We propose a multi-time scale energy management framework for a smart photovoltaic (PV) system that can calculate optimized schedules for battery operation, power purchases, and appliance usage. A smart PV system is a local energy community that includes several buildings and households equipped with PV panels and batteries. However, due to the unpredictability and fast variation of PV generation, maintaining energy balance and reducing electricity costs in the system is challenging. Our proposed framework employs a model predictive control approach with a physics-based PV forecasting model and an accurately parameterized battery model. We also introduce a multi-time scale structure composed of two-time scales: a longer coarse-grained time scale for daily horizon with 15-minutes resolution and a shorter fine-grained time scale for 15-minutes horizon with 1-second resolution. In contrast to the current single-time scale approaches, this alternative structure enables the management of a necessary mix of fast and slow system dynamics with reasonable computational times while maintaining high accuracy. Simulation results show that the proposed framework reduces electricity costs up 48.1% compared with baseline methods. The necessity of a multi-time scale and the impact on accurate system modeling in terms of PV forecasting and batteries are also demonstrated.