## 1 Introduction  
As global warming accelerates, environmental regulations are being introduced and enforced around the world. In the United States, the National Highway Traffic Safety Administration has developed corporate average fuel economy standards that require an average fuel efficiency of 23.2 km/L by 2025. In Europe, the Euro 7 regulations for vehicle emissions also require nitrogen oxide emission to be lowered to 40-120 mg/kWh by 2025 [1]. In response to strong environmental regulations, the automobile industry is making great efforts to electrify the vehicle powertrain. Hybrid vehicles that provide propulsion power from two or more sources are typical examples of electrified powertrains. Hybrid electric vehicles (HEVs) are generally powered by both an internal combustion engine (ICE) and an electric motor (EM). HEV can reduce fuel consumption by operating ICE and EM in high-efficiency areas. When combined with an ICE-based powertrain, an EM can support the ICE in low-speed sections that are associated with low efficiencies and can regenerate energy during braking. As a result, an effective energy management strategy (EMS) that reflects driving and vehicle conditions is important to improve the efficiency of a HEV. The study of EMSs of HEVs can be classified into two main categories. The first comprises rule-based EMSs, in which rules based on the experience of engineers determine the power distribution ratio between the ICE and EM. The advantage of a rule-based EMS is its intuitiveness. However, due to its innate rigidity, it is impossible to design a rule that anticipates every possible driving situation, and fuel efficiency may deteriorate in situations that are not reflected in the rule [2; 3]. In addition, a rule-based EMS requires considerable time and resources as intensive parameter-tuning is required to anticipate diverse driving situations [4]. The second category comprises optimization-based EMSs, which use optimization theories to determine the power distribution ratio between the ICE and EM. An optimization-based EMS can determine the power distribution ratio closest to a global optimal solution. However, such a strategy can be complicated and require lengthy computational times due to complex optimization problems. Various studies have been conducted on optimization-based EMSs because the hardware performance of the hybrid control unit (HCU) has been increased and fuel consumption can be minimized by using a global optimal solution in various driving situations. In an optimization-based EMS, an optimal control problem (OCP) can be defined and calculated by using optimization theory. Depending on how it is defined, or which optimization theory is applied, an OCP can be divided into offline and online optimization algorithms. An offline optimization algorithm produces a global optimal solution that can minimize fuel consumption in all expected driving conditions, assuming that all conditions are known in advance. The most representative algorithm is dynamic programming (DP), which is based on the principle of optimality proposed by [PERSON]. After splitting the complex problem into several simpler subproblems, the solution of the subproblems is sequentially obtained in a recursive manner, so that a complete algorithm obtains the global optimal solution. Although DP can identify a global optimal solution, it is difficult to apply such solutions to a real-world HCU because DP requires lots of computational resources. A global optimal solution obtained by DP is used primarily in offline analysis in the development of other EMS options, such as rule-based designs or as machine-learning training data [5]. Online optimization produces a solution that can minimize fuel consumption for a given driving condition in a limited time. In general, it can be applied to a real-world HCU by reducing the computational load by using a simplification method (e.g., linearization) or local optimization.