This paper deals with a control strategy of a grid-connected marine current energy conversion system driven by a 5-phase permanent magnet synchronous generator. A supercapacitor is used as an energy storage device in order to smooth the output power and improve the low-voltage ride-through capability of the energy conversion system. The proposed approach is based on a model predictive approach with finite control set, where the future system states are predicted and evaluated to choose the optimal control action. Simulation results are carried out to validate the efficiency of the proposed control strategy. The system is supposed to be located in the Raz-de-Seine site in Bretagne, France. The output power fluctuation caused by the swell effect is efficiently smoothed by the supercapacitor energy storage system. The low-voltage ride-through performances are tested in two different fault scenarios when the grid voltage reduces to 50 and 100% of the rated value.