I have been able to resolve this issue by replacing the battery in this laptop. This is the battery I purchased
After replacing the battery in the laptop, I have not had the random shutdown on battery power since. Here are the instructions to replacing the battery in this unit.
1. First you need to get the right screwdriver for the screws in the bottom of the laptop. They look like tiny little stars. I found this Kobalt screwdriver set at home depot that had the correct bit. It's the smallest one of the star bits. Here is the picture..
2. There are 12 screws in the bottom of the ASUS UX51V. 2 of them are hiding under the black plastic feet in the rear of the laptop. Use a knife to pry off the rubber feet. You can glue them back on later if needed.
3. After unscrewing, the bottom plate removes easily. At this point the battery is exposed and there are a number of screws holding it in. Be sure to remember the position of these screws after you've unscrewed them as some of them vary in size.
4. Before removing the battery you have to disconnect it from the main board. Simply LIFT UP on the little tag that is attached to the connector piece and it will disconnect easily. Be sure not to pull towards you or away from you. The connection disconnects in an upward fashion.
5. Simply replace the battery, make sure it is seated under the lip and that the peg/notches are in place and then screw it back in. Replace the bottom plate and screw it back in.
6. That's all!
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I love my Asus Zenbook UX51V, I even left a solid 5 star review on Newegg for it. However, after a few months of owning it there is a BIG problem which protential buyers need to know about.
The UX51V will randomly shutdown ( not BSOD, BlueScreen or Error). It seems to happen ONLY when I'm running off battery and almost always when I'm streaming some video from youtube or another site. Please note this is not a battery running out of power issue, the laptop shut's down with plenty of battery life left. I can turn it back on without a problem most of the time. Sometimes I have to wait a few minutes before it wil let me turn it back on without crashing and shutting down instantly again. If you have this laptop and you are experiencing this problem you may notice you are getting a Critical level (Kernel-Power) error in your Windows/System event log. (Accessible via Computer Management). "The system has rebooted without cleanly shutting down first. This error could be caused if the system stopped responding, crashed, or lost power unexpectedly."
As far as I know there are no known fixes yet or any official statement from Asus acknowledging this problem, despite their being numerous reports of it from other customers..
http://meieden.hubpages.com/hub/A-Zenbook-Users-Experiences-with-Asus-Laptop-Customer-Support-and-RMA-Service
http://technologicaltheologian.blogspot.com/2012/10/zenbook-prime-and-unexpected-shutdown.html
http://www.techsupportforum.com/forums/f299/asus-zenbook-crashes-on-battery-windows-8-a-685977.html
Some notable possible work-arounds posted by others include the following:
FYI if you have the problem that the computer shuts down when you unplug the charger you should try the latest driver for the Intel GPU. It has a crash fix for exactly that.
I have disabled VT-d in the BIOS, along with Intel’s shutdown power management. I have not yet seen the shutdown problem. Note that you don’t need VT-d, as it is a virtualization technology, and it is doubtful that most people will use that on an ultrabook with 4GB of RAM.
Good luck! Please comment if you are also having this problem and have any insight/frustrations to share.
Thanks for reading.