All modern alarm systems come complete with a back-up battery, which will continue to run the alarm system without power for many hours.
When power has been restored to the property, the alarm system will recharge the battery – in most cases to the same condition it was before the power failure occurred.
Should the alarm system be monitored back-to-base, the alarm may send a signal to the 24 hour control room that the power is off. Appropriate action can then be taken.