This paper is organized as follows. In section 2, we describe the electrical power system applications that are the basis for this work. In section 3, we briefly review direct solution techniques for factorization and forward reduction/backward substitution, and we review the literature concerning general parallel LU and Choleski factorization algorithms. This is followed by a theoretical derivation of the available parallelism in both the factorization and forward reduction/backward substitution phases when solving block-diagonal-bordered form sparse matrices. Paramount to exploiting the advantages of this parallel linear solver is the process of ordering the irregular sparse power system matrices into this form in a manner that balances the workload among multi-processors. In section 5, we describe the three-step preprocessing phase used to generate matrix ordering for block-diagonal-bordered matrices with uniformly distributed processing load. In this section, we introduce pseudo-factorization and we review minimum degree ordering and pigeon-hole load balancing algorithms. We present the node-tearing algorithm developed to order matrices into block-diagonal-bordered form in section 6. In section 7, we describe our block-diagonal-bordered sparse LU and Choleski algorithms that has been implemented on the CM-5. Analysis of the performance of these ordering techniques are presented in section 8 for actual power system network matrices from the Boeing-Harwell series, the Electrical Power Research Institute (EPRI), and an electrical utility, the Niagara Mohawk Power Corporation. We present our conclusions concerning parallel block-diagonal-bordered direct linear solvers for electrical power system applications in section 9.