There are no HP gains or loses, only a perceived gain or loss, due to its effects on the ability of the engine to accelerate and decelerate more or less quickly depending upon how much mass is added or removed. A flywheel is nothing more than a storage device, which changes the rate of delivery of power to the rear wheels, by parasitically storing some of the kinetic energy produced by the engine. It has no effect on fuel, CR, ignition burning, mixture, gas flow, nothing. Purely external, after all power has already been generated. It is just a big hunk of metal that acts as a parasitic loss exterior to the motor. The engine produces exactly the same power as before a change to the flywheel is made (except at extreme low rpm where the power pulse dampening effect is needed) Think of it as a capacitor in a DC electrical circuit; which charges up and stores energy for use later and for dampening of the baseline supply voltage.
The beauty of these things is their simplicity, which is one reason they have been explored as sources of storage in alternative energy systems, such as hydro, solar etc. They don't make too much sense in transportation systems because the system would have to lug around all that mass, but in a stationary system, they can be of some use. The only two significant losses are frictional in the bearing, and frictional in the air it spins through. Plus any losses incurred during the spinning up of the flywheel by the supply system. The air friction can be fairly easily
resolved by containing the storage system in a vacuum/semi vacuum, and the bearings already have plenty of technology already available to draw from to reduce friction there. I have had a very small peripheral input casually discussing these things with a utility commission member out on one of our Pacific Islands, where the desire to get away from oil is desired. A small community with hydro or solar power readily available could use these, when scaled properly, to augment a solar or hydro system to deliver power at night. And they are absolutely quiet, not like a Tesla coil in the neighborhood would be...
The beauty of them is that they are so scalable, and are so easily attached to generators, pumps, the grid, etc. But God help anyone nearby if one let go or broke into pieces, they need to spin at enormous rpms to contain enough energy at reasonable size to be useful. The containment structures have to be extremely strong.
It's too bad that it hasn't caught on more quickly, but the emphasis and efforts in alternative generation have been dominated by photovoltaics, for better or worse. When I was an Engineering student intern at the D.O.E. Northeast Solar Energy Center in Boston in the late 70's, things were in pretty good balance, we promoted photo, fluid, passive, pretty much everything equally, as it wasn't quite clear yet where the industry was going to go. I remember we used to hold training classes for plumbers, electricians, carpenters, roofers, etc. to try to promote the development of fluid based rooftop systems. I always thought that the problem was that there were just too many trades involved in installing these systems, and the costs would never be low enough for them to make sense. Like doing a bathroom in the house, it takes every single trade and costs more per square foot than any other part of the structure. Everyone who shows up has a basic nut to crack to make any money, and the costs add up very quickly. Turns out to be mostly true, and photovoltaics won out in the end for residential use. Although it is amazing that, 40 years later, they still don't make sense without our tax dollars subsidizing their use.
Anyway, that's enough for me, I've said my piece!
Back to re-plumbing all the ethanol-damaged fuel lines in the trunk and under the car. Oh joy.