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

TL;DR

The paper addresses the lack of reliable guarantees in ML-based voltage risk assessment by introducing a probabilistic framework built on Gaussian Process regression. It combines a topology-aware Vertex-Degree Kernel (VDK) with a network-swipe Active Learning strategy to learn the voltage-load mapping efficiently in large power systems. The authors derive explicit error bounds and conservatism guarantees that connect GP predictive variance to risk estimates, enabling Monte Carlo–level reliability with far fewer AC power flow evaluations. Across IEEE 118-, 500-, and 1354-bus systems, the approach achieves MAE below 10^{-3} p.u., reproduces MC-level voltage risk with roughly 15x fewer ACPF computations and 120x faster evaluation times, while maintaining conservative risk bounds. This work demonstrates that probabilistic guarantees and scalable, topology-aware learning can deliver trustworthy, fast voltage risk analytics suitable for real-time operation and planning.

Abstract

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.

Figures (7)

• Figure 1: Relationship between required number of samples $N$ and estimation error $\varepsilon$ for MCS-based voltage risk assessment (VRA). The proposed approach replaces these $N$ ACPF solutions with $4N$ GP model evaluations to achieve an error of comparable order. The key advantage arises from the GP’s closed-form evaluation, where the time per GP evaluation $T_{\text{GP}}$ is significantly smaller than the time per ACPF solution $T_{\text{solve}}$, leading to an overall speedup on the order of $T_{\text{solve}} / T_{\text{GP}} \gg 1$. Note that learning the GP surrogate requires only $N_{\text{train}} \ll N$, thus reducing the overall computational burden.
• Figure 2: A part of 118-Bus system showing the idea of vertex degree kernel construction via working over loads of a given nodes and its immediate neighbors. In Eq. \ref{['eq:VDK']}, we use $\mathbf{x}_b$ to represents all the load variables in $b$-th sub-kernel as $k_1(\mathbf{x}_1,\cdot)= k_1(\mathbf{s}_1,\mathbf{s}_2,\mathbf{s}_3,\cdot)$. For brevity, second input element in kernel function is represented using $(\cdot)$.
• Figure 3: Description of steps within $(t+1)^\text{th}$ iteration of network-swipe Algorithm \ref{['alg:gp_al']} for AL. At each step, only loads at a fixed graph distance from $j$ are treated as variables (colored in black) for information maximization, while others are kept fixed (colored in blue). For example, $\overline{\mathbf{x}}_{\mathcal{D}_1} = \{\mathbf{s}_2, \mathbf{s}_3\}$. Once those loads are optimized, their values are updated before the next step. In second step, updated output of the first step $\widehat{\mathbf{x}}^{t+1}_{\mathcal{D}_0}$ are used along with values from previous iteration at all other loads $\widehat{\mathbf{x}}^t_{\mathcal{D}_{2\dots d}}$, as given in Eq. \ref{['eq:network_swipe']}.
• Figure 4: Flowchart of risk measure calculation using AL-VDK based voltage learning. Here, $\mathcal{H}_ = \{h_m(\Delta^i_V)\}_{i=1}^{N}$ and $\widehat{F}_s$ is the cumulative density function (CDF) of violation $h_m(\Delta_V)$ random variable. This is then used directly to estimate probability of voltage violation $\widehat{\mathbb{P}}\{h_m(\mathbf{s})> 0\}$.
• Figure 5: Comparison of MAE performance of different methods, demonstrating the efficiency and low sample-complexity of AL-VDK, on three different nodes of 118-Bus system. GP, VDK and AL-VDK results are of 50 trials, and DNN results are of 10 trials. AL-VDK uses significantly fewer samples as dictated by Algorithm \ref{['alg:gp_al']}. AL-VDK training samples for all 50 trials are within 43 -- 48, 43 -- 47 and 42 -- 47 for nodes 21, 44, and 95 respectively.

Theorems & Definitions (6)

• Theorem 1
• Corollary 1.1
• Theorem 2
• proof
• proof
• proof