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Advances in Electrical and Computer Engineering Volume 8, Number 2, 2008

Modeling and Sizing of Supercapacitors

Dorin PETREUS, Daniel MOGA, Ramona GALATUS, Radu A. MUNTEANUTechnical University of Cluj Napoca

str. C. Daicoviciu nr. 15, RO-400020 Cluj-Napoca

[email protected]

Abstract Faced with numerous challenges raised by the

requirements of the modern industries for higher power and

higher energy, supercapacitors study started playing an

important role in offering viable solutions for some of these

requirements. This paper presents the surface redox reactions

based modeling in order to study the origin of high capacity of

EDLC (electrical double-layer capacitor) for better

understanding the working principles of supercapacitors. Some

application-dependent sizing methods are also presented since

proper sizing can increase the efficiency and the life cycle of the

supercapacitor based systems.

Index Terms electrochemical capacitor, modeling, sizing,

pseudo-capacitance

I. INTRODUCTIONElectrochemical capacitors have achieved substantial

acceptance in the electronics industry, replacing backupbatteries in many CMOS memory applications. Many of thefirst commercially available electrochemical capacitors weredirected toward such applications and, consequently, wereof limited size and power performance. As will be shown inthe next sections, these limitations are not inherent in thetechnology, so larger, higher-voltage capacitors with greatly

enhanced power performance have been available forseveral years from some suppliers and currently are beingdeveloped by others [18].

Different names like double layer capacitors,supercapacitors, ultracapacitors, power capacitors,gold capacitors, power cache were used to indicatedifferent types of capacitors displaying high values ofcapacity, but the first mention of a double-layerelectrochemical capacitor realization is made just in 1957, ina patent application of General Electric (Becker). [4]

In 1966, SOHIO (Standard Oil Company, Cleveland,Ohio) patented a similar device having porous carbon

electrode but with a higher energy density than that ofGeneral Electric. Further patents described improvedSOHIO developed solutions in the following years, but theirlicenses are transferred to NEC which produced for thefirst time commercial devices under the namesupercapacitor. The first high power electrochemicalcapacitors were produced in the years 1980 by PinnacleResearch Institute (PRI) for military applications like laserweapon and guided missiles with ruthenium/tantalum oxideelectrodes under the name PRI ultracapacitor. [4]

Nowadays, electrical double-layer capacitors (EDLC) aregaining more attention due to their practical potential inapplications areas with increasing power demands. Due toreversible electrochemical energy storage these capacitorscan be recharged very quickly with best long cycle life.Energy storage is by means of static charge rather than of anelectro-chemical process (inherent to common battery).

New applications are emerging in fields like automotiveengineering1, rail traction, telecommunication,uninterruptible power supplies, renewable energy resources,industrial electronics and medical engineering.

Recently, EDLC have become a topic of some interest inthe green energy world where their ability to quicklyimmerse up energy makes them particularly suitable forregenerative braking applications, whereas batteries havedifficulty in this applications due to slow charging times.

ELDC are electrochemical capacitors that have an

unusually high energy density when compared to commoncapacitors, typically of the order of thousands of timesgreater than a high-capacity electrolytic capacitor. Threemain factors determine how much electrical energy acapacitor can store: the surface area of the electrodes, theirdistance from each other, and the dielectric constant of thematerial separating them. In EDLC, the effective thicknessof the "dielectric" is exceedingly thinon the order ofnanometersand that, combined with the very large surfacearea, is responsible for their high capacitances in practicalsizes. At low working voltages, by using EDLC acapacitance of several farads is obtained.

That means a significant improvement compared with anelectrolytic capacitor having the same size.[6]When a supercapacitor is charged, there is no chemical

reaction, energy being stored as a charge or concentration ofelectrons on the surface of a material. It is capable of veryfast charges and discharges, and apparently is able to gothrough a large number of cycles without degradation -providing long life cycle.

Supercapacitors can be an excellent replacement forcommon batteries where applications requests power burst,quick charging, temperature stability, and excellent safetyproperties (immunity to shock and vibration). A taxonomyof supercapacitors is presented in figure 6 according to [7].

