Optimal power flow over tree networks

The optimal power flow (OPF) problem is critical to power system operation but it is generally non-convex and therefore hard to solve. Recently, a sufficient condition has been found under which OPF has zero duality gap, which means that its solution can be computed efficiently by solving the convex dual problem. In this paper we simplify this sufficient condition through a reformulation of the problem and prove that the condition is always satisfied for a tree network provided we allow over-satisfaction of load. The proof, cast as a complex semi-definite program, makes use of the fact that if the underlying graph of an n × n Hermitian positive semi-definite matrix is a tree, then the matrix has rank at least n-1.