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The ac optimal power flow (OPF) problem is one of the most fundamental problems in power systems engineering. For the past few decades, researchers have been relying on unproven heuristics to tackle OPF. The hardness of OPF stems from two issues: 1) nonconvexity and 2) combinatorial constraints (e.g., discrete power extraction 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 article presents a polynomial-time approximation algorithm to solve the convex-relaxed OPF with discrete demands as combinatorial constraints, which has a provably small parameterized approximation ratio (also known as polynomial-time approximation scheme (PTAS) algorithm). Together with the sufficient conditions on the exactness of the convex relaxation, we provide an efficient approximation algorithm to solve OPF with discrete demands, when the underlying network is radial with a fixed size and one feeder. The running time of the PTAS is O(n<sup>4m</sup>/<sup>&#x03F5;</sup>T), where T is the time required to solve a convex relaxation of the problem, and m and &#x03F5; are fixed constants. Based on prior hardness results of OPF, our PTAS is among the best achievable in theory. Simulations show that our algorithm can produce close-to-optimal solutions in practice.

Combinatorial Optimization of AC Optimal Power Flow With Discrete Demands in Radial Networks