# Unsupervised Deep Learning for AC Optimal Power Flow via Lagrangian Duality

arXiv: https://arxiv.org/abs/2212.03977 · HTML: https://mlsanity.com/paper/2212.03977/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 19 references that are in the map.
- First posted on arXiv 7 Dec 2022.

## Builds on (most cited first, 11 of 11)

- [DC3: A learning method for optimization with hard constraints](https://mlsanity.com/paper/2104.12225/) (2021)
- [Predicting AC Optimal Power Flows: Combining Deep Learning and Lagrangian Dual Methods](https://mlsanity.com/paper/1909.10461/) (2019)
- [Optimal Power Flow Using Graph Neural Networks](https://mlsanity.com/paper/1910.09658/) (2019)
- [DeepOPF: A Feasibility-Optimized Deep Neural Network Approach for AC Optimal Power Flow Problems](https://mlsanity.com/paper/2007.01002/) (2020)
- [Smart-PGSim: Using Neural Network to Accelerate AC-OPF Power Grid Simulation](https://mlsanity.com/paper/2008.11827/) (2020)
- [DeepOPF-V: Solving AC-OPF Problems Efficiently](https://mlsanity.com/paper/2103.11793/) (2021)
- [Solutions of DC OPF are Never AC Feasible](https://mlsanity.com/paper/1912.00319/) (2019)
- [High-Fidelity Machine Learning Approximations of Large-Scale Optimal Power Flow](https://mlsanity.com/paper/2006.16356/) (2020)
- [Convex Relaxation of Optimal Power Flow, Part II: Exactness](https://mlsanity.com/paper/1405.0814/) (2014)
- [Learning an Optimally Reduced Formulation of OPF through Meta-optimization](https://mlsanity.com/paper/1911.06784/) (2019)
- [A novel augmented Lagrangian method of multipliers for optimization with general inequality constraints](https://mlsanity.com/paper/2106.15044/) (2021)

## Built on by (most cited first, 3 of 3)

- [Physics-Informed Graph Neural Networks for Robust AC-Optimal Power Flow](https://mlsanity.com/paper/2410.04818/) (2024)
- [Scalable Self-Supervised Learning for Multiphase AC-OPF in Distribution Systems with Topology Reconfiguration](https://mlsanity.com/paper/2608.25095/) (2026)
- [Learning Samples Importance: Parameterizing Dual Variables in Everywhere Learning](https://mlsanity.com/paper/2609.36310/) (2026)

## Abstract (arXiv)

Non-convex AC optimal power flow (AC-OPF) is a fundamental optimization problem in power system analysis. The computational complexity of conventional solvers is typically high and not suitable for large-scale networks in real-time operation. Hence, deep learning based approaches have gained intensive attention to conduct the time-consuming training process offline. Supervised learning methods may yield a feasible AC-OPF solution with a small optimality gap. However, they often need conventional solvers to generate the training dataset. This paper proposes an end-to-end unsupervised learning based framework for AC-OPF. We develop a deep neural network to output a partial set of decision variables while the remaining variables are recovered by solving AC power flow equations. The fast decoupled power flow solver is adopted to further reduce the computational time. In addition, we propose using a modified augmented Lagrangian function as the training loss. The multipliers are adjusted dynamically based on the degree of constraint violation. Extensive numerical test results corroborate the advantages of our proposed approach over some existing methods.