# Torque Distribution Based on Dynamic Programming Algorithm for Four In-Wheel Motor Drive Electric Vehicle Considering Energy Efficiency Optimization

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- A torque distribution method with the comprehensive goals of optimal torque distribution and energy efficiency considering economy through energy efficiency is proposed in this paper.
- (2)
- The DP control algorithm is utilized for toque distribution between the front and rear in-wheel motors to obtain optimal torque distribution and energy efficiency in the 4IWMD EV.
- (3)
- The proposed torque distribution based on the DP algorithm for the 4IWMD electric vehicle considering energy efficiency optimization is effectively verified through simulation and experiment under the NEDC, WLTC and IM240 driving cycles.

## 2. IWMD Electric Vehicle Model

#### 2.1. Vehicle Dynamics Model

_{c}is vehicle curb weight, m

_{em}is total mass of the in-wheel motors, m

_{bat}is battery mass, m

_{fl}is the mass of the front left in-wheel motor, m

_{fr}is the mass of the front right in-wheel motor, m

_{rl}is the mass of the rear left in-wheel motor and m

_{rr}is the mass of the rear right in-wheel motor [23].

_{x}. Figure 1 shows the free body diagram of the 4IWMD electric vehicle on a slope, with the resistance forces that act on the vehicle [24,25].

_{t}) is derived through the following equations:

_{a}is the acceleration resistance, F

_{G}is the grade resistance, F

_{r}is the vehicle rolling resistance, F

_{w}is the aerodynamic resistance, C

_{r}represents the rolling resistance coefficient, C

_{a}represents the aerodynamic resistance coefficient, ρ is the density of air and A is the frontal area of the 4IWMD electric vehicle.

_{wheel}is radius of the drive wheel.

#### 2.2. In-Wheel Motor Model

_{m}is the output power of the in-wheel motor, T

_{m}is the output torque of the in-wheel motor and ω

_{m}is the speed of the in-wheel motor.

_{dem}represents the power demanded by the power system of the 4IWMD electric vehicle, and P

_{loss}

_{.m}is the power loss of the in-wheel motor, especially as a result of motor heat losses and mechanical losses.

_{out}and P

_{in}are the output and input power of the in-wheel motor, respectively [26].

#### 2.3. Battery Model

_{aux}is the auxiliary power demanded by the vehicle.

_{bat}represents the battery capacity, η

_{SOC}represents the coulomb efficiency, U

_{OCV}represents the battery’s open circuit voltage, R

_{int}represents the battery’s polarization internal resistance, R

_{t}represents the battery’s ohmic internal resistance and P

_{m}represents the required power of the in-wheel motor. R

_{int}and U

_{OCV}are the function of the battery’s SOC as a variable [27]. Equation (15) is the broadened expression of Equation (14).

## 3. Torque Distribution Strategies

#### 3.1. Torque Optimization Approach

_{f}represents the torque of the front axle motor, T

_{r}represents the torque of the rear axle motor and T

_{req}represents the torque demand of the 4IWMD electric vehicle.

_{f}represents the efficiency of the front axle motor, η

_{r}represents the efficiency of the rear axle motor and n represents the equivalent speed of the axle motor.

#### 3.2. Torque Distribution Based on DP

_{i}is an allowable decision elected at the state α, x

_{i}is the state adjacent to α that is replaced by the application of u

_{i}at α, h is the final state, ${J}_{a{x}_{i}}$ is the cost to move from α to x

_{i}, ${J}_{{x}_{i}h}^{\ast}$ is the minimum cost to reach the final state h from x

_{i}, ${C}_{a{x}_{i}h}^{\ast}$ is the minimum cost to go from α to h via x

_{i}, ${J}_{\alpha h}^{\ast}$ is the minimum cost to go from α to h (by any allowable path), ${u}^{\ast}(\alpha )$ is the optimal decision (control) at α.

_{dem}of the 4IWMD electric vehicle are the state variables in the range of actual domain [t

_{0}, t

_{f}] of power system of the 4IWMD. The speed of the 4IWMD electric vehicle can be determined according to the driving cycle utilized for the optimization. Therefore, the state variable is noted as x(t) = [T

_{dem}(t), n]′, meanwhile the vehicle demand power is utilized as the control variable, which is noted as u(t) = [P

_{dem}(t)] discrete state. The powertrain of the 4IWMD electric vehicle can then be described by the following Equation (24).

