## Introduction

3record stores for all swim, bike, or run activities mean maximal power (MMP) for predefined durations ranging from a few seconds up to a multiple hours. This information is useful to quickly see the difficulty of sustained efforts and is plotted on each activity view page.

MMP data can also be used to estimate fitness parameters such as CP or W' that are defined in the about metrics page. This is done by fitting the data to some pre-defined model of power-duration relationship. Different models can be used as explained below.

## Mean Maximal Power

Mean maximal power is defined as the maximal average power that was sustained for a certain duration. It is firt calculated for each activity. The data is then merged (via a max operation) for each week, from which last year reference is taken and plots on the CP page are constructed.

## Power-duration models

Most of the power-duration models are based on the Monod 2-parameter CP model which basically defines the critical power. It is however overly simplified and implies an infinite power at short durations and ability to sustain CP for infinite durations. More advanced models avoid this by using more complex formulae and more parameters.

The models available in 3record are listed below. For each, the values of the extracted parameters using least square fitting is displayed in the CP analysis page, even though it could be argued that the fit should be done on maximal datapoints only (ie. not using ordinary least square) to respect the maximal operation of MMP. FTP value displayed is the power predicted by the model for 1 hour duration.

### Monod 2-parameter CP

As mentioned above, Monod 2-parameter CP model [1, Monod 2P] is the most simple one. It has the simple formulation of a 1/x curve

$P(t) = CP + \frac{W'}{t}$ where $CP$ is the critical power and $W'$ coresponds to an anaerobic work capacity (that is time limited).

It has as main inconvenients the facts that power is supposedly infinite at short duration and long-term fatigue is not included as CP could be maintained whitout limits.

### Monod 3-parameter CP

Monod 3-parameter CP model [2, Monod 3P] is an improvement from the 2-parameter model to remove the first issue, power tending towards infinity at short durations, via the introduction of a time constant $\tau$

$P(t) = CP + \frac{W'}{t+\tau}$

CP is still supposed to be maintainable for infinitely-long durations though.

### Ward-Smith

Ward-Smith model [3, Ward-Smith] is similar to the 3-parameter CP model but introduces the maximal power $P_\text{max}$ instead of using the $W'$ parameter that has the units of energy

$P(t) = P_\text{max}\cdot \exp\left(-t/\tau\right) + CP\cdot \left(1-\exp\left(-t/\tau\right)\right)$

Note that it also supposes CP is maintainable for infinitely-long durations.

### Morton

Morton model [4, Morton] uses both $P_\text{max}$ and $W'$ for a more complex anaerobic expression (with feedback)

$P(t) = CP + \frac{W' - CP\cdot \tau\cdot \left(1-\exp\left(-t/\tau\right)\right)}{t+\frac{W'}{P_\text{max}-CP}}$

Note that it also supposes CP is maintainable for infinitely-long durations.

All models up to here are reviewed in [5, Morton rev].

### Alvarez

Alvarez model [6, Alvarez] uses both two time constants and feedback expressions for both aerobic and anaerobic parts

$P(t) = \frac{P_\text{max}}{t}\left(\tau_1\cdot\left(1-\exp\left(-t/\tau_1\right)\right) - \tau_{1a}\cdot\left(1-\exp\left(-t/\tau_{1a}\right)\right)\right) + \frac{CP}{t}\left(\tau_2\cdot\left(1-\exp\left(-t/\tau_2\right)\right) - \tau_{2a}\cdot\left(1-\exp\left(-t/\tau_{2a}\right)\right)\right)$

### Veloclinic

Veloclinic model [7, Veloclinic] uses similar expressions for both aerobic and anaerobic parts of the form

$P_i(t) = \frac{W'_i}{t}\cdot\left(1-\exp\left(-t/\tau_{ia}\right)\right)\cdot\left(1-\exp\left(-t/\tau_{ib}\right)\right)^{1/\alpha_i}$ For simplicity of model fitting, it was assumed though that $\tau_{1b}=10$ seconds and $\alpha_2\rightarrow\infty$ (droping effectively the second feedback for the aerobic part) $P(t) = \frac{W'_1}{t}\cdot\left(1-\exp\left(-t/\tau_1\right)\right)\cdot\left(1-\exp\left(-t/10\right)\right)^{1/\alpha}+\frac{W'_2}{t}\cdot\left(1-\exp\left(-t/\tau_2\right)\right)$

Note that other similar expressions could be obtained making other asusmptions.

