<p>In this paper, we have developed a numerical design methodology for computing optimal pacing strategies for the individual time trial discipline in professional cycling. These strategies minimise the finishing time for a given cyclist racing on a given course by optimising how their power output is distributed along the course. The method is based on a finite element formulation and adjoint sensitivity analysis is used to minimise the finishing time subjected to a physiological constraint based on the principle of normalised power. We apply the method to four hypothetical courses of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12283_2025_493_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="36" /> </InlineMediaObject> <EquationSource Format="TEX">$2\;\textrm{km}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mspace width="0.277778em" /> <mtext>km</mtext> </mrow> </math></EquationSource> </InlineEquation> simulating various gradients and wind conditions. A parameter-dependent simulation showed between <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12283_2025_493_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$0.45\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0.45</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12283_2025_493_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$2.84\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2.84</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> improvements in finishing times compared to benchmark pacing strategies. The method is also applied on a real-world course and the results are compared to the pacing strategy of professional cyclist and ITT specialist Martin Toft Madsen. The optimised strategy is <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12283_2025_493_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$1.2\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1.2</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> faster over <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12283_2025_493_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$21.3\;\textrm{km}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>21.3</mn> <mspace width="0.277778em" /> <mtext>km</mtext> </mrow> </math></EquationSource> </InlineEquation>. We believe that the method presented here constitutes a promising framework for efficient computation of optimal pacing strategies and with further research and a more accurate physiological model; this could prove an important tool for strategising in professional cycling.</p>

A numerical design methodology for optimal pacing strategy in the individual time trial discipline of cycling

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2025年3月13日

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Asker Friis Bach,
Joe Alexandersen,
Christian Bach Lundgaard

摘要

In this paper, we have developed a numerical design methodology for computing optimal pacing strategies for the individual time trial discipline in professional cycling. These strategies minimise the finishing time for a given cyclist racing on a given course by optimising how their power output is distributed along the course. The method is based on a finite element formulation and adjoint sensitivity analysis is used to minimise the finishing time subjected to a physiological constraint based on the principle of normalised power. We apply the method to four hypothetical courses of $2\;\textrm{km}$ $2 km$ simulating various gradients and wind conditions. A parameter-dependent simulation showed between $0.45\%$ $0.45 %$ and $2.84\%$ $2.84 %$ improvements in finishing times compared to benchmark pacing strategies. The method is also applied on a real-world course and the results are compared to the pacing strategy of professional cyclist and ITT specialist Martin Toft Madsen. The optimised strategy is $1.2\%$ $1.2 %$ faster over $21.3\;\textrm{km}$ $21.3 km$ . We believe that the method presented here constitutes a promising framework for efficient computation of optimal pacing strategies and with further research and a more accurate physiological model; this could prove an important tool for strategising in professional cycling.