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352 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 3, MAY 2004
Optimal Control of Parallel Hybrid Electric Vehicles
Antonio Sciarretta, Michael Back, and Lino Guzzella
Abstract—In this paper, a model-based strategy for the real-time
load control of parallel hybrid vehicles is presented. The aim is to
develop a fuel-optimal control which is not relying on the a priori
knowledge of the future driving conditions (global optimal con-
trol), but only upon the current system operation. The methodology
developed is valid for those problem that are characterized by hard
constraints on the state—battery state-of-charge (SOC) in this ap-
plication—and by an arc cost—fuel consumption rate—which is
not an explicit function of the state. A suboptimal control is found
with a proper definition of a cost function to be minimized at each
time instant. The ”instantaneous” cost function includes the fuel
energy and the electrical energy, the latter related to the state con-
straints. In order to weight the two forms of energy, a new def-
inition of the equivalence factors has been derived. The strategy
has been applied to the “Hyper” prototype of DaimlerChrysler, ob-
tained from the hybridization of the Mercedes A-Class. Simulation
results illustrate the potential of the proposed control in terms of
fuel economy and in keeping the deviations of SOC at a low level.
Index Terms—Road vehicle control, cost optimal control, fuel op-
timal control, suboptimal control, dynamic programming.
I. INTRODUCTION
C
OMPARED to conventional internal combustion engine
vehicles, hybrid electric vehicles (HEV’s) represent an ef-
fective way to substantially reduce fuel consumption. This ca-
pability basically is due to: 1) the possibility of downsizing the
engine, 2) the ability of the rechargeable storage system to re-
cover energy during braking phases (regenerative braking), and
3) the fact that an an additional degree of freedom is available
to satisfy the power demands from the driver, since power can
be split between thermal and electrical paths.
This third point also means that the performance of a par-
allel HEV system is strongly dependent on the control of this
power split. Controllers that are based on the minimization of
the fuel consumption seem to be one step ahead of heuristic con-
trol strategies that are based upon simple rules or maps [1]. The
former, also known as optimal controllers, in fact provide more
generality and reduce the need for heavy tuning of the control
parameters.
Several algorithms have been proposed in technical literature
for global optimization, based on the a priori knowledge of the
Manuscript received May 24, 2002; revised December 11, 2002. Manuscript
received in final form April 30, 2003. Recommended by Associate Editor I.
Kolmanovsky.
A. Sciarretta is with the Measurement and Control Laboratory, Swiss
Federal Institute of Technology (ETH), Zurich, Switzerland (e-mail: sciar-
retta@imrt.mavt.ethz.ch).
M. Back is with the Research and Technology, REM/EP-Powertrain
Control Department, DaimlerChrysler AG, Esslingen, Germany (e-mail:
michael.m.back@daimlerchrysler.com).
L. Guzzella is with the Measurement and Control Laboratory, Swiss
Federal Institute of Technology (ETH), Zurich, Switzerland (e-mail:
guzzella@imrt.mavt.ethz.ch).
Digital Object Identifier 10.1109/TCST.2004.824312
future driving conditions or of a scheduled driving cycle [2]–[5].
These results, though not suitable for real-time control, fulfill an
acknowledged function as a basis of comparison for evaluating
the quality of other control strategies.
On the other hand, various attempts have been made to de-
velop a real-time control strategy for the power split based on
an instantaneous optimization [6]–[12]. Two main aspects are
involved: 1) no—or only limited—
a priori knowledge of the
future driving conditions is available during the actual opera-
tion and 2) the self-sustainability of the electrical path has to be
guaranteed. This latter point is due to the fact that the storage
system is not expected to be recharged by an external device,
but rather during the vehicle operation by the fuel conversion
device and by means of regenerative braking.
A real-time control strategy based on an instantaneous opti-
mization needs a definition of the cost function to be minimized
at each instant. Such a function has to depend only upon the
system variables at the current time. Since the main control goal
is the minimization of the fuel consumption, it is clear that this
quantity has to be included in the cost function. However, based
on the requirements of electrical self-sustainability, the varia-
tions in the stored electrical energy (or state-of-charge, SOC)
have to be taken into account as well.
To deal with such aspects, various approaches have been pro-
posed in the literature. In some cases, a tuning parameter, which
is adjusted according to the current SOC deviation by means of
a PID controller, is introduced into the cost function minimiza-
tion [6]. In other cases, the cost function is the sum of all losses
in the electrical and thermal paths [7].
Another, more promising approach was used in [8]–[11]. It
consists of evaluating the instantaneous cost function as a sum
of the fuel consumption and an equivalent fuel consumption re-
lated to the SOC variation equivalent consumption minimiza-
tion strategy (ECMS). In this case, it is clearly recognized that
the electrical energy and the fuel energy are not directly com-
parable, but an equivalence factor is needed. The equivalence
between electrical energy and fuel energy is basically evaluated
by considering average energy paths leading from the fuel to the
storage of electrical energy. If the overall efficiencies of the elec-
trical and thermal paths were rigorously constant, such an equiv-
alence would be theoretically exact. Since efficiencies vary with
the operating point, this approach only allows the use of average
values.
