Analysis and Design of an Optimal Energy Management and Control System for Hybrid Electric Vehicles

Yuan Zhu, Yaobin Chen, and Quanshi Chen

Abstract This paper presents a preliminary design and analysis of an optimal energy management and control system for a parallel hybrid electric vehicle using hybrid dynamic control system theory and design tools. The vehicle longitudinal dynamics is analyzed. The practical operation modes of the hybrid electric vehicle are introduced with regard to the given power train configuration. In order to synthesize the vehicle continuous dynamics and the discrete transition between the vehicle operation modes, the hybrid dynamical system theory is applied to reformulate such a complex dynamical system in which the interaction of discrete and continuous dynamics are involved. A dynamic programming-based method is developed to determine the optimal power split between both sources of energy. Computer simulation results are presented and demonstrate the effectiveness of the proposed design and applicability and practicality of the design in real-time implementation. Copyright 2002 EVS19 Key words: Energy management, optimization, dynamic programming 1. Introduction Hybrid electric vehicles have attracted tremendous attention as a commercially viable alternative to either traditional vehicles or electric vehicles. Their acceptance is primarily due to their multiple power sources that provide flexibility in optimal power distribution, while satisfying the performance requirements. The effective operations of hybrid vehicles depend largely upon the sophisticated design of vehicle system controller (VSC) with optimal energy management strategy that commands each subsystem to its best for the overall system efficiency. Due to the complexity of the hybrid electric vehicles, the design of the energy management strategy (EMS) poses a considerable challenge to engineers. This paper presents a preliminary design and analysis of an optimal energy management and control system for parallel hybrid electric vehicles using hybrid dynamic control system theory and design tools. Hybrid electric vehicles (HEV) are generally described as vehicles with a main power unit, which converts fuel energy to electric and/or mechanical energy, and a bi-directional energy storage system. The hybrid electric vehicles can potentially improve fuel economy, emissions over the conventional CVT vehicles. This is because: 1) the engine size can be reduced with the ‘same’ vehicle performance due to the dual power sources, 2) the engine operation can be better optimized since the engine can be stopped if operational conditions are not favorable to the fuel economy and emissions due to the dual power sources, 3) the kinetic energy during braking can be captured and stored in the battery through regenerative braking. In order to achieve this goal, it is very essential to have an energy management strategy that determines the rates at which energy is released form fuel in the main power unit and energy drawn from the energy storage system to meet the following two constraints. The first constraint is that the demand for motive power must always be satisfied up to a fixed limit, and the second constraint is that the state of charge (SOC) of the energy storage system is maintained within a preferred range. Within these constraints, it is highly desirable that the strategy obtains the maximum fuel economy. The objective of the energy management strategy (real time optimizer) for HEVs are to control and operate the powertrain system with the specified operational conditions and constraints such that the fuel consumption and emission can be minimized, while the specified performance of vehicle

acceleration, deceleration and NVH (noise, vibration, harshness) can be achieved and maintained, and the battery life can be maximized. In this work, a power split power train system configuration [1][2] of a hybrid electric vehicle (HEV) is considered, as shown in Figure 1. In this configuration, there are two power sources that are connected to the driveline: 1) a combination of engine and generator subsystems using a planetary gear set to connect to each other, and 2) the electric motor subsystem. The battery subsystem is the energy storage system for the generator and the motor.

