Engine over heating, should fan be on if engine is not?

Yes, the fan(s) will even run with a non-running cool engine if the a/c is on.

 

My guess is the fan control is defective or not getting the correct signal. A diagnostic tool would help isolate. Does the a/c activate it with a non-running ICE (engine)?

 

I can imagine that because the engine is not running, there is no need for additional cooling (as the coolant temperature will not rise when the engine is not running). No idea if that is a strategy of the Prius.

Out of curiosity: does the fan stop when the engine stops? I can not imagine the fan will come on when the engine is stopped because it would be strange that the coolant temperature rises when the engine is not running.

 

If you put on the ignition ON, does the fan also cones on? If it does, you got a problem with the fan relay or the wiring shorting with the ignition ON circuitry. That isn't supposed to be like that. You need to have that looked into, because it would with time, results into frequent fan motor replacements, and noise coming from the fan bearings itself.

 

What's telling you it's overheating: some OBD monitor? What temps are you seeing?

 

Sorry, editted last post: what temps?

 

if A/C is on they turn on. Normal ICE vehicles both fans come on if A/C is on to transmit more heat energy away from the condenser and radiator since it has to do twice the work to keep both systems cool. I would also look at Engine Temp and inverter temp. Quick check when vehicle is on remove cap from inverter coolant and check for flow. You will see "current" like in a slow moving river in the coolant. If you see nothing then inverter pump is out. Looks like someone has some puzzle pieces to figure out.
Plus you have to account the Hot Water Storage Tank as well since the three way valve for that only works at certain times. Check those temps after a good 10 min drive to ensure your vehicle is not overheating killer. Keep it simple and stray away from electrical being your issue since if you start chasing too many issues, the only issue will be you.