Power System Modeling, Computation, and Control

JOE H. CHOW (周祖康), PHD, FIEEE, NAE, is Institute Professor of Electrical, Computer, and Systems Engineering at Rensselaer Polytechnic Institute, Troy, NY, USA. JUAN J. SANCHEZ-GASCA, PHD, FIEEE, is a Technical Director at GE Energy Consulting, Schenectady, NY, USA.

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 ΔTe2 Under Constant Field Voltage 272

10.3.2 ΔTe2 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 ΔPe 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