The book provides a comprehensive taxonomy of non-symmetrical eigenvalues problems as applied to power systems. The book bases all formulations on mathematical concept of “matrix pencils” (MPs) and considers both regular and singular MPs for the eigenvalue problems. Each eigenvalue problem is illustrated with a variety of examples based on electrical circuits and/or power system models and controllers and related data are provided in the appendices of the book. Numerical methods for the solution of all considered eigenvalue problems are discussed. The focus is on large scale problems and, hence, attention is dedicated to the performance and scalability of the methods. The target of the book are researchers and graduated students in Electrical & Computer Science Engineering, both taught and research Master programmes as well as PhD programmes and it Book explains eigenvalue problems applied into electrical power systems Explains numerical examples on applying the mathematical methods, into studying small signal stability problems of realistic and large electrical power systems. Includes detailed and in-depth analysis including non-linear and other advanced aspects Provides theoretical understanding and advanced numerical techniques essential for secure operation of power systems Comprehensive set of illustrative examples that support theoretical discussions
Author(s): Federico Milano, Ioannis Dassios, Muyang Liu, Georgios Tzounas
Publisher: CRC Press
Year: 2020
Language: English
Pages: 406
City: Boca Raton
Cover
Half Title
Title Page
Copyright Page
Dedication
Series Page
Contents
List of Figures
List of Tables
List of Mathematical Statements
List of Examples
Authors
Acronyms
Notation
Preface
I: Introduction
1. Power System Outlines
1.1. Differential Equations
1.2. Discrete Maps
1.3. Power System Models
1.3.1. Nodes
1.3.2. Transmission System
1.3.3. Loads
1.3.4. Synchronous Machine
1.3.5. Primary Frequency Control
1.3.6. Automatic Voltage Regulator
1.3.7. Power System Stabilizer
1.4. Equilibria
1.5. Linearization
1.6. Lyapunov Stability Criterion
1.7. De nition of Small-Signal Stability
2. Mathematical Outlines
2.1. Matrix Pencils
2.2. Taxonomy of Eigenvalue Problems
2.3. Canonical Forms of Matrix Pencils
II:
Linear Eigenvalue Problems
3. Differential Equations with Regular Pencil
3.1. Formulation
3.2. Solutions and Eigenvalue Analysis
3.3. Applications to Electrical Circuits and Systems
4. Explicit Differential-Algebraic Equations
4.1. Formulation
4.2. Power Systems Modeled as Explicit DAEs
5. Implicit Differential-Algebraic Equations
5.1. Formulation
5.2. Power Systems Modeled as Implicit DAEs
5.3. Floquet Multipliers
5.4. Lyapunov Exponents
6. Differential Equations with Singular Pencil
6.1. Formulation
6.2. Singular Power System Models
7. Mobius Transform
7.1. Formulation
7.2. Special Cases
7.3. Applications of the MŁobius Transform
8. Participation Factors
8.1. Classical Participation Factors
8.1.1. Residues
8.1.2. Power Flow Modal Analysis
8.2. Generalized Participation Factors
8.3. Participation Factors of Algebraic Variables
III:
Non-Linear Eigenvalue Problems
9. Polynomial Eigenvalue Problem
9.1. Formulation
9.2. Quadratic Eigenvalue Problem
9.3. Fractional Calculus
10. Delay Differential Equations
10.1. Formulation
10.1.1. Retarded Delay Differential Equations
10.1.2. Neutral Delay Differential Equations
10.2. Pade Approximation
10.3. Quasi-Polynomial Mapping-based Rootnder
10.4. Spectrum Discretization
10.4.1. Infnite-Dimensional Eigenvalue Problem
10.4.2. Approximated Eigenvalue Computation
10.4.3. Methods for Sparse Delay Matrices
10.5. Newton Correction
11. Systems with Constant Delays
11.1. Delay Differential-Algebraic Equations
11.2. Retarded Index-1 Hessenberg Form DDAEs
11.3. Retarded Non-Index-1 Hessenberg Form DDAEs
11.4. Modeling Transmission Lines as a Continuum
11.5. Descriptor Transform for NDDEs
11.6. Neutral DDAEs
11.7. Delay Compensation in Power Systems
12. Systems with Time-Varying Delays
12.1. Analysis of Time-Varying Delay Systems
12.2. RDDEs with Distributed Delays
12.3. RDDEs with Time-Varying Delays
12.4. Effect of Time-Varying Delays on Power Systems
12.4.1. Simplified Power System Model
12.4.2. Stability Margin of Delay Models
12.4.3. Power System Model with a Gamma-Distributed Delay
12.5. Realistic Delay Models for Power System Analysis
12.5.1. Physical Structure of a WAMS
12.5.2. WAMS Delay Model
IV:
Numerical Methods
13. Numerical Methods for Eigenvalue Problems
13.1. Introduction
13.2. Algorithms
13.2.1. Vector Iteration Methods
13.2.2. Rayleigh-Ritz Procedure
13.2.3. Schur Decomposition Methods
13.2.4. Krylov Subspace Methods
13.2.5. Contour Integration Methods
14. Open Source Libraries
14.1. Overview
14.1.1. LAPACK
14.1.2. ARPACK
14.1.3. SLEPc
14.1.4. FEAST
14.1.5. z-PARES
14.1.6. Summary of Library Features
14.2. Left Eigenvectors
14.3. Spectral Transforms
15. Large Eigenvalue Problems
V:
Appendices
A. Three-Bus System
A.1. Network Data
A.2. Static Data
A.3. Dynamic Data
A.4. Reduction to OMIB
B. LEP Matrices
C. GEP Matrices
Bibliography
Index
