# A Novel Adaptive Active Disturbance Rejection Control Strategy to Improve the Stability and Robustness for a Wind Turbine Using a Doubly Fed Induction Generator

Authors: Hammadi Laghridat; Ahmed Essadki; Maha Annoukoubi; Tamou Nasser
Journal: Journal of Electrical and Computer Engineering
 (2020)

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## Abstract

A novel and robust active disturbance rejection control (ADRC) strategy for variable speed wind turbine systems using a doubly fed induction generator (DFIG) is presented in this paper. The DFIG is directly connected to the main utility grid by stator, and its rotor is connected through a back-to-back three phase power converter (AC/DC/AC). Due to the acoustic nature of wind and to ensure capturing maximum energy, a control strategy to extract the available maximum power from the wind turbine by using a maximum power point tracking (MPPT) algorithm is presented. Moreover, a pitch actuator system is used to control the blades’ pitch angle of the wind turbine in order to not exceed the wind turbine rated power value in case of strong wind speeds. Furthermore, the rotor-side converter is used to control the active and reactive powers generated by the DFIG. However, the grid-side converter is used to control the currents injected into the utility grid as well as to regulate the DC-link voltage. This paper aims to study and develop two control strategies for wind turbine system control: classical control by proportional integral (PI) and the proposed linear active disturbance rejection control (LADRC). The main purpose here is to compare and evaluate the dynamical performances and sensitivity of these controllers to the DFIG parameter variation. Therefore, a series of simulations were carried out in the MATLAB/Simulink environment, and the obtained results have shown the effectiveness of the proposed strategy in terms of efficiency, rapidity, and robustness to internal and external disturbances.


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## Body

## 1. Introduction

The global strategy of energy has been oriented to the development of renewable energy resources (wind energy, solar energy, hydroenergy, etc.), which has many advantages as they are less concerned about the environmental degradation on the one hand, and on the other hand, their energy production cost is very low compared to the conventional resources. In recent years, wind energy has been classified as one of the most important and promising renewable energy sources; it is based on the variation of the wind speed. The wind turbine converts this wind into the aerodynamic power, which is converted into electrical power by using of electric generators such as doubly fed induction generator (DFIG), squirrel cage induction generator (SCIG), or permanent magnetic synchronous generator (PMSG).In a variable speed wind energy conversion system, which is based on doubly fed induction generators, the DFIG stator side is directly connected to the utility grid and the rotor side is connected through a back-to-back three phase power converter (AC/DC/AC). Currently, this configuration type is the most widely used for variable speed wind energy conversion systems; its main advantage is as follows: it can operate in both subsynchronous and supersynchronous speeds [1]. And more importantly, it operates with a smaller power capacity converter (rotor- and grid-side converters) compared to the fully rated power converter schemes.During the last decades, the control of variable speed wind energy conversion systems has become an important factor to efficiency enhancement and to increase the energetics yield. Moreover, it can be split in three parts:(i)
Maximum power point tracking (MPPT) algorithms are needed to improve the energy efficiency and to extract the maximum power available from the wind turbine. In this context, several research studies have been conducted in the literature. Among them, in [2], the authors deal with a review of maximum power point tracking algorithms for wind energy systems. It presents a comparison of different maximum power point algorithms by presenting the simulation of three selected control methods (optimal torque control, tip speed ratio control, and perturbation and observation control P&O) in terms of validity, efficiency, and speed of response.(ii)
Pitch angle control is used to ensure that the extracted wind turbine system power does not exceed its rated value in case of the instantaneous changing nature of the wind; a pitch actuator system is used to regulate the blades’ pitch angle of the wind turbine. In [3], the authors have designed a novel PI and PID-based pitch control techniques used for large wind turbine systems with the synthesis of the PI parameters.(iii)
The control of the rotor-side converter and the grid-side converter: the first converter is used to control the stator powers (active and reactive powers) independently, and the second converter is used to control power exchange with the utility grid at rotor side as well as to regulate the DC-bus voltage. Noteworthy, several control techniques for the rotor- and grid-side converters have been discussed in the literature. Among them, in [4], the authors have proposed to control DFIG-based variable speed wind turbine by PI controller, in [5], the authors proposed a sliding mode control of doubly fed induction generator, in [6], the authors presented a backstepping control of DFIG generators for wide-range variable speed wind turbines, in [7], the authors discussed the modeling and fuzzy logic control of DFIG-based wind energy conversion systems, and in [8], the authors have proposed to control a doubly fed induction generator for wind energy conversion systems by RST controller.Another control strategy has been proposed and utilized. Since the last 10 years, it is known as the active disturbance rejection controller (ADRC). Moreover, it is used for the applications in several areas. The main advantages of this controller lie on the real-time rejection of internal and external disturbances based on an observer, i.e., internal disturbances (as the variation of the internal parameters or the modeling errors) and external disturbances (like the voltage dips and frequency droops). We propose in this paper the application of wind turbine control by ADRC strategy; the first application is used to control the rotor-side converter to regulate the stator powers. Furthermore, it is used to control the grid-side converter to regulate the active and reactive powers injected into the utility grid as well as to control the DC-bus voltage.In this work, the main contributions are the development and comparison of two control methods in terms of efficiency, rapidity, and robustness, and also the response of the internal behaviour of wind energy conversion system based on DFIG. However, this paper is structured as follows. Section 2 presents the dynamic modeling of wind turbine system based on DFIG. Section 3 details the mathematical theory of active disturbance rejection controller. The maximum power point tracking algorithm and the blade orientation system control are given in Section4. The control of wind energy conversion system using DFIG through two control methods (PI controller and linear ADRC controller) is demonstrated in Section 5. The remainder of the paper presents the simulation results and discussion.


