Preprints, 8th
8th IFAC
IFAC International
International Symposium
Symposium on
on
Preprints,
Preprints,
8th
IFAC
International
Symposium on
Advances
in
Control
Preprints,
IFAC
International
on
Advances 8th
in Automotive
Automotive
Control Symposium
Preprints,
8th
IFAC
International
Symposium
on
Advances
Automotive
Control
June
19-23,in
2016.
Norrköping,
Sweden
Available
online at www.sciencedirect.com
Advances
in
Automotive
Control
June
19-23,
2016.
Norrköping,
Sweden
Advances
in2016.
Automotive
Control
June
19-23,
Norrköping,
Sweden
June
June 19-23,
19-23, 2016.
2016. Norrköping,
Norrköping, Sweden
Sweden
ScienceDirect
IFAC-PapersOnLine 49-11 (2016) 657–664
Cycle
Beating
-- An
Analysis
of
the
Cycle
Beating
An
Analysis
of
Cycle
Beating
An
Analysis
of
the
Cycle Beating
- An Vehicle
AnalysisTesting
of the
the
Boundaries
During
Boundaries
During
Vehicle
Testing
Boundaries
During
Vehicle
Testing
Boundaries
During Vehicle
Testing
∗
∗
Kristoffer Ekberg
Ekberg ∗∗ ,, Lars
Lars Eriksson
Eriksson ∗∗ and
and Martin
Martin Sivertsson
Sivertsson ∗∗∗
Kristoffer
∗ , Lars Eriksson ∗ and Martin Sivertsson ∗
Kristoffer
Ekberg
∗ , Lars Eriksson ∗ and Martin Sivertsson ∗
Kristoffer
Ekberg
Kristoffer Ekberg , Lars Eriksson and Martin Sivertsson
∗
∗ Vehicular Systems, Dept. of Electrical Engineering, Linköping
∗ Vehicular Systems, Dept. of Electrical Engineering, Linköping
∗ Vehicular Systems, Dept. of Electrical Engineering, Linköping
∗ University,
Vehicular
Dept.
of
Engineering,
Linköping
SE-581
83
Linköping,
Sweden,
{kristoffer.ekberg,
University,
SE-581
83
Linköping,
Sweden,
{kristoffer.ekberg,
Vehicular Systems,
Systems,
Dept.
of Electrical
Electrical
Engineering,
Linköping
University,
SE-581
83
Linköping,
Sweden,
{kristoffer.ekberg,
University,
SE-581
83
Linköping,
Sweden,
{kristoffer.ekberg,
lars.eriksson}@liu.se,
sivertsson.martin@gmail.com
lars.eriksson}@liu.se,
sivertsson.martin@gmail.com
University,
SE-581
83
Linköping,
Sweden,
{kristoffer.ekberg,
lars.eriksson}@liu.se, sivertsson.martin@gmail.com
sivertsson.martin@gmail.com
lars.eriksson}@liu.se,
lars.eriksson}@liu.se, sivertsson.martin@gmail.com
Abstract: Todays
Todays vehicle
vehicle industry
industry is
is strictly
strictly controlled
controlled by
by environmental
environmental legislations.
legislations. The
The
Abstract:
Abstract:
Todays
vehicle
industry
is
strictly
controlled
by
environmental
legislations.
The
Abstract:
Todays
vehicle
industry
is
strictly
controlled
by
environmental
legislations.
The
vehicle
industry
is
spending
much
money
on
reducing
the
fuel
consumption
and
fulfilling
the
vehicle
industry
is
spending
much
money
on
reducing
the
fuel
consumption
and
fulfilling
the
Abstract:
Todays
vehicle
industry
is
strictly
controlled
by
environmental
legislations.
The
vehicle
industry
is
spending
much
money
on
reducing
the
fuel
consumption
and
fulfilling
the
vehicle
industry
is
spending
much
money
on
reducing
the
fuel
consumption
and
fulfilling
the
emission
requirements
to
make
sales
possible
in
different
regions
in
the
world.
Before
introducing
emission
requirements
to
make
sales
possible
in
different
regions
in
the
world.
Before
introducing
vehicle
industry
is
spending
much
money
on
reducing
the
fuel
consumption
and
fulfilling
the
emission
requirements
toitmake
sales
possible
in
different
regions
in
thecycles
world.
Before
introducing
emission
requirements
sales
possible
regions
in
world.
a
vehicle
on
the
market,
is
tested
according
to
standardized
driving
to
specify
the
vehicle
a vehicle
vehicle on
on
the market,
market,to
itmake
is tested
tested
according
to different
standardized
driving
cycles
toBefore
specifyintroducing
the vehicle
vehicle
emission
requirements
toit
make
salesaccording
possible in
in
different
regions
in the
thecycles
world.
Before
introducing
a
the
is
to
standardized
driving
to
specify
apollutant
vehicle on
on
the market,
market,
it
is tested
tested
according
to standardized
standardized
driving
cycles to
to from
specify
the
vehicle
pollutant
emissions
and it
fuel
consumption.
These
cycles allow
allowdriving
some deviation
deviation
from
thethe
reference
emissions
and
fuel
consumption.
These
cycles
some
the
reference
a
vehicle
the
is
according
to
cycles
specify
the
vehicle
pollutant
emissions
and
fuel
consumption.
These
cycles
allow
some
deviation
from
the
reference
pollutant
emissions
fuel
consumption.
These
cycles
deviation
reference
vehicle
speed
during
tests,
e.g.
NEDC
allows
deviations
of
±2
km/h
and
±1
s.
This
paper
vehicle speed
speed
duringand
tests,
e.g.
NEDC allows
allows
deviations
of some
±2 km/h
km/h
and from
±1 s.
s.the
This
paper
pollutant
emissions
and
fuel e.g.
consumption.
Thesedeviations
cycles allow
allow
some
deviation
from
the
reference
vehicle
during
tests,
NEDC
of
±2
and
±1
This
paper
vehicle
speed
during
tests,
e.g.
NEDC
allows
deviations
of
±2
km/h
and
±1
s.
This
paper
uses
dynamic
programming
to
find
fuel
optimal
velocity
profiles,
given
the
allowed
deviations
uses
dynamic
programming
to
find
fuel
optimal
velocity
profiles,
given
the
allowed
deviations
vehicle
speed
during
tests,
e.g.
NEDC
allows
deviations
of
±2
km/h
and
±1
s.
This
paper
uses
dynamic
programming
to
find fuel
optimal
velocity
profiles,
given
the
allowed
deviations
uses
programming
to
optimal
velocity
profiles,
the
deviations
of
±2
km/h
and
±1
from
reference
speed
during
drive
cycle
test.
By
taking
advantage
of
of ±2
±2dynamic
km/h and
and
±1 sss from
from reference
reference
speed
during
drive cycle
cycle
test.given
By taking
taking
advantage
of the
the
uses
dynamic
programming
to find
find fuel
fuel
optimal
velocity
profiles,
given
the allowed
allowed
deviations
of
km/h
±1
speed
during
drive
test.
By
advantage
of
of
±2 km/h
km/h
and ±1
±1
s from
from
reference speed
speed
during
drive cycle
cycle
test.
