**4. Energy-saving oriented torque allocation strategy of distributed drive electric vehicles**

### **4.1 The overall control framework**

The four independently driven in-wheel motors endow more potential to enhance the multi-performance control requirement of distributed drive electric vehicles. The hierarchical control scheme can balance the multi-objectives through the layered control methods while simplifying the system complexity. Hence, this section introduces a dual MPC (model predictive control)-based hierarchical scheme to ensure energy conservation and stability control. The control framework details in **Figure 21**.

In the upper layer, the total torque inputs are generated according to the driver's speed control requirement. Then using the energy-efficiency map obtained from the dynamometer, the optimal torque inputs are distributed to the front and rear axles. Such design can realize the energy saving through guaranteeing the in-wheel motors work in a high-efficiency zone.

In the lower layer, the additional direct yaw moment control is generated by the differential torque inputs of left and right in-wheel motors, which aims to ensure vehicle handling stability. Considering that the additional torque inputs would degrade the energy-saving performance, a relaxation factor is designed to prevent excessive control inputs based on guaranteeing the vehicle safety. Note that *β γ* phase plane is used to represent the vehicle stability margins.

### **4.2 The upper layer torque allocation strategy**

This section allocates the torque inputs to the front and rear axles according to the motor efficiency map as shown in **Figure 22**. This is a PD18 RAM in-wheel motor. The design principle is to enable the in-wheel motors to work in a high-efficiency zone. Meanwhile, the system scheme should also take the vehicle longitudinal stability performance into account. Here, we build the wheel dynamics model to represent the rotational motion with the torque control inputs.

**Figure 21.** *Overall diagram of the proposed control strategy.*

### *4.2.1 Wheel dynamics model*

Through lumping the left and right wheels to the axle, the vehicle longitudinal motion can be represented as follows considering the tire slip ratio.

$$J\_w \dot{w}\_i = T\_{wi} - R\_\epsilon F\_{xi} \tag{33}$$

$$F\_{xi} = k\_i \lambda\_{wi}, \lambda\_{wi} = \frac{w\_i R\_\varepsilon - V\_{xi,w}}{V\_{xi,w}}, (i = f, r) \tag{34}$$

Combining Eq. (33) and Eq. (34), the tire rotational motion can be expressed by

$$\dot{\lambda}\_{wi} = -\frac{R\_e^2}{J\_{wi} V\_{xi,w}} k\_i \lambda\_{wi} + \frac{R\_e}{J\_{wi} V\_{xi,w}} T\_{wi} \tag{35}$$

where *Vxi*,*<sup>w</sup>* and *wi* are the longitudinal speed and angular speed of wheel *i*, respectively. *Re* and *Jw* represent rolling radius and inertia moment around *y* axis of the wheel, respectively. *ki* and *λwi* donate the tire longitudinal stiffness and slip ratio, respectively. *Twi* and *Fxi* represent the torque input and tire longitudinal force, respectively. Then the state space equation of the wheel motion is given by

$$
\dot{\mathbf{x}} = A\mathbf{x} + Bu\tag{36}
$$

$$\text{where } \varkappa = \lambda\_{wi}, \iota = T\_{wi}, A = -\frac{R\_\epsilon^2}{\int\_{ui} V\_{xi,w}} k\_i, B = \frac{R\_\epsilon}{\int\_{ui} V\_{xi,w}}.$$

