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Control strategies for hybrid vehicles:
Synthesis & Evaluation
Technical Subject Area of the paper : 08
S. DELPRAT (*), T.M. GUERRA (*), J. RIMAUX (**)
(*) LAMIH UMR CNRS 8530, Université de Valenciennes et du Hainaut Cambrésis
le Mont Houy 59313 Valenciennes CEDEX 9, France
E-mail : {guerra, delprat}@univ-valenciennes.fr
(**) PSA Peugeot Citroën, DRIA/SARA/EEES, Route de Gizy,
78943 Vélizy Villacoublay CEDEX, E-mail : Janette.Rimaux@mpsa.com
Abstract-Control strategies for hybrid vehicles are
algorithms that choose at each sampling time the power split
between the engine and the motor. In simulation, a global
optimization algorithm based on the optimal control theory [1,2]
can compute the power split between the engine and the motor
that minimizes the fuel consumption. Analyzing the results
obtained by this algorithm allows formulating a new control
strategy. Then, some criterions for the evaluation of real time
control strategy are proposed. The key idea is to express the
control strategy performance as the distance between the real
time control strategy results and the minimal fuel consumption
curve provided by the global optimization algorithm.
I. Introduction
Hybrid vehicles are expected to be less polluting and to have
lower fuel consumption than conventional vehicles. Control
strategies are algorithms that choose at each sampling time, the
power split between the engine and the motor in order to reduce the
fuel consumption and emissions.
Some strategies have been proposed. They can be divided into
two families. The first one contains strategies related to simulation
purpose. In this case, the main goal is not to derive realistic solutions
but global ones. This goal can be achieved using different global
optimization algorithms, simulated annealing [3], dynamic
programming [4,5] optimal control theory [1,2]. The second family
will contain strategies able to manage the powertrain for real time
control. The results (the operating point chosen at each time) must
be realistic ones, just think at the state of charge variation, the
management of the IC engine on/off, the different torques values
and rates etc. Nevertheless, the ideas allowing deriving such a
control are often coming from various domains : some of them can
use fuzzy logic, trying to implement an expertise in the controller
[6,7], other ones can take advantages from the first family [8], other
ones try to use some knowledge about the energetic transfers [9,10]
etc. The problem begins when we want to quantify and to compare
all these approaches. Can we address the following questions: what
is a good real time strategy? What is the best one?
This paper tries to partially answer at these crucial questions. For
the first question, the paper presents a new strategy, DOCS (Derived
from Optimal Control Strategy results). That one uses the
experience of the results given by global optimization using optimal
control theory [1,2]. Obviously, if we try to answer the second
question, we need to define some criterions. The second part of the
paper will be devoted to these points. At last we will show how to
use the different proposed criterions trying to answer the second
question. II. Global optimization algorithm
This first section will quickly describe the works done by the
authors in the first family of control strategy [1,2,3].
The presented work is related to a particular prototype built at
the LAMIH (Laboratoire d’Automatique et de Mécanique
Industrielles et Humaines, Valenciennes, France) in
collaboration with PSA Peugeot Citroën, ADEME (Agence
de l’Environnement et de la Maîtrise de l'énergie) and the
FEDER (Fond Européen pour le DÉveloppement Régional)
which is presented in [1,3]. Let us quote that the purpose can be
extended to any other parallel hybrid vehicle.
Battery
Fuel Tank
Electric
Motor
Internal
Combustion
Engine
Gear Box
Reductor
Clutch
Coupling
Figure 1 : Mechanical arrangement
In simulation, the vehicle speed can be assessed to follow a
speed cycle, i.e. a controller computes the torque at the wheel
()
w
Tt
according to the error between the speed set point
provided by the speed cycle and the simulated vehicle speed
()
wt
ω
.
For this study, algorithm results are provided for the
'Routier n°1' speed cycles, figure 2. It is a 3 minutes length
speed cycle in between city and highway driving conditions.
0100 200 300 400 500 600 700 800 900
0
20
40
60
80
Time (s)
ω
w
(km/h)
'Routier n°1' speed cy cle
0100 200 300 400 500 600 700 800 900
-3000
-2000
-1000
0
1000
2000
Time (s)
C
w
(Nm)
Figure 2 : 'Routier n°1' speed cycle
Notations: th is used for the IC engine, e for the electric
motor, w for the wheels, k for the gear number, T for
torque,
ω
for speeds, gb for the gearbox, red for the
reductor,
η
for efficiency, and t for time.
