A Fast and Robust Variable Metric Method for Optimum Power Flows

The Han-Powell algorithm has proved to be extremely fast and robust for small Optimum Power Problems [1]. There is every reason to believe its performance could be extended to large problems, provided its one serious disadvantage is eliminated. This disadvantage stems from its use of nonsparse approximations to certain Hessian matrices. These matrices are of dimension (m x m) where m is the number of decision variables. Since all the network variables are retained by the algorithm as decision variables, the Hessians quickly get too big to be conveniently accommodated. One remedy is to add a variable-reduction-procedure. Berna, Locke and Westerberg [2] have developed one such procedure. It helps but does not completely eliminate the difficulties. This paper develops another reduction procedure with concepts borrowed from the fields of network dissection and parallel processing. The computations are arranged in two nested loops. The inner loop eliminates n < m of the variables by satisfying the problem\´s n equality constraints to a tolerance that is tightened as the problem\´s solution is approached. The outer loop applies the Han-Powell algorithm to the reduced problem. Besides eliminating the need for dealing with unwieldly Hessians, this "reduced method" appears to be as robust as the original Han-Powell algorithm and converges at least as fast for small problems. The method has not yet been tested on large problems, but it is reasonable to expect that it will perform as well on them.