In Carmarthenshire, 793 homes were cut off along with 272 in Pembrokeshire, 207 in Rhondda Cynon Taff and 150 in Swansea.
All areas have power back, including parts of Ceredigion and Newport.
Localised flooding has been reported in Pontypool, Cowbridge and Llanishen in Cardiff, as well as on the M4.
The weather warning from the Met Office is for rain and wind of up to 70mph.
One flood warning is in place on the Lower Dee Valley in Wrexham and several flood alerts have been issued across Wales.
South Wales Police has warned of standing water on the M4 between junctions 35 (Pencoed) and 36 (Sarn) and have dealt with three crashes in the area.
The M48 Severn Bridge in Monmouthshire has been closed to high-sided vehicles due to wind, as has the Britannia Bridge at Anglesey.
Also on Anglesey, ferry services between Holyhead and Dublin have been cancelled while phone lines were brought down in Llansadwrn and Penmynydd.
The main road between Beaumaris and Menai Bridge remained closed on Wednesday after a landslide closed it on Monday evening.
In Powys, a fallen tree blocked the A40 at Trecastle, but this has now been cleared.
Natural Resources Wales (NRW) said a "significant" amount of rainfall was expected with rivers peaking late morning and early afternoon on Wednesday.
While south Wales is due to be worst hit, NRW is also expecting to issue flood warnings in north Wales.
NRW duty tactical manager Donna Littlechild said: "Anyone driving should be particularly careful as there will be a lot of water on the roads."
North Wales has suffered in recent days from previous bouts of rain with homes being evacuated and chaos on the roads.
Northern Ireland and Scotland are expected to bear the brunt of the storm.
Over the weekend the A55 dual carriageway in north Wales was shut for hours due to flooding.
Drivers were forced to abandon cars while homes at Anglesey and Gwynedd were evacuated.
Natural Resources Minister Carl Sargeant said Â£1m would be made available for local authorities to carry out immediate repairs and maintenance to river and drainage systems.
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More than 1,400 properties were left without power on Wednesday morning after Storm Frank hit Wales but they were reconnected by the end of the day.