So, the implementation of W'bal in GoldenCheetah has been a bit of a challenge.
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…
Below, in the spirit of the great continental philosophers, is a dialectic that attempts to synthesize the typical arguments that arise when debating a polarized training approach.
It is not intended to serve as an introduction to Polarized training, there are many of those in-print and online. I think that Joe Friel's blog post is a good intro for us amateurs.
Elite athletes have been shown in a number of
studies to train in a polarized manner  There
is more than one way to skin a cat.Elite athletes
adopt plans that include high-volumes of low intensity and
low-volumes of high-intensity.
Elite athletes have also been shown to train in a
Polarized Zones are between LT1/VT1 and LT2/VT2
LT1/VT1 and LT2/VT2 can be identified using a
number of field based approaches 
You can follow guidelines on
mapping LT1/LT2 to cycling power to make it useful for
Polarized zones are har…
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;
comment on you Blog post on optimising the Wbal model. I've done
thinking and I defn think it can be done without visiting the
as the Skiba formula can be decomposed further, i.e. From your blog I believe the integral part of the equation is:
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
of exponential functions allow the for…