Power management at thermal power station
Power grid managers continuously monitor the load drawn to make sure that they don't overload the grid. If too much power is being drawn, they will contact power plants and ask them to produce more power. If too little is being drawn, the opposite will happen.

It does effect the stations: think of what happens to a car engine. When you're in neutral and push a little on the pedal, the engine revs really high with very little input. However, climbing up a steep hill requires you to push the pedal in much further. Revving the engine really high isn't good for it mechanically, and you get terrible mileage, which translates to wasted fuel. Similar things happen to power plants, albeit the actual process is much more complicated than I described.

This might be of interest to you: Britain from Above (Tea-time Britain)

It doesn't really explain too much the consequences of too much/too low power supply, but does provide a view of what happens behind the scene to keep the grid working.