Every time I have had this happen in a modern car in ended up being the battery going - I did not know why this was the case -- because the alternator was never the problem.

I always felt that the battery just started the car but have learned that it is actually much more important than that in todays autos -- it is used to smooth the voltage sine wave out.

This is why so many vital electrical parts are connected directly to the battery -- it works like a battery back up on a computer.

This may not be your problem but keep a close watch on the battery.