Torque Optimal Allocation Strategy of All-Wheel Drive Electric Vehicle Based on Difference of Efﬁciency Characteristics Between Axis Motors

energies

Article Torque Optimal Allocation Strategy of All-Wheel Drive Electric Vehicle Based on Difference of Efﬁciency Characteristics between Axis Motors

Xiaogang Wu 1,2 , Dianyu Zheng 1, Tianze Wang 1,2 and Jiuyu Du 2,*

1 School of Electrical and Electronic Engineering, Harbin University of Science and Technology, Harbin 150080, Heilongjiang, China; [email protected] (X.W.); [email protected] (D.Z.); [email protected] (T.W.) 2 State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing 100084, China * Correspondence: [email protected]

Received: 25 February 2019; Accepted: 19 March 2019; Published: 22 March 2019

Abstract: All-wheel drive is an important technical direction for the future development of pure electric vehicles. The difference in the efficiency distribution of the shaft motor caused by the optimal load matching and motor manufacturing process, the traditional torque average distribution strategy is not applicable to the torque distribution of the all-wheel drive power system. Aiming at the above problems, this paper takes the energy efficiency of power system as the optimization goal, proposes a dynamic allocation method to realize the torque distribution of electric vehicle all-wheel drive power system, and analyzes and verifies the adaptability of this optimization algorithm in different urban passenger vehicle working cycles. The simulation results show that, compared with the torque average distribution method, the proposed method can effectively solve the problem that the difference of the efficiency distribution of the two shaft motors in the power system affects the energy consumption of the power system. The energy consumption rate of the proposed method is reduced by 5.96% and 5.69%, respectively, compared with the average distribution method under the China urban passenger driving cycle and the Harbin urban passenger driving cycle.

Keywords: electric vehicle; all-wheel drive; dynamic planning; torque distribution; drive efﬁciency; drive efﬁciency

1. Introduction As a new type of powertrain structure, all-wheel-drive electric vehicles can be equipped with drive motors on the front and rear two-wheel drive shafts of electric vehicles, or by installing power wheels such as hubs or wheel-side motors at the wheels to drive the vehicles. As a unique power system, the structure can realize ﬂexible distribution of torque and improve the power, economy. and stability of electric vehicles [1,2]. In the research of powertrain control of all-wheel-drive electric vehicles, the literature [3] developed a kind of anti-interference and model for the four-wheel-drive hybrid vehicle equipped with dry dual-clutch transmission, taking full advantage of the dual clutch participation mode conversion process. Robust control algorithm for parametric mode-switching to coordinate torque between different power sources and clutches to minimize vehicle longitudinal jitter. The literature [4] proposed a hierarchical control system for electric vehicles driven by four-wheel independent motors, which implemented nonlinear model predictive control at the upper level to solve the problem of nonlinear multi-input and multi-output over-drive, and realized driving and regenerative braking control of independent motors at the lower level through PID algorithm. The literature [5,6] proposed a method

Energies 2019, 12, 1122; doi:10.3390/en12061122 www.mdpi.com/journal/energies Energies 2019, 12, 1122 2 of 16 for estimating the longitudinal force and side slip angle of four-wheel independent driving electric vehicles based on observer iteration and information fusion. The electric drive wheel model is introduced into the vehicle modeling process and used for longitudinal force estimation. The longitudinal force reconstruction equation is obtained by model decoupling. The Luenberger observer and the high-order sliding mode observer are combined to design the longitudinal force observer. The Kalman filter is used to suppress the effects of noise. By estimating the longitudinal force, the estimation strategy is proposed based on observer iteration and information fusion. Literature [7] discusses the influence of motor output power limit, road friction coefficient and wheel torque response on stability control, and the influence of motor idling loss on torque distribution method. Based on this, a vehicle dynamics control method based on vehicle steady state is proposed. Under normal driving cycles, the energy efficiency of the power system is improved by the torque distribution between the front and rear wheels. Literature [8] proposes a hierarchical control algorithm for the all-wheel drive system. The upper layer is composed of the supervision-expected motion tracking controller and the lower layer is composed of the optimized controller. In the upper controller, the desired vehicle motion is calculated by considering the transient cornering characteristics. In the desired motion tracking controller, in order to track the desired vehicle motion, the virtual control input is determined in a sliding mode-controlled manner. In the lower control allocation, the cost function is minimized as the optimization target, and the tire dissipating energy under driving cycles is effectively reduced. Literature [9] proposes a new electric powertrain structure with hub motor at the front wheel and clutch at the back of the transmission to provide maximum traction to control hub motor to cycle near optimum slip point. Based on the front wheel’s cycle mode, the desired wheel speed of the rear wheel is defined. The rear wheel is controlled by controlling clutch torque to track the defined speed. Based on this, a sliding mode controller using the nonlinear characteristics of the tire is proposed, which can realize its function without relying on feedback error. Literature [10] solves the optimization problem based on nonlinear four-wheel vehicle model to create reference sideslip Angle and yaw rate to minimize energy consumption due to wheel skid. A subsequent linear quadratic controller calculates the yaw moment around the vertical axis of the vehicle to follow the desired reference value. A torque distribution algorithm that considers the energy efficiency characteristics of the motor is proposed to provide the required propulsive force and yaw torque with minimal loss of power. In [11], a control strategy with switched drive mode is proposed. The control strategy is based on two vehicle speed estimation algorithms. The vehicle speed estimation algorithm based on the unscented Kalman filter is designed as a four-wheel drive mode condition, and is based on the wheel speed. The vehicle speed estimation algorithm is designed as two-wheel drive, and the vehicle speed smoothing algorithm is applied in the two vehicle speed estimation switching processes. Literature [12] proposed a layered control strategy for four-wheel drive hybrid vehicles. In the high-level controller, the signal phase and timing information and the optimal cruising speed are combined to produce the target speed of the HEV. A model predictive controller is proposed that focuses on tracking target speed and associated expected control variables for each individual vehicle for predicting the optimal speed. In the lower controller, a dynamic planning strategy is developed to optimize the global energy management using the predicted speed. Literature [13] developed a three-layer hierarchical control system. In the upper layer, the integrated two-degree-of-freedom (DOF) linear model is used to calculate the equivalent yaw moment for vehicle stability. Due to actuator limitations, the intermediate layer solves the linear quadratic regulator (LQR) problem by Weighted Least Squares (WLS) method to optimally distribute wheel torque. At the lowest level, the Slip Ratio Controller (SRC) is applied to redistribute the actual torque based on the sliding mode method. The literature [14] also proposes a two-layer controller for the four-wheel-drive electric vehicle. The upper controller consists of a speed tracking controller, a yaw moment controller and four-wheel slip controllers, which are used to calculate the expected value of the traction force. The expected value of the pendulum torque and the corresponding net torque input for the four wheels. Energies 2019, 12, 1122 3 of 16

In the lower controller, a torque distribution strategy based on the dynamic load of the tire and an optimal torque distribution strategy based on the minimum objective function are designed to control the motor drive torque or the regenerative brake torque. In summary, in the research of powertrain control of all-wheel-drive electric vehicles, the focus is on establishing the motor loss model and the wheel adhesion optimization model, designing the multi-layer controller structure, the upper layer decision target torque, and the lower layer through the control motor. In the target torque distribution decision process, most of them rely on the torque distribution method of the motor based on the dynamic load change of the tire. In the process of torque distribution decision, the local optimization algorithm is used to control the strategy. Although the local optimal solution can be obtained, the local optimization requires specific qualification conditions, and it is impossible to obtain the optimal energy efficiency of the electric vehicle electric drive system under the whole driving cycle, resulting in a lack of globality under known driving cycles. In the actual all-wheel drive system, the characteristic parameters of multiple motors often have certain differences due to manufacturing processes and the like, and it provides an intractable problem for torque distribution optimization. In this paper, a two-axis motor all-wheel drive system for electric vehicles is considered. Under the premise of the difference of efficiency distribution of front and rear axle motors, the power system efficiency is taken as the optimization goal, and a torque allocation strategy based on dynamic programming (DP) algorithm is proposed. The strategy realizes the optimal torque distribution of the two-axle motor all-wheel drive system under different passenger vehicle driving cycles.