In the real-time control strategy proposed in [12], the equiv-
alent fuel consumption is evaluated under the assumption that
every variation in the SOC will be compensated in the future
by the engine running at the current operating point. The equiv-
alent fuel consumption therefore changes both with the opera-
tion point and with the power split control, and its evaluation
requires an additional, inner loop in the instantaneous optimiza-
1063-6536/04$20.00 © 2004 IEEE
SCIARRETTA et al.: OPTIMAL CONTROL OF PARALLEL HYBRID ELECTRIC VEHICLES 353
Fig. 1. Schematic of the system architecture.
tion procedure, or the prior storage of the results in a look-up
table.
In this paper, a control of the ECMS-type is presented. It is
based on a new method for evaluating the equivalence factor be-
tween fuel and electrical energy. This method does not require
the assumption of the average efficiencies of the parallel paths,
and it is based on a coherent definition of system self-sustain-
ability. The ECMS is valid for different system architectures and
types of machines involved. Here it has been applied to a pro-
totypical system under development at DaimlerChrysler, based
on the hybridization of the mass-production Mercedes A-Class.
For such a system, a validation of the ECMS is presented. It
includes the comparison of control performance with that ob-
tained with conventional control strategies, the robustness with
respect to the variation of control parameters, and the system
behavior in case of steady operating point.
II. T
HE MODEL OF THE HYBRID POWERTRAIN
The architecture of the parallel hybrid powertrain considered
here is sketched in Fig. 1. It is the Hyper prototypical vehicle,
based on the mass-production A-Class A 170 CDI [13]. The
thermal path consists of a front-wheel driven powertrain, with
a 1700 cm
diesel engine that yields 44 kW maximum power
(66 kW in the mass-production setup), and a five-speed, auto-
mated manual gearbox. The electrical path includes a 6.5 Ah, 20
kW NiMH-battery pack from Panasonic and a motor/generator
connected to the rear wheels via a second five-speed gearbox.
The two gearboxes are connected in such a way that they shift
simultaneously.
A. The Common Path
The model of the system has been developed according to a
quasistatic approach [14]–[16] that allows a backward simula-
tion of the powertrain. The force necessary to drive the vehicle
at the speed
is calculated as
(1)
The force
includes the contributions due to drag, road
slope
, and rolling resistance , which is a fifth-order poly-
nomial function of the vehicle speed
(2)
The essential data obtained from bench testing are summarized
in the Appendix.
The rotational speed and torque required at the wheels are
calculated including the effect of vehicle’s acceleration
,as
(3)
(4)
At the wheel axle, the basic relationship is the balance
of torques: . The con-
trol variable is the torque split factor
that regulates the
torque distribution among the parallel paths. It is defined as
. The value therefore means
that all the (positive) torque needed at the wheels is provided
by the electrical path, or that all the (negative) torque available
at the wheels from regenerative braking is driven entirely
to the electrical path. When
, it means that all the
torque needed at the wheels is provided by the fuel path. When
, no control is needed and remains undefined.
B. The Fuel Path
Given the torque and speed required at the wheels, the se-
lected value of
defines the torque and speed output of the
fuel path. Going backward in the fuel path, i.e., inverting the
“causality chain,” engine speed and torque are computed and
finally, using an “engine map,” the fuel consumption of the en-
gine. The fuel path is then followed according to the “causality
chain,” evaluating the power actually available at the output
stage of each block, and finally at the wheels. With this back-
ward-forward procedure, the power available at a certain stage
of the path can differ from the power required, due to limitations
in the upstream portion of the path.
The relationships used in both directions include the
input–output (I/O) model of the transmission
(5)
(6)
the limits for the input variables of the engine map, and the
mapped fuel consumption
(7)
The function
obtained from engine bench tests, and the other
essential data are reported in the Appendix. The gear number
is either prior scheduled or evaluated by a module that sim-
ulates a gearbox controller. Notice that in (6)–(7), the losses in
the transmission shifting have been neglected.
C. The Electrical Path
Once the torque available at the fuel path output stage is
known, the torque balance allows the calculation of the speed
and torque required from the electrical path. Going backward in
the electrical path, motor speed and torque are computed and,
using a “motor map,” the power out of battery. The electrical
path is then followed according to the “causality chain,” eval-
uating the power actually available at the output stage of each
block and, finally, at the wheels. With this backward-forward
procedure, the power available at a certain stage of the path can
differ from the power required, due to limitation in the upstream
portion of the path. If the combination of the two paths at the
354 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 3, MAY 2004
wheels produces extra power, then dissipative (friction) braking
is needed.
The relationships used in both directions for the electrical
path include the I/O model of the transmission (the losses have
been neglected)
(8)
(9)
the limits for the input variables of the motor map, and the
mapped motor input power (see the Appendix for the data)
(10)
For the battery, an equivalent circuit model has been adopted.
The relationships used for the evaluation of the current
and
then the battery’s SOC include the circuit equation
(11)
the current limits, and the SOC variations
(12)
where the open circuit voltage
is a tabulated function of
the SOC.
Three cumulative quantities are introduced for control pur-
poses: they are the fuel energy use,
, the electrical energy
use,
, that is the variation (positive or negative) of the elec-
trical energy stored, and the mechanical energy delivered at the
wheels,
. Their definitions are
(13)
(14)
(15)
being
the low heating value of the fuel.
