That will happen. The power companies and the local governments are looking for excuses to jack up the rates.
I have no problem with government charging a road tax to users of electric cars. If those of us who use gasoline are subject to a road tax for the same reason then those who drive electric cars should also be expected to pay for the roads as well.
Unfortunately, all of us will be expected to pay for higher rates for power because of these toys. I doubt the electric companies will distinguish between EV charging and other types of electrical use.
The power companies want to build more power plants and sell more juice.