# Robust Active Disturbance Rejection Control Approach to Maximize Energy Capture in Variable-Speed Wind Turbines

Authors: Horacio Coral-Enriquez; John Cortés-Romero; Germán A. Ramos
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper proposes an alternative robust observer-based linear control technique to maximize energy capture in a 4.8 MW horizontal-axis variable-speed wind turbine. The proposed strategy uses a generalized proportional integral (GPI) observer to reconstruct the aerodynamic torque in order to obtain a generator speed optimal trajectory. Then, a robust GPI observer-based controller supported by an active disturbance rejection (ADR) approach allows asymptotic tracking of the generator speed optimal trajectory. The proposed methodology controls the power coefficient, via the generator angular speed, towards an optimum point at which power coefficient is maximum. Several simulations (including an actuator fault) are performed on a 4.8 MW wind turbine benchmark model in order to validate the proposed control strategy and to compare it to a classical controller. Simulation and validation results show that the proposed control strategy is effective in terms of power capture and robustness.


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## Body

## 1. Introduction

The use of wind energy has a history of over a hundred years. Its applications include agriculture, milling, water pumping, and power production. In the 1970s, this technology started developing as an experimental technology. Nowadays, the conversion of wind energy into electrical energy by wind turbines is a mature technology that exhibits the highest growth rates among the renewable energy sources [1] and can be considered as the most promising option for replacing a significant part of the electricity produced by conventional sources [2].The main objective of wind turbines is to convert efficiently wind energy into electrical power. There are wind turbines available in a different number of configurations (vertical axis, horizontal axis, fixed speed, variable speed, etc.). The most used type for large-scale power production is the variable-speed horizontal-axis wind turbine (HAWT) with a two- or three-blade rotor [3]. HAWTs are commonly equipped with blade-pitch actuators, generator torque control and many sensors for use in real time control [4]. HAWTs operate in different control regions, and the region considered in the present work is a low-to-medium wind speeds operation (also called partial load operation or operation in region 2) where the main objective is to extract the maximum power from the wind.Modern wind turbines are machines that require big efforts when maximizing wind energy capture, not only because of their highly nonlinear aerodynamics, but also because of the high efficiency required even when model uncertainties, external perturbations, or system faults are present. As a consequence, the efficiency of both power capture and power generation is strongly dependent on the selected control method [5]. This situation provides a motivation to consider new alternative control techniques that improve the performance of HAWT without any structural change.A large number of control schemes to find the best way of solving the energy capture maximization problem for wind turbines at low-to-medium wind speeds have been proposed (see, e.g., [6–12]). The control techniques range from standard torque control [6], disturbance tracking control [7], maximum power point tracking [8], and aerodynamic torque feedforward [9] to complex nonlinear strategies [5, 10–12]. Most of these techniques deal with the wind turbine complexity using linearization techniques or nonlinear control. Following a different approach, some of the active-disturbance-rejection- (ADR-) based techniques allow linear control solutions for some class of uncertain complex nonlinear systems and could offer a linear, simpler, and robust solution to the wind energy capture maximization problem. This is the case of the ADR philosophy-based technique called generalized proportional integral (GPI) Control [13] and its GPI observer-based control extensions [14, 15].Generalized proportional integral (GPI) control technique was started in 2000 by Fliess et al. [13, 16] and involves in its design the active rejection of disturbances. The dual counterpart of the generalized proportional integral controller, called GPI observer, was introduced in [15], in the context of sliding mode observers for flexible robotics systems. The nonsliding version appears in [14] applied to chaotic systems synchronization. The GPI control strategies have been adapted, extended, and applied successfully in areas other than wind energy, such as induction motor control [17], chaotic systems control [18], and power converters control [19]. Therefore, it is interesting to adapt, evaluate, and determine the scope of this control method in partial load wind turbine operation.GPI observer-based control of nonlinear uncertain systems is very much related to methodologies known as disturbance accommodation control (DAC) and active disturbance rejection control (ADRC). These approaches deal with the problem of cancelling, from the controller’s actions, endogenous and exogenous unknown additive disturbance inputs affecting the system. Perturbation effects are made available via a suitable linear or nonlinear estimation. The reader is invited to read the works by Johnson [20], Han [21], and Gao et al. [22, 23].This work presents an alternative linear control technique based on GPI observers to maximize wind energy capture in variable-speed wind turbines operating at partial load. The proposed strategy uses a GPI observer to reconstruct the aerodynamic torque in order to provide a generator speed optimal trajectory to a robust GPI observer-based controller that regulates the power coefficient, via the generator torque, towards an optimum point at which the power coefficient is maximum. It is expected that the proposed GPI observer-based control technique adds robustness to the system and solves the control problem through linear active estimation and rejection of nonlinearities and perturbations of the wind energy conversion system (WECS).This paper is organized as follows. In Section2, a wind turbine dynamic model for control purposes is presented. Section 3 formulates the control problem and presents the proposed methodology to solve it. Section 4 describes the benchmark and the tests model used to validate the proposed methodology. Section 5 presents the simulations and validations results of the proposed control strategy. Finally, Section 6 contains the conclusions and suggestions for further work.


## 2. Wind Turbine Model

Consider a wind turbine represented by a two-mass mechanical system as shown in Figure1(a), where the aerodynamic power captured by the rotor is given by [24]
(1)Pa(t)=12ρπR2Cp(λ,β)υ(t)3,
whereRis the rotor radius,ρis the air density, υ(t) is the wind velocity, and Cp(λ,β) is the power coefficient which denotes the aerodynamic efficiency of the wind energy conversion system. The power coefficient curve Cp(λ,β) of the wind turbine considered in this work is shown in Figure 1(b). This curve was taken from the benchmark model published in [25]. The power coefficient Cp(λ,β) depends on both the blade pitch angleβand the tip-speed ratio λ. The latter is defined as
(2)λ=ωr(t)Rυ(t),
where ωr(t) is the rotor angular speed. The aerodynamic power Pa(t) can also be expressed in function of both the aerodynamic torque τa(t) and the rotor speed ωr(t) as follows [24]:
(3)Pa(t)=ωr(t)τa(t).
The aerodynamic torque τa(t) is given by
(4)τa(t)=12ρπR3Cq(λ,β)υ(t)2,
where Cq is the torque coefficient and is defined as follows:
(5)Cq(λ,β)=Cp(λ,β)λ.
The mechanical system of the wind turbine is modeled using Newton’s laws, and the following system is derived [25]:
(6)ddtxwt(t)=Awtxwt(t)+Bwtτg(t)+Fwtτa(t),y(t)=Cwtxwt(t)
with
(7)xwt(t)=[ωr(t)ωg(t)θΔ(t)],Awt=[-Bdt-BrJrBdtJrNg-KdtJrBdtηdtJgNg-(Bg+(Bdtηdt/Ng2))JgKdtηdtJgNg1-1Ng0],Bwt=[0-1Jg0],Fwt=[1Jr00],Cwt=[010],
where ωg(t) is the generator side angular speed, θΔ(t) is the torsion angle of the drive train, Jr is the moment of inertia of the low speed shaft, Jg is the moment of inertia of the high speed shaft, Kdt is the torsion stiffness of the drive train, Bdt is the torsion damping coefficient of the drive train, Br is the viscous friction of the low speed shaft, Bg is the viscous friction of the high speed shaft, Ng is the gear ratio, ηdt is the efficiency of the drive train, and τg(t) is the generator torque.Mechanical and aerodynamical characteristics of the wind turbine.
(a)
(b)The power converter and generator dynamics are given by [25]
(8)ddtτg(t)=-αgcτg(t)+αgcτg,ref(t),(9)Pg(t)=ηgωg(t)τg(t),
respectively, where τg,ref(t) is the desired generator torque (control input), Pg(t) is the produced power by the generator, αgc denotes the dynamic coefficient of the generator/converter, and ηg is the generator efficiency.