By taking
taking
advantage
of the
the
allowed
deviation,
the
fuel consumption
consumption
canduring
be reduced
reduced
by up
uptest.
to 16.56
16.56
% according
according
to model
model
allowed
deviation,
the
fuel
can
be
by
to
%
to
of
±2
and
s
reference
drive
By
advantage
of
the
allowed
deviation,
the
fuel
consumption
can
be
reduced
by
up
to
16.56
%
according
to
model
allowed
deviation,
the
consumption
can
reduced
up
to
%
to
results,
NEDC
if
gear
selections
are
unrestricted
(i.e.
using
automatic
gearbox),
and
results, running
running
NEDC
if
gear
selections
are be
unrestricted
(i.e.
using
automatic
gearbox),
and
allowed
deviation,
the fuel
fuel
consumption
can
be
reduced by
by(i.e.
up using
to 16.56
16.56
% according
according
to model
model
results,
running
NEDC
if
gear
selections
are
unrestricted
automatic
gearbox),
and
results,
running
NEDC
if
gear
selections
are
unrestricted
(i.e.
using
automatic
gearbox),
and
up
to
5.90
%
if
changing
gears
according
to
the
specifications
in
the
drive
cycle.
Two
different
up
to
5.90
%
if
changing
gears
according
to
the
specifications
in
the
drive
cycle.
Two
different
results,
running
NEDC
if
gear
selections
are
unrestricted
(i.e.
using
automatic
gearbox),
and
up
to
5.90 %
if
changing
gears
according
to
the
specifications
in
the
drive
cycle.
Two
different
up
to
if
gears
to
specifications
in
drive
optimization
goals
are
investigated,
minimum
amount
of
mass
fuel
consumed
and
best
mileage.
optimization
goals
are investigated,
investigated,
minimum
amount
of mass
mass fuel
fuel
consumed
and Two
best different
mileage.
up
to 5.90
5.90 %
% goals
if changing
changing
gears according
according
to the
the
specifications
in the
the
drive cycle.
cycle.
Two
different
optimization
are
minimum
amount
of
consumed
and
best
optimization goals
goals are
are investigated,
investigated, minimum
minimum amount
amount of
of mass
mass fuel
fuel consumed
consumed and
and best
best mileage.
mileage.
optimization
mileage.
© 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Keywords:
Keywords: Dynamic
Dynamic Programming,
Programming, Cycle
Cycle Beating.
Beating.
Keywords:
Dynamic
Programming,
Cycle
Beating.
Keywords:
Dynamic
Programming,
Cycle
Keywords: Dynamic Programming, Cycle Beating.
Beating.
1.
E/ECE/TRANS/505/Rev.1/Add.82/Rev.5
1. INTRODUCTION
INTRODUCTION
E/ECE/TRANS/505/Rev.1/Add.82/Rev.5 (2015),
(2015), the
the drdr1.
INTRODUCTION
E/ECE/TRANS/505/Rev.1/Add.82/Rev.5
(2015),
the
dr1.
E/ECE/TRANS/505/Rev.1/Add.82/Rev.5
(2015),
the
drive
cycle
is
displayed
in
Figure
1.
The
NEDC
defines
1. INTRODUCTION
INTRODUCTION
ive
cycle
is
displayed
in
Figure
1.
The
NEDC
defines
E/ECE/TRANS/505/Rev.1/Add.82/Rev.5
(2015),
the
drive speed
cycle is
is
displayedthat
in isFigure
Figure
1. The
The
NEDC
defines
ive
cycle
displayed
in
1.
NEDC
defines
the
trajectory
supposed
to
followed
and
the speed
speed
trajectory
that
isFigure
supposed
to be
be
followed
and
ive
cycle is
displayedthat
in is
1. The
NEDC
defines
Todays
concern
of
the
environment
has
resulted
in
difthe
trajectory
supposed
to
be
followed
and
Todays
concern
of
the
environment
has
resulted
in
difthe
speed
trajectory
that
is
supposed
to
be
followed
and
which
gears
that
should
be
selected.
The
cycle
consists
Todays
concern
of
the
environment
has
resulted
in
difwhich
gears
that
should
be
selected.
The
cycle
consists
the
speed
trajectory
that
is
supposed
to
be
followed
and
Todays
concern
of
the
environment
has
resulted
in
different
standards
and
regulations
concerning
vehicle
emiswhich
gears
that
should
be
selected.
The
cycle
consists
ferent standards
standards
andthe
regulations
concerning
vehicleinemisemisTodays
concern of
environment
has resulted
dif- which
gears
that
be
The
consists
of
standstills
and
accelerations.
ferent
and
regulations
concerning
vehicle
of constant
constant
speeds,
standstills
and constant
constant
accelerations.
which
gearsspeeds,
that should
should
be selected.
selected.
The cycle
cycle
consists
ferent
standards
and
concerning
vehicle
emissions,
regulations
are
for
and
deof
constant
speeds,
standstills
and
constant
accelerations.
sions, these
these
regulations
are drivers
drivers
for improving
improving
and
de- of
ferent
standards
and regulations
regulations
concerning
vehicle
emisconstant
speeds,
standstills
and
constant
accelerations.
The
behavior
of
the
cycle
can
be
hard
to
follow
precisely
in
sions,
these
regulations
are
drivers
for
improving
and
deThe
behavior
of
the
cycle
can
be
hard
to
follow
precisely
in
of
constant
speeds,
standstills
and
constant
accelerations.
sions,
these
regulations
are
drivers
for
improving
and
developing
better
vehicles.
The
regulations
have
a
large
imThe
behavior
of the
the
cycle
can be
befrom
hardregions
to follow
follow
precisely
in
veloping
better
vehicles.are
Thedrivers
regulations
have aa large
large
imsions,
these
regulations
for improving
and imde- The
behavior
of
cycle
can
hard
to
precisely
in
a
vehicle
due
to
rapid
changes
with
constant
veloping
better
vehicles.
The
regulations
have
a
vehicle
due
to
rapid
changes
from
regions
with
constant
The
behavior
of
the
cycle
can
be
hard
to
follow
precisely
in
veloping
better
vehicles.
The
regulations
have
a
large
impact
on
the
technical
development
concerning
automotive
a vehicle
vehicle due
due to
to rapid
rapid changes
changes
from regions
regions
with
constant
pact on
on the
the
technical
development
concerning
automotive
veloping
better
vehicles.
The regulations
have automotive
a large im- aacceleration
from
with
speed.
The
question
pact
technical
development
concerning
acceleration
totoregions
regions
with constant
constant
speed. with
The constant
question
aacceleration
vehicle dueto
rapid changes
from regions
with
constant
pact
on
the
technical
development
concerning
automotive
industry
(Eriksson
and
Nielsen
(2014)).
To
sell
vehicles
to
regions
with
constant
speed.
The
question
industry
(Eriksson
and
Nielsen
(2014)).
To
sell
vehicles
pact
on
the
technical
development
concerning
automotive
acceleration
to
regions
with
constant
speed.
The
question
raised
here
is,
how
much
can
much
can
you
gain
in
industry
(Eriksson
and
Nielsen
(2014)).
To
sell
vehicles
raised
here
is,
how
much
can
much
can
you
gain
in terms
terms
acceleration
to
regions
with
constant
speed.
The
question
industry
(Eriksson
and
Nielsen
(2014)).
To
sell
vehicles
in
regions,
they
have
fulfill
regional
reraised
here
is, how
how
much
can much
much
can you
you
gain in
in
terms
in different
different
regions,and
theyNielsen
have to
to
fulfill the
the
regional
re- raised
industry
(Eriksson
(2014)).
