Maximum power point tracking controller using Lyapunov theorem of wind turbine under varying wind conditions

ABSTRACT


INTRODUCTION
Renewable energy has become one of the most promising energy sources as a result of the advancements made in semiconductor technology and modern control techniques over the past few decades. Wind energy is one of the best renewable energy sources. As renewable energy becomes more prevalent, there is a growing interest in intelligently controlling wind turbines or wind plants to reduce the cost of wind energy. This can be accomplished by positioning the wind turbines to extract more wind energy, which is the focus of ongoing research.
In order to maintain the optimal blade tip speed ratio in order to achieve the maximum wind energy under both low and high wind speeds, a number of control strategies have been proposed over the past decade, the wind turbine Zhang et al. [1] proposed the fuzzy logic controller to control the wind wheel's rotation moment and the generator's reverse moment, Zhang et al. [2] Utilized the fuzzy logic controller to control the individual pitch angle of the turbine in order to guarantee a higher value for the power coefficient Cp, and thus the high aerodynamic torque. A new pitch controller based on the theory of generalized predictive control is proposed in Zhang et al. [3] to improve the quality of variable speed constant frequency power output in wind turbines. For maximum energy extraction from variable speed wind turbines, Calderaro  [4] proposed a data-driven design methodology able to generate a Takagi-Sugeno-Kang (TSK) fuzzy model, combined with genetic algorithms (GA) and recursive least-squares (LS) optimization methods for model parameter adaptation. Matthew and Saravanakumar [5] proposed a nonlinear controller, namely double integral sliding mode controller (DISMC), for the single mass model of a wind turbine at partial load region (below rated wind speed) to address the issue of optimal power extraction for variable-speed wind energy conversion systems (VSWECS) at partial load. Ullah et al. [6] proposed the linear active disturbance rejection control to control the output power and rotor speed of the wind turbine for variable pitch and variable speed wind turbine. Arya and Dewan [7] applied the H-infinity controller for speed control of variable speed wind turbine to solve the issue of the variation in rotor speed caused by the load charge. To achieve precise pitch control, the adaptive backstepping pitch angle control for wind turbines based on a servo-valve-controlled hydraulic motor was proposed in Yin et al. [8].
The proposed controller that is based on a Lyapunov theorem (the TSR_ LT controller) has as its goal the extraction of the maximum amount of power available from the wind. The proposed controller is primarily based on the definition of error speed, which is the difference between the optimal speed and the generator speed. This is done to ensure that the error will converge toward zero and that the system will be stable as a whole. It is essential to make certain that the Lyapunov energy function has a negative value by performing an action on the electromagnetic torque, which acts as a substitute for a command virtually.
The remaining sections are organized as: The modelling of the system is presented in section 2. The theory of turbine control and the proposed controller applied to our system were presented in section 3. The simulation results obtained and the discussion of these results were presented in section 4, and we concluded this work in section 5.

MODELING OF SYSTEM 2.1. Mathematical model for wind turbine
The wind power is converted into aerodynamic power and aerodynamic torque according to the Betz's law [9]- [11]. The aerodynamic power is given by (1).
where is the wind speed, air density, tip speed ratio area of the turbine blades in m 2 , wind turbine radius, and power coefficient.
where Ω is the turbine shaft speed. The turbine torque is the ratio of the aerodynamic power to the turbine shaft speed: The mechanical equation of the generator is given as (5) [15]: where is the electromagnetic torque, is the total moment of inertia and is the coefficient of viscous friction, Ω is the generator shaft speed, and is the generator torque. Where the shaft speed and torque of the generator are given by (6).
From (4), it is observed that the aerodynamic torque mainly depends on the value of the power coefficient and the wind speed So for each wind speed there is only one maximum torque and consequently only one maximum power point this point is configured by the and optimal ; this can be seen in the nonlinear torque-speed characteristic curve of a turbine shown in Figures 1 and 2. From Figure 1, we can see that for each wind speed, there is a maximum point. Two quantities define this ultimate point C_P^max and λ_opt, this can be seen that in the Figure 2, the objective of the control to reach these points is to extract the total amount of power available from the wind.

TURBINE CONTROL
There are numerous control techniques utilized to reach the maximum power point (MPPT). These control techniques include: tip speed ratio and optimal torque control. In this field, there are several literature searches done [10], [16]- [18].

Tip speed ratio control
The tip speed ratio (TSR) control method requires maintaining the TSR at an optimal value, to extract the maximum power in the wind speed [15], [19]- [21]. This method relies on the knowledge of wind speed and turbine is required, in order to keep the turbine operating in the maximum power point, Figure 3 represent the Tip speed ratio control method.

Optimal torque control
This control is based on the calculation of an optimal reference torque . According to the optimal and the , the only variable in the reference torque is the wind speed. The error between this torque and the generator torque is regulated via a regulator Figure 4. Several regulators are used in the literature [19], [22]- [26].

