Modeling of Working Machines Synergy in the Process of the Hybrid Electric Vehicle Acceleration

Abstract

There are many different mathematical models that can be used to describe relations between energy machines in the power-split hybrid drive system. Usually, they are created based on simulations or measurements in bench (laboratory) conditions. In that sense, however, these are the idealized conditions. It is not known how the internal combustion engine and electrical machines work in real road conditions, especially during acceleration. This motivated the authors to set the goal of solving this research problem. The solution was to implement and develop the model predictive control (MPC) method for driving modes (electric, normal) of a hybrid electric vehicle equipped with a power-split drive system. According to the adopted mathematical model, after determining the type of model and its structure, the measurements were performed. There were carried out as road tests in two driving modes of the hybrid electric vehicle: electric and normal. The measurements focused on the internal combustion engine and electrical machines parameters (torque, rotational speed and power), state of charge of electrochemical accumulator system and equivalent fuel consumption (expressed as a cost function). The operating parameters of the internal combustion engine and electric machines during hybrid electric vehicle acceleration assume the maximum values in the entire range (corresponding to the set vehicle speeds). The process of the hybrid electric vehicle acceleration from 0 to 47 km/h in the electric mode lasted for 12 s and was transferred into the equivalent fuel consumption value of 5.03 g. The acceleration of the hybrid electric vehicle from 0 to 47 km/h in the normal mode lasted 4.5 s and was transferred to the value of 4.23 g. The hybrid electric vehicle acceleration from 0 to 90 km/h in the normal mode lasted 11 s and corresponded to the cost function value of 26.43 g. The presented results show how the fundamental importance of the hybrid electric vehicle acceleration process with a fully depressed gas pedal is (in these conditions the selected driving mode is a little importance).

1. Introduction

The effective synergy of the internal combustion engine and electrical machines, as well as the use of energy recovery during recuperative braking, allows the hybrid electric vehicle (HEV) to reduce its fuel consumption. Another undoubted advantage associated with the greater energy efficiency is the lower level of toxic emissions [1,2]. The improvement of economic and ecological indicators depends on an effective energy management system. To date, various models of power flow management strategies have been used in hybrid electric vehicles. Early energy management systems were based on the predicted operating conditions of the hybrid electric vehicle drive system (heuristic hypotheses). Control strategy of the internal combustion engine and electric engine operation assumed the use of electric mode from the state of the stationary car to the given vehicle speed. Above this speed, the torque of the electric motor decreased, and the shortage of torque, to overcome the resistance to movement, was gratified by the internal combustion engine [3,4]. The priority of the energy management system was to minimize the performance indicator described as the mass of fuel consumed in time [5,6,7].

Barsali et al. proposed statistical optimization as an energy management strategy based on minimizing the fuel consumption [5]. This approach did not require the detailed knowledge of the actual power demand, but only the knowledge of its average value. The accuracy of the method used resulted in its development in an energy management strategy for the electric power system ensuring reduction of fuel consumption and toxic emissions, based on dynamic programming (DP) [8,9,10,11]. The concept presented by Lin et al. [10] was based on stochastic-dynamic programming (SDP). The authors’ approach assumed a deterministic vehicle model, but also a driver’s stochastic power demand. The optimization was not based on a specific driving cycle but on the basis of the likelihood of demand for specific power in general driving conditions. Due to the considerable computational time requirements, the numerical optimization methods have been replaced by analytical optimization methods [3,12]. The method limiting the calculation expenditure assumed the introduction of the Hamiltonian function. The energy management strategy also used the theory of optimal control described by the Euler–Lagrange equation.

A different approach to the topic of energy management in hybrid vehicles could be found in the works of Paganelli et al. [13,14]. They present algorithms for the equivalent consumption minimization strategy (ECMS) for both hybrid drive systems and hybrid electric vehicles equipped with fuel cells. The strategy for controlling the value of equivalent fuel consumption was also used in the literature [15,16] and was presented as a cost function defined in the form of the sum of fuel and electricity consumption. This value, however, depended on a priori information on driving conditions or on the conditions of the adopted driving cycle. The modification of the ECMS optimization algorithm used for power-split hybrid systems was undertaken by Liu and Peng [17]. They used the SDP and ECMS strategy and the algorithms they used allowed them to improve the dynamic properties of the hybrid electric vehicle while reducing fuel consumption. They also built an automatic model of the power-split hybrid system in which the applied methodology proposed the automatic generation of dynamic systems equations. The algorithms show the cooperation of the internal combustion engine with electric machines, ensuring lower fuel consumption (as part of experiments according to a specific driving cycle) [18].

Another real-time algorithm used for decreasing the fuel consumption was the particle swarm optimization (PSO) [19,20,21,22,23,24]. This method has been applied in hybrid electric vehicles (HEVs) and plug-in HEVs [19,20]. The main goal of the algorithm was to optimize the control strategy in order to achieve the lowest fuel consumption. The development of the algorithm used by Hwang and Chen improved the fuel consumption by 9.4% in relation to the base control model [24].

