Power System Control and Stability 3rd Edition by Paul M. Anderson

CONTENTS xiii

FOREWORD

xv

PREFACE

xvii

ABOUT THE AUTHORS PART I CHAPTER 1

1.1 1.2 1.3

1.4

1.5

1.6

2.4

2.5

2.6 2.7 2.8

2.9 2.10

3

POWER SYSTEM STABILITY

Introduction 3 Requirements of a Reliable Electrical Power Service Statement of the Problem 5 1.3.1 Definition of Stability 5 1.3.2 Classification of Stability Problems 6 1.3.3 Description of Stability Phenomenon 6 Effect of Impact on System Components 7 1.4.1 Loss of Synchronism 8 1.4.2 Synchronous Machine During a Transient 8 Methods of Simulation 10 1.5.1 Linearized System Equations 10 1.5.2 Large System with Nonlinear Equations 11 Planning and Operating Standards 11

CHAPTER 2

2.1 2.2 2.3

INTRODUCTION

4

19

THE ELEMENTARY MATHEMATICAL MODEL

Swing Equation 19 Units 21 Mechanical Torque 22 2.3.1 Unregulated Machines 22 2.3.2 Regulated Machines 24 Electrical Torque 26 2.4.1 Synchronous Torque 26 2.4.2 Other Electrical Torques 27 Power-Angle Curve of a Synchronous Machine 27 2.5.1 Classical Representation of a Synchronous Machine in Stability Studies 2.5.2 Synchronizing Power Coefficients 29 Natural Frequencies of Oscillation of a Synchronous Machine 30 System of One Machine Against an Infinite Bus: The Classical Model 31 Equal Area Criterion 37 2.8.1 Critical Clearing Angle 38 2.8.2 Application to a One-Machine System 39 2.8.3 Equal Area Criterion for a Two-Machine System 39 Classical Model of a Multimachine System 40 Classical Stability Study of a Nine-Bus System 42 2.10.1 Data Preparation 43 2.10.2 Preliminary Calculations 45

28

v

vi 2.11 2.12

CONTENTS

Shortcomings of the Classical Model Block Diagram of One Machine 53

CHAPTER 3

3.1 3.2

3.3

3.4 3.5

3.6

CHAPTER 4

4.4 4.5 4.6 4.7

4.8 4.9

4.10 4.11 4.12

61

SYSTEM RESPONSE TO SMALL DISTURBANCES

Introduction 61 Types of Problems Studied 62 3.2.1 System Response to Small Impacts 62 3.2.2 Distribution of Power Impacts 62 The Unregulated Synchronous Machine 63 3.3.1 Demagnetizing Effect of Armature Reaction 64 3.3.2 Effect of Small Changes of Speed 65 Modes of Oscillation of an Unregulated Multimachine System Regulated Synchronous Machine 73 3.5.1 Voltage Regulator with One Time Lag 73 3.5.2 Governor with One Time Lag 75 Distribution of Power Impacts 76 3.6.1 Linearization 77 3.6.2 A Special Case: t = 0+ 78 3.6.3 Average Behavior Prior to Governor Action (t = t1)

PART II

4.1 4.2 4.3

51

66

79

ELECTRICAL AND ELECTROMAGNETIC DYNAMIC PERFORMANCE 91

THE SYNCHRONOUS MACHINE

Introduction 91 Park’s Transformation 91 Flux Linkage Equations 94 4.3.1 Stator Self-Inductances 94 4.3.2 Rotor Self-Inductances 95 4.3.3 Stator Mutual Inductances 95 4.3.4 Rotor Mutual Inductances 95 4.3.5 Stator-to-Rotor Mutual Inductances 95 4.3.6 Transformation of Inductances 96 Voltage Equations 97 Formulation of State-Space Equations 99 Current Formulation 100 Per-Unit Conversion 101 4.7.1 Choosing a Base for Stator Quantities 102 4.7.2 Choosing a Base for Rotor Quantities 103 4.7.3 Comparison with Other Per-Unit Systems 104 4.7.4 The Correspondence of Per-Unit Stator EMF to Rotor Quantities Normalizing the Voltage Equations 108 Normalizing the Torque Equations 113 4.9.1 The Normalized Swing Equation 114 4.9.2 Forms of the Swing Equation 114 Torque and Power 115 Equivalent Circuit of a Synchronous Machine 117 The Flux Linkage State-Space Model 119 4.12.1 The Voltage Equations 120 4.12.2 The Torque Equation 120 4.12.3 Machine Equations with Saturation Neglected 121 4.12.4 Treatment of Saturation 123

107

CONTENTS

4.13

4.14 4.15

4.16

Load Equations 124 4.13.1 Synchronous Machine Connected to an Infinite Bus 124 4.13.2 Current Model 126 4.13.3 The Flux Linkage Model 127 Subtransient and Transient Inductances and Time Constants 131 4.14.1 Time Constants 133 Simplified Models of the Synchronous Machine 136 4.15.1 Neglecting Damper Windings: The Eq (One-Axis) Model 137 4.15.2 Voltage Behind Subtransient Reactance: The E Model 142 4.15.3 Neglecting λd and λq for a Cylindrical Rotor Machine: The Two-Axis Model 4.15.4 Neglecting Amortisseur Effects and λd and λq Terms: The One-Axis Model 4.15.5 Assuming Constant Flux Linkage in the Main Field Winding 154 Parameter Determination for Generator Dynamic Models 155

CHAPTER 5

5.1 5.2 5.3 5.4

5.5 5.6 5.7 5.8 5.9

6.1 6.2 6.3

6.4

6.5 6.6 6.7 6.8 6.9

7.1 7.2

SIMULATION OF MULTIMACHINE SYSTEMS

Introduction 239 Statement of the Problem

153

169

199

LOAD MODELING

Introduction 199 Static Load Models 200 Induction Motor Loads 203 6.3.1 Model Development of a Three-Phase Induction Machine 6.3.2 Representing Induction Machines in Stability Simulations 6.3.3 Stalled Motor Operation 215 Single-Phase Motors 216 6.4.1 Scroll Compressors 218 6.4.2 Point-on-Wave Effects 219 6.4.3 Dynamic Phasors 219 Power Electronic Loads 221 Self-Restoring Loads 224 Distributed Energy Resources 225 Composite Load Models 227 Data Development 229 6.9.1 Component Based 230 6.9.2 Measurement Based 232

CHAPTER 7

150

165

THE SIMULATION OF SYNCHRONOUS MACHINES

Introduction 165 Steady-State Equations and Phasor Diagrams 165 Machine Connected to an Infinite Bus Through a Transmission Line 168 Machine Connected to an Infinite Bus with Local Load at Machine Terminal 5.4.1 Special Case: The Resistive Load, Z L = RL + j0 170 5.4.2 General Case: Z L Arbitrary 171 Determining Steady-State Conditions 172 5.5.1 Machine Connected to an Infinite Bus with Local Load 173 Examples 174 Initial Conditions for a Multimachine System 182 Determination of Machine Parameters from Manufacturers’ Data 183 Digital Simulation of Synchronous Machines 188 5.9.1 Digital Computation of Saturation 189 5.9.2 Updating λAD 192

