new full-frequency-range supercapacitor model with easy

112 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 1, JANUARY 2013

New Full-Frequency-Range Supercapacitor ModelWith Easy Identification Procedure

Vincenzo Musolino, Luigi Piegari, Member, IEEE, and Enrico Tironi

Abstract—Energy storage has become a key issue for achievinggoals connected with increasing the efficiency of both producersand users. In particular, supercapacitors currently seem to beinteresting devices for many applications because they can supplyhigh power for a significant amount of time and can be rechargedmore quickly than electrochemical batteries. In different applica-tions, a combination of the two devices, batteries and supercapaci-tors, could be used to develop a high-efficiency storage system witha high dynamic performance. Because of the diffusion of super-capacitors, a good model is needed to represent their behavior indifferent applications. Some models have been proposed that char-acterize supercapacitors working at either low or high frequencies.In this paper, the authors present a full-frequency-range modelthat can be used to represent all of the phenomena that involvesupercapacitors. Moreover, to realize a simple and useful tool, theauthors present a simple procedure to identify the parametersof the model that can be used to characterize a supercapacitorbefore use.

Index Terms—Double-layer capacitors, parameter identifica-tion methods, supercapacitor efficiency, supercapacitor perfor-mance, supercapacitors.

I. INTRODUCTION

IN RECENT YEARS, the interest in storage systems hasbeen growing because their use is becoming more and

more widespread in different applications. For each kind ofapplication, it is possible to identify the technologies that wouldbe appropriate for realizing an optimal storage system [1].Moreover, the use of more than one technology to fit the goalsof specific applications represents a real solution to achievea high-performance and high-efficiency storage unit [2]–[6].For example, various applications that use supercapacitors andelectrochemical batteries to realize hybrid storages have beenreported [7]–[11].

Because of the wide variety of applications that use super-capacitors, it is important to realize a model that can representtheir behavior in all practical operations. Recently, the differentmodels of supercapacitors have been developed to match theseexigencies. The proposed models differ because they representa supercapacitor in a particular application and are designed forease of use in that application.

Manuscript received August 4, 2010; revised February 14, 2011 andOctober 31, 2011; accepted January 29, 2012. Date of publicationFebruary 10, 2012; date of current version September 6, 2012.

V. Musolino is with Dimac Red, 20853 Biassono, Italy (e-mail: [email protected]).

L. Piegari and E. Tironi are with the Department of Electrical Engineering,Politecnico di Milano, 20133 Milan, Italy (e-mail: [email protected];[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2012.2187412

Different authors have described methods to model super-capacitor devices with different resistive–capacitive (RC) net-works. These models have been obtained through analyses inthe time domain or frequency domain. In [12], Zubieta andBonert proposed a model with three RC branches connectedin parallel. An additional parallel resistor accounts for the self-discharge phenomenon. In this model, the bigger capacitanceis not constant but rather linear with voltage. This model isvery good for low frequencies, but it is not accurate at highfrequencies. In [13]–[15], Belhachemi et al. proposed a modelto address the limits of the model proposed in [12]. Theypresented a nonlinear transmission line. However, this modelis complex, and parameter identification is not easy. How-ever, the idea of separating the time constants of the differentbranches is interesting and is used in this paper. In [16]–[18],Gualous et al. proposed simple transmission lines focusingon the effect of temperature on model parameters. In [19],Rojat et al. introduced a multipenetrability model to estimatethe supercapacitor behaviors in the different states of health. In[20], Zhang et al. analyzed the quality of different RC modelsfor power electronic applications. In [21], Du considered aparallel branch RC equivalent circuit and proposed a parameteridentification procedure that was mainly based on least-squareerror minimization.

In this paper, the authors propose a model that is able torepresent the behavior of supercapacitors at both low and highfrequencies. This new model, described in the next section, is anextension of the model presented in [22]. Moreover, this modelcan be used to take into account the self-discharge and redistri-bution phenomena. The model is not completely new becauseit was obtained by joining different models, but it is usefulnot only for dynamic simulations but also for evaluating theefficiency of a device. In addition, a very simple, yet innovative,parameter identification procedure is proposed to avoid the useof least-square error minimization. Previous methods requiredthe results of accurate tests, while the proposed procedureprovides good parameter estimation using only a few simplemeasurements.

