Torque-based Optimal Acceleration Control for Electric Vehicle

CHINESE JOURNAL OF MECHANICAL ENGINEERING Vol. 27, No. 2, 2014

・319・

DOI: 10.3901/CJME.2014.02.319, available online at www.springerlink.com; www.cjmenet.com; www.cjmenet.com.cn

Torque-based Optimal Acceleration Control for Electric Vehicle LU Dongbin and OUYANG Minggao* State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing 100084, China Received July 18, 2013; revised December 12, 2013; accepted December 31, 2013

Abstract: The existing research of the acceleration control mainly focuses on an optimization of the velocity trajectory with respect to a criterion formulation that weights acceleration time and fuel consumption. The minimum-fuel acceleration problem in conventional vehicle has been solved by Pontryagin’s maximum principle and dynamic programming algorithm, respectively. The acceleration control with minimum energy consumption for battery electric vehicle(EV) has not been reported. In this paper, the permanent magnet synchronous motor(PMSM) is controlled by the field oriented control(FOC) method and the electric drive system for the EV(including the PMSM, the inverter and the battery) is modeled to favor over a detailed consumption map. The analytical algorithm is proposed to analyze the optimal acceleration control and the optimal torque versus speed curve in the acceleration process is obtained. Considering the acceleration time, a penalty function is introduced to realize a fast vehicle speed tracking. The optimal acceleration control is also addressed with dynamic programming(DP). This method can solve the optimal acceleration problem with precise time constraint, but it consumes a large amount of computation time. The EV used in simulation and experiment is a four-wheel hub motor drive electric vehicle. The simulation and experimental results show that the required battery energy has little difference between the acceleration control solved by analytical algorithm and that solved by DP, and is greatly reduced comparing with the constant pedal opening acceleration. The proposed analytical and DP algorithms can minimize the energy consumption in EV’s acceleration process and the analytical algorithm is easy to be implemented in real-time control. Keywords: permanent magnet synchronous motor(PMSM), field oriented control(FOC), efficiency model, electric vehicle, energy optimal acceleration

1

Introduction

Vehicle speed control is an important topic for both conventional cruise control systems and recently developed adaptive cruise control(ACC) systems[1–2]. With aid of an on board road slope database in combination with a GPS unit, information about the road geometry ahead can be extracted. The velocity trajectory that weights trip time and fuel consumption can be optimized by using this look-ahead road information. A dynamic programming approach is taken to obtain minimizing fuel consumption solutions for a number of driving scenarios on shorter road sections in Ref. [3]. Optimal speed profiles for heavy trucks driving important test road profiles are presented in Ref. [4]. The method is based on an analytic solution for linear road segments, and the continuous connection of such solutions. This reduces the dimension of the problem significantly compared with other methods. Predictive cruise control is investigated through computer simulations[5], but * Corresponding author. E-mail: ouymg@tsinghua.edu.cn This project is supported by US-China Clean Energy Research Collaboration: Collaboration on Cutting-edge Technology Development of Electric Vehicle(Program of International S&T Cooperation, Grant No. 2010DFA72760) © Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2014

constructing an optimizing controller that works on board in a real environment puts additional demands on the system in terms of robustness and complexity. A predictive cruise controller is developed where discrete DP is used to numerically solve the optimal control problem in Ref. [6]. An improved dynamic programming algorithm[7] where the search space is reduced by a preprocessing algorithm is realized and evaluated in actual experiments. The speed control in the acceleration process is the key part of the cruise control. The application of Pontryagin’s maximum principle and dynamic programming for vehicle acceleration control with minimum fuel consumption are described in Ref. [8]. With the maximum principle, the consumption model yields optimal accelerations with a linearly decreasing acceleration as a function of the velocity. Dynamic programming is implemented with a backward recursion on a specially chosen distance grid. This grid enables the calculation of realistic gear shifting behavior during vehicle accelerations. These two algorithms used to solve the optimal acceleration problem are complicated. The acceleration control with minimum energy consumption for battery electric vehicle(EV) has not been reported. In this paper, a torque-based acceleration control

