Tightening QC Relaxations of AC Optimal Power Flow through Improved Linear Convex Envelopes

AC optimal power flow (AC OPF) is a fundamental problem in power system operations. Accurately modeling the network physics via the AC power flow equations makes AC OPF a challenging nonconvex problem. To search for global optima, recent research has developed a variety of convex relaxations that bound the optimal objective values of AC OPF problems. The well-known QC relaxation convexifies the AC OPF problem by enclosing the non-convex terms (trigonometric functions and products) within convex envelopes. The accuracy of this method strongly depends on the tightness of these envelopes. This paper proposes two improvements for tightening QC relaxations of OPF problems. We first consider a particular nonlinear function whose projections are the nonlinear expressions appearing in the polar representation of the power flow equations. We construct a convex envelope around this nonlinear function that takes the form of a polytope and then use projections of this envelope to obtain convex expressions for the nonlinear terms. Second, we use certain characteristics of the sine and cosine expressions along with the changes in their curvature to tighten this convex envelope. We also propose a coordinate transformation that rotates the power flow equations by an angle specific to each bus in order to obtain a tighter envelope. We demonstrate these improvements relative to a state-of-the-art QC relaxation implementation using the PGLib-OPF test cases. The results show improved optimality gaps in 68 these cases.