## 2. Dynamic Modeling of Wind Turbine System

In this section, the DFIG variable speed wind energy conversion system control method is presented (active and reactive powers control). The wind energy conversion systems topology based on a DFIG is the most commonly used structure for large variable wind speed power plants [9]. As shown in Figure 1, the DFIG wind turbine generates the active power form the generator stator side (stator power). Moreover, it also generates or absorbs active power from the generator rotor side (rotor power) depending on the slip.The wind energy conversion system consists of the following main components: turbine, gearbox, DFIG generator with power converters (AC/DC/AC), DC-link, filter, transformer, and the grid.Figure 1
Structure of a wind turbine system using a DFIG.


### 2.1. Wind Energy Conversion System Mechanical Model

The turbine captures the wind kinetic energy and converts it to a torque that rotates the rotors blades (mechanical energy). The formula of aerodynamic powerPaero available from the turbine rotor can be given by(1)Paero=Cpλ,βPw=12ρAtCpλ,βw3,where Pw is the wind power; Cp is the power coefficient; β and λ represent the blades’ pitch angle and the turbine tip speed ratio, respectively; At = πR2 is the turbine blade area; w is the wind speed; and ρ represents the air density [10].The aerodynamic torqueTaero is given by the following expression:(2)Taero=12ωturρAtCpλ,βw3,where ωtur is the turbine speed and R is the turbine blade radius.The wind turbine aerodynamic is characterized by a power coefficient; it is a nonlinear equation which depends on λ and β; it is expressed by(3)Cpλ,β=k1k21λj−k3β−k4exp−k5/λj+k6λ,where(4)1λj=1λ+k7β−k8β3+1,λ=Rωturw,with k1 = 0.5176, k2 = 166, k3 = 0.4, k4 = 5, k5 = 21, k6 = 0.0068, k7 = 0.08, and k8 = 0.035.As shown in Figure2, the aerodynamic coefficient has a maximum power point (point M) when β = 0◦ and λ = 8.1 (with Cpmax=0.48).Figure 2
The aerodynamic coefficient characteristics of the wind turbine system.The gearbox torqueTg and mechanical angular speed ωmec are given by(5)Tg=TaeroG,ωmec=Gωtur,where G is the gearbox ratio.The system mechanical equation is(6)Jdωmecdt=Tg−Tem−fωmec,where Tem and f are the DFIG electromagnetic torque and the friction coefficient, respectively. J represents the total inertia which can be calculated by(7)J=JturG2+JDFIG,where JDFIG and Jtur are the inertias of the aggregated generator and the turbine.


### 2.2. Dynamic Model of DFIG

In order to model the DFIG, it is assumed that the stator and rotor windings are placed sinusoidally and symmetrically, and the magnetical saturation effects of all the windings are neglectable. Figure3 illustrates the block of the generator phase in dq frame reference.Figure 3
Schematic diagram of the DFIG equivalent circuit [11].The stator and rotor windings equations are presented by (8)–(11):(8)dΨsddt=−Rsisd+ωsΨsq+vsd,(9)dΨsqdt=−Rsisq−ωsΨsd+vsq,(10)dΨrddt=−Rrird+ωrΨrq+vrd,(11)dΨrqdt=−Rrirq−ωrΨrd+vrq,where vsd and vsq are the stator voltages in the direct-quadrature frame and vrd and vrq are the rotor voltages in the direct-quadrature frame. isd and isq are the direct and quadrature stator currents, ird and irq are the direct and quadrature rotor currents, Ψsd and Ψsq are the direct-quadrature frame stator fields, and Ψrd and Ψrq are the direct-quadrature frame rotor fields. Also, Rs and Rr are the stator and the rotor phase resistances; ωr and ωs are the rotor electrical speed and the stator electrical speed, respectively.The equations of stator and rotor fields are(12)Ψsd=Lm+Lsσisd+Lmird,Ψsq=Lm+Lsσisq+Lmirq,Ψrd=Lm+Lrσird+Lmisd,Ψrq=Lm+Lrσirq+Lmisq,where Lm is the mutual inductance and Lsσ and Lrσ are the stator leakage inductance and the rotor leakage inductance, respectively [9].The electrical active and reactive powers (Ps and Qs) delivered by generator stator side and generator rotor side can be written as follows:(13)Ps=vsdisd+vsqisq,Qs=vsqisd−vsdisq,Pr=vrdird+vrqirq,Qr=vrqird−vrdirq.The electromagnetic torque equation is then(14)Tem=−pLmLsΨsdird+Ψsqirq.