### Extended CP

Extended CP model [8, Extended CP] uses a lot more parameters than the others to try to fit the data. Its shape can consequently be more complicated but it can be harder to give some sense to each parameter. It is supposed to represent the 3 components of ATP, glycolysis, and oxydation with capacity, response time and max power for each

$P_i(t) = P_\text{aa}\cdot\left(1.20-0.20\exp\left(-t/60\right)\right)\cdot\exp\left(-\lambda_\text{Paa,dec}\cdot t/60\right) + CP_{e}\cdot \left(1-\exp\left(-\tau_\text{del}\cdot t/60\right)\right) \cdot \left(1-\exp\left(-\lambda_\text{CPe,del}\cdot t/60\right)\right) \cdot \left(1-\gamma_\text{CPe,dec}\cdot\exp\left(-\tau_\text{CPe,dec,del}/\left(t/60\right)\right)\right) \cdot \left( 1 + \tau_\text{e}/\left(t/60\right)\right)$

## Anaerobic work capacity: dynamic W'bal model

Using the simplest of the above models which is the Monod 2P, it is easy to prove that a constant effort over the critical power will consume an amount of energy above threshold equal to W' (it is in fact a rectangle on the power-duration chart that will go from the left up to the point where it touches the mean-max curve); hence the term of work capacity, see [9, Skiba W']. From there, we can extrapolate that every joule (energy, or power multiplied by time) consumed above threshold is taken from this reserve. Adding some recovery method to fill again this tank when effort is low, and a dynamic model can be found to give the evolution of the energy reserve and how much is available at a certain point during an activity. [10, GoldenCheetah W'bal] explains in detail how this model has been implemented originally in Golden Cheetah. Even though the implementation is not identical, concepts still apply to 3record as well.

Mathematically, and as every data is discrete in steps of 1 second, we can express W'bal differentially from the previous value as:

$W'\text{bal}(t)=W'\text{bal}(t-1)-\left(P(t)-CP\right)\quad\text{when}\quad P(t)>=CP$ $W'\text{bal}(t)=\left(W'\text{bal}(t-1)-W'\right)\cdot\left(1-\frac{CP-P(t)}{W'}\right)+W'\quad\text{when}\quad P(t)

How to use the W'bal plot (visible on each activity) is described in quantitative terms in our blog (in French), focussing on race pace analysis: http://www.besse.fr.nf/news/264-Wbal-un-outil-pour-la-gestion-de-course

## References

References in text

[1, Monod 2P]
The work capacity of a synergic muscular group, H. Monod, J. Scherrer (1965).
[2, Monod 3P]
A constant which determines the duration of tolerance to high-intensity work, BJ. Whipp et al. (1982).
[3, Ward-Smith]
A mathematical theory of running, based on the first law of thermodynamics, and its application to the performance of world-class athletes, AJ. Ward-Smith (1985).
[4, Morton]
A 3-parameter critical power model, RH. Morton et al. (1996).
[5, Morton rev]
The critical power and related whole-body bioenergetic models, RH. Morton (2005).
[6, Alvarez]
An improved Peronnet-Thibault mathematical model of human running performance, J. Alvarez-Ramirez (2002).
[7, Veloclinic]
Derivation of the veloclinic Mean Maximal Power Duration Models, Veloclinic. http://veloclinic.com/wp-content/uploads/2014/04/PowerModelDerivation-1.pdf.
[8, Extended CP]