To sell
vehicles
here
is,
much
can
can
gain
terms
of
fuel
consumed
by
the
vehicle,
if
taking
advantage
of
the
in
different
regions,
they
have
to
fulfill
the
regional
reof
fuel
consumed
by
the
vehicle,
if
taking
advantage
of
the
raised
here
is,
how
much
can
much
can
you
gain
in
terms
in
different
regions,
they
have
to
fulfill
the
regional
requirements
regarding
how
much
pollutant
emissions
the
of fuel
fuel consumed
consumed
byinthe
the
vehicle,
if taking
taking
advantage
of the
the
quirements
regarding
how have
muchtopollutant
pollutant
emissions
the
in
different regarding
regions, they
fulfill theemissions
regional the
re- of
by
vehicle,
if
advantage
of
allowed
deviation
vehicle
speed?
The
investigation
is
quirements
how
much
allowed
deviation
in
vehicle
speed?
The
investigation
is
of
fuel
consumed
by
the
vehicle,
if
taking
advantage
of
the
quirements
regarding
how
much
pollutant
emissions
the
vehicle
releases.
To
fulfill
the
emission
legislations
today,
allowed
deviation
in vehicle
vehicle
speed?
The
investigation
is
vehicle releases.
releases.
To fulfill
fulfill
the
emission
legislations
today,
quirements
regarding
howthe
much
pollutant
emissions
the allowed
deviation
in
speed?
The
investigation
is
made
using
both
specified
gear
changes
and
freely
selected
vehicle
To
emission
legislations
today,
made
using
both
specified
gear
changes
and
freely
selected
allowed
deviation
in
vehicle
speed?
The
investigation
is
vehicle
releases.
To
fulfill
the
emission
legislations
today,
electric
hybrids
are
introduced
on
the
market,
using
both
made
using
both
specified
gear
changes
and
freely
selected
electric
hybrids
are
introduced
on
the
market,
using
both
vehicle releases.
To introduced
fulfill the emission
legislations
today,
both
gear
selected
gears,
to
the
from
gear
selections
on
electric
hybrids
are
on the
the market,
using
both made
gears, using
to investigate
investigate
the impact
impact
from and
gear freely
selections
on
made
using
both specified
specified
gear changes
changes
and
freely
selected
electric
hybrids
are
on
both
an
machine
and
engine.
By
gears,
to
investigate
the
impact
from
gear
selections
on
an electric
electric
machine
and aaa combustion
combustion
engine. using
By downdownelectric
hybrids
are introduced
introduced
on the market,
market,
using
both gears,
to
investigate
the
impact
from
gear
selections
on
the
savings
in
terms
of
fuel
consumption.
During
vehicle
an
electric
machine
and
combustion
engine.
By
downthe
savings
in
terms
of
fuel
consumption.
During
vehicle
gears,
to
investigate
the
impact
from
gear
selections
on
an
electric
machine
and
a
combustion
engine.
By
downsizing
the
internal
combustion
engine
and
adding
a
electhe savings
savings in
in terms
terms
ofNEDC
fuel consumption.
consumption.
During
vehicle
sizing
the internal
internal
combustion
engine and
and
adding
elec- the
an
electric
machinecombustion
and a combustion
engine.
By aadownfuel
During
simulations,
full
(full
low
speed
sizing
the
engine
adding
elecsimulations,
both
fullof
NEDC
(full cycle)
cycle) and
and
lowvehicle
speed
the
savings inboth
terms
ofNEDC
fuel consumption.
During
vehicle
sizing
the
internal
combustion
engine
and
adding
aa electric
machine
with
aa battery,
energy
can
be
regenerated
simulations,
both
full
(full
cycle)
and
low
speed
tric
machine
with
battery,
energy
can
be
regenerated
sizing
the
internal
combustion
engine
and
adding
elecsimulations,
both
full
NEDC
(full
cycle)
and
low
section
of
NEDC
(city
cycle)
are
investigated.
Low
speed
tric
machine
with
a
battery,
energy
can
be
regenerated
section
of
NEDC
(city
cycle)
are
investigated.
Low
speed
simulations,
both
full
NEDC
(full
cycle)
and
low
tric
machine
with
aa Hybrid
battery,
energy
can
be
regenerated
and
used
if
needed.
Electric
Vehicles
(HEV)
are
section
of
NEDC
(city
cycle)
are
investigated.
Low
speed
and
used
if
needed.
Hybrid
Electric
Vehicles
(HEV)
are
tric
machine
with
battery,
energy
can
be
regenerated
of
NEDC
are
investigated.
Low
is
to
visualize
different
to
and used
used
if needed.
needed.
Hybrid of
Electric
Vehicles
(HEV)
are section
is displayed
displayed
to cycle)
visualize
the
different solutions
solutions
to
section
of
NEDC (city
(city
cycle)
arethe
investigated.
Low speed
speed
and
if
Hybrid
Electric
Vehicles
(HEV)
are
used
to
the
released
emissions.
Wang
section
is
displayed
to
visualize
the
different
solutions
to
used used
to reduce
reduce
the amount
amount
of
released
emissions.
Wang
and
if needed.
Hybrid of
Electric
Vehicles
(HEV)
are section
is
displayed
to
visualize
the
different
solutions
to
the
problem,
depending
on
which
optimization
goal
that
is
used
to
reduce
the
amount
released
emissions.
Wang
the
problem,
depending
on
which
optimization
goal
that
is
section
is
displayed
to
visualize
the
different
solutions
to
used
to
reduce
the
amount
of
released
emissions.
Wang
and
Lukic
(2012)
have
used
dynamic
programing
to
find
the problem,
problem,
depending
on
which
optimization
goal
that
is
and Lukic
Lukic
(2012)
have
used of
dynamic
programing
toWang
find the
used
to reduce
thehave
amount
released
emissions.to
depending
on
which
optimization
goal
that
is
selected
and
if
choosing
gears
according
to
cycle
or
freely
and
(2012)
used
dynamic
programing
find
selected
and
if
choosing
gears
according
to
cycle
or
freely
the
problem,
depending
on
which
optimization
goal
that
is
and
Lukic
(2012)
have
used
dynamic
programing
to
find
optimal
control
strategies
for
different
configurations
of
selected and
and
if choosing
choosing
gears
according
to cycle
cycle
or freely
freely
optimal
control
strategies
fordynamic
differentprograming
configurations
of selected
and
Lukic
(2012)strategies
have usedfor
to find
if
to
or
selecting
gears.
The
cycle
is
in
1,
optimal
control
different
configurations
of
selectingand
gears.
The full
fullgears
cycleaccording
is displayed
displayed
in Figure
Figure
1,
selected
if choosing
gears
according
to cycle
or freely
optimal
control
for
different
of
HEV’s.
programing
been
used
selecting
gears.
The
full
cycle
is
displayed
in
Figure
1,
HEV’s. Dynamic
Dynamic
programing
have
beenconfigurations
used by
by several
several
optimal
control strategies
strategies
for have
different
configurations
of selecting
gears.
The
full
cycle
is
displayed
in
Figure
1,
one
city
cycle
is
displayed
in
Figure
2,
in
both
figures
the
HEV’s.
Dynamic
programing
have
been
used
by
several
one
city
cycle
is
displayed
in
Figure
2,
in
both
figures
the
selecting
gears.
