Utility companies will lower the voltage that is supplied in times like these and hope for the best. They sometimes do it grid by grid in an attempt to conserve energy so that equipment doesn't blow.
Typically voltage to most outlets are 110volts +/- 10% (min.99v-max.121v) Sometimes it can drop even lower in times like this and in doing so the compressor strains itself because it is pulling electric as hard as it can to attempt to keep the compressor running. They do have overload protection but they don't always work. Sometimes from pulling so hard, the wires/components can seriously overheat and just melt away and burn the contacts and sometimes worse.
If you have a 220v A/C unit and a transformer blows on the pole, then you can lose power to one side of a plug and the compressor won't be able to turn on because it will only have 110v.
So if your A/C isn't or can't opperate with the compressor on, then it is best to shut it off because you can do more damage to it if it isn't already done and try in a few hours.
110v units can also lose a "leg" (of power) and not work properly if the neutral leg from the transformer is blown but if it is digital it may still display power and just not work. I don't think you can damage it tho unless (in the case of old, faulty or bad wiring) the ground is used to complete the circuit which can be dangerous.