III. T
HE ECMS
As mentioned in the Introduction, for each time
with a time
step
, the control approach used in this paper, ECMS, finds
the value of the control variable
by minimizing a cost func-
tion
, defined as
(16)
The quantities
and are the fuel energy use
and the electrical energy use in the interval
. Both are func-
tions of the control variable
and of the driving conditions,
which are assumed to be constant over the time
. The elec-
trical energy use is weighted by an equivalence factor
that
varies with time.
The evaluation of the equivalence factor
represents the
core of the ECMS. This parameter influences the system be-
havior as follows: if
is too large, the use of the electrical
energy tends to be penalized and the fuel consumption increases.
If, on the contrary,
is too small, then the electrical energy
use is overly favored and the battery SOC decreases.
The expression for
is derived below. The derivation is
subdivided in three steps. First, two constant equivalence fac-
tors,
and , are introduced in order to evaluate the fuel
equivalent of positive and negative electrical energy use at the
Fig. 2. Plot of the dependency , MVEG-A cycle.
end of a drive cycle. Then, by introducing a probability factor
, the variable equivalence factor to be used during the
cycle is evaluated as a function of
and . Finally, the
evaluation of
during real-time operation is presented.
A. The Fuel Equivalent of Electrical Energy Variations
at the End of a Drive Cycle
The procedure described in the following allows the evalu-
ation of the fuel equivalent of the electrical energy use for a
given HEV system over a given drive cycle. The procedure re-
quires running the model for various constant values of the con-
trol variable, in the range
given by the upper and
lower bounds for the SOC that are admissible during the system
operation. When the power required at the wheels is negative
(regenerative braking), it is always absorbed by the electrical
path, regardless of the value of
that therefore is meaningful
only in case of positive power required at the wheels. At the
end of each run, the values of the fuel energy use
and of
the electrical energy use
over the cycle are collected. These
values represent the final (i.e., at final time
of the cycle) values
of the cumulative quantities
and introduced in the
previous section. The couples of
and for the different
runs are plotted as in Fig. 2, which refers to the MVEG-A (also
known as NEDC) cycle.
The pure thermal case
is outlined in the plot. The fuel
energy used in this case,
, is also the energy that would be
used to drive the cycle if no electrical path were present. The
electrical energy use in the pure thermal case,
, is not zero,
due to regenerative braking power (that gives a negative con-
tribution) and idle losses in the electrical path (i.e., the losses
when the power at the output stage of the electrical path is zero,
which give a positive contribution to
). Since it is a constant
for a given cycle, the quantity
does not depend on the con-
trol law and, since no fuel is needed to obtain it,
does not
affect the corresponding fuel energy use either. In the following,
it will thus be called “net free energy.”
For the MVEG-A cycle, as well as for many other drive cycles
(regulatory and proprietary), it has been observed that the pure
thermal case separates the curve
in two branches,
SCIARRETTA et al.: OPTIMAL CONTROL OF PARALLEL HYBRID ELECTRIC VEHICLES 355
which are nearly linear in the range of interest. The slopes of
the straight lines that fit the data are labeled
and , being
these coefficients, as it will be shown below, the quantities that
weight the electrical energy when “discharged” (positive), re-
spectively, “recharged” (negative). An analytical approximation
for the curve of Fig. 2 is therefore
(17)
The significance of this relationship is the following: if
is
greater than
, it means that some electrical energy
has been used to partially drive the vehicle, saving a certain
amount of fuel energy with respect to
. On the other hand, if
is smaller than , it means that in addition to a certain
amount of fuel energy has been used to store some electrical
energy
.
The dependency
has been used to define the fuel
energy equivalent
of a generic variation of the elec-
trical energy stored after a drive cycle. The basic idea is based
upon the required self-sustainability of the electrical path:
has to be zero for a null , otherwise it has to represent the
fuel energy that will compensate in the future the variation
.
This future compensation has not to account for the net free en-
ergy
. Therefore, the energy required for the compensation
is
. Moreover, the fuel energy used for the com-
pensation has to be evaluated as a difference with respect to the
quantity
, which still is necessary to drive the vehicle over
the cycle.
These requirements are fulfilled by the following definition
of the fuel energy equivalent:
(18)
It is evident that, with such a definition,
for a positive
is the fuel energy that should be used in a future cycle, in addi-
tion to
, to recharge the battery of a quantity in addition to
the net free energy (Fig. 3(a)). Similarly, for
negative,
is the fuel energy that could be saved in a future cycle, with re-
spect to
, by using the amount subtracted from the net
free energy [see Fig. 3(b)].
Combining (18) and (19), one finally obtains
(19)
The fuel energy equivalent as defined in (20) has the advan-
tage of being a linear function of the electrical energy use. It
does not depend on the net free energy and may immediately
be correlated, through
, to the SOC deviations. Some further
considerations can better explain the meaning of the equivalence
factors
and .
• If the efficiencies of the parallel paths were constant, to-
gether with a perfect linearity of the relationship
, then an unambiguous definition of would
arise [17]. The fact that the relationship mentioned shows
a substantial linearity in a number of cases of practical in-
terest ensures that the definition (20) is sufficiently accu-
rate to represent reality.
