Efficient energy management in prosumer households is key to alleviating grid stress in an energy transition marked by electric vehicles (EV), renewable energies and battery storage. However, it is unclear how households optimize prosumer EV charging. Here we study real-world data from 90 households on fixed-rate electricity tariffs in German-speaking countries to investigate the potential of Deep Reinforcement Learning (DRL) and other control approaches (Rule-Based, Model Predictive Control) to manage the dynamic and uncertain environment of Home Energy Management (HEM) and optimize household charging patterns. The DRL agent efficiently aligns charging of EV and battery storage with photovoltaic (PV) surplus. We find that frequent EV charging transactions, early EV connections and PV surplus increase optimization potential. A detailed analysis of nine households (1 hour resolution, 1 year) demonstrates that high battery capacity facilitates self optimization; in this case further algorithmic control shows little value. In cases with relatively low battery capacity, algorithmic control with DRL improves energy management and cost savings by a relevant margin. This result is further corroborated by our simulation of a synthetic household. We conclude that prosumer households with optimization potential would profit from DRL, thus benefiting also the full electricity system and its decarbonization.