ϑ
is the engine
state (1 stands for 'on' and 0 for 'off'). ∆ is the sampling
period. 1. Mechanical modeling
According to the mechanical arrangement, the relation
between the speeds is:
() () ()
()
() ()
()
wth e
ttRkt tRkt
ωω ω
ρ
==⋅
(1)
and the torques :
() ()
()
() ()
()
.
weredthgb
Tt Rkt Tt Tt
ρ
ηη
=⋅⋅ +⋅ (2)
Of course the efficiencies depend on the torque signs.
Torques are limited by mechanical constraints:
()
()
() ()
()
_min _max
ee eee
TtTtTt
ωω
≤≤ (3)
() ()
()
_max
0th th th
Tt T t
ω
≤≤ (4)
Speeds are also limited:
()
_max
0ee
t
ωω
≤≤ (5)
()
th_min _ max
th th
t
ωωω
≤≤ (6)
The ratio of the electric motor reductor ensures that both
the motor and the engine achieve their maximal speed
simultaneously, so a unique constraint is enough:
() ()
kt Kt
∈(7)
With
()
Kt the set of admissible gear numbers at the
sample t.
Clearly, according to (1) and (2), if
()
wt
ω
and
()
w
Tt
are
known, only a torque (
()
th
Tt
or
()
e
Tt) and the gear number
()
kt need to be chosen to define the powertrain operating
point. 2. Energetic consumption modeling
The battery is considered as a dynamical system, with x
the state of charge:
( ) () () ()
()
1,
ee e
xt xt PT t t
ω
+= + ⋅∆ (8)
() ()
()
,
ee e
PTt t
ω
represents the power required (including
battery losses) to produce the torque
()
e
Tt at speed
()
et
ω
.
Usually e
P is given by a map, so this expression allows to
approximate easily different kind of electric drive (different
battery size, and motor, etc.).
With
() ()
()
,
th th
mT t t
ω
the instantaneous fuel consumption
of the IC engine required to produce the torque
()
th
Tt
at
speed
()
th t
ω
, the total fuel consumption over N samples is:
() ()
()
()
1
0,
N
th th
t
JmTt tt
ωϑ
−
=
=⋅⋅∆
∑(9)
m
is also given by a map.
3. Optimal control
For detailed explanation about global optimization control
algorithms based on the optimal control theory applied to the
hybrid vehicle, the reader may refer to [1,2,3].
For a given speed cycle and a given prototype, the global
optimization algorithm, figure 3, allows to find the operating
points
() () ()
()
,,
th
Ttkt t
ϑ
that minimize the total fuel
consumption (9). The electric energy consumption is taken
into account by a constraint on the state of charge:
() ()
0
xN x Soc
−=∆ (10)
With Soc
∆ a desired overall state of charge variation. This
constraint is quite convenient because it provides a direct way
for the implementation of the mono-criterion performance
evaluation. Let's us quote that, if 0
Soc
∆=
, then the
propelling of the vehicle is only due to the fuel, the battery is
only used has an energy buffer which allows to improve the
global powertrain efficiency. In this particular case, the
comparison with conventional vehicles is possible.
The global optimization algorithm is :
Choose an initial value for λ(0)
The solution that minimizes
is chosen in
Calculate the sets of solutions to reduced
problems : Ωon(t) ∀t∈[1..N]
|X(0)-X(N)-Soc |<∈
Adjust λ(0) according to the value of
x(N)-x(0)-Soc
No End
Yes
()
()
()
() ( ) () ()
()
,0,
th th
DktT t t PktT t
ϑλ
⋅⋅∆− ⋅ ⋅∆
() ()
off on
tt
Ω∪Ω
Calculate the final SOC x(N)
Figure 3 : Global optimization algorithm [1,2,3]
III. Synthesis of a real time control strategy
A. Analysis of global optimization results
It is not possible to give all the results obtained on many
speed cycles with many overall state of charge variation and
so on…The explanations are restricted to a particular speed
cycle: the 'Routier n°1' speed cycle.
