Representation and estimation of the power coefficient in wind energy conversion systems

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José Genaro González-Hernández, Ph. D.
Rubén Salas-Cabrera, Ph. D.


This paper aims at summarizing various methods used for representing and estimating the power coefficient in wind turbines, such as exponential, sinusoidal and polynomial models, as well as mathematical tools known as state observers. We present an exhaustive bibliographic review of the models used to calculate the power coefficient, given that this type of studies are scarce nowadays. In addition, we propose models that can be satisfactorily used for various analyzes of wind energy conversion systems, such as the representation by a polynomial function of fourth degree and the models based on the stochastic probability function. The relevance of this work is supported by the advantages and disadvantages of the various models and estimators of the power coefficient, which are presented at the end of the article in a comparative table with the purpose of offering to the reader a general summary. Ultimately, this review aims at helping researchers, students, university professors and those who wish to venture into this field, even though they do not have much experience, to establish a quick synthesized understanding of the different models and representations of the power coefficient.


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[1] F. Manzano-Agugliaro, A. Alcayde, F. G. Montoya, A. Zapata-Sierra, and C. Gil, "Scientific production of renewable energies worldwide: An overview," Renewable and Sustainable Energy Reviews, vol. 18, pp. 134-143, Feb. 2013. DOI:

[2] H. H. Chen, and A. H. I. Lee, "Comprehensive overview of renewable energy development in Taiwan," Renewable and Sustainable Energy Reviews, vol. 37, pp. 215-228, Sep. 2014. DOI:

[3] L. Tripathi, A. K. Mishra, A. Kumar-Dubey, C. B. Tripathi, and P. Baredar, "Renewable energy: An overview on its contribution in current energy scenario of India," Renewable and Sustainable Energy Reviews, vol. 60, pp. 226-233, Jul. 2016. DOI:

[4] S. Yong-Duan, L. Peng, U. Wei, and Q. Ming, "An overview of renewable wind energy conversion system modeling and control," Measurament and Control, vol. 43 (7), pp. 203-208, 2010. DOI:

[5] A. K. Hussein, "Applications of nanotechnology in renewable energies—A comprehensive overview and understanding," Renewable and Sustainable Energy Reviews, vol. 42, pp. 460-476, Feb. 2015. DOI:

[6] R. Bedard, P. T. Jacobson, M. Previsic, W. Musial, and R. Varley, "An Overview of Ocean Renewable Energy Technologies," Oceanography, vol. 23 (2), pp. 22-31, 2010. DOI:

[7] M. Dyrholm, S. Sawyer, L. Fried, C. Gill, I. Prosser, S. Shukla, L. Quiao, and L. Livzeniece, "Global Wind Energy Council Report," GWEC, Abr 2018. Available:

[8] R. Vennell, "Exceeding the Betz limit with tidal turbines", Renewable Energy, vol. 55, pp. 277-285, Jul. 2013. DOI:

[9] S. P. Farthing, "Technical note: Betz limit, not an exact optimum," Wind Engineering, vol. 37 (1), pp. 105-110, Feb. 2013. DOI:

[10] S. Zekai, "Modified wind power formulation and its comparison with Betz limits," International Journal of Energy Research, vol. 37 (8), pp. 959-963, Jun. 2013. DOI:

[11] R.M. Ochieng, and R.O. Ochieng, "A power series formulation of the Betz's criterion and equation in a wind turbine," International Journal of Energy, Environment and Economics, vol. 23 (2), pp. 291-299, 2015.

[12] V. L. Okulov, and J. N. Sørensen, "Refined Betz limit for rotors with a finite number of blades," Wind Energy, vol. 11 (4), pp. 415-426, 2008. DOI:

[13] Y. Venkata, W. Bin, C. S. Paresh, K. Samir, and M. Mehdi, "High Power Wind Energy Conversion Systems: State-of-the-Art and Emerging Technologies," Proc. of IEEE, vol. 103 (5), pp. 740-778, May. 2015. DOI:

[14] R. Mittal, K. S. Sandhu, and D. K. Jain, "An Overview of Some Important Issues Related to Wind Energy Conversion Systems (WECS)," International Journal of Environmental Science and Development, vol. 1 (4), pp. 351-363, Oct. 2010. DOI:

[15] E. C. Millán, El sistema eólico, diseño aerodinámico. Madrid, ESP: Ed. EOI, 2007.

