About 1,600 Kent homes were still without electricity on Friday evening after gales damaged power lines.
Waves crashed over the harbour wall in Dover during the storms
EDF Energy said it was "making good progress with the repair work" but new faults were being reported.
The storms in the South East led to ferry cancellations from Dover and Operation Stack was brought in on the M20 to back up waiting freight traffic.
Kent County Council (KCC) has set up a petition calling for an end to Stack, where lorries queue on the motorway.
The operation sees the M20 closed between junctions eight and nine, and 11 and 12, with non-freight traffic diverted onto other roads.
KCC is asking for the immediate implementation of a "Quick Moveable Barrier" scheme, which would see two-way traffic maintained on the northbound carriageway of the M20 and lorries parked on the southbound carriageway.
But the council has called on the government to fund a permanent solution - a dedicated lorry park for at least 2,000 vehicles.
Operation Stack was lifted by Kent Police on Friday lunchtime.
EDF Energy admitted some of its customers could remain without power into the early part of next week.
"Helicopters have been surveying the network to locate faults and direct engineers to them as quickly as possible," a statement said.
A seaside shelter was blown over at Hampton, near Herne Bay
There were blackouts in areas including Tunbridge Wells, Sittingbourne, Strood, Penshurst and Goudhurst.
Spokeswoman Helen Galley said: "While we're getting on with restoring power to customers, new faults are being reported to us.
"Trees damaged by the storms and left in an unstable condition are causing problems on overhead lines."
The gale force winds saw roads and railway lines blocked by debris and fallen trees.
The QEII Bridge over the Thames and the new bridge linking Kent to the Isle of Sheppey, were both closed during Thursday.