Proposed controller design
The speed error can be defined as (7): where the reference speed is given by (8): From (5) and (8), the dynamic speed error is defined as (9).
The proposed Lyapunov function is defined as (10).
The Lyapunov function's derivative can be calculated using (9): To ensure the system's stability, the proposed Lyapunov function is defined as positive; it is necessary to ensure the function's negativity by selecting the optimal virtual control [27].
In this case, the virtual control is electromagnetic torque: where is the constant positive gain. To justify the system's stability according to the Lyapunov theorem, the dynamic of the Lyapunov function must be negative. If we replaced the (12) into (11), we have: From (13), we can say that the system is stable. Figure 5 represents the proposed controller.

SIMULATION RESULTS
The proposed controller's simulation was carried out in the MATLAB Simulink environment. The proposed TSR LT controller's speed regulation performance will be compared to the TSR PI controller in this section. The goal of this regulation is to extract as much wind energy as possible in order to generate as much electricity as possible. In this regard, the wind turbine speed must be continuously adjusted in response to wind speed variations. The wind speed profile is represented in Figure 6 with 14% and 16% density. Table 1 lists the wind energy conversion systems (WECS's) parameters.   Figures 11 and 12, we compare the critical point and for the two winds speed profiles, 16% and 14%, in the two controllers, tip speed ratio-proportional integral (TSR_PI) and tip speed ratio-Lyapunov theorem (TSR_LT). From the results obtained, we notice that the TSR_LT controller gives good performance in maximum power point tracking, which justifies good maintenance and stability of its value of = 0.48 and = 8.1 for any wind speed.  Figure 11. WTGS performance based on 14% turbulence intensity wind speed Figure 12. WTGS performance based on 16% turbulence intensity wind speed

CONCLUSION
To operate the wind energy conversion system based on the Lyapunov theorem, we proposed a tip speed ratio control based on a permanent synchronous generator at the maximum power point and ensure the system's overall stability under all operating conditions. In comparison to a traditional PI controller, the TSR LT controller reached the maximum power point quickly and has good stability for a wide range of wind speed densities. Based on the simulation results and a comparison with the PI controller-based tip speed ratio control, we concluded that the Lyapunov theorem-based tip speed ratio control is very efficient, and we recommend using artificial intelligence in future work.

Brahim Brahmi
In 2019, he received a Ph.D. in Engineering from the École de Technologie Supérieure (ETS) in Montreal, Quebec, Canada. Nonlinear Control and Robotics are the topics of my thesis and specialization. He recently joined the Electrical and Computer Department at Miami University in the United States as an assistant professor, and he is also a member of the Electrical Department at College Ahuntsic in Canada. He worked as a postdoctoral research fellow in the Musculoskeletal Biomechanics Research Lab at McGill University's Mechanical Engineering Department from July 2019 to July 2020. He is a Control and Energy Management Lab member at the GREPCI-Lab, ETS, Montreal, QC, Canada, and an associate researcher at the Winsonsin-Milaukee University bio-robotics research lab in the United States. Nonlinear and adaptive control, bio-robotics, rehabilitation robots, fundamental motion control concepts for nonholonomic/underactuated vehicle systems, haptics systems, intelligent and autonomous control of unmanned systems, intelligent systems, and machine learning are among his research interests. He is a frequent referee and associate editor for a number of International Journals in Control and Robotics. He can be contacted at email: brahim.brahmi@collegeahuntsic.qc.ca.

Mohamed Horch
was born in AinTemouchent, Algeria, on January 22,1990. He received the M.Sc. and Ph.D degree in Electrical Engineering from the University of Tlemcen, Algeria in 2013 and 2018 respectively. He is a member of Automatique Laboratory of Tlemcen (LAT). He is working as an assistant professor in Higher National School of Electrical and Energetic Engineering of Oran, Algeria. His research interest on electrical machines drives, process control, power electronics and renewable energies systems. He can be contacted at email: mohamed.horch@mail.univ-tlemcen.dz and ResearchGate ID: https://www.researchgate.net/profile/Horch-Mohamed-2.

Mohamed Serraoui
was born in SAIDA (Algeria) on November 8, 1987. He obtained a diploma of Master degree in Electrical Engineering from University of SAIDA (Dr MolayTaher) in July 2011. He received the PhD degree from the Electrical Engineering Institute of The University of Sciences and Technology of Bechar Algeria in 2017. He is currently work in SADEG SONELGAZ SPA the National Society for electricity and gaz in Algeria as studies engineer. His fields of interest include renewable energies and the integration of artificial intelligence in photovoltaic systems. He can be contacted at email: serrmed@gmail.com and ResearchGate: https://www.researchgate.net/profile/Mohamed-Serraoui.