II. SUPERCAPACITOR MODELING

Supercapacitor devices consist of two electrodes with anion-permeable separator between them. The electrodes are im-mersed in an electrolyte solution, as shown in Fig. 1.

The charge separation process, which requires a voltagedifference across the electrodes, takes place on the twoelectrode–electrolyte interfaces. Thus, supercapacitors are usu-ally known as double-layer capacitors. The double layer is also

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MUSOLINO et al.: NEW FULL-FREQUENCY-RANGE SUPERCAPACITOR MODEL 113

Fig. 1. Internal structure of a supercapacitor.

called the Helmholtz layer after the physicist who introducedthe model to explain this phenomenon in 1853 [23], [24].

The dynamic behavior of a supercapacitor is strongly relatedto the ion mobility of the electrolyte used and the porosityeffects of the porous electrodes. As shown in the next sections,it is clear, based on observations, that the capacitance starts todecrease from 0.1 to 0.2 Hz and becomes null for frequenciesat hundreds of hertz or a few kilohertz, depending on thecell technology. The real part of the supercapacitor impedancedecreases with frequency; its value is reduced by half from dcto very high frequencies (some kilohertz). This reduction in thecapacitance with an increase in the charge/discharge frequencyis mainly the result of ion inertia. Thus, it is impossible touse the full capacitance of the device at high frequencies inpractice. Instead, the reduced resistance can be explained bya reduction in friction losses as ion migration slows in theelectrolyte solution. A deeper explanation of these phenomenacan be found in the theory of porous electrodes [25].

To represent this behavior in a frequency range of tens ofmillihertz to hundreds of hertz, Buller et al. [26] proposed amodel that presents the following impedance in the frequencydomain:

Zp(jω, V ) =Ri + jωLi +τ(V ) coth

(√jωτ(V )

)C(V )

√jωτ(V )

=Rp(ω) − 1jωC(V, ω)

(1)

where Ri is the resistance at an infinite frequency, Li is theleakage inductance, ω is the angular frequency, V is the voltageshown in Fig. 2, C is the dc capacitance value, and τ is,dimensionally, a time. The leakage inductance is usually verysmall (tens of nanohertz) and is neglected here. This modelis not able to model the self-discharge phenomenon and isnot accurate at very low frequencies. Thus, the authors [22]proposed a model that integrated the model proposed by Bullerand the model proposed by Zubieta. This model is composed ofthree parallel branches: The first one is equal to the impedancegiven in (1), the second one is a series RC branch representingthe recombination phenomena after fast charge or discharge,and the third one is a leakage resistance that takes into accountthe self-discharge phenomena. This model is very accurate for

Fig. 2. Proposed supercapacitor model.

simulating the phenomena occurring in some tens of secondsbut is not capable of achieving good results for a self-dischargeof some weeks. For this reason, after a set of experimental tests,whose main results are shown hereinafter, the authors extendedthe model proposed in [22] to obtain a model that is capable ofrepresenting a supercapacitor’s performance across the wholefrequency range, including the redistribution and self-dischargephenomena. Furthermore, all of the parameters are taken di-rectly from the information contained in the manufacturer’sdatasheet or estimated using very simple tests at a constantcurrent. The equivalent circuit of this model is shown in Fig. 2.

In the model shown in Fig. 2, the first branch, representingthe impedance Zp given in (1), models the fast dynamic, whilean indefinite number of parallel branches can represent the slowdynamic demonstrated by Zubieta. It is worth noting that thenumber of RC parallel groups in the first branch should theoret-ically be infinite. However, it has been experimentally verifiedthat five groups are enough to obtain an accurate model. Allthe 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 beinfinite. However, as shown in Section IV, two parallel branchesare sufficient to achieve accurate results. Finally, a resistorRleak is added to model the self-discharge phenomenon.

The proposed model has been validated through experimen-tal tests made on different kinds of cells. Some of the results arereported in the following. Moreover, the proposed procedure toestimate the parameters of the model is presented, and its resultsare compared with the experimental data.

