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. We illustrate our results on a problem of planning of a realistic distribution feeder on one day, taking into account the uncertainty associated with distributed solar production, and diffuse storage systems. We show theoretically a vanishing relaxation gap for this problem, when the installed solar capacity does not exceed local consumption, independently of the scenarios of solar production that are considered. Then, we illustrate numerically the quality of the a posteriori bound on the relaxation gap for higher installed capacities of distributed generation. Scenario trees for solar production are constructed from a stochastic model, by a quantile-based algorithm.