A New HEV Power Distribution Algorithm Using Nonlinear Programming

Abstract

An equivalent consumption minimization strategy (ECMS) is one of the most powerful and practical ways to improve the fuel efficiency of hybrid electric vehicles (HEVs). In an ECMS, it is important to determine the optimal equivalent factor to reach a global optimal solution. The optimal equivalent factor is determined by driving conditions. Previous studies have used an adaptive ECMS (A-ECMS) to determine the appropriate equivalent factor according to changing driving conditions. An A-ECMS adjusts the equivalent factor by controlling the battery’s state of charge (SOC) to follow a reference SOC trajectory. It is therefore critical to identify a reference SOC trajectory that reflects real-world driving conditions. These conditions, which are composed of the HEV’s nonlinear dynamics and complex constraints, can be formulated into a nonlinear optimal control problem (NOCP). Here, we propose applying nonlinear programming (NLP) to an A-ECMS. The NLP-based ECMS algorithm can be divided into two parts: the use of an NLP to solve an NOCP to obtain the reference SOC trajectory and the application of an NLP solution (the result of the first part) to an A-ECMS. Simulation results demonstrate that the proposed NLP-based ECMS closely resembles a global optimal solution for dynamic programming in a relatively brief calculation time.

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 Richard Bellman. 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. 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. The equivalent factor is determined by NLP, and the torque of the engine and motor is determined by the A-ECMS scheme using the equivalent factor. Therefore, the OCP size of the NLP can be reduced, and stable control of the vehicle is possible, even if the NLP has derived an infeasible solution. As mentioned above, it is important to obtain the reference SOC trajectory accurately and rapidly, and NLP can provide rapid and accurate calculations. To reduce NLP computation time, an optimal system model and constraints for an HEV with a P2 structure (TMED) was designed. Results of simulations showed that the proposed NLP-based ECMS closely resembled the global optimal solution produced by DP in a relatively short calculation time.

The main contributions of this paper are as follows. First, a suitable required torque preprocessing method for a P2 structure is proposed. In this process, the required torque demanded by the ICE and EM according to the change in gear ratio is calculated. Secondly, a method for calculating the reference SOC trajectory using NLP is proposed. System models, constraints, and cost functions were designed to reflect the characteristics of the HEV with a P2 structure and driving conditions. To reduce the computational time, the NLP model was simplified. Finally, an A-ECMS that can efficiently follow the reference SOC trajectory is proposed as an optimization-based EMS resembling the global optimal solution of DP.

This paper consists of the following structure. Section 2 presents a system model of a HEV with a P2 structure, Section 3 presents analysis result of the NLP-based ECMS, Section 4 discusses the significance of our findings, and Section 5 is discussion of the result.

2. System Model

In this paper, the NLP-based ECMS was applied to HEVs with a P2 structure, which includes a four-cylinder ICE, a permanent-magnet synchronous motor (PMSM), a lithium-ion battery, and a five-speed automatic transmission. The configuration of the P2 hybrid powertrain is shown in Figure 1. An HEV with a P2 powertrain was simulated by using Autonomie [37], a forward-looking simulation tool developed by Argonne. The NLP-based ECMS algorithm was designed to use Autonomie’s data. In addition, an Autonomie simulation environment was used to analyze the performance of the NLP-based ECMS.

Table 1. Specification of HEV.

Component	Parameter	Value
Engine (4-cylinder)	Displacement Maximum power	2.2 [L] 85.0 [kW]
Motor (PMSM)	Maximum power	29.2 [kW]
Battery (Lithium-ion)	Capacity Nominal voltage	7.03 [Ah] 324 [V]
Transmission (5-speed automatic)	Gear ratio	2.563/1.552/1.022/0.727/0.52
Final drive	Gear ratio	4.438
Wheel	Wheel radius	0.30115 [m]
Chassis	Mass	1680 [kg]

lists the specifications of the major components of the Autonomie model.