[16] F. D. Bianchi, H. de-Batista, and R. J. Mantz, Wind Turbine Control Systems Principles, Modeling and Gain Scheduling Design, Chapter 2, Luxemburgo, LUX: Springer, 2007.

[17] V. Reyes, J. J. Rodríguez, O. Carranza, and R. Ortega, "Review of mathematical of both the power coefficient and the torque coefficient in wind turbines," in IEEE 24th International Symposium on Industrial Electronics, Río de Janeiro, Brazil, 2015, pp. 1458-1463. DOI:

[18] A. Ahmad, D. Ahmed, and F. Karim, “Design and modeling of lowspeed axial flux permanent magnet generator for wind based microgeneration systems,” in International Conference on Robotics and Emerging Allied Technologies in Engineering (iCREATE), Islamabad, Pakistan, 2014, pp. 51-57.

[19] G. Bustos, F. Milla, D. Saez, L. S. Vargas, H. Zareipour, and A. Nuñez, “Comparison of fixed speed wind turbines models: a case study,” in 38th Annual Conference on IEEE (IECON), Montreal, QC, 2012, pp. 961-966. DOI:

[20] G. Boukettaya, O. Naifar, and A. Ouali, “A vector control of a cascaded doubly fed induction generator for a wind energy conversion system,” in 11th International Multi-Conference on Systems, Signals & Devices (SSD), Barcelona, Spain, 2014, pp. 1-7.

[21] X. Cai, G. Shi, Z. Wang, L. Yao, and M. Zhu, “Generalized average model of DC wind turbine with consideration of electromechanical transients,” in IECON 2013 - 39th Annual Conference of the IEEE, Vienna, 2013, pp. 1638-1643.

[22] D. Llano, R. McMahon, and M. Tatlow, “Control algorithms for permanent magnet generators evaluated on a wind turbine emulator test-ring,” in 7th IET International Conference on Power Electronics, Machines and Drives (PEMD), Mancester, UK, 2014, pp. 1-7. DOI:

[23] J. Chen, and D. Jiang, “Study on modeling and simulation of non-grid-connected wind turbine,” in WNWEC 2009 World Non-Grid-Connected Wind Power and Energy Conference, Nanjing, China, 2009, pp. 1-5. DOI:

[24] J. Chen, H. Wu, M. Sun, and W. Jiang, “Modeling and simulation of directly driven wind turbine with permanent magnet synchronous generator,” in IEEE Innovative Smart Grid Technologies - Asia (ISGT Asia), Tianjin, China, 2012, pp. 1-5.

[25] Q. Shi, G. Wang, L. Fu, L. Yuan, and H. Huang, “State-space averaging model of wind turbine with PMSG and its virtual inertia control,” in IECON 2013 - 39th Annual Conference of the IEEE Industrial Electronics Society, Vienna, Austria, 2013, pp. 1880-1886. DOI:

[26] Y. Ming, L. Gengyin, M. Zhou, and C. Zhao, “Modeling of the wind turbine with a permanent magnet synchronous generator for integration,” in IEEE Power Engineering Society General Meeting, Tampa, FL, 2007, pp. 1-6.

[27] S. K. Bagh, P. Samuel, R. Sharma, and S. Banerjee, “Emulation of static and dynamic characteristics of a wind turbine using Matlab/Simulink,” in 2nd International Conference on Power, Control and Embedded Systems (ICPCES), Allahabad, India, 2012, pp. 1-6. DOI:

[28] F. Gao, D. Xu, and Y. Lv, “Hybrid automaton modeling and global control of wind turbine generator,” in Proceedings of Seventh International Conference on Machine Learning and Cybernetics, Kunming, China, 2008, pp. 1991-1997.

[29] Y. Guo, S. H. Hosseini, C. Y. Tang, and J. N. Jiang, “An approximate model of wind turbine control systems for wind farm power control,” in 2011 IEEE Power and Energy Society, San Diego, CA, 2011, pp. 1-7. DOI:

[30] B. Hamane, M. L. Doumbia, M. Bouhamida, and M. Benghanem, “Control of wind turbine based on DFIG using Fuzzy-PI and sliding mode controllers,” in Ninth International Conference on Ecological Vehicles and Renewable Energies (EVER), Monte-Carlo, Monaco, 2014, pp. 1-8. DOI:

[31] Y. Guo, S. H. Hosseini, J. N. Jiang, C. Y. Tang, and R.G. Ramakumar, “Voltage/Pitch control for maximization and regulation of active/reactive powers in wind turbines with uncertainties,” in 49th IEEE Conference on Decision and Control (CDC), Atlanta, GA, 2010, pp. 3956-3963. DOI:

[32] A. B. Cultura, and Z. M. Salameh, “Modeling and simulation of a wind turbine-generator system,” in IEEE Power and Energy Society General Meeting, San Diego, CA, 2011, pp. 1-7. DOI:

[33] S. Duman, I. H. Altas, N. Yorukeren, and A. M. Sharaf, “A novel FACTS based on modulated power filter compensator for wind-grid energy systems,” in IEEE 5th International Symposium on Power Electronics for Distributed Generation Systems (PEDG), Galway, Ireland, 2014, pp. 1-7. DOI:

[34] P. Aree, and S. Lhaksup, “Dynamic simulation of self-excited Induction Generator feeding motor load using matlab/Simulink,” in 11th International Conference on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology (ECTICON), Nakhon Ratchasima, Tailand, 2014, pp. 1-6. DOI:

[35] Z. Jin, and X. Ma, “Semi-definite programming for power output control in a wind energy conversion system,” IEEE Transactions on sustainable energy, vol. 5 (2), pp. 466-475, Jan. 2014. DOI:

[36] R. Cao, L. Lu, Z. Xie, X. Zhang, and S. Yang, “A dynamic wind turbine simulator of the wind turbine generator system,” in International Conference on Intelligent System design and engineering application, Kuala Lumpur, Malaysia, 2012, pp. 967-970.

[37] R. II. Ovando, J. Aguayo, and M. Cotorogea, "Emulation of a low power wind turbine with a DC motor in Matlab/Simulink," in Power Electronics Specialists Conference, 2007, pp. 859-864.

[38] S. Khajuria, and J. Kaur, “Implementation of pitch control of wind turbine using Simulink (Matlab),” International Journal of Advanced Research in Computer Engineering & Technology, vol. 1 (4), pp. 196-200, 2012.

[39] S. Janakiraman, R. Kotti, and W. Shireen, “Adaptive sensorless maximum power point tracking control for PMSG wind energy conversion systems,” in Workshop on Control and modeling for Power Electronics (COMPEL), Santander, Spain, 2014, pp. 1-8.

[40] I. Nouira, and A. Khedher, “A contribution to the design and the installation of an universal platform of a wind emulator using a DC motor,” International Journal of renewable energy research, vol. 2 (4), pp. 797-804, 2012.

[41] E. Madjid, S. Mekhilef, and F. Merahi, “DC-Voltage regulation of a five levels neutral point clamped cascaded for wind energy conversion system,” in The 2014 International Power Electronics Conference, Hiroshima, Japan, 2014, pp. 560-566.

[42] Q. Bin, L. Pengcheng, Z. Wanli, and W. Xin, “Sliding mode control of pitch angle for direct driven PM wind turbine,” in 26th Chinese Control and Decision Conference (CCDC), Changsha, China, 2014, pp. 2447-2452. DOI:

[43] I. Moussa, A. Bouallegue, and A. Kehedher, “Design and Implementation of constant wind speed turbine emulator using Matlab/simulink and FPGA,” in Ninth International Conference Ecological Vehicles and Renewable Energies (EVER), Monaco, 2014, pp. 1-8. DOI:

[44] W. Li, H. Ma, D. Xu, and W. Zhang, “Research on Wind Turbine Emulation based on DC Motor,” in Second IEEE conference on industrial electronics and applications, Harbin, China, 2007, pp. 2589-2593. DOI:

[45] O. Carranza, E. Miranda, R. Ortega, and J. J. Rodríguez, “Emulador de un aerogenerador de baja potencia utilizando un Generador Síncrono de Imán Permanente,” in Reunión de Verano de Potencia y aplicaciones industriales, Acapulco México, 2014.

