  Distribution grids are challenged by rapid voltage fluctuations induced by
variable power injections from distributed energy resources (DERs). To regulate
voltage, the IEEE Standard 1547 recommends each DER inject reactive power
according to piecewise-affine Volt/VAR control rules. Although the standard
suggests a default shape, the rule can be customized per bus. This task of
optimal rule design (ORD) is challenging as Volt/VAR rules introduce nonlinear
dynamics, and lurk trade-offs between stability and steady-state voltage
profiles. ORD is formulated as a mixed-integer nonlinear program (MINLP), but
scales unfavorably with the problem size. Towards a more efficient solution, we
reformulate ORD as a deep learning problem. The idea is to design a DNN that
emulates Volt/VAR dynamics. The DNN takes grid scenarios as inputs, rule
parameters as weights, and outputs equilibrium voltages. Optimal rule
parameters can be found by training the DNN so its output approaches unity for
various scenarios. The DNN is only used to optimize rules and is never employed
in the field. While dealing with ORD, we also review and expand on stability
conditions and convergence rates for Volt/VAR dynamics on single- and
multi-phase feeders. Tests showcase the merit of DNN-based ORD by benchmarking
it against its MINLP counterpart.
