This paper proposes a new control and power management strategy for a grid-connected microgrid, which includes a hybrid renewable energy sources (HRES) system and a three-phase load. The HRES system consists of a photovoltaic (PV), a battery storage system (BSS), a super-capacitor (SC) and a solid oxide fuel cell (SOFC). The dynamic model of each of these units is described. The PV is the main energy source, while the SC and the BSS due to their various power densities are considered to provide a steady and transient load demand, respectively. For increasing the reliability of the system, SOFC source is selected to keep the BSS completely charged. All these units with different DC-DC converters are connected in parallel to a common DC bus. Then, a three-phase voltage source inverter (VSI) is employed to convert the DC voltage to AC. To maintain the power balance and appropriate load-sharing, an adaptive fractional fuzzy sliding mode control (AFFSMC) strategy for VSI-based HRES system is presented. The controller is able to track the pre-defined instruction precisely and quickly in the microgrid. For stable performance of the control strategy under load variation, a fractional order-based sliding surface is considered. Moreover, fractional adaptive rules-based fuzzy sets are employed to accurately estimate the uncertain parameters in the microgrid. The simulation results demonstrate the effectiveness and capability of an AFFSMC strategy under various faults and different loading conditions. Moreover, the proposed control strategy is compared with the conventional PI controller.