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.
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Sir Bruce Forsyth, Sir Cliff Richard and the Prime Minister have been among those paying tribute to Cilla Black following news of her death.
TV favourite Cilla Black, who has died aged 72, gave viewers a "lorra, lorra laughs" during her long career.
Some rain in the north. Very warm sunshine in the south