Sounds like a short circuit and "magic smoke", hopefully it didn't damage anything.
There's another thing I wanted to explain. You don't actually *know* that the power supply is at fault. You're guessing that it is because that's what you read, but that's not necessarily the case - you haven't probed it with a multimeter to check or stress tested it with another PS3 or a test load.
Here's how this works - the PSU gets mains voltage out of the wall which goes through a bunch of components (some kind of switching setup, transformer, inductors, capacitors etc.) and ends up with a kinda-sorta stable 5V, 12V and control signals. Those two power lines then are processed again on the motherboard through "mini supplies" in the form of VRM's before they actually hit any of the logic components, like the CPU and GPU.
Any component on that power distribution track can be at fault and you wouldn't really know the difference just by looking at the symptoms. For all you know, one of the MOSFETS could be overheating, shutting down, the console stops getting power and abruptly shuts down - that won't give you a prompt. It's *usually* the power supply since that's the part of the unit that does the heavy lifting, but you don't know that for sure without checking.
Long story short, there's a reason why repair stores charge money for repairs - troubleshooting isn't a matter of following a guide, you have to actually investigate the parts, look for bulging capacitors, black smears on PCB's, burnt or broken tracks etc. - what you can do as a consumer with no gear or experience is replace the part and hope for the best. If you've never worked with electronics before, a power supply is the last thing you should touch because half of it operates at high voltage, and not just when it's plugged in - capacitors hold charge even after you disconnect it. You can get hurt in a hurry if you touch the wrong connections.
EDIT: Your EADP-185AP power supply costs £12 on eBay, or around €14. Ask yourself if it's worth the headache. Power supplies are consumables - if you had the right equipment and know-how you could try and fix it, by all means, but it sounds like you're poking in the dark.