Negotiations are continuing between site managers at Didcot power station and environmental campaigners chained to equipment at the Oxfordshire plant.
A team from Greenpeace also climbed a 650ft (200m) tower on Thursday and painted "Blair's Legacy" on a chimney.
Operator Npower has called for the protest to stop and launched an inquiry into the security breach.
Campaigners want the coal-fuelled plant to close as they believe it makes a massive contribution to climate change.
Thames Valley Police confirmed some of the protesters spent the night at the power station and remained there on Friday morning.
Power generation at the site has been reduced, but Npower said it has not stopped.
"I think they've made their point now, we must get on and continue with the business"
John Rainford, site manager, told BBC News: "We requested them first thing this morning to end their protest, to come down safely - health and safety is our foremost intention.
"I think they've made their point now, we must get on and continue with the business. We can have this debate outside of this but this is no good to anybody."
On Thursday, a spokesman for Npower confirmed 15 protesters had gained entry in the early hours, while Greenpeace said 30 were present.
One man was arrested on the site on Thursday on suspicion of aggravated trespass.
A Greenpeace spokesman said: "The campaigners are demanding the government phases out this kind of coal-fired power station and instead backs localised or decentralised power generation, which is much more efficient."
Kevin Akhurst, managing director of generation and renewables at Npower, said: "Regarding Greenpeace's point of view, we fully support clean energy and we're at the forefront of developing it.
"Of our three coal stations Didcot A and Tilbury are already scheduled to close over the next five to 10 years, as are many coals stations in the UK, but they can't just be switched off overnight.
"Some coal is needed to maintain secure and affordable energy and for now stations like Didcot meet that need."
In July, campaigners from Reclaim Power climbed a 131ft (40m) lighting tower at the site to unveil a banner which read "Climate Crime".
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