Robust MPPT Observer-Based Control System for Wind Energy Conversion System With Uncertainties and Disturbance

The problem of tracking the maximum power point for the wind energy conversion system (WECS) is taken into consideration in this paper. The WECS in this article is simultaneously affected by the uncertainties and the arbitrary disturbance that cause the WECSs to be much more challenging to control. A new method to synthesize a polynomial disturbance observer for estimating the aerodynamic torque, wind speed, and electromagnetic torque without using sensors is proposed in this paper. Unlike the previous methods, in this work, both the uncertainties and the disturbance are estimated, then estimations of the uncertainties and disturbance are transmitted to the Linear Quadratic Regulator (LQR) controller for eliminating the influences of the uncertainties and disturbance; and tracking the optimal power point of WECS. It should be noted that the uncertainties in this work are time-varying and both uncertainties and disturbance do not need to satisfy the bounded constraints. The wind speed and aerodynamic torque are arbitrary and unnecessary to fulfill the low-varying constraint or $r^{th}$ time derivative bound. On the basis of Lyapunov methodology and the sum-of-square technique (SOS), the main theorems are derived to design the polynomial disturbance observer. Finally, the simulation results are provided to demonstrate the effectiveness and merit of the proposed method.


I. INTRODUCTION
Nowadays, air pollution and global warming caused by using fossil fuels are increasingly becoming a serious problem in the whole world that affects not only human health but also economic development. Due to this reason, discovering environmentally friendly energy resources to replace the fossil fuels is a pressing issue. Among new renewable energies developed in recent years, wind energy is considered as an efficient and free-pollution renewable resource. Recently, there is a fast-growing number of studies focusing on wind energy conversion system (WECS) [1]- [7]. For example, The associate editor coordinating the review of this manuscript and approving it for publication was R. K. Saket .
Wu et al. [1], the sliding mode control was applied for tracking the Maximum Power Point of the Low-Power WECS and improving the performance of the system. The new approach to synthesize the MPPT (maximum power point tracking) scheme based on dc-link voltage has been investigated for the small scale WECS with a fast-changing wind speed condition in paper [2]. Hodzic and Tai [3] have developed the novel purb and observe (P&O) algorithm to track the optimal power point and decrease the effects of harmonic distortion of the wind energy conversion system. Moreover, the MPPT algorithm for the offshore wind turbine system has been presented in [6] in which the active-rectifier d-axis current was employed to control the whole system. Although there are plenty of studies concentrating on WECS in the past few years, up to now, studying WECS is still a promising land for researchers.
Moreover, to design the controller for tracking optimal power point, the information of wind speed and other state variables of the WECS is necessary for sending back to the controller. Regarding obtaining the wind speed information, it can be measured directly by anemometers; however, using anemometers maybe obtain an inaccurate result and it also causes to increase the cost to build the system. In order to overcome the mentioned-above limitations, the wind speed can be estimated by employing the observer [9]- [16] instead of using the anemometers. For instance, the nonlinear disturbance observer and the sliding mode control were studied to estimate the aerodynamic torque, wind speed [9], and track the optimal power point and the high-order observer form was proposed to estimate the aerodynamic torque and wind speed [10]. The polynomial observer has been proposed by Vu and Do [11] to aerodynamic torque, electromagnetic torque, and stator current in axis-d in which aerodynamic torque did not need to satisfy any constraint. Moreover, a nonlinear observer was designed for WECS to estimate the rotor speed, position, and the turbine torque in paper [12] where the frequency of the grid was time-varying and unknown. The polynomial observer combining with fault-tolerant control was studied for the WECS to eliminate the influence of faults as well as tracking the maximum power point of WECS [13]. The sliding mode observer-based nonlinear controller has been developed by Hussain and Mishra [14] to estimate the wind speed, aerodynamic torque, and rotor speed of WECS. Although the observer in previous papers can estimate the aerodynamic torque and wind speed very well, there still exist several drawbacks. For example, the aerodynamic torque and wind speed must satisfy the constraint that the first-derivative of aerodynamic torque and wind speed is equal to zero [9]; or the aerodynamic torque and wind speed still need to fulfill the constraint that the r th time derivative of aerodynamic torque and wind speed must be bounded [10].
