112
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 1, JANUARY 2013
New Full-Frequency-Range Supercapacitor Model
With Easy Identification Procedure
Vincenzo Musolino, Luigi Piegari, Member, IEEE, and Enrico Tironi
Abstract—Energy storage has become a key issue for achieving
goals connected with increasing the efficiency of both producers
and users. In particular, supercapacitors currently seem to be
interesting devices for many applications because they can supply
high power for a significant amount of time and can be recharged
more quickly than electrochemical batteries. In different applications, a combination of the two devices, batteries and supercapacitors, could be used to develop a high-efficiency storage system with
a high dynamic performance. Because of the diffusion of supercapacitors, a good model is needed to represent their behavior in
different applications. Some models have been proposed that characterize supercapacitors working at either low or high frequencies.
In this paper, the authors present a full-frequency-range model
that can be used to represent all of the phenomena that involve
supercapacitors. Moreover, to realize a simple and useful tool, the
authors present a simple procedure to identify the parameters
of the model that can be used to characterize a supercapacitor
before use.
Index Terms—Double-layer capacitors, parameter identification methods, supercapacitor efficiency, supercapacitor performance, supercapacitors.
I. I NTRODUCTION
I
N RECENT YEARS, the interest in storage systems has
been growing because their use is becoming more and
more widespread in different applications. For each kind of
application, it is possible to identify the technologies that would
be appropriate for realizing an optimal storage system [1].
Moreover, the use of more than one technology to fit the goals
of specific applications represents a real solution to achieve
a high-performance and high-efficiency storage unit [2]–[6].
For example, various applications that use supercapacitors and
electrochemical batteries to realize hybrid storages have been
reported [7]–[11].
Because of the wide variety of applications that use supercapacitors, it is important to realize a model that can represent
their behavior in all practical operations. Recently, the different
models of supercapacitors have been developed to match these
exigencies. The proposed models differ because they represent
a supercapacitor in a particular application and are designed for
ease of use in that application.
Manuscript received August 4, 2010; revised February 14, 2011 and
October 31, 2011; accepted January 29, 2012. Date of publication
February 10, 2012; date of current version September 6, 2012.
V. Musolino is with Dimac Red, 20853 Biassono, Italy (e-mail: v.musolino@
dimacred.it).
L. Piegari and E. Tironi are with the Department of Electrical Engineering,
Politecnico di Milano, 20133 Milan, Italy (e-mail: luigi.piegari@polimi.it;
enrico.tironi@polimi.it).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIE.2012.2187412
Different authors have described methods to model supercapacitor devices with different resistive–capacitive (RC) networks. These models have been obtained through analyses in
the time domain or frequency domain. In [12], Zubieta and
Bonert proposed a model with three RC branches connected
in parallel. An additional parallel resistor accounts for the selfdischarge phenomenon. In this model, the bigger capacitance
is not constant but rather linear with voltage. This model is
very good for low frequencies, but it is not accurate at high
frequencies. In [13]–[15], Belhachemi et al. proposed a model
to address the limits of the model proposed in [12]. They
presented a nonlinear transmission line. However, this model
is complex, and parameter identification is not easy. However, the idea of separating the time constants of the different
branches is interesting and is used in this paper. In [16]–[18],
Gualous et al. proposed simple transmission lines focusing
on the effect of temperature on model parameters. In [19],
Rojat et al. introduced a multipenetrability model to estimate
the supercapacitor behaviors in the different states of health. In
[20], Zhang et al. analyzed the quality of different RC models
for power electronic applications. In [21], Du considered a
parallel branch RC equivalent circuit and proposed a parameter
identification procedure that was mainly based on least-square
error minimization.
In this paper, the authors propose a model that is able to
represent the behavior of supercapacitors at both low and high
frequencies. This new model, described in the next section, is an
extension of the model presented in [22]. Moreover, this model
can be used to take into account the self-discharge and redistribution phenomena. The model is not completely new because
it was obtained by joining different models, but it is useful
not only for dynamic simulations but also for evaluating the
efficiency of a device. In addition, a very simple, yet innovative,
parameter identification procedure is proposed to avoid the use
of least-square error minimization. Previous methods required
the results of accurate tests, while the proposed procedure
provides good parameter estimation using only a few simple
measurements.
