Grid connected battery

Hello everyone,

I am currently comparing an off-grid battery and a grid connected battery. For the off-grid connected battery, I am allowing it to charge from the PV aswell as discharge at all times. For the grid connected battery, my strategy is to "shift" the purchased electricity to occur during the night, when the electricity is cheaper, rather than during the day. To do this, I am allowing the battery to charge from the grid during the night but not discharge. By doing this, the battery gets charged up during the night and it can then discharge during the day which will reduce the amount of electricity needed from the grid.

However, when comparing the total amount of purchased electricity between the off-grid battery and the grid connected battery I notice a slight increase in pruchased electricity for the grid connected one. Of course, I am purchasing electricity from the grid to charge the battery, but shouldn't this purchased electricity simply be "shifted" from day time to night time, i.e the annual purchased electricity should be more or less the same? Why am I purchasing more electricity with the grid connected battery?

Thanks for your help.

Best regards,
Albin