Do power plants supply their own electricity?
Nuclear power plants in particular have extremely critical needs for power even when they are not generating, as decay heat requires circulating cooling water through not only the reactor long after shutdown, but also possibly some of the recently spent fuel stored in pools of water outside the reactor.
As a result they are not only able to draw from the grid, but have local diesel backup generators.
One of the fundamental causes of the disaster at Fukushima was that the diesel generators were flooded by the tsunami, leading to loss of power for active cooling.
(The backup generators are sized for emergency cooling only.  Very rarely have they been officially sufficient for normal operation in the sense of being able to perform a &quot;Black Start&quot;.  Some searching will find commentary from plant engineers who think they could get things slowly started if authorized in a dire need by using limited pumping and natural convection, but it would be well outside normal procedures and precautions)