Effect of Variable CdA on Combined CdA and Crr Solutions in Field Testing

Contents

The Chung (Virtual Elevation, or just VE) method is the most commonly known field testing method in cycling. This method, like other regression methods going back to automobile coast-downs, often uses the relationship of force change with speed, and the characteristic differences between rolling drag and aerodynamic drag, to calculate a CdA and Crr value from a joint data set. This methodology heavily leans on a constant CdA with speed, which is highly unlikely in most cycling scenarios. For background on this see The Case for a Variable CdA in Cycling.

Since the fundamental equations rely on a constant CdA with speed, and multiple different types of evidence suggest that actual data does not have this characteristic, the poor fitting from this mismatch creates an inherent bias. For a rising CdA with speed, the CdA value is biased high and the Crr biased low; for the more common descending CdA with speed, CdA is biased low and Crr is biased high.

It needs to be noted that not all VE tests use a simultaneous solve; sometimes either CdA or Crr is fixed. But this article will focus on the simultaneous solve since the knowledge of variable CdA has not been incorporated into the online field testing advice.

To put values to the bias created by a constant CdA equation with actual aerodynamic data, a VE test was simulated: hill climbs at three powers, each followed by a coast-down, with three variable CdA profiles from wind tunnel data.

The table and plots from the simulation show that the variable CdA data in a VE test that solves for both CdA and Crr has an invisible and inherent bias. These combinations saw a Crr bias of 8.7% to 61.8% and a CdA bias of −0.8% to −8.3%. This is under ideal conditions with no wind, no instrument noise, and no drift. The bias effect is dependent on the test speeds and drag profile, but the rides were chosen to represent ideal VE testing conditions, and the drag profiles chosen should be typical of no wind or low wind conditions. Other conditions would be expected to show larger biases, especially in wind, which gives larger differences in airspeed.
When looking at the plots of the different simulations, the scale is critically important. These runs have near perfect data with VE deviations in centimetres; the patterns that are seen will be completely obscured by the typical noise in the realm of ±0.5 to 1 m for a similar length run with actual data.
In addition to the bias which brings errors to absolute measurements, the speed makeup of the test becomes a factor in the measured outcome. In these tests, the Crr error differs by up to 29 percentage points across different speeds (Grappe 2009 at 225 / 300 / 375 W, +8.7% on set 1 and +38.1% on set 3), meaning that for comparison testing, any difference in airspeed becomes a source of error.
There is no speed combination that removes the bias.

Solving for CdA and Crr together also costs precision, which this article does not assess; the second widget below, which has a force error option, gives an idea of the effect.

This article is not intended to be a complete error analysis of Virtual Elevation, but an explanation of why a variable CdA undermines the purported accuracy of some methodologies including Virtual Elevation when both CdA and Crr are simultaneously solved using a variable speed and force comparison.

Details of the force model and the simulation are set out in the sections below.

Force model

A force model representing a simple form of the equation will be discussed first. This model will only use two points for illustrative purposes. In actual testing, a large number of data points are collected across a sweep of speeds. Those knowledgeable in VE will note that it is an energy balance and not a force balance; however, in the more detailed analysis later in the article, it can be seen that the outcome can be largely predicted from the force model.

CdA and Crr force relationship

The force model is the resistive force on a flat road with no wind, the sum of the rolling resistance and the aerodynamic drag:

\[F = m \, g \, \text{Crr} + \tfrac{1}{2} \, \rho \, \text{CdA} \, v^2\]

where m is the mass of rider and bike, g is gravity, ρ is the air density and v is the speed.

Pressing play on the widget will cycle through the relationship between CdA and Crr. It can be seen that an overall force increase on both points is emblematic of a Crr increase, while CdA appears as a larger increase in force at higher speeds and a smaller one at lower speeds.

It should be noted that two points are all that is needed to make this calculation; however, not all combinations of the two points are possible. When more than two points are used, the curve is fit to the cloud of points, which is similar in principle, but uses a weighting of all the points and does not necessarily pass through all the points.