_{dem}represents the in-wheel motor torque, T

_{m}

_{,min}represents the minimum torque, T

_{m}

_{,max}represents the maximum torque of the in-wheel motor, n

_{m}

_{,min}and n

_{m}

_{,max}represent the minimum and maximum speed of the in-wheel motor, respectively, P

_{dem}

_{,max}represents the maximum output power of the in-wheel motor.

_{f}represents the output torque value of the front axle in-wheel motor, T

_{r}represents the output torque value of the rear axle in-wheel motor, η

_{f}(T

_{f},n) represents the efficiency of the front axle in-wheel motor and η

_{r}(T

_{r},n) represents the efficiency of the rear axle in-wheel motor [32].

_{i}represents the in-wheel motor torque and i = 1, 2, 3, 4 represent each of the four motors (front left, front right, rear left, rear right), respectively. T

_{toll}represents the required total torque for propulsion and braking and K

_{r}is the torque distribution coefficient.

_{1}(at T

_{toll}> T

_{1}). Meanwhile, at other instances of T

_{toll}< T

_{1}, the torque is distributed to the in-wheel motors as indicated below:

_{1}. However, it only distributes the total required torque to the vehicle by the rear wheels, as it enhances the optimal performance of the in-wheel motor and electric vehicle overall, as the in-wheel motor operates with a higher torque.

_{i}(k + 1) represents the motor torque for the next node and T

_{toll}(k) is the required motor torque.

## 4. Simulation Results and Analysis

#### 4.1. WLTC Driving Cycle

_{toll}) is different from the required total torque to navigate through the driving cycles, as the latter is the total torque which the control algorithm utilizes to navigate through the driving cycle, considering the control parameters, constraints and objectives. Therefore, it is the result calculated by the torque distribution control algorithm, using the total desired torque by vehicle dynamics and in–wheel motor speed for the front and rear in-wheel motors.

