This paper deals with the behaviour of battery-supported residential photovoltaic (PV) systems connected to smart grids, with a special focus on the interface with electricity distributors. In particular, this paper is aimed at the assessment of the benefits resulting from two predictive control strategies managing the charge of the electrochemical storage unit, in comparison with the conventional control strategy and applied to various combinations of photovoltaic system and battery pack sizes in a parametric analysis. The proposed control strategies dispatch electricity to the battery pack or national grid depending on the state of charge and differ in the shape of the profile of exported electricity, hence adapting to different interface profiles required at the electricity distributor’s side. In particular, the research shows that, by the proposed control strategies, the frequencies of occurrence of high values of exported electricity may be decreased by almost 100%, whereas medium-high values of exported electricity by 50%. Moreover, both of the control strategies are shown to be able to lower the yearly quadratic effective imbalance coefficient by around 15% in case of PV systems sized for Zero Energy Buildings (ZEBs), thus showing opportunities in the mitigation of electricity grid stresses from PV systems.