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For a current controlled grid-connected inverter, the grid voltage distortion will give rise to the injected grid current harmonics. Besides, for the three-phase grid-connected inverter, under the unbalanced grid voltage condition, the injected grid currents will become unbalanced. This paper aims at weakening the effect of the grid voltages on the injected grid currents for a three-phase grid-connected inverter with an LCL filter. The full-feedforward scheme of the grid voltages is investigated in this paper. The full-feedforward function of the grid voltages in the stationary α-β frame for a three-phase grid-connected inverter with an LCL filter is proposed. This full-feedforward function consists of three parts, which is proportional part, derivative part and second derivative part. With the proposed full-feedforward scheme of the grid voltages, approximately zero steady-state error of the injected grid currents at line frequency can be achieved even with proportional-integral (PI) compensators as the injected grid current regulators in the stationary α-β frame. The injected grid current distortion and unbalance caused by the grid voltages are also greatly reduced. At last, the effectiveness of the proposed full-feedforward scheme is verified by the experimental results.