.1. Implicit Assumptions in LP ........................ 139
5.3. LP Solution Techniques ................................... 140
5.3.1. Graphical Method .................................. 140
5.3.2. Matrix Approach to LP ............................. 142
5.3.3. Simplex Method .................................... 144
5.3.4. Lemma of Matrix Inversion ......................... 146
5.3.5. Revised Simplex Method ............................ 150
5.4. Duality in LP ............................................ 153
5.5. Khun-Tucker Conditions in LP ............................. 156
5.5.1. Case 1: LP and KKT Conditions for Problems
with Equality Constraints ......................... 156
5.5.2. Case 2: KKT Applied to the Dual LP Problem ........ 158
5.5.3. Case 3: KKT Applied to LP Problems with
Equality Constraints .............................. 160
5.6. Mixed-Integer Programming ................................ 162
5.6.1. Branch-and-Bound Technique for Binary
Integer Programming Problems ...................... 164
5.7. Sensitivity Methods for Postoptimization in LP ........... 168
5.7.1. Case 1: Perturbation in the Parameters b1 ......... 169
5.7.2. Case 2: Perturbation in the Cost Coefficients
cj ................................................ 169
5.7.3. Case 3: Perturbation in the Coefficient aij ....... 170
5.7.4. Case 4: Injection of New Constraints .............. 170
5.7.5. Case 5: Injection of New Variables ................ 170
5.7.6. Sensitivity Analysis Solution Technique for
Changes in Parameters bi .......................... 170
5.7.6.1. Solution Methodology ..................... 172
5.7.6.2. Implementation Algorithm ................. 174
5.7.6.3. Duality in Postoptimal Analysis .......... 177
5.8. Power Systems Applications ............................... 179
5.9. Illustrative Examples .................................... 180
5.10.Conclusion ............................................... 188
5.11.Problem Set .............................................. 189
References ............................................... 196
Chapter 6 Interior Point Methods .............................. 197
6.1. Introduction ............................................. 197
6.2. Karmarkar's Algorithm .................................... 199
6.3. Projective-Scaling Method ................................ 200
6.4. Dual Affine Algorithm .................................... 202
6.5. Primal Affine Algorithm .................................. 203
6.6. Barrier Algorithm ........................................ 204
6.7. Extended IP Method for LP Problems ....................... 205
6.8. FI Sequence .............................................. 206
6.8.1. Optimality Condition .............................. 209
6.9. Extended Quadratic Programming Using IP Method ........... 211
6.10.Illustrative Examples .................................... 216
6.11.Conclusions .............................................. 227
6.12.Problem Set .............................................. 228
References ............................................... 230
Chapter 7 Nonlinear Programming ............................... 233
7.1. Introduction ............................................. 233
7.2. Classification of NLP Problems ........................... 233
7.2.1. NLP Problems with Nonlinear Objective Function
and Linear Constraints ............................ 233
7.2.2. Quadratic Programming ............................. 234
7.2.3. Convex Programming ................................ 234
7.2.4. Separable Programming ............................. 234
7.3. Sensitivity Method for Solving NLP Variables ............. 235
7.3.1. Procedure for Solving the NLP Problem ............. 236
7.4. Algorithm for Quadratic Optimization ..................... 240
7.5. Illustrative Example (Barrier Method for Solving NLP) .... 241
7.5.1. Algorithm for Recursive Process ................... 242
7.5.1.1. Analytical Forms ......................... 245
7.5.1.2. Penalty Vectors .......................... 246
7.5.2. Computer Implementation ........................... 247
7.6. Illustrative Examples .................................... 248
7.7. Conclusion ............................................... 257
7.8. Problem Set .............................................. 257
References ............................................... 262
Chapter 8 Dynamic Programming ................................. 263
8.1. Introduction ............................................. 263
8.2. Formulation of a Multistage Decision Process ............. 264
8.2.1. Representation of a Multistage Decision Process ... 264
8.2.2. Types of Multistage Decision Problems ............. 266
8.3. Characteristics of DP .................................... 266
8.4. Concept of Suboptimization and the Principle of
Optimality ............................................... 267
8.5. Formulation of DP ........................................ 269
8.6. Backward and Forward Recursion ........................... 274
8.6.1. Minimum Path Problem .............................. 275
8.6.2. Single Additive Constraint and Additively
Separable Return Problem .......................... 279
8.6.3. Single Multiplicative Constraint, Additively
Separable Return Problem .......................... 280
8.6.4. Single Additive Constraint, Multiplicatively
Separable Turn Problem ............................ 283
8.7. Computational Procedure in DP ............................ 283
8.8. Computational Economy in DP .............................. 285
8.9. Systems with More than One Constraint .................... 285
