Multi-Objective Energy Management Strategy Based on PSO Optimization for Power-Split Hybrid Electric Vehicles

Abstract

The hybrid electric vehicle is equipped with an internal combustion engine and motor as the driving source, which can solve the problems of short driving range and slow charging of the electric vehicle. Making an effective energy management control strategy can reasonably distribute the output power of the engine and motor, improve engine efficiency, and reduce battery damage. To reduce vehicle energy consumption and excessive battery discharge at the same time, a multi-objective energy management strategy based on a particle swarm optimization algorithm is proposed. First, a simulation platform was built based on a compound power-split vehicle model. Then, the ECMS (Equivalent Consumption Minimization Strategy) was used to realize the real-time control of the model, and the penalty function was added to modify the objective function based on the current SOC (State of Charge) to maintain the SOC balance. Finally, the key parameters of ECMS were optimized by using a particle swarm optimization algorithm, and the effectiveness of the control strategy was verified under the WLTC (Worldwide Light-Duty Test Cycle) and the NEDC (New European Driving Cycle). The results show that under the WLTC test cycle, the overall fuel consumption of the whole vehicle was 6.88 L/100 km, which was 7.7% lower than that before optimization; under the NEDC test cycle, the fuel consumption of the whole vehicle was 5.88 L/100 km, which was 9.8% lower than that before optimization.

1. Introduction

In the automotive industry, research on energy saving and environmental protection has taken a key position. To improve the energy efficiency of vehicles, HEV (hybrid electric vehicles) have been developed [1]. HEV adopts a hybrid drive of an internal combustion engine and an electric motor. The introduction of an electric motor makes the engine work away from the low-efficiency range and improves the utilization of energy. The internal combustion engine can solve the problems of short driving range and slow charging of pure electric vehicles. HEV can not only ensure the driving range, but also ensure the working efficiency of the engine, achieving energy saving and environmental protection [2]. The energy management problem of HEV is to solve how to distribute the power output of two power sources when the HCU (hybrid control unit) receives the driver’s pedal command. The energy management of hybrid electric vehicles is to distribute the output power of the two power sources reasonably, according to the characteristics of the two power sources. On the premise of meeting the power demand, the engine should work in the high-efficiency range as much as possible to reduce the fuel consumption, and an excessive charge and discharge should be avoided to reduce the loss of battery and other components [3,4]. The EMS (energy management control strategy) of HEV generally includes a rule-based control strategy and an optimization-based control strategy [5,6].

The rule-based control strategy is first applied to the energy control of HEV because of its convenience and simplicity. The rule-based control strategy is mainly based on engineering experience, test calibration, or some off-line optimization, and a series of control rules are formulated by setting a logic threshold. Generally, the SOC (state of charge) of battery, engine speed, and torque as well as vehicle demand torque and speed are selected as the control variables of a threshold value. The working map of the engine and the motor is worked out by an off-line optimization algorithm to make the vehicle work in the high-efficiency range [7,8]. The rule-based control strategy includes the control strategy based on definite rules and the control strategy based on fuzzy rules. Poursamad A [9] took the vehicle demand torque, current engine speed, and current SOC of the battery as the input variables, used the fuzzy rule control algorithm to formulate the output torque of the engine, and optimized the parameters of the membership function offline. Banvait [10] proposed a RBS (Rule-Based Strategy) energy management strategy to increase the gasoline mileage of the PHEV (Plug-In Hybrid Electric Vehicle) by 16% compared with the Prius control strategy. The advantage of the rule-based strategy lies in its simplicity, real-time applicability, and robustness to driving cycle differences. However, their performance optimization (i.e., fuel economy) cannot be guaranteed by predefined rules and calibrated parameters, which is the main defect of the rule-based method [11,12].

The EMS of HEV based on the optimization algorithm generally consists of three parts: establishing the optimization problem, defining cost function, and finding an optimal solution based on the optimization algorithm. According to the time step of the optimization problem, the control strategy can be divided into the global optimization control strategy and the instantaneous optimization control strategy. DP (dynamic programming) and ECMS (Equivalent Consumption Minimization Strategy) are the main energy management control strategies for HEV. The DP algorithm belongs to the global optimization control strategy, and the ECMS algorithm belongs to the real-time optimization control strategy.

The DP algorithm sets the vehicle fuel economy as the control objective under the global condition [13]. By dividing the problem into several subproblems, the state variables, control variables, and optimization models are established to gradually obtain the global optimal solution. CC Lin [14] applied the DP algorithm to the EMS of HEV, taking engine fuel consumption and pollutant emissions as optimization objectives, and calculated the global optimal solution under a specific UDDSHDV (Urban Dynamometer Driving Schedule for Heavy Duty Vehicles) condition. Rui Wang [15] used the DP algorithm to obtain the optimal solution of three kinds of HEV structures: parallel, series, and hybrid. Domenico Bianchi [16] and Heeyun Lee [17] summarized the decision rules from the results of the DP algorithm and applied them to the rule control strategy. Bader. B [18] made the optimal solution of the DP algorithm into a multi-dimensional table under different working conditions and carried out online control of the real vehicle through online value checking. Based on the statistical method, Heeyun Lee [19] proposed a stochastic DP algorithm to realize online calculation.

