Peter W. Sauer obtained his BS in Electrical Engineering from the University of Missouri at Rolla in 1969, and the MS and PhD degrees in Electrical Engineering from Purdue University in 1974 and 1977 respectively. He served as a facilities design engineer in the U.S. Air Force from 1969 to 1973. He is currently the Grainger Professor of Electrical Engineering at the University of Illinois, Urbana-Champaign where he has been since 1977. His main work is in modeling and simulation of power systems with applications to steady-state and transient stability analysis. He served as the program director for power systems at the National Science Foundation from 1990 to 1991. He was a cofounder of PowerWorld Corporation and the Power Systems Engineering Research Center (PSERC). He is a registered Professional Engineer in Virginia and Illinois, a Fellow of the IEEE, and a member of the U.S. National Academy of Engineering. M. A. Pai is Professor Emeritus in Electrical and Computer Engineering at the University of Illinois, Urbana-Champaign. He received his BE degree from Univ. of Madras, India in 1953, MS and PhD degrees from University of California, Berkeley in 1957 and 1961 respectively. He was with the Indian Institute of Technology, Kanpur, India from 1963 to 1981 and at the University of Illinois, Urbana-Champaign, from 1981 to 2003. His research interests are in dynamics and stability of power systems, smart grid, renewable resources and power system computation. He is the author of several text books and research monographs in these areas. He is a Fellow of IEEE, I.E. (India) and the Indian National Science Academy. Joe H. Chow is Professor of Electrical, Computer, and Systems Engineering at Rensselaer. He received his BS degrees in Electrical Engineering and Mathematics from the University of Minnesota, Minneapolis, in 1974, and his MS and PhD degrees from the University of Illinois, Urbana-Champaign, in 1975 and 1977. He worked in the power systems business at General Electric Company in 1978 and joined Rensselaer in 1987. His research interests include power system dynamics and control, voltage stability analysis, FACTS controllers, synchronized phasor measurements and applications, and integration of renewable resources. He is a fellow of IEEE, and past recipient of the Donald Eckman Award from the American Automatic Control Council, the Control Systems Technology Award from the IEEE Control Systems Society, and the Charles Concordia Power Systems Engineering Award from the IEEE Power and Energy Systems Society.

• Preface xiiiAbout the Companion Website xv1 Introduction 11.1 Background 11.2 Physical Structures 21.3 Time-Scale Structures 31.4 Political Structures 41.5 The Phenomena of Interest 51.6 New Chapters Added to this Edition 52 Electromagnetic Transients 72.1 The Fastest Transients 72.2 Transmission Line Models 72.3 Solution Methods 122.4 Problems 173 Synchronous Machine Modeling 193.1 Conventions and Notation 193.2 Three-Damper-Winding Model 203.3 Transformations and Scaling 213.4 The Linear Magnetic Circuit 293.5 The Nonlinear Magnetic Circuit 353.6 Single-Machine Steady State 403.7 Operational Impedances and Test Data 443.8 Problems 494 Synchronous Machine Control Models 534.1 Voltage and Speed Control Overview 534.2 Exciter Models 534.3 Voltage Regulator Models 584.4 Turbine Models 624.4.1 Hydroturbines 624.4.2 Steam Turbines 644.5 Speed Governor Models 674.6 Problems 705 Single-Machine Dynamic Models 715.1 Terminal Constraints 715.2 The Multi-Time-Scale Model 745.3 Elimination of Stator/Network Transients 765.4 The Two-Axis Model 815.5 The One-Axis (Flux-Decay) Model 835.6 The Classical Model 845.7 Damping Torques 865.8 Single-Machine Infinite-Bus System 905.9 Synchronous Machine Saturation 945.10 Problems 1006 Multimachine Dynamic Models 1016.1 The Synchronously Rotating Reference Frame 1016.2 Network and R-L Load Constraints 1036.3 Elimination of Stator/Network Transients 