Power System Control and Stability

VIJAY VITTAL, PhD, is an Ira A. Fulton Chair Professor at the School of Electrical, Computer and Energy Engineering at Arizona State University. He is a Fellow of IEEE and a member of the U.S. National Academy of Engineering and has more than 35 years experience in teaching and research related to power system dynamics and control. JAMES D. MCCALLEY, PhD, is an Anson Marston Distinguished Professor at Iowa State University. Dr. McCalley, a Fellow of the IEEE, was an industry engineer from 1985-1990 performing dynamic analysis of the Western US Interconnection. He has been on the faculty at Iowa State University since 1992 performing research and instruction in power system planning and dynamic analysis. PAUL M. ANDERSON, PhD, served as a professor of engineering at Iowa State University, Arizona State University, and as a visiting professor at Washington State University. Dr. Anderson passed away in 2011. A. A. FOUAD was Anson Marston Distinguished Professor Emeritus of Engineering at Iowa State University. He had more than 40 years experience in power system dynamics in teaching, research, and in industry. Dr. Fouad passed away in 2017.

• Foreword xiiiPreface xvAbout the Authors xviiPart I IntroductionChapter 1 Power System Stability 31.1 Introduction 31.2 Requirements of a Reliable Electrical Power Service 41.3 Statement of the Problem 51.3.1 Definition of Stability 51.3.2 Classification of Stability Problems 61.3.3 Description of Stability Phenomenon 61.4 Effect of Impact on System Components 71.4.1 Loss of Synchronism 81.4.2 Synchronous Machine During a Transient 81.5 Methods of Simulation 101.5.1 Linearized System Equations 101.5.2 Large System with Nonlinear Equations 111.6 Planning and Operating Standards 11Chapter 2 The Elementary Mathematical Model 192.1 Swing Equation 192.2 Units 212.3 Mechanical Torque 222.3.1 Unregulated Machines 222.3.2 Regulated Machines 242.4 Electrical Torque 262.4.1 Synchronous Torque 262.4.2 Other Electrical Torques 272.5 Power-Angle Curve of a Synchronous Machine 272.5.1 Classical Representation of a Synchronous Machine in Stability Studies 282.5.2 Synchronizing Power Coefficients 292.6 Natural Frequencies of Oscillation of a Synchronous Machine 302.7 System of One Machine Against an Infinite Bus: The Classical Model 312.8 Equal Area Criterion 372.8.1 Critical Clearing Angle 382.8.2 Application to a One-Machine System 392.8.3 Equal Area Criterion for a Two-Machine System 392.9 Classical Model of a Multimachine System 402.10 Classical Stability Study of a Nine-Bus System 422.10.1 Data Preparation 432.10.2 Preliminary Calculations 452.11 Shortcomings of the Classical Model 512.12 Block Diagram of One Machine 53Chapter 3 System Response to Small Disturbances 613.1 Introduction 613.2 Types of Problems Studied 623.2.1 System Response to Small Impacts 623.2.2 Distribution of Power Impacts 623.3 The Unregulated Synchronous Machine 633.3.1 Demagnetizing Effect of Armature Reaction 643.3.2 Effect of Small Changes of Speed 653.4 Modes of Oscillation of an Unregulated Multimachine System 663.5 Regulated Synchronous Machine 733.5.1 Voltage Regulator with One Time Lag 733.5.2 Governor with One Time Lag 753.6 Distribution of Power Impacts 763.6.1 Linearization 773.6.2 A Special Case: t = 0+ 783.6.3 Average Behavior Prior to Governor Action (t = t1) 79Part II Electrical and Electromagnetic Dynamic PerformanceChapter 4 The Synchronous Machine 914.1 Introduction 914.2 Park’s Transformation 914.3 Flux Linkage Equations 944.3.1 Stator Self-Inductances 944.3.2 Rotor Self-Inductances 954.3.3 Stator Mutual Inductances 954.3.4 Rotor Mutual Inductances 954.3.5 Stator-to-Rotor Mutual Inductances 954.3.6 Transformation of Inductances 964.4 Voltage Equations 974.5 Formulation of State-Space Equations 994.6 Current Formulation 1004.7 Per-Unit Conversion 1014.7.1 Choosing a Base for Stator Quantities 1024.7.2 Choosing a Base for Rotor Quantities 1034.7.3 Comparison with Other Per-Unit Systems 1044.7.4 The Correspondence of Per-Unit