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 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
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
[13]. The structural matrices, BCSSTK13 to BCSSTK32, are
frequently used in papers to benchmark parallel sparse linear
algorithms
[19,20,21,25,29,33,46,47,55,56,57,58,64].
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: Relative Frequency Histogram of Edges per Graph Node
: 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 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
[7,38,54]. 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 sections 7.1
and 7.2 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.