Basic bias effect

The force model above assumes one CdA at every speed. To see what happens when that is not the case, take three drag curves that all pass through a CdA of 0.300 m² at 11.11 m/s (40 km/h): one constant, one falling by 0.008 m² for each m/s of airspeed, and one rising by the same amount (Figure 1). The slope is chosen to make the effect easy to see. The two points are those of the widget above, 5 m/s and 11.11 m/s, and the curves meet at the faster one, so the force there is the same for all three.

**Figure 1.** CdA against airspeed for three drag curves through 0.300 m² at 11.11 m/s (40 km/h): constant, falling by 0.008 m² for each m/s and rising by 0.008 m² for each m/s. The dots are the CdA of each curve at the two points, P₁ at 5 m/s and P₂ at 11.11 m/s, where the three meet.

Each curve gives a force at the two points, with the same rider and bike (85 kg), the same Crr of 0.003 and the same air density of 1.225 kg/m³. A constant CdA and Crr are then solved from the two forces, in the same way as in the widget. Figure 2 shows the force curve of each solution, and Table 1 gives the values.

**Figure 2.** Resistive force against speed. Black: the constant curve, CdA 0.300 m² and Crr 0.003. Dots: the force of the falling (blue) and rising (orange) curves at P₁, and the force at P₂, the same for all three (black). Dotted: the force curve of the constant CdA and Crr solved from each curve's two forces. Where a dotted curve meets the left axis is its rolling force, against the true rolling force in green.

Table 1. CdA of each drag curve at the two points, and the constant CdA and Crr solved from the two forces. True Crr is 0.003 throughout.

Drag curve	CdA at 5 m/s (m²)	CdA at 11.11 m/s (m²)	Force at 5 m/s (N)	Force at 11.11 m/s (N)	Solved CdA (m²)	Solved Crr	Crr error
Constant	0.3000	0.3000	7.09	25.18	0.3000	0.00300	0.0%
Falling, −0.008 m² for each m/s	0.3489	0.3000	7.84	25.18	0.2876	0.00413	+37.5%
Rising, +0.008 m² for each m/s	0.2511	0.3000	6.35	25.18	0.3124	0.00187	−37.5%

The force at 11.11 m/s is the same for all three curves, as the CdA there is the same. The only difference is at 5 m/s, where the falling curve has more drag than the constant one and the rising curve has less. With a constant CdA, the only thing that can raise the force at the low speed point while leaving the high speed point alone is more rolling resistance, so the extra drag of the falling curve is read as Crr. The higher Crr then takes force away from the aerodynamic share at both points, and the solved CdA drops.

A CdA that falls with speed biases Crr high and CdA low.
A CdA that rises with speed biases Crr low and CdA high.

The solved CdA is not an average of the two actual values. For the falling curve it is 0.2876 m², below the CdA at either point (0.3489 and 0.3000 m²), and for the rising curve it is 0.3124 m², above both. The Crr error, at 37.5%, is far larger than the change in force that caused it: the falling curve differs from the constant one by 0.75 N at 5 m/s.

CdA and Crr errors

Before the next widget is introduced, the variable drag wind tunnel data from Grappe 2009 and Brown 2023 will be briefly covered.details The plot below shows the Grappe data overlaid as a CdA offset with the Brown mannequin data with and without skinsuit at multiple turbulence levels. As most VE tests are conducted in calm conditions, the low turbulence Ix 2.5% was chosen for the Brown curves. Ix is the turbulence intensity of the air stream in the wind tunnel: the size of the fluctuations in the airspeed as a percentage of the mean airspeed.

**Figure 3.** CdA relative to its value at 11.11 m/s (40 km/h) against airspeed. Brown et al. (2023) mannequin, bare (left) and in a skinsuit (right), at six turbulence intensities (Ix 1.4 to 9.1%), with Grappe (2009) overlaid.

Figure 4 shows the three curves used in the rest of this article: Grappe, and Brown bare and skinsuit at Ix 2.5%, each as an offset from a CdA of 0.300 m² at 40 km/h. The wind tunnel data stop at 15 km/h (Grappe) and 27 km/h (Brown), so below those speeds the curves follow assumed points (red circles). The points continue the rise in CdA as speed falls, which is the most likely course: every curve is still rising at its lowest measured speed, and the Reynolds effects described in The Case for a Variable CdA in Cycling do not stop where a wind tunnel test happens to. There is no foundation for holding CdA flat below the last measured point.

