*4.2. Analysis of Optimal Front/Rear Torque Distribution*

In this section, the effectiveness of the algorithm presented in this study will be analyzed by comparing two cases: one in which the input voltage changes are taken into account and one in which they are not. Figure 11 illustrates the results of the optimal distribution ratios of driving torques to the front and rear wheels corresponding to each voltage at the input terminal according to the respective motor speed and driving torque required by the driver.

As illustrated in Figure 5, when a driver requires a driving torque that is below that of peak efficiency at the rotating speed of driving, the optimal distribution ratio (*γopt*) of the driving torque to the front and rear wheels becomes zero. In contrast, when the required driving torque exceeds the peak efficiency, the ratio becomes *γopt* = 0.5. As a result, the driving torque is distributed uniformly to the front and rear wheels. However, since the driving torque required by a driver is limited to four times the maximum torque of the unit driving system, the optimal distribution ratio of the driving torque to the front and rear wheels will be limited to *γopt* = 0.5 when the driving torque required by a driver exceeds 50% of the available maximum torque, due to the limit set for the system. As illustrated in the map showing the efficiency characteristics of the unit driving system (Figure 5), this phenomenon is attributed to the rapid decrease in the efficiency characteristics in the low-torque domain, resulting in minimized driving in that domain.

**Figure 11.** Optimal front/rear distribution ratios with regard to the speed and desired torque: (**a**) at *Vdclink* = 75 V, (**b**) at *Vdclink* = 90 V (*Vnorm*), and (**c**) at *Vdclink* = 105 V.

The effects of the voltage variation at an input terminal on the optimal distribution of the driving torque to the front and rear wheels were verified at a system input terminal voltage of *Vdclink* = 75 V. The driving situation is assumed that motor speed is 2000 rpm (approx. 23 km/h) and the torque required by the driver is 40% of the entire driving torque. The operating point of the front and rear wheels and the power consumption for the optimal distribution ratio of the driving torque to the front and rear wheels derived at *Vdclink* = 90 V (*Vnorm*), with a system input terminal voltage of *Vdclink* = 75 V, are illustrated in Figure 12.

When applying the optimal driving torque distribution to both the front and rear wheels, derived at the input terminal voltage of *Vdclink* = 90 V (*Vnorm*), approximately 76 [W] more power was used, as compared to what was when applying the optimal driving torque distribution derived from the case when the input terminal voltage was *Vdclink* = 75 V. This result shows that consideration of the input terminal voltage is necessary to optimize system efficiency.

**Figure 12.** The difference of power consumption on optimal distribution with *Vdclink*; (**a**) The desired electric power of front and rear traction motor with respect to the distribution ratio, (**b**) Total electric power with respect to the distribution ratio, (**c**) Operating points of front and rear traction motor.

#### **5. Simulation Results**

To verify the algorithm presented in this paper, simulation model for a batterypowered electric shuttle bus in the longitudinal direction and driving simulation of the vehicle were carried out according to the Manhattan Bus Cycle as shown in Figure 13. The Manhattan Bus Cycle uses a maximum driving speed of 40.9 km/h, peak acceleration/deceleration of 0.2 g, and driving time of 1089 s with an average driving speed of 11 km/h; these are similar to the operating conditions and specifications of the target vehicle [17]. The simulation model built with Matlab/Simulnk is shown in Figure 14. All parameters used in the simulation are described as Tables 1 and 2 in Section 2. The battery model based on the internal resistance of the battery, Equation (25), is used to calculate the *Vdclink* of the driving system. The *Vdclink* of the driving system can be derived by Kirchhoff's current law [16]. Here, no voltage drop and losses between the battery output terminal and the input terminal of the driving system is assumed. And the state of charge (SOC) can be calculated by Equation (26) which is the ratio of the charged current over the full charged capacity.

$$V\_{dclink} = \ \ \ V\_{oc} - I\_b \mathcal{R}\_i \ \ \ \ \ P\_{batt} = \ \ \ V\_{dclink} I\_b \tag{25}$$

$$\text{SOC}\_{k} = \text{SOC}\_{k-1} + \frac{\Delta I\_{\text{b}\_{k}}}{Q\_{0}} \tag{26}$$

**Figure 13.** Manhattan Bus Cycle [17].

**Figure 14.** Simulation model: M1 to M4: In-wheel Motor.

Here, *Voc*, *Ib*, *Ri*, *SOCk*, *SOCk*−1, Δ*Ibk* and *Q*<sup>0</sup> represent open circuit voltage of the battery, current of the circuit, internal resistance of the battery, state of charge at time interval *k*, state of charge at time interval *k* − 1, variation of the current at time interval *k* − 1 and nominal battery capacity of the battery, respectively.

