# Research on Parameter Optimization Design Method for Dual-Motor Coupled Drive System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Configuration of DMCDS

#### 2.2. Modeling and Mode Analysis

#### 2.2.1. Dynamics Modeling of DMCDS

_{ms}, ω

_{mr}, and ω

_{c}are the output speeds of motors EM_S and EM_R, and the carrier, respectively; T

_{ms}, T

_{mr}, and T

_{c}are the output torque of motors EM_S and EM_R, and the planet carrier; J

_{s}, J

_{r}, J

_{c}, and J

_{p}represent the equivalent moments of inertia of the sun gear, ring gear, carrier, and planetary gear, respectively; i

_{m}, r

_{w}, and k denote the main reduction ratio, rolling radius of the wheels, and the planetary gear ratio, respectively; and v

_{a}denotes the vehicle speed.

#### 2.2.2. Driving Mode Analysis

_{ms}denotes the equivalent moment of inertia of motor EM_S, T

_{w}denotes the load torque on the driving wheel, and m

_{s}denotes the vehicle weight.

_{mr}denotes the equivalent moment of inertia of motor EM_R.

#### 2.3. Parameter Optimization of DMCDS

#### 2.3.1. Mathematical Models

- (1)
- Vehicle model

_{w}, T

_{w}, and J

_{w}are the speed, torque, and moment of inertia of the wheel, respectively; f

_{r}is the tire rolling resistance coefficient; C

_{D}is the aerodynamic drag coefficient; A

_{f}is the vehicular frontal area; and λ is the slip ratio of the driving wheel. The values of the vehicle parameters are displayed in Table 2.

- (2)
- Motor model

_{maxs}and P

_{maxr}are the peak power of motors EM_S and EM_R, respectively; and P

_{maxb}denotes the peak power of the baseline motor EMB, which equals half of the maximum vehicle power.

_{maxs}, T

_{maxr}, and T

_{maxb}are the peak output torque of motors EM_S, EM_R, and EMB, respectively.

- (3)
- Battery model

- (4)
- Efficiency model

_{a}and Z

_{b}represent the number of teeth, and the ± symbol indicates external (+) and internal gear pairs (−). In terms of the planetary gear set, there are different control strategies and energy losses for different design parameters and driving modes, so the efficiency model should be established for each of the three driving modes.

_{r(s−c)}indicates the efficiency when the ring gear is fixed and power is input into the sun gear and output from the planet carrier; η

_{c(s−r)}is the efficiency when the planet carrier is fixed with power input into the sun gear and output from the ring gear; η

_{s(r−c)}denotes the efficiency when the sun gear is fixed and power is input into the ring gear and output from the planet carrier; and η

_{s,r−c}denotes the efficiency when power is input into the ring gear and sun gear and output from the planet carrier.

#### 2.3.2. Optimization Problem

- (5)
- Inner-layer optimization

_{t}is the current state variables, u

_{t}is the current decision variables, x

_{t}

_{+1}denotes the state variables at the next time step, J

^{*}(x(t)) denotes the optimal value function from stage t to the terminal state, and L(x(t), u(t)) is the stage cost function of the system.

_{1}, and the current operating mode of the drive system, denoted as x

_{2}. The power allocation ratio increment is denoted as the decision variable u1, while the command for mode switching serves as the decision variable u

_{2}. The power allocation ratio of the two motors represents the ratio of the output power of motor EM_R to the total required power and can be expressed as follows:

_{mr}is the output power of motor EM_R, and η

_{c}represents the transmission efficiency of the planetary gear.

_{1}and x

_{2}are as follows:

_{2}takes values from the set {−1, 0, 1}, corresponding to the M1S mode, DMC mode, and M1R mode, respectively; and u

_{2}is restricted to the set {−1, 0, 1}, signifying the downshift, neutral, and upshift, respectively.

_{mR}, L

_{mS}, and L

_{gear}represent the power losses of motors EM_R and EM_S, and the planetary gear mechanism, respectively; and η

_{mr}and η

_{ms}denote the operational efficiencies of motors EM_R and EM_S, respectively.

_{ms}

_{_min}, T

_{ms}

_{_max}and T

_{mr}

_{_min}, T

_{mr}

_{_max}denote the minimum and maximum torque of motors EM_S and EM_R at the current speed, respectively; and n

_{ms}

_{_min}, n

_{ms}

_{_max}and n

_{mr}

_{_min}, n

_{mr}

_{_max}denote the minimum and maximum speed of motors EM_S and EM_R, respectively.

- (6)
- Outer-layer component parameter optimization

_{1}(p) is the energy losses of the DMCDS, corresponding to the inner optimization objective; J

_{2}(p) is the overall cost of the electric system; C

_{mot}and C

_{pe}are the costs associated with the motor and controller, respectively; and T

_{max}is the peak torque of the motors. γ

_{1}and γ

_{2}are the weighting factors, γ

_{1},γ

_{2}∈ [0,1], and γ

_{1}+ γ

_{2}= 1. To achieve a more reasonable weight distribution, and recognizing that the two objective sub-functions carry distinct physical meanings, we introduce the objective expectations J

_{1}

^{N}and J

_{2}

^{N}as normalization factors.

