Hi Frank,
I have acer aspire 4736z laptop and I have exactly the "opposite" problem as yours. I.e. when my laptop will run on AC power but will abruptly shut down on battery. On AC power it works fine, but the moment the AC power is switched off, the laptop will run for a few minutes (random time, but less than 10-12 mins) and shut down without any warning etc.
I thought the it was some problem with the battery, hence got a new one. But realized that with new battery too the same problem remained. Took my laptop to a local repair shop, they reinstalled the OS. But the problem remained. It works on the battery in the BIOS setup mode or even on the windows Safe mode, but the moment the OS is loaded it shuts off on battery.
It is not related to heating issue as it runs perfectly on AC power, it is only on the battery that this problem comes.
I visited another repair guy and he said that it was some hardware issue. But I am yet to decide whether to give my laptop to him for dissection.
If you have come across some solution or have got your machine repaired kindly advice on the course of action.
Thank You