Alternative modes of operation for wind energy conversion systems and the generalised Lambert W-function

Alternative modes of operation for wind energy conversion systems and the generalised Lambert W-function

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When a wind energy conversion system (WECS) based on a doubly fed induction generator is operating in a different mode than maximum power tracking, there exist two different modes of operation. Here, it is shown that such modes satisfy the torque balance condition between the WECS and the electric network, which is described by a transcendental equation in terms of the desired angular velocities. The analytic solution to such equation is the newly found generalised Lambert W-function. Using a real WECS, the authors make an analysis of the lower and upper torque modes of operation. Finally, the authors propose an effective formula to estimate the upper solution which is universally applicable to this class of WECS.


    1. 1)
      • 1. Nikkhah, S., Rabiee, A.: ‘Optimal wind power generation investment, considering voltage stability of power systems’, Renew. Energy, 2018, 115, pp. 308325.
    2. 2)
      • 2. Al-Sharafi, A., Sahin, A.Z., Ayar, T., et al: ‘Techno-economic analysis and optimization of solar and wind energy systems for power generation and hydrogen production in Saudi Arabia’, Renew. Sust. Energy Rev., 2017, 69, pp. 3349.
    3. 3)
      • 3. Cardenas, R., Perez, M.A., Clare, J.C.: ‘Guest editorial control and grid integration of mw-range wind and solar energy conversion systems’, IEEE Trans. Ind. Electron., 2017, 64, (11), pp. 87868789.
    4. 4)
      • 4. Bricma, Z., Cepin, M.: ‘Estimating the additional operating reserve in power systems with installed renewable energy sources’, Int. J. Electr. Power Energy Syst., 2014, 63, pp. 654664.
    5. 5)
      • 5. Patel, M.R.: ‘Wind and solar power systems: design, analysis, and operation’ (CRC, Boca Raton, FL, USA, 2006).
    6. 6)
      • 6. Dadhania, A., Venkatesh, B., Nassif, A., et al: ‘Modeling of doubly fed induction generators for distribution system power flow analysis’, Int. J. Electr. Power Energy Syst., 2013, 53, pp. 576583.
    7. 7)
      • 7. Cardenas, R., Pena, R., Alepuz, S., et al: ‘Overview of control systems for the operation of DFIGS in wind energy applications’, IEEE Trans. Ind. Electron., 2013, 60, (7), pp. 27762798.
    8. 8)
      • 8. Bianchi, F., Battista, H.D., Mantz, R.: ‘Wind turbine control systems: principles, modelling and gain scheduling design’ (Springer Verlag, Berlin, Germany, 2007).
    9. 9)
      • 9. Bubshait, A.S., Mortezaei, A., Simões, M.G., et al: ‘Power quality enhancement for a grid connected wind turbine energy system’, IEEE Trans. Ind. Appl., 2017, 53, (3), pp. 24952505.
    10. 10)
      • 10. Jafarian, M., Ranjbar, A.: ‘The impact of wind farms with doubly fed induction generators on power system electromechanical oscillations’, Renew. Energy, 2013, 50, pp. 780785.
    11. 11)
      • 11. Mitra, A., Chatterjee, D.: ‘A sensitivity based approach to assess the impacts of integration of variable speed wind farms on the transient stability of power systems’, Renew. Energy, 2013, 60, pp. 662671.
    12. 12)
      • 12. Diaz, M., Cardenas, R., Espinoza, M., et al: ‘Control of wind energy conversion systems based on the modular multilevel matrix converter’, IEEE Trans. Ind. Electron., 2017, 64, (11), pp. 87998810.
    13. 13)
      • 13. Yang, B., Zhang, X., Yu, T., et al: ‘Grouped grey wolf optimizer for maximum power point tracking of doubly-fed induction generator based wind turbine’, Energy Convers. Manage., 2017, 133, pp. 427443.
    14. 14)
      • 14. Meng, W., Yang, Q., Sun, Y.: ‘Guaranteed performance control of DFIG variable-speed wind turbines’, IEEE Trans. Control Syst. Technol., 2016, 24, (6), pp. 22152223.
    15. 15)