Electrochemical capacitors have significant advantagesfor deployment in renewable energy resources systems, likethe photovoltaic systems. Some of these are listed belowaccording to [18]:

- lack of maintenance - In contrast to the batterymaintenance, capacitors require no maintenance. Thisgreatly reduces system cost over time and allows the storagesystem to be located in places impractical for chemicalbattery systems (e.g., buried).

-longevity- Because capacitors store charge physically

1 As indicated in [18], one early Russian application of electrochemicalcapacitors was for starting vehicles in cold climates. In Siberia, the cold-weather advantages of electrochemical capacitors over chemical batterieswere quickly apparent.

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rather than chemically, cycling has virtually no effect ontheir capacity or longevity. Twenty-year life is easilyachieved by proper selection of materials and control ofoperating parameters.

-environmentally benign-Capacitors do not employ toxicmaterials, and thus present no environmental threat inmanufacture, transport, or disposal. They do not outgass inuse and present no threat of explosion.

-high discharge rate capability-Capacitors can bedischarged at very high rates without damage. High rates,however, reduce the delivered energy of the unit.

A comparison of properties of rechargeable batteries andelectrochemical capacitors is presented in Table 1 (after[12]).

TABLE 1Property Battery Electrochemical

capacitorStoragemechanism

Chemical Physical

Powerlimitation

Reaction kinetics,mass transport

Electrolyteconductivity

Energystorage

High (bulk) Limited(surface area)

Charge rate Kinetically limited High, same asdischarge rate

Cycle lifelimitations

Mechanical stability,chemical reversibility

Side reactions

II. ORIGINOFTHEHIGHCAPACITANCE,MODELINGSTUDY

In addition to high-surface-area carbons, even highercapacitance can be achieved by using redox-active materials

such as metal oxides and conducting polymers, able to storeelectrochemical charge using highly reversible surface redoxreactions, commonly described as pseudocapacitance inthe literature [12]. The chargestorage performance can bedrastically increased using these fast, reversible surfaceprocesses.

The high value of capacitance that is attainable throughEDLC is a result of double-layer capacitance andpseudocapacitance:

pscaplayerdouble CCC += (1)

The origin of high capacitance is studied using a

thermodynamic approach: the simulation of pseudo-capacitance by electroabsorbtion isotherms.The electric charge is stored performing adsorption and

desorption of the ions in electrolysis liquid to electrode.Considering an analogy with a conventional capacitor (seeFigure 1), Helmoltz proposed already in 1853 a model toestimate the double-layer capacitance:

4

AC layerdouble = (2)

where C is the double-layer capacitance, A is the surfacearea, is the relative dielectric constant of the medium between the two layers (the electrolyte), and is the

distance between the two layers (from the electrode surfaceto the centre of ion layers).

More elaborate (and accurate) models of the double- layercapacitance can be found in [1]. For a larger double-layer

capacitance it would be mandatory to produce a thin, highsurface-area electrical double-layer with a combination ofhigh surface area (about 200m2/g) with extremely smallcharge separation (~10) [3].

Regular double-layer capacitance arises from the potential-dependence of the surface density of chargesstored electrostatically (non-Faradaically) at the interfacesof the capacitor electrodes [3]. A double-layer capacitordevice must employ two double layers, one at each electrodeinterface, working one against the other on charge ordischarge, as emphasized in [3] -see Figure 2.

Pseudocapacitance arises in some electrosorbtion processes and in redox reactions at electrode surfaces oroxide films [3], and is Faradaic in origin involving thepassage of charge across the double layer2,3.

Figure 1. Schematic of a conventional capacitor (adapted from [7]).

Figure 2. Schematic of an electrochemical double-layer capacitor (adaptedfrom [7]).

2 capacitance arises due to the special relation that originates fromthermodynamic reasons between the extent of charge (q) and the changeof potential (V), so that a derivative d(q)/ d(V), which is equivalent to a

capacitance can be formulated and experimentally measured [3]3 Another type of pseudocapacitance is the capacitive component of theWarburg impedance associated with diffusion-controlled processes whenexamined under ac modulation. As pointed out in [3], this pseudo-capacitance has no value for storage purposes and is significant only for thefrequency response of electrochemical reactions involving diffusion.

