Mrinal K Pal. Power system stability
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Lecture Notes
on
Power System Stability
Mrinal K Pal
iii
CONTENTS
Preface ix
Chapter 1 Fundamentals of Power Flow and Power Limits
Representation of Transmission Lines 1-1
Equations of a transmission line 1-1
Equivalent circuit of a transmission line 1-3
Y - ∇ transformation 1-5
Per Unit System 1-6
Base quantities 1-6
Change of bases 1-7
Per unit and percent admittance 1-8
Power Limits 1-9
Circle Diagram 1-11
Basic Power Flow Calculation 1-12
Direct solution 1-12
Iterative procedure (Newton’s method) 1-13
Selection of starting values 1-14
Steady State Stability 1-14
Steady state stability limit 1-15
Steady state stability criterion 1-17
Steady state stability of a two-machine system 1-17
General Network Equations 1-20
Nodal formulation in terms of bus admittance matrix 1-20
General equations for real and reactive power 1-21
Elimination of network nodes by matrix method 1-22
Application of the Steady-State Stability Criterion to an n Machine System 1-23
Network Calculations Using Bus Impedance Matrix 1-24
Adding a new bus to the network 1-25
Adding a new line between two existing buses 1-26
Adding a new line between an existing bus and neutral (reference bus) 1-27
Calculation of Short-Circuit Current 1-28
Representation of Tap-changing Transformers and Phase-shifting Transformers 1-29
Effect of phase shift on power flow 1-30
References 1-32
Chapter 2 Stability Basics
Stability of Dynamic Systems 2-1
Stability definitions 2-1
Power system stability definitions 2-2
Comments 2-2
Small-disturbance stability 2-2
Large-disturbance stability 2-3
Linear and Non-Linear Systems 2-3
Linearization of non-linear systems 2-3
CONTENTS
iv
Solution of linear differential equations with constant coefficients 2-5
Evaluation of the arbitrary constants 2-6
Criterion for stability of a linear system 2-6
Concepts of damping ration and natural frequency 2-7
The State Variables of a Dynamic Systems 2-9
The state vector differential equation 2-11
Solution of the equation xAx
=
&
2-13
Liapunov’s Direct Method of Stability Analysis 2-17
Stability and instability theorems 2-17
Illustration of positive definite functions and their closedness 2-18
Stability determined by linear approximation 2-19
Extent of asymptotic stability 2-20
Construction of Liapunov functions 2-20
Comments of Certain Terms in Common Use 2-21
References 2-23
Chapter 3 Synchronous Machine Stability Basics
The Swing Equation 3-1
Selection of units 3-3
Mechanical Torque 3-4
Unregulated machine 3-4
Regulated machine 3-5
Small Disturbance Performance of Unregulated Synchronous Machine System 3-6
Single machine connected to infinite bus 3-6
Modes of Oscillation of an Unregulated Multi-Machine System 3-8
Division of Suddenly Applied Load Among Generators in the System 3-15
Transient Stability by Equal Area Criterion 3-19
Effect of damping 3-22
Equal area criterion for a two-machine system 3-22
Transient Stability by Liapunov’s Method 3-23
References 3-26
Chapter 4 Numerical Solution of the Transient Stability Problem
Numerical Solution of Differential Equations in Power System Stability Studies 4-1
Euler’s method 4-2
The modified Euler’s method 4-3
Runge-Kutta method 4-3
Predictor-Corrector method 4-4
Implicit multi-step integration 4-5
Solution of Multi-Machine Transient Stability Problem
Using Classical Machine Model 4-5
Single-machine-infinite bus system 4-6
Multi-machine system 4-7
Handling of constant current load 4-10
Solution of Faulted Networks 4-14
Analysis of Unsymmetrical Faults 4-16
Single line-to-ground fault 4-17
CONTENTS
v
Line-to-line fault 4-17
Double line-to-ground fault 4-17
One open conductor 4-18
References 4-19
Chapter 5 Synchronous Machines
Voltage Relations 5-2
Flux-Linkage Relations 5-3
Inductance Relations 5-4
Armature self-inductances 5-4
Armature mutual inductances 5-5
Rotor self-inductances 5-5
Rotor mutual inductances 5-5
Mutual inductances between stator and rotor circuits 5-5