2. All-Wheel Drive Power System Structure and Working Mode

2.1. Vehicle Dynamic Model The structure of the all-wheel-drive electric vehicle power system studied in this paper is shown in Figure1. The main structure includes two front and rear shaft motors, a motor controller, a power battery and a vehicle controller. The power battery provides energy for driving the shaft motor, and the shaft motor directly drives the vehicle. Each axis motor is equipped with a motor controller that independently controls its operating state. At the start, the output torque of the shaft motor is calculated by the vehicle controller and the information is transmitted to the power battery management system and the motor controller. The drive state of the all-wheel drive electric vehicle of the shaft motor is mainly the front wheel (rear wheel) independent drive, the four-wheel simultaneous drive and the two-wheel/four-wheel switch drive control mode. In the full-time two-wheel drive mode, the vehicle controller sends a control signal to drive the front or rear axle motor to run, and the power battery provides energy to drive the electric vehicle. In the full-time four-wheel drive mode, the vehicle controller sends a control signal to drive the front/rear axle motors to run together, and the power battery provides energy to drive the electric vehicle. The time-sharing two-wheel four-wheel switching mode is when the power demand of the electric vehicle is less than a certain threshold, the control signal is sent by the vehicle controller, and the two-wheel drive is used to provide driving energy to the front or rear wheel motor by the power battery. When the demand power of the electric vehicle is greater than a predetermined threshold, the four-wheel drive is used to provide energy to the two shaft motors by the power battery, which not only satisfies the power demand but also improves the working efficiency of the motor. Therefore, the main control method of the all-wheel-drive electric vehicle can be summarized as follows: In order to improve the working efficiency of the drive motor, according to the change of the required power, it is decided to adopt the two-wheel drive or the four-wheel drive mode switching control drive mode [15,16]. Energies 2018, 11, x FOR PEER REVIEW 4 of 17 wheel drive mode, where the rear wheel is in the follow-up state, and we define k as the torque distribution characteristic, k = 1. Figure 1b represents the rear wheel drive mode, where the front wheel is in the follow-up state, and we define k = 0. At this time, the vehicle controller calculates the required power and the required torque according to the vehicle speed. The motor controller controls the speed and torque of the shaft motor according to the motor demand speed and torque command sent by the vehicle controller, so that the output meets the demand. Power requirements. When the vehicle brakes, the shaft motor is in the power generating state, and the braking power can be Energies 2019, 12, 1122 4 of 16 transmitted to the power battery through the motor controller for recovery.

wheelVehicle wheel wheelVehicle wheel power controller controller power

Power Power Energies 2018Shaft ,motor 11, x FOR PEERBattery REVIEW Shaft motor Shaft motor Battery Shaft motor 4 of 17 Motor Motor Motor Motor Controller Controller Controller Controller Battery Battery wheel drive mode, wheremanagement the rear wheel is in the follow-up state, andmanagement we define k as the torque distribution wheelcharacteristic,system k = 1. Figure wheel 1b represents the rearwheel wheel drivesystem mode, where wheel the front wheel is inMechanical the connection follow-upElectrical connection state,Signal and connection we define k = 0. At this time,Mechanical connectionthe vehicleElectrical connection controllerSignal connection calculates the required(a) Front power wheel and drive the moderequired of operation torque according (k = 1) to the( bvehicle) Rear wheel speed. drive The modemotor of controller operation (controlsk = 0) the speed and torque of theFigure shaft 1. motor Front/rear according wheel wheel toindependent independent the motor drive drivedemand mode. mode. speed and torque command sent by the vehicle controller, so that the output meets the demand. Power requirements. When the 2.2. Working Mode vehicleFigure brakes, 2 shows the shaft the motorpower isflow in theof poweran all-whee generatingl-drive state,automotive and the powertrain braking power in all-wheel can be transmittedsimultaneousThe all-wheel-drive to drivethe power mode. battery electric In this through vehicletime, the the power torque motor system distributioncontroller can work for characteristic recovery. independently k = 0.5. on It the can front/rear be seen fromwheels Figure and 2 drivethat when all the the tworequired modes power simultaneously. is not satisfied Figureby one1 axis shows moto ther, the power required flow power of an is sharedall-wheel-drive by thewheel two vehicle shaftVehicle operatingmotors. At in wheel this front/rear time, th wheele controller independent calculateswheel drive theVehicle mode. required It can power wheel be seen and from the requiredFigure1 that torque the according all-wheel-drivepower controller to the vehicle electric speed vehicle and can allocates work in two two-wheel before and independentcontroller after.power The drivingoutput torque mode ofwhen each the of required the shaft power motors can and be satisfied the two under motor the controllers condition ofsend one the shaft required motor. Figure speed1 aand represents torque Power Power commandShaft motoraccording to theBattery vehicle controller,Shaft motor and control Shaftthe motor speed and torqueBattery of the twoShaft motorfront and the front wheel driveMotor mode, whereMotor the rear wheel is in the follow-upMotor state, andMotor we define k as the Controller Controller Controller Controller reartorque shaft distribution motors to make characteristic,Battery the outputk =meet 1. Figure the de1mandb represents power requirements. the rear wheelBattery When drive the mode, vehicle where brakes, the management management thefront shaft wheel motor iswheel in is the in follow-up thesystem power state, generating wheel and we state, define andk = 0.the At braking thiswheel time, power the vehicle systemcan be controller transmitted wheel calculates to the powerthe required battery power through and the the motor required controller torque for according recovery. to the vehicle speed. The motor controller Mechanical connection Electrical connection Signal connection Mechanical connection Electrical connection Signal connection controls the speed and torque of the shaft motor according to the motor demand speed and torque (a) Front wheel drive mode of operation (k = 1) (b) Rear wheel drive mode of operation (k = 0) command sent by the vehicle controller, so that the output meets the demand. Power requirements. When the vehicle brakes, theFigurewheel shaft 1. motorFront/rear is in wheel theVehicle power independent generating drive mode. state,wheel and the braking power can be transmitted to the power battery throughpower thecontroller motor controllerpower for recovery. FigureFigure 22 showsshows thethe powerpower flowflow ofof anan all-wheel-driveall-wheel-drive automotiveautomotive powertrainpowertrain inin all-wheelall-wheel simultaneoussimultaneous drive drive mode. mode. In In this this time, time, the the torque torque distribution distribution characteristic characteristic kk = 0.5. 0.5. It It can can be be seen seen from Figure 2 that when the required power is notPower satisfied by one axis motor, the required power is from Figure2 that whenShaft motor the required power is notBattery satisfied by one axis motor,Shaft motor the required power sharedis shared by bythe the two two shaft shaft motors. motors. At At thisMotor this time, time, th thee controller controllerMotor calculates calculates the the required required power power and and the the requiredrequired torque according toto thethe vehiclevehicleController speed speed and and allocates allocatesController two two before before and and after. after. The The output output torque torque of Battery ofeach each of theof the shaft shaft motors motors and theand two the motor two motor controllers controllers send the send required the required speed and speed torque and command torque management commandaccording according to the vehicle to the controller, vehiclewheel controller, and control and thesystem control speed the and speed torque and wheel of thetorque two of front the two and front rearshaft and rearmotors shaft to motors make the to make output the meet output the demandmeet the powerdemand requirements. power requirements. When the When vehicle the brakes, vehicle the brakes, shaft themotor shaft is inmotor the power is in the generatingMechanical power connection generating state, andElectrical the state, braking connection and the power brakingSignal can connection be power transmitted can be to transmitted the power batteryto the powerthrough battery the motor through controller the motor for recovery. controller for recovery. Figure 2. All-wheel simultaneous drive mode (k = 0.5).

3. Power System Modeling wheelVehicle wheel In order to study the torque distributionpower controllerstrategy of all-wheelpower drive power system, this paper carries out the power system modeling of all-wheel drive electric vehicle in Matlab/Simulink environment, and carries out simulation analysis of energy efficiency on this basis. Power Shaft motor Battery Shaft motor Motor Motor Controller Controller Battery management wheelsystem wheel

Mechanical connection Electrical connection Signal connection FigureFigure 2. All-wheelAll-wheel simultaneous simultaneous drive drive mode ( kk = 0.5). 0.5).

3. Power System Modeling In order to study the torque distribution strategy of all-wheel drive power system, this paper carries out the power system modeling of all-wheel drive electric vehicle in Matlab/Simulink environment, and carries out simulation analysis of energy efficiency on this basis.

