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Block-Diagonal-Bordered Power System Matrices

Power system distribution networks are generally hierarchical with limited numbers of high-voltage lines transmitting electricity to connected local networks that eventually distribute power to customers. In order to ensure reliability, highly interconnected local networks are fed electricity from multiple high-voltage sources. Electrical power grids have graph representations which in turn can be expressed as matrices --- electrical buses are graph nodes and matrix diagonal elements, while electrical transmission lines are graph edges which can be represented as non-zero off-diagonal matrix elements. We show that it is possible to identify the hierarchical structure within a power system matrix using only the knowledge of the interconnection pattern by tearing the matrix into partitions and coupling equations that yield a block-diagonal-bordered matrix. Node-tearing-based partitioning identifies the basic network structure that provides parallelism for the majority of calculations within the direct solution of a linear system.

In this paper we examine the applicability of parallel direct block-diagonal-bordered sparse solvers for real power system applications that require either the solution of symmetric positive definite sparse matrices or location symmetric sparse matrices that result from solving problems relating to power systems networks. Variations of this technique could be used to solve other power system sparse linear systems such as those that result from solving linearized differential-algebraic equations that result from transient stability analysis or small-signal stability assessments. The implementations we describe in this paper work directly with the equations resulting from the power systems network, the smallest class of power system matrix.

The implementations we developed can be used to solve symmetric positive definite load flow analysis Jacobian matrices or position symmetric network matrices from transient stability analysis. In spite of only examining direct linear solver implementations that solve relatively small network-related matrices, we have been able to obtain good parallel speedups. We expect that even better performances would be possible for parallel implementations designed to solve a single system of linear equations that represent a combination of the generator dynamical equations and network equations in a linearized form of the differential-algebraic equations from transient stability analysis or small-signal analysis. For these problems, there is additional parallel calculations with no additional parallel communications overhead.



next up previous contents
Next: Block-Diagonal-Bordered Direct Linear Up: Introduction Previous: Introduction



David P. Koester
Sun Oct 22 15:31:10 EDT 1995