Proceedings of the 17th World Congress
The International Federation of Automatic Control
Seoul, Korea, July 6-11, 2008
Optimal Hybridization in Two Parallel
Hybrid Electric Vehicles using Dynamic
Programming ⋆
Olle Sundström ∗,∗∗ Lino Guzzella ∗ Patrik Soltic ∗∗
∗
Department of Mechanical and Process Engineering
ETH Zurich 8092 Zurich, Switzerland
∗∗
EMPA Material Science and Technology
Überlandstrasse 129 8600 Dübendorf, Switzerland
Abstract: This study explores different hybridization ratios of two types of parallel hybrid
electric vehicles, a torque assist parallel hybrid and a full parallel hybrid, with equal powerto-weight ratio. The powertrain consist of an internal combustion engine, an electric motor,
and a NiMH battery. The different hybridization ratios are compared by their optimal fuel
consumption for eight different drive cycles. The optimal fuel consumption is determined using
dynamic programming for each of the different hybridization ratios. In the full parallel hybrid
the engine and motor can be decoupled while in the torque assist hybrid the engine and motor
are always mechanically connected. Results show that there are not only lower fuel consumption
for the full hybrid but the need for hybridization is lower than in the torque assist hybrid for
all eight cycles. The hybridization ratio where a full hybrid have the same fuel consumption as
the optimal torque assist hybrid can differ as much as 51%.
1. INTRODUCTION
The interest for hybrid electric vehicles, i.e. vehicles including an internal combustion engine, an electric motor
and a battery as power sources, is increasing among automotive manufacturers and researchers. Hybrid electric
vehicles provide opportunities to reduce fuel consumption
and hence reduce carbon dioxide emissions through brake
energy recuperation, engine start-stop operation, and engine operating point shifting. This study will focus on a
parallel hybrid electric vehicle where the electric motor
and combustion engine are mechanically linked. Parallel
hybrid vehicles can be divided into two separate classes:
full parallel hybrids and torque assist hybrids. In a torque
assist hybrid the motor and engine are always coupled
while in a full hybrid the engine and motor can be decoupled. When dimensioning hybrid powertrains the problem
always exist how to optimize the components without
knowing how they will be used. In other words if the energy
management strategy only utilizes the electric path rarely
there will be a lower need for hybridization. If the energy
management utilizes the electric path excessively there will
be a greater need for hybridization.
There have been previous studies on dimensioning of the
drive train components in a parallel hybrid electric vehicle like Lukic and Emadi [2004] and Holder and Gover
[2006]. However, previous studies focusing on dimensioning
uses a predefined energy management strategy, usually
a rule-based strategy parameterized in different ways.
When using a parameterized rule-based energy manage⋆ This work is a part of the CLEVER project which is a common
project of ETH, EMPA, Volkswagen and Bosch. The project is
supported in part by the Swiss Federal Office of Energy, the Swiss
Federal Office of Environment, and Novatlantis.
978-1-1234-7890-2/08/$20.00 © 2008 IFAC
ment strategy the different vehicle designs will benefit
differently from the strategy and thus unavoidably giving a
biased results, only valid for the specific parametrization.
In Desbois-Renaudin et al. [2004] the authors study the
energy losses for different hybridization ratios in a full
parallel hybrid vehicle using optimal control methods.
However the comparison between full and torque assist
parallel hybrid is never made.
The aim of this study is to quantify the hybridization
needs, i.e. the optimal dimensioning of the power train
components, in two types of parallel hybrid electric vehicles and to exclude the influence of the control strategy
on component sizing. The paper is structured as follows:
Section 1.1 describes the method used to solve the design/control problem, Section 2 describes the parallel hybrid vehicle model and model scaling, Section 3 describes
the dynamic programming algorithm, Section 4 shows the
optimal hybridization of the parallel hybrid vehicle and
finally Section 5 discusses the results and future work.
