Increased capacity associated with renewable energy sources has created a need for improved methods for controlling power flows from inverter-based generation. This research provides a comparative study of finite-control-set model predictive current control (FCS-MPC-based) with respect to conventional proportional-integral-based (PI-based) synchronous current control for a three-phase voltage source inverter (VSI). The inverter is accompanied by an inductive-capacitive-inductive (LCL) filter to attenuate pulse width modulation (PWM) switching harmonics. However, an LCL filter introduces a resonance near to the control stability boundary, giving rise to substantial complexity from a control perspective. In order to avoid potential instability caused by the resonance, active damping can be included in the PI-based current control. Though properly designed active damping can improve inverter stability, in practice the robustness of standard PI control is not attainable due to variability in the grid inductance at the point of common coupling (PCC). This is due to impedance variations causing large shifts in the LCL resonance frequency. Weak grid conditions (i.e., a low short-circuit ratio) and a correspondingly high line impedance are particularly susceptible to LCL induced resonance instabilities. As an approach to operate with grid impedance variations and weak grid conditions, FCS-MPC has the potential to produce superior performance compared to PI-based current control methods. This comparative study indicates that FCS-MPC has improved resonance damping and fast dynamic capability in a system with renewable energy sources under weak grid conditions. Detailed results from MATLAB/SimPower are presented to validate the suggested FCS-MPC method where it is robust to uncertainty in the grid impedance variations. Overall results indicate an improvement over conventional PI-based current control methods.