2.1. Internal Combustion Engine

An EMS is designed to minimize fuel consumption of an HEV. The ICE of an HEV is modeled as quasistationary, represented by a brake-specific fuel consumption (BSFC) map configured by using the ICE’s steady-state test data, as shown in Figure 2. A BSFC map is formulated as a function of torque ($T_{eng}$) and speed ($w_{eng}$), as shown in Equation (1). The relationship between BSFC and the power ($P_{eng}$) is shown in Equation (2). $Q_{LHV}$ is the lower heating value. We have

$$bsfc=f\left(T_{eng}, w_{eng}\right)$$

(1)

$$P_{eng}=bsfc\times Q_{LHV}.$$

(2)

The quasistationary model cannot consider the transient response of the ICE. The transient response of an ICE adversely affects fuel consumption. The transient response of an ICE has complex dynamics, which are difficult to reflect in an EMS accurately. The transient response of the ICE is simplified as a first-order delay function to reflect it in the EMS, as shown in Equation (3). As a result, when the input torque ($T_{e,cmd}$) is determined, the actual torque ($T_{eng}$) passes through the low-efficiency area of the BSFC map:

$${\dot{T}}_{eng}=\frac{1}{{\tau }_{eng}}\left(T_{e,cmd}-T_{eng}\right).$$

(3)

2.2. Electrical Motor

An HEV is being driven while maintaining the SOC at a certain level. The SOC is determined by the current of the EM, when excluding the influence of accessories. The current of the EM is determined by power. The power of the EM was represented in a quasistationary model in the form of a map. The power map was expressed as a function of torque ($T_{mot}$) and speed ($w_{mot}$), as shown in Equation (4). The power of an EM also has a nonlinear characteristic, as shown in Figure 3. We have

$$P_{mot}=f\left(T_{mot}, w_{mot}\right)$$

(4)

Because the power map of an EM is also quasistationary, it cannot consider a transient response. A transient response is simplified as a first-order delay function, as shown in Equation (5). The time delay (${\tau }_{eng}$, ${\tau }_{mot}$) should be decided by reflecting how quickly the EM is compared to the ICE. As a result, the input torque ($T_{m,cmd}$) passes through the low-efficiency area of the power map when the actual torque is determined. We have

$${\dot{T}}_{mot}=\frac{1}{{\tau }_{mot}}\left(T_{m,cmd}-T_{mot}\right).$$

(5)

2.3. Battery

The battery is a complex nonlinear system that is affected by temperature, voltage, and SOC. It is difficult to build a model that accurately represents the battery’s dynamic characteristics in an EMS because the formula is complicated and takes a long time to compute. Therefore, the internal resistance model, which is one of the control-oriented models, was applied to the EMS. This model calculates the battery SOC by defining the equivalent circuit expressed by internal resistance ($R_{int}$), open-circuit voltage ($V_{oc}$), and EM power ($P_{mot}$). The formula is shown in Equation (6). In this formula, $R_{int}$ and $V_{oc}$ are functions of SOC:

$$I_{mot}=\frac{V_{oc}-\sqrt{V_{oc}{}^2-4·R_{int}·P_{mot}}}{2·R_{int}}.$$

(6)

The SOC of the battery can be calculated integrating the current used by the EM, as shown in Equation (7). This formula is calculated by the initial SOC ($SO{C}_{init}$) and battery capacity ($Q_{bat}$):

$$SOC=SO{C}_{init}-\frac{1}{Q_{bat}}·\int I_{mot}\left(t\right)dt.$$

(7)

3. NLP-Based ECMS Algorithm

In this paper, the NLP-based ECMS algorithm, a combination of NLP and an A-ECMS, is proposed. A conceptual diagram of the NLP-based ECMS algorithm is shown in Figure 4. Based on the reference SOC trajectory obtained by NLP, the A-ECMS operates to minimize fuel consumption of the HEV. The process of performing the NLP-based ECMS algorithm is as follows. First, the required torque demanded by ICE and EM is calculated according to the driving condition. The gear ratio must be properly considered when calculating the required torque, because the gear ratio of the P2 powertrain is a discrete variable. Secondly, the reference SOC trajectory corresponding to the required torque through NLP is calculated. NLP can cause long calculation times and infeasible solutions if the system model and constraints are poorly designed. Thirdly, the A-ECMS determines the control input so that the measured SOC can follow the reference SOC trajectory.

