Scottish and Southern Electricity Networks (SSEN) said that Storm Malik caused “significant damage” to its network and had left around 18,000 customers without supply as of Sunday at 11am.
Across the UK, around 80,000 customers have been affected by power cuts.
Rural Aberdeenshire is among the hardest hit areas and some customers in Angus, the Highlands, the Moray Coast and Perthshire are still waiting for supplies to be restored.
SSEN warned that the double blow of Storms Malik and Corrie could mean that some customers, particularly those in rural Aberdeenshire, may have to wait until the early part of next week before supply problems are fixed.
Richard Gough, of SSEN, said: “Our teams continue to make good progress restoring power to customers impacted by yesterday’s extreme weather event, with a further 5,000 customers restored overnight and into this morning.
“Whilst we expect to restore power to a significant number of homes still impacted by Storm Malik throughout the course of today, due to the extent of damage caused, coupled with the expected impact of Storm Corrie, we expect the full restoration of customer supplies from both storms to extend into the early part of next week.
“We are therefore reminding all customers who remain off supply that they may want to consider making alternative arrangements, where possible.”
The Met Office has issued an amber and yellow weather warning for wind across northern parts of Scotland from Sunday into Monday morning.
It states that “flying debris is likely and could lead to Injuries or danger to life” while there may be some damage to trees and buildings.
The Met Office said: “Storm Corrie will bring a spell of very windy weather later today and during Monday morning, particularly across northern parts of the UK.
“Damaging winds are likely to lead to travel disruption in places.”
Storm Corrie will likely cause some travel disruption and generate some “large and dangerous waves around the coasts”, the forecasters add.