• The linearity of the curve expressed by (18) arises even
if the efficiencies of the parallel paths vary in a nonlinear
(a) (b)
Fig. 3. Fuel energy equivalent
as the energy spent or saved to
compensate a variation
(grey arrow) positive (left) or negative (right). The
black arrow points to
.
way depending on the operating point. This may be inter-
preted as an averaging effect that is due to the large number
of operating points included in a drive cycle.
• The difference between the numerical values of
and
arises from the losses in the reversible (i.e., electrical)
path of the hybrid system. If the efficiency of the electrical
path was unitary, an equivalence factor valid for the whole
range of operation would result.
• A straightforward but tedious analysis [17] shows that the
average efficiencies of the thermal and electrical paths
over the drive cycle can be computed from the equivalence
factors (and not vice versa as in [8]–[10]):
(20)
The use of the equivalence factors
and in the ECMS
will be discussed in the next section. The other possible appli-
cation of the definition (20) is the conversion of the SOC de-
viations into an equivalent fuel consumption, thus allowing the
comparison between different fuel economy data obtained with
different control strategies, under different conditions, etc. This
makes the method an alternative to the test procedures that are
currently used to evaluate the actual fuel economy of HEV’s
taking into account SOC deviations as well. These procedures
are based on the measurement of the steady-state fuel economy
not affected by SOC variations [19], obtained with several rep-
etitions of a drive cycle. The advantage of the proposed method
is that it requires only one cycle repetition.
B. The Variable Equivalence Factor
As stated above, the electrical energy use at the end of a
driving cycle
can be converted into an equivalent fuel en-
ergy by the equivalence factor
, if it is positive, or ,ifit
is negative.
During real-time operation, the ECMS uses those values of
and that are representative of the current driving condi-
tions. This can be done by storing in the controller certain values
that are typical of urban, extra-urban, etc. drive cycles.
Since the quantities
and are assumed to be known at
every time
during the cycle, they may be used to evaluate the
356 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 3, MAY 2004
Fig. 4. Sketch of the quantities that lead to the evaluation of and .
equivalence factor that converts the quantity into
an equivalent fuel energy (17). But the use of
rather than
depends on the final sign of , which cannot be known
during the real-time operation since the future values of
are not known.
Therefore,
cannot be replaced with certainty by or
by
, but is evaluated instead by introducing the estimated
probability that
will be positive,
(21)
If
it thus follows that and if it
follows that
. The problem of the evaluation of
is therefore transferred to the evaluation of .
Moreover, during real-time operation, the concept of “drive
cycle” is generalized as “mission”. A mission is defined here
as a trait of the vehicle’s route characterized by a given value
of the required energy at the wheels (”energy horizon”), a
quantity that is independent of the control law. The actual ve-
hicle operation therefore consists of a succession of missions,
such that, if
is the duration of the generic mission, it is
(22)
Consequently, the quantity
will be deemed now as the elec-
trical energy use over a mission.
C. The Probability of End-of-Mission Energy Use Being
Positive
The probability
is calculated as
(23)
once having estimated the two quantities (see Fig. 4)
and
, the maximum positive and negative values of that can
result at the end of the mission, given the current value of
.
The quantity
is given by the sum of three terms: 1)
the current value of
, 2) the electrical energy that would
be used for the traction with the system driven at a constant
(see Fig. 2) from till the end of the mission, and 3)
a negative term due to the free energy that will be stored from
till the end of the mission. The term 2) is calculated as
. In fact, is the mechanical energy that
still has to be delivered before the end of the mission. When
it is multiplied by the split factor
, the mechanical energy
provided at the output stage of the electrical path is obtained.
To derive the energy at the input stage, the average efficiency
of the electrical path introduced in the previous section (21) is
used. The term 3) is evaluated in the ECMS assuming a constant
ratio
between the free energy as a function of time and .
This parameter
is calculated for various drive cycles, as the
ratio
. In future work, this definition may be extended
to include variations of altitude (potential energy) and of kinetic
energy, although this latter quantity is usually negligible when
compared with electrical energy that can be stored in batteries.
The final expression for
is
(24)
The (negative) quantity
is also given by three terms: 1)
the current value of
, 2) the electrical energy that would be
recharged during the traction with the system driven at constant
(see Fig. 2) from till the end of the mission, and 3)
a negative term due to the free energy, evaluated as above since
it is independent of the control law. The term 2) is calculated as
, having applied the same considerations
as above. The final expression for
is therefore
(25)
The resulting equation for
is
(26)
with
limited between 0 and 1. For simplicity, (27) has been
implemented with
.
The assumption behind (22) is confirmed by the analysis of
the results found in [17] for the global optimization, performed
offline with the Dynamic Programming technique using de-
tailed information of the drive cycle. It was shown in [17] how
the global optimal control law is almost coincident, in several
common situations, with that obtained by using (17) with a
constant value of
that provides a null final deviation of
the SOC. This effective equivalence factor is shown in [17] to
be bounded between
and , which is in agreement with
(22). Unfortunately such a control, though optimal and easy to
implement, would not be practical for two main reasons.
• The control performance is very sensitive to the variations
of the effective equivalence factor. It, therefore, should be
estimated with very high precision
• Such a control strategy has no feedback from the actual
state of the system, while in (27) an inherent feedback is
included, depending on
.
For these reasons, the approach summarized by (22) seems to
be more appropriate.
