# Learning Power Flow with Confidence: A Probabilistic Guarantee Framework for Voltage Risk

arXiv: https://arxiv.org/abs/2308.07867 · HTML: https://mlsanity.com/paper/2308.07867/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 44 references that are in the map.
- First posted on arXiv 15 Aug 2023.

## Builds on (most cited first, 9 of 9)

- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [Physics-Guided Deep Neural Networks for Power Flow Analysis](https://mlsanity.com/paper/2002.00097/) (2020)
- [Bayesian Active Learning with Fully Bayesian Gaussian Processes](https://mlsanity.com/paper/2205.10186/) (2022)
- [Additive Gaussian Processes Revisited](https://mlsanity.com/paper/2206.09861/) (2022)
- [Risk-Aware Control and Optimization for High-Renewable Power Grids](https://mlsanity.com/paper/2204.00950/) (2022)
- [Gaussian Process Learning-based Probabilistic Optimal Power Flow](https://mlsanity.com/paper/2004.07757/) (2020)
- [Physics-guided Residual Learning for Probabilistic Power Flow Analysis](https://mlsanity.com/paper/2301.12062/) (2023)
- [Inferring power system dynamics from synchrophasor data using Gaussian processes](https://mlsanity.com/paper/2105.12608/) (2021)
- [A Data-Driven Uncertainty Quantification Method for Stochastic Economic Dispatch](https://mlsanity.com/paper/2109.08195/) (2021)

## Built on by (most cited first, 2 of 2)

- [Data-Efficient Strategies for Probabilistic Voltage Envelopes under Network Contingencies](https://mlsanity.com/paper/2310.00763/) (2023)
- [Safety Screening for Voltage Control in Active Distribution Grids via Distributionally Robust Conformal Screening](https://mlsanity.com/paper/2608.30889/) (2026)

## Abstract (arXiv)

The absence of formal performance guarantees in machine learning (ML) has limited its adoption for safety-critical power system applications, where confidence and interpretability are as vital as accuracy. In this work, we present a probabilistic guarantee for power flow learning and voltage risk estimation, derived through the framework of Gaussian Process (GP) regression. Specifically, we establish a bound on the expected estimation error that connects the GP's predictive variance to confidence in voltage risk estimates, ensuring statistical equivalence with Monte Carlo-based ACPF risk quantification. To enhance model learnability in the low-data regime, we first design the Vertex-Degree Kernel (VDK), a topology-aware additive kernel that decomposes voltage-load interactions into local neighborhoods for efficient large-scale learning. Building on this, we introduce a network-swipe active learning (AL) algorithm that adaptively samples informative operating points and provides a principled stopping criterion without requiring out-of-sample validation. Together, these developments mitigate the principal bottleneck of ML-based power flow, its lack of guaranteed reliability, by combining data efficiency with analytical assurance. Empirical evaluations across IEEE 118-, 500-, and 1354-bus systems confirm that the proposed VDK-GP achieves mean absolute voltage errors below 1E-03 p.u., reproduces Monte Carlo-level voltage risk estimates with 15x fewer ACPF computations, and achieves over 120x reduction in evaluation time while conservatively bounding violation probabilities.