CHAPTER 6

vii

239

203 213

239

viii

CONTENTS

7.3

Matrix Representation of a Passive Network 240 7.3.1 Network in the Transient State 242 7.3.2 Converting to a Common Reference Frame 243 Converting Machine Coordinates to System Reference 244 Relation Between Machine Currents and Voltages 245 System Order 249 Machines Represented by Classical Methods 249 Linearized Model for the Network 252 Hybrid Formulation 258 Network Equations with Flux Linkage Model 260 Total System Equations 262 Alternating Solution Method 264 7.12.1 Nonlinear Loads 265 7.12.2 Network–Machine Interface 268 Simultaneous Solution Method 275 Design of Numerical Solvers 277

7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12

7.13 7.14

CHAPTER 8

8.1 8.2 8.3 8.4 8.5 8.6

8.7

8.8 8.9

Introduction 281 Fundamentals of Linear System Stability 282 Linearization of the Generator State-Space Current Model 284 Linearization of the Load Equation for the One-Machine Problem Linearization of the Flux Linkage Model 293 State Matrix for Multimachine Systems 298 8.6.1 Formulation of the State Matrix 298 8.6.2 Representation of Static Loads in the State Matrix 300 Simplified Linear Model 312 8.7.1 The E Equation 312 8.7.2 Electrical Torque Equation 313 8.7.3 Terminal Voltage Equation 314 8.7.4 Summary of Equations 315 8.7.5 Effect of Loading 318 8.7.6 Comparison with Classical Model 320 Block Diagrams 321 State-Space Representation of Simplified Model 322

CHAPTER 9

9.1 9.2 9.3

9.4

281

SMALL-SIGNAL STABILITY ANALYSIS

288

EXCITATION SYSTEMS

Simplified View of Excitation Control 325 Control Configurations 327 Typical Excitation Configurations 328 9.3.1 Primitive Systems 328 9.3.2 Type DC Excitation Control Systems with DC Generator-Commutator Exciters 332 9.3.3 Type AC Excitation Control Systems with Alternator-Rectifier Exciters 332 9.3.4 Type AC Excitation Control Systems with Alternator-SCR Exciter Systems 334 9.3.5 Type ST Excitation Control Systems with Compound-Rectifier Exciter Systems 335 9.3.6 Type ST Excitation Control System with Compound-Rectifier Exciter Plus Potential-Source-Rectifier Exciter 336 9.3.7 Type ST Excitation Control Systems with Potential-Source-Rectifier Exciter 336 Excitation Control System Definitions 337 9.4.1 Voltage Response Ratio 339 9.4.2 Exciter Voltage Ratings 341 9.4.3 Other Specifications 342

325

CONTENTS

9.5

9.6

9.7

9.8

9.9

9.10

9.11 9.12

Voltage Regulator 344 9.5.1 Electromechanical Regulators 344 9.5.2 Early Electronic Regulators 345 9.5.3 Rotating Amplifier Regulators 345 9.5.4 Magnetic Amplifier Regulators 346 9.5.5 Digital Excitation Systems 348 Exciter Buildup 348 9.6.1 The DC Generator Exciter 348 9.6.2 Linear Approximations for DC Generator Exciters 356 9.6.3 The AC Generator Exciters 358 9.6.4 Solid-State Exciters 359 9.6.5 Buildup of a Loaded DC Exciter 360 9.6.6 Normalization of Exciter Equations 360 Limiting and Protection for Excitation Control Systems 361 9.7.1 Modeling Amplifier Limits 361 9.7.2 Control Limiters and Associated Protection 362 9.7.3 Volts per Hertz Protection 365 Excitation System Response 365 9.8.1 Noncontinuously Regulated Systems 365 9.8.2 Continuously Regulated Systems 369 State-Space Description of the Excitation System 379 9.9.1 Simplified Linear Model 381 9.9.2 Complete Linear Model 382 Computer Representation of Excitation Systems 389 9.10.1 Type DC1: DC Commutator Exciter 390 9.10.2 Type AC Systems: Alternator Supplied Rectifier Excitation Systems 393 9.10.3 Type AC1 System: Field-Controlled Alternator-Rectifier Excitation System 394 9.10.4 Type ST1 System: Controlled Rectifier System with Terminal Potential Supply Only 9.10.5 Type ST2 System: Static with Terminal Potential and Current Supplies 397 9.10.6 Type DC3 System: Noncontinuous Acting 399 Typical System Constants 400 The Effect of Excitation on Generator Performance 400

CHAPTER 10

10.1 10.2 10.3

10.4

10.5 10.6

10.7

THE EFFECT OF EXCITATION ON STABILITY

Introduction 409 10.1.1 Transient Stability and Small-Signal Stability Considerations 410 Effect of Excitation on Generator Power Limits 411 Effect of the Excitation System on Transient Stability 415 10.3.1 The Role of the Excitation System in Classical Model Studies 415 10.3.2 Increased Reliance on Excitation Control to Improve Stability 417 10.3.3 Parametric Study 419 10.3.4 Reactive Power Demand During System Emergencies 421 Effect of Excitation on Small-Signal Stability 421 10.4.1 Examination of Small-Signal Stability by Routh’s Criterion 421 10.4.2 Further Considerations of the Regulator Gain and Time Constant 424 10.4.3 Effect on the Electrical Torque 425 Root-Locus Analysis of a Regulated Machine Connected to an Infinite Bus 426 Approximate System Representation 432 10.6.1 Approximate Excitation System Representation 432 10.6.2 Estimate of Gx(s) 433 10.6.3 The Inertial Transfer Function 437 Supplementary Stabilizing Signals 439 10.7.1 Block Diagram of the Linear System 439

ix

395

409

x

CONTENTS

10.8 10.9 10.10 10.11

10.12

CHAPTER 11

11.1

11.2

13.3

13.4 13.5

14.1 14.2

487

VOLTAGE STABILITY

497

DYNAMIC PERFORMANCE AND MODELING OF FLEXIBLE AC TRANSMISSION SYSTEM (FACTS) COMPONENTS

Introduction 503 Static VAr System 503 13.2.1 Stability Characteristics of an SVS 506 13.2.2 Positive-Sequence Transient Stability Model for SVS Thyristor-Controlled Series Compensation 511 13.3.1 Operating Modes of a TCSC 512 13.3.2 Equipment Characteristics and Limiting Conditions 13.3.3 TCSC Model for Transient Stability Studies 515 Static Synchronous Compensator 517 13.4.1 Statcom Model for Transient Stability Studies 519 High Voltage DC Transmission 519

