SA Power Networks says about 12,000 properties in Adelaide remain without electricity.
They lost supply when trees and branches fell across power lines in severe wind gusts in the past day or so.
Repair crews are continuing their efforts to reconnect the affected properties in Adelaide suburbs and foothills areas.
Paul Roberts from SA Power Networks says the damage to lines is extensive and there is a chance some people might spend another day without any electricity.
"We're not talking minor repairs here or minor restoration jobs, we're talking about major rebuilding of sections of the network which have been smashed by trees and tree limbs brought down," he said.
"We're going to be trying everything we can to get it back on today [but] we can't make that guarantee yet.
"It will really will depend on progress and we'll certainly update people during the day."
Mr Roberts says groups of affected customers are being dealt with first, which might mean single customers whose supply is out, when neighbours are unaffected, will face a longer wait.
He says that is due to the sheer volume of reconnections needed, as it has been the most extensive damage in three decades.
Blackout compensation should increase: Darley
Independent John Darley says improved compensation is needed for people affected by blackouts.
He says the current Guaranteed Service Levy payment provides $90 if power is out for more than 12 hours, $140 if it exceeds 15 hours, $185 for more than 18 hours and $370 for more than a day.
But Mr Darley says there are many exclusions for such things as transmission or generator failure.
"The exclusions for payments for service interruptions is so broad that SA Power Networks could argue that any disruption in service is caused by a transmission failure," he said.
"The compensation should be increased. These days, with the cost of living increasing, the price of food and all that sort of business, plus the inconvenience, I think they should be adjusted."
The MLC said he would ask the Essential Services Commission to better define when payments were to be made.
He also plans to put fresh legislation to Parliament after a previous bid failed to get support from either Labor or the Liberals.
© ABC 2014
16:23 EDT Tropical Cyclone Kate has only caused minor damage on Cocos Islands after passing to the north of the tiny community, more than 2,000 kilometres off the Western Australian coast.