You cut all wires coming out of the supply keeping only the various 12 volts wires which are soldered together and also 1 wire giving 5 volts .

locate the Pulse Width Modulator which drives the switching transistors .It is the brain of the supply ! Generally it is a small digital circuit (reference TL 494 or something else ending with 494 having 16 pins ) .

To increase the output voltage you only have to put a resistor between pin 1 and 7 .

The value has to be adjusted in order to get the 13.8 volts at the output .(the value is between 10 and 20 Kohms )¹⁾

It should be noted that for some power supply ,to have a good regulation of the output , it is necesary to load the 5 volt circuit with a resistor in order to have some current anywhere between 0.5 and 1 amp .