I'm not sure what power plants and such do. Apparently they provide electricity and such but I don't see electricity mentioned as a resource or a pre-requisite for building anything(unlike uranium, oil, etc..). It's not mentioned as a game concept or anywhere(that I have found) in the Civilopedia.
I reached a point where I am given the option to build the Three Gorges Dam which provides power to all of my cities on a single continent. Seems impressive, I just have no idea what that will do.
thanks for your time!