In Part 3's exploration of mathematical thresholds we were looking at methods of approximating a rider's power curve, which allows us to estimate maximal power for a given duration. However, that's rarely how we ride (or run, except on the flattest courses). The models discussed assume that an athlete stays at precisely the same power for the duration of their effort, but in the real world only TTers even come close. So this post will look at ways of dealing with variation.

## Estimating Physiological Cost

*Normalised Power, xPower, Weighted Average Power*

**Normalised Power**was developed by Andrew Coggan. Simply put, it is the 4th root of the mean of the 4th power of a rolling 30s mean power. Unfortunately it somehow ended up trademarked by Training Peaks, though it seems to be appearing in other devices now (it was missing from Golden Cheetah for a while).

**xPower**was developed by Philip Skiba. It takes the 4th root of the mean of the 4th power of a rolling exponentially-weighted moving average, with a 25-second time constant.

**Weighted Average Power**is the term used on Strava. I believe it is identical to xPower.

Imagine an hour's session where you trundle along at 75% of your FTP. That'd be a fairly reasonable aerobic workout and not too hard. Alternating 10 minutes at 100% and 10 minutes at 50% for an hour would probably feel a bit harder. And repeating a minute at 150% with 3 minutes rest at 50% for an hour would be harder still. Yet if we just use a simple average, all these work out at 75% FTP.

The purpose of these weighted measures is to answer the question

*"What would be the equivalent power sustained constantly for the exercise duration?"*. It's difficult to exactly define "equivalent" as it's clear the workouts described above will have quite different physiological effects in different systems, but we can take a stab at some rules for how an ideal measure should behave:

- For any duration it should not be possible for it to read higher than the athlete's best average power for that duration done at an even pace (ie the power curve I explored in Part 3)
- For any duration where the athlete has "given it everything" it should be at, or close to, the athlete's power curve for that duration.

Ultimately, then, these metrics are a useful guide - and almost certainly better than just using a simple average of power - but they may well overestimate the effect of short, hard bursts.

4x4 workout showing W'bal, NP, xPower |

## Estimating Recovery

*W'Bal (Integral/Skiba formula), W'Bal (Differential/Froncioni & Clarke)*

### Skiba's Integral W'Bal

### Froncioni & Clarke's Differential W'Bal

**Above CP:**

**Below CP:**

In part 3 we discussed various models; and almost all since the original 1965 Monod/Scherrer paper include W', a measure of how much work can be done above critical power before exhaustion. However, some time after an effort - even a maximal one where all the W' is exhausted - it is possible for the athlete to go again. There has therefore been a good deal of work from people trying to predict the recovery of W'bal (ie how much W' is left). This kind of information could be vital in a massed-start cycle race, where an athlete needs to predict whether they have enough in reserve to push with a group before attempting to recover in the pack.

The models given above allow W'bal to recover in an exponential manner (ie, quickly at first but gradually getting slower) after a heavy effort when some of the W'bal has been used. The first (integral) formula has an explicitly stated τ, or time-constant, after which a fixed proportion of the W' should have reappeared. The athlete can then use any recovered W'bal in further surges. If W'bal reaches zero, the model predicts that the athlete should be exhausted, and limited to a power of no more than CP thereafter.

An ideal model should therefore:

- Allow no recovery above CP
- Above CP, use up W'bal equal to the work done above CP
- Below CP, allow recovery of W'bal
- Recovery should in some way relate to how far below CP an athlete recovers

Philip Skiba has done a lot of work on his W'bal formula, and claims to be able to "predict exhaustion of an athlete to within just a handful of seconds". Unfortunately with his original formula that's not true in the way most of us would use the word "predict", in that τ

_{W'}is calculated from the entire ride's data, including parts that haven't happened yet. Software and devices claiming to calculate it live therefore must be doing something slightly different (or incorporate a flux capacitor) - strategies I've seen include using a predetermined τ, or using τ from the ride so far (with or without updating previous calculations). Additionally, the formula doesn't take account of how far below CP an athlete is recovering at - recovery happens at a constant rate determined by τ, even while the athlete is going for a sprint.

Froncioni & Clarke's differential formulation doesn't suffer these main issues and in more recent articles Dr Skiba appears to suggest it is preferred to his original formulation. However, it gives very different results - W'bal falls off much more quickly during a burst, and recovers more quickly afterwards. This is a result of it having an "implicit" τ of W'/CP which is normally about 1-2 minutes, compared with the original formula which usually makes τ about 5 times as long.

Despite my issues with Skiba's integral formula, I struggle to work out which of the two I prefer. The integral is "obviously wrong" in that the future affects the present, and by allowing recovery during efforts it disagrees with the original Monod/Scherrer model that it's supposed to be based on. However, the charts it produces chime well with my own feelings of exhaustion, compared with the differential one which predicts I should "bounce back" within minutes.

2x20 test showing W'bal, NP, xPower |

## Some Links/References

- W'bal ConnectIQ app for Garmin devices (supports both equations)
- https://www.trainingpeaks.com/blog/what-is-normalized-power/
- http://perfprostudio.com/webhelp/studio/scr/BikeScore.htm
- Skiba PF, Chidnok W, Vanhatalo A, Jones AM. Modeling the Expenditure and Reconstitution of Work Capacity above Critical Power: Medicine & Science in Sports & Exercise. 2012 Aug;44(8):1526–32.
- Skiba PF, Fulford J, Clarke DC, Vanhatalo A, Jones AM. Intramuscular determinants of the ability to recover work capacity above critical power. European Journal of Applied Physiology. 2015 Apr;115(4):703–13.

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