Earlier Kim et al. [25] developed another energy management system in hybrid vehicles called the model predictive control (MPC). It was a strategy for optimal torque distribution for a parallel hybrid drive system. The cost function used in it was minimized on the basis of telemetric estimation of the vehicle speed. Borhan et al. [26] used the MPC in a power-split hybrid drive system equipped with an ultra-capacitor as the energy storage system. In parallel, Moura et al. [27] developed supervision to minimize the fuel consumption of a vehicle equipped with a series hybrid drive system by stochastic optimization of the control process using the Markov chain.

The MPC method was also used to determine the control inputs for economic driving (better fuel economy) of HEVs [28] and it depended largely on information from the management of Intelligent Transportation System (ITS). Yang and Zhu [29,30] developed the MPC system using linear quadratic tracking (LQT) consisting of controlling the power distribution of a power-split hybrid system to track the expected power demand set by a driver. The LQT controller minimized the cost function and at the same time maintained the battery level at the desired level.

An alternative development of the MPC strategy was the use of an energy management system known as nonlinear model predictive control (NMPC). The method used allowed an improvement in the amount of fuel consumption obtained in the NEDC cycle by 8.8% (compared to the factory control of the power-split hybrid system) [31].

The model of power-split hybrid systems was also presented by Cao, Peng and He. It was characterized by high accuracy; however, the created model was tested experimentally based on specific driving cycles, such as the new european driving cycle (NEDC) or the highway fuel economy driving schedule (HWYCOL) [32].

The above literature review brings the current state of knowledge related to the different mathematical models of the power-split hybrid drive system. They were created based on simulations or observations and measurements in bench (laboratory) conditions [17,18,29,30,31,32]. In the sense; however, these are idealized conditions. It is not known how the internal combustion engine and electrical machines work in real road conditions, especially during acceleration in various driving modes. It is not known whether the selection of the driving mode is of a significance importance during acceleration. It is also unknown in which driving mode the acceleration of the hybrid electric vehicle is more energy efficient. Those scientific considerations led the authors to set the goal of solving this research problem. The solution to the research problem was the implementation and development of the MPC method for driving modes (electric, normal) of a hybrid electric vehicle equipped with a power-split drive system. It was carried out in accordance with the research program, which assumed:

➢

specification of the components assembled in the tests (experimental part);

➢

creating a mathematical model of a power-split hybrid vehicle;

➢

presentation of the results;

➢

discussion of the results and conclusions.

2. Experimental Part

The experiments included the identification of the test object, a polygon stand and the test apparatus. The test object was a third generation Toyota Prius vehicle. The basic vehicle parameters are summarized in

Table 1. (a) Vehicle parameters; (b) Vehicle parameters.

(a)
Total mass	1630 kg (1445 kg + 185 kg—driver and passenger)
Dynamic wheel radius	0.29 m
Frontal area	1.62 m2
Rolling resistance coefficient	0.0084—tire energy class C
Air drag coefficient	0.25
Drive system type HEV	power-split hybrid
Internal Combustion Engine Specification
Ignition type	spark ignition
Capacity	1.8 dm3
Number of cylinders	4
Max. power	73 kW
Max. power rotational speed	5200 rpm
Max. torque	142 Nm
(b)
Max. torque rotational speed	4000 rpm
Mass moment of inertia	0.18 kg m2
Generator (MG1) Specification
Type	three-phase synchronous alternating current (AC)
Function	generator, internal combustion engine (ICE) starter
Rated voltage	500 V
Maximum output power	42 kW
Max torque	45 Nm
Current at max torque	75 A
Max. rotational speed	10,000 rpm
Mass moment of inertia	0.023 kg m2
Electric Motor (MG2) Specification
Type	three-phase synchronous AC
Function	generator, wheel drive
Rated voltage	500 V
Maximum output power	60 kW
Maximum torque	207 Nm
Current at max torque	230 A
Max. rotational speed	13,000 rpm
Mass moment of inertia	0.012 kg m2
High Voltage Battery and Inverter Specification
Battery type	NiMH
Nom. voltage	201.6 V
Capacity	6.5 Ah

a,b [19,33].

The polygon stand was an asphalt road located outside the city. The operation parameters of the internal combustion engine and electric machines have been tested using the test equipment (Gutmann Mega Macs diagnostic program and Dynomet chassis dynamometer). The specific results of the given parameters were obtained in Section 4.

3. Model of a Power-Split Hybrid Drive System

A model of a power-split electric-combustion drive system is shown in Figure 1. It consists of the following components: traffic conditions (desired speed), driver, driving modes, gas pedal, powertrain controller, internal combustion engine, generator, electric motor, high voltage battery, inverter, planetary gear, main gear, wheels and current speed. The driver, who selects the operating mode of the system (electric, normal) and presses on the gas pedal which transfers into the power of the vehicle (the power demand is stochastic), is the variable controlling the hybrid system. The drive system controller receives signals from a series of sensors, which include, among others, gas pedal position sensors and wheel speed sensor, and monitors the operation of the vehicle with the implemented energy management system.