3.1. Required Torque Preprocessing Algorithm

The required torque preprocessing algorithm determined the required torque used to calculate the reference SOC trajectory in NLP. The required torque preprocessing algorithm consisted of three steps. In first step, the speed was calculated along with the acceleration of the wheel by using the vehicle-speed and road-grade profiles. Inertial force, grade resistance force, air resistance force, rolling resistance torque, and final drive efficiency were reflected. In the second step, the gear ratio trajectory was determined by using the gear shift rule, which is a function of the speed and acceleration determined in the first step. A map for gear shift rules is provided in Figure 5. In third step, the required torque was determined by the gear ratio and transmission efficiency. The algorithm calculated the gear ratio and the torque which will be distributed to the EM and ICE.

3.2. Nonlinear Programming

The NLP-based ECMS algorithm calculated the reference SOC trajectory. NLP then calculated the solution for a nonlinear optimal control problem (NOCP) by using a solver. The NOCP was derived from the vehicle model and driving conditions. The IPOPT solver calculated the solution for the NOCP. The solution for the NOCP, the result of IPOPT, was the reference SOC trajectory. The more accurately and efficiently the NOCP was defined, the higher the quality of the solution provided by NLP derived from the solver [38,39,40,41,42].

The NOCP can be explained in three parts. The first part involves a nonlinear system dynamics model. The state vector, input vector, and known-disturbance vector of the HEV model is shown in Equation (8). The system dynamics equations of the components are represented by state space by using vectors, as shown in Equation (9). As part of the simplification work to accelerate the NLP computation, the speed of the EM was treated as a known disturbance. This is because the solution of the NLP, which was used indirectly in the form of the SOC, was determined to satisfy the required torque derived from the EM speed. We have

$$x=\left[\begin{array}{c}{T}_{eng}\\ T_{mot}\\ SOC\end{array}\right], u=\left[\begin{array}{c}{T}_{cmd, eng}\\ T_{cmd, mot}\end{array}\right], \mathrm{and} d=w_{mot}=w_{ref}$$

(8)

$$\dot{x}=\left[\begin{array}{c}{\dot{T}}_{eng}\\ {\dot{T}}_{mot}\\ \dot{SOC}\end{array}\right]=\left[\begin{array}{c}\frac{1}{{\tau }_{eng}}·\left(-T_{eng}+T_{cmd, eng}\right)\\ \frac{1}{{\tau }_{mot}}·\left(-T_{mot}+T_{cmd,mot}\right)\\ -\frac{1}{Q_{bat}}·\left(\frac{V_{oc}-\sqrt{V_{oc}{}^2-4·R_{int}·T_{mot}·w_{mot}}}{2·R_{int}}\right)\end{array}\right]=f\left(x,u,d\right).$$

(9)

The second part involves nonlinear constraints. The system model was used in NLP to predict how the HEV would behave within the time horizon. That is, the system model operated as a nonlinear constraint when the NLP calculated the solution. The NLP-based ECMS includes constraints of the time horizon by applying the system model to a fourth order Runge–Kutta method, which is a discretization method. The fourth-order Runge–Kutta method is shown in Equation (10). We have

$$\begin{gathered} x\left[k+1\right]=x\left[k\right]+\frac{T_{step}}{6}·\left(k_1+2·k_2+2·k_3+k_4\right) \\ {k}_1=f\left(x\left[k\right], u\left[k\right], d\left[k\right]\right) \\ {k}_2=f\left(x\left[k\right]+0.5·T_{step}·k_1, u\left[k\right], d\left[k\right]\right) \\ {k}_3=f\left(x\left[k\right]+0.5·T_{step}·k_2, u\left[k\right], d\left[k\right]\right) \\ {k}_4=f\left(x\left[k\right]+T_{step}·k_3, u\left[k\right], d\left[k\right]\right). \end{gathered}$$

(10)

