About 30,000 households have spent five nights without power so far since gusts of nearly 100mph hit parts of the UK.

The energy companies said they had reconnected about a million homes.

Energy regulator Ofgem said it would examine the speed of energy companies' responses and the resilience of the UK power network in extreme weather.

Business Secretary Kwasi Kwarteng said Storm Arwen resulted in the biggest disruption since 2005 and the scale of the restoration was "enormous", with the worst-affected areas in the north of England and north-east Scotland.

Mr Kwarteng said he had been assured that the "overwhelming majority" of disconnected customers would have power back by Thursday or Friday.

SP Manweb, which operates the electricity network in North Wales, said all homes had supplies restored by 18:30 GMT on Wednesday.

The Energy Networks Association (ENA) said 97% of homes that had lost power in the storm had been reconnected, with companies working at 4,500 damaged sites.

ENA director Ross Easton said they had continued to make progress overnight and restored power to 12,000 more homes across the UK, "against some challenging conditions".

"While this number is increasing all the time, the remaining 30,000 homes are in some of the worst-hit and often remote areas of the country," he said.

Some homes - mainly isolated rural homes or groups of houses - will not be reconnected until at least the end of the week, the energy companies said.

They said the damage had been "catastrophic", with 100 poles snapped in half at one site.

In the Commons, Mr Kwarteng said the storm was an event "the like of which we haven't seen for 60 years", but he said such severe weather could become more common because of climate change.

He said the Department of Business, Energy and Industrial Strategy would be looking at the lessons of the storm, to improve the resilience of power system.

He said he wanted to reassure people "who are exhausted, who are worried and who are angry" that energy companies and the government are working hard to restore power.

Welfare centres and hot food are being provided in some affected places, with energy companies working with emergency services, local councils and the British Red Cross.

But Dr Lily Fulton-Humble, who was at home near Alnwick in Northumberland with a seven-week-old baby and a sick toddler, said her family was "losing the stamina" to endure further nights without power.

"It's pretty cold - and when you're feeding a baby every two hours it's even colder," she said.

She said communication from Northern Powergrid during the outage had been "pretty appalling".

Energy companies said they were working at 4,500 sites to carry out repairs

Irene Amiet told BBC Radio 5 Live her family in Pendleton, East Lancashire had spent five nights without electricity or water - because the house was supplied by an electric pump from a well.

She said they had been trying to keep warm with log fires and were fetching water directly from the well, boiling it on the gas hob.

Her children had to go to cafés after school to charge their devices and do their homework.

The electricity company has told them their power could be restored by Friday, but Ms Amiet said they were running low on wood and temperatures were set to drop below freezing again.

Linda and Paul Dunk, who are in their 70s and live near the village of Torphins in Aberdeenshire, said they had been wearing five layers of clothes and cooking on a camping stove. They do not know when they will be reconnected.

"Slowly this building is getting colder and colder," Mrs Dunk said. "We're desperate really."

Mr Kwarteng said some customers calling the centralised 105 number for help over power cuts had faced "unacceptable" waits of up to two hours, but those delays had now been reduced to 10 or 15 minutes.

Conservative MP Richard Holden said a secondary school in his North West Durham had been without power for days while a rural surgery lost £10,000 of flu vaccines when its fridges stopped working.

He said some isolated communities had been told they could face very long waits to be reconnected and asked for reassurance that they they would all have power before Christmas.

Mr Kwarteng said it was "unacceptable" for them to be without electricity for so long, adding: "I'll do everything I can to make sure that that doesn't happen."

Liberal Democrat MP Tim Farron said some of his Cumbrian constituents had been told they would not be reconnected until 8 December, but Electricity North West said they now expected all their customers have power restored by Friday.

Customers are eligible for compensation, with payments of £70 for the first 24 hours of power loss - or 48 hours if conditions were severe - plus a further £70 for each extra 12 hours without electricity.

However, there is a cap of £700 for payouts, so customers can only get that amount even if they end up without power for a full seven days.