[46] M. D. Arifujjaman, M. T. Iqbal, and J. E. Quaicoe, "Maximum power extraction from a small wind turbine emulator using a DC-DC converter controlled by a microcontroller,” in 4th International Conference on Electrical and Computer Engineering ICECE, Dhaka, Bangladesh, 2006, pp. 213-216. DOI:

[47] M. L. Corradini, G. Ippoliti, and G. Orlando, "A robust observer for detection and estimation of icing in wind turbines," in IECON 2016 - 42nd Annual Conference of the IEEE Industrial Electronics Society, Florence, Italy, 2016, pp. 1894-1899. DOI:

[48] C. Caruana, A. Al-Durra, and F. Blaabjerg, "Observer-based scheme for tuning the control of variable speed wind turbines operating in hostile environments," IET Renewable Power Generation, vol. 10 (3), pp. 418-425, Feb. 2016. DOI:

[49] J. Sandoval-Moreno, G. Besançon, and J. J. Martinez, "Observer-based maximum power tracking in wind turbines with only generator speed measurement," in 2013 European Control Conference (ECC), Zurich, Switzerland, 2013, pp. 478-483. DOI:

[50] B. Yap, L. Dodson, and K. Busawon, "Online estimation of the power coefficient in wind energy conversion systems," in 2012 2nd International Symposium on Environment Friendly Energies and Applications, Newcastle, UK, 2012, pp. 458-463. DOI:

[51] A. J. Mahdi, W. H. Tang, and Q.H. Wu, "Estimation of tip speed ratio using an adaptive perturbation and observation method for wind turbine generator systems," in IET Conference on Renewable Power Generation (RPG 2011), Edinburgh, Scotland, 2011, pp. 1-6. DOI:

[52] K. Busawon, M. Jovanovic, and L. Dodson, "Power Coefficient Estimation in a Wind Energy Conversion System," in 12th International Power Electronics and Motion Control Conference, Portoroz, Slovenia, 2006, pp. 1873-1877.

[53] K. Busawon, L. Dodson, and M. Jovanovic, "Estimation of the power coefficient in a wind conversion system," in Proceedings of the 44th IEEE Conference on Decision and Control, Seville, Spain, 2005, pp. 3450-3455. DOI:

[54] R. Aubrée, F. Auger, and P. Dai, "A new low-cost sensorless MPPT algorithm for small wind turbines," in First International Conference on Renewable Energies and Vehicular Technology, Hammamet, Tunisa, 2012, pp. 305-311. DOI:

[55] A. Monroy, and L. Alvarez-Icaza, "Real-time identification of wind turbine rotor power coefficient," in 45th Conference on Decision & Control, San Diego, USA, 2006, pp. 3690-3695. DOI:

[56] G. T. Son, H. J. Lee, and J.W. Park, “Estimation of Wind Turbine Rotor Power Coefficient Using RMP Model,” in Proc. Industry Applications Society Annual Meeting, Houston, USA, 2009, pp. 1-8 DOI:

[57] P. F. Odgaard, R. Nielsen, and C. Damgaard, "On-line estimation of wind turbine power coefficients using unknown input observers," in IFAC Proceedings Volumes (IFAC-PapersOnline), 2008. DOI:

[58] A. B. Asghar, and X. Liu, "Estimation of wind turbine power coefficient by adaptive neuro-fuzzy methodology," Neurocomputing, vol. 238, pp. 227-233, May. 2017. DOI:

[59] A. Monroy, and L. Alvarez-Icaza, "Wind turbine power coefficient real-time identification," International Journal of Modelling, Identification and Control, vol. 6 (3), pp. 181-187, 2009. DOI:

[60] G. Hafidi, and J. Chauvin, "Wind speed estimation for wind turbine control," in Proceedings of the IEEE international conference on control applications, Dubrovnik, Croatia, Oct. 2012, pp. 1111–1117. DOI:

[61] H. Kala, and K. S. Sandhu, "Effect of change in power coefficient on the performance of wind turbines with different dimensions," in International Conference on Microelectronics, Computing and Communication, MicroCom, Durgapur, India, 2016, pp. 1-4. DOI:

[62] R. S. Pukale, K. U. Jadhav, and A. G. Thosar, "Data collection of variable wind speed to study the change in power and power coefficient," in International Conference on Energy Efficient Technologies for Sustainability ICEETS, Nagercoil, India, 2013. DOI:

[63] J. Dai, D. Liu, L. Wen, and X. Long, "Research on power coefficient of wind turbines based on SCADA data," Renewable Energy, vol. 86, pp. 206-215, Feb. 2016. DOI:

[64] I.M. De-la-Rosa-Alejandre, "Observador no lineal para la máquina de imanes permanentes," Master thesis, Madero Institute of Technology, Mexico, 2013.

[65] D. Jena, and S. Rajendran, "A review of estimation of effective wind speed based control of wind turbines," Renewable and Sustainable Energy Reviews, vol. 43, pp. 1046-1062. DOI:


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