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Hello fellow cyclists,

There are services out there that can calculate power curves based on your bike ride (when training with a power meter).

This curve shows the time to exhaustion (t) based on the power P. So let's assume that my maximum power is 1000 Watts, then this formula will. calculate how long it would take to be totally exhausted (probably not that long for me).

The formula I keep seeing is this: P(t) = W’/t + CP

This formula looks extremely easy and yet I have trouble understanding what those values are. Are there any calculation examples out there? If I understand this correctly, CP is the critical power. This is the power that can be sustained (theoretically) infinitely. I have to provide this value, so let's just assume this is 200 Watts.

What is W'? According to the internet, this is the work capacity that is currently available. How can I calculate this?

Are there any example calculations?

Example activity (not real numbers, just for an example calculation):

Duration 1h, 3600 seconds

FTP = 200 Watts

CP = 160 Watts

Power During Ride: 3600 data points, every even data point is 150 Watts and every odd one is 120 Watts.

I also now that there are way more complicated formulas out there, but I would like to understand this simpler one first.

TL;DR: What is W' and how can I calculate it based on my bike ride.

Any help is appreciated.

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There are a couple of different versions of the CP model. You are looking at the earliest, simplest, two-parameter model proposed by Monod and Scherrer in 1965. While there have been modifications and expansions of the original model since then, the basic simple model has enough features that you can understand it. Although CP has been described as "the power that could be sustained indefinitely," in practice it works fairly well from durations of perhaps 3 minutes out to around half an hour -- additional parameters have been proposed to address durations before and after that.

If you have been riding with a power meter for a while you've undoubtedly seen a curve as you described, showing maximal average power in watts over durations in time. The simplest way to understand the Monod-Scherrer model is to notice that a watt is a Joule per second. So, to get Joules of work, multiply the power in watts by a duration in seconds. Monod and Scherrer noticed that if you plot Joules on the y-axis and time in seconds on the x-axis, the relationship is nearly linear for durations from around 3 minutes out to around half an hour. A linear relationship like that can be approximated by a line with an intercept and a slope. The slope is what they called CP, and the intercept is what we now call W'.

If Joules(t) = W' + cp*t, then an equivalent formulation is the one you've already seen. Since a watt is just a Joule per second, just divide the equation by time on both sides to get watts(t) = W'/t + cp. In this formulation, if you plot watts against time, CP will be the asymptote and W' will be a constant that determines a hyperbola. You can see why some people prefer the intercept and slope formulation rather than the asymptote and hyperbola formulation.

Note that because W' is the y-intercept, it describes an amount of energy in Joules; because CP is a slope in the regression of Joules on seconds, it describes power in watts. Thus, if CP were 200 watts and W' were 15000 Joules (= 15 kilojoules), then a prediction for the amount of work you could do in 600 seconds (= 10 minutes) would be 15000 + 200 * 600 = 13500 Joules over 600 seconds, or an average of 13500/600 = 225 watts. You can think of W' as a "stock" of energy (like a storage battery) that you can expend in addition to the steady state power you can sustainably produce (that is, CP). You can expend that W' quickly or slowly but once the battery is run down you have to recharge it. In this example, you were expending it over 600 seconds, so another way to think of it is this: if your CP were 200 watts and you have 15000 Joules of stored energy, your baseline is 200 watts and you can add to that 15000/600 = 25 watts for a total of 225. If you expended it over 300 seconds (= 5 minutes) your baseline is still 200 but you can add to that 15000/300 = 50 watts, so your predicted maximum power for 5 minutes would be 250 watts. This alternative way of calculating predicted maximum power is Watts(t) = W'/t + cp.

In either formulation, if you have values for W' and CP, you can calculate the estimated power for any duration from around 30 seconds up to around an hour. (If you use the Joules formulation, you just have to divide by the duration to get watts). That's how the CP curves that you've seen are calculated: using values for W' and CP, you can create a curve and plot it showing predicted power at any duration. Note that with only two parameters, the model implies a power duration curve that has an asymptote not only at CP, but also one with the y-axis at zero duration. This is why models with more parameters have been developed -- to deal with the asymptote at zero and the asymptote at long durations.

From a practical perspective, typically your curve of time to exhaustion will not be "smooth" if you base it on just a single ride -- usually, you combine the maximal durations for several rides over a few days or weeks. Also, to get reliable estimates of W' and CP you'll want to make sure you've put out maximal efforts. A typical procedure may be to do maximal efforts at, say, 3 minutes, somewhere in the range of 5 to 8 minutes, and perhaps somewhere in the range of 15 to 20 minutes. That will give you 3 data points. You want these efforts to be maximal for the targeted duration, and typically that means you want them to be relatively steady -- there will always be some variability in your efforts but you don't want the efforts to be repeated sprints of 30 seconds followed by 30 seconds of recovery. Aim for the maximal output you can sustain. I don't recommend that you do these 3 efforts in one ride or even over 3 consecutive days -- space them out over a few days.

That procedure will give you 3 data points of maximal power and durations. Convert to Joules, and calculate the regression slope and intercept. A side effect of this is that you will have the usual regression diagnostics, with standard errors for the slope and intercept and an R^2 so you can evaluate goodness-of-fit. If you do more tests, you can use more than 3 data points for your regression.

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  • Thank you for your detailed answer. However, to be honest, I still don't quite understand what W' is. Is it the accumulated power up until that point or is it just the max. power I have recorded during my intervals? Lets say I did those 4 intervals on separate days: my max 3 min power is 270 Watts, 5 minutes is 250 Watts and 20 minutes is 200 Watts. My Critical Power is 180 Watts. How would an example calculation look like for these values? – Sascha S Apr 1 at 3:30
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    I've expaned the description of how to interpret W' and CP, and added an example with W' = 15 kJ and CP = 200 watts. See if it helps. For the specific example in your comment (180 secs = 270, 300 secs = 250, 1200 secs = 200), CP = 186 watts and W' = 17027 Joules. If you do the regression, note that the standard error on the estimated W' is large relative to the estimated CP. – R. Chung Apr 1 at 15:00
  • Hello again, thank you for the additional explanation. – Sascha S Apr 1 at 17:35
  • I don't know if I a just not capable of seeing the obvious...but where did the value for 17027 Joules come from?the problem is that I somehow only see values that I need to calculate so I can use the formulae but not really a starting point. So...the 17027 Joules are based on which number exactly? Sorry again to bother you... – Sascha S Apr 1 at 17:48
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    Ah. Joules are watts*seconds, so the Joules are 48.6 kJ, 75 kJ, and 240 kJ. Do a regression of Joules on seconds and the slope will be 186 watts and the y-intercept will be 17027 Joules. See anonymous.coward.free.fr/temp/cp-w'.png – R. Chung Apr 1 at 21:24
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You can watch Marks videos here that will explain W' vs CP in detail, he has a few different videos here but this is one https://vimeo.com/283303558

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    Please edit your question to provide a summary of the off-site link. Otherwise we can end up with bitrot and a useless answer. – Criggie Apr 2 at 20:02
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    Not to beat a dead horse, but wWe prefer answers on this site to be self-contained](bicycles.stackexchange.com/help/how-to-answer). That way, the answer is still valid if the link dies. Please use the edit button to summarize the information is contained in the video within the body of your answer. Otherwise, it is likely to be downvoted, flagged for moderator intervention, and possibly deleted. – jimchristie Apr 3 at 0:57

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