I have seen where BA have not been maintained and the valve regulated lead acid battery (VRLA) has failed that the alarm sounds with a power failure, as two batteries one in alarm and one in bell box, and loss of power causes bell to sound.
However in real terms unless connected to a call centre the alarm is next to useless, and if connected to a call centre it has to be on a contract and maintained.
My son for a time worked for an alarm company, if the alarm sounded the call centre would phone him, and he had to attend, he was to observe from outside, and if he thought there were intruders call the police, in the two years he worked for them, he never found an intruder, he didn't want to for good reason. And he was unlikely to arrive within 15 minutes, more like 30 minutes by which time well gone.
It seems if the alarm was designed with zones, and multi zones were activated police could be called, but the days of alarms connected direct to police stations are well gone, the call centre has to phone police, to stop nuisance calls.
However they do give some piece of mind when at home, that bump or bang down stairs you realise unlikely to be an intruder as alarm has not sounded, saves picking up some thing heavy and showing how brave you are to your wife.
But the power used by an intruder alarm is very small, and can come from any convenient point.
I have only ever had an extruder alarm, designed to tell us when mother had gone on walk about, so we could persuade her to return.