### 2.3. Dynamic Model of DC-Link and Filter

A commonly used dynamic Park model of the grid side can be expressed by (15)–(19):(15)vid=Rfifd+Lfdifddt−ωgLfifq+vgd,(16)viq=Rfifq+Lfdifqdt+ωgLfifd+vgq,where vfd, vfq and vid, viq are direct component and quadrature component of the filter voltage and the source inverter voltage; ifd and ifq are the filter current components in d-q frame; and Rf and Lf are the resistance and the inductance of the filter.The active and reactive filter powers injected or absorbed from the grid can be written as follows:(17)Pf=vfdifd+vfqifq,(18)Qf=vfqifd−vfdifq.The DC-bus voltage equation can be written as follows:(19)idc=Cdvdcdt=ir−if,where the currents ir and if are the rotor-side current and the filter current, respectively; vdc is the DC-link voltage; idc is the DC-bus current; and C is the capacitor [12].


## 2.1. Wind Energy Conversion System Mechanical Model

The turbine captures the wind kinetic energy and converts it to a torque that rotates the rotors blades (mechanical energy). The formula of aerodynamic powerPaero available from the turbine rotor can be given by(1)Paero=Cpλ,βPw=12ρAtCpλ,βw3,where Pw is the wind power; Cp is the power coefficient; β and λ represent the blades’ pitch angle and the turbine tip speed ratio, respectively; At = πR2 is the turbine blade area; w is the wind speed; and ρ represents the air density [10].The aerodynamic torqueTaero is given by the following expression:(2)Taero=12ωturρAtCpλ,βw3,where ωtur is the turbine speed and R is the turbine blade radius.The wind turbine aerodynamic is characterized by a power coefficient; it is a nonlinear equation which depends on λ and β; it is expressed by(3)Cpλ,β=k1k21λj−k3β−k4exp−k5/λj+k6λ,where(4)1λj=1λ+k7β−k8β3+1,λ=Rωturw,with k1 = 0.5176, k2 = 166, k3 = 0.4, k4 = 5, k5 = 21, k6 = 0.0068, k7 = 0.08, and k8 = 0.035.As shown in Figure2, the aerodynamic coefficient has a maximum power point (point M) when β = 0◦ and λ = 8.1 (with Cpmax=0.48).Figure 2
The aerodynamic coefficient characteristics of the wind turbine system.The gearbox torqueTg and mechanical angular speed ωmec are given by(5)Tg=TaeroG,ωmec=Gωtur,where G is the gearbox ratio.The system mechanical equation is(6)Jdωmecdt=Tg−Tem−fωmec,where Tem and f are the DFIG electromagnetic torque and the friction coefficient, respectively. J represents the total inertia which can be calculated by(7)J=JturG2+JDFIG,where JDFIG and Jtur are the inertias of the aggregated generator and the turbine.


## 2.2. Dynamic Model of DFIG

In order to model the DFIG, it is assumed that the stator and rotor windings are placed sinusoidally and symmetrically, and the magnetical saturation effects of all the windings are neglectable. Figure3 illustrates the block of the generator phase in dq frame reference.Figure 3
Schematic diagram of the DFIG equivalent circuit [11].The stator and rotor windings equations are presented by (8)–(11):(8)dΨsddt=−Rsisd+ωsΨsq+vsd,(9)dΨsqdt=−Rsisq−ωsΨsd+vsq,(10)dΨrddt=−Rrird+ωrΨrq+vrd,(11)dΨrqdt=−Rrirq−ωrΨrd+vrq,where vsd and vsq are the stator voltages in the direct-quadrature frame and vrd and vrq are the rotor voltages in the direct-quadrature frame. isd and isq are the direct and quadrature stator currents, ird and irq are the direct and quadrature rotor currents, Ψsd and Ψsq are the direct-quadrature frame stator fields, and Ψrd and Ψrq are the direct-quadrature frame rotor fields. Also, Rs and Rr are the stator and the rotor phase resistances; ωr and ωs are the rotor electrical speed and the stator electrical speed, respectively.The equations of stator and rotor fields are(12)Ψsd=Lm+Lsσisd+Lmird,Ψsq=Lm+Lsσisq+Lmirq,Ψrd=Lm+Lrσird+Lmisd,Ψrq=Lm+Lrσirq+Lmisq,where Lm is the mutual inductance and Lsσ and Lrσ are the stator leakage inductance and the rotor leakage inductance, respectively [9].The electrical active and reactive powers (Ps and Qs) delivered by generator stator side and generator rotor side can be written as follows:(13)Ps=vsdisd+vsqisq,Qs=vsqisd−vsdisq,Pr=vrdird+vrqirq,Qr=vrqird−vrdirq.The electromagnetic torque equation is then(14)Tem=−pLmLsΨsdird+Ψsqirq.


