The application of the fractional calculus for modeling electrochemical double layer capacitors is a novel way to get simpler and precise models. On using the impedance spectroscopy method, experimental results for different values have been obtained. In this paper, several classical mathematical models are studied and a different method is introduced in order to get a model from electrochemical double layer capacitors. This method is based on distinct models with fractional elements, and some parameters of the models are fitted to the experimental data, with minimal error. Finally, a Havriliak–Negami function based model is proposed. It achieves excellent fitting to the whole frequency interval analyzed.

1.
Conway
,
B. E.
, 1999,
Electrochemical Supercapacitors: Scientific Fundamentals and Technological Applications
,
Kluwer Academic
,
Dordrecht
/
Plenum
,
New York
.
2.
Podlubny
,
I.
, 1999,
Fractional Differential Equations
,
Academic
,
San Diego
.
3.
Axtell
,
M.
, and
Bise
,
M. E.
, 1990, “
Fractional Calculus Applications in Control Systems
,”
Proceedings of the IEEE 1990 National Aerospace and Electronics Conference
,
New York
.
4.
Podlubny
,
I.
, 1999, “
Fractional-Order Systems and PIλDμ Controllers
,”
IEEE Trans. Autom. Control
0018-9286,
44
, pp.
208
214
.
5.
Vinagre
,
B. M.
, 2001, “
Modelado y Control de Sistemas Dinámicos Caracterizados por Ecuaciones Integro-Diferenciales de Orden Fraccional
,” Ph.D. Thesis.
6.
Petrás
,
I.
,
Podlubny
,
I.
,
O’Leary
,
P.
,
Dorcak
,
L.
, and
Vinagre
,
B.
, 2002, “
Analogue Realization of Fractional Order Controllers
,” FBERG, Technical University of Kosice.
7.
Oldham
,
K. B.
, and
Spanier
,
J.
, 1974, “
The Fractional Calculus: Integration and Differentiations of Arbitrary Order
,”
Mathematics in Science and Engineering
,
Academic
,
NewYork
, Vol.
111
.
8.
Bisquert
,
J.
, and
Compte
,
A.
, 2001, “
Theory of the Electrochemical Impedance of Anomalous Diffusion
,”
J. Electroanal. Chem.
0022-0728,
499
(
1
), pp.
112
120
.
9.
Sokolov
,
I. M.
,
Klafter
,
J.
, and
Blumen
,
A.
, 2002, “
Fractional Kinetics
,”
Phys. Today
0031-9228,
55
(
11
), pp.
48
55
.
10.
Sornette
,
A.
, 2004,
Critical Phenomena in Natural Sciences (Chaos, Fractals, Self-Organization and Disorder: Concepts and Tools)
,
2nd ed.
,
Springer-Verlag
,
Berlin
.
11.
Haba
,
T. C.
,
Martos
,
T.
,
Ablart
,
G.
, and
Bidan
,
P.
, 1998, “
Composants Électroniques a Impedance Fractionnaire
,”
ESAIM Proceedings of the Conference on Fractional Differential Systems: Models, Methods and Applications
, Vol.
5
, pp.
99
109
.
12.
Matignon
,
D.
, and
D’Andréa-Novel
,
D.
, 1997, “
Observer-Based Controllers for Fractional Differential Systems
,”
Proceedings of the 36th Conference on Decision and Control
,
San Diego, CA
.
13.
Graca Marcos
,
M.
,
Duarte
,
F. B. M.
, and
Tenreiro Machado
,
J. A.
, 2007, “
Fraccional Dynamics in the Trajectory Control of Redundant Manipulators
,”
Commun. Nonlinear Sci. Numer. Simul.
1007-5704, in press.
14.
Oustaloup
,
A.
,
Sabatier
,
J.
, and
Moreau
,
X.
, 1998, “
From Fractal Robustness to the Crone Approach
,”
ESAIM Proceedings of the Conference on Fractional Differential Systems: Models, Methods and Applications
, Vol.
5
, pp.
177
192
.
15.
Vinagre
,
B.
,
Feliu
,
V.
, and
Feliu
,
J.
, 1997, “
Frequency Domain Identification of a Flexible Structure With Piezoelectric Actuators Using Irrational Transfer Function Model
,”
Proceedings of the 36th Conference on Decision and Control
,
San Diego, CA
, pp.
1278
1280
.
16.
Mbodje
,
B.
,
Montseny
,
G.
,
Audounet
,
J.
, and
Benchimol
,
P.
, 1994, “
Optimal Control for Fractional Damped Flexible Systems
,”
IEEE Conference on Control Applications
, Vol.
2
, pp.
1329
1333
.
17.
Montseny
,
G.
,
Audounet
,
J.
, and
Matignon
,
D.
, 1997, “
Fracctional Integridifferencial Boundary Control of the Euler-Bernouille Beam
,”
Proceedings of the 36th Conference on Decision and Control
,
San Diego, CA
, pp.
4973
4978
.
18.
Tenreiro Machado
,
J. A.
,
Jesus
,
I. S.
,
Galhano
,
A.
, and
Bohaventura Cunha
,
J.
, 2006, “
Fraccional Order Electromagnetics
,”
Signal Process.
0165-1684,
86
(
10
), pp.
2637
2644
, special issue on fraccional calculus applications in signals and systems.
19.
Tarasov
,
V. E.
, and
Zaslavsky
,
G. M.
, 2006, “
Fraccional Dynamics of Systems With Long-Range Interaction
,”
Commun. Nonlinear Sci. Numer. Simul.
1007-5704,
11
(
8
), pp.
885
898
.
20.
Korabel
,
N.
,
Zaslavsky
,
G. M.
, and
Tarasov
,
V. E.
, 2007, “
Coupled Oscillators With Power-Law Interaction and Their Fraccional Dynamics Analogues
,”
Commun. Nonlinear Sci. Numer. Simul.
1007-5704,
12
(
8
), pp.
1405
1417
.
21.
Tarasov
,
V. E.
, and
Zaslavsky
,
G. M.
, 2007, “
Conservations Laws and Hamilton’s Equations for Systems With Long-Range Interaction and Memory
,”
Commun. Nonlinear Sci. Numer. Simul.
1007-5704, in press.
22.
Jesus
,
I. S.
,
Tenreiro Machado
,
J. A.
, and
Boaventura Cunha
,
J.
, 2006, “
Fractional Electrical Dynamics in Fruits and Vegetables
,”
FDA 2006, Second IFAC Workshop on Fractional Differentiation and Its Applications
,
Porto, Portugal
.
23.
Haschka
,
M.
,
Ruger
,
B.
, and
Krebs
,
V.
, 2004, “
Identification of the Electrical Behavior of a Solid Oxide Fuel Cell in the Time-Domain
,”
Proceedings of Fractional Differentiation and Its Applications 2004 Conference
,
Bordeaux, France
, pp.
327
333
.
24.
Miller
,
K. S.
, and
Ross
,
B.
, 1993,
An Introduction to the Fractional Calculus and Fractional Differential Equations
,
Willey
,
New York
.
25.
Lorenzo
,
C. F.
, and
Hartley
,
T. T.
, 1998, “
Initialization, Conceptualization, and Application in the Generalized Fractional Calculus
,” National Aeronautics and Space Administration, NASA Report No. TP-1998-208415.
26.
Carlson
,
G. E.
, and
Halijak
,
C. A.
, 1961, “
Simulation of the Tradicional Derivate Operador and the Fraccional Integral Operador
,”
Proceedings of the Central State Simulation Council Meeting Kansas State University Bulletin
, Vol.
45
.
27.
Westerlund
,
S.
, and
Ekstam
,
L.
, 1999, “
Capacitor Theory
,”
IEEE Trans. Dielectr. Electr. Insul.
1070-9878,
1
(
5
), pp.
826
839
.
28.
Burke
,
A.
, 2000, “
Ultracapacitors: Why, How, and Where Is the Technology
,”
J. Power Sources
0378-7753,
91
, pp.
37
50
.
29.
Conway
,
B. E.
, 1999,
Electrochemical Supercapacitors: Scientific Fundamentals and Technological Applications
,
Kluwer Academic
,
Dordrecht
/
Plenum
,
New York
.
30.
Buller
,
S.
,
Karden
,
E.
,
Kok
,
D.
, and
Doncker
,
R. W.
, 2002, “
Modeling the Dynamic Behaviour of Supercapacitors Using Impedance Spectroscopy
,”
IEEE Trans. Ind. Appl.
0093-9994,
38
(
6
), pp.
1622
1626
.
31.
Vladikova
,
D.
, 2004, “
The Technique of the Differential Impedance Analysis. Part I: Basics of the Impedance Spectroscopy
,”
Proceedings of the International Workshop on Advance Techniques for Energy Sources Investigation and Testing
,
Sofia, Bulgaria
.
32.
Conway
,
B. E.
, 2002, “
Power Limitations of Supercapacitor Operation Associated With Resistance and Capacitance Distribution in Porous Electrode Devices
,”
J. Power Sources
0378-7753,
105
, pp.
169
181
.
33.
Riu
,
D.
,
Retiere
,
N.
, and
LinZen
,
D.
, 2004, “
Half-Order Modelling of Supercapacitors
,”
Industry Applications Conference 2004, 39th Annual Meeting, Conference Record of the 2004 IEEE
, Vol.
4
, pp.
2550
2554
.
34.
Quintana
,
J. J.
,
Ramos
,
A.
, and
Nuez
,
I.
, 2006, “
Identification of the Fractional Impedance of Ultracapacitors
,”
Proceedings of the Second IFAC Workshop on Fraccional Differentiation and Its Applications
,
Porto, Portugal
.
35.
Liu
,
S. H.
, and
Liu
,
A. J.
, 1986, “
Anomalous Diffusion on and Elastic Vibrations of Two Square Hierarchical Lattices
,”
Phys. Rev. B
0163-1829,
34
(
1
), pp.
343
346
.
36.
Walter
,
G. W.
, 1986, “
A Review of Impedance Plot Methods Used for Corrosion Performance Analysis of Painted Metals
,”
Corros. Sci.
0010-938X,
26
(
9
), pp.
681
703
.
37.
Muralidharan
,
V. S.
, 1997, “
Warburg Impedance-Basics Revisited
,”
Anti-Corros. Methods Mater.
0003-5599,
44
(
1
), pp.
26
29
.
38.
Baert
,
D. H.
, and
Vervaet
,
A.
, 2004, “
A Fast Methods for the Measurement of Electrical Capacitance for the Estimation of Battery Capacity
,”
IEICE Trans. Commun.
0916-8516,
E87-B
(
12
), pp.
3478
3484
.
39.
Nigmatullin
,
R. R.
, and
Osokin
,
S. I.
, 2003, “
Signal Processing and Recognition of True Kinetic Equations Containing Non-Integer Derivates From Raw Dielectric Data
,”
Signal Process.
0165-1684,
83
, pp.
2433
2453
.
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