# Modeling the AC Power Flow Equations with Optimally Compact Neural Networks: Application to Unit Commitment

arXiv: https://arxiv.org/abs/2110.11269 · HTML: https://mlsanity.com/paper/2110.11269/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 33 references that are in the map.
- First posted on arXiv 21 Oct 2021.

## Builds on (most cited first, 15 of 15)

- [The Power Grid Library for Benchmarking AC Optimal Power Flow Algorithms](https://mlsanity.com/paper/1908.02788/) (2019)
- [PowerModels.jl: An Open-Source Framework for Exploring Power Flow Formulations](https://mlsanity.com/paper/1711.01728/) (2017)
- [ReLU Networks as Surrogate Models in Mixed-Integer Linear Programs](https://mlsanity.com/paper/1907.03140/) (2019)
- [Physics-Guided Deep Neural Networks for Power Flow Analysis](https://mlsanity.com/paper/2002.00097/) (2020)
- [A Survey on Deep Neural Network Compression: Challenges, Overview, and Solutions](https://mlsanity.com/paper/2010.03954/) (2020)
- [Learning Optimal Power Flow: Worst-Case Guarantees for Neural Networks](https://mlsanity.com/paper/2006.11029/) (2020)
- [Verification of Neural Network Behaviour: Formal Guarantees for Power System Applications](https://mlsanity.com/paper/1910.01624/) (2019)
- [Solutions of DC OPF are Never AC Feasible](https://mlsanity.com/paper/1912.00319/) (2019)
- [Neural Networks for Encoding Dynamic Security-Constrained Optimal Power Flow](https://mlsanity.com/paper/2003.07939/) (2020)
- [Strong NP-hardness of AC power flows feasibility](https://mlsanity.com/paper/1512.07315/) (2015)
- [Contextual Reserve Price Optimization in Auctions via Mixed-Integer Programming](https://mlsanity.com/paper/2002.08841/) (2020)
- [Approximating Trajectory Constraints with Machine Learning -- Microgrid Islanding with Frequency Constraints](https://mlsanity.com/paper/2001.05775/) (2020)
- [Encoding Frequency Constraints in Preventive Unit Commitment Using Deep Learning with Region-of-Interest Active Sampling](https://mlsanity.com/paper/2102.09583/) (2021)
- [Global Solution Strategies for the Network-Constrained Unit Commitment Problem with AC Transmission Constraints](https://mlsanity.com/paper/1801.07218/) (2018)
- [Sequential Relaxation of Unit Commitment with AC Transmission Constraints](https://mlsanity.com/paper/1809.09812/) (2018)

## Built on by (most cited first, 17 of 17)

- [When Deep Learning Meets Polyhedral Theory: A Survey](https://mlsanity.com/paper/2305.00241/) (2023)
- [Topology-aware Graph Neural Networks for Learning Feasible and Adaptive ac-OPF Solutions](https://mlsanity.com/paper/2205.10129/) (2022)
- [Neur2SP: Neural Two-Stage Stochastic Programming](https://mlsanity.com/paper/2205.12006/) (2022)
- [Closing the Loop: A Framework for Trustworthy Machine Learning in Power Systems](https://mlsanity.com/paper/2203.07505/) (2022)
- [Learning Optimal Power Flow Value Functions with Input-Convex Neural Networks](https://mlsanity.com/paper/2310.04605/) (2023)
- [Global Performance Guarantees for Neural Network Models of AC Power Flow](https://mlsanity.com/paper/2211.07125/) (2022)
- [KCLNet: Physics-Informed Power Flow Prediction via Constraints Projections](https://mlsanity.com/paper/2506.12902/) (2025)
- [Power Flow Analysis Using Deep Neural Networks in Three-Phase Unbalanced Smart Distribution Grids](https://mlsanity.com/paper/2401.07465/) (2024)
- [Deep Learning for Two-Stage Robust Integer Optimization](https://mlsanity.com/paper/2310.04345/) (2023)
- [Scalable Exact Verification of Optimization Proxies for Large-Scale Optimal Power Flow](https://mlsanity.com/paper/2405.06109/) (2024)
- [DiffOPF: Diffusion Solver for Optimal Power Flow](https://mlsanity.com/paper/2510.14075/) (2025)
- [Multi-fidelity power flow solver](https://mlsanity.com/paper/2205.13362/) (2022)
- [Physics-Informed Heterogeneous Graph Neural Networks for DC Blocker Placement](https://mlsanity.com/paper/2405.10389/) (2024)
- [Adaptive Informed Deep Neural Networks for Power Flow Analysis](https://mlsanity.com/paper/2412.02659/) (2024)
- [Emission-Aware Optimization of Gas Networks: Input-Convex Neural Network Approach](https://mlsanity.com/paper/2209.08645/) (2022)
- [Enriching Neural Network Training Dataset to Improve Worst-Case Performance Guarantees](https://mlsanity.com/paper/2303.13228/) (2023)
- [DCEmbed: Scalable Optimization over Neural Surrogates](https://mlsanity.com/paper/2609.33879/) (2026)

## Abstract (arXiv)

Nonlinear power flow constraints render a variety of power system optimization problems computationally intractable. Emerging research shows, however, that the nonlinear AC power flow equations can be successfully modeled using Neural Networks (NNs). These NNs can be exactly transformed into Mixed Integer Linear Programs (MILPs) and embedded inside challenging optimization problems, thus replacing nonlinearities that are intractable for many applications with tractable piecewise linear approximations. Such approaches, though, suffer from an explosion of the number of binary variables needed to represent the NN. Accordingly, this paper develops a technique for training an "optimally compact" NN, i.e., one that can represent the power flow equations with a sufficiently high degree of accuracy while still maintaining a tractable number of binary variables. We show that the resulting NN model is more expressive than both the DC and linearized power flow approximations when embedded inside of a challenging optimization problem (i.e., the AC unit commitment problem).