Energies 2019, 12, 1122 5 of 16

3.1. Vehicle Dynamic Model When the electric vehicle is driving on the road, the traction motor needs to overcome the running resistance (Ft), including rolling resistance (Ff), air resistance (Fw), slope resistance (Fi) and acceleration resistance (Fj), which satisfy the following formula:

Ft = Ff + Fw + Fi + Fj (1)

Ff = f mg cos α (2)

Fi = mg sin α (3) 1 F = C Aρv2 (4) w 2 d

Fj = ma (5) where f is the rolling resistance coefficient, m is the mass of the electric vehicle, g is the acceleration of 2 gravity, taking 9.8 m/s , α is the slope of the road. This paper assumes α = 0, Cd is the air resistance coefficient, and A is the windward area of the electric vehicle. The ρ is the air density, and v is the vehicle speed. The traction motor in Equation (1) overcomes the running resistance (Ft), and the output power Pm of the electric motor is: Pm = Ft × v (6) According to the power balance characteristics of electric vehicles, the relationship is satisfied at time t. Pdem = Pm + Ploss,m (7)

Among them, Pdem is the power demanded by the power system, and Ploss,m is the power loss of the motor.

3.2. Motor Model In order to reduce the computational burden of the system, the model of the shaft motor established in this paper ignores the dynamic characteristics of the motor. Because of the loss of efficiency in the process of mutual conversion between electric power and mechanical power, a static map can be used as a model for driving the motor [17]. At the same time, the driving motor must meet the conditions as shown in Equations (8)–(10). Since this paper considers the torque distribution strategy when there is a difference in the efficiency of the shaft motor, the front and rear wheels use motors with the same output power and different efficiency curves.

Pm(t) = Tm(t)ωm(t) (8)

Tm(t) ∈ [Tm,min(ωm(t)), Tm,max(ωm(t))] (9)

ωm(t) ∈ [0, ωm,max(t)] (10)

Among them, Tm and ωm are the output torque and speed of the drive motor. Energies 2018, 11, x FOR PEER REVIEW 6 of 17

ωωmm,max()tt∈[] 0, () (10)

Among them, Tm and ωm are the output torque and speed of the drive motor.

Energies 2019, 12, 1122 6 of 16 3.3. Power Battery Model The simplified model of the battery equivalent circuit selected in this paper is shown in Figure 3.3.3. PowerThe equation Battery Modelof battery SOC (State of Charge) is as shown in Equations (11) and (12) [18,19]. The simpliﬁed model of the battery equivalent circuit selected in this paper is shown in Figure3. The equation of battery SOC (State of Charge) is˙ as shownIk() in Equations (11) and (12) [18,19]. SOC =− (11) Q . I(k)bat SOC = − (11) Qbat 2 ˙ U() SOC−−+q U () SOC4() R () SOC R P() k OCV2 OCV int t m SOC. =−η UOCV(SOC) − UOCV(SOC) − 4(Rint(SOC) + Rt)Pm(k) (12) SOC = −ηSOC ()+ (12) SOC 2()RSOCRQint t bat 2(Rint(SOC) + Rt)Qbat I Q U wherewhereis I is the the battery battery current, current, A; A;bat Qbatis is the the battery battery capacity, capacity, Ah; Ah;ηSOC ηSOCis is the the coulomb coulomb efﬁciency; efficiency;OCV UOCV R R isis the the open open circuit circuit voltage voltage of of the the battery, battery, V; V;int Rintand and tRaret are the the polarization polarization internal internal resistance resistance and and R U ohmicohmic internal internal resistance resistance of of the the battery, battery,Ω ;Ω;int Rintand and OCVUOCVare are the the function function of of the the battery battery SOC SOC as as a a P variable;variable; mPmis is the the required required power power of of the the drive drive motor, motor, kW. kW.

Rin t Rt ib

UOCV

Figure 3. Battery equivalent circuit simpliﬁed model. Figure 3. Battery equivalent circuit simplified model.

According to reference [20], the battery discharge efﬁciency ηdis in the model is calculated using EquationAccording (13). to reference [20], the battery discharge efficiency ηdis in the model is calculated using Equation (13). s ! U − IR 1 4R P = ocv dis = + − dis bat ( ) ≥ ηdis 1 1 2 , Pmotor k 0 (13) Uocv 2 Uocv UIR− 1 4 RP η ==+−≥ocv dis11 dis bat ,Pk() 0 (13) Among them, Rdis isdis the internal resistance of the battery2 discharge. motor UUocv2  ocv 3.4. Operating Conditions Among them, Rdis is the internal resistance of the battery discharge. Because this paper wants to verify the effectiveness of the proposed allocation strategy in different passenger3.4. Operating car driving Conditions cycles, it chooses the passenger vehicle urban driving cycles and the Harbin urban passenger driving cycle (HUPDC). The road conditions of the vehicle are used as the cycle conditionsBecause of the this simulation paper wants model. to Theverify selected the effectiv the Chinaeness urban of the passenger proposed driving allocation cycle (CUPDC)strategy in isdifferent shown in passenger Figure4. car The driving HUPDC cycles, is shown it chooses in Figure the passenger5[ 21]. Table vehicle1 compares urban driving the characteristic cycles and the valuesHarbin of urban the two passenger urban passenger driving cycle cars driving (HUPDC). cycle. The It canroad be conditions seen from of Table the 1vehicle that there are used are large as the differencescycle conditions in the main of the characteristic simulation values model. of theThe selected selected two the passenger China urban vehicle passenger driving cycles, driving which cycle can(CUPDC) basically is provide shown different in Figure types 4. ofThe passengers HUPDC foris shown the torque in distributionFigure 5 [21]. strategy Table of1 thecompares proposed the all-wheel-drivecharacteristic values electric of vehicle. the two urban passenger cars driving cycle. It can be seen from Table 1 that there are large differences in the main characteristic values of the selected two passenger vehicle driving cycles,Table which 1. Comparison can basically of main provide characteristic different values types under of twopassengers different drivingfor the cycles.torque distribution strategy of the proposed all-wheel-drive electric vehicle. Highest Maximum Average Average Speed Maximum Driving Cycle Acceleration Deceleration Acceleration (km/h) Speed (km/h) (m/s2) (m/s2) (m/s2) The CUPDC 23.138 74 2.294 −2.593 0.439 The HUPDC 16.98 46 1.944 −2.77 0.499 Energies 2018, 11, x FOR PEER REVIEW 7 of 17

EnergiesEnergies2019 2018, ,12 11,, 1122 x FOR PEER REVIEW 77 of 1617 80

7080

6070

5060

4050 v/km/h 3040 v/km/h 2030

1020

100 0 200 400 600 800 1000 1200 t/s 0 0 200 400 600 800 1000 1200 Figure 4. The China urbant/s passenger driving cycle. FigureFigure 4.4.The The ChinaChina urbanurban passengerpassenger drivingdriving cycle.cycle. 50