1056.3.1 Generalization of Network and Load Dynamic Models 1106.3.2 The Special Case of “Impedance Loads” 1126.4 Multimachine Two-Axis Model 1136.4.1 The Special Case of “Impedance Loads” 1156.5 Multimachine Flux–Decay Model 1166.5.1 The Special Case of “Impedance Loads” 1176.6 Multimachine Classical Model 1186.6.1 The Special Case of “Impedance Loads” 1196.7 Multimachine Damping Torques 1206.8 Multimachine Models with Saturation 1216.8.1 The Multimachine Two-Axis Model with Synchronous Machine Saturation 1236.8.2 The Multimachine Flux-Decay Model with Synchronous Machine Saturation 1246.9 Frequency During Transients 1266.10 Angle References and an Infinite Bus 1276.11 Automatic Generation Control (AGC) 1297 Multimachine Simulation 1357.1 Differential-Algebraic Model 1357.1.1 Generator Buses 1367.1.2 Load Buses 1377.2 Stator Algebraic Equations 1387.2.1 Polar Form 1387.2.2 Rectangular Form 1387.2.3 Alternate Form of Stator Algebraic Equations 1397.3 Network Equations 1407.3.1 Power-Balance Form 1407.3.2 Real Power Equations 1417.3.3 Reactive Power Equations 1417.3.4 Current-Balance Form 1427.4 Industry Model 1497.5 Simplification of the Two-Axis Model 1537.5.1 Simplification #1 (Neglecting Transient Saliency in the Synchronous Machine) 1537.5.2 Simplification #2 (Constant Impedance Load in the Transmission System) 1547.6 Initial Conditions (Full Model) 1587.6.1 Load-Flow Formulation 1587.6.2 Standard Load Flow 1597.6.3 Initial Conditions for Dynamic Analysis 1607.6.4 Angle Reference, Infinite Bus, and COI Reference 1657.7 Numerical Solution: Power-Balance Form 1657.7.1 SI Method 1657.7.2 Review of Newton’s Method 1657.7.3 Numerical Solution Using SI Method 1667.7.4 Disturbance Simulation 1677.7.5 PE Method 1687.8 Numerical Solution: Current-Balance Form 1687.8.1 Some Practical Details 1707.8.2 Prediction 1717.9 Reduced-Order Multimachine Models 1717.9.1 Flux-Decay Model 1717.9.2 Generator Equations 1727.9.3 Stator Equations 1727.9.4 Network Equations 1727.9.5 Initial Conditions 1727.9.6 Structure-Preserving Classical Model 1737.9.7 Internal-Node Model 1777.10 Initial Conditions 1797.11 Conclusion 1807.12 Problems 1808 Small-Signal Stability 1838.1 Background 1838.2 Basic Linearization Technique 1848.2.1 Linearization of Model A 1858.2.2 Differential Equations 1858.2.3 Stator Algebraic Equations 1868.2.4 Network Equations 1868.2.5 Linearization of Model B 1938.2.6 Differential Equations 1948.2.7 Stator Algebraic Equations 1948.2.8 Network Equations 1948.3 Participation Factors 1948.4 Studies on Parametric Effects 1988.4.1 Effect of Loading 1988.4.2 Effect of KA 2008.4.3 Effect of Type of Load 2018.4.4 Hopf Bifurcation 2038.5 Electromechanical Oscillatory Modes 2058.5.1 Eigenvalues of A and A𝜔 2078.6 Power System Stabilizers 2098.6.1 Basic Approach 2098.6.2 Derivation of K1 − K6 Constants 2098.6.3 Linearization 2118.6.4 Synchronizing and Damping Torques 2158.6.5 Damping of Electromechanical Modes 2158.6.6 Torque-Angle Loop 2198.6.7 Synchronizing Torque 2218.6.8 Damping Torque 2218.6.9 Power System Stabilizer Design 2218.6.10 Frequency-Domain Approach 2228.6.11 Design Procedure Using the Frequency-Domain Method 2238.7 Conclusion 2278.8 Problems 2279 Energy Function Methods 2339.1 Background 2339.2 Physical and Mathematical Aspects of the Problem 2339.3 Lyapunov’s Method 2369.4 Modeling Issues 2379.5 Energy Function Formulation 2389.6 Potential Energy Boundary Surface (PEBS) 2419.6.1 Single-Machine Infinite-Bus System 2419.6.2 Energy Function for a Single-Machine Infinite-Bus System 2449.6.3 Equal-Area Criterion and the Energy Function 2479.6.4 Multimachine PEBS 2499.6.5 Initialization of VPE(𝜃) and its Use in PEBS Method 2529.7 The Boundary Controlling u.e.p (BCU) Method 2549.7.1 Algorithm 2569.8 Structure-Preserving Energy Functions 2599.9 Conclusion 2609.10 Problems 26010 Synchronized PhasorMeasurement 26310.1 Background 26310.2 Phasor Computation 26410.2.1 Nominal Frequency Phasors 26410.2.2 