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.
Matrices representing power system networks are some of the most sparse matrices encountered throughout the academic or industrial community. Figure 2 illustrates the proportion of graph nodes with a particular number of graph edges or the number of non-zero values in a matrix row or column for five separate power system matrices:
In this relative frequency histogram, the most frequently occurring number of edges per node is only 2! Table 1 provides additional data to illustrate that power system matrices are both relatively small in size and also have the fewest average edges per node of available matrices. In this table, all data except that from EPRI and Niagara Mohawk are from the Boeing-Harwell series [10]. The structural matrices, BCSSTK13 to BCSSTK32, are frequently used in papers to benchmark parallel sparse linear algorithms [14,15,16,18,20,22,31,32,37,38,39,40,41,45]. For power system matrices, the average number of edges per node is less than two while for many of the structural matrices, the average number of nodes per edge is greater than ten. Also the number of nodes in power system matrices are limited when compared to the Boeing-Harwell structural matrices.
Figure 2: Electrical Power System Networks --- Relative Frequency Histogram of Edges per Graph Node
Table 1: Comparison of Power System Matrices and Boeing-Harwell Structural Matrices
While power systems matrices are extremely sparse, they are also irregular, with the larger matrices having some nodes with greater than twenty edges. The histogram presented in figure 2 has been truncated at ten edges per node to emphasize the high incidence of edges with less than three nodes. As a result of the degree of sparsity and irregularity in these matrices, developing parallel sparse linear solvers for power systems application has proven to be a challenge [36,4]. Nevertheless, by developing parallel algorithms that actively address the irregular nature of the graphs with explicit load-balancing and by making all necessary communications as balanced, regular, and as asynchronous as possible, we will show in section 8 that our block-diagonal-bordered approach to addressing linear solvers for power system applications can yield respectable speedups even for as many as 32 processors.