Efficient multiprocessing approaches to the execution of digital computer programs, which analyse power system transient stability, have been investigated. Different program sections are found to have greatly varying levels of effect on overall execution performance. The most demanding need which emerges is for a very efficient practical method to solve large sparse sets of linear equations. Without a satisfactory scheme, the maximum gain in performance over single processor operation is severely restricted. A suitable algorithm has been developed and is described. To validate the effectiveness of proposed algorithms, practical multiprocessing hardware has been built. The equipment has also allowed evaluation of hardware requirements, in particular the capability of the inter-processor communication network. The parallel processor developed enables efficient program development and testing in an environment which is research oriented yet very closely resembles possible practical implementations. The results of an execution simulation are combined with practical performance measurements to determine limits to the number of processors usefully employed, and the gains in performance over single processor operation achieveable. When compared with other algorithms for solving linear equations, the one developed is shown to run very efficiently. To further improve performance, novel methods of mixing execution of the linear equation solutions and other sections of a transient stability analysis program have been practically implemented.