The
full
cycle
is
displayed
in
Figure
1,
HEV’s.
Dynamic
programing
have
been
used
by
several
authors
(for
example
Luu
et
al.
(2010)
and
Wollaeger
one city
city cycle
cycle
is allowed
displayed
in Figure
Figure
2, in
in both
both
figuresgear
the
authors Dynamic
(for example
example
Luu et
et have
al. (2010)
(2010)
and by
Wollaeger
HEV’s.
programing
been used
several one
is
displayed
in
2,
figures
the
reference
speed,
speed
deviation
specified
authors
(for
Luu
al.
and
Wollaeger
reference
speed,
allowed
speed
deviation
and
specified
gear
one
city cycle
is allowed
displayed
in Figure
2, inand
both
figuresgear
the
authors
(for
example
Luu
et
al.
(2010)
and
Wollaeger
et
al.
(2012))
to
solve
vehicle
optimization
problems
conreference
speed,
speed
deviation
and
specified
et
al.
(2012))
to
solve
vehicle
optimization
problems
conauthors
(for
example
Luu
et
al.
(2010)
and
Wollaeger
speed,
allowed
speed
deviation
and
specified
gear
selections
are
displayed.
et al.
al. (2012))
(2012))
to
solve vehicle
vehicle
optimization
problems
con- reference
selections
are
displayed.
reference
speed,
allowed
speed
deviation
and
specified
gear
et
to
solve
optimization
problems
concerning
speed
trajectories
and
fuel
consumption.
Mensing
selections are
are displayed.
displayed.
cerning
speed to
trajectories
and optimization
fuel consumption.
consumption.
Mensing
et
al. (2012))
solve vehicle
problems
con- selections
cerning
speed
trajectories
and
fuel
Mensing
cerning
speed
trajectories
and
fuel
et
dynamic
to
optimal
et al.
al. (2011)
(2011)
uses
dynamic
programing
to find
find Mensing
optimal selections are displayed.
cerning
speed uses
trajectories
andprograming
fuel consumption.
consumption.
Mensing
et
al.
(2011)
uses
dynamic
programing
to
find
optimal
et
al.
uses
programing
to
optimal
ECO-driving
Vehicles
are
according
to
1.1 Contributions
Contributions
ECO-driving
trajectories.
Vehicles
are tested
tested
according
to 1.1
et
al. (2011)
(2011) trajectories.
uses dynamic
dynamic
programing
to find
find
optimal
1.1
Contributions
ECO-driving
trajectories.
Vehicles
are
tested
according
to
1.1
ECO-driving
trajectories.
Vehicles
are
tested
according
to
standardized
drive
cycles
to
make
it
possible
to
compare
1.1 Contributions
Contributions
standardized
drive
cycles
to
make
it
possible
to
compare
ECO-driving
trajectories.
Vehicles
are
tested
according
to
standardized
drive
cycles to
to make
it possible
possible
toincompare
compare
standardized
drive
cycles
it
different
manufactures.
Vehicles
tested
specific
This paper
paper uses
uses aaa dynamic
dynamic programing
programing approach
approach to
to show
show
different vehicle
vehicle
manufactures.
Vehicles
testedto
incompare
specific This
standardized
drive
cycles to make
make
it possible
toin
This
paper
uses
dynamic
programing
approach
to
show
different
vehicle
manufactures.
Vehicles
tested
specific
This
paper
uses
aa dynamic
programing
approach
to
show
different
vehicle
manufactures.
Vehicles
tested
in
specific
how
fuel
economy
is
affected
by
smart
driving,
within
drive
cycles
can
today
be
driven
by
robots
to
achieve
how
fuel
economy
is
affected
by
smart
driving,
within
drive
cycles
can
today
be
driven
by
robots
to
achieve
This
paper
uses
dynamic
programing
approach
to
show
different
vehicle
manufactures.
Vehicles
tested
in
specific
how
fuel
economy
is
affected
by
smart
driving,
within
drive
cycles
can
today
be
driven
by
robots
to
achieve
fuel
by
within
drive
cycles
today
robots
to
the
limits
the
drive
The
contribution
is
good
following,
but
NEDC
(New
European
Drive
the legal
legal
limits of
of is
theaffected
drive cycle.
cycle.
The driving,
contribution
is
good reference
reference
following,
but driven
NEDC by
(New
European
Drive how
how
fuel economy
economy
is
affected
by smart
smart
driving,
within
drive
cycles can
can
today be
be
driven
by
robots
to achieve
achieve
the
legal
limits
of
the
drive
cycle.
The
contribution
is
good
reference
following,
but
NEDC
(New
European
Drive
the
legal
limits
of
the
drive
cycle.
The
contribution
is
good
reference
following,
but
NEDC
(New
European
Drive
a
quantification
how
much
the
fuel
consumption
can
Cycle)
allows
some
deviation
in
vehicle
speed
during
vehia
quantification
how
much
the
fuel
consumption
can
Cycle)
allows
some
deviation
in
vehicle
speed
during
vehithe
legal
limits
of
the
drive
cycle.
The
contribution
is
good
reference
following,
but
NEDC
(New
European
Drive
a quantification
quantification
of how
how
much the
the
fuel
consumption
can
Cycle)
allows
some deviation
deviation
in vehicle
vehicle
speed
during
vehi- abe
of
much
consumption
can
Cycle)
allows
some
in
during
vehireduced
taking
advantage
of
the
deviation
cle
Regulations
concerning
the
drive
NEDC
bequantification
reduced by
by
taking
advantage
of fuel
the allowed
allowed
deviation
cle testing.
testing.
Regulations
concerning
the speed
drive cycle
cycle
NEDC
of how
much the
fuel
consumption
can
Cycle)
allows
some deviation
in vehicle
speed
during
vehi- abe
reduced
by
taking
advantage
of
the
allowed
deviation
cle
testing.
Regulations
concerning
the
drive
cycle
NEDC
be
reduced
by
taking
advantage
of
the
allowed
deviation
cle
testing.
Regulations
concerning
the
drive
cycle
NEDC
speed
during
drive
paper
allows
aa deviation
in
speed
following
of
km/h
in reference
reference
speed
during
drive cycle
cycle
testing,
the
paper
allows
deviation
in reference
reference
speedthe
following
of ±2
±2NEDC
km/h in
be
reduced by
taking
advantage
of thetesting,
allowed the
deviation
cle
testing.
Regulations
concerning
drive cycle
in
reference
speed
during
drive
cycle
testing,
the
paper
allows
a
deviation
in
reference
speed
following
of
±2
km/h
speed
drive
testing,
paper
allows
in
following
of
km/h
also
show
gear
if
and
±1
during
decelerations
constant
alsoreference
show optimal
optimal
gear selections
selections
if gears
gears
arethe
selected
and
±1aa sssdeviation
during accelerations,
accelerations,
decelerations
and
constant
in
reference
speed during
during
drive cycle
cycle
testing,are
theselected
paper
allows
deviation
in reference
reference speed
speed
followingand
of ±2
±2
km/h in
also
show
optimal
gear
selections
if
gears
are
selected
and
±1
during
accelerations,
decelerations
and
constant
also
show
optimal
gear
selections
if
gears
are
selected
and
±1
s
during
accelerations,
decelerations
and
constant
freely.
