# Data Driven Methods for Finding Coefficients of Aerodynamic Drag and Rolling Resistance of Electric Vehicles

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Context

_{d}) and rolling resistance (C

_{r}). The coefficient of aerodynamic drag is typically found using computational fluid dynamics (with expensive software licences) or wind tunnel testing, and the coefficient of rolling resistance is found using tedious testing and complex mathematics. This study aims to find an alternate and effective way to determine the coefficients required to model a vehicle’s energy consumption.

#### 1.2. Literature Review

#### 1.2.1. Mathematical Modelling of Vehicles

#### 1.2.2. Optimisation

_{d}and C

_{r}forces from data obtained using such a method is approximate and, therefore, not as accurate as it could be.

_{d}and C

_{r}, which are needed for modelling vehicles. This research is necessary, as an accurate and more straightforward method will be beneficial for determining these coefficients. MATLAB establishes proof of the concept for this research. However, the programming can be performed using open-source software to reduce the cost of finding these coefficients.

#### 1.2.3. Existing Noteworthy Research

## 2. Materials and Methods

#### 2.1. Data Collection

#### 2.2. Data Pre-Processing

_{d}and C

_{r}values using the optimisation algorithm (data used for optimisation). The remaining data could then be used to validate the coefficients found to determine the accuracy of the process (data used for validation).

#### 2.3. Modelling

#### 2.4. Optimisation

_{d}and C

_{r}that could be used in Equations (1) and (2).

_{d}and C

_{r}to produce Figure 5a. The aim was to calculate these C

_{d}and C

_{r}values by minimising the error between the blue (real energy) and the red (modelled energy) data lines. An RMSE confidence bound can also be seen in Figure 5a. This confidence interval can provide a user with the statistical mean forecasted risk quantification. This may be useful to users when making energy decisions regarding the model output with the expected associated risk.

_{d}value examples) with the frontal areas of different vehicle types to serve as a reference [29,30,31,32,33].

_{d}value of the fourth-generation solar car (Sunchaser 4) was recently announced [35]. The Aurora Vehicle Association (headquartered in Melbourne, Australia) began developing solar cars in 1987 [33]. The drag and rolling resistance coefficients of their 2007 SEV can be seen in Table 6 and Table 7. This is a good starting point for this research as to what the coefficient of aerodynamic drag is expected to be after optimisation. This also serves as a validation of the values found in this work.

_{r}value examples) found for different tyres that correspond to their contact surface with specified constant tyre pressures [22,36,37].

## 3. Results and Discussion

_{d}and C

_{r}values, and the data used for validation was used to determine if these coefficients are realistic. The percentage error between the real and modelled energies after validation confirmed how realistic the coefficients are.

#### 3.1. Calculating and Validating the Unknown Coefficients

_{d}and C

_{r}components.

_{d}and C

_{r}values found from the minimisation process are 0.0044 and 0.0131, respectively. However, this shows that the coefficients are far off from what was expected according to typical values, as seen in Table 6 and Table 7. The RMSE confidence bounds also show this. The final value of the modelled energy is outside the statistical mean forecasted error. There are a few factors at play here in the inaccuracy of the coefficients. Specifically, the non-dynamic model of the electric motor, different weights of the SEV drivers during the trip, variations in tyre pressure and bearing wear, crosswinds, road surface, and other unmodelled components contributed to this.

_{r}value. Bearing wear also plays a role.

_{d}parameter. Although wind was considered in creating the model, it was difficult to determine the full effects of these wind gusts and crosswinds. For example, during heavy cross winds, the amount of tyre scrub increased dramatically as the vehicle underwent micro-sliding from left to right rather than just forward or backward (as a result of the car’s lightweight design). The micro-sliding caused significant wear to the tyres and affected their rolling resistance forces.

_{d}and C

_{r}. The optimiser without constraints tried to compensate for all discrepancies in the energy comparison by manipulating the C

_{d}and C

_{r}values accordingly.

_{d}and C

_{r}, which can be seen in Table 6 and Table 7. The graph of the minimisation of the RMSE between the two energies can be seen in Figure 8a. This is from the same optimisation data set from Days 2, 4, and 6 used above.

