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 xviiAbout the Companion Website xxi1 Introduction 11.1 Electrification 11.2 Generation, Transmission, and Distribution Systems 21.2.1 Central Generating Station Model 21.2.2 Renewable Generation 41.2.3 Smart Grids 51.3 Time Scales 51.3.1 Dynamic Phenomena 51.3.2 Measurements and Data 51.3.3 Control Functions and System Operation 71.4 Organization of the Book 7Part I System Concepts 92 Steady-State Power Flow 112.1 Introduction 112.2 Power Network Elements and Admittance Matrix 122.2.1 Transmission Lines 122.2.2 Transformers 132.2.3 Per Unit Representation 142.2.4 Building the Network Admittance Matrix 142.3 Active and Reactive Power Flow Calculations 162.4 Power Flow Formulation 192.5 Newton-Raphson Method 212.5.1 General Procedure 212.5.2 NR Solution of Power Flow Equations 222.6 Advanced Power Flow Features 272.6.1 Load Bus Voltage Regulation 272.6.2 Multi-area Power Flow 282.6.3 Active Line Power Flow Regulation 292.6.4 Dishonest Newton-Raphson Method 302.6.5 Fast Decoupled Loadflow 302.6.6 DC Power Flow 312.7 Summary and Notes 31Appendix 2.A Two-winding Transformer Model 32Appendix 2.B LU Decomposition and Sparsity Methods 36Appendix 2.C Power Flow and Dynamic Data for the 2-area, 4-machine System 39Problems 423 Steady-State Voltage Stability Analysis 473.1 Introduction 473.2 Voltage Collapse Incidents 483.2.1 Tokyo, Japan: July 23, 1987 483.2.2 US Western Power System: July 2, 1996 483.3 Reactive Power Consumption on Transmission Lines 493.4 Voltage Stability Analysis of a Radial Load System 553.4.1 Maximum Power Transfer 593.5 Voltage Stability Analysis of Large Power Systems 613.6 Continuation Power Flow Method 643.6.1 Continuation Power Flow Algorithm 663.7 An AQ-Bus Method for Solving Power Flow 673.7.1 Analytical Framework for the AQ-Bus Method 693.7.2 AQ-Bus Formulation for Constant-Power-Factor Loads 703.7.3 AQ-Bus Algorithm for Computing Voltage Stability Margins 713.8 Power System Components Affecting Voltage Stability 733.8.1 Shunt Reactive Power Supply 743.8.2 Under-Load Tap Changer 763.9 Hierarchical Voltage Control 793.10 Voltage Stability Margins and Indices 803.10.1 Voltage Stability Margins 803.10.2 Voltage Sensitivities 813.10.3 Singular Values and Eigenvalues of the Power Flow Jacobian Matrix 823.11 Summary and Notes 82Problems 834 Power System Dynamics and Simulation 874.1 Introduction 874.2 Electromechanical Model of Synchronous Machines 884.3 Single-Machine Infinite-Bus System 904.4 Power System Disturbances 944.4.1 Fault-On Analysis 944.4.2 Post-Fault Analysis 964.4.3 Other Types of Faults 984.5 Simulation Methods 984.5.1 Modified Euler Methods 994.5.1.1 Euler Full-Step Modification Method 1004.5.1.2 Euler Half-Step Modification Method 1014.5.2 Adams-Bashforth Second-Order Method 1014.5.3 Selecting Integration Stepsize 1024.5.4 Implicit Integration Methods 1044.5.4.1 Integration of DAEs 1054.6 Dynamic Models of Multi-Machine Power Systems 1064.6.1 Constant-Impedance Loads 1074.6.2 Generator Current Injections 1084.6.3 Network Equation Extended to the Machine Internal Node 1084.6.4 Reduced Admittance Matrix Approach 1094.6.5 Method for Dynamic Simulation 1094.7 Multi-Machine Power System Stability 1144.7.1 Reference Frames for Machine Angles 1154.8 Power System Toolbox 1174.9 Summary and Notes 119Problems 1195 Direct Transient Stability Analysis 1235.1 Introduction 1235.2 Equal-Area Analysis of a Single-Machine Infinite-Bus System 1245.2.1 Power-Angle Curve 1245.2.2 Fault-On and Post-Fault Analysis 1265.3 Transient Energy Functions 1275.3.1 Lyapunov Functions 1285.3.2 Energy Function for Single-Machine Infinite-Bus Electromechanical Model 1285.4 Energy Function Analysis of a Disturbance Event 1315.5 Single-Machine Infinite-Bus Model Phase Portrait and Region of Stability 1355.6 Direct Stability Analysis using Energy Functions 1385.7 Energy Functions for Multi-Machine Power Systems 1395.7.1 Direct Stability Analysis for Multi-Machine Systems 1425.7.2 Computation of Critical Energy 1435.8 Dynamic Security Assessment 1465.9 Summary and Notes 146Problems 1476 Linear Analysis and Small-Signal Stability 1496.1 Introduction 1496.2 Electromechanical Modes 1506.3 Linearization 1516.3.1 State-Space Models 1516.3.2 Input-Output Models 1526.3.3 Modal Analysis and Time-Domain Solutions 1526.3.4 Time Response of Linear Systems 1546.3.5 Participation Factors 1566.4 Linearized Models of Single-Machine Infinite-Bus Systems 1576.5 Linearized Models of Multi-Machine Systems 1606.5.1 Synchronizing Torque Matrix and Eigenvalue Properties 1626.5.2 Modeshapes and Participation Factors 1626.6 Developing Linearized Models of Large Power Systems 1646.6.1 Analytical Partial Derivatives 1656.6.2 Numerical Linearization 1696.7 Summary and Notes 171Problems 171Part II Synchronous Machine Models and their Control Systems 1757 Steady-State Models and Operation of Synchronous Machines 1777.1 Introduction 1777.2 Physical Description 1777.2.1 Amortisseur Bars 1797.3 Synchronous Machine Model 1797.3.1 