1.1 Method
To exclude the influence of the control strategy on component sizing an optimal control method is used. By using
this method all different designs are evaluated based on
their optimal performance and therefore compared on an
equal basis. This method has been used to determine the
effect of battery size on total energy losses in a fuel cell
hybrid electric vehicle by Sundstroem and Stefanopoulou
[2007]. Since the considered system is highly nonlinear
and is valid under multiple complex constraints Bellman’s
dynamic programming algorithm (Bellman [1957]) is a
suitable method to compute the optimal control input. The
optimal hybridization is compared using eight different
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Torque Assist Parallel Hybrid
vehicle speed [km/h]
SoC
Battery
Electric motor
.
mfuel
Engine
Gearbox
Clutch
u
v
a
Electric motor
Gearbox
Clutch
u
2nd
3rd
4th
5th
6th
Fig. 2. Gear switching strategy when considering the
manual gearbox.
Battery
Clutch
1st
gear
SoC
Engine
50
0
i
Full Parallel Hybrid
.
mfuel
100
v
a
i
Fig. 1. Two types of parallel hybrid electric vehicle configurations: torque assist parallel hybrid (top) and full
parallel hybrid (bottom).
drive cycles. The torque assist hybrid and the full hybrid
configurations are shown in Fig. 1. Note that the full
hybrid requires an electrically controlled clutch between
the engine and motor and that the mechanically actuated
clutch between the motor and gearbox in the full hybrid
is not necessary if the gearbox is automated.
2. PARALLEL HYBRID VEHICLE MODEL
The parallel hybrid electric vehicle model is a quasi static,
i.e. non causal, discrete model where the signals flow from
the drive cycle through the power train one way. The
modeling follows the theories in Guzzella and Sciarretta
[2007]. The parallel hybrid electric vehicle model can be
described as
xk+1 = f (xk , u, v, a, i),
(1)
where xk is the state of charge in the battery, u is the
torque split factor (further described in Section 2.3), v is
the vehicle speed, a is the vehicle acceleration and i is the
gear number. Throughout this study a time step of one
second has been used. The model is separated into the
subsystems; vehicle, gearbox, internal combustion engine,
electric motor, and a battery. The equations describing
the subsystems and their scaling equations are shown in
the following section. The model assumes no extra fuel
consumption during starting of the combustion engine,
isothermal conditions and no energy losses during gear
shifting.
2.1 Vehicle
The vehicle model is based on a midsized vehicle with
the mass of m0 = 1503 kg equiped with a 1.6l internal
combustion engine (totally m0 + mice = 1611 kg). The
total mass of the vehicle is
mveh = m0 + mice + mem + mbatt ,
(2)
where the engine mass mice , motor mass mem , and battery
mass mbatt are all depending on the specific component
sizes.
The inputs to the vehicle model are the speed vveh and
the acceleration aveh which is given by the drive cycle.
The outputs of the model are wheel rotational speed ωveh ,
wheel rotational acceleration ∆ωveh , and the wheel torque
Tveh . The outputs are determined from the inputs using
the following equations
ωveh = vveh /rwheel
∆ωveh = aveh /rwheel ,
(3)
where rwheel is the wheel radius and
Tveh = (Ff + Fa + Fi ) · rwheel ,
(4)
where the vehicle air drag force is
2
Fa = 0.5 · ρair · cd · A · vveh
,
(5)
and the inertial force is
Fi = (mveh + mrot ) · aveh ,
(6)
with mrot is the moment of inertia of the drivetrain
converted into an equivalent mass. The friction force is
cr2
Ff = mveh · g · (cr0 + cr1 · vveh
).
(7)
The friction coefficients cr0 ,cr1 , cr2 , and cd have been identified using vehicle coastdown experiments. The vehicle
model assumes no wheel slip.
2.2 Gearbox
The gearbox is a six gear manual transmission with the
same gear ratios for all configurations. The inputs are the
wheel speed, wheel acceleration, wheel torque, and gear
number i. The outputs are the crankshaft rotational speed
and acceleration together with the crankshaft torque. The
gearbox efficiency is constant for all gears ηgb = 0.95 and
thus the torque on the clutch side of the gearbox is
T
veh
Tveh ≥ 0
ηgb · γ(i)
(8)
Tgb = T
veh · ηgb Tveh < 0
γ(i)
where γ(i) is the gear ratio for each of the gears (including
the final drive). The rotational speed of the crankshaft is
ωc = γ(i) · ωveh and ∆ωc = γ(i) · ∆ωveh . (9)
The New European Drive Cycle (NEDC) includes a gear
shifting strategy and since NEDC is used for rating of
vehicles the given gear shifting strategy is used. The gear
shifting strategy is, for all other drive cycles, given by a
speed dependent shifting policy, shown in Fig. 2. When
the vehicle speed exceeds the speed of the gray boxes in
Fig. 2 there will be an upshift and when the speed is lower
than the speed in the box then there will be a downshift.
