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

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Pothen, Alex Halappanavar, Mahantesh Huang, Zhenyu Yeung, Yu-Hong
Copyright Year	2017
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 three power grids with PARDISO, a parallel direct solver, and CHOLMOD, a direct solver with the ability to modify the Cholesky factors of the matrix. We show that our augmented algorithms outperform PARDISO (by two orders of magnitude) and CHOLMOD (by a factor of up to 5). Further, our algorithms scale better than CHOLMOD as the number of updated elements increases. The solutions are computed with high accuracy. Our algorithms are capable of computing $N-k$ contingency analysis on a 778,000-bus grid, updating a solution with $k=20$ elements in 16 milliseconds on an Intel Xeon processor.
Sponsorship	Battelle. Office of Science. National Science Foundation
Starting Page	S809
Ending Page	S827
Page Count	19
File Format	PDF
ISSN	10648275
DOI	10.1137/16M1082755
e-ISSN	10957197
Issue Number	5
Volume Number	39
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2017-10-26
Access Restriction	Subscribed
Subject Keyword	iterative methods matrix updates direct methods Iterative methods for linear systems Complexity and performance of numerical algorithms Direct methods for linear systems and matrix inversion Sparse matrices power flow analysis augmented matrix sparse matrices contingency analysis
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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