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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. 

It is useful to quantify electrical distribution system resilience based on historical performance. This paper systematically extracts resilience curves from historical utility outage data, extracts resilience metrics such as duration, average recovery rates, and maximum number of simultaneously outaged components, and examines the statistics of these resilience metrics for small, medium, and large events. The resilience metrics and their typical variabilities are expected to be helpful in predicting and bounding the likely outcomes of future resilience events. For example, we can calculate the restoration time that will be achieved with 95% confidence.