  Time-varying renewable energy generation can result in serious
under-/over-voltage conditions in future distribution grids. Augmenting
conventional utility-owned voltage regulating equipment with the reactive power
capabilities of distributed generation units is a viable solution. Local
control options attaining global voltage regulation optimality at fast
convergence rates is the goal here. In this context, novel reactive power
control rules are analyzed under a unifying linearized grid model. For
single-phase grids, our proximal gradient scheme has computational complexity
comparable to that of the rule suggested by the IEEE 1547.8 standard, but it
enjoys well-characterized convergence guarantees. Adding memory to the scheme
results in accelerated convergence. For three-phase grids, it is shown that
reactive injections have a counter-intuitive effect on bus voltage magnitudes
across phases. Nevertheless, when our control scheme is applied to unbalanced
conditions, it is shown to reach an equilibrium point. Yet this point may not
correspond to the minimizer of a voltage regulation problem. Numerical tests
using the IEEE 13-bus, the IEEE 123-bus, and a Southern California Edison
47-bus feeder with increased renewable penetration verify the convergence
properties of the schemes and their resiliency to grid topology
reconfigurations.