Flux Linkage and Voltage Equations 1817.3.2 Stator (Armature) Self and Mutual Inductances 1837.3.3 Mutual Inductances between Stator and Rotor 1837.3.4 Rotor Self and Mutual Inductances 1847.4 Park Transformation 1857.4.1 Electrical Power in dq0 Variables 1887.5 Reciprocal, Equal Lad Per-Unit System 1897.5.1 Stator Base Values 1897.5.2 Stator Voltage Equations 1907.5.3 Rotor Base Values 1917.5.4 Rotor Voltage Equations 1917.5.5 Stator Flux-Linkage Equations 1927.5.6 Rotor Flux-Linkage Equations 1927.5.7 Equal Mutual Inductance 1927.6 Equivalent Circuits 1967.6.1 Flux-Linkage Circuits 1967.6.2 Voltage Equivalent Circuits 1977.7 Steady-State Analysis 1997.7.1 Open-Circuit Condition 1997.7.2 Loaded Condition 2017.7.3 Drawing Voltage-Current Phasor Diagrams 2027.8 Saturation Effects 2047.8.1 Representations of Magnetic Saturation 2057.9 Generator Capability Curves 2077.10 Summary and Notes 209Problems 2098 Dynamic Models of Synchronous Machines 2138.1 Introduction 2138.2 Machine Dynamic Response During Fault 2138.2.1 DC Offset and Stator Transients 2158.3 Transient and Subtransient Reactances and Time Constants 2168.4 Subtransient Synchronous Machine Model 2218.5 Other Synchronous Machine Models 2278.5.1 Flux-Decay Model 2278.5.2 Classical Model 2288.6 dq-axes Rotation Between a Generator and the System 2298.7 Power System Simulation using Detailed Machine Models 2308.7.1 Power System Simulation Algorithm 2318.8 Linearized Models 2328.9 Summary and Notes 234Problems 2359 Excitation Systems 2379.1 Introduction 2379.2 Excitation System Models 2389.3 Type DC Exciters 2399.3.1 Separately Excited DC exciter 2399.3.2 Self-Excited DC Exciter 2439.3.3 Voltage Regulator 2449.3.4 Initialization of DC Type Exciters 2459.3.5 Transfer Function Analysis 2469.3.6 Generator and Exciter Closed-Loop System 2489.3.7 Excitation System Response Ratios 2519.4 Type AC Exciters 2529.5 Type ST Excitation Systems 2549.6 Load Compensation Control 2579.7 Protective Functions 2599.8 Summary and Notes 259Appendix 9.A Anti-Windup Limits 260Problems 26110 Power System Stabilizers 26510.1 Introduction 26510.2 Single-Machine Infinite-Bus System Model 26610.3 Synchronizing and Damping Torques 27110.3.1 ΔTe2 Under Constant Field Voltage 27210.3.2 ΔTe2 With Excitation System Control 27310.4 Power System Stabilizer Design using Rotor Speed Signal 27510.4.1 PSS Design Requirements 27610.4.2 PSS Control Blocks 27710.4.3 PSS Design Methods 27910.4.4 Torsional Filters 28410.4.5 PSS Field Tuning 28710.4.6 Interarea Mode Damping 28710.5 Other PSS Input Signals 28810.5.1 Generator Terminal Bus Frequency 28810.5.2 Electrical Power Output ΔPe 28810.6 Integral-of-Accelerating-Power or Dual-Input PSS 28910.7 Summary and Notes 293Problems 29311 Load and Induction Motor Models 29511.1 Introduction 29511.2 Static Load Models 29611.2.1 Exponential Load Model 29611.2.2 Polynomial Load Model 29711.3 Incorporating ZIP Load Models in Dynamic Simulation and Linear Analysis 29811.4 Induction Motors: Steady-State Models 30311.4.1 Physical Description 30411.4.2 Mathematical Description 30411.4.2.1 Modeling Equations 30411.4.2.2 Reference Frame Transformation 30611.4.3 Equivalent Circuits 30811.4.4 Per-Unit Representation 31011.4.5 Torque-Slip Characteristics 31111.4.6 Reactive Power Consumption 31311.4.7 Motor Startup 31411.5 Induction Motors: Dynamic Models 31511.5.1 Initialization 31811.5.2 Reactive Power Requirement during Motor Stalling 32011.6 Summary and Notes 323Problems 32412 Turbine-Governor Models and Frequency Control 32712.1 Introduction 32712.2 Steam Turbines 32812.2.1 Turbine Configurations 32812.2.2 Steam Turbine-Governors 33112.3 Hydraulic Turbines 33312.3.1 Hydraulic Turbine-Governors 33712.3.2 Load Rejection of Hydraulic Turbines 33812.4 Gas Turbines and Co-Generation Plants 33912.5 Primary Frequency Control 34212.5.1 Isolated Turbine-Generator Serving Local Load 34312.5.2 Interconnected Units 34712.5.3 Frequency Response in US Power Grids 34912.6 Automatic Generation Control 35112.7 Turbine-Generator Torsional Oscillations and Subsynchronous Resonance 35612.7.1 Torsional Modes 35612.7.2 Electrical Network Modes 36312.7.3 SSR Occurrence and Countermeasures 36512.8 Summary and Notes 366Problems 367Part III Advanced Power System Topics 37113 High-Voltage Direct Current Transmission Systems 37313.1 Introduction 37313.1.1 HVDC System Installations and Applications 37513.1.2 HVDC System Economics 37713.2 AC/DC and DC/AC Conversion 37713.2.1 AC-DC Conversion using Ideal Diodes 37813.2.2 Three-Phase Full-Wave Bridge Converter 37913.3 Line-Commutation Operation in HVDC Systems 38313.3.1 Rectifier Operation 38313.3.1.1 Thyristor Ignition Delay Angle 38313.3.1.2 Commutation Overlap 38513.3.2 Inverter Operation 38813.3.3 Multiple Bridge Converters 38913.3.4 Equivalent Circuit 38913.4 Control Modes 39113.4.1 Mode 1: Normal Operation 39213.4.2 Mode 2: Reduced-Voltage Operation 39313.4.3 Mode 3: Transitional Mode 39413.4.4 System Operation Under Fault Conditions 39613.4.5 Communication Requirements 39613.5 