IFAC Workshop ICPS'07
2007, July 09-11
Cluj-Napoca, Romania
ON BOARD ENERGY SYSTEM BASED ON BATTERIES AND SUPERCAPACITORS
C. Lungoci1, M.Becherif2, A. Miraoui3, E. Helerea1
1
Electrical Engineering Department, UNITBV, Brasov, ROMANIA
2
SeT, UTBM, Belfort (cedex) 90010, FRANCE
3
L2ES, UTBM, Belfort (cedex) 90010, FRANCE
Abstract: The on-board energy system permits the storage/delivery energy in electrical
vehicles. It can use different energy devices working together, like batteries and
supercapacitors. The energetic performances of batteries make them efficient, but for
supplying pulsating loads, supercapacitors must be added.
In this paper, the design of the system is presented and the dynamic model is described
using the Port-Hamiltonian formalism. The Passivity-Based Control strategy is proposed
according to achieve the best energy management of the system.
Simulation results of the system processes are presented in order to prove that the model
and the control strategy applied can ensure the global system stability. Copyright ©
2007IFAC.
Keywords: Electrical vehicle, Supercapacitors, Batteries, Passivity-Based Control.
1. INTRODUCTION
The Passivity Based Control strategy is focused on
applied for the control of the DC bus voltage and
supercapacitors current. This control strategy
analyzes the characteristics of the system from the
point of view of energy and combines the highest
energy flow management to system stability (Yang,
et al, 2004).
Finally, simulation results are presented, while
conclusions and indications about the future work
are discussed.
In the automotive fields, a great importance is
accorded to the optimization of the on-board energy
system, with the aim to increase the efficiency of the
motor drive. Most recent researches proved that it is
suitable to use a hybrid solution for the on-board
systems architecture, by combining many energy
storage/delivery devices (Becherif, et al., 2007;
Becherif, et al., 2006; Becherif, 2006).
In papers developed by Lungoci, et al. (2006), and
Lungoci, and Helerea (2006) it was studied a model
of an energy supply-system, including energy
devices, converters and a DC motor. The proposed
energy devices were batteries - which deliver a
constant energy and the supercapacitors - which
ensure the picks of power during the transient.
This paper comes to advance the researches by using
another simplified equivalent circuit model for the
supercapacitors. Also, according to obtain a high
energy management of the system and to reduce the
energy losses, bidirectional converters are proposed.
The new design of the system is presented and the
dynamical model is given. The Port-Hamiltonian
theory is used to exhibit important physical
properties.
2. DESIGN AND MODEL OF THE SYSTEM
In (Lungoci, et al. 2006) the general architecture of
the on-board energy system is explained. It consists
of batteries, the boost and bidirectional converters,
supercapacitors and the motor as load.
In the new proposed design, the boost is replaced by
a bidirectional converter. In this way, during braking
phases of the motor, the energy can be stored in both
storage devices: batteries and supercapacitors, which
leads to benefits in terms of energy management of
the system (Camara, et al, 2006).
The used model for the pack of supercapacitors is
based on the first order model, containing an internal
resistance in series with a varying capacity.
-197-
di sc
dt
di
Lm m
dt
V 0 R sc i sc (1 U sc )V dc
L sc
The structure of the system is presented in the Fig.1:
(3)
R m i m V dc k e p :
The control variables U cb , U sc are reported to the
converters and each arm of them is controlled by
complementarities law, as follows:
Fig. 1. On board energy system structure.
According to simulations in temporal domain of the
systems based on supercapacitors, the variable
capacity represented is the load capacity part which
depends on the load voltage of supercapacitors V0 .
The charge of supercapacitors follows the relation:
C sc
C 0 kV0 ! 0
Tb 1 ½
Ucb 1: ®
¾,Tb
¯Tb 0¿
Tb 0½
Ucb 0 : ®
¾,Tb
¯Tb 1 ¿
°Tsc 1 ½°
Usc 1: ®
¾, Tsc
°̄Tsc 0°¿
Tsc 0½
Usc 0 : ®
¾,Tsc
¯Tsc 1 ¿
Eb Rb ib
(1)
(4)
closed, Tsc open
open, Tsc closed
x 1
1
> x 4 (1 U cb ) x 2 @
Cs
x 2
1
>E b R b x 2 (1 U cb ) x 1 @
Lb
x 3
1
>x 4 x 7 (1 U sc ) x 6 @
C dc
x 4
1
>x 1 x 3 @
L dc
x 5
x 6
1
>x 5 R sc x 6 (1 U sc ) x 3 @
L sc
x 7
1
> R m x 7 x 3 k e p : @
Lm
(2)
The equivalent circuit of motor is composed by a
resistance in series with an internal inductance and
an electromotive force.