### *4.2.2 Energy-saving controller design*

The LTV-MPC (linear time varying model predictive control) is employed to handle the uncertain model parameter of longitudinal velocity. The Eq. (36) is required to be discrete first in the predictive controller. The Δ*T* is the sampling time. Then the discrete equation is expressed as

$$\varkappa(k+1) = A'\varkappa(k) + B'u(k)\tag{37}$$

where *<sup>A</sup>*<sup>0</sup> <sup>¼</sup> *eA*Δ*<sup>T</sup>*, *<sup>B</sup>*<sup>0</sup> <sup>¼</sup> <sup>Ð</sup> ð Þ *<sup>k</sup>*þ<sup>1</sup> <sup>Δ</sup>*<sup>T</sup> <sup>k</sup>*Δ*<sup>T</sup> <sup>e</sup>A k* ½ � ð Þ <sup>þ</sup><sup>1</sup> <sup>Δ</sup>*T*�*<sup>t</sup> Bdt*. The vehicle state and torque control input at time *k* are represented by *x k*ð Þ and *u k*ð Þ, respectively. It should be noted that in the LTV-MPC design, *Vxi*,*<sup>w</sup>* in the parameter matrices is updating at different sampling time. The LTV-MPC can guarantee the model accuracy, thereby avoiding the invalid direct yaw moment control inputs.

In the upper layer of the torque allocation strategy, the driver's longitudinal velocity control requirement is satisfied first by the total torque control input. Here, a PI controller is employed to describe the driver longitudinal speed-tracking intention. Hence, the total in-wheel motor torque input is calculated by

$$T\_{wd} = K\_P e\_v + K\_I \int e\_v dt \tag{38}$$

where *ev* denotes the speed-tracking deviation. *Ki* and *Kp* represent the integral and proportional coefficients. Then the following cost function is designed to realize the total torque control.

*Dynamics Modeling and Characteristics Analysis of Distributed Drive Electric Vehicles DOI: http://dx.doi.org/10.5772/intechopen.111908*

$$J\_1 = \sum\_{t=1}^{N\_p} \rho \left( 2T\_{uf}(t+k|k) + 2T\_{wr}(t+k|k) - T\_{wd}(k) \right)^2 \tag{39}$$

where *Tw*, min ≤ *Twi* ≤*Tw*, max . *Tw*, max and *Tw*, min are the admitted maximal and minimal torque control inputs, respectively. *Twf* and *Twr* represent the torque control inputs of front and rear in-wheel motors. *ρ* and *Np* denote the weight coefficient and predictive horizon, respectively. Note that the predictive horizon is equal to the control horizon in this paper. Next, the energy-saving control has a priority in the upper layer. The specific method is to guarantee a higher efficiency zone for the motors. Hence, we establish a mapping function between the vehicle speed and energy efficiency. Based on the motor efficiency map in **Figure 22**, The most energyefficient torque control at the current speed is selected as the reference value *Twi*,*<sup>r</sup>*ð Þ *i* ¼ *f*,*r* to optimize the control inputs. Then the obtained optimal torque inputs of front and rear axles are evenly distributed to the left and right in-wheel motors. Moreover, the optimization objective of a smaller tire slip ratio is also added to the cost function and expressed by

$$J\_2 = \sum\_{t=1}^{N\_p} (I\_2^1 + I\_2^2) \tag{40}$$

$$J\_2^1 = \sum\_{t=1}^{N\_p} \left[ \hbar\_1 (T\_{wf}(k+t|k) - T\_{wf,r}(k))^2 + \hbar\_2 (T\_{wr}(k+t|k) - T\_{wr,r}(k))^2 \right] \tag{41}$$

$$J\_2^2 = \sum\_{t=1}^{N\_p} \left( a\_1 \lambda\_{wf}^2(k+t|k) + a\_2 \lambda\_{wr}^2(k+t|k) \right) \tag{42}$$

where *α*1, *α*2, ℏ1, and ℏ<sup>2</sup> are the weighting coefficients. Considering the vehicle longitudinal stability, the front axle has a priority to satisfy the high-efficiency zone. When approaching the tire force limitation, a small slip of the rear wheel would lead

