Comparison of Numerical Methods and Open-Source Libraries for Eigenvalue Analysis of Large-Scale Power Systems

Abstract

This paper discusses the numerical solution of the generalized non-Hermitian eigenvalue problem. It provides a comprehensive comparison of existing algorithms, as well as of available free and open-source software tools, which are suitable for the solution of the eigenvalue problems that arise in the stability analysis of electric power systems. The paper focuses, in particular, on methods and software libraries that are able to handle the large-scale, non-symmetric matrices that arise in power system eigenvalue problems. These kinds of eigenvalue problems are particularly difficult for most numerical methods to handle. Thus, a review and fair comparison of existing algorithms and software tools is a valuable contribution for researchers and practitioners that are interested in power system dynamic analysis. The scalability and performance of the algorithms and libraries are duly discussed through case studies based on real-world electrical power networks. These are a model of the All-Island Irish Transmission System with 8640 variables; and, a model of the European Network of Transmission System Operators for Electricity, with 146,164 variables.

Complex moment-based methods for differential eigenvalue problems

“…These linear systems can be independently solved so that the eigensolvers have good scalability, as demonstrated in [21,19]. For this reason, complex moment-based eigensolvers have attracted considerable attention, particularly in physics [26,21], materials science [19,9,22], power systems [34], data science [16] and so on. Currently, there are several methods, including direct extensions of Sakurai and Sugiura's approach [28,11,10,12,14,18,15], the FEAST eigensolver [26] developed by Polizzi, and its improvements [30,6,21].…”

Complex moment-based methods for differential eigenvalue problems

“…These linear systems can be independently solved so that the eigensolvers have good scalability, as demonstrated in [2,3]. For this reason, complex moment-based eigensolvers have attracted considerable attention, particularly in physics [1,2], materials science [3][4][5], power systems [17], data science [18], and so on. Currently, there are several methods, including direct extensions of Sakurai and Sugiura's approach [19][20][21][22][23][24][25], the FEAST eigensolver [1] developed by Polizzi, and its improvements [2,26,27].…”

Magnetic Normal Mode Calculations in Big Systems: A Highly Scalable Dynamical Matrix Approach Applied to a Fibonacci-Distorted Artificial Spin Ice

“…For comparison, the largest ever eigenvalue analysis with a direct algorithm to date (as far as we know) was the solution of a 10 6 problem, half the number of our independent variables, carried out in 2014 by the Japanese K computer in Riken. To be able to obtain this result, the K computer includes 88,000 processors that draw a peak power of 12.6 MW, while its operation costs annually US$10 million [37]. Many different modes are found in the lowest frequency spectrum of the system (0-20 GHz); some of the typical ones are plotted in Figure 4.…”