The transition to sustainable energy systems relies heavily on green hydrogen as a clean energy carrier. However, its production from solar photovoltaics is fundamentally challenged by the resource's inherent intermittency, leading to fluctuating output and significant energy curtailment. This study investigates the performance of a battery-buffered solar-to-hydrogen unit designed to overcome this limitation. A comprehensive dynamic system model, integrating mathematical representations of a PV panel, a battery energy storage system, and an alkaline water electrolyzer, was developed. The model simulates the supervisory power management logic over a 48-hour period under realistic diurnal conditions. Simulation results demonstrate that the integration of the BESS successfully decouples hydrogen generation from solar variability. The system maintained a stable power supply of 150 W to the electrolyzer, enabling a constant production rate of 0.6 L/min during all operational periods, including nighttime. This resulted in a total yield of 111.1 grams of hydrogen over 48 hours. The control strategy achieved an overall solar energy utilization rate exceeding 95%, with the battery operating effectively within its 20-90% state-of-charge limits. These findings provide a quantitative validation for the critical role of battery buffering in enhancing the reliability and efficiency of standalone solar-to-hydrogen systems, confirming that this integrated architecture is a viable pathway for transforming intermittent renewable resources into a steady, dispatchable source of green hydrogen.