ECMS is a real-time optimal energy management control strategy, which is often used in real-time control of HEV. In this algorithm, the battery energy consumption is equivalent to the engine fuel consumption by the equivalent factor, and then the energy distribution scheme with the minimum equivalent fuel consumption is found according to the current SOC value, driving conditions, and other conditions. According to the principle of minimum instantaneous equivalent fuel consumption, the output power of the engine and motor is distributed [20,21]. Kessels J [22] proposed a fuzzy adjustment ECMS aiming at the complex relationship between the fuel cell, SOC, and control variables of HEV. However, the cost factor of this method largely depended on expert experience. Kazemi H [23] used an off-line optimization algorithm to optimize the equivalent factor under different driving conditions and then used condition identification to realize the adaptive control of minimum equivalent fuel consumption under different driving conditions. Wang Y [24] gave the battery optimization model of energy management considering the health of the battery and used the PSO (particle swarm optimization) algorithm, but did not consider the minimum equivalent fuel consumption. Liu H [25] used a shooting algorithm to solve the initial value of the equivalent factors, and then used the PI (Proportional Integral) controller to track the reference SOC to obtain the real-time equivalent factor. Liu X [26] established the relationship model of SOC, vehicle acceleration, and equivalent factor to solve the problem where the battery cannot continue to discharge when the SOC is lower than the target value and it was difficult to adjust the engine operating point to the high-efficiency zone during the acceleration process.

Based on the current research, using the optimization algorithm to obtain the optimal equivalent factor is an effective direction of ECMS research. In PHEV, the SOC of the battery varies widely, and its initial SOC may be very different. Under different initial conditions and complex driving conditions, it is difficult to determine the optimal equivalent factor. To solve this problem, this paper proposed a real-time energy management strategy of power-split HEV. The PSO algorithm was used for multi-objective optimization while reducing vehicle energy consumption and maintaining battery balance.

The remainder of this paper is structured as follows. In Section 2, the mathematical model of the key parts and vehicle dynamics model are established, the co-simulation platform is built, and the real vehicle test data are collected to verify the model. In Section 3, the DP algorithm is used to solve the global optimal solution of vehicle fuel consumption under the NEDC and the WLTC cycles, respectively, to minimize vehicle fuel consumption under the battery power conservation mode. It provides a reference for the follow-up real-time control strategy. In Section 4, equivalent factors of the ECMS are optimized and a SOC balance strategy is added. The PSO algorithm is used to optimize the parameters of the ECMS so that the SOC can be balanced as much as possible and the fuel consumption can be reduced. Finally, our conclusions are presented in Section 5.

2. Model

2.1. Compound Power-Split System Model

The PHEV (plug-in hybrid electric vehicle) studied in this paper adopted the compound power-split system, which added two locking clutches and two clutches based on the improved Simpson double row planetary gear mechanism. The system could switch gears by opening and closing the locking clutch and clutch. The system structure diagram is shown in Figure 1. In Figure 1, ENG represents the engine, while E1 and E2 represent the small motor and large motor, respectively. B1 and B2 represent the brakes, C0 and C1 represent the clutches. The output shaft was connected to the main transmission. Brake B1 could lock the planet frame of the front planetary gear and the gear ring of the back planetary gear. Brake B2 could lock the output shaft of the small motor E1. The clutch C0 was installed between the engine and the planet frame of the front planetary gear. The clutch C1 was installed between the engine and the large motor E2.

The equivalent lever of the compound power-split system is shown in Figure 2. Multiple working modes can be obtained by the combination of different clutch states and brake states.

Table 1. The working modes of the compound power-split system.

Gear Number	Clutch C0	Clutch C1	BrakeB1	Brake B2	Transmission Ratio	Mode Type
G1	○	○	●	○	fixed	pure electric
G2	○	○	○	●	fixed	pure electric
G3	○	○	○	○	not fixed	pure electric
G4	○	●	●	○	fixed	series hybrid
G5	●	○	●	○	fixed	parallel hybrid
G6	○	●	○	●	fixed	series hybrid
G7	●	○	○	○	not fixed	compound split hybrid
G8	○	●	○	○	not fixed	output split hybrid
G9	●	●	○	○	fixed	series hybrid

showed the working modes of the PHEV equipped with a hybrid power distributary system under different clutch states and brake states. To facilitate the distinction, the different modes were recorded at different gears. The black dot ● in Table 1 means that the clutch/brake was closed and the white dot ○ means that the clutch/brake was separated.

2.2. Vehicle Physical Model

To build the physical model of the vehicle, the basic parameters of the vehicle module were required. The vehicle studied in this paper was a PHEV equipped with a new hybrid power distributary system above-mentioned. The key parameters of the vehicle are shown in

Table 2. The key parameters of the vehicle components.

Parts	Parameter Name	Unit	Value
vehicle	curb weight	kg	2200
	windward area	m2	2.7
	drag coefficient	-	0.37
	wheel rolling radius	m	0.353
	main reduction ratio	-	3.8
	tire moment of inertia	-	1.195
transmission case	front planetary gear ratio	-	−2.96
	rear planetary gear ratio	-	−2
engine	maximum power	kW	106
	maximum speed	r/min	5500
	maximum torque	N·m	255
	engine moment of inertia	kg·m2	0.15
motorE1	rated power	kW	40
	maximum speed	r/min	9000
	maximum torque	N·m	100
motorE2	rated power	kW	70
	maximum speed	r/min	10,500
	maximum torque	N·m	70
power battery	battery capacity	Ah	37
	cell voltage	V	3.6
	number of cells	-	96
	maximum discharge power	kW	90

.

2.2.1. Engine Model

The engine model in this paper was a static model, which was based on the fuel consumption rate of the engine at the corresponding torque and speed obtained by the bench test under the steady-state engine conditions. The calculation formula of engine fuel consumption $F_{\mathit{fuel}}$ is as follows:

$$F_{fuel}=\frac{T_{eng}\cdot n_{eng}}{9550\cdot 3600}\times b\left(n_{eng},T_{eng}\right)$$

(1)

where the unit of $F_{fuel}$ is in grams; $T_{eng}$ is the engine output torque; $n_{eng}$ is the engine output speed; $b\left(n_{eng},T_{eng}\right)$ is the specific fuel consumption. $b\left(n_{eng},T_{eng}\right)$ could be obtained by interpolation in the engine fuel consumption map in Figure 3.

