  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 extraction constraints). The recent advances in providing
sufficient conditions on the exactness of convex relaxation of OPF can address
the issue of non-convexity. To complete the understanding of OPF, this paper
presents a polynomial-time approximation algorithm to solve the convex-relaxed
OPF with discrete demands as combinatoric constraints, which has a provably
small parameterized approximation ratio (also known as 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 PTAS is $O(n^{4m/\epsilon}T)$, where $T$ is the
time required to solve a convex relaxation of the problem, and $m, \epsilon$
are fixed constants. Based on prior hardness results of OPF, our PTAS is among
the best achievable in theory. Simulations show our algorithm can produce
close-to-optimal solutions in practice.
