A partitioning method for finding eigenvalues and eigenvectors
Danuta Joanna Stadnicki · 1991
The need to compute eigenvalues and eigenvectors of very large matrices arises in many areas of science and engineering. One such example is in power systems. The successful operation of a power system requires that the system be stable for a wide range of the operating conditions. The purpose of this research was to study small-signal stability of large power systems. In the presented approach the total system is first divided into subsystems, and the eigenvalues and the eigenvectors of the subsystems are computed using the standard library routines. Each subsystem represents some physical part of the entire system. The subsystems are next interconnected, and the interactions between them are studied. Only the critical eigenvalues are computed. The eigenvalues of the total system are classified as local or global. Local eigenvalues are mainly influenced by one eigenvalue of a subsystem, and the change between the subsystem and total system eigenvalues is small. For global eigenvalues, several eigenvalues of subsystems interact with each other, and the change between subsystem and total system eigenvalues may be large. Eigenvalues of subsystems are examined to determine which ones interact with each other and may move substantially. Those which are found to interact with each other are studied together using a method based on the invariant subspaces. The invariant subspaces method solves for a group of eigenvalues and corresponding eigenvectors at the same time. It is similar to Newton's method, but it overcomes many problems which Newton's method has, such as: convergence to the same final eigenvalue starting from different eigenvalues of subsystems, poor convergence properties in the case of close eigenvalues, and problems with the solution of a large linear equation using iterative methods. The invariant subspaces method is capable of handling cases of close or equal eigenvalues, it has better convergence properties, especially when the global eigenvalues are studied, and the large set of linear equations, generated by it, is easier to solve.