Alarm turns on after power failure
Most alarms are plugged into a transformer, which is plugged into the wall. Easiest way is to find that, and disconnect it. 





If there is no transformer, and it is hard-wired directly to mains power, then you have a couple options:


If it's on its own circuit, turn off the breaker (or disconnect fuse) 
Turn off the breaker to it. Open the panel, and then disconnect the mains wires, and cap them off with wire nuts, before turning the breaker back on. The wires should look like your regular house wiring (in contrast to the small cat3 wiring the alarm sensors use). 


It could also be hardwired to an external transformer (like a doorbell is), though I've personally never seen an alarm installed that way. If so, same options as before: 


If it's on its own circuit, turn off the breaker (or disconnect fuse)
Disconnect the wires from the transformer, or inside the panel. They should be black and red or black and white, and marked something like "Pwr in" or "12V DC" or "24V AC". In the pic below, the power wires are the thicker wires on the left side: two go to the battery, and two go to the transformer/mains.