The basic battery sizing formula is simple, but each input matters. Start with the energy you need in kWh. Multiply by the number of days you want the battery to cover. Then divide by usable depth of discharge and efficiency so the battery is not forced to deliver more than it safely can.
For example, if you use 12kWh per day, want one day of autonomy, use 85% depth of discharge and 95% efficiency, the calculation before margin is 12 ÷ 0.85 ÷ 0.95 = about 14.85kWh. Add a 10% safety factor and the recommended battery becomes about 16.34kWh. At 48V, that is about 340Ah.
Why usable capacity is lower than label capacity
A battery labeled 10kWh may not deliver the full 10kWh in normal operation. The BMS may reserve energy to protect cells, the inverter loses energy during conversion, and a backup reserve may be kept for outages. Temperature, battery age and high load current can also reduce available energy.
Daily energy versus backup energy
If you are sizing an off-grid system, daily energy use may include everything. If you are sizing emergency backup, do not use full monthly home consumption unless you truly want whole-home backup. Instead, add only essential loads such as refrigerator, lights, fans, router, security, medical equipment and selected outlets.