Preface xvii
About the Companion Website xxi
1 Introduction 1
1.1 Electrification 1
1.2 Generation, Transmission, and Distribution Systems 2
1.2.1 Central Generating Station Model 2
1.2.2 Renewable Generation 4
1.2.3 Smart Grids 5
1.3 Time Scales 5
1.3.1 Dynamic Phenomena 5
1.3.2 Measurements and Data 5
1.3.3 Control Functions and System Operation 7
1.4 Organization of the Book 7
Part I System Concepts 9
2 Steady-State Power Flow 11
2.1 Introduction 11
2.2 Power Network Elements and Admittance Matrix 12
2.2.1 Transmission Lines 12
2.2.2 Transformers 13
2.2.3 Per Unit Representation 14
2.2.4 Building the Network Admittance Matrix 14
2.3 Active and Reactive Power Flow Calculations 16
2.4 Power Flow Formulation 19
2.5 Newton-Raphson Method 21
2.5.1 General Procedure 21
2.5.2 NR Solution of Power Flow Equations 22
2.6 Advanced Power Flow Features 27
2.6.1 Load Bus Voltage Regulation 27
2.6.2 Multi-area Power Flow 28
2.6.3 Active Line Power Flow Regulation 29
2.6.4 Dishonest Newton-Raphson Method 30
2.6.5 Fast Decoupled Loadflow 30
2.6.6 DC Power Flow 31
2.7 Summary and Notes 31
Appendix 2.A Two-winding Transformer Model 32
Appendix 2.B LU Decomposition and Sparsity Methods 36
Appendix 2.C Power Flow and Dynamic Data for the 2-area, 4-machine System 39
Problems 42
3 Steady-State Voltage Stability Analysis 47
3.1 Introduction 47
3.2 Voltage Collapse Incidents 48
3.2.1 Tokyo, Japan: July 23, 1987 48
3.2.2 US Western Power System: July 2, 1996 48
3.3 Reactive Power Consumption on Transmission Lines 49
3.4 Voltage Stability Analysis of a Radial Load System 55
3.4.1 Maximum Power Transfer 59
3.5 Voltage Stability Analysis of Large Power Systems 61
3.6 Continuation Power Flow Method 64
3.6.1 Continuation Power Flow Algorithm 66
3.7 An AQ-Bus Method for Solving Power Flow 67
3.7.1 Analytical Framework for the AQ-Bus Method 69
3.7.2 AQ-Bus Formulation for Constant-Power-Factor Loads 70
3.7.3 AQ-Bus Algorithm for Computing Voltage Stability Margins 71
3.8 Power System Components Affecting Voltage Stability 73
3.8.1 Shunt Reactive Power Supply 74
3.8.2 Under-Load Tap Changer 76
3.9 Hierarchical Voltage Control 79
3.10 Voltage Stability Margins and Indices 80
3.10.1 Voltage Stability Margins 80
3.10.2 Voltage Sensitivities 81
3.10.3 Singular Values and Eigenvalues of the Power Flow Jacobian Matrix 82
3.11 Summary and Notes 82
Problems 83
4 Power System Dynamics and Simulation 87
4.1 Introduction 87
4.2 Electromechanical Model of Synchronous Machines 88
4.3 Single-Machine Infinite-Bus System 90
4.4 Power System Disturbances 94
4.4.1 Fault-On Analysis 94
4.4.2 Post-Fault Analysis 96
4.4.3 Other Types of Faults 98
4.5 Simulation Methods 98
4.5.1 Modified Euler Methods 99
4.5.1.1 Euler Full-Step Modification Method 100
4.5.1.2 Euler Half-Step Modification Method 101
4.5.2 Adams-Bashforth Second-Order Method 101
4.5.3 Selecting Integration Stepsize 102
4.5.4 Implicit Integration Methods 104
4.5.4.1 Integration of DAEs 105
4.6 Dynamic Models of Multi-Machine Power Systems 106
4.6.1 Constant-Impedance Loads 107
4.6.2 Generator Current Injections 108
4.6.3 Network Equation Extended to the Machine Internal Node 108
4.6.4 Reduced Admittance Matrix Approach 109
4.6.5 Method for Dynamic Simulation 109
4.7 Multi-Machine Power System Stability 114
4.7.1 Reference Frames for Machine Angles 115
4.8 Power System Toolbox 117
4.9 Summary and Notes 119
Problems 119
5 Direct Transient Stability Analysis 123