CHAPTER 14

463

481

Modeling Requirements for Voltage Instability Analysis 487 Voltage Instability Analysis Using Time Domain Simulation 489 Dynamic VAr Planning and Optimization 493 12.3.1 Trajectory Sensitivity Analysis 493 12.3.2 Formulation of the VAr Optimization Problem 495 12.3.3 Implementation of the Dynamic VAr Optimization Approach 12.3.4 Application of Dynamic VAr Optimization Approach 499

CHAPTER 13

13.1 13.2

DYNAMIC MODELING AND REPRESENTATION OF RENEWABLE ENERGY RESOURCES

Wind Turbine Generators 463 11.1.1 Type 1 WTGs 465 11.1.2 Type 2 WTGs 466 11.1.3 Type 3 WTGs 467 11.1.4 Type 4 WTGs 479 Photovoltaic Solar Plant Modeling 480 11.2.1 Generic Model of PV Solar Plant 480 11.2.2 Modified Generic Model of PV Solar Plant

CHAPTER 12

12.1 12.2 12.3

440

10.7.2 Approximate Model of the Complete Exciter-Generator System 10.7.3 Lead Compensation 442 Linear Analysis of the Stabilized Generator 446 PSS Tuning in Multimachine Power Systems 448 Alternate Types of PSS 449 Digital Computer Transient Stability Studies 450 10.11.1 Effect of Fault Duration 452 10.11.2 Effect of the Power System Stabilizer 457 Some General Comments on the Effect of Excitation on Stability 459

503

509

513

POWER SYSTEM PROTECTION AND MONITORING ASSOCIATED WITH POWER SYSTEM STABILITY

Introduction 525 Power System Protection Functions Associated with Transient Stability Analysis 14.2.1 Bulk Transmission Line Out-of-Step Protection 527 14.2.2 Generator Out-of-Step Protection 533

525

527

CONTENTS

14.3

14.4

14.2.3 Undervoltage Load Shedding 533 14.2.4 Underfrequency Load Shedding 534 Special Protection Schemes 535 14.3.1 Generation Rejection and Load Shedding 535 14.3.2 Controlled Islanding and Load Shedding 535 Synchrophasor-Based Monitoring of Power System Stability 537 14.4.1 Online Dynamic Security Assessment Using Synchrophasor Measurements and Decision Trees 537 14.4.2 Island Formation Prediction Scheme Supported by PMU Measurements 539 14.4.3 Real-Time Voltage Security and Oscillation Monitoring Using PMU Measurements

PART III CHAPTER 15

15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8

16.1 16.2

16.3 16.4 16.5 16.6 16.7 16.8 16.9

16.10

17.1 17.2 17.3 17.4 17.5 17.6 17.7

545

SPEED GOVERNING

571 577

STEAM TURBINE PRIME MOVERS

Introduction 577 Power Plant Control Modes 579 16.2.1 The Turbine-Following Control Mode 579 16.2.2 The Boiler-Following Control Mode 579 16.2.3 The Coordinated Control Mode 580 Thermal Generation 581 A Steam Power Plant Model 582 Steam Turbines 583 Steam Turbine Control Operations 590 Steam Turbine Control Functions 592 Steam Generator Control 604 Fossil-Fueled Boilers 605 16.9.1 Drum-Type Boilers 606 16.9.2 Once-Through Boilers 613 16.9.3 Computer Models of Fossil-Fueled Boilers Nuclear Steam Supply Systems 620 16.10.1 Boiling Water Reactors 620 16.10.2 Pressurized Water Reactors 620

CHAPTER 17

540

MECHANICAL DYNAMIC PERFORMANCE

The Flyball Governor 546 The Isochronous Governor 551 Incremental Equations of the Turbine 553 The Speed Droop Governor 556 The Floating Lever Speed Droop Governor 561 The Compensated Governor 564 Electronic Governors 570 Governor Models for Transient Stability Simulations

CHAPTER 16

xi

HYDRAULIC TURBINE PRIME MOVERS

Introduction 627 The Impulse Turbine 627 The Reaction Turbine 629 Propeller-Type Turbines 631 The Deriaz Turbine 632 Conduits, Surge Tanks, and Penstocks Hydraulic System Equations 639

633

617

627

xii

CONTENTS

17.8 17.9 17.10 17.11 17.12

Hydraulic System Transfer Function 644 Simplifying Assumptions 647 Block Diagram for a Hydro System 649 Pumped-Storage Hydro Systems 650 Representation of Hydro Turbines and Governors in Stability Studies

CHAPTER 18

18.1 18.2

18.3

651

COMBUSTION TURBINE AND COMBINED-CYCLE POWER PLANTS

Introduction 655 The Combustion Turbine Prime Mover 655 18.2.1 Combustion Turbine Control 657 18.2.2 Off-Nominal Frequency and Voltage Effects 658 18.2.3 Nonlinear Governor Droop Characteristic 659 18.2.4 Recent Advances in Modeling Gas Turbines 660 The Combined-Cycle Prime Mover 663 18.3.1 Fuel and Air Controls 664 18.3.2 The Gas Turbine Power Generation 668 18.3.3 The Steam Turbine Power Generation 669 18.3.4 Recent Development in Modeling Combined-Cycle Plants

655

671

APPENDIX A

673

APPENDIX B

675

APPENDIX C

685

APPENDIX D

695

APPENDIX E

727

APPENDIX F

737

APPENDIX G

759

APPENDIX H

767

APPENDIX I

775

APPENDIX J

783

INDEX

793

CHAPTER

1

POWER SYSTEM STABILITY

1.1 INTRODUCTION Since the industrial revolution the worldwide demand for and consumption of energy has increased steadily. The invention of the induction motor by Nikola Tesla in 1888 signaled the growing importance of electrical energy in the industrial world as well as its use for artificial lighting. A major portion of the energy needs of a modern society is supplied in the form of electrical energy. Industrially developed societies need an economical and reliable supply of electrical power, and the demand on the North American continent has steadily grown and stabilized since 2015. Very complex power systems have been built to satisfy this demand. Increasingly the demand is being met by renewable resources consisting primarily of wind and solar generation. The trend in electric power production is toward an interconnected network of transmission lines, linking generators and loads into large integrated systems, some of which span entire continents. Indeed, in the United States and Canada, generators located thousands of miles apart operate in parallel. This vast enterprise of supplying electrical energy presents many engineering problems that provide the engineer with a variety of challenges. The planning, construction, and operation of such systems become exceedingly complex. Some of the problems stimulate the engineer’s managerial talents; others tax his/her knowledge and experience in system design. The entire design must be predicated on automatic control and not on the slow response of human operators. To be able to predict the performance of such complex systems, the engineer is forced to seek ever more powerful tools of analysis and synthesis. This book is concerned with some aspects of the design problem, particularly the dynamic performance of interconnected power systems. Characteristics of the various components of a power system during normal operating conditions and during disturbances will be examined, and effects on the overall system performance will be analyzed. Emphasis will be given to the transient behavior in which the system is described mathematically by ordinary differential equations. Detailed analysis and modeling of synchronous generators is introduced. In this edition of the book, an enhancement of the synchronous machine model including a fictitious G-winding is introduced to extend the accuracy of the model. The previous edition of the book primarily focused on the modeling of generation associated with synchronous generators. Additionally, since the previous edition of the book was published, a range of new devices for generating and control of electric energy have been developed. This edition incorporates these changes as identified below. Significant new material on modeling and analyzing renewable energy sources, primarily consisting of wind and solar, is included. These resources are interconnected to the electric grid via power electronic interfaces. Salient features of the power electronic interface and the associated dynamic characteristics are also introduced and detailed. Power System Control and Stability, Third Edition. Vijay Vittal, James D. McCalley, Paul M. Anderson, and A. A. Fouad. © 2020 by The Institute of Electrical and Electronic Engineers, Inc. Published 2020 by John Wiley & Sons, Inc.

