The Smart Tray simply needs any 12 volts, you can use it from a power supply or with the adaptor, it makes no difference at all.
The Smart Tray "needs" power because it is actually doing something. It measures the voltage and has powered "Cut off" so each cell stops discharging. It's very different from most of the trays that are "out there".
Never tested the fridge ideas, but in theory it's for long term storage, I don't know anything official about that though.
Novak Electronics Tech Support