1

Abstract Increase power demands on automotive are making current 12/14 V electrical systems inadequate. The total power demand will triple in some cars from 800 W today to an average of 2500 W and into kW range for peak demand in the future. Supercapacitor can be used for energy storage and for peak power requirement in order to increase the efficiency and the life cycle of the system. However, the sizing of energy storage with supercapacitors is very important for embedded applications, because of the weight and the volume of the system. This paper presents a sizing method of a pack of supercapacitors. The proposed method is based on the power and energy demand. In order to validate this method, a model of the Maxwell cell of supercapacitors is presented and implemented in the SIMPLORER software.

1. Introduction In the last years, a great attention has been focused on double layer capacitors or supercapacitors in the United States, Europe and Japan. Much of this work is directed towards transportation. Potential applications concern short time uninterrupted power supplies and peak load in combination with batteries or fuel cell [4, 5, 6]. The power density of supercapacitors is considerably higher than that of batteries, and the energy density is higher than that of electrolytic capacitors. Supercapacitors store high level of energy in a small volume and then release this energy in power burst. Because electrical charges only move between conducting materials in supercapacitors move electrical charges between conducting materials, rather than perform any chemistry reaction, the supercapacitors maintain cycle ability far longer than batteries. Supercapacitors can be used in numerous applications, in electric or hybrid vehicles in order to provide peak power for improved acceleration, for energy recovery, in parallel with the vehicle battery during start up of an internal combustion engine with the purpose of decreasing the size and the power of the battery, in fuel cell vehicles in order to reduce the power and therefore the cost of the fuel cell. To use supercapacitors in hybrid vehicle, it’s necessary to size supercapacitors, in order to simulate and to optimize the system. For this reason, we present in this paper a sizing method of supercapacitors. This method is based on power and energy demand. In the first one, we propose a model of Maxwell BCAP0010. The model has been implemented in Simplorer software. Experimental and simulation results are compared. A sizing algorithm of supercapacitors is presented and validated by simulation results.

2. Supercapacitor modelling

The unit cell of a supercapacitor is based on the double layer capacitance. The elementary structure of supercapacitor consists of two activated carbon electrodes and a separator that prevents physical contact of the electrodes but allows ion transfer between them. Energy is stored in the double layer capacitor as a charge separation in the double layer formed at the interface between the solid electrode material surface and the liquid electrolyte in the pores of the electrodes. When a DC voltage is applied, the electric double-layer is formed to store electrical energy [3]. The double layer capacitance is proportional to the surface area of the electrode and inversely proportional to the thickness of the double layer. This thickness is in the order of a few angstroms. The supercapacitor capacitance is given by

))2(

1

)1(

1(

1

dldl CC+

where Cdl(1) et Cdl(2) are the

electric double layer capacitance at the two electrodes. These parameters can be determined by Helmholtz, Gouy and Chapman theories. In fact, practical situations are more complicated. Measured capacitance of activated carbon shows a non linear relationship with their surface area because of the activated carbon types used and their treatments. To establish an equivalent electric circuit for supercapacitor which takes into account these problems, Conway [8], and other authors [6, 9, 10, 11], proposed an equivalent circuit based on transmission-line model, which involves distributed capacitance Ci and resistance Ri. Ri and Ci can be considered as resistance and capacitance of the pores with a certain pore size. However, in power electronic applications, the supercapacitor electric behaviour can be described by an equivalent electric circuit with RC branches [1, 12]. This model is not complex and the simulation time is reduced compared with the model of transmission line. Fig. 2 shows the equivalent circuit of supercapacitor used in this study. The purpose of this equivalent circuit is to provide a model of the terminal behavior of the double-layer capacitor in power electronics circuits. Therefore, the following requirements have been set before formulating the equivalent circuit structure:

* The model should be as simple as possible and should describe the terminal behavior of the supercapacitor over a range of a few minutes with sufficient accuracy.

