There are plenty of guides on the internet showing how to repair a slim when it doesn’t show a red light upon plugging the power in. Most guides lead you to adjusting the two pots in the psu itself. Do not do this, the power supply is NOT at fault.
As Baileyscream once said, it is the RSX chip creating a increased current draw or short and the ps3 is preventing powerup.
There is nothing wrong with the slim PSU‘s in general. Yet they are being replaced and seemingly curing the problem. However this psu will eventually repeat the same problem, as it gets hotter with age it over powers and triggers the failsafe.
Reflow the RSX chip to cure the problem, again, don’t adjust the pots or you might find you fry your monitors HDMI input (as I did).