Stator EMF to Rotor Quantities 1074.8 Normalizing the Voltage Equations 1084.9 Normalizing the Torque Equations 1134.9.1 The Normalized Swing Equation 1144.9.2 Forms of the Swing Equation 1144.10 Torque and Power 1154.11 Equivalent Circuit of a Synchronous Machine 1174.12 The Flux Linkage State-Space Model 1194.12.1 The Voltage Equations 1204.12.2 The Torque Equation 1204.12.3 Machine Equations with Saturation Neglected 1214.12.4 Treatment of Saturation 1234.13 Load Equations 1244.13.1 Synchronous Machine Connected to an Infinite Bus 1244.13.2 Current Model 1264.13.3 The Flux Linkage Model 1274.14 Subtransient and Transient Inductances and Time Constants 1314.14.1 Time Constants 1334.15 Simplified Models of the Synchronous Machine 1364.15.1 Neglecting Damper Windings: The E’q (One-Axis) Model 1374.15.2 Voltage Behind Subtransient Reactance: The E” Model 1424.15.3 Neglecting λd and λq for a Cylindrical Rotor Machine: The Two-Axis Model 1504.15.4 Neglecting Amortisseur Effects and λd and λq Terms: The One-Axis Model 1534.15.5 Assuming Constant Flux Linkage in the Main Field Winding 1544.16 Parameter Determination for Generator Dynamic Models 155Chapter 5 The Simulation of Synchronous Machines 1655.1 Introduction 1655.2 Steady-State Equations and Phasor Diagrams 1655.3 Machine Connected to an Infinite Bus Through a Transmission Line 1685.4 Machine Connected to an Infinite Bus with Local Load at Machine Terminal 1695.4.1 Special Case: The Resistive Load, ZL = RL + j0 1705.4.2 General Case: ZL Arbitrary 1715.5 Determining Steady-State Conditions 1725.5.1 Machine Connected to an Infinite Bus with Local Load 1735.6 Examples 1745.7 Initial Conditions for a Multimachine System 1825.8 Determination of Machine Parameters from Manufacturers’ Data 1835.9 Digital Simulation of Synchronous Machines 1885.9.1 Digital Computation of Saturation 1895.9.2 Updating λAD 192Chapter 6 Load Modeling 1996.1 Introduction 1996.2 Static Load Models 2006.3 Induction Motor Loads 2036.3.1 Model Development of a Three-Phase Induction Machine 2036.3.2 Representing Induction Machines in Stability Simulations 2136.3.3 Stalled Motor Operation 2156.4 Single-Phase Motors 2166.4.1 Scroll Compressors 2186.4.2 Point-on-Wave Effects 2196.4.3 Dynamic Phasors 2196.5 Power Electronic Loads 2216.6 Self-Restoring Loads 2246.7 Distributed Energy Resources 2256.8 Composite Load Models 2276.9 Data Development 2296.9.1 Component Based 2306.9.2 Measurement Based 232Chapter 7 Simulation of Multimachine Systems 2397.1 Introduction 2397.2 Statement of the Problem 2397.3 Matrix Representation of a Passive Network 2407.3.1 Network in the Transient State 2427.3.2 Converting to a Common Reference Frame 2437.4 Converting Machine Coordinates to System Reference 2447.5 Relation Between Machine Currents and Voltages 2457.6 System Order 2497.7 Machines Represented by Classical Methods 2497.8 Linearized Model for the Network 2527.9 Hybrid Formulation 2587.10 Network Equations with Flux Linkage Model 2607.11 Total System Equations 2627.12 Alternating Solution Method 2647.12.1 Nonlinear Loads 2657.12.2 Network–Machine Interface 2687.13 Simultaneous Solution Method 2757.14 Design of Numerical Solvers 277Chapter 8 Small-Signal Stability Analysis 2818.1 Introduction 2818.2 Fundamentals of Linear System Stability 2828.3 Linearization of the Generator State-Space Current Model 2848.4 Linearization of the Load Equation for the One-Machine Problem 2888.5 Linearization of the Flux Linkage Model 2938.6 State Matrix for Multimachine Systems 2988.6.1 Formulation of the State Matrix 2988.6.2 Representation of Static Loads in the State Matrix 3008.7 Simplified Linear Model 3128.7.1 The E' Equation 3128.7.2 Electrical Torque Equation 3138.7.3 Terminal Voltage Equation 3148.7.4 Summary of Equations 3158.7.5 Effect of Loading 3188.7.6 Comparison with Classical Model 3208.8 Block Diagrams 3218.9 State-Space Representation of Simplified Model 322Chapter 9 Excitation Systems 3259.1 Simplified View of Excitation Control 3259.2 Control Configurations 3279.3 Typical