II. S UPERCAPACITOR M ODELING
Supercapacitor devices consist of two electrodes with an
ion-permeable separator between them. The electrodes are immersed in an electrolyte solution, as shown in Fig. 1.
The charge separation process, which requires a voltage
difference across the electrodes, takes place on the two
electrode–electrolyte interfaces. Thus, supercapacitors are usually known as double-layer capacitors. The double layer is also
0278-0046/$31.00 © 2012 IEEE
MUSOLINO et al.: NEW FULL-FREQUENCY-RANGE SUPERCAPACITOR MODEL
Fig. 1.
113
Internal structure of a supercapacitor.
called the Helmholtz layer after the physicist who introduced
the model to explain this phenomenon in 1853 [23], [24].
The dynamic behavior of a supercapacitor is strongly related
to the ion mobility of the electrolyte used and the porosity
effects of the porous electrodes. As shown in the next sections,
it is clear, based on observations, that the capacitance starts to
decrease from 0.1 to 0.2 Hz and becomes null for frequencies
at hundreds of hertz or a few kilohertz, depending on the
cell technology. The real part of the supercapacitor impedance
decreases with frequency; its value is reduced by half from dc
to very high frequencies (some kilohertz). This reduction in the
capacitance with an increase in the charge/discharge frequency
is mainly the result of ion inertia. Thus, it is impossible to
use the full capacitance of the device at high frequencies in
practice. Instead, the reduced resistance can be explained by
a reduction in friction losses as ion migration slows in the
electrolyte solution. A deeper explanation of these phenomena
can be found in the theory of porous electrodes [25].
To represent this behavior in a frequency range of tens of
millihertz to hundreds of hertz, Buller et al. [26] proposed a
model that presents the following impedance in the frequency
domain:
τ (V ) coth
jωτ (V )
Zp (jω, V ) = Ri + jωLi +
C(V ) jωτ (V )
= Rp (ω) −
1
jωC(V, ω)
(1)
where Ri is the resistance at an infinite frequency, Li is the
leakage inductance, ω is the angular frequency, V is the voltage
shown in Fig. 2, C is the dc capacitance value, and τ is,
dimensionally, a time. The leakage inductance is usually very
small (tens of nanohertz) and is neglected here. This model
is not able to model the self-discharge phenomenon and is
not accurate at very low frequencies. Thus, the authors [22]
proposed a model that integrated the model proposed by Buller
and the model proposed by Zubieta. This model is composed of
three parallel branches: The first one is equal to the impedance
given in (1), the second one is a series RC branch representing
the recombination phenomena after fast charge or discharge,
and the third one is a leakage resistance that takes into account
the self-discharge phenomena. This model is very accurate for
Fig. 2. Proposed supercapacitor model.
simulating the phenomena occurring in some tens of seconds
but is not capable of achieving good results for a self-discharge
of some weeks. For this reason, after a set of experimental tests,
whose main results are shown hereinafter, the authors extended
the model proposed in [22] to obtain a model that is capable of
representing a supercapacitor’s performance across the whole
frequency range, including the redistribution and self-discharge
phenomena. Furthermore, all of the parameters are taken directly from the information contained in the manufacturer’s
datasheet or estimated using very simple tests at a constant
current. The equivalent circuit of this model is shown in Fig. 2.
In the model shown in Fig. 2, the first branch, representing
the impedance Zp given in (1), models the fast dynamic, while
an indefinite number of parallel branches can represent the slow
dynamic demonstrated by Zubieta. It is worth noting that the
number of RC parallel groups in the first branch should theoretically be infinite. However, it has been experimentally verified
that five groups are enough to obtain an accurate model. All
the elements of the first branch depend on the three parameters:
Ri , C, and τ ; the last two are voltage-dependent parameters.
For slow dynamics, the number of parallel branches should be
infinite. However, as shown in Section IV, two parallel branches
are sufficient to achieve accurate results. Finally, a resistor
Rleak is added to model the self-discharge phenomenon.
The proposed model has been validated through experimental tests made on different kinds of cells. Some of the results are
reported in the following. Moreover, the proposed procedure to
estimate the parameters of the model is presented, and its results
are compared with the experimental data.