8.10.Conversion of a Final Value Problem into an Initial
Value Problem ............................................ 288
8.11.Illustrative Examples .................................... 289
8.12.Conclusions .............................................. 296
8.13.Problem Set .............................................. 297
References ............................................... 301
Chapter 9 Lagrangian Relaxation ............................... 303
9.1. Introduction ............................................. 303
9.2. Concepts ................................................. 304
9.3. Subgradient Method for Setting the Dual Variables ........ 305
9.4. Setting tk ............................................... 313
9.4.1. Case 1: Subgradient Method with tk = l for
All к ............................................. 313
9.4.2. Case 2: Subgradient Method with tk = 1, 0.5,
0.25 .............................................. 314
9.4.3. Case 3: Subgradient Method with tk = 1, 1/3,
1/9 ............................................... 315
9.5. Comparison with LP-Based Bounds .......................... 317
9.6. Improved Relaxation ...................................... 318
9.7. Summary of Concepts ...................................... 319
9.8. Past Applications ........................................ 321
9.9. Summary .................................................. 322
9.9.1. Overview .......................................... 322
9.9.2. Algorithm of Solution Using Lagrangian
Relaxation Approach ............................... 323
9.9.3. Power System Application: Scheduling in Power
Generation Systems ................................ 324
9.9.3.1. Model .................................... 324
9.9.3.2. Relaxation and Decomposition of the
Model .................................... 326
9.9.3.3. Solution Technique ....................... 328
9.10.Illustrative Examples .................................... 329
9.11.Conclusions .............................................. 330
9.12.Problem Set .............................................. 332
References ............................................... 333
Chapter 10 Decomposition Method ............................... 335
10.1.Introduction ............................................. 335
10.2.Formulation of the Decomposition Problem ................. 335
10.3.Algorithm of the Decomposition Technique ................. 338
10.4.Illustrative Example of the Decomposition Technique ...... 339
10.5.Conclusions .............................................. 345
10.6.Problem Set .............................................. 346
References ............................................... 348
Chapter 11.State Estimation ................................... 351
11.1.Historical Perspective of State Estimation ............... 351
11.1.1.Conventional State Estimation ..................... 353
11.1.2.Generalized State Estimation ...................... 353
11.2.Simple Mathematical Background ........................... 355
11.2.1.Definition of Static State Estimation ............. 355
11.3.State Estimation Techniques .............................. 357
11.3.1.Method ............................................ 357
11.3.1.1.Least Squares Estimation (LSE) ........... 358
11.3.1.2.Weighted Least Square Estimation ......... 360
11.4.Applications to Power Network ............................ 362
11.4.1.State Estimation in Power Systems ................. 362
11.4.1.1.WLSs Estimator ........................... 364
11.4.2.Statistical Properties of State Estimator
Outputs ........................................... 366
11.4.2.1.Decoupled WLS and DC Models .............. 367
11.4.2.2.Including Equality Constraints ........... 368
11.4.2.3.Necessary Solution Conditions ............ 371
11.4.3.Model Parameter Identification—Sources
of Inaccuracy ..................................... 371
11.4.4.State Estimation in Deregulated Environment ....... 371
11.4.4.1.Network Real-Time Modeling ............... 372
11.4.4.2.Impact of the Changing Marketplace ....... 372
11.5.Illustrative Examples .................................... 373
11.6.Conclusion ............................................... 375
11.7.Problem Set .............................................. 376
References ............................................... 380
Chapter 12 Optimal Power Flow ................................. 383
12.1.Introduction ............................................. 383
12.2.OPF—Fuel Cost Minimization ............................... 386
12.2.1.Modeling Issues ................................... 386
12.2.2.Mathematical Description of the Objective
Functions and Constraints for Cost Minimization ... 387
12.3.OPF—Active Power Loss Minimization ....................... 389
12.3.1.Modeling Issues for Loss Minimization ............. 390
12.3.2.Mathematical Description of the Objective
Functions and Constraints for Loss Minimization ... 391
12.4.OPF—VAr Planning ......................................... 393
12.4.1.Modeling Issues for VAr Planning Type I Problem ... 395
12.4.2.Mathematical Description of the Objective and
Constraints for Type I Problem for VAr Planning ... 396
12.4.3.Type II Problem for VAr Planning .................. 397
12.4.3.1.Control Variables ........................ 397
12.4.3.2.Constraints .............................. 398
12.4.3.3.Assumptions .............................. 398
12.4.4.Mathematical Description of the Objective
and Constraints for Type II Problem for VAr
Planning .......................................... 398
12.4.4.1.Mathematical Notation .................... 399
