The use of photovoltaic (PV) energy in combination with heat pump systems for heating and cooling of residential buildings can lead to renewable energy self-consumption, reducing the energy required from the grid and the carbon footprint of the building uses. However, energy storage technologies and control strategies are essential to enhance the self-consumption level. This paper proposes and analyzes a new control strategy for the operation of a modulating air-source heat pump, based on the actual PV availability. The solar energy surplus is stored as thermal energy by the use of water tanks and the activation of the thermal capacitance of the building. The efficacy of the control strategy is evaluated considering different rule-based strategies, and different boundary conditions. The effect of climate data, building insulation level and thermal inertia are investigated and compared. The results show the efficacy of the proposed strategy to decrease up to 17% the amount of electricity purchased from the grid and to increase the self-consumption by 22%, considering a high-insulated building in Bolzano, Northern Italy. The thermal mass activation is found effective to increase the self-consumption of the system. Nonetheless, the achievable energy reduction depends largely on the building characteristics and the boundary conditions.