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W'bal optimisation by a mathematician !

So the integral computation for W'bal was expensive.

I tried to optimise from a domain and  programming perspective, where Dave Waterworth, a mathematician found a much more elegant and fast reformulation.

This means W'bal can EASILY be computed as you ride.

To explain the math here are his words;

I posted a comment on you Blog post on optimising the Wbal model. I've done some more thinking and I defn think it can be done without visiting the previous samples as the Skiba formula can be decomposed further, i.e. From your blog I believe the integral part of the equation is:





Basically this takes a weighted sum of preceding W'exp samples where the weight decays at a rate determined by tau, older samples are weighted less than newer ones. We can approximate as a sum provided tau is large compared to Ts (the sample rate):





Basic properties of exponential functions allow the for…

W'bal its implementation and optimisation

So, the implementation of W'bal in GoldenCheetah has been a bit of a challenge.

The Science I wanted to explain what we've done and how it works in this blog post, but realised that first I need to explain the science behind W'bal, W' and CP.

W' and CP How hard can you go, in watts, for half an hour is going to be very different to how hard you can go for say, 20 seconds. And then thinking about how hard you can go for a very long time will be different again. But when it comes to reviewing and tracking changes in your performance and planning future workouts you quickly realise how useful it is to have a good understanding of your own limits.

In 1965 two scientists Monod and Scherrer presented a ‘Critical Power Model’ where the Critical Power of a muscle is defined as ‘the maximum rate of work that it can keep up for a very long time without fatigue’. They also proposed an ‘energy store’ (later to be termed W’, pronounced double-ewe-prime) that represented a finit…