This book is developed from very basic electrical engineering concepts and builds them step-by-step to the final concept of power flow control in complex transmission systems. Chapter 1 explains the origin of modern power flow controllers and its evolution in the last three decades. Chapter 2 explains various topics of the book in simple engineering terms and corroborates the theory with relevant mathematics. The characteristic equations of various power flow controllers, including their equivalent compensating impedances, are developed. Chapter 3 gives the “nuts-and-bolts” of modeling in the Electromagnetic Transients Program (EMTP) and explains the basic differences of modeling various Power Flow Controllers (PFCs). Chapter 4 describes the transformer-based PFCs and sets some baseline examples for comparison with other PFCs in the following chapters. Chapter 5 describes some early PFCs that use mechanical switches and sets some baselines for comparison in the following chapters. Chapter 6 presents the motive and evolution of the Sen Transformer and its wide variety of applications.

Kalyan K. Sen, PhD, PE (PA & NY), MBA, IEEE Fellow, is President and Chief Technology Officer at Sen Engineering Solutions, Inc. He was a key member of the FACTS development team at Westinghouse Science & Technology Center, where he developed some of the basic concepts of FACTS technology. He is an IEEE Distinguished Lecturer, and is the co-author of ntroduction to FACTS Controllers: Theory, Modeling, and Applications.

Mey Ling Sen, MEE, IEEE Member, is Chief Operating Officer at Sen Engineering Solutions, Inc. Previously, she was a consultant engineer at the Westinghouse Electro-Mechanical Division Technology Center. Ms. Sen is the co-inventor of the Sen transformer, which is the most efficient, reliable, and cost-effective SMART power flow controller (SPFC).

Authors’ Biographies xiii

Foreword xv

Nomenclature xix

Preface xxv

Acknowledgments xxix

About the Companion Website xxxi

1 Smart Controllers 1

1.1 Why is a Power Flow Controller Needed? 1

1.2 Traditional Power Flow Control Concepts 5

1.3 Modern Power Flow Control Concepts 14

1.4 Cost of a Solution 22

1.4.1 Defining a Cost-Effective Solution 22

1.4.2 Payback Time 24

1.4.3 Economic Analysis 24

1.5 Independent Active and Reactive PFCs 26

1.6 SMART Power Flow Controller (SPFC) 39

1.6.1 Example of an SPFC 40

1.6.2 Justification 41

1.6.3 Additional Information 41

1.7 Discussion 42

2 Power Flow Control Concepts 45

2.1 Power Flow Equations for a Natural or Uncompensated Line 60

2.2 Power Flow Equations for a Compensated Line 63

2.2.1 Shunt-Compensating Voltage 67

2.2.1.1 Power Flow at the Modified Sending End with a Shunt-Compensating Voltage 70

2.2.1.2 Power Flow at the Receiving End with a Shunt-Compensating Voltage 73

2.2.1.3 Exchanged Power by a Shunt-Compensating Voltage 79

2.2.1.4 Representation of a Shunt-Compensating Voltage as a Shunt-Compensating Impedance 79

2.2.2 Series-Compensating Voltage as an Impedance Regulator, Voltage Regulator, and Phase Angle Regulator (Asymmetric) 80

