The festival promoter told The Independent that it was not fair for licence fee money to be spent on the free event.
"In a special time of austerity, I'm sure that taxpayers don't want to see their money spent on festivals," Power said.
"If the BBC is giving something out for free, then we can't compete. It's really p*ssed me off. A lot of people aren't willing to put their head above the parapet, to say that this isn't fair. It isn't right."
Promoter Harvey Goldsmith added: "Competition is competition. I don't know why the BBC is doing this, but that's life."
Radio 1 boss Ben Cooper told the Evening Standard: "We're going into an area that I don't think any commercial operator would have gone into after the unrest of last year.
"That is the job of the BBC. I don't think we are taking ticket sales from competing festivals because people aren't being charged."
Of the inclusion of American Jay-Z and Barbadian Rihanna on the bill, Cooper added: "We feel this is a special year, an Olympic year, so we wanted to get the best artists in the world, as well as British artists.
"I think, why can't young people have their culture of popular music celebrated in a way that is of scale?
"Why can't young people in this Olympic year have an event that they can attend and they can enjoy and feel is for them?"