3.1. Torque Transmission Dynamics and Vehicle Movement

The third generation Toyota Prius vehicle is equipped with two planetary gears. The kinematic diagram of the system for the normal driving mode is shown in Figure 2. The torque generated by the internal combustion engine is transmitted to the yoke of the satellite wheels and then to the ring gear of the planetary gear.

After the transmission through the planetary gear, the torque, using the intermediate gear and main gear, drives the car’s axle shafts. The MG1 generator, connected to the sun gear, acts as a starter and is used to charge the battery while driving. The MG2 electric machine, connected to the sun gear of the second planetary gear (transferring torque to the ring gear by the immobilized satellite yoke) was designed to assist the internal combustion engine in generating power, but also to provide energy recovery during braking [33].

In order to implement the mathematical model describing the dynamic scheme of torque transmission in the power-split hybrid drive system, the following assumptions were made:

➢

all shaft connections are rigid so there is no slippage between the drive components (no power loss);

➢

the moments of inertia of the internal combustion engine and electric machines are related to the moments of inertia of the yokes of satellites, sun wheels and crown gear;

➢

the moment of inertia of the main gear pinion is not taken into account;

➢

only longitudinal forces acting on the vehicle during travel are taken into account.

The drive system model (Figure 2) for the normal driving mode is described by the following equations [33]:

(a)

Angular velocities:

$${\omega }_{C1}=\frac{R_1}{S_1+R_1}{\omega }_R+\frac{S_1}{S_1+R_1}{\omega }_{S1}$$

(1)

$${\omega }_{C2}=\frac{R_2}{S_2+R_2}{\omega }_R+\frac{S_2}{S_2+R_2}{\omega }_{S2}$$

(2)

The yoke of the satellites of the second planetary gear is locked, therefore ${\omega }_{C2}=0$.

$$0=\frac{R_2}{S_2+R_2}{\omega }_R+\frac{S_2}{S_2+R_2}{\omega }_{S2}$$

(3)

$$\frac{R_2}{S_2+R_2}{\omega }_R=-\frac{S_2}{S_2+R_2}{\omega }_{S2}$$

(4)

$${\omega }_R=-\frac{S_2}{R_2}{\omega }_{S2}$$

(5)

$${\omega }_{S2}={\omega }_M,{\omega }_{C1}={\omega }_E,{\omega }_{S1}={\omega }_G$$

(6)

$${\omega }_R=-\frac{S_2}{R_2}{\omega }_M$$

(7)

$${\omega }_E=\frac{R_1}{S_1+R_1}{\omega }_R+\frac{S_1}{S_1+R_1}{\omega }_G$$

(8)

$${\omega }_E=-\frac{R_1}{S_1+R_1}\cdot \frac{S_2}{R_2}{\omega }_M+\frac{S_1}{S_1+R_1}{\omega }_G$$

(9)

$${\omega }_M=-\frac{R_2}{S_2}{\omega }_R$$

(10)

$${\omega }_G=\frac{{\omega }_E\left(R_1+S_1\right)}{S_1}+\frac{R_1}{S_1}\frac{S_2}{R_2}\cdot {\omega }_M$$

(11)

(b)

Mass moments of inertia and torques of the system:

$$J_E\cdot \stackrel{•}{{\omega }_E}=T_E-F_1\left(R_1+S_1\right)$$

(12)

$$J_G\cdot \stackrel{•}{{\omega }_G}=T_G+F_1{S}_1$$

(13)

$$J_M\cdot \stackrel{•}{{\omega }_M}=T_M-F_2{S}_2$$

(14)

$$J_R\cdot \stackrel{•}{{\omega }_R}=F_1{R}_1+F_2{R}_2-T_R=\frac{R_1}{R_1+S_1}{T}_E+\frac{R_2}{S_2}\cdot T_M-T_R$$

(15)

(c)

Vehicle inertia:

$$m\cdot \stackrel{•}{v_{veh}}=\frac{T_R{f}_d+T_{brk}}{r_d}-\frac{1}{2}{\rho }_A\cdot C_d{A}_f{v}_{veh}^2-mg\left(f_r\mathit{cos}\alpha +\mathit{sin}\alpha \right)$$

(16)

(d)

Vehicle speed:

$$v_{veh}=\frac{{\omega }_R\cdot r_d}{f_d}$$

(17)

