*2.5. Battery Model*

The electric power required by the M/Gs was provided by the battery, and the total power of the battery was estimated through the motor's power and efficiency, as presented in Equation (2).

$$P\_{\rm butt} = T\_{\rm MG1} \alpha\_{\rm MG1} \eta\_{\rm MG1}^k \eta\_{\rm c1}^k + T\_{\rm MG2} \alpha\_{\rm MG2} \eta\_{\rm MG2}^k \eta\_{\rm c2}^k \tag{2}$$

*Pbatt* is the power of the battery, η*<sup>c</sup>* is the efficiency of the electric converter, *TMG* is the M/G torque, ω*MG* is the speed of the M/G, η*MG* is the efficiency of the M/G, and *k* indicates the energy path. *k* = −1 indicates that the battery was discharged, and *k* = 1 indicates that the battery was charged.

The battery module was composed as an equivalent circuit, including open circuit voltage and a battery pack, as shown in Figure 7. *Ibatt* represents the battery current, and *Pbatt* is the battery output or input power.

**Figure 7.** Diagram of battery equivalent circuit.

The relationship between battery SOC and current *Ibatt* is as follows.

$$\dot{SOC} = -\frac{I\_{\text{batt}}}{Q\_{\text{max}}} \tag{3}$$

*Qmax* is the ampere-hour capacity of the battery at the current rate *Ibatt*. The battery current, open circuit voltage, and internal resistance would vary according to the battery power. The relationship between battery output or input power and current is presented as follows.

$$P\_{\text{batt}} = V\_{\text{OC}} I\_{\text{batt}} - I\_{\text{batt}}^2 R\_{\text{batt}} \tag{4}$$

$$I\_{batt} = -\frac{V\_{OC} - \sqrt{V\_{OC}^2 - 4P\_{batt}R\_{batt}}}{2R\_{batt}}\tag{5}$$

$$\dot{SOC} = -\frac{V\_{OC} - \sqrt{V\_{OC}^2 - 4\left(T\_{MG1}\omega\_{MG1}\eta\_{MG1}^k \eta\_{c1}^k + T\_{MG2}\omega\_{MG2}\eta\_{MG2}^k \eta\_{c2}^k\right)}}{2R\_{\text{hult}}Q\_{\text{max}}} \tag{6}$$

where *Voc* is the battery open circuit voltage and *Rbatt* is the battery's internal resistance.

## *2.6. Controller Model*

### 2.6.1. Rule-Based Controller

The controller module had three main functions: the first one was the control of switching mode, which decided the timing of switching between the first and the second mode according to the driving cycle; the second one was based on the driving condition and battery SOC to determine the controlled engine speed and engine torque according to the rules established in the controller; the third one was engine state control logic. To achieve fuel saving, the engine would be turned off when it was not required to provide driving power.
