  Recently, we demonstrated success of a time-synchronized state estimator
using deep neural networks (DNNs) for real-time unobservable distribution
systems. In this letter, we provide analytical bounds on the performance of
that state estimator as a function of perturbations in the input measurements.
It has already been shown that evaluating performance based on only the test
dataset might not effectively indicate a trained DNN's ability to handle input
perturbations. As such, we analytically verify 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.