Using the equations from (1) to (17) with the above-mentioned assumptions (J_R ≈ 0) the simplified version of the model is as follows:

$$\left[J_E+J_G{\left(\frac{R_1+S_1}{S_1}\right)}^2\right]\cdot {\stackrel{•}{\omega }}_E-\left[J_G\frac{R_1\cdot S_2\cdot \left(R_1+S_1\right)}{R_2\cdot S_1^2}\right]\cdot {\stackrel{•}{\omega }}_M=T_E+T_G\cdot \frac{\left(R_1+S_1\right)}{S_1}$$

(18)

$$\begin{array}{c}[J_E\cdot \frac{R_1}{R_1+S_1}]{\stackrel{•}{\omega }}_E+[J_M\frac{R_2}{S_2}-m\cdot {\left(\frac{r_d}{f_d}\right)}^2\cdot \frac{S_2}{R_2}]\cdot {\stackrel{•}{\omega }}_M\hfill \\ =T_E\cdot \frac{R_1}{R_1+S_1}+T_M\cdot \frac{R_2}{S_2}+\frac{T_{brk}}{f_d}+\frac{1}{2}\cdot {\rho }_A\cdot C_d\cdot A_f\cdot \frac{S_2{}^2\cdot {\omega }_M{}^2\cdot r_d^3}{R_2{}^2\cdot f_d^3}\hfill \\ -mg\cdot \frac{r_d}{f_d}(f_r\mathit{cos}\alpha +\mathit{sin}\alpha )\hfill \end{array}$$

(19)

The drive system model for the electric mode is described by the following relationships [33]:

(a)

Angular velocities:

The first two relations are the same as the (1) and (2) but the yoke of the satellites of the first and second planetary gears is locked, so:

$$0=\frac{R_1}{S_1+R_1}{\omega }_R+\frac{S_1}{S_1+R_1}{\omega }_{S1}$$

(20)

$${\omega }_{S1}=-\frac{R1}{S_1}\cdot {\omega }_R$$

(21)

$$0=\frac{R_2}{S_2+R_2}{\omega }_R+\frac{S_2}{S_2+R_2}{\omega }_{S2}$$

(22)

$$\frac{S_2}{S_2+R_2}{\omega }_{S2}=-\frac{R_2}{S_2+R_2}{\omega }_R$$

(23)

$${\omega }_R=-\frac{S_2}{R_2}{\omega }_{S2}$$

(24)

$${\omega }_{S1}={\omega }_G,{\omega }_{S2}={\omega }_M$$

(25)

$${\omega }_R=-\frac{S_2}{R_2}{\omega }_M$$

(26)

$${\omega }_E=0$$

(27)

$${\omega }_G=-\frac{R}{S_1}\cdot {\omega }_R=\frac{R}{S_1}\frac{S_2}{R_2}{\omega }_M$$

(28)

(b)

Mass moments of inertia and torques of the system (J_E = 0, T_E = 0):

$$J_G\cdot \stackrel{•}{{\omega }_G}=T_G+F_1{S}_1$$

(29)

$$J_M\cdot \stackrel{•}{{\omega }_M}=T_M-F_2{S}_2$$

(30)

$$J_R\cdot \stackrel{•}{{\omega }_R}=F_2{R}_2-T_R=\frac{R_2}{S_2}\cdot T_M-T_R$$

(31)

$$\begin{array}{c}\left[J_M\cdot \frac{R_2}{S_2}-m\cdot {\left(\frac{r_w}{f_d}\right)}^2\cdot \frac{S_2}{R_2}\right]{\stackrel{•}{\omega }}_M\hfill \\ =T_M\cdot \frac{R_2}{S_2}+\frac{T_{brk}}{f_d}+\frac{1}{2}\cdot {\rho }_A\cdot C_d\cdot A_f\cdot \frac{S_2{}^2\cdot {\omega }_M{}^2\cdot r_d^3}{R_2{}^2\cdot f_d^3}-mg\cdot \frac{r_d}{f_d}\left(f_r\mathit{cos}\alpha +\mathit{sin}\alpha \right)\hfill \end{array}$$

(32)

The inertia and speed of the vehicle have been determined according to the earlier formulas.

3.2. High Voltage Battery Level

The full hybrid drive system model assumed the efficient use of an electricity source (high voltage battery). The state of available electricity is a reflection of the battery charge level, which is determined by the relationship below [16,17,29,30]:

$$SOC=\frac{Q_{max}-Q_{used}}{Q_{max}}$$

(33)

The battery capacity used is described by the following relationship:

$$Q_{used}=\{\begin{array}{c}{{\displaystyle \int }}_0^t{I}_{bat}\cdot dt\to discharging,propulsion\\ {{\displaystyle \int }}_0^t{I}_{bat}{\eta }_{coulomb}dt\to charging,recuperation\end{array}$$

(34)