In practice, the systems are always impacted by the uncertainties and the disturbance [17]- [19]. The uncertainties may origin from the parameter error and/or the modeling error. Both the uncertainties and system disturbance will make the controller design more challenging and even cause to degrade the performance of the system. The WECS is not an exception, it is also strongly affected by the disturbance and the uncertainties. Recently, there are several studies paying attention to the control of the WECS with the existence of the uncertainties and/or disturbance [9], [20]- [23]. In paper [9], the time-varying uncertainty affected WECS; however, the uncertainty of this case was only impacted to the mechanical part of WECS and the system disturbance was not considered in this work. The WECS was modeled in terms of the uncertain T-S fuzzy model and then the robust observer-based controller was designed to estimate the wind speed [20]. Nonetheless, the uncertainties considered by Sung et al. [20] must satisfy the bounded constraints. The high-order observer was also synthesized for the WECS with the effects of the uncertainties to estimate the wind speed [21]. However, the uncertainties in the paper [21] were time-unvarying and this work omitted the effects of the system disturbance. The active disturbance rejection controller was synthesized for the WECS in papers [22] and [23], but these two papers only consider the existence of the disturbance and did not concern the influence of the uncertainties. With the above analyses, it is seen that the previous papers either deal with the uncertainties [9], [20], [21] or take into consideration the disturbance [22], [23]. To the best of our knowledge, up to now, there is not any paper considering the WECS system with the existence of both uncertainties and disturbance simultaneously.
In recent years, a new format to model the system called ''polynomial linear system'' was investigated in many papers [23]- [28]. The polynomial linear system, actually, is an extended format of the linear system. However, the difference between the polynomial linear system and the linear system is that the matrices of the polynomial linear system are varying matrices and are expressed in terms of polynomial form while the matrices of the linear system are constant. Employing the polynomial linear system to model the nonlinear system will decrease the number of linearization terms and reduce the modeling error as well. Recently, with the aid of the SOS tool of Matlab [29], solving the conditions expressed under polynomial format to design controller and observer becomes easy and efficient. Therefore, in this paper, the WECS will be represented under the framework of the polynomial linear system and then the polynomial observer is synthesized for this system.
On the basis of the above discussions, we are inspired to develop a new method for designing a robust disturbance observer-based controller to track the optimal power point of WECS that is simultaneously influenced by time-varying uncertainties and the disturbance. The contributions of this paper are presented as follows i) Unlike the previous papers that ignored the effects of the uncertainties and disturbance of WECS system [10]- [16], or merely considered the impacts of the uncertainties [9], [20], 21] or disturbance [18], [19], the WECS system in our paper is simultaneously influenced by both the uncertainties and disturbance. ii) The uncertainties in this paper are time-varying uncertainties that more relax than the constant uncertainties do in paper [21]. In addition, the uncertainties are unnecessary to fulfill the bounded constraints that are mandatory in the paper [20]. iii) The WECS is represented in terms of the polynomial linear system that will assist to decrease the modeling errors with respect to the method used T-S fuzzy model in [20]. iv) The wind speed and aerodynamic torque in this article are more relaxed with respect to the previous studies, because they do not need to fulfill the low-varying constraint as in paper [9], and r th time derivative of aerodynamic torque and wind speed must be bounded as in paper 10], [20]. v) The robust polynomial disturbance observer-based controller that is the combination of disturbance observer and LQR controller is synthesized to estimate aerodynamic torque, wind speed, electromagnetic torque to track the maximum power point. Moreover, both the uncertainties and the disturbance are also estimated and then this information is transmitted to the LQR controller to eliminate completely the impacts of the uncertainties and the system disturbance. The remains of this paper are structured as follows. The mathematical model of WECS and the problem description are presented in Section 2. Section 3 provides the method to design LQR controller. The polynomial disturbance observer synthesis for WECSs is shown in Section 4. The stability analysis of the closed-loop WECS system is mentioned in Section 5. Section 6 will provide the simulation. Finally, the conclusions are presented in Section 7.
Notations: T and −1 indicate the transpose and inverse of matrix , respectively.
> 0 ( < 0) means that is the positive(negative) definite matrix. I is an identity matrix.