III. PARAMETER I DENTIFICATION
Different parameter identification procedures have been proposed in the literature. However, the possibility of estimating
114
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 1, JANUARY 2013
parameters with very simple tests and with the information
contained in the manufacturer’s datasheets is a very important point for the usability of the model. It is clear that the
parameter values that give the best match between the model
shown in Fig. 2 and the experimental data can be obtained
with least-square error minimization. However, this procedure
is too complex to be performed every time. In fact, it requires
the execution of experimental tests at many frequencies to
obtain the experimental curve of impedance versus frequency.
Thus, a new easier procedure is proposed in the following. The
parameters of the first branch and the parallel branches are
identified separately as discussed, in detail, in the following.
A. First-Branch Parameter Identification
The first branch of the model shown in Fig. 2 is the impedance given by (1). In this expression, C and τ are the
functions of the voltage. During the experimental tests, it was
verified that C and τ vary almost linearly with the voltage
C(V ) = C0 + k V
τ (V ) = τ0 + kτ V.
(2)
From (1), the following result:
lim e{Zp } = lim Rp = Ri +
ω→0
ω→0
= Ri +
lim m{ωZp } = −
ω→0
∞
2τ (V )
2 π 2 C(V )
n
n=1
τ (V )
= Rdc
3C(V )
1
C(V )
(3)
where Rp indicates the real part of the impedance of the first
branch Zp .
Therefore, the capacitance can be evaluated, for each voltage
V , using a dc test. In particular, it is possible to use the same
charging test at a constant current as described in [27]. In contrast, to experimentally evaluate Rp at zero frequency, it is not
possible to use a dc test. Indeed, at frequencies lower than some
tens of millihertz, the parallel branches cannot be neglected, and
Rp is not directly measurable. The supercapacitor’s datasheet
gives the discharge time Tdc to which the rated capacitance
refers. According to the manufacturer’s recommendation [28],
Tdc has to be used to estimate the dc resistance. Working
at frequency fdc = 1/Tdc , the recombination phenomena do
not occur. Thus, the electrochemical capacitor can be modeled
using only the first branch of Fig. 2. A resistance measurement
at fdc allows the estimation of Rdc . Typically, fdc has a value
of some tens of millihertz.
For high frequencies, when the supercapacitor behaves like a
resistance, Rp becomes constant. Some measurements at high
frequencies, when no capacitive behavior is shown, are enough
for estimating the high-frequency resistance Ri . However, this
value is also available on the device datasheet. Thus, given Rdc
and Ri from (3), we get the following:
τ (V ) ≈ 3C(V )(Rdc − Ri ).
(4)
From (4), it is possible to estimate all of the parameters of the
first branch for a fixed voltage. Repeating the tests at different
voltages makes it possible to estimate a value for the parameter
k that takes into account the variability of the capacitance with
the voltage.
B. Parallel-Branch Parameter Identification
First, the number of branches must be chosen as a compromise between the required accuracy and the consequent complexity. The branches represent the supercapacitor’s behavior
over a long time scale. Thus, to model the phenomena that occur
at different times, their time constants should be quite different.
It is therefore possible to consider different transient phenomena without interference. After a fast charge, all of the energy is
stored in the first branch. Then, leaving the supercapacitor in an
open circuit causes the energy to be redistributed in the parallel
branches, after which it dissipates in the leakage resistance. If
the parallel branches with greatly different time constants are
chosen, it is possible to split the discharge time into n + 1
intervals. In interval k, only the branches from 1 to k − 1 are
involved in the energy transfer. All of the other branches are in
a steady state and are not affected by the phenomenon because
they have much higher time constants. In the n + 1 interval, all
of the energy is dissipated on the leakage resistance.
To make the parameter identification procedure easy, it is
important for the choice of the time intervals to be free and
independent of the cell under consideration. The only restriction in the choice of the time intervals is that they have to be
sufficiently different from each other that, in each transient,
all of the other branches can be considered to be in the steady
state. In the analysis of the experimental data, it was observed
that a good solution in the choice of the time intervals is to set
them equidistant on a logarithmic scale. The logarithmic scale
is obtained by choosing the intervals as follows:
I1 : τ 0 < t < M τ 0 = τ 1
I2 : τ1 < t < M τ1 = τ2
...
