Convex envelopes of optimal power flow with branch flow model in rectangular form

Optimal power flow (OPF) is an important tool for economic dispatch and other power system applications. Due to the non-convex nature of OPF formulations, convex relaxation serves as a bridge linking OPF formulation variants with efficient polynomial-time algorithms to obtain a global optimum. This paper proposes: 1) an OPF model based on branch flow model in rectangular form, and 2) two convex envelopes for this OPF model. First, by introducing explicit redundant variables for bi-directional power flows, and the products of bus voltages for each branch, OPF is formulated into a non-convex quadratic programming (QP) model. For an n-bus, l-branch power system, the non-convex part of this QP model is collectively expressed by 2n quadratic terms and 4l bilinear terms. Second, primal and dual convex envelopes for these terms are derived to convert the non-convex QP model into a convex one. Third, the tightness of primal and dual convex envelopes is validated via a spatial branch-and-bound framework using IEEE 14-bus system. Numerical studies show that the proposed convex envelopes are tight to get an optimum with small optimality gap and slight bus power mismatches.