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If a property y is proportional to the charge passed and isrelated to the potential by an equation of the form:

RT

VF

Key

y=

1

then, from a thermodynamic point of view, a pseudocapacitance arises [3] . An example for the quantity y is theextent of the fractional coverage of an electrode surface

generated by the charge required for deposition of adatoms.There are two types of reactions that can involve a charge

transfer that is voltage dependent: surface redox reactions oradsorption of ions. A model for study of the electrodepotential influence on the extent of fractional coverage, , ofthe absorbed species is proposed in [3] using Frumkinsisotherm, as follows:

gieKC

2

1=

(3)

where, Ci is the species concentration and K is a constantthat depends on V (K=f(V)).

The Langmuir isotherm (6) applies for the case when theinteraction parameter, g, is null (g = 0) (4), at adsorptionequilibrium:

i

i

CK

CK

+

=

1 (4)

with:

dV

dqC m

= (5)

where qm is the charge on the metal surface that can beexpressed as:

10 )1( qqq m += (6)

it is obtained:

( ) ( )

mpscap

RT

qqC

=

1201 (7)

where q1 is the charged associated with the coverage of theabsorbed species, , and q0 is the charged associated withthe uncovered surface, R universal gas constant, Ttemperature in Kelvin, m maximum amount of adsorbedspecies.

If g>0, corresponding to pairwaise adsorbate repulsion inthe layers, Temkins equation is used instead of Langmuir

equation:)ln(max iCK= (8)

where max corresponds to maximum coverage surface, withthe normalization max = 1.

The influence of the interaction parameter on the pseudocapacitance is presented in Figure 3. Figure 4presents the pseudocapacitance dependence by temperaturein [-20C, +60C], (using g = 1), in an interval which iscommonly specified by producer as normal operationconditions. In Figure 5 is presented the user interface, andthe simulations results for both, the Langmuir and Temkins

Isotherms.The pseudocapacitance corresponding to Temkinisotherm is expressed by:

( ) ( )))1(21(

1201

=

gRT

qqC

mpscap (9)

The models presented in this section are useful ingrasping some of the basic principles of the electrochemicalsupercapacitors (see Figure 6). More complex models areneeded for a closer and more in-depth characterization ofthese devices.

Carbon black supercapacitors and carbon black-supportednoble metal oxides pseudo-supercapacitors, as well as porous coatings, consisting of noble metaloxides onactivated titanium anodes are indicated in [13] as typicalexamples of porous electrochemical systems, and the factthat the capacitive properties of these materials are stronglyinfluenced by the real surface area, porosity and particle sizeand distribution is acknowledged.

Generally speaking, results of methodical experimentsshould provide better understanding, but, as mentioned in[16], the electrochemical measurements do not provideenough insight into the structural and electronic changes

associated with the redox processes, despite theirextraordinary sensitivity. Scanning electron microscopy(SEM), transmission electron microscopy (TEM) andsurface-area measurements are frequently used toinvestigate the electrode micro and macro structure. Figure 7reproduces a cross section of an electrode and currentcollector together wiht a plan view of an electrode surface asin [17].

Pseudocapacitance according to Temkin Isotherm

Pseudocap

ce

[F]

acitan

Electrode Potential [V]

Figure 3. Pseudocapacitance values directly dependent on interactionparameter.

Pseudocapacitance according to Temkin Isotherm

Pseudocapacitance

[F]


Figure 4. Pseudocapacitance values indirect dependent on temperaturevariation.

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Coverage according to Langmuir Isotherm

Coverage[100%]


Coverage according to Temkin Isotherm

Cove

rage[100%]


Pseudocapacitance according to Temkin IsothermPseudocapacitance according to Langmuir Isotherm

Pseudocapacitance[F]

Pseudocapacitance[F]

Electrode Potential [V] Electrode Potential [V]

Figure 5. Simulation interface. Graphical representation of coverage and pseudocapacitance vs. electrode potential using:a) Langmuir isotherm, and b) Temkin isotherm.