2.2.2.1 Power Flow at the Sending End with a Series-Compensating Voltage 92

2.2.2.2 Power Flow at the Receiving End with a Series-Compensating Voltage 95

2.2.2.3 Power Flow at the Modified Sending End with a Series-Compensating Voltage 100

2.2.2.4 Exchanged Power by a Series-Compensating Voltage 109

2.2.2.5 Additional Series-Compensating Voltages 126

2.2.2.5.1 Phase Angle Regulator (Symmetric) 126

2.2.2.5.2 Reactance Regulator 129

2.2.2.5.2.1 Reactance Control Method 137

2.2.2.5.2.2 Voltage Control Method 139

2.2.2.6 Representation of a Series-Compensating Voltage as a Series-Compensating Impedance 145

2.2.2.6.1 Equivalent Impedance of a Voltage Regulator (VR) 152

2.2.2.6.2 Equivalent Impedance of a Phase Angle Regulator (Asymmetric) 154

2.2.2.6.3 Equivalent Impedance of a Phase Angle Regulator (Symmetric) 157

2.2.2.6.4 Equivalent Impedance of a Reactance Regulator 160

2.2.3 Comparison Between Series- and Shunt-Compensating Voltages 165

2.3 Implementation of Power Flow Control Concepts 168

2.3.1 Voltage Regulation 168

2.3.1.1 Direct Method 168

2.3.1.2 Indirect Method 170

2.3.2 Phase Angle Regulation 173

2.3.2.1 Single-core Phase Angle Regulator 173

2.3.2.2 Dual-core Phase Angle Regulator 176

2.3.3 Series Reactance Regulation 178

2.3.3.1 Direct Method 178

2.3.3.2 Indirect Method 178

2.3.4 Impedance Regulation 179

2.3.4.1 Unified Power Flow Controller (UPFC) 181

2.3.4.2 Sen Transformer (ST) 183

2.4 Interline Power Flow Concept 185

2.4.1 Back-to-Back SSSC 186

2.4.2 Multiline Sen Transformer (MST) 188

2.4.3 Back-to-Back STATCOM 192

2.4.4 Generalized Power Flow Controller 194

2.5 Figure of Merits Among Various PFCs 196

2.5.1 VR 196

2.5.2 PAR (sym) 196

2.5.3 PAR (asym) 198

2.5.4 RR 202

2.5.5 IR 204

2.5.6 RPI, LI, and APR of a PFC 206

2.6 Comparison Between Shunt-Compensating Reactance and Series-Compensating Reactance 228

2.6.1 Shunt-Compensating Reactance 230

2.6.1.1 Restoration of Voltage at the Midpoint of the Line 230

2.6.1.2 Restoration of Voltage at the One-Third and Two-Third Points of the Line 232

2.6.1.3 Restoration of Voltage at the One-Fourth, Half, and Three-Fourth Points of the Line 233

2.6.1.4 Restoration of Voltage at n Points of the Line 235

2.6.2 Series-Compensating Reactance 239

2.7 Calculation of RPI, LI, and APR for a PAR (sym), a PAR (asym), a RR, and an IR in a Lossy Line 242

2.7.1 PAR (sym) 245

2.7.2 PAR (asym) 246

2.7.3 RR 248

2.7.4 IR 249

2.8 Sen Index of a PFC 253

3 Modeling Principles 255

3.1 The Modeling in EMTP 255

3.1.1 A Single-Generator/Single-Line Model 259

3.1.2 A Two-Generator/Single-Line Model 264

3.2 Vector Phase-Locked Loop (VPLL) 277

3.3 Transmission Line Steady-State Resistance Calculator 280

3.4 Simulation of an Independent PFC, Integrated in a Two-Generator/Single-Line Power System Network 281

4 Transformer-Based Power Flow Controllers 297

4.1 Voltage-Regulating Transformer (VRT) 297

4.1.1 Voltage Regulating Transformer (Shunt-Series Configuration) 298

4.1.2 Two-Winding Transformer 315

4.2 Phase Angle Regulator (PAR) 322

4.2.1 PAR (Asymmetric) 322

4.2.2 PAR (Symmetric) 332

5 Mechanically-Switched Voltage Regulators and Power Flow Controllers 341

5.1 Shunt Compensation 341

5.1.1 Mechanically-Switched Capacitor (MSC) 341

5.1.2 Mechanically-Switched Reactor (MSR) 353

5.2 Series Compensation 354

5.2.1 Mechanically-Switched Reactor (MSR) 354

5.2.2 Mechanically-Switched Capacitor (MSC) with a Reactor 363

5.2.3 Series Reactance Emulator 369

6 Sen Transformer 375

6.1 Existing Solutions 377

6.1.1 Voltage Regulation 383

6.1.2 Phase Angle Regulation 385

6.2 Desired Solution 386

6.2.1 ST as a New Voltage Regulator 389

6.2.2 ST as an Independent PFC 392

6.2.3 Control of ST 394

6.2.3.1 Impedance Emulation 395

6.2.3.2 Resistance Emulation 396

