Applications of Methods of Numerical Linear Algebra in Engineering
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José de Jesús Rubio, Luis Arturo Soriano, Wen Yu, "Dynamic Model of a Wind Turbine for the Electric Energy Generation", Mathematical Problems in Engineering, vol. 2014, Article ID 409268, 8 pages, 2014. https://doi.org/10.1155/2014/409268
Dynamic Model of a Wind Turbine for the Electric Energy Generation
Abstract
A novel dynamic model is introduced for the modeling of the wind turbine behavior. The objective of the wind turbine is the electric energy generation. The analytic model has the characteristic that considers a rotatory tower. Experiments show the validity of the proposed method.
1. Introduction
Researchers are often trying to improve the total power of a wind turbine. The dynamic model of a wind turbine plays an important role in some applications as the control, classification, or prediction.
Some authors have proposed the equations to model the dynamic behavior of the wind turbine as shown in [1–5].
This paper presents a novel dynamic model of a wind turbine; the first dynamic model is for the wind turbine, the second is for the tower, and both are related. Because the rotatory tower can turn, it may help the wind turbine to increase the air intake. Some companies propose wind turbines that consider rotatory blades; however, in this study, a rotatory tower is proposed instead of the rotatory blades because the first dynamic model is easier to obtain than the second.
The proposed model is linear in the states as the works analyzed by [6–10] show. The technique provides an acceptable approximation of the wind turbine behavior.
The paper is structured as follows. In Section 2, the dynamic model of a windward wind turbine of three blades with a rotatory tower is introduced. In Section 3, the simulations of the dynamic model are compared with the real data obtained by a real wind turbine prototype. Finally, in Section 4, the conclusion and future research are detailed.
2. Dynamic Model of the Wind Turbine with a Rotatory Tower
This section is divided into four parts: the first is the description of the mechanic model, the second is the description of the aerodynamic model, the third is the description of the electric model, and, finally, the fourth is the combination of the aforementioned models to obtain the final dynamic model.
The dynamic model of the wind turbine is, first, the equations that represent the change between the wind energy and mechanic energy and, second, the equations that represent the change between the mechanic energy and electric energy.
2.1. The Mechanic Model
A windward wind turbine of three blades with a rotatory tower is considered. First, the Euler Lagrangian method [11, 12] is used to obtain the model that represents the change from the wind energy to mechanic energy for the wind turbine and the change from the electric energy to mechanic energy for the tower. The masses are concentrated at the center of mass. Consider the lateral view of Figure 1 and upper view of Figure 2.
From Figures 1 and 2, it can be seen that where , , , , is the angular position of the tower motor in rad, is the angular position of a wind turbine blade in rad, and is the length to the center of the wind turbine blade in . Consequently, the kinetic energy , and potential energy , are given as where is the tower mass in kg, is the blade mass in kg, is the acceleration gravity in m/s^{2}, is the constant length of the tower in m, and is the length to the center of the tower in m. The torques and are given as where is the torque of the generator moved by the blade in kg m^{2} rad/s^{2}, is the torque of the motor used to move the tower in kg m^{2} rad/s^{2}, and are the spring effects presented when the blade is near to a stop in kg m^{2}/s^{2}, and and are the shock absorbers in kg m^{2} rad/s. Using the Euler Lagrange method in [11, 12] gives the following equations: The angular position of the blade 1 is related to the angular position of the blades 2 and 3 as follows: where and are the angular positions for the blades 2 and 3, respectively. Then, using (4), it yields the equations for blades 2 and 3 as a function of as follows: where , , , and are the torques applied to move the blades 2 and 3, respectively. Adding , , and , respectively, it gives Now, adding the three equations of (4), (6), and (7) and using (8), (9), and (10), it gives where and . From Figures 1 and 2, of (11) is defined as follows: where and and are the force of the air received by the three blades. Equation (12) describes the assumption that the air goes in one direction; if , then the maximum air intake moves the blades of the wind turbine, but if the tower turns to the left or to the right and changes, then the wind turbine turns and the air intake decreases. From Figures 1 and 2 and [12], of (11) is defined as follows: where is a motor magnetic flux constant of the tower in Wb and is the motor armature current of the tower in A. Equations (11), (12), and (13) are the main equations of the mechanic model that represents the wind turbine and tower.
2.2. Aerodynamic Model
The aerodynamic model is the dynamic model of the torque applied to the blades. The mechanic power captured by the wind turbine is given by [2–5] where is the air density in Kg/m^{3}, is the area swept by the rotor blades in m^{2} with radius in m, is the wind speed in m/s, and is the performance coefficient of the wind turbine, whose value is a function of the tip speed ratio , defined as [2–5] For the purpose of simulation, the following model of is presented [2–5]: where and the coefficients are , , , , , , and is the blade pitch angle in rad. Using (12), (14), (15), (16), and (17) gives the dynamic model of the torque applied to the wind turbine blades as follows: where is defined in (12), is defined in (15), is defined in (16), and is defined in (17).