## 2.3. Dynamic Model of DC-Link and Filter

A commonly used dynamic Park model of the grid side can be expressed by (15)–(19):(15)vid=Rfifd+Lfdifddt−ωgLfifq+vgd,(16)viq=Rfifq+Lfdifqdt+ωgLfifd+vgq,where vfd, vfq and vid, viq are direct component and quadrature component of the filter voltage and the source inverter voltage; ifd and ifq are the filter current components in d-q frame; and Rf and Lf are the resistance and the inductance of the filter.The active and reactive filter powers injected or absorbed from the grid can be written as follows:(17)Pf=vfdifd+vfqifq,(18)Qf=vfqifd−vfdifq.The DC-bus voltage equation can be written as follows:(19)idc=Cdvdcdt=ir−if,where the currents ir and if are the rotor-side current and the filter current, respectively; vdc is the DC-link voltage; idc is the DC-bus current; and C is the capacitor [12].


## 3. Mathematical Theory of Active Disturbance Rejection Control

The active disturbance rejection control (ADRC) strategy is a novel adaptive robust strategy; it was proposed by Jingqing Han in 1995 [13, 14]. The main aim of this strategy is to estimate and compensate in real time the various external and internal disturbances; it is based on an extended state observer (ESO). In this work, we have developed and applied a linear ADRC in order to reduce the model complexity and design compared to the nonlinear ADRC.


### 3.1. Linear ADRC Design

We consider a nonlinear process (m-dimensional) variable in time, where u(t) and y(t) are the input and output signals, respectively; it can be written as follows:(20)ymt=dintyt,y1t,y2t,…,ym−1t,ut+dextt+b0ut,where dint(y(t), y(1)(t), y(2)(t), …, y(m−1)(t), u(t)) represents the nonlinear internal dynamic of the system, which is assumed to be unknown, and dext(t) and b0 represent the external disturbances and the known system parameter, respectively.If we bring together all the disturbances affecting the system to be controlled:d(t) = dint(t) + dext(t). Wherein, the system equation is rewritten in the following form:(21)ymt=dt+b0ut.Instead of looking for a model ofd(t), ADRC offers another way that significantly reduces the dependence of the control on accurate system modeling, as the proposed strategy consists in estimating f(t) and then cancelling it in real time using an appropriate control signal u(t).As a result, the canonical form of linear ADRC is given by(22)y˙t=dt+b0ut.


### 3.2. Extended State Observer (ESO)

The extended state observer is a Luenberger observer that estimates the termf(t) and then compensates the effect of disturbances d(t) on the system; therefore, we consider an extended state vector with an additional state representing the total disturbances of the system:(23)x=x1x2…xm−1xmxm+1t=yy˙…ymdtt.The system representation in matrix form:(24)x˙=Ax+Bu+Ed˙,y=Cx,with(25)Am+1,m+1=010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1000⋯0,Bm+1,1=00⋯b00T,Cm+1,1=10⋯00,Em+1,1=00⋯01T.The Luenberger state observer (ESO model) can be designed as(26)x^˙=Ax^+Bu+Ly−y^,y^=Cx^,where L=β1β2T and x^=x^1x^2T represent the observer gain vector and the estimated variables vector, respectively [15].The error betweenx and x^ can be given by(27)ε=x−x^.Therefore, the dynamics of error estimation is(28)ε˙=A−LCε,where(29)A−LC=−β110⋯0−β201⋯0⋮⋮⋮⋱⋮−βn00⋯1−βm+100⋯0.In order to ensure asymptotic convergence of the estimation error (ɛ ⟶ 0 when t ⟶ ∞), the observer requires that the parameters of the gains vector L are chosen in such a way that (A − LC) forms a Hurwitz matrix, that is to say, the poles of its polynomial characteristic PESO(s) of (A − LC) are all with strictly negative real parts [16].(30)PESOs=detsIm+1−A−LC=sm+1+β1sm+β2sm−1+⋯+βms+βm+1.Ultimately, placing all observer poles at one location is also known as “bandwidth parameterization” [13]. Taking into account all these constraints, the cutoff pulse of the extended state observer ωo is chosen to have a suitable stabilization time; its (m + 1) poles are placed to −ωo.We can compute the necessary observer gains for the common pole location from its characteristic polynomial:(31)PESOs=sm+1+β1sm+β2sm−1+β3sm−2+⋯+βms+βm+1=s+ωobsm+1.The general expression of observer’s gains is given by(32)βi=m+1!m+1−i!i!ωobsi.From equation (32), the extended state observer gains for a first-order plant are given by(33)β1β2=2ωobsωobs2,and ωobs is chosen in order to ensure both fast dynamics and stability of the observer.


### 3.3. The Control Law

By taking into consideration the estimated variables, a control law is determined to eliminate the effect of total disturbances acting on the system:(34)ut=u0t−x^m+1tb0.Then, system 30 becomes(35)yt=dt−x^m+1t+u0t.Ifx^m+1t≈ft, then the system is(36)yt=u0t.To implement the disturbance rejection, the actual state feedback control law is designed by(37)u0t=Kprt−x^t,where r(t) is the reference signal. Since x^1t,…,x^mt is a correct estimation of y(t), …, y(m−1)(t), the design of a linear active disturbance rejection controller can be presented by(38)x^˙t=A−LCx^t+But+Lyt,ut=Kpr^t−x^t,x^1t=y^t,x^2t=f^t.Figure4 presents the active disturbance rejection control strategy and corrector structure linear ADRC.Figure 4
Linear ADRC structure [17].The tuning of the proportional gain of the linear controller is given bykp=ωc=4/tr, where tr is the process desired time [18]. And for ωobs, it is taken in a range of ωobs = 3 ∼ 10ωc.