#### 4.2. NEDC Driving Cycle

#### 4.3. Customized IM240 Driving Cycle

#### 4.4. Energy Saving Analysis

## 5. Experimental Validation

#### 5.1. WLTC Drive Cycle

#### 5.2. NEDC Drive Cycle

#### 5.3. Customized IM240 Driving Cycle

#### 5.4. Energy Saving Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Li, B.; Goodarzi, A.; Khajepour, A.; Chen, S.K.; Litkouhi, B. An optimal torque distribution control strategy for four-independent wheel drive electric vehicles. Veh. Syst. Dyn.
**2015**, 53, 1172–1189. [Google Scholar] [CrossRef] - Joa, E.; Park, K.; Koh, Y.; Yi, K.; Kim, K. A tyre slip-based integrated chassis control of front/rear traction distribution and four-wheel independent brake from moderate driving to limit handling. Veh. Syst. Dyn.
**2018**, 56, 579–603. [Google Scholar] [CrossRef] - Park, G.; Han, K.; Nam, K.; Kim, H.; Choi, S.B. Torque Vectoring Algorithm of Electronic-Four-Wheel Drive Vehicles for Enhancement of Cornering Performance. IEEE Trans. Veh. Technol.
**2020**, 69, 3668–3679. [Google Scholar] [CrossRef] - Deng, H.; Zhao, Y.; Feng, S.; Wang, Q.; Lin, F. Torque Vectoring Algorithm Based on Mechanical Elastic Electric Wheels with Consideration of the Stability and Economy. Energy
**2021**, 219, 119643. [Google Scholar] [CrossRef] - Chatzikomis, C.; Zanchetta, M.; Gruber, P.; Sorniotti, A.; Modic, B.; Motaln, T.; Blagotinsek, L.; Gotovac, G. An energy-efficient torque-vectoring algorithm for electric vehicles with multiple motors. Mech. Syst. Signal Process.
**2019**, 128, 655–673. [Google Scholar] [CrossRef] - Debada, E.; Marcos, D.; Montero, C.; Camacho, E.F.; Bordons, C.; Ridao, M.A. Torque distribution strategy for a four In-wheel fully electric car. Jorn. Autom.
**2015**, 517–525. Available online: https://idus.us.es/handle/11441/92138 (accessed on 6 September 2022). - Mokhiamar, O.; Abe, M. How the four wheels should share forces in an optimum cooperative chassis control. Control Eng. Pract.
**2006**, 14, 295–304. [Google Scholar] [CrossRef] - Yong, L.; Deng, H.; Xing, X.; Jiang, H. Review on torque distribution strategies for four in-wheel motor drive electric vehicles. IOP Conf. Ser. Mater. Sci. Eng.
**2018**, 394, 042041. [Google Scholar] - Wang, Y.; Su, Y. A research for brake strategy based on fuzzy control in pure electric vehicles. In Proceedings of the 2015 4th International Conference on Computer Science and Network Technology (ICCSNT), Harbin, China, 19–20 December 2015; pp. 689–693. [Google Scholar]
- Wang, B.; Huang, X.; Wang, J.; Guo, X.; Zhu, X. A robust wheel slip control design for in-wheel-motor-driven electric vehicles with hydraulic and regenerative braking systems. In Proceedings of the 2014 IEEE American Control Conference, Portland, OR, USA, 4–6 June 2014; pp. 3225–3230. [Google Scholar]
- Prajeesh, K.; Beevi, M.W. An Efficient Regenerative Braking System for BLDCM driven Electric Vehicles. In Proceedings of the 2018 4th International Conference for Convergence in Technology (I2CT), Mangalore, India, 27–28 October 2018; pp. 1–5. [Google Scholar]
- Hannan, M.A.; Hoque, M.M.; Mohamed, A.; Ayob, A. Review of energy storage systems for electric vehicle applications: Issues and challenges. Renew. Sustain. Energy Rev.
**2017**, 69, 771–789. [Google Scholar] [CrossRef] - Tie, S.F.; Tan, C.W. A review of energy sources and energy management system in electric vehicles. Renew. Sustain. Energy Rev.
**2013**, 20, 82–102. [Google Scholar] [CrossRef] - Lu, D.; Ouyang, M.; Gu, J.; Li, J. Torque distribution algorithm for a permanent brushless DC hub motor for four-wheel drive electric vehicles. J. Tsinghua Univ. (Sci. Technol.)
**2012**, 52, 451–456. [Google Scholar] - Yang, L.; Zhang, J.W.; Guo, K.; Wu, D. Optimized Torque Distribution Algorithm to Improve the Energy Efficiency of 4WD Electric Vehicle. SAE Tech. Pap.
**2014**, 1–10. [Google Scholar] [CrossRef] - Peng, H.; Hu, S.J. Traction/Braking Force Distribution for Optimal Longitudinal Motion during Curve Following. Veh. Syst. Dyn.
