Combinatorial Optimization of AC Optimal Power Flow in Radial Distribution Networks

The AC Optimal power flow (OPF) problem is one of the most fundamental problems in power systems engineering. For the past decades, researchers have been relying on unproven heuristics to tackle OPF. The hardness of OPF stems from two issues: (1) non-convexity and (2) combinatoric constraints (e.g., discrete power injection/ejection constraints). The recent advances in providing sufficient conditions on the exactness of convex relaxation of OPF can address the issue of nonconvexity. To complete the understanding of OPF, this paper presents a polynomial-time approximation algorithm to solve the convex relaxed OPF with combinatoric constraints, which has a provably small parameterized approximation ratio (also known as PTAS algorithm). Together with the sufficient conditions on the exactness of convex relaxation, complete insights can be provided on the efficiency of solving OPF in general. This paper also discusses fundamental hardness results of OPF to show that our PTAS is among the best achievable in theory. Simulations show our algorithm can produce close-to-optimal solutions in practice.