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The increase should be less than your gas bill. If your utility burns gas to generate your electricity it'll use less gas to heat your house with a heat pump than you'd use in a gas furnace.

BTW gas furnaces don't work without electricity so switching to electric heat reduces you from two points of failure to one.

Here is a source showing the electricity generation profile for each state:

https://nei.org/resources/statistics/state-electricity-gener...

Gas heat still needs electricity to work.