* The parameters of the proposed model should be determined using measurements at the supercapacitor terminals. Based on the desire for a simple model and the experience from measurements, a model of two branches is proposed. This provides two different time constants to model the different charge transfers, which provides sufficient accuracy to describe the terminal behavior of the supercapacitor.

Contribution to the sizing of supercapacitors and their applications F.Rafik*, H.Gualous**, R. Gallay***, M.Karmous*, A. Berthon**

* Ecole d'ingénieurs Arc, CH-2400 LE LOCLE, Switzerland **L2ES-UTBM-Univ.Franche comté, Rue T.MIEG, F90010 BELFORT, France ***MAXWELL Technologies, CH-1728 Rossens, Switzerland

Fouad.rafik@he-arc.ch

2

Supercapacitor BCAP0010 - 2600 Farad / 2.5V

To reflect the voltage dependence of the capacitance, the first branch is modeled as a voltage dependent differential capacitor C1. It’s consists of a constant capacitor and a capacitor whose value varies linearly with the voltage Vc [1]: CT = C0 + K * Vc. R1 is the equivalent serial resistance (ESR). The R1C1 branch dominates the immediate behavior of the supercapacitor in the time range of seconds. The R1C1 cell is the main branch, which determines energy evolution during charge and discharge cycles in power electronics applications (charge and discharge in a few seconds). It’s called a fast branch. The R2C2 cell is the slow branch; it completes the first cell in long time range in order of a few minutes and describes the internal energy distribution at the end of the charge (or discharge). Rp is the equivalent parallel resistance. The later has only impact on long term storage performances since it is a leakage effect. Rp can be neglected during fast charge/discharge of the supercapacitor. A series inductance may be added for pulse applications, but measurements show that it is so small (some nH) that it can be neglected in power electronic applications. The parameters of the proposed model with two RC branches can be identified carrying out a single fast current controlled charge. The parameters are identified by charging the supercapacitor, from zero to rated voltage and by observing the terminal voltage during the internal charge redistribution, the current of charge being constant (Ich). The approach to determine the different parameters is based on the fact that the two equivalent branches have distinctly different time constants. Therefore, the transient process of each branch can be observed independently by measuring the terminal voltage as function of time. It assumed that the response to the fast controlled charging process is determined only by the parameters of the fast or immediate branch R1C1. After the external charging stops, all charge is in the capacitors of the immediate branch. Then, the charge redistributes itself to the second branch R2C2. Zubieta and Bonert [1] have established an experimental method to determine the parameters of the supercapacitor equivalent circuit (figure 1). L1 2

Rp Cv

R1

C2V1C0

R2

Vc

The characteristics of the supercapacitor unit cell used in this study are listed in table below:

In the first we have determined the parameter of Maxwell BCAP0010 model equivalent circuit exposed previously. The supercapacitor is charged at constant current (in order of 100A). Figure 2 shows the voltage evolution versus time. The characteristics of this cell are listed in table 1 [Maxwell data].

0

0.5

1

1.5

2

2.5

3

0 20 40 60 80 100Time (s)

Vo

ltag

e (V

)

The supercapacitor charge curve at constant current shows that its capacitance is variable according to the voltage [1]. Therefore, the analysis of this supercapacitor profile charge makes it possible to model it by a resistance in series with two capacitances composed by a constant capacitance and a capacitance which value varies linearly with the voltage. In power electronic applications supercapacitor are used to provide the peak power in transient state (short time), the slow R2C2 branch can be neglected figure 1. Hence, the simplified model is proposed in figure 3:

ccT VKCC ⋅+= 0 Where R: is the equivalent serial resistance.

CT: is the equivalent supercapacitor capacitance.

