Dual ${M}$ -Convex Variable Subsets Family and Extremum Analysis for the OPF Problem

The optimal power flow (OPF) model is a central optimization problem in power system network. In this paper, we propose a novel approach to solve the OPF problem that has a convex objective function and non-convex feasible domain due to the constraints. Based on the concept of abstract convex analysis, we construct the dual
${M}$
-convex subsets family of original variables by using the variable separation method, followed by the analysis of the extremum according to the infimum base of the
${M}$
-convex subsets. It is challenging to obtain an explicit mapping function among the separated variables due to the nonlinear equality constraints. We, therefore, use the theorem of implicit function and the differential function to do duality analysis on separated variables. Based on the min–max principle of the primal-dual problem of OPF, we derive the condition of the complementary factor which leads the Lagrange dual problem to maximum, and make the variable separation that results in a Minkowski-type dual
${M}$
-convex subset. We then can obtain a local minimum using the principle of abstract convex optimization, which will be the global optimal solution under the Karush–Kuhn–Tucker conditions. We evaluate the proposed approach on several IEEE systems. The simulation results indicate that the approach is feasible and effective to deal with the non-convex OPF problem with a non-convex feasible domain.