ring

sun

ring

brakebrake

Motor

planetarplanetaryengine

&

o.w.co.w.c

N3N3

N1N1

NN5

N2N2

NN4

high voltage bushigh voltage bus

TMU/Transaxle

battery & controller

Battery & Controller

Gen

Figure 1: Power Split Power Train System Configuration The planetary gear set can also be viewed as a power split device that splits the engine output power to the driveline and to the generator. From the viewpoint of the electrical path (series hybrid), the portion of the power from the engine to the generator can be converted into the electric energy. Then the electric motor draws the electric power provided by the battery and the generator to propel the vehicle. From the viewpoint of the mechanical path (parallel hybrid), another portion of the power from the engine to the carrier to the ring gear to counter shaft can be used to drive the vehicle without energy transformation. The two power paths can provide propulsion to the vehicle simultaneously and independently. As described above, by controlling the generator appropriately, the planetary gear set can serve as a pseudo continuous variable transmission (pseudo-CVT) between the engine and the ring gear that is eventually connecting to the driven wheels. 2. Analysis of Vehicle Dynamics The fundamental law from which vehicle dynamics analyses begin is Newton's Second Law. When this law applies to rotational systems, the Newton's Second Law is described as follows: the sum of the torques acting on a body about a given axis is equal to the production of its rotational moment of inertia and the rotational acceleration about that axis. As depicted in Figure 1, included in the transaxle is a planetary gear device, which makes the engine, the generator and motor mechanically connected and delivers torque to the wheels through a differential and half shafts. Assuming 100% mechanical efficiency, rigid gears and no gear lash or dead band, then a wet of dynamic equations based on the planetary gear model can be obtained as follows.

Starting with the engine, the engine torque delivered through the one-way clutch and the carrier to the sun gear and ring gear can be described as

ringsuneeeJ τττω −−=⋅ & (1)

where Je is the moment of inertia of the engine and the carrier, ωe the engine rotational speed, eτ the engine torque at a given speed, sunτ the torque at the sun gear, and ringτ the torque at the ring gear. The torque delivered at the ring gear is used to drive the driveshaft, axle shaft and the tires. The motor torque can provide auxiliary torque when the vehicle needs a high torque to accelerate the vehicle. Hence the dynamic equation for the ring gear can be easily obtained below.

tireMringrr NNNN

NNJ τττω

25

34

1

3 −+=⋅ & (2)

where Jr is the moment of inertia of the ring gear, N1~N5 gears, the motor, axle shafts and the tires (as seen from the ring gear side), ωr the ring gear angular speed, ringτ the torque at the ring gear, Mτ the

motor torque at a given speed, tireτ the torque generated by the tractive force, and N1~N5 are the numbers of the gear teeth. Similarly, the torque delivered to the sun gear can be used to drive the generator. The expression for this is:

gsunggJ ττω +=⋅ & (3) where Jg is the moment of inertia of the sun gear and the generator, ωg the generator rotational speed,

sunτ the torque at the sun gear, and gτ is the generator torque. A hybrid electric vehicle is made up of many components distributed within its exterior envelope. Yet, for many of the more elementary analyses applied to it, all components move together. Thus it can be represented as one lumped mass. When Newton's Second Law is applied to the lumped mass, one obtains

ceresistire FR

vm tan1

−=⋅ τ& (4)

where m is the mass of the vehicle, v is the vehicle speed, R the rolling radius of the tires, tireτ the torque generated by the tractive force, and is the total road load, including the rolling resistance forces and aerodynamic drag force on the vehicle. Typically,

,

ceresisF tan

2tan 5.0 AvCmgfF Drceresis λ+=

where fr is the rolling resistance coefficient, λ the air density, CD the aerodynamic drag coefficient, A is the frontal area of the vehicle, and g is the acceleration of gravity Both the generator and the motor in this power train system can work in a generative mode or motoring mode. Suppose the generator is working in a generative mode and the motor is in a motoring mode, the following equations can be obtained.

memMMemgggbatP 2_2_ /ητωητω ⋅+⋅= (5)

>−

<−

=

0/

0

arg

arg

battbatt

edischbat

battbatt

echbat

PC

UP

PC

UP

COSη

η

& (6)

where is the battery power desired, batP gτ is the generator torque, ωg the generator rotational speed,

e2mg _η the efficiency of the generator when it converts the mechanical energy into electric energy, ωM

is the motor rotational speed, Mτ the motor torque at a given speed, mem 2_η the efficiency of the motor

when it converts the electric energy into mechanical energy, SOC the battery state of charge, P the battery power desired, C the battery capacity,

bat

batt ech argη the efficiency when charging the battery,

eargdischη the efficiency when discharging the battery, and U is the battery terminal voltage. When the working status changes, the equation (5) should change in accordance with the working status. Once the planetary is designed, the relationship among the rotational speeds of the carrier gear, the sun gear and the ring gear is determined as equation (7).

