PaperlayerView exact revision on arXiv

Source-linked AI summary

Hard-Constrained Probabilistic Factor Graph Neural Network for Distribution System State Estimation under Non-Gaussian Uncertainty

M. Furqan Azam, Marta Vanin, Chris Hermans, Geert Deconinck

arXiv:2609.35246v1eess.SYcs.SY

TL;DR

Distribution system state estimation needs to handle sparse measurements and uncertainty while respecting physical constraints, but hard-constrained probabilistic approaches remain underexplored, particularly for unbalanced networks. HCP-PINN combines a probabilistic factor graph with a differentiable constrained estimation layer. It reports lower errors and faster inference than numerical and learning-based baselines, with improved robustness to reconfiguration and out-of-distribution conditions versus standard GNNs.

  • Problem

    Hard-constrained probabilistic DSSE remains largely unexplored, especially for unbalanced three-phase networks, despite limited observability and the need to model uncertainty.

    Source: §1 · pp. 4, 7, 9, 10

  • Method

    HCP-PINN uses a probabilistic factor-graph neural prior and generalized likelihood objective, then a differentiable constrained MAP layer to enforce equality constraints and measurement consistency.

    Source: §1 · pp. 9–10, §3 · p. 14

  • Results

    Errors fell approximately 58% versus WLS/LAV and 85% versus soft-PINN/GNN; inference was approximately 35× faster without batching and up to 125× faster at batch size 16.

    Source: §6 · p. 50

  • Takeaways & Limitations

    HCP-PINN improves robustness to network reconfiguration and out-of-distribution operating conditions compared with standard GNNs.

    Source: §6 · p. 50

    Universal acceptance thresholds remain challenging under network parameter errors or prosumer additions; voltage-error levels are empirical targets, not strict criteria.

    Source: §5.7 · p. 46

Abstract

from arXiv · show

Robust and accurate state estimation is fundamental for the reliable operation and monitoring of active distribution networks. Conventional numerical estimators, such as weighted least squares, are computationally slower and often suffer from convergence issues in the presence of sparse measurements affected by non-Gaussian noise. Physics-informed neural networks have recently emerged as a promising alternative by incorporating physical principles through residual-based penalty terms in the objective, which can improve robustness to noise and computational efficiency. However, such penalty-based approaches do not guarantee strict enforcement of physical constraints during inference and provide limited support for principled uncertainty modeling. To address these limitations, we propose a novel Hard-Constrained, Physics-Informed Factor Graph Neural Network (HCP-PINN) that formulates distribution system state estimation as a probabilistic constrained estimation problem on a factor graph. The proposed approach explicitly models non-Gaussian (pseudo-)measurement uncertainty through flexible, closed-form likelihood-based loss functions. It further incorporates a differentiable optimization-based estimation layer that strictly enforces nonlinear equality constraints and measurement consistency, producing physically feasible state estimates during both training and inference. To the best of our knowledge, this is the first PINN-based DSSE framework to combine hard physical constraints with non-Gaussian uncertainty modeling for unbalanced three-phase networks. Numerical experiments demonstrate that the proposed method delivers more accurate and physically consistent state estimates than deep learning benchmarks and numerical estimators, while exhibiting greater robustness to network model errors and improved computational efficiency under realistic measurement scenarios.

Hard-Constrained Probabilistic Factor Graph Neural Network for Distribution System State Estimation under Non-Gaussian Uncertainty

HCP-PINN frames distribution system state estimation as a hard-constrained probabilistic factor-graph problem, combining uncertainty modeling with an estimation layer that enforces physical consistency. It is presented as an accurate and computationally efficient alternative to numerical estimators, penalty-based PINNs, and data-driven approaches.

1. Introduction

The introduction motivates DSSE methods that can handle sparse coverage, non-Gaussian uncertainty, and physical constraints without relying on soft penalties. It presents HCP-PINN as a probabilistic factor-graph approach with a constrained estimation layer and geometric surrogate gradients, evaluated against numerical and learning-based alternatives.

