This paper includes the study of a control strategy for the connection of PMSG-based wind turbines during unbalanced faults in weak grids. The wind turbine front-end converter is controlled in the stationary frame by the direct calculation of current references. In this way, it is possible to keep a reasonably smooth generated power during the fault, in order to reduce the fault impact on the wind turbine mechanical components. The effects on the mechanical parts of the wind turbine have been studied by using the NREL 5 MW reference wind turbine, modeled with the NREL FAST code. The effects of highly distorted network voltages during the fault and the limitations of the current control loop dynamics are thoroughly studied by means of detailed PSCAD simulations.