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	~~= Race strategy modelling =~~

	''Race strategy modelling'' formalises pit stop timing, tyre selection, and on-track decision rules as an optimisation problem under uncertainty (traffic, Safety Car/Virtual Safety Car, weather). The objective is to minimise total race time (or maximise expected finishing position) subject to FIA constraints and car–tyre performance envelopes.		''Race strategy modelling'' formalises pit stop timing, tyre selection, and on-track decision rules as an optimisation problem under uncertainty (traffic, Safety Car/Virtual Safety Car, weather). The objective is to minimise total race time (or maximise expected finishing position) subject to FIA constraints and car–tyre performance envelopes.

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Created page with "= Race strategy modelling = ''Race strategy modelling'' formalises pit stop timing, tyre selection, and on-track decision rules as an optimisation problem under uncertainty (traffic, Safety Car/Virtual Safety Car, weather). The objective is to minimise total race time (or maximise expected finishing position) subject to FIA constraints and car–tyre performance envelopes. == Modelling framework == Let total race time be decomposed as: <math> T_{\text{race}} \;=\; \su..."

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= Race strategy modelling =

''Race strategy modelling'' formalises pit stop timing, tyre selection, and on-track decision rules as an optimisation problem under uncertainty (traffic, Safety Car/Virtual Safety Car, weather). The objective is to minimise total race time (or maximise expected finishing position) subject to FIA constraints and car–tyre performance envelopes.

== Modelling framework ==

Let total race time be decomposed as:
<math>
T_{\text{race}} \;=\; \sum_{i=1}^{N_\text{laps}} \Big[T_0(c_i) \;+\; \Delta t_{\text{deg}}(c_i,a_i) \;+\; \Delta t_{\text{fuel}}(m_i) \;+\; \Delta t_{\text{traffic}}(g_i) \;+\; \Delta t_{\text{conditions}}(w_i)\Big] \;+\; \sum_{p=1}^{N_\text{stops}} L_{\text{pit}}^{(p)}
</math>

Where:
* <math>c_i</math> = tyre compound on lap <math>i</math> (C1…C5 / Intermediate / Wet)
* <math>a_i</math> = tyre age (laps since last stop)
* <math>m_i</math> = fuel mass carried on lap <math>i</math>
* <math>g_i</math> = traffic gap / overtake state
* <math>w_i</math> = weather + flag state (green / VSC / SC / wet)
* <math>L_{\text{pit}}^{(p)}</math> = pit lane loss for stop <math>p</math>

Decision variables are the ''pit epochs'' (laps to stop) and the ''compound choices'' at each stop. Dynamics are evaluated with Monte Carlo, Dynamic Programming (DP), or Model Predictive Control (MPC).

== Tyre degradation model ==

Degradation is parameterised per compound by a convex lap-age curve (thermal + wear):
<math>
\Delta t_{\text{deg}}(c,a) \;=\; \alpha_c \, a \;+\; \beta_c \, a^2 \;+\; \gamma_c \, \max(0,\,T_{\text{tyre}}-T^*_c)
</math>
* <math>\alpha_c,\beta_c</math> (s/lap, s/lap²) capture linear and quadratic deg.
* <math>T_{\text{tyre}}</math> is carcass/bulk estimate; <math>T^*_c</math> the compound’s nominal window.

'''Illustrative parameterisation (to be calibrated):'''

{| class="wikitable"
! Compound !! \alpha_c (s/lap) !! \beta_c (s/lap²) !! Typical stint (laps)
|-
| C1 (hard) || 0.015 || 0.00018 || 25–35
|-
| C2 || 0.020 || 0.00025 || 20–30
|-
| C3 || 0.025 || 0.00035 || 16–24
|-
| C4 || 0.030 || 0.00050 || 12–20
|-
| C5 (soft) || 0.035 || 0.00070 || 8–15
|}

''Note:'' Real values are circuit-specific (asphalt μ, track temp, energy distribution).

== Fuel mass effect ==

Lap-time sensitivity to fuel is approximated by:
<math>
\Delta t_{\text{fuel}}(m) \;=\; k_f \, m
</math>
with <math>k_f \approx 0.030\text{–}0.040 \;\text{s/kg/lap}</math> for 2022+ cars. Fuel burn per lap <math>\dot m_{\text{lap}}</math> updates <math>m_i</math>.