Off-Nominal Frequency Phasors 26510.2.3 Post Processing 26910.2.4 Positive-Sequence Signals 27110.2.5 Frequency Estimation 27210.2.6 Phasor Data Accuracy 27410.2.7 PMU Simulator 27510.3 Phasor Data Communication 27610.4 Power System Frequency Response 27710.5 Power System Disturbance Propagation 28010.5.1 Disturbance Triggering 28510.6 Power System Disturbance Signatures 28510.6.1 Generator or Load Trip 28610.6.2 Oscillations 28710.6.3 Fault and Line Switching 28810.6.4 Shunt Capacitor or Reactor Switching 28910.6.5 Voltage Collapse 28910.7 Phasor State Estimation 28910.8 Modal Analyses of Oscillations 29310.9 Energy Function Analysis 29610.10 Control Design Using PMU Data 29910.11 Conclusions and Remarks 30110.12 Problems 30211 Power SystemToolbox 30511.1 Background 30511.2 Power Flow Computation 30611.2.1 Data Requirement 30611.2.2 Power Flow Formulation and Solution 30811.2.3 Nonconvergent Power Flow 31111.3 Dynamic Simulation 31111.3.1 Dynamic Models and Per-Unit Parameter Values 31211.3.2 Initialization 31311.3.3 Network Solution 31411.3.4 Integration Methods 31611.3.5 Disturbance Specifications 31711.4 Linear Analysis 32111.5 Conclusions and Remarks 32411.6 Problems 324A IntegralManifolds for Model Reduction 327A.1 Manifolds and Integral Manifolds 327A.2 Integral Manifolds for Linear Systems 328A.3 Integral Manifolds for Nonlinear Systems 336Bibliography 341Index 353

Peter W. Sauer obtained his BS in Electrical Engineering from the University of Missouri at Rolla in 1969, and the MS and PhD degrees in Electrical Engineering from Purdue University in 1974 and 1977 respectively. He served as a facilities design engineer in the U.S. Air Force from 1969 to 1973. He is currently the Grainger Professor of Electrical Engineering at the University of Illinois, Urbana-Champaign where he has been since 1977. His main work is in modeling and simulation of power systems with applications to steady-state and transient stability analysis. He served as the program director for power systems at the National Science Foundation from 1990 to 1991. He was a cofounder of PowerWorld Corporation and the Power Systems Engineering Research Center (PSERC). He is a registered Professional Engineer in Virginia and Illinois, a Fellow of the IEEE, and a member of the U.S. National Academy of Engineering. M. A. Pai is Professor Emeritus in Electrical and Computer Engineering at the University of Illinois, Urbana-Champaign. He received his BE degree from Univ. of Madras, India in 1953, MS and PhD degrees from University of California, Berkeley in 1957 and 1961 respectively. He was with the Indian Institute of Technology, Kanpur, India from 1963 to 1981 and at the University of Illinois, Urbana-Champaign, from 1981 to 2003. His research interests are in dynamics and stability of power systems, smart grid, renewable resources and power system computation. He is the author of several text books and research monographs in these areas. He is a Fellow of IEEE, I.E. (India) and the Indian National Science Academy. Joe H. Chow is Professor of Electrical, Computer, and Systems Engineering at Rensselaer. He received his BS degrees in Electrical Engineering and Mathematics from the University of Minnesota, Minneapolis, in 1974, and his MS and PhD degrees from the University of Illinois, Urbana-Champaign, in 1975 and 1977. He worked in the power systems business at General Electric Company in 1978 and joined Rensselaer in 1987. His research interests include power system dynamics and control, voltage stability analysis, FACTS controllers, synchronized phasor measurements and applications, and integration of renewable resources. He is a fellow of IEEE, and past recipient of the Donald Eckman Award from the American Automatic Control Council, the Control Systems Technology Award from the IEEE Control Systems Society, and the Charles Concordia Power Systems Engineering Award from the IEEE Power and Energy Systems Society.

