## 1 Introduction  
Hybrid electric vehicles (HEVs) have become a research focus of the automotive industry, with their technology developing and evolving at a fast pace over the last two decades. They have the potential to reduce fuel consumption and emissions in comparison to conventional vehicles while eliminating the range anxiety issues that electric vehicles (EVs) usually have due to the limited capacity of their battery pack [1, 2]. In brief, HEVs combine more than one power source, for example, an internal combustion engine (ICE) and a chemical battery-electric motor. Based on the components' connection and energy flow routes, HEVs can typically be classified as series hybrids, parallel hybrids, series-parallel hybrids, or complex hybrids, and different power sources can be combined through electrical and/or mechanical coupling [3]. In addition, a particular type of HEV called a plug-in hybrid electric vehicle (PHEV) features a large-capacity battery pack, which can be charged from a power grid and covers a certain pure electric driving mileage [4]. The idea behind fuel saving with an HEV is that the ICE can be operated within an efficient range with the assistance of an electric power source. Therefore, the energy management strategy (EMS), which governs the energy distribution between different power sources, becomes the core essential factor of HEV technology [5]. Generally, EMSs can be classified into two main categories: rule-based and optimization-based [6]. For series hybrid electric vehicles (SHEVs), there are two commonly used rule-based EMSs: the thermostat control strategy (TCS) and the power follower control strategy (PFCS). The TCS operates the ICE with a fixed power at the most efficient point, and a state of charge (SOC) fluctuation window is set to determine the engine's on/off status [7]. The TCS can provide the highest engine efficiency. However, the battery's energy loss is relatively large; therefore, the best overall fuel efficiency may not be guaranteed. Furthermore, there is no relationship between engine behavior and driving action, which will result in poor drivability. On the contrary, under the PFCS, the engine output power is made to follow the power demands, and the operation of the battery can be significantly improved. However, the ICE may not always operate within an efficient range; thus, the overall fuel economy could be worse compared to that achieved with the TCS [8]. For PHEVs, since the battery has a relatively large capacity and can achieve zero fuel consumption if the travel distance is shorter than the all-electric range (AER), the charge-depleting-charge-sustaining (CD/CS) strategy, as a rule-based EMS, is widely used. Here, the vehicle uses all-electric energy in the charge-depleting (CD) mode, until the SOC falls below some certain threshold; then, in the charge-sustaining (CS) mode, the SOC is maintained at a relatively stable level by governing the power split between the ICE and the battery [9]. The CD/CS strategy has a simple control logic for real-time implementation; however, its fuel economy is not optimal for long-distance travel [10]. Optimization-based EMSs aim to solve an optimal control problem by minimizing the cost function of fuel consumption and can be classified into two categories: global optimization and real-time optimization. The representative approaches to global optimization include dynamic programming (DP) [11], Pontryagin's minimum principle (PMP) [12], and metaheuristic methods, such as the genetic algorithm (GA) [13] and particle swarm optimization (PSO) [14]. The global optimization method guarantees that the absolute optimal solution will be found, but it is based on the requirement of a priori driving cycle information, and the computational load is fairly high, which makes it unsuitable for real-time implementation. One of the most extensively employed real-time optimization methods is the equivalent consumption minimization strategy (ECMS), which was first proposed by [PERSON] [15].