Figure 6. Taxonomy of supercapacitors (according to [7]).

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The electrode thickness and internal area, the porositiesand particle sizes of the active material, as well as thematerials selected for the electrolyte and electrode are thedesign elements that can be more accurately investigatedwith microstructural models such as that described in [17].

Figure 7. Carbon particle sizes on the order of a micron as shown by theSEM images and cross section of an electrode and current collector.

III. SIMPLE SIZING METHODSThe supercapacitors have usually to work in a series of

discharge and charge cycles. Recharge current depends onthe supercapacitance value and on the internal losses. Thenumber of supercapacitor cells required can be determinedby the system variables such as allowable change in voltage(max and min voltage), current (or power) and requiredcharge/discharge time.

The supercapacitors voltage profile has a capacitive andresistive component. The capacitive component is in relationwith the charge/discharge voltage and the resistivecomponent represents the voltage variation due to internalresistance (ESR equivalent series resistance). A simplemodel for a EDLC can be represented by a capacitance with

an equivalent series resistance (ESR) and an equivalentparallel resistance (EPR) (Figure 2) 4.

This can be described by:

iRdtC

iv

t

+= 0

(10)

4As suggested in [4], the series resistor ESR (accounting for thedissipation during charging and discharging) can be determinedfrom the time constant S corresponding to a discharge through aload R, and the parallel resistor EPR (accounting for the self-

discharge phenomenon which became apparent after long times -days) and can be computed from the self discharge time constant

P: RC

ESR S = ,

CEPR P

=

where:- v is allowable change in voltage in [Volts],- i is current in [Amps],- R is equivalent internal series resistance (ESR) in

[Ohms],- tis charge or discharge time is [sec], and- Cis capacitance in [Farads].

Figure 8.Simple EDLC equivalent electric model.

The equivalent serial resistance (ESR) models the lossesby Joule effects.

The equivalent parallel resistance (EPR) represents thecurrent loses that influences the energy storage. The IEC62391 standard [8], [9], [10], which has been written to beadapted to high series resistance supercapacitormeasurement, defines that the mean value should becalculated between 80% and 40% of the nominal voltagevalue.

Internal resistance, ESR, blocks instant discharge of thesupercapacitor. In the equivalent model of supercapacitor[15] the parallel resistance has a visible effect only at very

low frequency (below the milihertz range). It is responsiblefor the capacitors self discharge time.Considering this model, the equivalent parameters of a

supercapacitor system (pack), based on a number ofindividual supercapacitors groups connected in series or inparallel, can be described as:

cellsnumberserial

cellsnumberparallelCC celsystem

__

__= (11)

The equivalent resistance of supercapacitor system is:

cellsnumberparallel

cellsnumberserialRR celsystem

__

__= (12)

Assuming constant charge/discharge current, i (that canbe associated to an average value, as indicated in (13)), thefollowing algorithm based on this simple model can beapplied (see also [4], [5], [6]):

Step 1: Supercapacitor technical properties

- Voltage variation during the charge/discharge cycle,calculated as difference between known operatingvoltage (Vop often Vmax of the system) and minimalvoltage (Vmin) required by the system (often 50% Vmax ).- The time constant t of the self charge/discharge isequal to: CEPRP =

- Power P (assuming constant power,

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)maxminminmax IVIVP ==

- Specific application energy requirement specification(W)- Current value (constant):

2minmax IIIi average

+== (13)

Step 2: Cells of supercapacitor used

- serial cells number:

cellindividualV

Vcellsnumberserial

_

max__ = (14)

Usually V individual_cell = 2.5V. (15)

- parallel cells number iteration:

Loop: parallel_number_cells = 1If after the computation results an inadequatecapacitance for the specific application, it needs tomodify:a. the number of parallel cels, parallel_number_cells= parallel_number_cells + 1 and then return tocomputation loop , or: b. the individual cell type and then return to thebeginning, at Step 1.

In some situations using a lower number of serial cellswith higher voltage supply is a good option but is a

compromise between performance and life-cycle of thesystem.System resistance calculation can be done with (12).