**Figure 4.** CdA against airspeed for the three drag curves as simulated: 0.300 m² at 40 km/h plus each source's offset. Grappe (2009); Brown et al. (2023), Fig. 21, bare and skinsuit at turbulence intensity 2.5%. The control is a constant 0.300 m². Dots are measured points. The red circles are assumed points added below the lowest measured speed, the same for all three curves. The shaded band is the range of speeds on the analysed legs of the rides simulated in the VE model section, on the hill of Figure 5: 6.2 to 42.3 km/h. Below the lowest measured speed CdA comes from the assumed points: on that hill 7 to 17% of the analysed time on the Grappe rides (below 15 km/h) and 30 to 53% on the Brown rides (below 27 km/h).

The widget below brings in a lot more complexity.

The CdA and Crr sliders have been removed and replaced with a slider for the airspeed of each measurement point.
Force error options on the left allow some experimental error to be introduced into the process.
There is now the option of including variable CdA in the equation and applying the graphs from above.

Pressing play on the widget will cycle through different options.
With the constant CdA:

The 0.1 and 0.2 N force errors can be combined with different speed combinations to see the effects.
Higher speeds give more accurate CdA numbers with lower Crr accuracy.
Lower speeds give more accurate Crr numbers with lower CdA accuracy.

With variable CdA and no force error:

All speed options have considerable CdA errors, measured against the mean CdA between the two speeds.
Crr errors are much larger than CdA errors.
The decreasing CdA with speed biases CdA low and Crr high.

Adding a force error just makes the situation worse.

VE uses an energy balance which has a slightly different weighting between different speeds, and uses more points in the calculation, but the same fundamental weaknesses apply to any method that solves for CdA and Crr together. The next section will conduct simulations using a standard VE methodology.

VE model

For this analysis a hill was created for a hill climb and coast-down for an idealised VE test. The elevation profile is shown below. On this course the following power profiles were used, with each hill climb combined with a coast-down:

125 / 200 / 275 W
175 / 250 / 325 W
225 / 300 / 375 W

The rides were produced with a simulator, so that the true CdA and Crr are known and the analysis method can be tested against them.

As a note, since this is a simulation, all elevation and power meter numbers have perfect precision. The differences seen in the controls come from the conversion to 1 s time steps. A hill was chosen to create the speed differences, so that the format is that of a typical VE test. Different power numbers on a flat profile could also have been used.

The format of three different hill climb speeds with coast-downs has been recommended in podcasts by Robert Chung.details

**Figure 5.** Elevation and gradient of the simulated hill. Each climb rides east from gate A to gate B: 1021.8 m, rising 21.48 m, an average of 2.10% and at most 5.00%. The coast-down rolls back from gate B to gate A. The blue bands are where the analyser's own gates fell over the 12 rides on this hill, 45 to 75 m inside gate A and gate B.

Two steeper hills are used alongside this one, giving three (Figure 6). They are called set 1, set 2 and set 3. Set 3 is the hill of Figure 5: from gate A to gate B it rises 21.5 m at an average of 2.1%. Set 2 rises 33.7 m over the same distance, an average of 3.3%, and set 1 rises 43.6 m over 892 m, an average of 4.9%. All three share gate A and the road as far as the lowest point, and differ on the climb from there to gate B. A steeper hill gives slower climbs and a faster coast-down. The 125 W climb falls to 4.5 km/h on set 1 and 6.7 km/h on set 2, and to no less than 9.9 km/h on set 3, where the slowest point on the analysed legs (6.2 km/h) is on the coast-down. The coast-down reaches 61.7 km/h on set 1 and 52.1 km/h on set 2, and no ride on set 3 goes above 42.3 km/h.