_{d}and C

_{r}values were 0.13 and 0.0059, respectively. It can be noted that the value found for C

_{d}does reach the lower bound of its constraint. This is due to the fact that the optimisation algorithm was still attempting to minimise the error towards the RMSE value found when no constraints were added. This is acceptable for this research as the lower bound of the constraint is a realistic value for C

_{d}. As confirmation of this, the lower bound coefficient value is justified by the results of a CFD analysis found for the same vehicle performed by the members of the TUT solar car team [35]. A part of the modelled energy profile still exceeded the RMSE confidence interval, most likely because of the unmodelled components, as discussed above. However, it still provides a statistical error range for consideration by the user (in this case, the energy manager, or when this method is applied to other EVs, it might be the vehicle driver). The confidence interval was intentionally omitted from the optimisation results (Figure 8a), as visual error quantification is more applicable to the validation results.

_{d}and C

_{r}using the optimisation data, it was then possible to test them on the validation data from Days 3, 5, and 7. The values of 0.13 for the C

_{d}component and 0.0059 for the C

_{r}component were inserted into the model of the SEV. With all model components known, the energy according to the model could be compared against the real (actual) energy. This was to validate the coefficients. The energy comparison can be seen in Figure 8b.

#### 3.2. Discussion of the Parameters That Affect the Model’s Performance

#### 3.3. Final Results for Optimisation and Validation

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Examples: (

**a**) energy comparison between real and modelled data; (

**b**) power comparison between real and modelled data.

**Figure 7.**Using no constraints: (

**a**) minimising the RMSE between real and modelled energy to determine C

_{d}and C

_{r}for the optimisation data set (days 2, 4, and 6); (

**b**) validation of these coefficients on the second data set (days 3, 5, and 7).

**Figure 8.**Using constraints: (

**a**) minimising the RMSE between real and modelled energy to determine C

_{d}and C

_{r}for the optimisation data set (days 2, 4, and 6); (

**b**) validation of these coefficients on the second data set (days 3, 5, and 7).

**Figure 9.**Environmental conditions on Days 2, 4, and 6: (

**a**) elevation; (

**b**) temperature; (

**c**) air density; (

**d**) humidity.

**Figure 10.**Environmental conditions on Days 3, 5, and 7: (

**a**) elevation; (

**b**) temperature; (

**c**) air density; (

**d**) humidity.

Parameter | SI Unit | Point of Collection |
---|---|---|

Air density | (kg/m^{3}) | Calculated |

SV velocity | km/h | Measured |

GPS elevation | m | Measured |

GPS heading | Degrees | Measured |

GPS latitude | Degrees | Measured |

GPS longitude | Degrees | Measured |

Humidity | % | Measured |

Incline of the road | Degrees | Calculated |

Pressure | Pa | Measured |

Temperature | °C | Measured |

Time | h:m:s | Measured |

Wind direction | Degrees | Measured |

Wind velocity | km/h | Measured |

Parameter | SI Unit | Point of Collection |
---|---|---|

Battery voltage | V | Measured |

Motor current | A | Measured |

Motor energy | kWh | Calculated |

Power | W | Calculated |

Time | h:m:s | Measured |

SEV velocity | km/h | Measured |

Day | Town Start | Town End | Road Type | Distance (km) | Velocity (km/h) ^{1} | Energy Consumed (kWh) ^{2} |
---|---|---|---|---|---|---|

1 | Pretoria, SA | Vryburg, SA | Flat | 406 | - | - |

2 | Vryburg, SA | Upington, SA | Descent | 394 | 66 | 3.75 |

3 | Upington, SA | Grunau, NM | Ascent | 182 | 68 | 1.93 |

4 | Grunau, NM | Keetmanshoop, NM | Hilly | 155 | 66 | 1.47 |

5 | Keetmanshoop, NM | Mariental, NM | Hilly | 236 | 74 | 2.61 |

6 | Mariental, NM | Windhoek, NM | Ascent | 268 | 64 | 3.03 |

7 | Windhoek, NM | Swakopmund, NM | Descent | 434 | 65 | 2.78 |

^{1}Average.

^{2}Battery energy consumed.

Day | Temperature (°C) ^{1} | Air Density (kg/m^{3}) ^{1} | Humidity (%) ^{1} | Wind Speed (km/h) ^{2} | Elevation (m) ^{3} |
---|---|---|---|---|---|

1 | 16–34 | 0.993–1.045 | 28–74 | 6, (137) | 1374–1187 |

2 | 25–35 | 0.995–1.041 | 22–53 | 8, (128) | 1210–820 |

3 | 28–36 | 1.006–1.065 | 22–49 | 7, (30) | 811–1110 |

4 | 24–34 | 0.992–1.041 | 26–46 | 10, (45) | 1110–935 |

5 | 25–33 | 1.010–1.057 | 32–41 | 10, (28) | 935–1105 |

6 | 23–30 | 0.936–1.036 | 38–79 | 10, (44) | 1109–1705 |

7 | 20–27 | 0.984–1.194 | 50–80 | 15, (102) | 1698–21 |

^{1}Minimum to maximum.