## 3. Active Disturbance Rejection Design for Wind Turbine Control

The following assumptions in relation to the system (1)–(9) are stated.Assumption 1.
All the parameters of the WECS are known.Assumption 2.
The pair(Awt,Cwt) is completely observable.Assumption 3.
The generator angular speedωg(t) as well as the generated generator torque τg(t) is available to be used in the control system.Assumption 4.
For sufficiently large positive integerp, the disturbance input τa(t) exhibits uniformly absolute bounded time derivative of order p. This condition assures the existence of an unknown but finite constant, Kτa, such that supt≥0|τa(p)(t)|≤Kτa.


### 3.1. Problem Formulation

For a partial load-operating regime (operation in region 2), the main control objective is the maximization of wind power capture. This objective has a strong relation with the wind turbine power coefficient curveCp(λ,β), which has a unique maximum point that corresponds to the optimal capture of the wind power:
(10)Cp(λopt,βopt)=Cpopt,
where
(11)λopt=ωgopt(t)RNgυ(t).
Accordingly, in order to maximize wind power capture, the blade pitch angle β is fixed to its optimal value βopt, and in order to maintain λ at its optimal value λopt, the generator speed must be adjusted to track the optimal reference ωgopt(t), given by
(12)ωgopt(t)=NgλoptRυ(t).
Then, it is desired to force the output ωg(t) to accurately track the given trajectory ωgopt(t), independently of the aerodynamic torque input and possible unmodeled perturbation inputs in the WECS, using the desired generator torque τg,ref(t) as the control input and the generator angular speed ωg(t) as the feedback signal.


### 3.2. GPI Observer Design for Aerodynamic Torque Estimation

In order to obtain the optimal referenceωgopt(t), (4), (5), and (12) are combined. The wind velocity υ(t) is easily represented as a function of τa(t) and Cp using (4) and (5):
(13)υ(t)=2λτa(t)ρπR3Cp(λ,β).
Then, by replacing (13) in (12), setting λ and Cp to its optimal values, and changing τa(t) to its estimated version τ^a(t), the following expression is obtained for the optimal reference trajectory:
(14)ω^gopt(t)=NgλoptR2λoptτ^a(t)ρπR3Cpopt.
According to (14), it is necessary to estimate the aerodynamic torqueτa(t). For that purpose, an extended Luenberger-like linear observer is developed, here referred as GPI observer. The proposed observer uses an approximated internal model of the unknown input disturbance to compose an augmented model for the plant and the disturbance input. Inherent to this kind of observer, a state estimation is also provided. This estimation will be used in the establishment of the GPI observer-based control for ωgopt(t) tracking.Given a positive integerp, the unknown input perturbation τa(t) can be modeled by the approximation of its internal model given by
(15)dpτa(t)dtp≈0.
Consider the following disturbance states, related to (15):
(16)xd(t)=[τa(t)τ˙a(t)⋯τa(p-2)(t)τa(p-1)(t)]T,
where its corresponding dynamics is given by
(17)ddtxd(t)=Adxd(t)+Bdτa(p)(t),τa(t)=Cdxd(t)
with
(18)Ad=[010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1000⋯0],Bd=[00⋮01],Cd=[100⋯0],
where xd(t)∈ℝp×1, Ad∈ℝp×p, Bd∈ℝp×1, and Cd∈ℝ1×p.Now, the disturbance statesxd(t) can be added to the system state vectorxwt(t)to form the following augmented system:
(19)ddtx(t)=Ax(t)+Bτg(t)+Baτa(p)(t),y(t)=Cx(t)
with
(20)x(t)=[xwt(t)xd(t)],A=[AwtFwtCd0Ad],B=[Bwt0],Ba=[0Bd],C=[Cwt0],
where x(t)∈ℝ(p+3)×1,A∈ℝ(p+3)×(p+3),B,Ba∈ℝ(p+3)×1, andC∈ℝ1×(p+3).The next step is to design a GPI observer for the composite system in (19) regarding the approximated internal model given in (15). The estimated augmented state vectorx(t)contains a real-time estimate of xd(t), which is used along withCdto recoverτa(t).Remark 5.
ADR-GPI observer-based controllers use an internal model approximation of the perturbation functions to reconstruct and reject the perturbations. Under this disturbance model approximation setting, several authors have applied it to different areas. Parker and Johnson used a first-order perturbation approximation to model wind speed perturbations in a wind turbine operating in region 3 [26]. Freidovich and Khalil [27] used a first-order perturbation model approximation to estimate the model uncertainty and disturbance on a nonlinear system. Zhao and Gao also used a first-order internal model disturbance approximation to estimate the resonance in two-inertia systems [28] and a first- and second-order approximation to estimate the nonlinearities of an actuator [29]. Zheng et al. also used disturbance model approximation applied to disturbance decoupling control [30].Remark 6.
The parameterp is related to the complexity of the signal to estimate, as in the case of Taylor polynomial approximation. A first-order perturbation model approximation means that the internal model approximation is capable to converge towards a constant disturbance. Equation (15) is a more generalized extension of the internal model perturbation function which provides extra information and increases the ability to track different types of disturbances. For example, p=2 allows convergence to a disturbance with a constant derivative, p=3 allows convergence to a disturbance with a constant acceleration, and so forth.Theorem 7.
Supposing that Assumptions1–4 are valid, for the augmented WECS represented by (19) and (20), the following GPI observer is proposed:
(21)ddtx^(t)=Ax^(t)+Bτg(t)+α(y-Cx^(t)),
where x^(t)=[x^wt(t)Tx^d(t)T]T=[ω^r(t)ω^g(t)θ^Δ(t)τ^a(t)τ^˙a(t)⋯τ^a(p-2)(t)τ^a(p-1)(t)]T is the augmented system state estimation and α=[αp+3αp+2⋯α2α1]Tare the observer gains. The GPI observer (21) asymptotically and exponentially reconstructs the system states ωr(t),ωg(t),θΔ(t), and the perturbation inputs τa(t),τ˙a(t),…,τa(p-1) forcing the state estimation errore~x(t)=x(t)-x^(t)to converge towards the interior of a disk centered in the origin of the corresponding estimation error phase space, as long as the set of coefficients {αp+3,…,α2,α1} is chosen in such a way that the characteristic polynomial defined by
(22)det(sI-A+αC)=0
is a Hurwitz polynomial, with roots located to the left of the imaginary axis of the complex plane.Proof.
By subtracting the proposed observer (21) from the augmented system state equation (19), the following estimation error dynamics is obtained:
(23)e~˙x(t)=(A-αC)e~x(t)+Baτa(p)(t)=Aae~x(t)+Baτa(p)(t),
where the eigenvalues of Aa=(A-αC) can be placed as desired by selecting the gain vector α.
In order to obtain an ultimate bound fore~x(t), letQ∈ℝ(p+3)×(p+3)be a constant, positive definite symmetric matrix. The proper stable character of the matrixAaimplies the existence of a positive definite matrix P∈ℝ(p+3)×(p+3)such that AaTP+PAa=-Q. Consider the Lyapunov function candidateV(e~x)=(1/2)e~xT(t)Pe~x(t). The time derivative ofVsatisfies
(24)V˙(e~x,t)=12[e~xT(-Q)e~x]+BaTPe~xτa(p)(t).
ForQ=I, that is, an identity matrix,V˙(e~x,t)satisfies
(25)V˙(e~x,t)=12[e~xT(-Q)e~x]+BaTPe~xτa(p)(t)≤-12∥e~x∥22+∥BaT∥2∥P∥2∥e~x∥2Kτa<0.
Given that∥BaT∥2=1and according to (25),V˙(e~x,t)is strictly negative if
(26)∥e~x∥2>2Kτa∥P∥2.
Therefore,V˙(e~x,t)is strictly negative outside the following disc:
(27)Dx={e~x(t)∈ℝp+3,∥e~x∥2≤2Kτa∥P∥2}.
Consequently, a uniform ultimate bounded (UUB) result was obtained regarding the estimation error phase variablese~x(t).Remark 8.
GPI observers are bandwidth limited by the roots location of the estimation error characteristic polynomial. Generally, the larger the observer bandwidth is, the more accurate the estimation will be. However, a large observer bandwidth will increase noise sensitivity. Then, the selection of the roots of the estimation error characteristic polynomial affects the bandwidth of the GPI observer and also the influence of measurement noises on the estimations. Therefore, GPI observers are usually tuned in a compromise between disturbance estimation performance (set by the internal model approximation degree) and noise sensitivity.


