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An iterative method for solving complex-symmetric systems arising in electrical power modeling. (English) Zbl 1082.65039

Heavily relying on work by D. Gremban, G. L. Miller and M. Zagha [Performance evaluation of a new parallel preconditioner, in Proceedings of the International Parallel Processing Symposium, IEEE Computer Society, Los Alamitos, CA 65–69 (1995)] and on the dissertation of K. D. Gremban [Combinatorial Preconditioners for Sparse, Symmetric, Diagonally Dominant Linear Systems, Ph.D. thesis, Carnegie Mellon University, Pittsburgh, PA (1996)], the authors consider the systems of linear equations arising from statically modelling large power systems. The solution of those equations is the vector of node voltages.
Compared to the work of Gremban et al. [loc. cit.], here two contributions are made: 1) the network graph based (namely: support tree) preconditioner is extended to allow for complex weights of the graph edges (which means AC networks) and for widely varying weights (more closely: two classes of weights, corresponding to functioning and faulty edges, where the impedances differ by several magnitudes); 2) using the projection onto the nullspace of that part of the system matrix which corresponds to the functioning edges, the preconditioned equations can be solved with higher accuracy.
The proofs and algorithms use considerations on the network graph (e.g. the determination of the nullspace needs a search of connected subgraphs). A convincing series of numerical examples is provided. The solution by Matlab’s A\b command is usually much faster but sometimes (e.g. on faulty systems) delivers only low solution accuracy.

MSC:

65F10 Iterative numerical methods for linear systems
78A55 Technical applications of optics and electromagnetic theory
65F50 Computational methods for sparse matrices
78M25 Numerical methods in optics (MSC2010)
65F35 Numerical computation of matrix norms, conditioning, scaling

Software:

Matlab; MESHPART
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