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 various convex relaxations that bound the optimal objective values of AC OPF problems. The QC relaxation convexifies the AC OPF problem by enclosing the non-convex terms within convex envelopes. The QC relaxation's accuracy 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 polytope-shaped convex envelope around this nonlinear function and derive convex expressions for the nonlinear terms using its projections. Second, we use sine and cosine expression properties, along with changes in their curvature, to tighten this convex envelope. We also propose a coordinate transformation to tighten the envelope by rotating power flow equations based on individual bus-specific angles. We compare these enhancements to a state-of-the-art QC relaxation method using PGLib-OPF test cases, revealing improved optimality gaps in 68% of the cases.