This paper presents a method for the optimal siting and sizing of energy storage systems (ESSs) in active distribution networks (ADNs) to achieve their dispatchability. The problem formulation accounts for the uncertainty inherent to the stochastic nature of distributed energy sources and loads. Thanks to the operation of ESSs, the main optimization objective is to minimize the dispatch error, which accounts for the mismatch between the realization and prediction of the power profile at the ADN connecting point to the upper layer grid, while respecting the grid voltages and ampacity constraints. The proposed formulation relies on the so-called Augmented Relaxed Optimal Power Flow (AR-OPF) method: it expresses a convex full AC optimal power flow, which is proven to provide a global optimal and exact solution in the case of radial power grids. The AR-OPF is coupled with the proposed dispatching control resulting in a two-level optimization problem. In the first block, the site and size of the ESSs are decided along with the level of dispatchability that the ADN can achieve. Then, in the second block, the adequacy of the ESS allocations and the feasibility of the grid operating points are verified over operating scenarios using the Benders decomposition technique.