Circuits, Systems, and Signal Processing

, Volume 38, Issue 5, pp 1907–1922 | Cite as

Rates and Effects of Local Minima on Fractional-Order Circuit Model Parameters Extracted from Supercapacitor Discharging Using Least Squares Optimization

  • Todd J. FreebornEmail author
  • Ahmed S. Elwakil


Optimization routines are widely used to numerically determine a set of model parameters that best fit collected experimental data. One recent application of these methods is to extract the fractional-order circuit model parameters that accurately characterize the transient behavior of discharging supercapacitors. However, the variability that these methods introduce to the extracted model parameters must be understood to determine if changes in model parameters are artifacts of the optimization routine or are representative of physical changes in the device under study. In this work, the variability of supercapacitor fractional-order model parameters is quantified when extracted using a nonlinear least squares optimization applied to simulated data (with 0.1–3% noise) and experimental data of their discharging behavior. These results indicate that the local minima problem occurs at a rate of \(<1\%\) for this application, though this problem can be overcome using at least 10 independent executions of the solver to each dataset. The variability of the model parameters using 1000 executions of the solver with 1 and 10 iterates is quantified for both simulated and experimental data, with the 10 iterates showing at least an order of magnitude reduction in the range of returned parameters.


Supercapacitors Fractional calculus Fractional-order circuits Optimization 


  1. 1.
    A. Adhikary, S. Sen, K. Biswas, Practical realization of tunable fractional order parallel resonator and fractional order filters. IEEE Trans. Circuit Syst. I 63(8), 1142–1151 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. Allagui, T.J. Freeborn, A.S. Elwakil, B. Maundy, Reevaluation of performance of electric double-layer capacitors from constant current charge/discharge and cyclic voltammetry. Sci. Rep. 6, 38568 (2016)CrossRefGoogle Scholar
  3. 3.
    A. Allagui, A.S. Elwakil, B. Maundy, T.J. Freeborn, Spectral capacitance of series and parallel combinations of supercapacitors. ChemElectroChem 9(3), 1429–1436 (2016)CrossRefGoogle Scholar
  4. 4.
    A. Allagui, T.J. Freeborn, A.S. Elwakil, M.E. Fouda, B.J. Maundy, A.G. Radwan, Z. Said, M.A. Abdelkareem, Review of fractional-order electrical characterization of supercapacitors. J. Power Sources 400, 457–467 (2018)CrossRefGoogle Scholar
  5. 5.
    G. Brando, A. Dannier, A. Del Pizzo, Grid connection of wave energy converter in heaving mode operation by supercapacitor storage technology. IET Renewable Power Gener. 10(1), 88–97 (2016)CrossRefGoogle Scholar
  6. 6.
    D. Brunelli, C. Moser, L. Thiele, L. Benini, Design of a solar-harvesting circuit for batteryless embedded systems. IEEE Trans. Circuit Syst. I 56(11), 2519–2528 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Burke, Ultracapacitors: why, how, and where is the technology. J. Power Sources 90(1), 37–50 (2000)MathSciNetCrossRefGoogle Scholar
  8. 8.
    J. Cao, A. Emadi, A new battery/ultracapacitor hybrid energy storage system for electric, hybrid, and plug-in hybrid electric vehicles. IEEE Trans. Power Electron. 27(1), 122–132 (2012)CrossRefGoogle Scholar
  9. 9.
    A. Castaings, W. Lhomme, R. Trigui et al., Practical control schemes of a battery/supercapacitor system for electric vehicle. IET Electr. Syst. Transp. 6, 20–26 (2016)CrossRefGoogle Scholar
  10. 10.
    D. Cressey, The DIY electronics transforming research. Nature 544, 125–126 (2017). CrossRefGoogle Scholar
  11. 11.
    A. Dzieliński, G. Sarwas, D. Sierociuk, Comparison and validation of integer and fractional order ultracapacitor models. Adv. Differ. Equ. (2011).
  12. 12.
    A.S. Elwakil, Fractional-order circuits and systems: an emerging interdisciplinary research area. IEEE Circuits Syst. Mag. 10(4), 40–50 (2010)CrossRefGoogle Scholar
  13. 13.
    M.E. Fouda, A.S. Elwakil, A.G. Radwan, A. Allagui, Power and energy analysis of fractional-order electrical energy storage devices. Energy 111, 785–792 (2016)CrossRefGoogle Scholar
  14. 14.
    T.J. Freeborn, Estimating supercapacitor performance for embedded applications using fractional-order models. Electron. Lett. 52(17), 1478–1480 (2016)CrossRefGoogle Scholar
