According to the National Grid, 10,845 properties in Bristol and Somerset are without power, while 2,256 are facing outages in Gloucestershire.
SSE, which supplies electricity to Wiltshire, said 721 properties in the county have no power.
Matt Turner from the National Grid said restoring power this weekend has been a “mammoth task” for engineers.
“We are very aware that some customers have been without power for longer than they’d hoped.
“The high winds can really present challenges when making repairs and the treacherous conditions do make things a little bit more difficult.
“Coupled with the volume of incidents, it can be really challenging.”
Since the storm began on Saturday morning, the supplier has restored power to about 750,000 properties across the South West.
“We’re under no illusion that there’s still plenty more to do, but our teams will be out there again today working around the clock,” he added.