### 3.3. Robust GPI Observer-Based Control Design for Energy Capture Maximization

Based on (6), the generator angular speedωg(t)satisfies the following dynamics:
(28)dωg(t)dt=BdtηdtJgNgωr(t)-Bg+(Bdtηdt/Ng2)Jgωg(t)+KdtηdtJgNgθΔ(t)-1Jgτg(t).
Then, by reorganizing and lumping together some terms of (28), the following simplified system (typical of the ADR philosophy) can be obtained:
(29)ω˙g(t)=κτg(t)+Δ(t)+φ(t)
with
(30)κ=-1Jg,φ(t)=BdtηdtJgNgωr(t)-Bg+(Bdtηdt/Ng2)Jgωg(t)+KdtηdtJgNgθΔ(t),
where κ is a known constant, φ(t) is a state dependent input perturbation, and Δ(t) is an input perturbation function that lumps together all the uncertainty associated to the system. The perturbationΔ(t)contains the rest of the system dynamics (actuator), including some unmodeled dynamics, disturbances of additive nature, actuator faults, parameter variations, and nonlinear effects of the WECS.In relation to the simplified system (29), the following assumption is stated.Assumption 9.
For a sufficiently large positive integerm, the disturbance inputΔ(t)exhibits a uniformly absolute bounded time derivative of order m. This condition assures the existence of an unknown but finite constant,KΔ, such that supt≥0|Δ(m)(t)|≤KΔ.Assumption 10.
The unknown input perturbationΔ(t)can be modeled by the approximation of its internal model given by
(31)dmΔ(t)dtm≈0.
Consider the following disturbance states, related to (31),
(32)xΔ(t)=[Δ(t)Δ˙(t)⋯Δ(m-2)(t)Δ(m-1)(t)]T,
where their corresponding dynamics is given by:
(33)ddtxΔ(t)=AΔxΔ(t)+BΔΔ(m)(t),Δ(t)=CΔxΔ(t)
with
(34)AΔ=[010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1000⋯0],BΔ=[00⋮01],CΔ=[100⋯0],
where xΔ(t)∈ℝm×1, AΔ∈ℝm×m, BΔ∈ℝm×1, and CΔ∈ℝ1×m.
Then, it is possible to augment the simplified system in (29) with the unknown input perturbation state vector xΔ(t); thus,
(35)ddtxc(t)=Acxc(t)+Bc1τg(t)+Bc2φ(t)+Bc3Δ(m)(t),y(t)=Ccxc(t)
with
(36)xc(t)=[ωg(t)xΔ(t)],Ac=[0CΔ0AΔ],Bc1=[κ0],Bc2=[10],Bc3=[0BΔ],Cc=[10],
where xc(t)∈ℝ(m+1)×1,Ac∈ℝ(m+1)×(m+1),Bc1,Bc2,Bc3∈ℝ(m+1)×1, andCc∈ℝ1×(m+1).
It is desired that the generator angular speedωg(t)accurately tracks the optimal reference trajectory ωgopt(t), with tracking error defined by eω(t)=ωg(t)-ωgopt(t)absolutely bounded by a small quantity ε; that is, supt≥0|eω(t)|≤ε. Then, based on (29), (31), and (35), the following GPI observer-based control is proposed.Theorem 11.
Given Assumptions9 and 10, the estimation of the perturbation functionΔ(t), denoted asΔ^(t), is given by the following GPI observer:
(37)ddtx^c(t)=Acx^c(t)+Bc1τg(t)+Bc2φ^(t)+σ(y(t)-Ccx^c(t)),(38)Δ^(t)=[0CΔ]x^c(t)
with,
(39)φ^(t)=BdtηdtJgNgω^r(t)-Bg+(Bdtηdt/Ng2)Jgω^g(t)+KdtηdtJgNgθ^Δ(t),
where x^c(t)=[ω^g(t)Δ^(t)Δ^˙(t)⋯Δ^(m-2)(t)Δ^(m-1)(t)]T is the estimated system state vector, σ=[σm+1σm⋯σ2σ1]Tis the observer gains, andφ^(t) is the estimation ofφ(t)reconstructed by using the states of the aerodynamic torque observer given in (21) that asymptotically and exponentially reconstructs the perturbation input Δ(t), forcing the state estimation error e~xc(t)=xc(t)-x^c(t) to converge towards the interior of a disk centered in the origin of the corresponding estimation error phase space, provided the set of coefficients: {σm+1,…,σ2,σ1}, which are chosen in such a way that the polynomial PΔ(s), in the complex variables, defined by
(40)PΔ(s)=sm+1+σm+1sm+⋯+σ2s+σ1
is a Hurwitz polynomial, with roots located to the left of the imaginary axis of the complex plane.Proof .
By subtracting the proposed GPI observer (37) from the augmented system state equation (35), the following estimation error dynamics is obtained:
(41)e~˙xc(t)=(Ac-σCc)e~xc(t)+Bc3Δ(m)(t)=Aace~xc(t)+Bc3Δ(m)(t),
where the roots ofdet(sI-Aac)=sm+1+σm+1sm+⋯+σ2s+σ1can be placed as desired by selecting the gain vector σ.