The gearbox model assumes no energy losses during gear
shifting.
2.3 Torque Split
The torque split strategy in the hybrid vehicle, or more
generally the energy management strategy (EMS), determines how the torque demand is split between the electric
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[kg/h]
16
88
160
−100
0
−20
70
0
−150
0
1000
2000
3000
4000
rotational speed [rpm]
5000
0
6000
Fig. 3. Internal combustion engine fuel consumption map;
measured (solid) and simulated (dotted). The drag
mean effective torque is shown in dashed.
motor and the internal combustion engine. The EMS refers
to the control problem of determining the torque split factor at every time step. Let us define the torque split factor
as the continuous variable u ∈ [−1 1]. The torque split
factor can assign negative values (more torque is provided
by the internal combustion engine than is demanded to
recharge the battery with u = −1 being full recharge),
zero u = 0 (torque is only provided using the internal
combustion engine), positive values (torque is provided
from both internal combustion engine and electric motor)
and one u = 1 (all torque is provided by the electric motor
or full brake energy recuperation).
The total torque demand required by the electric motor
and the internal combustion engine will be different when
considering a torque assist parallel hybrid compared to a
full parallel hybrid. In the full hybrid the second clutch
between the internal combustion engine and the electric
motor allows the vehicle to run pure electric without
dragging the internal combustion engine. A torque assist parallel hybrid on the other hand will always drag
the internal combustion engine even when the vehicle is
driving pure electrically since the motor and engine are
always directly coupled. Therefore when considering the
torque assist configuration the total torque demanded from
combustion and/or the electric motor is
Tdem = Tice0 + Tem0 + Tgb ,
(10)
where Tice0 is the engine drag torque, Tem0 is the electric
motor drag torque, and Tgb is the demanded torque by
the gearbox. In contrast when considering the full parallel
hybrid the total demanded torque depends on the torque
split factor as follows
Tgb + Tem0
u = 1 and Tem,max > Tdem
Tdem =
Tgb + Tem0 + Tice0 otherwise
(11)
The electric motor torque demand and the internal combustion engine torque demand is determined using the
total torque demand Tdem , the torque split factor u, the
battery current limits, the motor torque limits, and the
engine torque and speed limits.
1000
2000
3000
4000
5000
rotational speed [rpm]
6000
7000
Fig. 4. Electric motor efficiency map with maximum and
minimum motor torque.
2.4 Internal Combustion Engine
The internal combustion engine model is based on the
Willans approximation, i.e. the brake mean effective pressure pbmep is an affine function of the fuel mean effective
pressure pf mep
pbmep ≈ e(ωc ) · pf mep − pbmep0 (ωc )
(12)
where e(ωc ) is the internal efficiency and pbmep0 (ωc ) is
the drag mean effective pressure. The engine drag torque
(including the inertial torque) is then given as
Vd · pbmep0 (ωc )
Tice0 = Jice · ∆ωc +
(13)
4·π
with the inertia Jice and the displacement Vd . The fuel
consumption is calculated using
Tice · ωc
∆mf =
,
(14)
e(ωc ) · Qlhv
where Qlhv is the lower heating value of gasoline. The
rotational speed dependent variables e(ωc ) and pme0 (ωc )
have been fitted to measurement data from an engine
dynamometer with a naturally aspirated 1.6 liter gasoline
direct injection engine. A comparison between the measured and simulated fuel consumption map is shown in
Fig. 3. Figure 3 shows that the model estimates the fuel
consumption well. The mass of the internal combustion
engine is approximated as
mice = Vd · cdm/dV ,
(15)
with the constant cdm/dV = 67.6 [kg/l].