The flowchart of the ECMS is sketched in Fig. 5. At each time
with a time step , the strategy runs the following steps.
• The vehicle speed and acceleration are measured. Using
(1)–(5), the torque and speed required at the wheels are
evaluated.
• Tentative values of the control variables
are applied in
the range from
(limited by the engine and
generator power) to
(pure electrical mode), with a
step
.
• For each tentative value of
, the model is run and,
through (6)–(13), the fuel and electrical energy use
are computed, and the cumulative
SCIARRETTA et al.: OPTIMAL CONTROL OF PARALLEL HYBRID ELECTRIC VEHICLES 357
Fig. 5. Flowchart of the ECMS.
TABLE I
S
PECIFIC FUEL CONSUMPTION (LITERS/100 km) W
ITH THE
PRESENT MODEL
AND AS
DECLARED BY DAIMLERCHRYSLER FOR
VARIOUS REGULATORY
DRIVE
CYCLES AND FOR
WARM,COLD START
quantities in (16) are updated. The energy use can
be also related to the measurable SOC.
• Equation (27) is applied to evaluate
and then (22)
for
. This requires values for , and to be
memorized. The energy horizon
is user-defined, and
it determines the duration of the successive missions.
• The cost function
for the tentative value is com-
puted as in (17).
• The control value
is chosen as the tentative value
which yields the minimal value of
.
IV. VALIDATION OF THE
ECMS
Before presenting a validation of the control strategy pro-
posed, the reliability of the model developed is demonstrated by
comparing the fuel consumption for the conventional arrange-
ment (i.e., without the hybrid equipment) with the data obtained
from experiments and confirmed by the simulation tool SFVq
(Simulation von Fahrleistung und Verbrauch—quasistationär)
used at DaimlerChrysler [18]. The comparison, made for three
regulatory drive cycles with hot start, is shown in Table I.
The capabilities of the proposed control strategy will be
shown by comparing the fuel economy and charge sustain-
ability obtained with the ECMS with those offered by 1)
pure thermal operation, 2) the ”parallel hybrid electric assist”
strategy derived from the literature, and 3) the global optimal
control strategy, calculated off-line using Dynamic Program-
ming techniques.
As stated in Section III-A, once the values of
and
have been derived over the desired cycle, it is possible to com-
TABLE II
V
ALUES OF THE
EQUIVALENCE
FACTORS FOR
DIFFERENT CYCLES
Fig. 6. Pure thermal mode—Battery SOC (above) and power distribution
(below) for the urban (left) and the extra-urban (right) portion of the MVEG-A
cycle.
pare different control strategies. For the MVEG-A cycle, the
values calculated are
(see Fig. 2) and, if
the whole cycle is defined as a mission,
MJ. Values for other regulatory cycles are shown in Table II.
The general trend is that urban cycles exhibit a larger difference
between the two equivalence factors, with higher values of
and lower values of . However, the variability of these pa-
rameters is limited, and their average value appears to be almost
constant.
The control performance considered in the following includes
1) the specific fuel consumption (SFC) (liters/100 km), 2) the
final battery SOC reached, starting from a value of 0.7, and 3)
the equivalent specific fuel consumption
eSFC (liters/100 km).
This latter figure is the sum of the fuel energy use and of the the
electrical energy use weighted by the equivalence factor
or
, according to its sign.
A. Pure Thermal Mode
Fig. 6 shows the behavior of the vehicle driven in the
pure thermal mode over the MVEG-A cycle. For this case,
it is
, except when 1) the required power at the
wheels would be too high for the engine alone and 2) in case
of regenerative braking, in which the regenerative power is
collected with
. In the plot, the computed trace of
the battery SOC is shown above. The lower plot shows the
mechanical power at the wheels (bold line) and the portion
provided by the electrical path (solid line), for the urban (ECE)
and, separately, the highway (EUDC) portions. The two curves
358 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 3, MAY 2004
TABLE III
S
PECIFIC
FUEL CONSUMPTION
SFC (L
ITERS/100 KM), F
INAL SOC (I
NITIAL
0.7) AND
EQUIVALENT
SPECIFIC FUEL
CONSUMPTION eSFC (LITERS/100
KM)
FOR THE
MVEG-A CYCLE.
are superimposed in case of regenerative braking, except when
the regenerative braking power exceeds generator capability to
accept it, which is the case of the EUDC portion (right).
The control performance evaluated for the pure thermal mode
are shown in Table III. The fuel consumption is lower than the
value obtained for the conventional arrangement, resulting from
an increase of 1.92 MJ of fuel energy due to the additional mass
and a reduction of 2.70 MJ due to the suppression of the idle
consumption by turning the engine off. An additional amount
of 0.91 MJ is stored in the battery and can be used to reduce the
fuel consumption further.
B. ”Parallel Hybrid Electric Assist” Control
It is clear that the pure thermal mode does not fully exploit
the potential of the hybrid configuration in reducing the fuel
consumption. This can only be achieved with a control strategy
that is oriented toward the minimization of fuel consumption. A
strategy used here as a basis of comparison is the “parallel hy-
brid electric assist” as implemented in [12]. The basic idea is to
turn the engine on when 1) the vehicle speed is higher than a set
point, or 2) if the vehicle is accelerating, or 3) if the battery SOC
is lower than a minimum value. When the engine is on, all the
torque required at the wheels has to be provided by the engine
path
, and an additional (positive or negative) torque is
required that is proportional to the current SOC deviation. The
minimum torque that the engine can provide is given by a speci-
fied fraction of the maximum torque at the current engine speed.