5.1 Introduction 123
5.2 Equal-Area Analysis of a Single-Machine Infinite-Bus System 124
5.2.1 Power-Angle Curve 124
5.2.2 Fault-On and Post-Fault Analysis 126
5.3 Transient Energy Functions 127
5.3.1 Lyapunov Functions 128
5.3.2 Energy Function for Single-Machine Infinite-Bus Electromechanical Model 128
5.4 Energy Function Analysis of a Disturbance Event 131
5.5 Single-Machine Infinite-Bus Model Phase Portrait and Region of Stability 135
5.6 Direct Stability Analysis using Energy Functions 138
5.7 Energy Functions for Multi-Machine Power Systems 139
5.7.1 Direct Stability Analysis for Multi-Machine Systems 142
5.7.2 Computation of Critical Energy 143
5.8 Dynamic Security Assessment 146
5.9 Summary and Notes 146
Problems 147
6 Linear Analysis and Small-Signal Stability 149
6.1 Introduction 149
6.2 Electromechanical Modes 150
6.3 Linearization 151
6.3.1 State-Space Models 151
6.3.2 Input-Output Models 152
6.3.3 Modal Analysis and Time-Domain Solutions 152
6.3.4 Time Response of Linear Systems 154
6.3.5 Participation Factors 156
6.4 Linearized Models of Single-Machine Infinite-Bus Systems 157
6.5 Linearized Models of Multi-Machine Systems 160
6.5.1 Synchronizing Torque Matrix and Eigenvalue Properties 162
6.5.2 Modeshapes and Participation Factors 162
6.6 Developing Linearized Models of Large Power Systems 164
6.6.1 Analytical Partial Derivatives 165
6.6.2 Numerical Linearization 169
6.7 Summary and Notes 171
Problems 171
Part II Synchronous Machine Models and their Control Systems 175
7 Steady-State Models and Operation of Synchronous Machines 177
7.1 Introduction 177
7.2 Physical Description 177
7.2.1 Amortisseur Bars 179
7.3 Synchronous Machine Model 179
7.3.1 Flux Linkage and Voltage Equations 181
7.3.2 Stator (Armature) Self and Mutual Inductances 183
7.3.3 Mutual Inductances between Stator and Rotor 183
7.3.4 Rotor Self and Mutual Inductances 184
7.4 Park Transformation 185
7.4.1 Electrical Power in dq0 Variables 188
7.5 Reciprocal, Equal Lad Per-Unit System 189
7.5.1 Stator Base Values 189
7.5.2 Stator Voltage Equations 190
7.5.3 Rotor Base Values 191
7.5.4 Rotor Voltage Equations 191
7.5.5 Stator Flux-Linkage Equations 192
7.5.6 Rotor Flux-Linkage Equations 192
7.5.7 Equal Mutual Inductance 192
7.6 Equivalent Circuits 196
7.6.1 Flux-Linkage Circuits 196
7.6.2 Voltage Equivalent Circuits 197
7.7 Steady-State Analysis 199
7.7.1 Open-Circuit Condition 199
7.7.2 Loaded Condition 201
7.7.3 Drawing Voltage-Current Phasor Diagrams 202
7.8 Saturation Effects 204
7.8.1 Representations of Magnetic Saturation 205
7.9 Generator Capability Curves 207
7.10 Summary and Notes 209
Problems 209
8 Dynamic Models of Synchronous Machines 213
8.1 Introduction 213
8.2 Machine Dynamic Response During Fault 213
8.2.1 DC Offset and Stator Transients 215
8.3 Transient and Subtransient Reactances and Time Constants 216
8.4 Subtransient Synchronous Machine Model 221
8.5 Other Synchronous Machine Models 227
8.5.1 Flux-Decay Model 227
8.5.2 Classical Model 228
8.6 dq-axes Rotation Between a Generator and the System 229
8.7 Power System Simulation using Detailed Machine Models 230
8.7.1 Power System Simulation Algorithm 231
8.8 Linearized Models 232
8.9 Summary and Notes 234
Problems 235
9 Excitation Systems 237
9.1 Introduction 237
9.2 Excitation System Models 238
9.3 Type DC Exciters 239
9.3.1 Separately Excited DC exciter 239
9.3.2 Self-Excited DC Exciter 243
9.3.3 Voltage Regulator 244