2.5 Electric Motor
The model for the electric motor is generated based on
detailed simulation data of a 24 kW motor. The motor
drag torque is
Tem0 = Jem · ∆ωc
(16)
with the motor inertia Jem . To determine the electric
power needed from or supplied to the battery a map,
Γ(ω, T ), derived from the detailed simulations, is used
Pem = Γ(ωc , Tem )
(17)
The efficiency map of the 24 kW electric motor
Tem · ωc
Tem ≥ 0
ηem (ωc , Tem ) = Γ(ωc , Tem )
(18)
Γ(ωc , Tem ) Tem < 0
Tem · ωc
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Vd [liter] dashed
is shown in Fig. 4.
Pem,tot = Pem + Paux
(19)
where Paux = 350 W is a constant auxiliary power demand
by the . The battery current Ibatt is calculated using
p
2 −4·R
Voc − Voc
int · Pem,tot
, (20)
Ibatt (Pem,tot ) =
2 · Rint
where Voc is the open circuit voltage of the battery and
Rint is the battery’s internal resistance. Both the open
circuit voltage and the internal resistance are functions of
the state of charge and the number of modules, and cells
per module, used in the battery pack. The battery current
is limited to, Imin ≤ Ibatt ≤ Imax , where Imin and Imax
depends on the capacity of the battery. The battery’s state
of charge (SoC) xk is calculated using
−Ibatt ηbatt (Ibatt )
xk+1 =
+ xk
(21)
3600 · Qbatt
where ηbatt is the battery charging efficiency
1.0 Ibatt ≥ 0
ηbatt (Ibatt ) =
(22)
0.9 Ibatt < 0,
and Qbatt is the battery capacity.
2.7 Model Scaling
The internal combustion engine, electric motor and battery models have to be scaled to allow for different hybridization ratios. The combustion engine model is scaled
using the displacement volume Vd according to the equations in Section 2.4. The stroke-to-bore ratio and the mean
piston speed have been kept constant for all engine sizes.
In the electric motor model the maximum torque and
efficiency have been scaled using a linear dependency on
the rated maximum power. The mass and inertia of the
motor have also been scaled using a linear relationship
of the maximum power. In the battery model the open
circuit voltage has been kept constant while the internal
resistance and battery mass have been scaled based on the
battery capacity. The battery maximum power is given
by the maximum power of the electric motor and hence
the maximum battery current and therefore the battery
capacity.
The total maximum power of the vehicle Ptot,max is the
maximum power of the two power sources together
Ptot,max = max (Tice,max (ω) + Tem,max (ω)) · ω, (23)
ω
where Tice,max is the maximum torque of the engine and
Tem,max is the maximum torque of the electric motor. Let
ω o = arg max (Tice,max (ω) + Tem,max (ω)) · ω,
(24)
ω
50
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
60
Q batt [Ah]
The battery pack consists of multiple modules in parallel
and in series which are each modeled as a voltage source in
series with a resistance. The battery model is based on an
ADVISOR model of a 6.5 Ah NiMH battery. The battery
input/output power is the total power supplied to (or by)
the electric motor Pem,tot
40
20
0
total mass m veh [kg]
2.6 Battery
100
2
P em,max [kW] solid
17th IFAC World Congress (IFAC'08)
Seoul, Korea, July 6-11, 2008
2000
1800
1600
hybridization ratio [-]
Fig. 5. Engine displacement, motor maximum power, vehicle total mass, and battery capacity with changing
hybridization ratio.
then the hybridization ratio is defined as
Pem,max (ω o )
HR =
(25)
Ptot,max
When optimizing the hybridization ratio in the vehicle it
is only interesting to compare ratios with similar performance. Performance can be defined in several ways such
as time from 0-100 km/h, top speed and gradability. The
time from 0-100 km/h is strongly related to the maximum power-to-weight ratio which include both engine
and electric motor size. This study therefore compare
different hybridization ratios with equivalent maximum
power-to-weight ratio (≈ 67 W/kg). The different vehicle
characteristics are shown in Fig. 5. Since the total vehicle
mass is increasing with increasing hybridization the total
maximum power also increases to maintain equal powerto-weight ratio.