3

4

CHAPTER 1

POWER SYSTEM STABILITY

A joint effort by the IEEE Power and Energy Society (PES) and CIGRÉ (International Council on Large Electric Systems) has led to a systematic definition of power system stability and its various manifestations. These new definitions are presented and discussed. The topic of voltage stability, which has gained significance around the world, is introduced. Large-signal voltage stability analysis is also examined. Power system stabilizers have played a critical role in stabilizing power system oscillations. Major advances in designing and tuning power system stabilizers for multimachine systems have been designed and implemented. These new advancements are introduced and described. Flexible AC transmission system (FACTS) devices have played a significant role in shaping and altering power system dynamic behavior, and so FACTS devices and their modeling and representation are introduced and presented. The chapter on small-signal stability analysis has been expanded to include a detailed treatment of multimachine small-signal stability analysis, together with an extensive description of the formulation of the A matrix for multimachine systems. Additionally, new material on modeling and incorporating dynamic loads is included. Advanced models for induction machines, motor drives, and performance models for single-phase induction machines are also introduced. A new chapter on representation of protections systems that are critical in transient stability analysis has also been introduced.

1.2 REQUIREMENTS OF A RELIABLE ELECTRICAL POWER SERVICE The interconnected power system is made up of an elaborate and complex interconnection of power system components. When we refer to the reliability and security of the electrical power system, we are primarily interested in what is referred to as the “bulk electric system” (BES). According to the definition provided by the North American Electric Reliability Corporation (NERC), the BES consists of all transmission elements operated at 100 kV or higher and active and reactive power resources connected at 100 kV or higher. Any interruptions in the BES are considered very serious, as many users are affected, and can result in significant economic impacts. Thus, much effort is invested in avoiding interruptions to the BES. The reliability of the interconnected BES is broadly defined using the following two terms: Adequacy – The ability of the electric systems to supply the aggregate electrical demand and energy requirements of customers at all times taking into account scheduled and reasonably expected unscheduled outages of system elements. Security – The ability of the electric systems to withstand sudden disturbances such as electric short circuits or unanticipated loss of system elements. The following are among the more important requirements of a reliable electric power service: • Voltage and frequency must be held within close tolerances. • Synchronous generators must be kept running in synchronism with adequate capacity to meet the load demand. • The “integrity” of the BES must be maintained to avoid cascading outages. In the United States, NERC serves as the electric reliability organization (ERO) for the Federal Energy Regulatory Commission (FERC). The main goal of NERC is to augment the reliability of the BES in the electricity systems of North America. NERC is composed of eight regional reliability councils and encompasses all the electric power systems in the United States and Canada and a small portion of Mexico. NERC sets the reliability standards and criteria for planning and operating the interconnected network in the United States and Canada. When designing and operating the interconnected electric power network, it is necessary to take into consideration the dynamic performance of the system because power systems are subjected to changes (small and large). It is also important that

1.3 STATEMENT OF THE PROBLEM

5

when these changes are completed and the system settles to new operating conditions, no constraints are violated. In other words, not only should the new operating conditions be acceptable (as revealed by steady-state analysis), but also the system must survive the transition to the new conditions. The study of the survival of the transition requires dynamic analysis. An important aspect of reliability is the ability of the system to withstand sudden changes called disturbances. A disturbance could be a sudden change in a system parameter or operating condition. A disturbance is characterized as “small” when the equations governing system behavior can be linearized for the purpose of analysis. The analysis of system stability behavior when subjected to small disturbances is referred to as small-signal stability analysis. On the other hand, a disturbance is characterized as “large” when the equations governing the system behavior cannot be linearized for the purpose of analysis. The analysis of system behavior when subjected to large disturbances is referred to as transient stability analysis.

1.3 STATEMENT OF THE PROBLEM Over the years, several textbooks, papers, and documents have provided definitions of power system stability. There also exist strict analytical definitions of stability for nonlinear systems. These earlier efforts did not fully encompass existing needs, practical experiences, and understanding of the problems at hand. In 2000, a joint IEEE PES and CIGRÉ effort was initiated to define and classify power system stability. The task force created for this effort produced a comprehensive document [1], which addresses the issue of definition and classification of power system stability from first principles, while paying close attention to important practical aspects related to the problem. This section of the book will borrow extensively from the PES and CIGRÉ publication to define and classify power system stability without reinventing the wheel.

1.3.1 Definition of Stability Definition [1]: Power system stability is the ability of an electric power system, for a given initial operating condition, to regain a state of operating equilibrium after being subjected to a physical disturbance, with most system variables bounded so that practically the entire system remains intact. Power systems are nonlinear systems that operate with significant time varying changes: loads, generation output, network topologies, and operating parameters. When subjected to changes or disturbances, the stability of the system is dependent on two factors: the initial operating condition and the severity of the disturbance. As an analogy, consider a pitcher filled with water. The level of water in the pitcher is analogous to the operating condition in the power system. Now consider a pebble being dropped in this pitcher. The size of the pebble corresponds to the severity of the disturbance. The stability problem in this case is defined by water spilling out of the pitcher when the pebble is dropped. Consider the pitcher to be half full, and a small pebble is dropped. In this case, no water will spill over and the system at hand will be stable. Now consider two alternatives: (a) with the pitcher being half full, a large-sized pebble is dropped, causing water to be spilled, and (b) with the pitcher being filled close to the brim, either the same small pebble considered earlier or a moderate-sized pebble is dropped, which causes water to spill out of the pitcher. It should be noted that in case (a) the system becomes unstable due to the severity of the disturbance (large-sized pebble) and in case (b) the system becomes unstable due to a highly stressed operating condition when subjected to a disturbance that is not so severe. Hence, this analogy illustrates that both the operating condition and the severity of the disturbance can impact the stability behavior of the system. Based on the above definition, it should be noted that stability analysis deals with the examination of the property of system dynamics around an equilibrium set. Stability analysis consists of examining whether the power system attains a new equilibrium state with system integrity preserved following a disturbance. In [1] system integrity is characterized by the statement “with practically all

6

CHAPTER 1

POWER SYSTEM STABILITY

generators and loads connected through a single contiguous transmission system.” This allows for some generators and loads to be disconnected due to protective relay actions to isolate faulted elements or intentional tripping so as to maintain the integrity of the bulk electric system.

