Fractional non-linear modelling of ultracapacitors

Commun Nonlinear Sci Numer Simulat 15 (2010) 1327–1337

Contents lists available at ScienceDirect

Commun Nonlinear Sci Numer Simulat

journal homepage: www.elsevier .com/locate /cnsns

Fractional non-linear modelling of ultracapacitors

Nicolas Bertrand a,b, Jocelyn Sabatier b,*, Olivier Briat a, Jean-Michel Vinassa a

a Laboratoire IMS, Groupe COFI, CNRS UMR 5218, Université Bordeaux 1, 351 cours de la liberation, 33405 Talence, Franceb Laboratoire IMS, Groupe LAPS/CRONE, CNRS UMR 5218, Université Bordeaux 1, 351 cours de la liberation, 33405 Talence, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 April 2009Accepted 30 May 2009Available online 7 June 2009

PACS:02.30.Yy82.47.Uv

Keywords:UltracapacitorsImpedance spectroscopyAnalytical modellingHEV applicationsFractional differentiation

1007-5704/$ - see front matter � 2009 Elsevier B.Vdoi:10.1016/j.cnsns.2009.05.066

* Corresponding author. Tel.: +33 5 40 00 26 13;E-mail addresses: nicolas.bertrand@ims-bordeaux

[email protected] (J.-M. Vinassa

In this paper, it is demonstrated that an ultracapacitor exhibits a non-linear behaviour inrelation to the operating voltage. A set of fractional order linear systems resulting from afrequency analysis of the ultracapacitor at various operating points is first obtained. Then,a non-linear model is deduced from the linear systems set, so that its Taylor linearizationaround the considered operating points (for the frequency analysis), produces the linearsystem set. The resulting non-linear model is validated on a Hybrid Electric Vehicle(HEV) application.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Because of their high specific power and long lifetime, ultracapacitors (UC) are interesting elements for electric storage inapplications such as Hybrid Electric Vehicles (HEV). In order to study their performance evolutions throughout their lives, itis necessary to develop accurate models. Theses models must be able to fit with the various current profile imposed in HEVapplications. In theses applications, the power requirements are mainly composed of pulse current demands but also of restperiods. So, an UC model must reproduce these two types of behaviours in order to obtain the best approximation of the UCstate of life.

In UC modelling field, two classes of models have been proposed. In the first one, voltage dependency is taken into ac-count. Theses models are often electrical circuits in which component values vary with the UC voltage [1,2]. Voltage depen-dency is based on the component values identification around a bias voltage. In the second class of models, long rangephenomena observed during UC relaxation are taken into account through the introduction of fractional differentiation inmodelling [3,4] but voltage dependency is not addressed in theses models.

In this paper, a new fractional order modelling is proposed that closes the gap between the two classes of models previouslydefined. The proposed modelling is based on the Taylor approximation of a non-linear system around a set of operating points.Using such an approximation, a non-linear system can be converted into a set of linear models whose dynamic behaviour isequivalent to the non-linear system dynamic behaviour around the operating points considered for the approximation. In this

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fax: +33 5 56 37 15 45..fr (N. Bertrand), [email protected] (J. Sabatier), [email protected] (O. Briat),

).

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1328 N. Bertrand et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1327–1337

paper, an inverse strategy is adopted. A set of fractional linear model that represents an ultracapacitor behaviour arounddifferent operating points is obtained. Then the linear models are gathered in a unique non-linear fractional order model, so thatits Taylor approximation produces the set of linear models. The non-linear fractional model obtained is validated with variouscurrent profiles.

2. Ultracapacitor non-linear behaviour

Studies [5] have shown that the dynamic behaviour of an UC depends on the operating voltage. This voltage dependencyis revealed by the time or the frequency response of an UC.

2.1. Non-linear behaviour demonstration in time domain

In order to demonstrate the voltage dependency of UCs in time domain, tests have been done on a 2600 F Maxwell UC(M26) at temperature 25 �C. The M26 element is submitted to a series of charges with a current of 80 A. The current appliedand the UC voltage response are shown in Fig. 1a. Time between two current pulses is long enough to consider the UC relaxedwithout suffering of self-discharge phenomenon.

