Residual Power Flow for Neural Solvers

TL;DR

Residual Power Flow (RPF) reframes AC power flow feasibility as a continuous residual minimisation problem derived from Kirchhoff's laws, avoiding bus-type-induced asymmetries. A neural solver learns the RPF mapping from operating conditions to voltage states, enabling fast, differentiable predictions that feed into a Predict-then-Optimise (PO) pipeline for AC-feasible PF, quasi-steady state, and AC-OPF tasks. The approach shows that residual-based learning can improve robustness to infeasible conditions and deliver accurate solutions on the IEEE 9-bus test system, with demonstrated flexibility across multiple tasks without retraining. By combining a foundational, physics-grounded formulation with neural speed, the work offers a practical path to scalable, versatile power-flow solvers suitable for real-time operation and planning. The residual-centric perspective also opens connections to broader topics such as state estimation and dynamic simulations, where infeasibility quantification and differentiable optimization are advantageous.

Abstract

The energy transition challenges operational tasks based on simulations and optimisation. These computations need to be fast and flexible as the grid is ever-expanding, and renewables' uncertainty requires a flexible operational environment. Learned approximations, proxies or surrogates -- we refer to them as Neural Solvers -- excel in terms of evaluation speed, but are inflexible with respect to adjusting to changing tasks. Hence, neural solvers are usually applicable to highly specific tasks, which limits their usefulness in practice; a widely reusable, foundational neural solver is required. Therefore, this work proposes the Residual Power Flow (RPF) formulation. RPF formulates residual functions based on Kirchhoff's laws to quantify the infeasibility of an operating condition. The minimisation of the residuals determines the voltage solution; an additional slack variable is needed to achieve AC-feasibility. RPF forms a natural, foundational subtask of tasks subject to power flow constraints. We propose to learn RPF with neural solvers to exploit their speed. Furthermore, RPF improves learning performance compared to common power flow formulations. To solve operational tasks, we integrate the neural solver in a Predict-then-Optimise (PO) approach to combine speed and flexibility. The case study investigates the IEEE 9-bus system and three tasks (AC Optimal Power Flow (OPF), power-flow and quasi-steady state power flow) solved by PO. The results demonstrate the accuracy and flexibility of learning with RPF.

Figures (11)

• Figure 1: Proposed approach: The power system task is formulated as an optimisation requiring task specifications and the operating conditions. The sub-task considers the power flow equations with the neural solver approximating the RPF.
• Figure 2: Kirchhoff's current and voltage law.
• Figure 3: Single line diagram of the IEEE 9-bus system.
• Figure 4: Comparison of the angle variables for and the .
• Figure 5: Comparison of the prediction between the proposed formulation (orange) and the formulation (blue). The filled part of the boxplots represent the range of the 25th to 75th percentile, the whiskers indicate the 1.5-fold of the inter-quartile range and all points beyond are considered outliers represented as crosses.