**Figure 22.** *The upper layer torque allocation strategy.*

to the vehicle instability. Therefore, ℏ<sup>1</sup> is endowed with a higher value. Furthermore, to ensure the driver's longitudinal control intention, a logical judgment is also added. If the optimized torque of the rear wheel is not consistent with the driver's control intention, the control inputs are set as 0. Through combing Eq. (39)-Eq. (42), the control objective function is represented by

$$J = J\_1 + J\_2 \tag{43}$$

### **4.3 The lower layer of direct yaw moment control strategy**

The lower layer develops the direct yaw moment control (DYC) to enhance the vehicle handling stability based on the differential torque control inputs of left and right in-wheel motors. To improve the energy efficiency, the relaxation factor is introduced to prevent the excessive yaw moment control inputs. Here, the *β* � *γ* phase plane is used to represent the vehicle stability region.

### *4.3.1 The vehicle dynamics modeling*

A two degree-of-freedom (2-DoF) vehicle model is adopted to describe the vehicle lateral dynamics characteristics. Assuming that the vehicle runs with a small yaw angle and steering input, the vehicle model is expressed as

$$\begin{aligned} m V\_x \left( \dot{\beta} + \dot{\phi} \right) &= F\_{\mathcal{Y}} + F\_{\mathcal{Y}^r} \\ I\_x \dot{\gamma} &= l\_f F\_{\mathcal{Y}} - l\_r F\_{\mathcal{Y}^r} + M\_c \end{aligned} \tag{44}$$

where *Iz* is the vehicle inertia moment of the yaw motion. *lf* and *lr* represent the distances from the front and rear axles to the vehicle gravity, respectively. *β* and *ϕ* denote the vehicle sideslip angle and yaw angle, respectively. *γ* ¼ *φ*\_ , *Fyi* ¼ 2*Ciαi*. *γ* represents the vehicle yaw rate. The tire slip angle *α<sup>i</sup>* generates the lateral force *Fyi*. *Mc* is the additional direct yaw moment control input.

The tire slip is further written as

$$\begin{cases} a\_{\mathcal{f}} = \delta\_{\mathcal{f}} - \frac{l\_{\mathcal{f}} \chi}{V\_{\mathcal{x}}} - \beta \\\\ a\_{\mathcal{r}} = \frac{l\_{\mathcal{r}} \chi}{V\_{\mathcal{x}}} - \beta \end{cases} \tag{45}$$

where *δ<sup>f</sup>* is the driver steering input. Through combing Eq. (44) and Eq. (45), we can obtain

$$
\dot{\xi} = \overline{A}\xi + \overline{B}\nu + \overline{C}\delta\_{\overline{f}} \tag{46}
$$

$$\begin{aligned} \text{where } \boldsymbol{\xi} = \left[ \boldsymbol{\beta}, \boldsymbol{\eta} \right]^{T}, \overline{\boldsymbol{A}} &= \begin{bmatrix} -\frac{\mathbf{C}\_{f} + \mathbf{C}\_{r}}{m\mathbf{V}\_{x}} \frac{\mathbf{C}\_{r}\mathbf{l}\_{r} - \mathbf{C}\_{f}\mathbf{l}\_{f}}{m\mathbf{V}\_{x}^{2}} - 1\\ \mathbf{C}\_{r}\mathbf{l}\_{r} - \mathbf{C}\_{f}\mathbf{l}\_{f} - \frac{\mathbf{C}\_{r}\mathbf{l}\_{f}^{2} + \mathbf{C}\_{r}\mathbf{l}\_{r}^{2}}{I\_{x}\mathbf{V}\_{x}} \end{bmatrix}, \overline{\mathbf{B}} &= \left[ \mathbf{0}\mathbf{1}/I\_{x}\right]^{T}, \overline{\mathbf{C}} = \begin{bmatrix} \frac{2\mathbf{C}\_{f}}{m\mathbf{V}\_{x}} \frac{2\mathbf{C}\_{f}\mathbf{l}\_{f}}{I\_{x}} \end{bmatrix}^{T}, \\\ \boldsymbol{\nu} &= \mathbf{M}\_{\boldsymbol{\epsilon}}. \end{aligned}$$

