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The fractional-order state-space averaging modeling of the Buck–Boost DC/DC converter in discontinuous conduction mode and the performance analysis

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Abstract

In this paper, the fractional-order state-space averaging model of the Buck–Boost DC/DC converter in discontinuous conduction mode is presented based on the theory of fractional calculus. The order of the model can be considered as an extra parameter, and this extra parameter has a significantly influence on the performance of the model. The quiescent operation point of the fractional-order model is not only related to the switching time period, the duty ratio, and the inductance value, but also related to the inductor order. The amplitude of the output voltage, the inductor current, and the inductor current ripple increase with the decreasing of the inductor order. Subsequently, some low-frequency characteristics, such as the line-to-output transfer function and the control-to-output transfer function of the fractional-order model, are derived from alternating current components of the fractional-order state-space averaging model. And the results suggest that transfer functions are not only relative to the inductor order, but also have intimate correlation to the capacitor order. Fractional-order models can increase the flexibility and degrees of freedom by means of fractional parameters. Numerical and circuit simulations are presented to demonstrate the correctness of the fractional-order model and the efficiency of the proposed theoretical analysis. These simulation results indicate that the fractional-order model has a certain theoretical and practical significance for the design and performance analysis of DC/DC converter.

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References

  1. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  2. Podlubny, I., Petras, I., Vinagre, B.M., O’Leary, P., Dorcak, L.: Analogue realizations of fractional-order controllers. Nonlinear Dyn. 29(1–4), 281–296 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Nimmo, S., Evans, A.K.: The effects of continuously varying the fractional differential order of chaotic nonlinear systems. Chaos Soliton Fract. 10(7), 1111–1118 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Vinagre, B.M., Petras, I., Podlubny, I., Chen, Y.Q.: Using fractional order adjustment rules and fractional order reference models in model-reference adaptive control. Nonlinear Dyn. 29(1–4), 269–279 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ahmad, W.M., Harb, A.M.: On nonlinear control design for autonomous chaotic systems of integer and fractional orders. Chaos Soliton Fract. 18(4), 693–701 (2003)

    Article  MATH  Google Scholar 

  6. Chen, D.Y., Zhang, R.F., Sprott, J.C., Ma, X.Y.: Synchronization between integer-order chaotic systems and a class of fractional-order chaotic system based on fuzzy sliding mode control. Nonlinear Dyn. 70(2), 1549–1561 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Wang, X.Y., Zhang, X.P., Ma, C.: Modified projective synchronization of fractional-order chaotic systems via active sliding mode control. Nonlinear Dyn. 69(1–2), 511–517 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  8. Raynaud, H.F., Zergainoh, A.: State-space representation for fractional order controllers. Automatica 36(7), 1017–1021 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chen, D.Y., Wu, C., Iu, H.H.C., Ma, X.Y.: Circuit simulation for synchronization of a fractional-order and integer-order chaotic system. Nonlinear Dyn. 73(3), 1671–1686 (2013)

    Article  MathSciNet  Google Scholar 

  10. Wang, X.Y., Wang, M.J.: Dynamic analysis of the fractional-order Liu system and its synchronization. Chaos 17(3), 033106-1–033106-6 (2007)

  11. Jonscher, A.K.: Dielectric Relaxation in Solids. Chelsea Dielectric Press, London (1983)

    Google Scholar 

  12. Westerlund, S., Ekstam, L.: Capacitor Theory. IEEE Trans. Dielectr. Electr. Insul. 1(5), 826–839 (1994)

    Article  Google Scholar 

  13. Jesus, I.S., Machado, J.A.T.: Development of fractional order capacitors based on electrolyte processes. Nonlinear Dyn. 56(1–2), 45–55 (2009)

    Article  MATH  Google Scholar 

  14. Westerlund, S.: Dead Matter has Memory! Causal Consulting. Kalmar, Sweden (2002)

    Google Scholar 

  15. Machado, J.A.T., Galhano, A.M.S.F.: Fractional order inductive phenomena based on the skin effect. Nonlinear Dyn. 68(1–2), 107–115 (2012)

    Article  Google Scholar 

  16. Petras, I.: A note on the fractional-order Chua’s system. Chaos Soliton Fract. 38(1), 140–147 (2008)

    Article  Google Scholar 

  17. Haba, T.C., Ablart, G., Camps, T., Olivie, F.: Influence of the electrical parameters on the input impedance of a fractal structure realised on silicon. Chaos Soliton Fract. 24(2), 479–490 (2005)

    Article  Google Scholar 

  18. Haba, T.C., Loum, G.L., Ablart, G.: An analytical expression for the input impedance of a fractal tree obtained by a microelectronical process and experimental measurements of its non-integral dimension. Chaos Soliton Fract. 33(2), 364–373 (2007)

    Article  Google Scholar 

  19. Angulo, F., Ocampo, C., Olivar, G., Ramos, R.: Nonlinear and nonsmooth dynamics in a DC–DC Buck converter: two experimental set-ups. Nonlinear Dyn. 46(3), 239–257 (2006)

    Article  MATH  Google Scholar 

  20. Plesko, H., Biela, J., Luomi, J., Kolar, J.W.: Novel concepts for integrating the electric drive and auxiliary DC–DC converter for hybrid vehicles. IEEE Trans. Power Electr. 23(6), 3025–3034 (2008)

