Properties in Leeds, Wakefield, York, Bradford, Halifax, Doncaster, Hull and the Yorkshire Dales have been affected by the outage, according to Northern Powergrid.
Engineers are on site at many of the locations, some of which are affecting more than 300 properties in a single area.
More than 110 homes in Leeds' LS19 area have lost power, along with a further 50 properties in the LS24 postcode.
Elsewhere in West Yorkshire about 120 homes in Halifax and Bradford also suffered a power cut after "an unexpected problem with electricity cables" serving the area.
Gusts are forecasted to reach up to 50mph.
North Powergrid said on Twitter it is working to resolve the power problems in Doncaster, where dozens of homes have also been affected.