## 3.1. Linear ADRC Design

We consider a nonlinear process (m-dimensional) variable in time, where u(t) and y(t) are the input and output signals, respectively; it can be written as follows:(20)ymt=dintyt,y1t,y2t,…,ym−1t,ut+dextt+b0ut,where dint(y(t), y(1)(t), y(2)(t), …, y(m−1)(t), u(t)) represents the nonlinear internal dynamic of the system, which is assumed to be unknown, and dext(t) and b0 represent the external disturbances and the known system parameter, respectively.If we bring together all the disturbances affecting the system to be controlled:d(t) = dint(t) + dext(t). Wherein, the system equation is rewritten in the following form:(21)ymt=dt+b0ut.Instead of looking for a model ofd(t), ADRC offers another way that significantly reduces the dependence of the control on accurate system modeling, as the proposed strategy consists in estimating f(t) and then cancelling it in real time using an appropriate control signal u(t).As a result, the canonical form of linear ADRC is given by(22)y˙t=dt+b0ut.


## 3.2. Extended State Observer (ESO)

The extended state observer is a Luenberger observer that estimates the termf(t) and then compensates the effect of disturbances d(t) on the system; therefore, we consider an extended state vector with an additional state representing the total disturbances of the system:(23)x=x1x2…xm−1xmxm+1t=yy˙…ymdtt.The system representation in matrix form:(24)x˙=Ax+Bu+Ed˙,y=Cx,with(25)Am+1,m+1=010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1000⋯0,Bm+1,1=00⋯b00T,Cm+1,1=10⋯00,Em+1,1=00⋯01T.The Luenberger state observer (ESO model) can be designed as(26)x^˙=Ax^+Bu+Ly−y^,y^=Cx^,where L=β1β2T and x^=x^1x^2T represent the observer gain vector and the estimated variables vector, respectively [15].The error betweenx and x^ can be given by(27)ε=x−x^.Therefore, the dynamics of error estimation is(28)ε˙=A−LCε,where(29)A−LC=−β110⋯0−β201⋯0⋮⋮⋮⋱⋮−βn00⋯1−βm+100⋯0.In order to ensure asymptotic convergence of the estimation error (ɛ ⟶ 0 when t ⟶ ∞), the observer requires that the parameters of the gains vector L are chosen in such a way that (A − LC) forms a Hurwitz matrix, that is to say, the poles of its polynomial characteristic PESO(s) of (A − LC) are all with strictly negative real parts [16].(30)PESOs=detsIm+1−A−LC=sm+1+β1sm+β2sm−1+⋯+βms+βm+1.Ultimately, placing all observer poles at one location is also known as “bandwidth parameterization” [13]. Taking into account all these constraints, the cutoff pulse of the extended state observer ωo is chosen to have a suitable stabilization time; its (m + 1) poles are placed to −ωo.We can compute the necessary observer gains for the common pole location from its characteristic polynomial:(31)PESOs=sm+1+β1sm+β2sm−1+β3sm−2+⋯+βms+βm+1=s+ωobsm+1.The general expression of observer’s gains is given by(32)βi=m+1!m+1−i!i!ωobsi.From equation (32), the extended state observer gains for a first-order plant are given by(33)β1β2=2ωobsωobs2,and ωobs is chosen in order to ensure both fast dynamics and stability of the observer.


## 3.3. The Control Law

By taking into consideration the estimated variables, a control law is determined to eliminate the effect of total disturbances acting on the system:(34)ut=u0t−x^m+1tb0.Then, system 30 becomes(35)yt=dt−x^m+1t+u0t.Ifx^m+1t≈ft, then the system is(36)yt=u0t.To implement the disturbance rejection, the actual state feedback control law is designed by(37)u0t=Kprt−x^t,where r(t) is the reference signal. Since x^1t,…,x^mt is a correct estimation of y(t), …, y(m−1)(t), the design of a linear active disturbance rejection controller can be presented by(38)x^˙t=A−LCx^t+But+Lyt,ut=Kpr^t−x^t,x^1t=y^t,x^2t=f^t.Figure4 presents the active disturbance rejection control strategy and corrector structure linear ADRC.Figure 4
Linear ADRC structure [17].The tuning of the proportional gain of the linear controller is given bykp=ωc=4/tr, where tr is the process desired time [18]. And for ωobs, it is taken in a range of ωobs = 3 ∼ 10ωc.


