Toward improved uses of the conjugate gradient method for power system applications

The conjugate gradient method has been suggested as a better alternative to direct methods for the solution of certain large sparse linear systems Ax=b; where A is symmetric and positive definite. Efficiency considerations often require that the conjugate gradient method be accelerated by preconditioning (a linear transformation of A). One of the most widely used preconditioners is based on the incomplete LU factors of A. Positive definite preconditioner matrices assure convergence. However, the incomplete factorization for a symmetric and positive definite matrix is not necessarily positive definite. This paper provides significant theoretical insights into the conjugate gradient method for matrices arising from several classes of power systems problems. The paper also presents a new preconditioner (based on a one-time complete factorization) that is guaranteed to be positive definite.