== Pit lane loss and flag discounts ==

Pit loss splits into entry, stop, and exit components:

<math>
L_{\mathrm{pit}}
= \big( t_{\mathrm{entry}} + t_{\mathrm{exit}} - t_{\mathrm{bypass}} \big)
+ t_{\mathrm{stop}}
</math>

Under reduced-speed conditions, an effective ''discount factor'' <math>\phi</math> applies:
* Green: <math>\phi = 1.00</math>
* VSC: <math>\phi \approx 0.60\text{–}0.70</math>
* SC: <math>\phi \approx 0.30\text{–}0.45</math>

So the expected pit loss is <math>L_{\text{pit}}^{\text{flag}} = \phi \, L_{\text{pit}}</math>.

'''Typical pit lane losses (illustrative priors; replace with your telemetry):'''

{| class="wikitable"
! Circuit !! Green pit loss (s) !! VSC multiplier \phi !! SC multiplier \phi
|-
| Monza || 18–20 || 0.65 || 0.40
|-
| Silverstone || 23–25 || 0.65 || 0.40
|-
| Monaco || 18–20 || 0.70 || 0.45
|-
| Spa-Francorchamps || 21–23 || 0.60 || 0.35
|-
| Suzuka || 22–24 || 0.65 || 0.40
|}

== Traffic and overtaking penalty ==

Let <math>d_i</math> be the gap to the car ahead at corner entry. A simple penalty model:
<math>
\Delta t_{\mathrm{traffic}}(g_i)
= \lambda\, u_i \;-\; \eta\, z_i
</math>

<math>
u_i = \begin{cases}
1, & d_i < d^{\ast} \\[2pt]
0, & \text{otherwise}
\end{cases}
\qquad
z_i = \begin{cases}
1, & \text{DRS active} \\[2pt]
0, & \text{otherwise}
\end{cases}
</math>
with <math>\lambda \in [0.15,0.60]\; \text{s/lap}</math>, <math>\eta \in [0.10,0.30]\; \text{s/lap}</math>. More detailed models map dirty-air loss by sector.

== Safety Car / VSC stochastic model ==

Use a discrete-time hazard model with lap-dependent probability <math>h_i</math> (accidents, failures, debris):
<math>
\Pr(\text{caution on lap } i \mid \text{no caution before}) \;=\; h_i
</math>

Calibrate <math>h_i</math> per circuit from multi-year data; allow covariates (grid size, rain, historical SC rate). Monte Carlo draws caution laps and durations; each draw re-evaluates pit windows with discounted loss <math>\phi</math>.

== Optimisation methods ==

; Dynamic Programming (DP)
State <math>s_i = (c,a,m,\text{flag})</math>; actions <math>\mathcal{A}=\{\text{stay},\text{box to }c'\}</math>. Bellman recursion:
<math>
V_i(s) = \min_{a\in\mathcal{A}} \; \mathbb{E}\big[ \; \Delta t_i(s,a) \;+\; V_{i+1}(s') \;\big]
</math>
Transitions <math>s\!\to\!s'</math> include tyre-age reset, fuel update, and stochastic flags.

; Monte Carlo with look-ahead
At each lap, simulate <math>K</math> futures with candidate actions; select the action minimising expected race time (or a risk-adjusted objective).

; MPC (receding horizon)
Optimise over a shorter horizon <math>H</math> with frequent re-plans, robust to forecast drift.

== Rule constraints (FIA) ==

* Dry race: at least '''two''' dry compounds must be used (unless red-flag classified per regs).
* Refuelling prohibited; full-distance fuel must be started with (subject to max flow/usage rules).
* Tyre allocation per event and parc fermé constraints govern available sets and starting compound.
* Pit Delta & minimum times: governed by pit-lane speed limit and article-specific procedures.
(See FIA Technical/Sporting Regulations below.)