The Runge–Kutta method, which requires a relatively large amount of computation, was applied rather than the Euler method, which has a small amount of computation, because the step time ($T_{step})$ can be relatively large. A large step time can predict longer intervals using the same number of receiving horizons. Equation (10) can be reformulated to equality constraints, as shown in Equation (11). The first nonlinear constraint was derived from a system dynamics model. We have

$$g_1\left(x\right)=x\left[k+1\right]-\left\{x\left[k\right]+\frac{T_{step}}{6}·\left(k_1+2·k_2+2·k_3+k_4\right)\right\}=0.$$

(11)

If the described Runge–Kutta method is used, the initial value of the state vector is important because it is accumulated from the initial value. That means the initial values of the state vector are also equality constraints. The initial values of engine torque, motor torque, and SOC were set as shown in Equation (12). Because the SOC must be situated within a certain range at the end of driving, the final value was also important for the SOC. The final value of the SOC was also set to an equality constraint as in Equation (13). We have

$$\left\{\begin{array}{c}{g}_2\left(x\right)=x_1\left(0\right)=T_{eng0}=0\\ g_3\left(x\right)=x_2\left(0\right)=T_{mot0}=0\\ g_4\left(x\right)=x_3\left(0\right)=SO{C}_0=0.6\end{array}\right.$$

(12)

$$g_5\left(x\right)=x_3\left(T_{fin}\right)=SO{C}_{fin}.$$

(13)

The maximum/minimum torque is determined by the speed ($w_{ref}$). Because the speed ($w_{ref}$) is a known disturbance, the maximum/minimum torque can be known in advance and is an inequality constraint, as shown in Equation (14):

$$\left\{\begin{array}{c}{g}_6\left(x\right)\to f_{eng,min}\left(w_{ref}\right)\le T_{eng}\le f_{eng,max}\left(w_{ref}\right)\\ g_7\left(x\right)\to f_{mot,min}\left(w_{ref}\right)\le T_{mot}\le f_{mot,max}\left(w_{ref}\right)\end{array}\right..$$

(14)

The third part involves a cost function. The algorithm includes a term for fuel consumption and a term for tracking the required torque by using the sum of ICE torque ($T_{eng})$ and EM torque ($T_{mot})$. The cost function of an NLP-based ECMS is shown in Equation (15). $Q_{fuel}$ is the weight cost of fuel consumption ($cos{t}_{fuel}$), and $Q_{trq}$ is the weight cost of the reference torque ($cos{t}_{trq}$). We have

$$cost=Q_{fuel}·cos{t}_{fuel}+Q_{trq}·cos{t}_{trq}.$$

(15)

The cost of fuel consumption can be calculated by using Equation (16). Surface-fitting for a fuel-consumption rate map was performed to include fuel consumption in the cost function of NLP. To derive a polynomial with an appropriate degree, various models were analyzed as shown in Figure 6. We have

$$cos{t}_{fuel}=sum\left(p_{20}·T_{eng}^2+p_{10}·T_{eng}+p_{11}·T_{eng}·w_{ref}+p_{01}·w_{ref}\right).$$

(16)

The cost for tracking the required torque using the sum of ICE torque ($T_{eng}$) and EM torque ($T_{mot}$) can be calculated by using Equation (17). The required torque is the reference torque given to the ICE and EM. The cost function was designed to use reference torque, not speed, to decrease the computational time by reducing the size of the system model. We have

$$cos{t}_{trq}=sum\left\{{\left(T_{eng}+T_{mot}-T_{ref}\right)}^2\right\}.$$

(17)

The constraint and cost functions were defined in the form of an OCP for receding horizons. The defined problem was solved by an NLP solver. The IPOPT solver, which is a rapid and powerful NLP solver, was used for the NLP-based ECMS algorithm.

3.3. Adaptive Equivalent Consumption Minimization Strategy

The NLP-based ECMS algorithm was controlled by an A-ECMS based on the reference SOC trajectory, which is the result of NLP. An A-ECMS computes the cost function for input grids and obtains the optimal power distribution ratio, which minimize the cost function. A schematic diagram of A-ECMS consisting of five steps is shown in Figure 7.