Multi-terminal HVDC Systems 39713.6 Harmonics and Reactive Power Requirement 39813.6.1 Harmonic Filters 39813.6.2 Reactive Power Support 39913.7 AC-DC Power Flow Computation 40113.8 Dynamic Models 40613.8.1 Converter Control 40613.8.2 DC Line Dynamics 40813.8.3 AC-DC Network Solution 40913.9 Damping Control Design 41113.10 Summary and Notes 416Problems 41614 Flexible AC Transmission Systems 42114.1 Introduction 42114.2 Static Var Compensator 42214.2.1 Circuit Configuration and Thyristor Switching 42214.2.2 Steady-State Voltage Regulation and Stability Enhancement 42314.2.2.1 Voltage Stability Enhancement 42414.2.2.2 Transient Stability Enhancement 42714.2.3 Dynamic Voltage Control and Droop Regulation 42914.2.4 Dynamic Simulation 43314.2.5 Damping Control Design using SVC 43514.3 Thyristor-Controlled Series Compensator 44114.3.1 Fixed Series Compensation 44214.3.2 TCSC Circuit Configuration and Switching 44214.3.3 Voltage Reversal Control 44414.3.4 Mitigation of Subsynchronous Oscillations 44514.3.5 Dynamic Model and Damping Control Design 44614.4 Shunt VSC Controllers 45114.4.1 Voltage-Sourced Converters 45114.4.1.1 Three-Phase Full-Wave VSCs 45314.4.1.2 Three-Level Converters 45514.4.1.3 Harmonics 45514.4.2 Static Compensator 45814.4.2.1 Steady-State Analysis 45814.4.2.2 Dynamic Model 45914.4.3 VSC HVDC Systems 46314.4.3.1 Steady-State Operation 46314.4.3.2 Dynamic Model 46614.5 Series and Coupled VSC Controllers 46914.5.1 Static Synchronous Series Compensation 46914.5.1.1 Steady-State Analysis 46914.5.2 Unified Power Flow Controller 47114.5.2.1 Steady-State Analysis 47114.5.3 Interline Power Flow Controller 47514.5.3.1 Steady-State Analysis 47514.5.4 Dynamic Model 47814.5.4.1 Series Voltage Insertion 47914.5.4.2 Line Active and Reactive Power Flow Control 48014.6 Summary and Notes 480Problems 48115 Wind Power Generation and Modeling 48715.1 Background 48715.2 Wind Turbine Components 48915.3 Wind Power 49115.3.1 Blade Angle Orientation 49215.3.2 Power Coefficient 49415.4 Wind Turbine Types 49615.4.1 Type 1 49615.4.2 Type 2 49715.4.3 Type 3 49815.4.4 Type 4 49815.5 Steady-State Characteristics 49915.5.1 Type-1Wind Turbine 49915.5.2 Type-2Wind Turbine 50115.5.3 Type-3Wind Turbine 50215.6 Wind Power Plant Representation 50515.7 Overall Control Criteria for Variable-Speed Wind Turbines 51015.8 Wind Turbine Model for Transient Stability Planning Studies 51315.8.1 Overall Model Structure 51315.8.2 Generator/Converter Model 51415.8.3 Electrical Control Model 51515.8.4 Drive-Train Model 51715.8.5 Torque Control Model 51915.8.6 Aerodynamic Model 52015.8.7 Pitch Controller 52215.9 Plant-Level Control Model 52615.9.1 Simulation Example 52615.10 Summary and Notes 527Problems 52816 Power System Coherency and Model Reduction 53116.1 Introduction 53116.2 Interarea Oscillations and Slow Coherency 53216.2.1 Slow Coherency 53416.2.2 Slow Coherent Areas 53616.2.3 Finding Coherent Groups of Machines 54116.3 Generator Aggregation and Network Reduction 54416.3.1 Generator Aggregation 54516.3.2 Dynamic Aggregation 54816.3.3 Load Bus Elimination 55116.4 Simulation Studies 55516.4.1 Singular Perturbations Method 55616.5 Linear Reduced Model Methods 55716.5.1 Modal Truncation 55816.5.2 Balanced Model Reduction Method 55916.6 Dynamic Model Reduction Software 55916.7 Summary and Notes 560Problems 560References 563Index 577

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 xviiAbout the Companion Website xxi1 Introduction 11.1 Electrification 11.2 Generation, Transmission, and Distribution Systems 21.2.1 Central Generating Station Model 21.2.2 Renewable Generation 41.2.3 Smart Grids 51.3 Time Scales 51.3.1 Dynamic Phenomena 51.3.2 Measurements and Data 51.3.3 Control Functions and System Operation 71.4 Organization of the Book 7Part I System Concepts 92 Steady-State Power Flow 112.1 Introduction 112.2 Power Network Elements and Admittance Matrix 122.2.1 Transmission Lines 122.2.2 Transformers 132.2.3 Per Unit Representation 142.2.4 Building the Network Admittance Matrix 142.3 Active and Reactive Power Flow Calculations 162.4 Power Flow Formulation 192.5 Newton-Raphson Method 212.5.1 General Procedure 212.5.2 NR Solution of Power Flow Equations 222.6 Advanced Power Flow Features 272.6.1 Load Bus Voltage Regulation 272.6.2 Multi-area Power Flow 282.6.3 Active Line Power Flow Regulation 292.6.4 Dishonest Newton-Raphson Method 302.6.5 Fast Decoupled Loadflow 302.6.6 DC Power Flow 312.7 Summary and Notes 31Appendix 2.A Two-winding Transformer Model 32Appendix 2.B LU Decomposition and Sparsity Methods 36Appendix 2.C Power Flow and Dynamic Data for the 2-area, 4-machine System 39Problems 423 Steady-State Voltage Stability Analysis 473.1 Introduction 473.2 Voltage Collapse Incidents 483.2.1 Tokyo, Japan: July 23, 1987 483.2.2 US Western Power System: July 2, 1996 483.3 Reactive Power Consumption on Transmission Lines 493.4 Voltage Stability Analysis of a Radial Load System 553.4.1 Maximum Power Transfer 593.5 Voltage Stability Analysis of Large Power Systems 613.6 Continuation