In : τn−1 < t < M τn−1 = τn
(5)
where τ0 is the time necessary to extinguish the transient of
the first branch over the second one. So, τ0 is greater than five
times the time constant of the first branch already identified.
This constant usually ranges from some seconds to some tens
of seconds.
In the set of equations in (5), it is necessary to choose a value
for M so that each transient can be considered independent
from the others. As is well known, exponential transients can
be considered extinguished after five times the time constant
because 99.3% of the steady-state value is reached at this time.
Thus, to guarantee that each transient caused by the parallel
branches is finished before the beginning of the next one, it is
necessary for the time constant of the k branch to be more than
five times the time constant of the k − 1 branch. Thus, M > 5.
From (5), for a fixed time window Tw during which the selfdischarge phenomenon has to be analyzed, the parameter M
MUSOLINO et al.: NEW FULL-FREQUENCY-RANGE SUPERCAPACITOR MODEL
Fig. 3.
115
Equivalent circuit of two parallel branches.
can be easily evaluated as
M=
n−1
Tw
.
τ0
(6)
The proposed identification procedure estimates the parameters of the equivalent circuit using only n voltage measurements
made at the instants given by (5).
The equivalent circuit for each interval is shown in Fig. 3.
At time t = 0, the charged capacitance is indicated with the
subscript II, while the discharged capacitance is indicated with
the subscript I. The sum of the resistances of the two branches
is indicated by Req . It is worth noting that when considering
the first interval, the capacitance CI is not constant. The capacitance varies when the voltage changes according to (2).
If V0 and Vf indicate the initial and final voltages of the
considered interval, respectively, for the circuit in Fig. 3, it is
possible to write
CII =
V02 − Vf2
V0 − Vf
CI + k
Vf
2Vf
(7)
where k is different from zero only for the capacitance of the
first branch as discussed in Section II.
Using the charge conservation law, all of the capacitances
of the parallel branches can be estimated iteratively, and they
do not depend on the resistances. The resistances are easily
calculated by knowing the time constant and capacitances.
Taking into account (5) and (7)
Cρ = k
Rρ =
Vτ2ρ−1 − Vτ2ρ
τρ
Cρ
Rleak =
1
n−1
2Vτρ
τn
n−1
ρ−1
Vτρ−1 − Vτρ +
Cj
Vτρ
j=1
.
(8)
Cj
j=1
IV. E XPERIMENTAL R ESULTS
Different experimental tests were conducted at the Department of Electrical Engineering, Politecnico di Milano, Milan,
Italy. The main goals of these tests were as follows.
1) Validate the proposed model over the whole frequency
range and for the different kinds of cells.
2) Validate the proposed method for parameter
identification.
To fulfill the first goal, cells with different technologies
and sizes were tested. In particular, the cells with different
origins were taken into account. In the following, the results
Fig. 4. Experimental hardware setup.
are reported for two 150-F cells from the old and new generations. The cells were manufactured by Maxwell and were the
BCAP0140 and BCAP0150. The change in the rated capacitance from 140 to 150 F was caused by the increase in the rated
voltage from 2.5 to 2.7 V. According to (2), the capacitance is
a function of the voltage. Therefore, the rated capacitance of
the new cell was greater than the old one without significantly
changing the size of the cell.
Measurements were used to analyze the supercapacitor performances in a frequency range of dc up to 1 kHz. The dc measurements focused on the redistribution and leakage phenomena
and were conducted by analyzing the open-circuit voltage of the
device, while electrochemical impedance spectroscopy (EIS)
was used to analyze the frequency response of the device in
the frequency range of 0.01−103 Hz. It is worth noting that
EIS was only used to validate the model; it is not necessary for
parameter identification.
The equipment used to realize the frequency analysis consisted of an AB-class power amplifier driven by a signal generator. The signal generator was fully remote controlled via a GPIB
standard interface. The cell voltage and current were measured
using a 16-bit multifunction National Instruments acquisition
board (Fig. 4).
For the first branch of the model [i.e., the impedance Zp
given in (1), (2), and (4)], three EIS at different polarization
voltages were realized. In Figs. 6 and 7, the capacitances
and resistances versus frequency obtained with the model are
compared with the experimental data for the BCAP0140 and
BCAP0150 cells.