The first step determined the number of grids for the control input candidates ($u_{grid}$) for the power distribution ratio, as described by Equation (18). The A-ECMS in this paper used 201 grids (N = 201). We have

$$u_{grid}=u_{N\times 1}=\left[-1, 1\right].$$

(18)

The second step calculated the torque candidate of the ICE and EM. The required torque ($T_{req}$), calculated by a preprocessing algorithm, was distributed to the EM torque candidates ($T_{mot, grid}$) by the input grid. The engine torque candidates ($T_{eng, grid}$) were based on the difference between the required torque and EM torque candidates. The formulas are found in Equations (19) and (20). In this process, the maximum/minimum torque of the EM and the ICE was considered as shown in Equation (21):

$$T_{mot, grid}=T_{req}·u_{grid}$$

(19)

$$T_{eng, grid}=T_{req}-T_{mot,grid}$$

(20)

$$subj. to. \left\{\begin{array}{c}{T}_{mot,min}\le T_{mot,grid}\le T_{mot,max}\\ T_{eng,min}\le T_{eng,grid}\le T_{eng,max}\end{array}\right..$$

(21)

The third step calculated the power candidate of the ICE and EM. The ICE power candidates ($P_{eng, grid}$) were obtained from the BSFC (${\dot{m}}_{bsfc}$) and lower heating value ($Q_{LHV}$). The EM power candidates ($P_{mot, grid}$) were obtained from the internal resistance ($R_{int}$), open-circuit voltage ($V_{oc}$), and EM current ($I_{mot}$). Internal resistance and open-circuit voltage are functions of the battery’s SOC, and the EM current can be calculated by using Equation (22). SOC candidates were calculated from the EM power candidates because the EM power candidates affected the SOC. The formulas are as follows Equations (23)–(25):

$$I_{mot,grid}=\raisebox{1ex}{$\left(V_{oc}-\sqrt{V_{oc}{}^2-4·R_{int}·T_{mot,grid}·w_{mot}}\right)$}\!\left/ \!\raisebox{-1ex}{$2·R_{int}$}\right.$$

(22)

$$P_{mot, grid}=V_{oc}·I_{mot,grid}-R_{int}·I_{mot,grid}^2$$

(23)

$$P_{eng grid}=Q_{LHV}·{\dot{m}}_{bsfc}\left(T_{eng,grid}, w_{eng}\right)$$

(24)

$$SO{C}_{ grid}=(-1/Q_{bat})·I_{mot,grid}.$$

(25)

The fourth step calculates the equivalent factor ($S_{eq}$), which was used to equalize the power candidates of the ICE and EM. This process determines whether to use power from the EM or the ICE. An A-ECMS regulates the equivalent factor, so that the battery’s SOC follows the reference SOC trajectory, which can be calculated by NLP. If the battery’s SOC exceeded the reference trajectory, the equivalent factor was decreased. If the battery’s SOC was less than reference trajectory, the equivalent factor was increased. This is a form of a proportional–integral–derivative controller ($S_{pid}$), as shown in Equations (26) and (27). Equation (28) was used to prevent the equivalent factor from changing rapidly [33,38]. We have

$$SO{C}_{err}=SO{C}_{ref}-SO{C}_{grid}$$

(26)

$$S_{pid}=S_0+K_p·SO{C}_{err}+K_i·\int SO{C}_{err}+K_d·\frac{dSO{C}_{err}}{dt}$$

(27)

$$S_{eq}=S_{pid}-l_1·tan\left(-\frac{\pi }{2}·\frac{l_s·SO{C}_{err}}{\Delta SOC}\right).$$

(28)

The final step calculates the cost function and derives the optimal input, which is the power-split ratio between the ICE and EM that minimizes the cost function from the candidates. The cost function ($J_{ecms}$) of an A-ECMS reflecting the equivalent factor is described by Equation (29). The power candidates of the EM were equalized by using the equivalent factor as the power candidates of the ICE. We have

$$J_{ecms}=S_{eq}·P_{mot,grid}+P_{eng,grid}.$$

(29)

