# Embedding Power Flow into Machine Learning for Parameter and State Estimation

arXiv: https://arxiv.org/abs/2103.14251 · HTML: https://mlsanity.com/paper/2103.14251/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 44 references that are in the map.
- First posted on arXiv 26 Mar 2021.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Optimal Power Flow Using Graph Neural Networks](https://mlsanity.com/paper/1910.09658/) (2019)
- [A Review of Graph Neural Networks and Their Applications in Power Systems](https://mlsanity.com/paper/2101.10025/) (2021)
- [Real-time Power System State Estimation and Forecasting via Deep Neural Networks](https://mlsanity.com/paper/1811.06146/) (2018)
- [Fault Location in Power Distribution Systems via Deep Graph Convolutional Networks](https://mlsanity.com/paper/1812.09464/) (2018)
- [Physics-Informed Graphical Neural Network for Parameter & State Estimations in Power Systems](https://mlsanity.com/paper/2102.06349/) (2021)
- [Real-time Faulted Line Localization and PMU Placement in Power Systems through Convolutional Neural Networks](https://mlsanity.com/paper/1810.05247/) (2018)
- [Data-Driven Learning-Based Optimization for Distribution System State Estimation](https://mlsanity.com/paper/1807.01671/) (2018)
- [Monitoring and Optimization for Power Grids: A Signal Processing Perspective](https://mlsanity.com/paper/1302.0885/) (2013)
- [Online Learning of Power Transmission Dynamics](https://mlsanity.com/paper/1710.10021/) (2017)
- [Physics-Informed Deep Neural Network Method for Limited Observability State Estimation](https://mlsanity.com/paper/1910.06401/) (2019)
- [PMU-Based Estimation of Dynamic State Jacobian Matrix and Dynamic System State Matrix in Ambient Conditions](https://mlsanity.com/paper/1706.01114/) (2017)

## Built on by (most cited first, 6 of 6)

- [Robust Power System State Estimation using Physics-Informed Neural Networks](https://mlsanity.com/paper/2507.05874/) (2025)
- [Graph Neural Networks for Electric and Hydraulic Data Fusion to Enhance Short-term Forecasting of Pumped-storage Hydroelectricity](https://mlsanity.com/paper/2404.03368/) (2024)
- [Constrained Diffusion Models for Synthesizing Representative Power Flow Datasets](https://mlsanity.com/paper/2506.11281/) (2025)
- [Towards Model Reduction for Power System Transients with Physics-Informed PDE](https://mlsanity.com/paper/2110.14066/) (2021)
- [Machine Learning for Electricity Market Clearing](https://mlsanity.com/paper/2205.11641/) (2022)
- [Mixing Artificial and Natural Intelligence: From Statistical Mechanics to AI and Back to Turbulence](https://mlsanity.com/paper/2403.17993/) (2024)

## Abstract (arXiv)

Modern state and parameter estimations in power systems consist of two stages: the outer problem of minimizing the mismatch between network observation and prediction over the network parameters, and the inner problem of predicting the system state for given values of the parameters. The standard solution of the combined problem is iterative: (a) set the parameters, e.g. to priors on the power line characteristics, (b) map input observation to prediction of the output, (c) compute the mismatch between predicted and observed output, (d) make a gradient descent step in the space of parameters to minimize the mismatch, and loop back to (a). We show how modern Machine Learning (ML), and specifically training guided by automatic differentiation, allows to resolve the iterative loop more efficiently. Moreover, we extend the scheme to the case of incomplete observations, where Phasor Measurement Units (reporting real and reactive powers, voltage and phase) are available only at the generators (PV buses), while loads (PQ buses) report (via SCADA controls) only active and reactive powers. Considering it from the implementation perspective, our methodology of resolving the parameter and state estimation problem can be viewed as embedding of the Power Flow (PF) solver into the training loop of the Machine Learning framework (PyTorch, in this study). We argue that this embedding can help to resolve high-level optimization problems in power system operations and planning.