2.3. The Electric Model
Now, analyze the change from the mechanic to electric energy for the wind turbine generator and the electric energy to mechanic energy for the tower motor. Figure 3 shows the wind turbine generator and Figure 4 shows the tower motor.
In [12, 13], they presented the model of a motor, and similarly, from Figure 3, a model for the generator is obtained by using the Kirchhoff voltage law. Therefore, the dynamic models of the motor and generator are as follows: where is the motor back emf constant in V s/rad, is the generator back emf constant in V s/rad, is the motor armature resistance in , is the generator armature resistance in , is the motor armature inductance in H, is the generator armature inductance in H, is the motor armature voltage in V, is the generator armature voltage in V, is the motor armature current in A, and is the generator armature current in A. For the generator of this paper, . Thus, (19) becomes
2.4. The Final Dynamic Model
Thus, (11), (13), and (18) that represent the change between the wind energy and mechanic energy and (20) that represents the change between the mechanic energy and electric energy are considered as the wind turbine dynamic model with a rotatory tower.
Define the state variables as , , , , , , the inputs as and , and the output as . Consequently, the dynamic model of (11), (13), (15), (18), and (20) becomes where is defined in (16) and is defined in (17).
Remark 1. The model of this research considers a direct current motor and a direct current generator. This wind turbine is proposed to be used on the house roof for the daily usage to feed a light as a renewable source. If more than one wind turbine is used, more electricity is generated. The proposed model could be extended to other kinds of machines by changing (20).
Remark 2. Note that the dynamic model states of the wind turbine with a rotatory tower (21) is a linear system [6–10].
3. Experiments to Validate the Dynamic Model
Figure 5 shows the prototype of a wind turbine with a rotatory tower which is considered for the simulations of the dynamic model. This prototype has three blades with a rotatory tower which does not use a gear box. Table 1 shows the parameters of the prototype. The parameters and are obtained from the wind turbine blades. The parameters , , and are obtained from the tower motor. The parameters , , , and are obtained from the wind turbine generator. The parameters , , , and are obtained from [2–5].

The dynamic model of the wind turbine with a rotatory tower is given by (21) with the parameters of Table 1. are considered as the initial conditions for the plant states , , , , , and . The root mean square error (RMSE) is used, and it is given as [13–15] where , , or and , , and are the real data of , , and , respectively, , .
3.1. Example 1: The First Behavior
Figures 6 and 7 show the wind turbine inputs, outputs, and states. Table 2 shows the RMSE for the errors.

From Figures 6 and 7, it can be seen that the dynamic model has good behavior described as follows: (1) from s to s, both inputs are fed; consequently, the tower moves far of the maximum air intake, the generator current is decreased, and the wind turbine blades stop moving; (2) from s to s, both inputs are not fed; consequently, current is not generated and both the tower and wind turbine blades do not move; (3) from s to s, both inputs are fed, but the air intake is positive and tower voltage is negative; consequently, the tower returns to the maximum air intake, the generator current is increased, and the wind turbine blades move; (4) from s to s, both inputs are not fed; consequently, current is not generated and the tower and wind turbine blades do not move. The dynamic model is a good approximation of the wind turbine behavior because the signals of the first are near to the signals of the second. From Table 2, it is shown that the dynamic model is a good approximation of the real process because the RMSE is near to zero.
3.2. Example 2: The Second Behavior
Figures 8 and 9 show the wind turbine inputs, outputs, and states. Table 3 shows the RMSE for the errors.

From Figures 8 and 9, it can be seen that the dynamic model has good behavior described as follows: (1) from s to s, the input air is fed and the tower input is not fed; consequently, the tower remains in the maximum air intake, the generator current is maximum, and the wind turbine blades have motion; (2) from s to s, the air is not fed and the tower input is fed; consequently, current is not generated, the tower moves far of the maximum air intake, and the wind turbine blades do not have motion; (3) from s to s, the air is fed and the tower input is not fed; consequently, the tower does not move, the generator current is minimum, and the wind turbine blades almost do not move; (4) from s to s, the air is not fed and the tower input is fed with a negative voltage; consequently, current is not generated, the tower returns to the maximum air intake, and the wind turbine blades do not have motion. The dynamic model is a good approximation of the wind turbine behavior because the signals of the first are near to the signals of the second. From Table 2, it is shown that the dynamic model is a good approximation of the real process because the RMSE is near to zero.
