Hybrid Vehicle (City Bus) Optimal Power Management for Fuel Economy Benchmarking

doi:10.4236/lce.2013.41005

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Low Carbon Economy, 2013, 4, 45-50

http://dx.doi.org/10.4236/lce.2013.41005 Published Online March 2013 (http://www.scirp.org/journal/lce) 45

Hybrid Vehicle (City Bus) Optimal Power Management for

Fuel Economy Benchmarking

Boukehili Adel, Youtong Zhang, Chengqun Ni, Jutang Wei

Low Emission Vehicle Research Laboratory, Beijing Institute of Technology, Beijing, China.

Email: [email protected]

Received January 5th, 2013; revised February 12th, 2013; accepted Febru ary 20th, 2013

ABSTRACT

In this paper a global optimization method (dynamic programming) is used to find the optimal power management in

hybrid electric city bus for the obj ective to reduce the fuel consu mption. Knowing that when using a global op timization

method the results cannot be used in real-time con trol; because we need to know the entire vehicle speed in advance to

perform the optimization, but in spite of that this method is very useful to make a benchmark for hybrid electric city

buses fuel economy and to judge the effectiveness and improve real-time control strategies. Finally results of optimal

power management are shown and discussed.

Keywords: Hybrid Vehicle; Power Management; Fuel Economy; Optimization; Dynamic Programming

1. Introduction

The study of ground vehicles has taken a tremendous in-

terest in recent years due to the increased price of fuel

and emission stringent laws. In this way, Hybrid Electric

Vehicles (HEV) seems to be the most promising short-

term solution and is under enthusiastic development by

many automotive companies. An HEV adds an electric

motor to the conven tional powertrain, which helps to im-

prove fuel economy by engine downsizing, load leveling,

and regenerative braking. A downsized engine has better

fuel efficiency and smaller heat loss. The reduced engine

power is compensated by the electric motor. Load level-

ing can be achieved by adding the electric motor, which

enables the engine to operate more efficiently, indepen-

dent from the road load. Regenerative braking allows the

electric machine to capture part of the vehicle kinetic

energy.

Power management strategies for parallel HEVs can

be classified into three categories. The first type uses he u-

ristic control techniques such as control rules [1], fuzzy

logic [2,3] or neural networks [4] for estimation and con-

trol algorithm development. The second approach is b as ed

on static optimization methods [5-7]. Generally, electric

power is translated into an equivalent amount of fuel rate

in order to calculate the overall fuel cost. The optimiza-

tion scheme and figures out the proper split between the

two energy sources using steady-state efficiency maps.

The third type of HEV control algrithms considers the

dynamic nature of the system when performing the opti-

mization [8,9]. Furthermore, the optimization is with re-

spect to a time horizon, rather than for an instant in time.

In general, power split algorithms resulting from dyna-

mic optimization approaches are more accurate, but are

computationally more intensive.

In this paper we use the dynamic programming me-

thod to solve the problem of optimal power management

in a HEV, for that reason the reference speed should be

known in advance to solve the problem; thus we use a

simple reference speed (not a normalized drive cycle),

this reference speed contains linear acceleration and de-

celeration and constant speed in order to facilitate the in-

terpretation.

2. System Specification

2.1. System Structure and Modeling

The hybrid vehicle structure is a parallel single shaft to-

pology, which utilizes a PMSM motor placed before the

transmission and coupled with the diesel engine via clu-

tch. The engine, motor and battery are modeled using ex-

perimental data (efficiency maps for engine and motor)

and an equivalent electric circuit for the battery with ex-

perimental data.

2.2. Problem Formulation

In this paper we seek to find the optimal power split be-

tween engine and electric motor in HEV in order to

achieve minimum fuel consumption, this is a problem of

optimal control; for that reason we need to define the cri-

terion of optimization the constraints and the state equa-

Hybrid Vehicle (City Bus) Optimal Power Management for Fuel Economy Benchmarking

tion.

