# Battery Electric Vehicles: How Many Gears? A Technical–Economic Analysis

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## Abstract

**:**

## 1. Introduction and Literature Analysis

- It makes a comparison between vehicles that rather closely reproduce two existing cars on the market (based on published data about them);
- It allow the reader to reproduce a portion of the simulations through a freely available simulator and the publication of the simulation code and to adapt them to their needs, changing vehicle parameters such as the mass, drag coefficient, efficiencies, etc. (see Section 5) and the test cycle (target speed vs. time).

## 2. Materials and Methods

#### 2.1. Vehicle Data

^{®}by creating a matrix where each element represents the drive’s efficiency for each coordinate in the rotational speed–torque plane. The following figure shows the efficiency maps for the electric drive of vehicle N. (Figure 1).

#### 2.2. Modelica Language and Simulation Tools

#### 2.3. The Single-Speed BEV Simulation Model

#### 2.3.1. Submodel “Battery”

#### 2.3.2. Submodel “eleDrive”

- flange_a is the mechanical flange connected to the rest of the powertrain and, in particular, to the first flange of model “lossyGear”;
- The model determines the output mechanical power using the product of angular speed ω and torque τ. The block “toElePow” contains the efficiency map of the electric drive, e.g., the maps shown in Figure 1 and Figure 3; considering the operation point, as determined by the torque and speed, it applies the corresponding efficiency and determines the electrical power that must be drawn from the battery. The resistor with the red “P” is a non-linear resistor, which receives the electrical DC power as input and draws that power from the two input pins of “eleDrive”;
- The electric drive inertia (which is equal to the inertia of the electric motor rotor) is considered through the submodel “inertia”.

#### 2.3.3. Submodel “lossyGear” (from MSL)

- Regarding the single-speed total gear ratio group, an efficiency of 98% was considered for each gearset due to the gear meshing of the two single-stage gear sets.
- Regarding the differential, an overall efficiency of 97% was considered, due to the engagement of the differential inner wheels and the lubrication losses.
- The model also takes into account the bearing friction related to the shafts on which the gear wheels are mounted. The friction torque denoted as T was evaluated externally to the model using the following equation:

#### 2.3.4. Submodel “Inertias” (from the MSL)

#### 2.3.5. Submodel “Mass” (from MSL)

#### 2.3.6. Submodel “dragForce” (Created by us)

- f is the rolling friction coefficient, usually between 0.011 and 0.014 in modern cars;
- mg is the vehicle weight;
- $\rho $ is the air density;
- C
_{x}is the aerodynamic drag coefficient which, in modern cars, is usually between 0.20 and 0.35; - S is the vehicle frontal area.

#### 2.3.7. Submodel “Driver” (Created by Us)

#### 2.4. The Two-Speed BEV Simulation Model

- Submodel “basicStrategy” (from library [20])

- Submodel “clutchControl” (from library [20])

- Submodel “Dual-clutch gearbox”

#### 2.4.1. Gear Ratio Choice

^{®}[23] through an optimization process: the efficiency of the transmission was written as a function of the transmission ratio τ, from which a cost function was then formulated. Subsequently, for each time pointover the entire driving cycle, the value of τ that minimized this cost function was determined, maximizing the transmission efficiency point-by-point throughout the entire cycle. Figure 9 shows the trend of the optimal τ during the first 500 s of the WLTC cycle for vehicle T: the graph in the figure is the result of the optimization process described earlier that was carried out in Matlab

^{®}.

