Scalable Global Optimization for AC-OPF via Quadratic Convex Relaxation and Branch-and-Bound

Abstract: The Optimal Power Flow (OPF) problem is central to the reliable and efficient operation of power systems, yet its non-convex nature poses significant challenges for finding globally optimal solutions. While convex relaxation techniques such as Quadratic Convex (QC) relaxation have shown promise in providing tight lower bounds, they typically do not guarantee global optimality. Conversely, global optimization methods like the Branch and Bound (B&B) algorithm can ensure optimality but often suffer from high computational costs due to the large search space involved. This paper proposes a novel B&B-assisted QC relaxation framework for solving the AC-OPF problem that leverages the strengths of both approaches. The method systematically partitions the domains of key OPF variables, specifically, voltage magnitudes and voltage angle differences, into two equal subintervals at each iteration. The QC relaxation is then applied to each subregion to compute a valid lower bound. These bounds are compared against an upper bound obtained from a feasible AC-OPF solution identified at the outset. Subregions that yield lower bounds exceeding the upper bound are pruned from the search, eliminating non-promising portions of the feasible space. By integrating the efficiency of the QC relaxation with the global search structure of the B&B algorithm, the proposed method significantly reduces the number of subproblems explored while preserving the potential to reach the global optimum. The algorithm is implemented using the PowerModels.jl package and evaluated on a range of PGLib-OPF benchmark cases. Results demonstrate that this hybrid strategy improves computational tractability and solution quality, particularly for large OPF instances.