There was severe disruption on LNER's routes running along the East Coast Main Line from London King's Cross to north-east England and Scotland.
Around 10 LNER trains have been taken out of service due to damage to their pantographs, which is mounted on the roof to collect power through overhead wires.
LNER managing director David Horne said the operator is working hard to identify the cause of the problem, which has affected 30% of its electric trains.
The Department for Transport-owned company has a fleet of approximately 30 InterCity 225 electric trains. It is introducing new electric and bi-mode Azuma trains next month.
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An LNER spokesman said: "Due to disruption on the LNER route, we're advising customers not to travel today.
"This is due to damage to some of our electric trains and is being investigated by our engineering teams and Network Rail.
"We are sorry for the inconvenience this will cause to customers."
LNER passengers are able to use their tickets on alternative routes, including those run by TransPennine Express, Virgin Trains, East Midlands Trains, Northern Rail and Govia Thameslink Railway.
They can also travel on Wednesday or claim a refund.