6.2.3.3 Reactance Emulation 396

6.2.3.4 Closed-Loop Power Flow Control 397

6.2.3.5 Open-Loop Power Flow Control 398

6.2.4 Simulation of ST Integrated in a Two-Generator/One-Line Power System Network 425

6.2.5 Simulation of ST Integrated in a Three-Generator/Four-Line Power System Network 439

6.2.6 Testing of ST 453

6.2.7 Limited-Angle Operation of ST 485

6.2.8 ST Using LTCs with Lower Current Rating 498

6.2.9 ST with a Two-Core Design 501

6.3 Comparison Among the VRT, PAR, UPFC, and ST 510

6.3.1 Power Flow Enhancement 510

6.3.2 Speed of Operation 511

6.3.3 Losses 512

6.3.4 Switch Rating 512

6.3.5 Magnetic Circuit Design 513

6.3.6 Optimization of Transformer Rating 513

6.3.7 Harmonic Injection into the Power System Network 515

6.3.8 Operation During Line Faults 515

6.4 Multiline Sen Transformer 516

6.4.1 Basic Differences Between the MST and BTB-SSSC 519

6.5 Flexible Operation of the ST 520

6.6 ST with a Shunt-Compensating Voltage 522

6.7 Limited Angle Operation of the ST with Shunt-Compensating Voltages 526

6.8 MST with Shunt-Compensating Voltages 531

6.9 Generalized Sen Transformer 532

6.10 Summary 533

Appendix A Miscellaneous 535

A.1 Three-Phase Balanced Voltage, Current, and Power 535

A.2 Symmetrical Components 538

A.3 Separation of Positive-, Negative-, and Zero-Sequence Components in a Multiple Frequency Composite Variable 544

A.4 Three-Phase Unbalanced Voltage, Current, and Power 547

A.5 d-q Transformation (3-Phase System, Transformed into d-q axes; d-axis Is the Active Component and q-axis Is the Reactive Component) 551

A.5.1 Conversion of a Variable Containing Positive-, Negative-, and Zero-Sequence Components into d-q Frame 556

A.5.2 Calculation of Instantaneous Power into d-q Frame 560

A.5.3 Calculation of Instantaneous Power into d-q frame for a Three-Phase, Three-Wire System 560

A.6 Fourier Analysis 566

A.7 Adams-Bashforth Numerical Integration Formula 569

Appendix B Power Flow Equations in a Lossy Line 571

B.1 Power Flow Equations for a Natural or Uncompensated Line 575

B.2 Power Flow Equations for a Compensated Line 582

B.2.1 Shunt-Compensating Voltage 583

B.2.1.1 Power Flow at the Modified Sending End with a Shunt-Compensating Voltage 584

B.2.1.2 Power Flow at the Receiving End with a Shunt-Compensating Voltage 587

B.2.1.3 Exchanged Power by a Shunt-Compensating Voltage 590

B.2.1.4 Representation of a Shunt-Compensating Voltage as a Shunt-Compensating Impedance 590

B.2.2 Series-Compensating Voltage as an Impedance Regulator, Voltage Regulator, and Phase Angle Regulator (Asymmetric) 591

B.2.2.1 Power Flow at the Sending End with a Series-Compensating Voltage 596

B.2.2.2 Power Flow at the Receiving End with a Series-Compensating Voltage 600

B.2.2.3 Power Flow at the Modified Sending End with a Series-Compensating Voltage 606

B.2.2.4 Exchanged Power by a Series-Compensating Voltage 615

B.2.2.5 Additional Series-Compensating Voltages 624

B.2.2.5.1 Phase Angle Regulator (Symmetric) 624

B.2.2.5.2 Reactance Regulator 628

B.2.2.6 Representation of a Series-Compensating Voltage as a Series-Compensating Impedance 631

B.2.2.6.1 Equivalent Impedance of a Voltage Regulator (VR) 635

B.2.2.6.2 Equivalent Impedance of a Phase Angle Regulator (Asymmetric) 636

B.2.2.6.3 Equivalent Impedance of a Phase Angle Regulator (Symmetric) 638

B.2.2.6.4 Equivalent Impedance of a Reactance Regulator 640

B.2.2.7 RPI, LI, and APR of a PFC 640

B.3 Descriptions of the Examples in Chapter 2 644

Appendix C Modeling of the Sen Transformer in PSS^®E 647

C.1 Sen Transformer 647

C.2 Modeling with Two Transformers in Series 648

C.3 Relating the Sen Transformer with the PSSE^®E Model 649

C.4 Chilean Case Study 650

C.5 Limitations – PSS^®E Two-Transformer Model 654

C.6 Conclusion 655

References 657

Index 669