Hopefully, this is for informative purposes only - as I will be taking mine in for that iMac G5 Repair Extension Program and hope to resolve this issue that way.
Model: Refurb 1.6GHz (pre-iSight) iMac G5
Early history: Fan ran out of control, shut down due to overheating: logic board replacement fixed (?) it. Later, OS patch also addressed it.
Later history: Even plugged into UPS, after power outage iMac would not restart until power cord was unplugged and reattached.
Current history: Upgraded to OS X 10.5.1, Energy Saver set to "Restart automatically after power failure". After clean install of 10.5, it shuts down (and restarts) several times a day, but never while I was working on it - 1:45am, 2:15am, and 3:20am this morning alone before I turned off the auto restart. This was
definitely not happening before 10.5 was installed. Contributing factor may be lack of that OS patch due to doing a clean install.
Also, the Sys log in Console shows "AppleSMU -- shutdown cause = -110" during each of those three shutdowns mentioned.
Resolution: Taking it in for iMac G5 Repair Extension Program - just makes it in under the 3 year deadline. Will post results. I'll be using the program mentioned here though:
http://www.apple.com/support/imac/repairextensionprogram/