In this paper we present an adaptive robust optimization framework for the day-ahead scheduling of Active Distribution Networks (ADNs) where the controlled devices are distributed Energy Storage Systems (ESSs). First, the targeted problem is formulated using a two-stage optimization approach. The first-stage decisions determine the amount of import/export energy from the external grid at each hour, as well as energy exchanges for each ESS. The second stage deals with the intra-day control, given the first stage decisions. In order to effectively consider the impacts of uncertainties, the problem is transformed into a `min-max-min' formulation. Here, we minimize the first (day-ahead scheduling) and second stage (intra-day operation) costs while uncertainties are maximally affecting the cost function of the second stage. The Benders dual cut algorithm is employed for the solution of the optimization problem. IEEE 34 bus standard network is the benchmark grid for assessing the performances and effectiveness of the developed robust optimization process.