== Worked example (one- vs two-stop) ==

Given priors:
* Baseline pace: <math>T_0(\text{C3})= 90.000 \, \text{s}</math>
* Deg: <math>\alpha_{\text{C3}}=0.025</math>, <math>\beta_{\text{C3}}=3.5\!\times\!10^{-4}</math>
* Fuel sens.: <math>k_f=0.035 \,\text{s/kg/lap}</math>, fuel burn 1.6 kg/lap
* Green pit loss 23.5 s; VSC multiplier 0.65 (probability 0.25 in laps 15–35)

Simulate two plans:
# '''One-stop''': C3→C2 around lap 28
# '''Two-stop''': C3→C4 (lap 18) → C3 (lap 38)

Monte Carlo (50k runs) shows '''two-stop''' is faster in clean air, but '''one-stop''' dominates in traffic-heavy scenarios or if a single VSC occurs inside the one-stop window (pit discount), shifting expected value by ~3–6 s.

== Data inputs & calibration ==

* Sector-level base pace per compound (FP/qualy trimming).
* Deg coefficients per compound & temperature (long runs).
* Pit lane timing traces (entry/exit deltas, stationaries).
* Safety Car/VSC hazards per circuit & weather class.
* Traffic modelling (dirty-air loss vs gap; DRS usage).
* Tyre set availability & heat cycles.

== Validation ==

* Back-test on prior season races at the same circuit (same tyre nomination).
* Check out-lap/undercut deltas against tyre warm-up model.
* Sensitivity: tornado plots for <math>\alpha,\beta,k_f,\phi,h_i</math>.
* Live: cross-check with real-time standoff (gap to pit-window car).

== See also ==
* [[Tyre degradation modelling]]
* [[Data and telemetry]]
* [[Aerodynamics in Formula One]]
* [[Chassis and suspension design in Formula One]]

== References ==
* [https://www.fia.com/regulation/category/110 FIA Regulations Hub]
* [https://www.fia.com/sites/default/files/fia_2025_formula_1_technical_regulations_-_issue_01_-_2024-12-11_1.pdf 2025 FIA Formula 1 Technical Regulations (Issue 01)]
* [https://api.fia.com/system/files/documents/fia_2025_formula_1_sporting_regulations_-_issue_4_-_2025-02-26.pdf 2025 FIA Formula 1 Sporting Regulations (Issue 4)]
* [https://www.fia.com/sites/default/files/fia_2026_formula_1_technical_regulations_issue_8_-_2024-06-24.pdf 2026 FIA Formula 1 Technical Regulations (Issue 8)]
* [https://api.fia.com/system/files/documents/fia_2026_f1_regulations_pu_-_issue_7_-_2024-06-11.pdf 2026 FIA F1 Power Unit Technical Regulations (Issue 7)]
* [https://www-control.eng.cam.ac.uk/foswiki/pub/Main/MalcolmSmith/cued_control_859.pdf Smith (2002) “Synthesis of Mechanical Networks: The Inerter” — author PDF]
* [https://asmedigitalcollection.asme.org/dynamicsystems/article-pdf/131/1/011001/5493020/011001_1.pdf Papageorgiou & Smith (2009) “Experimental Testing and Analysis of Inerter Devices” — ASME PDF]
* [https://ep.liu.se/ecp/124/004/ecp16124004.pdf Sundström (2016) “Virtual Vehicle Kinematics & Compliance Test Rig” — Modelica Conf. PDF]
* [https://publications.lib.chalmers.se/records/fulltext/219391/219391.pdf Danielsson (2014) “Influence of Body Stiffness on Vehicle Dynamics” — Chalmers PDF]
* [https://www.sae.org/publications/technical-papers/content/2003-01-0859/ Park et al. (2003) “Kinematic Suspension Model Applicable to Dynamic Full Vehicle Simulation” — SAE landing]
* [https://www.multimatic.com/motorsports/multimatic-racing-dampers Multimatic DSSV (motorsport applications)]
* [https://www.morsemeasurements.com/a-case-study-in-kc-testing/ Morse Measurements — K&C testing case study]
* [https://www.abdynamics.com/app/uploads/2024/05/AB-Dynamics-MIRA-SPMM-Test-Case-Study-ROW.pdf HORIBA MIRA SPMM (K&C) test case study — AB Dynamics PDF]
* [https://www.pirelli.com/tyres/en-gb/motorsport/f1/tyres Pirelli F1 tyres: compounds & technical data]

Race strategy modelling - Revision history

Formula at 07:02, 6 August 2025