Usually, a high capacitance and low resistance system ispreferred [19].

IV. POWERTRANSFERBASED SIZINGThis section will present further consideration for

dimensioning the supercapacitors based on the specificpower needs of the application where they are intended to beused. The approach is mainly the one described in [11],starting with a pre-dimensioning and introducing then thestored energy versus delivered power diagrams (Ragoneplots).

Figure 9 presents a simplified model of the super-capacitor that will be used in this section, characterized bythe values of the parameters , .supC supCR

Denoting with the energy stored in the

supercapacitor at a certain moment, this can be expressed(neglecting the series resistance) as:

*

supCW

2sup

*supsup 2

1CC VCW = (16)

If the allowable limits of the voltage between which thedischarging process takes place are considered, then the

variation of the energy is ( 22sup* minsupmaxsupsup 21

CCC VVCW = ) .

Figure 9. Simplified model (after [11]).

For example, if an excursion of 2maxsupC

V is allowed, the

energy variation will be 2sup*

maxsupsup 8

3CC VCW = ,

accounting for 75% of the stored energy.Considering the needs of the system to which the

supercapacitor has to deliver its charge, as defined by thevalue needW , some simple calculations [11] lead to the

dimensioning of the capacity through = :*supC

W stocW

2supmaxsup

3

8

C

need

V

WC

= . (17)

Consider now the powers and the energies delivered bythe supercapacitor (the suffix i stands for internal), asnoted in figure 10.

Figure 10. The energies an the powers associated to the supercapacitorsmodel considered (after [11]).

In order to plot the stored energy versus delivered power

one starts from the energy stored: 2sup*

supsup 21

iCCVCW = .

Observing that and that

, the expression of becomes:

supsupsupsup CCCCIRVV

i+=

supsupsup CCCIVP = *

supCW

( )

++

= 2sup

2sup

22sup*

supsupsupsupsup

sup

sup 2

1

2

1CCCCC

C

C ICRPCRPI

CW (18)

This formula is useful for tracing the stored energy versusdelivered power plots for a given supercapacitor at differentcurrent regimes. As one can see in the figures 11 to 13, a

parabola is obtained for each value of the current. These parabolas touch the horizontal axis when , i.e.,0*

sup=CW

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for the case (meaning that the

recharging of with a current will be effective

only if the power received by the supercapacitor is larger

than ).

2supsupsup CCC

IRP =

supC supCI

2supsup CC

IR

The result of (18) can be rewritten in terms of the voltage

value, as follows:

( )

++

= 2supsup

22

sup2

*

supsupsupsup

sup

sup

sup 2

1

2

1CCCC

C

C

C VCPCRPV

CRW

By limiting the voltage trip across the supercapacitors, theextremities of the parabolas are also limited, as can be seenin figures 11 to 13.

A final remark on the shape of the curves in these figuresis that the right side extremities of the parabolas are equally

limited to the value 2sup*

maxsupmaxsup 2

1iCC

VCW = . That can be

easily observed considering (16) and the fact that is

the upper limit of .

maxsupCV

maxsup iCV

The above mentioned equations were applied for asupercapacitor with the following parameters: nominalcapacity Csup=350F, nominal voltage U=14V, and weightm=24kg (produced by the Russian company Econd Ltd).The results obtained for operating currents of 20A, 40A,60A, and 80A are presented in figures 11, 12 and 13.

V. CONCLUSIONIn order to approximate the coverage of the adsorbed

species, , and the pseudocapacitance, some simple models(based on Frumkin isotherm, Langmuir isotherm andThemkin isotherm) were simulated in Matlab and a friendlyuser interface for studying with graphical representationsthese interactions was developed. More accurate modelingmethods for supercapacitors have to take into account the pseudocapacitance variation as a function by electrodepotential (voltage on the capacitor).

A better understanding of the processes characteristic toelectrochemical supercapacitors can be gained only bydeveloping more accurate models. If these models willprovide enough information on factors that are critical to the

rational design of materials composing the supercapacitors,then new generations of devices with optimal performancecharacteristics will appear in coming years.

