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Race strategy modelling

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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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)} }

Where:

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_i} = tyre compound on lap Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} (C1…C5 / Intermediate / Wet)
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_i} = tyre age (laps since last stop)
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_i} = fuel mass carried on lap Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i}
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_i} = traffic gap / overtake state
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_i} = weather + flag state (green / VSC / SC / wet)
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{\text{pit}}^{(p)}} = pit lane loss for stop Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p}

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): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta t_{\text{deg}}(c,a) \;=\; \alpha_c \, a \;+\; \beta_c \, a^2 \;+\; \gamma_c \, \max(0,\,T_{\text{tyre}}-T^*_c) }

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_c,\beta_c} (s/lap, s/lap²) capture linear and quadratic deg.
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{\text{tyre}}} is carcass/bulk estimate; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^*_c} the compound’s nominal window.

Illustrative parameterisation (to be calibrated):

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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta t_{\text{fuel}}(m) \;=\; k_f \, m } with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_f \approx 0.030\text{–}0.040 \;\text{s/kg/lap}} for 2022+ cars. Fuel burn per lap Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot m_{\text{lap}}} updates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_i} .

Pit lane loss and flag discounts

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{\mathrm{pit}} = \big( t_{\mathrm{entry}} + t_{\mathrm{exit}} - t_{\mathrm{bypass}} \big) + t_{\mathrm{stop}} }

Under reduced-speed conditions, an effective discount factor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} applies:

  • Green: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi = 1.00}
  • VSC: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi \approx 0.60\text{–}0.70}
  • SC: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi \approx 0.30\text{–}0.45}

So the expected pit loss is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{\text{pit}}^{\text{flag}} = \phi \, L_{\text{pit}}} .

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

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_i} be the gap to the car ahead at corner entry. A simple penalty model: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta t_{\mathrm{traffic}}(g_i) = \lambda\, u_i \;-\; \eta\, z_i }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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} } with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda \in [0.15,0.60]\; \text{s/lap}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta \in [0.10,0.30]\; \text{s/lap}} . More detailed models map dirty-air loss by sector.

Safety Car / VSC stochastic model

Use a discrete-time hazard model with lap-dependent probability Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_i} (accidents, failures, debris): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Pr(\text{caution on lap } i \mid \text{no caution before}) \;=\; h_i }

Calibrate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_i} 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  .

Optimisation methods

Dynamic Programming (DP)

State Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_i = (c,a,m,\text{flag})} ; actions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{A}=\{\text{stay},\text{box to }c'\}} . Bellman recursion: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_i(s) = \min_{a\in\mathcal{A}} \; \mathbb{E}\big[ \; \Delta t_i(s,a) \;+\; V_{i+1}(s') \;\big] } Transitions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s\!\to\!s'} include tyre-age reset, fuel update, and stochastic flags.

Monte Carlo with look-ahead

At each lap, simulate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} futures with candidate actions; select the action minimising expected race time (or a risk-adjusted objective).

MPC (receding horizon)

Optimise over a shorter horizon Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} 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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_0(\text{C3})= 90.000 \, \text{s}}
  • Deg: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_{\text{C3}}=0.025} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta_{\text{C3}}=3.5\!\times\!10^{-4}}
  • Fuel sens.: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_f=0.035 \,\text{s/kg/lap}} , 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:

  1. One-stop: C3→C2 around lap 28
  2. 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha,\beta,k_f,\phi,h_i} .
  • Live: cross-check with real-time standoff (gap to pit-window car).

See also

References

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