Following the same idea of the proof of Theorem7, letQc=I∈ℝ(m+1)×(m+1)be a constant, positive definite symmetric matrix; then, a positive definite matrixPc∈ℝ(m+1)×(m+1)exists such thatAacTPc+PcAac=-Qc. Consider the Lyapunov function candidateV(e~xc)=(1/2)e~xcT(t)Pce~xc(t). The time derivative ofV, that is, V˙(e~xc,t), is strictly negative outside the disc:
(42)Dxc={e~xc(t)∈ℝm+1,∥e~xc∥2≤2KΔ∥Pc∥2}.
Consequently, a uniform ultimate bounded (UUB) result was obtained regarding the estimation error phase variablese~xc(t).Theorem 12.
Assume that there is an accurate estimation ofφ(t), Δ(t), τa(t), and τ˙a(t); then, for the simplified system (29), the following control law is proposed:
(43)τg(t)=1κ[ω^˙gopt(t)-kc(ωg(t)-ω^gopt(t))-φ^(t)-Δ^(t)]
with
(44)ω^˙gopt(t)=Ngλopt2R2λoptρπR3Cpopt1τ^a(t)τ^˙a(t),
whereτ^a(t)andτ^˙a(t)are provided by the aerodynamic torque observer given in (21),φ^(t) is reconstructed by the estimated states of the aerodynamic torque observer given in (21), andΔ^(t)is provided by the GPI disturbance observer given in (37) and (38).
Such law asymptotically and exponentially forces the closed loop system tracking erroreω(t)to converge towards the interior of a disk of radius as small as desired centered in zero, provided that the coefficient iskc>0.Proof.
By replacing (43) in (29), the following dynamics is obtained:
(45)ω˙g(t)-ω^˙gopt(t)+kc(ωg(t)-ω^gopt(t))=-φ^(t)-Δ^(t)+Δ(t)+φ(t).
Then, by defining some estimation errors: e~ωopt=ωgopt(t)-ω^gopt(t),e~Δ(t)=Δ(t)-Δ^(t), and e~φ(t)=φ(t)-φ^(t) and by replacing them in (45), the following control system tracking error dynamics is obtained:
(46)(ω˙g(t)-ω˙gopt(t))+kc(ωg(t)-ωgopt(t))=-e~˙ωopt(t)-kce~ωopt(t)+e~Δ(t)+e~φ(t),e˙ω(t)+kceω(t)=-e~˙ωopt(t)-kce~ωopt(t)+e~Δ(t)+e~φ(t).
Therefore, as long askc>0and the estimation errors e~ωopt(t), e~Δ(t), and e~φ(t) are ultimately bounded by the GPI observers (21) and (37), the tracking error dynamicseω(t)will remain stable and bounded since the right side of (46) is also bounded.Since the relative order of the WECS is one and system order is three, zero dynamics come into play and in consequence are analyzed. Considering the third-order system dynamics defined in (7), the zero dynamics is given byωr(t)andθΔ(t)withωg(t)set to zero:
(47)[ω˙r(t)θ˙Δ(t)]=[-Bdt+BrJr-KdtJr10]︸Azero[ωr(t)θΔ(t)]+[1Jr0]τa(t).
Then, the internal dynamics is now given by the eigenvalues of Azero (48). In this work, the eigenvalues are stable since the parameters of the WECS Bdt,Br,Kdt, andJrare positive:
(48)|sI-Azero|=|s+Bdt+BrJrKdtJr-1s|=s2+Bdt+BrJrs+KdtJr.
Some troubles may arise in the control system when the internal dynamics is poorly damped; however, in the studied case, such internal dynamics does not lead to any problem. In cases where the internal dynamics became problematic, some methods are used [31].Remark 13.
Note that for the GPI observer-based control strategy defined in (43) and (37), the energy capture maximization depends on the accurate reconstruction of the optimal reference trajectory ωgopt(t).


## 3.1. Problem Formulation

For a partial load-operating regime (operation in region 2), the main control objective is the maximization of wind power capture. This objective has a strong relation with the wind turbine power coefficient curveCp(λ,β), which has a unique maximum point that corresponds to the optimal capture of the wind power:
(10)Cp(λopt,βopt)=Cpopt,
where
(11)λopt=ωgopt(t)RNgυ(t).
Accordingly, in order to maximize wind power capture, the blade pitch angle β is fixed to its optimal value βopt, and in order to maintain λ at its optimal value λopt, the generator speed must be adjusted to track the optimal reference ωgopt(t), given by
(12)ωgopt(t)=NgλoptRυ(t).
Then, it is desired to force the output ωg(t) to accurately track the given trajectory ωgopt(t), independently of the aerodynamic torque input and possible unmodeled perturbation inputs in the WECS, using the desired generator torque τg,ref(t) as the control input and the generator angular speed ωg(t) as the feedback signal.


