Convexification of Generalized Network Flow problem with application to power systems

This paper is concerned with the minimum-cost flow problem over an arbitrary flow network. In this problem, each node is associated with some possibly unknown injection, each line has two unknown flows at its ends related to each other via a nonlinear function, and all injections and flows need to satisfy certain box constraints. This problem, named generalized network flow (GNF), is highly non-convex due to its nonlinear equality constraints. Under the practical assumption of monotonicity and convexity of the flow and cost functions, a convex relaxation is proposed, which always finds the optimal injections. This relaxation may fail to find optimal flows because the mapping from injections to flows might lead to an exponential number of solutions. However, once optimal injections are found in polynomial time, other techniques can be used to find a feasible set of flows corresponding to the injections. A primary application of this work is in optimization over power networks. Recent work on the optimal power flow (OPF) problem has shown that this non-convex problem can be solved efficiently using semidefinite programming (SDP) after two approximations: relaxing angle constraints (by adding virtual phase shifters) and relaxing power balance equations to inequality constraints. The results of this work prove two facts for the OPF problem: (i) the second approximation (on balance equations) is not needed in practice under a very mild angle assumption, and (ii) if the SDP relaxation fails to find a rank-one solution, the optimal injections (and not flows) may still be recovered from an undesirable high-rank solution.