Co-operative Energy has already paid most of the compensation after customers were affected by a new IT system in March last year.
The system's introduction led to direct debits being blocked and lengthy waits for new customers.
The company has apologised and said it had improved the service.
Regulator Ofgem got involved after complaints levels were running 50% higher for Co-operative Energy than any other supplier.
Ofgem said these complaints were caused by customers being unable to log into their accounts online, leaving them unable to submit meter readings or check their bill. Bills were delayed and direct debit updates stopped.
In addition, new customers also experienced delay transferring to the supplier. It also took too long to resolve a significant number of customer complaints, Ofgem said.
In response, Co-op Energy stopped marketing its products, improved the service and has already paid £1.6m in compensation.
Martin Crouch, Ofgem senior partner said: "While customers experienced unacceptable levels of service, Co-operative Energy has done the right thing by taking responsibility for the situation and paying out compensation to those people affected.
"If trust is to be restored in the energy market suppliers must make amends to their customers when things go wrong. We want all suppliers to constantly ensure customers are treated fairly."
Ben Reid, chief executive of Midcounties Co-operative which runs Co-op Energy. said: "As the UK's only member-owned energy supplier, we put our customers at the heart of everything we do and their interests first.
"We have apologised to those customers who were affected by the problems we experienced when we introduced a new IT system last year. The system issues are now resolved and we have made significant improvements to our service."
On 1 October, Co-operative Energy increased prices for both gas and electricity by 3%.
One of the UK's smaller energy suppliers is paying £1.8m to its customers after bills were delayed and online accounts were blocked.