In Fig. 1b all the charge profiles are gathered on the same graph, time and voltage offsets being suppressed. The compar-ison reveals that the higher is the operating voltage, the lower is the charge sweep rate. Given that the current profile is thesame in each UC charge, Fig. 1b demonstrates a non-linear behaviour of the UC.

2.2. Non-linear behaviour demonstration in frequency domain

Frequency responses resulting from impedance spectroscopies around various operating voltages demonstrate a differentbehaviour of the UC according to the operating voltage. As it is presented in Fig. 2, the difference is very important on the realpart of the UC impedance on the whole frequency range and on the imaginary part in low frequency.

The frequency analysis thus highlights the operating voltage dependency of the UC dynamic behaviour and thus its non-linear dynamic behaviour.

2.3. UC non-linear modelling

In most case, UC non-linear modelling [1],[2] consists in the approximation of the UC by an electrical circuit for variousoperating voltages. A lookup table is then applied on each component of the circuit to take into account the voltage depen-

0 50 100 150 200 250 3000.2

0.4

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2.2

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ge (V

)

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10

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50

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80

90

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(A)

0 1 2 3 4 5 6 70

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0.3

Time (s)

Volta

ge (V

)

Charge1Charge2Charge3Charge4Charge5Charge6Charge7Charge8

CurrentVoltage

a b

Fig. 1. (a) Current profile and UC voltage response; (b) charge profiles, time and voltage offset suppressed.

10-3

10-2

10-1

100

101

102

0.4

0.6

0.8

Frequency (Hz)

Impe

danc

e R

ealP

art (

m Ω

)

10-3

10-2

10-1

100

101

102-20

-15

-10

-5

0

Frequency (Hz)Impe

danc

e Im

agin

ary

Part

(m Ω

)

Bias Voltage 0VBias Voltage 1.25VBias Voltage 2.7V

Bias Voltage 0VBias Voltage 1.25VBias Voltage 2.7V

Fig. 2. Experimental frequency response for various operating voltage.

u(t) uc(t)

R

C(uc)

Fig. 3. RC circuit with voltage dependent capacitance.

N. Bertrand et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1327–1337 1329

dency. However such an approach is not valid in all cases. This can be demonstrated on the R–C circuit of Fig. 3. The non-linear differential equation associated to the electrical circuit is given by:

duc

dt¼ � 1

RCðucÞuc þ u with CðucÞ ¼ C0 þ A � uc: ð1Þ

It is supposed that relation (1) results from the capacitance variation fitting for various operating voltages, so that aroundvoltage uc0 the following equation holds:

dduc

dt¼ � 1

RCðuc0Þduc þ du: ð2Þ

However linearization of (2) around the operating voltage uc0 leads to the following differential equation:

duc

dt¼ 1

C0 þ a � uc01� a � uc0

C0 þ a � uc0

� �� duc þ du: ð3Þ

This highlights that a simple fitting of the component variation does not produce a valid modelling of the real system. Onecan note however, that relations (2) is closed to relation (3) for high values of capacitance C0.

The method introduced in two next sections permits non-linear models involving UC physical variables. This model isbased on UC fractional linear modelling at various operating voltages and pseudo-integration. This last method is nowpresented.

3. First-order Taylor approximation and pseudo-integration

The first-order Taylor approximation permits to obtain local models of a non-linear system. This section provides a meth-od [6] to obtain a non-linear model from a set of fractional linear models that represents the non-linear system at variousoperating points.


3.1. From a fractional non-linear system to a set of fractional linear models

Taylor expansion of a multi-variable non-linear function is defined by:

f ðaÞ ¼ f ðbÞ þ Dðf ÞðbÞ � ða� bÞ þ Oka� bk: ð4Þ

in which D(f)(b) represents the Jacobian matrix of function f at point b.Let now considers the following non-linear system:

dcqðtÞdtc

¼ f ðqðtÞ; uðtÞÞ yðtÞ ¼ hðqðtÞ;uðtÞÞ; ð5Þ

supposed at rest before time t = 0.For a trajectory q0(t) produced by the input u0 (t), (5) becomes:

dcq0ðtÞdtc

¼ f ðq0ðtÞ; u0ðtÞÞ y0ðtÞ ¼ hðq0ðtÞ; u0ðtÞÞ: ð6Þ

The linear approximation of system (5) around trajectory [q0 (t), u0(t)] can be obtained using relation (4) applied to func-tions f(�) and h(�). The Jacobian matrix corresponding to the function f(�) is thus:

Dðf Þ ¼ ðDq : DuÞ ¼ofoq

:ofou

� �: ð7Þ

The same can be done for function h(�), thus leading to the linear approximation of system (5):

dcqðtÞdtc¼ f ðq0;u0Þ þ Dðf Þðq0; u0Þ

dq

� � �du

24 35

y ¼ hðq0;u0Þ þ DðhÞðq0;u0Þdq

� � �du

264375

8>>>>>>>><>>>>>>>>:: ð8Þ

Using changes of variable:

dq ¼ qðtÞ � q0ðtÞ du ¼ uðtÞ � u0ðtÞ dy ¼ yðtÞ � y0ðtÞ: ð9Þ

and taking into account relation (6), a linear fractional model which depends on the operating trajectory (q0, u0) isobtained:

dcdqdtc¼ Dqðf Þðq0ðtÞ;u0ðtÞÞdqðtÞ þ Duðf Þðq0ðtÞ; u0ðtÞÞduðtÞ

dyðtÞ ¼ DqðhÞðq0ðtÞ; u0ðtÞÞdqðtÞ þ DuðhÞðq0ðtÞ; u0ðtÞÞduðtÞ

(: ð10Þ

This method can be done for various operating trajectories. A set of local model is then obtained that represents the initialnon-linear system around the considered trajectories. The next section presents a method to obtain an approximated non-linear model based on the set of local models.

3.2. From a set of fractional linear models to a fractional non-linear model

Let s denotes the Laplace variable. If all the coefficients are supposed time-invariant in system (10) that behaves like afractional order integrator in low frequency, its transfer function is defined by:

dydu¼ k1 þ k2

1þ b1sc þ � � � þ bn�1sðn�1Þc

a1sc þ � � � þ an�1sðn�1Þc þ snc : ð11Þ

In (11), coefficients bk and ak depend on the operating trajectory [q0(t),u0(t),y0(t)]. Given relation (6), a relation exists be-tween q0(t) and u0(t). A state space description of (11) is thus:

dcdz1dtc¼ dz2

..

.

dcdzn�1dtc¼ dzn

dcdzndtc¼ �a1ðu0; y0Þdz2 � � � � � an�1ðu0; y0Þdzn þ k2ðu0; y0Þdu

8>>>>>><>>>>>>:: ð12Þ

dy ¼ dz1 þ b1ðu0; y0Þdz2 þ � � � þ bn�1ðu0; y0Þdzn þ k1ðu0; y0Þdu


Pseudo-integration [6] of relation (12) produces the non-linear fractional model:

dcz1dtc¼ z2

..

.

dczn�1dtc¼ zn

dczndtc¼ �a1ðu; yÞz2 � � � � � an�1ðu; yÞzn þ k2ðu; yÞu

8>>>>>><>>>>>>:: ð13Þ

y ¼ z1 þ b1ðu; yÞz2 þ � � � þ bn�1ðu; yÞzn þ k1ðu; yÞu

It is easy to check that Taylor approximation of model (13) around an operating trajectory [q0(t),u0(t),y0(t)] producesmodel (12). The integration method previously presented thus provides a way to represent a set of linear models by a uniquenon-linear model. To obtain an accurate non-linear model it is important to use the integration method with a large numberof local models obtained for different operating points. The next section presents a method to obtain these local models.

4. Ultracapacitor linear models

Studies [7] have shown that a linear UC model which takes physical variables of the cell into account can be obtained.Others [8] show that the rough aspect of the electrode must be considered in order to obtain an accurate behaviour inlow frequency.

4.1. Assumptions

A double layer capacitor cell consists in two identical porous electrodes with a separator between them and electrolytethroughout. As usually done [9], assumptions made in developing the present model are summarized in five points.

� There is not faradic process at the electrode/electrolyte interface.� The porous electrode is assumed to be only one pore due to electrode symmetry.� The physical variables of each slide of the dissecting pore are different from a slide to another. Physical variables include

conductivity of electrode and electrolyte, double layer capacitor of the electrode/electrolyte interface.� There is not contact between current collector and electrolyte. The current crossing the current collector thus reach elec-

trolyte thought electrode.� The electrode/electrolyte interface is assumed to be rough.

