# Deep Statistical Solver for Distribution System State Estimation

arXiv: https://arxiv.org/abs/2301.01835 · HTML: https://mlsanity.com/paper/2301.01835/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 54 references that are in the map.
- First posted on arXiv 4 Jan 2023.

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

- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Stability Properties of Graph Neural Networks](https://mlsanity.com/paper/1905.04497/) (2019)
- [A Unified View on Graph Neural Networks as Graph Signal Denoising](https://mlsanity.com/paper/2010.01777/) (2020)
- [pandapower - an Open Source Python Tool for Convenient Modeling, Analysis and Optimization of Electric Power Systems](https://mlsanity.com/paper/1709.06743/) (2017)
- [Graphs, Convolutions, and Neural Networks: From Graph Filters to Graph Neural Networks](https://mlsanity.com/paper/2003.03777/) (2020)
- [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)
- [Stability of Graph Convolutional Neural Networks to Stochastic Perturbations](https://mlsanity.com/paper/2106.10526/) (2021)
- [Power to the Relational Inductive Bias: Graph Neural Networks in Electrical Power Grids](https://mlsanity.com/paper/2109.03604/) (2021)
- [Bayesian State Estimation for Unobservable Distribution Systems via Deep Learning](https://mlsanity.com/paper/1811.02756/) (2018)
- [State Estimation in Electric Power Systems Leveraging Graph Neural Networks](https://mlsanity.com/paper/2201.04056/) (2022)
- [Physics-Aware Neural Networks for Distribution System State Estimation](https://mlsanity.com/paper/1903.09669/) (2019)
- [A Survey on State Estimation Techniques and Challenges in Smart Distribution Systems](https://mlsanity.com/paper/1809.00057/) (2018)
- [Urban MV and LV Distribution Grid Topology Estimation via Group Lasso](https://mlsanity.com/paper/1611.01845/) (2016)
- [Distributed Nonlinear State Estimation in Electric Power Systems using Graph Neural Networks](https://mlsanity.com/paper/2207.11465/) (2022)

## Built on by (most cited first, 10 of 10)

- [PowerFlowNet: Power Flow Approximation Using Message Passing Graph Neural Networks](https://mlsanity.com/paper/2311.03415/) (2023)
- [Spatiotemporal Covariance Neural Networks](https://mlsanity.com/paper/2409.10068/) (2024)
- [Heterogeneous Graph Neural Networks for Short-term State Forecasting in Power Systems across Domains and Time Scales: A Hydroelectric Power Plant Case Study](https://mlsanity.com/paper/2507.06694/) (2025)
- [SenseFlow: A Physics-Informed and Self-Ensembling Iterative Framework for Power Flow Estimation](https://mlsanity.com/paper/2505.12302/) (2025)
- [Physics-Informed Inductive Biases for Voltage Prediction in Distribution Grids](https://mlsanity.com/paper/2509.25158/) (2025)
- [Time and Frequency Domain-based Anomaly Detection in Smart Meter Data for Distribution Network Studies](https://mlsanity.com/paper/2504.18231/) (2025)
- [Learning a Generalized Model for Substation Level Voltage Estimation in Distribution Networks](https://mlsanity.com/paper/2510.16063/) (2025)
- [Graph Machine Learning: An Opportunity for Power Systems](https://mlsanity.com/paper/2608.16494/) (2026)
- [Mycelium: A Generalizable Cross-Grid Multi-Task Model for Electrical Distribution Systems](https://mlsanity.com/paper/2609.33120/) (2026)
- [Sensor-Aware Joint Data Fusion and State Estimation under Sparse Multi-rate Distribution Grid Sensing](https://mlsanity.com/paper/2610.15156/) (2026)

## Abstract (arXiv)

Implementing accurate Distribution System State Estimation (DSSE) faces several challenges, among which the lack of observability and the high density of the distribution system. While data-driven alternatives based on Machine Learning models could be a choice, they suffer in DSSE because of the lack of labeled data. In fact, measurements in the distribution system are often noisy, corrupted, and unavailable. To address these issues, we propose the Deep Statistical Solver for Distribution System State Estimation (DSS2), a deep learning model based on graph neural networks (GNNs) that accounts for the network structure of the distribution system and for the physical governing power flow equations. DSS2 leverages hypergraphs to represent the heterogeneous components of the distribution systems and updates their latent representations via a node-centric message-passing scheme. A weakly supervised learning approach is put forth to train the DSS2 in a learning-to-optimize fashion w.r.t. the Weighted Least Squares loss with noisy measurements and pseudomeasurements. By enforcing the GNN output into the power flow equations and the latter into the loss function, we force the DSS2 to respect the physics of the distribution system. This strategy enables learning from noisy measurements, acting as an implicit denoiser, and alleviating the need for ideal labeled data. Extensive experiments with case studies on the IEEE 14-bus, 70-bus, and 179-bus networks showed the DSS2 outperforms by a margin the conventional Weighted Least Squares algorithm in accuracy, convergence, and computational time, while being more robust to noisy, erroneous, and missing measurements. The DSS2 achieves a competing, yet lower, performance compared with the supervised models that rely on the unrealistic assumption of having all the true labels.