The dynamics of the system are described by the
following:
dVs
idc (1 U cb )ib
dt
di
Lb b Vb (1 U cb )Vs
dt
dV
C dc dc idc im (1 U sc )i sc
dt
di
Ldc dc Vs Vdc
dt
dV
(C 0 kV0 ) 0 i sc
dt
open, Tb closed
To obtain the state space model of the system,
equations system (3) can be rewritten into the state
equations:
where k is the coefficient representing the
dependence on voltage (Rafik, et al, 2007).
The battery is represented by its electromotive force
e.m.f. in serial with its internal resistance. The
mathematical model is:
Vb
closed, Tb open
Cs
(5)
1
x6
C 0 kx 5
where the state variables vector is:
(3)
x
>x1
>Vs
x2
x3
ib Vdc
x4
idc
x5
V0
x6
i sc
and control variables vector is: u
x7 @
T
im @
T
>U cb
U sc @ .
T
The principal goal is to maintain a desired value of
the DC bus voltage, which imposes to have a
constant energy delivered by the battery, without big
fluctuations. Moreover, for minimizing energy losses
-198-
in the system, it is necessary to feed back the energy
during the braking phases of the motor, especially
into the supercapacitors. In order to accomplish
these tasks, the output variables vector is proposed
to be:
y
>y1
y2 @
>x3
T
x6 @
>Vdc
T
where:
H
system,
isc @
T
and the desired equilibrium point is defined by:
ib ct1 ;Vdc
®
¯V0 ct 3 ; i sc
Vd ; idc
ct 2 ;
0; im
ct 4 ; :
1 T
x Qx is the scalar energy function of the
2
Q
diag ^C s , Lb , C dc , Ldc , C sc , Lsc , Lm `,
J
R
is the interconnection matrix,
J T
T
R ! 0 is the damping matrix and G is
matrix of proper dimensions.
The energy H of the (passive) uncontrolled system
decreases until a minimal value which usually
doesn’t correspond to the desired equilibrium point.
The PBC technique aims to develop a state feedback
u that shapes the total energy function of the system
to obtain closed-loop energy with a minimum in the
desired equilibrium point x .
All the computations are made after the following
x x x , in order to have a
variable change ~
steady state regulation error equal to zero. The
desired energy function of the system is:
½
¾
:d ¿
where Vd is the desired DC bus voltage and : d is
the desired motor speed.
Generally the internal resistance rb is very low, so
the Joules losses on it are negligible compared to the
power delivered by the battery. In these conditions,
after calculations, the desired equilibrium state space
vector is:
x >x1 x2 x3 x4 x5 x6 x7@
Hd
T
1 ~T ~
x Qx
2
(7)
T
ª Vd (Vd ke p:d)
V k p: º
V k p:
Vd d e d V0 0 d e d »
«Vd
Rm ¼
Rm
Rm
¬ Eb
The new error dynamics model of the system is:
~
x1
1 ~
> x4 x4 (1Ucb )(~x2 x2 )@
Cs
~
x2
1
>Eb Rb~x2 Rb x2 (1Ucb)(~x1 x1)@
Lb
1 ~ ~
>x 4 x 7 (1 U sc ) ~x 6 @
C dc
1 ~ ~
>x1 x3 @
Ldc
1
~
x6
C0 k (~
x5 x5 )
1 ~
>x5 x5 Rsc~x6 (1Usc )(~x3 x3 )@
Lsc
1
> Rm (~x7 x7 ) ~x3 x3 ke p:@
Lm
k (~
x5 x5 ) C sc >0
and the equilibrium control vector is:
u
>U
U sc @
T
cb
ª Eb
V º
R
b (Vd ke p:d ) 1 0 »
«1
Vd ¼
¬ Vd Eb Rm
T
~
x 3
~
x 4
Using the PCH: Port Controlled Hamiltonian
representation, the PBC: Passivity Based Control
technique will be implemented for the system
control.
~
x 5
~
x6
3. PASSIVITY BASED CONTROL STRATEGY
~
x7
From the system theory perspective, the concept of
energy is lost; hence it is necessary to adopt another
viewpoint on dynamical systems. In the PCH theory,
systems are considered as energy-transformation
devices which we interconnect to achieve the desired
behaviour (Ortega, et al, 2000). The classical state
variables become energy variables and the effect of
the input and output signals are to modify the system
total energy. In this way, the energy system
properties are explicitly shown in their mathematical
description (Machelli, 2003).