1.3.2 Classification of Stability Problems Power system stability as the definition above suggests is a unique characteristic of system dynamics. However, this characterization precludes a proper analysis and understanding of the phenomenon because a power system can undergo different forms of instabilities. Additionally, due to the large size and complexity of the stability problems, we make simplified assumptions in order to analyze specific types of problems and use appropriate degrees of detail for system representation and analysis methods that are suited to the problem at hand. As a result, a stability analysis of the system, which includes identifying key factors that impact stability, as well as determining appropriate methods to mitigate the detrimental impacts of the instability, can be facilitated by a systematic classification of the stability problems [1, 2]. Figure 1.1 (obtained from [1]) depicts the classification of the power system stability problems, including identifying the various categories and subcategories. An excellent description of each form the stability phenomenon is provided in [1]. A brief description of the phenomenon addressed in this book is provided next.

1.3.3 Description of Stability Phenomenon Rotor Angle Stability This stability phenomenon is associated with the response of interconnected synchronous machines that remain in synchronism after being subjected to a disturbance. The physics of the associated problem will be discussed and developed in detail in the chapters that follow. Simply put, a synchronous machine remains in synchronism following a disturbance when the balance between output electromagnetic torque and input mechanical torque can be restored/maintained. When this balance is lost, instability manifests itself as increasing or decreasing angular swings of generators, resulting in loss of synchronism with other interconnected generators. Subsequent chapters in this book deal with the development of models and analysis techniques related to both electromagnetic torque and mechanical torque. Mathematical models for synchronous

Power system stability

Rotor angle stability

Small-disturbance angle stability

Frequency stability

Transient stability

Voltage stability

Large-disturbance voltage stability

Short term

Short term Short term

Long term

Figure 1.1 Classification of power system stability [1].

Small-disturbance voltage stability

Long term

1.4 EFFECT OF IMPACT ON SYSTEM COMPONENTS

7

machines that accurately represent the developed electromagnetic torque and mechanical torque will be derived. The change in electromagnetic torque of a synchronous machine following a disturbance can be decomposed into: Synchronizing torque, which is in phase with the rotor angle deviation. Damping torque, which is in phase with the speed deviation. Both components of torque are essential in maintaining system stability. Lack of sufficient synchronizing torque leads to aperiodic instability, and lack of sufficient damping torque leads to oscillatory instability. Rotor angle stability is associated with both large disturbances and small disturbances. Largedisturbance rotor angle stability, also referred to as transient stability, deals with the ability of the power system to remain in synchronism when subjected to large disturbances. An analysis in this case necessitates the nonlinear representation of all the associated components of the power system. Small-disturbance rotor angle stability is associated with maintaining synchronism when subjected to small disturbances. The analysis of small-disturbance rotor angle stability is primarily conducted by linearizing the system equations. Voltage Stability This stability phenomenon relates to the power system’s ability to maintain steady voltages at all buses in the system after being subjected to a disturbance at a given operating point. Akin to rotor angle stability, voltage stability is associated with the ability to restore/maintain balance between load supply and load demand in the power system. Voltage instability manifests itself as a progressive fall or rise of voltages at certain buses. A loss of voltage stability could result in localized tripping of load and transmission outages due to protective relaying action, resulting in cascading outages or loss of synchronism in generators caused by field current limit violations. Similar to rotor angle stability, voltage stability can also be classified as large-disturbance voltage stability and small-disturbance voltage stability. Large-disturbance voltage stability pertains to the ability of the system to maintain steady voltages when subjected to large disturbances, such as short circuits, generation tripping, or transmission outages. The system and load characteristics and the interactions with continuous and discrete controls and protection systems greatly impact large-disturbance voltage stability. As in the case of large-disturbance rotor angle stability, large-disturbance voltage stability analysis requires the nonlinear analysis of system response over a sufficient amount of time to capture the effect of the discrete control and dynamic load components, including motors and excitation field current limiters. This analysis needs to be carefully conducted with all requisite components appropriately modeled. Small-disturbance voltage stability is associated with the ability of the system to maintain steady voltages when subjected to small disturbances, including changes in system load. The behavior and characteristics of loads and continuous and discrete controls greatly influence this behavior. As in the case of small-disturbance rotor angle stability, the study of small-disturbance voltage stability can be analyzed by linearizing the system equations. However, this linearization cannot account for tap changers, limiters, deadbands, and delays. As a result, in many instances, small-disturbance voltage stability analysis would involve the use of both nonlinear and linear approaches in a complementary manner.

1.4 EFFECT OF IMPACT ON SYSTEM COMPONENTS In this section, a survey of the effect of impacts is made to estimate the elements that should be considered in a stability study. A convenient starting point is to relate an impact to a change in power somewhere in the network. Our “test” stimulus will be a change in power, and we will use the point of impact as our reference point. The following effects, in whole or in part, may be felt. The system

8

CHAPTER 1

POWER SYSTEM STABILITY

frequency will change because, until the input power is adjusted by the machine governors, the power change will go to or come from the energy in the rotating masses. The change in frequency will affect the loads, especially the motor loads. A common rule of thumb used among power system engineers is that a decrease in frequency results in a load decrease of equal percentage, i.e., load regulation is 100%. The network bus voltages will be affected to a lesser degree unless the change in power is accompanied by a change in reactive power.

1.4.1 Loss of Synchronism Any unbalance between the generation and load initiates a transient that causes the rotors of the synchronous machines to “swing” because net accelerating (or decelerating) torques are exerted on these rotors. If the net torques are sufficiently large to cause some of the rotors to swing far enough so that one or more machines “slip a pole,” then synchronism is lost. To assure stability, a new equilibrium state must be reached before any of the machines experience this condition. Loss of synchronism can also occur in stages, e.g., if the initial transient causes an electrical link in the transmission network to be interrupted during the swing. This creates another transient, which when superimposed on the first may cause synchronism to be lost. Let us now consider a severe impact initiated by a sizable generation unbalance, say, excess generation. The major portion of the excess energy will be converted into kinetic energy. Thus, most of the machine rotor angular velocities will increase. A lesser part will be consumed in the loads and through various losses in the system. However, an appreciable increase in machine speeds may not necessarily mean that synchronism will be lost. The important factor here is the angle difference between machines, where the rotor angle is measured with respect to a synchronously rotating reference. This is illustrated in Figure 1.2 in which the rotor angles of the machines in a hypothetical four-machine system are plotted against time during a transient. In case (a) all the rotor angles increase beyond π radians, but all the angle differences are small, and the system will be stable if it eventually settles to a new angle. In case (b) it is evident that the machines are separated into two groups where the rotor angles continue to drift apart. This system is unstable.