If voltage dependency is not taken into account, spatial discretization of the porous electrode in n layers leads to intro-duce three impedances denoted zEk(s),zSk(s) and zIk(s). As shown in the electric representation of the porous electrode ofFig. 4, theses impedances are, respectively, linked to electrode, electrolyte and electrode/electrolyte interface slides.

Impedances zEk(s) and zSk(s) are resistances and result respectively from electrode and electrolyte conductivity. Due torough aspect, the equivalent impedance of the interface electrode/electrolyte is a resistance in series with the double layercapacitance.

4.2. The linear UC model

Let Z(s) the impedance of the electrical circuit in Fig. 4. For the computation of Z(s), Görh [10] introduced the followingimpedances:

ZEðsÞ ¼Xn

k¼1

zEkðsÞ; ZSðsÞ ¼Xn

k¼1

zSkðsÞ ð14Þ

Fig. 4. Half porous electrode model taking rough interface into account.


and

1ZIðsÞ

¼Xnþ1

k¼1

CDLðkÞ � sCDLðkÞ � rIðkÞ � sþ 1

: ð15Þ

Values of CDL(k) and rIk differ from one slide to another due to surfaces and volumes differences in each slide (due to rough-ness). It allows then to approximate ZI(s) by a ‘‘Constant Phase Element” (CPE) impedance [11] with a non-integer order c.

zIðkÞðsÞ ¼CDLðkÞ � rIðkÞ � sþ 1

CDLðkÞ � sand ZIðsÞ ¼

1Q � sc : ð16Þ

To simplify Z(s) expression, five variables must be introduced:

Z1ðsÞ ¼ ½ðZEðsÞ þ ZSðsÞÞ � ZIðsÞ�1=2 ð17Þ

Z2ðsÞ ¼ZSðsÞ � ZEðsÞðZSðsÞ þ ZEðsÞÞ

; ð18Þ

q1ðsÞ ¼ZEðsÞ

ðZSðsÞ þ ZEðsÞÞ; ð19Þ

q2ðsÞ ¼ZSðsÞ

ðZSðsÞ þ ZEðsÞÞ; ð20Þ

TðsÞ ¼ tanhðZSðsÞ þ ZEðsÞÞ

ZIðsÞ

� �1=2" #

: ð21Þ

Influence of wires, current collectors and separator were until now ignored. These elements can be taken into account bya resistance R in series with the pore model. The behaviour of the porous electrode model can thus be modelled by the fol-lowing impedance:

ZðsÞ ¼ Rþ Z2ðsÞ þ Z1ðsÞ1þ 2 � q1ðsÞ � q2ðsÞ � ½ð1� ðTðsÞÞ

2Þ1=2 � 1�TðsÞ : ð22Þ

The rough nature of the interface electrode/electrolyte modifies the model proposed by Görh [10] in the followingway:

ZUCðsÞ ¼ Rþ ZS:ZE

ðZS þ ZEÞþ 2:ZE:ZS

ðZS þ ZEÞ:Q :ðsÞ2c þZS � 2:ZS:ZE þ ZE

ððZS þ ZEÞ:Q :ðsÞcÞ1=2 : coth½ððZS þ ZEÞ:Q :ðsÞcÞ1=2�: ð23Þ

Despite the approximation done in relation (22), most of the parameters of impedance (23) depend on UC physicalvariables.

Parameters in (23) results from its frequency response fitting with the UC frequency response. Comparison of the result-ing model and a 2600 F element frequency response is presented in Fig. 5. The results show a similar behaviour of the model

10-2

10-1

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101

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0.8

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danc

e R

ealP

art (

m Ω

)

10-2

10-1

100

101

-20

-15

-10

-5

0

5

Frequency (Hz)

Impe

danc

e Im

agin

ary

Pa

rt (m

Ω)

Impedance Real Part M26 ElementImpedance Real Part Model

Impedance Imaginary Part M26 ElementImpedance Imaginary Part Model

Fig. 5. Comparison between frequency response of a 2600 F cell and the fractional order model.


with the UC on the whole frequency range. Model (23) is thus validated. The frequency response of Fig. 5 was obtained byimpedance spectroscopy for an operating voltage V0. Given that the small signal amplitude applied to the UC during theimpedance spectroscopy, model obtained is thus valid only near V0 and is equivalent to model (10) obtained by Taylorapproximation.