Two
different
optimization
goals
are
analyzed,
High
speed
traveling,
see
E/ECE/324/Rev.1/Add.82/Rev.5freely.
Two
different
optimization
goals
are
analyzed,
High
speed
traveling,
see
E/ECE/324/Rev.1/Add.82/Rev.5also
show
optimal
gear
selections
if
gears
are
selected
and
±1
s
during
accelerations,
decelerations
and
constant
freely. Two
Two different
different optimization
optimization goals
goals are
are analyzed,
analyzed, High
High
speed traveling,
traveling, see
see E/ECE/324/Rev.1/Add.82/Rev.5E/ECE/324/Rev.1/Add.82/Rev.5- freely.
speed
speed traveling, see E/ECE/324/Rev.1/Add.82/Rev.5- freely. Two different optimization goals are analyzed, High
Copyright
© 2016,
2016
669
2405-8963 ©
IFAC (International Federation of Automatic Control)
Copyright
2016 IFAC
IFAC
669 Hosting by Elsevier Ltd. All rights reserved.
Copyright
©
2016
IFAC
669
Copyright
©
2016
IFAC
669
Peer
review
under
responsibility
of
International
Federation
of
Automatic
Copyright © 2016 IFAC
669Control.
10.1016/j.ifacol.2016.08.095
IFAC AAC 2016
658
June 19-23, 2016. Norrköping, Sweden
(1) Low Fuel as possible, using manual gearbox with gear
selections specified by the drive cycle.
(2) High Mileage as possible, using manual gearbox with
gear selections specified by the drive cycle.
(3) Low Fuel as possible, using automatic gearbox with
free possibility to select gears.
(4) High Mileage as possible, using automatic gearbox
with free possibility to select gears.
Cycle speed and Speed limits
150
Velovity [km/h]
Kristoffer Ekberg et al. / IFAC-PapersOnLine 49-11 (2016) 657–664
Vref
Vmin
100
Vmax
50
0
0
200
400
6
Gear [-]
600
800
1000
1200
Time [s]
Gear selections
2.1 Vehicle model
To calculate the fuel consumption for a specific vehicle
driving according to the drive cycle, a vehicle model is
implemented. The model calculates the amount of fuel
needed for a certain energy requirement at the wheels.
During simulations, vehicle data according to Table 1 is
used. The gearing setup γ is used for both the specified
and unspecified gear changes.
4
2
0
0
200
400
600
800
1000
1200
Time [s]
Fig. 1. Full NEDC, displays speed in km/h, time in seconds
and gear number. Speed trajectory is displayed in top
figure and which gears to select is displayed in bottom
figure. Gear number 0 corresponds to neutral gear.
Cycle speed and Speed limits
Velovity [km/h]
60
Vref
Vmin
40
Vmax
20
0
0
50
3
Gear [-]
100
150
200
150
200
Time [s]
Gear selections
2
1
0
0
50
100
Time [s]
Fig. 2. Section from NEDC, city cycle, displays speed
in km/h, time in seconds and gear number. Speed
trajectory is displayed in top figure and which gears
to select is displayed in bottom figure. Gear number
0 corresponds to neutral gear.
Mileage and Low Fuel, with and without restrictions on the
gear changes made in the vehicle. Drive cycle simulations
are performed on both low speed part of NEDC cycle and
full NEDC cycle.
2. CYCLE BEATING
Table 1. Vehicle data used during simulations.
Parameter
Value
Unit
ρair
ρf uel
Cd
Cr
A
g
m
rw
Jw
γ
ηgearbox
Je
Te,max
Ne,max
Vd
e
pme0
Hl
1.18
737.2
0.32
0.015
2.31
9.81
1500
0.30
0.6
0 13.0529 8.1595 5.6651 4.2555 3.2623
0.98
0.2
115
5000
1.497 × 10−3
0.4
1 × 105
44.6 × 106
kg/m3
kg/m3
−
−
m2
m/s2
kg
m
kg m2
−
−
kg m2
N m
rpm
m3
−
Pa
J/kg
The forces acting on a vehicle traveling on flat road are
described in equations (1a), (1b) and (1c) (see Guzzella
and Sciarretta (2007)):
1
(1a)
Fdrag = ρair Cd Av 2
2
Jw
(1b)
Facc = a(m + 2 )
rw
Froll = Cr mg
(1c)
The energy dissipated during one time step, from time tn
to tn+1 , with the assumption that the vehicle acceleration
a is constant, is described by:
! tn+1
F vdt
(2)
E=
tn
Cycle beating refers to performing well while fulfilling the
specifications of the cycle. By using a vehicle model to
simulate the forces acting on the vehicle during driving,
the required fuel consumption is calculated. This paper
uses dynamic programing to find the least fuel consuming
driving strategies. The analysis consists of four cases.
The cases have different setups and different optimization
goals. Optimizing the speed trajectory within the limit of
±2 km/h and ±1 s from reference speed, to achieve as
670
If using equations (1a), (1b) and (1c) together with (2),
following energy equations can be derived:
1
2
Edrag = ρair Cd A(vn+1
+ vn2 )(vn+1 + vn )∆t
(3a)
8
Jw
a
(3b)
Eacc = (m + 2 )(vn+1 + vn )∆t
2
rw
1
Eroll = Cr mg(vn+1 + vn )∆t
(3c)
2
IFAC AAC 2016
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Kristoffer Ekberg et al. / IFAC-PapersOnLine 49-11 (2016) 657–664
Energy required at the wheels is Ereq = Edrag + Eacc +
Eroll .
659
ω1 θ1
ω2 θ2
T
2.2 Gearbox and clutch model
To extend the analysis of the drive cycle, a gearbox model
and clutch model are implemented. These models are used
to analyze the use of free gear selections. The models takes
following actions into account:
(1) Changes in kinetic energy due to engine rotation.
(2) Dissipated energy in clutch during gear changes.
(3) Start/stop model, restricts the engine speed to 800
rpm during starts and stops.
The total amount of energy required by the combustion
engine is described in equation (4).
Eice = Egearbox + Eengine + Eclutch
(4)
Energy required by gearbox (Egearbox ), with gearbox efficiency ηgearbox and gear ratio γ (se values in Table 1) is
displayed in equation (5).
Ereq
Egearbox =
(5)
ηgearbox
The kinetic rotational energy inside the engine is included
in the analysis, to enable the use of engine braking, the
kinetic rotational energy in the engine at current and next
time step is expressed in equations (6a) and (6b).
Je
vcurr
2
, where ωice0 =
Eeng0 = ωice0
ginit
(6a)
2
rw
vnext
Je
2
, where ωice1 =
Ggrid
(6b)
Eeng1 = ωice1
2
rw
Where Ggrid represents all possible gear changes from the
current time step with gear, ginit to next time step. The
cost to perform a gear change, is both the change in engine
rotational speed (which can be both positive and negative)
and the energy dissipated in the clutch. The kinetic energy
needed to change the engine speed from current time step
to next time step (Eengine ) is described in equation (7).
Eengine = Eeng1 − Eeng0
(7)
The energy dissipated in the clutch (Eclutch ) is described
in equation (14). Eclutch will always be positive, since the
clutch, independent of if there is a upshift or downshift
maneuver, will dissipate heat when performing clutch lock
maneuver.