• Preface xiiiAbout the Companion Website xv1 Introduction 11.1 Background 11.2 Physical Structures 21.3 Time-Scale Structures 31.4 Political Structures 41.5 The Phenomena of Interest 51.6 New Chapters Added to this Edition 52 Electromagnetic Transients 72.1 The Fastest Transients 72.2 Transmission Line Models 72.3 Solution Methods 122.4 Problems 173 Synchronous Machine Modeling 193.1 Conventions and Notation 193.2 Three-Damper-Winding Model 203.3 Transformations and Scaling 213.4 The Linear Magnetic Circuit 293.5 The Nonlinear Magnetic Circuit 353.6 Single-Machine Steady State 403.7 Operational Impedances and Test Data 443.8 Problems 494 Synchronous Machine Control Models 534.1 Voltage and Speed Control Overview 534.2 Exciter Models 534.3 Voltage Regulator Models 584.4 Turbine Models 624.4.1 Hydroturbines 624.4.2 Steam Turbines 644.5 Speed Governor Models 674.6 Problems 705 Single-Machine Dynamic Models 715.1 Terminal Constraints 715.2 The Multi-Time-Scale Model 745.3 Elimination of Stator/Network Transients 765.4 The Two-Axis Model 815.5 The One-Axis (Flux-Decay) Model 835.6 The Classical Model 845.7 Damping Torques 865.8 Single-Machine Infinite-Bus System 905.9 Synchronous Machine Saturation 945.10 Problems 1006 Multimachine Dynamic Models 1016.1 The Synchronously Rotating Reference Frame 1016.2 Network and R-L Load Constraints 1036.3 Elimination of Stator/Network Transients 1056.3.1 Generalization of Network and Load Dynamic Models 1106.3.2 The Special Case of “Impedance Loads” 1126.4 Multimachine Two-Axis Model 1136.4.1 The Special Case of “Impedance Loads” 1156.5 Multimachine Flux–Decay Model 1166.5.1 The Special Case of “Impedance Loads” 1176.6 Multimachine Classical Model 1186.6.1 The Special Case of “Impedance Loads” 1196.7 Multimachine Damping Torques 1206.8 Multimachine Models with Saturation 1216.8.1 The Multimachine Two-Axis Model with Synchronous Machine Saturation 1236.8.2 The Multimachine Flux-Decay Model with Synchronous Machine Saturation 1246.9 Frequency During Transients 1266.10 Angle References and an Infinite Bus 1276.11 Automatic Generation Control (AGC) 1297 Multimachine Simulation 1357.1 Differential-Algebraic Model 1357.1.1 Generator Buses 1367.1.2 Load Buses 1377.2 Stator Algebraic Equations 1387.2.1 Polar Form 1387.2.2 Rectangular Form 1387.2.3 Alternate Form of Stator Algebraic Equations 1397.3 Network Equations 1407.3.1 Power-Balance Form 1407.3.2 Real Power Equations 1417.3.3 Reactive Power Equations 1417.3.4 Current-Balance Form 1427.4 Industry Model 1497.5 Simplification of the Two-Axis Model 1537.5.1 Simplification #1 (Neglecting Transient Saliency in the Synchronous Machine) 1537.5.2 Simplification #2 (Constant Impedance Load in the Transmission System) 1547.6 Initial Conditions (Full Model) 1587.6.1 Load-Flow Formulation 1587.6.2 Standard Load Flow 1597.6.3 Initial Conditions for Dynamic Analysis 1607.6.4 Angle Reference, Infinite Bus, and COI Reference 1657.7 Numerical Solution: Power-Balance Form 1657.7.1 SI Method 1657.7.2 Review of Newton’s Method 1657.7.3 Numerical Solution Using SI Method 1667.7.4 Disturbance