Power Flow Method 643.6.1 Continuation Power Flow Algorithm 663.7 An AQ-Bus Method for Solving Power Flow 673.7.1 Analytical Framework for the AQ-Bus Method 693.7.2 AQ-Bus Formulation for Constant-Power-Factor Loads 703.7.3 AQ-Bus Algorithm for Computing Voltage Stability Margins 713.8 Power System Components Affecting Voltage Stability 733.8.1 Shunt Reactive Power Supply 743.8.2 Under-Load Tap Changer 763.9 Hierarchical Voltage Control 793.10 Voltage Stability Margins and Indices 803.10.1 Voltage Stability Margins 803.10.2 Voltage Sensitivities 813.10.3 Singular Values and Eigenvalues of the Power Flow Jacobian Matrix 823.11 Summary and Notes 82Problems 834 Power System Dynamics and Simulation 874.1 Introduction 874.2 Electromechanical Model of Synchronous Machines 884.3 Single-Machine Infinite-Bus System 904.4 Power System Disturbances 944.4.1 Fault-On Analysis 944.4.2 Post-Fault Analysis 964.4.3 Other Types of Faults 984.5 Simulation Methods 984.5.1 Modified Euler Methods 994.5.1.1 Euler Full-Step Modification Method 1004.5.1.2 Euler Half-Step Modification Method 1014.5.2 Adams-Bashforth Second-Order Method 1014.5.3 Selecting Integration Stepsize 1024.5.4 Implicit Integration Methods 1044.5.4.1 Integration of DAEs 1054.6 Dynamic Models of Multi-Machine Power Systems 1064.6.1 Constant-Impedance Loads 1074.6.2 Generator Current Injections 1084.6.3 Network Equation Extended to the Machine Internal Node 1084.6.4 Reduced Admittance Matrix Approach 1094.6.5 Method for Dynamic Simulation 1094.7 Multi-Machine Power System Stability 1144.7.1 Reference Frames for Machine Angles 1154.8 Power System Toolbox 1174.9 Summary and Notes 119Problems 1195 Direct Transient Stability Analysis 1235.1 Introduction 1235.2 Equal-Area Analysis of a Single-Machine Infinite-Bus System 1245.2.1 Power-Angle Curve 1245.2.2 Fault-On and Post-Fault Analysis 1265.3 Transient Energy Functions 1275.3.1 Lyapunov Functions 1285.3.2 Energy Function for Single-Machine Infinite-Bus Electromechanical Model 1285.4 Energy Function Analysis of a Disturbance Event 1315.5 Single-Machine Infinite-Bus Model Phase Portrait and Region of Stability 1355.6 Direct Stability Analysis using Energy Functions 1385.7 Energy Functions for Multi-Machine Power Systems 1395.7.1 Direct Stability Analysis for Multi-Machine Systems 1425.7.2 Computation of Critical Energy 1435.8 Dynamic Security Assessment 1465.9 Summary and Notes 146Problems 1476 Linear Analysis and Small-Signal Stability 1496.1 Introduction 1496.2 Electromechanical Modes 1506.3 Linearization 1516.3.1 State-Space Models 1516.3.2 Input-Output Models 1526.3.3 Modal Analysis and Time-Domain Solutions 1526.3.4 Time Response of Linear Systems 1546.3.5 Participation Factors 1566.4 Linearized Models of Single-Machine Infinite-Bus Systems 1576.5 Linearized Models of Multi-Machine Systems 1606.5.1 Synchronizing Torque Matrix and Eigenvalue Properties 1626.5.2 Modeshapes and Participation Factors 1626.6 Developing Linearized Models of Large Power Systems 1646.6.1 Analytical Partial Derivatives 1656.6.2 Numerical Linearization 1696.7 Summary and Notes 171Problems 171Part II Synchronous Machine Models and their Control Systems 1757 Steady-State Models and Operation of Synchronous Machines 1777.1 Introduction 1777.2 Physical Description 1777.2.1 Amortisseur Bars 1797.3 Synchronous Machine Model 1797.3.1 Flux Linkage and Voltage Equations 1817.3.2 Stator (Armature) Self and Mutual Inductances 1837.3.3 Mutual Inductances between Stator and Rotor 1837.3.4 Rotor Self and Mutual Inductances 1847.4 Park Transformation 1857.4.1 Electrical Power in dq0 Variables 1887.5 Reciprocal, Equal Lad Per-Unit System 1897.5.1 Stator Base Values 1897.5.2 Stator Voltage Equations 1907.5.3 Rotor Base Values 1917.5.4 Rotor Voltage Equations 1917.5.5 Stator Flux-Linkage Equations 1927.5.6 Rotor Flux-Linkage Equations 1927.5.7 Equal Mutual Inductance 1927.6 Equivalent Circuits 1967.6.1 Flux-Linkage Circuits 1967.6.2 Voltage Equivalent Circuits 1977.7 Steady-State Analysis 1997.7.1 Open-Circuit Condition 1997.7.2 Loaded Condition 2017.7.3 Drawing Voltage-Current Phasor Diagrams 2027.8 Saturation Effects 2047.8.1 Representations of Magnetic Saturation 2057.9 Generator Capability Curves 2077.10 Summary and Notes 209Problems 2098 Dynamic Models of Synchronous Machines 2138.1 Introduction 2138.2 Machine Dynamic Response During Fault 2138.2.1 DC Offset and Stator Transients 2158.3 Transient and Subtransient Reactances and Time Constants 2168.4 Subtransient Synchronous Machine Model 2218.5 Other Synchronous Machine Models 2278.5.1 Flux-Decay Model 2278.5.2 Classical Model 2288.6 dq-axes Rotation Between a Generator and the System 2298.7 Power System Simulation using Detailed Machine Models 2308.7.1 Power System Simulation Algorithm 2318.8 Linearized Models 2328.9 Summary and Notes 234Problems 2359 Excitation Systems 2379.1 Introduction 2379.2 Excitation System Models 2389.3 Type DC Exciters 2399.3.1 Separately Excited DC exciter 2399.3.2 Self-Excited DC Exciter 2439.3.3 Voltage