    Article  Google Scholar 

  21. Lin, B.R., Dong, J.Y.: New zero-voltage switching DC–DC converter for renewable energy conversion systems. IET Power Electron. 5(4), 393–400 (2012)

    Article  MathSciNet  Google Scholar 

  22. Marchesoni, M., Vacca, C.: New DC–DC converter for energy storage system interfacing in fuel cell hybrid electric vehicles. IEEE Trans Power Electr. 22(1), 301–308 (2007)

    Article  Google Scholar 

  23. Radwan, A.G., Fouda, M.E.: Optimization of fractional-order RLC filters. Circ. Syst. Signal Process. 32(5), 2097–2118 (2013)

    Article  MathSciNet  Google Scholar 

  24. Dzielinski, A., Sierociuk, D.: Ultracapacitor modelling and control using discrete fractional order state-space model. Acta Montan. Slovaca 13(1), 136–145 (2008)

    Google Scholar 

  25. Chen, Y.Q., Luo, Y.: Discussion on: Simple fractional order model structures and their applications in control system design. Eur. J. Control 16(6), 695–696 (2010)

  26. Jalloul, A., Trigeassou, J.C., Jelassi, K., Melchior, P.: Fractional order modeling of rotor skin effect in induction machines. Nonlinear Dyn. 73(1–2), 801–813 (2013)

    Article  Google Scholar 

  27. Galvao, R.K.H., Hadjiloucas, S., Kienitz, K.H., Paiva, H.M., Afonso, R.J.M.: Fractional order modeling of large three-dimensional RC networks. IEEE Trans. Circuits Syst. I Regul. Pap. 60(3), 624–637 (2013)

    Article  MathSciNet  Google Scholar 

  28. Wang, F.Q., Ma, X.K.: Modeling and analysis of the fractional order buck converter in DCM operation by using Fractional Calculus and the circuit-averaging technique. J. Power Electron. 13(6), 1008–1015 (2013)

    Article  Google Scholar 

  29. Wang, F.Q., Ma, X.K.: Transfer function modeling and analysis of the open-loop Buck converter using the fractional calculus. Chin. Phys B 22(3), 030506-1–030506-8 (2013)

  30. Petras, I.: Fractional Order Nonlinear Systems-Modeling, Analysis and Simulation. Springer Publication, New York (2011)

    Book  MATH  Google Scholar 

  31. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  32. Recherches sur La Conductibilit Des Corps Cristallises. Annales de chimie et de physique 18, 203–269 (1889)

  33. Erickson, R.W.: Fundamentals of Power Electronics. Chapman and Hall, New York (1997)

    Book  Google Scholar 

  34. Charef, A., Sun, H.H., Tsao, Y.Y., Onaral, B.: Fractal system as represented by singularity function. IEEE Trans Autom. Contr. 37(9), 1465–1470 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  35. Oustaloup, A., Levron, F., Mathieu, B., Nanot, F.M.: Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans. Circuits I 47(1), 25–39 (2000)

    Article  Google Scholar 

  36. Chen, Y.Q., Moore, K.L.: Discretization schemes for fractional-order differentiators and integrators. IEEE Trans. Circuits Syst. I Regul. Pap. 49(3), 363–367 (2002)

    Article  MathSciNet  Google Scholar 

  37. Chen, Y.Q., Vinagre, B.M., Podlubny, I.: Continued fraction expansion approaches to discretizing fractional order derivatives—an expository review. Nonlinear Dyn. 38(1–4), 155–170 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  38. Aoun, M., Malti, R., Levron, F., Oustaloup, A.: Numerical simulations of fractional systems: an overview of existing methods and improvements. Nonlinear Dyn. 38(1–4), 117–131 (2004)

    Article  MATH  Google Scholar 

  39. Radwan, A.G., Salama, K.N.: Passive and active elements using fractional L beta C alpha circuit. IEEE Trans. Circuits Syst. I Regul. Pap. 58(10), 2388–2397 (2011)

    Article  MathSciNet  Google Scholar 

  40. Wang, F.Q., Ma, X.K.: Fractional order Buck–Boost converter in CCM: modelling, analysis and simulations. Int. J. Electron. 101(12), 1671–1682 (2014)

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Authors and Affiliations

  1. State Key Laboratory of Electrical Insulation and Power Equipment, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Chaojun Wu, Gangquan Si, Yanbin Zhang & Ningning Yang

  2. School of Electrical Engineering, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

    Chaojun Wu, Gangquan Si, Yanbin Zhang & Ningning Yang

Authors
  1. Chaojun Wu
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  2. Gangquan Si
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  3. Yanbin Zhang
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  4. Ningning Yang
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Corresponding author

Correspondence to Gangquan Si.

Additional information

This paper is supported by the National Science Foundation for Post-doctoral Scientists of China (Grant No.: 2013M530426).

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Wu, C., Si, G., Zhang, Y. et al. The fractional-order state-space averaging modeling of the Buck–Boost DC/DC converter in discontinuous conduction mode and the performance analysis. Nonlinear Dyn 79, 689–703 (2015). https://doi.org/10.1007/s11071-014-1695-4

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  • DOI: https://doi.org/10.1007/s11071-014-1695-4

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