Operations director for National Grid Stephen Joseph said teams were making “significant progress” in restoring power supplies despite the challenges.
Engineers from as far as Lincolnshire have been sent to the county to assist, the BBC understands.
Joseph said: “Since the height of the storm, we’ve restored power to over 53,000 customers across Cornwall, which is a massive effort from the hundreds of engineers, not just from the Cornwall area but from across the midlands, Wales and the wider South West.
“We are still seeing the challenges from the incredibly high number of trees fallen down across our network.
“We did a significant amount of tree cutting yesterday and throughout the night to ensure the safety of our people [which] remains a priority to us.
“As those trees are being removed, we are getting our customers restored as quickly as possible.”
He said teams were still finding issues across the network, including fallen trees and broken cables.