*2.4. Controller Module*

In this study, two sets of controllers were established, the rule-based control module for baseline HEV model and the Equivalent Consumption Minimization Strategy (ECMS) module for the optimized HEV model. These control modules were combined with the engine switch strategy to further determine the operating time of the ICE. The individual controllers were built with Matlab functions.

Based on the torque required for the vehicle driving and the battery SOC, the rule-based controller, heuristic method (if-then-else), determined the speed and torque of ICE. The ESCM with object function

developed the working state of the engine. After determining the status of ICE, the mode switch control module would switch between different eCVT and fixed-gear modes.

### *2.5. Transmission Module*

The transmission includes two planetary gear sets, two motor/generators, and four clutches. Based on the vehicle driving condition, the mode switch module would determine the mode of operation, mode 1 for first eCVT mode and mode 2 for second eCVT mode. For mode 1, the motor speeds and torques of MG1 and MG2 were simulated by Equations (5)–(8). Equations (9)–(12) were for mode 2 operation.

$$
\omega\_{\rm MG1} = \frac{1}{i\_1} \omega\_\varepsilon - \frac{(1 - i\_1)(1 + i\_2)}{i\_1 i\_2} \omega\_{\rm out} \tag{5}
$$

$$
\omega\_{\rm MC2} = \frac{1 + i\_2}{i\_2} \omega\_{\rm out} \tag{6}
$$

$$T\_{\rm MG1} = -i\_1 T\_{\rm \varepsilon} \tag{7}$$

$$T\_{\rm MC2} = -(1 - i\_1)T\_c + \frac{i\_2}{1 + i\_2}T\_{\rm out\_{\prime}} \tag{8}$$

$$
\omega\_{\rm MG1} = -\frac{i\_2}{1 - i\_1 - i\_1 i\_2} \omega\_{\rm \varepsilon} + \frac{(1 - i\_1)(1 + i\_2)}{1 - i\_1 - i\_1 i\_2} \omega\_{\rm out},\tag{9}
$$

$$
\omega\_{\rm MC,2} = \frac{1}{1 - i\_1 - i\_1 i\_2} a \iota\_\varepsilon - \frac{i\_1 (1 + i\_2)}{1 - i\_1 - i\_1 i\_2} a\_{\rm out} \iota\_\prime \tag{10}
$$

$$T\_{\rm MG1} = -i\_1 T\_c + \frac{1}{1 + i\_2} T\_{out\prime} \tag{11}$$

$$T\_{\rm MC2} = -(1 - i\_1)T\_c + \frac{i\_2}{1 + i\_2}T\_{\rm out} \tag{12}$$

where,

$$i\_1 = \frac{R\_{\rm S1}}{R\_{\rm R1}} \,\prime\tag{13}$$

$$i\_2 = \frac{R\_{\rm S2}}{R\_{\rm R2}},\tag{14}$$

ω*e*, ωMG1, ω*MG2*, ω*out, Te*, *T*MG1, *T*MG2, and *Tout* are the rotational speeds and torques of the engine, two motors, and transmission output. *R*R1, *R*R2, *R*S1 and *R*S2 are the radii of ring gear 1 and 2 and of sun gear 1 and 2, respectively. *i1, i2* are the radius ratio of sun gear to ring gear for gear train 1 and 2, respectively.

In the simulation, the rotational inertia of engine, *Ie*; inertia of ring gear 1 and 2, *I*R1 and *I*R2; inertia of carrier 1 and 2, *I*C1, and *I*C2; inertia of motor/generator 1 and 2, *I*MG1 and *I*MG2; and inertia of sun gear 1 and 2, *I*S1 and *I*S2; are all considered. In mode 1 case, the general force-acceleration matrix can be written as shown in Equation (15). Similarly, Equation (16) is the case of mode 2.