A selection of the control parameters as reported in [12] led to
the results shown in Fig. 7 and to the performance data listed in
Table III. The comparison with the pure thermal case shows that
both fuel consumption and charge sustainability are improved.
Actually, the battery SOC is kept closer to the initial value, be-
cause the electrical motor is activated both during urban and
during extra-urban portions of the cycle.
C. Global Optimal Control
The global optimal control law yields the minimal fuel con-
sumption with no SOC variations at the end of the cycle. This
control law was obtained using the Dynamic Programming tech-
niques, whose details may be found in [17]. The results obtained
for the MVEG-A cycle are shown in Fig. 8. As discussed in
[17], it must be noticed that these results are affected by an
unavoidable uncertainty due to the discretization of the state-
space needed by the algorithm, and the consequent truncations.
Having used a
, the uncertainty calculated
for the SFC was around 2.5%. It means that the global optimal
specific fuel consumption can range from 3.11 to 3.19 liters/100
Fig. 7. ”Parallel hybrid electric assist”—Battery state-of-charge (above) and
power distribution (below) for the urban (left) and the extra-urban (right)
portions of the MVEG-A cycle.
Fig. 8. Global optimization-Battery state-of-charge (above) and power
distribution (below) for the urban (left) and the extra-urban (right) portion of
the MVEG-A cycle.
km. In any case, a large improvement is gained with respect to
the ”electric assist” control strategy in terms of fuel consump-
tion, due to the fact that the energy otherwise stored in the bat-
tery is used to sustain the system at low load operating points.
The effective equivalence factor that provides a self-sustaining
control law (see the discussion in Section III.C) has been found
to be
and to provide an SFC of 3.18 liters/100 km.
This value is in the range given above and can be therefore
considered as a good estimation of the “true” global optimum,
which could be precisely evaluated only in absence of the trun-
cation errors of the Dynamic Programming algorithm.
D. ECMS
The ECMS has been implemented with the values for
and
the energy horizon
derived from the cycle analysis, and an
SCIARRETTA et al.: OPTIMAL CONTROL OF PARALLEL HYBRID ELECTRIC VEHICLES 359
Fig. 9. ECMS-Battery state-of-charge (above) and power distribution (below)
for the urban (left) and the extra-urban (right) portion of the MVEG-A cycle.
TABLE IV
S
PECIFIC FUEL CONSUMPTION SFC (L
ITERS/100 km), FINAL
SOC (INITIAL
0.7)
AND EQUIVALENT SPECIFIC
FUEL CONSUMPTION
eSFC (LITERS/100
km)
FOR THE ECE CYCLE
TABLE V
S
PECIFIC FUEL CONSUMPTION
SFC (LITERS/100 km), F
INAL SOC (INITIAL
0.7) AND EQUIVALENT SPECIFIC FUEL CONSUMPTION eSFC (LITERS/100 km)
FOR THE JP10-15 CYCLE
incremental step of the control variable, . The result is
very close to the global optimal one (Table III), both in terms
of fuel consumption and of sustainability of the battery charge
(Fig. 9).
Similar results have been found for different drive cycles, like
the ECE cycle (Table IV) or the Japanese 10–15 cycle (Table V).
The capabilities of the proposed approach are even more evident
in an urban-like drive cycle (ECE) than in the MVEG-A cycle.
Actually, a reduction of more than 30% seems to be achievable
with the ECMS with respect to the pure thermal operation on
the ECE cycle. This is mainly due to the increase in overall
efficiency achieved at low-load operating points. In a combined
cycle like the Japanese 10–15, still a substantial improvement is
possible with the ECMS.
E. Multiple Cycle Repetitions
The results presented above in terms of equivalent fuel con-
sumption are based on the use of the equivalence factors eval-
Fig. 10. ECMS-Battery SOC for 5 ECE cycle repetitions.
TABLE VI
C
OMPARISON BETWEEN STEADY-STATE SPECIFIC FUEL CONSUMPTION ssSFC
(L
ITERS/100 km) AND EQUIVALENT SPECIFIC FUEL CONSUMPTION eSFC
(L
ITERS/100 km)
FOR THE
MVEG-A
AND
ECE CYCLES
uated for the current drive cycle. A test for the correctness of
this approach consists of comparing the quantity eSFC calcu-
lated after one cycle with the specific fuel consumption that is
obtained when enough repetitions of the cycle are conducted.
In this latter case, the SOC tends to a cyclic steady-state (see
Fig. 10, for the ECE cycle). The fuel consumption (labeled as
ssSFC, in liters/100 km) is no longer affected by SOC variations
and thus represents a good estimation of the equivalent fuel con-
sumption. This is the conceptual procedure for evaluating fuel
consumption of HEV’s, though it is hardly adopted due to its
long test times required [19].