Note that I_bat > 0 (discharging), I_bat < 0 (charging). Apart from the charging/discharging current, the influence of uneven charging of individual cells/modules on its SOC should also be taken into account. Modern batteries use the battery charge equalizers (BCE) systems. Their task is to divide the current from the “overcharged” battery cell to the “undercharged” battery cell. This affects the extension of the battery life, but also improves the efficiency of the entire power-split hybrid system (due to the maintenance of a constant electric capacity of the battery). This issue is described in more detail in the literature [34,35,36,37]. Next, in correlation to the Equation (34), the derivative of the battery level can be represented by the formula:

$$\stackrel{•}{SOC}=-\frac{I_{bat}}{Q_{max}}$$

(35)

The power of a high-voltage battery:

$$P_{bat}=U_{bus}\cdot I_{bat}=\left(V_{oc}-I_{bat}\cdot r_{bat}\right)\cdot I_{bat}=V_{oc}{I}_{bat}-I_{bat}^2\cdot r_{bat}$$

(36)

Ultimately, the current for charging or discharging the battery takes the form:

$$\stackrel{•}{SOC}=-\frac{V_{OC}-\sqrt{V_{OC}^2-4r_{bat}{P}_{bat}}}{2r_{bat}{Q}_{max}}$$

(37)

The electric power of the P_bat battery is used during various driving modes (normal, electric), either during drive or recuperation, to cover the demand for power generated by the generator and/or the electric motor according to the following equation:

$$P_{bat}=T_G\cdot {\omega }_G+T_M\cdot {\omega }_M=T_{des}{\omega }_M-T_E\cdot {\omega }_E$$

(38)

T_des is the torque obtained by pressing the accelerator or brake pedal. It is worth noting that the battery power is positive when it is discharged and negative when it is being charged.

3.3. Energy Management Strategy

At this point the MPC type energy management was used, which allows optimization of the torque distribution value (between the internal combustion engine and electric machines), the substitute fuel consumption at the battery level is maintained as expected. The energy management system is characterized by the following limitations:

(a)

Internal combustion engine torque:

$$T_E^{min}\le T_E\le T_E^{max}$$

(39)

(b)

Angular velocity of internal combustion engine:

$${\omega }_E^{min}\le {\omega }_E\le {\omega }_E^{max}$$

(40)

(c)

Torque of the MG1 electric machine:

$$0\le T_G\le T_G^{max}$$

(41)

(d)

Angular speed of the MG1 electric machine:

$$0\le {\omega }_G\le {\omega }_G^{max}$$

(42)

(e)

Torque of the MG2 electric machine:

$$0\le T_M\le T_M^{max}$$

(43)

(f)

Angular speed of the MG2 electric machine:

$$0\le {\omega }_M\le {\omega }_M^{max}$$

(44)

(g)

High voltage battery power:

$$P_{bat}^{min}\le P_{bat}\le P_{bat}^{max}$$

(45)

(h)

Battery state of charge:

$$SO{C}^{min}\le SOC\le SO{C}^{max}$$

(46)

The model described in Section 3.1 and Section 3.2 is nonlinear so it has been linearized for specific conditions. The linear version of the model (for the normal and electric driving modes) was presented using the matrix [30]:

$$\left[\begin{array}{cc}{E}_{11}& -E_{12}\\ E_{21}& E_{22}-{\rho }_A\cdot C_d\cdot A_f\cdot \frac{S_2^2\cdot r_d^3}{R_2^2\cdot f_d^3}\end{array}\right]\cdot \left[\begin{array}{c}\stackrel{•}{{\omega }_E}\\ \stackrel{•}{{\omega }_M}\end{array}\right]=\left[\begin{array}{ccc}1& \frac{R_1}{R_1+S_1}& 0\\ \frac{R_1}{R_1+S_1}& 0& \frac{R_2}{S_2}\end{array}\right]\left[\begin{array}{c}{T}_E\\ T_G\\ T_M\end{array}\right]$$

(47)

where:

$$\begin{gathered} E_{11}=J_E+J_G{\left(\frac{R_1+S_1}{S_1}\right)}^2,E_{12}=\left[J_G\frac{R_1\cdot S_2\cdot \left(R_1+S_1\right)}{R_2\cdot S_1^2}\right],E_{21}=\left[J_E\cdot \frac{R_1}{R_1+S_1}\right],E_{22}=\left[J_M\frac{R_2}{S_2}-m\cdot {\left(\frac{r_d}{f_d}\right)}^2\cdot \frac{S_2}{R_2}\right] \\ {E}_D=E_{11}\left(E_{22}-\rho \cdot C_d\cdot A_f\cdot \frac{S_2^2\cdot r_d^3}{R_2^2\cdot f_d}\right)+E_{12}\cdot E_{21} \end{gathered}$$

For the electric driving mode T_E = 0, ω_E = 0.

