While the opera, based on Danish fairy tale writer Hans Christian Andersen's unrequited love for a Swedish singer, is not due to be staged until next year, Costello appeared on stage at the new Copenhagen Opera House to perform 10 songs from the work.
Costello was commissioned by the Royal Danish Opera to write a new piece celebrating the bicentenerary of the country's literary great, who wrote amongst other stories The Snow Queen and The Emperor's New Clothes.
"I wanted to find a way to connect Andersen the man to the Anglophone world, rather than Andersen the writer," Costello told BBC World Service's The Music Review programme.
"Obviously everybody knows his tales - but they sometimes know them in very bowdlerised versions. Not everybody's heard them in good and accurate translations, because they've been handed on so much since the Victorian era."
'Good songs'
The story of The Secret Arias is Andersen's "infatuation and idealisation" of Jenny Lind, the Swedish soprano.
Andersen continuously became infatuated with both men and women throughout his life, but "kept the sexual dimension at a safe distance".
"A lot of modern biographers concentrate on pinning a label on him - whether he's gay or straight - to me that's of no concern," Costello added.
"To me what's of concern is that he seemed to do this wilfully - and one of the ways that he drove himself on was to be in a constant state of emotional flux. Jenny Lind is the most famous person on who he fixated in this way."
Lind, however, only returned the affection platonically, describing Andersen as being like a brother.
One key part of the opera is Lind's trip to the US, masterminded by the showman PT Barnham.
The opera shows Andersen's reflections on this, and the secret songs he might have written for her.
Costello, who first came to prominence in the late 1970s and had his biggest hit in 1979 with Oliver's Army, sings the parts of both Barnham and Andersen.
He said he had tried to find a different music thread for each of the characters.
The company's artistic director, Casper Beck Holten, said he appointed Costello because he had written "some remarkable songs."
"If there's one thing opera's about, it's writing good songs," he added.
"He also has a very special sense of narrative in songs, of telling stories with music and words.
"Finally, he's one of the musicians in the world who has shown the broadest musical styles - so I thought it was obvious really. I was surprised no-one had asked him before."
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