A graphical representation of the clutch model is displayed
in Figure 3. The description of the energy dissipated
in the clutch during a clutch maneuver is described in
Bhandari (2007). The rotational speeds in the clutch can
be described by the incoming and outgoing shaft torques,
see equation (8).
T1
T2
θ̈1 =
and θ̈2 =
(8)
J1
J2
The rotational speeds θ̇i are received by integrating the
above functions from time t = 0 to time t = tclutch . The
initial rotational speeds att t = 0 are ω1 and ω2 .
! tclutch
T1
T1
θ̇1 =
dt =
tclutch + ω1
(9a)
J
J1
1
!0 tclutch
T2
T2
dt =
tclutch + ω2
(9b)
θ̇2 =
J
J2
2
0
671
J1
J2
Fig. 3. Clutch discs, index 1 corresponds to disc connected
to driveline, index 2 corresponds to engine disc. The
torque T indicates the torque delivered between the
two clutch discs.
The rotational speed difference between the incoming and
outgoing disc in the clutch is described as:
∆θ̇ = θ̇1 − θ̇2 =⇒
T2
T1
tclutch + ω1 − tclutch − ω2
∆θ̇ =
J1
J2
Torque balance T = −T1 = T2 gives:
J1 + J2
)T tclutch + ω1 − ω2
∆θ̇ = −(
J1 J2
The clutch lockup is completed when the rotational
difference ∆θ̇ equals 0, equation (11) gives:
J1 + J 2
0 = −(
)T tclutch + ω1 − ω2 =⇒
J1 J2
ω 1 − ω2 J 1 J2
tclutch =
T
J1 + J2
(10a)
(10b)
(11)
speed
(12a)
(12b)
The energy dissipated in the clutch during a clutch lockup
is described in equation (13b).
! tclutch
T ∆θ̇dt =⇒
(13a)
Eclutch =
0
Eclutch
1 J 1 J2
=
(ω1 − ω2 )2
2 J1 + J2
(13b)
2
By introducing J1 = Jv = γ12 (mrw
+ Jw ), J2 = Je . A gear
change is assumed to take place during one time step, ∆t,
therefore the rotational speeds ω1 = ωice0 and ω2 = ωice1 ,
when a gear change is performed. If no gear change is
performed, Eclutch = 0.
"
0
if ∆θ̇ = 0
Eclutch = 1 Jv Je
(14)
2
(ω
−
ω
)
if
∆θ̇ ̸= 0
ice0
ice1
2 Jv +Je
2.3 Combustion engine model
Total mass fuel consumed during one time step can be
calculated using an engine model based on Willans approximation (see Guzzella and Sciarretta (2007)), the mass fuel
consumed during one time step is described in equation
(15).
wice
pme0 Vd
∆mf =
(Tice +
+ Je ω̇ice )∆t
(15)
Hl e
4π
Equation (15) is rewritten with equation (16) to form the
expression in equation (17a). In both equations (17a) and
(17b) the inertia Je in equation (15) is included in Eice .
The mass fuel consumed during one time step from tn to
tn+1 , is used to form the cost function in the dynamic
programming algorithm.
IFAC AAC 2016
660
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Kristoffer Ekberg et al. / IFAC-PapersOnLine 49-11 (2016) 657–664
ωice1 − ωice0
2ω̇ice
, where ω̇ice =
2
2
ωice1
− ωice0
∆t
(16)
During starts, stops and standstills, the engine speed ωice0
and ωice1 are restricted to a constant speed of 800 rpm.
During starting maneuvers, the rotational speed is kept
at 800 rpm until the rotational speeds of the engine
and gearbox (clutch incoming and outgoing rotational
speeds) equals each other. This restriction is implemented
to include the energy dissipated during starting maneuver
when clutch is slipping and first gear is engaged.
2
w2 − wice0
pme0 Vd
∆mf = ice1
) (Acceleration)
(Tice +
Hl e2ẇice
4π
(17a)
wice0
pme0 Vd
(Tice +
)∆t (Constant speed)
∆mf =
Hl e
4π
(17b)
Tice = Eice
2.4 Dynamic Programing Solving Optimization problem
The drive cycle is driven with the specified gearbox and
clutch model in all four cases, two of these where the gears
are selected freely by the optimization, and two where the
gear changes are according to the specified drive cycle.
Dynamic programing (Bertsekas (2000)) is used to find
the best driving pattern to reduce the fuel consumption
as much as possible.Two different optimization goals are
investigated, one where the total mass fuel is minimized
and one where the mean fuel consumption is minimized.
When the free gear selection mode is simulated, the vehicle
is free to select gears as long as the cycle requirements
and vehicle restrictions are fulfilled. The problem size
changes from 1D to 2D, when changing from specified gear
changes to free gear changes, a graphical representation
of the different gear and speed possibilities are displayed
in Figure 4. A backward algorithm is used, solving the
problem from the final states at time tend to the states at
starting time tstart . The restrictions in the vehicle model
are listed in Table 2.
Table 2. Constraints on the vehicle model.
Constraints
γ ̸= 0 if vavg > 0
ω̇e < ω̇e,max
Tice < Tice,max
Ne ≤ Ne,max
If γωw < ωe,min =⇒ γ = 1 st gear.
!T
JHM = min ! 0T
JLF = min
2.5 Optimization problem
The two different optimization problems solved during
the simulation cases are displayed in equations (HM) and
(LF).
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vopt dt
0
" T
ṁf dt
(HM)
(LF)
0
where mf is the mass of fuel consumed, v is the vehicle
speed. Equation (HM) for High Mileage and equation (LF)
is the problem to solve for Low Fuel during the cycle.
The problem consists of three states, vehicle speed, driven
distance and selected gear. The states driven distance and
vehicle speed are linked to each other. The vehicle speed
is defined by the drive cycle and the allowed deviation
from the speed reference, the driven distance is solved
with the knowledge of the vehicle speed and the time step
length. Since the total driven distance is not specified by
the drive cycle, the state driven distance is unconstrained.
By solving the problem with dynamic programing, a twodimensional grid at each time step is needed, consisting of
all possible gears and all possible vehicle speeds from tn
to tn+1 .
3. RESULTS
Results for the european city cycle are displayed in Figures
A.1, A.3 and A.5. Results for the full cycle are displayed in
Figures A.2 , A.4 and A.6. The figures displays the optimal
speed trajectory for each case, fuel flow to combustion
engine, selected gear and deviation from cycle reference
speed. The results in Figures A.1 and A.2 (when using
specified gear changes) are used as references when calculating the savings in terms of fuel consumption. All
the simulations are summarized and displayed in Table
3, where the savings in fuel consumption is calculated
according to equation (18), where ∅ref is fuel consumption (l/100 km) for reference simulation and ∅case is fuel
consumption for the case that is to be compared.
ηsavings
Fig. 4. Graphical representations of the dynamic programing algorithm. At each time step from tn to tn+1 , the
cost for each opportunity of selecting speed, v, and
gear γ, is calculated.
