They have to shut down during the earthquake to ensure they maintain coolable geometries. Cooling is easier at lower power. Some Fukushima reactors did have so

They have to shut down during the earthquake to ensure they maintain coolable geometries. Cooling is easier at lower power. Some Fukushima reactors did have some self-powered safety features, like a little steam turbine that comes from the boiling coolant and is hooked to a coolant pump. These features are at least a little related to why some reactors failed at different times than others in Fukushima. But seriously, there is enough decay heat to do more of this so it's a pretty good point.