II. SYSTEM DESCRIPTION AND PROBLEM STATEMENT A. MATHEMATICAL MODEL OF WECS
In practice, the power obtained by the wind turbine (WT) is described in the following equation: in which P a is the aerodynamics power, v indicates the wind speed; ρ and R denote the WT rotor radius and the air density, respectively. C p (λ, β) is for the power coefficient which is a nonlinear function dependent on the tip-speed ratio λ and the pitch angle β of the blades. The tip-speed ratio λ is calculated as follows where ω r is the rotor speed of the WT. Let us define T a to be the aerodynamic torque of WT, then substituting (1) into (2), one obtains From (1), it is obvious that the power P a is proportional to the power coefficient C p , therefore, the optimal power point is obtained when the C p is achieved its maximum value at an optimal tip-speed ratio λ opt . With a specific blade angle, this optimal ratio is constant. The maximum power is obtained by tracking the optimal reference of turbine rotor speed The turbine is connected to the generator via a gearbox with ratio n gb , thus the relations of speed and torque between two sides are determined as (5) in which ω is the mechanical angular speed of the generator, and T gs is the equivalent aerodynamic torque applied to the generator. The mathematical model of a permanent magnet synchronous generator (PMSG) is presented as where the parameters are explained in Table 1.
The electromagnetic torque can be calculated by, in which K = 3/2 ψ m P.
Combining equations (5), (6), and (7), the dynamic equations of a PMSG are expressed as: 96468 VOLUME 9, 2021 Assume that WECS system is impacted by the uncertainties and the system disturbance, then (8) is modified in the following for where d(t) is the disturbance, γ 1 and γ 2 are the coefficient of the disturbance; are the time-varying uncertainties that may come from the modeling error or parameter errors.
Remark 1: The uncertainties in this paper are time-varying, it means that it is a more general case with respect to the constant uncertainties in the paper [17].

B. PROBLEM STATEMENT
It is assumed that the WECS system is simultaneously influenced by the time-varying uncertainties and the system disturbance. It will cause difficulties to design a controller for WECS. In addition, the upper/lower bound values of the uncertainties do not know, hence the previous methods in papers [9]- [16], [20]- [23] are unable to apply to design the controller for this case. Additionally, the WECS in this work also assumes that 1. Both ω and i d are measured. 2. i q is unavailable, therefore T e is not known either. 3. Wind speed v and aerodynamic torque T a are unavailable.
The T a is arbitrary signals, thus, the approaches in papers [9], [10], [20] are failed to design a controller for WECS. Owing to these reasons, in this paper a polynomial observer is proposed to estimate electromagnetic torque T e , aerodynamic torque T a , the time-varying uncertainties, and the system disturbance d(t) without employing the sensors. The information of these signals will be transmitted to the LQR controller for tracking maximum power point purposes.

III. LINEAR QUADRATIC OPTIMAL CONTROL DESIGN
To track the maximum power point of WECS, we need to design the controller to track the optimal reference of turbine rotor speed.
Let us define thatω is the tracking error of rotor speed, T e is the tracking error of the electromagnetic torque of the generator, and u qc and u dc are the compensating terms of control input, then the system (9) can be rewritten as follows: wherẽ For the sake of simplification, we assign two slack terms which include both time-varying uncertainties and the disturbance in the following forms: and then where ω ref and T e,ref are the speed reference of the generator and the reference of electromagnetic torque. Remark 2: It is seen that two variables φ 1 (t) and φ 2 (t) consist of the information of both the time-varying uncertainties and disturbance, therefore, when the φ 1 (t) and φ 2 (t) are estimated, it means that the information of the uncertainties and disturbance are also estimated.
Denote (10) can be expressed under the state-space framework as follows,ẋ in which In this section, Linear Quadratic Regulator (LQR) controller is selected to control the system (12) in order to make the vector of state variables approach zero. It means that the tracking errors also converge towards zero. VOLUME 9, 2021 To design the LQR controller, firstly, let us consider the cost function as follows: (13) in which Z ∈ R 3×3 and R ∈ R 2×2 denote the positive-definite weigh-matrices. The matrices Z and R are typically selected as the following diagonal matrices: with where t si is considered as expected settling time of x i , τ is a control parameter, and x imax and u imax are upper bounds of |x i | and |u i |, respectively. The optimal controller form for system (12) is where K u is the optimal controller gain. The optimal LQR controller gain is obtained by the following steps: Step 1: Solving the following algebraic Riccati Equation to obtain the positive-definite matrix Step 2: Determine the optimal controller gain by following the formula

IV. OBSERVER SYNTHESIS
In this section, a polynomial disturbance observer will be synthesized to estimate the aerodynamic torque T a , wind speed, electromagnetic torque T e , the uncertainties, and the system disturbance d(t). However, before starting designing the polynomial disturbance observer, the system mathematical model of the WECS needs to be modified in the following steps.

