  This paper address the problem of voltage regulation in power distribution
networks with deep penetration of distributed energy resources (DERs) without
any explicit communication between the buses in the network. We cast the
problem as an optimization problem with the objective of minimizing the
distance between the bus voltage magnitudes and some reference voltage profile.
We present an iterative algorithm where each bus updates the reactive power
injection provided by their DER. The update at a bus only depends on the
voltage magnitude at that bus, and for this reason, we call the algorithm a
local control algorithm. We provide sufficient conditions that guarantee the
convergence of the algorithm and these conditions can be checked a priori for a
set of feasible power injections. We also provide necessary conditions
establishing that longer and more heavily loaded networks are inherently more
difficult to control. We illustrate the operation of the algorithm through case
studies involving 8-,34- and 123-bus test distribution systems.
