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Recently, we demonstrated the success of a time-synchronized state estimator using deep neural networks (DNNs) for real-time unobservable distribution systems. In this paper, we provide analytical bounds on the performance of the state estimator as a function of perturbations in the input measurements. It has already been shown that evaluating performance based only on the test dataset might not effectively indicate the ability of a trained DNN to handle input perturbations. As such, we analytically verify the robustness and trustworthiness of DNNs to input perturbations by treating them as mixed-integer linear programming (MILP) problems. The ability of batch normalization in addressing the scalability limitations of the MILP formulation is also highlighted. The framework is validated by performing time-synchronized distribution system state estimation for a modified IEEE 34-node system and a real-world large distribution system, both of which are incompletely observed by micro-phasor measurement units. 

Recently, there has been a major emphasis on developing data-driven approaches involving machine learning (ML) for high-speed static state estimation (SE) in power systems. The emphasis stems from the ability of ML to overcome difficulties associated with model-based approaches, such as handling of non-Gaussian measurement noise. However, topology changes pose a stiff challenge for performing ML-based SE because the training and test environments become different when such changes occur. This paper circumvents this challenge by formulating a graph neural network (GNN)-based time-synchronized state estimator that considers the physical connections of the power system during the training itself. The results obtained using the IEEE 118-bus system indicate that the GNN-based state estimator outperforms both the model-based linear state estimator and a data-driven deep neural network-based state estimator in the presence of non-Gaussian measurement noise and topology changes, respectively.