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YLU Dongbin, et al: Torque-based Optimal Acceleration Control for Electric Vehicle Y

with minimum energy consumption for electric vehicle(EV) is presented. In EV, the accurate torque control for traction motor can be achieved by field oriented control(FOC) method. The permanent magnet synchronous motor (PMSM) drive system is modeled to favor over a detailed consumption map. The optimal torque versus speed curve in the acceleration process is obtained by analytical algorithm, which is much simpler than maximum principle and DP algorithms. In the time limitation case, the electric motor torque demanded is set up by adding a penalty function to fast track the target vehicle speed. To assess the performance of the above analytical algorithm, the dynamic programming(DP) solutions for different acceleration time are used as benchmarks for comparisons. The organization of this paper is as follows. In section 2, the vehicle, the motor, the inverter and the battery model are described. In section 3, the analytical model is proposed to solve the optimal acceleration problem. The optimal acceleration control solved by DP in different acceleration time is described in section 4. The simulation and experiment results are shown in section 5, and the conclusions of this research are in section 6.

2

The flux equation can be torque expressed as ïìï d = Ld idt + f , í = L i . ïïî q q qt

(3)

The electromagnetic torque equation is Te = p ( d iqt - q idt ) = p[ f iqt + ( Ld - Lq )idt iqt ],

(4)

where ud and uq are the d- and q-axis terminal voltages, respectively; id and iq are the d- and q-axis armature currents, respectively; idi and iqi are the equivalent d- and q-axis iron consumption currents, respectively; idt and iqt are the equivalent d- and q-axis torque currents, respectively; ψd and ψq are the d- and q-axis stator flux-linkages, respectively; ψf is the magnet flux-linkage; Ld and Lq are the d- and q-axis inductances, respectively; ω is the electrical angular velocity; and Ra is the armature resistance; and p is the number of pole pairs.

Dynamic Modeling

2.1 Vehicle model The force needed when driving on a sloped road in an EV can be expressed as ìïTtq ig i0T 1 dv ïï = mgf cos  + CD  Av 2 + mg sin  +  m , ïí r 2 dt ïï ïïîTtq = Te - Tm , (1)

where Ttq, Te and Tm are the motor output, electromagnetic and mechanical torque, respectively; ig is the ratio of transmission; i0 is the ratio of the reducer; ηT is the efficiency of the transmission system; r is the wheel radius; m is the mass of the vehicle; f is the rolling resistance coefficient; α is the slope angle; CD is the dimensionless coefficient; A is the cross-sectional area exposed to flow; v is the vehicle velocity; δ is the vehicle rotating mass conversion factor; dv/dt is the vehicle acceleration; and ρ is the air density. 2.2 Efficiency model of PMSM Considering the iron loss, and the d- and q-axis equivalent circuit model of PMSM[9–10] is shown in Fig. 1. The d-axis stator current id and q-axis stator current iq are divided by the iron consumption current idi, iqi and the torque current idt, iqt. In the steady-state case, the voltage balance equation can be expressed as ìud = Ra id -  q , ï ï í ï ïuq = Ra iq +  d . î

(2)

Fig. 1. Efficiency model of PMSM

The surface mounted permanent magnet synchronous motor(SPMSM) will be studied in this paper and the research methods of this paper are also adapted to study the interior PMSM(IPMSM) and induction motor(IM). In the SPMSM, there is Ld=Lq=L and the troque equation can be simplified as Eq. (5). The troque generated by the SPMSM is proportional to the q-axis current and has no relationship with the d-axis current. Te = p ( d iqt - q idt ) = p f iqt .

(5)

Fig. 2 is the power flow diagram of the PMSM in driving mode. The input power of the PMSM can be deduced from Eqs. (2) and (3) and the efficiency model in Fig. 1 and is expressed as Pin = ud id + uq iq = ( Ra id -  q )id + ( Ra iq +  d )iq = Ra (id + iq ) + 2

2

 2 ( d 2 + q 2 ) Ri

+  f iqt .

