Deep Reinforcement Learning (DRL) proved to be successful for solving complex control problems and has become a hot topic in the field of energy systems control, but for the particular case of thermal energy storage (TES) systems, only a few studies have been reported, all of them with a complexity degree of the TES system far below the one of this study. In this paper, we step forward through a DRL architecture able to deal with the complexity of an innovative hybrid energy storage system, devising appropriate high-level control operations (or policies) over its subsystems that result optimal from an energy or monetary point of view. The results show that a DRL policy in the system control can reduce the system operating costs by more than 50%, as compared to a rule-based control (RBC) policy, for cooling supply to a reference residential building in Mediterranean climate during a period of 18 days. Moreover, a robustness analysis was carried out, which showed that, even for large errors in the parameters of the system simulation models corresponding to an error multiplying factors up to 2, the average cost obtained with the original model deviates from the optimum value by less than 3%, demonstrating the robustness of the solution over a wide range of model errors.