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AC optimal power flow (AC~OPF) is a challenging non-convex optimizationproblem that plays a crucial role in power system operation and control.Recently developed convex relaxation techniques provide new insights regardingthe global optimality of AC~OPF solutions. The quadratic convex (QC) relaxationis one promising approach that constructs convex envelopes around thetrigonometric and product terms in the polar representation of the power flowequations. This paper proposes two methods for tightening the QC relaxation.The first method introduces new variables that represent the voltage magnitudedifferences between connected buses. Using "bound tightening" techniques, thebounds on the voltage magnitude difference variables can be significantlysmaller than the bounds on the voltage magnitudes themselves, so constraintsbased on voltage magnitude differences can tighten the relaxation. Second,rather than a potentially weaker "nested McCormick" formulation, this paperapplies "Meyer and Floudas" envelopes that yield the convex hull of thetrilinear monomials formed by the product of the voltage magnitudes andtrignometric terms in the polar form of the power flow equations. Comparison toa state-of-the-art QC implementation demonstrates the advantages of theseimprovements via smaller optimality gaps.

Improving QC Relaxations of OPF Problems via Voltage Magnitude Difference Constraints and Envelopes for Trilinear Monomials