Simulation 1677.7.5 PE Method 1687.8 Numerical Solution: Current-Balance Form 1687.8.1 Some Practical Details 1707.8.2 Prediction 1717.9 Reduced-Order Multimachine Models 1717.9.1 Flux-Decay Model 1717.9.2 Generator Equations 1727.9.3 Stator Equations 1727.9.4 Network Equations 1727.9.5 Initial Conditions 1727.9.6 Structure-Preserving Classical Model 1737.9.7 Internal-Node Model 1777.10 Initial Conditions 1797.11 Conclusion 1807.12 Problems 1808 Small-Signal Stability 1838.1 Background 1838.2 Basic Linearization Technique 1848.2.1 Linearization of Model A 1858.2.2 Differential Equations 1858.2.3 Stator Algebraic Equations 1868.2.4 Network Equations 1868.2.5 Linearization of Model B 1938.2.6 Differential Equations 1948.2.7 Stator Algebraic Equations 1948.2.8 Network Equations 1948.3 Participation Factors 1948.4 Studies on Parametric Effects 1988.4.1 Effect of Loading 1988.4.2 Effect of KA 2008.4.3 Effect of Type of Load 2018.4.4 Hopf Bifurcation 2038.5 Electromechanical Oscillatory Modes 2058.5.1 Eigenvalues of A and A𝜔 2078.6 Power System Stabilizers 2098.6.1 Basic Approach 2098.6.2 Derivation of K1 − K6 Constants 2098.6.3 Linearization 2118.6.4 Synchronizing and Damping Torques 2158.6.5 Damping of Electromechanical Modes 2158.6.6 Torque-Angle Loop 2198.6.7 Synchronizing Torque 2218.6.8 Damping Torque 2218.6.9 Power System Stabilizer Design 2218.6.10 Frequency-Domain Approach 2228.6.11 Design Procedure Using the Frequency-Domain Method 2238.7 Conclusion 2278.8 Problems 2279 Energy Function Methods 2339.1 Background 2339.2 Physical and Mathematical Aspects of the Problem 2339.3 Lyapunov’s Method 2369.4 Modeling Issues 2379.5 Energy Function Formulation 2389.6 Potential Energy Boundary Surface (PEBS) 2419.6.1 Single-Machine Infinite-Bus System 2419.6.2 Energy Function for a Single-Machine Infinite-Bus System 2449.6.3 Equal-Area Criterion and the Energy Function 2479.6.4 Multimachine PEBS 2499.6.5 Initialization of VPE(𝜃) and its Use in PEBS Method 2529.7 The Boundary Controlling u.e.p (BCU) Method 2549.7.1 Algorithm 2569.8 Structure-Preserving Energy Functions 2599.9 Conclusion 2609.10 Problems 26010 Synchronized PhasorMeasurement 26310.1 Background 26310.2 Phasor Computation 26410.2.1 Nominal Frequency Phasors 26410.2.2 Off-Nominal Frequency Phasors 26510.2.3 Post Processing 26910.2.4 Positive-Sequence Signals 27110.2.5 Frequency Estimation 27210.2.6 Phasor Data Accuracy 27410.2.7 PMU Simulator 27510.3 Phasor Data Communication 27610.4 Power System Frequency Response 27710.5 Power System Disturbance Propagation 28010.5.1 Disturbance Triggering 28510.6 Power System Disturbance Signatures 28510.6.1 Generator or Load Trip 28610.6.2 Oscillations 28710.6.3 Fault and Line Switching 28810.6.4 Shunt Capacitor or Reactor Switching 28910.6.5 Voltage Collapse 28910.7 Phasor State Estimation 28910.8 Modal Analyses of Oscillations 29310.9 Energy Function Analysis 29610.10 Control Design Using PMU Data 29910.11 Conclusions and Remarks 30110.12 Problems 30211 Power SystemToolbox 30511.1 Background 30511.2 Power Flow Computation 30611.2.1 Data Requirement 30611.2.2 