Global Performance Guarantees for Neural Network Models of AC Power Flow

TL;DR

This work tackles the problem of rigorously verifying neural network surrogates for AC power flow by deriving global performance guarantees. It introduces Sequential Targeted Tightening (STT), a procedure that alternates SDP relaxations with selective Sherali–Adams cuts to tightly bound the maximum deviation between NN predictions and ground-truth power-flow equations. Applied to learned piecewise-linear power-flow mappings across 14–200 bus systems, STT typically delivers substantially tighter worst-case bounds than a state-of-the-art MIQP solver (Gurobi 11.0) within comparable time budgets, albeit with scalability and numerical conditioning challenges. The framework provides a practical certifiability pathway for deploying ML surrogates in power system operations and suggests extensions to broader learned-physics verification problems.

Abstract

Machine learning, which can generate extremely fast and highly accurate black-box surrogate models, is increasingly being applied to a variety of AC power flow problems. Rigorously verifying the accuracy of the resulting black-box models, however, is computationally challenging. This paper develops a tractable neural network verification procedure which incorporates the ground truth of the non-linear AC power flow equations to determine worst-case neural network prediction error. Our approach, termed Sequential Targeted Tightening (STT), leverages a loosely convexified reformulation of the original verification problem, which is an intractable mixed integer quadratic program (MIQP). Using the sequential addition of targeted cuts, we iteratively tighten our formulation until either the solution is sufficiently tight or a satisfactory performance guarantee has been generated. After learning neural network models of the 14, 57, 118, and 200-bus PGLib test cases, we compare the performance guarantees generated by our STT procedure with ones generated by a state-of-the-art MIQP solver, Gurobi 11.0. We show that STT often generates performance guarantees which are far tighter than the MIQP upper bound.

Figures (7)

• Figure 1: The value $e^{\star}$ represents the worst-case overestimation of the surrogate model ${\hat{f}}(x)$, with respect to the ground truth function $f(x)$, within the feasible space estimated by the surrogate: $y_{\rm min}\le{\hat{f}}(x)\le y_{\rm max}$.
• Figure 2: Four iterations of the STT algorithm are plotted. For each iteration, the largest 200 (out of 158,404 total) constraint violations are depicted (i.e., where \ref{['eq: Omega_ge0']} is violated). At each iteration, the 8 largest constraint violations (gray box) are identified and enforced at the next iteration. After four iterations, the degree of constraint violation becomes minimal.
• Figure 3: 500 voltage magnitude solutions are plotted for the 57 bus PGLib system, where each solution is associated with a single iteration of Alg. \ref{['algo:data_collect']} and a corresponding maximization of \ref{['eq: data']}. The upper and lower voltage profile outlines form an approximate epigraph of the feasible voltage space.
• Figure 4: The prediction of a trained NN (trained on data from the PGLib 14 bus system) is plotted against the ground truth power flow data for a single test point (i.e., a point not used to train the NN).
• Figure 5: Performance guarantees (worst case underestimation of active power injection at load buses) generated by ($i$) Gurobi's MIQP solver (red) through BaB iterations, and ($ii$) 10 iterations of STT algorithm (blue).

Theorems & Definitions (7)

• Remark 1
• Remark 2
• Lemma 1
• proof
• Definition 1: Sequential Targeted Tightening (STT)
• Theorem 1
• proof