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

Abstract

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.