## 3.2. GPI Observer Design for Aerodynamic Torque Estimation

In order to obtain the optimal referenceωgopt(t), (4), (5), and (12) are combined. The wind velocity υ(t) is easily represented as a function of τa(t) and Cp using (4) and (5):
(13)υ(t)=2λτa(t)ρπR3Cp(λ,β).
Then, by replacing (13) in (12), setting λ and Cp to its optimal values, and changing τa(t) to its estimated version τ^a(t), the following expression is obtained for the optimal reference trajectory:
(14)ω^gopt(t)=NgλoptR2λoptτ^a(t)ρπR3Cpopt.
According to (14), it is necessary to estimate the aerodynamic torqueτa(t). For that purpose, an extended Luenberger-like linear observer is developed, here referred as GPI observer. The proposed observer uses an approximated internal model of the unknown input disturbance to compose an augmented model for the plant and the disturbance input. Inherent to this kind of observer, a state estimation is also provided. This estimation will be used in the establishment of the GPI observer-based control for ωgopt(t) tracking.Given a positive integerp, the unknown input perturbation τa(t) can be modeled by the approximation of its internal model given by
(15)dpτa(t)dtp≈0.
Consider the following disturbance states, related to (15):
(16)xd(t)=[τa(t)τ˙a(t)⋯τa(p-2)(t)τa(p-1)(t)]T,
where its corresponding dynamics is given by
(17)ddtxd(t)=Adxd(t)+Bdτa(p)(t),τa(t)=Cdxd(t)
with
(18)Ad=[010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1000⋯0],Bd=[00⋮01],Cd=[100⋯0],
where xd(t)∈ℝp×1, Ad∈ℝp×p, Bd∈ℝp×1, and Cd∈ℝ1×p.Now, the disturbance statesxd(t) can be added to the system state vectorxwt(t)to form the following augmented system:
(19)ddtx(t)=Ax(t)+Bτg(t)+Baτa(p)(t),y(t)=Cx(t)
with
(20)x(t)=[xwt(t)xd(t)],A=[AwtFwtCd0Ad],B=[Bwt0],Ba=[0Bd],C=[Cwt0],
where x(t)∈ℝ(p+3)×1,A∈ℝ(p+3)×(p+3),B,Ba∈ℝ(p+3)×1, andC∈ℝ1×(p+3).The next step is to design a GPI observer for the composite system in (19) regarding the approximated internal model given in (15). The estimated augmented state vectorx(t)contains a real-time estimate of xd(t), which is used along withCdto recoverτa(t).Remark 5.
ADR-GPI observer-based controllers use an internal model approximation of the perturbation functions to reconstruct and reject the perturbations. Under this disturbance model approximation setting, several authors have applied it to different areas. Parker and Johnson used a first-order perturbation approximation to model wind speed perturbations in a wind turbine operating in region 3 [26]. Freidovich and Khalil [27] used a first-order perturbation model approximation to estimate the model uncertainty and disturbance on a nonlinear system. Zhao and Gao also used a first-order internal model disturbance approximation to estimate the resonance in two-inertia systems [28] and a first- and second-order approximation to estimate the nonlinearities of an actuator [29]. Zheng et al. also used disturbance model approximation applied to disturbance decoupling control [30].Remark 6.
The parameterp is related to the complexity of the signal to estimate, as in the case of Taylor polynomial approximation. A first-order perturbation model approximation means that the internal model approximation is capable to converge towards a constant disturbance. Equation (15) is a more generalized extension of the internal model perturbation function which provides extra information and increases the ability to track different types of disturbances. For example, p=2 allows convergence to a disturbance with a constant derivative, p=3 allows convergence to a disturbance with a constant acceleration, and so forth.Theorem 7.
Supposing that Assumptions1–4 are valid, for the augmented WECS represented by (19) and (20), the following GPI observer is proposed:
(21)ddtx^(t)=Ax^(t)+Bτg(t)+α(y-Cx^(t)),
where x^(t)=[x^wt(t)Tx^d(t)T]T=[ω^r(t)ω^g(t)θ^Δ(t)τ^a(t)τ^˙a(t)⋯τ^a(p-2)(t)τ^a(p-1)(t)]T is the augmented system state estimation and α=[αp+3αp+2⋯α2α1]Tare the observer gains. The GPI observer (21) asymptotically and exponentially reconstructs the system states ωr(t),ωg(t),θΔ(t), and the perturbation inputs τa(t),τ˙a(t),…,τa(p-1) forcing the state estimation errore~x(t)=x(t)-x^(t)to converge towards the interior of a disk centered in the origin of the corresponding estimation error phase space, as long as the set of coefficients {αp+3,…,α2,α1} is chosen in such a way that the characteristic polynomial defined by
(22)det(sI-A+αC)=0
is a Hurwitz polynomial, with roots located to the left of the imaginary axis of the complex plane.Proof.
By subtracting the proposed observer (21) from the augmented system state equation (19), the following estimation error dynamics is obtained:
(23)e~˙x(t)=(A-αC)e~x(t)+Baτa(p)(t)=Aae~x(t)+Baτa(p)(t),
where the eigenvalues of Aa=(A-αC) can be placed as desired by selecting the gain vector α.
In order to obtain an ultimate bound fore~x(t), letQ∈ℝ(p+3)×(p+3)be a constant, positive definite symmetric matrix. The proper stable character of the matrixAaimplies the existence of a positive definite matrix P∈ℝ(p+3)×(p+3)such that AaTP+PAa=-Q. Consider the Lyapunov function candidateV(e~x)=(1/2)e~xT(t)Pe~x(t). The time derivative ofVsatisfies
(24)V˙(e~x,t)=12[e~xT(-Q)e~x]+BaTPe~xτa(p)(t).
ForQ=I, that is, an identity matrix,V˙(e~x,t)satisfies
(25)V˙(e~x,t)=12[e~xT(-Q)e~x]+BaTPe~xτa(p)(t)≤-12∥e~x∥22+∥BaT∥2∥P∥2∥e~x∥2Kτa<0.
Given that∥BaT∥2=1and according to (25),V˙(e~x,t)is strictly negative if
(26)∥e~x∥2>2Kτa∥P∥2.
Therefore,V˙(e~x,t)is strictly negative outside the following disc:
(27)Dx={e~x(t)∈ℝp+3,∥e~x∥2≤2Kτa∥P∥2}.
Consequently, a uniform ultimate bounded (UUB) result was obtained regarding the estimation error phase variablese~x(t).Remark 8.
GPI observers are bandwidth limited by the roots location of the estimation error characteristic polynomial. Generally, the larger the observer bandwidth is, the more accurate the estimation will be. However, a large observer bandwidth will increase noise sensitivity. Then, the selection of the roots of the estimation error characteristic polynomial affects the bandwidth of the GPI observer and also the influence of measurement noises on the estimations. Therefore, GPI observers are usually tuned in a compromise between disturbance estimation performance (set by the internal model approximation degree) and noise sensitivity.


