This paper  proposes a novel scheme for  the  current controller  for the  grid-side converter (GSC) of permanent-magnet synchronous generator (PMSG) wind turbines to eliminate the high-order harmonics in the grid currents under grid voltage disturbances. The voltage unbalance and harmonics in three-phase systems cause grid current distortions. In order to mitigate the input  current distortions, multi-loop current controllers are applied, where  the  positive-sequence component  is regulated  by proportional-integral (PI)  controllers, and  the  negative-sequence and  high-order harmonic  components  are regulated  by proportional-resonance (PR)  controllers.  For extracting  the positive/negative-sequence  and  harmonic components of the grid voltages and currents without a phase delay or magnitude reduction, composite observers are applied, which give faster and more precise estimation results. In addition, an active damping method using PR controllers to damp the grid current component of the resonant frequency is employed to improve  the  operating  stability  of  VSCs with  inductor-capacitor-inductor (LCL) filters. The validity of the proposed method is verified by simulation and experimental results.