## 3 Optimization-Based Strategies
### Instantaneous Optimization
#### 3.1.1 Equivalent Consumption Minimization Strategy (ECMS)  
The equivalent consumption minimization strategy (ECMS), as an instantaneous optimization algorithm, can obtain a near-optimal control strategy and can be implemented online due to its smaller requirements in terms of storage memory and computation time compared with DDP-based strategies. The core idea of ECMS is to convert electricity consumption of electric machines into fuel consumption by using an equivalent factor (EF), and adds this to the actual fuel consumption of the engine to obtain the equivalent fuel consumption at each moment. The equivalent fuel consumption is chosen as the objective function to solve the optimal energy distribution of HEVs at the minimum. The objective function can be expressed as:  
\[minJ(t)=min\int_{0}^{t}\dot{m}_{\alpha q}(\tau)d\tau \tag{1}\]  
\[\dot{m}_{eq}(t)=\dot{m}_{eng}(t)+\dot{m}_{bat\_eq}(t) \tag{2}\]  
\[\dot{m}_{bat\_eq}(t)=\gamma\dot{m}_{batt}(t)=\left\{ \begin{array}{l}\gamma. \frac{P_{bat}(t)}{Q_{br}}\eta_{clg}(t)P_{batt}<0\\ \gamma.\frac{P_{bat}(t)}{Q_{br}}\frac{1}{\eta_{clg}(t)}P_{batt}\geq 0\end{array} \right. \tag{3}\]  
where, \(\dot{m}_{eq}\) is the total equivalent fuel consumption rate of the HEV, \(\dot{m}_{eng}\) is the actual fuel consumption rate of the engine, \(\dot{m}_{bat\_eq}\) is the equivalent fuel consumption rate of the battery, \(P_{batt}\) is battery power, \(Q_{br}\) is fuel calorific value, \(\eta_{clg}\) is battery charging efficiency, \(\eta_{dis}\) is battery discharging efficiency, and \(\gamma\) is the equivalent factor, whose value is positive during the battery discharging process, while being negative during the battery charging process. It is worth noting that the results of ECMS are very sensitive to the equivalent factor, which is influenced by driving conditions, battery SOC, driving style, road gradient, and so forth. If the equivalent factor is too large, the strategy will prefer to use fuel, which results in increasing fuel consumption and higher battery power. Conversely, the strategy will tend to use electricity, which leads to excessive power consumption and lower battery power. Generally, the equivalent factor can be calculated by the efficiency of oil-electric conversion \(\eta_{fuel\_batt}\), which is defined as the efficiency of converting engine's chemical energy into electrical energy to store in the battery through generator. The efficiency \(\eta_{fuel\_batt}\) can be formulated as:  
\[\eta_{fuel\_batt}=\eta_{e}\eta_{gc}\eta_{clg} \tag{4}\]  
where, \(\eta_{e}\) is engine efficiency, \(\eta_{gc}\) is generator efficiency. Considering the average effect of a large number of different operation points during the driving cycle, the efficiency \(\eta_{fuel\_batt}\), \(\eta_{e}\), \(\eta_{gc}\), \(\eta_{clg}\) can be calculated by \(\overline{\eta}_{fuel\_batt}\), \(\overline{\eta}_{e}\), \(\overline{\eta}_{cg}\), respectively. Where \(\overline{\eta}_{e}\) is the average efficiency of the engine, \(\overline{\eta}_{gc}\) is the average efficiency of the generator, \(\overline{\eta}_{clg}\) is the average charging efficiency of the battery, and \(\overline{\eta}_{fuel\_batt}\) is the average conversion efficiency, which satisfies \(\overline{\eta}_{fuel\_batt}=\overline{\eta}_{e}\overline{\eta}_{gc}\overline{ \eta}_{chg}\). In the end, the equivalent factor can be calculated by the following formula:  
\[\gamma=\left\{ \begin{array}{l}\frac{\overline{\eta}_{en}\overline{\eta}_{ clg}}{\overline{\eta}_{e}}P_{batt}<0\\ \frac{1}{\overline{\eta}_{fuel\_batt}}P_{batt}\geq 0\end{array} \right. \tag{5}\]  
where \(\overline{\eta}_{m}\) is the average efficiency of the motor, \(\overline{\eta}_{dis}\) is the average discharging efficiency of the battery. In addition, the instantaneous fuel consumption rate of the engine \(\dot{m}_{eng}\) can be calculated by the interpolation of engine torque \(T_{e}\) and engine speed \(w_{e}\), and the battery power \(P_{batt}\) can be calculated by the change of SOC and the battery's terminal voltage.