Transformation of Equations 5-6
Per Unit System 5-8
Choice of rotor base currents 5-14
Per unit voltage equations 5-15
Per unit flux-linkage equations 5-16
Power and Torque 5-16
Steady State Operation 5-17
Locating the d and q axes 5-19
Open-circuit operation 5-19
Reactive Power 5-20
Subtransient and Transient Reactances and Time Constants 5-20
Synchronous Machine Models for Stability Studies 5-24
Model 1 -- One damper winding on each axis 5-26
Model 2 -- One equivalent damper winding on the q axis only 5-31
Model 3 -- no damper winding 5-32
Saturation 5-32
Saturation in cylindrical rotor machines 5-33
Saturation in salient pole machines 5-34
Inclusion of the effect of saturation in machine modeling 5-35
Network Model in DQ Reference Frame 5-37
Transformation from machine to network reference frame 5-38
Stability Computation in Multi-Machine Systems 5-39
An iterative method for handling saliency 5-42
Generator Capability Curve 5-43
Stability Limit at Constant Field Voltage 5-44
Reactive power limit at P = 0 5-48
Self Excitation 5-48
References 5-52
Chapter 6 Effect of Excitation Control on Stability
Effect of Excitation on Generator Power Limit 6-1
Effect of Excitation Control on Small-Disturbance Performance of
Synchronous Machines 6-3
CONTENTS
vi
Synchronizing power coefficient with constant flux linkage 6-7
Steady-state synchronizing power coefficient 6-7
Evaluation of damping torque 6-9
No voltage regulator action 6-9
Including voltage regulator action 6-11
Effect of Damper Windings 6-15
One damper winding on the q axis, no damper winding on the d axis 6-15
Evaluation of damping torque 6-17
No voltage regulator action 6-17
Effect of voltage regulator 6-18
One damper winding on the d axis, two damper windings on the q axis 6-19
Evaluation of damping torque 6-21
No voltage regulator action 6-21
Effect of voltage regulator 6-22
Supplementary Stabilizing Signals 6-22
References 6-25
Chapter 7 Representation of Loads in Stability Studies
Load Characteristics 7-1
Load Modeling Concepts 7-1
Static Model for Stability Studies 7-2
Discussion 7-3
Motor Load 7-5
Induction Motor 7-5
Induction Motor Representation in Stability Studies 7-12
Initial condition calculation 7-15
Generic Load Model 7-16
References 7-17
Chapter 8 Small Disturbance Stability
State Space Representation of Power System 8-3
Single machine-infinite bus system 8-3
State model of the multi-machine system 8-5
An Efficient Method of Deriving the State Model 8-8
State model using flux linkages as state variables 8-8
State model for machine model 2 8-13
Mixed machine models 8-14
State model for systems containing induction motor loads 8-16
Singularity of the coefficient matrix 8-18
Procedure for Small-Disturbance Stability Studies 8-18
Eigen-sensitivity Analysis 8-19
System Performance 8-20
References 8-23
Chapter 9 Turbine-Generator Shaft Torsionals
A Basic Analysis of the Shaft Torque Problem 9-2
Modes and Mode Shapes in a Turbine-Generator Shaft Torsional System 9-7
CONTENTS
vii
Development of modal equivalents 9-9
Damping 9-12
Evaluation of Torsional Stress and Fatigue in Turbine-Generator Shafts 9-13
Estimation of fatigue 9-14
Effect of damping 9-15
Series Compensation and Subsynchronous Stability 9-16
Transient torque amplification 9-16
Self excitation 9-17
Torsional interaction 9-17
Development of Detailed Electrical System Model 9-19
Small Disturbance Stability Study of Turbine-Generator Shaft Torsional System 9-23
Estimation of Electrical Damping in Torsional Interaction 9-27
Self Excitation 9-35
Control of Subsynchronous Oscillation Problem 9-36
References 9-43
Chapter 10 Voltage Stability
Steady-State Analysis of the Voltage-Reactive Power Problem 10-2
Relationship between power flow and voltage drop 10-2
Approximate relationships 10-3
Power limit without specific voltage support at the receiving end 10-5
Power transfer at constant sending and receiving end voltage 10-9
Effect of load characteristics 10-10
Effect of reactive support at the receiving end by shunt capacitors 10-11