**2007**, 26, 301–320. [Google Scholar] [CrossRef] [Green Version] - Wu, D.; Tian, S. Torque distribution strategy of pure electric bus with double motors driving by front and rear axles. J. Jiangsu Univ. (Nat. Sci. Ed.)
**2021**, 42, 634–641. [Google Scholar] - Li, S.; Ding, X.; Yu, B. Optimal control strategy of efficiency for dual motor coupling drive system of pure electric vehicle. J. Jiangsu Univ. (Nat. Sci. Ed.)
**2022**, 43, 1–7. [Google Scholar] - Wang, B. Study on Experiment Platform of Four-Wheel-Independent-Drive Ev and Its Driving Force Control System; Tsinghua University: Beijing, China, 2009. [Google Scholar]
- Fu, X.; Yang, F.; Huang, B.; He, Z.; Pei, B. Coordinated control of active rear wheel steering and four wheel independent driving vehicle. J. Jiangsu Univ. (Nat. Sci. Ed.)
**2021**, 42, 497–505. [Google Scholar] - Zhao, X.; Guo, G. Braking torque distribution for hybrid electric vehicles based on nonlinear disturbance observer. Proc. Inst. Mech. Eng. Part D J. Automob. Eng.
**2019**, 233, 3327–3341. [Google Scholar] [CrossRef] - Li, Y.; Adeleke, O.P.; Xu, X. Methods and applications of energy saving control of in-wheel motor drive system in electric vehicles: A comprehensive review. J. Renew. Sustain. Energy
**2019**, 11, 062701. [Google Scholar] [CrossRef] - Li, Z.; Song, X.; Chen, X.; Xue, H. Dynamic Characteristics Analysis of the Hub Direct Drive-Air Suspension System from Vertical and Longitudinal Directions. Shock Vib.
**2021**, 2021, 8891860. [Google Scholar] [CrossRef] - Kühlwein, J. Driving resistances of light-duty vehicles in Europe: Present situation, trends, and scenarios for 2025. Communications
**2016**, 49. Available online: https://theicct.org/publication/driving-resistances-of-light-duty-vehicles-in-europe-present-situation-trends-and-scenarios-for-2025/ (accessed on 8 September 2022). - Jazar, R.N. Vehicle Dynamics: Theory and Application; Springer: Berlin/Heidelberg, Germany, 2017; pp. 287–288. [Google Scholar]
- Hong, J.; Yu, Z.; Hongtao, X.; Zhongxing, L. Sequential diagnosis method for bearing fault of in-wheel motor based on CDI and AHNs. J. Jiangsu Univ. (Nat. Sci. Ed.)
**2021**, 42, 15–21. [Google Scholar] - Wu, D.; Tian, S. New control strategy of motor for pure electric vehicle based on TLGI technology. J. Jiangsu Univ. (Nat. Sci. Ed.)
**2021**, 42, 9–14. [Google Scholar] - Li, Y.; Zhang, B.; Xu, X. Robust control for permanent magnet in-wheel motor in electric vehicles using adaptive fuzzy neural network with inverse system decoupling. Trans. Can. Soc. Mech. Eng.
**2018**, 42, 286–297. [Google Scholar] [CrossRef] - Zhang, J.; Wang, T.; Wang, L.; Zou, X.; Song, W. Optimization control strategy of driving torque for slope-crossing of pure electric vehicles. J. Jiangsu Univ. (Nat. Sci. Ed.)
**2021**, 42, 506–512. [Google Scholar] - Fan, L.; Ma, Z. Fuzzy comprehensive evaluation method for symmetry degree of mechanical structure symmetry. Trans.-Can. Soc. Mech. Eng.
**2017**, 41, 337–353. [Google Scholar] [CrossRef] - Liang, H.; Yue, M.; Yu, W.; Zhi, J.; Peng, Y. Research on Torque Optimization Allocation Strategy about Multi-wheel Vehicles. In Innovative Techniques and Applications of Modelling, Identification and Control; Springer: Singapore, 2018; pp. 63–92. [Google Scholar] [CrossRef]
- Wu, X.; Zheng, D.; Du, J.; Liu, Z.; Zhao, X. Torque Optimal Allocation Strategy of All-wheel Drive Electric Vehicle. Energies
**2019**, 12, 1122. [Google Scholar] [CrossRef] - Chang, H.H.; Chia, W.C. Permanent Magnetic Brushless DC Motor Magnetism Performance depends on Different Intelligent Controller Response. Trans. Can. Soc. Mech. Eng.
**2020**, 45, 287–296. [Google Scholar] [CrossRef] - Li, J.; He, R. Optimization design and performance analysis of dual-rotor in-wheel motor based on parameter sensitivity. J. Jiangsu Univ. (Nat. Sci. Ed.)
**2020**, 41, 640–647. [Google Scholar] - Wu, Z.; Wu, Y.; He, S.; Xiao, X. Hierarchical fuzzy control based on spatial posture for a support-tracked type in-pipe robot. Trans. Can. Soc. Mech. Eng.
**2020**, 44, 133–147. [Google Scholar] [CrossRef]