III. PARAMETER IDENTIFICATION

Different parameter identification procedures have been pro-posed in the literature. However, the possibility of estimating


parameters with very simple tests and with the informationcontained in the manufacturer’s datasheets is a very impor-tant point for the usability of the model. It is clear that theparameter values that give the best match between the modelshown in Fig. 2 and the experimental data can be obtainedwith least-square error minimization. However, this procedureis too complex to be performed every time. In fact, it requiresthe execution of experimental tests at many frequencies toobtain the experimental curve of impedance versus frequency.Thus, a new easier procedure is proposed in the following. Theparameters of the first branch and the parallel branches areidentified 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 im-pedance given by (1). In this expression, C and τ are thefunctions of the voltage. During the experimental tests, it wasverified that C and τ vary almost linearly with the voltage

C(V ) =C0 + k V

τ(V ) = τ0 + kτV. (2)

From (1), the following result:

limω→0

�e{Zp} = limω→0

Rp = Ri +∞∑

n=1

2τ(V )n2π2C(V )

= Ri +τ(V )

3C(V )= Rdc

limω→0

�m{ωZp} = − 1C(V )

(3)

where Rp indicates the real part of the impedance of the firstbranch Zp.

Therefore, the capacitance can be evaluated, for each voltageV , using a dc test. In particular, it is possible to use the samecharging test at a constant current as described in [27]. In con-trast, to experimentally evaluate Rp at zero frequency, it is notpossible to use a dc test. Indeed, at frequencies lower than sometens of millihertz, the parallel branches cannot be neglected, andRp is not directly measurable. The supercapacitor’s datasheetgives the discharge time Tdc to which the rated capacitancerefers. According to the manufacturer’s recommendation [28],Tdc has to be used to estimate the dc resistance. Workingat frequency fdc = 1/Tdc, the recombination phenomena donot occur. Thus, the electrochemical capacitor can be modeledusing only the first branch of Fig. 2. A resistance measurementat fdc allows the estimation of Rdc. Typically, fdc has a valueof some tens of millihertz.

For high frequencies, when the supercapacitor behaves like aresistance, Rp becomes constant. Some measurements at highfrequencies, when no capacitive behavior is shown, are enoughfor estimating the high-frequency resistance Ri. However, thisvalue 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 thefirst branch for a fixed voltage. Repeating the tests at differentvoltages makes it possible to estimate a value for the parameterk that takes into account the variability of the capacitance withthe voltage.

B. Parallel-Branch Parameter Identification

First, the number of branches must be chosen as a compro-mise between the required accuracy and the consequent com-plexity. The branches represent the supercapacitor’s behaviorover a long time scale. Thus, to model the phenomena that occurat different times, their time constants should be quite different.It is therefore possible to consider different transient phenom-ena without interference. After a fast charge, all of the energy isstored in the first branch. Then, leaving the supercapacitor in anopen circuit causes the energy to be redistributed in the parallelbranches, after which it dissipates in the leakage resistance. Ifthe parallel branches with greatly different time constants arechosen, it is possible to split the discharge time into n + 1intervals. In interval k, only the branches from 1 to k − 1 areinvolved in the energy transfer. All of the other branches are ina steady state and are not affected by the phenomenon becausethey have much higher time constants. In the n + 1 interval, allof the energy is dissipated on the leakage resistance.

To make the parameter identification procedure easy, it isimportant for the choice of the time intervals to be free andindependent of the cell under consideration. The only restric-tion in the choice of the time intervals is that they have to besufficiently different from each other that, in each transient,all of the other branches can be considered to be in the steadystate. In the analysis of the experimental data, it was observedthat a good solution in the choice of the time intervals is to setthem equidistant on a logarithmic scale. The logarithmic scaleis obtained by choosing the intervals as follows:

I1 : τ0 < t < Mτ0 = τ1

I2 : τ1 < t < Mτ1 = τ2

. . .

In : τn−1 < t < Mτn−1 = τn (5)

where τ0 is the time necessary to extinguish the transient ofthe first branch over the second one. So, τ0 is greater than fivetimes the time constant of the first branch already identified.This constant usually ranges from some seconds to some tensof seconds.

