The more variability there is in the grid, the more the grid need to invest in balancing, reserve energy and transmission. Each of those are expensive to build out and maintain. They are usually paid through taxes and grid fees, and here in Sweden you will generally pay more for those than for the energy that you consume.
Any saving on the production side will only effect part of the bill, and the total bill can go up even as the average wholesale price goes down.
I was more thinking of increasing throughput on existing power lines. That's a lot of work/money that'll need to be spent if we want to switch cars and heating to electric.
A house might have a typical peak power demand of 1kWH. Now? It might peak at 10. I'm making up these numbers by the way.
Everywhere that I know of, you pay for the grid through your bill.