9.3.4 Initialization of DC Type Exciters 245
9.3.5 Transfer Function Analysis 246
9.3.6 Generator and Exciter Closed-Loop System 248
9.3.7 Excitation System Response Ratios 251
9.4 Type AC Exciters 252
9.5 Type ST Excitation Systems 254
9.6 Load Compensation Control 257
9.7 Protective Functions 259
9.8 Summary and Notes 259
Appendix 9.A Anti-Windup Limits 260
Problems 261
10 Power System Stabilizers 265
10.1 Introduction 265
10.2 Single-Machine Infinite-Bus System Model 266
10.3 Synchronizing and Damping Torques 271
10.3.1 Te2 Under Constant Field Voltage 272
10.3.2 Te2 With Excitation System Control 273
10.4 Power System Stabilizer Design using Rotor Speed Signal 275
10.4.1 PSS Design Requirements 276
10.4.2 PSS Control Blocks 277
10.4.3 PSS Design Methods 279
10.4.4 Torsional Filters 284
10.4.5 PSS Field Tuning 287
10.4.6 Interarea Mode Damping 287
10.5 Other PSS Input Signals 288
10.5.1 Generator Terminal Bus Frequency 288
10.5.2 Electrical Power Output Pe 288
10.6 Integral-of-Accelerating-Power or Dual-Input PSS 289
10.7 Summary and Notes 293
Problems 293
11 Load and Induction Motor Models 295
11.1 Introduction 295
11.2 Static Load Models 296
11.2.1 Exponential Load Model 296
11.2.2 Polynomial Load Model 297
11.3 Incorporating ZIP Load Models in Dynamic Simulation and Linear Analysis 298
11.4 Induction Motors: Steady-State Models 303
11.4.1 Physical Description 304
11.4.2 Mathematical Description 304
11.4.2.1 Modeling Equations 304
11.4.2.2 Reference Frame Transformation 306
11.4.3 Equivalent Circuits 308
11.4.4 Per-Unit Representation 310
11.4.5 Torque-Slip Characteristics 311
11.4.6 Reactive Power Consumption 313
11.4.7 Motor Startup 314
11.5 Induction Motors: Dynamic Models 315
11.5.1 Initialization 318
11.5.2 Reactive Power Requirement during Motor Stalling 320
11.6 Summary and Notes 323
Problems 324
12 Turbine-Governor Models and Frequency Control 327
12.1 Introduction 327
12.2 Steam Turbines 328
12.2.1 Turbine Configurations 328
12.2.2 Steam Turbine-Governors 331
12.3 Hydraulic Turbines 333
12.3.1 Hydraulic Turbine-Governors 337
12.3.2 Load Rejection of Hydraulic Turbines 338
12.4 Gas Turbines and Co-Generation Plants 339
12.5 Primary Frequency Control 342
12.5.1 Isolated Turbine-Generator Serving Local Load 343
12.5.2 Interconnected Units 347
12.5.3 Frequency Response in US Power Grids 349
12.6 Automatic Generation Control 351
12.7 Turbine-Generator Torsional Oscillations and Subsynchronous Resonance 356
12.7.1 Torsional Modes 356
12.7.2 Electrical Network Modes 363
12.7.3 SSR Occurrence and Countermeasures 365
12.8 Summary and Notes 366
Problems 367
Part III Advanced Power System Topics 371
13 High-Voltage Direct Current Transmission Systems 373
13.1 Introduction 373
13.1.1 HVDC System Installations and Applications 375
13.1.2 HVDC System Economics 377
13.2 AC/DC and DC/AC Conversion 377
13.2.1 AC-DC Conversion using Ideal Diodes 378
13.2.2 Three-Phase Full-Wave Bridge Converter 379
13.3 Line-Commutation Operation in HVDC Systems 383
13.3.1 Rectifier Operation 383
13.3.1.1 Thyristor Ignition Delay Angle 383
13.3.1.2 Commutation Overlap 385
13.3.2 Inverter Operation 388
13.3.3 Multiple Bridge Converters 389
13.3.4 Equivalent Circuit 389
13.4 Control Modes 391
13.4.1 Mode 1: Normal Operation 392
13.4.2 Mode 2: Reduced-Voltage Operation 393
13.4.3 Mode 3: Transitional Mode 394
13.4.4 System Operation Under Fault Conditions 396
13.4.5 Communication Requirements 396
13.5 Multi-terminal HVDC Systems 397
