Storing it is extremely expensive, at least with any currently available technology.
This is apparent looking at how much energy prices change throughout the day / month based on usage and cost of production, and how predictable those changes are.
If storing energy were cost effective you could make a killing buying low and selling high on the national grid.
This giant (world's largest!) and very expensive thing provides short term-relief against grid outages. The batteries last for ~1 hour and it can still only provide electricity for something like 50k homes.
(It's used for more economically useful things, like stabilizing the grid in emergencies.)
The periods between high and low/zero wind are in the order of 200-300 hours (at least where I live).
> For one common naive case: Storing the energy in Li-Ion batteries in e.g. a Tesla Powerwall: $437/kWh.
How did you come up with that number? Total cost of the battery divided by its capacity?
Since you do not dispose of the battery upon first discharge, the true cost should be amortized across the thousands of cycles it would go through during its lifetime.
Batteries are still far from economical in many situations, but there are many situations where they now make sense. They are also getting cheaper all the time, and as they do, so will the range of applications increase.
Texas was 5 minutes away from a total grid failsafe collapse if the grid frequency had continued to drop, and it just happened that pretty much all of North America was undergoing high usage. Uncontrolled energy are not sustainable at scale.
Wind $26-$54 Nuclear $129-$198