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

TL;DR

The paper tackles the non-convex AC-OPF problem by integrating quadratic convex (QC) relaxations with a Branch-and-Bound (B&B) global search. It introduces auxiliary variables and convex envelopes to reformulate the OPF into a QC-OPF problem and then uses a B&B scheme that halves variable bounds (voltage magnitudes and angle differences) to systematically explore subregions, pruning those that cannot beat the incumbent or an AC-OPF upper bound. The hybrid QC-B&B approach yields tighter lower bounds than QC alone and improves proximity to the global optimum on PGLib-OPF cases, while managing computational costs through selective branching and aggressive pruning. This method demonstrates scalable, high-quality solutions for large OPF instances, offering a practical pathway toward globally near-optimal operation in power systems.

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.

Figures (12)

• Figure 1: Overview of the BB-QC-OPF framework. The flowchart illustrates how the Branch-and-Bound algorithm iteratively splits a single variable at each level, solving the QC-OPF problem within updated bounds for each child node. The figure highlights how variables' bounds are inherited from parent to child nodes throughout the branching process.
• Figure 2: Feasible space of the cyclic three bus system from narimani2018empirical, with $0.9 < V_1, V_2, V_3 < 1.1$ and $-2\pi < \Delta \theta_1, \Delta \theta_2, \text{and}~ \Delta \theta_3 < 2\pi$.
• Figure 3: Feasible space of cyclic three bus system from narimani2018empirical, with $0.9<V_1<1$, $0.9<V_2<1.1$, $0.9<V_3<1.1$, and $-2\pi < \Delta \theta_1, \Delta \theta_2, \text{and} \Delta \theta_3 < 2\pi$.
• Figure 4: Feasible space of cyclic three bus system from narimani2018empirical, with $0.9<V_1<1$, $0.9<V_2<1$, $0.9<V_3<1.1$, and $-2\pi < \Delta \theta_1, \Delta \theta_2, \text{and} \Delta \theta_3 < 2\pi$.
• Figure 5: Feasible space of cyclic three bus system from narimani2018empirical, with $0.9<V_1<1$, $0.9<V_2<1$, $1<V_3<1.1$, and $-2\pi < \Delta \theta_1, \Delta \theta_2, \text{and}~ \Delta \theta_3 < 2\pi$.