From an analysis of Figs. 5 and 6, it is clear that the model
and experimental data agree. For the two cells, the reference
parameters of the first branch obtained using the classical
method of least-square minimization are reported in Table I.
Observing Figs. 5 and 6, it is possible to see how both
the capacitance and resistance are practically constant in a
frequency range of some tens of millihertz up to some hundreds
of millihertz. These values correspond to the dc values reported
on the datasheet (see Section III). This behavior was also
verified experimentally using cells with different capacitances
(i.e., Maxwell PC10, old and new generations, 10 F, with a
rectangular shape despite the cylindrical shape of BCAP0140
and BCAP0150), and it always appeared in the same frequency
range. This can be explained by taking into account that,
116
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 1, JANUARY 2013
TABLE I
PARAMETERS OF THE F IRST B RANCH
TABLE II
C OEFFICIENTS OF D ETERMINATION
TABLE III
PARAMETERS OF THE F IRST B RANCH E STIMATED W ITH THE S IMPLIFIED
P ROCEDURE AND E STIMATION E RRORS
Fig. 5. Equivalent capacitance and resistance versus frequency at different
polarization voltages for BCAP0140: Comparison between model and experimental data.
Fig. 7. Self-discharge of BCAP0140: Comparison between the experimental
data and four-branch model.
Fig. 6. Equivalent capacitance and resistance versus frequency at different
polarization voltages for BCAP0150: Comparison between model and experimental data.
over the low frequencies of tens of millihertz, the dynamic
behavior of supercapacitors is mainly related to the electrolyte
properties. The redistribution phenomenon, which is caused by
the irregularity of the porous carbon structure, takes place for
lower frequencies. Therefore, it is necessary to consider the
effect of the parallel branches of the model.
To evaluate the correspondence of the proposed model with
the experimental data, the coefficient of determination R2 was
evaluated. Its definition is as follows:
2
j (yj,ex − yj,mod )
2
(9)
R =1− 2
j (yj,ex − ȳex )
where the suffixes ex and mod refer to experimental- and
model-based data, respectively, and the mean value is indicated
with an overbar. For the data shown in Figs. 5 and 6, the
coefficients of determination are reported in Table II.
The parameters of the first branch, identified as suggested in
Section III, with the proposed simplified procedure are reported
in Table III, together with the estimation errors which are
indicated in parentheses. The estimation errors are always lower
than 5%, which confirms that the proposed simplified method
of identification can be used.
To prove the suitability of the approximated procedure of (8),
self-discharge tests of the two cells were performed. In Figs. 7
and 8, the experimental curves are compared with the results
given by a four-branch model, while in Figs. 9 and 10, the
experimental test results are compared with those of a numerical simulation of a six-branch model. The parameters obtained
by (8) and used in the simulations leading to Figs. 7–10 are
reported in Tables IV and V. A time interval of τ0 = 50 s was
chosen, and M was chosen such that τn ≈ 5 · 106 s because of
the time required to watch the system. Thus, from (6), M = 45
and 10 for the 4 and 6 branches, respectively.
MUSOLINO et al.: NEW FULL-FREQUENCY-RANGE SUPERCAPACITOR MODEL
Fig. 8. Self-discharge of BCAP0150: Comparison between the experimental
data and four-branch model.
117
Fig. 11. Experimental current versus time for a test at 70 A.
Fig. 9. Self-discharge of BCAP0140: Comparison between the experimental
data and six-branch model.
Fig. 12.
70 A.
BCAP0140 experimental and simulated voltages during a test at
Fig. 10. Self-discharge of BCAP0150: Comparison between the experimental
data and six-branch model.
TABLE IV
PARAMETERS OF A F OUR -B RANCH E QUIVALENT M ODEL (M = 45)
TABLE V
PARAMETERS OF A S IX -B RANCH E QUIVALENT M ODEL (M = 10)
Fig. 13. Experimental current versus time for a test at 110 A.
It is evident that its value is weakly dependent on the cell but
strongly dependent on the number of branches used to model
the system. The fit with the six-branch model seems to achieve
good interpolation results.
A. Experimental Results in Time Domain
It is worth noting that the choice of the time constants,
according to (5) and (6), is independent of the cells and depends
only on the observation time and the number of branches. Thus,
with the same time constants, two different cells have been
modeled. The coefficient of determination R2 was evaluated as
a measure of the goodness of the fit and is shown in the figures.
