A very common failure, which seems to crop up on some of the earlier PS2 models, is where the power supply fails for whatever reason. When switched on there is no life and often, the internal fuses (or sometimes the fuse in the power lead) blow.
Very often, the fault lies with a very simple component on the power board. The usual component is a diode, of which there are four similar ones. Often though, the same diode fails each time. This is the third of the largish diodes, arrowed in the following picture.
