Prius headlight goes off after some time

"Whatever the voltage is supposed to be at the bulb when running........it should be a fairly simple matter to test for the presence of that voltage, right ??"
The voltage at the HID bulb is around 15-20kV to start the arc. My guess is that not many people have the ability to test for that voltage without getting a serious zap (I happen to have a high voltage probe originally intended to test old color TV tubes, anode voltage of 30kV).

This is why the Prius with HID lights has two big yellow warning stickers advising the presence of high voltage, located on the black plastic radiator cover, near the headlight assemblies.

 

With that kind of voltage, putting one probe to ground and getting the other one anywhere near the bulb socket might give enough indication to know if it's working or not. Is this AC or DC voltage ?

And my advice always is: If you don't REALLY understand the thing you are working on......you probably should stop and seek the services of a professional. Many times that turns out to be cheaper in the long run anyway.

 

I don't know whether the high voltage is AC or DC. In either case, I don't think the testing can be as simple as you suggest, because a failure is not necessarily binary: either on or off. A ballast failure might result in a lower voltage than spec.

If the OP is confident that he has tested two authentic Philips bulbs in the socket and both are intermittent, then I agree with nh70's suggestion that the 12V supply to the ballast should be checked. It will probably be OK because the OP has never reported that if he turned on the switch, the lights failed to light up upon the first try. He only has reported that the light turns off after a while.

Then the ballast will be the final item to suspect. I suggest to the OP that if he gets to that point, he install the correct Toyota part because if that is not successful and he goes back to the dealer for help, they will insist on installing only the correct Toyota parts. This of course will require that the headlight assembly be removed for access to the ballast at the bottom, which also requires that the front bumper cover be removed.