In order to test the suitability of the model and the influence
of the parameter identification procedure, some experimental
tests were performed. In order to show the usability of the
model over a wide frequency range, a pulse load condition was
used for the tests. Using a controlled dc source (EA-PSI 8240170) and a controlled electronic load (EA-EL 9400-150), the
pulsing currents of different values were realized. The results
of some of the tests performed are reported in the following.
In particular, the results for 70 and 110 A are shown. The
118
Fig. 14.
110 A.
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 1, JANUARY 2013
BCAP0140 experimental and simulated voltages during a test at
Fig. 15. Experimental current versus time for a test at 70 A.
Fig. 16.
70 A.
BCAP0150 experimental and simulated voltages during a test at
experimental currents and voltages of the BCAP0140 cell are
shown in Figs. 11–14, while the currents and voltages for the
BCAP0150 cell are shown in Figs. 15–18.
The simulation was conducted by injecting an experimentally
measured current into the cells. A good agreement between the
experimental and numerical data was found by comparing the
simulated and measured voltage profiles (the maximum error
was lower than 1.5%).
B. Analysis of Experimental Results
In this analysis, the performances of the cells were compared
and related to the parameters of the proposed model. Studying
(1), it is possible to relate the knee of the capacitance trend to
the parameter τ (V ). In particular, at an angular frequency equal
Fig. 17.
Experimental current versus time for a test at 110 A.
Fig. 18.
110 A.
BCAP0150 experimental and simulated voltages during a test at
to the inverse of τ , the capacitance value is 97.8% of the dc
value. At angular frequencies a decade greater, the capacitance
is 46% of the dc value.
Taking into account (4), it is possible to state that to increase
the dynamic of a device with a rated capacitance C(V ), it
is necessary to design a cell with a low value of τ . Thus,
it is necessary to reduce the difference between Rdc and Ri .
From Table I, it is clear that, in the new cells, the difference
between these resistances has been reduced by increasing the
high-frequency resistance Ri . This explains why the supercapacitance is shown up to higher frequencies in the new cells.
However, the increase in Ri implies a higher voltage drop
during a fast transient, as shown in Figs. 13 and 14 and 17
and 18.
From a physical point of view, the resistance Rdc is more
related to the behavior of the electrode in terms of the conductivity and geometry of the pores, while the resistance Ri is
more related to the behavior of the electrolyte. To improve the
dynamic of the device, both aspects have to be considered. In
contrast, the increase in capacitance was achieved by increasing
the rated voltage [the capacitance is an increasing function of
the voltage, as given in (2)].
Finally, from these considerations, it can be seen that reducing the resistance of the device and, consequently, improving
the efficiency, are not always related to a better dynamic. In
particular, a good compromise is reducing the resistance at low
frequency Rdc and increasing the Ri -to-Rdc -resistance ratio.
Thus, it is possible to improve the dynamic performances and
reduce the losses in the typical frequency range of these devices
(1–2 Hz).
MUSOLINO et al.: NEW FULL-FREQUENCY-RANGE SUPERCAPACITOR MODEL
Fig. 19. Ragone plot of tested cells.
In order to compare the performances of the two cells,
some discharge tests at a constant power were performed. The
Ragone plot obtained by these experimental tests is shown in
Fig. 19, which also shows a comparison between the experimental and simulated data.
The Ragone plot shows that the performances of the two
cells are very similar. Indeed, the new cell (BCAP0150) has an
increased capacitance (because of its higher rated voltage), but
the increase in its weight is such that its specific energy results
are lower than those of BCAP0140. The experimental values
shown in Fig. 19 are in agreement with the data reported in the
literature.
V. C ONCLUSION
A new model has been proposed that is able to represent
the behavior of supercapacitors over the entire frequency range,
from dc to tens of kilohertz. Through the experimental tests, it
was shown that supercapacitors do not behave like capacitances
at higher frequencies.
The proposed model was able to simulate the self-discharge
phenomenon for tens of days. Indeed, by observing the experimental results, it was verified that the voltage across a
supercapacitor decreases from the rated value to half the rated
value over a period of one month.
