CP - Mean Max NP Curve

djconnel

I suggested an exponent of 12, because this corresponds to a 6% reduction in power with a factor of 2 duration incease, which seemed fairly conservative. An exponent of 26 would correspond to 2.7% decrease.

With an infinite exponent, the model becomes particularly simple:

NP = max { P* }
where P* is smoothed power.

For exponential smoothing with time constant tau, a rider can then sustain constant power P for time t:

P = FTP / (1 - exp(-t / tau))

which for small t approaches a limiting case:

P -> FTP * tau / t (for small t)

However, to maximize average power for an interval t, an impulse of area FTP * tau followed by a steady FTP results in an average power:

P = FTP (1 + tau / t)

which is the critical power model. These results suggests tau should be close to the ratio of anaerobic capacity and aerobic power from the critical power model, if a large exponent is used.

You need to consider in what % of rides the rider is ready to produce a personal best. For every NP buster there should be many other rides where CP60 is overestimated for the rider in his current condition, yet this still falls shy of his FTP derived under ideal conditions (taper, hydration, warm-up, weather...)

acoggan

...assuming, of course, that you're primarily interested in modeling the early, large non-linearity in the power-duration relationship, and not the long-term decay in power that occurs subsequently.

acoggan

Agreed - hence the reason that I've focussed on NP busters in trying to understand the limitations of the current algorithm.

djconnel

Right -- to "bring NP busters back into line" the smoothing time constant needs to be increased. If it's increased too far, hard, short efforts are undervalued. However, a larger exponent will increase the value of these efforts, as long as smoothed power is still able to exceed FTP . For example, if I'm at 75% FTP, and I sprint for 10 seconds, with a time constant of 60 seconds, I'd need to average approximately 75% FTP + 25% FTP * 60 / 10 = 225% of FTP for the ten seconds to bring smoothed power up to FTP . My peak 5 second time (in the time I've been recording, which is around 2 weeks...) is around 270% FTP (FTP is estimated from histograms), so I'd find this hard to do. I suspect the smoothing time constant should be the ratio of anaerobic capacity and aerobic power, based on the maximal power inferred by NP for short times, so different riders would have different optimal time constants. My time constant would be relatively low, as I have relatively poor anaerobic capacity compared to my aerobic power.

djconnel

I agree. I think an infinite exponent is a bit absurd.

frenchyge

Judging from your example, I'm guessing you mean neuromuscular power rather than anaerobic capacity? Neuromuscular power is typically associated with efforts <30sec.

djconnel

I mean AWC (anaerobic work capacity) from the critical power model:

peak available work = AWC + CP x Duration

where CP = "critical power".

tau =~ AWC/CP should approximate the critical power model for short durations, I believe, where tau is the smoothing time using exponential smoothing.

Dan

frenchyge

Heh, ok. Another departure between mathematics and physical reality.

Despite what the model suggests, one can't expend their entire anaerobic work capacity in the course of a short sprint. The rate at which the energy can be expended in such a short timeframe is limited by one's neuromuscular power.

djconnel

The critical power model applies to efforts in a relative limited range of durations. For example, one paper which "validates" the model is Pringle and Jones, Eur J Appl Physiology (2002) 88: 214-226. Subjects rode an exercise bike at constant power, for different powers, to exhaustion. Only data corresponding to 2-15 minutes were used to validate the model.

Indeed, for my own data, I estimate my FTP using my max NP versus time curve @ 250 watts. Then I plot (max power for time - 250 watts) * time, versus time, I get a ramp-up to 60 seconds (where it maxes out at 6 kJ), then it hovers in this regime (4.4-6.9 kJ) until 15 minutes, then it falls off (the intermediate drops suggest for these durations my max was below my potential). At 15 seconds, it reaches 5 kJ. This suggests, from my data, the critical power model is ballpark from 15 seconds to around 15 minutes. Less than 15 seconds, and I can't expend my anaerobic capacity. More than 15 minutes, and my aerobic capacity may start to drop, or the quality of my efforts may do so, or both. Still, that's a factor of 60 in time, not bad.

