For a blackout, the diesel generators kicked in (and could completely power the platform).
Emergency power would be expected to work independently of mains power.
The most likely limitation of a typical power plant operating disconnected from the grid would be if it could “black start” or not. Cogen plants would normally have this capability. Essentially the problem is whether you have enough power available from backup generation to bring all of the plant auxiliary systems online and get the generators initially started and producing power. Once started you set the governor to maintain 60.0Hz (if in North America) and begin slowly bringing on load so you can ramp the generation to match it.
The easiest solution to a future blackout would be to match the load in your island area to the output of your plant and then disconnect from the rest of the grid before or at the time the blackout occurs. This way you have avoided the black start situation completely. The blackout detection, disconnection, and load balancing steps can be automated to occur nearly instantly if the system is designed well.
The main issue for Berkeley is that the cogeneration plant doesn’t seem to be sized to accommodate their islanded load. This is especially problematic if you lose a generator while islanded because then you will have instant frequency collapse if you were already on the edge of your plant’s capabilities.
Cogeneration is designed to produce heat (in chillier climates) and the process makes electricity as well while doing so, because why not?
* https://en.wikipedia.org/wiki/Cogeneration
Emergency / standby generators do nothing most of the time, and only kick in when needed:
I remember some time ago when the cogeneration plant failed and took out power in one building where many biological samples were stored at -80.