While it sounds strange, you may save money, even if your bills go up! In a period of regular price rises, switching often doesn't mean you actually pay less. If you're saving 10% when all energy prices are increased by 20%, you'll still pay more than you were, but you'll be paying less than if you hadn't switched.
The other complication is the way energy firms set direct debit levels - often direct debit amounts aren't linked to the amount of energy you use. When you move to a new supplier, it has little usage history for you. So sometimes it sets the direct debit amount over and above what you're using, to make sure it’s covering itself.
Don't use the direct debit amount as an indicator of what you're saving - it could be higher than before, or your old firm may have set your direct debit too low (if you had to pay a lump sum when you switched away, then that'll be the case).