EMB145 Captain reports wing anti-ice failure upon entering clouds at FL190. A request to climb to FL210 is denied by ATC. The Captain starts climbing and informs ATC of his actions.

Narrative

We were given clearance to FL190. We were also given clearance to deviate in whichever direction we needed to maneuver around multiple thunderstorms. The First Officer and I agreed an the best path around the weather and set the course. As we approached an area of rapidly rising clouds; approximately FL200; the First Officer requested a climb to FL210. Center said that they were unable FL210...FL 190 would be our final altitude. Immediately upon entering the clouds we encountered moderate icing and moderate turbulence. The aircraft ice system activated...all but the wing anti-ice. With the red 'anti-ice fail' and amber 'wing anti-ice' fail on the EICAS; I asked the First Officer to request FL210 due to anti-ice failure. ATC rejected our request for higher and said lower was available. I began the climb to FL210. Exercising our Emergency Authority. The First Officer informed them that we could not descend because of the weather below and we were initiating a climb to get out of the cloud layer and icing. He also informed them that FL210 would put us safely in the clear. At that time we were cleared to FL210. At FL200 we were on top and proceeded to run the checklists for the wing anti-ice failure.