*Dynamics Modeling and Characteristics Analysis of Distributed Drive Electric Vehicles DOI: http://dx.doi.org/10.5772/intechopen.111908*

To facilitate the MPC design, the vehicle lateral dynamics model is discrete as follows.

$$
\xi(k+1) = \overline{A}'\xi(k) + \overline{B}'\nu(k) + \overline{C}\,\delta\_{\overline{f}} \tag{47}
$$

The system matrices are obtained by

$$\overline{A}' = e^{\overline{A}\Delta T}, \overline{B}' = \int\_{k\Delta T}^{(k+1)\Delta T} e^{\overline{A}[(k+1)\Delta T - t]} \overline{B}dt, \overline{C}' = \int\_{k\Delta T}^{(k+1)\Delta T} e^{\overline{A}[(k+1)\Delta T - t]} \overline{\mathbf{C}}dt\tag{48}$$

Due to the uncertain model parameter of vehicle longitudinal velocity, the LTV-MPC is also adopted in the lower layer.

### *4.3.2 The vehicle yaw motion control design*

For the vehicle yaw motion control, the sideslip angle and yaw rate are treated as important indices to represent the handling stability performance. In this paper, the steady yaw rate response and small value of sideslip angle are used as the reference value. Hence, the reference yaw motion can be represented by

$$\begin{cases} \beta\_{ref} = 0\\ \gamma\_{ref} = \frac{V\_{\infty}}{l\_f + l\_r + \frac{mV\_{\infty}^2(C\_r l\_r - C\_f l\_f)}{2C\_f C\_r (l\_f + l\_r)}} \delta\_f\\ \end{cases} \tag{49}$$

The cost function for the MPC design can be expressed as

$$\overline{J} = \sum\_{t=1}^{N\_p} \left[ \lambda\_1 \left( \beta(t+k|k) - \beta\_{\rm ref}(k) \right)^2 + \lambda\_2 \left( \gamma(t+k|k) - \gamma\_{\rm ref}(k) \right)^2 + \lambda\_3 \nu^2 \right] \tag{50}$$

$$|\Theta\_1 \chi - \Theta\_2 \beta| \le \sigma\_1 \tag{51}$$

$$|\Phi\_1 \chi - \Phi\_2 \beta| \le \sigma\_2 \tag{52}$$

where *M*min ≤ *Mc* ≤ *Mmax*. *Mmax* and *Mmin* represent the admitted maximal and minimal yaw moment control input, respectively. *λ*1, *λ*2, and *λ*<sup>3</sup> denote the weighting coefficients.

The Eq. (51) and Eq. (52) is widely used as the envelop control to describe the vehicle stability margin [1]. However, as shown in **Figure 23**, the direct yaw moment control input would also have an effect on the vehicle stability performance. Hence, in this work, the slack factors Φ<sup>1</sup> and Φ<sup>2</sup> in Eq. (53) and Eq. (54) are introduced to permit the vehicle runs out of the traditional stability boundaries to some extent.

$$|\Theta\_1 \chi - \Theta\_2 \beta| \le \sigma\_1 + \Psi\_1 \tag{53}$$

$$|\Phi\_1 \chi - \Phi\_2 \beta| \le \sigma\_2 + \Psi\_2 \tag{54}$$

Furthermore, considering that the yaw moment control input would also have an effect on the energy saving performance, a small DYC control should be given when

