Numerical Investigation on the Solution of Ill-Conditioned Load Flow Linear Equations

Solving load flow problems via Krylov subspace iterative methods is not a new subject in the power system industry. The early methods worked with symmetric positive definite matrices only, but new ones have been developed for general-purpose matrices, such as the generalized minimum residual method (known by the acronym GMRES). When compared with direct methods, their implementation demands too much effort and their performance is unaffordable without proper preconditioning and other strategies. However, important numerical issues that may justify the low (or high) efficiency and robustness of such methods have been usually uncovered in the power systems literature. This paper investigates some of these issues based on a multiuse incomplete LU preconditioner (known by the acronym ILU) focusing on ill- and very ill-conditioned real power systems aiming to provide important insights into ILU-GMRES performance and into the dilemma: iterative versus direct methods.