A key factor is the relative size of the generator compared to the grid. Large generators generally run in some type of voltage control mode to help supply vars to the grid and regulate voltage. But for small cogeneration systems, the generator too small to appreciably influence the grid voltage so running in voltage control mode can be problematic. The AVR will alternate between max and min excitation settings for small grid voltage changes. For these smaller units, some type of power factor or VAR control mode is generally used. This is often a point of friction between merchant power plants and grid operators and requirements are often spelled out interconnection agreements.
In island mode, the generator needs to be in voltage control (or at least one of them for multiple units).