3. DYNAMIC PROGRAMMING
Let the discrete model (1) be reduced to
xk+1 = fk (xk , uk ),
k = 0, 1, ..., N − 1
(26)
with the state of charge xk ∈ Sk and the torque split
factor uk ∈ Ck . Furthermore assume that the drive cycle
is known in advance and that the particular driving speed,
acceleration and gear number at instance k are included
in the model function f hence the subscript k.
Let then π = {µ0 , µ1 , ...µN −1 } be a torque split strategy
for the particular drive cycle and vehicle model. Further
let the cost of using π with the initial state x(0) = x0 be
Jπ (x0 ) = gN (xN ) +
N
−1
X
gk (xk , µk (xk )),
(27)
k=0
with gN (xN ) being the final cost which is zero for
SoC(N ) = SoC(0) = 0.6 and infinite otherwise, thus
forcing a charge sustaining solution. The final and initial
state of charge has been chosen 0.6 since this is between the
limits of the battery. Changing the final and initial value
would have a small effect on the overall fuel consumption,
if the state of charge trajectory is within the boundaries
of the battery, since the state of charge only have a small
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NEDC
100
50
0
5
time [min]
10
0.65
Fig. 6. The drive cycle speed profiles used.
state of charge (SoC) [−]
0.6
effect on battery characteristics. The cost function gk is the
fuel consumption of the combustion engine, see equation
(14). The optimal trajectory π o is the trajectory that
minimizes Jπ
J o (x0 ) = min Jπ (x0 )
(28)
π∈Π
Bellmans principle of optimality (Bellman [1957]) states
that an optimal input trajectory for a discrete decision
problem going from t = 0 to t = T is also optimal for
the subproblem going from t = n to t = T . Based on
the principle of optimality, dynamic programming is the
algorithm, which proceeds backward in time from N − 1
to 0,
uk ∈Uk (xk )
0.5
0.45
0.4
0
infeasible
stand still
max recharge
(1) End cost calculation step
JN (xN ) = gN (xN )
(29)
(2) Intermediate calculation step
Jk (xk ) = min
gk (xk , uk ) + Jk+1 (fk (xk , uk ))
0.55
Full Parallel Hybrid
0
20
200
400
600
time [s]
800
1000
electric
10
15
time [min]
torque assist
5
thermal
0
recharge
50
0
10
HWFET
150
FTP-72
100
5
100
50
150
15
CADC Highway
0
150
CADC
100
10
0.6
50
0
5
100
50
0
0.65
0
150
CADC Road
100
speed [km/h]
5
state of charge (SoC) [−]
0
CADC Urban
100
Torque Assist Parallel Hybrid
150
Fig. 7. The optimal input map for the torque assist
hybrid (top) and the full hybrid (bottom) with 20%
hybridization ratio driving the NEDC (middle) and
the optimal state of charge trajectory (black).
(30)
The optimal torque split factor is given by the argument
that minimizes the right side of equation (30) for each xk
and k. Before using the dynamic programming algorithm
the input space C = [−1 1] and state space S = [0.4 0.7]
must be limited and discretized. Further the state of charge
upper and lower boundaries have been precalculated before using the dynamic programming algorithm. The state
of charge boundaries are important to know exactly because the optimal state of charge trajectory will, in the
end of the drive cycles, tangent the boundaries since it
is optimal to recuperate as much as possible during the
final braking phase of the drive cycle. Further the dynamic
programming optimizations have been implemented in a
way that allows the utilization of a computer cluster and
distributed calculations.