For sake of convenience, the results will be represented in
the
()
,
th th
C
ω
plane. Iso-efficiency curves and the optimal
torque curve
": that ensures the maximal efficiency
are also drawn. For example, figure 4 represents results
obtained with
()
-5
0 -5.62e
λ
= corresponding to a
1.49 l/100kmand an overall state of charge variation
() ()
017%
xN x
−=−
.
0
0.05
0.1
0.15
0.2
0.25
0.3
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500
0
20
40
60
80
100
120
ωth (tr/mn)
Tth (Nm)
IC engine efficiency
Figure 4 : IC engine operating point for 17%Soc
∆=−
Another trial, figure 5, with
()
-5
0 -7.97e
λ
= allows
reaching
() ()
00.35%xN x
−= with a 5.0 /100lkm fuel
consumption.
Figure 5 : IC engine operating points for 0.35%Soc
∆=
The IC engine operating points remains into high
efficiency area, even for different overall state of charge
variations. This behavior has been also be confirmed by
different experiments on different speed cycles. Of course,
sometimes, the choice of the operating point may be limited
by mechanical constraints, according to the driving cycle and
the prototype mechanical characteristics. In this case, the IC
engine torque is chosen as close as possible with the optimal
torque, but with respect to the different constraints.
The IC engine torque being often chosen into high
efficiency area, different overall state of charge variation can
be reach mainly by acting on the IC engine state, figure 6.
0100 200 300 400 500 600 700 800
0
0.2
0.4
0.6
0.8
1
1.2
θ
∆
Soc=-17.0495 %
0100 200 300 400 500 600 700 800
0
0.2
0.4
0.6
0.8
1
1.2
Time(s)
θ
∆
Soc=0.34958 %
Figure 6 : Comparison of the IC engine state for
different overall state of charge variations
Figure 6 shows the choice of IC engine state for the studied
experiments. The smallest (resp. highest) the overall state of
charge variation is, the most often (resp. less) IC engine is
switched off. This remark has also been confirmed by other
experiments with other driving cycles.
The same kinds of analysis on different cycles and for
different overall state of charge variations shows that the
highest admissible gear number is chosen.
B. Proposition of a real time control strategy : DOCS
According to the previous remarks, a new real time control
strategy can be proposed: DOCS (Derived from Optimal Control
Strategy results). It is based on two rules that have been
designed to obtain a behavior similar to the optimal control
algorithm:
• The chosen IC engine torque is the optimal torque curve
to maximize IC efficiency (or closest as possible with
respect to mechanical constraints):
() ()
()
th opti th
Tt T t
ω
=.
• The IC engine state
()
t
ϑ
is used to control the state of
charge. When the state of charge reach a lowest bound
min
x, the IC engine is switch on
()
()
1t
ϑ
=, and when the
IC engine state reach the upper bound max
x, the IC engine
is switched off
()
()
0t
ϑ
=. The IC engine may be
switched on if the motor alone can not provide the
requested torque at the wheel:
() ()
()
()
_max
wee redbv
Tt T t Rk
ωηη
>⋅⋅⋅
.
• The highest admissible gear number should be chosen, but
this can leads to huge number of gearshifts because the
requested torque at the wheel
()
w
Tt
may have quick and
important variations. To avoid this problem, it is chosen
according using a map that limits the number of gearshifts
and ensures a good driving comfort [13].
For example, results obtained on the Routier n°1 speed
cycle are shown figure 7. The corresponding overall state of
charge variation is 2.06%
− and the fuel consumption is
6.05 /100
lkm
. For this experiment, the SOC target was
0.8% with a 5%
±hysteresis for the IC engine management.
0100 200 300 400 500 600 700 800
0
50
ω
r
(km/h)
Cycle Rout ier n°1
0100 200 300 400 500 600 700 800
0
50
100
C
th
(Nm)
0100 200 300 400 500 600 700 800
1
1.5
2
k(t)
0100 200 300 400 500 600 700 800
77
78
79
80
Soc(%)
0100 200 300 400 500 600 700 800
0
0.5
1
1.5
θ
Temps (s )
Figure 7 : DOCS results on the 'Routier n°1'
speed cycle
IV. Evaluation of real time control strategies
A. Proposition of criterion for the control strategy
performance evaluation
Evaluation of real time control strategy is a multi-criterion
problem: the performance needs to be expressed in term of
fuel and electric energy consumption. The classical 'Soc
correction routine' reduces this evaluation to the mono-
criterion case by considering different control strategy results
with the same overall state of charge variation, i.e. all the real
time control strategy must provides solutions such that
() ()
0
xN x Soc
−=∆
, with Soc
∆ a fixed electric energy
consumption, usually 0
Soc
∆=
. In this case, the fuel
consumption obtained using different algorithms can be
compared. This method does not provide an absolute
evaluation of the performances, but just a ranking of the
strategy results for a particular overall state of charge
variation.