Impedance spectroscopy for various V0 produces a set of linear models on which the integration method of Section 3 canbe applied to obtain a non-linear model of the M26 element.

5. An ultracapacitor non-linear model

The interest in developing a fractional non-linear UC state space model is to take into account simultaneously the voltagedependency and also long range phenomenon as UC relaxation.

5.1. Non-linear UC model

Due to voltage dependency, all the parameters of impedance (23) depend on operating voltage. Impedance (23) thusbecomes:

ZUCðsÞ¼RV0 þZSV0 �ZEV0

ðZSV0 þZEV0 Þþ 2 �ZEV0 �ZSV0

ðZSV0 þZEV0 Þ �QV0� ðsÞ2cV0

þ ZSU�2 �ZSU �ZEUþZEU

ðZSV0 þZEV0Þ �QV0� ðsÞcV0

� �1=2 �coth ZSV0 þZEV0

� ��Q V0

� ðsÞcV0� �1=2h i

:

ð24Þ

To obtain a rational model, coth function is approximated using the Pade approximation method [3]. A Pade approximantis used because it gives better approximations than those obtained though Taylor series expansion. Moreover it may stillwork where the Taylor series does not converge. The approximated impedance is defined as the following equation:

eZUCðsÞ ¼ K1ðV0Þ þ K2ðV0Þ �b3ðV0Þs3cðV0Þ þ b2ðV0Þs2cðV0Þ þ b1ðV0ÞscðV0Þ þ 1

s4cðV0Þ þ a3ðV0Þs3cðV0Þ þ a2ðV0Þs2cðV0Þ þ a1ðV0ÞscðV0Þ: ð25Þ

Fig. 6 is a comparison of ZUC(s) and eZUCðsÞ frequency responses. This comparison highlights the validity of relation (25). Eq.(25) is similar to Eq. (11) but its coefficients only depend on the output of the UC (voltage).

Integration method of Section 3 applied to relation (25) (similar to relation (11)) thus produces the non-linear model

Impedance Real Part M26 ElementImpedance Real Part Approximated Model

Impedance Imaginary Part M26 ElementImpedance Imaginary Part Approximated Model

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10-2

10-1

100

101

Frequency (Hz)

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danc

e R

eal P

art (

mΩ

)

-20

-15

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-5

0

5

Impe

danc

e Im

agin

ary

Part

(mΩ

)

Fig. 6. Comparison between frequency response of a 2600 F cell and the approximated model.

0 1 2 3

200

400

600

α3

0 1 2 30

1

2x 10-3

β3

0 1 2 33.63.8

44.24.4

x 10-4

k1

0 1 2 30

100

200

Voltage (V)

Voltage (V)

Voltage (V)

Voltage (V)

Voltage (V)

k2

0 1 2 30.008

0.01

0.012

0.014

1-γ

Fig. 7. Comparison of parameter variations with voltage and their corresponding polynomial approximations.


dcz1dtc¼ z2

dcz2dtc¼ z3

dcz3dtc¼ z4

dcz4dtc¼ �a1ðVÞz2 � a2ðVÞz3 � a3ðVÞz4 þ k2ðVÞI

8>>>>>><>>>>>>:: ð26Þ

V ¼ z1 þ b1ðVÞz2 þ b2ðVÞz3 þ b3ðVÞz4 þ k1ðVÞI

Eq. (26) reveals that analytic expressions of parameters a(�) and b(�) are required. More the evolution of these parametersis respected and more the non-linear and linear models will have the same response around an operating voltage. As it isshown in Fig. 7, a polynomial expression can approximate the parameter evolution. (a1,a2) and (b1,b2) are not presented be-cause they have the same evolution as coefficients a3(�) and b3(�) but with an additional offset. A nine-order polynomialexpression is used in Fig. 7 to model parameter variations.

Using the method described in the previous section, it can be demonstrated analytically that around an operating voltagethe non-linear and the linear model have the same behaviour.