A. MODIFIED WECS SYSTEM
The WECS system model (9) is able to be rewritten in the following form (19) Suppose that the Electromagnetic torque T e is not measured, thus, the output of the system (19) is represented by From (19) and (20), the mathematical model of WECS is rewritten in terms of the polynomial linear system γ 1 and γ 2 are the disturbance coefficient. Assumption 1: suppose that the uncertainty satisfies the matching condition A (t)x = H ϑ(t) where ϑ(t) is an arbitrary signal with an appropriate dimension.
Under Assumption 1, the system is modified as follows Let us define ϕ (t) = [ϑ (t) + d (t)] then we The system (23) can be modified as follows Let us define: then (24) becomes From (19)- (25), it can see that the system (21) has been modified to the polynomial system (25)

B. DISTURBANCE POLYNOMIAL OBSERVER SYNTHESIS
In this section, the modified model (25) is used to replace the system (21) to design the disturbance for estimating the unknown state T e , aerodynamic torque T a , uncertainties, and the disturbance d(t).
The estimation of T a and ϕ (t) are computed by the following formulasT It should be noted that the system (21) and the observer (26) consist of the polynomial matrices. Therefore, the LMI technique in paper [17]- [19] is unable to apply to find the observer gains. Because of this reason, in this work, the Sum-Of-Square (SOS) tool is used to design the polynomial disturbance observer (26). To help readers easily understand, the definitions of SOS are presented by the following two propositions.
Proposition 1 [30]: for a given arbitrary function h(x (t)), this function is called Sum-Of-Square if only if it can be decomposed to the form h (x (t)) = n i=1 [b i (x (t))] 2 , in which b i (x (t)) is expressed in the polynomial form in x (t). If the function h(x (t)) is an SOS, we can conclude that h(x (t)) ≥ 0, unfortunately, the converse is not guaranteed.
Proposition 2 [30]: Let us consider a polynomial symmetric matrix (x) ∈ R n×n in x and a vector v ∈ R n does not depend on x, it concludes that then (x) ≥ 0 for all x if only if v T (x)v is expressed under the SOS framework.
Taking the derivative both sides of Eq.(40) It is seen that if the condition (33) of Theorem 1 holds, then N T (ω) Q + QN (ω) < 0, hence, it infers thatV (e (t)) < 0 and the estimation error e (t) converges towards zero asymptotically.
Denote the estimation error of the disturbance Combining (34) and (44), we have If the conditions (30) and (31) of Theorem 1 fulfill, then (45) is written in the following form From Eq. (46), it is clear that if e (t) → 0 when t → ∞ then we can conclude that eφ (t) → 0.
The proof of Theorem 1 is completed. From the condition (33), it is seen that this is a nonlinear polynomial matrix inequality (BPMI) which is hard to solve by SOS Tool in MATLAB. Owing to this reason, the condition (33) must be transformed to the Polynomial Linear Matrix Inequality (PLMI) which is presented in the next steps.

Proof:
From the condition (29) of Theorem 1, it infers that From (27), we have Substituting (57) into (58) yields From (33), it implies that Substituting (59) into (60), one obtains then (61) becomes From (63), it is obvious that the PBMI (33) has been converted to the PLMI successfully by Theorem 2. The observer gains will be easily determined by resolving the PLMIs conditions of Theorem by SOS tool of Matlab. The proof is completed.
To help readers easily understand, the procedure to determine the observer gains of the disturbance observer (26) is briefly presented as follows.
Step 3: Based on the observer gains in Step 2, constructing the observer (26) to estimate thex(t), andφ(t) Step 4: The estimationT a andφ (t) (that consist of uncertainties and disturbance) are estimated by the following for-mulasT

V. OBSERVER-BASED CONTROL SCHEME AND CLOSED-LOOP STABILITY ANALYSIS A. OBSERVER-BASED CONTROLLER
With the estimated information of speed reference, aerodynamic torque, time-varying uncertainties, and the disturbance, the tracking error and compensating terms becomễ Under Assumption 1, the estimation of uncertainties and disturbanceφ 1 (t),φ 2 (t) are computed as followŝ The controller framework (16) is written as follows Then, from (31), the following equations are achieved, wherẽ e = e T aė T aë T a e T e e φ 1 e φ 2