  • DSSE is complicated by limited measurement coverage and observability, non-Gaussian errors, and the computational cost of numerical estimators.Source: §1 · pp. 4–5
  • Data-driven GNNs can violate physical constraints and miss higher-order power-flow dependencies, while PINN penalty terms do not guarantee exact constraint satisfaction at inference.Penalty-weight tuning can also affect generalization under challenging conditions.Source: §1 · pp. 6–7
  • Hard-constrained methods for state estimation remain largely unexplored, especially for unbalanced three-phase networks; the embedded layer should enforce physics while minimizing measurement residuals.Source: §1 · pp. 7–8
  • HCP-PINN uses factor nodes to preserve higher-order state-measurement dependencies and a generalized likelihood objective for non-Gaussian uncertainty.Source: §1 · pp. 9–10
  • A differentiable constrained MAP layer enforces power-balance constraints, while geometric surrogate gradients avoid solver unrolling during backpropagation.Source: §1 · pp. 8–10
  • Case studies on balanced and unbalanced networks report better estimation accuracy and constraint satisfaction than WLS, LAV, soft-constrained PINNs, and data-driven GNNs.The evaluations include partial observability, non-Gaussian noise, network parameter errors, reconfigurations, and out-of-distribution loading scenarios.Source: §1 · p. 10

2. Problem Formulation

The section formulates distribution system state estimation as inference over bus voltage states from nonlinear, noisy measurements. It develops likelihood-based MLE and MAP objectives, then imposes exact physical constraints and represents the probabilistic structure as a factor graph.

2. Problem Formulation

  • The state collects bus voltage magnitudes and phase angles, while nonlinear AC measurement functions map these states to network quantities and real and pseudo-measurements.The equations use a single-phase equivalent representation for readability.Source: §2 · p. 11
  • Assuming measurements are conditionally independent given the state, the joint likelihood yields an MLE objective that minimizes the sum of measurement negative log-likelihood terms.Each term is determined by the probability density of its measurement error.Source: §2 · p. 12
  • When state prior information is available, its negative log-prior augments the MLE objective to form the MAP estimate.The resulting constrained MAP formulation is the basis of HCP-PINN.Source: §2 · pp. 12–13
  • The constrained MAP formulation enforces exact zero-injection conditions while measurement likelihoods model uncertainty and equality constraints ensure physical consistency.Under Gaussian noise with known variances, the likelihood objective reduces to WLS; under Laplace noise, it yields LAV.Source: §2 · pp. 12–13
  • Conventional WLS-based DSSE uses iterative Gauss–Newton-type updates, and sparse noisy measurements can make estimation ill-conditioned and computationally demanding.Source: §2 · p. 13
  • A factor graph represents bus states as variable nodes and measurement, prior, or physical-constraint factors, with sparse local connections that motivate message passing.Each measurement factor connects only to the state variables used by its measurement function.Source: §2 · p. 13

3. Proposed DSSE Methodology

HCP-PINN combines a probabilistic factor graph neural network with the differentiable SMAP layer, which refines learned state priors using measurement likelihoods and equality constraints. Its end-to-end training uses likelihood-based losses and a geometric surrogate-gradient method.

3. Proposed DSSE Methodology

  • HCP-PINN combines a probabilistic FGNN prior with constrained MAP refinement, trained end-to-end using likelihood-based losses and geometric surrogate gradients.The SMAP layer combines the learned prior, measurement likelihood, and explicit equality constraints.Source: §3 · pp. 13–14

3.1. Probabilistic Factor Graph Neural Network

  • The probabilistic FGNN represents the grid as a bipartite graph of state-variable and measurement-likelihood nodes, with neural message passing approximating belief propagation.Shared message and update functions support generalization across networks of varying sizes and topologies.Source: §3.1 · pp. 15–18

3.2. Differentiable Estimation Layer SMAP

  • SMAP combines the FGNN prior and measurement likelihood, then uses fixed-iteration equality-constrained Gauss–Newton updates to produce a physically constrained estimate.For non-Gaussian measurement uncertainty, it replaces the quadratic residual term with the corresponding negative log-likelihood and solves by iteratively reweighted least squares.Source: §3.2 · pp. 18–21

3.3. Geometric Surrogate-Gradient Method

  • The geometric surrogate-gradient method avoids full solver unrolling by constructing the backward signal from the constraint geometry at the refined estimate.It discards the gradient component normal to the feasible set and backpropagates the tangential component.Source: §3.3 · pp. 21–23
  • The geometric-surrogate backward method reduces training computation and memory costs while achieving comparable estimation accuracy with fewer SMAP refinement iterations.The method is presented as an alternative to exact solver unrolling.Source: §3.3 · p. 23

4. Experimental Setup

The experiments evaluate HCP-PINN on balanced and unbalanced distribution networks under varied measurement coverage and noise, against numerical and learning-based benchmarks. The setup also specifies staged training and metrics for estimation accuracy and probabilistic calibration.

