Well, no.

First, increased demand drives increased prices. This is the least controversial axiom of modern economic theory. So if you add a huge power consumer to a market, all consumers in that market will have to pay more. You can mitigate that some if that new, big consumer builds their own power facility, but the fact still remains that the local price in fuel (oil, coal, etc) or materials for renewable generators (turbins, solar panels, etc) will increase. Again, because demand increased.

Second, its one thing for things to cost more in a market that has a booming economy and plenty of high paying jobs. Home prices in the Bay Area are horrifying, but the poverty line for a family of 4 is $80k, which sort of grounds things. If energy costs go up by $100/year in the Bay Area, nobody notices. But if energy costs suddenly skyrocket in Great Falls, Montana (poverty line for 4: $33k) or similar that lacks a vibrant economy, the residents don't have much choice but to tighten their belts over the suddenly larger electric bill that has done basically nothing to actually revitalize their economy.