Short Version
If you have an Electric Kiwi outage or fault call them on 0800 843 5494.
They may ask you to call your lines company. Our lines (Wellington Electricity) company only takes emergency faults directly, otherwise it's meant to go via the power company. However, if you call WE and hit the emergency line they will accept a fault report. I'm not sure how other lines companies work.
The Problem
The phone number on the Electric Kiwi website listed under "Wellington" is answered by someone who says they can't help with Wellington power outages. The only way myself and a friend could get power restored is to call the Wellington Electric Emergency Fault line on 0800 248 148.
They will tell you to call your power provider, but Electric Kiwi doesn't seem to provide an our of hours fault service. So you'll have to tell WE that and ask them very nicely to fix your power. If you have young children or similar tell them that.
Long Version
We moved to Electric Kiwi (EK) a couple of days ago. This morning our power went out at around 5:15am.
As far as I'm aware EK hadn't provided any information about what to do in the event of faults. I went to their website on my phone to try to find a way to contact them. The website says
So I called Wellington Electricity (WE). They say "if this is not an emergency regarding potential loss of life or public danger please call your power company".
So I go back to the EK website. At 5:20am probably not entirely awake or in a particularly good mood I couldn't see any way to call EK so I called WE back, went to their emergency line, and they did let me report a fault on the emergency line because we have a small child.
Looking into it more, when you scroll down some more on the EK website you see this.
That phone number under the block labeled "Wellington Electricity" is actually the Electric Kiwi fault line. When you call it they give you options to be put through to your local lines company, or hit 0 to talk to their faults and outages service. I believe it is a 24 hour line. That one phone number works for any of their customers in any area, yet it's well hidden.
The Electric Kiwi website really needs to say more clearly how to contact them in the event of a fault or outage. Since everything is online, no printed bills to refer to, they really need to make it much easier to find a way to call them.
If you're an EK customer I suggest you go to their web page and put their contact number and the number for your local lines company into your phone. The number is 0800 843 5494.
Update
Later, a friend called the number listed on the Electric Kiwi website for a fault in Wellington on a Sunday. Electric Kiwi told them they didn't provide a faults service in Wellington, and to call the lines company. The lines company told my friends they had to log the problem with their power company - Electric Kiwi. As a result there's effectively no faults service for Wellington Electric Kiwi customers outside business hours.
Fortunately Wellington Electricity will go around their standard process when you tell them things like "we have a young baby that needs to be kept warm, bottles washed, and food heated".