Constant current and constant voltage control of inverter
Most of the inverters on the grid are based on energy storage in an inductance, either in a discrete inductor, or the inductance of a transformer. 

The purpose of the outer loop is to control the flow of power to the load. The purpose of the inner loop is to control the cycle by cycle energy contained in the energy storage element. 

This means the natural thing to do is to control the current.