**Figure 23.** *Effect of yaw-moment control on the vehicle stability region.*

the vehicle has enough stability region. Therefore, a relaxation factor *ϑ* is adopted to dynamically adjust the weighting coefficients *λ*<sup>1</sup> and *λ*2. Here, the relaxation factor *ϑ* can be calculated by

$$\theta = \frac{1}{2} \left( \frac{|\overline{\sigma}\_1|}{\sigma\_{1,m}} + \frac{|\overline{\sigma}\_2|}{\sigma\_{2,m}} \right) \times (\overline{w} - \underline{w}) + \underline{w} \tag{55}$$

where *w* ¼ 0*:*5, *w*, *σ<sup>i</sup>*,*<sup>m</sup>* ¼ *σ<sup>i</sup>* þ Ψ*i*ð Þ *i* ¼ 1, 2 . Then the weighting coefficients are rewritten as

$$
\overline{\lambda}\_1 = \theta \lambda\_1, \overline{\lambda}\_2 = \theta \lambda\_2 \tag{56}
$$

Then the optimal direct yaw moment control inputs are evenly allocated to the left and right in-wheel motors. The total torque inputs for each in-wheel motor are represented by

$$\begin{cases} T\_{w\mathcal{J}} = T\_{w\mathcal{J}} - \frac{\mathcal{M}\_c}{t\_w} R\_\epsilon \\\\ T\_{w\mathcal{J}r} = T\_{w\mathcal{J}} + \frac{\mathcal{M}\_c}{t\_w} R\_\epsilon \\\\ T\_{w,rl} = T\_{w,r} \\\ T\_{w,rr} = T\_{w,r} \end{cases} \tag{57}$$

### **4.4 Test results**

Here, as shown in **Figure 24**, the hardware-in-the-loop test is conducted to verify the control effect. A high-fidelity distributed drive electric vehicle built by the commercial software Carsim is embedded into the PXI, which provides a real-time

*Dynamics Modeling and Characteristics Analysis of Distributed Drive Electric Vehicles DOI: http://dx.doi.org/10.5772/intechopen.111908*

**Figure 24.** *HIL bench test.*

simulator. The control strategy is downloaded in the calculation platform dSPACE by the code generation technology. The calculated torque inputs are sent to the PXI through the CAN bus. The DDEV model will execute the control command. Then the vehicle states are regarded as the feedback signal and transmitted to the dSPACE to calculate the optimal control inputs. The driver steering behavior is obtained by the driving simulator. The U-turn maneuver is selected as the test condition in the HIL test. In addition, to demonstrate the proposed method (PC), the traditional torque allocation combined linear quadratic regulator (TLQ) and the proposed controller without the relaxation factor (WRF) are set as the comparative tests. The traditional torque allocation method can be represented by

$$T\_{wgl} = T\_{wfr} = \frac{l\_r}{l\_f + l\_r} \frac{T\_{wd}}{2}, \\ T\_{w,rl} = T\_{w,rr} = \frac{l\_f}{l\_f + l\_r} \frac{T\_{wd}}{2} \tag{58}$$

Considering the limitation of the tire force, **Figure 25** shows the reference vehicle speed. The proposed method behaves with good performance to track the desired speed. The vehicle lateral dynamics response is shown in **Figures 26** and **27**. It can be seen that the proposed method can significantly guarantee the vehicle reference yaw rate tracking performance compared with the TLQ method, while reducing the sideslip angle. Owing to the superiority to handle the uncertain model parameter, the proposed method is effective to enhance vehicle handling stability under the large-curvature road driving condition. However, the tracking error of the TLQ method is a little large. In addition, as observed from the vehicle *β* � *γ* phase

**Figure 25.** *Vehicle speed tracking performance.*

**Figure 26.** *Vehicle yaw rate.*

plane in **Figure 28**, the proposed method can guarantee a more safe state by the DYC control. In contrast, the vehicle runs out of the stability margins with the TLQ method. It also proves the proposed method works to balance the multi-performance control.