An A-ECMS is a quasistationary method that uses an input grid. Because the transient response of the ICE was not considered, the ICE torque changed rapidly or was turned on and off frequently. To prevent this problem, Equations (30) and (31) were used to reflect the cost function. Equation (30) is a cost function for engine torque change in previous ($T_{eng}$) and current steps ($T_{eng,grid}$). Equation (31) is the cost of starting an engine that was in EV mode in the previous step ($u_{cmd}$):

$$J_{trq}=\sqrt{{\left(T_{eng,grid}·w_{eng}-T_{eng}·w_{eng}\right)}^2}$$

(30)

$$J_{str}=\left(u_{grid}\ne 1\right)·\left(u_{cmd}=1\right).$$

(31)

In addition, the torque candidates of the ICE and EM are constrained by maximum/minimum torque conditions. If the engine torque and motor torque do not satisfy the required torque because of the constraints, the vehicle cannot follow the speed profile. To prevent this problem, Equation (32) was applied to reflect the cost function by using the difference between the power components (EM and ICE) and the required power:

$$J_{ref}=abs\left\{\left(T_{req}-T_{mot}-T_{eng}\right)·w_{gb}\right\}.$$

(32)

The cost function of A-ECMS ($J$) is shown in Equation (33), and the optimal solution of A-ECMS ($u^{*}$) can be obtained from Equation (34). Each term that constitutes the cost function must be tuned by using a weight ($W_1, W_2, W_3)$. When setting the weight, it is important to prevent contamination of the original cost function ($J_{ecms}$) with the ICE power candidate, EM power candidate, and equivalent factor. We have

$$J=J_{ecms}+W_1·J_{trq}+W_2·J_{str}+W_3·J_{ref}$$

(33)

$$u^{*}=\underset{u_{grid}}{minimize} J.$$

(34)

4. Results

In this paper, the performance of an NLP-based ECMS was validated in a simulation environment by using Autonomie, a forward-looking simulation tool developed by Argonne. An HEV model (P2 powertrain) and an NLP-based EMS were connected in a Simulink environment, as shown in Figure 8. The FTP-75 driving cycle was used to carry out the simulation and verify the performance. Simulations were run on a desktop computer equipped with an AMD Ryzen 5 5600X 6-Core Processor (3.70 GHz) and 16 GB of RAM.

Dynamic programming was used to compare the performance of the NLP-based ECMS. As mentioned above, DP was used to calculate a global optimal solution. That is, an optimal solution that can minimize fuel consumption for a given speed profile. If NLP-based ECMS ensures the HEV follows the optimal trajectory, the result of the NLP-based ECMS will get close to that of DP.

In this paper, the reductions in fuel consumption in the NLP-based ECMS were compared with the results of a rule-based EMS, DP, and DP-based ECMS. The computational times of each EMS were also compared.

Section 4.1 describes the DP model, which is a comparative group, and analyzes the results of the DP model. Section 4.2 analyzes the performance of NLP’s reference SOC trajectory calculation. Section 4.3 analyzes the performance of the NLP-based ECMS by using the HEV vehicle model of Autonomie.

4.1. Comparison Group (Global Optimal Solution/Dynamic Programming)

First, we analyzed the results of deriving the reference SOC trajectory of the NLP-based ECMS. Because an NLP-based ECMS derives the reference SOC trajectory by using NLP, the results of NLP and DP could be compared. The DP setting as a comparison target is shown in

Table 2. Settings for comparison groups (DP).

Case	Input Grid	State Grid	Step Time	Calculation Time
0 (reference)	201	601	0.1 [s]	2481.9 [s]
1	201	61	0.1 [s]	210.7 [s]
2	21	601	0.1 [s]	210.1 [s]
3	21	61	0.1 [s]	89.2 [s]
4	201	61	1.0 [s]	20.4 [s]
5	21	601	1.0 [s]	20.8 [s]
6	21	61	1.0 [s]	8.8 [s]

. The input grid involves the power-split ratio, which ranges from −1 to 1. The state grid is for the SOC, which ranges from 0.3 to 0.9. As the number of input grids and state grids increased, the calculation time increased. As the step time decreased, the calculation time increased significantly.