1.4.2 Synchronous Machine During a Transient During a transient, the system, when seen by a synchronous machine, causes the machine terminal voltage, rotor angle, and frequency to change. The impedance seen “looking into” the network at the machine terminal may also change. The field-winding voltage will be affected by: 1. Induced currents in the damper windings (or rotor iron) due to sudden changes in armature currents. The time constants for these currents are usually on the order of less than 0.1 s and are often referred to as “subtransient” effects. 2. Induced currents in the field winding due to sudden changes in armature currents. The time constants for this transient are on the order of seconds and are referred to as “transient” effects. 3. Change in rotor voltage due to change in exciter voltage if activated by changes at the machine terminal. Both subtransient and transient effects are observed. Since the subtransient effects decay very rapidly, they are usually neglected, and only the transient effects are considered important. Note that the behavior discussed above depends on the network impedance as well as the machine parameters. The machine output power will be affected by the change in the rotor-winding EMF and rotor position, in addition to any changes in the impedance “seen” by the machine terminals. However,

1.4 EFFECT OF IMPACT ON SYSTEM COMPONENTS

9

(a)

Rotor angle, rad

π 3π/4 π/2

A B C D

π/4

Time, s (b)

Rotor angle, rad

π 3π/4

A B

π/2 π/4

C D

Time, s Figure 1.2 Response of a four-machine system during a transient: (a) stable system and (b) unstable system.

until the speed changes to the point where it is sensed and corrected by the governor, the change in the output power will come from the stored energy in the rotating masses. The important parameters here are the kinetic energy in MW-s per-unit MVA (usually called H) or the machine mechanical time constant τj, which is twice the stored kinetic energy per MVA. When the impact is large, the speeds of all machines change so that they are sensed by their speed governors. Machines under load frequency control will correct for the power change. Until this correction is made, each machine’s share will depend on its regulation or droop characteristic. Thus, the controlled machines are the ones responsible for maintaining the system frequency. The dynamics of the transition period, however, are important. The key parameters are the governor dynamic characteristics. In addition, the flow of the tie lines may be altered slightly. As a result, some machines are assigned the requirement of maintaining scheduled flow in the ties. Supplementary controls are provided to these machines, the basic functions of which are to permit each control area to supply a given load. The responses of these controls are relatively slow, and their time constants are on the order of seconds. This is appropriate since the scheduled economic loading of machines is secondary in importance to stability.

10

CHAPTER 1

POWER SYSTEM STABILITY

1.5 METHODS OF SIMULATION If we look at a large power system with its numerous machines, lines, and loads and consider the complexity of the consequences of any impact, we may tend to think it is hopeless to attempt analysis. Fortunately, however, the time constants associated with the phenomena may be appreciably different, allowing concentration on the key elements affecting the transient and the area under study. The first step in a stability study is to construct a mathematical model of the system during the transient. The elements included in the model are those affecting the acceleration (or deceleration) of the machine rotors. The complexity of the model depends on the type of transient and system being investigated. Generally, the components of the power system that influence the electrical and mechanical torques of the machines should be included in the model. These components are: 1. The network before, during, and after the transient. 2. The loads and their characteristics. 3. The parameters of the synchronous machines. 4. The excitation systems of the synchronous machines. 5. The mechanical turbine and speed governor. 6. Other important components of the power plant that influence the mechanical torque. 7. Renewable generation associated with both wind and solar resources. (The modeling and representation of this type of generation is important new content in this edition of the book.) 8. Network control, devices including FACTS devices, such static VAr systems and thyristorcontrolled series capacitors (TCSC) and their associated controls. 9. Other supplementary controls, such as tie-line controls, deemed necessary in the mathematical description of the system. The basic ingredients for the solution are, thus, the knowledge of the initial conditions of the power system prior to the start of the transient and the mathematical description of the main components of the system that affect the transient behavior of the synchronous machines. The number of power system components included in the study and the complexity of their mathematical description will depend on many factors. In general, however, differential equations are used to describe the various components. Study of the dynamic behavior of the system is contingent on the nature of these differential equations.

1.5.1 Linearized System Equations If the system equations are linear (or have been linearized), the techniques of linear system analysis are used to study dynamic behavior. The most common method is to simulate each component by its transfer function. The various transfer function blocks are connected to represent the system under study. The system performance may then be analyzed by such methods as root-locus plots, frequency domain analysis (Nyquist criteria), and Routh’s criterion. The above methods have been frequently used in studies pertaining to small systems or a small number of machines. For larger systems the state-space model has been used more frequently in connection with system studies described by linear differential equations. Stability characteristics may be determined by examining the eigenvalues of the A matrix, where A is defined by the equation x = Ax + Bu

(1.1)

where x is an n vector denoting the states of the system and A is a coefficient matrix. The system inputs are represented by the r vector u, and these inputs are related mathematically to differential equations by an n × r matrix B. This description has the advantage that A may be time varying and u may be used to represent several inputs if necessary.

1.6 PLANNING AND OPERATING STANDARDS

11

1.5.2 Large System with Nonlinear Equations The system equations for a transient stability study are usually nonlinear. Here the system is described by a large set of coupled nonlinear differential equations of the form x = f x, u, t

(1.2)

where f is an n vector of nonlinear functions. Determining the dynamic behavior of the system described by (1.2) is a more difficult task than that of the linearized system of (1.1). Usually time solutions of the nonlinear differential equations are obtained by numerical methods with the aid of digital computers, a method typically used in power system stability studies. Stability of synchronous machines is usually decided by behavior of their rotor angles, as discussed in Section 1.4.1. More recently, modern theories of stability of nonlinear systems have been applied to the study of power system transients to determine the stability of synchronous machines without obtaining time solutions. Such efforts, while they seem to offer considerable promise, are not in common use because of the limitations associated with applying these methods to detailed models of power system components. Both linear and nonlinear equations will be developed in following chapters.

1.6 PLANNING AND OPERATING STANDARDS A question that commonly arises in the context of stability studies is, “Why is there a need to conduct stability studies?” The answer to this question is that stability studies are needed to determine if the system satisfies planning and operation guidelines or the standards for the area in which the system operates. These guidelines or standards are typically set up by regulatory bodies with jurisdiction over the grid in a specific geographical area or country. In the United States, Canada, and a small portion of the Baja peninsula, this responsibility lies with the NERC, which serves as the ERO subject to oversight by the FERC. NERC has an extensive list of reliability standards available at http://www.nerc.com/pa/Stand/ Reliability%20Standards%20Complete%20Set/RSCompleteSet.pdf. These standards cover aspects of planning standards for the BES and include operating standards as well. Other countries and interconnections also have similar grid codes or standards. Stability studies are conducted either in a planning horizon or operating horizon to make sure the BES meets the planning and operating standards or codes, respectively. The reliability criteria or grid codes are used in the power system planning function to make decisions on size, type, and timing of generation and transmission facilities. They are also used to design the transmission network to withstand normal and (prescribed) emergency conditions. This would include examination of the design subjected to short circuits, loss of major system components, and other large disturbances. In the power system operating function, the reliability criteria are primarily used to: • Establish the most economical operation conditions under normal operating conditions. • Operate the system such that if an unscheduled event occurs, it does not result in uncontrolled (or cascading) outages. • Establish safe operating limits for all probable situations. • In the new market environment, the hourly day-ahead schedules obtained via an auction mechanism will also be examined to see if the operating conditions associated with the schedules satisfy the reliability criteria. An example of a typical reliability criteria or grid code is provided in Table 1.1 (obtained from the NERC Standard TPL-001-4 – Transmission System Planning Performance Requirements). This

