  The classical optimal power flow problem optimizes the power flow in a power
network considering the associated flow and operating constraints. In this
paper, we investigate optimal power flow in the context of utility-maximizing
demand response management in distribution networks, in which customers'
demands are satisfied subject to the operating constraints of voltage and
transmission power capacity. The prior results concern only elastic demands
that can be partially satisfied, whereas power demands in practice can be
inelastic with binary control decisions, which gives rise to a mixed integer
programming problem. We shed light on the hardness and approximability by
polynomial-time algorithms for optimal power flow problem with inelastic
demands. We show that this problem is inapproximable for general power network
topology with upper and lower bounds of nodal voltage. Then, we propose an
efficient algorithm for a relaxed problem in radial networks with bounded
transmission power loss and upper bound of nodal voltage. We derive an
approximation ratio between the proposed algorithm and the exact optimal
solution. Simulations show that the proposed algorithm can produce
close-to-optimal solutions in practice.
