Regulators should refer the "broken" energy market for a full competition inquiry after figures showed the Big Six companies received more than 5.5 million complaints last year, watchdog Which? has said.
The consumer group said the Office of Fair Trading and Ofgem should refer the market to the Competition and Markets Authority for investigation into whether competition was working for consumers and how it could be increased to deliver trust, transparency and fair prices.
Which? has launched a campaign calling for "six fixes" to address trust in the market after the first full year of official figures from energy companies revealed they drew 5,579,665 complaints from customers.
Complaints ranged from issues with bills and metering to customer service, switching and payments.
Of the 5,579,665 complaints in 2013, Npower received 1,383,650 - the most of all six companies.
They were followed by EDF with 1,240,005, British Gas with 1,235,550 and E.ON with 929,230.
SSE and Scottish Power received 482,582 and 308,648 complaints respectively.
The fourth quarter of last year drew the highest total number of complaints, when 1,492,065 were received.
Which? said the latest complaints figures from the six companies - who handle 97% of the market - were "yet another indication that the energy market is not working for its customers".
The watchdog's latest figures show that eight in 10 consumers are worried about rising energy prices and just one in five trust gas and electricity suppliers.
Which? said it wanted to see an industry "that works for customers as well as shareholders, where prices are kept as low as possible and people can trust companies to give them a good deal".
Aside from the competition investigation, the watchdog's new campaign calls for energy companies' supply and generation businesses to be separated and for simple pricing and swifter switching to be introduced to help consumers find the best deal.
It said the Government must control costs added to consumers' bills while energy companies must make sure that charges for not paying by direct debit are cost reflective and should not hold onto high levels of customers' cash when they're in credit.
Which? executive director Richard Lloyd said: "Millions of people are unhappy with the service they receive from the suppliers which, combined with low levels of trust, is yet more evidence that more must be done to fix the broken energy market.
"Next month we want the regulators to refer the energy market to the Competition and Markets Authority and launch a full scale inquiry. This is the first and most important step towards a more radical reform of the energy market, giving hard-pressed consumers the confidence that they are paying a fair price."
Npower said: "We know we let many of our customers down following issues with our new billing system and, right now, our priority is to put these issues right.
"Since we issued a public apology to our customers at the end of last year, we have been reporting to Ofgem. We are making good progress and will soon be able to give an update on where we have got to. Looking at Which's proposals, there are many elements that we support and several that we are already doing."
A spokesman for Ofgem said: "The number of complaints is far too high and this reflects both poor company performance in 2013, especially from Npower, and low levels of trust. The onus is now on suppliers to use our reforms to re-build customer trust.
"Customer service is extremely important and Npower's performance is of deep concern to us. We have made it clear that Npower must improve its record on complaints and we have ordered them to report their progress to us regularly. Npower has accepted its failings, apologised to its customers and committed to making sure that no customer will be left out of pocket.
"We haven't ruled out further action on Npower complaints. We take supplier performance on complaint handling very seriously and in 2011 we fined British Gas and Npower £2.5 million and £2 million respectively for poor performance in this area."
He added: "We are already putting in place the most radical reforms to the energy retail market since competition began to make it simpler, clearer and fairer for consumers. As part of this, we have set suppliers enforceable standards of conduct so that they treat customers fairly. We therefore expect suppliers to raise their game on customer service, and if they don't, our reforms are making it easier for consumers to vote with their feet and switch.
"Together with the Office of Fair Trading and the Competition Markets Authority we are carrying out the first annual assessment of competition in the energy market. The assessment will be completed at the end of March and we will report the findings shortly after."
Trade association Energy UK said: "No industry wants unhappy customers but when you are dealing with nearly 27 million customers things can occasionally go wrong. Energy companies put a great deal of effort into resolving any problems their customers may have and most complaints are fixed within a couple of working days and with no more than a phone call.
"Energy companies are already making a number of changes which put customer service at the heart of what they do. They are making it easier to compare and find the best deal available - most recent figures show that around 1.3 million customers switched in the last quarter of 2013 alone. They are working to make switching easier and quicker and new rules also mean the industry is cutting the number of tariffs and making energy bills clearer to understand.
"No one should be afraid to heat their homes and help is always available. Our members want to make sure people get the support they need. There are many schemes helping consumers make their homes more energy efficient, cut usage and keep warm. If you are worried about your energy bills you should contact your energy provider directly or speak to the free, confidential Home Heat Helpline on 0800 336699."
A spokesman for the Department of Energy and Climate Change said: "Energy bills are a big concern for many people, which is why we've been working to reform the energy market, increase competition to keep prices down and make it easier for people to shop around and switch supplier.
"Changes to the Energy Company Obligation mean customers will save £50 on average on their bills and by extending and simplifying the Green Deal we will help more families to have warmer, more energy efficient homes and lower bills."
Shadow energy secretary Caroline Flint said: "This is yet more evidence that Britain's energy market is not working for families and businesses and is causing a cost-of-living crisis. Under David Cameron people are paying more than ever for their gas and electricity, but all too often the customer service they receive is not good enough.
"That's why the next Labour government will undertake the biggest overhaul of our energy market since privatisation. Our plans will break up the big energy companies, put an end to their secret deals and make tariffs simpler and fairer. And until these reforms kick in, we will put a stop to unfair price rises by freezing energy bills until 2017, saving the average household £120."