  Implementing accurate Distribution System State Estimation (DSSE) faces
several challenges, among which the lack of observability and the high density
of the distribution system. While data-driven alternatives based on Machine
Learning models could be a choice, they suffer in DSSE because of the lack of
labeled data. In fact, measurements in the distribution system are often noisy,
corrupted, and unavailable. To address these issues, we propose the Deep
Statistical Solver for Distribution System State Estimation (DSS$^2$), a deep
learning model based on graph neural networks (GNNs) that accounts for the
network structure of the distribution system and for the physical governing
power flow equations. DSS$^2$ leverages hypergraphs to represent the
heterogeneous components of the distribution systems and updates their latent
representations via a node-centric message-passing scheme. A weakly supervised
learning approach is put forth to train the DSS$^2$ in a learning-to-optimize
fashion w.r.t. the Weighted Least Squares loss with noisy measurements and
pseudomeasurements. By enforcing the GNN output into the power flow equations
and the latter into the loss function, we force the DSS$^2$ to respect the
physics of the distribution system. This strategy enables learning from noisy
measurements, acting as an implicit denoiser, and alleviating the need for
ideal labeled data. Extensive experiments with case studies on the IEEE 14-bus,
70-bus, and 179-bus networks showed the DSS$^2$ outperforms by a margin the
conventional Weighted Least Squares algorithm in accuracy, convergence, and
computational time, while being more robust to noisy, erroneous, and missing
measurements. The DSS$^2$ achieves a competing, yet lower, performance compared
with the supervised models that rely on the unrealistic assumption of having
all the true labels.