Excitation Configurations 3289.3.1 Primitive Systems 3289.3.2 Type DC Excitation Control Systems with DC Generator-Commutator Exciters 3329.3.3 Type AC Excitation Control Systems with Alternator-Rectifier Exciters 3329.3.4 Type AC Excitation Control Systems with Alternator-SCR Exciter Systems 3349.3.5 Type ST Excitation Control Systems with Compound-Rectifier Exciter Systems 3359.3.6 Type ST Excitation Control System with Compound-Rectifier Exciter Plus Potential-Source-Rectifier Exciter 3369.3.7 Type ST Excitation Control Systems with Potential-Source-Rectifier Exciter 3369.4 Excitation Control System Definitions 3379.4.1 Voltage Response Ratio 3399.4.2 Exciter Voltage Ratings 3419.4.3 Other Specifications 3429.5 Voltage Regulator 3449.5.1 Electromechanical Regulators 3449.5.2 Early Electronic Regulators 3459.5.3 Rotating Amplifier Regulators 3459.5.4 Magnetic Amplifier Regulators 3469.5.5 Digital Excitation Systems 3489.6 Exciter Buildup 3489.6.1 The DC Generator Exciter 3489.6.2 Linear Approximations for DC Generator Exciters 3569.6.3 The AC Generator Exciters 3589.6.4 Solid-State Exciters 3599.6.5 Buildup of a Loaded DC Exciter 3609.6.6 Normalization of Exciter Equations 3609.7 Limiting and Protection for Excitation Control Systems 3619.7.1 Modeling Amplifier Limits 3619.7.2 Control Limiters and Associated Protection 3629.7.3 Volts per Hertz Protection 3659.8 Excitation System Response 3659.8.1 Noncontinuously Regulated Systems 3659.8.2 Continuously Regulated Systems 3699.9 State-Space Description of the Excitation System 3799.9.1 Simplified Linear Model 3819.9.2 Complete Linear Model 3829.10 Computer Representation of Excitation Systems 3899.10.1 Type DC1: DC Commutator Exciter 3909.10.2 Type AC Systems: Alternator Supplied Rectifier Excitation Systems 3939.10.3 Type AC1 System: Field-Controlled Alternator-Rectifier Excitation System 3949.10.4 Type ST1 System: Controlled Rectifier System with Terminal Potential Supply Only 3959.10.5 Type ST2 System: Static with Terminal Potential and Current Supplies 3979.10.6 Type DC3 System: Noncontinuous Acting 3999.11 Typical System Constants 4009.12 The Effect of Excitation on Generator Performance 400Chapter 10 The Effect of Excitation on Stability 40910.1 Introduction 40910.1.1 Transient Stability and Small-Signal Stability Considerations 41010.2 Effect of Excitation on Generator Power Limits 41110.3 Effect of the Excitation System on Transient Stability 41510.3.1 The Role of the Excitation System in Classical Model Studies 41510.3.2 Increased Reliance on Excitation Control to Improve Stability 41710.3.3 Parametric Study 41910.3.4 Reactive Power Demand During System Emergencies 42110.4 Effect of Excitation on Small-Signal Stability 42110.4.1 Examination of Small-Signal Stability by Routh’s Criterion 42110.4.2 Further Considerations of the Regulator Gain and Time Constant 42410.4.3 Effect on the Electrical Torque 42510.5 Root-Locus Analysis of a Regulated Machine Connected to an Infinite Bus 42610.6 Approximate System Representation 43210.6.1 Approximate Excitation System Representation 43210.6.2 Estimate of Gx(s) 43310.6.3 The Inertial Transfer Function 43710.7 Supplementary Stabilizing Signals 43910.7.1 Block Diagram of the Linear System 43910.7.2 Approximate Model of the Complete Exciter-Generator System 44010.7.3 Lead Compensation 44210.8 Linear Analysis of the Stabilized Generator 44610.9 PSS Tuning in Multimachine Power Systems 44810.10 Alternate Types of PSS 44910.11 Digital Computer Transient Stability Studies 45010.11.1 Effect of Fault Duration 45210.11.2 Effect of the Power System Stabilizer 45710.12 Some General Comments on the Effect of Excitation on Stability 459Chapter 11 Dynamic Modeling and Representation of Renewable Energy Resources 46311.1 Wind Turbine Generators 46311.1.1 Type 1 WTGs 46511.1.2 Type 2 WTGs 46611.1.3 Type 3 WTGs 46711.1.4 Type 4 WTGs 47911.2 Photovoltaic Solar Plant Modeling 48011.2.1 Generic Model of PV Solar Plant 48011.2.2 Modified Generic Model of PV Solar Plant 481Chapter 12 Voltage Stability 