Inspiron 15R - random complete shutdowns

Go figure - my official warranty expired on September 5, and on September 6 I begin having this problem.

My Inspiron 15R randomly shuts itself down after a period of non-use.  I'm not talking about going into sleep or hibernation mode.  No, it will be working fine, and I'll go into another room to eat dinner or watch TV, and when I come back it's completely shut down.  I have to reboot from a cold state.  When I do, I get various error messages from whatever program was running at the time it shut itself down.  For example, if I had Chrome open, then when I reopen Chrome I get the message stating that Chrome did not shut down properly, and am offered the chance to reopen the tabs that were open when the computer shut down.

The computer is ALWAYS set to full performance.  I use it as a desktop unit 95% of the time, never unplug it.  It is set to NEVER shut down (with the exception of the screen), for any reason, unless I specifically do a shutdown.

The problem I had previously was something to do with the system not recognizing when the AC power cable was plugged in.  (The cable would be plugged in, as it generally is 24/7/365, but the system would be running off the battery.)  A Dell tech came out and replaced the motherboard completely, and it's worked perfectly since they came out.  Now I have this problem, and I'm not sure if it is related or not.

Given my experience with this particular Dell unit, it's highly unlikely that I will purchase another Dell when it comes time to upgrade or replace.