The inertia of a generator is not free.
As the generator leads or lags, the current has to go up if real power stays constant.-which it by default does because real power feeding the generator stays as constant as the stream pressure at the turbine.
By "not free", do you mean the cost of the generator is increased above the base cost if it has to provide grid stabilization in addition to the normal generation capacity? Because I was making the implicit assumption that a generator is big and has a lot of inertia by default, and engineers wouldn't add cost by making it bigger than it needs to be without a strong justification.
The stabilizing effect of the inertia of the generator is simply that it takes energy to change its speed.
Synchronous condensers, which are just synchronous motors free wheeling, also add inertia to the grid. Induction motors do not, because they only dump power into the grid if the frequency falls below 59,58,57 hz faster than the load causes the induction motor to slow down.
So anyhow, if synchronous condensers add inertia, what makes you think they are free? They have static losses and when current (real or not) is drawn from them, due to frequenxy and voltage changes, do you not see there is i^2R losses?
The same for the generator.
Lets assume the grid has a 1 second time constant due to the inertia of the generators, which isnt very much. So say that inertia is equal to 62% of 1 second of total grid power consumption.
If solar makes up 10% of the grid, and provided power via a trained oscillator with a 1 minute time constant, the grid's inertia would be able to be increased by the energy stored in the capacitors of those inverters.
It does not mean the inverters are capable of dumping out all their energy quickly, but they dont need to. We are talking small, .95 leading or lagging power factor changes over the course of a few seconds. The contribution of losses due to the current rising from 100 to 102% is very small, but it is by no means free.
Now im not sure this would work without utility control over the inverters.
Because if you have the inverters always wanting to push the grid back to 60hz, the problem is the inverters dont know if they should export leading or lagging power to do so, unless they also were synchronized to a gps clock, and had knowledge of what the voltage should be.