Regulator 2449.3.4 Initialization of DC Type Exciters 2459.3.5 Transfer Function Analysis 2469.3.6 Generator and Exciter Closed-Loop System 2489.3.7 Excitation System Response Ratios 2519.4 Type AC Exciters 2529.5 Type ST Excitation Systems 2549.6 Load Compensation Control 2579.7 Protective Functions 2599.8 Summary and Notes 259Appendix 9.A Anti-Windup Limits 260Problems 26110 Power System Stabilizers 26510.1 Introduction 26510.2 Single-Machine Infinite-Bus System Model 26610.3 Synchronizing and Damping Torques 27110.3.1 ΔTe2 Under Constant Field Voltage 27210.3.2 ΔTe2 With Excitation System Control 27310.4 Power System Stabilizer Design using Rotor Speed Signal 27510.4.1 PSS Design Requirements 27610.4.2 PSS Control Blocks 27710.4.3 PSS Design Methods 27910.4.4 Torsional Filters 28410.4.5 PSS Field Tuning 28710.4.6 Interarea Mode Damping 28710.5 Other PSS Input Signals 28810.5.1 Generator Terminal Bus Frequency 28810.5.2 Electrical Power Output ΔPe 28810.6 Integral-of-Accelerating-Power or Dual-Input PSS 28910.7 Summary and Notes 293Problems 29311 Load and Induction Motor Models 29511.1 Introduction 29511.2 Static Load Models 29611.2.1 Exponential Load Model 29611.2.2 Polynomial Load Model 29711.3 Incorporating ZIP Load Models in Dynamic Simulation and Linear Analysis 29811.4 Induction Motors: Steady-State Models 30311.4.1 Physical Description 30411.4.2 Mathematical Description 30411.4.2.1 Modeling Equations 30411.4.2.2 Reference Frame Transformation 30611.4.3 Equivalent Circuits 30811.4.4 Per-Unit Representation 31011.4.5 Torque-Slip Characteristics 31111.4.6 Reactive Power Consumption 31311.4.7 Motor Startup 31411.5 Induction Motors: Dynamic Models 31511.5.1 Initialization 31811.5.2 Reactive Power Requirement during Motor Stalling 32011.6 Summary and Notes 323Problems 32412 Turbine-Governor Models and Frequency Control 32712.1 Introduction 32712.2 Steam Turbines 32812.2.1 Turbine Configurations 32812.2.2 Steam Turbine-Governors 33112.3 Hydraulic Turbines 33312.3.1 Hydraulic Turbine-Governors 33712.3.2 Load Rejection of Hydraulic Turbines 33812.4 Gas Turbines and Co-Generation Plants 33912.5 Primary Frequency Control 34212.5.1 Isolated Turbine-Generator Serving Local Load 34312.5.2 Interconnected Units 34712.5.3 Frequency Response in US Power Grids 34912.6 Automatic Generation Control 35112.7 Turbine-Generator Torsional Oscillations and Subsynchronous Resonance 35612.7.1 Torsional Modes 35612.7.2 Electrical Network Modes 36312.7.3 SSR Occurrence and Countermeasures 36512.8 Summary and Notes 366Problems 367Part III Advanced Power System Topics 37113 High-Voltage Direct Current Transmission Systems 37313.1 Introduction 37313.1.1 HVDC System Installations and Applications 37513.1.2 HVDC System Economics 37713.2 AC/DC and DC/AC Conversion 37713.2.1 AC-DC Conversion using Ideal Diodes 37813.2.2 Three-Phase Full-Wave Bridge Converter 37913.3 Line-Commutation Operation in HVDC Systems 38313.3.1 Rectifier Operation 38313.3.1.1 Thyristor Ignition Delay Angle 38313.3.1.2 Commutation Overlap 38513.3.2 Inverter Operation 38813.3.3 Multiple Bridge Converters 38913.3.4 Equivalent Circuit 38913.4 Control Modes 39113.4.1 Mode 1: Normal Operation 39213.4.2 Mode 2: Reduced-Voltage Operation 39313.4.3 Mode 3: Transitional Mode 39413.4.4 System Operation Under Fault Conditions 39613.4.5 Communication Requirements 39613.5 Multi-terminal HVDC Systems 39713.6 Harmonics and Reactive Power Requirement 39813.6.1 Harmonic Filters 39813.6.2 Reactive Power Support 39913.7 AC-DC Power Flow Computation 40113.8 Dynamic Models 40613.8.1 Converter Control 40613.8.2 DC Line Dynamics 40813.8.3 AC-DC Network Solution 40913.9 Damping Control Design 41113.10 Summary and Notes 416Problems 41614 Flexible AC Transmission Systems 42114.1 Introduction 42114.2 Static Var Compensator 42214.2.1 Circuit Configuration and Thyristor Switching 42214.2.2 Steady-State Voltage Regulation and Stability Enhancement 42314.2.2.1 Voltage Stability Enhancement 42414.2.2.2 Transient Stability Enhancement 42714.2.3 Dynamic Voltage Control and Droop Regulation 42914.2.4 Dynamic Simulation 43314.2.5 Damping Control Design using SVC 43514.3 Thyristor-Controlled Series Compensator 44114.3.1 Fixed Series Compensation 44214.3.2 TCSC Circuit Configuration and Switching 44214.3.3 Voltage Reversal Control 44414.3.4 Mitigation of Subsynchronous Oscillations 44514.3.5 Dynamic Model and Damping Control Design 44614.4 Shunt VSC Controllers 45114.4.1 Voltage-Sourced Converters 45114.4.1.1 Three-Phase Full-Wave VSCs 45314.4.1.2 Three-Level Converters 45514.4.1.3 Harmonics 45514.4.2 Static Compensator 45814.4.2.1 Steady-State Analysis 45814.4.2.2 Dynamic Model 45914.4.3 VSC HVDC Systems 46314.4.3.1 Steady-State Operation 46314.4.3.2 Dynamic Model 46614.5 Series and Coupled VSC Controllers 46914.5.1 Static Synchronous Series Compensation 46914.5.1.1 Steady-State Analysis 46914.5.2 Unified Power Flow Controller 47114.5.2.1 Steady-State Analysis 47114.5.3 Interline Power Flow Controller 47514.5.3.1 Steady-State Analysis 47514.5.4 Dynamic Model 47814.5.4.1 Series Voltage Insertion 47914.5.4.2 Line Active and Reactive Power Flow Control 48014.6 Summary and Notes 480Problems 48115 Wind Power Generation and Modeling 48715.1 Background 48715.2 Wind Turbine Components 48915.3 Wind Power 49115.3.1 Blade Angle Orientation 49215.3.2 Power Coefficient 49415.4 Wind Turbine Types 49615.4.1 Type 1 49615.4.2 Type 2 49715.4.3 Type 3 49815.4.4 Type 4 49815.5 Steady-State Characteristics 49915.5.1 Type-1Wind Turbine 49915.5.2 Type-2Wind Turbine 50115.5.3 Type-3Wind Turbine 50215.6 Wind Power Plant Representation 50515.7 Overall Control Criteria for Variable-Speed Wind Turbines 51015.8 Wind Turbine Model for Transient Stability Planning Studies 51315.8.1 Overall Model Structure 51315.8.2 Generator/Converter Model 51415.8.3 Electrical Control Model 51515.8.4 Drive-Train Model 51715.8.5 Torque Control Model 51915.8.6 Aerodynamic Model 52015.8.7 Pitch Controller 52215.9 Plant-Level Control Model 52615.9.1 Simulation Example 52615.10 Summary and Notes 527Problems 52816 Power System Coherency and Model Reduction 53116.1 Introduction 53116.2 Interarea Oscillations and Slow Coherency 53216.2.1 Slow Coherency 53416.2.2 Slow Coherent Areas 53616.2.3 Finding Coherent Groups of Machines 54116.3 Generator Aggregation and Network Reduction 54416.3.1 Generator Aggregation 54516.3.2 Dynamic Aggregation 54816.3.3 Load Bus Elimination 55116.4 Simulation Studies 55516.4.1 Singular Perturbations Method 55616.5 Linear Reduced Model Methods 55716.5.1 Modal Truncation 55816.5.2 Balanced Model Reduction Method 55916.6 Dynamic Model Reduction Software 55916.7 Summary and Notes 560Problems 560References 563Index 577

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 xviiAbout the Companion Website xxi1 Introduction 11.1 Electrification 11.2 Generation, Transmission, and Distribution Systems 21.2.1 Central Generating Station Model 21.2.2 Renewable Generation 41.2.3 Smart Grids 51.3 Time Scales 51.3.1 Dynamic Phenomena 51.3.2 Measurements and Data 51.3.3 Control Functions and System Operation 71.4 Organization of the Book 7Part I System Concepts 92 Steady-State Power Flow 112.1 Introduction 112.2 Power Network Elements and Admittance Matrix 122.2.1 Transmission Lines 122.2.2 Transformers 132.2.3 Per Unit Representation 142.2.4 Building the Network Admittance Matrix 142.3 Active and Reactive Power Flow Calculations 162.4 Power Flow Formulation 192.5 Newton-Raphson Method 212.5.1 General Procedure 212.5.2 NR Solution of Power Flow Equations 222.6 Advanced Power Flow Features 272.6.1 Load Bus Voltage Regulation 272.6.2 Multi-area Power Flow 282.6.3 Active Line Power Flow Regulation 292.6.4 Dishonest Newton-Raphson Method 302.6.5 Fast Decoupled Loadflow 302.6.6 DC Power Flow 312.7 Summary and Notes 31Appendix 2.A Two-winding Transformer Model 32Appendix 2.B LU Decomposition and Sparsity Methods 36Appendix 2.C Power Flow and Dynamic Data for the 2-area, 4-machine System 39Problems 423 Steady-State Voltage Stability Analysis 473.1 Introduction 473.2 Voltage Collapse Incidents 483.2.1 Tokyo, Japan: July 23, 1987 483.2.2 US Western Power System: July 2, 1996 483.3 Reactive Power Consumption on Transmission Lines 493.4 Voltage Stability Analysis of a Radial Load System 553.4.1 Maximum Power Transfer 593.5 Voltage Stability Analysis of Large Power Systems 613.6 Continuation Power Flow Method 643.6.1 Continuation Power Flow Algorithm 663.7 An AQ-Bus Method for Solving Power Flow 673.7.1 Analytical Framework for the AQ-Bus Method 693.7.2 AQ-Bus Formulation for Constant-Power-Factor Loads 703.7.3 AQ-Bus Algorithm for Computing Voltage Stability Margins 713.8 Power System Components Affecting Voltage Stability 733.8.1 Shunt Reactive Power Supply 743.8.2 Under-Load Tap Changer 763.9 Hierarchical Voltage Control 793.10 Voltage Stability Margins and Indices 803.10.1 Voltage Stability Margins 803.10.2 Voltage Sensitivities 813.10.3 Singular Values and Eigenvalues of the Power Flow Jacobian Matrix 823.11 Summary and Notes 82Problems 834 Power System Dynamics and Simulation 874.1 Introduction 874.2 Electromechanical Model of Synchronous Machines 884.3 Single-Machine Infinite-Bus System 904.4 Power System Disturbances 944.4.1 Fault-On Analysis 944.4.2 Post-Fault Analysis 964.4.3 Other Types of Faults 984.5 Simulation Methods 984.5.1 Modified Euler Methods 994.5.1.1 Euler Full-Step Modification Method 1004.5.1.2 Euler Half-Step Modification Method 1014.5.2 Adams-Bashforth Second-Order Method 1014.5.3 Selecting Integration Stepsize 1024.5.4 Implicit Integration Methods 1044.5.4.1 Integration of DAEs 1054.6 Dynamic Models of Multi-Machine Power Systems 1064.6.1 Constant-Impedance Loads 1074.6.2 Generator Current Injections 1084.6.3 Network Equation Extended to the Machine Internal Node 1084.6.4 Reduced Admittance Matrix Approach 1094.6.5 Method for Dynamic Simulation 1094.7 Multi-Machine Power System Stability 1144.7.1 Reference Frames for Machine Angles 1154.8 Power System Toolbox 1174.9 Summary and Notes 119Problems 1195 Direct