ṁf dt
case
= 100
∅ref − ∅case
∅ref
(18)
The vehicle speed trajectory is optimized within the limit
of ±2 km/h and ±1 s from reference speed, to achieve as
Low Fuel consumption as possible, and as High Mileage as
possible, using both manual and automatic gearbox with
free possibility to select gears. The Low Fuel consumption
optimization corresponds to comparing the fuel consumed
by the nominal cycle distance in the drive cycle, with
the fuel actually consumed. Since the vehicle speed and
nominal cycle speed are unconnected, the solution ends
up with minimizing the total amount of fuel used during
the drive cycle. The High Mileage optimization uses the
driven distance and amount of fuel consumed, since the
vehicle speed is allowed to deviate from the cycle speed
reference, also the driven distance will vary. All solutions
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where a deviation in vehicle speed is allowed, have one
thing in common, the optimal solution is to extend the
time in fuel-cut (e.g. where the engine brakes) compared
to reference simulations. Due to the speed variance from
the reference speed, the total driven distances are different
for the different simulations.
3.1 Low speed part of NEDC
Speed deviations not allowed (optimizing for Low Fuel)
If the deviations in vehicle speed had been forbidden,
the gear changes had been the only degree of freedom
if using automatic gearbox. In Figure A.1 the solutions
for specified gear changes and optimized gear changes are
displayed. The optimal solution when gear selections are
unrestricted, is to select highest gear possible, to reduce
the fuel flow and thereby reduce fuel consumed (the fuel
consumption is reduced from 6.79 to 5.49 l/100 km).
Optimizing for High Mileage When vehicle speed is allowed to deviate from reference speed, the problem has two
degrees of freedom (when gear changes are unrestricted),
gear changes and vehicle speed. In Figure A.3 the solutions
for the High Mileage optimizations are displayed. The
optimal solution when the gear selections are unrestricted,
is to extend the driven distance, from 1018 m (in reference
case) to 1042 m. When using specified gear changes, the
optimal solution is to shorten the driven distance, from
1018 m (in reference case) to 971 m, the driven distance is
only extended when using fuel cut during engine braking.
Also the solver uses highest gear possible when optimized
gear changes are used.
Optimizing for Low Fuel When vehicle speed is allowed
to deviate from reference speed, the problem has two
degrees of freedom (when gear changes are unrestricted),
gear changes and vehicle speed. In Figure A.5 the solutions
for the Low Fuel optimizations are displayed. It is visible
that the optimal solution is to keep the vehicle speed as low
as possible, to keep the fuel flow low. The lowest allowed
speed is kept along the cycle, except during engine braking,
where the possibility to run on low or no fuel is present.
3.2 Full NEDC
The low speed part of NEDC is previously displayed, to
show the different optimal solutions found by the dynamic
programming algorithm. The full NEDC cycle is also
simulated, to investigate how the allowed deviation in
vehicle speed may be used during a full drive cycle test.
Speed deviation not allowed (optimizing for Low Fuel) In
Figure A.2 it is visible that the optimal solution for the low
speed parts of the drive cycle looks similar to the city cycle
simulation (see Figure A.1). The solver uses the ability
to change gears, by selecting the highest gear possible,
to reduce the fuel flow and thereby the fuel consumed.
Investigating the gear selections pattern in Figure A.2, at
around 900s, the solver changes from 5th to 4th and down
to 3rd gear for a short moment and then up to 5th gear
again, during this time the vehicle does not require any fuel
flow. The optimal solution at that moment is therefore to
store kinetic energy in the engine (changing down to 3rd
673
661
gear), then use the kinetic energy stored in the engine
when needed (changing from 3rd to 5th gear).
Optimizing for High Mileage Studying the vehicle velocity profile in Figure A.4, it shows that the optimal
solution for both specified and optimized gear selections is
to increase the driven distance by driving faster than the
reference profile at some occasions (such as accelerating
earlier than the reference profile). It is also visible that
the driven distance when using specified gear changes is
shorter (10790 m) compared to optimal gear selections
(11306 m). The optimal solution when gear selections are
unrestricted, is to choose the highest gear possible while
fulfilling the vehicle constraints, the use of higher gear
reduces the fuel flow.
Optimizing for Low Fuel As for the city cycle optimized
for LF, the optimal vehicle speed is following the low speed
boundary, except during roll outs when engine braking
is possible (which cuts the fuel flow). Even though the
highest gear is selected when possible (running the case
when gear changes are unrestricted), the optimal vehicle
speed stays below the reference speed (on the lower vehicle
speed boundary condition). This is visible in Figure A.6.
3.3 Summary Results
Table 3 summarizes all the results for the different cycles.
Table 3. Results from all the different simulation cases. The case ”LF (vref )” follows the
specified vehicle speed exactly.
Case
Description
Savings
%
City Cycle
Reference
LF
HM
City Cycle
LF (vref )
LF
HM
Full Cycle
Reference
LF
HM
Full Cycle
LF (vref )
LF
HM
Manual
4.71%
8.40 %
Automatic
19.15 %
24.60 %
29.60 %
Manual
3.61 %
5.90 %
Automatic
9.67 %
13.28 %
16.56 %
Mass Fuel
g
Mean Fuel
l/100 km
Distance
m
50.95
44.14
44.53
6.79
6.47
6.22
1018
925
971
41.22
35.52
36.72
5.49
5.12
4.78
1018
941
1042
495.57
450.32
456.63
6.10
5.88
5.74
11021
10394
10790
448.01
408.25
424.28
5.51
5.29
5.09
11021
10469
11306
3.4 Parameter sensitivity analysis
A parameter sensitivity analysis has been performed, investigating the change in fuel consumption, if changing
parameters vehicle mass m, tire rolling resistance cr and
vehicle frontal area A. The analysis is performed for the
reference case in Figure A.2, with the full NEDC cycle,
where both the vehicle speed and gear changes are specified. The result of the parameter sensitivity analysis is
displayed in Table 4. The result is displayed as the difference in fuel consumption, calculated according to Equation
Kristoffer Ekberg et al. / IFAC-PapersOnLine 49-11 (2016) 657–664
Table 4. Parameter sensitivity analysis. The
top percentage shows the size of the change in
the parameters A, m and cr . The result shows
the relative difference in fuel consumption,
compared to the reference case in Figure A.2,
using specified gear selections (increased fuel
consumption in negative numbers).
Parameter
10%
5%
A
m
cr
-1.743
-4.504
-2.410
-0.872
-2.250
-1.205
−5%
0.871
2.249
1.204
−10%
1.742
4.495
2.409
4. CONCLUSION
Results show that by optimizing the driving pattern within
the allowed deviation from the drive cycle reference velocity of ±2 km/h and ±1 second, the fuel consumption
of the driven vehicle can be clearly reduced. A dynamic
programming, backward algorithm has been used, to find
the optimal solutions for High Mileage and Low Fuel over
a specified drive cycle, with allowed deviations in vehicle
speed. The use of free possibility to change gears was
also examined. The results in table 3, show that the fuel
consumption can be clearly reduced by taking advantage
of the allowed deviation in vehicle speed. Optimization
results shows that it is optimal to use engine braking when
possible, to enable fuel cut, independent of if the optimization goal is High Mileage or Low Fuel. The possibility
to deviate from reference speed leads to different driven
distances for the different cases. To increase High Mileage,
the driven distance is increased, while fuel flow is kept low.
When optimizing for Low Fuel, the vehicle speed is made
as low as possible, to lower the fuel flow in the engine.
Comparing the savings of fuel consumption in Table 3 with
the changes of fuel consumption in Table 4, it shows that
the savings when taking advantage of the allowed speed
deviation is generally larger than a 10% decrease of any
of the parameters frontal area A, rolling resistance cr or
vehicle mass m used during simulations.
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of Engines and Drivelines. John Wiley and Sons Ltd,
United Kingdom.
