I am not a lawyer so I don't know for sure, but my feeling is this is going to vary from state to state or possibly even from town to town or county to county.
I think in places where there is no statute specifically addressing electricity, it is going to be up to the opinions of police officers, judges, caseworkers, etc. to assess whether or not the living conditions are dangerous, deplorable or neglectful. I think it would be one of those things that they would look at on a case-by-case basis. If your home is clean, has food and running water, and the kids are able to bathe, do their homework, have clean clothes, are warm in the winter, etc., then that would be a much different situation than a home that is a squalor, dirty, without food or a way to prepare food, or a way to bathe or stay warm.
You have the generator, so it's not like you are completely without electricity. You can prepare food, bathe, stay warm, etc. If your kids are safe and well cared for, I don't know what the rub would be. But again, I'm not a lawyer and laws differ from place to place. Hopefully, local authorities could give you a more definite answer.