Power Flow Formulation and Solution 30811.2.3 Nonconvergent Power Flow 31111.3 Dynamic Simulation 31111.3.1 Dynamic Models and Per-Unit Parameter Values 31211.3.2 Initialization 31311.3.3 Network Solution 31411.3.4 Integration Methods 31611.3.5 Disturbance Specifications 31711.4 Linear Analysis 32111.5 Conclusions and Remarks 32411.6 Problems 324A IntegralManifolds for Model Reduction 327A.1 Manifolds and Integral Manifolds 327A.2 Integral Manifolds for Linear Systems 328A.3 Integral Manifolds for Nonlinear Systems 336Bibliography 341Index 353

Peter W. Sauer obtained his BS in Electrical Engineering from the University of Missouri at Rolla in 1969, and the MS and PhD degrees in Electrical Engineering from Purdue University in 1974 and 1977 respectively. He served as a facilities design engineer in the U.S. Air Force from 1969 to 1973. He is currently the Grainger Professor of Electrical Engineering at the University of Illinois, Urbana-Champaign where he has been since 1977. His main work is in modeling and simulation of power systems with applications to steady-state and transient stability analysis. He served as the program director for power systems at the National Science Foundation from 1990 to 1991. He was a cofounder of PowerWorld Corporation and the Power Systems Engineering Research Center (PSERC). He is a registered Professional Engineer in Virginia and Illinois, a Fellow of the IEEE, and a member of the U.S. National Academy of Engineering. M. A. Pai is Professor Emeritus in Electrical and Computer Engineering at the University of Illinois, Urbana-Champaign. He received his BE degree from Univ. of Madras, India in 1953, MS and PhD degrees from University of California, Berkeley in 1957 and 1961 respectively. He was with the Indian Institute of Technology, Kanpur, India from 1963 to 1981 and at the University of Illinois, Urbana-Champaign, from 1981 to 2003. His research interests are in dynamics and stability of power systems, smart grid, renewable resources and power system computation. He is the author of several text books and research monographs in these areas. He is a Fellow of IEEE, I.E. (India) and the Indian National Science Academy. Joe H. Chow is Professor of Electrical, Computer, and Systems Engineering at Rensselaer. He received his BS degrees in Electrical Engineering and Mathematics from the University of Minnesota, Minneapolis, in 1974, and his MS and PhD degrees from the University of Illinois, Urbana-Champaign, in 1975 and 1977. He worked in the power systems business at General Electric Company in 1978 and joined Rensselaer in 1987. His research interests include power system dynamics and control, voltage stability analysis, FACTS controllers, synchronized phasor measurements and applications, and integration of renewable resources. He is a fellow of IEEE, and past recipient of the Donald Eckman Award from the American Automatic Control Council, the Control Systems Technology Award from the IEEE Control Systems Society, and the Charles Concordia Power Systems Engineering Award from the IEEE Power and Energy Systems Society.