B. STABILITY ANALYSIS
From (65) and (66), we can have where From (65) and (12), one obtainṡ It is seen that the system (68) is considered a closed-loop system with an observer-based controller.
Lemma 1 [30]: For a given system with the following form suppose that the systemẏ = s (y) has an asymptotically stable equilibrium at y = 0. Ifż = f (z, 0) has an asymptotically stable equilibrium at z = 0, then (69) has an asymptotically stable equilibrium at (z, y) = (0, 0). Theorem 3: Under Lemma 1, LQR controller design in Section 2 and observer design in Section 2, the tracking error x and estimation errorẽ with observer (26) and controller (65) converge towards zero asymptotically. Proof: If there exist the observer gains of the observer (26) that satisfy the conditions of Theorem 2 in Section 3 then disturbance observer (26) successfully estimates the state T e , aerodynamic torque T a , time-varying, and system disturbance. It means that the estimation errorẽ → 0 when t → ∞. Due to this reason, the system (68) is modifieḋ According to the LQR controller design procedure in Section 2, it is clear that system (70) is asymptotically stable equilibrium at zero. Therefore, under Lemma 1, it is concluded that the closed-loop control system (68) with disturbance observer-based controller is stable at the tracking error x(t) and estimation errorẽ(t) is asymptotically stable at zero.

VI. RESULTS AND DISCUSSION
To prove the effectiveness of the proposed method, in this section, the system WECS with parameters illustrated in Table 1. The structure of the WECS with a robust polynomial disturbance observer-based controller is described in Fig. 1. Suppose that the system is impacted by the time-varying uncertainties (71) and the random system disturbance is shown in Fig. 2. The waveform of the wind speed is shown in Fig. 3. Following the procedure for synthesizing the polynomial disturbance observer in Theorem 2, the observer gains are VOLUME 9, 2021  computed as follows.  Fig. 6 and Fig. 7, respectively. Fig. 4 shows that the mechanical angular speed ω can be tracked the signal reference very well and the tracking   error converges towards zero. In addition, the simulation results in Figs. [5][6][7] show that the estimation of aerodynamic torqueT a , the estimations of electromagnetic torqueT e , and estimation of stator currentsî d in d-axis are able to approach real states T a , T e , and i d , respectively; and the estimation  errors e Ta , e Te , and e id converge towards zero asymptotically as well. With these results, it is obvious that the proposed polynomial disturbance observer and the LQR controller still work efficiently to estimate T a , T e , and i d , of WECS system even with the effects of the time-varying uncertainties and disturbance. In addition, both the time-varying uncertainties and the disturbance are estimated by the observer and send back to the LQR controller to eliminate the impacts of the uncertainties and the disturbance; and track the optimal power point very well.
Discussion 2: It should be noted that the method in [13] was proposed for WECS with the effects of the faults, while in this paper, WECS system is affected both time-varying uncertainties and disturbance. Therefore, the method in [13] is impossible to apply for WECS in this work.

VII. CONCLUSION
A new approach to design a polynomial disturbance observerbased controller to track the maximum power point of WECS has been studied in this article. The system WECS is impacted VOLUME 9, 2021 by the time-varying uncertainties and the disturbance. The robust polynomial disturbance observer is designed to estimate the aerodynamic torque, electromagnetic torque, wind speed, uncertainties as well as disturbance. The estimation information of these parameters, uncertainties and the disturbance is transmitted to the LQR controller to eliminate the influences of both uncertainties and disturbance and track the maximum power point. The conditions for the polynomial disturbance observer expressed under the polynomial framework are derived in main theorems. Finally, the obtained simulation results have proved that the robust disturbance observer-based controller can operate efficiently and the proposed method of this paper is successful to control the WECS. VAN His research interests include the field of advanced control system theories, electric machine drives, renewable energy conversion systems, uninterruptible power supplies, electromagnetic actuator systems, targeted drug delivery systems, and nanorobots. He received the Best Research Award from Dongguk University, in 2014. He was the Lead Guest Editor for the Mathematical Problems in Engineering Special Issue on ''Advanced control methods for systems with fast-varying disturbances and applications.'' He is currently an Associate Editor of IEEE ACCESS. VOLUME 9, 2021