2.2.1. Criterion

The criterion of optimization also known as the cost

function or the objective function is the function that we

seek to minimize, which is the fuel consumption in this

case

 



Minimize ,



CTeiwei Ts



 (1)

2.2.2. Cons tr a i nts

In a parallel single shaft hybrid powertrain topology, the

sum of engine and motor torque must be instantaneously

equal to the torque demand described in the engine shaft.

and the engine and motor speeds are proportional to the

wheel speed by the final drive and gearbox ratios. Also

we must constrain the engine and motor torque to make

sure that they do not exceed their maximum torques and

finally constrain the battery state of charge to remain be-

tween two limits denoted as SOCmax and SOCmin. Con-

straining the battery SOC in this way helps to prolong its

life, the constraints are described by the equations below:













TdiTe iTm i





















ifitiw iwe iwmi 





,min ,maxwewe iwe





,min ,maxTeTe iTe





,min ,maxwmwm iwm





,min ,maxTmTm iTm





SOC1 SOCSOC2i

2.2.3. St ate Equation

The state equation gives the variation of the energy sto red

in the battery (X) as a function of the electric power fur-

nished by this battery. In discrete time this variation is

described by





















iXiPewmiTmi Ts (2)

2.2.4. Limit Condition

In order to be able to perform the optimization the zone

of acceptable solution must be closed, which leads to

constraining the battery SOC to converge to a known

limit, this limit is described by SOCfinal, in our article a

limit condition u sed is described by:

SOCfinal SOCinitial 80% (3)

3. Principal of the Method of Dynamic

Programming

Dynamic Programming (DP) is a powerful mathematical

technique developed to solve dynamic optimization pro-

blems. The advantage is that it can easily handle the con-

straints and nonlinearity of the problem while obtaining a

globally optimal solution. The DP technique is based on

Bellman’s Principle of Optimality, which states that the

optimal policy can be obtained if we first solve a one

stage sub problem involving only the last stage and then

gradually extend to sub-problems involving the last two

stages, last three… etc. until the entire problem is solved

(backward method). In this manner, the overall dynamic

optimization problem can be decomposed into a sequen ce

of simpler minimization problems [10].

In HEV the sequence of choices represents the power

split between the internal combustion engine and the

el e c t r i c moto r a t s u c c e s s i v e t i me s t e p s . T h e ob j ec t i v e func-

tion can be fuel consumption, emissions, or any other de-

sign objective. The set of choices at each instant is de-

termined by considering the state of each powertrain com-

ponent and the total power requested by the driver. Given

the current vehicle speed and the driver’s demand (ac-

celerator position); the controller determines the total

power that should be delivered to the wheels. Then, using

maps of the components and feedback on their present

state, it also determines the maximum and minimum

power that each energy source can deliver. If the power

demand equals or exceeds the total available power from

both sources, there is no choice to be made: each of them

should be used at the maximum of its capabilities. Oth-

erwise, there are infinite combinations such that the sum

of the power from engine and motor equals the power

demand. In most algorithms, including dynamic program-

ming, instead of considering this continuum of solutions,

a discrete number is selected and evaluated. The number

of solution candidates that can be considered is a com-

promise between the computational capabilities and the

accuracy of the result: in fact, the minimum cost may not

exactly coincide with one of the selected points, but the

closer these are to each other, the better the approxima-

tion of the optimal solution. Once the grid of possible

power splits, or solution candidates, is created associat-

ing a cost to each of the solution candidates, the optimal

cost is calculated for each grid point, and stored in a ma-

trix of costs. When the entire cycle has been examined,

the path with the lowest total cost represents the optimal

solution (Figure 1).

3.1. Torque Demand Calculation

As we said before, this method requires the knowledge of

the whole reference speed in advance to precede the op-

timization; thus; after knowing the speed, we can calcu-

late the power demand and also the torque demand (since

both engine and motor run at the same speed) in each

sample time using the wheel speed and its derivatives as

shown below:

Energy of the power source (Engine and Motor)

Hybrid Vehicle (City Bus) Optimal Power Management for Fuel Economy Benchmarking 47

Figure 1. A grid with small number of discretization, the

elementary costs and the optimal path are shown.

d EsTd wet



Energy of rotation of different inertia plus energy of

translation of the vehicle (kinetic energy)

2222

1111

12 123

2222

EkEkJweJwtJwM V

Energy to overcome resistant forces

ddErFaerVtFrollVt



FaerCdA V









rollMCrCr V 

So after neglecting the energy lost by friction and

damping we get:

12Es =Ek+Ek+Er

That means:



222

111

d123

222

12d2

Td wetJweJwtJw

CAVVt

CrCrVVtM V



 















The derivation by time gives us the equation below



dd111

d12 3

dd222

1 2d

Td wetJweJwtJw

VCAVV

MCrCrVVt





 







 













After derivation and arrangement we get this last dif-

ferential equation



22 22

232

321

JMRifJitif Jt

Td it if

CrRwCdARwR MCr

it if



 























This last equation gives a relation between wheel an-

gular speed





w and torque demand:



.TdTe Tm

3.2. Creation of the Grid of Acceptable Solution

If we have the reference speed in advance, we can know

the wheel speed





wi at any sample time; this means

that we can know the torque demand











TdiTe iTm i

at any sample time, but the goal in this paper is to decide

how much is and in each sample time.

