The parallel block-diagonal-bordered direct solvers we implemented in this research, address the most difficult power systems applications to implement on multi-processor architectures --- solutions pertaining only to power system networks. Parallel block-diagonal-bordered sparse linear solver algorithms can readily be extended to applications that have power systems networks as a small portion of a larger matrix, for example, the entire system of linearized differential-algebraic equations (DAEs) encountered in transient stability analysis or small-signals analysis applications. These applications add many natural blocks of linearized differential equations that significantly increase the size of the matrix and the data density. The linearized differential equations are less-sparse than the network equations and may require pivoting to ensure numerical stability. Pivoting for this matrix would be limited to within diagonal-blocks to place limits on fillin, but the efficient static data structures would need to be replaced by less-efficient dynamic linked-list-based data structures. Any of these modifications would increase computational workload --- work that does not require interprocessor communications. As a result, any modifications to algorithms to include these additional features would improve problem granularity and parallel speedup on the Thinking Machines CM-5 and on future SPPs.
In addition to simply implementing a version of these parallel block-diagonal-bordered linear solvers for transient stability or small signals analysis Jacobian solutions, there is a rich area for research to incorporate the concepts of these parallel linear solvers more closely with DAE solvers used to solve the differential equations and non-linear algebraic equations found in power systems simulations [34]. This research has examined general parallel block-diagonal-bordered sparse linear solvers; meanwhile, there are research opportunities to examine a more tight coupling of parallel block-diagonal-bordered sparse linear solvers with DAE solvers in power systems applications.
The parallel block-diagonal-bordered Gauss-Seidel solver we implemented in this research, does not include a preconditioner to speed convergence of the iterative solver. Future research could examine effective and efficient ways to add preconditioners such as incomplete LU factorization to improve overall iterative linear solver performance.