Grade variability and climbing power

I've looked at this matter before, but one factor which I've seen continually neglected in all of the climbing power analysis estimates is the effect of grade variability. Road grade on climbs is almost never constant: it varies about a mean in some fashion. Yet the estimates are almost always done assuming constant speed, constant power.

Now these estimates end up remarkably accurate anyway. Why? Because the grade variability effect is negligible? Well, no. It's because you're canceling one mistake with another. For example, you neglect grade variability, which always increases power, but you also neglect drafting, which always decreases power.

How does grade variability increase power? It's because grade variability typically results in speed variability and speed variability yields variability results in variability in wind resistance and wind resistance, by virtue of being superlinear, is increased more by increases in speed than it is decreased by decreases in speed. So if instead of maintaining a constant speed v, if instead I am v + Δv for half the climbing time, then v - Δv for the 2nd half of the climbing time, the average wind resistance power is increased, assuming still air and no drafting:

[ (v + Δv)³ + (v - Δv)³ ] / 2 - v³ =
[ v³ + 3 v²Δv + 3 v (Δv)² + (Δv)³ - 3 v²Δv + 3 v (Δv)² - Δv³ ] / 2 - v³ =
3 v (Δv)²

So the fractional increase in wind resistance power from grade variability, for a given average speed, is:

ΔP_w/P_w = 3 (Δv)² / v²

Note this is just the aerodynamic portion of the power.

This is very simple, but it's in terms of speed, not grade. You can ride a variable grade at a constant speed, and in this case the average power will be calculated using the average grade as is typically done. However, this would result in a variable power, and according to Coggan's normalized power approach this would result in a lower power than could be attained by riding at a constant power, going faster on the flatter portions and slower on the steeper portions. Indeed, I think it's clear that this is almost always done. Essentially nobody rides climbs at constant speed.

So I need a relationship between grade and speed. The key here is that I neglect inertia, which tends to make speed more constant than otherwise. But inertia is only sustained over short distances, so I'm assuming grade variations to a resolution of no better than 10 meters distance or so.

Speed versus grade is a nonlinear problem: a cubic equation. So to estimate this effect you use linear analysis: linearize the nonlinear function.

Back in 2009 I analyzed the effect of grade on VAM, and a part of that calculation was the effect of grade on speed. This calculation begins with the following approximation for power, which is fairly standard in still air:

P = f_m m s + f_w s³
where f_w is the coefficient of power on speed-cubed, f_m is the coefficient of power on mass (gravity times rolling resistance coefficient plus grade), m is mass, and s is speed. It's not hard to go from this equation to the following using the chain rule:

∂s / ∂grade = ‒ m s g / (f_m m + 3 f_w s²)
where f_m = ( grade + C_RR ) × g, where grade is the road grade, C_RR is the coefficient of rolling resistance, and g is the acceleration of gravity (not to be confused with road grade).

The result of this is if there is a variation in grade σ²_grade, there will be a corresponding variation in speed σ²_s:

σ²_s = σ²_grade × [m s g / (f_m m + 3 f_w s²) ]²

So the result is, recognizing that the average value of (Δv)² ≡ σ²_s:

ΔP_w/P_w = 3 σ²_grade × [m g / (f_m m + 3 f_w s²) ]²

This equation has too many constants and it's hard to get a grasp for what it means. However, if I define mass-proportional and wind-resistance unitless fractions of retarding force (and of power) α_m (mass-proportional) and α_w (wind resistance proportional), then I can write this as follows:

ΔP_w/P_w = 3 σ²_grade × [m g s / (α_m + 3 α_w) P ]²

But α_m + α_w = 1, so this can be slightly simplified:

ΔP_w/P_w = 3 σ²_grade × [m g s / (1 + 2 α_w) P ]²
given that ΔP = ΔP_w, since climbing power depends only on the average VAM (and is insensitive to speed fluctuations while rolling resistance power depends only on the average speed.

Then if I define α_CRR is the fraction of the mass (or weight)-proportional power which is due to rolling resistance, I can multiply and divide by grade, then observe the numerator is a power due to grade, and I can simplify it further:

ΔP_w/P_w = 3 (σ_grade / grade)² × [ (1 - α_CRR) (1 - α_w) / (1 + 2 α_w) ]²

This is all in unitless quantities and so is easier to grasp. It works for everything except zero or small average grade (for which you should use one of the previous forms).

The nice thing about unitless quantities is they tend to be more universal without requiring specific estimates for a given rider. For example, suppose we're dealing with a grade which varies 20% about the mean (for example, σ_grade is 1.4% with a mean grade of 7%), and 15% of the power goes into wind resistance, and rolling resistance coefficient is around 0.4% (so responsible for around 0.4% / (7.0% + 0.4%) ≈ 5% of the mass-proportional power), then I get a 1.5% increase of aerodynamic power relative to the assumption of constant speed.

If I want to convert this to fraction increase in total power, I need to multiply by the fraction of total power which is aerodynamic power:

ΔP/P = ( ΔP_w/P_w ) ( P_w / P ) = ( ΔP_w/P_w ) α_w

I then get:

ΔP/P = 3 (σ_grade / grade)² × α_w [ (1 - α_CRR) (1 - α_w) / (1 + 2 α_w) ]²

So going from that 1.5% I need to multiply by my assumed 15% total power from wind resistance and that brings me to 0.23%, so a small fraction of the total.

I think this is a typical example which implies a persistent underestimation of climbing power in time trial like efforts of 0.7% just considering the effect of grade variability. Of course there's other sources of variability which will have similar effect, for example variations in speed due to non-uniform efforts, such as in mass-start races where tactics come into play (going more conservatively at the base of a climb, attacking toward the finish, etc). But the result ends up being fairly minor assuming I didn't make any errors.

An interesting aspect of this formula is that variations in a grade of a given fraction have zero influence in limits both where grade is zero (because fractional variations of near-zero are near-zero) and also in the limit of no wind resistance (because for climbing power variations in speed average out). It's only in the middle range: climbing hills fast enough that wind resistance is still a factor, that the grade variations are significant.

In summary, here's a description of the parameters I used in this analysis:

parameter	units	formula	description
fm	acceleration	(grade + CRR)×gravity	mass (and speed)-proportional power
fw	mass/distance	1/2 ρCDA2	speed-cubed-proportional power
CRR	1		coefficient of rolling resistance (for example 0.4%)
CD	1		coefficient of wind resistance (for example 80%)
ρ	mass/volume		mass-density of air (for example 1.1 kg/m3)
A	area		effective cross-sectional area of bike + rider (for example 0.4 meters2)
αRR	1	CRR/(CRR + grade)	fraction of mass-proportional power due to rolling resistance
αw	1	1 / (1 + fmm / fws2)	fraction of power due to wind resistance
αm	1	1 / (1 + fws2 / fmm)	fraction of power due to mass-proportional power