(6)

CHINESE JOURNAL OF MECHANICAL ENGINEERING where Ri is the equivalent iron consumption resistance, which can be obtained by experiment[10]. In the right side of the equation, the first part is copper loss PCu, the second part is iron loss PFe, and the third part is the electromagnetic power Pe. The electromagnetic power, which is the sum of the mechanical loss Pm, stray loss Ps and output power Pout, can be expressed as

 f iqt = Pm + Ps + Pout .

Fig. 2.

(7)

In SPMSM, the maximum torque/current control can be realized by controlling the d-axis current id=0 which is applied to the traction motor of this research. According to the equivalent circuit and Eqs. (2)–(6), the relations of power loss, electromagnetic torque and the electric angular velocity can be expressed as ìï éæ  2 L2 ù2 ö T ïï  ÷ ç e f ê ú , + ïï PCu = Ra êçç 2 + 1÷÷÷ ú ç  p R R è ø ïï f i ûú ëê i ïï 2 2ù éæ 2 í Te ï 2 êç  L ÷ö + çæ L Te ÷÷ö ú ÷ ï +   ç ç ê ú f÷ ïï ÷ø çè p f p f ÷÷ø ú êèçç Ri ïï ë û. ïï PFe = Ri ïî

(8)



(9)

The maximum torque of the PMSM in drive mode needs to satisfy the following conditions: ïìïTedrv = Temax , if Pe £ Pe n , ï í ïïTedrv = Pe n p, if Pe > Pe n , ïî 

(11)

where Vds and Vak represent the on-state voltage drops of a MOSFET and diode, respectively; Vf is the diode voltage drops at the zero-current condition; Rds and Rak are the resistive elements of MOSFET and diode; and I is the device current. Eq. (12) expresses the conduction losses for the MOSFET and the anti-parallel diode, respectively: ìï æ 1 mi ö 2 ïï P ç + cos  ÷÷ , ç c-MOSFTET = I m Rds ç ÷ ïï è 8 3π ø í ïï æ1 m ö æ1 m ö 1 ïï Pc-D = I mVf çç - i cos  ÷÷ + I m 2 Rak çç - i cos  ÷÷ , ÷ ç ç èπ 4 ø è 8 3π ø÷ 2 ïïî (12)

The input power can be deduced as 2 éæ  2 L2 ö T  f ùú ÷ ç e ê ÷ Pin = Te + Ra êçç 2 + 1÷ + + ÷ø p f p Ri úúû êëçè Ri 2ù éæ 2 ö2 æ Te Te ÷ö ú 2 êç  L ÷ ÷÷ ú  êçç + f ÷÷ + çç L p f p f ÷ø ú êèç Ri ø÷ çè ë û. Ri

have well-developed component models and an inverter switching algorithm. Otherwise, an accurate evaluation can only be obtained by actual tests. However, by making some assumptions and simplifying the device models, the analytical model can be applied to the inverter efficiency evaluation[11–13]. For conduction loss evaluation, a simplified device models is employed: a pure resistor for power MOSFETs and a voltage source in series with a resistor for Insulated Gate Bipolar Transistors(IGBTs) and diodes. In this paper, the loss of a three-phase MOSFET-based full-bridge inverter is discussed, and the space vector pulse width modulation(SVPWM) is adopted by this inverter. (1) Conduction loss. The simplified models for the power MOSFET and diode are expressed as ì Vds = IRds , ï ï í ï ï îVak = Vf + IRak ,

Power flow diagram of the PMSM

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(10)

where Temax is maximum electromagnetic torque, and Pen is the rated power of the PMSM. 2.3 Efficiency model of inverter To accurately evaluate the inverter efficiency, one must

where mi is the per unit fundamental stator voltage, φ is the load power factor angle, and Im is the peak value of the sinusoidal wave. In this paper, the conduction loss of the inverter contains the conduction loss of the MOSFETs and the anti-parallel diode. Because the SVPWM is adopted and the MOSFET conducts current bidirectionally, the diodes conduct only in the dead time. Considering the dead time effect, the conduction loss of the MOSFETs and the anti-parallel diodes in the three-phase inverter can be simplified as ì æ 2t ö ï 3 ï ïï Pc-MOSFET s = I m 2 Rds ççç1- d ÷÷÷ , çè 2 tc ø÷ ï ï í ï æ6 3 2 ö÷ 2td ï ç ï ïï Pc-D s = ççè π Vf I m + 2 I m Rak ø÷÷ t , c ï î

(13)

where td is the dead time, and tc is the period time of the SVPWM. The conduction loss of the inverter is Pc = Pc-MOSFET s + Pc-Ds .