As shown in Table VI, after five repetitions of the MVEG-A
cycle, a steady-state specific fuel consumption of 4.14 liters/100
km is obtained for the ”electric assist” strategy and of 3.22
liters/100 km for the ECMS. These values are quite close to the
equivalent specific fuel consumption calculated with the equiv-
alent factor analysis (Table III). Similar results have been ob-
tained for the ECE cycle (Table VI). This confirms the accuracy
of the approach used for the evaluation of the equivalent factors.
F. Sensitivity of the Control Performance to Variations in
Energy Horizon
The results shown in Tables III–VI were obtained using for
the energy horizon
the integral (16) over the entire cycle.
To check whether this choice is critical for the strategy, some
simulations have been run with arbitrarily selected values for
. The resulting variations in control performance are shown
in Fig. 11 for the ECE cycle and the MVEG-A cycle. In both
cases, the variations of the equivalent fuel consumption are lim-
ited in a range of few percent.
G. Road Drive Cycle
A drive cycle actually performed on the road was recorded
around the city of Stuttgart, Germany. This cycle, labeled AMS
includes highway and extra-urban portions, with altitude varia-
tions. The trace of vehicle speed is shown in Fig. 12.
360 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 3, MAY 2004
Fig. 11. Equivalent specific fuel consumption eSFC obtained with different
values of the energy horizon
for the MVEG-A and ECE cycles.
Fig. 12. Vehicle speed trace of the road test AMS.
The fuel consumption calculated for the conventional ar-
rangement is 4.13 liters/100 km. For the hybrid vehicle in the
pure thermal mode, there is an increase up to 4.70 liters/100
km, with the battery charged to its limit. In applying the ECMS,
the altitude variations are taken into account in
. Due to the
long duration of the cycle, it was unpractical to define the
energy horizon as the “cycle”
. On the contrary, the value
set is
MJ, which is of the same order of magnitude as
those typical of the regulatory cycles, after having considered
the results of the previous subsection. The SFC calculated is
4.34 liters/100 km and the final SOC is 0.69 (
liters/100 km). It is clear that the contribution of high-load,
high-fuel-efficiency operative points is dominant and conse-
quently the improvement achievable in fuel consumption is
smaller than in the regulatory cycles.
H. Steady Operating Point
While the results shown above deal with test cycles, it is worth
discussing the response of the ECMS for steady operation at a
constant operating point. For this purpose the following simula-
tion test was set up. A constant operating point was selected for
the system, such that the power required at the wheels is 0.58
kW only, and the engine is at idle (
rad/s,
Nm) in the pure thermal case. The plot of the equivalence factors
is shown in Fig. 13. The dots represent data obtained with dif-
ferent values of the control variable
, from the pure electrical
mode
to the maximal engine torque at the operating
speed. This corresponds to a value
, with an engine
power of 6.67 kW, due to
Nm. Evidently, the two
curve branches at both sides of the point
are nonlinear.
This is expected because the engine and electrical machine map
outputs vary with their operating points, which depend on
.
Actually, the situation of Fig. 2 contained an ”averaging” effect,
due to the several operating points included in a cycle, which is
no longer present in this case. To deal with this scenario, only
Fig. 13. Plot of the equivalent factors (above) and efficiency of the fuel path
(below) for a steady operating point.
a limited set of possible control variable values has been de-
fined. This set includes the pure thermal point
, the
pure electrical point
, and the point of maximum en-
gine efficiency. As shown in Fig. 13, the latter corresponds to
. The computation of the slopes of the two secant lines
yields
and , that is to say, .
It is possible to prove analytically [17] that in such a case, the
optimal sequence of
leads to a ”duty cycle.” The system is
alternatively driven as purely electrical
and with max-
imum recharging
. This is somehow intuitive, because
the pure thermal mode is at an engine operating point with very
low efficiency. If, on the contrary, a constant operating point was
selected such that
, then a purely thermal steady state
would be obtained [17].
The ECMS was presented so far for the usual case of
. To deal with the new scenario, and keeping the same defini-
tion of
and of the other quantities involved in the evaluation
of the equivalence factor, it is necessary to invert the role of
and and rewrite (22) in the more general way
(27)
With this new notation, the ECMS yields the expected result
with the above values for and (and ,
MJ). The SOC variations are shown in Fig. 14, together with the
power distribution and the engine efficiency, for two repetitions
of the “cycle” of 200 s.
I. Preventing Frequent Engine Starts/Stops
Fig. 14 clearly shows that the ECMS would force the engine
to a very rapid succession of starts/stops. Since this behavior is
unacceptable for many reasons, a correction has been added to
the control strategy in order to prevent the engine status from
being changed too frequently. When the cost function
is evaluated for each possible control variable value , a varia-
tion
in the equivalence factor is applied if would lead to a
change in engine status. The quantity
takes into considera-
tion the additional fuel energy related to engine starts/stops. It is
SCIARRETTA et al.: OPTIMAL CONTROL OF PARALLEL HYBRID ELECTRIC VEHICLES 361
Fig. 14. Duty Cycle with ECMS-Control results.
Fig. 15. Duty Cycle with ECMS and prevention of frequent engine
starts/stops.
possible to estimate such a considering the fuel equivalent
energy that is required to accelerate the engine from rest to idle
speed
: , the value
being the engine’s inertia and the efficiency of the starter
motor. Notice the use of
to weight the electrical energy use
required, since this is a positive quantity. Increasing the value
of
, the number of saw-tooth peaks in Fig. 14 decreases.