**Figure 6.** Elevation and gradient of the three hills. Set 3 is the hill of Figure 5. Gate A is the same on all three, and west of the lowest point they are the same road. From gate A to gate B the hills average 4.9% (set 1), 3.3% (set 2) and 2.1% (set 3). Speeds on the analysed legs are 4.5 to 61.7 km/h on set 1, 6.7 to 52.1 km/h on set 2 and 6.2 to 42.3 km/h on set 3. Below the drag curve's lowest measured speed CdA comes from the assumed points of Figure 4: on set 1, set 2 and set 3 that is 28 to 54%, 36% and 7 to 17% of the analysed time on the Grappe rides (below 15 km/h) and 56 to 70%, 62% and 30 to 53% on the Brown rides (below 27 km/h).

All three power profiles were ridden on set 3, 125 / 200 / 275 W on sets 1 and 2, and 225 / 300 / 375 W on set 1 as well. Each combination was ridden with a constant CdA as a control and with the three drag curves of Figure 4, 24 rides in all.

Every ride with a drag curve spends part of its time below the slowest wind tunnel measurement of that curve, so part of each result rests on the assumed points of Figure 4: 7 to 54% of the analysed time on the Grappe rides and 30 to 70% on the Brown rides. The bias does not depend on them: the Grappe rides on set 3 spend only 7 to 17% of their analysed time below the measured range and still return a Crr 32 to 38% high.

The rides are simulated with no noise: rider and bike 85 kg, Crr 0.003, air density 1.225 kg/m³, no wind, one sample per second. A ride is three climbs from gate A to gate B, one at each power of the profile, each followed by a coast-down back to gate A. Each ride was then put through the Virtual Elevation Analyzer, a web analyser for VE, solving for CdA and Crr together.

Variable power and hill profiles

Figure 7 and Table 2 give the error in the CdA and Crr that the analyser returned for each ride. The control rides come back within 0.2% of the true values, so the errors on the other 18 rides come from the drag curve: Crr 8.7% to 61.8% high and CdA 0.8% to 8.3% low.

**Figure 7.** CdA error (top) and Crr error (bottom) returned by the VE analyser on each hill, against power profile, one line for each drag curve. A hill has a point only for the power profiles ridden on it: two on set 1, one on set 2 and three on set 3. The values are those of Table 2. The two rows have different vertical scales.

Table 2. CdA and Crr returned by the VE analyser, solving for both together, against the true values for the 24 rides: the three power profiles on set 3, 125 / 200 / 275 W on sets 1 and 2, and 225 / 300 / 375 W on set 1. True CdA is the average over the analysed legs, weighted by the energy lost to drag. Errors are relative to the true values. The rides are simulated with no noise. The last column links to the ride's analyser plots in the appendix.