(14)

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(2) Switching loss. According to Ref. [14], the switching loss of hard switch circuit is ìï ïï Psw-on = kon I m fs , π ïïí ïï koff I m fs , ïï Psw-off = π ïî

(15)

where Psw-on and Psw-off are the turn on loss and the turn off loss, respectively; fs is the switching frequency; and kon and koff can be obtained by the device datasheet or test data. The switching loss of the three-phase inverter is Psw = 3( Psw-o n + Psw-off ).

Rs, an RC network (Rt//Ct) and a DC source with voltage uoc(OCV) that is a function of SOC are connected in series. ub is defined as the battery terminal voltage and ib is the outflow current.

Fig. 3.

Generalized ECM for lithium batteries

Using Kirchhoff’s law, the dynamics of the ECM shown in Fig. 3 can be expressed as

(16)

uc = -

The total loss of the three-phase inverter is the sum of the conduction loss and the switching loss, which can be expressed as

(17)

where the relationship between Im and iq, Te is as Im =

PbR = ( Rs + Rt )ib2 .

(21)

2 æç12td Vf 3kon fs 3koff fs ö÷ + + ç ÷÷´ 3 ççè πtc π π ø÷ æ 2td ÷öù Te  f ùú éê 2 Rak td çç1÷÷ú ´ R + + + ds ççè p f Ri úúû êêë tc tc ÷øúúû 2 éæ  2 L2 ö÷ T  f ùú e êçç ÷ (19) êçç R 2 + 1÷÷ p + R ú . ø f i úû êëè i

Pinv =

2.4 Battery model The electric model of the battery, which is described by a circuit that is composed of the basic elements, such as resistor and capacitor, is called an ECM. It is widely used to analyze the dynamic properties of the battery voltage and current[14–16]. The error in analyzing the battery charge and discharge properties does not exceed 5%[16], which satisfies the engineering requirement. Typically, the architecture of the circuit is composed of a fundamental ohmic resistor and one or more RC networks connected in series to simulate both the transient and steady responses of the battery. One of the ECMs used to simulate cell performance is illustrated in Fig. 3, where an ohmic resistor with resistance

(22)

The energy consumption in a battery can also be expressed by the electromagnetic torque and electrical angular speed of the PMSM as

ö T  f ùú 2 2 éêæç  2 L2 e çç 2 + 1÷÷÷ iq = . (18) + ê ÷ø p f 3 3 êëçè Ri Ri úúû

The loss of the inverter can be expressed by the electromagnetic torque and electrical angular speed of the PMSM as

éæ  2 L2 ö÷ êçç ÷÷ 1 + êçç R 2 ø÷ êëè i

ub = uo c - Rs ib - uc ,

(20)

where uc is defined as the voltage across the RC network, as observed in Fig. 3. In the steady case, the power consumption in a battery can be expressed as

æ12t V 3k f 3k f ö Pinv = Pc + Psw = ççç d f + on s + off s ÷÷÷ I m + çè πtc π π ø÷ é 3R t æ öù ê ak d + 3 R çç1- 2td ÷÷ú I 2 , ds ç ÷ m ê t çè 2 tc ø÷úûú ëê c

1 1 uc + ib , Ct Rt Ct

éu - u 2 - 4( R + R )( P + P ) ù 2 ê oc oc s t in c ú û . PbR = ë 4( Rs + Rt )

(23)

where Pin and Pinv are functions of the electromagnetic torque and electrical angular velocity of the PMSM respectively, as shown in Eqs. (9) and (19)

3

Optimal Acceleration Control Solved by Analytical Algorithm

3.1 Optimal energy acceleration control The optimal energy acceleration control of the electric vehicle is defined that the output energy of the battery is minimum when the electric vehicle accelerates from one speed to a higher speed. The target function is as follows:

òt

tf

Pbat ( (t ), Te (t )) d t ,

(24)

(v0 + a(t ) t ) d t = vf (t ),

(25)

Jv =

0

òt

tf 0

Pb at (， Te ) = Pin (， Te )+Pinv (， Te )+PbR (， Te ).