The PCH representation of the system is the
following:
x
( J R)
y
wH
G
wx
wH
Gu
wx
C0
(8)
The gradient of the desired energy variable is:
wHd
wx
and:
(6)
T
-199-
>Cs~x1
T
Lb~
x2 Cdc~
x3 Ldc~
x4 Csc~
x5 Lsc~
x6 Lm~
x7 @
1Ucb
1
ª
0
0
« 0
CsLb
CsLdc
«
«(1Ucb) Rb
0
0
0
2
« CsLb
Lb
«
1
« 0
0
0
0
«
CdcLdc
« 1
1
0
0
0
J R «
CdcLdc
« CsLdc
«
0
0
0
0
« 0
«
(1Usc)
1
«
0
0
« 0
CdcLsc
CscLsc
«
1
« 0
0
0
0
«
CdcLm
¬
º
0 »
»
0
0 »
»
»
1Usc 1 »
CdcLsc CdcLm»
»
0
0 »
»
»
1
0 »
CscLsc
»
Rsc
»
0 »
2
Lsc
»
Rm »
0
2
Lm »¼
T
dH d dH d
dt
dx
T
dH
d Rd
dx
0
(12)
dH d
d0
dx
and the damping matrix:
Rm º
2»
Lm »¼
ª Rb
R rV
0 0 0 sc 2 d
diag«0
2
Lsc
«¬ Lb
Rd
t0
(13)
In this way, the controlled system behaves as a
dissipative element, the energy system decreases and
the system reach the configuration corresponding to
minimum energy in the desired equilibrium point x .
The system configuration becomes:
~
x
T
dH d
dH
( J d Rd ) d
dx
dx
~
x
º
ª 1
« C ( x4 (1 U cb ) x2 ) »
s
»
«
« 1 ( E R x (1 U ) x ) »
1
b 2
cb
»
« Lb b
»
0
wH d «
»
( J R)
«
wx «
0
»
»
«
0
»
«
1
«
( x5 (1 U sc ) x3 ) »
»
« Lsc
»¼
«¬
0
4. SIMULATIONS
The results obtained with the proposed controller are
reported in the figure below.
In order to limit the current requirements during the
motor start, the DC bus reference voltage profile is
created using many staircases. It is considered the
initial voltage of the supercapacitors having
V (t 0) 12V (full charge). The controller
coefficient r has been chosen in
supply/recover the maximum of energy.
The control law u is developed such that the
resulting closed-loop PCH dynamics will be given
by:
order
to
24.5
( J d Rd )
wH d
wx
24
(9)
V b (V)
~
x
23.5
23
Rd T t 0 the new
22.5
interconnection and damping matrices.
In this order, one of the possible choices for the
feedback laws is:
40
with J d
T
J d and Rd
0
1
2
3
4
5
time(s)
6
7
8
9
10
0
1
2
3
4
5
time(s)
6
7
8
9
10
U cb
U cb
U sc
U sc r~
x6 , r ! 0
i b (A)
20
0
10)
-20
where r is a design parameter.
The global stabilization around desired equilibrium
point x is achieved if: the desired energy function
is a negative semi-definite function:
dH d
d0
dt
Fig. 2. Voltage Vb and current ib on battery.
Fig.2 presents the variation of the battery voltage
Vb and the current ib . According to the changes on
(11)
the DC bus reference voltage, the terminal voltage of
the battery indicates small variations, which are
smoothed by the presence of supercapacitors. The
current moves in opposition of voltage fluctuations.
It can be observed that:
-200-
.
100
V d , V dc (V)
80
60
40
20
0
2
4
6
8
10
6
8
10
time(s)
10
i m (A)
8
6
4
2
0
0
2
4
time(s)
Fig. 5. Control signals U cb and U sc .
Fig. 3. DC bus voltage Vdc , reference voltage Vd
Fig.5 presents the control signals of the bidirectional
U cb is related to the bidirectional
converters.
and motor current i m .
battery
current i m . The two characteristics show that the
bidirectional converter control at the output of the
supercapacitors. The control values are in the set
[0, 1].
In the fig.6 the power of the batteries Pb ,
DC bus voltage and the motor current have the same
allure. One can see that the DC Bus voltage is
tracking its reference with no steady state error and
very small overshoot.
converter
and
U sc
Fig.3 shows the DC voltage Vdc and the motor
represent
the
supercapacitors Psc and the motor required power
Pm are lay out.
12
700
V sc (V )
11.999
600
11.998
500
11.997
400
0
1
2
3
4
5
time(s)
6
7
8
9
P b , P sc , P m (W)
11.996
10
10
0
300
200
i sc (A)
100
0
-10
-100
-20
0
1
2
3
4
5
time(s)
6
7
8
9
10
-200
0
Fig. 4. Voltage Vsc and current i sc on
1
2
3
4
5
time(s)
6
7
8
9
10
Fig. 6. Batteries power Pb , Supercapacitors
supercapacitors.
power Psc and Motor power Pm .
