Power systems matrices are both irregular and the sparsest matrices available to the academic and industrial communities. As a result, research into efficient sparse linear solvers for power systems applications has not met with the same success as research into efficient general parallel sparse linear solvers [7,54]. The state of parallel direct linear solver development in the power systems community has yielded solvers sufficiently inefficient that sequential algorithms are used to factor and triangular solve equations that may be formulated in parallel [7]. While efficient parallel linear solvers have not been reported in the power systems community journals to solve the special very sparse irregular power systems network matrices, there has been significant research into efficient general sparse linear solvers for general matrices, always larger and less sparse than power systems network matrices [8,9,10,19,20,21,25,29,46,47,55,56,57,58,64].
In the research presented in this thesis, we have developed specialized, efficient parallel sparse linear solvers for linear systems derived from power systems networks. The performance of our parallel linear solvers is significantly better than the performance of linear solvers reported in the power systems literature [7,54]. In order to develop efficient parallel linear solvers, we have utilized state-of-the-art research into:
Research by others into general parallel linear solvers has answered
many of the outstanding questions concerning the development of
techniques for efficient parallel algorithms to solve
general sparse matrices from the structural analysis community. The
research presented in this thesis presents efficient
parallel direct and iterative linear solvers that yield efficient
performance on power systems network linear equations for either
symmetric positive definite matrices or position symmetric matrices
that do not require pivoting to ensure numerical stability.
Developing linear solvers for power systems applications that require
pivoting remains as future research, as has the optimization of the
use of preconditioning techniques with iterative solvers. Another
power systems application problem that remains for future research,
is the development of new, more efficient differential-algebraic
equation (DAE) solvers that would utilize the parallel
block-diagonal-bordered linear solvers and address the entire
linearized block-diagonal-bordered DAE matrix [34]. A summary
of the research to date and future research is presented in
figure
Figure: Summary of Completed and Future Research