*2.8. Battery Module*

This study used a battery equivalent circuit, as shown in Figure 7, to establish the battery module [13]. The model provides the information of the open circuit voltage, output voltage, battery current, required power of battery, battery SOC, SOC changing rate, battery internal resistance, and battery capacity. The battery was mainly to support the power required for the motor/generator in order to keep the system in high fuel efficiency. The proposed model was adequate for the fuel consumption optimization.

**Figure 7.** Battery equivalent circuit.

In Figure 7, *Voc* is the open circuit voltage, *Req* is the internal equivalent resistance, and *VL* is the output voltage. The power output of battery can be represented in terms of the electric current, *Ibatt*, as shown in Equation (19). The total output of battery, *VocIbatt*, includes the power required for the system, *Pem\_batt*, and the power consumed by the internal resistance of the battery, *ReqI 2 batt*. The power required can also represented in terms of motors' power, as shown in Equation (20).

$$P\_{cm\\_batt} = V\_{oc}I\_{batt} - R\_{c\eta}I\_{batt\prime}^2 \tag{19}$$

$$P\_{\rm em\\_batt} = T\_{\rm MC1} \alpha\_{\rm MC1} \eta\_{\rm MC1}^K \eta\_{\rm MC1}^K + T\_{\rm MC2} \alpha\_{\rm MC2} \eta\_{\rm MC2}^K \eta\_{\rm conv}^K \tag{20}$$

where η*MG1* and η*MG2* are the efficiency of the motor/generator 1 and 2, η*con* is the motor controller efficiency. The battery SOC can be calculated by accumulating the charged and discharged current. The relationship between battery SOC changing rate, battery capacity *Qmax* and current *Ibatt* is as follows:

$$\text{S\dot{O}C}(t) = -\frac{I\_{\text{bttt}}}{Q\_{\text{max}}}.\tag{21}$$

From Equation (19), the battery current can be derived as follows:

$$I\_{batt} = \frac{V\_{\rm oc} + \sqrt{V\_{\rm oc}^2 - 4R\_{\rm cq}P\_{cm\\_batt}}}{2R\_{\rm cq}}\tag{22}$$

The open circuit voltage and internal equivalent resistance are function of SOC. From Equations (21) and (22), the SOC rate can be obtained as follows:

$$\text{SOC}(t) = -\frac{V\_{\text{cc}}(\text{SOC}) + \sqrt{V\_{\text{cc}}^2(\text{SOC}) - 4R\_{\text{batt}}(\text{SOC})P\_{\text{cm\\_batt}}(t)}}{2R\_{\text{batt}}(\text{SOC})Q\_{\text{max}}}.\tag{23}$$

The internal resistance of battery was based on the curve shown in Figure 8. A portion of the battery power output was provided for the driving system, and the other was consumed by the internal resistance. The efficiency of battery can be calculated, as shown in Equation (24).

$$
\eta\_{\text{batt}} = P\_{m\\_batt} / V\_{\text{oc}} \mathbf{I}\_{\text{batt}}.\tag{24}
$$

**Figure 8.** Internal resistance of battery while charging/discharging.

In this study, the SOC of battery was limited in the range between 0.4 and 0.6 since the battery had relatively less energy loss due to the battery internal resistance while considering for both charging and discharging states.

### **3. Energy Management Strategy**

## *3.1. Optimization*

The objective of the optimization problem was to minimize the fuel consumption and satisfy the following requirements for the HEV system: (1) To meet the demand of vehicle driving condition, and (2) to be constrained within the operation limits of the system components, as shown in Equations (25)–(30). The goal of the optimization, the cost function *J*, is expressed numerically in finite time, as shown in Equation (25). For the hybrid powertrain system with charge-sustaining control, the initial battery SOC and the final state should remain the same. In other words, the power loss of the system must be compensated by the engine. The power required for the vehicle is provided through engine and motors, as shown in Equation (26). Equation (27)–(30) define the SOC controlled limits, battery power output limits, engine power output limits, and motor power output limits, respectively. The optimization problem is defined as the following:

Objective:

$$\text{min} = \left\{ J = \int \frac{t\_f}{t\_0} \dot{m}\_{f\varepsilon}(t) dt \right\} \tag{25}$$

Subject to

$$P\_{req}(t) = P\_c(t) + P\_{cm}(t),\tag{26}$$

$$\text{SOC}\_{\text{min}} \le \text{SOC}(t) \le \text{SOC}\_{\text{-max}} \tag{27}$$

$$P\_{em\\_batt\\_min} \le P\_{cm\\_batt}(t) \le P\_{em\\_batt\\_max} \tag{28}$$

$$P\_{\mathcal{E}\\_\text{min}} \le P\_{\mathcal{E}}(t) \le P\_{\mathcal{E}\\_\text{max}} \tag{29}$$

$$P\_{\rm cm\\_min} \le P\_{\rm cm}(t) \le P\_{\rm cm\\_max\\_\prime} \tag{30}$$

where *t*, *J*, *mfc(t)*, *Pbatt*, *Pe*, *Pem,* and *Preq* are time, cost function, engine fuel rate, battery power, electric motor power, engine power, and vehicle power required, respectively. In this study, the SOC was limited between 0.4 and 0.6. Battery power and electric motor power were constrained between −60 kW and 60 kW. The engine power was between 0 kW and 157 kW.

### *3.2. Rule-Based Control Strategy*

According to the understanding of the system architecture and the efficiency of each element, the output energy of each driving element is defined based on the different road conditions. The basic principle is to meet the driving force required during vehicle travelling, while the control rule should keep the engine and motor/generators in the high operating efficiency range as long as possible to achieve the best fuel consumption and the lowest emissions. This study applied a rule-based controller for the baseline HEV model, and the heuristic was applied, as shown in Figure 9. The fuel economy of this controller would be compared with the manufacture data to verify the accuracy the HEV model. The rule-based strategy is listed in Table 2. Based on different SOC and required torque output, engine operation conditions are provided.

**Figure 9.** Heuristic controller.


**Table 2.** Rule-based strategy.
