A Quadratically-Constrained Convex Approximation for the AC Optimal Power Flow

TL;DR

The paper presents QCAC, a quadratically-constrained convex approximation for AC-OPF that avoids common simplifying assumptions and preserves the sparsity of the original AC constraints. It reformulates AC-OPF in lifted rectangular coordinates as a difference-of-convex problem and applies a first-order Taylor convexification of the concave terms around a base point $V_i^{re}, V_i^{im}$ to obtain an inner convex approximation when feasible. Across PowerGrid Library instances up to 30,000 buses, QCAC achieves competitive optimality gaps while improving distance-to-feasibility relative to TS, SOC, and SDP relaxations, and scales favorably due to its sparse, matrix-free convexification. Case studies in ORPF and PV hosting capacity illustrate practical advantages, with faster solve times and robust performance under topology changes. The approach supports warm-starts and learning-based base-point predictions, enabling decentralized and real-time applications in large-scale power systems.

Abstract

We introduce a quadratically-constrained approximation (QCAC) of the AC optimal power flow (AC-OPF) problem. Unlike existing approximations like the DC-OPF, our model does not rely on typical assumptions such as high reactance-to-resistance ratio, near-nominal voltage magnitudes, or small angle differences, and preserves the structural sparsity of the original AC power flow equations, making it suitable for decentralized power systems optimization problems. To achieve this, we reformulate the AC-OPF problem as a quadratically constrained quadratic program. The nonconvex terms are expressed as differences of convex functions, which are then convexified around a base point derived from a warm start of the nodal voltages. If this linearization results in a non-empty constraint set, the convexified constraints form an inner convex approximation. Our experimental results, based on Power Grid Library instances of up to 30,000 buses, demonstrate the effectiveness of the QCAC approximation with respect to other well-documented conic relaxations and a linear approximation. We further showcase its potential advantages over the well-documented second-order conic relaxation of the power flow equations in two proof-of-concept case studies: optimal reactive power dispatch in transmission networks and PV hosting capacity in distribution grids.

Figures (5)

• Figure 1: Illustration of difference-of-convex function (nonconvex) constraint (black), and first-order Taylor series linearization (blue) and proposed convex approximation (red) around $x=1$. Note that the proposed convexification is tight for $x \ge 0$ whereas the first-order Taylor series linearization is tight only at the linearization point $x=1$.
• Figure 2: Correlation of voltage magnitudes, phase angles, and active and reactive power generation for a randomly selected instance of the 30000-bus system
• Figure 3: Empirical cumulative distributions of absolute errors
• Figure 4: Time performance profile. QCAC OPF model (Solid). AC OPF model (Dashed).
• Figure 5: PV hosting capacity: Maximum slack-variable penalty trade-off curve.