13.6 Harmonics and Reactive Power Requirement 398
13.6.1 Harmonic Filters 398
13.6.2 Reactive Power Support 399
13.7 AC-DC Power Flow Computation 401
13.8 Dynamic Models 406
13.8.1 Converter Control 406
13.8.2 DC Line Dynamics 408
13.8.3 AC-DC Network Solution 409
13.9 Damping Control Design 411
13.10 Summary and Notes 416
Problems 416
14 Flexible AC Transmission Systems 421
14.1 Introduction 421
14.2 Static Var Compensator 422
14.2.1 Circuit Configuration and Thyristor Switching 422
14.2.2 Steady-State Voltage Regulation and Stability Enhancement 423
14.2.2.1 Voltage Stability Enhancement 424
14.2.2.2 Transient Stability Enhancement 427
14.2.3 Dynamic Voltage Control and Droop Regulation 429
14.2.4 Dynamic Simulation 433
14.2.5 Damping Control Design using SVC 435
14.3 Thyristor-Controlled Series Compensator 441
14.3.1 Fixed Series Compensation 442
14.3.2 TCSC Circuit Configuration and Switching 442
14.3.3 Voltage Reversal Control 444
14.3.4 Mitigation of Subsynchronous Oscillations 445
14.3.5 Dynamic Model and Damping Control Design 446
14.4 Shunt VSC Controllers 451
14.4.1 Voltage-Sourced Converters 451
14.4.1.1 Three-Phase Full-Wave VSCs 453
14.4.1.2 Three-Level Converters 455
14.4.1.3 Harmonics 455
14.4.2 Static Compensator 458
14.4.2.1 Steady-State Analysis 458
14.4.2.2 Dynamic Model 459
14.4.3 VSC HVDC Systems 463
14.4.3.1 Steady-State Operation 463
14.4.3.2 Dynamic Model 466
14.5 Series and Coupled VSC Controllers 469
14.5.1 Static Synchronous Series Compensation 469
14.5.1.1 Steady-State Analysis 469
14.5.2 Unified Power Flow Controller 471
14.5.2.1 Steady-State Analysis 471
14.5.3 Interline Power Flow Controller 475
14.5.3.1 Steady-State Analysis 475
14.5.4 Dynamic Model 478
14.5.4.1 Series Voltage Insertion 479
14.5.4.2 Line Active and Reactive Power Flow Control 480
14.6 Summary and Notes 480
Problems 481
15 Wind Power Generation and Modeling 487
15.1 Background 487
15.2 Wind Turbine Components 489
15.3 Wind Power 491
15.3.1 Blade Angle Orientation 492
15.3.2 Power Coefficient 494
15.4 Wind Turbine Types 496
15.4.1 Type 1 496
15.4.2 Type 2 497
15.4.3 Type 3 498
15.4.4 Type 4 498
15.5 Steady-State Characteristics 499
15.5.1 Type-1Wind Turbine 499
15.5.2 Type-2Wind Turbine 501
15.5.3 Type-3Wind Turbine 502
15.6 Wind Power Plant Representation 505
15.7 Overall Control Criteria for Variable-Speed Wind Turbines 510
15.8 Wind Turbine Model for Transient Stability Planning Studies 513
15.8.1 Overall Model Structure 513
15.8.2 Generator/Converter Model 514
15.8.3 Electrical Control Model 515
15.8.4 Drive-Train Model 517
15.8.5 Torque Control Model 519
15.8.6 Aerodynamic Model 520
15.8.7 Pitch Controller 522
15.9 Plant-Level Control Model 526
15.9.1 Simulation Example 526
15.10 Summary and Notes 527
Problems 528
16 Power System Coherency and Model Reduction 531
16.1 Introduction 531
16.2 Interarea Oscillations and Slow Coherency 532
16.2.1 Slow Coherency 534
16.2.2 Slow Coherent Areas 536
16.2.3 Finding Coherent Groups of Machines 541
16.3 Generator Aggregation and Network Reduction 544
16.3.1 Generator Aggregation 545
16.3.2 Dynamic Aggregation 548
16.3.3 Load Bus Elimination 551
16.4 Simulation Studies 555
16.4.1 Singular Perturbations Method 556
16.5 Linear Reduced Model Methods 557
16.5.1 Modal Truncation 558
16.5.2 Balanced Model Reduction Method 559
16.6 Dynamic Model Reduction Software 559
16.7 Summary and Notes 560
Problems 560
References 563
Index 577