(26)

When the electric vehicle accelerate from v0 to vf , the increasing kinetic energy is Ek =

1 m(vf 2 - v0 2 ). 2

(27)

CHINESE JOURNAL OF MECHANICAL ENGINEERING

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The increasing energy Ek is only related to the initial speed v0 and the final speed vf, and equals to the work of the acceleration force. If the two speeds are certain, the increasing energy is constant. The above optimal problem can also be described as

k =

Ek

òt

tf

=

Pbat ( (t ), Te (t )) d t

0

1 m(vf 2 - v02 ) 2

òt

tf

Pbat ( (t ), Te (t )) d t

. (28)

0

Fig. 4.

The discretion of Eq. (28) is

k j =

DEk , Pbat j ( (t ), Te (t )) t j

(29)

where ΔEk, Pbatj and tj are the increasing kinetic energy, battery output power and time in every step, respectively. The total battery output energy can be expressed as n

Ebat =

å éêë Pbat j ( (t ), Te (t )) j =1

t j ùúû =

n

DE

å k jk .

(30)

j =1

From Eq. (30), if the maximum efficiency of output energy in every step is achieved, the minimum battery output energy in every increasing kinetic energy can be obtained, the sum of which is the optimal battery output energy. Thus, the global optimal problem can be turned into one which to seek the maximum kinetic energy efficiency. According to the vehicle resistance equation, the kinetic energy output efficiency can be expressed as éT i i  æ öù 1 k = ê tq g 0 T -çççmgf cos + CD  Av2 + mg sin  ÷÷÷ú v ê r è øúû 2 ë é Pbat (, Te ) = êë(Te -Tm )ig i0T -r ´ ù æ æ r ÷ö2 ÷ö çç 1 ÷÷ + mg sin  ÷÷÷úú ççmgf cos  + CD  Aççç ÷ú çè pig i0 ÷÷ø 2 ççè ø÷úû pig i0 Pbat (, Te ),

Motor torque-speed curve in optimal energy acceleration at different slope angles

3.2 Acceleration control considering acceleration time In section 3.1, the optimum acceleration control with minimum battery energy to accelerate to the target speed is discussed. However, the acceleration time is not considered, without which the driver’s demand may not be met in some situation. This section will study the optimal acceleration control with the acceleration time. As the given target speed is vf(t), the problem can be described to solve the electromagnetic torque to make the following objective function minimum: Jv = ò

tf t0

( Pbat ( (t ), Te (t )) + kv (vf (t ) - v(t ))2 ) d t,

òt

tf

(v(t0 ) + a (t ) t ) d t = vf (t ),

(32)

(33)

0

where kv is the weighting factor of penalty function, and larger kv can shorten the acceleration time and track the target vehicle speed more fast. In accordance with section 3.1, the discretion equation can be expressed as é Ttqig i0T

æ öù 1 -ççmgf cos  + CD  Av2 + mg sin  ÷÷÷ú v èç øúû 2 ë r 2 ( Pbat (, Te ) + kv (vf - v) ) = éêë(Te -Tm )ig i0T -r ´

k = êê

(31)

where ηk is a function of Te and ω, and the optimum electromagnetic torque Te* at any speed can be obtained to achieve the maximum kinetic energy output efficiency. The optimum electromagnetic torque curve changes with the change of road condition. With the assumption that the rolling resistance coefficient and the drag coefficient are constant, the optimal electromagnetic torques at different slope angles are shown in Fig. 4. It can be seen from Fig. 4, the larger the slope angle is, the closer the optimum torque curve is to the maximum motor electromagnetic torque.

