The new power cuts came just days after supplies were restored after last month's Storm Arwen caused widespread damage to the network.
Utility giant SSEN said engineers had restored power to more than 8,000 customers following severe conditions.
As of 8am, about 1,000 remained without power across northern Scotland, with 700 of these in the Grampian region.
The Scottish Environment Protection Agency issued eight flood alerts and three flood warnings.
Some rail services are also affected, with cancellations affecting a few routes in the west of Scotland and speed restrictions in force for others.
CalMac also said certain timetables had been adjusted after some ferry services were cancelled.
An SSEN Distribution spokesman said: “Engineers have restored power to over 8,000 customers following severe wind gusts and heavy snowfall brought by Storm Barra.
“As of 8am, around 1,000 customers remain without power across the north of Scotland, with 700 of these in the Grampian area.
“Our teams are working hard to restore supplies to our impacted customers as soon as possible this morning and expect to reconnect all customers impacted by Storm Barra later today.
“We are working closely with our local resilience partners to provide support to our priority service register customers and gain access to faults blocked by heavy snow and fallen trees.”
On the railways, ScotRail said services between Stranraer, Kilmarnock and Glasgow Central may be cancelled or revised on Wednesday due to the impact of Storm Barra the previous day.
Caledonian MacBrayne said amended timetables will operate on some ferry routes due to disruption the day before.
Meanwhile, Braemar Mountain Rescue Team spent 16 hours helping a walker in difficulty in snowy conditions at the Fords of Avon refuge overnight.
They tweeted: “A very challenging night at times.
“Huge thanks to the couple that found him initially and came for help.”
The walker was recovered safe and well.
READ MORE: Storm Arwen: Military being sent to north-east Scotland to help thousands without power
Storm Barra moved in from the west on Tuesday, and a yellow weather warning for wind is in place from midnight until 6pm on Wednesday for the west coast of Wales and south-west England.
Some schools in Dumfries and Galloway were forced to close because of the weather.
Stranraer Academy was shut after the wind caused structural damage to the roof, the council said, and Drummore Primary closed after trees were blown down and the main road leading to the village, the A716, was flooded by the sea and left covered with debris..
Some primary schools in Aberdeenshire decided to remain closed today due to lack of power and heating, including Bracoden, Rhynie and Strathdon schools.
Despite previous disruption, Met Office forecasters said Storm Barra was not expected to cause as much chaos as Storm Arwen.
Spokeswoman Nicky Maxey said: "Storm Barra will bring strong winds and heavy rainfall to many parts of the UK today. We may see some snow on the higher ground, too.
"It is unlikely to be as impactful as Storm Arwen last week but there will be blustery conditions so people should still be prepared."