4. Experimental Setup

  • The evaluation covers balanced MV networks and realistic unbalanced LV feeders, comparing HCP-PINN with WLS, LAV, soft-constrained PINN, and supervised GNN methods.The study varies measurement noise and observability while assuming known topology and line parameters during training.Source: §4 · pp. 23–25

4.1. Dataset Generation

  • The dataset combines hourly load and generation scenarios with incomplete real measurements and noisy real and pseudo-measurements, yielding 4,320 scenarios split 60/20/20 for training, validation, and testing.Load profiles receive 15% standard-deviation perturbations, and reactive injections assume a 0.98 power factor.Source: §4.1 · pp. 26–27

4.2. Non-Gaussian Uncertainty Models

  • Gaussian, Laplacian, and two-component GMM noise model baseline, heavy-tailed, and multimodal measurement errors, respectively.Pseudo-measurements receive larger, predominantly non-Gaussian errors than real measurements.Source: §4.2 · pp. 27–28, Figure 9 · p. 28, Table 4 · p. 28

4.3. Training Configurations and Hyperparameters

  • Training proceeds from an unconstrained FGNN trained with an NLL objective to end-to-end constrained training with SMAP and geometric surrogate-gradient backpropagation.The staged procedure provides a stable initialization and mitigates unstable gradients early in training.Source: §4.3 · pp. 28–29

4.4. Benchmark Methods

  • The benchmarks include variance-weighted WLS, absolute-residual LAV, penalty-based PINN, and fully supervised GNN trained on ground-truth states.The GNN is an optimistic reference because complete state labels are generally unavailable in practical LV grids.Source: §4.4 · pp. 29–30

4.5. Evaluation Metrics

  • RMSE evaluates state-estimation accuracy, while CRPS evaluates probabilistic predictions, with lower CRPS indicating sharper, well-calibrated estimates.The RMSE reference states are AC power-flow ground truth and estimated states.Source: §4.5 · pp. 30–31

5. Numerical Results and Discussion

Across the balanced-network tests, HCP-PINN achieves lower estimation errors than the numerical and learning-based benchmarks, including on unbalanced three-phase feeders. It also runs substantially faster than WLS and LAV, especially with batched evaluation on the larger feeder.

5.1. Estimation Accuracy under Gaussian Noise

  • HCP-PINN consistently achieves lower voltage, phase-angle, and line-loading RMSE than WLS, LAV, PINN, and GNN across balanced MV networks under Gaussian noise.Its predictions also align more closely with ground truth under partial observability, and it recovers monitored and unmonitored 70-bus voltages despite high noise.Source: §5.1 · p. 31, Figure 10 · p. 32, Figure 11 · p. 32
  • HCP-PINN maintains lower per-phase estimation errors on the LV2-43 and LV9-258 feeders, though phase-angle accuracy varies on the larger feeder.On LV9-258, it outperforms WLS and LAV in voltage magnitude and line loading; GNN and PINN show weaknesses in derived quantities or voltage estimates.Source: §5.1 · p. 33, Table 7 · p. 34
  • On LV9-258, batched HCP-PINN inference reaches approximately 123× and 134× speedups over WLS and LAV, respectively.Batch size 16 reduces per-sample inference time to 2.1 ms on LV2-43 and 6.2 ms on LV9-258; PINN and GNN remain faster but less accurate.Source: §5.1 · pp. 34–35, Figure 12 · p. 35

5.2. Performance under Non-Gaussian Uncertainty

HCP-PINN maintains comparatively accurate estimates across heavy-tailed and multimodal measurement noise, increasing Gaussian noise, and the tested bad-data scenario. Its robustness in the latter case is limited to the evaluated scenario; multiple simultaneous gross errors remain untested.