æ öù æ ö2 çççmgf cos  + 1 C  Açç r ÷÷ + mg sin  ÷÷÷úú ÷÷ D çç pi i ÷÷÷ çç ú 2 è g 0ø çè ø÷úû 2ù é æ r ÷÷ö ú ê ç ç pig i0 ê Pbat (, Te ) + kv çvf ÷ ú. çè ê pig i0 ÷÷ø ú êë úû

(34)

From the above equation, the electromagnetic torques in different kv can be obtained on the level road, as shown in Fig. 5. The motor speed curves corresponding to the time are shown in Fig. 6. The acceleration control at different acceleration time can be achieved by controlling kv and a good acceleration control can be achieved. The weighting factor kv only reflects the case to follow the reference speed and it cannot control acceleration time accurately.

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YLU Dongbin, et al: Torque-based Optimal Acceleration Control for Electric Vehicle Y where sj is the j-step distance of EV traveled and is the system state transfer variable, Pbatj is the j-step battery output power. The system constraint is Tebrk £ Tej £ Tedrv ,

(38)

where Tej is the system control variable, and Tebrk is the maximum regenerative braking torque. The discretion state transition model is as Fig. 5.

Motor torque-speed curves in different kv

s j +1 = s j +

v j +12 - v j 2 2a j

,

(39)

where aj is the vehicle acceleration and is expressed as 1 êé (Te j - Tm )ig i0T m êë r æ öù ççmgf cos  j + 1 CD  Av j 2 + mg sin  j ÷÷ú . ÷øú çè 2 û aj =

Fig. 6.

4

Motor speed curve in different kv

The optimum electromagnetic torque on the level road solved by DP is shown in Fig. 7, which is similar to that solve by analytical model(AM).

Optimal Acceleration Control Solved by Dynamic Programming

4.1 Optimal energy acceleration control under no constraint The analytical nature enables one to get a physical insight in the control problem. Dynamic programming(DP) is very easy to implement and can easily handle all sorts of constraints and dynamics. The optimal energy acceleration control is solved by the analytical model in section 3.1. The problem will be solved by dynamic programming in this section, and, indirectly, to prove the conclusions of section 3.1. The increasing kinetic energy in every step is ΔEk, and Ek=NΔEk, the speed in every step can be expressed as 2 Ekj

vj =

m

(35)

,

where Ekj is the vehicle kinetic energy in step j. The cost function is defined as N

Jv =

å Ebatj ,

(36)

j =1

where Ebatj is the j-step output energy of the battery, and can be expressed as Ek j =

s j +1 - s j v j + v j +1

Pbat j ,

(40)

(37)

Fig. 7. Torque-speed curve comparisons between analytical model and dynamic programming

The algorithm solved by analytical model can quickly seek the optimum value. However, if the vehicle’s load conditions change, the global optimum value cannot be solved. The DP algorithm is used to solve optimum problem in dynamic system. The drawback is that the computation consumption is large and hard to use in real-time computation. 4.2 Optimal energy acceleration control under time constraint Acceleration control under no constraint consumes large time to the target speed. In this section, the optimal energy acceleration control under time constraint is discussed. The acceleration time is divided into N steps with the step size Δt, and ta=NΔt. The cost function is defined as follows: N

Jv =

å Ebat j , j =1

(41)

CHINESE JOURNAL OF MECHANICAL ENGINEERING Ebat j = t Pbat j .

(42)

The system constraint is Tebrk £ Tej £ Tedrv .

(43)

The discretion state transition model is

v j +1 = v j + a j t.

(44)

Fig. 8 shows the optimum electromagnetic torque at different time constraint on the level road conditions.

Table 1.

Specifications of surface permanent magnet synchronous motor

Motor parameter Rated phase voltage Un/V Rated speed nr/(r • min–1) Rated torque Ten/(N • m) Pole pairs p Armature resistance Ra/Ω d-axis inductance Ld/H q-axis inductance Lq/H Magnet flux linkage ψf/Wb Iron consumption resistance Ri/Ω

Table 2.

