When I read your first post, I was thinking that perhaps a cooling fan inside might be bad and making that noise. Depending on the processor, fans can be in the power supply, in the case, and on the processor chip itself. If there is one on the processor chip and it fails, you risk the chip overheating and damage.
Then when I read your second post, things look worse. SMOKE is not a good thing. It means that something in your circuitry is pulling way too much current, overheating, and subsequently smoking while it destroys itself. By the way, any amount (tiny or not) of smoke is a bad sign.
When you purchased the replacement power supply, did you ensure that the specifications MATCHED your PC? Not all power supplies are the same. If you didn't do this, do it now and don't power up again. Check that the new PS output voltages and pin assignments are identical to the old one. If you put the wrong voltage onto a bus, things smoke.
Were the motherboards the same? Does the new PS match the new MB? Look at the old MB closely. Do you see any signs of overheating (discoloration, bulging, burnt marks, etc.) on any of the components? Check the processor chips especially.
I don't know what could have caused your PC to turn on all by itself, unless you had last done a "restart", and when you shut it down did not change the selection to "shut down", but simply clicked on the OK button and walked away before the shut down cycle finished.