• Preface xiiiAbout the Companion Website xv1 Introduction 11.1 Background 11.2 Physical Structures 21.3 Time-Scale Structures 31.4 Political Structures 41.5 The Phenomena of Interest 51.6 New Chapters Added to this Edition 52 Electromagnetic Transients 72.1 The Fastest Transients 72.2 Transmission Line Models 72.3 Solution Methods 122.4 Problems 173 Synchronous Machine Modeling 193.1 Conventions and Notation 193.2 Three-Damper-Winding Model 203.3 Transformations and Scaling 213.4 The Linear Magnetic Circuit 293.5 The Nonlinear Magnetic Circuit 353.6 Single-Machine Steady State 403.7 Operational Impedances and Test Data 443.8 Problems 494 Synchronous Machine Control Models 534.1 Voltage and Speed Control Overview 534.2 Exciter Models 534.3 Voltage Regulator Models 584.4 Turbine Models 624.4.1 Hydroturbines 624.4.2 Steam Turbines 644.5 Speed Governor Models 674.6 Problems 705 Single-Machine Dynamic Models 715.1 Terminal Constraints 715.2 The Multi-Time-Scale Model 745.3 Elimination of Stator/Network Transients 765.4 The Two-Axis Model 815.5 The One-Axis (Flux-Decay) Model 835.6 The Classical Model 845.7 Damping Torques 865.8 Single-Machine Infinite-Bus System 905.9 Synchronous Machine Saturation 945.10 Problems 1006 Multimachine Dynamic Models 1016.1 The Synchronously Rotating Reference Frame 1016.2 Network and R-L Load Constraints 1036.3 Elimination of Stator/Network Transients 1056.3.1 Generalization of Network and Load Dynamic Models 1106.3.2 The Special Case of “Impedance Loads” 1126.4 Multimachine Two-Axis Model 1136.4.1 The Special Case of “Impedance Loads” 1156.5 Multimachine Flux–Decay Model 1166.5.1 The Special Case of “Impedance Loads” 1176.6 Multimachine Classical Model 1186.6.1 The Special Case of “Impedance Loads” 1196.7 Multimachine Damping Torques 1206.8 Multimachine Models with Saturation 1216.8.1 The Multimachine Two-Axis Model with Synchronous Machine Saturation 1236.8.2 The Multimachine Flux-Decay Model with Synchronous Machine Saturation 1246.9 Frequency During Transients 1266.10 Angle References and an Infinite Bus 1276.11 Automatic Generation Control (AGC) 1297 Multimachine Simulation 1357.1 Differential-Algebraic Model 1357.1.1 Generator Buses 1367.1.2 Load Buses 1377.2 Stator Algebraic Equations 1387.2.1 Polar Form 1387.2.2 Rectangular Form 1387.2.3 Alternate Form of Stator Algebraic Equations 1397.3 Network Equations 1407.3.1 Power-Balance Form 1407.3.2 Real Power Equations 1417.3.3 Reactive Power Equations 1417.3.4 Current-Balance Form 1427.4 Industry Model 1497.5 Simplification of the Two-Axis Model 1537.5.1 Simplification #1 (Neglecting Transient Saliency in the Synchronous Machine) 1537.5.2 Simplification #2 (Constant Impedance Load in the Transmission System) 1547.6 Initial Conditions (Full Model) 1587.6.1 Load-Flow Formulation 1587.6.2 Standard Load Flow 1597.6.3 Initial Conditions for Dynamic Analysis 1607.6.4 Angle Reference, Infinite Bus, and COI Reference 1657.7 Numerical Solution: Power-Balance Form 1657.7.1 SI Method 1657.7.2 Review of Newton’s Method 1657.7.3 Numerical Solution Using SI Method 1667.7.4 Disturbance Simulation 1677.7.5 PE Method 1687.8 Numerical Solution: Current-Balance Form 1687.8.1 Some Practical Details 1707.8.2 Prediction 1717.9 Reduced-Order Multimachine Models 1717.9.1 Flux-Decay Model 1717.9.2 Generator Equations 1727.9.3 Stator Equations 1727.9.4 Network