The grid side converter (GSC) needs proper controller design when connected to the weak grid, otherwise it becomes unstable due to the overlook of grid impedance dynamic. Although the phase locked loop (PLL) will cause the instability of the interconnected system that consists of GSC and inductive grid due to its negative damping characteristic, the improper current controller design will also contribute to the instability due to the lagging effect of its integral controller. In this paper, GSC controlled by the current controller based on model predictive control (MPC) strategy is investigated under the weak grid. The results show the impedance of GSC near fundamental frequency applying the MPC current controller has a higher amplitude than applying the conventional control strategy with PI current controller, which benefits the stability of the interconnected system that consists of GSC and weak grid. Simulations are carried out to verify the effectiveness of MPC current controller in the weak grid.