A furious row has broken out today between the energy regulator and the "big six" power companies after Ofgem in effect accused the utilities of profiteering by increasing their profit per dual-fuel customer by 733% – from £15 to £125 – through a slew of price rises.

Alistair Buchanan, the Ofgem chief executive, said a "radical break" was needed in the way that energy companies operated to rebuild trust among householders and other power users.

"When consumers face energy bills at around £1,345 [per annum] they must have complete confidence that this price is set by companies competing in a fully competitive market. At the moment that is not the case," he said.

Ofgem said it was determined to ensure that companies provided simple tariffs and clearer bills in the future to help consumers baffled by the 400 different tariffs currently provided.

The latest initiative comes after a hail of criticism of the big six at the political party conferences amid rising concerns about the number of householders falling into fuel poverty.

But the figures used by the regulator were immediately criticised by the energy firms, which believe that Buchanan is responding to what they see as unjustified political pressure.

Phil Bentley, managing director of British Gas, said Ofgem's report was misleading. "Their methodology is flawed, excluding, as it does, the discounts we give our customers and the benefits they receive from fixed price contracts, as well as understating our commodity costs.

"In 2010 alone, this methodology overstated industry profits by 100% compared with Ofgem's own analysis of audited accounts."

Volker Beckers, chief executive of RWE npower, said his company made just £1.50 profit on every £100 spent, while making a loss per average customer between 2004 and 2009.

"These are not the figures associated with an industry that is profiteering or uncompetitive, and despite this challenging financial climate, RWE still managed to invest over £1bn into new, more efficient energy infrastructure for the UK in each of the last three years," he argued.

But the energy companies are fighting a wave of attacks over the way they have been operating with fines for doorstep mis-selling and criticism over the slow speed with which they respond to a fall ion wholesale prices.

Yesterday, an undercover investigation by the consumer body Which? revealed that the number of energy tariffs available to householders is so vast, and the options so complex, that staff at energy companies have no idea which is the best deal.

Which? called each of the six major energy suppliers 12 times in a week to get advice on the cheapest deal. Despite being asked clearly for the lowest cost option in each case, in nearly a third of the calls the firms failed to offer their cheapest tariff. Staff also gave questionable advice about potential savings, cashback deals and fixed prices.

Britain's energy minister, Chris Huhne, pledged last month "to get tough with the big six energy companies", while Ed Miliband, the Labour party leader, pledged to abolish a "rigged" energy market that has allowed the companies to achieve market dominance.

Ofgem has already threatened energy companies with a formal referral to the Competition Commission if they do not transform their pricing structure to stop confusing customers.

It has promised to come up with detailed proposals for reforming the energy market for business customers before the end of next month, further discussions in December on how to bring new entrants into the market and in the new year will reveal the findings of an independent inquiry into energy company accounts.

Buchanan is aware that a looming £200bn investment drive to green the UK's energy supply would only put further pressure on bills, adding further momentum to the need for reform.

"That is why a radical break with the past is needed. Ofgem's tariff reforms offer the quickest way to create a market where consumers can have confidence that prices are set by effective competition. Suppliers have told Ofgem they want to restore confidence in the industry and now they have the chance to do so."