This paper also introduced some optimal sizingalgorithms of a supercapacitor pack, based on power andenergy requirement for specific applications. Thesealgorithms were used as a basis of simulation in storageapplications of the electric power, based on supercapacitors,for the practical applications of terrestrial transportation.[15]

Figure 11. Stored energy versus delivered power.



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REFERENCES [11] Langlois O., Conception dun rseau de secours lectrique pourlaronautique, These, Laboratoire dElectrotechnique etdElectronique Industrielle de lENSEEIHT, juin 2006[1] Bard, A.J., and L.R. Faulkner, Electrochemical Methods:

Fundamentals and Applications, A.J. Bard and L.R. Faulkner (eds.),John Wiley & Sons, Inc., New York, 2001

[12] Miller J.R. and Simon P.,Electrochemical Capacitors for EnergyManagement , Science, August 2008:Vol. 321. no. 5889, pp. 651 652[2] Belhachemi F, Ral S, Davat B. A physical based model of power

electric double-layer supercapacitors", IEEE-IAS'00, 2000, Roma. [13]Naoi K. and Simon P., New Materials and New Configurations forAdvanced Electrochemical Capacitors, Interface, The ElectrochemicalSociety, Spring 2006

[3] Conway B. E. , Electrochemical Supercapacitors: ScientificFundamentals and Technological Applications (KluwerAcademic/Plenum, New York, 1999). [14] Panic V. V., Supercapacitive characteristics of electrochemically

active porous materials, J. Serb. Chem. Soc. 73 (6) 661664 (2008)[4] D'Arco, Salvatore (2006) Lelettronica di potenza per la gestione di

sistemi di generazione distibuita. Tesi di Dottorato, Universit degliStudi di Napoli Federico II.

[15] Petreus D., Energetic optimized electrical systems for terrestrialtransport using batteries and supercondensators (TRANS-SUPERCAP). Project nr. 3393/200, contract nr. D2-1_018/2007[5] Dougal R.A., Gao L, Liu S. Ultracapacitor model with automatic

order selection and capacity for dynamic system simulation, J PowerSources, 2004; pp. 126:250-7

[16] Scherson D.A., Palencsr A., Batteries and ElectrochemicalCapacitors, Interface, The Electrochemical Society, Spring 2006

[6] Halber D., Researchers fired up over new battery, MIT Newsletter,8th February, 2006.

[17] Verbrugge, M.W. and Liub P., Microstructural Analysis andMathematical Modeling of Electric Double-Layer Supercapacitors,Journal of The Electrochemical Society, 152 (5) D79-D87 (2005)[7] Halper M. S., Ellenbogen J.C., Supercapacitors: A Brief

Overview,MITRE Nanosystems Group,March 2006, The MITRECorporation, McLean, Virginia, USA

[18] Wohlgemuth J., Miller J., Sibley L.B., Investigation of Synergy between Electrochemical Capacitors,Flywheels, and Batteries inHybrid Energy Storage for PV Systems, CONTRACTOR REPORTSAND99-1477, Sandia National Laboratories, June 1999

[8] IEC 62391-1. Fixed electric double layer capacitors for use inelectronic equipment - Part 1: Generic Specification, Ed. 1. 2006.

[9] IEC 62391-2. Fixed electric double layer capacitors for use inelectronic equipment - Part 2: Sectional specification - Electric doublelayer capacitors for power application, Ed. 1. 2006.

[19] Yoshida A, Imoto K, Yoneda H, Nishino A. An electric double-layercapacitor with high capacitance and low resistance, IEEETransactions on components, hybrids, and manufacturing technology,

1992,15(1), pp. 133-8.[10] IEC 62391-2-1. Fixed electric double layer capacitors for use in

electronic equipment - Part 2-1: Blank detail specification - Electricdouble layer capacitors for power application, Assessment level EZ,Ed. 1. 2006

[20] Zubieta L. Bonert R, Dawson F., Considerations in the design ofenergy storage systems using double-layer capacitors Researchersfired up over new battery, IPEC, Tokyo 2000; pp. 1551.

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