Iwan Ward, a councillor for Boncath and Clydau in Pembrokeshire, said residents there had been without power since 02:00 GMT on Saturday and they had been told it would not be restored until 22:00 on Thursday.
He said some people were without phones, with more people now using digital phones which rely on wifi.
“This is the 21st Century and people are in complete blackout,” he said.
Roisin Quinn of National Grid said its engineers and support teams had been “working around the clock to maintain power across the network”.
She thanked customers for their patience, adding that “we are doing everything we can to get power restored across the region as soon as possible”.
The Porth Wen Solar Farm in Anglesey was also damaged in the storm, with repair work and the replacement of damaged panels expected to carry on into early next year, EDF Renewables UK said.
“We are currently assessing the extent of the damage and conducting a controlled clean up.
“Once the initial recovery efforts are complete and the damage is fully assessed, we will carry out a full investigation and when safe to do so, resume generation,” a spokesperson said.
