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.

翻译：实现高精度配电网状态估计（DSSE）面临多重挑战，其中主要包括系统可观测性不足和网络拓扑密度过高。尽管基于机器学习模型的数据驱动方案可作为备选方案，但因标记数据匮乏而难以在DSSE场景中有效应用——事实上，配电网量测数据常伴随噪声、误差甚至缺失。针对上述问题，本文提出面向配电网状态估计的深度统计求解器（DSS²），该深度学习模型基于图神经网络（GNN）构建，能够兼顾配电网拓扑结构及其物理支配的潮流方程。DSS²利用超图表征配电网异构组件，并通过以节点为中心的消息传递机制更新其潜层表征。提出弱监督学习范式，通过学习-优化框架使DSS²在含噪量测与伪量测条件下优化加权最小二乘损失。通过强制GNN输出满足潮流方程并将该约束融入损失函数，迫使DSS²遵循配电网物理规律。该策略使模型具备从噪声量测中学习的能力——既充当隐式去噪器，又免除了理想标记数据的依赖。在IEEE 14节点、70节点及179节点网络上的大量案例研究表明：DSS²在精度、收敛性和计算效率上均显著超越传统加权最小二乘算法，同时对噪声、异常及缺失量测具有更强鲁棒性。相较于依赖全标签不现实假设的有监督模型，DSS²虽性能稍逊但已具备竞争性优势。