In total, three simulations corresponding to each case were carried out. All simulation was conducted with time interval of 10 ms. To compare the performance of the presented algorithm for varying input terminal voltages, a low–voltage condition (instead of the nominal voltage of 90 V) and an initial condition where the battery SOC was 20% were used. As described in Section 2, the target vehicle is a battery-powered electric vehicle for last-mile mobility. Once the target vehicle is charged, the mission of the target vehicle is to drive the predefined for root without additional charge of the battery. So, the simulation condition assumes that the battery of the vehicle is depleted to 20% of SOC. Here, the initial value of the input terminal voltage of the driving system was 82 V. The conditions employed for the simulation are as follows: a case with an uniform distribution, a case with an optimal distribution of the driving torque based on a nominal voltage of 90 V, and a case with the optimal distribution of the driving torque reflecting the real-time voltage of the input terminal. The following additional conditions were assumed for the simulation: inverter and decelerator efficiencies of 95% and an average electric component load of 400 W.

Table 6 presents the analysis results. The designed vehicle was considered in the driving simulation carried out based on the Manhattan Bus Cycle. In the case of uniform distribution of the required driving torque by the driver to both the front and rear wheels, the energy efficiency was 4.58 km/kWh, whereas the energy efficiency for the case where the optimal driving torque distribution was applied to both the front and rear wheels without considering the effect of the input terminal voltage was 4.83 km/kWh. This indicates an increase in efficiency of approximately 5.3%, compared to the uniform distribution. Furthermore, for the final case of applying the optimal distribution of the driving torque to both the front and rear wheels while considering the input terminal voltage (as presented in this study), the energy efficiency was found to be 4.86 km/kWh, indicating an improvement of approximately 6.0% compared to the case with uniform distribution. This shows that an additional 0.7% improvement in the energy efficiency was achieved by taking the input terminal voltage into account.


**Table 6.** Simulation results.

Figure 15 shows the operating points of the front and rear wheels under the conditions of the three simulations listed in Table 6. Due to the uniform distribution of the driving torque to the left and right single-axle torque, the left and right sides thereof represent the operating points of the front-right (FR) and rear-right (RR) wheels. In addition, the operating points were marked on the efficiency map of the 90 V condition to generalize the marking of operating points on the efficiency map, despite the fact that the efficiency map varied with changes in the input terminal voltage. Figure 15a illustrates the uniform distribution of the driving torque, wherein the operating points of FR and RR are identical. The operating point in (b) represents the results of allocating the driving torque to the front and rear wheels according to the driving speed of the vehicle and the driving torque required by the driver by following (b) presented in Figure 11. Meanwhile, (c) represents the results of applying the driving torque derived by applying the method presented in Figure 10, according to varying the input terminal voltage between (a) and (b) in Figure 11, to both the front and rear wheels. Altogether, the operating points of (b) and (c) in Figure 15 appear to be similar to each other. As illustrated in Figure 15c, the operating point varied with the application of the optimal distribution ratio of the driving torque as the input terminal voltage varied. Due to the accumulation of changes in the operating points, an additional 0.7% improvement in efficiency was secured.

Figure 16 represents the results of the simulations corresponding to (a)–(c) in Figure 15. In terms of the distribution ratio, when the input voltage is taken into account, it is operated at a better efficiency point than the nominal voltage of *Vdclink* = 90 V (*Vnorm*) by delaying transition to the uniform distribution. As a result, the accumulated consumption of energy decreased, ultimately resulting in an increased final energy efficiency.

**Figure 15.** Operating points: (**a**) Uniform distribution, (**b**) Optimization@*Vnorm* (90 V), and (**c**) Optimization@*Vdclink*.

**Figure 16.** Simulation results.

### **6. Conclusions**

In this paper, a method for improving the energy efficiency was presented by considering the input terminal voltage of the driving system of a battery-powered electric shuttle bus equipped with a decentralized driving system according to the battery's SOC. The proposed algorithm was verified by conducting simulations of the vehicle driving efficiency according to the Manhattan Bus Cycle. The conclusions are as follows:


**Author Contributions:** Conceptualization, I.-G.J.; methodology, I.-G.J.; software, I.-G.J.; validation, I.-G.J., C.-S.L.; data curation, I.-G.J., C.-S.L.; writing—original draft preparation, I.-G.J.; writing review and editing, S.-H.H.; visualization, I.-G.J., C.-S.L.; supervision, S.-H.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** This research was supported by the MSIT (Ministry of Science and ICT), Korea, under the ITRC (Information Technology Research Center) support program (IITP-2020-2018-0-01426) supervised by the IITP (Institute for Information & Communications Technology Planning & Evaluation).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