## 3.3. Robust GPI Observer-Based Control Design for Energy Capture Maximization

Based on (6), the generator angular speedωg(t)satisfies the following dynamics:
(28)dωg(t)dt=BdtηdtJgNgωr(t)-Bg+(Bdtηdt/Ng2)Jgωg(t)+KdtηdtJgNgθΔ(t)-1Jgτg(t).
Then, by reorganizing and lumping together some terms of (28), the following simplified system (typical of the ADR philosophy) can be obtained:
(29)ω˙g(t)=κτg(t)+Δ(t)+φ(t)
with
(30)κ=-1Jg,φ(t)=BdtηdtJgNgωr(t)-Bg+(Bdtηdt/Ng2)Jgωg(t)+KdtηdtJgNgθΔ(t),
where κ is a known constant, φ(t) is a state dependent input perturbation, and Δ(t) is an input perturbation function that lumps together all the uncertainty associated to the system. The perturbationΔ(t)contains the rest of the system dynamics (actuator), including some unmodeled dynamics, disturbances of additive nature, actuator faults, parameter variations, and nonlinear effects of the WECS.In relation to the simplified system (29), the following assumption is stated.Assumption 9.
For a sufficiently large positive integerm, the disturbance inputΔ(t)exhibits a uniformly absolute bounded time derivative of order m. This condition assures the existence of an unknown but finite constant,KΔ, such that supt≥0|Δ(m)(t)|≤KΔ.Assumption 10.
The unknown input perturbationΔ(t)can be modeled by the approximation of its internal model given by
(31)dmΔ(t)dtm≈0.
Consider the following disturbance states, related to (31),
(32)xΔ(t)=[Δ(t)Δ˙(t)⋯Δ(m-2)(t)Δ(m-1)(t)]T,
where their corresponding dynamics is given by:
(33)ddtxΔ(t)=AΔxΔ(t)+BΔΔ(m)(t),Δ(t)=CΔxΔ(t)
with
(34)AΔ=[010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1000⋯0],BΔ=[00⋮01],CΔ=[100⋯0],
where xΔ(t)∈ℝm×1, AΔ∈ℝm×m, BΔ∈ℝm×1, and CΔ∈ℝ1×m.
Then, it is possible to augment the simplified system in (29) with the unknown input perturbation state vector xΔ(t); thus,
(35)ddtxc(t)=Acxc(t)+Bc1τg(t)+Bc2φ(t)+Bc3Δ(m)(t),y(t)=Ccxc(t)
with
(36)xc(t)=[ωg(t)xΔ(t)],Ac=[0CΔ0AΔ],Bc1=[κ0],Bc2=[10],Bc3=[0BΔ],Cc=[10],
where xc(t)∈ℝ(m+1)×1,Ac∈ℝ(m+1)×(m+1),Bc1,Bc2,Bc3∈ℝ(m+1)×1, andCc∈ℝ1×(m+1).
It is desired that the generator angular speedωg(t)accurately tracks the optimal reference trajectory ωgopt(t), with tracking error defined by eω(t)=ωg(t)-ωgopt(t)absolutely bounded by a small quantity ε; that is, supt≥0|eω(t)|≤ε. Then, based on (29), (31), and (35), the following GPI observer-based control is proposed.Theorem 11.
Given Assumptions9 and 10, the estimation of the perturbation functionΔ(t), denoted asΔ^(t), is given by the following GPI observer:
(37)ddtx^c(t)=Acx^c(t)+Bc1τg(t)+Bc2φ^(t)+σ(y(t)-Ccx^c(t)),(38)Δ^(t)=[0CΔ]x^c(t)
with,
(39)φ^(t)=BdtηdtJgNgω^r(t)-Bg+(Bdtηdt/Ng2)Jgω^g(t)+KdtηdtJgNgθ^Δ(t),
where x^c(t)=[ω^g(t)Δ^(t)Δ^˙(t)⋯Δ^(m-2)(t)Δ^(m-1)(t)]T is the estimated system state vector, σ=[σm+1σm⋯σ2σ1]Tis the observer gains, andφ^(t) is the estimation ofφ(t)reconstructed by using the states of the aerodynamic torque observer given in (21) that asymptotically and exponentially reconstructs the perturbation input Δ(t), forcing the state estimation error e~xc(t)=xc(t)-x^c(t) to converge towards the interior of a disk centered in the origin of the corresponding estimation error phase space, provided the set of coefficients: {σm+1,…,σ2,σ1}, which are chosen in such a way that the polynomial PΔ(s), in the complex variables, defined by
(40)PΔ(s)=sm+1+σm+1sm+⋯+σ2s+σ1
is a Hurwitz polynomial, with roots located to the left of the imaginary axis of the complex plane.Proof .
By subtracting the proposed GPI observer (37) from the augmented system state equation (35), the following estimation error dynamics is obtained:
(41)e~˙xc(t)=(Ac-σCc)e~xc(t)+Bc3Δ(m)(t)=Aace~xc(t)+Bc3Δ(m)(t),
where the roots ofdet(sI-Aac)=sm+1+σm+1sm+⋯+σ2s+σ1can be placed as desired by selecting the gain vector σ.
Following the same idea of the proof of Theorem7, letQc=I∈ℝ(m+1)×(m+1)be a constant, positive definite symmetric matrix; then, a positive definite matrixPc∈ℝ(m+1)×(m+1)exists such thatAacTPc+PcAac=-Qc. Consider the Lyapunov function candidateV(e~xc)=(1/2)e~xcT(t)Pce~xc(t). The time derivative ofV, that is, V˙(e~xc,t), is strictly negative outside the disc:
(42)Dxc={e~xc(t)∈ℝm+1,∥e~xc∥2≤2KΔ∥Pc∥2}.
Consequently, a uniform ultimate bounded (UUB) result was obtained regarding the estimation error phase variablese~xc(t).Theorem 12.
Assume that there is an accurate estimation ofφ(t), Δ(t), τa(t), and τ˙a(t); then, for the simplified system (29), the following control law is proposed:
(43)τg(t)=1κ[ω^˙gopt(t)-kc(ωg(t)-ω^gopt(t))-φ^(t)-Δ^(t)]
with
(44)ω^˙gopt(t)=Ngλopt2R2λoptρπR3Cpopt1τ^a(t)τ^˙a(t),
whereτ^a(t)andτ^˙a(t)are provided by the aerodynamic torque observer given in (21),φ^(t) is reconstructed by the estimated states of the aerodynamic torque observer given in (21), andΔ^(t)is provided by the GPI disturbance observer given in (37) and (38).
Such law asymptotically and exponentially forces the closed loop system tracking erroreω(t)to converge towards the interior of a disk of radius as small as desired centered in zero, provided that the coefficient iskc>0.Proof.
By replacing (43) in (29), the following dynamics is obtained:
(45)ω˙g(t)-ω^˙gopt(t)+kc(ωg(t)-ω^gopt(t))=-φ^(t)-Δ^(t)+Δ(t)+φ(t).
Then, by defining some estimation errors: e~ωopt=ωgopt(t)-ω^gopt(t),e~Δ(t)=Δ(t)-Δ^(t), and e~φ(t)=φ(t)-φ^(t) and by replacing them in (45), the following control system tracking error dynamics is obtained:
(46)(ω˙g(t)-ω˙gopt(t))+kc(ωg(t)-ωgopt(t))=-e~˙ωopt(t)-kce~ωopt(t)+e~Δ(t)+e~φ(t),e˙ω(t)+kceω(t)=-e~˙ωopt(t)-kce~ωopt(t)+e~Δ(t)+e~φ(t).
Therefore, as long askc>0and the estimation errors e~ωopt(t), e~Δ(t), and e~φ(t) are ultimately bounded by the GPI observers (21) and (37), the tracking error dynamicseω(t)will remain stable and bounded since the right side of (46) is also bounded.Since the relative order of the WECS is one and system order is three, zero dynamics come into play and in consequence are analyzed. Considering the third-order system dynamics defined in (7), the zero dynamics is given byωr(t)andθΔ(t)withωg(t)set to zero:
(47)[ω˙r(t)θ˙Δ(t)]=[-Bdt+BrJr-KdtJr10]︸Azero[ωr(t)θΔ(t)]+[1Jr0]τa(t).
Then, the internal dynamics is now given by the eigenvalues of Azero (48). In this work, the eigenvalues are stable since the parameters of the WECS Bdt,Br,Kdt, andJrare positive:
(48)|sI-Azero|=|s+Bdt+BrJrKdtJr-1s|=s2+Bdt+BrJrs+KdtJr.
Some troubles may arise in the control system when the internal dynamics is poorly damped; however, in the studied case, such internal dynamics does not lead to any problem. In cases where the internal dynamics became problematic, some methods are used [31].Remark 13.
Note that for the GPI observer-based control strategy defined in (43) and (37), the energy capture maximization depends on the accurate reconstruction of the optimal reference trajectory ωgopt(t).