()Te i()Tm i

To make that choice we should define the grid of all

possible solution that satisfy the constraints discussed be-

fore, for that we suppose that and XX

are maximum

and minimum battery charge that can possibly be achi-

eved.











iXiPewmiTmiTs



 











1(),



iXiPewmiTmiTs



 

By the same logic and if we start from the last point

(backward) we can find the max i mu m b a t t e r y ener g y that

make the system converge to SOC limit described in (3).











biXbiPewmi TmiTs















biXbiPewmi TmiTs





It is obvious that the grid of possible solution is lim-

ited by:

minmax(,,1) XXXbX



maxmin( ,,2)

XXbX

After defining the zone of possible solution (Figure 2)

and using a sample time (Ts = 1 s) and a sample of bat-

tery energy (dx = 500 J) to make a mesh, knowing that

the number of samples of battery energy in a time

()ni

()ti iTs



 is described by the integer value plus one

Figure 2. Zone of possible solution over the reference speed.

Hybrid Vehicle (City Bus) Optimal Power Management for Fuel Economy Benchmarking

cussed in Equation (4).

of : ()ni











max min

iX i



 4. Results and Discussion

After optimization; the optimal torque split between the

engine and motor is finally found. Figures 3 and 4 show

the engine torque distribution and motor torque distribu-

tion all over the speed reference (Figure 5).

In our case , in order to make the fuel economy

easy to interpret. 0X

3.3. Optimal Trajectory Calculation We notice that the motor torque is engaged at start of

the vehicle which is very understandable since the engine

efficiency is too poor at low power and we believe that

the motor drag the engine with it until a better operating

point and then only the engine torque is used.

Suppose that is a point in the mesh and

is the cost to bring the system from the point

to the final point

(, (iTsXj

))

()

(,iTs (Xj(, (0)NTsX X

 

min ,

jji

Qi CijTs

 Also we notice that at low speed coasting, only motor

torque is used which is also understandable for the same

reason as before, since we know that when coasting at

low speed, the power demand is low; which makes the

engine not efficient; thus the use of motor is more eco no-

mic (in fuel consumption). But when coasting at higher

speed we notice that the engine torque is used.

where is the specific fuel consumption at the

sample time and for different engine torque corre-

sponding to different motor torque that makes the battery

energy to vary from Xmin to Xmax by a step of dx (knowing

that always we have .

(, )Cij ()ti

()()()TdiTe iTm i

Ce()iIn another hand let be the elementary cost to

jlAnother result we found is that when the bus is decal-

erating part of the power is stocked in the battery by re-

generative braking when the motor become generator and

acts as brake by using a negative torque; which makes

the battery SOC to increase (another profit of hybrid ve-

hicle over conventional vehicle) (Fi gure 6).

bring the battery from the point to the

point and using Bellman’s principle of

optimality we can finally find that

((1), ())iTsXl

(, (iTsXj

 





1min Ce

jjl

QiQi i





Finally the fuel consumption of the bus can be easily

integrated from the engine Map, because in Figure 3, we

have the engine torque and since we know the speed at

the wheel which is proportional to engine speed by the

transmission and final drive ratios, the fuel rate can be

This means that if we start from the last point and by

recurrence until the first point we can solve the problem

backwardly.

In this paper a program is made using MATLAB (M

file) to seek the optimal path based on the relation dis-

Figure 3. Engine torque distr ibution over the referenc e spe ed.

Hybrid Vehicle (City Bus) Optimal Power Management for Fuel Economy Benchmarking 49

Figure 4. Motor torque distribution over the speed reference.

Figure 5. Transmission ratio distribution over the reference speed.

Figure 6. Battery SOC distribution over the reference speed.

found using the engine Map; thus by time integration we

can find the fuel consumption. In other hand since we

have battery SOCfinal equal to SOCinitial (3), this means

that no fuel equivalent has to be transformed to electric

energy. After in tegration we found a fuel consumptio n of

(25.2 L/100 Km) which is an optimal value (benchmark)

for this bus that cannot be reached by a real-time control

strategy.