#### 2.4.2. Gear-Shift Strategy

#### 2.5. The CVT BEV Simulation Model

- Gear Ratio: Regarding the DCT model, the gear ratios selected for each vehicle are reported in Table 2. As for the CVT model, the transmission ratio can vary continuously within the range of 4.2 to 1.9 for vehicle T and 3 to 0.9 for vehicle N.
- Friction: The DCT model considers viscous losses due to the action of the lubricant on the clutch discs. The CVT model takes into account the friction torque due to belt slip on the pulleys. This torque is proportional to the transmitted torque and the gear ratio.
- Controller: In the DCT model, the controller chooses the most efficient transmission ratio considering the vehicle speed and the percentage of the accelerator at that specific driving moment. Regarding the CVT model, the gear ratio is selected by a table that imposes the most efficient transmission ratio for the specific driving conditions at each simulation instant.
- Gear-Shift Strategy: The trend of the optimal transmission ratio during the WLTC cycle was obtained using the method already shown in Section 2.3.1 for selecting the gears of the two-speed model. A post-processing activity was conducted on the obtained profile to achieve a new transmission ratio trend. Losses associated with the CVT occur both when the transmission ratio changes but also with a fixed gear ratio. Therefore, the τ profile was adjusted by attempting to eliminate unnecessary gear shifts between transmission ratios close to each other, which do not bring efficiency benefits to the transmission but introduce losses. The optimal profile was found through a trial-and-error process by eliminating gear shifts between two transmission ratios that are closer to each other by a certain fixed value Δτ.

#### 2.6. The WLTC

## 3. Results

#### 3.1. Vehicle Efficiency

#### 3.2. The Single-Speed Ratio for Maximum Efficiency

#### 3.3. Consumption Evaluation at Equal Performances

## 4. Discussion

- 1.
- The improved efficiency of the new transmissions allows, for the same driving range, the vehicles to be equipped with a battery pack with a lower energy capacity, saving on this component. The capacity ${C}_{Bn}$ of the new battery is obtained by multiplying the original vehicle’s range R by its energy consumption in Wh/km with the new transmission. The new battery capacities are, therefore, shown in Table 7.

- 2.
- The lower vehicle consumption due to the new transmissions clearly reduces the amount of energy consumed and therefore the related costs over the entire lifespan of the vehicle. Specifically, given the value ${L}_{v}$ for the vehicle’s lifespan, the relative efficiency of the charging process ${\eta}_{c}$, and the cost of electrical energy ${c}_{E}$, the cost related to the consumed energy ${E}_{{L}_{v}}$ during the vehicle’s lifespan is obtained with Formula (4):$${E}_{{L}_{v}}=\frac{{L}_{v}}{R{\eta}_{c}}{C}_{Bn}{c}_{B}$$

- For both vehicles, the first column shows the capital cost (CC), which is the sum of the savings on the battery, which are higher in the case of the highest considered quotation (e.g., 200 €/kWh), and the additional costs of the transmissions, which are fixed for each considered technology (e.g., DCT or CVT); this affects the initial cost of the vehicle.
- The second column shows the running cost (RC), representing the savings due to lower energy consumption during the vehicle’s lifespan.
- The sum of these two items determines the total cost (TC), shown in the third column.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 17.**Comparison between the operating points of the original gear ratio and DCT (

