  The problem of state estimation for unobservable distribution systems is
considered. A deep learning approach to Bayesian state estimation is proposed
for real-time applications. The proposed technique consists of distribution
learning of stochastic power injection, a Monte Carlo technique for the
training of a deep neural network for state estimation, and a Bayesian bad-data
detection and filtering algorithm. Structural characteristics of the deep
neural networks are investigated. Simulations illustrate the accuracy of
Bayesian state estimation for unobservable systems and demonstrate the benefit
of employing a deep neural network. Numerical results show the robustness of
Bayesian state estimation against modeling and estimation errors and the
presence of bad and missing data. Comparing with pseudo-measurement techniques,
direct Bayesian state estimation via deep learning neural network outperforms
existing benchmarks.