      • 15. Liu, J., Meng, H., Hu, Y., et al: ‘A novel MPPT method for enhancing energy conversion efficiency taking power smoothing into account’, Energy Convers. Manage., 2015, 101, pp. 738748.
    16. 16)
      • 16. Bossoufi, B., Karim, M., Lagrioui, A., et al: ‘Observer backstepping control of DFIG-generators for wind turbines variable-speed: FPGA-based implementation’, Renew. Energy, 2015, 81, pp. 903917.
    17. 17)
      • 17. Taraft, S., Rekioua, D., Aouzellag, D., et al: ‘A proposed strategy for power optimization of a wind energy conversion system connected to the grid’, Energy Convers. Manage., 2015, 101, pp. 489502.
    18. 18)
      • 18. Tang, C., Guo, Y., Jiang, J.: ‘Nonlinear dual-mode control of variable-speed wind turbines with doubly fed induction generators’, IEEE Trans. Control Syst. Technol., 2011, 19, (4), pp. 744756.
    19. 19)
      • 19. Lopez-Garcia, I., Espinosa-Perez, G., Cardenas, V.: ‘Power control of a doubly-fed induction generator connected to the power grid’, Int. J. Control, 2017,, pp. 123,
    20. 20)
      • 20. Mezö, I., Baricz, Á.: ‘On the generalization of the Lambert W-function’, Trans. Am. Math. Soc., 2017, 369, (11), pp. 79177943.
    21. 21)
      • 21. Kaźmierkowski, M.P., Krishnan, R.: ‘Control in power electronics: selected problems’ (Academic Press, San Diego, CA, USA, 2002).
    22. 22)
      • 22. Bhatt, P., Roy, R., Ghoshal, S.: ‘Dynamic participation of doubly fed induction generator in automatic generation control’, Renew. Energy, 2011, 36, (4), pp. 12031213.
    23. 23)
      • 23. Krause, P., Wasynczuk, O., Sudhoff, S.D., et al: ‘Analysis of electric machinery and drive systems’, vol. 75 (John Wiley & Sons, Hoboken, NJ, USA, 2013).
    24. 24)
      • 24. Heier, S.: ‘Grid integration of wind energy: onshore and offshore conversion systems’ (John Wiley & Sons, Chichester, UK, 2014).
    25. 25)
      • 25. Kundur, P., Balu, N.J., Lauby, M.G.: ‘Power system stability and control’, vol. 7 (McGraw-Hill, New York, 1994).
    26. 26)
      • 26. Wasynczuk, O., Man, D., Sullivan, J.: ‘Dynamic behavior of a class of wind turbine generators during random wind fluctuations’, IEEE Trans. Power Appar. Syst., 1981, PAS-100, (6), pp. 28372845.
    27. 27)
      • 27. Slootweg, J., De Haan, S., Polinder, H., et al: ‘General model for representing variable speed wind turbines in power system dynamics simulations’, IEEE Trans. Power Syst., 2003, 18, (1), pp. 144151.
    28. 28)
      • 28. Şahin, A.D.: ‘Progress and recent trends in wind energy’, Prog. Energy Combust. Sci., 2004, 30, (5), pp. 501543.
    29. 29)
      • 29. Santos-Martin, D., Arnaltes, S., Amenedo, J.R.: ‘Reactive power capability of doubly fed asynchronous generators’, Electr. Power Syst. Res., 2008, 78, (11), pp. 18371840.
    30. 30)
      • 30. Lopez-Garcia, I.: ‘Control basado en pasividad de generadores de induccion con rotor devanado’. PhD thesis, Universidad Nacional Autonoma de Mexico. Programa de Posgrado en Ingenieria, 2012.
    31. 31)
      • 31. López-Garca, I., Espinosa-Pérez, G., Siguerdidjane, H., et al: ‘On the passivity-based power control of a doubly-fed induction machine’, Int. J. Electr. Power Energy Syst., 2013, 45, (1), pp. 303312.
    32. 32)
      • 32. Scott, T.C., Mann, R., Martinez Ii, R.E.: ‘General relativity and quantum mechanics: towards a generalization of the Lambert W function a generalization of the Lambert W function’, Appl. Algebra Eng. Commun. Comput., 2006, 17, (1), pp. 4147.
    33. 33)
      • 33. Scott, T.C., Fee, G., Grotendorst, J.: ‘Asymptotic series of generalized Lambert W function’, ACM Commun. Comput. Algebra, 2014, 47, (3/4), pp. 7583.
    34. 34)
      • 34. Mezö, I., Keady, G.: ‘Some physical applications of generalized Lambert functions’, Eur. J. Phys., 2016, 37, (6), p. 065802.
    35. 35)
      • 35. Maignan, A., Scott, T.C.: ‘Fleshing out the generalized Lambert W function’, ACM Commun. Comput. Algebra, 2016, 50, (1/2), pp. 4560.

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