45 50

40 45

3540

30 35

2530 v/km/h 20 25 v/km/h 15 20

1015

5 10

0 50 200 400 600 800 1000 1200 1400 t/s 0 0 200 400 600 800 1000 1200 1400 Figure 5. The Harbin urban passenger driving cycle. Figure 5. The Harbin urbant/s passenger driving cycle. 4. Torque Optimization AllocationFigure 5. The Method Harbin Based urban on passenger Difference driving in Efficiencycycle. Characteristics between ShaftsTable 1. Comparison of main characteristic values under two different driving cycles. Table 1. ComparisonAverage of main characteristicMaximum valuHighestes under two differentMaximum driving cycles.Average 4.1. Torque Optimization Problem Driving Cycle Speed Speed Acceleration Deceleration Acceleration Average Maximum Highest Maximum Average In this paper, the optimal(km/h) operation of(km/h) the drive motor(m/s is2) improved(m/s by2) the optimal(m/s distribution2) DrivingThe CUPDC Cycle 23.138Speed Speed 74 Acceleration 2.294 Deceleration−2.593 Acceleration0.439 of the drive torque, so that the drive motor operates in the high2 efficiency2 zone at a certain2 speed. The HUPDC (km/h) 16.98 (km/h) 46 1.944(m/s ) −(m/s2.77 ) 0.499(m/s ) Therefore, theThe torqueCUPDC optimal distribution 23.138 control 74 can be 2.294expressed as a−2.593 problem of determining0.439 the torque distributionThe HUPDC coefficient k 16.98of the front and 46 rear wheels 1.944 (axes). The− coefficient2.77 k is0.499 defined as the 4. Torque Optimization Allocation Method Based on Difference in Efficiency Characteristics torque distribution characteristic between the front and rear wheels (axis) after the start of the control, between Shafts as4. shownTorque in Optimization Equation (14), Allocation and the boundary Method conditionBased on isDifference as shown in in Efficiency Equation (15). Characteristics between Shafts 4.1. Torque Optimization Problem T k = f (14) T + T 4.1. TorqueIn this Optimizationpaper, the optimal Problem operation of the drivef motorr is improved by the optimal distribution of the drive torque, so that the drive motor operates in the high efficiency zone at a certain speed. In this paper, the optimal operation of Tthef + driveTr = Tmotorreq is improved by the optimal distribution Therefore, the torque optimal distribution control can be expressed as a problem of determining the of the drive torque, so that the drive motor 0 operates≤ T ≤ T inreq the high efficiency zone at a certain speed. torque distribution coefficient k of the front and rearf wheels (axes). The coefficient k is defined as(15) the Therefore, the torque optimal distribution control0 ≤ Tr can≤ T reqbe expressed as a problem of determining the torque distribution characteristic between the front and rear wheels (axis) after the start of the control, torque distribution coefficient k of the front 0and≤ krear≤ 1 wheels (axes). The coefficient k is defined as the as shown in Equation (14), and the boundary condition is as shown in Equation (15). torque distribution characteristic between the front and rear wheels (axis) after the start of the control, as shown in Equation (14), and the boundary condition is as shown in Equation (15). Energies 2019, 12, 1122 8 of 16

where Tf is the torque value of the front axle motor, Tr is the torque value of the rear axle motor, and Treq is the demand torque of the vehicle. When k = 0, it is expressed as a separate rear wheel drive. When k = 1, it is expressed as a separate front wheel drive. When k = 0.5, it is expressed as a four-wheel torque average distribution mode. The driving energy utilization efﬁciency under the condition of the shaft motor driving can be expressed as the Equation (16).

k (1 − k) maxη = + (16) ηf(kTdem, n) ηr((1 − k)Tdem, n) where ηf is the efficiency of the front axle motor, ηr is the efficiency of the rear axle motor. And n is the corresponding of the axle motor speed. The boundary condition satisfies the Equation (17).  n < n  max  0 < T < T f dem (17)  0 < Tr < Tdem   Tdem < Tmax

Under the interference of other factors, the overall efﬁciency of the system can be obtained, and the energy consumption of the shaft motor drive system under driving cycles can be simpliﬁed as follows:

Z t E = (Pdem × η)dt (18) 0

4.2. Application of DP Algorithm

In the range of the actual domain [t0, tf], the state variables in the optimization problem of the all-wheel drive power system are the vehicle speed and the torque Tdem. Since the vehicle speed can be determined according to the operating driving cycles, the state variable is recorded as x(t) = [Tdem(t), n]’, and the vehicle demand power is used as the control variable, which is recorded as u(t) = [Pdem(t)], discrete state. The underlying all-wheel-drive electric vehicle powertrain can be described by the equation of state (19) [22]. dx = f (x(k), u(k)) (19) dt where f is the equation of the electric vehicle power system, consisting of Equation (1) to Equation (7). The constraints of the state space are shown in expression (20).

0 ≤ Pdem ≤ Pdem,max T ≤ T (t) ≤ T m,min dem m,max (20) SOCmin ≤ SOC(t) ≤ SOCmax nmotor,min ≤ n(t) ≤ nmotor,max where Tdem is the shaft motor torque; Tm,min and Tm,max represent the minimum and maximum torque of the shaft motor respectively; nmotor,min and nmotor,max represent the minimum and maximum speed of the shaft motor respectively. Pdem,max is the maximum output power of the shaft motor. In this paper, the energy efﬁciency of the electric drive system is taken as the objective function, and the objective function is shown in Equation (21).

N nT nT N nkT n(1 − k)T J = min [ f + r ] = min [ dem + dem ] (21) ∑ ( ) ( ) ∑ ( ) (( − ) ) i=0 ηf Tf, n ηr Tr, n i=0 ηf kTdem, n ηr 1 k Tdem, n where Tf is the output torque value of the front axle motor, Tr is the output torque value of the rear axle motor, ηf(Tf,n) is the efﬁciency of the front axle motor, ηr(Tr,n) is the efﬁciency of the rear axle motor. Energies 2018, 11, x FOR PEER REVIEW 9 of 17

≤≤ 0 PPdem dem,max

TTtT≤≤dem () m,min m,max ≤≤() (20) SOCmin SOC t SOC max

nntnmotor, min≤≤() motor, max

where Tdem is the shaft motor torque; Tm,min and Tm,max represent the minimum and maximum torque of the shaft motor respectively; nmotor,min and nmotor,max represent the minimum and maximum speed of the shaft motor respectively. Pdem,max is the maximum output power of the shaft motor. In this paper, the energy efficiency of the electric drive system is taken as the objective function, and the objective function is shown in Equation (21). NNnT nT nkT n(1− k ) T J =+=+min [fr ] min [dem dem ] ηη η η− (21) Energies 2019, 12, 1122 ii==00f(,)Tn f r (,) Tn r f ( kTn dem ,) r ((1) kT dem ,) n 9 of 16

where Tf is the output torque value of the front axle motor, Tr is the output torque value of the rear axle motor, ηf(Tf,n) is the efficiency of the front axle motor, ηr(Tr,n) is the efficiency of the rear axle Since the DP adopts the numerical solution, the time and system state are first discretized, and the motor. calculation gridSince of thethe DP torque adopts distribution the numerical ratiosolution, state the istime divided and system along state the are time first discretized, direction and of the driving cycle. Accordingthe calculation to the grid known of the driving torque distribution cycle, the ratio vehicle state modelis divided is usedalong the to calculatetime direction the of power the demand power anddriving speed cycle. of the According driving to the cycle known along driving the cycle, time the direction. vehicle model According is used to calculate to the the constraints power of the demand power and speed of the driving cycle along the time direction. According to the constraints motor, the systemof the motor, reachable the system boundary reachable ofboundary the whole of the driving whole driving cycle cycle is obtained is obtained from from thethe initial initial state and the terminationstate and state the termination of the system state of respectively. the system respec Withintively. Within the reachable the reachable boundary boundary range, range, the the system constraintssystem are met, constraints and the are forward met, and the function forwardis function calculated is calculated according according to theto the designed designed objective objective function. function. Obtain the shaft motor torque distribution ratio state matrix for the entire driving cycle. Obtain the shaft motor torque distribution ratio state matrix for the entire driving cycle. Finally, through Finally, through the recursive call method, the final state is reversed to the initial state, the traversal the recursiveoptimization call method, process theis completed, final state the isoptimal reversed torque todistribution the initial trajectory state, of the the traversalshaft motor optimizationis process is completed,obtained, and the the optimalcalculation torque result is distribution output. The calculation trajectory plan of of the the shaft DP method motor is is shown obtained, in and the calculationFigure result 6 is[23]. output. The calculation plan of the DP method is shown in Figure6[23].

State space

K=1 DP path

K=0.5 Torque distribution factor distribution Torque

K=0.1 K=0 i=1 i=2 i=3 i=N-2 i=N-1 N Time/s

Figure 6.FigureSchematic 6. Schematic diagram diagram of of dynamic dynamic planni planningng control control method method calculation. calculation.

The processThe of process using of theusing DP the method DP method to to realize realize th thee torque torque distribution distribution of the hub of themotor hub is shown motor is shown in Figure 7. in FigureEnergies7. 2018, 11, x FOR PEER REVIEW 10 of 17

Figure 7.FigureDynamic 7. Dynamic planning planning method method to to achieveachieve shaft shaft motor motor torque torque distribution distribution flow chart. ﬂow chart.

5. Simulation and Result Analysis In order to verify the effectiveness of the dynamic planning allocation strategy, this paper simulates the all-wheel-drive electric vehicle under different driving cycles in passenger car cities in China. The simulation prototype parameters are shown in Table 2.