2.2.2. Motor Model

The PHEV adopted two permanent magnet synchronous motors. The modeling of the motor was based on the motor efficiency map. The motor efficiency map was obtained by measuring the input voltage Um and current Im of the motor under steady-state operation. At the same time, the output torque Tm and speed nm of the motor were measured. The efficiency maps of the two motors are shown in Figure 4.

The calculation relationship between the electrical power P_e and the mechanical power P_m of the current motor working point is as follows:

$$Pm=\{\begin{array}{c}{P}_e\cdot {\eta }_m, n_{mot}\cdot T_{mot}>0\\ P_e/{\eta }_m, n_{mot}\cdot T_{mot}\ge 0\end{array}$$

(2)

where ${\eta }_m$ is the motor efficiency, which could be obtained from Figure 4a,b; $T_{mot}$ is the motor torque; and $n_{\mathit{mot}}$ is the motor speed. When $n_{mot}\cdot T_{mot}$ > 0, the motor is driving, generating energy. When $n_{mot}\cdot T_{mot}$ < 0, the motor is in a recovery state, recovering energy.

2.2.3. Battery Model

The battery model only involved the SOC, maximum charge, and discharge power parameters, the Rint model [27], which is easy to calculate, was selected, as shown in Figure 5. The maximum discharge power and charging power of the battery pack studied in this paper were 90 Kw and 70 Kw, respectively.

The SOC of the battery was calculated by the current integration method, and the formula is as follows:

$$\Delta \mathrm{SOC}=\frac{{{\displaystyle \int }}_{\mathrm{t}}^{}\frac{{\mathrm{U}}_{\mathrm{oc}}-\sqrt{{\mathrm{U}}_{\mathrm{oc}}^2-4{\mathrm{R}}_{\mathrm{int}}\cdot {\mathrm{P}}_{\mathrm{req}}}}{2{\mathrm{R}}_{\mathrm{int}}}\mathrm{dt}}{{\mathrm{C}}_{\mathrm{batt}}\cdot 3600}$$

(3)

where P_req is the electric power requirement. When P_req > 0, the battery is in a discharged state; when P_req < 0, the battery is in a charged state. t represents time using unit s. ${\mathrm{C}}_{\mathrm{batt}}$ represents the battery capacity using unit Ah.

2.3. Co-Simulation and Verification of AMESim and MATLAB Platform

AMESim was used to build the physical model of the hybrid system, and the control strategy built by MATLAB/Simulink was analyzed and verified.

The control signal of the HCU during the actual vehicle test process was collected, and then the control signal was imported into the Simulink model to control the physical model of the AMESim vehicle. Finally, the simulation results were compared with the actual results collected from the actual vehicle.

The initial value of SOC was set to 0.3. As shown in

Table 3. The comparison of the fuel and the power consumption between the simulation results and actual results.

Driving Circle	Hybrid Mode	Oil Consumption (L/km)	Power Consumption (kW·h)
WLTC	Actual test	6.82	0.1654
	Simulation	6.76	0.1661
	error	0.91%	0.42%
NEDC	Actual test	5.25	0.6319
	Simulation	5.2	0.6342
	error	0.88%	0.37%

, the simulation results were compared with the actual results of fuel consumption and power consumption in hybrid mode. The errors were all within 1%, which proved the rationality of the simulation model.

As shown in Figure 6a,b, the speed of the simulation vehicle model was compared with the real vehicle speed under the WTLC and the NEDC cycles. In Figure 6a,b, the purple line is the actual vehicle speed, and the yellow line is the simulation vehicle speed. It could be seen that the simulation vehicle speed was consistent with the real vehicle speed. Considering the test environment and other factors, the simulation results were within the error tolerance range, which indicated reasonableness of the co-simulation model.

3. ECMS Real-Time Energy Management Control Strategy

The ECMS (Equivalent Consumption Minimization Strategy) algorithm is an energy management algorithm proposed for the CS (Charge Sustaining Circle) mode of the PHEV. The energy consumed by the vehicle in the ECMS algorithm comes from the fuel energy, and the energy consumed by the battery needs to be replenished in the subsequent driving process. Therefore, the key to the ECMS algorithm is the conversion relationship between power consumption and fuel consumption, which is called the equivalent factor. The instantaneous energy consumption of the battery and the engine were unified by the equivalent factor as the optimization objective [28]. The optimization objective function is as follows:

$${\dot{m}}_{fuel}\left(t\right)={\dot{m}}_{eng}\left(t\right)+{\dot{m}}_{bat}\left(t\right)$$

(4)

$$m_{bat}\left(t\right)=P_{bat}\left(t\right)\cdot \xi$$

(5)

where $\xi$ is the equivalent factor with unit kg/kW·h; ${\dot{m}}_{fuel}\left(t\right)$ is the instantaneous total fuel consumption rate of the system with unit kg/h; ${\dot{m}}_{eng}\left(t\right)$ is the instantaneous fuel consumption rate of the engine with unit kg/h. ${\dot{m}}_{bat}\left(t\right)$ is the instantaneous equivalent fuel consumption rate of battery (kg/h). When the battery is charged, ${\dot{m}}_{bat}<0$, the total fuel consumption rate of the system is less than that of the engine; in contrast, when the battery is discharged, ${\dot{m}}_{bat}>0$, the total fuel consumption rate of the system is greater than that of the engine. It could be found that when the equivalence factor $\xi$ was larger, the battery tended to charge and the engine fuel consumption increased; when the equivalence factor $\xi$ was smaller, the battery tended to discharge and the engine fuel consumption decreased.

3.1. Equivalent Factors

The equivalent factors had a greater impact on the results of the ECMS algorithm. The empirical coefficient method and deduction calculation method were used to calculate the equivalent factor. The empirical coefficient method calculated the equivalent factor based on empirical judgments. This method relies on experience and is not easy to extend to other vehicles. The deduction calculation method takes the future working state of the power components as input to calculate the equivalent factor, which has a large amount of calculation and it is not easy to realize real-time online control. Therefore, based on the offline optimization results of the DP algorithm, the dynamic parameters of the vehicle’s dynamic components were optimized, then the equivalent factor was determined.