A very simple procedure to estimate the model parameters
was also proposed. The model and identification procedure
were validated using a set of tests on different manufactured
cells in the frequency and time domains. These showed that
very good results could be obtained when simulating both the
dynamic response and the very slow phenomena that occur
inside supercapacitor devices.
Finally, the model was used to determine the effect of parameter variations on the dynamic response of supercapacitors.
R EFERENCES
[1] J. A. Martinez, “Modeling and characterization of energy storage devices,” in Proc. IEEE Power Energy Soc. Gen. Meeting, Jul. 24–29, 2011,
pp. 1–6.
[2] M. B. Camara, H. Gualous, F. Gustin, A. Berthon, and B. Dakyo, “DC/DC
converter design for supercapacitor and battery power management in
hybrid vehicle applications—Polynomial control strategy,” IEEE Trans.
Ind. Electron., vol. 57, no. 2, pp. 587–597, Feb. 2010.
[3] S. Lemofouet and A. Rufer, “A hybrid energy storage system based on
compressed air and supercapacitors with maximum efficiency point tracking (MEPT),” IEEE Trans. Ind. Electron., vol. 53, no. 4, pp. 1105–1115,
Jun. 2006.
119
[4] S. Vazquez, S. M. Lukic, E. Galvan, L. G. Franquelo, and J. M. Carrasco,
“Energy storage systems for transport and grid applications,” IEEE Trans.
Ind. Electron., vol. 57, no. 12, pp. 3881–3895, Dec. 2010.
[5] W. Greenwell and A. Vahidi, “Predictive control of voltage and current
in a fuel cell–ultracapacitor hybrid,” IEEE Trans. Ind. Electron., vol. 57,
no. 6, pp. 1954–1963, Jun. 2010.
[6] A. M. Pernia, S. A. Menendez, M. J. Prieto, J. A. Martinez, F. Nuno,
and V. Ruiz, “Power supply based on carbon ultracapacitors for remote
supervision systems,” IEEE Trans. Ind. Electron., vol. 57, no. 9, pp. 3139–
3147, Sep. 2010.
[7] J. Dixon, I. Nakashima, E. F. Arcos, and M. Ortuzar, “Electric vehicle
using a combination of ultracapacitors and ZEBRA battery,” IEEE Trans.
Ind. Electron., vol. 57, no. 3, pp. 943–949, Mar. 2010.
[8] M. B. Camara, F. Gustin, H. Gualous, and A. Berthon, “Energy management strategy of hybrid sources for transport applications using supercapacitors and batteries,” in Proc. 13th EPE-PEMC, 2008, pp. 1542–1547.
[9] D. Iannuzzi and P. Tricoli, “Integrated storage devices for ropeway plants:
Useful tools for peak shaving,” in Proc. 33rd Annu. Conf. IEEE IECON,
Tapei, Taiwan, 2007, pp. 396–401.
[10] P. F. Riberio, B. K. Johnson, M. L. Crow, A. Arsoy, and Y. Liu, “Energy
storage system for advanced power applications,” Proc. IEEE, vol. 89,
no. 12, pp. 1744–1756, Dec. 2001.
[11] M. Pagano and L. Piegari, “Hybrid electrochemical power sources for onboard applications,” IEEE Trans. Energy Convers., vol. 22, no. 2, pp. 450–
456, Jun. 2007.
[12] L. Zubieta and R. Bonert, “Characterization of double-layer capacitors
for power electronics applications,” IEEE Trans. Ind. Appl., vol. 36, no. 1,
pp. 199–205, Jan./Feb. 2000.
[13] F. Belhachemi, S. Rael, and B. Davat, “A physical based model of power
electric double-layer supercapacitors,” in Proc. IEEE Ind. Appl. Conf.,
2000, vol. 5, pp. 3069–3076.
[14] N. Bertrand, J. Sabatier, O. Briat, and J. Vinassa, “Embedded fractional nonlinear supercapacitor model and its parametric estimation
method,” IEEE Trans. Ind. Electron., vol. 57, no. 12, pp. 3991–4000,
Dec. 2010.
[15] N. Rizoug, P. Bartholomeus, and P. Le Moigne, “Modeling and characterizing supercapacitors using an online method,” IEEE Trans. Ind.
Electron., vol. 57, no. 12, pp. 3980–3990, Dec. 2010.
