The growing Electric Vehicles' (EVs) popularity among commuters creates new challenges for the smart grid. The most important of them is the uncoordinated EV charging that substantially increases the energy demand peaks, putting the smart grid under constant strain. In order to cope with these peaks the grid needs extra infrastructure, a costly solution. We propose an Adaptive Management of EV Storage (AMEVS) algorithm, implemented through a learning agent that acts on behalf of individual EV owners and schedules EV charging over a weekly horizon. It accounts for individual preferences so that mobility service is not violated but also individual benefit is maximized. We observe that it reshapes the energy demand making it less volatile so that fewer resources are needed to cover peaks. It assumes Vehicle-to-Grid discharging when the customer has excess capacity. Our agent uses Reinforcement Learning trained on real world data to learn individual household consumption behavior and to schedule EV charging. Unlike previous work, AMEVS is a fully distributed approach. We show that AMEVS achieves significant reshaping of the energy demand curve and peak reduction, which is correlated with customer preferences regarding perceived utility of energy availability. Additionally, we show that the average and peak energy prices are reduced as a result of smarter energy use.