Comparative Evaluation of SDP, SOCP, and QC Convex Relaxations for Large-Scale Market-Based AC Optimal Power Flow

Abstract: The alternating current optimal power flow (ACOPF) problem is central to modern power system operations, determining how electricity is generated and transmitted to maximize social welfare while respecting physical and operational constraints. However, the nonlinear and non-convex nature of AC power flow equations makes finding globally optimal solutions computationally intractable for large networks. Convex relaxations - including semidefinite programming (SDP), second-order cone programming (SOCP), and quadratic convex (QC) formulations - provide tractable alternatives that can yield provably optimal or near-optimal solutions under appropriate conditions. This paper presents a comprehensive comparative study of multiple ACOPF relaxations applied to market-based welfare maximization. We implement DCOPF, Shor's SDP relaxation (complex and real-valued forms), chordal SDP, Jabr's SOCP relaxation, and QC relaxations in a unified, solver-native framework using the MOSEK Fusion API, eliminating modeling overhead present in high-level frameworks such as CVXPY. To address the practical challenge of missing or overly conservative angle difference bounds required by QC relaxations, we employ quasi-Monte Carlo sampling with Sobol sequences to empirically estimate tighter bounds. We evaluate these relaxations on subnetworks of varying sizes derived from the ARPA-E dataset, systematically comparing solution quality, runtime, and memory consumption. Our results demonstrate the trade-offs between relaxation tightness and computational efficiency, providing practical guidance for selecting appropriate formulations based on network scale and solution requirements.