reg ωρ

ωρ

ρω 11−

+= (7)

where ωg is the generator rotational speed, ωe the engine rotational speed, ωr the ring gear angular

speed, andring

sun

NN

=ρ is the planetary gear ratio.

Assuming the pinion moment of inertia is negligible when compared with those of Je, Jg and Jr, the torque relationship between the ring gear and the sun gear can be approximated by

ρττ ⋅= ringsun (8)

where sunτ is the torque at the sun gear, ringτ the torque at the ring gear, and ρ is the planetary gear ratio. Now the rotational speed of the ring gear and the motor are related to that of the wheels by the gear ratios. Assuming the tires are not slipping, then equation (9) can be obtained.

52

41

25

34

NNNN

NNNN

Rv

Mr ωω == (9)

where v is the vehicle speed, R the rolling radius of the tires, ωr the ring gear angular speed, and ωM is the motor rotational speed. Finally, combining equations (1)-(9) yields the following state equation.

)10(

)1(

)1()1(

])1[()1(

)1()1(

2

25

34

22

tan25

3422

22

1

3

1

32

mNN

RNNJJ

JJJJJJ

FNN

RNNJJ

J

JJNN

JJ

JNN

JJJ

rr

rggere

ceresiseg

g

M

g

e

egeg

grgr

r

e

+=′

′+++′=

++

+

−

++−+

+′++′=

ρρα

ρρρ

τττ

ρρρρ

ρρρρ

ωω

α&

&

where τe, τg and τM are input continuous variables, ωe, and ωr are continuous state variables. It should be pointed out that we have made extensive use of the “quasi-static” assumption that substantially

simplifies the modeling of the engine, the motor and generator. This “quasi-static” assumption assumes that the internal combustion engine fuel consumption at any given time instant is static function of engine speed and engine torque. 3. Analysis of the Different Vehicle Operation Modes It should be pointed out that the vehicle dynamic analysis above only considered one of the practical operation modes. There exist some other operation modes we are interested in. In fact, there are many factors that make the transition possible, such as one-way clutch between the engine and the carrier, the vehicle braking, the generator braking, the engine shutoff and so on. All of these make the hybrid electric vehicle system exhibit simultaneously several kinds of dynamic behavior, such as continuous-time dynamics, discrete-time dynamics, jump phenomena, switching and logic commands, discrete events and the like. For the purpose of optimal energy management system design, the vehicle operation modes are described below in terms of the energy management system. Standstill mode (Vehicle idle charging) When the vehicle is stopped at a standstill, the engine usually stops. However, if the engine, the battery and/or other components need to be warmed up, or the battery is in low state of charge (SOC), the engine is on and working near the idle speed to satisfy these power request. In most conditions, especially for the purpose of dynamic optimization, we should simplify the system control and just let the engine be off. Vehicle creeping Vehicle creeping mode is when the vehicle speed is under 20km/h. The one-way clutch locks to prevent the engine from rotating reversely. In this mode, the vehicle dynamics is different when compared to the equation (10), as shown in equation (11) below

ceresisg

Mrg

r FNN

RNNNNJ

J tan25

34

1

32

' −−=⋅

+

ρτ

τωρ

& (11)

where, τg and τM are input continuous variables (excluding τe because the engine is locked by the one-way clutch, so the engine torque is not available for the purpose of the control), and ωr is continuous state variable (excluding ωg because the generator speed is proportional to the ring gear speed). In this mode, the strategy is proposed as follows:

• If the battery SOC is not below the lower limit and the power demand is not large, the vehicle is driven by the motor.

