## 1 Introduction
By introducing an equivalent factor (EF), the ECMS transforms electric energy from the battery into an equivalent quantity of fuel consumption and minimizes the total fuel consumption at each instant, which is the sum of the real fuel consumption by the ICE and the weighted equivalent of fuel consumption by the electric motor. Inspired by the PMP [16], the EF can be assumed to be a constant if the battery's SOC remains unchanged; therefore, it is suitable for real-time implementation in PHEVs in charge-sustaining mode. However, the optimal EF is highly dependent on the driving cycle condition, and an EF that is optimal for one particular cycle may be inapplicable to other driving cycles [17; 18]. Therefore, various adaptive ECMSs have been proposed to obtain an online-adjusted EF that can be adapted to actual driving conditions: the feedback correction of the EF with respect to target SOC tracking [19], EF correction with look-up tables using the information of vehicle speed and trip length [20], neural network- and learning-based adaption [21], etc. Besides the ECMS, some recent studies have applied the Koopman operator to linearize the high nonlinearities of the powertrain system and combined it with model predictive control (MPC) in a data-driven framework to solve the local optimal problem of EMS [22; 23; 24]. This approach reduces the computational complexity and guarantees the real-time optimization of the system; however, it requires sufficient data to train the linearized model to ensure accuracy. In summary, studies on optimization-based EMSs are largely focused on fuel economy optimization; by including weighted terms in the cost function--such as SOC sustainability, battery health, and ICE start/stop and emissions--a multi-objective optimization problem can be formed to achieve a comprehensive solution for EMSs. However, few studies have considered the noise, vibration, and harshness (NVH) issues that occur with ICEs, which means that the optimized EMS is not applicable in real-world implementations. Specifically, the NVH of ICEs is relatively high when operating within an efficient range under high loads; it cannot be masked by the road and wind noises if the vehicle is driving slowly. Hence, the ICE should not operate under low vehicle speed conditions, such as in downtown areas or in heavy traffic [25]. However, the battery's SOC is not sustainable if this situation persists for a long time; thus, the ICE has to operate if the SOC has achieved a suitably low threshold, but at a lower load under the NVH constraint. Therefore, the NVH constraint results in a trade-off between fuel economy and SOC sustainability in the charge-sustaining mode. Homogeneous charge compression ignition (HCCI) combustion is another promising technology for improving fuel economy and reducing emissions that has been widely investigated in recent decades. Compared with conventional spark ignition (SI) combustion, HCCI combustion shows a significant improvement in fuel efficiency under part-load conditions owing to its unthrottled operation, which can reduce pumping loss [26]. However, HCCI combustion has the drawback of a limited operating range due to possible misfire at low loads and knock at high loads. Therefore, in conventional ICE vehicles, it is difficult to operate the engine purely in the HCCI mode; thus, an SI/HCCI dual-mode engine with a mode transition control strategy is required. Furthermore, the combustion timing control and load transition are also challenging due to the auto-ignition mechanism [27]. If we are to retain the advantageous high efficiency under part-load conditions that is achieved with HCCI, as well as the flexibility of power distribution that is achieved with HEVs, then the integration of a dual-mode combustion engine in an HEV is a promising solution. In [28], an SI/HCCI engine was applied in a mild HEV to achieve the torque assist from the integrated starter-generator (ISG), which enables a longer residence time in HCCI combustion.