Guzzella, L. and Sciarretta, A. (2007). Vehicle Propulsion
Systems. Springer, New York.
Luu, H.T., Nouvelière, L., and Mammar, S. (2010). Dynamic programming for fuel consumption optimization
on light vehicle. In Prep. IFAC symp. Advances in
Automotive Control. AAC2010.
Mensing, F., Trigui, R., and Bideaux, E. (2011). Vehicle
trajectory optimization for application in eco-driving.
In Vehicle Power and Propulsion Conference (VPPC),
2011 IEEE, 1–6. doi:10.1109/VPPC.2011.6042993.
Wang, R. and Lukic, S. (2012). Dynamic programming
technique in hybrid electric vehicle optimization. In
Electric Vehicle Conference (IEVC), 2012 IEEE International, 1–8. doi:10.1109/IEVC.2012.6183284.
Wollaeger, J., Kumar, S., Onori, S., Filev, D., Ozguner,
U., Rizzoni, G., and Di Cairano, S. (2012). Cloudcomputing based velocity profile generation for minimum fuel consumption: A dynamic programming based
solution. In American Control Conference (ACC), 2012,
2108–2113. doi:10.1109/ACC.2012.6314931.
Appendix A. OPTIMIZATION RESULTS
20
1
20
40
60
100
120
140
160
180
Specified Gear
Optimal Gear
0
0
20
40
60
80
100
120
140
160
180
140
160
180
160
180
Gear selections
6
4
2
0
0
20
40
60
80
100
120
Difference from reference velocity
1
Velocity [km/h]
80
Fuel Consumption Spec/Opt 6.79/5.49 l/100 km
0.5
Gear [-]
674
0
×10-3
This work was supported by the Vinnova Industry Excellence Center: LINK-SIC Linköping Center for Sensor
Informatics and Control.
Bertsekas, D.P. (2000). Dynamic Programming and Optimal Control, volume 2. Athena Scientific, Bellmonth,
Massachusetts.
Bhandari, V.B. (2007). Design of Machine Elements. Tata
McGraw-Hill.
E/ECE/324/Rev.1/Add.82/Rev.5E/ECE/TRANS/505/Rev.1/Add.82/Rev.5
(2015).
Regulation no. 83 uniform provisions concerning the
approval of vehicles with regard to the emission of
pollutants according to engine fuel requirements.
Boundary
Boundary
Specified Gear
Optimal Gear
40
0
ACKNOWLEDGEMENTS
REFERENCES
Driven distance Spec/Opt 1018/1018 m
60
Velocity [km/h]
(18). If comparing best saving in fuel consumption in Table
4 with for example Full Cycle - Manual - LF in Table 3, one
of the smaller savings due to the allowed speed deviation
corresponds the biggest change of mass in table 4.
Fuel Flow [kg/s]
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0.5
0
-0.5
-1
0
20
40
60
80
100
120
140
Time [s]
Fig. A.1. Exact speed reference following with cycle specified gear changes and automatic gear changes. Optimizing for LF with specified vehicle speed and both
specified and optimal gear changes. It shows in figure
that fuel cut is used with both specified and optimal
gear changes.
IFAC AAC 2016
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Velocity [km/h]
100
50
0
0
200
Fuel Flow [kg/s]
400
600
800
1
0
200
400
800
200
400
600
800
0
0
200
Velocity [km/h]
0
400
600
800
600
0
200
400
600
-5
200
400
600
Velocity [km/h]
80
100
120
140
160
20
40
60
100
120
140
160
0
Gear [-]
20
40
60
80
100
120
40
140
160
Velocity [km/h]
Velocity [km/h]
-5
60
80
100
120
140
120
140
160
180
0
20
40
60
80
100
120
140
160
180
140
160
180
160
180
Gear selections
2
0
20
40
60
80
100
120
Difference from reference velocity
0
40
100
4
0
180
5
20
80
Specified Gear
Optimal Gear
Difference from reference velocity
0
60
Fuel Consumption Spec/Opt 6.47/5.12 l/100 km
6
2
0
20
0.5
180
4
0
1
Gear selections
6
Gear [-]
80
0
×10-3
Specified Gear
Optimal Gear
0
20
0
0.5
0
Boundary
Boundary
Specified Gear
Optimal Gear
40
180
Fuel Flow [kg/s]
Velocity [km/h]
Fuel Flow [kg/s]
60
Driven distance Spec/Opt 925/941 m
60
Fuel Consumption Spec/Opt 6.22/4.78 l/100 km
×10-3
1
40
1000
Fig. A.4. Optimizing for HM, full cycle, when velocity
deviations are allowed. Both specified and optimal
gear changes. The optimal gear changes case has
longer driven distance than specified gear case.
Driven distance Spec/Opt 971/1042 m
20
800
Time [s]
Boundary
Boundary
Specified Gear
Optimal Gear
0
1000
0
0
20
0
800
5
1000
Fig. A.2. Exact speed reference following with cycle specified gear changes and automatic gear changes. Optimizing for LF with specified vehicle speed and both
specified and optimal gear changes. It shows in figure
that fuel cut is used with both specified and optimal
gear changes.
40
1000
2
Time [s]
60
800
Difference from reference velocity
-0.5
200
400
4
0
1000
0.5
0
1000
Gear selections
Difference from reference velocity
1
800
1
6
2
600
0.5
1000
4
0
400
Fuel Consumption Spec/Opt 5.74/5.09 l/100 km
Specified Gear
Optimal Gear
2
Gear [-]
Gear [-]
600
200
1.5
Gear selections
6
-1
0
×10-3
0.5
0
50
0
1.5
0
100
1000
Specified Gear
Optimal Gear
2
Boundary
Boundary
Specified Gear
Optimal Gear
Fuel Consumption Spec/Opt 6.1/5.51 l/100 km
×10-3
Velocity [km/h]
Driven distance Spec/Opt 10790/11306 m
Boundary
Boundary
Specified Gear
Optimal Gear
Fuel Flow [kg/s]
Velocity [km/h]
Driven distance Spec/Opt 11021/11021 m
663
160
180
5
0
-5
0
Time [s]
20
40
60
80
100
120
140
Time [s]
Fig. A.3. Optimizing for HM, city cycle, when velocity
deviations are allowed. Both specified and optimal
gear changes. The optimal gear changes case has
longer driven distance than specified gear case.
675
Fig. A.5. Optimizing for LF, city cycle, when velocity
deviations are allowed. Both specified and optimal
gear changes. The optimal gear changes case has
longer driven distance than specified gear case.
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Velocity [km/h]
Driven distance Spec/Opt 10394/10469 m
Boundary
Boundary
Specified Gear
Optimal Gear
100
50
0
0
Fuel Flow [kg/s]
×10-3
200
400
600
800
1000
Fuel Consumption Spec/Opt 5.88/5.29 l/100 km
Specified Gear
Optimal Gear
2
1.5
1
0.5
0
0
200
400
6
Gear [-]
600
800
1000
800
1000
Gear selections
4
2
0
0
200
400
600
Velocity [km/h]
Difference from reference velocity
5
0
-5
0
200
400
600
800
1000
Time [s]
Fig. A.6. Optimizing for LF, full cycle, when velocity
deviations are allowed. Both specified and optimal
gear changes. The optimal gear changes case has
longer driven distance than specified gear case.
676