In the set of equations in (5), it is necessary to choose a valuefor M so that each transient can be considered independentfrom the others. As is well known, exponential transients canbe considered extinguished after five times the time constantbecause 99.3% of the steady-state value is reached at this time.Thus, to guarantee that each transient caused by the parallelbranches is finished before the beginning of the next one, it isnecessary for the time constant of the k branch to be more thanfive times the time constant of the k − 1 branch. Thus, M > 5.

From (5), for a fixed time window Tw during which the self-discharge phenomenon has to be analyzed, the parameter M


Fig. 3. Equivalent circuit of two parallel branches.

can be easily evaluated as

M = n−1

√Tw

τ0. (6)

The proposed identification procedure estimates the parame-ters of the equivalent circuit using only n voltage measurementsmade 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 thesubscript II, while the discharged capacitance is indicated withthe subscript I. The sum of the resistances of the two branchesis indicated by Req. It is worth noting that when consideringthe first interval, the capacitance CI is not constant. The capac-itance varies when the voltage changes according to (2).

If V0 and Vf indicate the initial and final voltages of theconsidered interval, respectively, for the circuit in Fig. 3, it ispossible to write

CII =V0 − Vf

VfCI + k

V 20 − V 2

f

2Vf(7)

where k is different from zero only for the capacitance of thefirst branch as discussed in Section II.

Using the charge conservation law, all of the capacitancesof the parallel branches can be estimated iteratively, and theydo not depend on the resistances. The resistances are easilycalculated by knowing the time constant and capacitances.Taking into account (5) and (7)

Cρ = kV 2

τρ−1− V 2

τρ

2Vτρ

+Vτρ−1 − Vτρ

Vτρ

ρ−1∑j=1

Cj

Rρ =τρ

Cρ

Rleak =τn

1n−1

n−1∑j=1

Cj

. (8)

IV. EXPERIMENTAL RESULTS

Different experimental tests were conducted at the Depart-ment 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 frequencyrange and for the different kinds of cells.

2) Validate the proposed method for parameteridentification.

To fulfill the first goal, cells with different technologiesand sizes were tested. In particular, the cells with differentorigins 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 genera-tions. The cells were manufactured by Maxwell and were theBCAP0140 and BCAP0150. The change in the rated capaci-tance from 140 to 150 F was caused by the increase in the ratedvoltage from 2.5 to 2.7 V. According to (2), the capacitance isa function of the voltage. Therefore, the rated capacitance ofthe new cell was greater than the old one without significantlychanging the size of the cell.

Measurements were used to analyze the supercapacitor per-formances in a frequency range of dc up to 1 kHz. The dc mea-surements focused on the redistribution and leakage phenomenaand were conducted by analyzing the open-circuit voltage of thedevice, while electrochemical impedance spectroscopy (EIS)was used to analyze the frequency response of the device inthe frequency range of 0.01−103 Hz. It is worth noting thatEIS was only used to validate the model; it is not necessary forparameter identification.

The equipment used to realize the frequency analysis con-sisted of an AB-class power amplifier driven by a signal genera-tor. The signal generator was fully remote controlled via a GPIBstandard interface. The cell voltage and current were measuredusing a 16-bit multifunction National Instruments acquisitionboard (Fig. 4).

For the first branch of the model [i.e., the impedance Zp

given in (1), (2), and (4)], three EIS at different polarizationvoltages were realized. In Figs. 6 and 7, the capacitancesand resistances versus frequency obtained with the model arecompared with the experimental data for the BCAP0140 andBCAP0150 cells.

From an analysis of Figs. 5 and 6, it is clear that the modeland experimental data agree. For the two cells, the referenceparameters of the first branch obtained using the classicalmethod of least-square minimization are reported in Table I.

Observing Figs. 5 and 6, it is possible to see how boththe capacitance and resistance are practically constant in afrequency range of some tens of millihertz up to some hundredsof millihertz. These values correspond to the dc values reportedon the datasheet (see Section III). This behavior was alsoverified experimentally using cells with different capacitances(i.e., Maxwell PC10, old and new generations, 10 F, with arectangular shape despite the cylindrical shape of BCAP0140and BCAP0150), and it always appeared in the same frequencyrange. This can be explained by taking into account that,


Fig. 5. Equivalent capacitance and resistance versus frequency at differentpolarization voltages for BCAP0140: Comparison between model and experi-mental data.