Transient Stability Analysis 1235.1 Introduction 1235.2 Equal-Area Analysis of a Single-Machine Infinite-Bus System 1245.2.1 Power-Angle Curve 1245.2.2 Fault-On and Post-Fault Analysis 1265.3 Transient Energy Functions 1275.3.1 Lyapunov Functions 1285.3.2 Energy Function for Single-Machine Infinite-Bus Electromechanical Model 1285.4 Energy Function Analysis of a Disturbance Event 1315.5 Single-Machine Infinite-Bus Model Phase Portrait and Region of Stability 1355.6 Direct Stability Analysis using Energy Functions 1385.7 Energy Functions for Multi-Machine Power Systems 1395.7.1 Direct Stability Analysis for Multi-Machine Systems 1425.7.2 Computation of Critical Energy 1435.8 Dynamic Security Assessment 1465.9 Summary and Notes 146Problems 1476 Linear Analysis and Small-Signal Stability 1496.1 Introduction 1496.2 Electromechanical Modes 1506.3 Linearization 1516.3.1 State-Space Models 1516.3.2 Input-Output Models 1526.3.3 Modal Analysis and Time-Domain Solutions 1526.3.4 Time Response of Linear Systems 1546.3.5 Participation Factors 1566.4 Linearized Models of Single-Machine Infinite-Bus Systems 1576.5 Linearized Models of Multi-Machine Systems 1606.5.1 Synchronizing Torque Matrix and Eigenvalue Properties 1626.5.2 Modeshapes and Participation Factors 1626.6 Developing Linearized Models of Large Power Systems 1646.6.1 Analytical Partial Derivatives 1656.6.2 Numerical Linearization 1696.7 Summary and Notes 171Problems 171Part II Synchronous Machine Models and their Control Systems 1757 Steady-State Models and Operation of Synchronous Machines 1777.1 Introduction 1777.2 Physical Description 1777.2.1 Amortisseur Bars 1797.3 Synchronous Machine Model 1797.3.1 Flux Linkage and Voltage Equations 1817.3.2 Stator (Armature) Self and Mutual Inductances 1837.3.3 Mutual Inductances between Stator and Rotor 1837.3.4 Rotor Self and Mutual Inductances 1847.4 Park Transformation 1857.4.1 Electrical Power in dq0 Variables 1887.5 Reciprocal, Equal Lad Per-Unit System 1897.5.1 Stator Base Values 1897.5.2 Stator Voltage Equations 1907.5.3 Rotor Base Values 1917.5.4 Rotor Voltage Equations 1917.5.5 Stator Flux-Linkage Equations 1927.5.6 Rotor Flux-Linkage Equations 1927.5.7 Equal Mutual Inductance 1927.6 Equivalent Circuits 1967.6.1 Flux-Linkage Circuits 1967.6.2 Voltage Equivalent Circuits 1977.7 Steady-State Analysis 1997.7.1 Open-Circuit Condition 1997.7.2 Loaded Condition 2017.7.3 Drawing Voltage-Current Phasor Diagrams 2027.8 Saturation Effects 2047.8.1 Representations of Magnetic Saturation 2057.9 Generator Capability Curves 2077.10 Summary and Notes 209Problems 2098 Dynamic Models of Synchronous Machines 2138.1 Introduction 2138.2 Machine Dynamic Response During Fault 2138.2.1 DC Offset and Stator Transients 2158.3 Transient and Subtransient Reactances and Time Constants 2168.4 Subtransient Synchronous Machine Model 2218.5 Other Synchronous Machine Models 2278.5.1 Flux-Decay Model 2278.5.2 Classical Model 2288.6 dq-axes Rotation Between a Generator and the System 2298.7 Power System Simulation using Detailed Machine Models 2308.7.1 Power System Simulation Algorithm 2318.8 Linearized Models 2328.9 Summary and Notes 234Problems 2359 Excitation Systems 2379.1 Introduction 2379.2 Excitation System Models 2389.3 Type DC Exciters 2399.3.1 Separately Excited DC exciter 2399.3.2 Self-Excited DC Exciter 2439.3.3 Voltage Regulator 2449.3.4 Initialization of DC Type Exciters 2459.3.5 Transfer Function Analysis 2469.3.6 Generator and Exciter Closed-Loop System 2489.3.7 Excitation System Response Ratios 2519.4 Type AC Exciters 2529.5 Type ST Excitation Systems 2549.6 Load Compensation Control 2579.7 Protective Functions 2599.8 Summary and Notes 259Appendix 9.A Anti-Windup Limits 260Problems 26110 Power System Stabilizers 26510.1 Introduction 26510.2 Single-Machine Infinite-Bus System Model 26610.3 Synchronizing and Damping Torques 27110.3.1 ΔTe2 Under Constant Field Voltage 27210.3.2 ΔTe2 With Excitation System Control 27310.4 Power System Stabilizer Design using Rotor Speed Signal 27510.4.1 PSS Design Requirements 27610.4.2 PSS Control Blocks 27710.4.3 PSS Design Methods 27910.4.4 Torsional Filters 28410.4.5 PSS Field Tuning 28710.4.6 Interarea Mode Damping 28710.5 Other PSS Input Signals 28810.5.1 Generator Terminal Bus Frequency 28810.5.2 Electrical Power Output ΔPe 28810.6 Integral-of-Accelerating-Power or Dual-Input PSS 28910.7 Summary and Notes 293Problems 29311 Load and Induction Motor Models 29511.1 Introduction 29511.2 Static Load Models 29611.2.1 Exponential Load Model 29611.2.2 Polynomial Load Model 29711.3 Incorporating ZIP Load Models in Dynamic Simulation and Linear Analysis 29811.4 Induction Motors: Steady-State Models 30311.4.1 Physical Description 30411.4.2 Mathematical Description 30411.4.2.1 Modeling Equations 30411.4.2.2 Reference Frame Transformation 30611.4.3 Equivalent Circuits 30811.4.4 Per-Unit Representation 31011.4.5 Torque-Slip Characteristics 31111.4.6 Reactive Power Consumption 31311.4.7 Motor Startup 31411.5 Induction Motors: Dynamic Models 