The linearization of the battery charge level to the operating point was then presented in accordance with the following relations [30]:

$$\dot{SOC}={\Gamma }_{sG}{T}_G+{\Gamma }_{sM}{T}_M+{\Gamma }_{s{\omega }_E}{\omega }_E+{\Gamma }_{s{\omega }_M}{\omega }_M$$

(48)

$${\Gamma }_{sG}=\left[{\omega }_E\left(R_1+S_1\right)+\left(\frac{R_1\cdot S_2}{R_2}\right)\cdot {\omega }_M\right]/S_1/\Omega$$

(49)

$${\Gamma }_{sM}={\omega }_M/\Omega$$

(50)

$${\Gamma }_{s\omega E}=T_G\cdot \left(R_1+S_1\right)/S_1/\Omega$$

(51)

$${\Gamma }_{s\omega M}=T_M/\Omega$$

(52)

$$\Omega =-Q_{max}\sqrt{V_{OC}^2-4r_{bat}\left[T_G\cdot \frac{\left[{\omega }_E\left(R_1+S_1\right)+\left(R_1\cdot \frac{S_2}{R_2}\right)\cdot {\omega }_M\right]}{S_1}+T_M\cdot {\omega }_M\right]}$$

(53)

The executive dynamics of the system (internal combustion engine, generator, electric motor) was implemented in accordance with the following relationship [30]:

$$\stackrel{•}{T_E}=-\frac{T_E}{{\tau }_E}+\frac{T_{Edes}}{{\tau }_E},\stackrel{•}{T_G}=-\frac{T_G}{{\tau }_G}+\frac{T_{Gdes}}{{\tau }_G}\stackrel{•}{,T_M}=-\frac{T_M}{{\tau }_M}+\frac{T_{Mdes}}{{\tau }_M}$$

(54)

The relations between the required torque values (internal combustion engine, electric motor) and the system inputs (internal combustion engine fuel consumption and equivalent high voltage battery fuel consumption) have been described using the equations [30]:

$$b=\frac{\stackrel{•}{m}}{P_E}$$

(55)

$$\stackrel{•}{m}=b\cdot P_E=b\cdot T_{Edes}\cdot {\omega }_E=\frac{T_{Edes}\cdot {\omega }_E}{{\eta }_E\cdot W_u}$$

(56)

Calorific value of the fuel (petrol) equaled to $44\times {10}^6$ J/kg. The equivalent fuel consumption (calculated from the energy consumed from a high-voltage battery) was determined from the following formula:

$$\stackrel{•}{m_{eq}}=C_{bat}\cdot P_{bat}=C_{bat}\cdot V_{oc}\cdot I_{bat}$$

(57)

Finally, it took the form:

$$\stackrel{•}{m_{eq}}=\frac{C_{bat}}{2r_{bat}}\left[V_{oc}^2-V_{oc}\cdot \sqrt{V_{oc}^2-4r_{bat}{P}_{bat}}\right]$$

(58)

The value of the equivalent energy consumption coefficient C_bat for batteries has been calculated and it was equal to 0.0000227. It was determined on the basis of the equivalent fuel consumption (the ratio of the battery power to the calorific value of the fuel). In accordance with the literature [16] the value of the coefficient b1 was adopted and the values of the a1 coefficient have been calculated (from 0.0000227 to 0.0000237).

The main purpose of the presented energy management system was to minimize the fuel consumption of the internal combustion engine and the equivalent fuel consumption (corresponding to the energy consumption of the high-voltage battery). The parameter describing these relationships was the cost function presented according to the formula [17]:

$$J={\sum }_{s=0}^{N-1}(fue{l}_s+{\alpha }_E{\Delta }_{SOC}^2)\to min$$

(59)

$${\Delta }_{SOC}=\{\begin{array}{c}SO{C}_s-SO{C}_{d\hspace{1em}}SO{C}_s<SO{C}_d\\ 0SO{C}_k\ge SO{C}_d\end{array}$$

(60)

Using the previous relationships, the cost function can assume the following form:

$$J={{\displaystyle \int }}_0^{N\cdot t_d}\rho ({\left[\stackrel{•}{m}\left(T_{E\_des},\tau \right)\right]}^2+{\left[{\stackrel{•}{m}}_{eq}\left(T_{G\_des},T_{M\_des},\tau \right)\right]}^2)\}d\tau \to min$$

(61)

Taking into account the expected torque in the cost function, which should correspond to the value of the output torque of the system, the relation (61) takes the form of [17,30]:

$$J={\int }_0^{N\cdot t_d}{\left[T_{pre}\left(\tau \right)-T_R\right]}^2+\rho \left({\left[\stackrel{•}{m}\left(T_{E\_des},\tau \right)\right]}^2+{\left[{\stackrel{•}{m}}_{eq}\left(T_{G\_des},T_{M\_des},\tau \right)\right]}^2\right)\}d\tau \to min$$

(62)

Because the cost function is nonlinear, according to the Equation (60), the output control must be converted into categories m and m_eg:

$$T_{E\_des}=\frac{{\eta }_e\cdot W_u}{{\omega }_E}\cdot \stackrel{•}{m}$$

(63)