Normal System

P2 Single Contingency

SLG N/A SLG SLG SLG

1. Opening of a line section w/o a fault7 2. Bus section fault 3. Internal breaker fault8 (non-bus-tie-breaker) 4. Internal breaker fault (bus-tie breaker)

3ϕ

N/ A

Fault Type2

4. Shunt device6 5. Single pole of a DC line

3. Transformer5

2. Transmission circuit

Loss of one of the following: 1. Generator

Normal System

P1 Single Contingency

Event1

None

Initial Condition

Normal System

P0 No Contingency

Category

EHV, HV EHV HV EHV HV EHV, HV

EHV, HV

EHV, HV

BES Level3

No9 No9 Yes No9 Yes Yes

No9

No

Interruption of Firm Transmission Service Allowed4

No12 No Yes No Yes Yes

No12

No

NonConsequential Load Loss Allowed

i. The response of voltage sensitive load that is disconnected from the system by end-user equipment associated with an event shall not be used to meet steady state performance requirements. Stability Only: j. Transient voltage response shall be within acceptable limits established by the planning coordinator and the transmission planner.

h. Planning event P0 is applicable to steady state only.

g. System steady state voltages and post-contingency voltage deviations shall be within acceptable limits as established by the planning coordinator and the transmission planner.

e. Planned system adjustments such as transmission configuration changes and re-dispatch of generation are allowed if such adjustments are executable within the time duration applicable to the Facility Ratings. Steady State Only: f. Applicable facility ratings shall not be exceeded.

d. Simulate normal clearing unless otherwise specified.

c. Simulate the removal of all elements that protection systems and other controls are expected to automatically disconnect for each event.

b. Consequential load loss as well as generation loss is acceptable as a consequence of any event excluding P0.

Steady State and Stability: a. The system shall remain stable. Cascading and uncontrolled islanding shall not occur.

Steady State and Stability Performance Planning Events

TABLE 1.1 NERC Standard TPL-001-4 – Transmission System Planning Performance Requirements

Loss of generator unit followed by system adjustments9

Normal System

Normal System

P4 Multiple contingency (Fault plus stuck breaker10)

P5 Multiple contingency (Fault plus relay failure to operate)

Initial Condition

P3 Multiple Contingency

Category

TABLE 1.1 (Continued)

5. Bus section

4. Shunt device6

3. Transformer5

2. Transmission circuit

Delayed fault clearing due to failure of non-redundant relay13 protecting the faulted element to operate as designed, for one of the following: 1. Generator

5. Bus section 6. Loss of multiple elements caused by a stuck breaker10 (bus-tie breaker) attempting to clear a fault on the associated bus

4. Shunt device6

3. Transformer5

2. Transmission circuit

Loss of multiple elements caused by a stuck breaker10 (non-bustie breaker) attempting to clear a fault on one of the following: 1. Generator

4. Shunt device6 5. Single pole of a DC line

3. Transformer5

2. Transmission circuit

Loss of one of the following: 1. Generator

Event1

SLG

SLG

SLG

SLG

3ϕ

Fault Type2

EHV HV

EHV, HV

EHV HV

EHV, HV

BES Level3

No9 Yes

Yes

No9 Yes

No9

Interruption of Firm Transmission Service Allowed4

(Continued )

No Yes

Yes

No Yes

No12

NonConsequential Load Loss Allowed

Normal System

4. Single pole of a DC line

3. Shunt device6

2. Tranformer5

1. Transmission circuit

Event1

Steady State and Stability Performance Extreme Events

2. Loss of bipolar DC line

Los of: 1. Any two adjacent (vertically or horizontally) circuits on common structure11

3. Shunt device6 4. Single pole of a DC line

Loss of one of the following Loss one of the following: followed by system 1. Transmission circuit adjustments.9 2. Transformer5

Initial Condition

SLG

SLG

3ϕ

Fault Type2

11

e. Loss of a large load or major load center.

d. Loss of all generating units at a generating station.

c. Loss of a switching station or substation (loss of one voltage level plus transformers).

b. Loss of all transmission lines on a common right-of-way .

a. Loss of a tower line with three or more circuits.11

b. 3Ø fault on transmission circuit with stuck breaker10 or a relay failure13 resulting in delayed fault clearing.

a. 3Ø fault on generator with stuck breaker10 or a relay failure13 resulting in delayed fault clearing.

2. Local or wide area events affecting the transmission system such as:

Yes

Yes

2. Local area events affecting the transmission system such as:

Yes

Yes

No

No9

Stability 1. With an initial condition of a single generator, Transmission circuit, single pole of a DC line, shunt device, or transformer forced out of service, apply a 3Ø fault on another single generator, Transmission circuit, single pole of a different DC line, shunt device, or transformer prior to system adjustments.

EHV, HV

HV

EHV

BES Level3

NonConsequential Load Loss Allowed

Interruption of Firm Transmission Service Allowed4

b. Simulate normal clearing unless otherwise specified. Steady State 1. Loss of a single generator, transmission circuit, single pole of a DC Line, shunt device, or transformer forced out of service followed by another single generator, transmission circuit, single pole of a different DC line, shunt device, or transformer forced out of service prior to system adjustments.

Steady State and Stability For all extreme events evaluated: a. Simulate the removal of all elements that protection systems and automatic controls are expected to disconnect for each contingency.

P7 Multiple contingency (Common structure)

P6 Multiple contingency (Two overlapping singles)

Category

TABLE 1.1 (Continued)

e. 3Ø internal breaker fault. f. Other events based upon operating experience, such as consideration of initiating events that experience suggests may result in wide area disturbances.

ii. Loss of the use of a large body of water as the cooling source for generation.

iii. Wildfires.

8. An internal breaker fault means a breaker failing internally, thus creating a system fault which must be cleared by protection on both sides of the breaker.

7. Opening one end of a line section without a fault on a normally networked transmission circuit such that the line is possibly serving load radial from a single source point.

6. Requirements which are applicable to shunt devices also apply to FACTS devices that are connected to ground.

(Continued )

5. For non-generator step up transformer outage events, the reference voltage, as used in footnote 1, applies to the low-side winding (excluding tertiary windings). For generator and generator step up transformer outage events, the reference voltage applies to the BES connected voltage (high-side of the generator step up transformer). Requirements which are applicable to transformers also apply to variable frequency transformers and phase shifting transformers.

4. Curtailment of conditional firm transmission service is allowed when the conditions and/or events being studied formed the basis for the conditional firm transmission service.

