When mentioning people, animals, or places after a verb, we use "a", so you would say "vive a Madrid".

Another example is "I call my mother" - "Llamo a mi madre".

The "a Madrid" here is not the "a personal" mentioned. The "a" here is implying it's a destination, hence "to" as she does not live there yet. Once she's there, "Angela vive en Madrid" will make sense. 

That "a" has no name, but I call it "Moving A" in class, it's like the "a" that lives with ir - Ir "a". and it refers to -go "to"- Not to be confused with the "Personal A" which we use to interact with creatures like pets and people.