• If the battery SOC is above the lower limit, the power demand is very large, that is, the pedal depression is large during acceleration; the vehicle is driven by the generator and the motor.

• If the battery SOC is below the lower limit, the engine is on for charging the battery. In fact, the vehicle will work under the power split mode in this scenario.

Power split mode (vehicle hybrid drive) In section 2, the dynamics of this mode has been analyzed. The engine, the generator and the motor will cooperate to achieve the maximum fuel economy. In this mode, there are more possibility and flexibility for the vehicle system controller to improve the fuel economy with regard to other vehicle modes. Depending on the battery SOC and the power demand, the power train can be operated in three different operations: positive split, parallel, and negative split.

1) Positive split The generator should work as the generative mode. If the battery requires (when the battery SOC is below the lower limit) to be charged and the charging is allowed (when the battery SOC is not above the upper limit) based on the power demand, the control strategy is to operate the engine at a higher power than the power demand to achieve higher engine efficiency.

2) Parallel

The generator brake is engaged and the vehicle is driven by the engine and the motor. The generator does not rotate during this time. Generally speaking, in this mode the vehicle speed is not too low because the engine speed becomes very low according to equation (7). In this case, the vehicle dynamics would be changed to the following.

ceresisMe

re

r FNN

RNNNNJJ tan

25

34

1

32

'

1)1(−+

+=⋅

+

+ τρ

τω

ρ& (12)

where τe and τM are input continuous variables (excluding τg because the generator is locked by the generator brake); ωr is continuous state variable (excluding ωe because the engine speed is proportional to the ring gear speed)

3) Negative split When the battery SOC becomes too high, the generator operates as the motoring mode. With the generator rotating in the negative direction, the engine speed becomes much lower. This will benefit if the vehicle speed is very high.

Regeneration Mode When the brake pedal is applied, the strategy first reduces the engine output for a while. If the brake pedal is depressed consistently, the fuel supply is cut off to stop the engine. To protect the batteries, the regeneration energy is limited according to their charge acceptability. Furthermore, there exist some other vehicle operation modes though they are relatively less important for the energy management system design. For example, the engine start control can be a vehicle operation mode. In this mode, the generator starts up the engine in a short period of time. From the point of view of hybrid control, we can consider the engine start-up as a discrete event with the instantaneous state changes, i.e., the engine speed is changed from zero to the idle speed during a transient period of time. To deal with this complex system, we are inclined to combine the continuous controllers with the discrete controllers using hybrid dynamical system theory. Therefore we will synthesize the control system and reformulate this problem in the framework of hybrid dynamical system theory. 4. Structure of Vehicle Control System A high-level, simplified structure of the vehicle control system is shown in Figure 2. The vehicle control system consists of four major subsystems - the driver, the driver evaluator, the vehicle system control and the transaxle management unit. 1) Driver The Driver model is used to simulate the human behaviors. In this model there is a PID controller that determines the braking pedal position and accelerator pedal position. 2) Driver Evaluator

The Driver Evaluator model is used to translate the driver behaviors, such as the pedal positions, into the desired wheel torque for the vehicle. The desired driver wheel torque value comes from a torque lookup table based on the accelerator pedal position, the brake pedal position and the vehicle speed. 3) Vehicle System Control A finite state machine in the Vehicle System Control block determines the vehicle operating mode. Meanwhile, the engine ON/OFF status, the generator brake engagement control and so on are determined upon the determination of the vehicle operating mode. The power based strategy block, which is one part of the Vehicle System Control block, is used to distribute the driver power desired over the two power sources, the engine and the battery. In terms of optimization, the power based control strategy is the key to improve the vehicle fuel economy. 4) Transaxle Management Unit The transaxle configuration of the hybrid electric vehicles is different from that of the conventional vehicles. Compared to the conventional vehicle, the hybrid electric vehicle has one motor and one generator besides the engine. The coordinated control of these three machines presents a big challenge to the vehicle system controller. We need to select the corresponding strategy to implement the control of each component because of their different characteristics and mechanical connections. For example, the objective of the generator speed control is to adjust the engine working speed. So a closed loop PID controller is adopted in the generator speed control. If the generator speed, which is calculated from the engine speed desired and the vehicle speed, is less than 200 rpm, the generator brake is engaged to prevent the generator from working at the low efficiency area. As for the engine controller, the throttle position is determined by the engine power desired based on the engine BSFC (best specific fuel consumption) curve. The motor control is a simple open loop torque control.

