A more powerful PSU also produces less heat than one with lower specs, at the same load. (switched PSU's anyway)
Just for the record: there is a pin on the connector named PS_ON (Green wire, pin 14 on the ATX connector. See the
pinout for the ATX connector.)
That pin is a signal from the motherboard to the PSU to go out of standby mode and turn on all voltages (PS_ON low) or vice versa (PS_ON high). This signal is a common source of trouble as different manufacturers have implemented it slightly differently. Some motherboard implementations of power save mode logic and BIOS are also simply buggy.
What one can try if suspecting a miscommunication on PS_ON, after making sure that the PSU isn't simply overloaded (check the voltages) is to simply shorthen this pin to nearest GND (black wires, 13 or 15 is nearest). You don't have to cut any wires. It's just a pull up resistor there so shortening the pin to GND won't damage anything. This will inhibit the motherboard, electrically, from shutting down the PSU.