My local water district here in NorCal raised it's rates 2 years ago during the height of the last drought as well...
Nest has a program that gives the power company a small amount of control, in return for which you get a bill credit (and a free or discounted Nest).
https://nest.com/energy-partners/#rush-hour
And, of course, there's peak pricing, which is more what you're suggesting. But power companies are a regulated monopoly, so there are typically restrictions on how much they can increase the price. If they can lower peak consumption within their regulated prices, they can defer on infrastructure investments and thus save money.
Diesels are the industry term for small generators that can start within a few seconds during high load. They can run on diesel or natural gas.