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Scalable Exact Verification of Optimization Proxies for Large-Scale Optimal Power Flow

Rahul Nellikkath, Mathieu Tanneau, Pascal Van Hentenryck, Spyros Chatzivasileiadis

TL;DR

The paper tackles the challenge of formally verifying neural-network-based proxies for solving large-scale DC-OPF by transforming NN policies into MILP formulations and computing worst-case violations. It introduces two scalable acceleration strategies: gradient-based adversarial search (PGA) to rapidly identify high-violation inputs for warm-start primal bounds, and advanced bound-tightening via $oldsymbol{ extalpha},oldsymbol{ extbeta}$-CROWN to tighten big‑M bounds for ReLUs, improving dual bounds. A complementary approach uses primal-warm starts and selective cutting planes to further tighten the MILP relaxations. Empirical results on IEEE 300, goc793, pegase1k, and pegase2k DC-OPF proxies show near-linear scaling and convergence where prior methods fail, demonstrating that formal verification can be feasibly applied to industry-scale power grids and increase trust in ML-based OPF solutions.

Abstract

Optimal Power Flow (OPF) is a valuable tool for power system operators, but it is a difficult problem to solve for large systems. Machine Learning (ML) algorithms, especially Neural Networks-based (NN) optimization proxies, have emerged as a promising new tool for solving OPF, by estimating the OPF solution much faster than traditional methods. However, these ML algorithms act as black boxes, and it is hard to assess their worst-case performance across the entire range of possible inputs than an OPF can have. Previous work has proposed a mixed-integer programming-based methodology to quantify the worst-case violations caused by a NN trained to estimate the OPF solution, throughout the entire input domain. This approach, however, does not scale well to large power systems and more complex NN models. This paper addresses these issues by proposing a scalable algorithm to compute worst-case violations of NN proxies used for approximating large power systems within a reasonable time limit. This will help build trust in ML models to be deployed in large industry-scale power grids.

Scalable Exact Verification of Optimization Proxies for Large-Scale Optimal Power Flow

TL;DR

The paper tackles the challenge of formally verifying neural-network-based proxies for solving large-scale DC-OPF by transforming NN policies into MILP formulations and computing worst-case violations. It introduces two scalable acceleration strategies: gradient-based adversarial search (PGA) to rapidly identify high-violation inputs for warm-start primal bounds, and advanced bound-tightening via -CROWN to tighten big‑M bounds for ReLUs, improving dual bounds. A complementary approach uses primal-warm starts and selective cutting planes to further tighten the MILP relaxations. Empirical results on IEEE 300, goc793, pegase1k, and pegase2k DC-OPF proxies show near-linear scaling and convergence where prior methods fail, demonstrating that formal verification can be feasibly applied to industry-scale power grids and increase trust in ML-based OPF solutions.

Abstract

Optimal Power Flow (OPF) is a valuable tool for power system operators, but it is a difficult problem to solve for large systems. Machine Learning (ML) algorithms, especially Neural Networks-based (NN) optimization proxies, have emerged as a promising new tool for solving OPF, by estimating the OPF solution much faster than traditional methods. However, these ML algorithms act as black boxes, and it is hard to assess their worst-case performance across the entire range of possible inputs than an OPF can have. Previous work has proposed a mixed-integer programming-based methodology to quantify the worst-case violations caused by a NN trained to estimate the OPF solution, throughout the entire input domain. This approach, however, does not scale well to large power systems and more complex NN models. This paper addresses these issues by proposing a scalable algorithm to compute worst-case violations of NN proxies used for approximating large power systems within a reasonable time limit. This will help build trust in ML models to be deployed in large industry-scale power grids.
Paper Structure (14 sections, 14 equations, 3 figures, 7 tables)

This paper contains 14 sections, 14 equations, 3 figures, 7 tables.

Table of Contents

  1. Introduction
  2. DNN Verification for OPF Proxies
  3. DC Optimal Power Flow
  4. Neural Networks for Optimal Power Flow Approximation
  5. Worst-Case Guarantees for Neural Networks
  6. Worst-Case Guarantees for Constraint Violations
  7. Acceleration Techniques
  8. $\alpha,\beta{-}\text{CROWN}$ for Tighter Big-M Bounds
  9. Gradient Descent for Strong Primal Solutions
  10. Numerical results
  11. Average Neural Network Performance
  12. The Impact of the Primal Acceleration
  13. The Impact of the Dual Acceleration
  14. Conclusion

Figures (3)

  • Figure 1: Illustration of the neural network architecture to predict the optimal generation active power outputs $\mathbf{\hat{p}}^{\text{g}}$ using the active power demand $\mathbf{p}^{\text{d}}$ as input: There are K hidden layers in the neural network with $N_k$ neurons each. Where k = 1, ...,K.
  • Figure 2: Proposed Method for accelerating the NN exact verification for computing worst-case guarantees
  • Figure 3: Illustrates the feasible region for Relaxed ReLU. Here $y = ReLU(X)$. A tighter bound for the ReLU results in a tighter relaxation.