Fig.4 illustrates the voltage on supercapacitors
Vsc and the current i sc . The initial condition is that
Studying the picks of storage devices power it can be
seen that at the t 2.034 s time, when appears this
pick power demand, the power values are:
Pm 320.6W , Pb 199.5W , Psc 110.7W ,
the pack of the supercapacitors is charged. At the
motor start, the supercapacitors deliver energy and
the voltage decreases. In steady state the current is
null, then in the braking motor regime,
supercapacitors recover energy and the voltage
increases.
so 34% percents of the motor power is ensured by
the supercapacitors and the rest, by the batteries.
The energy delivered by the supercapacitors is
E sc 1076.4 J and the recovered energy is
E sc
1360.8 J . It can be seen that the potential
for brake energy recovery is highly exploited due to
-201-
the presence of supercapacitors and the system is
able to take up the power.
Becherif, M., M.Y. Ayad and A. Miraoui (2006).
Modeling and Passivity-Based Control of Hybrid
Sources. In: Proceedings of 41St IEEE/IAS
Annual Meeting, pp.1134-1139, USA.
Becherif, M. (2006),
Passivity-based control of hybrid sources: Fuel
cell and Battery. In: 11th IFAC Symposium on
Control in Transportation Systems (CTS’06),
Netherlands.
Camara, M.B., H. Gualous, F. Gustin and A.
Berthon (2006).
Control strategy of hybrid sources for transport
applications using supercapacitors and
batteries.
In: Power Electronics and Motion Control
Conference, IPEMC ’06 CES/IEEE, Volume
1, pp.1-5.
Lungoci, C., E. Helerea and A. Munteanu (2006).
On an energy supply combined system used in
electric vehicle. In: Bulletin of Transilvania
University, Series A, pp.213-218, Brasov.
Lungoci, C. and E. Helerea (2006).
Modeling and simulation of the energy supplymotor system for an electric vehicle. In:
Proceedings
of
Second
International
Symposium, Pollack University, Pecs.
Macchelli, A. (2003).
Port Hamiltonian Systems – A unified
approach for modelling and control finite and
infinite dimensional physical systems. In:
Ph.D. Thesis, Dept. of Electronics, Computer
Science and Systems, University of Bologna.
Ortega, R., A.J.van der Schaft, I. Mareels and B.
Maschke (2000).
Energy Shaping Revisited. In: Proceedings of
the 2000 IEEE, International Conference on
Control Applications, Anchorage, Alaska,
USA.
Rafik, F., H. Gualous, M. Karmous and A. Berthon
(2007).
Frequency, thermal and voltage supercapacitor
characterization and modelling. In: Journal of
Power Sources 165, pp.928 – 934.
Yang, J. M., J. Wu, P. Dong and J. H. Yang (2004).
Passivity-based control in Wind Turbine for
Maximal
Energy
Capture.
In:
IEEE
International Conference on Electric Utility
Deregulation, Restructuring and Power
Technologies, Hong Kong.
5. CONCLUSIONS
This paper deals with a modern on-board energy
system based on batteries and supercapacitors. The
hybridisation of the supercapacitors with storage
batteries allows an optimised system design
according to mission requirements. The presence of
the supercapacitors supports a higher lifetime of the
batteries and allows having a right sizing batteries in
terms of weight and volume. In this way, the entire
system can be designed at the low values of power,
the power sizing is independent from the energy
sizing and that improves the efficiency, volume,
weight and costs.
A new design of the on-board energy system is
proposed, and then the state space model and the
Controlled Port Hamiltonian form of the system are
presented in order to obtain its dynamical and
physical properties. The Passivity Based Control is
applied, using the energy considerations and
following to obtain a system with favourable
dynamic characteristics. The global stability of the
system is proved and simulation results come to
demonstrate the viability of this new model of the
on-board energy system.
Simulations show that the presented architecture is
more viable and the output controllers used are
satisfying. The Passivity Based Control provides a
simple linear control laws with few measurements to
achieve the desired aims even with an under
actuated nonlinear system (only the measure of the
supercapacitor current is needed to achieve the
described goal).
In a next work, a comparison between the Passivity
Based Control strategy and a classical controller will
be treated, in order to find the best control strategy
for the system requirements.
REFERENCES
Becherif, M., M.Y. Ayad, A. Djerdir and A. Miraoui
(2007). Electrical train feeding by association of
supercapacitors,
photovoltaic
and
wind
generators. In: Proceedings of IEEE-ICEE,
pp.55-60, Capri-Italy.
-202-