4. Conclusion
In this paper, a dynamic model of a wind turbine with a rotatory tower was introduced; the simulations using the parameters of a prototype showed that the proposed dynamic model has an acceptable approximation of the wind turbine behavior. The proposed dynamic model could be used on control, prediction, or classification. As a future research, the modeling will be improved using other interesting methods as the least squares in [16–26], neural networks in [14, 27–32], or fuzzy systems in [15, 33–35].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are grateful to the editor and the reviewers for their valuable comments and insightful suggestions, which helped to improve this paper significantly. The authors thank the Secretaría de Investigación y Posgrado, Comisión de Operación y Fomento de Actividades Académicas del IPN, and Consejo Nacional de Ciencia y Tecnología for their help in this paper.
References
 L.L. Fan and Y.D. Song, “Neuroadaptive modelreference faulttolerant control with application to wind turbines,” IET Control Theory & Applications, vol. 6, no. 4, pp. 475–486, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 A. Žertek, G. Verbič, and M. Pantoš, “Optimised control approach for frequencycontrol contribution of variable speed wind turbines,” IET Renewable Power Generation, vol. 6, no. 1, pp. 17–23, 2012. View at: Publisher Site  Google Scholar
 E. B. Muhando, T. Senjyu, A. Yona, H. Kinjo, and T. Funabashi, “Disturbance rejection by dual pitch control and selftuning regulator for wind turbine generator parametric uncertainty compensation,” IET Control Theory and Applications, vol. 1, no. 5, pp. 1431–1440, 2007. View at: Publisher Site  Google Scholar
 C. Y. Tang, Y. Guo, and J. N. Jiang, “Nonlinear dualmode control of variablespeed wind turbines with doubly fed induction generators,” IEEE Transactions on Control Systems Technology, vol. 19, no. 4, pp. 744–756, 2011. View at: Publisher Site  Google Scholar
 R. Vepa, “Nonlinear, optimal control of a wind turbine generator,” IEEE Transactions on Energy Conversion, vol. 26, no. 2, pp. 468–478, 2011. View at: Publisher Site  Google Scholar
 M. Dehghan and M. Hajarian, “Modied AOR iterative methods to solve linear systems,” Journal of Vibration and Control, 2012. View at: Publisher Site  Google Scholar
 M. Hajarian, “Matrix form of the BiCGSTAB method for solving the coupled Sylvester matrix equations,” IET Control Theory & Applications, vol. 7, no. 14, pp. 1828–1833, 2013. View at: Publisher Site  Google Scholar
 M. Hajarian, “Effcient iterative solutions to general coupled matrix equations,” International Journal of Automation and Computing, vol. 10, no. 5, pp. 481–486, 2013. View at: Publisher Site  Google Scholar
 M. Hajarian, “Matrix iterative methods for solving the Sylvestertranspose and periodic Sylvester matrix equations,” Journal of the Franklin Institute, vol. 350, no. 10, pp. 3328–3341, 2013. View at: Publisher Site  Google Scholar  MathSciNet
 J. J. Rubio, G. Ordaz, M. JiménezLizárraga, and R. I. Cabrera, “General solution of the NavierStokes equation to describe the dynamics of a homogeneous viscous fluid in an open tube,” Revista Mexicana de Física, vol. 59, no. 3, pp. 217–223, 2013. View at: Google Scholar  MathSciNet