When the battery is discharged, the output energy of the battery ${\mathrm{E}}_{\mathrm{dis}}$ is

$${\mathrm{E}}_{\mathrm{dis}}={\mathrm{P}}_{\mathrm{bat},\mathrm{dis}}\cdot \mathrm{t}=\frac{{\mathrm{P}}_{\mathrm{mot}}}{{\mathsf{\eta }}_{\mathrm{mot}}}\cdot \mathrm{t}$$

(6)

where ${\mathrm{P}}_{\mathrm{bat},\mathrm{dis}}$ is the discharge power of the battery; ${\mathrm{P}}_{\mathrm{mot}}$ is the power of the motor; ${\mathsf{\eta }}_{\mathrm{mot}}$ is the efficiency of the motor; and t is the time.

The energy consumed by each battery discharge will be supplemented by the engine or braking energy recovery in the future, so the fuel consumption of ${\mathrm{E}}_{\mathrm{dis}}$ corresponding to future charging is

$${\mathrm{C}}_{\mathrm{E},\mathrm{chg}}=\frac{{\mathrm{C}}_{\mathrm{fuel},\mathrm{chg}}}{{\mathrm{E}}_{\mathrm{all}\_\mathrm{chg}}}\cdot {\mathrm{E}}_{\mathrm{dis}}$$

(7)

where ${\mathrm{C}}_{\mathrm{E},\mathrm{chg}}$ is the fuel consumption of future charging corresponding to ${\mathrm{E}}_{\mathrm{dis}}$; ${\mathrm{C}}_{\mathrm{fuel},\mathrm{chg}}$ is the fuel consumption generated by charging the battery for all engines; and ${\mathrm{E}}_{\mathrm{all}\_\mathrm{chg}}$ is the charging energy for all batteries including engine charging and braking energy recovery. Among them:

$${\mathrm{C}}_{\mathrm{fuel},\mathrm{chg}}=\frac{1}{{\mathrm{H}}_{\mathrm{fuel}}}\underset{\mathrm{chg}}{\overset{}{{\displaystyle \int }}}\frac{{\mathrm{P}}_{\mathrm{bat},\mathrm{c}\mathrm{h}\mathrm{g}}\left(\mathrm{t}\right)}{{\mathsf{\eta }}_{\mathrm{eng}}}\mathrm{dt}$$

(8)

$${\mathrm{E}}_{\mathrm{all}\_\mathrm{c}\mathrm{h}\mathrm{g}}=\underset{\mathrm{c}\mathrm{h}\mathrm{g}+\mathrm{regen}}{\overset{}{{\displaystyle \int }}}{\mathrm{P}}_{\mathrm{bat},\mathrm{c}\mathrm{h}\mathrm{g}}\left(\mathrm{t}\right)\mathrm{dt}=\underset{\mathrm{c}\mathrm{h}\mathrm{g}}{\overset{}{{\displaystyle \int }}}\frac{{\mathrm{P}}_{\mathrm{bat},\mathrm{c}\mathrm{h}\mathrm{g}}\left(\mathrm{t}\right)}{{\mathsf{\eta }}_{\mathrm{eng}}}\mathrm{dt}+\underset{\mathrm{regen}}{\overset{}{{\displaystyle \int }}}\frac{-{\mathrm{P}}_{\mathrm{req}}\left(\mathrm{t}\right)}{{\mathsf{\eta }}_{\mathrm{mot}}}\mathrm{dt}$$

(9)

where ${\mathrm{P}}_{\mathrm{req}}\left(\mathrm{t}\right)$ is the power required by the vehicle; ${\mathsf{\eta }}_{\mathrm{eng}}$ is the fuel efficiency of the engine; ${\mathrm{P}}_{\mathrm{bat},\mathrm{chg}}\left(\mathrm{t}\right)$ is the battery charging power; and ${\mathrm{H}}_{\mathrm{fuel}}$ is the low heating value of the fuel.

Assuming that the ratio of the braking energy recovery charge to the engine charge is ${\mathrm{R}}_{\mathrm{chg}\_\mathrm{regen}}$, then ${\mathrm{R}}_{\mathrm{chg}\_\mathrm{regen}}$ is:

$${\mathrm{R}}_{\mathrm{c}\mathrm{h}\mathrm{g}\_\mathrm{regen}}=\frac{{{\displaystyle \int }}_{\mathrm{regen}}^{}\frac{-{\mathrm{P}}_{\mathrm{req}}\left(\mathrm{t}\right)}{{\mathsf{\eta }}_{\mathrm{mot}}}\mathrm{dt}}{{{\displaystyle \int }}_{\mathrm{c}\mathrm{h}\mathrm{g}}^{}\frac{{\mathrm{P}}_{\mathrm{bat},\mathrm{c}\mathrm{h}\mathrm{g}}\left(\mathrm{t}\right)}{{\mathsf{\eta }}_{\mathrm{eng}}}\mathrm{dt}}$$

(10)

According to the above expression, it can be deduced that:

$${\mathrm{C}}_{\mathrm{E},\mathrm{dis}}=\frac{{\mathrm{P}}_{\mathrm{mot}}}{{\mathrm{H}}_{\mathrm{fuel}\cdot }{\mathsf{\eta }}_{\mathrm{mot}}}\cdot \frac{1}{1+{\mathrm{R}}_{\mathrm{c}\mathrm{h}\mathrm{g}\_\mathrm{regen}}}={\mathrm{P}}_{\mathrm{bat},\mathrm{dis}}\cdot \frac{1}{{\mathrm{H}}_{\mathrm{fuel}}\cdot \left(1+{\mathrm{R}}_{\mathrm{c}\mathrm{h}\mathrm{g}\_\mathrm{regen}}\right)}$$

