In a power outage situation, a solar + storage system will:
1. Use the generated solar power to both power the home, and fill the storage (powerwalls or other batteries)
2. If solar is enough to power home and fill batteries, then when the batteries are full the solar will shut off (because the power has no place to go), and the home will drain some energy from the batteries.
3. Once batteries are drained a bit, the solar will kick back on and fill the battery again, then shut off.
Cycle repeats until sun goes down, then batteries power home overnight, draining, till PV comes back on with the sun.