48712.1 Modeling Requirements for Voltage Instability Analysis 48712.2 Voltage Instability Analysis Using Time Domain Simulation 48912.3 Dynamic VAr Planning and Optimization 49312.3.1 Trajectory Sensitivity Analysis 49312.3.2 Formulation of the VAr Optimization Problem 49512.3.3 Implementation of the Dynamic VAr Optimization Approach 49712.3.4 Application of Dynamic VAr Optimization Approach 499Chapter 13 Dynamic Performance and Modeling of Flexible AC Transmission System(Facts) Components 50313.1 Introduction 50313.2 Static VAr System 50313.2.1 Stability Characteristics of an SVS 50613.2.2 Positive-Sequence Transient Stability Model for SVS 50913.3 Thyristor-Controlled Series Compensation 51113.3.1 Operating Modes of a TCSC 51213.3.2 Equipment Characteristics and Limiting Conditions 51313.3.3 TCSC Model for Transient Stability Studies 51513.4 Static Synchronous Compensator 51713.4.1 Statcom Model for Transient Stability Studies 51913.5 High Voltage DC Transmission 519Chapter 14 Power System Protection and Monitoring Associated With Power System Stability 52514.1 Introduction 52514.2 Power System Protection Functions Associated with Transient Stability Analysis 52714.2.1 Bulk Transmission Line Out-of-Step Protection 52714.2.2 Generator Out-of-Step Protection 53314.2.3 Undervoltage Load Shedding 53314.2.4 Underfrequency Load Shedding 53414.3 Special Protection Schemes 53514.3.1 Generation Rejection and Load Shedding 53514.3.2 Controlled Islanding and Load Shedding 53514.4 Synchrophasor-Based Monitoring of Power System Stability 53714.4.1 Online Dynamic Security Assessment Using Synchrophasor Measurements and Decision Trees 53714.4.2 Island Formation Prediction Scheme Supported by PMU Measurements 53914.4.3 Real-Time Voltage Security and Oscillation Monitoring Using PMU Measurements 540Part III Mechanical Dynamic PerformanceChapter 15 Speed Governing 54515.1 The Flyball Governor 54615.2 The Isochronous Governor 55115.3 Incremental Equations of the Turbine 55315.4 The Speed Droop Governor 55615.5 The Floating Lever Speed Droop Governor 56115.6 The Compensated Governor 56415.7 Electronic Governors 57015.8 Governor Models for Transient Stability Simulations 571Chapter 16 Steam Turbine Prime Movers 57716.1 Introduction 57716.2 Power Plant Control Modes 57916.2.1 The Turbine-Following Control Mode 57916.2.2 The Boiler-Following Control Mode 57916.2.3 The Coordinated Control Mode 58016.3 Thermal Generation 58116.4 A Steam Power Plant Model 58216.5 Steam Turbines 58316.6 Steam Turbine Control Operations 59016.7 Steam Turbine Control Functions 59216.8 Steam Generator Control 60416.9 Fossil-Fueled Boilers 60516.9.1 Drum-Type Boilers 60616.9.2 Once-Through Boilers 61316.9.3 Computer Models of Fossil-Fueled Boilers 61716.10 Nuclear Steam Supply Systems 62016.10.1 Boiling Water Reactors 62016.10.2 Pressurized Water Reactors 620Chapter 17 Hydraulic Turbine Prime Movers 62717.1 Introduction 62717.2 The Impulse Turbine 62717.3 The Reaction Turbine 62917.4 Propeller-Type Turbines 63117.5 The Deriaz Turbine 63217.6 Conduits, Surge Tanks, and Penstocks 63317.7 Hydraulic System Equations 63917.8 Hydraulic System Transfer Function 64417.9 Simplifying Assumptions 64717.10 Block Diagram for a Hydro System 64917.11 Pumped-Storage Hydro Systems 65017.12 Representation of Hydro Turbines and Governors in Stability Studies 651Chapter 18 Combustion Turbine and Combined-Cycle Power Plants 65518.1 Introduction 65518.2 The Combustion Turbine Prime Mover 65518.2.1 Combustion Turbine Control 65718.2.2 Off-Nominal Frequency and Voltage Effects 65818.2.3 Nonlinear Governor Droop Characteristic 65918.2.4 Recent Advances in Modeling Gas Turbines 66018.3 The Combined-Cycle Prime Mover 66318.3.1 Fuel and Air Controls 66418.3.2 The Gas Turbine Power Generation 66818.3.3 The Steam Turbine Power Generation 66918.3.4 Recent Development in Modeling Combined-Cycle Plants 671Appendix A 673Appendix B 675Appendix C 685Appendix D 695Appendix E 727Appendix F 737Appendix G 759Appendix H 767Appendix I 775Appendix J 783Index 793