[16] H. Gualous, D. Bouquain, A. Berthon, and J. M. Kauffmann, “Experimental study of supercapacitor serial resistance and capacitance variations with temperature,” J. Power Sources, vol. 123, no. 1, pp. 86–93,
Sep. 2003.
[17] F. Rafik, H. Gualous, R. Gallay, M. Karmous, and A. Berthon, “Contribution to the sizing of supercapacitors and their applications,” in Proc. 1st
ESSCAP, Belfort, France, 2004.
[18] P. Venet, Z. Ding, G. Rojat, and H. Gualous, “Modelling of the supercapacitors during self-discharge,” EPE J., vol. 17, no. 1, pp. 6–10,
2007.
[19] A. Hammar, P. Venet, R. Lallemand, G. Coquery, and G. Rojat, “Study
of accelerated aging of supercapacitors for transport applications,” IEEE
Trans. Ind. Electron., vol. 57, no. 12, pp. 3972–3979, Dec. 2010.
[20] Y. C. Zhang, L. Wei, X. Shen, and H. Liang, “Study of supercapacitor in
the application of power electronics,” WSEAS Trans. Circuits Syst., vol. 8,
no. 6, pp. 508–517, Jun. 2009.
[21] L. Du, “Study on supercapacitor equivalent circuit model for power electronics applications,” in Proc. PEITS, 2009, vol. 2, pp. 51–54.
[22] E. Tironi and V. Musolino, “Supercapacitor characterization in power
electronic applications: Proposal of a new model,” in Proc. Int. Conf.
Clean Elect. Power, 2009, pp. 376–382.
[23] R. Kotz and M. Carlen, “Principles and applications of electrochemical capacitors,” Electrochim. Acta, vol. 45, no. 15/16, pp. 2483–2498,
May 2000.
[24] J. Auer and J. Miller, “Ultracapacitor-based energy management strategies
for eCVT hybrid vehicles,” in Proc. IEEE Conf. Automotive Electron.,
2007, pp. 1–11.
[25] R. De Levie, “Electrochemical response of porous and rough electrodes,”
in Advances in Electrochemistry and Electrochemical Engineering, vol. 6.
New York: Wiley-Interscience, 1967.
[26] S. Buller, E. Karden, D. Kok, and R. W. De Doncker, “Modeling the
dynamic behavior of supercapacitors using impedance spectroscopy,” in
Conf. Rec. IEEE IAS Annu. Meeting, 2001, vol. 4, pp. 2500–2504.
[27] R. Faranda, M. Gallina, and D. T. Son, “A new simplified model of doublelayer capacitors,” in Proc. ICCEP, 2007, pp. 706–710.
[28] Mawell Application Note, Document 1007239 Rev 1. [Online]. Available:
http://www.maxwell.com/products/ultracapacitors/docs/1007239_
R E P R E S E N T A T I V E_T E S T_P R O C E E D U R E_C U S T O M E R_
EVALUATIONS.PDF
120
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 1, JANUARY 2013
Vincenzo Musolino received the M.S. degree
in electrical engineering from the Politecnico di
Milano, Milan, Italy, in 2007, where he is currently
working toward the Ph.D. degree in electrical engineering in the Department of Electrical Engineering.
He is also currently a Design Engineer with Dimac
Red, Biassono, Italy. His research interests include
electrical storage devices, power electronics, and
distributed generation.
Luigi Piegari (M’04) was born in 1975. He received
the M.S. and Ph.D. degrees in electrical engineering
from the University of Naples Federico II, Naples,
Italy, in 1999 and 2003, respectively.
He is currently an Assistant Professor with the
Department of Electrical Engineering, Politecnico di
Milano, Milan, Italy. His current research interests
include electrical machines, high-efficiency power
electronic converters, renewable energy sources, and
electrical supplies and cableway plants.
Enrico Tironi received the M.S. degree in electrical
engineering from the Politecnico di Milano, Milan,
Italy, in 1972.
In 1972, he joined the Department of Electrical
Engineering, Politecnico di Milano, where he is currently a Full Professor. His areas of research include
power electronics, power quality, and distributed
generation.
Dr. Tironi is a member of Italian Standard Authority (C.E.I.), Italian Electrical Association (A.E.I.),
and Italian National Research Council (C.N.R.)
group of Electrical Power System.