The Equations (56) and (58) take the following form:

$$T_{des}=\frac{{\stackrel{•}{m}}_{eq}}{C_{bat}\cdot {\omega }_M}-\frac{r_{bat}\cdot {\stackrel{•}{m}}_{eq}{}^2}{V_{oc}^2\cdot C_{bat}{}^2\cdot {\omega }_M}+\frac{T_E\cdot {\omega }_E}{{\omega }_M}$$

(64)

A linear model of the same operational point is shown below:

$$\stackrel{•}{T_E}=-\frac{1}{{\tau }_E}{T}_E+{\Gamma }_{e\omega e}\cdot {\omega }_E+{\Gamma }_{em}\cdot \stackrel{•}{m}$$

(65)

$$\stackrel{•}{T_G}={\Gamma }_{GTE}\cdot T_E-\frac{1}{{\tau }_G}{T}_G$$

(66)

$$\stackrel{•}{T_M}={\Gamma }_{MTE}\cdot T_E-\frac{1}{{\tau }_M}{T}_M+{\Gamma }_{M{\omega }_E}\cdot {\omega }_E+{\Gamma }_{M{\omega }_M}\cdot {\omega }_M+{\Gamma }_{Meq}\cdot \stackrel{•}{m_{eq}}$$

(67)

where:

$${\Gamma }_{e\omega e}=-\frac{1}{{\tau }_E}\cdot \frac{{\eta }_e\cdot W_u\cdot \stackrel{•}{m_0}}{{\omega }_{E0}^2},{\Gamma }_{em}=\frac{1}{{\tau }_E}\cdot \frac{{\eta }_e\cdot W_u}{{\omega }_{E0}}$$

(68)

$${\Gamma }_{GTE}=-\frac{1}{{\tau }_G}\cdot \frac{S_1}{R_1+S_1}$$

(69)

$${\Gamma }_{MTE}=-\frac{1}{{\tau }_M}\cdot \left(\frac{R_1}{S_1+R_1}\cdot \frac{S_2}{R_2}-\frac{{\omega }_{E0}}{{\omega }_{M0}}\right)$$

(70)

$${\Gamma }_{M\omega E}=\frac{1}{{\tau }_M}\cdot \frac{T_{E0}}{{\omega }_{M0}}$$

(71)

$${\Gamma }_{M{\omega }_M}=-\frac{1}{{\tau }_M\cdot {\omega }_{M0}^2}\cdot \left(\frac{\stackrel{•}{m_{eq0}}}{C_{bat}}-\frac{r_{bat}\cdot \stackrel{•}{m_{eq0}^2}}{V_{0C}^2\cdot C_{bat}^2}+T_{E0}\cdot {\omega }_{E0}\right)$$

(72)

$${\Gamma }_{M_{eq}}=-\frac{1}{{\tau }_M\cdot C_{bat}\cdot {\omega }_{M0}^{}}\cdot \left(1-2\frac{r_{bat}}{V_{0C}^2\cdot C_{bat}^{}}\right)$$

(73)

Then the linear model was a follows:

$$\{\begin{array}{c}\dot{x}=A_{c}x+B_c\mathit{u}\\ y=C_{c}x+D_c\mathit{u}\end{array},\mathit{u}=\left[\begin{array}{c}\stackrel{•}{m}\\ \stackrel{•}{m_{eq}}\end{array}\right]\mathit{y}=T_R$$

(74)

where:

$${\mathit{A}}_{\mathit{c}}=\left[\begin{array}{cccccc}-\frac{1}{{\tau }_E}& 0& 0& {\Gamma }_{e\omega e}& 0& 0\\ {\Gamma }_{GTE}& -\frac{1}{{\tau }_G}& 0& 0& 0& 0\\ {\Gamma }_{MTE}& 0& -\frac{1}{{\tau }_M}& {\Gamma }_{M\omega E}& {\Gamma }_{M\omega M}& 0\\ \frac{E_{22}}{E_D}-\rho \cdot C_d\cdot A_f\cdot \frac{S_2^2\cdot r_d^3}{R_2^2\cdot f_d^3\cdot E_D}+\frac{E_{12}}{E_D}\left(\frac{R_1}{R_1+S_1}\right)& \frac{E_{22}}{E_D}-\rho \cdot C_d\cdot A_f\cdot \frac{S_2^2\cdot r_d^3}{R_2^2\cdot f_d^3\cdot E_D}\left(\frac{R_1}{R_1+S_1}\right)& \frac{E_{12}}{E_D}\frac{R_2}{S_2}& 0& 0& 0\\ -\frac{E_{21}}{E_D}+\frac{E_{11}}{E_D}\left(\frac{R_1}{R_1+S_1}\right)& -\frac{E_{21}}{E_D}\left(\frac{R_1}{R_1+S_1}\right)& \frac{E_{11}\cdot R_2}{E_D\cdot S_2}& 0& 0& 0\\ 0& {\Gamma }_{sG}& {\Gamma }_{sM}& {\Gamma }_{s\omega e}& {\Gamma }_{s\omega M}& 0\end{array}\right]$$