**Figure 9.**Torque distribution to front and rear in–wheel motors in WLTC: (

**a**) FLC algorithm; (

**b**) DP algorithm.

**Figure 11.**Distribution of motor working points of the front and rear motors under WLTC driving cycle: (

**a**) Front motor; (

**b**) Rear motor.

**Figure 14.**Torque distribution to front and rear electric motors in NEDC: (

**a**) FLC algorithm; (

**b**) DP algorithm.

**Figure 16.**Distribution of motor working points of the front and rear motors under NEDC driving cycle: (

**a**) Front motor; (

**b**) Rear motor.

**Figure 19.**Torque distribution to front and rear electric motors using FLC and DP in custom IM240 driving cycle: (

**a**) FLC algorithm; (

**b**) DP algorithm.

**Figure 20.**Distribution of motor working points of the front motor and rear motor under custom IM240 driving cycle: (

**a**) Front motor; (

**b**) Rear motor.

**Figure 23.**Torque distribution to front motor and rear motor obtained in experiment and FLC simulation of WLTC driving cycle: (

**a**) Front motor; (

**b**) Rear motor.

**Figure 24.**Torque distribution to front motor and rear motor obtained in experiment and DP simulation under WLTC driving cycle: (

**a**) Front motor; (

**b**) Rear motor.

**Figure 25.**Torque distribution to front motor and rear motor obtained in experiment and FLC simulation of NEDC driving cycle: (

**a**) Front motor; (

**b**) Rear motor.

**Figure 26.**Torque distribution to front motor rear motor obtained in experiment and simulation with DP algorithm under NEDC driving cycle: (

**a**) Front motor; (

**b**) Rear motor.

**Figure 27.**Torque distribution to front motor and rear motor obtained in experiment and simulation with FLC algorithm under custom IM240 driving cycle: (

**a**) Front motor; (

**b**) Rear motor.

**Figure 28.**Torque distribution to front motor and rear motor obtained in experiment and simulation with DP algorithm under custom IM240 driving cycle: (

**a**) Front motor; (

**b**) Rear motor.

**Figure 29.**Energy consumption in the three drive cycles using FLC-based and DP-based torque distribution during experiment.

Vehicle Parameter | Symbol | Value (Unit) |
---|---|---|

Curb weight | M | 1270 kg |

Coefficient of rolling friction | C_{r} | 0.017 |

Cross-sectional area | A | 1.97 m^{2} |

Aerodynamic drag coefficient | C_{D} | 0.35 |

Rolling radius | R | 0.31 m |

In-Wheel Motor Parameter | Value (Unit) |
---|---|

Rated voltage | 72 V |

Rated current | 110 A |

Maximum speed | 1200 rpm |

Rated power | 8 kW |

Rated frequency | 50 Hz |

Maximum torque | 250 Nm |

FLC-Based Torque Distribution Energy Consumption (kWh/100 km) | DP-Based Torque Distribution Energy Consumption (kWh/100 km) | Improvement in Energy Consumption (%) | |
---|---|---|---|

WLTC | 10.01 | 7.74 | 22.68 |

NEDC | 9.89 | 7.84 | 20.73 |

Custom IM240 | 9.11 | 7.12 | 21.84 |

Energy Consumption with FLC Algorithm (kWh/100 km) | Energy Consumption with DP Algorithm (kWh/100 km) | Improvement in Energy Consumption (%) | |
---|---|---|---|

WLTC | 9.992 | 7.605 | 23.89 |

NEDC | 14.690 | 11.293 | 23.12 |

Custom IM240 | 4.529 | 3.487 | 23.01 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Adeleke, O.P.; Li, Y.; Chen, Q.; Zhou, W.; Xu, X.; Cui, X.
Torque Distribution Based on Dynamic Programming Algorithm for Four In-Wheel Motor Drive Electric Vehicle Considering Energy Efficiency Optimization. *World Electr. Veh. J.* **2022**, *13*, 181.
https://doi.org/10.3390/wevj13100181

**AMA Style**

Adeleke OP, Li Y, Chen Q, Zhou W, Xu X, Cui X.
Torque Distribution Based on Dynamic Programming Algorithm for Four In-Wheel Motor Drive Electric Vehicle Considering Energy Efficiency Optimization. *World Electric Vehicle Journal*. 2022; 13(10):181.
https://doi.org/10.3390/wevj13100181

**Chicago/Turabian Style**

Adeleke, Oluwatobi Pelumi, Yong Li, Qiang Chen, Wentao Zhou, Xing Xu, and Xiaoli Cui.
2022. "Torque Distribution Based on Dynamic Programming Algorithm for Four In-Wheel Motor Drive Electric Vehicle Considering Energy Efficiency Optimization" *World Electric Vehicle Journal* 13, no. 10: 181.
https://doi.org/10.3390/wevj13100181