## 4. MPPT Algorithms and Pitch Angle Control

### 4.1. MPPT Algorithms (Optimal Torque Control)

As mentioned previously in the introduction, the maximum power point tracking algorithm is a technique which is used to extract the maximum power from the turbine for different wind speeds. As described in [19], the available MPPT algorithms can be classified as either with or without wind sensors. The authors in [2] have demonstrated the superiority of the optimal torque control (OTC) method in terms of simplicity and accuracy (because it does not measure the wind speed directly). The principle of this method is to maintain the tip speed ratio to an optimum value in order to maximize the power coefficient (λ = λopt; Cp=Cpmax).The maximum aerodynamic power can be rearranged as follows:(39)Paeromax=12ρAtCpmaxλopt3R3G3ωmec3.The optimal aerodynamic torque can be expressed by(40)Taeroopt=Koptωmec2,where the optimal torque coefficient is(41)Kopt=12ρAtCpmaxλopt3R3G3.


### 4.2. Pitch Angle Control

The blade orientation system is essentially used to limit the power extracted by adjusting the blade pitch angleβ. The blade positioning mechanism consists in orienting the blades at a reference βref angle by means of a hydraulic or electrical system. The choice of this angle generally comes from an external loop to regulate either the turbine speed or the generated mechanical power. With such a system, the blades are rotated by a device called the pitch control [20].The reference of the pitch angleβ comes from the mechanical power control Pt, regulated around its nominal power. As long as the speed of the wind does not reach its nominal value (around 12 m/s), the blade orientation mechanism does not intervene and β is 0. Beyond that, the regulation intervenes and the angle varies between 0° and 10°.The equation of pitch angleβ can be expressed by(42)β=11+Tβsβref,where s is a Laplace operator and Tβ is the time constant of the pitch servo. The pitch angle control scheme is depicted in Figure 5.Figure 5
Block diagrams of the blade pitch angle control.Generally, the pitch angle rateβ˙ varies around 10 degrees, and the angle β˙ varies between 0 and 45 degrees [3].


## 4.1. MPPT Algorithms (Optimal Torque Control)

As mentioned previously in the introduction, the maximum power point tracking algorithm is a technique which is used to extract the maximum power from the turbine for different wind speeds. As described in [19], the available MPPT algorithms can be classified as either with or without wind sensors. The authors in [2] have demonstrated the superiority of the optimal torque control (OTC) method in terms of simplicity and accuracy (because it does not measure the wind speed directly). The principle of this method is to maintain the tip speed ratio to an optimum value in order to maximize the power coefficient (λ = λopt; Cp=Cpmax).The maximum aerodynamic power can be rearranged as follows:(39)Paeromax=12ρAtCpmaxλopt3R3G3ωmec3.The optimal aerodynamic torque can be expressed by(40)Taeroopt=Koptωmec2,where the optimal torque coefficient is(41)Kopt=12ρAtCpmaxλopt3R3G3.


## 4.2. Pitch Angle Control

The blade orientation system is essentially used to limit the power extracted by adjusting the blade pitch angleβ. The blade positioning mechanism consists in orienting the blades at a reference βref angle by means of a hydraulic or electrical system. The choice of this angle generally comes from an external loop to regulate either the turbine speed or the generated mechanical power. With such a system, the blades are rotated by a device called the pitch control [20].The reference of the pitch angleβ comes from the mechanical power control Pt, regulated around its nominal power. As long as the speed of the wind does not reach its nominal value (around 12 m/s), the blade orientation mechanism does not intervene and β is 0. Beyond that, the regulation intervenes and the angle varies between 0° and 10°.The equation of pitch angleβ can be expressed by(42)β=11+Tβsβref,where s is a Laplace operator and Tβ is the time constant of the pitch servo. The pitch angle control scheme is depicted in Figure 5.Figure 5
Block diagrams of the blade pitch angle control.Generally, the pitch angle rateβ˙ varies around 10 degrees, and the angle β˙ varies between 0 and 45 degrees [3].


## 5. The Novel Adaptive Active Disturbance Rejection Control Strategy of Wind Turbine System

The control strategies of the variable speed wind turbine system based on a DFIG connected into grid are developed in this section. The first strategy is the classical one based on the proportional integral (PI) controller, and the other one is the novel strategy based on the linear ADRC. We began with the classical control strategy by the proportional integral.


### 5.1. Wind Turbine System Control by PI

The rotor-side converter controllers are used to control the stator active and reactive powers. Furthermore, the grid-side converter controllers are used to regulate the DC-link voltage and to control the grid powers (active and reactive) as shown in Figure6.Figure 6
The PI control strategy of the DFIG used in WTS.