Motor torque-speed curve of different acceleration time in optimal energy acceleration

The motor speed at different time corresponding to the time is shown in Fig. 9.

Fig. 9.

5

constraint

Value 21.6 500 30 23 0.031 7.6´10–5 7.6´10–5 0.020 4 0.006ω+1.5

Parameters of electric vehicle and battery

Vehicle parameter Curb weigh m/kg Wheel radius r/m Frontal area A/m2 Dimensionless coefficient CD Rolling resistance coefficient f Battery Voltage Ubat/V Battery rated capacity C/(A • h)

Fig. 8.

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Value 660 0.25 1.4 0.4 0.014 53 100

The PMSM accelerates from 0 to 500 r/min in three different control strategies, and the advantages and disadvantages are discussed. The three acceleration control strategies are: the constant accelerator pedal opening control, acceleration control solved by analytical model in section 3, and acceleration control solved by DP in section 4. The constant accelerator pedal opening control is often used by driver. The output torque is reduced corresponding to the increasing speed due to the rate power. Fig. 10 shows the electromagnetic torques in 20%, 40%, 60%, 80% and 100% acceleration pedal opening. The acceleration time, the acceleration distance and the battery output energy are shown in Table 3.

Motor speed curve in different acceleration time

Simulation and Experiment Results

The optimal acceleration control algorithms of the PMSM are studied in simulation and experiment. The experiment are carried out by a four-wheel hub motor driven electric vehicle. 5.1 Simulation analysis based on Matlab/Simulink The surface permanent magnet synchronous motor and the inverter based on power MOSFET are used for the simulation and experiment, and the parameters are shown in Table 1. The vehicle and battery parameters are shown in Table 2.

Fig. 10.

Motor torque-speed curve in different acceleration pedal opening

The acceleration torques solved by analytical model in different kv are shown in Fig. 11. The acceleration distance

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and the battery output energy are shown in Table 4. Table 3.

Acceleration properties in different acceleration pedal opening

Accelerator pedal opening α/% 20 40 60 80 100

Acceleration time ta/s 252.05 42.47 20.72 13.94 10.39

Acceleration distance sa/m 2 115.8 375.60 169.05 110.69 81.06

Battery output energy Ebat/kJ 462.28 146.44 108.77 101.78 99.20

parts: the rolling resistance loss, the wind drag loss and the power loss of the electric drive system. A longer traveling distance, higher speed and the low efficiency of the electric drive system will increase the loss respectively. Therefore, the vehicle speed, traveling distance and the efficiency of the drive system should be taken into consideration in the optimal acceleration control. Table 5.

Acceleration properties solved by DP in different acceleration time Acceleration distance sa/m 82.73 89.09 94.74 98.95 101.79

Acceleration time ta/s 11 13 15 17 19.3 (Optimum)

Battery output energy Ebat/kJ 97.72 95.50 94.63 94.33 93.87

5.2 Experiment results The electric vehicle driven by four in-wheel PMSMs is used to verify the optimal acceleration control strategy, as shown in Fig. 13.

Fig. 11.

Motor torque-speed curve in different kv

Table 4.

Acceleration properties in different kv

Weighting factor kv 0 (Optimum) 0.5 2 5

Acceleration time ta/s 19.27 15.46 12.89 11.17

Acceleration distance sa/m 101.79 96.83 90.52 84.91

Battery output energy Ebat/kJ 93.87 94.23 95.54 97.83

The acceleration torques solved by DP algorithm under different acceleration time constraint are shown in Fig. 12. The acceleration distance and the battery output energy are shown in Table 5.

Fig. 12.

Motor torque-speed curve of different acceleration time in optimal energy acceleration

The energy consumption of EV can be divided by three

Fig. 13.

Four-wheel drive electric vehicle

In order to verify the conclusions, the motor and inverter efficiency models shall be verified first. Fig. 14 shows the motor and inverter efficiency calculated by analytical model and experiment data. The two results are almost the same. The motor and inverter efficiency model calculated by analytical model can reflect the actual motor and inverter efficiency.