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������ � ������ ������

���������������� ���� ����� �����

���������������� ������

�������������� ������

����� ��� �� ������������

������� �����

���������� �������������

R

C0 C(vc) = Kc.Vc Vc

If I

V

Figure 2: BCAP0010 charge at constant current (100A)

Figure 3: Simplified supercapacitor model

Figure 1: Equivalent circuit of supercapacitor

3

Supercapacitor parameters determination: From the curve obtained we identified the various elements forming the model. Resistance R is the cause of the fall of charging voltage at the initial moment:

IRUR .= =>

I

UR R=

C0 is directly given starting from the characteristic charges at constant current of the cell; it translates the initial slope

of the curve:

(0)t

vI

C 0

∆∆=

We calculate the value of C0 from the supercapacitor charge curve slope at constant current of the cell of Supercapacitors. In practice, we make an approximation of

the initial slope of V by a slope0

0�

t

�V

, we find in the

vicinity immediate of the initial moment (figure1). When the supercapacitor is charged at its nominal voltage, the constant KC can be expressed by the following relation [1]:

0 12

1

( . .( . ))

( . )c

I t C V R IK

V R I

∆ − ∆ −=∆ −

The model proposed by the manufacturer of this supercapacitor (BCAP0010) is composed by three RC branches figure 4 [2]: Where:

The proposed model in this study and the model with three RC branches are implemented in the Simplorer software. Figure 5 presents the simulation and experimental results. It’s clear that the results obtained using the model, where the capacitance varies linearly with the voltage, is in good agreement with the experimental ones. However, the supercapacitor charge profile of supercapacitor obtained

using the model composed by three RC branches can be considered as a good approximation. Then, the proposed model is validated and will be used for supercapacitors sizing.

0

0.5

1

1.5

2

2.5

3

-10 10 30 50 70 90Time (s)

Vo

ltag

e (V

)

Practical chargeModel createsMAXWELL Model

The energy stored in supercapacitor is given by the equation:

� ⋅= dQVW Where: VvCQ ⋅= )(

Using the expression of the supercapacitor equivalent capacitor the energy stored is:

20

1 4

2 3 cW C K V V� �= + ⋅ ⋅ ⋅� �� �

This energy expression is composed by classic term 1/2 C0V

2 and a second expression 1/3 KcV3. We have plotted in

figure 6 the evolution of the equivalent energetic capacitance and the energy stored according to the voltage of BCAP0010. The energetic capacitance varies from 1902 to 2590 F, when the voltage varies from 0 to 2,5V [4].

We write: VKC ⋅⋅+=34

C Cw 0

Cw is the “energetic” equivalent capacitance: the capacitance which is involved to determine the energy of the supercapacitor.

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 0.5 1 1.5 2 2.5 3

Voltage (V)

Capacitance (F)Energy (J)

y = 276x + 1902

y = 206.67x3 + 951x2 - 9E-10x + 1E-09

Cf = 2600F Cm = 250F Cs = 560F Rf = 0.68m� Rm = 0.8� Rs = 2.9� R = 3k�

Figure 6: Capacitance and energy evolution with the supercapacitor voltage

Figure 5: Supercapacitor charge at constant current (100A)

Figure 4: Model proposed by Maxwell

R Vc Rf

I

V Rm RS

Cf Cm Cs

4

Supercapacitor sizing method: To use supercapacitors for embarked applications, it’s necessary to optimise the size and the weight of the supercapacitors pack. The sizing method proposed in this study is based on power and energy requirement. To determine the energy we have used the expression of energy defined previously. In most applications, supercapacitors pack is charged and discharged between Vmax and 50% Vmax, where Vmax is the maximum voltage. This is due to the fact that when the supercapacitors pack is discharged from Vmax to 50%Vmax, 75% of the energy storage is used. The sizing method is presented on the following diagram (figure 7).

