Kerry Worrall from Greater Anglia said the operator hoped to have services moving again soon.
“We have just heard that power has been restored to the Hackney Downs area so it does mean over the next couple of hours we will be looking to reinstate some services as we are now able to use the lines via Southbury and Stratford,” Ms Worrall told BBC Essex.
“Hopefully as the morning continues, things will improve, but at the moment we are advising people that they will not be able to travel between London Liverpool Street and Broxbourne.”
Ms Worrall said she did not know how much of the cables were stolen.
A BTP spokesperson said: “Officers were called to the line near Clapton railway station at about 1.20GMT today (20 February) following reports of cable theft.
They encouraged anyone with information to contact them and added that “inquiries are ongoing”.
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