  Pronounced variability due to the growth of renewable energy sources,
flexible loads, and distributed generation is challenging residential
distribution systems. This context, motivates well fast, efficient, and robust
reactive power control. Real-time optimal reactive power control is possible in
theory by solving a non-convex optimization problem based on the exact model of
distribution flow. However, lack of high-precision instrumentation and reliable
communications, as well as the heavy computational burden of non-convex
optimization solvers render computing and implementing the optimal control
challenging in practice. Taking a statistical learning viewpoint, the
input-output relationship between each grid state and the corresponding optimal
reactive power control is parameterized in the present work by a deep neural
network, whose unknown weights are learned offline by minimizing the power loss
over a number of historical and simulated training pairs. In the inference
phase, one just feeds the real-time state vector into the learned neural
network to obtain the `optimal' reactive power control with only several
matrix-vector multiplications. The merits of this novel statistical learning
approach are computational efficiency as well as robustness to random input
perturbations. Numerical tests on a 47-bus distribution network using real data
corroborate these practical merits.
