Problem Chosen
A
2022
MCM/ICM
Summary Sheet
Team Control Number
2207343
Optimal Power Allocation − Ride to The Future
Who would have thought that the champion of the Tokyo Olympics cycling time trial was a
mathematician? Believe it or not, math does it. In this paper, we will build a mathematical model of
the power curve to help riders win races.
In Task 1, we build a power-duration model based on biological principles. This model has three
stages: Extreme, Severe, Heavy. After substituting the searched rider data into the model, we plot
the power curves of the sprinter and time trial specialist of different genders.
In Task 2, we first use the power curve in Task 1 to build the Human Energy Expenditure
Model, which describes the rider’s constraints on energy limits. We analyzed the dynamic bicycle
model to determine the P
out
− v relationship dur ing cycling. By analyzing the characteristics of the
track, we establish a piecewise nonlinear programming model with a wide range of applications for
straights, cur ves, and slopes. For large slope segments or shar p bends, we made detailed calculations
individually. Finally, the optimal output power cur ves of the r iders when riding on the three courses
in Task 1 are solved by Mathematica. The completion times are shown in the table below. Compared
with the champion, our model lags within 6%, reflecting the model’s accuracy.
Course Male sprinter Female sprinter
Male time
trial specialist
Female time
trial specialist
Self-designed 47.53min 48.58min 46.48min 48.20min
In Japan 62.68min 40.97min 56.03min 33.63min
In Belgium 58.15min 59.62min 53.03min 54.05min
After the model is solved, we perform sensitivity analysis in Task 3 and Task 4. For weather
factors, we focus on the effects of wind speed and direction. When the wind speed changes within
±2m/s and the direction changes within ±180
◦
, we get the change graph of the output power with
Python. We conclude that the model is robust as its sensitivity to crosswinds is within 1%. And it
has a 10% impact on headwinds, which provides us with ideas for team competition strategies.
We also tested the model’s sensitivity to rider deviations. Taking the Belgian track as an example,
we establish the riders’ actual output power function P
out
and add a random disturbance term
g
P
out
to
simulate the randomness of the rider. After calculation, the riders can finish the race before they run
out of energy. The time lag is within 2%.
Inspired by the model’s high sensitivity to headwind, we develop a team time trial strategy in
Task 5. The results are shown in the table below.
Stage Balance(48%) Storage(27%) Sprint(25%)
Formation Pyramid formation Linear array Shield formation
Finally, we write to Directeur Sportif with a race guidance for time trial specialists and sprinters.
Keywords: power curve, nonlinear prog ramming, dynamics model, formation planning
Summary