Power system transient stability analysis using the transient energy function method

Contents
Preface
Table of Symbols and Parameters
1. POWER SYSTEM TRANSIENT STABILITY
	1.1 Introduction
		1.1.1 Definitions
	1.2 Conventional Transient Stability Studies
	1.3 Reasons for Conducting Transient Stability Studies
	1.4 Changing Conditions in the North American Interconnection
	1.5 Stability Implications of Changing Conditions
	References
2. THE POWER SYSTEM MODEL
	2.1 The Power Network
	2.2 The Classical Power System Model
	2.3 Load Model
		2.3.1 Classical Model
		2.3.2 Nonlinear Load Model
	2.4 Center of Inertia Formulation
	2.5 Higher–Order Generator Model with Excitation Control
		2.5.1 Synchronous Generator Model
		2.5.2 Generator Currents–Constant Impedance Loads
		2.5.3 Generator Currents–Nonlinear Loads
		2.5.4 Excitation System Model
	References
3. TRANSIENT STABILITY ANALYSIS USING ENERGY FUNCTIONS – AN INTRODUCTION
	3.1 The Basic Idea
	3.2 An Example
	3.3 The Equal Area Criterion
	3.4 Multimachine Energy Functions
		3.4.1 Gorev\'s Energy Criteria
		3.4.2 Magnusson\'s Method of Transitional Energy
		3.4.3 Aylett\'s Energy Integral Criterion
		3.4.4 Tavora and Smith\'s Investigation of the Transient Energy
	3.5 The Modern Transient Energy Function
		3.5.1 Introduction
		3.5.2 The SCI Work
	3.6 Motivation for Current Direct Stability Work
	References
4. REVIEW OF STABILITY THEORY APPLIED TO THE TRANSIENT ENERGY FUNCTION METHOD
	4.1 Introduction
		4.1.1 Notational Conventions
		4.1.2 Nonlinear Physical Systems
		4.1.3 Well Posedness
	4.2 Autonomous Systems, Equilibrium Points
		4.2.1 Autonomy and Nonautonomy
		4.2.2 Equilibrium Points
	4.3 Stability in the Sense of Lyapunov
		4.3.1 Stability Definitions
		4.3.2 Asymptotic Stability
		4.3.3 Lyapunov\'s Theorems
		4.3.4 Invariance Theory
		4.3.5 Domain of Attraction
		4.3.6 The Potential Energy Boundary Surface
	4.4 Analytical Justification for the Controlling UEP and the PEBS
		4.4.1 Characterization of the Stability Boundary
		4.4.2 Characterization of the PEBS
		4.4.3 Concluding Remarks
	References
5. THE TRANSIENT ENERGY FUNCTION METHOD APPLIED TO THE CLASSICAL POWER SYSTEM MODEL
	5.1 Introduction
	5.2 The Transient Energy Function
		5.2.1 Derivation
		5.2.2 Description of Components
		5.2.3 Approximation of Dissipation Energy Component
		5.2.4 Energy Not Contributing to System Separation
		5.2.5 Corrected Kinetic Energy
	5.3 Transient Stability Assessment
	5.4 Concept of the Controlling Unstable Equilibrium Point
	References
6. COMPUTATIONAL ASPECTS OF THE TRANSIENT ENERGY FUNCTION METHOD FOR THE CLASSICAL GENERATOR MODEL
	6.1 TEF Procedure Outline
	6.2 Calculation of Initial Conditions
		6.2.1 Data Requirement
		6.2.2 Internal Voltage Calculations
		6.2.3 Calculation of Initial Nonlinear Load Current Component
	6.3 Construction of Admittance Matrices
		6.3.1 Predisturbance Network Admittance Matrix Y[sup(A)][sub(PR)]
		6.3.2 Disturbed Network Admittance Matrix Y[sub(F)]
		6.3.3 Postdisturbance Network Admittance Matrix Y[sub(PO)]
	6.4 Conditions at the End of the Disturbance
		6.4.1 Conditions at Fault Clearing
		6.4.2 Disturbances Other than Faults
	6.5 Determination of the Controlling UEP
	6.6 UEP Determination Using the Mode of Disturbance Procedure
		6.6.1 Selection of the Candidate Modes
		6.6.2 A Practical Mode of Disturbance Test
	6.7 UEP Determination Using the Exit Point Method
		6.7.1 Introduction
		6.7.2 Modified Exit Point Procedure
	6.8 Calculation of Equilibrium Points
		6.8.1 Problem Formulation
		6.8.2 Solution Algorithms
		6.8.3 Analytical Expressions for the Jacobian and Hessian Matrices
		6.8.4 Computational Issues
	6.9 Energy Margin Calculation
	References
7. APPLICATIONS OF THE TEF METHOD–CLASSICAL POWER SYSTEM MODEL
	7.1 Introduction
	7.2 Plant Mode versus Interarea Mode Stability
	7.3 Plant Mode Stability
		7.3.1 Degree of Stability (or Instability)
