It sounds like the power cord may have a broken conductor near the plug end that goes into the outlet. The cord may look fine but there could be a broken wire inside the insulating jacket that is making contact when the plug is moved around when reconnecting to the outlet. The next time the projector shuts off try wiggling the cord near the outlet plug and the projector itself to see if it comes back on..... if so, the cord needs replaced or a new end will need installed. If it is not the cord you will need to see if a thermal protector is present and has become weak. Another thing to consider is ventilation and air flow.... if the unit gets hot it will shut down. I'm still thinking it's the cord because the thermal protector takes a minute or two to reset itself and from what I gather, the projector comes back on as soon as you plug the cord back in.
It's a good thing I never had any sanity......I would have lost it by now.