In Northern Ireland, weather warnings have also been issued as the third storm of the season, Storm Clodagh, moved in.
The Met Office issued a yellow warning for severe gales with gusts up to 60mph (100km/h) for exposed areas.
A yellow warning means plan ahead and consider possible travel delays and disruption.
Wind and heavy rain across Northern Ireland is making driving conditions hazardous.
Police have advised drivers to slow down, use lights and drive with extra care.
As the weather deteriorated, a Christmas market in Magherafelt, County Londonderry, had to be cancelled as gales ripped through the stalls.
A tree fell on a house in Drumbeemore in County Armagh during high winds
The Irish meteorological office issued multiple weather warnings on Sunday as Storm Clodagh swept across the country.
South-westerly winds were expected to reach speeds of 60-80km/h (37-49mph) with gusts of 100-130km/h (62-80mph).
The storm is expected to move across Ireland and then on to Scotland.
In the Republic of Ireland, the areas hit by power cuts include Cratloe, County Limerick, where 449 customers were affected; Buncrana, County Donegal where there were 269 outages and 185 customers were without electricity in Clonminch, County Offaly.
More than 45,000 people were left without power when Storm Barney hit Ireland less than two weeks ago.

The Italian authorities concluded Claire Martin killed herself after suffering from post-natal depression.
Her parents, from Sutton-in-Ashfield in Nottinghamshire, believe she was murdered because "things don't add up".
Ashfield MP Gloria De Piero said there are "anomalies to explore".
Ms Martin, 30, lived with son and her in-laws in Grottaminarda, near Naples, while her partner worked in Germany.
In March 2012, Ms Martin suffered multiple stab wounds to her neck, struggled upstairs and died.
A murder investigation was launched, but 15 months later the Italian authority closed the case after it ruled she had committed suicide.
Following a recent Guardian newspaper article on her death, her father Ray Martin said: "Things don't add up.
"It's impossible to stab yourself multiple times in the neck and climb up stairs after wiping the blood clean off the knife."
Her mother Pat Martin said: "It's a continuing nightmare, nobody gives a damn."
Ms De Piero said forensic reports have been translated into English, but there are witness statements and other evidence only available in Italian.
The MP has asked for a meeting in October with the Italian government to "get the answers the Martin's need".
Mr Martin said: "We hope to meet with the magistrate, but I honestly don't know why they won't speak with us."
The Foreign Office said it would not comment on individual cases, but has provided consular assistance to the family.
It added: "We will continue to assist in any way we can, although we cannot interfere in the legal system of another country."

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Scotland scored twice in three minutes in the final quarter to win 2-1 in Glasgow.
James Carson gave Wales the lead from a penalty corner, but Ben Cosgrove and Alan Forsyth won it for the hosts.
Earlier the Wales women's team lost 3-0 to Italy to finish fourth at their EuroHockey Championships in Cardiff.
Chiara Tiddi, Celina Traverso and Federica Carta scored the Italian goals.
"We came out fighting for the bronze today but just couldn't hit the line," midfielder Sian French told BBC Wales Sport.
"We'll keep on pushing as a team for the next two years and we'll come back even stronger next time."
Because two teams get promoted from the B Division, Wales's men will compete in Europe's top tier at the next EuroHockey Championships in 2019.
On Friday, Wales women's hopes of rising into Europe's top tier were shattered in a 2-1 defeat by Russia.
Wales' men secured promotion to their top tier by beating France 4-3 in their semi-final in Glasgow.
Wales women had beaten Austria and Poland, but lost to Belarus on the way to the semi-finals.
Neither Wales side has been in the EuroHockey A Division since it was reduced to eight teams more than a decade ago.
The nation's women were playing in the third tier just four years ago.
Up to 3,500 people have been left without electricity in heavy storms in the Republic of Ireland.
[NEXT_CONCEPT]
An MP has written to Prime Minister David Cameron in her bid to establish how a suicide verdict was ruled after a British woman died in Italy in 2012 from stab wounds to the neck.
[NEXT_CONCEPT]
Wales men's hockey team suffered an agonising late defeat by Scotland in their EuroHockey Championships II final.