(11)

When the battery is discharged, the equivalent factor ${\mathsf{\xi }}_{\mathrm{dis}}$ is:

$${\mathsf{\xi }}_{\mathrm{dis}}=\frac{1}{{\mathrm{H}}_{\mathrm{fuel}}\cdot \left(1+{\mathrm{R}}_{\mathrm{c}\mathrm{h}\mathrm{g}\_\mathrm{regen}}\right)}$$

(12)

When the battery is charged, its charged energy ${\mathrm{E}}_{\mathrm{chg}}$ is consumed by the motor in the future, so its equivalent fuel quantity is:

$${\mathrm{C}}_{\mathrm{E},\mathrm{chg}}={\mathrm{C}}_{\mathrm{fuel},\mathrm{chg}}\cdot \overline{{\mathsf{\eta }}_{\mathrm{mot},\mathrm{dis}}}=\frac{1}{{\mathrm{H}}_{\mathrm{fuel}}}\underset{\mathrm{chg}}{\overset{}{{\displaystyle \int }}}\frac{{\mathrm{P}}_{\mathrm{bat},\mathrm{chg}}\left(\mathrm{t}\right)}{{\mathsf{\eta }}_{\mathrm{eng},\mathrm{chg}}}\mathrm{dt}\cdot \overline{{\mathsf{\eta }}_{\mathrm{mot},\mathrm{dis}}}\overline{{\mathsf{\eta }}_{\mathrm{mot},\mathrm{dis}}}$$

(13)

where $\overline{{\mathsf{\eta }}_{\mathrm{mot},\mathrm{dis}}}$ is the average efficiency of the motor when driving; and ${\mathsf{\eta }}_{\mathrm{eng},\mathrm{chg}}$ is the efficiency of the engine when the battery was charging. When the battery is charging, the equivalent factor ${\mathsf{\xi }}_{\mathrm{chg}}$ is:

$${\mathsf{\xi }}_{\mathrm{chg}}=\frac{\overline{{\mathsf{\eta }}_{\mathrm{mot},\mathrm{dis}}}}{{\mathrm{H}}_{\mathrm{fuel}}\cdot {\mathsf{\eta }}_{\mathrm{eng},\mathrm{chg}}}$$

(14)

Based on the above derivation, the equivalent fuel consumption rate ${\dot{\mathrm{m}}}_{\mathrm{bat}}\left(\mathrm{t}\right)$ of the battery in the optimization objective function of ECMS is:

$${\dot{\mathrm{m}}}_{\mathrm{bat}}\left(\mathrm{t}\right)=\{\begin{array}{ll}{\mathsf{\xi }}_{\mathrm{dis}}\cdot {\mathrm{P}}_{\mathrm{bat}}& {\mathrm{P}}_{\mathrm{bat}}>0\\ {\mathsf{\xi }}_{\mathrm{chg}}\cdot {\mathrm{P}}_{\mathrm{bat}}& {\mathrm{P}}_{\mathrm{bat}}<0\end{array}$$

(15)

The equivalent factors under the WLTC and the NEDC cycles are shown in

Table 4. The equivalent factors under the WLTC and the NEDC cycles.

Driving Circle	Discharge Equivalent Factor ${\mathsf{\xi }}_{\mathbf{dis}}$ (g/(kW·h))	Charging Equivalent Factor ${\mathsf{\xi }}_{\mathbf{chg}}$ (g/(kW·h))
WLTC	16.18	211.72
NEDC	30.42	213.35

:

3.2. Improved ECMS Objective Function

When the ECMS objective function is used alone to control, the balance of SOC cannot be well guaranteed, and the objective function needs to be modified based on the current SOC size, so a penalty function was introduced to control the battery charge balance.

The S-shaped curve was used as the penalty function of SOC. If the SOC is x, the penalty function of SOC is expressed as P(x). The expression of P(x) is as follows:

$$\mathrm{P}\left(\mathrm{x}\right)=1+\mathrm{a}\Delta {\mathrm{x}}^3+\mathrm{b}\Delta {\mathrm{x}}^4$$

(16)

where a and b are the penalty function coefficients. Initially, select a = 1 and b = 0.25.

Then, the optimization objective function of the modified ECMS algorithm is:

$${\dot{\mathrm{m}}}_{\mathrm{fuel}}\left(\mathrm{t}\right)={\dot{\mathrm{m}}}_{\mathrm{eng}}\left(\mathrm{t}\right)+{\dot{\mathrm{m}}}_{\mathrm{bat}}\left(\mathrm{t}\right)\cdot \mathrm{P}\left(\mathrm{x}\right)$$

(17)

Meet the following physical constraints:

$$\mathrm{nE}{2}_{\min }\le \mathrm{nE}2\le \mathrm{nE}{2}_{\max }$$

(18)

$$\mathrm{nE}{1}_{\min }\le \mathrm{nE}1\le \mathrm{nE}{1}_{\max }$$

(19)

$$\mathrm{TE}{2}_{\min }\le \mathrm{TE}2\le \mathrm{TE}{2}_{\max }$$

(20)

$$\mathrm{TE}{1}_{\min }\le \mathrm{TE}1\le \mathrm{TE}{1}_{\max }$$

(21)

$${\mathrm{nEng}}_{\min }\le \mathrm{nEng}\le {\mathrm{nEng}}_{\max }$$

(22)

$${\mathrm{Teng}}_{\min }\le \mathrm{Teng}\le {\mathrm{Teng}}_{\max }$$

(23)

$${\mathrm{SOC}}_{\min }\le \mathrm{SOC}\le {\mathrm{SOC}}_{\max }$$

(24)