_{sys}is the system efficiency, θ

_{max}is the maximum gradient, v

_{acc}is the vehicle speed at the end of acceleration, and t

_{acc}is the acceleration time.

#### 2.3.3. Optimization Process

## 3. Results and Discussion

_{ADE}is the average drive efficiency and P

_{w}is the vehicle demand power.

## 4. Conclusions

- (1)
- The selection of motor parameters and gear ratios exerts a substantial influence on the power losses and drive efficiency of the system. While keeping the system maximum output power unchanged, adjustments to the rated power, rated speed, and gear ratios can enhance the utilization efficiency of the high-efficiency region and effectively reduce electrical energy consumption.
- (2)
- The optimized motors exhibit an increase in rated speed and a decrease in peak torque, resulting in a substantial improvement in the utilization efficiency of the high-efficiency region. Compared to the prototype scheme, motors EM_R and EM_S experience an increase of 45% and 48%, respectively. Moreover, the optimized DMCDS achieves an average drive efficiency 2.5% and 4.2% higher than that of DMCDS-pro and SMDS, respectively, leading to DMCDS-opt possessing the lowest energy consumption of 16.95 kWh/100 km.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of DMCDS. 1. EM_S; 2. EM_R; 3. B1; 4. B2; 5. ring gear; 6. planetary gear mechanism; 7. main reduction gear; 8. differential mechanism.

**Figure 2.**Equivalent lever model in M1S mode. The red line in the figure represents the lever and the green line represents the power flow.

**Figure 3.**Equivalent lever model in M1R mode. The red line in the figure represents the lever and the green line represents the power flow.

**Figure 4.**Equivalent lever model in DMC mode. The red line in the figure represents the lever and the green line represents the power flow.

**Figure 8.**(

**a**) Motor working points in SMDS; (

**b**) working points of motors in DMCDS-pro; (

**c**) working points of EM_S in DMCDS-opt; (

**d**) working points of EM_R in DMCDS-opt.

Working States | Driving Modes | M1 | M2 | B1 | B2 |
---|---|---|---|---|---|

Park/Neutral | N/P | ○ | ○ | ○ | ○ |

Driving states | M1S | ● | ○ | ○ | ● |

M1R | ○ | ● | ● | ○ | |

DMC | ● | ● | ○ | ○ |

Parameter | Meaning | Value |
---|---|---|

m_{s} (kg) | Mass of vehicle | 1949 |

A_{f} (m^{2}) | Frontal area | 2.66 |

C_{D} | Air resistance coefficient | 0.4 |

fr | Tire rolling friction coefficient | 0.015 |

r_{w} (m) | Tire radius | 0.343 |

v_{max} (km/h) | Maximum velocity | 150 |

t_{acc} (s) | 0–100 km/h acceleration time | 9 |

Parameter | Lower Limit | Upper Limit |
---|---|---|

Rated power of EM_S (kW) | 30 | 60 |

Rated speed of EM_S (r/min) | 2500 | 4000 |

Rated power of EM_R (kW) | 30 | 60 |

Rated speed of EM_R (r/min) | 2500 | 4000 |

Planetary gear ratio | 1.5 | 4 |

Final drive ratio | 4 | 6.5 |

Optimization Parameter | Optimized Parameter Value | Prototype Parameter Value |
---|---|---|

Rated power of EM_S (kW) | 33.5 | 32 |

Rated speed of EM_S (r/min) | 3500 | 2250 |

Rated power of EM_R (kW) | 31.5 | 32 |

Rated speed of EM_R (r/min) | 4000 | 2250 |

Planetary gear ratio | 2.26 | 1.86 |

Final drive ratio | 5.15 | 4.93 |

Parameter | Value |
---|---|

Rated power of motor (kW) | 64 |

Peak power of motor (kW) | 106 |

Rated speed of motor (r/min) | 2250 |

First gear ratio | 3.27 |

Second gear ratio | 1.98 |

Final drive ratio | 2.826 |

Schemes | SMDS | DMCDS-pro | DMCDS-opt | |
---|---|---|---|---|

Indicator | ||||

High-efficiency region utilization (efficiency > 90%) | 6.6% | EM_R 30.2% | EM_R 43.8% | |

EM_S 8.6% | EM_S 12.8% | |||

Electricity consumption (kWh/100 km) | 18.18 | 17.58 | 16.95 |

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## Share and Cite

**MDPI and ACS Style**

Li, T.; Zhang, N.; Gao, X.; Pang, D.
Research on Parameter Optimization Design Method for Dual-Motor Coupled Drive System. *World Electr. Veh. J.* **2023**, *14*, 282.
https://doi.org/10.3390/wevj14100282

**AMA Style**

Li T, Zhang N, Gao X, Pang D.
Research on Parameter Optimization Design Method for Dual-Motor Coupled Drive System. *World Electric Vehicle Journal*. 2023; 14(10):282.
https://doi.org/10.3390/wevj14100282

**Chicago/Turabian Style**

Li, Tonghui, Nan Zhang, Xiaoyu Gao, and Daqian Pang.
2023. "Research on Parameter Optimization Design Method for Dual-Motor Coupled Drive System" *World Electric Vehicle Journal* 14, no. 10: 282.
https://doi.org/10.3390/wevj14100282