Steady-State Analysis of the Voltage-Reactive Power Problem:
Extension to Large Networks 10-13
Clarification of Certain Issues in Voltage Stability 10-21
Voltage Stability Limit Determined by Singularity of the Power Flow Jacobian --
A Reality Check 10-22
Voltage Stability: Effect of Load Characteristics 10-25
System and load model 10-26
Region of attraction 10-28
Large disturbance voltage stability with constant source voltage 10-30
Voltage stability limit when part of the load is static 10-31
Extension to large networks 10-33
Large disturbance voltage stability for mixed loads 10-34
Voltage stability for fast response load 10-35
Voltage Stability Using Static Load Models 10-36
Constant Power Dynamic Load with Fixed Generator Field Voltage 10-39
Induction Motor Load 10-40
Modification of the power flow Jacobian to account for
induction motor load 10-42
Effect of finite speed of voltage control 10-44
Stability analysis using detailed induction motor model 10-45
Voltage Stability Using Type-Two Load Model 10-46
Corrective Measures for Voltage Stability 10-49
Load shedding 10-50
CONTENTS
viii
Capacitor switching 10-53
Extending Voltage Stability Limit by SVC 10-55
LTC Operation and Voltage Stability 10-58
Voltage Stability Studies in Actual Power Systems 10-62
A Typical Voltage Collapse Scenario 10-63
References 10-64
Appendix A Review of Matrices
Appendix B Review of Selected Topics from Control Theory
Appendix C Block Diagram State Model
ix
PREFACE
The materials presented in this work are based on the notes used in courses taught sporadically to
graduate students and practicing engineers over a period of thirty-five years. The objective here
is to present the fundamental materials and concepts in power system stability in a tutorial form
The materials covered were dictated by the need of the course attendees. This is certainly not an
exhaustive treatment of all aspects of the subject found in the literature. Some of the topics
usually found in text books on power system stability are not covered. For example, detailed
descriptions and workings of excitation and governor control systems are not included. However,
the effect of excitation control on stability has been covered in some detail.
Chapter 1 includes some basic material considered important for developing a proper
understanding of the subject. In the days of the extensive use of commercially available large-
scale stability programs, the topics discussed in this chapter might be considered superfluous.
However, a good understanding of the basics not only helps in developing a clearer concept of
the subject, but also allows a quick estimate of what is to be expected from a computer run, thus
enabling a check on the proper working of the program as well as on the correctness and validity
of the data entered.
It will be noted that no mention of “FACTS” has been made anywhere in the book. This is
because the concepts and underlying principles involved in the application of devices e.g., series
and shunt capacitors, phase-shifting transformers, etc., are not new. It is the use of power
electronics, that provides fast and smooth continuous control, that is new. In the future newer
technologies will appear that will provide even better control. The basic principles and concepts
involved in applying these technologies, and the methods of analyses, are discussed throughout
the book. Anyway, it is not clear why a terminology like “FACTS” was adopted in the first
place, when these are really applications of power electronics.
The preparation of the book, especially typing the equations, was tedious and typos are
inevitable. I would be grateful if the readers would report them, and any other problem they
might encounter, to me for future correction. I can be reached at (732) 494 3716 and at
m.k.pal@ieee.org
.
I would like to thank Mr. Rondeep Pal for converting the document into the PDF format, and
providing help in many other ways.
Mrinal K Pal
Edison, New Jersey
June, 2007