Fig. 14.

Motor calculated and measured efficiency

Fig. 15 shows the q-axis current, phase current, motor speed, battery output current and battery terminal voltage

CHINESE JOURNAL OF MECHANICAL ENGINEERING of a single PMSM in different constant accelerator pedal opening controls. The electromagnetic torque is proportional to the q-axis current form Eq. (5). As the pedal opening increases, the required torque and q-axis current also increase. Meanwhile, the battery output current

Fig. 15.

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increases and the acceleration time decreases. The electromagnetic torque in constant power area is small in the constant accelerator pedal opening controls, which consumes much acceleration time and battery energy.

Acceleration control results in different constant accelerator pedal opening control

Fig. 16 shows the q-axis current, phase current, motor speed, battery output current and battery terminal voltage of a single PMSM in different kv. The acceleration torques solved by analytical model in different kv minimize the battery output energy in the entire acceleration process. When kv equals 0, the optimal acceleration control with

minimum battery output energy can be achieved. Larger kv can shorten the acceleration time, but consumes more battery energy. Fig. 17 shows the experiment results solved by DP algorithm. The optimal energy consumption under different acceleration time can be achieved.

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The energy consumptions in the above three acceleration control strategies are compared in Fig. 18. In the same acceleration time, the battery output energy in the constant accelerator pedal opening control is significantly greater than that in the acceleration control calculated by analytical model and DP algorithm. Although there are some differences in the torques calculated by analytical algorithm and DP solution, as shown in Fig. 11 and Fig. 12, the energy consumptions in both algorithms are almost the

Fig. 16.

same due to the high efficiency of PMSM. The acceleration torque can be easily calculated by the analytical model, and it can be used for real-time control. However, it is difficult to control the acceleration time accurately in analytical algorithm. The global optimal acceleration control can be achieved by DP algorithm with precise acceleration time constraint, but the DP algorithm consumes a large amount of computation time.

Acceleration control results in different kv

6 Conclusions (1) The electric drive system(including the PMSM, the inverter and the battery) is modeled based on the electromagnetic torque of PMSM under field oriented

control. The efficiency model of the electric drive system can reflect the measured efficiency to favor over a detailed consumption map. (2) Motivated by the benefits of torque-based control in electric vehicle, an optimal acceleration control that

CHINESE JOURNAL OF MECHANICAL ENGINEERING incorporates torque control has been proposed. The optimal acceleration control solved by analytical model can be implemented to minimize the battery energy consumption in unit kinetic energy increment, which is simple and

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effective. For the case of time-bound, in order to achieve a robust, less computationally expensive, a penalty function is introduced to change the vehicle speed tracking performance.

Fig. 17. Acceleration control results solved by DP

Fig. 18.

Battery consumed energy in different acceleration control strategies

(3) The global optimal acceleration control can be achieved by dynamic programming (DP) algorithm with precise acceleration time constraint, but the DP algorithm needs to consume a large amount of computation time. The DP solutions for different acceleration time are presented to assess the performance of the above analytical algorithm. (4) Experiment results show that the energy consumptions in analytical algorithm and DP solution are almost the same and the required battery energy with optimal acceleration control strategy is greatly reduced compared with that in the constant pedal opening condition.

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Biographical notes LU Dongbin, born in 1982, is currently a PhD candidate at State Key Laboratory of Automotive Safety and Energy, Tsinghua University, China. He received his bachelor degree in electrical engineering from Shandong University, China, in 2006 and MS degree in electrical and electronic engineering from Huazhong University of Science and Technology, China, in 2008. His research interests are modeling, design and control of the powertrain system for electric vehicle. Tel: +86-10-62785706; E-mail: hustludb@126.com OUYANG Minggao, born in 1958, is currently a professor at Department of Automotive Engineering, Tsinghua University, China. He received his PhD degree in mechanical engineering from the Technical University of Denmark, Lyngby, in 1993. His research interests include new energy vehicles, automotive powertrains, engine control systems, and transportation energy strategy and policy. E-mail: ouymg@tsinghua.edu.cn