^{2}Average, (gusts).

^{3}Start of day to end of day.

Parameter | Value | SI Unit |
---|---|---|

Frontal area of SEV | 0.8 | m^{2} |

SEV mass | 200 | kg |

Driver 1 mass | 80 | kg |

Driver 2 mass | 63 | kg |

Acceleration due to gravity | 9.81 | m/s^{2} |

Wheel diameter | 0.567 | m |

Specific gas constant (dry air) | 287.058 | J/kg·K |

Motor efficiency constant | 80 | % |

Vehicle Type | A (m^{2}) | C_{d} |
---|---|---|

TUT 2nd generation SEV | 1.27 | 0.23 |

TUT 3rd generation SEV | 1.03 | 0.16 |

TUT 4th generation SEV | 0.8 | 0.13 |

Aurora (2007) | 0.76 | 0.1 |

Toyota Corolla (typical passenger vehicle) | - | 0.27 |

Tesla Model S (electric passenger vehicle) | 2.34 | 0.208 |

Tesla Model X (electric SUV) | 2.59 | 0.24 |

Toyota Land Cruiser (4WD) | - | 0.35 |

Truck (Heavy duty) | ~3 | ~0.6 |

Tyre | C_{r} |
---|---|

Michelin solar car tyres | 0.0025 |

Maximum TUT SEV tyres | 0.0085 |

Aurora (2007) | 0.0027 |

Car tyre on tar | 0.018–0.02 |

Car tyre on gravel | 0.02–0.025 |

Car tyre on cobbles | 0.035–0.05 |

Car tyre on compact sand | 0.04–0.08 |

Car tyre on loose sand | 0.2–0.4 |

**Table 8.**Amount of time spent travelling on each day (refer to Figure 9).

Day | Colour | Time Range (s) | Time Range (h) | Time Length (h:m:s) |
---|---|---|---|---|

2 | Red | 0–21,285 | 0–5.9 | 5:54:45 |

4 | Blue | 21,286–29,632 | 5.9–8.2 | 2:19:07 |

6 | Green | 29,633–44,571 | 8.2–12.4 | 4:08:59 |

Total | - | 0–44,571 | 0–12.4 | 12:22:51 |

**Table 9.**Time spent travelling each day (refer to Figure 10).

Day | Colour | Time Range (s) | Time Range (h) | Time Length (h:m:s) |
---|---|---|---|---|

3 | Red | 0–9568 | 0–2.7 | 2:39:28 |

5 | Blue | 9569–20,941 | 2.7–5.8 | 3:09:33 |

7 | Green | 20,942–41,000 | 5.8–11.4 | 5:34:19 |

Total | - | 0–41,000 | 0–11.4 | 11:23:20 |

C_{d} | C_{r} | RMSE (Wh) | Final Value Error (Wh) | Final Value Error (%) | |
---|---|---|---|---|---|

Optimisation ^{1} | 0.0044 | 0.0131 | 88 | 18 | 0.22 |

Validation ^{1} | 0.0044 | 0.0131 | 232 | 531 | 7.25 |

Optimisation ^{2} | 0.13 | 0.0059 | 161 | 322 | 3.90 |

Validation ^{2} | 0.13 | 0.0059 | 217 | 301 | 4.12 |

^{1}Without constraints.

^{2}With constraints.

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**MDPI and ACS Style**

Van Greunen, R.; Oosthuizen, C.
Data Driven Methods for Finding Coefficients of Aerodynamic Drag and Rolling Resistance of Electric Vehicles. *World Electr. Veh. J.* **2023**, *14*, 134.
https://doi.org/10.3390/wevj14060134

**AMA Style**

Van Greunen R, Oosthuizen C.
Data Driven Methods for Finding Coefficients of Aerodynamic Drag and Rolling Resistance of Electric Vehicles. *World Electric Vehicle Journal*. 2023; 14(6):134.
https://doi.org/10.3390/wevj14060134

**Chicago/Turabian Style**

Van Greunen, Ryan, and Christiaan Oosthuizen.
2023. "Data Driven Methods for Finding Coefficients of Aerodynamic Drag and Rolling Resistance of Electric Vehicles" *World Electric Vehicle Journal* 14, no. 6: 134.
https://doi.org/10.3390/wevj14060134