For a constant power: P= Vmax.Imin=Vmin.Imax The average current is estimated in the following way:

)I (I

2

1 I minmaxmoy +⋅=

The maximum and minimum capacitances are:

( )paralel

series

moyc

W NN

IRVKCC ⋅

⋅−⋅+=

max0

max3

4

( )parallel

series

moyc

W NN

IRVKCC ⋅

⋅−⋅+=

min0

min3

4

Vmax and Vmin are respectively the maximum and minimum voltage of one supercapacitor cell Imax and Imin are respectively the maximum and minimum current of one supercapacitor cell Nseries: the series number supercapacitor cell Nparallel: the parallel number supercapacitor cell

The average capacitance equal:

)C (C2

1 C WminWmaxWmoy +⋅=

The energy is defined by:

])I .R-(V - )I .R-[(V . C . 21

W 2moyTmin

2moyTmaxWmoy=∆ *

To validate the sizing method proposed in this study, a pack of supercapacitor composed by 5 BCAP0010 is considered. The maximum voltage is 12V and the minimum is 6V. The supercapacitors pack is discharged at constant power of 1200W. Using the flow chart above, we estimate the energy stored by this pack. And by simulation we confirm the result found before. After charging the supercapacitors pack to his maximum voltage, we discharge it with a power of 1200W and we calculate the supply duration of supply of this power for until the discharging it. By applying the method of calculation above we find:

WmoyC 486F= And A 150 Imoy =

The energy provided by the pack is of: � W = 24930J The pack provided the power of 1200W during the time � t = 21 seconds.

Figure 7: Sizing supercapacitor pack diagram

5

From these results (figure8), we can deduce that the pack is discharged at 1200W during 22s. This measured result is in good agreement with the calculated one. The sizing method is therefore validated. If the equation of the capacitance used is unknown, the method used before can be simplified by simply using the value of the capacitance given by the manufacturer and by neglecting the losses dues to series resistance of supercapacitors. An approximation of the equation * gives:

])(V - )[(V C 2

1 W 2

min2

maxT=∆

CT : Total capacitance determined from the capacitance value provided by the manufacturer

Supercapacitors in the field of transport: In this paragraph, an application of supercapacitors in the field of transport is proposed. The structure of the system is composed by a pack of supercapacitors and a battery. DC/DC converter is used for voltage and current levels adaptation between the two sources and a 42V DC bus. Supercapacitors are used, in parallel with a battery in this 42V hybrid power source for automotive applications, for peak power requirement in order to increase the efficiency and the life cycle of the system, and to reduce the battery size. Figure 9 presents, it consists of a 42 V batteries with a pack of supercapacitors. This last one is sized to provide

2.5kW during 12s while the battery delivers the power in permanent regime. The supercapacitors pack is charged and discharged between 7V and 14V. The pack is charged at constant current. The regulation is realized by Hysteresis. The control of the 42V DC Bus is realized using a Proportional – Integral (PI) regulator. Using the model with two RC branches of supercapacitors (figure 3), we have simulated the charge and discharge from the 42V DC bus. Figure 10 shows the first results. The hybrid power source output voltage is 42V; this voltage is also the DC voltage of the on-board electrical network. The maximum supercapacitors voltage is fixed at 14 V. The pack of supercapacitors is charged with constant current. Because of the PC memory, the first charge of supercapacitors is realized, at high and constant current, from 0 to 14V. So, in operating system, the pack is sized to be charged and discharged between 100 % and 50 % of its maximum voltage. In order to validate the pack of supercapacitors autonomy as well as the performances of the regulation loop of the converter, we charge the supercapacitors to their maximum voltage, and we discharge them on the DC bus voltage with their maximum power. A 60A current is requested on the continuous bus, the converter keeps the voltage of the bus at a 42 V constant value. Figure 10 shows the results of simulation. The cycle of charge/discharge from the pack of supercapacitors, voltage and current of the DC bus voltage. The discharge is carried out with maximum current of 60A on the DC bus voltage. The supercapacitors guaranteed well the power of 2.5kW for the desired duration.

Figure 8: Current and voltage of a pack: Charge with constant current – discharge with a constant power

Figure 9: 42V automotive electric power net

Supercapacitors Supercapacitors DC/DC converter DC/DC Converter

36/42 V

42 V

Battery

6

In order to validate the performances of regulation of the voltage of the DC bus voltage; we request a variable current on this last. Figure 11 represents the voltage of the pack, the current of the DC bus voltage and its voltage which is controlled with 42V.