5.2. Performance under Non-Gaussian Uncertainty

  • HCP-PINN achieves the lowest errors across Gaussian, Laplacian, and bimodal GMM noise, while WLS degrades under Laplacian noise and LAV under multimodal noise.LAV is more robust than WLS under Laplacian noise, but this does not ensure lower estimation errors in nonlinear DSSE with limited observability.Source: §5.2 · pp. 35–37
  • Under indirect phase-angle estimation from noisy voltage and power measurements, GNN has substantially higher errors across the tested noise cases.No angle noise is applied directly because phase angles are not measured.Source: §5.2 · p. 36
  • On the 70-bus network, HCP-PINN voltage errors remain concentrated near zero under GMM and Laplacian noise, unlike the more dispersed WLS and shifted LAV errors.The comparison examines error concentration, spread, and tail behavior, not only RMSE.Source: §5.2 · p. 36

5.3. Robustness to High Noise and Outliers

  • As Gaussian noise increases under partial observability, HCP-PINN retains the lowest median errors and narrowest interquartile ranges for voltage magnitude and line loading.Its advantage over WLS and LAV is most pronounced at high noise, where their error distributions spread further and show more extreme outliers.Source: §5.3 · p. 37, Figure 15 · p. 37
  • In the tested bad-data case, HCP-PINN remains accurate across most buses, whereas corrupted injections substantially distort the WLS voltage profile.The test scales Bus 10 injections to three times their true values; performance under multiple simultaneous gross errors remains unevaluated.Source: §5.3 · pp. 37–38, Figure 16 · p. 38

5.4. Impact of Measurement Coverage

HCP-PINN remains accurate as measurement availability decreases and satisfies equality constraints to high numerical precision. Against constrained IPOPT on the 258-bus feeder, it provides comparable solution quality with lower inference time.

5.4. Impact of Measurement Coverage

  • As measurement availability decreases, HCP-PINN maintains comparatively small voltage-magnitude and line-loading errors, with its advantage over WLS and LAV clearer under minimal observability.Lower FAM means fewer real measurements and greater reliance on pseudo-measurements; HCP-PINN is trained at FAM = 0.5 and evaluated across three regimes.Source: §5.4 · pp. 39–40

5.5. Constraint Satisfaction and Physical Consistency

  • At zero-injection buses, HCP-PINN achieves zero power-balance residuals on both LV feeders, with reported mean and maximum residuals remaining very small.The mean residuals are 2.51 × 10^-11 and 2.72 × 10^-13, and maximum residuals remain below 1.83 × 10^-9 and 3.18 × 10^-11, respectively.Source: §5.5 · pp. 40–41, Table 8 · p. 40
  • On the three-phase 258-bus feeder, HCP-PINN closely reproduces constrained IPOPT solution quality and achieves about 25× speedup, with mean inference time of approximately 57 ms versus 1427 ms.Its reported power-balance residuals are 2.72 × 10^-13 and 2.64 × 10^-13, compared with 1.09 × 10^-11 and 8.43 × 10^-12 for IPOPT.Source: §5.5 · p. 41, Figure 18 · p. 41

5.6. Uncertainty Quantification

HCP-PINN’s uncertainty estimates remain well calibrated across evaluated networks and noise models, while its sensitivity study finds that impedance errors and new loads or PV units have limited impact on median voltage error. Incorrect phase assignments cause the strongest degradation, and the study assumes feeder connectivity is known.

5.6. Uncertainty Quantification

  • Under GMM noise on the 122-bus network, CRPS rises to 0.0045, although HCP-PINN retains reasonable probabilistic accuracy.For this network, CRPS is around 0.0010–0.0011 under Gaussian and Laplacian noise.Source: §5.6 · pp. 42–43
  • HCP-PINN’s predicted uncertainty intervals remain close to expected coverage across evaluated confidence levels, indicating generally calibrated predictions.The posterior covariance is approximated locally around the MAP solution using measurement-model curvature; calibration is assessed by comparing expected and observed coverage.Source: §5.6 · pp. 42–43

