Battery model impact on time-optimal co-design for electric racing cars: review and application

TL;DR

This paper investigates how battery modeling choices influence time-optimal co-design for electric racing cars by formulating a bi-level joint optimization of minimum race time and battery sizing within a space-domain OCP. It analyzes vehicle dynamics, powertrain, and three battery models (static $V_n$–$R$, SOC-dependent $V_{SoC}$–$R$, and SOC with RC dynamics $V_{SoC}$–$RC$), deriving convex reformulations feasible only for the simplest model and comparing their impact on sizing and control trajectories. A Gen 3 Formula E case study on the 2021 Rome ePrix demonstrates that simple models suffice for sizing while more complex models are needed to capture realistic control evolutions; convex relaxations yield results consistent with nonconvex formulations for sizing, albeit with potential non-physical energy profiles in some relaxations. The study provides practical guidelines for battery model selection in racing applications: use minimal models for sizing to leverage easy-datasheet parameters, and adopt richer battery models when analyzing dynamic control, while noting the trade-offs between convexity and physical fidelity.

Abstract

The sustainable mobility trend touches the racing world as well, from the hybridization of Formula 1 (F1) and Le Mans Hypercars to the fully electric Formula E racing class. In this scenario, the research community is studying how to push electric racing vehicles to their limit, combining vehicle dynamics and energy management, to successfully solve the minimum lap time problem. Recently, this class of problems has been enlarged towards optimal sizing, with a particular interest in batteries, which represent the main bottleneck for electric vehicle performance. In this work, starting from a thorough review of literature approaches, we define a general optimization framework of minimum lap and race time problems for electric vehicles, suitable to figure out the impact of different modeling choices on both problem structure and optimal variables profiles. Exploiting a case study on Generation 3 (Gen 3) of Formula E cars, we delve into the impact of battery models' complexity on both optimal sizing and optimal battery usage. We show how highly detailed models are necessary to study the evolution of both battery and vehicle control variables during the race, while, simple models are more than sufficient to address the battery sizing problem.

Figures (7)

• Figure 1: Comparison of sizing results for all the discussed non-convex battery models, as a function of the number of parallels $N_p$. A zoom around the optimum highlights the similarity among the obtained results.
• Figure 2: Comparison of SoC, current and voltage race profiles with $N_p =24$, for the three battery models with increasing complexity. For clarity, only the operating region envelopes are depicted for battery currents and voltages.
• Figure 3: Comparison of vehicle speed, in the fourteenth lap of the race, with $N_p =24$, for the three battery models with increasing complexity, a zoom of the current and voltage is given, to appreciate the difference between the models.
• Figure 4: Comparison of sizing results, as a function of the number of parallels $N_p$, between the convex and non-convex approaches for the simplest battery model ($\mathrm{V}_\mathrm{n}$--$\mathrm{R}$).
• Figure 5: Comparative analysis of the integration errors between the convex and non-convex formulation of the simplest battery model ($\mathrm{V}_\mathrm{n}$--$\mathrm{R}$). Race time discrepancies and distributions of vehicle speed differences are depicted as function $\mathrm{d}s$.