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

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.