The increasing penetration of renewable energy technologies causes major problems in the power network, as their generation cannot be totally predicted. Along with fluctuations in the generation of renewables due to weather uncertainties, storage is very important for mitigating several problems that may arise, affecting the stability and reliability of the grid. In particular, in recent years there has been an emphasis on residential storage applications (behind-the-meter storage), with the aim of increasing the energy self-consumption and therefore reducing electricity bills. The proposed model consists of a 3 kWp rooftop solar photovoltaic (PV) system connected to the grid through converters and a battery-supercapacitor hybrid energy storage system. The model is developed and simulated in the MATLAB/Simulink software environment, based on mathematical analysis and average modeling. The supercapacitor handles rapid changes that occur within 0.2 s, and this can relieve the battery stress and extend the battery lifetime. The building’s electricity demand is satisfied through the PV, hybrid energy storage and/or grid. A new filtration-based power management algorithm (PMA) is proposed here, prioritizing the utilization of the PV and battery-supercapacitor instead of the grid, thus achieving a reduced power exchange between the building and the grid and increasing the PV self-consumption and self-sufficiency of the building. The dynamic performance of the proposed model is verified through several simulations over short time periods (10–30 s) for different scenarios that could occur. The obtained results show that the model works properly and responds extremely fast during the different mode transitions, exhibiting a very fast DC-bus voltage regulation with a very small ripple voltage of up to 5 V (a maximum of ± 0.625%). Additionally, both battery and supercapacitor remain between their minimum and maximum limits. Finally, an effective power sharing is achieved between the PV, the battery-supercapacitor storage, the building load and the grid.