Spectral Transformation Algorithms for Computing Unstable Modes of Large Scale Power Systems

Abstract: In this paper we describe spectral transformation algorithms for the computation of eigenvalues with positive real part of sparse nonsymmetric matrix pencils $(J,L)$, where $L$ is of the form $\pmatrix{M&0\cr 0&0}$. For this we define a different extension of M\"obius transforms to pencils that inhibits the effect on iterations of the spurious eigenvalue at infinity. These algorithms use a technique of preconditioning the initial vectors by M\"obius transforms which together with shift-invert iterations accelerate the convergence to the desired eigenvalues. Also, we see that M\"obius transforms can be successfully used in inhibiting the convergence to a known eigenvalue. Moreover, the procedure has a computational cost similar to power or shift-invert iterations with M\"obius transforms: neither is more expensive than the usual shift-invert iterations with pencils. Results from tests with a concrete transient stability model of an interconnected power system whose Jacobian matrix has order 3156 are also reported here.