31511.5.1 Initialization 31811.5.2 Reactive Power Requirement during Motor Stalling 32011.6 Summary and Notes 323Problems 32412 Turbine-Governor Models and Frequency Control 32712.1 Introduction 32712.2 Steam Turbines 32812.2.1 Turbine Configurations 32812.2.2 Steam Turbine-Governors 33112.3 Hydraulic Turbines 33312.3.1 Hydraulic Turbine-Governors 33712.3.2 Load Rejection of Hydraulic Turbines 33812.4 Gas Turbines and Co-Generation Plants 33912.5 Primary Frequency Control 34212.5.1 Isolated Turbine-Generator Serving Local Load 34312.5.2 Interconnected Units 34712.5.3 Frequency Response in US Power Grids 34912.6 Automatic Generation Control 35112.7 Turbine-Generator Torsional Oscillations and Subsynchronous Resonance 35612.7.1 Torsional Modes 35612.7.2 Electrical Network Modes 36312.7.3 SSR Occurrence and Countermeasures 36512.8 Summary and Notes 366Problems 367Part III Advanced Power System Topics 37113 High-Voltage Direct Current Transmission Systems 37313.1 Introduction 37313.1.1 HVDC System Installations and Applications 37513.1.2 HVDC System Economics 37713.2 AC/DC and DC/AC Conversion 37713.2.1 AC-DC Conversion using Ideal Diodes 37813.2.2 Three-Phase Full-Wave Bridge Converter 37913.3 Line-Commutation Operation in HVDC Systems 38313.3.1 Rectifier Operation 38313.3.1.1 Thyristor Ignition Delay Angle 38313.3.1.2 Commutation Overlap 38513.3.2 Inverter Operation 38813.3.3 Multiple Bridge Converters 38913.3.4 Equivalent Circuit 38913.4 Control Modes 39113.4.1 Mode 1: Normal Operation 39213.4.2 Mode 2: Reduced-Voltage Operation 39313.4.3 Mode 3: Transitional Mode 39413.4.4 System Operation Under Fault Conditions 39613.4.5 Communication Requirements 39613.5 Multi-terminal HVDC Systems 39713.6 Harmonics and Reactive Power Requirement 39813.6.1 Harmonic Filters 39813.6.2 Reactive Power Support 39913.7 AC-DC Power Flow Computation 40113.8 Dynamic Models 40613.8.1 Converter Control 40613.8.2 DC Line Dynamics 40813.8.3 AC-DC Network Solution 40913.9 Damping Control Design 41113.10 Summary and Notes 416Problems 41614 Flexible AC Transmission Systems 42114.1 Introduction 42114.2 Static Var Compensator 42214.2.1 Circuit Configuration and Thyristor Switching 42214.2.2 Steady-State Voltage Regulation and Stability Enhancement 42314.2.2.1 Voltage Stability Enhancement 42414.2.2.2 Transient Stability Enhancement 42714.2.3 Dynamic Voltage Control and Droop Regulation 42914.2.4 Dynamic Simulation 43314.2.5 Damping Control Design using SVC 43514.3 Thyristor-Controlled Series Compensator 44114.3.1 Fixed Series Compensation 44214.3.2 TCSC Circuit Configuration and Switching 44214.3.3 Voltage Reversal Control 44414.3.4 Mitigation of Subsynchronous Oscillations 44514.3.5 Dynamic Model and Damping Control Design 44614.4 Shunt VSC Controllers 45114.4.1 Voltage-Sourced Converters 45114.4.1.1 Three-Phase Full-Wave VSCs 45314.4.1.2 Three-Level Converters 45514.4.1.3 Harmonics 45514.4.2 Static Compensator 45814.4.2.1 Steady-State Analysis 45814.4.2.2 Dynamic Model 45914.4.3 VSC HVDC Systems 46314.4.3.1 Steady-State Operation 46314.4.3.2 Dynamic Model 46614.5 Series and Coupled VSC Controllers 46914.5.1 Static Synchronous Series Compensation 46914.5.1.1 Steady-State Analysis 46914.5.2 Unified Power Flow Controller 47114.5.2.1 Steady-State Analysis 47114.5.3 Interline Power Flow Controller 47514.5.3.1 Steady-State Analysis 47514.5.4 Dynamic Model 47814.5.4.1 Series Voltage Insertion 47914.5.4.2 Line Active and Reactive Power Flow Control 48014.6 Summary and Notes 480Problems 48115 Wind Power Generation and Modeling 48715.1 Background 48715.2 Wind Turbine Components 48915.3 Wind Power 49115.3.1 Blade Angle Orientation 49215.3.2 Power Coefficient 49415.4 Wind Turbine Types 49615.4.1 Type 1 49615.4.2 Type 2 49715.4.3 Type 3 49815.4.4 Type 4 49815.5 Steady-State Characteristics 49915.5.1 Type-1Wind Turbine 49915.5.2 Type-2Wind Turbine 50115.5.3 Type-3Wind Turbine 50215.6 Wind Power Plant Representation 50515.7 Overall Control Criteria for Variable-Speed Wind Turbines 51015.8 Wind Turbine Model for Transient Stability Planning Studies 51315.8.1 Overall Model Structure 51315.8.2 Generator/Converter Model 51415.8.3 Electrical Control Model 51515.8.4 Drive-Train Model 51715.8.5 Torque Control Model 51915.8.6 Aerodynamic Model 52015.8.7 Pitch Controller 52215.9 Plant-Level Control Model 52615.9.1 Simulation Example 52615.10 Summary and Notes 527Problems 52816 Power System Coherency and Model Reduction 53116.1 Introduction 53116.2 Interarea Oscillations and Slow Coherency 53216.2.1 Slow Coherency 53416.2.2 Slow Coherent Areas 53616.2.3 Finding Coherent Groups of Machines 54116.3 Generator Aggregation and Network Reduction 54416.3.1 Generator Aggregation 54516.3.2 Dynamic Aggregation 54816.3.3 Load Bus Elimination 55116.4 Simulation Studies 55516.4.1 Singular Perturbations Method 55616.5 Linear Reduced Model Methods 55716.5.1 Modal Truncation 55816.5.2 Balanced Model Reduction Method 55916.6 Dynamic Model Reduction Software 55916.7 Summary and Notes 560Problems 560References 563Index 577