A primal-dual integrated nonlinear rescaling approach applied to the optimal reactive dispatch problem

Abstract

The objective of this paper is to propose a novel augmented Lagrangian approach for solving the Optimal Reactive Dispatch (ORD) problem in electric power systems, which consists in a large nonconvex Nonlinear Programming (NLP) problem. Given an NLP problem, its associated dual problem is solved by a proximal point method, whose distance between two points is calculated by means of a convex combination involving Bregman’s distance and Csiszar’s entropic φ-divergence distance pondered by a weighting factor, namely Integrated Distance Measure (IDM). Such dual proximal point method with the IDM leads to an equivalent augmented Lagrangian method with a new family of nonquadratic penalty functions for the primal problem, namely Integrated Nonlinear Rescaling (INLR) method. The concepts of integrated logarithmic modified barrier and integrated exponential functions, which are elements of the new family of penalty functions, are defined. A Primal-Dual Integrated Nonlinear Rescaling (PDINLR) method is proposed to solve convex and nonconvex NLPs. A small convex NLP problem is used to depict how such algorithm attains optima, and a small nonconvex NLP problem is used to show how the proposed approach outperforms other known methods in the area when a suitable value for the weighting factor is chosen. The effectiveness, efficiency and the robustness of the proposed PDINLR method with the integrated logarithmic modified barrier function are shown by the resolution of the ORD problem for the IEEE 30 and 300-bus test-systems.