DriverEvaluator

VehicleSystemControl(EMS)

TransaxleManagement

Unit

Driver

acc. pedalbrake pedal

driver torque desiredidle flag

engine speedengine torque

Gen speedengine torque desiredengine ON/OFF requesttotal torque desiredengine speed desired

battery SOCbattery system status

vehicle speed

Figure 2: the simplified vehicle control structure

5. HEV System Control Model Using Hybrid Dynamical System Theory In the literature [5], the term hybrid dynamical system is used to describe systems that incorporate both continuous and discrete dynamics. The area of hybrid dynamical systems is a new, fascinating discipline bridging control engineering, theoretical computer science and applied mathematics. However, in the area of hybrid dynamical systems, the main problem is the lack of formal mathematical tools for analysis and design of such systems. For convenience of discussions, the following notations and definitions are given with regard to hybrid dynamical systems [5]. Definition 5.1: A hybrid dynamical system H is a collection H=(Q,X,V,Init,f,Inv,E,R,Φ), where Q is a set of discrete variables and Q is countable; X is a set of continuous variables;

V is a finite collection of input variables and V=VDUVc, where VD contains discrete and Vc contains continuous variables;

XQInit ×⊆ is a set of initial states; nVXQf ℜ→××: is an input dependent vector field;

VXQInv ×→ 2: assigns to each q∈Q an input dependent invariant set; QQE ×⊂ is collection of discrete transition;

XVXER 2: →×× assigns to each e=(q, q') ∈E, x∈X and v∈V a reset relation; VXQ 2: →×φ assigns to each state a set of admissible inputs.

For the hybrid vehicle system under the consideration, the dynamical behavior can be described in the context of hybrid dynamical systems according to Definition 5.1 as follows. The dynamical behavior of the hybrid vehicle system described in the previous sections can be described by a hybrid dynamical system with eH

eH =(Q,X,V,Init,f,Inv,E,R,Φ) (13) where Q is a set of countable discrete state variables with q={q1, q2}∈Q, q1∈{Engine_On (1), Engine_Off (0)}, q2∈{ Vehicle_braking_on (1), Vehicle_brake_off (0)}; X is a set of continuous state variables defined by x={ωe, ωr, ωg}∈X. The state continuous variables may change subject to the different vehicle operating modes;; V is a finite collection of input variables and V=VDUVc. The continuous inputs are defined as vc={τe, τg ,τM}∈VC and the discrete inputs as vd∈{Generator_brake_engaged(1), Generator_brake_disengaged(0)} =VD;

XQInit ×⊆ is a set of initial states, Init={q0, x0}; nVXQf ℜ→××: is the input dependent vector field depending on the vehicle operating modes

which are given three different vehicle dynamics equations (10,11,12) corresponding to the different vehicle operating modes;

VXQInv ×→ 2: assigns to each q∈Q an input dependent invariant set given by

====

=1000

)( 1

dg

e

vifqif

qInvωω

QQE ×⊂ is collection of discrete transition. Because the discrete state is a finite number of values it is very convenient to represent the discrete transitions by a finite graph. Each node in the graph represents a discrete state value, q∈Q. The discrete transitions are illustrated as Figure 3. A run should be able to continue from every state, either by a discrete jump or by flowing along the vector field. If so, this hybrid dynamical system is non-blocking.

