## Wednesday, July 17, 2013

### simulating effect of grade variations on climbing VAM

One issue in using VAM to predict power is the effect of variation in grade. I looked at this back in November, 2009. Grade variation leads to speed variation (at constant power) and speed variation results in an increase in energy dissipated to wind resistance. The analytic result I got was:

Δp / p0
3 f [ <Δgrade²> / (grade0 + CRR)² ] × [ ( 1 - f ) / ( 1 + 2 f ) ]²

where f is the fraction of power from wind resistance, <Δgrade²> is the variance of grade with respect to time (this is to first order the variance with respect to position), and CRR is the coefficient of rolling resistance. Using iBike data from a ride up Old La Honda (iBike directly measures grade, rather than indirectly via altitude), I concluded at constant power up Old La Honda the grade variation would lead to a 0.44% increase in power.

I can convert this to the effect on VAM by multiplying by the fractional effect of power on VAM:

ΔVAM / VAM0
3 f [ <Δgrade²> / (grade0 + CRR)² ] × ( 1 - f )2 / ( 1 + 2 f )3

This is a bit indirect so it's important to review the assumptions:

1. A rider is riding at constant power up a climb. This yields a certain speed variation associated with a certain grade variation.
2. The speed variation is going to yield a certain power increase due to wind resistance relative to riding at a steady, average speed.
3. The rider is riding a constant power, so higher required power must be canceled with a reduction in average speed.
4. Lower average speed reduces average VAM.

The reason this came up is the recent widespread interest in the use of power analysis to assess the likelihood of rider doping in the Tour de France. If you estimate power using certain assumptions, it's important to assess the error introduced by those assumptions.