going home and plugging in for a recharge seems like it would be the standard thing just to keep things going.
Yes, I'm pretty sure we've seen this with the likes of May and Momo already.
It does seem that power outages would be a much bigger problem for AIs than they are for humans, though. For all we might joke about not being able to live without this or that piece of technology, a power outage isn't going to leave us
literally unconscious with no way to wake up no matter how long it lasts. Presumably most AIs would be very careful about making sure that they were at least plugged in before their batteries got too low, so that they could start charging whenever power was restored. Which would be fine for your run of the mill power outage, but might not work so well when critical infrastructure is destroyed in a natural disaster.