So, you’re saying we should size cables for the rating of the inverter, not the battery capacity?
Exactly. Think of this for a moment: Utility companies are busy along with corporation building giant capacity lithium storage facilities for nighttime energy demand. Ask yourself this: Do they use cables and wires many feet in diameter because the battery facility is really really big? No they do not.
The size of a wire in cross sectional area (AWG or gauge) and material determines the properties of resistance, voltage and current carrying capacity. Think of voltage as the force or impetus push for current to travel along the path of the wire. And think of current as the movement of electrons from the positive pole to the negative pole down the wire.
The inverter itself determines by its composition the amount of voltage it can receive as well as the amount of current that it will allow to flow into and out of it both to loads and batteries (if it play a role in charging those batteries).
The size of the battery bank you choose to install depends on factors such as your daily and peak load demands and the length of time you desire to be able to provide power to loads without a grid, generator or PV source. Of course you cannot place a load upon the inverter in excess of its rated output.
But you cannot achieve a rated output of the inverter unless you have the source of power, either the grid, generator or batteries. For example, suppose you have 1 single lithium 5.1 kW capacity battery. So you have 5,100 watts (assuming discharge to 100%) of power at an average of 51 volts. This amounts to 100 amps of current flow. Now you decide to use your toaster oven at 400 degrees plugged into a normal household outlet of 15 amps at 120 volts. The toaster oven draws 1600 watts of power. So 1600/120 = 13.33 amps. Now in this case, the current draw is well within the capacity of the battery to discharge (most server rack batteries are 50-100 amps per hour of discharge capacity). Your toaster would operate in this situation for 3.18 hours before the battery is depleted.
Now, think of having multiple loads operating with the single battery. Once those loads exceed the discharge capacity of the battery, either 50 or 100 amps per hours, the voltages will drop and inverter output will fall short of demand.
So you increase the number of batteries to 10 in parallel giving you a charge and discharge capacity of either 500 or 1000 amps depending on the model. But the inverter is a 12k watt inverter. It outputs 50 amps at 240 volts or 100 amps at 120 volts. The maximum current it allows is 250 amps in or out of the pack of batteries. So again, your batteries under full load would last 2-4 hours depending on battery spec at maximum output loads and demand. But the wire size only needs to allow for 250 amps of current flow into and out of the battery bank and 50 amps to the load panel at 240 volts which is 6 gauge wire.