Zombie flicks are not exactly known for their strict adherence to realism. Residential power would likely be the first to go, if only to reserve generator capacity and fuel for military, medical, and government facilities, and rich assholes' mansions.

Maintaining the electric grid takes an incredible amount of work and the tight cooperation of all of its facilities. If one or several large generator were to shut down, it might cause a cascading collapse. Executing a black start is a massive challenge at the best of times, and probably impossible in a post-collapse society.