Table 2. Basic parameters of the simulation sample car.

Frontal Wheel Centroid Front Rear Mass/kg Wheelbase/m Area/m2 Radius/m Height/m Track/m Track/m 1000 2.5 2.60 0.27 0.52 1.102 1.158

5.1. Comparative Analysis of Torque Distribution Strategies for Chinese Passenger Car under Different Cycles According to the calculation, the maximum demand torque of the simulation prototype car in the CUPDC is 676.11 N∙m, and the maximum demand power is 29.23 kW. This paper focuses on torque distribution in driving mode and therefore does not take account of the impact of braking conditions. The torque and power requirements during CUPDC are shown in Figure 8. Energies 2019, 12, 1122 10 of 16

Table 2. Basic parameters of the simulation sample car.

Frontal Wheel Centroid Front Rear Mass/kg Wheelbase/m Area/m2 Radius/m Height/m Track/m Track/m 1000 2.5 2.60 0.27 0.52 1.102 1.158

5.1. Comparative Analysis of Torque Distribution Strategies for Chinese Passenger Car under Different Cycles According to the calculation, the maximum demand torque of the simulation prototype car in the CUPDC is 676.11 N·m, and the maximum demand power is 29.23 kW. This paper focuses on torque distribution in driving mode and therefore does not take account of the impact of braking conditions.

The torqueEnergies and 2018 power, 11, x FOR requirements PEER REVIEW during CUPDC are shown in Figure8. 11 of 17

Figure 8.FigureThe demand8. The demand torque/power torque/power in in thethe China urban urban passenger passenger driving driving cycle (CUPDC). cycle (CUPDC).

AccordingAccording to the vehicleto the vehicle speed speed calculation calculation of of CUPDC, CUPDC, thethe required power power of the of thewhole whole vehicle vehicle can be obtained.can be Combined obtained. Combined with the with power the systempower system structure structure of the of the axle axle motor motor directly directly driving driving the the wheel, wheel, the torque of the shaft motor and the torque required by the power system can be further the torque of the shaft motor and the torque required by the power system can be further decoupled. decoupled. According to the boundary conditions and optimization objectives formulated by AccordingEquations to the boundary(20) and (21), conditions the ratio K andof the optimization output torque objectivesof the motor formulated of the two shafts byEquations before and (20) and (21), the ratioafter canK of be thecalculated output according torque to of the the DP motor algorith ofm, the and two then shafts the distributi beforeon and of output after cantorque be of calculated accordingthe to two the shafts DP can algorithm, be conducted. and Figure then the9 is the distribution result obtained of outputaccording torque to DP global of the optimization two shafts can be conducted.algorithm. Figure As9 is can the be result seen from obtained Figure according 9, torque distribution to DP global results optimization calculated based algorithm. on energy As can be efficiency as the optimization objective in this paper are mostly based on four-wheel simultaneous seen fromdriving, Figure with9, torque torque distributiondistribution coefficient results calculatedbetween 0.5 basedand 0.65. on Compared energy efficiencywith the average as the torque optimization objectivedistribution in this paper coefficient are mostly of 0.5, torque based distribution on four-wheel coefficient simultaneous can be optimized driving, according with to torque the power distribution coefficientdemand between of driving 0.5 and cycles. 0.65. Compared with the average torque distribution coefficient of 0.5, torque distribution coefficient can be optimized according to the power demand of driving cycles. This paper compares different torque distribution strategies of all-wheel-drive electric vehicles under the CUPDC. Figure 10 shows the torque distribution control strategy of the front and rear shafts motor and the operating point distribution of the shafts motor when different torque distribution strategies are adopted for the all-wheel-drive electric vehicle. Figure 11 shows the comparison results of shaft motor efficiency distribution under two different torque distribution strategies. By the comparison of Figures 10 and 11, we can see that the average torque distribution strategy, front axle and the electric motor shaft motor working in high efficient area number and location are consistent,

Figure 9. Torque distribution ratio map optimized by dynamic planning algorithm for the CUPDC.

This paper compares different torque distribution strategies of all-wheel-drive electric vehicles under the CUPDC. Figure 10 shows the torque distribution control strategy of the front and rear shafts motor and the operating point distribution of the shafts motor when different torque distribution strategies are adopted for the all-wheel-drive electric vehicle. Figure 11 shows the comparison results of shaft motor efficiency distribution under two different torque distribution strategies. By the comparison of Figure 10 and Figure 11, we can see that the average torque distribution strategy, front axle and the electric motor shaft motor working in high efficient area Energies 2018, 11, x FOR PEER REVIEW 11 of 17

Figure 8. The demand torque/power in the China urban passenger driving cycle (CUPDC).

According to the vehicle speed calculation of CUPDC, the required power of the whole vehicle can be obtained. Combined with the power system structure of the axle motor directly driving the Energieswheel,2019 the, 12 torque, 1122 of the shaft motor and the torque required by the power system can be 11further of 16 decoupled. According to the boundary conditions and optimization objectives formulated by Equations (20) and (21), the ratio K of the output torque of the motor of the two shafts before and soafter if by can changing be calculated the control according strategy, to the and DP then algorith realizem, the and change then the of thedistributi motoron working of output point, torque from of thethe point two ofshafts promoting can be conducted. the comprehensive Figure 9 efficiencyis the result of obtained power system, according thereby to DP lowering global optimization the energy consumptionalgorithm. As of can the electricbe seen drivefrom system.Figure 9, Compared torque distribution with the meanresults torque calculated distribution based on strategy, energy theefficiency DP algorithm as the adoptedoptimization is a global objective optimization in this paper method, are mostly so the based working on pointsfour-wheel of the simultaneous motor that improvedriving, the with global torque efficiency distribution interval coefficient are the main betw ones.een 0.5 In and the 0.65. interval Compared of 30% towith 90% the of average the efficiency torque distributiondistribution of coefficient the motor of working 0.5, torque point, distribution the proportion coefficient of the workingcan be optimized points of according the shaft motor to the of power the DPdemand global optimizationof driving cycles. torque distribution strategy is all increased.

Energies 2018, 11, x FOR PEER REVIEW 12 of 17 number and location are consistent, so if by changing the control strategy, and then realize the change of the motor working point, from the point of promoting the comprehensive efficiency of power system, thereby lowering the energy consumption of the electric drive system. Compared with the mean torque distribution strategy, the DP algorithm adopted is a global optimization method, so the working points of the motor that improve the global efficiency interval are the main ones. In the interval of 30% to 90% of the efficiency distribution of the motor working point, the proportion of the working points of the shaft motor of the DP global optimization torque distribution strategy is all increased.Figure 9. Torque distribution ratio map optimized by dynamic planning algorithm for the CUPDC. Figure 9. Torque distribution ratio map optimized by dynamic planning algorithm for the CUPDC. 400

0 0 0 0 0 0 5 Efficiency area distribution 3 4 2 5 6 7 7 This paper compares different torque distributionMaximum strategies external characteristic of curve all-wheel-drive electric vehicles under the CUPDC. Figure 10 shows the torque distDPribution distributed control strategy of the front and rear 0 300 8 Equally distributed shafts motor and the operating point distribution of the shafts motor when different torque

5 distribution strategies are adopted for the8 all-wheel-drive electric vehicle. Figure 11 shows the

0 5 0 0 0 0 0 7 7 3 4 5 2 6 7 9 comparison results of shaft200 motor efficiency8 distribution8 under two different torque distribution T/N·m strategies. By the comparison of 0 Figure 10 and Figure 11, we can see that the average torque 8 distribution strategy, front axle and the electric motor shaft motor working in high efficient area 5 100 8 89 89 87 87 87 5 0 85 0 0 7 0 0 0 7 85

5 6 3 4 2 85 80 80 80 75 7075 7750 7600 60 0 60 4050 4050 0 8 20304050 3020 3020 0 100 200 300 400 500 600 700 800 n/r/min (a) Distribution of working points of the front axle motor

440

0 0 0 0 0 0 3 2 4 5 5 6 Efficiency area distribution 7 7 400 0 8 Maximum external characteristic curve DP distributed 360 Equally distributed