**left**) and CVT (

**right**) on the electric drive efficiency map for vehicle T.

**Figure 18.**Comparison between the operating points of the original gear ratio and DCT (

**left**) and CVT (

**right**) on the electric drive efficiency map for vehicle N.

**Figure 19.**Comparison between the energy consumption of the original transmission and DCT (

**left**) and CVT (

**right**) for vehicle T in the four different phases of the WLTC cycle.

**Figure 20.**Comparison between the energy consumption of the original transmission and DCT (

**left**) and CVT (

**right**) for vehicle N in the four different phases of the WLTC cycle.

Parameter | Description | Values | Units | |
---|---|---|---|---|

T | N | |||

m | Vehicle mass | 2099 | 1643 | kg |

${\tau}_{t}$ | Total gear ratio | 9.734 | 7.938 | - |

${\tau}_{d}$ | Differential gear ratio | 3.12 | 4.353 | - |

${r}_{r}$ | Tire radius | 0.3525 | 0.3162 | m |

${C}_{x}$ | Aerodynamic drag coefficient | 0.24 | 0.29 | - |

S | Frontal area | 2.34 | 2.28 | ${\mathrm{m}}^{2}$ |

${P}_{max}$ | Maximum output power | 225 | 80 | kW |

${T}_{max}$ | Maximum output torque | 430 | 280 | Nm |

${\omega}_{max}$ | Maximum motor speed | 1571 | 1088 | rad/s |

$f$ | Rolling friction coefficient | 0.013 | - |

T | N | |
---|---|---|

1st | 4.12 | 1.82 |

2nd | 1.92 | 1.15 |

Data | Results | |||
---|---|---|---|---|

Vehicle | Gear Ratio * | Wh/km | 0–100 km/h (s) | |

T | Literature | 9.734 | 182.5 | 6.54 |

Efficient | 6.000 | 179.0 | 9.92 | |

N | Literature | 7.938 | 159.9 | 12.01 |

Efficient | 5.500 | 156.9 | 13.81 |

Results | |||
---|---|---|---|

Vehicle | Wh/km | 0–100 km/h (s) | |

T | Original | 182.5 | 6.54 |

DCT * | 172.4 (−5.57%) | 6.31 | |

CVT ** | 169.0 (−7.42%) | 5.50 | |

N | Original | 159.9 | 12.01 |

DCT | 153.9 (−3.76%) | 11.81 | |

CVT | 148.7 (−7.00%) | 10.91 |

Two-Speed (DCT) | CVT | |
---|---|---|

T (E-Class) | 0.87 | 1.34 |

N (B-Class) | 0.53 | 0.91 |

T | N | |
---|---|---|

DCT * | 56 (−4) | 23 (−1) |

CVT ** | 55 (−5) | 22 (−2) |

Parameter | Description | Values | Units |
---|---|---|---|

${c}_{B}$ | Battery cost | 150–200 | €/kWh |

${c}_{T}$ (E-Class) | Transmission cost (E-Class) | 12.5 | €/kW |

${c}_{T}$ (B-Class) | Transmission cost (B-Class) | 9.5 | €/kW |

${c}_{E}$ | Energy cost | 0.25–0.70 | €/kWh |

${L}_{v}$ | Vehicle’s lifetime range | 250,000 | km |

${\eta}_{c}$ | Charging efficiency | 0.9 | - |

T | N | |||||||
---|---|---|---|---|---|---|---|---|

Battery [€/kWh] | Energy [€/kWh] | CC | RC | TC | CC | RC | TC | |

DCT | 150 | 0.25 | +590 | −704 | −114 | +371 | −416 | −45 |

0.70 | +590 | −1970 | −1380 | +371 | −1166 | −795 | ||

200 | 0.25 | +423 | −704 | −281 | +352 | −416 | −91 | |

0.70 | +423 | −1970 | −1547 | +352 | −1166 | −841 | ||

CVT | 150 | 0.25 | +1009 | −939 | +70 | +611 | −775 | −164 |

0.70 | +1009 | −2628 | −1619 | +611 | −2169 | −1558 | ||

200 | 0.25 | +786 | −939 | −153 | +527 | −775 | −248 | |

0.70 | +786 | −2628 | −1842 | +524 | −2169 | −1642 |

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## Share and Cite

**MDPI and ACS Style**

Bertucci, E.; Bucchi, F.; Ceraolo, M.; Frendo, F.; Lutzemberger, G.
Battery Electric Vehicles: How Many Gears? A Technical–Economic Analysis. *Vehicles* **2024**, *6*, 71-92.
https://doi.org/10.3390/vehicles6010003

**AMA Style**

Bertucci E, Bucchi F, Ceraolo M, Frendo F, Lutzemberger G.
Battery Electric Vehicles: How Many Gears? A Technical–Economic Analysis. *Vehicles*. 2024; 6(1):71-92.
https://doi.org/10.3390/vehicles6010003

**Chicago/Turabian Style**

Bertucci, Emmanuele, Francesco Bucchi, Massimo Ceraolo, Francesco Frendo, and Giovanni Lutzemberger.
2024. "Battery Electric Vehicles: How Many Gears? A Technical–Economic Analysis" *Vehicles* 6, no. 1: 71-92.
https://doi.org/10.3390/vehicles6010003