Power utility Eskom on Wednesday called on consumers to switch off geysers, pool pumps and all non-e.
Berlin police spokesman Thomas Neuendorf says the tunnel led from an underground garage into the bank's safe deposit room.
Neuendorf told The Associated Press Monday that the tunnel was "very professional" and must have taken weeks or even months to complete. It was elaborately constructed and even had ceiling supports.
Police were alerted to the break-in early Monday when a security guard noticed smoke coming from the deposit room.
Neuendorf says police are still trying to determine what valuables were stolen from the deposit boxes.
Details of the heist called to mind the spectacular robbery of another Berlin bank in 1995. During that robbery thieves entered the bank through the door, took hostages and demanded a helicopter and ransom.
Police besieged and eventually stormed the safe room where the thieves had holed up only to find they had escaped through a tunnel dug by accomplices. Several of the thieves were later caught.