 F. L. Lewis, D. M. Dawson, and C. T. Abdallah, Control of Robot Manipulators, Theory and Practice, 2004.
 M. W. Spong, S. Hutchinson, and M. Vidyasagar, Robot Modeling and Control, John Wiley & Sons, New York, NY, USA, 2006.
 J. J. Rubio, J. Pacheco, J. H. PérezCruz, and F. Torres, “Mathematical model with sensor and actuator for a transelevator,” Neural Computing and Applications, vol. 24, no. 2, pp. 277–285, 2014. View at: Publisher Site  Google Scholar
 J. J. Rubio and J. H. PerezCruz, “Evolving intelligent system for the modelling of nonlinear systemswith deadzone input,” Applied Soft Computing B, vol. 14, pp. 289–304, 2014. View at: Publisher Site  Google Scholar
 J. J. Rubio, “Evolving intelligent algorithms for the modelling of brain and eye signals,” Applied Soft Computing B, vol. 14, pp. 259–268, 2014. View at: Publisher Site  Google Scholar
 S. Ding, R. Ding, and E. Yang, “A filtering based recursive least squares estimation algorithm for pseudolinear autoregressive systems,” Journal of the Franklin Institute, 2013. View at: Publisher Site  Google Scholar
 F. Ding, “Combined state and least squares parameter estimation algorithms for dynamic systems,” Applied Mathematical Modelling, vol. 38, no. 1, pp. 403–412, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 F. Ding, “Coupledleastsquares identification for multivariable systems,” IET Control Theory & Applications, vol. 7, no. 1, pp. 68–79, 2013. View at: Publisher Site  Google Scholar  MathSciNet
 F. Ding, “Decomposition based fast least squares algorithm for output error systems,” Signal Processing, vol. 93, no. 5, pp. 1235–1242, 2013. View at: Publisher Site  Google Scholar
 F. Ding, “Twostage least squares based iterative estimation algorithm for CARARMA system modeling,” Applied Mathematical Modelling, vol. 37, no. 7, pp. 4798–4808, 2013. View at: Publisher Site  Google Scholar  MathSciNet
 L. Han, F. Wu, J. Sheng, and F. Ding, “Two recursive least squares parameter estimation algorithms for multirate multipleinput systems by using the auxiliary model,” Mathematics and Computers in Simulation, vol. 82, no. 5, pp. 777–789, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 H. Hu and F. Ding, “An iterative least squares estimation algorithm for controlled moving average systems based on matrix decomposition,” Applied Mathematics Letters, vol. 25, no. 12, pp. 2332–2338, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 X. L. Li, L. C. Zhou, R. F. Ding, and J. Sheng, “Recursive leastsquares estimation for Hammerstein nonlinear systems with nonuniform sampling,” Mathematical Problems in Engineering, vol. 2013, Article ID 240929, 8 pages, 2013. View at: Publisher Site  Google Scholar
 S. Pan and J.S. Chen, “A leastsquare semismooth Newton method for the secondorder cone complementarity problem,” Optimization Methods & Software, vol. 26, no. 1, pp. 1–22, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 D. Q. Wang, F. Ding, and Y. Y. Chu, “Data filtering based recursive least squares algorithm for Hammerstein systems using the keyterm separation principle,” Information Sciences, vol. 222, pp. 203–212, 2013. View at: Publisher Site  Google Scholar  MathSciNet
 D. Wang, F. Ding, and L. Ximei, “Least squares algorithm for an input nonlinear system with a dynamic subspace state space model,” Nonlinear Dynamics, vol. 75, no. 12, pp. 49–61, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 J. Chen and R. F. Ding, “An auxiliarymodelbased stochastic gradient algorithm for dualrate sampleddata BoxJenkins systems,” Circuits, Systems, and Signal Processing, vol. 32, no. 5, pp. 2475–2485, 2013. View at: Publisher Site  Google Scholar  MathSciNet
 J. Chen, “Several gradient parameter estimation algorithms for dualrate sampled systems,” Journal of the Franklin Institute, vol. 351, no. 1, pp. 543–554, 2014. View at: Publisher Site  Google Scholar
 F. Ding, X. Liu, H. Chen, and G. Yao, “Hierarchical gradient based and hierarchical least squares based iterative parameter identification for CARARMA systems,” Signal Processing, vol. 97, pp. 31–39, 2014. View at: Publisher Site  Google Scholar
 M. Pratama, S. G. Anavatti, P. P. Angelov, and E. Lughofer, “PANFIS: a novel incremental learning machine,” IEEE Transactions on Neural Networks and Learning Systems, vol. 25, no. 1, pp. 55–68, 2014. View at: Publisher Site  Google Scholar
 J. H. PérezCruz, J. J. Rubio, J. Pacheco, and E. Soriano, “State estimation in MIMO non linear systems subject to unknown deadzones using recurrent neural networks,” Neural Computing and Applications, 2013. View at: Publisher Site  Google Scholar
 J. Sun, J.S. Chen, and C.H. Ko, “Neural networks for solving secondorder cone constrained variational inequality problem,” Computational Optimization and Applications, vol. 51, no. 2, pp. 623–648, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 A. Buchachia, “Dynamic clustering,” Evolving Systems, vol. 3, no. 3, pp. 133–134, 2012. View at: Publisher Site  Google Scholar
 E. Lughofer, “Sigle pass active learning with conflict and ignorance,” Evolving Systems, vol. 3, no. 4, pp. 251–271, 2012. View at: Publisher Site  Google Scholar
 L. Maciel, A. Lemos, F. Gomide, and R. Ballini, “Evolving fuzzy systems for pricing fixed income options,” Evolving Systems, vol. 3, no. 1, pp. 5–18, 2012. View at: Publisher Site  Google Scholar
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Copyright © 2014 José de Jesús Rubio et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.