The 28-year-old secured a loan move to Spain in January following a disappointing stint at Villa Park, where he was overlooked by manager Paul Lambert.
The Scot insists he is relishing the opportunity to start over after his frustrating time with Villa, particularly in the Primera Division.
"I can savour the frustrations I've been through," Hutton said in The Sun.
"I was training with the kids at Villa and thought 'What am I doing here?'
"To go from feeling as if I was wasting away at Villa to be playing Real Madrid the next minute makes it more special."