## 4. Benchmark Model

The simulations are carried out using a benchmark model for wind turbine control implemented in MATLAB/Simulink. This benchmark model was published by Odgaard et al. in [25] and can be used to evaluate both fault tolerant and classic control schemes in any region of operation of a wind turbine. The test bench model is based on a realistic nonlinear generic three-bladed horizontal-axis variable-speed wind turbine, containing sensors, actuators, system faults, tower shadow and wind shear effects, full converter coupling, and rated power at 4.8 MW.For wind speeds between 0 and 12.5 m/s, the turbine is controlled to operate in region 2. The wind profile used has an average hub-height wind speed of 8.68 m/s and a turbulence intensity of 12%. The test bench defines a standard torque control strategy for the operation in region 2 with the following control law:τg,ref(t)=koptωg2(t) with kopt=1.2171, Cpopt=0.4554, and λopt=8.0. The converter model has the following constraints: max torque gradient =1.25×104 N·m/s, min torque gradient =-1.25×104 N·m/s, max torque =3.6×104 N·m, and min torque =0 N·m.


## 5. Results and Discussion

The proposed aerodynamic torque GPI observer (21) and the GPI observer-based control (37)–(43) were implemented and tested on the nonlinear wind turbine benchmark model described previously. Figure 2 shows the system scheme of the proposed control strategy used for simulations. The parameters of the GPI observers and the control strategy were selected as follows:GPI observer for aerodynamic torque estimation:m=3,σ1=1000, and σ2=1028.22, σ3=1028.62, σ4=0.39, σ5=234.81, σ6=28.46.GPI observer-based control:p=3, kc=1, α1=39.38, α2=61.69, α3=35.30, and α4=8.66.Figure 2
Closed loop system schema of the proposed control strategy.


### 5.1. Aerodynamic Torque Estimation

The estimate of the aerodynamic torque obtained by the GPI observer in (21) is shown in Figure 3(a). The aerodynamic torque estimation error is shown in Figure 3(c). The estimation of the generator speed optimal reference trajectoryω^gopt(t)is shown in Figure 3(b), and its estimation error e~ωopt=ωgopt(t)-ω^gopt(t) can be detailed in Figure 3(d). Note that this estimation error has a small offset of -1.0811 rad/s, which in turn will cause a slight offset in the aerodynamic power coefficient control.Aerodynamic torque estimation results: (a) aerodynamic torque, (b) generator speed optimal trajectory, (c) aerodynamic torque estimation error, and (d) generator speed trajectory estimation error.
(a)
(b)
(c)
(d)


### 5.2. Energy Capture Maximization

The simulation results of the proposed control strategy are shown in Figure4. Under nominal conditions, the control system tracks the optimal reference trajectory to force the WECS power coefficientCpclose to its optimal value (see Figures 4(b) and 4(e)). Despite the external disturbances and nonlinearities of the benchmark model, the tracking error of the control system is near to zero as shown in Figure 4(c). In order to achieve a compromise between energy capture and dynamic loads on the low speed shaft, a medium performance on the control gainkcwas selected. Most of the fast fluctuations of the aerodynamic torque transferred to the generator speed optimal trajectory are not tracked (see Figure 4(b)).Simulation results of the proposed GPI observer-based control: (a) wind profile, (b) generator angular speed, (c) tracking error, (d) aerodynamic power captured, and (e) power coefficient.
(a)
(b)
(c)
(d)
(e)Figures4(d) and 4(e) detail the aerodynamic power captured by the GPI observer-based control and the evolution in time of the WECS power coefficient, respectively. It is observed that the captured aerodynamic power with the proposed control is greater than the power captured by the standard torque control defined in the benchmark. In addition, it is noted that the proposed control strategy forces the power coefficient to stay close to its optimal value Cpopt=0.4554, which allows better power capture.The performance of each control system is compared using an aerodynamic efficiency indexηaero[24]. It is defined as follows:
(49)ηaero(%)=∫titfPa(t)dt∫titfPaopt(t)dt,Paopt(t)=12ρπR2Cpoptυ(t)3.
The evaluation of the criteria defined in (49) stated that the aerodynamic efficiency obtained by the proposed control technique is 98.8%, while the efficiency of the classical controller (benchmark) is 95.6%.


