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.

Iterative methods built on Krylov subspaces have been little explored to date for the computation of eigenvalues and eigenvectors in small-signal stability analysis. Such computation is challenging and computationally expensive for matrices with a certain number of multiple and clustered eigenvalues, conditions that can be found in many dynamic state Jacobian matrices. The present paper aims to contribute with a block algorithm to perform small-signal stability analysis with this particular type of matrix, built on the Augmented Block Householder Arnoldi (ABHA) method. The advantages of using a block method lie on the fact that the searching subspace for approximate solutions is the sum of every Krylov subspace, and therefore, the solution is expected to converge in less iterations than an unblock method. The efficiency and robustness of the proposal are examined through numerical simulations using three power systems and two other methods: the conventional Arnoldi (unblock) and QR decomposition. The results indicate that the proposed numerical algorithm is more robust than the other two for handling dynamic state Jacobian matrices having a certain number of multiple and clustered eigenvalues.

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