Substations convert and distribute 220 to your neighborhood, from high tension wires from the power plants.
Line voltage may even go up from 110
What are you, 200 years old? How are your carbon filament lights doing these days?
125V and 240V+ is typical US line voltage. If you're seeing 110V and 220V, your (knob and tube?) wiring is ready to catch fire...
Batteries are one method, but flywheels can work well, too. They could spin up a flywheel to consume the excess energy, then release it later as-needed (e.g. a dark cloud). In fact, they can spin up a flywheel at nighttime, too, when they have excess production, to smooth out daytime use
Bull. Shit. Flywheels store power for a few short MINUTES. There will be no energy left an hour later.
There have been promises of long-term flywheel power storage for many years, decades even, but they've never been able to get those magnetic bearings to work, due to pesky little eddy currents. Without that, you've got a big heavy wheel spinning around at high speed on ball bearings. Shut off the power to your clothes drier, and measure how long it keeps spinning... That's about how long-term flywheel energy storage works here on Earth. Go up into space, where there's no gravity or atmosphere, and it does a better job.
this infrastructure will smooth out their plants for normal use, too.
It could, but they'd be stupid to use it for that purpose. Grid losses average 7%. You get that much loss out of the AC-DC then DC-AC conversion, and possibly many times more losses out of the charging/discharging cycle losses of any battery chemistry... Meanwhile, speeding-up/slowing-down a turbine has little or no loss.
Localizing the storage is far more efficient than sending it hundreds of miles,
No, storage is vastly LESS efficient, and that's even if you pretend the infrastructure and maintenance is free. See above.