XVXER 2: →×× assigns to each e=(q, q') ∈E, x∈X and v∈V a reset relation given by

→==

→==→

)10(0

)10(_),(

1

dvg

qe

eespeedIdle

xeRω

ω

VXQ 2: →×φ assigns to each state a set of admissible inputs. In each vehicle operating mode, the system input and state variables are subjected to constraints due to their physical limits and maximum operating capabilities. Hence it is necessary to impose certain inequality constraints on the state and control variables such as the engine speed ( eω ), the battery state of charge (SOC), the battery power ( ), the motor torque (batP Mτ ), the generator torque ( gτ ), the engine torque ( eτ ), the generator speed

( gω ), the motor speed ( Mω ), and etc.

Figure 3: Transition between the vehicle operating modes

6. Optimal EMS Design Using Dynamic Programming Based on the vehicle control system model described above, the design of optimal energy management control strategy can be formulated as an optimal control problem for a hybrid dynamical system with constraints on both control and state variables. It is, however, in general very difficult to solve such an optimization problem, if not impossible. It is known that the dynamic programming method has been proven very effective in tackling many complex dynamic optimization problems [10]. Two steps are usually followed when using the DP method. First the quantization and interpolation on the state and control variables are performed to obtain the optimal control solution. Then the problem is formulated as multi-stage decision problem, where the time variable is used to order the sequence according to the Bellman's principle of optimality. It is worth mentioning that the principle of optimality has been widely used in many application problems, such as a simple optimal path problem, job allocation problem, linear optimal control problem and so on. Mathematically, the principle of optimality for optimal decision-making at the kth step can be expressed as

))]1(())(),(([min))(( *1)(

* ++= + kxJkukxLkxJ kkuk , 0≤ k < N-1 (14)

Where L(.,.) is the objective functional (cost function) to be optimized. The recursive equation (14) is solved backwards from step N-1 to 0 in order to find the optimal control policy. Each of the minimizations is performed subject to the dynamic and static constraints and given driving cycles. Selecting a good and effective cost function is very important to the optimization problem. In this work, the following cost function was chosen based on the overall objectives of the problem

001

/

arg

arg

1

0

2

max_

<>

=

+⋅+= ∑

−

=

batt

batt

ech

edisch

N

ke

e

batteee

PP

dtPJ

ηηη

ωδηηητω &

(15)

This cost function represents a control strategy that determines the optimal split of the power sources (the engine and the battery) such that the total energy (fuel and battery energy) consumption is minimized while satisfying the motive power demand and vehicle driving performance. The cost function contains three components: 1) The first term on the right-hand side is the fuel consumption by the engine. This term only

represents the fuel consumption assuming the engine is rotating in a steady state. 2) The second term on the right-hand side is the battery energy consumption. In this term the battery

power is divided by the maximum engine efficiency in order to transform the electric energy consumption into the equivalent fuel consumption. The sum of the first two terms means the equivalent energy consumption in a unit of time, which is used to measure the effective fuel economy.

3) The third term on the right-hand side is used to compensate the extra fuel consumption for the engine acceleration when taking the engine dynamics into account. From the point of view of optimal control, this term is used as anti-jerk function. Here δ is the weight for the purpose of anti-jerk.

7. Simulation Results Numerous computer simulations were performed for several typical drive cycles to validate the proposed analysis and design of the optimal EMS control strategy. Some numerical results are presented and discussed for the EPA city cycle. Figure 4 presents the operating points of the ICE when the initial battery SOC is chosen to be 95%. The battery tends to discharge when the SOC is higher than 0.75. Then the battery SOC changes between 0.65 and 0.75. Figure 5 presents the operating points of the ICE when the initial battery SOC is chosen to be 45%. The battery tends to charge when the SOC is lower than 0.65. Then the battery SOC maintains between 0.65 and 0.75. The optimized strategy tends to keep the battery SOC within the range of 65% ~ 75%. On one hand, this leaves enough capacity to handle an extended period of the battery discharge (such as during a longtime acceleration) and enough “headroom” to absorb a long period of charging (such as during a long downhill). On the other hand, from the control point of view, the battery SOC is maintained near a balance point to ensure the system stability. Therefore, we have confidence that the optimization results are reliable.

