AMPS: An Augmented Matrix Formulation for Principal Submatrix Updates with Application to Power Grids

Abstract

We present AMPS, an augmented matrix approach to update the solution to a linear system of equations when the matrix is modified by a few elements within a principal submatrix. This problem arises in the dynamic security analysis of a power grid, where operators need to perform $N-k$ contingency analysis, i.e., determine the state of the system when exactly $k$ links from $N$ fail. Our algorithms augment the matrix to account for the changes in it, and then compute the solution to the augmented system without refactoring the modified matrix. We provide two algorithms---a direct method and a hybrid direct-iterative method---for solving the augmented system. We also exploit the sparsity of the matrices and vectors to accelerate the overall computation. We analyze the time complexity of both algorithms and show that it is bounded by the number of nonzeros in a subset of the columns of the Cholesky factor that are selected by the nonzeros in the sparse right-hand-side vector. Our algorithms are compared on th...