To obtain an absolute evaluation, the control strategy
performance can be defined as the distance between the
control strategy result and an optimal fuel consumption curve
provided by the global optimization algorithm.
On a given speed cycle, the presented algorithm can be
used to compute
()
min
FC Soc
∆ and
()
max
FC Soc
∆, the
minimal and maximal fuel consumption expressed as a
function of the overall state of charge variation Soc
∆[12].
NB :
()
max
FC Soc
∆ can be computed by the global
optimization algorithm using
() ()
()
,
th th
mC t t
ω
− as the IC
engine fuel consumption.
()
min
FC Soc
∆ and
()
max
FC Soc
∆ are only defined for
[]
min m
,ax
Soc Soc Soc
∆∈∆ ∆ with min
Soc
∆ (resp. max
Soc
∆) the
overall state of charge obtained by applying the minimal
(resp. maximal) IC engine torque with respect to the different
mechanical and architectural constraints (1)-(6)) during the
whole speed cycle.
-15 -10 -5 0 5 10 15 20
0
5
10
15
20
25
30 Cycle : Rout ier 1
∆
Soc
(%)
Conso (l/100km)
∆
Soc
min
∆
Soc
max
C
min
C
max
Figure 8 : Area of possible results
Any results obtained with any algorithm (in simulation or
in real-time control conditions) will fall into the area defined
by the two curve
()
min
FC Soc
∆ and
()
max
FC Soc
∆ and the
two lines min
Soc Soc
∆=∆ and max
Soc Soc
∆=∆ , figure 8.
This area can be considered as an intrinsic characteristic of
the speed cycle: The smallest this area is, the more the control
strategies are constrained for their choice of the controls
() ()
()
,
th
Ttkt. The extreme case is a speed cycle
corresponding to a maximal acceleration. For this particular
cycle, the area is reduced to a single point, all the strategies
would give the same result: they would apply the maximal
torque on both the engine and the motor.
At the opposite, when the area is large, the results obtained
with different control strategies mainly depend on the control
strategy performances.
We propose a criterion for the evaluation of this degree of
freedom allowed to the control strategy by the speed cycle:
() ()
()
max
min
max min
max min
Soc
Soc
cycle
FC Soc FC Soc d Soc
JSoc Soc
∆
∆
∆− ∆ ⋅∆
∆−∆
∫
(11)
The biggest cycle
J is, the more the strategies are free to
optimize the energy consumption. If 0
cycle
J=, control
strategies do not have any freedom for the choice of the
controls (for example, maximal acceleration speed cycle).
In simulation, for a given control strategy strat , let us note
()
strat
FC Soc
∆ the fuel consumption as a function of the
overall state of charge variations Soc
∆. The control strategy
performance can be expressed as the distance between
()
strat
FC Soc
∆ and the curve of minimal fuel consumption
computed by the global optimization algorithm
()
min
FC Soc
∆:
() ()
()
()
max
min
min
max min
Soc
strat
Soc
cycle
strat cycle
FC Soc FC Soc d Soc
JJ Soc Soc
∆
∆
∆− ∆ ∆
⋅∆ −∆
∫
(12)
The more (resp. less) efficient a control strategy is, and the
more 0
cycle
strat
J→ (resp. 1
cycle
strat
J→).
B. Evaluation of some real time control strategies
As an illustration of the proposed criterions, 5 control
strategies are evaluated:
• DOCS, the presented control strategy based on the
analysis of the global optimization algorithm results
•
λ
-Control: This control strategy is another adaptation of
the global optimization algorithm to real time control [8].
• Equivalent Consumption Minimization Strategy: The
electric energy consumption of the motor being expressed
as an equivalent fuel consumption, this strategy
minimizes the total fuel consumption, i.e. the motor
equivalent fuel consumption + the real engine fuel
consumption [10].
