Comparison of Various Trilinear Monomial Envelopes for Convex Relaxations of Optimal Power Flow Problems

Solutions to optimal power flow (OPF) problems provide operating points for electric power systems that minimize operational costs while satisfying both engineering limits and the power flow equations. OPF problems are non-convex and may have multiple local optima. To search for global optima, recent research has developed a variety of convex relaxations to bound the optimal objective values of OPF problems. Certain relaxations, such as the quadratic convex (QC) relaxation, are derived from OPF representations that contain trilinear monomials. Previous work has considered three techniques for relaxing these trilinear monomials: recursive McCormick (RMC) envelopes, Meyer and Floudas (MF) envelopes, and extreme-point (EP) envelopes. This paper compares the tightness and computational speed of relaxations that employ each of these techniques. Forming the convex hull of a single trilinear monomial, MF and EP envelopes are equivalently tight. Empirical results show that QC formulations using MF and EP envelopes give tighter bounds than those using RMC envelopes. Empirical results also indicate that the EP envelopes have advantages over MF envelopes with respect to computational speed and numerical stability when used with state-of-the-art second-order cone programming solvers.