320 5 8 280

5 0 0 7 0 0 0 0 7 0 6 7 3 4 5 8 240 2 8

T/N·m 200 89 91 5 8 2 400 9

300 7 8 92 5 0 91 7 8 200 0 91 7 0 8 0 0 0 0 9 6 5 89 4 3 2 89 87 87 100 5 87 85 85 8 85 80 80 75 7850 7075 70 50670 6050 6050 0 203040 302040 402030 0 100 200 300 400 500 600 700 800 n/r/min (b) Distribution of operating points of the rear axle motor

Figure 10.10. DistributionDistribution of of motor motor operating operating points points in different in different torque torque distribution distribution strategies strategies for CUPDC. for CUPDC. Energies 2019, 12, 1122 12 of 16

Energies Energies2018, 11 2018, x FOR, 11, x PEER FOR PEER REVIEW REVIEW 13 of 1713 of 17

40 40 35.23 36.15 35.57 Equally 35.23 36.1535.57 35.57 35 EquallyDP 35.57 35 DP 30 25.25 30 24.67 2525.25 24.67 25 20 20 15 Percentage(&) 15 10 Percentage(&) 10 5 1.43 1.60 2.52 2.01 0 5 2.52 0-301.43 30-60 1.60 60-80 80-90 90-100 2.01 0 Efficiency distribution % 0-30 30-60 60-80 80-90 90-100

Figure 11. ComparisonFigure 11. Comparison of motor of motor efﬁciency efficiencyEfficiency distribution distribution of different of different% torque distribution torque strategies distribution for strategies CUPDC. for CUPDC. Figure 11. Comparison of motor efficiency distribution of different torque distribution strategies for 5.2. Comparative Analysis of Torque Distribution Strategies for HUPDC 5.2. ComparativeCUPDC. Analysis of Torque Distribution Strategies for HUPDC After calculation, the maximum demand torque of the simulation prototype car in the HUPDC 5.2. Comparativeis 763.27 N Analysis∙m, and the of maximumTorque Distribution demand power Strategies is 16.72 for kW. HUPDC The torque and power requirements of After calculation,the simulation the prototype maximum running demand under HUPDC torque are shown of the insimulation Figure 12. prototype car in the HUPDC is 763.27 N·m, andAfter the calculation, maximum the maximum demand demand power istorque 16.72 of kW.the simulation The torque prototype and power car in the requirements HUPDC of the simulationis 763.27 prototype N∙m, and running the maximum under demand HUPDC power are is shown 16.72 kW. in The Figure torque 12 and. power requirements of the simulation prototype running under HUPDC are shown in Figure 12.

Figure 12. The demand torque/power of Harbin urban passenger driving cycle (HUPDC).

The required power of the whole vehicle can be obtained by calculating the speed of the HUPDC. The dynamic system structure of the wheel driven directly by the shaft motor can be further decoupled to calculate the speed of the shaft motor and the torque required by the power system. According to the boundary conditions and optimization objectives formulated by Equations (20) and Figure 12. The demand torque/power of Harbin urban passenger driving cycle (HUPDC). (21),Figure the ratio 12. K The of thedemand output torque/power torque of the motorof Harbin of the ur twoban shaftspassenger before driving and after cycle can (HUPDC). be calculated according to the DP algorithm, and then the distribution of output torque of the two shafts can be The requiredTheconducted. required power Figure power of the13 ofis wholethethe result whole vehicleobtained vehicle by cancan DP be beglobal obtained obtained optimization by calculating by algorithm. calculating the As speed can the be ofseen speed the from HUPDC. of the HUPDC. Figure 13, the torque distribution results calculated in this paper based on energy efficiency as the The dynamic system structure of the wheel driven directly by the shaft motor can be further The dynamic systemoptimization structure objective of are the mostly wheel based driven on four directly-wheel simultaneous by the shaft driving, motor and can thebe torque further decoupled to calculatedecoupled thedistribution speed to calculate of coefficient the the shaft isspeed between motor of 0.3–0.7.the and shaft Compared the moto torquer withand requiredthe average torque torque byrequired the distribution power by the coefficientpower system. system. According to According to the boundary conditions and optimization objectives formulated by Equations (20) and the boundary conditions and optimization objectives formulated by Equations (20) and (21), the ratio (21), the ratio K of the output torque of the motor of the two shafts before and after can be calculated K of the outputaccording torque to the of DP the algorithm, motor ofand the then two the shafts distribution before of andoutput after torque can of be the calculated two shafts accordingcan be to the DP algorithm,conducted. and Figure then the 13 is distribution the result obtained of output by DP torqueglobal optimization of the two algorithm. shafts can As becan conducted.be seen from Figure 13 is the resultFigure obtained 13, the torque by DP distribution global optimization results calculated algorithm. in this paper As canbased be on seen energy from efficiency Figure as 13 the, the torque distributionoptimization results calculated objective are in thismostly paper based based on four on-wheel energy simultaneous efficiency asdriving, the optimization and the torque objective are distribution coefficient is between 0.3–0.7. Compared with the average torque distribution coefficient mostly based on four-wheel simultaneous driving, and the torque distribution coefficient is between 0.3–0.7. Compared with the average torque distribution coefficient of 0.5, the torque distribution coefficient can be optimized according to different motor speed and torque demand. This paper compares different torque distribution strategies of all-wheel-drive electric vehicles under the Harbin urban passenger driving cycle. Figure 14 shows the different torque distribution strategies adopted for all-wheel-drive electric vehicles, including the strategy of average torque distribution, and the operating point distribution of the shaft motor when the torque distribution control strategy of the front and rear shaft motor is obtained according to DP calculation. Figure 15 shows the comparison results of shaft motor efficiency distribution under two different torque Energies 2018, 11, x FOR PEER REVIEW 14 of 17

of 0.5, the torque distribution coefficient can be optimized according to different motor speed and torque demand.

Energies 2019, 12, 1122 13 of 16 distribution strategies. By the comparison of Figures 14 and 15, we can see that the average torque distribution strategy, front and rear wheel motor working in high efficient area number and location are consistent, so if by changing the control strategy, and then realize the change of the motor working point, from the point of promoting the comprehensive efficiency of power system, thereby lowering the energyEnergies consumption 2018, 11, x FOR of PEER the REVIEW electric drive system. Compared with the mean torque14 distribution of 17 strategy, the DP optimal torque distribution strategy proposed in this paper improves the operating of 0.5, the torque distribution coefficient can be optimized according to different motor speed and point of theFigure front 13. and Torque rear distribution shaft motor ratio efficiency map optimized distribution by dynamic in planning the range algorithm of 30–80%. for HUPDC. torque demand. This paper compares different torque distribution strategies of all-wheel-drive electric vehicles under the Harbin urban passenger driving cycle. Figure 14 shows the different torque distribution strategies adopted for all-wheel-drive electric vehicles, including the strategy of average torque distribution, and the operating point distribution of the shaft motor when the torque distribution control strategy of the front and rear shaft motor is obtained according to DP calculation. Figure 15 shows the comparison results of shaft motor efficiency distribution under two different torque distribution strategies. By the comparison of Figure 14 and Figure 15, we can see that the average torque distribution strategy, front and rear wheel motor working in high efficient area number and location are consistent, so if by changing the control strategy, and then realize the change of the motor working point, from the point of promoting the comprehensive efficiency of power system, thereby lowering the energy consumption of the electric drive system. Compared with the mean torque distribution strategy, the DP optimal torque distribution strategy proposed in this paper improves the operating point of the front and rear shaft motor efficiency distribution in the range of 30–80%. Figure 13.FigureTorque 13. Torque distribution distribution ratio ratio map map optimized optimized by by dynamic dynamic planning planning algorithm algorithm for HUPDC. for HUPDC.