For the current application, the modified results are shown in
Fig. 15. The periods in which the engine is on
and in
which it is off
are now re-distributed in time, keeping
the ratio approximately the same as in the uncorrected case of
Fig. 14, such that the SOC is equally well sustained, but de-
creasing the number of engine starts. Notice that, even at the
same SOC, the ECMS may select a different control variable.
This is because the equivalence factor
not only depends on
the operating conditions, which are constant, and on the current
TABLE VII
N
UMERICAL VALUES OF THE
VEHICLE PARAMETERS
SOC, but also on the advance in the current mission, which is
variable with time.
V. C
ONCLUSION
Real-time power split control in hybrid electric vehicles can
be effectively designed using an instantaneous optimization
strategy. The ECMS presented in this paper is a strategy based
on the definition of the fuel equivalent of the electrical energy.
On this basis, an instantaneous cost function can be evaluated
and minimized by selecting a proper value for the torque split
control variable. The fuel equivalent is calculated online as
a function of the current system status and, in particular, of
quantities that are measurable on board. No predictions are
needed of the future, and only few control parameters are
required, which vary from one vehicle to another and as a
function of the driving conditions. The way to derive numerical
values for these parameters is discussed in detail in the paper.
With the ECMS, the capabilities of HEVs for the recupera-
tion of free energy and the optimization of power split are fully
outlined. For the MVEG-A cycle, the ECMS shows a potential
for reducing the fuel consumption by up to 30% of the conven-
tional value. In an urban cycle, such as the ECE, the reduction
calculated is even larger, around 50%. This performance does
not affect the charge sustainability, since SOC excursions over
the quoted cycles are limited to 2%. The correctness of the ap-
proach used was demonstrated by comparing the data on the
steady-state specific fuel consumption, which are not affected
by SOC excursions, and the equivalent specific fuel consump-
tion as evaluated by the ECMS. The difference between these
data is of few percent.
The robustness of the ECMS was proved with respect to the
variation of the energy horizon, one of the control parameters,
and for a steady system operating point, when a “duty-cycle” oc-
curs. For this latter case, an optimal control strategy can be de-
362 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 3, MAY 2004
Fig. 16. Contour map of the engine fuel consumption as a function of the
engine speed and torque. The bold curve is the maximum torque as a function
of speed for the tested setup. The axes are normalized to 100.
Fig. 17. Contour map of the electric machine input power
as a function of
the speed and torque. The bold curves are the maximum and minimum torque
as a function of speed. The axes are normalized to 100.
Fig. 18. Battery voltage as a function of the SOC, for the charging (solid) and
the discharging modes (dashdot). The axes are normalized to 100.
rived analytically as well. It was shown to be in complete quanti-
tative agreement with the ECMS results. Moreover, a fully qual-
itative agreement is obtained with the introduction of an addi-
tional cost term in order to prevent changes in engine status from
occurring too frequently. This extension is physically consistent
and can be applied also to the generally valid ECMS.
A
PPENDIX
NUMERICAL VALUES OF THE VEHICLE PARAMETERS
See Table VIII, Figs 16, 17, and 18.
R
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Antonio Sciarretta was born in Ortona, Italy, in
1970. He received the Laurea degree (cum laude)in
mechanical engineering and the Doctorate degree in
thermal machines from the University of L’Aquila,
Italy, in 1995 and 1999, respectively.
In 1999, he was a Postdoctoral Consultant at
Centro Ricerche Fiat (Fiat research center), Turin,
Italy. From 1999 to 2001, he was a Postdoctoral
Researcher at the University of L’Aquila, where his
interests were control-oriented modeling of engines,
injection control of spark-ignition engines, load
control of unthrottled engines. In August, 2001, he joined the Measurement
and Control Laboratory at ETH, Zurich, where his interests are modeling and
control of vehicle systems.
SCIARRETTA et al.: OPTIMAL CONTROL OF PARALLEL HYBRID ELECTRIC VEHICLES 363
Michael Back was born in Heidelberg, Germany,
in 1973. He received the Diploma in electrical
engineering from the University of Karlsruhe,
Germany, in 1999.
Since 1999, he has been working as a Doctorate
student at DaimlerChrysler, in cooperation with the
Control Systems Laboratory, University of Karl-
sruhe. There he is involved in projects concerned
with predictive powertrain control for different
drivetrain configurations.
Lino Guzzella was born in Zurich, Switzerland, in
1957. He received the Diploma in mechanical engi-
neering in 1981 and the Ph.D. degree in control en-
gineering in 1986 from the Swiss Federal Institute of
Technology (ETH) Zurich.
From 1987 to 1989, he was with the R&D-De-
partment of Sulzer Bros., Winterthur, Switzerland.
From 1989 to 1991, he was an Assistant Professor
for Automatic Control in the Electrical Engineering
Department of ETH Zurich. He then joined Hilti
R&D, Liechtenstein, where he was the head of the
Mechatronics Department from 1992 to 1993. He is currently Professor for
Thermotronics in the Mechanical Engineering Department of ETH-Zurich.
His research interests are modeling of dynamic systems, nonlinear and robust
control and applications of these ideas to thermal and especially automotive
systems.