5. Conclusion

In this paper we used the dynamic programming method

to solve the problem of optimal power management in a

hybrid city bus; first we calculated the torque demand at

each sample time all over the speed reference; then we

specified a zone of acceptable solution (if a solution is

not inside this zone that means at least one of the con-

straints discussed before is not satisfied) and finally we

used a sweep method (dynamic programming) to sweep

all the possible torque split and choose the solution that

gives minimum fuel consumption (optimal solution).

REFERENCES

[1] B. Adel, Y. T. Zhang and S. Sun, “Parallel HEV Control-

ler Modeling for Power Management,” The 25th World

Hybrid Vehicle (City Bus) Optimal Power Management for Fuel Economy Benchmarking

Battery, Hybrid and Fuel Cell Electric Vehicle Sympo-

sium & Exhibition, Shenzhen, 5-9 November 2010.

[2] L. Majdi, A. Ghaffari and N. Fatehi, “Control Strategy in

Hybrid Electric Vehicle using Fuzzy Logic Controller,”

IEEE Proceedings of the International Conference on Ro-

botics and Biomimetics, Guilin, 19-23 December 2009.

[3] N. J. Schouten, M. A. Salman and N. A. Kheir, “Fuzzy

Logic Control for Parallel Hybrid Vehicles,” IEEE Trans-

actions on Control Systems Technology, Vol. 10, No. 3,

2002.

[4] J. Moreno, M. E. Ortuzar and J. W. Dixon, “Energy-Ma-

nagement System for a Hybrid Electric Vehicle, Using

Ultracapacitors and Neural Networks,” IEEE Transac-

tions on Industrial Electronics, Vol. 53, No. 2, 2006, pp.

614-623. doi:10.1109/TIE.2006.870880

[5] S. Delprat, J. Lauber, T. M. Guerra and J. Rima ux, “Con-

trol of a Parallel Hybrid Powertrain: Optimal Control,”

IEEE Transactions on Vehicular Technology, Vol. 53, No.

3, 2004, pp. 872-881. doi:10.1109/TVT.2004.827161

[6] A. Sciarretta, M. Back and L. Guzzella, “Optimal Control

of Parallel Hybrid Electric Vehicles,” IEEE Transactions

on Control Systems Technology, Vol. 12, No. 3, 2004, pp.

352-363. doi:10.1109/TCST.2004.824312

[7] L. V. Perez, G. R. Bossio, D. Moitre and G. O. Garcia,

“Optimization of Power Management in a Hybrid Electric

Vehicle Using Dynamic Programming,” Mathematics and

Computers in Simulation, Vol. 7 3, No. 1-4, 2006, pp. 244-

254. doi:10.1016/j.matcom.2006.06.016

[8] A. Brahma, Y. Guezennec and G. Rizzoni, “Dynamic Op-

timization of Mechanical Electrical Power Flow in Paral-

lel Hybrid Electric Vehicles,” Proceedings of the 5th In-

ternational Symposium Advanced Vehicle Control, Ann

Arbor, 22-24 August 2000.

[9] C.-C. Lin, J. Kang, J. W. Grizzle and H. Peng, “Energy

Management Str ategy for a Paral l el Hy brid Elect ri c T ruck , ”

Proceedings of American Control Conference, Arlington,

June 2001, pp. 2878-2883.

[10] C.-C. Lin, J.-M. Kang, J. W. Grizzle and H. Peng, “En-

ergy Management Strategy for a Parallel Hybrid Electric

Truck,” American Control Conference, 2001.

Nomenclature

Te: Diesel Engine Torque

Tm: PMSM Motor Torque

Td: Torque demand

we: Engine Speed (equal to motor speed)

wt: Speed after Transmission

w: Wheel Speed

V: Vehicle speed

M: Vehicle mass (11000 kg)

J1: Sum of inertia moving at the same speed as the en-

gine

J2: Sum of inertia moving at the same speed as the

transmission

J3: Sum of inertia moving at the same speed as the wheel

if, it: final drive and transmission ratios

Ts: sample time

Cr1, Cr2; Cd: rolling and aerodynamic coefficien

SOC: state of charge of the battery of (34 Ah, 42 V)

Pe: electric power

C(Te, we): engine fuel consumption at torque Te and

speed we

X: energy stocked into the battery