0 0 0 0 0 0 This paper compares different2 3 4 torque distribution strategies of all-wheel-drive electric vehicles 5 6 5 Efficiency area distribution 7 7 under the Harbin urban400 passenger driving cycle.Maximum Figure external 14 characteristic shows curvethe different torque distribution DP distributed strategies adopted for all-wheel-drive electricEqually vehicles, distributed including the strategy of average torque

distribution, and the operating point distribution0 of the shaft motor when the torque distribution 300 8

control strategy of the front and rear shaft motor is obtained87 according to DP calculation. Figure 15

5 0 0 7 0 0 0 shows the comparison results0 of7 shaft motor efficiency distribution under two different torque 6 3 4 5 2 5 8 distribution strategies. By the comparison of Figure 14 and Figure 15, we can see that the average

T/N·m 200

torque distribution strategy, front and rear wheel motor89 working in high efficient area number and 0 7 8 8 location are consistent, so if by changing the control strategy, and then 9realize1 the change of the motor

working point, from the point of promoting5 the comprehensive efficiency of power system, thereby 100 8 5 7 0 89 7 89 lowering the energy consumption0 of the electric drive system. Compared with the mean torque 0 0 0 0 6 5 3 4 2 87 87 85 85 distribution strategy, the DP optimal torque85 distribution strategy proposed80 in this paper improves 80 75 80 7570 70 0 75 60 8 50670 6050 50 the operating point of the0 front and rear203040 shaft motor efficiency 302040 distribution 403020 in the range of 30–80%. 0 100 200 300 400 500 600 700 800 Energies 2018, 11, x FOR PEER REVIEW n/r/min 15 of 17

0 0 0 0 0 0 2 3 4 5 6 5 Efficiency area distribution 7 7 (a) Distribution400 of working pointsMaximum of external the characteristic front axle curve motor

DP distributed

0 0 Equally distributed 0 0 0 30 2 4 5 5 6 Efficiency area distribution 7 7 0 400 0 88 Maximum external characteristic curve 300 DP distributed 7 Equally distributed8 5 0 0 7 0 0 0 0 7 6 3 4 5 2 5 8 5 8 300

T/N·m 200

5 0 7 0 8 8 0 7 0 0 0 7 0 6 30 5 9 4 8 2 87 1

T/N·m 5 200 8 9 100 8 1 5 9 7 0 5 89 7 8 89 0 0 0 0 2 0 6 9 5 3 4 2 87 87 85 85 85 80 80 75 80 7570 70 0 75 60 8 7 70 6050 50 8 2030405060 302040 403020 100 0 92 05 0 100 200 300 400 91500 600 700 800 7 8 0 91 7 0 8 n/r/min 0 0 0 9 6 5 89 304 2 89 87 87 5 87 85 85 8 85 80 80 75 7850 7075 70 50670 6050 6050 (a) 0Distribution of203040 working points 302040 of the front axle 402030 motor 0 100 200 300 400 500 600 700 800 n/r/min

(b) Distribution of operating points of the rear axle motor

FigureFigure 14. Distribution 14. Distribution of motor of motor operating operating points points for different for different torque torque distribution distribution strategies strategies ofHUPDC. of HUPDC.

60 54.71 Equally 52.50 50 DP

40 32.43 31.57 30

20 Percentage(&) 13.43 12 10 1.36 1.60 0.28 0.28 0 0-30 30-60 60-80 80-90 90-100 Efficiency distribution %

Figure 15. Motor efficiency distribution in HUPDC.

5.3. Energy Consumption Analysis of Electric Drive System Under two different control strategies, the 100 km power consumption of the all-wheel drive power system is compared as shown in Figure 16. As a result of the optimization of the working point of the shaft motor, when the simulation prototype vehicle adopted the DP global optimization torque distribution strategy under China urban passenger cars driving cycle, the 100-kilometer energy consumption of the electric drive system was 12.585 kWh, which was reduced by 0.75 kWh compared to the average allocation rule. Under the HUPDC, when adopting the DP global optimization torque distribution strategy, the energy consumption of 100 km of electric drive system is 10.338 kWh, which is 0.588 kWh lower than that based on the average allocation rule. Energies 2018, 11, x FOR PEER REVIEW 15 of 17

0 0 0 0 0 30 2 4 5 5 6 Efficiency area distribution 7 7 400 0 8 Maximum external characteristic curve DP distributed Equally distributed

5 8 300

5 0 0 7 0 0 0 7 0 6 30 4 5 8 2 87

T/N·m 200 89 91 5 8 2 9

7 8 100 92 5 0 91 7 8 0 91 7 0 8 0 0 0 9 6 5 89 304 2 89 87 87 5 87 85 85 8 85 80 80 75 7850 7075 70 50670 6050 6050 0 203040 302040 402030 0 100 200 300 400 500 600 700 800 n/r/min

(b) Distribution of operating points of the rear axle motor

Energies 2019,Figure12, 1122 14. Distribution of motor operating points for different torque distribution strategies of 14 of 16 HUPDC.

60 54.71 Equally 52.50 50 DP

40 32.43 31.57 30

20 Percentage(&) 13.43 12 10 1.36 1.60 0.28 0.28 0 0-30 30-60 60-80 80-90 90-100 Efficiency distribution %

FigureFigure 15. 15.Motor Motor efﬁciency efficiency distribution in in HUPDC. HUPDC.

5.3. Energy5.3. Energy Consumption Consumption Analysis Analysis of Electricof Electric Drive Drive SystemSystem UnderUnder two differenttwo different control control strategies, strategies, thethe 100100 km power power consumption consumption of the of theall-wheel all-wheel drive drive powerpower system system is compared is compared as as shown shown in Figure Figure 16. 16 As. a As result a result of the ofoptimization the optimization of the working of the point working point ofof the shaft shaft motor, motor, when when the thesimulation simulation prototype prototype vehicle vehicleadopted adoptedthe DP global the DPoptimization global optimization torque torquedistribution distribution strategy strategy under under China China urban urban passenger passenger cars carsdriving driving cycle, cycle, the 100-kilometer the 100-kilometer energy energy consumption of the electric drive system was 12.585 kWh, which was reduced by 0.75 kWh compared consumption of the electric drive system was 12.585 kWh, which was reduced by 0.75 kWh compared to the average allocation rule. Under the HUPDC, when adopting the DP global optimization torque to thedistribution average allocation strategy, rule.the energy Under consumption the HUPDC, of 100 when km of adopting electric drive the system DP global is 10.338 optimization kWh, which torque distributionis 0.588 strategy, kWh lower the than energy that based consumption on the average of 100 allocation km of electric rule. drive system is 10.338 kWh, which is 0.588Energies kWh 2018 lower, 11, x thanFOR PEER that REVIEW based on the average allocation rule. 16 of 17

Figure 16. The energy consumption under different driving cycle and different control strategies. Figure 16. The energy consumption under different driving cycle and different control strategies. 6. Conclusions 6. Conclusions In this paper, the optimal torque distribution strategy for electric drive system is studied to In this paper, the optimal torque distribution strategy for electric drive system is studied to solve solvethe torque torque optimization optimization problem problem of the ofefficiency the efficiency difference difference of the axial of distributed the axial drive distributed motor. A drive motor.torque A torque distribution distribution method method based based on global on global optimized optimized dynamic dynamic programming programming is proposed. is proposed. ComparedCompared with with the averagethe average distributed distributed torque torque method,method, the the proposed proposed method method effectively effectively reduces reduces the the powerpower consumption consumption of the of the electric electric drive drive system system byby 100 km, km, among among which, which, 0.75 0.75 kWh kWh can be can reduced be reduced underunder the China the China urban urban passenger passenger cars cars driving driving cycle. cycle. In In the the Harbin Harbin urban urban passenger passenger cars driving cars driving cycle, cycle, 0.60 kWh0.60 waskWh reduced,was reduced, and and the the increase increase was was over over 5%.5%. The The results results show show that that the thetorque torque distribution distribution ratio of the front and rear shafts can be adjusted dynamically in real time to solve the problem of ratio of the front and rear shafts can be adjusted dynamically in real time to solve the problem of working point optimization caused by the difference in the efficiency distribution of the shafts. The workingproposed point optimizationmethod can be causedapplied byto the the optimization difference inof thetorque efficiency distribution distribution control under of the different shafts. The proposeddriving method cycles, can which be appliedcan effectively to the improve optimization the energy of torque efficiency distribution of the system control and underextend differentthe drivingdriving cycles, distance which of canelectric effectively passenger improve vehicles. the energy efficiency of the system and extend the driving distance of electric passenger vehicles. Author Contributions: X.W. and D.Z. conceived and designed the simulations; J.D. analyzed the data; D.Z. and T.W. wrote the paper.

Funding: This research received no external funding.

Acknowledgments: This work was sponsored through the National Natural Science Foundation of China (51877121); University Nursing Program for Young Scholars with Creative Talent in Heilongjiang Province (UNPYSCT-2016164); State Key Laboratory of Automotive Safety and Energy under Project No. KF1826.