5.7. Sensitivity to Network Parameter Errors and Grid Upgrades

  • With line-impedance perturbations up to ±20%, median voltage magnitude RMSE stays below 6.50 × 10−4 p.u. (0.163 V) for every phase.The reported error distributions remain largely unchanged relative to baseline; Fig. 20 summarizes the median, interquartile range, and 99th-percentile errors.Source: §5.7 · p. 43, Figure 20 · p. 45
  • Incorrect phase assignments produce the largest degradation: at Ph5, maximum median voltage RMSE reaches 1.32×10−3 p.u. (0.33 V), about 4.6× the Base case.Ph5 misassigns five of the feeder’s seven single-phase customers; the 99th-percentile RMSE reaches 1.43 V, while median error remains below 1 V.Source: §5.7 · p. 44
  • New loads and PV units at previously inactive buses leave median voltage RMSE near baseline, but PV additions increase upper-tail errors.For new loads, the 99th-percentile RMSE remains below 0.46 V; for new PV on Phase A, it rises from 0.32 V at baseline to 0.84 V for nPV3, still below 1 V.Source: §5.7 · pp. 44–46, Figure 20 · p. 45
  • The sensitivity tests assume feeder connectivity is known, and the authors treat approximately 0.5 V or lower as desirable and approximately 1 V as a practical upper bound, not a strict acceptance threshold.The study evaluates line-parameter errors and incorrect phase assignments because these are considered more prone to inaccuracies than main feeder topology.Source: §5.7 · pp. 43, 46

5.8. Generalization to Topological Changes and Operating Point Shifts

The study tests generalization to radial topology changes and increased customer loading using models trained on the original topology and nominal operating conditions. HCP-PINN has lower voltage RMSE and less degradation than the standard GNN under topology changes, and maintains low errors as loading increases.

5.8. Generalization to Topological Changes and Operating Point Shifts

  • Across tested radial reconfigurations, HCP-PINN consistently achieves lower voltage RMSE and less performance degradation than the standard GNN.The models are evaluated on up to four configurations per switch-change count after training only on the original 14-bus topology; HCP-PINN’s factor graph and constrained layer preserve topology-aware physical consistency.Source: §5.8 · pp. 46–47, Figure 21 · p. 47
  • As customer loads scale from α = 1.00 to 1.50, HCP-PINN maintains low median voltage error and a narrow interquartile range, while WLS and LAV errors remain much larger.The evaluation uses partial observability and normal measurement noise on the CIGRE 14-bus network.Source: §5.8 · p. 47
  • The standard GNN shows larger errors and dispersion at higher loading, whereas HCP-PINN maintains accuracy under shifted operating conditions.The paper attributes the GNN’s limited extrapolation to lacking explicit physical constraints, particularly under limited measurement coverage.Source: §5.8 · p. 48

5.9. Ablation Study of the HCP-PINN

The ablation study finds that the SMAP estimation layer contributes the largest improvement over the standalone FGNN, while hard constraints, the surrogate backward rule, and the probabilistic prior add complementary gains.

5.9. Ablation Study of the HCP-PINN

  • Removing hard equality constraints increases voltage magnitude and line loading errors and produces larger power balance residuals.The full model has the best overall performance across electrical quantities.Source: §5.9 · p. 48
  • Removing the geometric surrogate backward rule further degrades estimation accuracy, while the rule supplies the FGNN a learning signal tied to the constrained estimation objective.The ablation also finds poorer voltage, line-loading, and power-balance accuracy when the probabilistic FGNN prior is used only to initialize SMAP rather than retained in the MAP formulation.Source: §5.9 · p. 49
  • Adding SMAP to the standalone FGNN reduces voltage magnitude RMSE by approximately 56% and line loading RMSE by approximately 58%.Removing SMAP causes the most pronounced degradation in the sequential ablation.Source: §5.9 · pp. 48–49

6. Conclusions

HCP-PINN combines a probabilistic FGNN prior with a constrained SMAP estimation layer for DSSE under non-Gaussian uncertainty in balanced and unbalanced networks. The reported evaluations show improved accuracy, speed, and robustness over the compared estimators and learning baselines; joint estimation and model identification remain future work.

6. Conclusions

  • HCP-PINN formulates DSSE as constrained probabilistic optimization on a factor graph, with SMAP enforcing measurement consistency and equality constraints.The learned state distribution is embedded in the estimation objective rather than used only as an initializer.Source: §6 · p. 49
  • The conclusion reports improved robustness to network reconfiguration and out-of-distribution operating conditions compared with standard GNNs.The paper identifies joint state estimation and system identification, including line-parameter estimation and customer phase identification, as future work.Source: §6 · p. 50
  • Across test cases, HCP-PINN reduces state-estimation errors by approximately 58% versus WLS/LAV and 85% versus soft-PINN/GNN baselines.It also reports approximately 35× inference speedup in non-batched settings and up to 125× speedup with batch size 16 over numerical solvers on the 258-bus network.Source: §6 · p. 50

Related papers

Loading 2609.35246v1…