12.4.4.2.Mathematical Description of VAr
Planning ................................. 399
12.5.OPF—Adding Environmental Constraints ..................... 402
12.5.1.Modeling Issues for Environmental Constraint ...... 402
12.6.Commonly Used Optimization Technique in Linear
Programming (LP) ......................................... 403
12.6.1.LP ................................................ 404
12.6.1.1.Definition of LP Problem Structure ....... 406
12.6.1.2.LP Iteration ............................. 406
12.6.1.3.Selection of Variable to Enter Basis ..... 407
12.6.2.LP Applications in OPF ............................ 408
12.6.3.Interior Point .................................... 410
12.6.3.1.OPF Formulation (Method II) .............. 411
12.7.Commonly Used Optimization Techniques in Nonlinear
Programming .............................................. 415
12.7.1.NLP ............................................... 415
12.7.1.1.Finding the Descent Direction ............ 416
12.7.1.2.Finding the Step Length .................. 416
12.7.1.3.Treatment of the Constraints ............. 417
12.7.2.Sequential Quadratic Programming (SQP) ............ 418
12.7.3.Augmented Lagrangian Methods ...................... 420
12.7.4.Generalized Reduced Gradients ..................... 420
12.7.4.1.OPF Formulation Using QP Reduced
Gradient Method .......................... 422
12.7.5.Projected Augmented Lagrangian .................... 425
12.7.6.Discussion on Nonlinear OPF Algorithms ............ 426
12.7.6.1.Decomposition Strategies ................. 427
12.7.6.2.Adding Security Constraints .............. 427
12.8.Illustrative Examples .................................... 428
12.9.Conclusions .............................................. 434
12.10.Problem Set ............................................. 435
References ............................................... 437
Chapter 13 Pricing ............................................ 441
13.1.Introduction ............................................. 441
13.2.Marginal Pricing ......................................... 442
13.3.Marginal Costing ......................................... 443
13.4.Marginal Revenue ......................................... 444
13.5.Pricing Policies for Regulated Systems and Markets ....... 445
13.6.Pricing Methods .......................................... 446
13.6.1.Megawatt-Mile (MWM) Method ........................ 447
13.6.2.Modulus Method (MM) or Usage Method ............... 447
13.6.3.Zero Counterflow Method (ZCM) ..................... 448
13.6.4.Dominant Flow Method (DFM) ........................ 448
13.6.5.Alternative Pricing Methods ....................... 449
13.7.Economic Basis of Shadow Prices in Linear Programming
(LP) ..................................................... 449
13.7.1.Special Case of LP Problems with Two-Sided
Bounded Variables ................................. 451
13.7.2.Further Interpretation of Dual Shadow Prices
Variables ......................................... 451
13.8.LMP ...................................................... 452
13.8.1.Components of LMP ................................. 452
13.8.2.LMP in Energy Markets ............................. 455
13.8.2.1.Formulation for NLP Approximations ....... 455
13.8.2.2.Formulation for LP-Based OPF ............. 456
13.8.3.Computational Steps for LMP Using DC OPF .......... 457
13.8.4.Transmission Congestion Charges (TCCs) ............ 461
13.9.Alternative OPF Formulation for Pricing Using Duality
in LP .................................................... 462
13.9.1.Linearization of the OPF .......................... 463
13.9.2.LP Dual Construct ................................. 465
References ............................................... 467
Chapter 14 Unit Commitment .................................... 469
14.1.Introduction ............................................. 469
14.2.Formulation of Unit Commitment ........................... 471
14.2.1.Reserve Constraints ............................... 471
14.2.2.Modeling in Unit Commitment ....................... 472
14.2.3.Lagrangian Function for Unit Commitment ........... 473
14.3.Optimization Methods ..................................... 474
14.3.1.Priority List Unit Commitment Schemes ............. 474
14.3.2.Priority Criteria ................................. 475
14.3.2.1.Type I: Fuel Cost-Based Lists ............ 475
14.3.2.2.Type II: Incremental Fuel Cost-Based
List ..................................... 476
14.3.2.3.Type III: Incremental Fuel Cost with
Start-Up Cost-Based List ................. 476
14.3.2.4.Type IV: Dynamic Priority Lists .......... 477
14.3.3.Simple Merit-Order Scheme ......................... 477
14.4.Illustrative Example ..................................... 478
14.4.1.Lagrangian Relaxation Approach to Unit
Commitment ........................................ 478
14.4.2.Single Unit Relaxed Problem ....................... 480
14.4.3.Lagrangian Relaxation Procedure ................... 483
14.4.4.Searching for a Feasible Solution ................. 486
14.5.Updating λn(t) in the Unit Commitment Problem ............ 489
14.5.1.Case A: Updating λn(t) ............................ 489
14.5.2.Case B: Updating λn(t) ............................ 491
14.6.Unit Commitment of Thermal Units Using Dynamic
Programming .............................................. 493
14.6.1.Dynamic Programming Approaches to Unit
Commitment Problem ................................ 494
14.6.1.1.Backward Dynamic Programming Approach .... 494
14.6.1.2.Forward Dynamic Programming Approach ..... 494
