HELP! power outage problem

I have a panasonic 3 handset dec 6.0 model kx tga101s. My phone line goes dead when I have an electrical power outage. I don't expect the panasonic phone to work. However a plain old-fashioned corded phone doesn't work either...it's dead as a door nail. Everything works fine when electrical power is restored.
At&t has no suggestion for fixing the problem, which is particular irksome for me because my burglar alarm siren sounds every time the phone goes dead and the power company where I live seems to have a hard time keeping the power on.
Please, can someone help?