(75)

$$\mathit{x}=\left[\begin{array}{c}{T}_E\\ T_G\\ T_M\\ {\omega }_E\\ {\omega }_M\\ SOC\end{array}\right]$$

(76)

$${\mathit{B}}_{\mathit{c}}=\left[\begin{array}{cc}{\Gamma }_{em}& 0\\ 0& 0\\ 0& {\Gamma }_{Meq}\\ 0& 0\\ 0& 0\\ 0& 0\end{array}\right]$$

(77)

$${\mathit{C}}_{\mathit{C}}^{\mathit{T}}=\left[\begin{array}{c}\frac{R_1}{R_1+S_1}\\ 0\\ \frac{R_2}{S_2}\\ 0\\ 0\\ 0\end{array}\right]$$

(78)

$${\mathit{C}}_{\mathit{C}\mathit{E}\mathit{V}}^{\mathit{T}}=\left[\begin{array}{c}0\\ 0\\ \frac{R_2}{S_2}\\ 0\\ 0\\ 0\end{array}\right]$$

(79)

$${\mathit{D}}_{\mathit{c}}=0$$

(80)

Finally, the linear model takes the form, u and y the same as (74):

$$\begin{array}{c}\dot{\mathit{x}}=A_{c}x+B_c\mathit{u}\\ \mathit{y}=C_{c}x\end{array}$$

(81)

A further way of solving the problem (using the LQT controller) by converting the analytical model represented by the above equations for the discrete model can be found in the literature [30].

4. Results

The tests scenario was conducted according to the architecture in Figure 3.

The key simulation parameters of the test scenario are summarized in

Table 2. Key simulation parameters.

Item	Parameter	Value
Vehicle	Mass	1630 kg
Engine	Start delay	0.5 s
	Time constant	1 s
	Max torque	142 Nm
	Power output	73 kW
Motor	Max torque	207 Nm
	Power output	60 kW
Battery	SOC upper bound	0.75
	SOC lower bound	0.45
	SOC target	0.60

.

Characteristics of operational parameters of working machines for selected driving modes (normal and electric) based on the tests scenario are presented in Figure 4, Figure 5, Figure 6 and Figure 7.

Based on the above characteristics it can be seen that the acceleration of the hybrid electric vehicle in the normal driving mode with the accelerator pedal maximum pressed is spent by a greater decrease in the battery charge level than in the electric driving mode. In the electric mode the vehicle accelerates much slower which is associated with lower energy consumption. While accelerating in the normal driving mode, almost all operational parameters of working machines (torques, powers and rotational speeds) are set within the range of maximum values (corresponding to the given vehicle speed). The situation is different for the electric mode, but this is due to the fact that the driver does not fully press the gas pedal (no maximum power demand). The above considerations also reflect the results of the determined cost function (based on the LQT algorithm) presented below (Figure 8).

5. Discussion

The process of accelerating the hybrid electric vehicle from 0 to 47 km/h in electric mode (partial stroke of the accelerator pedal) lasts for 12 s and transfers into the cost function value (equivalent fuel consumption value) of 5.03 g. Accelerating the hybrid electric vehicle from 0 to 47 km/h in normal mode lasts 4.5 s and transfers into the cost function value of 4.23 g. The acceleration of the hybrid electric vehicle from 0 to 90 km/h in normal mode (the accelerator pedal fully pressed) lasts 11 s and corresponds to the value of the cost function of 26.43 g.

Based on Figure 8, it can be concluded that the hybrid electric vehicle accelerates faster in normal mode than in electric mode. Accelerating from 0 to 47 km/h in normal mode takes 7.5 s less than in electric mode (and corresponds to the lower value of the cost function). This demonstrates the importance of the hybrid electric vehicle acceleration process with fully depressed gas pedal (it is the most energy-efficient). In these conditions the selected driving mode is a little importance.

6. Conclusions

The results calculated on the basis of the relationship from the first part of the article prove its high reliability (the accuracy in the vast majority of results reaches 4%). The operating parameters of the internal combustion engine and electric machines during the hybrid electric vehicle acceleration assume the maximum values in the entire range (corresponding to the given car speeds).

The torque values of the internal combustion engine, corresponding to the external characteristic torque values (full power characteristics), correspond to a high efficiency of the entire powertrain of forty percent (the MPC energy management strategy and the LQT controller automatically “sets” the internal combustion engine into the range of a maximum efficiency). This is transferred into low values of the calculated cost function, which actually means lower fuel consumption.

The tests were preliminary; however, in the longer term, the road measurements of the operating parameters of the internal combustion engine and electric machines can be made, as well as the fuel consumption in steady states (constant values of the vehicle speed and torque loading the wheels of the car) and the measurements of the above parameters according to specific driving cycles (NEDC, WLTP, FTP etc.) using bench tests (chassis dynamometer).