## 1 Introduction
Pontryagin's minimum principle (PMP), model predictive control (MPC), and equivalent consumption minimization strategy (ECMS) are representative algorithms. A PMP performs local optimization by applying a numerical method to a Hamiltonian function, which considers the battery's state of charge (SOC) and the fuel consumption characteristics of the ICE [6; 7; 8]. A PMP needs to efficiently design the Hamiltonian function to reduce computational loads. An MPC obtains an optimal solution for a given driving condition on a finite horizon. For an infinite horizon, an OCP is defined by designing cost functions and constraints that represent the characteristics of the system. A numerical solution can be obtained by applying a solver, such as quadratic programming and nonlinear programming (NLP) to a defined OCP [9; 10]. An MPC needs to design system models and constraints carefully to reduce computational loads and create a coping mechanism for infeasible regions. Recently, research on the use of nonlinear-MPC (NMPC) by NLP solver has also been actively conducted. NMPC directly determines the torque of the engine and motor. NLP solvers used in NMPC include the sequential quadratic programming (SQP) method [11; 12], mixed-integer nonlinear programming (MINLP) method [13], and the interior point method [14]. NMPC is one of the most efficient and powerful online optimization methods. When designing an NMPC, two analyses are most important. The first is an analysis for model simplification to reduce long computational time. The second is an analysis of the infeasible region to design a strategy for the infeasible solution. An ECMS derives a solution that minimizes equivalent energy consumption, which is the equivalent energy consumption in the form of a cost function that reflects fuel energy and electrical energy. This cost function defines the equivalent factor, which confirms the relative importance of electrical energy and fuel energy, according to driving conditions. The equivalent factor is a therefore an important tuning factor when designing an ECMS. A traditional ECMS uses an equivalent factor with a fixed value [15; 16]. However, the equivalent factor can be sensitive to driving conditions [17; 18; 19; 20]. An adaptive-ECMS (A-ECMS) has been proposed to regulate the equivalent factor according to changes in driving conditions. An A-ECMS adjusts the equivalent factor so that the SOC can follow the reference SOC trajectory while considering driving conditions. To obtain the reference SOC trajectory, researchers have applied particle swarm optimization (PSO) [21; 22; 23], approximate DP [24; 25; 26], and neural networks (NN) [27; 28; 29; 30; 31]. An A-ECMS is one of the most suitable EMS options to use in real-world HCUs [32; 33; 34; 35; 36]. However, A-ECMS designs have some drawbacks. First, computational time must be minimized to operate in a real-world HCU. An A-ECMS compares the control input candidates presented in a complex solution space to determine the power distribution ratio. Because the calculation of many control input candidates should reflect the characteristics of the HEV, system models and constraints need to be efficiently designed for computation. If the system model and constraints are too complex or poorly designed, the computational time increases significantly. Determining the reference SOC trajectory accurately and rapidly is also challenging. An A-ECMS adjusts the equivalent factor according to the reference SOC trajectory. If the accuracy of the reference SOC trajectory is insufficient, fuel consumption cannot be minimized. Even an accurate reference SOC trajectory cannot be used for an A-ECMS if it requires a long computational time. Only accurate and rapid calculation of a reference SOC trajectory can maximize the performance of an A-ECMS. In this paper, an NLP-based ECMS is proposed. NLP-based ECMS does not directly determine the torque of the engine and motor.