When using the penalty function P(x), the SOC needs to be normalized, and the processing formula is as follows:

$$\Delta \mathrm{x}=\{\begin{array}{cc}1& x>{\mathrm{x}}_{\mathrm{high}}\\ \frac{\left({\mathrm{x}}_{\mathrm{high}}+{\mathrm{x}}_{\mathrm{low}}\right)/2-\mathrm{x}}{\left({\mathrm{x}}_{\mathrm{high}}-{\mathrm{x}}_{\mathrm{low}}\right)/2}& {\mathrm{x}}_{\mathrm{low}}<\mathrm{x}<{\mathrm{x}}_{\mathrm{high}}\\ -1& {\mathrm{x}}_{\mathrm{low}}<x\end{array}$$

(25)

where ${\mathrm{x}}_{\mathrm{low}}$ is the lower limit of SOC; and ${\mathrm{x}}_{\mathrm{high}}$ is the upper limit of SOC.

The function graph of P(x) is shown in Figure 7. When SOC = initial SOC, P(x) = 1; when SOC > initial SOC, 0 < P(x) < 1, this means that the equivalent fuel consumption of the power consumption conversion was reduced; when SOC < initial SOC, P(x) > 1, this means that the equivalent fuel consumption of power consumption conversion increased to avoid excessive battery discharge.

3.3. Feasible Intervals of Control Variables

The ECMS algorithm is different from the DP algorithm, which needs to consider the additional torque generated by the inertia moment of the engine and the motor. Due to the constraint of the angular acceleration, the choice of control variables was different from the DP algorithm. The rule of selecting ECMS control variables is to solve the minimum required parameters of the transmission dynamics equations on the premise that the required torque THo, the required speed nHo, and the angular acceleration of the transmission output end are known. The calculation of the angular acceleration at the gearbox output terminal is as follows:

$${\mathsf{\alpha }}_{\mathrm{out}}=\frac{2\mathsf{\pi }\left({\mathrm{nHo}}_{\mathrm{k}}-{\mathrm{nHo}}_{\mathrm{k}-1}\right)}{60\mathrm{t}}$$

(26)

The dynamic equations group of the gearbox is shown in Formula (27).

$$\left[\begin{array}{cccccccc}0& 0& -1& 0& 0& -{\mathrm{J}}_{\mathrm{PC}1}& 0& 0\\ 0& 0& 0& -1& 0& 0& -{\mathrm{J}}_{\mathrm{S}1}& 0\\ 1& 0& 0& 0& -1& 0& 0& -{\mathrm{J}}_{\mathrm{S}2}\\ 0& 1& 0& 0& 0& 0& 0& 0\\ 0& 1& 1& 1& 1& 0& 0& 0\\ 0& 0& 1& 1-\mathrm{a}& \mathrm{b}& 0& 0& 0\\ 0& 0& 0& 0& 0& \mathrm{a}-1& 1& 0\\ 0& 0& 0& 0& 0& \mathrm{b}& 0& 1\end{array}\right]\left[\begin{array}{c}{\mathrm{T}}_{\mathrm{E}2}\\ {\mathrm{T}}_{\mathrm{R}2}\\ {\mathrm{T}}_{\mathrm{PC}1}\\ {\mathrm{T}}_{\mathrm{S}1}\\ {\mathrm{T}}_{\mathrm{S}2}\\ {\mathsf{\alpha }}_{\mathrm{PC}1}\\ {\mathsf{\alpha }}_{\mathrm{S}1}\\ {\mathsf{\alpha }}_{\mathrm{S}2}\end{array}\right]=\left[\begin{array}{c}\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 1& -{\mathrm{J}}_{\mathrm{OUT}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -\mathrm{a}\\ 0& 0& 0& \mathrm{b}-1\end{array}\end{array}\right]\left[\begin{array}{c}{\mathrm{T}}_{\mathrm{eng}}\\ {\mathrm{T}}_{\mathrm{E}1}\\ {\mathrm{T}}_{\mathrm{out}}\\ {\mathsf{\alpha }}_{\mathrm{out}}\end{array}\right]$$

(27)

The corresponding control variables and their feasible ranges in different gears of the ECMS algorithm are shown in

Table 5. The control variables and their feasible ranges.

Gear Number	Control Variables	Feasible Ranges	Units
G1	TE1	[−100:4:100];0	$\mathrm{N}\cdot \mathrm{m}$
G2	nE1	[−9000:100:9000]	$\mathrm{r}/\min$
G3	TE1	[−100:4:100];0	$\mathrm{N}\cdot \mathrm{m}$
G4	nEng TE1	[800:50:5500] [−100:4:100];0	$\mathrm{r}/\min$ $\mathrm{N}\cdot \mathrm{m}$
G5	Teng nE1	[5:5:255] [−9000:100:9000]	$\mathrm{N}\cdot \mathrm{m}$ $\mathrm{r}/\min$
G6	Teng nE1	[5:5:255] [−9000:100:9000]	$\mathrm{N}\cdot \mathrm{m}$ $\mathrm{r}/\min$
G7	Teng TE2	[5:5:255] [−255:5:255];0	$\mathrm{N}\cdot \mathrm{m}$ $\mathrm{N}\cdot \mathrm{m}$
G8	Teng TE1	[5:5:255] [−100:4:100];0	$\mathrm{N}\cdot \mathrm{m}$ $\mathrm{N}\cdot \mathrm{m}$
G9	Teng TE1	[5:5:255] [−100:4:100];0	$\mathrm{N}\cdot \mathrm{m}$ $\mathrm{N}\cdot \mathrm{m}$

.

3.4. Algorithm Process and Result Analysis

After selecting the equivalent factors and the modified objective function in the ECMS algorithm, the ECMS algorithm could be solved. The ECMS algorithm process is shown in Figure 8.