$$
\begin{bmatrix}
\dot{\omega}\_{\varepsilon} \\
\dot{\omega}\_{\text{fuc}} \\
\dot{\omega}\_{\text{MG2}} \\
\dot{\omega}\_{\text{MG2}} \\
F\_{1} \\
F\_{2}
\end{bmatrix} = \begin{bmatrix}
I\_{\text{I}} + I\_{\text{R1}} & 0 & 0 & 0 & R\_{\text{R1}} & 0 \\
0 & I\_{\text{C2}} + \frac{r\_{\text{fuc}}^{2}}{K\_{f}^{2}}m & 0 & 0 & 0 & -R\_{\text{R2}} - R\_{\text{S2}} \\
0 & 0 & I\_{\text{MG1}} + l\_{\text{S1}} & 0 & -R\_{\text{S1}} & 0 \\
0 & 0 & 0 & I\_{\text{MG2}} + l\_{\text{C1}} + l\_{\text{S2}} & -R\_{\text{R1}} + R\_{\text{S1}} & R\_{\text{S2}} \\
R\_{\text{R1}} & 0 & -R\_{\text{S1}} & -R\_{\text{R1}} + R\_{\text{S1}} & 0 & 0 \\
0 & -R\_{\text{R2}} - R\_{\text{S2}} & 0 & R\_{\text{S2}} & 0 & 0
\end{bmatrix}^{-1} \begin{bmatrix}
T\_{\varepsilon} \\
T\_{\text{MG1}} \\
T\_{\text{MG2}} \\
0 \\
0
\end{bmatrix} \tag{15}
$$

$$
\begin{bmatrix}
\dot{w}\_{\rm t} \\
\dot{w}\_{\rm tot} \\
\dot{w}\_{\rm MG1} \\
\dot{w}\_{\rm RG2} \\
F\_{1} \\
F\_{2}
\end{bmatrix} = \begin{bmatrix}
I\_{t} + I\_{\rm R1} & 0 & 0 & 0 & R\_{\rm R1} & 0 \\
0 & I\_{\rm C2} + \frac{r\_{\rm I\rm H}^{2}}{R\_{f}^{2}}m & 0 & 0 & 0 & -R\_{\rm R2} - R\_{\rm S2} \\
0 & 0 & \hbar\_{\rm MC1} + I\_{\rm S1} + I\_{\rm R2} & 0 & -R\_{\rm S1} & R\_{\rm R2} \\
0 & 0 & 0 & \hbar\_{\rm MC2} + I\_{\rm C1} + I\_{\rm S2} & -R\_{\rm R1} + R\_{\rm S1} & R\_{\rm S2} \\
0 & R\_{\rm R1} & 0 & -R\_{\rm R1} & -R\_{\rm R1} + R\_{\rm S1} & 0 & 0 \\
0 & -R\_{\rm R2} - R\_{\rm S2} & R\_{\rm R2} & R\_{\rm S2} & 0 & 0
\end{bmatrix}^{-1} \begin{bmatrix}
T\_{\rm e} \\
T\_{\rm MG1} \\
T\_{\rm MG1} \\
T\_{\rm MG2} \\
0 \\
0
\end{bmatrix} \tag{16}
$$

where

$$
\Delta m = M + \frac{I\_{\text{turbell}}}{r\_{\text{fire}}^2}.\tag{17}
$$

*F1*, *F2*, and *Ftire* are the forces acting on the sun gear, ring gear, and tire, respectively. *Kf* is the final axle ratio, and *rtire* is the radius of tire. *Iwheel* is the total rotational inertia of the wheels [18].

### *2.6. Internal Combustion Engine Module*

The ICE module of this study was represented by a lookup table. Figure 5 shows the three-dimensional ICE fuel consumption rate. Through the controller module to determine the engine running state, the corresponding engine speed and torque could determine engine fuel consumption rate.

**Figure 5.** ICE fuel consumption rate.

### *2.7. Motor*/*Generator Module*

In AHS-II powertrain, there are two electric motor/generators, MG1 and MG2, which have same output power. The motor/generators are 60kW permanent magnet AC motors. In this study, MG1 and MG2 had same specifications. The motor/generators efficiency is a function of speed and output torque, as shown in Figure 6. This module was modeled with a lookup table. The motor/generator power calculation is shown in Equation (18).

$$P\_{\rm MG} = \omega\_{\rm MG} T\_{\rm MC} \tau\_{\rm MC}^K \begin{cases} \ K = 1\\ \ K = -1 \end{cases} \tag{18}$$

where *P*MG, ωMG, and *T*MG are motor/generators power, speed, and torque, respectively. If the speed and torque are in the same sign, the motor/generator works as a motor. If the speed and torque are in different sign, the motor/generator works as a generator, which transforms the mechanical energy into electricity and stores in the battery pack. ηMG is the efficiency of the motor/generator, and *K* is the power flow of the motor/generator. *K* = 1 is motoring, and *K* = −1 is generating.

**Figure 6.** The efficiency of motor/generator.