14.6.2.Case Study ........................................ 496
14.7.Illustrative Problems .................................... 501
14.8.Conclusions .............................................. 503
14.9.Problems ................................................. 504
References ............................................... 507
Chapter 15 Genetic Algorithms ................................. 509
15.1.Introduction ............................................. 509
15.1.1.General Structure of GAs .......................... 509
15.2.Definition and Concepts Used in Genetic Computation ...... 510
15.2.1.Evolutionary Algorithms ........................... 510
15.2.2.Genetic Programming ............................... 511
15.3.GA Approach .............................................. 512
15.3.1.GA Operators ...................................... 512
15.3.2.Major Advantages .................................. 513
15.3.3.Advantages of GAs over Traditional Methods ........ 514
15.4.Theory of GAs ............................................ 514
15.4.1.Continuous and Discrete Variables ................. 514
15.4.2.Constraints ....................................... 514
15.4.3.Multiobjective Decision Problems .................. 515
15.4.4.Other GA Variants ................................. 515
15.4.5.Coding ............................................ 516
15.4.6.Fitness ........................................... 516
15.4.7.Selection ......................................... 516
15.4.8.Crossover ......................................... 517
15.4.9.Parameters ........................................ 517
15.5.Schemata Theorem ......................................... 517
15.6.General Algorithm of GAs ................................. 520
15.7.Application of GAs ....................................... 521
15.7.1.Control System Engineering ........................ 521
15.7.2.Timetabling ....................................... 521
15.7.3.Job-Shop Scheduling ............................... 521
15.7.4.Management Sciences ............................... 522
15.7.5.Game Playing ...................................... 522
15.8.Application to Power Systems ............................. 522
15.8.1.GAs in the Unit Commitment Problem ................ 523
15.8.1.1.UCP Statement ............................ 524
15.8.1.2.GA Implementation in the GTS Algorithm ... 525
15.8.1.3.Proposed Algorithm ....................... 527
15.8.2.Load Shedding: A Model with GA .................... 531
15.8.2.1.Coding ................................... 533
15.8.2.2.Fitness .................................. 533
15.8.2.3.Initial Population ....................... 533
15.8.2.4.Genetic Operators ........................ 534
15.9.Illustrative Examples .................................... 534
15.10.Conclusions ............................................. 535
15.11.Problem Set ............................................. 536
References ............................................... 537
Chapter 16 Functional Optimization, Optimal Control,
and Adaptive Dynamic Programming ................... 539
16.1.System Performance Evaluation and Optimization
of Functionals ........................................... 539
16.1.1.Extremization of Functionals ...................... 539
16.1.2.Performance Measure ............................... 540
16.1.3.Theorems of Optimization of Constrained
Functionals ....................................... 541
16.1.4.Summary of Procedure for Optimizing Constrained
Functionals ....................................... 544
16.2.Solving the Optimal Control Problem ...................... 545
16.2.1.Continuous Optimum Principle ...................... 548
16.2.2.Formulation of the Problem ........................ 549
16.2.3.Theorems for the Pontryagin Maximum Principle
(PMP) ............................................. 551
16.2.4.Sufficiency Test and Some Special Cases
for the Optimum Principle ......................... 552
16.2.5.Use of the Optimum Principle for Special
Control Problems .................................. 554
16.2.6.Regulator Problem and Riccati Equation ............ 556
16.3.Selected Methods of Determining the Control Functions
for Convergence of Optimum Principle ..................... 558
16.3.1.Dynamic Programming Method ........................ 559
16.3.2.Principle of Optimality Is Used to Find u&42;(t)
(Richard Bellman's Method) ........................ 559
16.3.3.Relationship between Dynamic Programming
and the Minimum Principle ......................... 562
16.3.4.Section Summary ................................... 565
16.4.Adaptive Critics Design (ACD) and ADP .................... 565
16.4.1.Background to Complex Intelligent Networks ........ 565
16.4.2.From DP to Adaptive or "Approximate" Dynamic
Programming (ADP) ................................. 567
16.4.3.Critic Network Variants ........................... 569
16.5.Architecture of ACDs ..................................... 573
16.5.1.Critic Networks ................................... 574
16.5.2.Action Networks ................................... 575
16.5.3.ACDs Comparative Studies .......................... 576
16.5.4.Summary ........................................... 577
16.6.Typical Architectures of Variants or ADP (Critics
Illustrations) ........................................... 577
16.7.Applications of DSOPF to Power Systems Problems .......... 581
References ............................................... 593
Index ......................................................... 595
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