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 M 0 0 0. For this we deene a diierent extension of MM obius transforms to pencils that inhibits the eeect on iterations of the spurious eigenvalue at innnity. These algorithms use a technique of precondi-tioning the initial vectors by MM obius transforms which together with shift-invert iterations accelerate the convergence to the desired eigenvalues. Also, we see that MM 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 MM 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. RESUMO: Neste artigo, descrevemos algoritmos baseados em transformaa c~ oes espectrais para computaa c~ ao de autovalores com parte real positiva de pencils de matrizes esparsas e n~ ao sim etricas, (J; L), em que L e da forma M 0 0 0. Para isso deenimos uma extens~ ao das transformac~ oes de MM obius a pencils que inibe a atuaa c~ ao do autovalor innnito sobre as iteraa c~ oes. Esses algoritmos usam uma t ecnica 1 de precondicionamento dos vetores iniciais via transformadas de MM obius que junto com iteraa c~ oes tipo pot^ encia inversa com shift aceleram a converg^ encia para os autovalores desejados. Vemos tamb em que as transformadas de MM obius podem ser usadas com sucesso no processo de inibir a converg^ encia para um autovalor jj a