As residential adoption of renewable energy sources increases, optimizing rooftop photovoltaic systems (RTPVs) with Battery Energy Storage Systems (BESSs) is key for enhancing self-sufficiency and reducing dependence on the grid. This study introduces a novel methodology for sizing Home Energy Management Systems (HEMS), with the objective of minimizing the cost of imported energy while accounting for battery degradation. The battery model integrated nonlinear degradation effects and was evaluated in a real case study, considering different temporal data resolutions and various energy management strategies. For BESS capacities ranging from 1 to 5 kWh, the economic analysis demonstrated cost-effectiveness, with a Net Present Value (NPV) ranging from 54.53 € to 181.40 € and discounted payback periods (DPBs) between 6 and 10 years. The proposed HEMS extended battery lifespan by 22.47% and improved profitability by 21.29% compared to the current HEMS when applied to a 10 kWh BESS. Sensitivity analysis indicated that using a 5 min resolution could reduce NPV by up to 184.68% and increase DPB by up to 43.12% compared to a 60 min resolution for batteries between 1 and 5 kWh. This underscores the critical impact of temporal resolution on BESS sizing and highlights the need to balance accuracy with computational efficiency.