1st Edition

# Eigenvalue Problems in Power Systems

406 Pages 50 B/W Illustrations
by CRC Press

406 Pages 50 B/W Illustrations
by CRC Press

406 Pages 50 B/W Illustrations
by CRC Press

Also available as eBook on:

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:

• 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
• provides a comprehensive set of illustrative examples that support theoretical discussions

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 Definition 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 Möbius transform 7.1 Formulation 7.2 Special cases 7.3 Applications of the Möbius 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 Root-finder 10.4 Spectrum Discretization 10.4.1 Infinite-dimensional Eigenvalue Problem 10.4.2 Approximated Eigenvalue Computation 10.4.3 Methods for Sparse Delay 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 13.2.2 Rayleigh-Ritz Procedure 13.2.3 Schur Decomposition Methods 13.2.4 Krylov Subspace 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 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

### Biography

Federico Milano received from the University of Genoa, Italy, the ME and Ph.D. in Electrical Eng. in 1999 and 2003, respectively. From 2001 to 2002 he was with the University of Waterloo, Canada, as a Visiting Scholar. From 2003 to 2013, he was with the University of Castilla-La Mancha, Spain. In 2013, he joined the Univ. College Dublin, Ireland, where he is currently Professor of Power Systems Control and Protections and Head of Electrical Engineering. His research interests include power system modelling, control and stability analysis. A Fellow of the IEEE and the IET, he has been an editor for several journals published by IEEE, IET, Elsevier and Springer, including the IEEE Transactions on Power Systems and IET Generation, Transmission & Distribution. He has authored or co-authored about 150 peer reviewed journal and conference papers, several book chapters and three books (published by Springer, McGraw Hill and Cambridge University Press) and edited one book (published by IET).

Ioannis Dassios is currently a UCD Research Fellow at AMPSAS, University College Dublin, Ireland. He studied Mathematics and completed a two-year MSc in Applied Mathematics & Numerical Analysis at the Department of Mathematics, University of Athens, Greece with grade “Excellent” (highest mark in the Greek system). As a MSc student he received a travel grant to visit the Department of Mathematics at the University of North Texas, USA. In July 2013 he received his PhD in Mathematics from the Department of Mathematics, University of Athens, Greece, focusing on singular discrete dynamical systems and their applications. Since then, he has been a Post-Doctoral Researcher at MACSI (Mathematics Applications Consortium for Science and Industry), Department of Mathematics & Statistics, University of Limerick, Ireland and a Senior researcher at ERC/ESIPP, University College Dublin, Ireland. Previously, he worked as a Post-Doctoral Research & Teaching Fellow at the School of Mathematics, University of Edinburgh, UK, and for six months as a Post-Doctoral Research Associate at the Modelling and Simulation Centre, University of Manchester, UK. He has published 45 articles (16 single authored, and 29 co-authored with researchers from Universities and Industries) in internationally leading academic journals. He is an Editor in Applied Sciences, Signals, and has also served as a reviewer 333 times in 66 different journals.

Muyang Liu received from University College Dublin, Ireland, the ME in Electrical Energy Engineering in 2016. Since September 2016, she is a Ph.D. student candidate with University College Dublin. Her scholarship is funded through the SFI Investigator Award with title “Advanced Modelling for Power System Analysis and Simulation”. Her current research interests include small-signal and transient stability analysis of power systems modelled through functional differential-algebraic equations.

Georgios Tzounas received from National Technical University of Athens, Greece, the ME in Electrical and Computer Engineering in 2017. Since September 2017, he is Ph.D. candidate with University College Dublin. His scholarship is funded through the SFI Investigator Award with title “Advanced Modelling for Power System Analysis and Simulation.” His current research interests include eigenvalue problems as well as stability analysis and robust control of power system with inclusion of measurement delays.