Editing Race strategy modelling

= 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]