  15. 15.
    T.J. Freeborn, A. Allagui, A.S. Elwakil, Modelling supercapacitors leakage behaviour using a fractional-order model, Euro. Conf. Circuit Theor. Design, Catania, Italy (2017).
  16. 16.
    T.J. Freeborn, A.S. Elwakil, Variability of supercapacitor fractional-order parameters extracted from discharging behavior using least squares optimization, Int. Symp. Circuits Syst. (ISCAS), pp. 1510–1513, Baltimore, USA (2017)Google Scholar
  17. 17.
    T.J. Freeborn, B. Maundy, A.S. Elwakil, Measurement of supercapacitor fractional-order model parameters from voltage excited step response. IEEE J. Emerg. Sel. Top. Circuits Syst. 3(3), 367–376 (2013)CrossRefGoogle Scholar
  18. 18.
    R.K.H. Galvao et al., Fractional order modeling of large three-dimensional RC networks. IEEE Trans. Circuits Syst. I 60(3), 624–637 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    S.D. Jayasinghe, D.M. Vilathgamuwa, Flying super-capacitors as power smoothing elements in wind generation. IEEE Trans. Ind. Electron. 60(6), 2909–2918 (2013)CrossRefGoogle Scholar
  20. 20.
    R. Kopka, Estimation of supercapacitor energy storage based on fractional differential equations. Nanoscale Res. Lett. 12, 636 (2017). CrossRefGoogle Scholar
  21. 21.
    M.R. Kumar, S. Ghosh, S. Das, A hybrid optimization-based approach for parameter estimation and investigation of fractional dynamics in ultracapacitors. Circuits Syst. Signal Process. 35(6), 1949–1971 (2016)CrossRefGoogle Scholar
  22. 22.
    A. Oukaour, Electrical double-layer capacitors diagnosis using least square estimation method. Electric Power Syst. Res. 117, 69–75 (2014)CrossRefGoogle Scholar
  23. 23.
    J. Pegueroles-Queralt, F.D. Bianchi, O. Gomis-Bellmunt, A power smoothing system based on super-capacitors for renewable distributed generation. IEEE Trans. Ind. Electron. 62(1), 343–350 (2014)CrossRefGoogle Scholar
  24. 24.
    I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)zbMATHGoogle Scholar
  25. 25.
    J.J. Quintana, A. Ramos, I. Nuez, Modeling of an EDLC with fractional transfer functions using Mittag-Leffler equations, Math. Problems Eng., vol. 2013, Article ID 807034 (2013)Google Scholar
  26. 26.
    A.G. Radwan, A.S. Elwakil, A.M. Soliman, Fractional-order sinusoidal oscillators: design procedure and practical examples. IEEE Trans. Circuits Syst. I 55(7), 2051–2063 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    M.S. Sarafraz, M.S. Tavazoei, Realizability of fractional-order impedances by passive electrical networks composed of a fractional capacitor and RLC components. IEEE Trans. Circuits Syst. I 62(12), 2829–2835 (2015)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Z. Tianpei, W. Sun, Optimization of battery-supercapacitor hybrid energy storage station in wind/solar generation system. IEEE Trans. Sustain. Energy 5(2), 408–415 (2014)CrossRefGoogle Scholar
  29. 29.
    A.S. Weddell, G.V. Merrett, T.J. Kazmierski, B.M. Al-Hashimi, Accurate supercapacitor modeling for energy harvesting wireless sensor nodes. IEEE Trans. Circuit Syst. II 58(12), 911–915 (2011)CrossRefGoogle Scholar
  30. 30.
    H. Yang, Y. Zhang, Analysis of supercapacitor energy loss for power management in environmentally powered wireless sensor nodes. IEEE Trans. Power Electron. 28(11), 5391–5403 (2013)MathSciNetCrossRefGoogle Scholar
  31. 31.
    H. Yang, Y. Zhang, Estimation of supercapacitor energy using a linear capacitance for applications in wireless sensor networks. J. Power Sources 275, 498–505 (2015)CrossRefGoogle Scholar
  32. 32.
    L. Zhang, X. Hu, Z. Wang, F. Sun, D.G. Dorrell, Fractional-order modelling and state-of-charge estimation for ultracapacitors. J. Power Sources 314, 28–34 (2016). CrossRefGoogle Scholar
  33. 33.
    K. Zhou, D. Chen, X. Zhang, R. Zhou, H. Iu, Fractional-order three-dimensional \(\varDelta \times n\) circuit network. IEEE Trans. Circuits Syst. I 62(10), 2401–2410 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringUniversity of AlabamaTuscaloosaUSA
  2. 2.Department of Electrical and Computer EngineeringUniversity of SharjahSharjahUAE
  3. 3.Department of Electrical and Computer EngineeringUniversity of CalgaryCalgaryCanada
  4. 4.Nanoelectronics Integrated Systems Center (NISC)Nile UniversityCairoEgypt

Personalised recommendations