#### 5.1.1. Rotor-Side Converter Control

To control the rotor-side converter, we have applied the indirect stator field oriented control (ISFOC), in which the stator field vector Ψs is oriented along the axis d. Then, the stator field components are Ψsq = 0; Ψsd = Ψs [21].it is assumed for high power machine that the resistanceRs in negligible and that the stator field Ψs is constant when the grid is stable.The DFIG model can be written as(43)dirddt=−c1ird+ωrirq+c2vrd,(44)dirqdt=−c1irq−ωrird−ωrc3Ψs+c2vrq,where c1=Rr/σLr; c2=1/σLr; c3=Lm/σLsLr; ωr = ωg − pΩmecThe expression of the rotor angle is(45)θr=∫ωg−ωmecdt.The compensation and the coupling terms are(46)erd=1c2ωrirq,erq=1c2ωrird+c3c2ωrΨs.The electromagnetic torque equation is then(47)Tem=−32pLmLsΨsirq.The active and reactive powers can be described by(48)Ps=−32LmLsvgirq,(49)Qs=32−LmLsvgird+ΨsLsvg.From equations equations (47) and (49), we can derive the reference currents irqref and irdref to regulate the active power produced by the DFIG and to control the reactive power supplied or absorbed by the generator:(50)irqref=−23LspΨsLmPaeroopt,(51)irdref=23ΨsLm−LsvgLmQsref,where the torque Paeroopt is generated by MPPT and Qsref is the reference reactive power; it is given as zero in normal operations Qsref=0 to obtain a unit power factor [22].


#### 5.1.2. Grid-Side Converter Control

Among the most used techniques to control the grid-side converter, we can find the voltage oriented control (VOC) or the direct power control (DPC) techniques. However, as described in [3, 9–23], the VOC is considered to be more efficient due to lower energy losses and lower current distortion compared to direct power control (DPC). Hence, vgd = 0 and vgq = vg. Then, the power Pf and Qf are expressed in following equations:(52)Pf=32vgifq,Qf=32vgifd,and the filter currents are(53)difddt=−c4ifd+ωgifq+c5vid,(54)difqdt=−c4ifq−ωgifd+c5viq−vg,where c4=Rf/Lf and c5=1/Lf.The compensation and the coupling terms are(55)efd=−1c5ωgifq,efq=vg+1c5ωgifd.The direct reference currentifdref is(56)ifdref=23vgQfref,where Qfref is the reference reactive power; it is given as zero to obtain a unit power factor in grid side.Figure6 shows the block diagram of the DFIG wind energy conversion system by PI control strategy.Finally, there are many methods to tune the PI controller parameters (gainsKp and Ki) of rotor-side control and grid-side control. As described in [3], the compensation method for the closed loop system is not complex and it is effective, which is shown in Figure 7.Figure 7
The method to tune the PI controller parameters [24].C(s) is the PI controller. The transfer functions are Ir(s), V(s), and If(s) of the rotor currents, DC-bus voltage, and grid currents, respectively, which are(57)Cs=Kp+Kis;Irs=kr1+γrs;Vs=1Cs;Ifs=kf1+γfs,where: kr=1/Rr, kf=1/Rf, γr=σLr/Rr, and γf=Lf/Rf.The rotor-side converter PI controller parameters are(58)Kpir=3γrtrckr;Kiir=3trckr,where trc is the settling time for the rotor currents.The grid-side converter PI controller parameters:(i)
For the grid currents:(59)Kpif=3γftrgkf;Kiif=3trfkf.(ii)
For the DC-link voltage:(60)Kpvdc=2ζCω0;Kivdc=Cω02;ω0=3trvζ,where trf and trv are the settling time for the grid currents and the DC-link voltage, respectively and ζ is the damping coefficient; it is generally between 0 < ζ < 1 in terms of stability. Thus, in regulation loops of the grid side, we will take into consideration that the outer loop (vdc) has a slower response time than the dynamics of the inner loop (if) [25, 26].


### 5.2. Wind Turbine System Control by ADRC

As mentioned previously in the Section3, the active disturbance rejection control (ADRC) is a novel and robust control strategy. Similar to the control by PI, we elaborate the rotor-side converter and the grid-side converter controls by linear ADRC. Figure 8 shows the block diagram of the DFIG wind turbine system by ADRC control strategy connected into grid.Figure 8
The ADRC control strategy of the DFIG used in WECS.


#### 5.2.1. Linear ADRC Design for Rotor-Side Converter

To control rotor-side converter by ADRC, we have used two linear ADRC controllers which control the wind turbine system powers to extract the available maximum power from the wind turbine. Ultimately, to design the ADRC regulator on rotor side, we have achieved the rotor current regulations by two linear ADRC controllers, wherein equations (43) and (44) are adapted to the canonical form of linear ARDC:(61)dirdqdt=ftirdqt+b0irdquirdqt.(i)
The design of the linear ADRC controller ford-axis rotor current loop ird:(1)
Total (external and internal) disturbances affecting the rotor currentird:(62)ftird=−c1irq+ωrird+c2−b0irqvrq.(2)
Known parts of the controlled system parameters:(63)b0ird=c2.(3)
Control inputs of the rotor currentird:(64)uird=vrd.(ii)
The design of the linear ADRC controller forq-axis rotor current loop irq:(1)
Total (external and internal) disturbances affecting the rotor currentirq:(65)ftirq=−c1irq+ωrird−ωrc3Ψs+c2−b0isqvrq.(2)
Known parts of the controlled system parameters:(66)b0irq=c2.(3)
Control inputs of the rotor currentirq:(67)uirq=vrq.


