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Sparse Neural Approximations for Bilevel Adversarial Problems in Power Grids

Young-ho Cho, Harsha Nagarajan, Deepjyoti Deka, Hao Zhu

TL;DR

<3-5 sentence high-level summary>

Abstract

The adversarial worst-case load shedding (AWLS) problem is pivotal for identifying critical contingencies under line outages. It is naturally cast as a bilevel program: the upper level simulates an attacker determining worst-case line failures, and the lower level corresponds to the defender's generator redispatch operations. Conventional techniques using optimality conditions render the bilevel, mixed-integer formulation computationally prohibitive due to the combinatorial number of topologies and the nonconvexity of AC power flow constraints. To address these challenges, we develop a novel single-level optimal value-function (OVF) reformulation and further leverage a data-driven neural network (NN) surrogate of the follower's optimal value. To ensure physical realizability, we embed the trained surrogate in a physics-constrained NN (PCNN) formulation that couples the OVF inequality with (relaxed) AC feasibility, yielding a mixed-integer convex model amenable to off-the-shelf solvers. To achieve scalability, we learn a sparse, area-partitioned NN via spectral clustering; the resulting block-sparse architecture scales essentially linearly with system size while preserving accuracy. Notably, our approach produces near-optimal worst-case failures and generalizes across loading conditions and unseen topologies, enabling rapid online recomputation. Numerical experiments on the IEEE 14- and 118-bus systems demonstrate the method's scalability and solution quality for large-scale contingency analysis, with an average optimality gap of 5.8% compared to conventional methods, while maintaining computation times under one minute.

Sparse Neural Approximations for Bilevel Adversarial Problems in Power Grids

TL;DR

<3-5 sentence high-level summary>

Abstract

The adversarial worst-case load shedding (AWLS) problem is pivotal for identifying critical contingencies under line outages. It is naturally cast as a bilevel program: the upper level simulates an attacker determining worst-case line failures, and the lower level corresponds to the defender's generator redispatch operations. Conventional techniques using optimality conditions render the bilevel, mixed-integer formulation computationally prohibitive due to the combinatorial number of topologies and the nonconvexity of AC power flow constraints. To address these challenges, we develop a novel single-level optimal value-function (OVF) reformulation and further leverage a data-driven neural network (NN) surrogate of the follower's optimal value. To ensure physical realizability, we embed the trained surrogate in a physics-constrained NN (PCNN) formulation that couples the OVF inequality with (relaxed) AC feasibility, yielding a mixed-integer convex model amenable to off-the-shelf solvers. To achieve scalability, we learn a sparse, area-partitioned NN via spectral clustering; the resulting block-sparse architecture scales essentially linearly with system size while preserving accuracy. Notably, our approach produces near-optimal worst-case failures and generalizes across loading conditions and unseen topologies, enabling rapid online recomputation. Numerical experiments on the IEEE 14- and 118-bus systems demonstrate the method's scalability and solution quality for large-scale contingency analysis, with an average optimality gap of 5.8% compared to conventional methods, while maintaining computation times under one minute.

Paper Structure

This paper contains 19 sections, 25 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Adversarial Worst-case Load Shed (AWLS) problem:
  4. Optimal Value-function Reformulation of AWLS Problem
  5. Neural Approximation of Value-Function
  6. Model and training
  7. Exact MILP encoding of ReLU
  8. Preliminary NN Surrogate for the AWLS
  9. Physics-constrained Sparse NN Surrogate
  10. Physics-constrained NN (PCNN) Surrogate
  11. Sparsifying NN Surrogates using Area Partitioning
  12. Numerical Studies
  13. On Benchmarking the AWLS Problem
  14. Value-function Approximation Quality
  15. NN-based AWLS Approximation
  16. ...and 4 more sections

Figures (8)

  • Figure 1: The proposed NN model predicts the approximate optimal load shed $\hat{\eta}({\mathbf x})$ for any given topology ${\mathbf x}$ and active/reactive loads.
  • Figure 2: Optimality gap as a function of the slack-regularization parameter $\lambda$ on the IEEE 14- and 118-bus systems.
  • Figure 3: Optimal lower-level (NLP) solutions for IEEE 118 system under randomly sampled line interdictions. Distributions of total load shed are shown for (i) full system without partitioning (red), (ii) simple sum of the three area sheds (blue), and (iii) each area individually. Per-area distributions are significantly narrower, indicating spatial localization, while the area-sum (blue) closely tracks the full-system shed (red), confirming that area partitioning preserves aggregate shed.
  • Figure 4: Our architecture partitions the system into electrically coherent areas via spectral clustering (three shown for illustration), then uses per-area NNs with tie-line inputs to predict local load shedding, summing the results for the total system prediction.
  • Figure 5: Comparisons of the prediction errors (%) in approximating the load sheds for NN_Single and NN_Multi models.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Remark 3: Selecting the penalty $\lambda$
  • Remark 4
  • Remark 5