Vibration Performance, Stability and Energy Transfer of Wind Turbine Tower Via Pd Controller

1 Department of Mathematics and Statistics, Faculty of Science, Taif University, Taif, PO Box 888, Saudi Arabia. 2 Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf, PO Box 32952, Egypt. 3 Department of Mechanical Engineering, College of Engineering, Taif University, Taif, PO Box 888, Saudi Arabia. 4 Department of Mechanical Engineering, Faculty of Engineering, Assiut University, Assiut, PO Box 71516, Egypt. 5 Department of Electrical Engineering, College of Engineering, Taif University, Taif, PO Box 888, Saudi Arabia. * Corresponding Author: Y. S. Hamed. Email: eng_yaser_salah@yahoo.com. Received: 29 July 2019; Accepted: 11 August 2019. Abstract: In this paper, we studied the vibration performance, energy transfer and stability of the offshore wind turbine tower system under mixed excitations. The method of multiple scales is utilized to calculate the approximate solutions of wind turbine system. The proportional-derivative controller was applied for reducing the oscillations of the controlled system. Adding the controller to single degree of freedom system equation is responsible for energy transfers in offshore wind turbine tower system. The steady state solution of stability at worst resonance cases is studied and examined. The offshore wind turbine system behavior was studied numerically at its different parameters values. Furthermore, the response and numerical results were obtained and discussed. The stability is also analyzed using technique of phase plane and equations of frequency response. In addition, the numerical results and comparison between analytical and numerical solutions were obtained with MAPLE and MATLAB algorithms.

out for the effect of different parameters on the offshore wind turbine system behavior and responses of soil monopile tower system [Bisoi and Haldar (2014)]. The effective control approach based on an active observer of a standard model of large rotating wind turbines is studied [Shi and Patton (2015)]. Passive control technique is investigated for vibrations of spar and offshore wind turbine nacelle using tuned mass dampers [Nguyen Dinh and Basu (2015)]. The active control strategy effects is studied for a barge floating wind turbine type using a hybrid mass damper [Hu and He (2017)]. Mathematical analysis is studied for the dynamics of wind turbines with control and time scale simulations [Eisa, Stone and Wedeward (2018)]. An active control study to suppress the structural vibrations of wind turbines has been proposed [Fitzgerald, Sarkar and Staino (2018)]. The effect of seismic loads and environmental forces on the behavior of the offshore wind tower is performed with two different approaches [Dagli, Tuskan and Gokkus (2018)]. An analytical solutions, chaotic dynamics and stability of some systems under multi excitation forces such as a simply rectangular plate, MEMS gyroscope and a Cartesian manipulator systems are obtained and studied by Hamed et al. [Hamed (2014); Hamed, EL-Sayed and El-Zahar (2016) ;Hamed, Alharthi and AlKhathami (2018)]. The analysis detailed of some dynamical systems with different forces is founded in the books [Cartmell (1990) ;Nayfeh, and Balachandran (1995)]. In the present work, the PD controller was applied for reducing the oscillations of the controlled system and transfer the energy to the offshore wind turbine tower system. The stability at worst resonance cases is examined. Also, the offshore wind turbine system behavior was studied numerically at its different parameters values. In addition, the numerical results and comparison between analytical and numerical solutions were obtained.

Description of structure and governing equation of motion
The structural modal of wind turbine consists of hub, tower, blade and concentrated mass. The hub height of the wind turbine is 65 m with diameter 6 m, the blade length is 24 m and the tower carried the weight of the hub, nacelle, and the rotor blades which is 83,000 kg. The structural modal subjected to some external forces such as wind and wave forces ( , a H F F ) and earthquake force ( eqk F ). The offshore wind turbine tower model is shown in Fig. 1. The equation of motion of the single degree of freedom system is obtained from Dagli et al. [Dagli, Tuskan and Gokkus (2018)] and described by the following equations: The initial conditions of Eq. (1) are where     px dx is the PD controller.

Perturbation analysis
Multi-scale disturbance technique (MSPT) [Nayfeh (1985); Nayfeh and Mook (1995)] is performed to obtain approximate solutions for Eq. (2). Assuming that the solution is in the form: We presented the derivatives in the form: The time scales (3)-(4) into Eq.
(2) and equating the coefficients of ε leads to: The general solution of Eq. (5) has the form: the following are obtained: For the abounded solutions of Eq. (9), the coefficients of secular terms  0 exp( ) i T  must be removed and the general solution of Eq. (9) should be in form: (8) and (10) into Eq.
(7) and removing the secular terms from Eq. (7), the solution to this equation is as follows: the approximate solutions obtained, we extracted all the resonances and reported them as follows: (a) The primary resonance:

Stability analysis of the steady state solution
To stability examination, the analysis is studied to the first approximation. The solution depends only on 1 0 , T T and the stability is analyzed and studied for the solution at the primary resonance    . The detuning parameter  is presented as: From Eq. (8) the secular terms are removed and the solvability conditions for the first approximation are presented as: Substituting Eqs. (12) into (13) and eliminating the secular terms leads to solvability conditions for the first and second-order expansions as: Let's introduce the polar form as: where a and  are the amplitude and phase of the motion at the steady state. Using Eqs. (15) into (14) to obtain the imaginary and real parts, the following equations are obtained as follows: where       (16) and (17) as follows: By squaring both sides of Eqs. (18) and (19) and adding the results, we obtained the frequency response equation in the form:

Stability of nonlinear solution
To examine the stability of the nonlinear solutions, we takes For the steady state the Eqs. (22) and (13) become: The systems (24) and (25) can be expressed in a matrix form as follows Using the criterion of Routh-Hurwitz, the necessary and sufficient conditions for the system to be stable that the real parts of all roots of Eq. (28) are negative.

Results and discussion
The Runge-Kutta algorithm of fourth order is used to find the analytic results numerically for the equation of motion (2). Also, we examined the stability of the controlled system using the frequency response function and the effects of some different parameters on the behavior of the controlled system are also studied. Finally, we compared the analytical results with the numerical ones.

System behavior without control
The System behavior is studied numerically at the obtaining resonance cases from Eqs. (10) and (11). The Eq. (2) is integrated numerically at the system parameters: 9.73, 2.828, 67.17, 5.129, 15, 0.5 a H F F p d Fig. 1, indicate the phase plane and time histories of uncontrolled offshore wind turbine tower system at primary condition    . From this figure, we find that, the behavior of the uncontrolled system is nearly about 75% of the wind force a F , 975% of the wave force H F and phase plane showing multi limit cycle.  Fig. 2 represent time histories for the offshore wind turbine tower system after applying the proportional-derivative (PD) controller at resonance case   , . In this figure, the amplitude for the controlled system is nearly 6% of the wind force a F , 78% of the wave force H F . Then, the efficiency of the controller a E (uncontrolled system amplitude /controlled system amplitude) is about 12.5. Figs. 3. and 4. show the transfer of energy between uncontrolled and controlled modes for the offshore wind turbine tower system due to values of wind, wave forces and super harmonic resonance case

System behavior with control
. From these figures, we observed that energy is transferred from the uncontrolled system to the controlled system due to apply the PD controller with different values of wind and wave forces a F , H F and natural, excitation frequencies   , compared with its values in Fig. 1, so we can used these parameters to control the oscillation amplitude of the controlled system.

Response curves of the controlled system
In the section, we studied the different parameters effect and stability zone of the controlled system using frequency response curves. Also, using the numerical methods, the stability of nontrivial solutions is investigated for Eq. (20). In Fig. 5(a), the detuning parameter 1  effects on the behavior of the controlled system is shown. Figs. 5(b)-5(d).
show that the behavior of the controlled system is a monotonic decreasing functions in the damping coefficient  , the control parameter d and the natural frequency  , also the curves of the system is shifted to right with increasing the values of the control parameter p as shown in Fig. 5(e). The behavior of the controlled system is a monotonic increasing function in the wind amplitude force a F as shown in Fig. 5(f).

Comparison of analytical and numerical simulation
In this subsection, the comparison of numerical simulation for the controlled system of Eq.

Comparison of numerical solution and response curve
Figs. 8-9 indicate a comparison of the system response with applying the PD controller at the parameters values used for stability at primary resonance    . In these figures, the behavior of the controlled system is about 4 and 10 which is in a well agreement with the controlled system amplitude at 1 0   .

Conclusions
The behavior of the offshore wind turbine system with mixed excitations and PD controller is investigated. The approximate solutions, stability analysis and numerical integration are studied for the system behavior. The different parameters effect and comparison of analytical with numerical solutions are studied numerically. From this study, we included the following: 1. The behavior of the uncontrolled system is nearly about 75% of the wind force a F , 975% of the wave force H F and the phase plane showing multi-limit cycle.
2. The amplitude for the controlled system is nearly 6% of the wind force a F , 78% of the wave force H F and the efficiency of the controller a E is about 12.5.
3. The energy is transferred from the uncontrolled system to the controlled system due to apply the PD controller with different values of wind and wave forces a F , H F and natural, excitation frequencies   , .
4. The behavior of the controlled system is a monotonic decreasing functions in the damping coefficient  , the control parameter d and the natural frequency  .
5. The curves of the controlled system have a right shift to with increasing values of the control parameter p .
6. The behavior of the controlled system is a monotonic increasing function in the wind amplitude force a F . 7. The analytical results are well agreement with numerical simulation