In figure 11, we have represented the waveforms of the supercapacitor pack voltage, the DC bus current and voltage. It can be seen that when the load current value changes from 0 to 30A, 50A and 70A, the DC bus voltage is constant and regulated at 42V. Then, the pack of supercapacitors voltage varies between 14V and 7V.

Conclusion Thanks to the recent developments, the supercapacitors offer the possibility to store energy and to manage the electric power. In this paper, we developed a simplified model of the cell of BCAP0010 supercapacitors. A model was implemented in the SIMPLORER software. This model is valid for the low frequencies. Simulation results and experimental ones are in good agreement. This paper also introduced an optimal sizing method of supercapacitor pack. This last is based on the power and energy requirement. This sizing which must be very precise especially in the applications embarked in order to reduce the weight and the obstruction of the system. The method was validated by simulation of time of discharge of a pack forming cells of supercapacitors modelled before. This method is also validated using a 42V automotive electric power net. This application made it possible to simulate the operation of supercapacitors applied in the field of the surface transport by using the model created before, to simulate the power and energy provided by the pack like supervising its current and its voltage. This work will be used as a basis of simulation of the storage applications of the electric power based on the supercapacitors for the optimized practical applications. References: [1] L. Zubieta, R. Bonert , IEEE-IAS 98 (1998) 1149-1154. “Characterization of double-layer capacitors for power electronics applications”. [2] By John M. Miller, J-N-J Miller, plc Patrick J. McCleer, McCleer Power, Inc. and Mark Cohen, Maxwell Technologies: “In Ultracapacitors as Energy Buffers in a Multiple Zone Electrical Distribution System”. [3] Alfred Rufer, Philippe Barrade, Laboratoire d’électronique industrielle, Ecole Polytechnique Fédérale de Lausanne : ″Stockage d’énergie électrique par Supercondensateurs, solutions de l’électronique de puissance et applications″. [4] V. Hermann, A. Schneuwly and R. Gallay, montena components, Switzerland: “High Power Double-layer Capacitor Developments and Applications, 52th Meeting of the ISE, San Francisco, Sept 2001” [5] A. Burke, Ultracapacitor: why, how, and where is the technology, J. Power Sources 91 (2000) 37–50. [6] E.J. Dowgiallo, A.F. Burke, Ultracapacitors for electric and hybrid vehicles, in: Proceedings of the Electric Vehicle Conference, Florence, Italy, 1993. [7] R. Kötz, M. Carlen, Principles and applications of electrochemical capacitors, Electrochim. Acta 45 (2000) 2483–2498. [8] B.E. Conway, Electrochemical Supercapacitors, Kluwer Academic Publishers/Plenum Press, New York, 1999, pp. 335–452. [9] Deyang Qu, Hang Shi, Studies of activated carbons used in double-layer capacitors, J. Power Sources 74 (1998) 99–107. [10] Eckhard Karden, Stephan Buller, Rik W. De Doncker, A frequency-domain approach to dynamical modeling of electrochemical power sources, Electrochim. Acta 47 (2002) 1–10. [11] R. De Levie, in: P. Delahay (Ed.), Advances in Electrochemistry and Electrochemical Engineering, vol. 6, Wiley, New York, 1967, pp. 329–397. [12] B. L. Meng, H. Gualous, D. Bouquain, A. Djerdir, A. Berthon, J.M. Kauffmann, Thermal modeling and behavior of supercapacitors for electric vehicle applications, in: Proceedings of the Ninth European Conference on Power Electronics, EPE 2001, Graz, 27–29 August 2001, CDROM.

Figure 11: Simulation results of the hybrid power source

Figure 10: Simulation results of the hybrid power source. Charge / Discharge cycle

Scaps Voltage

42V-Bus voltage (V)

42V-Bus current (A)

Scaps Voltage (V)

42V-Bus voltage (V)

42V-Bus current (A)