4. RESULTS
The drive cycles used to compare different hybridization
ratios for the two configurations are shown in Fig. 6. When
using dynamic programming on the torque split strategy
problem, for a particular drive cycle and hybridization
ratio, the results is an optimal torque split map over time
and state of charge. To get the optimal state trajectory
and the minimum CO2 emission, the optimal torque split
map is used to determine the torque split during a forward
simulation of the vehicle for the same drive cycle. Figure
7 shows the optimal input map and the optimal state
of charge trajectory (when starting in x0 = 0.6) for the
NEDC of a torque assist hybrid and a full hybrid with
20% hybridization ratio. Note that the optimal strategy
is very different in the two types of hybrids. In the full
hybrid there are long periods of pure electric driving
(green) while in the torque assist hybrid the engine and
motor both supplies power (blue). The only time the
torque assist hybrid is using the motor solely is during
the braking and starting phases. The actual optimal input
trajectory is the torque split (color) exactly on the state of
charge trajectory (black line) in Fig. 7. The resulting CO2
emission for the 20% torque assist hybrid is 161.5 g/km
and for a full hybrid with the same hybridization is
118.7 g/km which is approximately 27% less.
In order to see the influence of hybridization on CO2
emissions the torque split problem is solved using dynamic
programming for hybridization ratios ranging from 5% to
75% with a step of 0.5%. The resulting CO2 emissions
for the eight different drive cycles are shown in Fig. 8. The
resulting optimal hybridization ratios and comparisons are
summarized in Table 1. Since the step in hybridization is
2.5% the numbers in Table 1 are only a rough estimate.
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CO2 emission [g/km]
Torque Assist Hybrid
NEDC
160
140
FTAo
CO2 emission [g/km]
TAo
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Cycle
NEDC
CADC U.
CADC R.
CADC H.
CADC
HWFET
FTP-72
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CADC Urban
200
180
160
140
120
0
CO2 emission [g/km]
Fo
120
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
120
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
CADC Highway
210
200
190
180
CO2 emission [g/km]
0
CO2 emission [g/km]
CO2 emission [g/km]
0.1
0.2
0.3
0.4
0.5
0.6
0.7
190
0.8
CADC
180
170
160
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
HWFET
150
140
130
REFERENCES
120
0
CO2 emission [g/km]
Hybridization Ratio (%)
TAo − Fo
FTAo
TAo − FTAo
7.5
14.4
45.1
17
13.3
48.7
16
14.5
41.5
3.5
27.9
17.1
7.5
19.1
29.9
2
49.2
12.3
14
14.2
50.8
8.5
20.1
25.4
We can conclude that for all eight drive cycles the optimal
hybridization is lower in a full parallel hybrid than in a
torque assist parallel hybrid. The difference in hybridization ratio between the optimal torque assist hybrid and
the full hybrid that has the same fuel consumption and
CO2 emission can be as high as ≈ 51% (see FTP-72 in
Table 1 and Fig. 8). What is more, the difference in CO2
emissions between the torque assist hybrid and the full
hybrid increases with decreasing hybridization ratio. For
a 20% hybrid driving the NEDC there is a 27% reduction
of the fuel consumption when including an extra electronically controlled clutch. Since this study does not consider
any energy losses during clutching and during starting
of the combustion engine, the results of the full parallel
hybrid is optimistic. This study assumes a predefined gear
switching strategy and fixed gear ratios therefore future
work will investigate the gear switching strategy’s and
ratios’ influence on the results. Future work also include
an analysis of the phenomenons that explain why the
hybridization requirements in a full hybrid is smaller than
in a torque assist hybrid.
130
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
FTP-72
160
140
120
100
Fo
52
45
40
41.5
41.5
59.5
51
37
5. CONCLUSION AND FUTURE WORK
140
150
TAo
59.5
62
56
45
49
61.5
65
45.5
CADC Road
150
0
CO2 emission [g/km]
Table 1. Optimal hybridization ratio for the
torque assist hybrid (TAo ), the full hybrid (Fo )
and the difference (TAo − Fo ). The hybridization ratio where the full hybrid has the same
CO2 emission as the optimum of the torque
assist hybrid (FTAo ) and the difference (TAo −
FTAo )
Full Hybrid
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
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190
180
170
160
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
hybridization ratio [-]
Fig. 8. Carbon dioxide emissions for the torque assist
hybrid (dashed) and the full hybrid (solid)
R. E. Bellman. Dynamic Programming. Princeton University Press, 1957.
M. Desbois-Renaudin, R. Trigui, and J. Scordia. Hybrid
powertrain sizing and potential consumption gains. In
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