On the nonconvex feasible region of optimal power flow: Theory, degree, and impacts

The non-convexity of the Optimal Power Flow (OPF) feasible region complicates the solution process and affects the applicability of various optimization techniques, which is crucial for understanding the OPF problem. This paper systematically investigates the non-convexity properties of the AC OPF feasible (power) injection region (FIR) and identifies key factors influencing its non-convexity from both analytical and numerical perspectives. Specifically, a necessary condition for FIR convexity and a sufficient condition for FIR non-convexity are derived. Based on these findings, it is concluded that the feasible region of ACOPF is inherently non-convex, with network losses playing a significant role. To avoid misjudgment of non-convexity, a non-convexity degree index for the FIR is introduced, and a numerical method to compute it is proposed. Numerical results on 9-bus and 57-bus systems indicate that the non-convexity degree of a lossless FIR is 0, whereas for a lossy FIR, it ranges from 70% to 100%. Furthermore, factors contributing to non-convexity and their impact on the location of the optimal solution and the effectiveness of convex relaxation methods (CRMs) are discussed. The numerical results demonstrate that for the same system, the optimality gap of CRMs can be as low as 0.02% in lossless networks but increases to 0.28% or more in lossy networks. These findings elucidate the relationship between network losses and the optimality gap of CRMs, providing deeper insights into the characteristics of the ACOPF problem.