Energy Secretary Ed Davey said he wants to bring Ofgem in line with other regulators such as Ofcom and the Financial Services Authority, who can ensure compensation goes to customers.
Currently energy suppliers voluntarily give cash to consumers following errors - but there is no obligation - while Ofgem has been able to fine companies, but that money has gone to the Government.
The redress could include pound-for-pound compensation, goodwill payments or public apologies.
More top news
Police in Beirut are investigating the sudden death of British embassy worker Rebecca Dykes, government sources said.
A tip from the CIA was sufficient to track down and detain a group of suspects planning to bomb the Kazan Cathedral and other sites.
The 12 Years A Slave director is filming an artwork as a 'lasting memorial' to the Grenfell Tower fire tragedy.