Pedaling, Fast and Slow: The Race Towards an Optimized Power Strategy

Abstract: With the advent of power-meters allowing cyclists to precisely track their power outputs throughout the duration of a race, devising optimal power output strategies for races has become increasingly important in competitive cycling. To do so, the track, weather, and individual cyclist's abilities must all be considered. We propose differential equation models of fatigue and kinematics to simulate the performance of such strategies, and an innovative optimization algorithm to find the optimal strategy. Our model for fatigue translates a cyclist's power curve (obtained by fitting the Omni-Power Duration Model to power curve data) into a differential equation to capture which power output strategies are feasible. Our kinematics model calculates the forces on the rider, and with power output models the cyclist's velocity and position via a system of differential equations. Using track data, including the slope of the track and velocity of the wind, the model accurately computes race times given a power output strategy on the exact track being raced. To make power strategy optimization computationally tractable, we split the track into segments based on changes in slope and discretize the power output levels. As the space of possible strategies is large, we vectorize the differential equation model for efficient numerical integration of many simulations at once and develop a parallelized Tree Exploration with Monte-Carlo Evaluation algorithm. The algorithm is efficient, running in $O(ab\sqrt{n})$ time and $O(n)$ space where $n$ is the number of simulations done for each choice, $a$ is the number of segments, and $b$ is the number of discrete power output levels. We present results of this optimization for several different tracks and athletes. As an example, the model's time for Filippo Ganna in Tokyo 2020 differs from his real time by just 18%, supporting our model's efficacy.