*Dynamics Modeling and Characteristics Analysis of Distributed Drive Electric Vehicles DOI: http://dx.doi.org/10.5772/intechopen.111908*

**Figure 27.** *Vehicle sideslip angle.*

**Figure 28.** *Vehicle yaw motion phase plane.*

**Figures 29** and **30** show the efficiency of front and rear in-wheel motors, respectively. It is clear that the efficiency of the front in-wheel motor with the proposed method is better than that of the rear in-wheel motor. This is because when allocating the torque inputs, the proposed method is first to guarantee the high-efficiency work zone for front in-wheel motors. Hence, the efficiency of the rear in-wheel motor with the proposed method is worse than that of the TLQ method. However, from the power consumption in **Figure 31**, the proposed method still performs better to ensure the energy-saving performance compared with the TLQ method. However, the test results also demonstrate the proposed controller can be effective to guarantee the prescribed performance. Due to the existence of the disturbance during the HIL tests, there would have some fluctuation in test results. However, the test results also

**Figure 29.** *Vehicle speed tracking performance.*

**Figure 30.** *Vehicle yaw rate.*

demonstrate the proposed controller can be effective to guarantee the prescribed performance. From **Table 1**, the proposed method can reduce energy consumption by 3.18% and 10.02% compared to the WRF method and e TLQ method, respectively.

Furthermore, the feasibility of the proposed energy-saving control method has been proved in the HIL test. This indicates that the proposed controller has great potential to improve the comprehensive performance of the vehicle. In the future, we would concentrate on more efficient ways of energy-saving optimization problems.

*Dynamics Modeling and Characteristics Analysis of Distributed Drive Electric Vehicles DOI: http://dx.doi.org/10.5772/intechopen.111908*


**Table 1.**

*Energy consumption/(kJ).*

**Figure 31.** *Vehicle sideslip angle.*

**Figure 32.** *Vehicle yaw motion phase plane.*

The proposed strategy can not only be limited to the distributed drive electric vehicles. Meanwhile, the emergency collision avoidance condition would be also considered. In addition, from **Figure 32**, the tire longitudinal slip ratio with the proposed method is significantly smaller than that of the TLQ method. This demonstrates the proposed method can also improve the vehicle longitudinal stability. The test results verify the proposed method can be effective to improve the energy-saving performance based on guaranteeing the vehicle stability.

### **5. Conclusions**

This chapter introduces the distributed drive electric vehicle from the viewpoint of the dynamics modeling, stability performance analysis, and energy-saving strategy. The conventional modeling method of DDEVs is detailed first. Then, the stability region of DDEVs is estimated by establishing a rational polynomial-based DDEV model and adopting the SOSP technique to find the maximal Lyapunov function for estimation. The resulting stability regions with different parameters are presented, and comparison shows that the additional DYC has an expanding effect on the stability region. This suggests that DDEVs have greater potential in terms of stability and safety compared to centralized drive vehicles. Finally, a torque vector control framework for DDEVs is proposed in this paper to reduce the energy-consumption on the basis of maintaining the vehicle stability. The LTV-MPC-based hierarchical strategy is adopted to realize the parallel control of energy-saving and handling stability. A relaxation factor is introduced to reduce the energy consumption caused by additional direct-yaw-moment control input through evaluating the vehicle stability performance.

The proposed stability analysis method also has some issues to solve, in which the developed mode decision theorem and division of drive stability regions are mainly based on the tire adhesion ellipse theorem. However, the nonlinearity of the vehicle dynamics model also has an influence on the stability performance. In future research, theorems of body stability including *γ β* phase diagram and *g g* diagram will be considered in the torque distribution method design.

Furthermore, the feasibility of the proposed energy-saving control method has been proved in the HIL test. This indicates that the proposed controller has great potential to improve the comprehensive performance of the vehicle. In the future, we would concentrate on more efficient ways of energy-saving optimization problems. The proposed strategy can not only be limited to the distributed drive electric vehicles. Meanwhile, the emergency collision avoidance condition would be also considered.

### **Abbreviations**


*Dynamics Modeling and Characteristics Analysis of Distributed Drive Electric Vehicles DOI: http://dx.doi.org/10.5772/intechopen.111908*

