Unexpected, unexplained shutdown

We are trying to figure out what causes our PC to go to sleep or even to shutdown without notice. This always happens overnight, when not in use. It is set to 'never sleep'. It runs Windows 7 Home Premium 64bit. When I look at the events, all it tells me is this critical error: “The system has rebooted without cleanly shutting down first. This error could be caused if the system stopped responding, crashed, or lost power unexpectedly.” That's at boot up. But the events did not record the reason for it to shutdown in the first place. They kept track of all sorts of things of the day before, but not why it went down. Actually, there is no BlueScreenofDeath involved...
We have LogMeIn running so we can connect by remote. LogMeIn keeps track of connection details. Unfortunately, it only keeps track of things since midnight of that day I believe. So, there, too, all is a blank slate. Puzzled. Any ideas where else I could look? Thanks in advance.

Before going to bed, do a clean boot. Follow Step 1 of the following tutorial:

Troubleshoot Application Conflicts by Performing a Clean Startup

Next morning: Did it still shutdown? Report back.

Thanks for the tips. The PC is actually about 2 years old. Don't know about the PSU's condition. But the temperature is within normal range. At idle, about 40 F. It's actually a work station, not at home. Can't shut it down intentionally because there are other programs running through it, like fax.
During my lunch time, it shutdown again. And I was only gone 1 hour. What I see in the event viewer yesterday evening and during lunch time is the VSS shutting down because it is idle. I stopped the service.
But I am also suspecting a Battery backup, which is older than the PC. For now I disconnected the USB from it. Perhaps it is giving the wrong signals to the PC. I'll be watching the situation and reporting back.