		7.3.2 Critical Clearing Time
		7.3.3 Power Limits
		7.3.4 Other Stability Information
	7.4 The Interarea Mode
		7.4.1 Analytical and Numerical Issues
		7.4.2 Stability Classification
		7.4.3 Power Limits
		7.4.4 Complexity of the System Dynamic Behavior
	7.5 Application in Transmission Planning
	7.6 Disturbance Other than Faults
		7.6.1 Opening of a Loaded Line
		7.6.2 Loss of Generation Disturbance
		7.6.3 Special Case – Fault Resulting in Power Deficiency
	7.7 Network Conditions During Transient
		7.7.1 Identifying \"Peak of the Swing\"
		7.7.2 Voltage at a Given Bus
		7.7.3 Apparent Impedance Seen by an Out–of–Step Relay
	7.8 Practical Problems
		7.8.1 Numerical Problems
		7.8.2 Efficiency Problems
		7.8.3 Reliability Problems
		7.8.4 Efficiency of a Run Stream
	References
8. MODELING IMPROVEMENTS IN THE TEF METHOD
	8.1 Introduction
	8.2 Modeling of Excitation Control Effects
		8.2.1 Synchronous Generator–Exciter Models
		8.2.2 One–Gain, One–Time–Constant Exciter Model
		8.2.3 Network Equations
		8.2.4 Modeling of the Generator Internal Impedance
		8.2.5 Effect of Excitation Control on the Transient Energy
		8.2.6 Modeling of Salient Pole Generators
	8.3 Modeling of Nonlinear Loads
		8.3.1 Introduction
		8.3.2 General Approach
		8.3.3 Determining I[sub(GL)] for Classical Generator Model
		8.3.4 Incorporation of Nonlinear Loads in the TEF Method
		8.3.5 Incorporation of the Nonlinear Load Model and Excitation Control in the TEF Method
	8.4 Incorporating Two–Terminal HVDC in the Transient Energy Function Method
		8.4.1 A Simplified Two–Terminal HVDC Model
		8.4.2 Interfacing the ac/dc Systems
		8.4.3 Incorporation of the HVDC Model in the TEF
		8.4.4 Procedure for Transient Stability Assessment
	8.5 Incorporation of Uniform Damping in TEF
		8.5.1 Introduction
		8.5.2 System Equations
		8.5.3 The Damping Energy
		8.5.4 Approximation of θ[sub(i)] as a Sine Curve
	8.6 Obtaining the Controlling UEP with Sparse Formulation
		8.6.1 Introduction
		8.6.2 Sparse Formulation of the Transient Energy Function
		8.6.3 Experience with Sparse Formulation
	8.7 General Comments on the Approximations in the TEF Method and on the Models Used
	References
9. ENERGY MARGIN SENSITIVITY TO CHANGES IN SYSTEM CONDITIONS
	9.1 Introduction
	9.2 Calculation of Sensitivity Factors
		9.2.1 Functional Dependence on System Parameters
	9.3 First–Order Sensitivity of ΔV
	9.4 Second–Order Sensitivity of ΔV
		9.4.1 Numerical Method
		9.4.2 Analytical Method
	9.5 Validation Studies
		9.5.1 First–Order Sensitivity Results
		9.5.2 Second–Order Sensitivity Results
	9.6 Determination of Interface Flow Stability Limits
	9.7 On–Line Derivation of Stability Limits
		9.7.1 Introduction
		9.7.2 Stability Limits for an Area in a Large Network
		9.7.3 Stability Limits for More Complex System Conditions
		9.7.4 Use of Sensitivity Methods to Track Security Trend
	References
10. ADVANCED APPLICATIONS OF THE TEF METHOD
	10.1 Introduction
	10.2 Corrective Actions
		10.2.1 Predisturbance Action
		10.2.2 Emergency Actions
	10.3 Expert Systems and Dynamic Security Assessment
	10.4 Parallel Computation
		10.4.1 Parallel FORTRAN Environment on the IBM 3090
		10.4.2 Modifying the TEF Program Code
		10.4.3 General Comments on Parallel Computation
	10.5 Alternate Transient Energy Functions
		10.5.1 Individual Machine Energy Function
		10.5.2 Transient Stability Program Output Analysis
		10.5.3 Individual Machine Energy Function for the Critical Generator and Cutset
		10.5.4 The Partial Energy Function
		10.5.5 Hybrid Method
		10.5.6 Modal–Based Transient Energy Function
	10.6 Emerging Applications
		10.6.1 A Framework for Reliability Computation
		10.6.2 Combining the Use of TEF and Time Solution Methods
		10.6.3 Interaction Between Various Modes of Oscillation
	10.7 Concluding Remarks
	References
Special References
Appendix
Index
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