In one aspect, a method for controlling the operation of a power generation system configured to supply power to an electrical grid may generally include monitoring a rotor speed of a generator of the power generation system and determining a gain scaling factor based on the rotor speed, wherein the gain scaling factor increases with decreases in the rotor speed across a range of rotor speeds. In addition, the method may include adjusting a regulator gain to be applied within a current regulator of the power generation system based on the gain scaling factor and applying the adjusted regulator gain within the current regulator in order to generate a voltage command signal for controlling a power converter of the power generation system.