• Loss Minimization Strategy : This strategy is based on the
powertrain losses minimization [9]
• Fuzzy control strategy: This strategy use a fuzzy logic
controller to keep the IC engine torque close to its optimal
value and also to manage the battery state of charge [13].
Three speed cycles with different criterion are considered,
figure 9:
• Urbain Fluide n°2 (UF2): fluid urban driving
conditions. 26.0
UF
J=.
• Routier n°3 (R3): between urban and highway driving
conditions. 34.52
R
J=
• Autoroute n°1 (A1): Highway driving conditions.
32.97
R
J=
0100 200 300 400 500 600 700 800 900 1000
0
50
100
ω
r
(km/h)
Urbain Fluide 2
0100 200 300 400 500 600 700 800 900 1000
0
50
100
ω
r
(km/h)
Routier 3
0100 200 300 400 500 600 700 800 900 1000
0
50
100
ω
r
(km/h)
Time(s)
Autorout e 1
Figure 9 : Speed cycles used for the evaluation
Figure 10 shows the fuel consumption curves obtained
with the considered control strategies and the area defined
by the minimal and maximal fuel consumption curves.
-40 -20 0
0
5
10
15
20
25
Cycle : Urbain Fluide 2
∆
Soc
(%)
Fuel cons umption (l/100km)
E.C.M.S.
LMS
l-control
Etat Mth
Fuzzy Control
-40 -20 0
0
5
10
15
20
25 Cycle : Routier 3
∆
Soc
(%)
Fuel cons umption (l/100km)
-40 -20 0
0
5
10
15
20
25 Cycle : Autoroute 1
∆
Soc
(%)
Fuel cons umption (l/100km)
Figure 10 : Results obtained using different control
strategies
The criterions for all the strategies are given figure 11. The
DOCS control strategy obtains best results for all the
different speed cycles. Ranking the others control strategy
is delicate because the different values of the criterion are
quite close. The results are so closed so that slightly
different tuning of the control strategy parameters may
produce a different ranking.
Control strategies results
0.0
20.0
40.0
60.0
80.0
100.0
(%)
CpleOpti 31.5 11.7 21.3
l-control 60.4 33.8 41.9
ECMS 64.9 48.6 46.5
LMS 70.7 61.5 54.4
FuzzyContro l 80.2 63.5 53.5
Urbain Fluide 2 Routier 3 Autoroute 1
Figure 11 : Criterion obtained for the considered
control strategies
Analyzing the ECMS, LMS,
λ
-control and FCS behavior
shows that the lower performance of these strategies is due
mainly to the SOC sustaining [11]. These strategies decrease
the IC engine torque to discharge (or stop charging the
battery). But reducing the torque often leads to a lower IC
engine efficiency. DOCS outperform these strategies because
it shuts the engine off rather than decreasing the IC engine
torque. For DOCS, two cases may be considered:
• The SOC is too low: The IC engine is set to the optimum
torque and so ensures the best efficiency for the
powertrain. To have a faster battery charging during high
torque at the wheel demand, it may be also possible to
choose an IC engine torque comprised between the
maximum torque and the optimal torque. This will not
decrease the efficiency too much because the efficiency
remains high in this area (cf. iso-efficiency curves on
figure 4).
• The SOC is too high: The powertrain is used in pure
electric mode. The efficiency of the motor and battery
being quite high compared to the IC engine one, this
mode ensure the best powertrain efficiency.
V. Conclusion
A global optimization algorithm based on the optimal
control theory allows to compute quickly, in simulation, a
minimal fuel consumption achievable for any overall state of
charge variation on a given speed cycle. Analyzing the
obtained results allows summarizing the behavior of the
global optimization to two rules. According to this, a new real
time control strategy DOCS is proposed. Of course, the
obtained rules depends on the prototype characteristics but
the same method can be applied with others prototypes and
then, different rules may be obtained.
A method for the evaluation of real time control strategy
performance has been proposed. Both energy sources are
considered by expressing the fuel consumption as function of
overall state of charge variation. The performance is defined
as the distance between the fuel consumption curve obtained
using the real time control strategy and the minimal fuel
consumption curve computed by the global optimization
algorithm.
Several real time control strategies have been evaluated.
The proposed control strategy, DOCS, provides better results.
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