On Friday, the storm brought high winds of up to 61 mph and pouring rain across southern England.
It also caused disruption to travel and power lines in the south and on the Isle of Wight.
Outage coverage provider Green Flag has forecast nine outages per minute on UK roads between Friday and Monday.
Mark Newberry, commercial director of Green Flag, said he expected more calls on Saturday, with outages increasing 10% from normal.
“Due to these weather conditions, we urge drivers to remain cautious and carry out the necessary safety checks before leaving for their journeys,” he said.
RAC Breakdown and the AA have also warned drivers to be extra careful over the weekend as some roads could become flooded.
RAC Breakdown spokesman Rod Dennis said: “Storm Alex will make road conditions miserable, if not downright dangerous for drivers by the end of this week, and they will have to be prepared for an ugly mix of surface spray, gusty winds and more than likely. some disturbances on the roads.
“Flooding is also a possibility, so drivers should remember to never attempt to cross the water unless they know for sure that it is shallow enough. “
The AA advised drivers to make sure their lights and wipers are working and the tires are in good condition before setting off, allow extra time for slower-speed trips and find an alternate route if they are faced with flooding.
Scottish and Southern Electricity Networks has experienced unplanned blackouts in more than 20 regions, including Portsmouth, Southampton and towns east of Reading, according to the customer’s interactive map.
The company said this was to be expected, however, calling the shortage “low impact”, adding that the cuts were concentrated in small areas, with power lines “generally holding up very well.”
Western Power Distribution has also recorded incidents in the southwest, affecting Devon, Cornwall, Dorset and Somerset.
Energy Networks Association, the industry body for gas and electricity providers, said the storm had not caused “significant disruption” but was monitoring the weather “very closely.”
Met Office spokeswoman Nicola Maxey said the highest wind speeds of 61 mph hit Berry Head in Devon and the Isle of Wight on Friday morning, making power lines “susceptible to damage “.
She added that power cuts were more likely in early fall because the trees still have their leaves, which means branches are more likely to fall in strong winds and disrupt cables.
Orange weather warnings for the heaviest rain likely to disrupt transport are in place for parts of Wales and parts of the West Midlands and south-west England until 6 p.m. Sunday.
The gale force winds brought by Storm Alex were expected to subside by the end of Friday, but yellow weather warnings for rain will remain in place for most of the UK over the weekend.
Ms Maxey said: “A second rain front comes to replace Storm Alex over the weekend, pushing from the east on Saturday morning and affecting the western areas later.
“The rain is slowly pushing north, but it will be relieved as it goes.
“On Saturday most of the country will be affected by the rain, and it seems the heaviest in the center, along the backbone of the country, from Aberdeen to Bristol. “
Autumn temperatures of 10-15 ° C are forecast over the weekend, with the coolest temperatures expected in Scotland.
Carol Holt, head of flooding at the Environment Agency, added that “widespread and persistent rains are likely to cause flooding” in some areas over the weekend.
Ms Holt said: ‘We urge people to stay away from swollen rivers and not to drive in flood water – it is often deeper than it looks and only 12 inches of water. current are enough to float your car.
Additional reports from the Press Association