Fig. 6. Equivalent capacitance and resistance versus frequency at differentpolarization voltages for BCAP0150: Comparison between model and experi-mental data.

over the low frequencies of tens of millihertz, the dynamicbehavior of supercapacitors is mainly related to the electrolyteproperties. The redistribution phenomenon, which is caused bythe irregularity of the porous carbon structure, takes place forlower frequencies. Therefore, it is necessary to consider theeffect of the parallel branches of the model.

To evaluate the correspondence of the proposed model withthe experimental data, the coefficient of determination R2 wasevaluated. Its definition is as follows:

R2 = 1 −∑

j(yj,ex − yj,mod)2∑j(yj,ex − yex)2

(9)

TABLE IPARAMETERS OF THE FIRST BRANCH

TABLE IICOEFFICIENTS OF DETERMINATION

TABLE IIIPARAMETERS OF THE FIRST BRANCH ESTIMATED WITH THE SIMPLIFIED

PROCEDURE AND ESTIMATION ERRORS

Fig. 7. Self-discharge of BCAP0140: Comparison between the experimentaldata and four-branch model.

where the suffixes ex and mod refer to experimental- andmodel-based data, respectively, and the mean value is indicatedwith an overbar. For the data shown in Figs. 5 and 6, thecoefficients of determination are reported in Table II.

The parameters of the first branch, identified as suggested inSection III, with the proposed simplified procedure are reportedin Table III, together with the estimation errors which areindicated in parentheses. The estimation errors are always lowerthan 5%, which confirms that the proposed simplified methodof 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. 7and 8, the experimental curves are compared with the resultsgiven by a four-branch model, while in Figs. 9 and 10, theexperimental test results are compared with those of a numeri-cal simulation of a six-branch model. The parameters obtainedby (8) and used in the simulations leading to Figs. 7–10 arereported in Tables IV and V. A time interval of τ0 = 50 s waschosen, and M was chosen such that τn ≈ 5 · 106 s because ofthe time required to watch the system. Thus, from (6), M = 45and 10 for the 4 and 6 branches, respectively.


Fig. 8. Self-discharge of BCAP0150: Comparison between the experimentaldata and four-branch model.

Fig. 9. Self-discharge of BCAP0140: Comparison between the experimentaldata and six-branch model.

Fig. 10. Self-discharge of BCAP0150: Comparison between the experimentaldata and six-branch model.

TABLE IVPARAMETERS OF A FOUR-BRANCH EQUIVALENT MODEL (M = 45)

TABLE VPARAMETERS OF A SIX-BRANCH EQUIVALENT MODEL (M = 10)

It is worth noting that the choice of the time constants,according to (5) and (6), is independent of the cells and dependsonly on the observation time and the number of branches. Thus,with the same time constants, two different cells have beenmodeled. The coefficient of determination R2 was evaluated asa measure of the goodness of the fit and is shown in the figures.

Fig. 11. Experimental current versus time for a test at 70 A.

Fig. 12. BCAP0140 experimental and simulated voltages during a test at70 A.


It is evident that its value is weakly dependent on the cell butstrongly dependent on the number of branches used to modelthe system. The fit with the six-branch model seems to achievegood interpolation results.

A. Experimental Results in Time Domain

In order to test the suitability of the model and the influenceof the parameter identification procedure, some experimentaltests were performed. In order to show the usability of themodel over a wide frequency range, a pulse load condition wasused for the tests. Using a controlled dc source (EA-PSI 8240-170) and a controlled electronic load (EA-EL 9400-150), thepulsing currents of different values were realized. The resultsof some of the tests performed are reported in the following.In particular, the results for 70 and 110 A are shown. The





experimental currents and voltages of the BCAP0140 cell areshown in Figs. 11–14, while the currents and voltages for theBCAP0150 cell are shown in Figs. 15–18.

The simulation was conducted by injecting an experimentallymeasured current into the cells. A good agreement between theexperimental and numerical data was found by comparing thesimulated and measured voltage profiles (the maximum errorwas lower than 1.5%).

