# Fast Calculation of Probabilistic Power Flow: A Model-based Deep Learning Approach

arXiv: https://arxiv.org/abs/1906.06017 · HTML: https://mlsanity.com/paper/1906.06017/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jun 2019.

## Built on by (most cited first, 15 of 15)

- [PowerFlowNet: Power Flow Approximation Using Message Passing Graph Neural Networks](https://mlsanity.com/paper/2311.03415/) (2023)
- [Controlling Smart Inverters using Proxies: A Chance-Constrained DNN-based Approach](https://mlsanity.com/paper/2105.00429/) (2021)
- [Power Failure Cascade Prediction using Graph Neural Networks](https://mlsanity.com/paper/2404.16134/) (2024)
- [PowerGNN: A Topology-Aware Graph Neural Network for Electricity Grids](https://mlsanity.com/paper/2503.22721/) (2025)
- [Towards Practical Physics-Informed ML Design and Evaluation for Power Grid](https://mlsanity.com/paper/2205.03673/) (2022)
- [Grid-SiPhyR: An end-to-end learning to optimize framework for combinatorial problems in power systems](https://mlsanity.com/paper/2206.06789/) (2022)
- [A Robust Data-driven Process Modeling Applied to Time-series Stochastic Power Flow](https://mlsanity.com/paper/2301.02651/) (2023)
- [Adaptive Informed Deep Neural Networks for Power Flow Analysis](https://mlsanity.com/paper/2412.02659/) (2024)
- [Towards Understanding the Unreasonable Effectiveness of Learning AC-OPF Solutions](https://mlsanity.com/paper/2111.11168/) (2021)
- [Improving Sample Efficiency of Deep Learning Models in Electricity Market](https://mlsanity.com/paper/2210.05599/) (2022)
- [Power Flow Approximations for Multiphase Distribution Networks using Gaussian Processes](https://mlsanity.com/paper/2504.21260/) (2025)
- [Homotopy-Guided Self-Supervised Learning of Parametric Solutions for AC Optimal Power Flow](https://mlsanity.com/paper/2511.11677/) (2025)
- [Test Time Training for AC Power Flow Surrogates via Physics and Operational Constraint Refinement](https://mlsanity.com/paper/2511.22343/) (2025)
- [Scale-Adaptive Power Flow Analysis with Local Topology Slicing and Multi-Task Graph Learning](https://mlsanity.com/paper/2601.01387/) (2026)
- [BOOST-RPF: Boosted Sequential Trees for Radial Power Flow](https://mlsanity.com/paper/2603.21977/) (2026)

## Abstract (arXiv)

Probabilistic power flow (PPF) plays a critical role in power system analysis. However, the high computational burden makes it challenging for the practical implementation of PPF. This paper proposes a model-based deep learning approach to overcome the computational challenge. A deep neural network (DNN) is used to approximate the power flow calculation and is trained according to the physical power flow equations to improve its learning ability. The training process consists of several steps: 1) the branch flows are added into the objective function of the DNN as a penalty term, which improves the approximation accuracy of the DNN; 2) the gradients used in the back propagation process are simplified according to the physical characteristics of the transmission grid, which accelerates the training speed while maintaining effective guidance of the physical model; and 3) an improved initialization method for the DNN parameters is proposed to improve the convergence speed. The simulation results demonstrate the accuracy and efficiency of the proposed method in standard IEEE and utility benchmark systems.