As shown in Figure 9 and Figure 10, the DP result changed significantly according to the setting for the input grid, the state grid, and the step time. Figure 9 displays the results of DP calculations for the SOC trajectory, which differs greatly depending on the case. Figure 10 depicts the result of DP calculations for ICE torque. If the grid and step time were set incorrectly, the calculated engine torque would include ripple.

Case 0 is the DP result with the largest grid and the smallest step time set. Case 0 had the longest calculation time because of the setting, but the solution was the closest to the global optimal. Case 0 was therefore used as the reference result, which is an ideal result that minimizes fuel consumption during the speed profile. Case 0 cannot be applied to the ECMS because it took too much time to calculate. Accordingly, Case 2, Case 5, and Case 6, which required relatively short computation times, were further compared. The results of Case 2, Case 5, and Case 6 were applied to the ECMS to implement a DP-based ECMS and compared it with the NLP-based ECMS proposed in this paper. Case 2, Case 5, and Case 6, which have relatively short calculation times, were also included in the comparison group. The results of Case 2, Case 5, and Case 6 were applied to the ECMS to design a DP-based ECMS and compared them with the NLP-based ECMS proposed in this paper.

4.2. Reference SOC Trajectory

The NLP-based ECMS calculated the reference SOC trajectory by using NLP. Accuracy and computational time were important factors. Accuracy indicates how close the results of NLP were to the global optimal solution.

The results of NLP are compared with DP models in the comparison group.

Table 3. Comparison of NLP results and DP results (1): calculation time and sum of error.

Case	Input Grid	State Grid	Step Time	Calculation Time	Sum of Error
0 (reference)	201	601	0.1 [s]	2481.9 [s]	0
2	21	601	0.1 [s]	210.1 [s]	45.187
5	21	601	1.0 [s]	20.8 [s]	508.71
6	21	61	1.0 [s]	8.8 [s]	717.21
NLP	-	-	-	2.6 [s]	156.77

compares the NLP and DP results. The calculation time is the time spent calculating the reference SOC trajectory. The sum of error is the difference between the global optimal solution and the reference SOC trajectory obtained by DP and NLP. As a result, NLP was the quickest and the most accurate to calculate the reference SOC trajectory.

Case 0, Case 2, Case 5, and Case 6 are results for the reference framework obtained by the DP model. Case 0 takes the longest time because it has the largest number of input/state grids and the smallest step size. Because the most accurate DP model is used, Case 0 is used as a reference because it is the closest result to a global optimal solution.

In order to be applied to DP based ECMS, the DP model of Case 2, Case 5, and Case 6 has been simplified. Fewer input/state girds and larger step sizes were used, which can reduce the computation time. Because the model is simplified, however, the results were moved away from the global optimal solution. The cost function of the DP is the fuel consumption of the HEV as shown in Equation (1). The cost function used in the DP of each case and the system model (the equations, parameters, maps, etc. shown in Section 2) are also identical.

The NLP case is a result obtained by NLP. The cost function and system models of NLP and DP are different. The cost function and system model of NLP are shown in Section 3.2. However, because the target vehicle is same, same vehicle parameters are used as shown in Section 2.

The tendency of the reference SOC trajectory is important because it determines the charging or discharging of the EM. As shown in Figure 11, the reference SOC trajectory obtained by NLP was similar to the global optimal solution. As shown in Figure 12, the states of ICE ($x1$) and EM torque ($x2$) were found by the NLP input to be first-order delay functions. Additionally, the states of ICE and EM torque were limited by constraints on the maximum/minimum ICE and EM torque.

4.3. NLP-Based ECMS

Figure 13 depicts the results of the simulation by applying the NLP-based ECMS to Autonomie’s HEV model. The NLP-based ECMS properly determined the ICE torque and EM torque so that the HEV follows the speed profile. In addition, the NLP-based ECMS appropriately determined the charging and discharging mode of the EM, so that the battery SOC follows the reference SOC trajectory calculated by NLP. An error occurred between the engine torque and motor torque and command because of the influence of clutch and transmission.

Table 4. Analysis of Autonomie simulation results: Fuel Improvement and Simulation Time.

