If someone can clarify if these scenarios are possible, I would be most grateful. All these scenarios are for off-grid use with ac being only used to supplement the solar power and not to feed back to the grid.
When PV is providing power
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Use PV+Battery for all loads. If exceeding the load capacity of the inverter/battery, then pull from the ac mains seamlessly. Otherwise ac is not used.
When PV is not providing power (low light/night)
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Use battery for all critical loads. If battery depleted switch to ac. Do not charge battery from ac, but charge next day from PV.
Use ac for non-critical loads. If ac power fails, switch to battery for these loads automatically.