Figure 4: Simulation results of EPA city cycle with initial SOC of 95%.

Figure 5: Simulation results of EPA city cycle with initial SOC of 45%

8. Conclusion A preliminary design and analysis of an optimal energy management and control system for a parallel hybrid electric vehicle was presented. By considering various possible operating modes, the problem was cast as an optimal control problem for a hybrid dynamical system using hybrid dynamic control system theory. A dynamic programming-based method was used to obtain the numerical solution. Computer simulation results illustrate the effectiveness of the proposed design. It is noted that the proposed optimization can be only implemented off-line. For real-time implementation of the proposed optimal strategy, an on-line scheme needs to be developed. One of the possible solutions would be to develop a fuzzy logic energy management system controller that approximates the off-line optimal strategy by using evolutionary computational approaches. References [1] Kozo Yamaguchi and Yoshinori Miyaishi, "Dual System – Newly Developed Hybrid System" EVS 13,

Japan, 1996 [2] K. Yamaguchi, S. Moroto, K. Kobayashi, M. Kawamoto and Y. Miyaishi "Development of a New Hybrid

System – Dual System," SAE Paper 960231 1996 [3] Chan-Chiao Lin, Jun-Mo Kang, J.W. Grizzle, and Huei Peng "Energy Management Strategy for a

Parallel Hybrid Electric Truck", American Control Conference, Washington D.C., 2001 [4] Alexey S. Matveev Alexy S. Matveev Andrey V. Savkin "Qualitative Theory of Hybrid Dynamical

Systems" , Birkhauser Boston, Feb. 2000 [5] John Lygeros "Hierarchical Hybrid Control of Large Scale Systems", PhD thesis, University of California

at Berkeley, 1996 [6] A.Brahma, Y.Guezennec and G. Rizzoni, “Optimal Energy Management in Series Hybrid Electric

Vehicles”, Proc. ACC, Arlington, VA, June 25-27, 2001 [7] Chan-Chiao Lin, Jun-Mo Kang, J.W. Grizzle and Huei Peng, “Energy Management Strategy for a Parallel

Hybrid Electric Truck”, Proc. ACC, Arlington, VA, June 25-27, 2001 [8] M. Salman, Niels J. Schouten, and Naim A. Kheir “Control Strategies for Parallel Hybrid Vehicles”, Proc.

ACC, Chicago, Illinois, June 2000 [9] Arthur E. Bryson "Dynamic Optimization" Addison Wesley Longman, Inc., California 1999 [10] R.E. Bellman, Dynamic programming, Princeton University Press, 1957 Affiliation

Yuan Zhu Department of Automotive Engineering, Tsinghua University, Beijing , 100084, China Tel: +86-10-6279-5045 Fax: +86-10-6278-6907 E-mail: zhuyuan98@mails.tsinghua.edu.cn Yaobin Chen Department of Electrical and Computer Engineering, Purdue University at Indianapolis, 723 West Michigan Street, SL 160, Indianapolis, Indiana 46202, U.S.A. Tel: 317-274-4032 Fax: 317-274-4493 E-mail: ychen@iupui.edu Dr. Chen received his B.S. degree from Southeast University (China) in 1982 and his M.S. and Ph.D. degrees from Rensselaer Polytechnic Institute (Troy, New York, U.S.A.) in 1986 and 1988, respectively all in electrical engineering. He is currently Professor of electrical and computer engineering and Director of Advanced Vehicle Technology Institute.

Prof. Quanshi Chen Department of Automotive Engineering, Tsinghua University, Beijing, China Tel: (86) 10-6278-5704 E-mail: chenqs@mails.tsinghua.edu.cn