There is no extra supply. That would violate a basic principle of grid stability as expressed in what we power engineers call the Swing Equation as well as the Equal Area Criterion:
Basically, this equation states that for the grid to be stable, the mechanical power of the generator as expressed by the mechanical characteristics of the generator motor must equal the electrical power generated. If either one of these things gets out of whack, the grid will go unstable and the generator will be out of step (a fancy way of saying that the generator is out of control).
There are protective measures to ensure that does not happen, namely relaying systems that will either cut the generator off the grid if it starts to run away, or to isolate faults on the electrical system that will act as a short circuit—those short circuits can cause the generator to spin out of control.
Bad things like equipment failure and faults occur in a system that has this balance out of control. A generator that produces excess power is a generator that is out of step/running away, and therefore would render the system unstable or cause massive equipment failure on the electrical grid due to too much power being forced into the system if permitted to continue to operate in such a manner.
The Equal Area Criterion basically states that electrical power generated must equal electrical power consumed in order for the grid to remain stable:
