- Research
- Open Access
Comparison and validation of integer and fractional order ultracapacitor models
- Andrzej Dzieliński^{1}Email author,
- Grzegorz Sarwas^{1} and
- Dominik Sierociuk^{1}
https://doi.org/10.1186/1687-1847-2011-11
© Dzieliński et al; licensee Springer. 2011
- Received: 15 December 2010
- Accepted: 14 June 2011
- Published: 14 June 2011
Abstract
In this article, the modeling of the ultracapacitor using different models of capacity part is shown. Two fractional order models are compared with the integer model of traditional capacitor. The identification was made using the diagram matching technique. Next, the derivation of time domain response of the ultracapacitor and system with the ultracapacitor are presented. The results of frequency domain identification were used to validate the response of the ultracapacitor in time domain. All theoretical results are compared with the response of the physical system with the ultracapacitor.
Keywords
- fractional calculus
- fractional order dynamic systems
- ultracapacitors modeling
Introduction
Ultracapacitors (aka supercapacitors) are electrical devices which are used to store energy and offer high power density that is not possible to achieve with traditional capacitors. Nowadays, ultracapacitors have many industrial applications and are used wherever a high current in a short time is needed. Thanks to a very complicated internal structure, they are able to store or yield a lot of energy in a short period of time. Many researchers started building a more or less complicated model to explain the capability of ultracapacitors. Numerous articles have presented the RC model (e.g., [1–5]), which is particularly accurate for low frequencies. Some authors describe ultracapacitors by the RC transmission line [4–6]. Also, the dynamic behavior of ultracapacitors has been modeled using the technique based on impedance spectroscopy in, e.g., [2]. In the papers, [7–9] a very efficient approach using fractional order calculus was presented and in [10, 11] ultracapacitor frequency domain modeling was introduced. In this article, modeling using three different models of ultracapacitors are compared. Two of them are fractional order. We validate the identified models from frequency domain [12] with the step response of this model in time domain. The time domain responses of the ultracapacitor and a system with the ultracapacitor are calculated. All theoretical results are compared with the results achieved from a physical system.
Fractional order differential calculus introduction
Fractional order differential calculus is only a generalization of integer order integral and differential calculus to real or even complex order. This idea has first been mentioned at the end of seventeenth century. There exist two (in fact three) main definitions of the fractional order integrals and derivatives: Riemann-Liouville (and Caputo) and Grünwald-Letnikov [13]. In this article, the Riemann-Liouville definition of the fractional order difference is used.
Definition of fractional order differ-integral
In this article, the following definition of the fractional derivative will be used:
where t ∈ ℝ, a ∈ ℝ, t > a and α ∈ ℝ is a fractional order of the differ-integral of the function f(t). For α > 0 m - 1 < α ≤ m, m ∈ ℕ and for α ≤ 0 m = 0.
When α > 0 the result of this function is equivalent to the fractional order derivative, for α < 0 to fractional order integral and for α = 0 to the function itself. This is why the above definition is called a differ-integral.
It is easy to see that the Laplace transformation of the Riemann-Liouville definition possesses the fractional order derivatives of initial conditions. Despite the difficulty in finding the physical meaning of these parameters, using the R-L definition to model the ultracapacitor is feasible, because the model presented in this article is based on the fractional order transfer function, where initial conditions are equal to zero.
Fractional order integrator
Frequency domain models of ultracapacitor
Ultracapacitors are large capacity and power density electrical energy storage devices. This large capacity is the effect of a very complicated internal structure. This structure also has a significant impact on the dynamic behavior of the ultracapacitor. Many authors use different RC models to describe the performance of the ultracapacitors but these models are correct only for a limited range of frequencies. A more effective approach is based on using the fractional order model which gives highly accurate results of modeling over a wider range of frequencies. In this article, we would like to show the advances of using fractional order model for ultracapacitor modeling.
Fractional order ultracapacitor model
where R _{ c } is the resistance of the ultracapacitor and i is the model index (i = 1, 2, or 3).
Frequency domain identification of ultracapacitors parameters
The modeling of the ultracapacitor presented in this article is based on the diagram matching. For modeling we used two experimental systems. The first one is to examine high-capacity ultracapacitors and the second one to examine low-capacity ultracapacitors.
Modeling high-capacity ultracapacitor
To model a high-capacity ultracapacitor the experimental setup contained high-capacity ultracapacitor connected to the DS1104 Control Card by the electronic interface based on the MOSFET power converter. The research was focused on the Maxwell^{®} ultracapacitor of nominal capacity 1500F/2.7V (BCAP1500) and 3000F/2.7V (BCAP3000).
The identification was based on Bode diagram matching. A Bode diagram of the models was tuned to the diagram of the ultracapacitor which was determined from measurements. As a result of this research, we obtained the parameters of the models.
Using the experimental setup to examine high-capacity ultracapacitors, we were able to use high current to charge and discharge ultracapacitors (current values of more than 100 A were used). Additionally, using this system we were also able to show the physical step response of the ultracapacitor.
Modeling low-capacity ultracapacitor
This setup is composed of the operational amplifier OPA544, matching resistor 180Ω and ultracapacitors produced by Panasonic^{®} of nominal capacity 0.047 F, 0.1 F, 0.33 F. The high current operational amplifier OPA 544 works in the voltage follower configuration.
Also in this case, the modeling of the ultracapacitor was based on Bode diagram matching and ultracapacitor models shown above.
Results of experiments
The ultracapacitor is an electrolytic capacitor and it can accept only positive voltages. In the case of using this type of setup with a current converter, to model high-capacity ultracapacitor, the input signal was a current sine wave. The ultracapacitors had an initial voltage depending on a signal frequency (u _{0}). Capacitor voltage in this case was equal to u _{ c } (t) = u _{0} + A _{ c } (ω)sin(ωt + φ _{ u } ), and input (capacitor) current was i(t) = A _{ i } (ω)sin(ωt + φ _{ i } ).
In the case of a system with a voltage follower, the configuration used a voltage input signal with a constant component u(t) = 2 + sin(ωt). Capacitor voltage (in steady state) in this case was equal to u _{ c } (t) = 2 + A _{ c } (ω)sin(ωt + φ _{ u } ) and capacitor current was i(t) = A _{ i } (ω)sin(ωt + φ _{ i } ).
where A _{ c } (ω) is the magnitude of ultracapacitor voltage and A _{ i } (ω) is the magnitude of current flowing through ultracapacitor for used frequency. The φ _{ i } (ω) and φ _{ u } (ω) are the phase shifts between input system voltage and current flowing through ultracapacitor and voltage of the ultracapacitor, respectively.
Identified parameters of 1500 F ultracapacitor
Model | C(F) | α | T |
---|---|---|---|
| 1467.3 | - | - |
| 1122.4 | 0.9 | - |
| 1336.9 | 0.3502 | 1.3163 |
Parameters of ultracapacitors
Capacitor | T | C(F) | R(Ω) | α | C _{real} (F) | R _{real} (Ω) |
---|---|---|---|---|---|---|
0.047 F | 5.1138 | 0.05 | 32 | 0.6 | 0.06 | 32 |
0.1 F | 13.6628 | 0.1 | 38 | 0.6 | 0.1 | 42 |
0.33 F | 52.4546 | 0.27 | 27 | 0.6 | 0.27 | 28 |
1500 F | 1.3163 | 1336.9 | 0.47m | 0.61 | 1189 | 0.47m |
The result of identification using fractional order model gave very good parameters which means that this model can very precisely model the ultracapacitors. Their parameters are equivalent to these obtained from real ultracapacitor by different methods, e.g., step response. The approach presented in this section is in fact analogous to the impedance spectroscopy. However in our case, we rather use Bode plot frequency response as opposed to Nyquist diagram.
Validation of time domain response of ultracapacitor
In this section, the validation of models with identified parameters in time domain are presented.
Time domain response derivation
Calculation of this time response of ultracapacitor model is however more complicated.
where the is the confluent hypergeometric function.
Step response of the system with ultracapacitor
for i = { 1, 2, 3}.
where is the one-parameter representation of the Mittag-Leffler function below.
where α > 0.
Calculation of the step response of RC quadripole with the ultracapacitor model for arbitrary α is not easy. Therefore, we present the resulting step response of this model for α = 0.5. The experiments performed revealed that for low-capacity ultracapacitors α is close enough to 0.5 to provide correct modeling of ultracapacitor dynamics.
and is a two-parameter Mittag-Leffler function. Parameters C, R _{ c } , T are the parameters of the ultracapacitor model and R is the resistance of RC quadripole's resistor.
where B = R _{ c } C and A = (R + R _{ c } )C.
for α, β > 0. Finally, from (26), (27), and (28) we obtained the step response of the system with the ultracapacitor.
Results of identification in time domain
These figures present that fractional order model gives the best result of the ultracapacitor modeling in frequency and also in time domain.
Conclusions and future works
Conclusions
In the article, the results of frequency and time domain modeling of ultracapacitors, using three different models, have been presented. The best description of the ultracapacitor dynamic was achieved using fractional model in the form which describes the dynamics of the ultracapacitor over a reasonable range of frequencies. This model of the ultracapacitor can be used in either time or frequency domains. The model proposed provides good results of modeling in time domain, in comparison with the other models considered. However, to cover even wider range of frequencies the intrinsic nonlinearities of the ultracapacitors have to be taken into account resulting in either a time-varying or nonlinear model.
Model is also better for modeling ultracapacitors then fractional order model from the point of view of physical interpretation. Because the model corresponds to the physical ultracapacitor we use the integer order in the denominator, otherwise if the order of ultracapacitor is in the range 0 < α < 1 then the capacity of such a device tends to infinity with time, which is not possible in the case of physical systems [12].
Future works
The model obtained with the technique presented may form the foundation for establishing the control design procedures. The scope of future works will also deal with building the state space model of the ultracapacitor which will be composed of the part responsible for the integer order capacitor and the fractional order part responsible for a better description of the behavior and all the specific phenomena of the ultracapacitors.
Declarations
Acknowledgements
This work was partially supported by the Polish Ministry of Science and Higher Education Grant number 4125/B/T02/2009/36 and the European Union in the framework of European Social Fund through the Warsaw University of Technology Development Programme (by Centre for Advanced Studies WUT).
Authors’ Affiliations
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