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: boukh.adel@yahoo.com
Received January 5th, 2013; revised February 12th, 2013; accepted Febru ary 20th, 2013
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
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-
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-
Copyright © 2013 SciRes. LCE
Hybrid Vehicle (City Bus) Optimal Power Management for Fuel Economy Benchmarking
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
 
Minimize ,
CTeiwei Ts
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
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
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)
Copyright © 2013 SciRes. LCE
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)
12 123
EkEkJweJwtJwM V
Energy to overcome resistant forces
FaerCdA V
rollMCrCr V 
So after neglecting the energy lost by friction and
damping we get:
12Es =Ek+Ek+Er
That means:
Td wetJweJwtJw
 
The derivation by time gives us the equation below
d12 3
1 2d
Td wetJweJwtJw
 
 
After derivation and arrangement we get this last dif-
ferential equation
22 22
JMRifJitif Jt
Td it if
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-
 
 
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)
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
()ti iTs
is described by the integer value plus one
Figure 2. Zone of possible solution over the reference speed.
Copyright © 2013 SciRes. LCE
Hybrid Vehicle (City Bus) Optimal Power Management for Fuel Economy Benchmarking
Copyright © 2013 SciRes. LCE
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 ,
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
jlAnother 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
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
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
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).
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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-
J2: Sum of inertia moving at the same speed as the
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
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