Hill	Power profile	Drag curve	True CdA (m²)	VE CdA (m²)	CdA error	True Crr	VE Crr	Crr error	VE plots
Set 1	125 / 200 / 275 W	Control (constant CdA)	0.3000	0.3001	0.0%	0.00300	0.00301	+0.2%	Figure 11
Set 1	225 / 300 / 375 W	Control (constant CdA)	0.3000	0.3001	0.0%	0.00300	0.00300	+0.2%	Figure 12
Set 2	125 / 200 / 275 W	Control (constant CdA)	0.3000	0.3000	0.0%	0.00300	0.00301	+0.2%	Figure 13
Set 3	125 / 200 / 275 W	Control (constant CdA)	0.3000	0.3000	0.0%	0.00300	0.00300	+0.1%	Figure 14
Set 3	175 / 250 / 325 W	Control (constant CdA)	0.3000	0.3000	0.0%	0.00300	0.00300	+0.1%	Figure 15
Set 3	225 / 300 / 375 W	Control (constant CdA)	0.3000	0.3000	0.0%	0.00300	0.00300	+0.1%	Figure 16
Set 1	125 / 200 / 275 W	Grappe 2009	0.3128	0.3087	−1.3%	0.00300	0.00338	+12.6%	Figure 17
Set 1	225 / 300 / 375 W	Grappe 2009	0.3120	0.3095	−0.8%	0.00300	0.00326	+8.7%	Figure 18
Set 2	125 / 200 / 275 W	Grappe 2009	0.3109	0.3017	−2.9%	0.00300	0.00363	+21.0%	Figure 19
Set 3	125 / 200 / 275 W	Grappe 2009	0.3113	0.2934	−5.8%	0.00300	0.00395	+31.7%	Figure 20
Set 3	175 / 250 / 325 W	Grappe 2009	0.3098	0.2917	−5.9%	0.00300	0.00405	+35.1%	Figure 21
Set 3	225 / 300 / 375 W	Grappe 2009	0.3087	0.2907	−5.8%	0.00300	0.00414	+38.1%	Figure 22
Set 1	125 / 200 / 275 W	Brown 2023 bare	0.2955	0.2844	−3.8%	0.00300	0.00408	+36.0%	Figure 23
Set 1	225 / 300 / 375 W	Brown 2023 bare	0.2959	0.2824	−4.6%	0.00300	0.00440	+46.8%	Figure 24
Set 2	125 / 200 / 275 W	Brown 2023 bare	0.3038	0.2878	−5.3%	0.00300	0.00414	+37.8%	Figure 25
Set 3	125 / 200 / 275 W	Brown 2023 bare	0.3112	0.2875	−7.6%	0.00300	0.00426	+42.2%	Figure 26
Set 3	175 / 250 / 325 W	Brown 2023 bare	0.3092	0.2836	−8.3%	0.00300	0.00451	+50.3%	Figure 27
Set 3	225 / 300 / 375 W	Brown 2023 bare	0.3075	0.2839	−7.7%	0.00300	0.00451	+50.3%	Figure 28
Set 1	125 / 200 / 275 W	Brown 2023 skinsuit	0.2851	0.2719	−4.6%	0.00300	0.00433	+44.3%	Figure 29
Set 1	225 / 300 / 375 W	Brown 2023 skinsuit	0.2863	0.2688	−6.1%	0.00300	0.00485	+61.8%	Figure 30
Set 2	125 / 200 / 275 W	Brown 2023 skinsuit	0.2952	0.2757	−6.6%	0.00300	0.00441	+46.9%	Figure 31
Set 3	125 / 200 / 275 W	Brown 2023 skinsuit	0.3089	0.2872	−7.0%	0.00300	0.00415	+38.5%	Figure 32
Set 3	175 / 250 / 325 W	Brown 2023 skinsuit	0.3069	0.2849	−7.2%	0.00300	0.00431	+43.6%	Figure 33
Set 3	225 / 300 / 375 W	Brown 2023 skinsuit	0.3053	0.2853	−6.5%	0.00300	0.00427	+42.4%	Figure 34

Understanding the plots

The VE plot from the Brown 2023 bare ride at 175 / 250 / 325 W on set 3 shows why the errors are not detectable via VE.

**Figure 27a.** Brown 2023 bare, 175 / 250 / 325 W. Virtual elevation and residuals at the analyser's answer. RMSE 0.04 m.

This plot, like all of these plots, has an exceptionally low RMSE. This is due to the simulator producing very clean data. The noise that does appear is related to the 1 Hz sampling rate. According to the standard metric of elevation matching, this is an almost perfect run — except it's not: it has a CdA value which is 8.3% low and a Crr value that is 50.3% high. The analyser can only fit a constant CdA, and a constant CdA with a higher Crr matches the elevation better (RMSE 0.04 m) than the true values do (0.25 m).

The next plot shows what the actual correct values are:

**Figure 27b.** Brown 2023 bare, 175 / 250 / 325 W. Virtual elevation and residuals with CdA and Crr set to the true values. RMSE 0.25 m.

Here the plot shows a higher 0.25 m RMSE value and a slightly higher VE Gain of 0.03 m. The VE Gain is the change in virtual elevation over the lap, which should be zero as each lap returns to its starting point. Still very good, but looking at the numbers or the plot, the fit appears inferior to the previous fit.

The convergence plot below shows how tight the optima of CdA and Crr are: it maps how well the laps close for every combination of the two values, with a star at the true values and a circle at the analyser's answer. The analyser gives its own range, within 5 cm of the best closure, as CdA 0.284 ± 0.003 m² and Crr 0.0045 ± 0.0002, and the true values lie about eight times as far from its answer as that. From this plot the user has no way of telling what the true values are.