A debate on hospital closures was interrupted after AMs were thrown into darkness at about 15:50 GMT.
Deputy Presiding Officer David Melding told members there had been a power failure in Cardiff before it happened again minutes later.
The discussion continued, after a break of about half an hour, when a back-up generator was turned on.
Conservative AM Darren Millar said it took the shine off the Senedd building's 10th anniversary celebrations.
Elin Jones, Plaid Cymru health spokeswoman, was delivering a speech when the lights in the Senedd went off.
Public Services Minister Leighton Andrews tweeted that the "power surge" had blown out "the computers and clocks".
Some time later Mr Melding restarted proceedings, apologising and saying there had been a power failure "in this part of Cardiff".
Less than 10 minutes later it happened again, interrupting Conservative AM Angela Burns.
Plaid Cymru AM Llyr Gruffydd joked on Twitter: "I've always said the assembly doesn't have enough power!"
But Mr Millar wrote: "Drama in the Senedd as power cuts bring proceedings to a halt; takes the shine off the celebrations to mark its 10 year anniversary."
Western Power Distribution said it had a fault on its network.
Proceedings in the assembly were disrupted twice on Wednesday because of power failures.