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This paper investigates the operation of a current controlled Voltage Source Converter (VSC) under weak grid conditions caused by large grid impedance. Three different current controller structures are investigated; the conventional decoupled PI-controller in the synchronously rotating reference frame, Proportional Resonant (PR) controllers in the stationary reference frame, and the phase current hysteresis controllers. The different control strategies are studied by simulation, and the results show how large grid impedances can influence the dynamic response of the system. It is further discussed how the interaction between the current controllers, the Phase Locked Loop (PLL) and the grid inductance can trigger instability when the voltage measurements are highly influenced by the operation of the converter. The results indicate that the tuning of the PLL and the way of utilizing the phase information is of large importance for the stability and dynamic response of the control system.