Having distinguished the difference between power and energy, I pose the question: what kind of power can this new approach yield? Is it practically useful or just theoretically amusing? If generating 288% more energy takes 288% longer to accomplish, you haven't really made any progress.
You have a good point about power being a consideration, but this point doesn't necessarily follow. If the plant is cheap enough more scale can overcome the lesser power output.
Power in watts x time = joules which is a measurement of energy.
I was careful to keep power in watts and not call it energy. It is assumed that time is the same on the input and output sides of the energy converter (electricity and chemical energy conversion to hydrogen). If you put in 1 watt of electrical power for 1 minute and get a flow of 3.88 watts of hydrogen power for 1 minute the energy gain (only counting the electrical energy being put in, not the chemical energy being put in) is an energy gain of 288% which is what the article claims. This is clearly possible and not simply a claim of getting free energy from some mysterious source in the universe... The energy gain is coming from the chemical energy of the acetic acid that is also consumed in the process along with the electricity.
I'm guessing the reference of energy gain regarding only the electricity going in is to show that there is much more involved than just electrolysis in the hydrogen generation. Electrolysis only would be less than 100% conversion ratio.
neverdem posted more detailed information and the overall energy conversion efficiency was claimed to be 82%. This is based on the electricity and acetic acid that is consumed to make the hydrogen. All of which sounds realistic and useful for the generation of hydrogen.