Magic or Madness by Justine Larbalestier

Reason Cansino has spent her whole life on the run from a grandmother who supposedly practices black magic. Now, she is forced to live with her grandmother and learns that she herself has inherited magical powers. But she faces a terrible dilemma. If she practices magic she will shorten her life by many years. If she doesn't use her magic powers, she will go mad. Determined to find a way to avoid either fate and to escape her grandmother, Reason steps through a portal which leads from sunny Australia into wintery New York. Only now she is being pursued not only by her grandmother, but also her grandfather who seems much, much worse.

This is a wonderful book, first in a trilogy.