B. Analysis of Experimental Results

In this analysis, the performances of the cells were comparedand related to the parameters of the proposed model. Studying(1), it is possible to relate the knee of the capacitance trend tothe parameter τ(V ). In particular, at an angular frequency equal



to the inverse of τ , the capacitance value is 97.8% of the dcvalue. At angular frequencies a decade greater, the capacitanceis 46% of the dc value.

Taking into account (4), it is possible to state that to increasethe dynamic of a device with a rated capacitance C(V ), itis 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 differencebetween these resistances has been reduced by increasing thehigh-frequency resistance Ri. This explains why the superca-pacitance is shown up to higher frequencies in the new cells.However, the increase in Ri implies a higher voltage dropduring a fast transient, as shown in Figs. 13 and 14 and 17and 18.

From a physical point of view, the resistance Rdc is morerelated to the behavior of the electrode in terms of the con-ductivity and geometry of the pores, while the resistance Ri ismore related to the behavior of the electrolyte. To improve thedynamic of the device, both aspects have to be considered. Incontrast, the increase in capacitance was achieved by increasingthe rated voltage [the capacitance is an increasing function ofthe voltage, as given in (2)].

Finally, from these considerations, it can be seen that reduc-ing the resistance of the device and, consequently, improvingthe efficiency, are not always related to a better dynamic. Inparticular, a good compromise is reducing the resistance at lowfrequency Rdc and increasing the Ri-to-Rdc-resistance ratio.Thus, it is possible to improve the dynamic performances andreduce the losses in the typical frequency range of these devices(1–2 Hz).


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. TheRagone plot obtained by these experimental tests is shown inFig. 19, which also shows a comparison between the experi-mental and simulated data.

The Ragone plot shows that the performances of the twocells are very similar. Indeed, the new cell (BCAP0150) has anincreased capacitance (because of its higher rated voltage), butthe increase in its weight is such that its specific energy resultsare lower than those of BCAP0140. The experimental valuesshown in Fig. 19 are in agreement with the data reported in theliterature.

V. CONCLUSION

A new model has been proposed that is able to representthe behavior of supercapacitors over the entire frequency range,from dc to tens of kilohertz. Through the experimental tests, itwas shown that supercapacitors do not behave like capacitancesat higher frequencies.

The proposed model was able to simulate the self-dischargephenomenon for tens of days. Indeed, by observing the ex-perimental results, it was verified that the voltage across asupercapacitor decreases from the rated value to half the ratedvalue over a period of one month.

A very simple procedure to estimate the model parameterswas also proposed. The model and identification procedurewere validated using a set of tests on different manufacturedcells in the frequency and time domains. These showed thatvery good results could be obtained when simulating both thedynamic response and the very slow phenomena that occurinside supercapacitor devices.

Finally, the model was used to determine the effect of para-meter variations on the dynamic response of supercapacitors.

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Vincenzo Musolino received the M.S. degreein electrical engineering from the Politecnico diMilano, Milan, Italy, in 2007, where he is currentlyworking toward the Ph.D. degree in electrical engi-neering in the Department of Electrical Engineering.

He is also currently a Design Engineer with DimacRed, Biassono, Italy. His research interests includeelectrical storage devices, power electronics, anddistributed generation.

Luigi Piegari (M’04) was born in 1975. He receivedthe M.S. and Ph.D. degrees in electrical engineeringfrom the University of Naples Federico II, Naples,Italy, in 1999 and 2003, respectively.

He is currently an Assistant Professor with theDepartment of Electrical Engineering, Politecnico diMilano, Milan, Italy. His current research interestsinclude electrical machines, high-efficiency powerelectronic converters, renewable energy sources, andelectrical supplies and cableway plants.

Enrico Tironi received the M.S. degree in electricalengineering from the Politecnico di Milano, Milan,Italy, in 1972.

In 1972, he joined the Department of ElectricalEngineering, Politecnico di Milano, where he is cur-rently a Full Professor. His areas of research includepower electronics, power quality, and distributedgeneration.

Dr. Tironi is a member of Italian Standard Author-ity (C.E.I.), Italian Electrical Association (A.E.I.),and Italian National Research Council (C.N.R.)

group of Electrical Power System.

new full-frequency-range supercapacitor model with easy

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