How accurately a physical model predicts time-trial times, and which input weighs most in the error

Time trial19 October 20266 min readBorja Alfaraz

Sensitivity

11 s

That is how much a 40 km time trial’s time changes for every 1 % error in CdA.

Model calculation: 300 W, CdA 0.24 m², 78 kg, 55.4 minutes. A 1 % power error is 12 seconds and an extra kilo, 2.

Summary. Physical cycling power models reproduce power measured on the road very accurately when the inputs are right¹, and they can predict performances². Prediction error almost always comes from the inputs: CdA, power, weight, rolling resistance and wind. We quantify how much each weighs over a 40 km time trial and propose how to reduce the error.

The bike equation is not the hard part: a validated model explains 97 % of the variation in power measured on the road¹. The hard part is knowing the inputs precisely. In a flat 40 km time trial, a 1 % CdA error moves the time by 11 seconds; 1 % of power, 12; a kilo of weight, 2; and half a thousandth of Crr, 19.

Which input weighs most in the error?

Input error	Change over 40 km
Power 3 % lower	+37 s
CdA 3 % higher	+31 s
Crr 0.0005 higher	+19 s
Power 1 % lower	+12 s
CdA 1 % higher	+11 s
1 kg more	+2 s

Table 1. Change in a flat 40 km time trial (55.4 min at 300 W) for each input error. Model calculation [1].

On the flat, power and CdA rule. Weight barely counts on the flat, but on a course with climbs its weight grows a lot (see hilly time trial). Wind, which is not in the table, can be the biggest error of all on a windy day (see out-and-back time trial in the wind).

Where do errors come from in practice?

CdA: measured on another day, with another helmet or in another position. A well-run field test keeps it within a few per cent³.
Power: what the rider can hold on race day is not their best test. Power meters also have error, of the order of 1 to 2 %.
Wind and density: they depend on the forecast and the actual time of passage.
Course: bends, roundabouts and braking that force losing and regaining speed.

How do you sharpen a prediction?

Measure CdA with the race equipment and position, and repeat the test (see Chung method errors).
Use the power sustainable for the event’s duration, not the power from a short test.
Calculate density and wind with the weather at the time of passage.
After the race, compare predicted and actual time by section. If the difference is systematic, it almost always points to an input, above all CdA.

Rule of thumb: if the prediction always misses in the same direction, check CdA; if it misses sometimes high and sometimes low, check the wind.

What is known and what is not

That the physical model is accurate with good inputs is well established^1,2, as is how much each equipment or position gain is worth in time⁴. What limits race-day prediction is input uncertainty, above all wind and the power the rider will be able to hold.

Frequently asked questions

Can you predict a time-trial time?

Yes, with a physical model and good CdA, power, weight and weather data. The error depends almost entirely on the quality of those inputs.

What affects a flat time-trial time most?

Power and CdA. Each 1 % of either moves the time by 11 to 12 seconds over 40 km.

To see how a real time trial changes as you move each input, try the demo on the Tour time trial.

References

Martin JC, Milliken DL, Cobb JE, McFadden KL, Coggan AR. Validation of a mathematical model for road cycling power. J Appl Biomech. 1998;14(3):276-91. doi:10.1123/jab.14.3.276
Olds TS, Norton KI, Lowe EL, Olive S, Reay F, Ly S. Modeling road-cycling performance. J Appl Physiol. 1995;78(4):1596-611. doi:10.1152/jappl.1995.78.4.1596
Lim AC, Homestead EP, Edwards AG, Carver TC, Kram R, Byrnes WC. Measuring changes in aerodynamic/rolling resistances by cycle-mounted power meters. Med Sci Sports Exerc. 2011;43(5):853-60. doi:10.1249/MSS.0b013e3181fcb140
Jeukendrup AE, Martin J. Improving cycling performance: how should we spend our time and money. Sports Med. 2001;31(7):559-69. doi:10.2165/00007256-200131070-00009