The plants should be self sufficient in terms of electrical power generation (i.e., they use what electricity they generate first to run the plant then put the rest on the grid. Second, when the rods go in if you loose control, the plant shuts down and cools down. This is a feature of Pressurized water reactors which are what the US uses. Chernobyl was a graphite moderated reactor, rods in mean more heat which means big molten pile.

Big take away, don't worry about the nuc plants.

Regards,
Bill