## 3 Optimization-Based Strategies
### Instantaneous Optimization
#### 3.1.1 Equivalent Consumption Minimization Strategy (ECMS)
Therefore, battery SOC and vehicle speed \(v\) can be chosen as the state variables, and \(T_{e},w_{e}\) or \(T_{m},w_{m}\) can be chosen as the control variables. Based on the above state variables and control variables, the main research areas of ECMS are shown in Figure 4, including the determination and optimization of the equivalent factor, and the solution of the ECMS algorithm with constraints. For a given driving cycle, the optimal equivalent factor can be obtained by repeatedly adjusting the equivalent factor until the terminal SOC is equal to the initial SOC in the offline calculation. Meanwhile, the equivalent factor should be adaptively adjusted due to uncertain future driving conditions. Therefore, the determination and adaptive adjustment of equivalent factor are the keys of ECMS. In [50], the optimal equivalent factor \(\lambda^{*}\) was determined by the full trajectory of the driver's demanded power, and the time-varying equivalent factor \(\lambda\) was set as an estimate of \(\lambda^{*}\) to achieve the adaptive ECMS, which was not needed in order to predict vehicle speed or perform horizon optimization. However, the optimal equivalent factor \(\lambda^{*}\) of ECMS can be calculated only if the whole driving cycle is known in advance. Instead of direct estimation of the optimal equivalent factor in ECMS, the optimal equivalent factor \(\lambda^{*}\) can be estimated through the estimation of the upper and lower bounds of the optimal equivalent factor, which are functions of the HEV's configuration and independent of the driving cycle. The idea of the upper and lower bounds of EF can be employed in designing adaptive ECMS (A-ECMS) [51], which can obtain a charge-sustaining solution and minimize the total fuel consumption. The adaptive ECMS (A-ECMS) strategy as an online control strategy is introduced to estimate \(\lambda^{*}\) at each moment, which can be categorized into instantaneous A-ECMS and predictive A-ECMS. Based on the theoretical analysis of [PERSON]'s Minimum Principle and introducing soft constraints inside the range, an ECMS-CESO (catch energy-saving opportunity) strategy for series HEVs as the instantaneous A-ECMS is introduced in [52], which can achieve near-optimal fuel economy without the need for predicting future driver demand. The cost function of ECMS-CESO is penalized when SOC exceeds the soft bounds and uses battery power, and its penalty factor is equivalent to the equivalent factor for the latter. A driving-style-oriented adaptive ECMS is developed in [53], which classifies drivers into six groups, from moderate to aggressive, using kernel density estimation and entropy theory. According to driving style, the EF is tuned based on the relationship between SOC and power demand to improve the fuel economy and charge sustainability of HEVs. An adaptive ECMS with velocity prediction as predictive A-ECMS was proposed to adaptively adjust the equivalence factor in real time [54]. The velocity predictor was constructed by neural network to forecast short-term future driving behaviors based on historical data, and the bisection method was utilized to guarantee the convergence of EF. By deriving the theoretical relationship between the optimal EF and future driving statistics, a strategy synthesized with predictive ECMS was proposed to develop the causal adaptation law to adjust equivalence factor [55]. In this work, the non-causal behavior of EF in various driving scenarios was analyzed using the DP-based extraction method, which obtains the equivalence factor corresponding to optimal SOC trajectory and the optimal control policy by using the DP algorithm. Figure 4: Depiction of the main research areas of ECMS. In addition, RB-ECMS, combining rule-based strategy and ECMS, was developed to further improve fuel economy while ensuring drivability and battery charge-sustaining for HEVs. The equivalent factor in ECMS can vary with power demand and battery SOC, which can be obtained online from a 2-dimensional mapping established offline, which can be optimized by intelligent algorithm.