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：实现精确的配电系统状态估计（DSSE）面临若干挑战，其中包括可观测性不足以及配电系统的高密度特性。虽然基于机器学习模型的数据驱动方法是一种可选方案，但在DSSE中因缺乏标注数据而效果不佳。实际上，配电系统中的量测量常伴有噪声、误差甚至缺失。为解决这些问题，我们提出了面向配电系统状态估计的深度统计求解器（DSS$^2$），这是一种基于图神经网络（GNN）的深度学习模型，能够兼顾配电系统的网络结构以及物理支配的潮流方程。DSS$^2$利用超图表征配电系统的异构组件，并通过以节点为中心的消息传递机制更新其隐层表示。我们提出一种弱监督学习方法，以学习-优化方式训练DSS$^2$，使其针对含噪量测和伪量测的加权最小二乘损失函数进行优化。通过将GNN输出强制纳入潮流方程，并将后者引入损失函数，我们迫使DSS$^2$遵从配电系统的物理规律。该策略使得系统能够从含噪量测中学习，发挥隐式去噪器的作用，并减轻了对理想标注数据的需求。在IEEE 14节点、70节点和179节点网络上的大量实验案例表明，DSS$^2$在精度、收敛性和计算时间上显著优于传统加权最小二乘算法，同时对噪声、误差和缺失量测具有更强的鲁棒性。尽管DSS$^2$的性能略低于依赖拥有全部真实标签这一不现实假设的监督模型，但其表现仍具有竞争力。