Equations 1727.9.5 Initial Conditions 1727.9.6 Structure-Preserving Classical Model 1737.9.7 Internal-Node Model 1777.10 Initial Conditions 1797.11 Conclusion 1807.12 Problems 1808 Small-Signal Stability 1838.1 Background 1838.2 Basic Linearization Technique 1848.2.1 Linearization of Model A 1858.2.2 Differential Equations 1858.2.3 Stator Algebraic Equations 1868.2.4 Network Equations 1868.2.5 Linearization of Model B 1938.2.6 Differential Equations 1948.2.7 Stator Algebraic Equations 1948.2.8 Network Equations 1948.3 Participation Factors 1948.4 Studies on Parametric Effects 1988.4.1 Effect of Loading 1988.4.2 Effect of KA 2008.4.3 Effect of Type of Load 2018.4.4 Hopf Bifurcation 2038.5 Electromechanical Oscillatory Modes 2058.5.1 Eigenvalues of A and A𝜔 2078.6 Power System Stabilizers 2098.6.1 Basic Approach 2098.6.2 Derivation of K1 − K6 Constants 2098.6.3 Linearization 2118.6.4 Synchronizing and Damping Torques 2158.6.5 Damping of Electromechanical Modes 2158.6.6 Torque-Angle Loop 2198.6.7 Synchronizing Torque 2218.6.8 Damping Torque 2218.6.9 Power System Stabilizer Design 2218.6.10 Frequency-Domain Approach 2228.6.11 Design Procedure Using the Frequency-Domain Method 2238.7 Conclusion 2278.8 Problems 2279 Energy Function Methods 2339.1 Background 2339.2 Physical and Mathematical Aspects of the Problem 2339.3 Lyapunov’s Method 2369.4 Modeling Issues 2379.5 Energy Function Formulation 2389.6 Potential Energy Boundary Surface (PEBS) 2419.6.1 Single-Machine Infinite-Bus System 2419.6.2 Energy Function for a Single-Machine Infinite-Bus System 2449.6.3 Equal-Area Criterion and the Energy Function 2479.6.4 Multimachine PEBS 2499.6.5 Initialization of VPE(𝜃) and its Use in PEBS Method 2529.7 The Boundary Controlling u.e.p (BCU) Method 2549.7.1 Algorithm 2569.8 Structure-Preserving Energy Functions 2599.9 Conclusion 2609.10 Problems 26010 Synchronized PhasorMeasurement 26310.1 Background 26310.2 Phasor Computation 26410.2.1 Nominal Frequency Phasors 26410.2.2 Off-Nominal Frequency Phasors 26510.2.3 Post Processing 26910.2.4 Positive-Sequence Signals 27110.2.5 Frequency Estimation 27210.2.6 Phasor Data Accuracy 27410.2.7 PMU Simulator 27510.3 Phasor Data Communication 27610.4 Power System Frequency Response 27710.5 Power System Disturbance Propagation 28010.5.1 Disturbance Triggering 28510.6 Power System Disturbance Signatures 28510.6.1 Generator or Load Trip 28610.6.2 Oscillations 28710.6.3 Fault and Line Switching 28810.6.4 Shunt Capacitor or Reactor Switching 28910.6.5 Voltage Collapse 28910.7 Phasor State Estimation 28910.8 Modal Analyses of Oscillations 29310.9 Energy Function Analysis 29610.10 Control Design Using PMU Data 29910.11 Conclusions and Remarks 30110.12 Problems 30211 Power SystemToolbox 30511.1 Background 30511.2 Power Flow Computation 30611.2.1 Data Requirement 30611.2.2 Power Flow Formulation and Solution 30811.2.3 Nonconvergent Power Flow 31111.3 Dynamic Simulation 31111.3.1 Dynamic Models and Per-Unit Parameter Values 31211.3.2 Initialization 31311.3.3 Network Solution 31411.3.4 Integration Methods 31611.3.5 Disturbance Specifications 31711.4 Linear Analysis 32111.5 Conclusions and Remarks 32411.6 Problems 324A IntegralManifolds for Model Reduction 327A.1 Manifolds and Integral Manifolds 327A.2 Integral Manifolds for Linear Systems 328A.3 Integral Manifolds for Nonlinear Systems 336Bibliography 341Index 353