Conflicts of Interest: The authors declare no conflict of interest.

Reference

1. Zhang, H.; Huang, X.; Wang, J.; Karimi, H.R. Robust energy-to-peak sideslip angle estimation with applications to ground vehicles. Mechatronics 2015, 30, 338–347. 2. Shuai, Z.; Zhang, H.; Wang, J.; Li, J.; Ouyang, M. Lateral motion control for four-wheel-independent-drive electric vehicles using optimal torque allocation and dynamic message priority scheduling. Control Eng. Pract. 2014, 24, 55–66. 3. Zhao, Z.; Lei, D.; Chen, J.; Li, H. Optimal control of mode transition for four-wheel-drive hybrid electric vehicle with dry dual-clutch transmission. Mech. Syst. Signal Process. 2018, 105, 68–89. 4. Zhou, H.; Jia, F.; Jing, H.; Liu, Z.; Güvenç, L. Coordinated longitudinal and lateral motion control for four wheel independent motor-drive electric vehicle. IEEE Trans. Veh. Technol. 2018, 67, 3782–3790. 5. Chen, T.; Chen, L.; Xu, X.; Cai, Y.; Jiang, H.; Sun, X. Estimation of longitudinal force and sideslip angle for intelligent four-wheel independent drive electric vehicles by observer iteration and information fusion. Sensors 2018, 18, 1268. 6. Chen, T.; Xu, X.; Chen, L.; Jiang, H.; Cai, Y.; Li, Y. Estimation of longitudinal force, lateral vehicle speed and yaw rate for four-wheel independent driven electric vehicles. Mech. Syst. Signal Process. 2018, 101, 377– Energies 2019, 12, 1122 15 of 16

Author Contributions: X.W. and D.Z. conceived and designed the simulations; J.D. analyzed the data; D.Z. and T.W. wrote the paper. Funding: This research received no external funding. Acknowledgments: This work was sponsored through the National Natural Science Foundation of China (51877121); University Nursing Program for Young Scholars with Creative Talent in Heilongjiang Province (UNPYSCT-2016164); State Key Laboratory of Automotive Safety and Energy under Project No. KF1826. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Zhang, H.; Huang, X.; Wang, J.; Karimi, H.R. Robust energy-to-peak sideslip angle estimation with applications to ground vehicles. Mechatronics 2015, 30, 338–347. [CrossRef] 2. Shuai, Z.; Zhang, H.; Wang, J.; Li, J.; Ouyang, M. Lateral motion control for four-wheel-independent-drive electric vehicles using optimal torque allocation and dynamic message priority scheduling. Control Eng. Pract. 2014, 24, 55–66. [CrossRef] 3. Zhao, Z.; Lei, D.; Chen, J.; Li, H. Optimal control of mode transition for four-wheel-drive hybrid electric vehicle with dry dual-clutch transmission. Mech. Syst. Signal Process. 2018, 105, 68–89. [CrossRef] 4. Zhou, H.; Jia, F.; Jing, H.; Liu, Z.; Güvenç, L. Coordinated longitudinal and lateral motion control for four wheel independent motor-drive electric vehicle. IEEE Trans. Veh. Technol. 2018, 67, 3782–3790. [CrossRef] 5. Chen, T.; Chen, L.; Xu, X.; Cai, Y.; Jiang, H.; Sun, X. Estimation of longitudinal force and sideslip angle for intelligent four-wheel independent drive electric vehicles by observer iteration and information fusion. Sensors 2018, 18, 1268. [CrossRef] 6. Chen, T.; Xu, X.; Chen, L.; Jiang, H.; Cai, Y.; Li, Y. Estimation of longitudinal force, lateral vehicle speed and yaw rate for four-wheel independent driven electric vehicles. Mech. Syst. Signal Process. 2018, 101, 377–388. [CrossRef] 7. Wu, D.; Ding, H.; Du, C. Dynamics characteristics analysis and control of FWID EV. Int. J. Automot. Technol. 2018, 19, 135–146. [CrossRef] 8. Joa, E.; Park, K.; Koh, Y.; Yi, K.; Kim, K. A tyre slip-based integrated chassis control of front/rear traction distribution and four-wheel independent brake from moderate driving to limit handling. Veh. Syst. Dyn. 2018, 56, 579–603. [CrossRef] 9. Han, K.; Choi, M.; Lee, B.; Choi, S.B. Development of a traction control system using a special type of sliding mode controller for hybrid 4wd vehicles. IEEE Trans. Veh. Technol. 2018, 67, 264–274. [CrossRef] 10. Koehler, S.; Viehl, A.; Bringmann, O.; Rosenstiel, W. Energy-efﬁciency optimization of torque vectoring control for battery electric vehicles. IEEE Intell. Transp. Syst. Mag. 2017, 9, 59–74. [CrossRef] 11. Xin, X.; Zhang, W.; Shen, C.; Zheng, H. Control strategy of four-wheel independent drive electric vehicle based on vehicle velocity estimation and switchover. Trans. Inst. Meas. Control 2017, 37, 965–975. [CrossRef] 12. Qiu, L.; Qian, L.; Zomorodi, H.; Pisu, P. Global optimal energy management control strategies for connected four-wheel-drive hybrid electric vehicles. IET Intell. Transp. Syst. 2017, 11, 264–272. [CrossRef] 13. Zhang, X.; Wei, K.; Yuan, X.; Tang, Y. Optimal torque distribution for the stability improvement of a four-wheel distributed-driven electric vehicle using coordinated control. J. Comput. Nonlinear Dyn. 2016, 11, 051017. [CrossRef] 14. Zhai, L.; Sun, T.; Wang, J. Electronic stability control based on motor driving and braking torque distribution for a four in-wheel motor drive electric vehicle. IEEE Trans. Veh. Technol. 2016, 65, 4725–4739. [CrossRef] 15. Shuai, Z.; Zhang, H.; Wang, J.; Li, J.; Ouyang, M. Combined AFS and DYC control of four-wheel-independent- drive electric over CAN network with time-varying delays. IEEE Trans. Veh. Technol. 2014, 63, 591–602. [CrossRef] 16. Wang, R.; Zhang, H.; Wang, J. Linear parameter-varying controller design for four-wheel independently actuated electric ground vehicles with active steering systems. IEEE Trans. Control Syst. Technol. 2014, 22, 1281–1296. 17. Du, J.; Chen, J.; Song, Z.; Gao, M.; Ouyang, M. Design method of a power management strategy for variable battery capacities range-extended electric vehicles to improve energy efﬁciency and cost-effectiveness. Energy 2017, 121, 32–42. [CrossRef] Energies 2019, 12, 1122 16 of 16

18. Xiong, R.; Zhang, Y.; He, H.; Zhou, X.; Pecht, M.G. A double-scale, particle-ﬁltering, energy state prediction algorithm for lithium-ion batteries. IEEE Trans. Ind. Electron. 2018, 65, 1526–1538. [CrossRef] 19. Xiong, R.; He, H.; Sun, F.; Liu, X.; Liu, Z. Model-based state of charge and peak power capability joint estimation of lithium-ion battery in plug-in hybrid electric vehicles. J. Power Sources 2013, 229, 159–169. [CrossRef] 20. Xu, L.; Ouyang, M.; Li, J.; Yang, F.; Lu, L.; Hua, J. Application of Pontryagin’s Minimal Principle to the energy management strategy of plugin fuel cell electric vehicles. Int. J. Hydrog. Energy 2013, 38, 10104–10115. [CrossRef] 21. Zheng, D.Y.; Wu, X.G.; Chen, H.; Du, J.Y. Construction of driving conditions of Harbin urban passenger cars. J. Highw. Transp. Res. Dev. 2017, 34, 101–107. [CrossRef] 22. Cheng, S.; Xu, L.; Wu, K.; Fang, C.; Hu, J.; Li, J.; Ouyang, M. Optimal warm-up control strategy of the PEMFC system on a city bus aimed at improving efﬁciency. Int. J. Hydrog. Energy 2017, 42, 11632–11643. [CrossRef] 23. Song, Z.; Hofmann, H.; Li, J.; Han, X.; Ouyang, M. Optimization for a hybrid energy storage system in electric vehicles using dynamic programing approach. Appl. Energy 2015, 139, 151–162. [CrossRef]

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