3. Bulk electric system (BES) level references include extra-high voltage (EHV) facilities, defined as greater than 300 kV, and high voltage (HV) facilities defined as the 300 kV and lower voltage systems. The designation of EHV and HV is used to distinguish between stated performance criteria allowances for interruption of firm transmission service and non-consequential load loss.

2. Unless specified otherwise, simulate normal clearing of faults. Single line to ground (SLG) or three-phase (3Ø) are the fault types that must be evaluated in Stability simulations for the event described. A 3Ø or a double line to ground fault study indicating the criteria are being met is sufficient evidence that a SLG condition would also meet the criteria.

1. If the event analyzed involves BES elements at multiple System voltage levels, the lowest system voltage level of the element(s) removed for the analyzed event determines the stated performance criteria regarding allowances for interruptions of firm transmission service and non-consequential load loss.

Steady State and Stability Performance Footnotes (Planning Events and Extreme Events)

b. Other events based upon operating experience that may result in wide area disturbances.

vi. Shutdown of a nuclear power plant(s) and related facilities for a day or more for common causes such as problems with similarly designed plants.

v. A successful cyber-attack.

iv. Severe weather, e.g., hurricanes, tornadoes, etc.

d. 3Ø fault on bus section with stuck breaker10 or a relay failure13 resulting in delayed fault clearing.

i. Loss of a large gas pipeline into a region or multiple regions that have significant gas-fired generation.

a. Loss of two generating stations resulting from conditions such as:

c. 3Ø fault on transformer with stuck breaker10 or a relay failure13 resulting in delayed fault clearing.

Steady State and Stability Performance Extreme Events

3. Wide area events affecting the transmission system based on system topology such as:

TABLE 1.1 (Continued)

13. Applies to the following relay functions or types: pilot (#85), distance (#21), differential (#87), current (#50, 51, and 67), voltage (#27 & 59), directional (#32, & 67), and tripping (#86, & 94).

12. An objective of the planning process is to minimize the likelihood and magnitude of non-consequential load loss following planning events. In limited circumstances, non-consequential load loss may be needed throughout the planning horizon to ensure that BES performance requirements are met. However, when non-consequential Load Loss is utilized under footnote 12 within the near-term transmission planning horizon to address BES performance requirements, such interruption is limited to circumstances where the non-consequential load loss meets the conditions shown in Attachment 1. In no case can the planned non-consequential load loss under footnote 12 exceed 75 MW for US registered entities. The amount of planned non-consequential load loss for a non-US registered entity should be implemented in a manner that is consistent with, or under the direction of, the applicable governmental authority or its agency in the non-US jurisdiction.

11. Excludes circuits that share a common structure (Planning event P7, Extreme event steady state 2a) or common right-of-way (extreme event, steady state 2b) for 1 mile or less.

10. A stuck breaker means that for a gang-operated breaker, all three phases of the breaker have remained closed. For an independent pole operated (IPO) or an independent pole tripping (IPT) breaker, only one pole is assumed to remain closed. A stuck breaker results in delayed fault clearing.

9. An objective of the planning process should be to minimize the likelihood and magnitude of interruption of firm transmission service following contingency events. Curtailment of firm transmission service is allowed both as a system adjustment (as identified in the column entitled “initial condition”) and a corrective action when achieved through the appropriate re-dispatch of resources obligated to re-dispatch, where it can be demonstrated that Facilities, internal and external to the transmission planner’s planning region, remain within applicable facility ratings and the re-dispatch does not result in any non-consequential load loss. Where limited options for re-dispatch exist, sensitivities associated with the availability of those resources should be considered.

TABLE 1.1 (Continued)

PROBLEMS

17

table provides a clear example of the extensive analysis that should be conducted in order to satisfy the standard. Table 1.1 shows that stability studies need to be conducted with the system in normal conditions (with all elements in service) and for multiple contingencies. The studies also involve cases that include stuck breakers and relay failures. The cases identified in Table 1.1 are the minimum requirement set by NERC. Additionally, each regional reliability council could set stricter requirements based on distinctive system characteristics in each regional jurisdiction. The nature of the stability studies to be conducted according to Table 1.1 is complex and involves the examination of a wide range of operating conditions and contingencies. The range and scope of the analysis to be conducted is comprehensive and requires careful consideration of an exhaustive list of probable contingencies and disturbances, thus making transient stability analysis a daunting task that requires meticulous attention to detail and careful choice of the model features to capture details of the phenomenon being examined.

PROBLEMS 1.1. Suggest definitions for the following terms: a. Power system reliability b. Power system security c. Power system stability 1.2. Distinguish between steady-state (dynamic) and transient stability according to a. The type of disturbance b. The nature of the defining equations 1.3. What is a tie line? Is every line a tie line? 1.4. What is an impact insofar as power system stability is concerned? 1.5. Consider the system shown in Figure P1.5 where a mass M is pulled by a driving force f(t) and is restrained by a linear spring K and an ideal dashpot B.

x K M

f(t)

B Figure P1.5 Write the differential equation for the system in terms of the displacement variable x and determine the relative values of B and K to provide critical damping when f(t) is a unit step function. 1.6. Repeat Problem 1.5 but convert the equations to the state-space form of (1.1). 1.7. Consider the rotational system shown in Figure P1.7 where two rotating masses are connected by a small shaft with rotational spring constant Kθ and viscous friction Bθ. Assume that J1 is a DC shunt motor and J2 is a DC shunt generator that is loaded suddenly at t = 0 by closing the switch. Measure all mechanical angles from a fixed reference. a. State all necessary assumptions and write the equations of the system. b. Repeat part (a) but let the angular reference move clockwise at a constant speed equal to the no load speed of the rotating system.

18

CHAPTER 1

POWER SYSTEM STABILITY

t=0 θ2 R en.

DC G

Ref.

θ1

Kθ J2 Bθ

J1 Figure P1.7

c. Sketch the torque speed characteristics for the motor and for the generator. d. Repeat part (a) using state-space notation. 1.8. A schematic representation of the turbine-generator unit is shown in Figure P1.8. The shaft is not rigid and has five masses associated with the various turbine stages, generator and exciter as shown. Tm1

HP

Tm2 δ1

IP K12

d1

Tm3 δ2

LP

Te δ3

K23 d2

GEN. K34

d3

Tex δ4

δ5

EXC.

K45 d4

d5

Figure P1.8 Tmi = mechanical torque applied to mass i Te, Tex = electrical torque of generator and exciter, respectively Kij = spring constant of shaft between masses i and j di = damping (mechanical) applied to mass i Hi = inertia constant associated with mass i δi = angular displacement of the shaft at mass i Write the equations of motion (swing equations) for the various masses of this system in matrix form. 䊏

REFERENCES 1. Kundur, P., Paserba, J., Ajjarapu, V. et al. (2004). Definition and Classification of Power System Stability. IEEE/CIGRE Joint Task Force on Stability Terms and Definitions Report. IEEE Trans. Power Syst. 19 (3): 1387–1401. 2. Kundur, P. (1994). Power System Stability and Control. New York: McGraw Hill.