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AC optimal power flow (AC OPF) is a fundamental problem in power systemoperations. Accurately modeling the network physics via the AC power flowequations makes AC OPF a challenging nonconvex problem. To search for globaloptima, recent research has developed a variety of convex relaxations thatbound the optimal objective values of AC OPF problems. The well-known QCrelaxation convexifies the AC OPF problem by enclosing the non-convex terms(trigonometric functions and products) within convex envelopes. The accuracy ofthis method strongly depends on the tightness of these envelopes. This paperproposes two improvements for tightening QC relaxations of OPF problems. Wefirst consider a particular nonlinear function whose projections are thenonlinear expressions appearing in the polar representation of the power flowequations. We construct a convex envelope around this nonlinear function thattakes the form of a polytope and then use projections of this envelope toobtain convex expressions for the nonlinear terms. Second, we use certaincharacteristics of the sine and cosine expressions along with the changes intheir curvature to tighten this convex envelope. We also propose a coordinatetransformation that rotates the power flow equations by an angle specific toeach bus in order to obtain a tighter envelope. We demonstrate theseimprovements relative to a state-of-the-art QC relaxation implementation usingthe PGLib-OPF test cases. The results show improved optimality gaps in 68% ofthese cases.

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