Set the initial SOC to 0.3, and carry out the simulation calculation of the ECMS algorithm. Figure 9a,b shows the SOC change curve and the fuel consumption and power consumption change curves under the WLTC cycle. The SOC was reduced by 2.4% at the end of the cycle. The engine fuel consumption of the whole cycle was 7.31 L/100 km. The comprehensive fuel consumption was 7.45 L/100 km by converting the power consumption of battery loss into the minimum fuel consumption rate of the engine.

Set the initial SOC to 0.3. The change curve of SOC and that of the fuel consumption and power consumption under the NEDC cycle are shown in Figure 9c,d, respectively. It can be seen from the figure that SOC decreased by 2.46% after the end of the whole cycle. The total fuel consumption of the whole cycle was 6.26 L/100 km. The comprehensive fuel consumption was 6.52 L/100 km.

Compared with the results of the DP algorithm, the fuel consumption was not reduced. This was caused by the improper selection of the penalty function parameters. Therefore, the parameters in the ECMS algorithm needed to be optimized.

4. Improved ECMS Algorithm for Multi-Objective Optimization

4.1. Particle Swarm Optimization Algorithm

The particle swarm optimization (PSO) algorithm is an intelligent random search algorithm proposed by Kennedy and Eberhart to simulate the migration and gathering behavior of birds in the foraging process [29]. The PSO algorithm has fewer parameters and is easy to implement, so it is widely used to solve multi-objective optimization problems.

In a D-dimensional search space, there is a particle swarm with N particles, and the position of the ith particle in the D-dimensional vector during the kth search is:

$${\mathrm{X}}_{\mathrm{i}}^{\mathrm{k}}=\left[{\mathrm{x}}_{\mathrm{i}1},{\mathrm{x}}_{\mathrm{i}2},{\mathrm{x}}_{\mathrm{i}3}\cdots {\mathrm{x}}_{\mathrm{iD}}\right],\mathrm{i}=1,2,3\dots \mathrm{N}$$

(28)

The “flying” speed of the ith particle in the kth search is:

$${\mathrm{V}}_{\mathrm{i}}^{\mathrm{k}}=\left[{\mathrm{v}}_{\mathrm{i}1},{\mathrm{v}}_{\mathrm{i}2},{\mathrm{v}}_{\mathrm{i}3}\cdots {\mathrm{v}}_{\mathrm{iD}}\right],\mathrm{i}=1,2,3\dots \mathrm{N}$$

(29)

The individual extreme value of the ith particle in the kth search is:

$${\mathrm{P}}_{\mathrm{i}}^{\mathrm{k}}=\left[{\mathrm{p}}_{\mathrm{i}1},{\mathrm{p}}_{\mathrm{i}2},{\mathrm{p}}_{\mathrm{i}3}\cdots {\mathrm{p}}_{\mathrm{iD}}\right],\mathrm{i}=1,2,3\dots \mathrm{N}$$

(30)

The global extremum value of the whole particle swarm in the kth search is:

$${\mathrm{g}}_{\mathrm{i}}^{\mathrm{k}}=\left[{\mathrm{p}}_{\mathrm{i}1},{\mathrm{p}}_{\mathrm{i}2},{\mathrm{p}}_{\mathrm{i}3}\cdots {\mathrm{p}}_{\mathrm{iD}}\right],\mathrm{i}=1,2,3\dots \mathrm{N}$$

(31)

The position update formula of the ith particle in the (k + 1)th search is:

$${\mathrm{X}}_{\mathrm{i}}^{\mathrm{k}+1}={\mathrm{X}}_{\mathrm{i}}^{\mathrm{k}}+{\mathrm{V}}_{\mathrm{i}}^{\mathrm{k}+1},\mathrm{i}=1,2,3\dots \mathrm{N}$$

(32)

The velocity update formula of the ith particle in the (k + 1)th search is:

$${\mathrm{V}}_{\mathrm{i}}^{\mathrm{k}+1}=\mathsf{\omega }\cdot {\mathrm{V}}_{\mathrm{i}}^{\mathrm{k}}+{\mathrm{c}}_1\cdot {\mathrm{r}}_1\cdot \left({\mathrm{p}}_{\mathrm{i}}^{\mathrm{k}}-{\mathrm{X}}_{\mathrm{i}}^{\mathrm{k}}\right)+{\mathrm{c}}_2\cdot {\mathrm{r}}_2\cdot \left({\mathrm{g}}^{\mathrm{k}}-{\mathrm{X}}_{\mathrm{i}}^{\mathrm{k}}\right),\mathrm{i}=1,2,3\dots \mathrm{N}$$

(33)

where $\mathsf{\omega }$ is the inertia weight; and ${\mathrm{c}}_1$ and ${\mathrm{c}}_2$ are the learning factors.

The learning factors determine how close the particle is to the individual extreme value and the global extreme value, and usually takes a value of 2. The inertia weight determined the inheritance of the current velocity of the particle from the previous velocity. A larger inertia weight is helpful to avoid local optima, and a smaller inertia weight is helpful in accelerating the algorithm convergence. Generally, the inertia weight that decreases with the number of searches is used to balance the search speed and global search capability of the algorithm [30]. Therefore, the following formula was used as the inertia weight:

$$\mathsf{\omega }={\mathsf{\omega }}_{\max }-\left({\mathsf{\omega }}_{\max }-{\mathsf{\omega }}_{\min }\right)\frac{\mathrm{k}}{{\mathrm{k}}_{\max }},\mathrm{i}=1,2,3\dots \mathrm{N}$$

(34)

where ${\mathsf{\omega }}_{\max }$ and ${\mathsf{\omega }}_{\min }$ are the maximum and minimum values of inertia weight; and ${\mathrm{k}}_{\max }$ is the total number of iterations.

