# Optimizing Torque Delivery for an Energy-Limited Electric Race Car Using Model Predictive Control

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## Abstract

**:**

## 1. Introduction

## 2. Vehicle Model

#### Identified Linear Time-Invariant Model

**$\mathit{A}$**,

**$\mathit{B}$**, $\mathit{C}$, and $\mathit{D}$ matrices are a function of the plant dynamics and were determined through the process of system identification (Appendix A).

## 3. MPC Design

#### 3.1. Horizons

#### 3.2. Set Point and Control Signal Weighting

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FS | Formula Student |

ICE | Internal combustion engine |

MPC | Model predictive controller |

PID | Proportional, integral, derivative |

LQR | Linear quadratic regulation |

RMSE | Root-mean-squared error |

NRMSE | Normalized root-mean-squared error |

IDLTI | Identified linear time-invariant |

LTI | Linear time-invariant |

## Appendix A

**$\mathit{A}$**,

**$\mathit{B}$**, $\mathit{C}$, and $\mathit{D}$ from the system identification process:

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**Figure 1.**Energy consumption comparison between a transient model developed using the AVL VSM software and a simplified lumped mass model.

**Figure 2.**Comparison between 3rd- and 4th-order linearized models with the nonlinear plant. The NRMSEs for all three inputs were greater than 98% for the 3rd-order model and greater 99% for the 4th-order model (Equation (5)).

**Figure 3.**Prediction horizons within boundary limits. Adapted from [27].

**Figure 4.**MPC block diagram. Energy consumption is optimized based on the feedback of the battery energy capacity, torque output, and vehicle velocity.

**Figure 5.**Acceleration torque request scenario. (

**a**) Speed profile; (

**b**) torque request; (

**c**) energy remaining in the battery.

**Figure 6.**Formula Student manoeuvre torque request scenario. (

**a**) Speed profile; (

**b**) torque request; (

**c**) energy remaining in the battery.

**Figure 7.**Reduced horizons scenario. (

**a**) Speed profile; (

**b**) torque request, (

**c**) energy remaining in the battery.

Parameter | Value | Units | Description |
---|---|---|---|

${\eta}_{t}$ | $0.9$ | - | Transmission efficiency |

e | $1.4$ | - | Rotational mass factor |

m | 300 | (kg) | Vehicle mass |

$\rho $ | $1.225$ | (kg/m^{3}) | Air density |

A | $2.2$ | (m^{2}) | Vehicle frontal area |

g | $9.81$ | (m/s^{2}) | Gravitational acceleration |

r | $0.203$ | (m) | Tire rolling radius |

${k}_{t}$ | $0.26$ | (A/N.m) | Torque constant |

${i}_{\theta}$ | $15.55$ | - | Final drive ratio |

${C}_{d}$ | $0.40$ | - | Coefficient of drag |

$\alpha $ | 0 | (deg) | Road inclination angle |

f | $0.015$ | - | Rolling resistance coefficient |

Variable | Weight ${}^{1}$ |
---|---|

Manipulated variable—torque request (N.m) | 0 |

Manipulated variable rate (N.m/Δt) | $0.1$ |

Target velocity (km/h) | 10 |

Energy consumption (kW) | 13 |

Torque output (N.m) | 3 |

^{1}dimensionless.

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**MDPI and ACS Style**

Maull, T.; Schommer, A.
Optimizing Torque Delivery for an Energy-Limited Electric Race Car Using Model Predictive Control. *World Electr. Veh. J.* **2022**, *13*, 224.
https://doi.org/10.3390/wevj13120224

**AMA Style**

Maull T, Schommer A.
Optimizing Torque Delivery for an Energy-Limited Electric Race Car Using Model Predictive Control. *World Electric Vehicle Journal*. 2022; 13(12):224.
https://doi.org/10.3390/wevj13120224

**Chicago/Turabian Style**

Maull, Thomas, and Adriano Schommer.
2022. "Optimizing Torque Delivery for an Energy-Limited Electric Race Car Using Model Predictive Control" *World Electric Vehicle Journal* 13, no. 12: 224.
https://doi.org/10.3390/wevj13120224