But point well taken: my derivation was for durations substantially less than the averaging time. Yet the critical power model doesn't apply to durations much below the averaging time (I derived 24 seconds from my data, but 75 seconds from data in the paper). So matching it for small times has limited validity.. a more careful comparison is needed, where the CP model has some validity.

acoggan

Or to put it another way: the assumption of the critical power model that there is a linear relationship between work and time isn't absolutely correct, i.e., you can't really fit a right hyperbola to the power-duration relationship. Nothing really new there...but it's worth recognizing that that is not the only assumption of the critical power paradigm that isn't absolutely correct (and this gets back to the original topic, i.e., how one might go about improving the normalized power algorithm). Specifically, the critical power approach assumes that aerobic energy production is completely available at its maximum rate from the very onset of exercise. This is not true, however, which means that at first you're actually "chewing through" your anaerobic capacity (note that I didn't say anaerobic work capacity) at a faster rate than calculated from the critical power approach. The result is that although anaerobic work capacity as calculated from the critical power model is reasonably well correlated with other laboratory-based measurements of anaerobic capacity (the physiological trait) such as maximial accumulated O2 deficit, it is always significantly lower, because some of the energy that is being produced anaerobically is credited to the aerobic energy system.

djconnel

I took my maximal power curve, based on quite limited data, and compared it to a maximal power curve derived from the NP model.

To derive the curve, I assumed an interval of time t at power P, followed by an "interval" at time 1 hour - t at power 0. If the initial interval was maximal, this should equal NP for all (P, t). Thus I could map out the effective maximal power curve for a given set of NP parameters. I assumed exponential smoothing.

Now, a limiting factor here is my maximal power curve isn't truly maximal. These intervals were embedded within longer rides, with other efforts, typically with non-optimal tapering, etc. I haven't yet done any formal tests. But the same sort of exercise could be conducted with a more mature maximal power curve.

The result of my manual fitting attempt: an exponent of 32 with a 32-second smoothing. If I had more anaerobic capacity, the smoothing constant would be longer. I had derived approx 25 seconds based on the critical power model, so the result is close. A lesson here is that the smoothing time should probably be tuned for each person. If it isn't, accessible power at intermediate durations will be either overestimated or underestimated. The latter case busts NP. Note there's an obvious bump which rises above the curve. So obviously it's not my "true" maximal curve. But I do capture the trend.

I also show on the curve the result for an exponent of 12. It predicts a much steeper power versus time. With n=12, even tweaking FTP and tau, I can't fit the data.

So basically, if I were to produce any efforts in excess of my maximal power curve, that would indicate either that my FTP was estimated too low (my aerobic capacity had improved), or my smoothing time constant had improved (anaerobic capacity had improved), or perhaps the exponent had changed (which would be more associated with long-term endurance: loss of aerobic power with increasing time).

For now, I'll stick with n=12.

Dan

frenchyge

You're defining the maximal power curve as a maximal power ride to exhaustion such that no power can then be output for the remainder of an hour? Unless the rider dies at the end of the effort, I'm pretty sure the body will be able to make some recovery before the end of an hour at zero power. As I said before, trying to improve the NP algorithm across the board would probably need to include some modelling of the body's energy recovery mechanisms.


I'll propose an alternate explanation for why you have a bump that busts NP: intervals and rests will allow a person to push the NP envelope further than a single maximal effort to death, because they leverage the body's ability to recover at least a portion of it's energy between efforts. The maximal NP curve probably constitutes an entirely different riding mode than the maximal AP curve.

djconnel

You're right about the bump -- the only way to get a bump in a maximal power curve is to span two intervals, separated by a rest period. A problem with my curve is I just haven't done intervals at enough durations. A few times up Old La Honda, in the 19:45 range, a hard climb of Diablo near the end of a hard ride, where I burned out around 45 minutes in and finished in around 1 hour, some sprints, plus some random stuff in group rides. Not enough quality data.

WRT rest maximizing power: Eddie B suggests pedalling essentially one-legged occasionally, resting the left, then later resting the right, for the reason you claim: that constant power is not optimal. I've not seen this recommended elsewhere. So I'm still of the opinion I could have averaged that bump power at relatively steady power for that duration.

In any case, a feature of high exponents is they essentially pick out regions of maximal smoothed power. An exponent of 32 means that if an interval has a power only 5% less, it counts only 19% as much. So basically it's estimating FTP on the hardest interval long enough to poke its nose above the smoothing. Even if you did 5 intervals at the same power, the multiplicity of intervals would boost NP only 5% versus one interval at the same power. This is all about estimating FTP and optimizing pacing, not about traning stress. Optimal pacing still involves non-uniform power for changing conditions, but that's more playing with power smoothing than trading off steady-state power with such a high exponent.

rmur17

yes I think that's the crux of the matter. My example showed that pretty clearly - 30min PB effort and after a single 5-min recovery spin I could ride at pretty decent tempo power for another 25-min -- probably more if I pushed it. And I'm sure we're all pretty much the same in this respect.

There's no way NP should take that 30-min max interval and tell me I'd have to just spin for 30-min afterwards.

Interesting thread though ...

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rmur