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Abstract-This is a concise critical survey of the theory and practice relating to the ordered Gaussian elimination on sparse systems. A new method of renumbering by clusters is developed, and its properties described. By establishing a correspondence between matrix pattems and directed graphs, a sequential binary partition is used to decompose the nodes of a graph into clusters. By appropriate ordering of the nodes within each cluster and by selecting clusters, one at a time, both optimal ordering and a useful form of matrix banding are achieved. Some results pertaining to the compatibility between optimal ordering for sparsity and the usual pivoting for numerical accuracy are included.

A b s t r a c tThe paper describes the implementation of a quadratic interior point method for optimal power flow. Optimal power f l o w involves the determination of the optimal of a given objective function subject to given c o n s t r a i n t s .The scheme developed solves the quadratic or linear optimization problem subject to linear constraints. The algorithm has been evaluated on a 14-bus system, and its accuracy and speed are d e m o n s t r a t e d . I. INTRODU~IONIn the normal daily operation of a power system, emergency conditions may develop due to forced or random events. In order to maintain an economic and secure power system, several options are available to power system operators and planners. The traditional approach utilizes a trial and error technique f o r providing sub-optimum c o r r e c t i v e s t r a t e g i e s f o r alleviating violations due to planned or unexpected contingencies. This approach does not in general guarantee an acceptable optimum solution.The development of an optimal strategy for maintaining power system operation is accomplished by using several optimization approaches.Almost three decades ago, Carpentier [ 13 introduced a generalized, nonlinear programming formulation of the economic dispatch problem. It was later named optimal power flow (OPF) by Dommel and Tinney [2]. The optimal power flow computation has received widespread attention.It is of current interest to many power utilities, and has been identified as one of the most important operational tools of the power

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