Case	Fuel Consumption	Fuel Improvement	Final SOC	Simulation Time
DP	0.512 [kg]	–	–	–
Ruel-based	0.595 [kg]	0.0 [%]	0.61 [-]	73.79 [s]
0	0.527 [kg]	11.4 [%]	0.61 [-]	2553.18 [s]
2	0.527 [kg]	11.4 [%]	0.61 [-]	281.49 [s]
5	0.533 [kg]	10.4 [%]	0.61 [-]	90.39 [s]
6	0.536 [kg]	9.9 [%]	0.61 [-]	81.00 [s]
NLP	0.531 [kg]	10.8 [%]	0.61 [-]	74.18 [s]

reveals that the NLP-based ECMS reduced fuel consumption in a manner similar to the Case-0 DP-based ECMS with the shortest simulation time. The Case-0 DP-based ECMS is the closest to the global optimal solution because the A-ECMS used the most accurate DP result. An NLP-based ECMS is an EMS that optimally controls HEV.

5. Discussion

An NLP-based ECMS that can determine an appropriate equivalent factor for an A-ECMS was proposed. The NLP-based ECMS used a reference SOC trajectory to control the equivalent factor. The NOCP was designed to calculate the reference SOC trajectory. The NOCP includes ICE torque, EM torque, and battery SOC as state variables, and the system models for these variables were applied in a simplified form to increase calculation speed. The NOCP also includes a cost function, which was designed through surface-fitting to the fuel-consumption-rate map. The solution closest to the global optimal with a short computation time through NLP-based ECMS was preferred. In addition, the proposed method was able to respond to infeasible solutions that occurred in NLP by using a strategy involving the equivalent factor. The performance of the proposed method, which was verified in an Autonomie simulation environment, was superior to those of a DP-based ECMS and rule-based EMS in terms of calculation time and fuel consumption.

The proposed NLP-based ECMS completed its tasks rapidly in a desktop-computer simulation. However, the hybrid or electric control unit of an actual vehicle likely has far fewer computational resources. To apply an NLP-based ECMS to a real-world environment, a solver that can be applied to embedded system is therefore required. In this paper, the possibility that NLP-based ECMS can reduce fuel consumption close to the global optimal solution (DP result) is confirmed. However, empirical research on NLP-based ECMS is also important. For empirical verification, it is necessary to compare NLP-based ECMS with other online optimization approaches, such as PSO [21,22,23] or NN-based method [27,28,29,30,31]. We are conducting a follow-up study for empirical verification of NLP-based ECMS. In conclusion, follow-up studies are required to apply NLP-based ECMS to embedded systems. NLP-based ECMS includes NLP, which is a challenge when applied to embedded systems. To apply NLP-based ECMS to embedded systems, two representative follow-up studies are needed.

The first is to improve the computational speed and computational stability of NLP. To improve the computational performance of NLP, it is necessary to simplify OCP. Simplifying OCP means softening constraints on road information (speed profile, grade profile, etc.) or vehicle models (degree of surface fitting polynomial for engine/motor map). It is important to determine the appropriate prediction/control horizon through this process. Improving computational stability means considering the infeasible region of the NLP. NLP can derive infeasible solutions because of the exceptional driving situation, like a cut-in.

The second is to develop a NLP solver suitable for embedded systems. Various studies have been actively conducted on the research subject [43,44,45], such as SQP or interior-point method. NLP requires a very large amount of computation because it uses numerical and recursive methods. Therefore, it is necessary to optimize the amount of computation through solution space analysis. The solution space is determined by an operation characteristic or a driving situation of the HEV.

The proposed NLP-based ECMS does not consider clutch and transmission effects. Because the simulation results show that clutch and transmission effects degrade the performance of the proposed method, the influence of clutch and transmission must be considered when defining the OCP in NLP. A MINLP method may be used because the clutch and transmission involve mathematically discrete variables.

In this paper, previously supplied driving conditions were assumed. However, it is impossible to accurately anticipate actual driving conditions. An EMS must therefore reflect changing driving conditions, which depend on traffic conditions.