### 5.3. Active Disturbance Rejection Evaluation

The benchmark model [25] contains faults which require the control system to be reconfigured to continue power generation, as well as very severe faults which require a safe and fast shutdown of the wind turbine. In this work, in order to evaluate the active disturbance rejection capability of the proposed GPI observer-based control strategy, a typical malfunction in the internal power converter control loops is used. As a consequence, this nonsevere fault must be accommodated in some way, and the wind turbine must continue its operation.The fault considered is an offset, denoted asΔτg, on the generator torque, which can be caused by an error in the initialization of the converter controller [32]. The converter offset is configured to Δτg=5000 N·m.Figure5 shows the closed loop performance of both the standard torque control and the proposed GPI observer-based control approach under the actuator fault. The fault occurs from 200 s to 400 s as seen in Figure 5(d). As the fault is active, the GPI observer (37), via the observer stateΔ^(t), estimates the perturbation function on line (see Figure 5(c)) and actively rejects the internal perturbation applying the GPI observer-based control law (43).Simulation results on power converter fault.
(a)
(b)
(c)
(d)It is observed in Figure5(a) that the perturbation is rejected by the proposed control and any captured aerodynamic power lost is appreciated. Figure 5(b) shows that the power coefficient of the WECS is still close to its optimal value. On the other hand, the standard torque control of the benchmark cannot handle the actuator fault, and much of the aerodynamic power is lost. The evaluation of the criteria defined in (49) stated that the aerodynamic efficiency obtained by the proposed control is 98.65%, while the efficiency of the classical controller (benchmark) is 85.88%.


## 5.1. Aerodynamic Torque Estimation

The estimate of the aerodynamic torque obtained by the GPI observer in (21) is shown in Figure 3(a). The aerodynamic torque estimation error is shown in Figure 3(c). The estimation of the generator speed optimal reference trajectoryω^gopt(t)is shown in Figure 3(b), and its estimation error e~ωopt=ωgopt(t)-ω^gopt(t) can be detailed in Figure 3(d). Note that this estimation error has a small offset of -1.0811 rad/s, which in turn will cause a slight offset in the aerodynamic power coefficient control.Aerodynamic torque estimation results: (a) aerodynamic torque, (b) generator speed optimal trajectory, (c) aerodynamic torque estimation error, and (d) generator speed trajectory estimation error.
(a)
(b)
(c)
(d)


## 5.2. Energy Capture Maximization

The simulation results of the proposed control strategy are shown in Figure4. Under nominal conditions, the control system tracks the optimal reference trajectory to force the WECS power coefficientCpclose to its optimal value (see Figures 4(b) and 4(e)). Despite the external disturbances and nonlinearities of the benchmark model, the tracking error of the control system is near to zero as shown in Figure 4(c). In order to achieve a compromise between energy capture and dynamic loads on the low speed shaft, a medium performance on the control gainkcwas selected. Most of the fast fluctuations of the aerodynamic torque transferred to the generator speed optimal trajectory are not tracked (see Figure 4(b)).Simulation results of the proposed GPI observer-based control: (a) wind profile, (b) generator angular speed, (c) tracking error, (d) aerodynamic power captured, and (e) power coefficient.
(a)
(b)
(c)
(d)
(e)Figures4(d) and 4(e) detail the aerodynamic power captured by the GPI observer-based control and the evolution in time of the WECS power coefficient, respectively. It is observed that the captured aerodynamic power with the proposed control is greater than the power captured by the standard torque control defined in the benchmark. In addition, it is noted that the proposed control strategy forces the power coefficient to stay close to its optimal value Cpopt=0.4554, which allows better power capture.The performance of each control system is compared using an aerodynamic efficiency indexηaero[24]. It is defined as follows:
(49)ηaero(%)=∫titfPa(t)dt∫titfPaopt(t)dt,Paopt(t)=12ρπR2Cpoptυ(t)3.
The evaluation of the criteria defined in (49) stated that the aerodynamic efficiency obtained by the proposed control technique is 98.8%, while the efficiency of the classical controller (benchmark) is 95.6%.


## 5.3. Active Disturbance Rejection Evaluation

The benchmark model [25] contains faults which require the control system to be reconfigured to continue power generation, as well as very severe faults which require a safe and fast shutdown of the wind turbine. In this work, in order to evaluate the active disturbance rejection capability of the proposed GPI observer-based control strategy, a typical malfunction in the internal power converter control loops is used. As a consequence, this nonsevere fault must be accommodated in some way, and the wind turbine must continue its operation.The fault considered is an offset, denoted asΔτg, on the generator torque, which can be caused by an error in the initialization of the converter controller [32]. The converter offset is configured to Δτg=5000 N·m.Figure5 shows the closed loop performance of both the standard torque control and the proposed GPI observer-based control approach under the actuator fault. The fault occurs from 200 s to 400 s as seen in Figure 5(d). As the fault is active, the GPI observer (37), via the observer stateΔ^(t), estimates the perturbation function on line (see Figure 5(c)) and actively rejects the internal perturbation applying the GPI observer-based control law (43).Simulation results on power converter fault.
(a)
(b)
(c)
(d)It is observed in Figure5(a) that the perturbation is rejected by the proposed control and any captured aerodynamic power lost is appreciated. Figure 5(b) shows that the power coefficient of the WECS is still close to its optimal value. On the other hand, the standard torque control of the benchmark cannot handle the actuator fault, and much of the aerodynamic power is lost. The evaluation of the criteria defined in (49) stated that the aerodynamic efficiency obtained by the proposed control is 98.65%, while the efficiency of the classical controller (benchmark) is 85.88%.


## 6. Conclusions

In this paper, a linear active disturbance rejection control strategy based on two GPI observers for maximum wind energy capture of variable-speed wind turbines operating at partial load has been proposed. In order to create the generator speed optimal trajectory towards an optimum point at which the WECS power coefficient is maximum, an ADR philosophy-based GPI observer was developed to estimate the aerodynamic torque and its first derivative. Then, an ADR philosophy-based GPI observer-based controller was designed, and it was able to absolutely and arbitrarily bound the generator speed tracking error.The proposed design strategy solved the control problem based on linear active estimation of possible nonlinearities and perturbations of the WECS, and these accurate estimations were used by a simplified linear control law, in which the captured wind energy was maximized.It was shown through simulation tests on a fully nonlinear benchmark model that the proposed dual GPI observer control strategy maximized the captured wind energy even when an actuator fault was applied. This is a demonstration of some robustness added by the GPI observer-based control.Since performance of wind turbines is significantly affected by the used control strategy, considering new alternative linear control strategies that can improve the performance of the WECS is a motivation to use, adapt, and evaluate linear GPI observer-based controllers to operate wind turbines at partial load.It is worth noting that the proposed control strategy is related to exact feedback linearization, but there are some important differences between both strategies which give advantages to GPI observer-based control, such as the following: (a) GPI observer-based control does not require system state measurements, (b) any mismatch between the system model and the real system is lumped together in a perturbation function that is estimated and rejected on line, (c) GPI observers are capable of estimating a certain number of perturbation function derivatives (useful to determineω˙gopt), and (d) ADR philosophy plays a very important role in GPI observer-based control since the internal model of the perturbation functions is taken into account in the design process.



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*Source: 396740-2013-07-10.xml*