Eighteen power plants have been shut for months, mostly for repairs and maintenance, leaving the capital Dhaka and other main cities in the dark for several hours every night. In rural areas power cuts can last for days.
Power generation dipped to 2,500 megawatt, nearly half of installed capacity, after the 250 megawatt unit at Ghorasal, east of Dhaka, tripped on Saturday, a power department official said.
"The time has come for them [the government] to face the people and explain," Abdur Razzak, a former minister and senior leader of the main opposition Awami League, said.
The power crisis has become a major election issue in the run-up to the parliamentary election which is due to be held in January.
Sunday's protests were called by the opposition Awami League led by Sheikh Hasina, a former prime minister.
Hundreds of people attacked two power distribution centres in northeastern Sylhet on Friday night in a protest over the lack of electricity.
More than 20 people, mostly farmers, were killed in clashes with police, mostly in northern Bangladesh earlier this year, during demonstrations demanding adequate power for irrigation.
At least 500 others have been injured in clashes across the country over the past two months.
Begum Khaleda Zia, the Bangladeshi prime minister, has accused the opposition of encouraging the violence for political gains.
The government has already sacked two junior power ministers to try to appease angry consumers,
"Give us another term in office and we will solve the power problem once for all," Khaleda told a rally in Sylhet.
A World Bank study earlier this year said that power failures had damaged Bangladesh's gross domestic product by around $1 billion a year.
In July, the World Bank said Bangladesh would need $10 billion investment over the next 10 years to alleviate the power shortage.