4.2. Optimization Objective Function and Decision Variables

It can be seen from the previous section that the results of the ECMS algorithm were mainly affected by the battery charge equivalent factor ${\mathsf{\xi }}_{\mathrm{chg}}$, the battery discharge equivalent factor ${\mathsf{\xi }}_{\mathrm{dis}}$, and the parameters a and b of the penalty function P(x). Therefore, the above parameters were selected as the decision variables of the particle swarm algorithm, that is, the search space dimension D = 4. The value range of each decision variable is shown in

Table 6. The decision variables range of the particle swarm optimization (PSO) algorithm.

Variable	Lower Limit	Upper Limit
${\mathsf{\xi }}_{\mathrm{chg}}$ (g/kW·h)	0	400
${\mathsf{\xi }}_{\mathrm{dis}}$ (g/kW·h)	0	400
a	0	1
b	0	1

.

To enable the ECMS algorithm to balance battery SOC and achieve better fuel economy, an optimization objective function was established as (35):

$$\mathrm{f}\left(\mathrm{x}\right)=\{\begin{array}{ll}\mathrm{fuel},& \left|\Delta \mathrm{SOC}\right|\le \Delta {\mathrm{SOC}}_1\\ 50,& \left|\Delta \mathrm{SOC}\right|>\Delta {\mathrm{SOC}}_1\end{array}$$

(35)

where $\Delta {\mathrm{SOC}}_1$ is the SOC constraint limit. When $\Delta \mathrm{SOC}$ was in the specified interval, the normal ECMS total fuel consumption was used as the fitness value. When $\Delta \mathrm{SOC}$ exceeded the specified interval, a larger value was artificially set as the fitness value. In this way, $\Delta \mathrm{SOC}$ was constrained, and engine fuel consumption was optimized.

4.3. Optimization Results and Analysis

Set the number of particles to 20, the maximum number of iterations to 100, and the optimal accuracy to 0.0001 to optimize the ECMS algorithm parameters under the WLTC and the NEDC cycles. The optimized ECMS algorithm parameters under the WLTC cycle were calculated by the PSO algorithm as listed in

Table 7. The parameters after the ECMS optimization.

Driving Circle	${\mathsf{\xi }}_{\mathbf{chg}}$ (g/kW·h)	${\mathsf{\xi }}_{\mathbf{dis}}$ (g/kW·h)	a	b	Fuel Consumption (L/100 km)	ΔSOC
WLTC	197.39	215.53	0.34	0.27	6.87	−0.2%
NEDC	224.05	218.8	1	0.1	5.73	−1.09%

. It can be seen from Table 7 that the optimized fuel consumption under the WLTC cycle was 6.87 L/100 km and the SOC was reduced by 0.2%. Under the NEDC cycle, the optimized fuel consumption was 5.73 L/100 km, and the SOC was reduced by 1.09%.

The iterative curves of fitness values are shown in Figure 10. By incorporating the above-optimized parameters in the ECMS algorithm, the SOC change curve under the WLTC and the NEDC cycles are shown in Figure 11a,b, respectively.

Figure 12a,b shows the comparison of the engine operating points before and after the PSO algorithm under the WLTC cycle. It can be seen that before the PSO algorithm, the engine mainly worked in a higher speed range, but the overall fuel consumption rate was relatively high; after the PSO optimization, the engine was mostly in a high-efficiency range. However, due to the WLTC cycle, to ensure the conservation of battery power, some engine operating points had to work in the low-efficiency range.

The comparison of the engine operating points before and after the PSO algorithm under the NEDC cycle is shown in Figure 12c,d. It can be seen that before the optimization, the working hours of the engine were greater and the working efficiency was lower. After the optimization, the number of working points of the engine in areas with high speed, large load, and better efficiency increased compared with before the optimization.

As shown in

Table 8. The results of energy consumption in different control strategies.

Control Strategy	Engine Fuel Consumption (L/100 km)	Δ $\mathbf{SOC}$ (%)	Comprehensive Fuel Consumption (L/100 km)
Not optimized under WLTC	7.31	−2.4%	7.45
After optimized under WLTC	6.87	−0.2%	6.88
Not optimized under NEDC	6.26	−2.46%	6.52
After optimized under NEDC	5.73	−1.09%	6.88

. The results of energy consumption in different control strategies, the fuel consumption and battery SOC change in the ECMS algorithm before and after the optimization, respectively.

5. Conclusions

(1) Based on the basic parameters of the whole vehicle, the simulation models of engine, motor, battery, gearbox, and other components were established based on the AMESim platform. The simulation model was verified by using INCA software to collect real vehicle test data. This model was used to study the real-time control strategy of ECMS under the WLTC and NEDC cycles.

(2) The ECMS algorithm can meet the real-time control of real vehicles, but it is difficult to select the parameters of equivalent fuel consumption factors. The equivalent factors of the ECMS algorithm under the WLTC and NEDC cycles were derived and calculated, and the SOC balance strategy was proposed based on a penalty function. For different gear modes, the control variables and their feasible ranges of the ECMS algorithm were selected. The ECMS algorithm was used to simulate and analyze the fuel consumption under the WLTC and the NEDC cycles. The comprehensive fuel consumption under the WLTC cycle was 7.45 L/100 km. Under the NEDC cycle, the comprehensive fuel consumption was 6.52 L/100 km. Compared with the DP control strategy, the result of the ECMS algorithm was not ideal, so it was necessary to optimize the parameters of the ECMS algorithm offline.

(3) For the ECMS algorithm, the optimization variables were selected and the optimization objective function was formulated, so that the fuel consumption was as small as possible and the SOC balance was ensured. After the multi-objective PSO optimization of the ECMS algorithm, the comprehensive fuel consumption was 6.88 L/100 km, reduced by 7.7% under the WLTC cycle. Under the NEDC cycle, the comprehensive fuel consumption was 5.88 L/100 km, reduced by 9.8%. In future research and application, more intelligent control algorithms can be considered.

6. Patents