To achieve a fully decarbonized power grid, a massive deployment of renewable energy resources will be needed, but because of the intermittent nature of their generation, their full potential will not be unleashed unless demand side flexibility plays a bigger role than today. Introducing energy storage at the residential level enables increasing load flexibility, as it allows end-customers to easily change their consumption profile and adapt to the grid requirements. As of today, energy storage for residential consumers represents a considerable investment that is not guaranteed to be profitable. Shared investment models in which a group of consumers jointly acquires energy storage have been proposed in the literature to increase the attractiveness of these devices. Such models naturally employ concepts of cooperative game theory. In this paper, we extend the state-of-the-art cooperative game for modeling the shared investment in storage by adding two crucial extensions: stochasticity of the load and discreetness of the storage device capacity. As our goal is to increase storage capacity in the grid, the number of devices that would be acquired by a group of players that cooperate according to our proposed scheme is compared to the number of devices that would be bought by consumers acting individually. Under the same criteria of customer profitability , simulations using real data reveal that our proposed scheme can increase the deployed storage capacity between 100% and 250%.