5.2. Simulink implementation

State space description (26) can be written as:

Xc½V � ¼ A½V � � X þ B½V � � I V ¼ C½V � � X þ D½V � � I ð27Þ

C[V]

D[V]

+

+ +

+

V B[V] Id[V,s]

A[V]

I

Fig. 8. State space non-linear model.

0 dBlog(ω)

0

−90°

ω ω hω1 ω’1 ω i ω’i ωN ω’N

( )dB

a jI ωγ( )dB

jI ωγ

1/2logη logη

logα 1/2logη

log(ω)

−γ90°

( )( )°ωγ jIaarg( )( )°ωγ jIarg ( )( )°ωγ jINarg

( )dB

N jI ωγ

Δ dB -20γ dB/décade

-20 dB/décadeΑ A’

B’Β

Fig. 9. Approximation of a fractional order integrator of transmittance Ic(s) by a transmittance IcNðsÞ constituted of N poles and zeros recursively distributed.


in which matrices A, B, C and D depend on the UC operating voltage V. For simulation purpose, a Simulink compatible rep-resentation of (26) is given in Fig. 8 Matrix Id[V,s] is a diagonal matrix whose all elements are fractional integrator 1/sc withc < 1. Values of matrices A, B, C, D change at each calculation step.

Fractional integrators in matrix Id[V,s] are approximated by a transmittance based on recursive distribution of poles andzeros as described in [12]. Fig. 9 is an illustration of such an approximation.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000.75

1

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2

2.25

2.5

Times (s)

Volta

ge (V

)

Measurement VoltageNon-linear Model Voltage

5485 5490 5495 5500 5505 5510 5515 5520 55251

1.25

1.5

1.75

2

2.25

2.5

Times (s)

Volta

ge (V

)

Measurement Voltage

Non-linear Model Voltage

0 500 1000 1500 2000 25002.2886

2.3286

2.3686

2.4086

2.4406

Times (s)

Volta

ge (V

)

Measurement Voltage

Non-linear Model Voltage

Fig. 10. Voltage responses for successive charges and discharges.

0 50 100 150 200 2501.2

1.6

2

2.4

2.8

Volta

ge (V

)

0 50 100 150 200 250-500

-250

0

250

500

Times (s)

Cur

rent

(A)

Measurement VoltageNon-linear Model Voltage

Current

Fig. 11. Voltage responses for a discontinuous current profile (model simulations and measurements).


6. Time simulation of the ultracapacitor

The non-linear model validation is done through a comparison between its time response and the 2600 F cell response.

6.1. Time response with a charge and discharge profile

Time response of the model must be similar to the UC one. Different tests have been done on a M26 UC. The current pro-file used here is composed of two charges and discharges at 80 A. The experimental results and the model time responseseem to be similar as shown in Fig. 10. The error between the simulated results from the model and the measurements isabout 5 mV after a 4-h test.

Fig. 10 highlights that the modelling method proposed produces a non-linear model that permit to take into account thevoltage dependency in charge and in discharge in a very accurate way. However an error appears during charge recovery. Themodel reaches rapidly the final voltage whereas the charge recovery of the element shows a continuous voltage decrease.This problem is under investigation.

6.2. Time response for a HEV profile

To observe the time response of the model, a specific current profile has been used. It is made of a combination of severalpulses which can be encountered in HEV applications. As shown in Fig. 11, the model response is in good agreement with theexperimental behaviour of the M26 cell.

During charges and discharges, the model response matches well the UC one. However, as depicted in the previous test,the model limits come from the approximation of the charge recovery phenomenon.

7. Conclusion

In this paper a set of fractional order linear model are used to characterize the dynamic behaviour of an ultracapacitoraround various operating points. These models match well the behaviour of a real large cell ultracapacitor in the frequencydomain. The voltage dependency of the UC is demonstrated in time and in frequency domain thus showing a non-linearbehaviour of the UC. A non-linear model is proposed. It is built with a set of linear models and an integration method.The time responses obtained highlight that the voltage dependency is well taken into account, but also show the model lim-its in case of slow stimulus that corresponds to the UC charge recovery. Efforts are actually made in order to model thecharge recovery and avoid the drift of the simulated voltage. The evolution of the c parameter (fractional order) that ad-dresses charge recovery phenomenon has to be estimated with more accuracy.

References

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Fractional non-linear modelling of ultracapacitors