SWP also said a car had been caught in flood waters between Capel Gwilym Road and Thornhill Road, north Cardiff, urging motorists to avoid the area and use alternative routes where possible.
Meanwhile, three schools in Powys closed for the afternoon on Wednesday because of the weather.
National Rail warned passengers of disruption across the TfW network on Wednesday.
It said: “Your journey will take significantly longer than normal.
“Please check your entire journey is still possible before setting off today.”
TfW, external also said multiple lines were flooded, including “major disruption” in mid and north east Wales, as well as delays in south Wales near Cardiff.
It added that flooding has led to the lines between Shrewsbury and Chester being blocked, with “a very limited” replacement bus service running.
Lines between Wrexham Central and Bidston, Merseyside, are also cancelled due to flooding, with disruption expected until 23:00 on Wednesday, while there are also cancellations between Hereford and Shrewsbury, TfW said.
North Wales Fire and Rescue Service said it received numerous reports of flooding in the Flintshire and Wrexham areas and warned people to keep away from fast flowing watercourses.