  We propose a generic multistage stochastic model for the Alternating Current
Optimal Power Flow (AC OPF) problem for radial distribution networks, to
account for the random electricity production of renewable energy sources and
dynamic constraints of storage systems. We consider single-phase radial
networks. Radial three-phase balanced networks (medium-voltage distribution
networks typically have this structure) reduce to the former case. This induces
a large scale optimization problem, which, given the non-convex nature of the
AC OPF, is generally challenging to solve to global optimality. We derive a
priori conditions guaranteeing a vanishing relaxation gap for the multi-stage
AC OPF problem, which can thus be solved using convex optimization algorithms.
We also give an a posteriori upper bound on the relaxation gap. In particular,
we show that a null or low relaxation gap may be expected for applications with
light reverse power flows or if sufficient storage capacities with low cost are
available. Then, we discuss the validity of our results when incorporating
voltage regulation devices. Finally, we illustrate our results on problems of
planning of a realistic distribution feeder with distributed solar production
and storage systems. Scenario trees for solar production are constructed from a
stochastic model, by a quantile-based algorithm.