#### 5.2.2. Linear ADRC Design for Grid-Side Converter

The grid-side converter is used to control the active power injected into the utility grid through the regulation of the DC-link voltage and to control the reactive power to achieve a unity power factor for wind variations. The voltage oriented control (VOC) based linear ADRC is used here and the controllers are designed as follows:(i)
The filter current controllers:Similar to the rotor-side control, the filter currents are adapted to canonical form of linear ADRC. Replacing equations (53) and (54) into equation (68):(68)difdqdt=ftifdqt+b0ifdquifdqt.(1)
Total (external and internal) disturbances affecting the rotor current (ird and irq):(69)ftifd=−c4ifd+ωgifq+c5−b0ifdvid,ftifq=−c4ifq+ωgifd−c5vg+c5−b0ifqviq.(2)
Known parts of the controlled system parameters:(70)b0ifd=−c5,b0ifq=−c5.(3)
Control inputs of the rotor currents (ird and irq):(71)uifd=vid,uifq=viq.(ii)
The DC-link voltage controller:In equation (19), motivated by the work in [3, 9–27], we neglect all the losses in converters, wherein, the powers Pr and Pf can be written as follows: Pr = irvdc and Pf = ifvdc.The DC-link voltage equation:(72)dvdc2dt=2CPr−3Cvgifq.Furthermore, we substituteU=vdc2:(73)dUdt=c6Pr−c7ifq,wherec6=2/C and c7=3/Cvg.After adaptation of the DC-Link voltage equation to the canonical form, Linear-ADRC design for DC-Link voltage regulation is as follows.(1)
Total (external and internal) disturbances affecting the DC-link voltagevdc:(74)ftvdc=−c6Pr+−k7−b0vdcifq.(2)
Known parts of the controlled system parameters:(75)b0vdc=c7.(3)
Control inputs of the DC-link voltagevdc:(76)uvdc=ifq.


## 5.1. Wind Turbine System Control by PI

The rotor-side converter controllers are used to control the stator active and reactive powers. Furthermore, the grid-side converter controllers are used to regulate the DC-link voltage and to control the grid powers (active and reactive) as shown in Figure6.Figure 6
The PI control strategy of the DFIG used in WTS.


### 5.1.1. Rotor-Side Converter Control

To control the rotor-side converter, we have applied the indirect stator field oriented control (ISFOC), in which the stator field vector Ψs is oriented along the axis d. Then, the stator field components are Ψsq = 0; Ψsd = Ψs [21].it is assumed for high power machine that the resistanceRs in negligible and that the stator field Ψs is constant when the grid is stable.The DFIG model can be written as(43)dirddt=−c1ird+ωrirq+c2vrd,(44)dirqdt=−c1irq−ωrird−ωrc3Ψs+c2vrq,where c1=Rr/σLr; c2=1/σLr; c3=Lm/σLsLr; ωr = ωg − pΩmecThe expression of the rotor angle is(45)θr=∫ωg−ωmecdt.The compensation and the coupling terms are(46)erd=1c2ωrirq,erq=1c2ωrird+c3c2ωrΨs.The electromagnetic torque equation is then(47)Tem=−32pLmLsΨsirq.The active and reactive powers can be described by(48)Ps=−32LmLsvgirq,(49)Qs=32−LmLsvgird+ΨsLsvg.From equations equations (47) and (49), we can derive the reference currents irqref and irdref to regulate the active power produced by the DFIG and to control the reactive power supplied or absorbed by the generator:(50)irqref=−23LspΨsLmPaeroopt,(51)irdref=23ΨsLm−LsvgLmQsref,where the torque Paeroopt is generated by MPPT and Qsref is the reference reactive power; it is given as zero in normal operations Qsref=0 to obtain a unit power factor [22].


### 5.1.2. Grid-Side Converter Control

Among the most used techniques to control the grid-side converter, we can find the voltage oriented control (VOC) or the direct power control (DPC) techniques. However, as described in [3, 9–23], the VOC is considered to be more efficient due to lower energy losses and lower current distortion compared to direct power control (DPC). Hence, vgd = 0 and vgq = vg. Then, the power Pf and Qf are expressed in following equations:(52)Pf=32vgifq,Qf=32vgifd,and the filter currents are(53)difddt=−c4ifd+ωgifq+c5vid,(54)difqdt=−c4ifq−ωgifd+c5viq−vg,where c4=Rf/Lf and c5=1/Lf.The compensation and the coupling terms are(55)efd=−1c5ωgifq,efq=vg+1c5ωgifd.The direct reference currentifdref is(56)ifdref=23vgQfref,where Qfref is the reference reactive power; it is given as zero to obtain a unit power factor in grid side.Figure6 shows the block diagram of the DFIG wind energy conversion system by PI control strategy.Finally, there are many methods to tune the PI controller parameters (gainsKp and Ki) of rotor-side control and grid-side control. As described in [3], the compensation method for the closed loop system is not complex and it is effective, which is shown in Figure 7.Figure 7
The method to tune the PI controller parameters [24].C(s) is the PI controller. The transfer functions are Ir(s), V(s), and If(s) of the rotor currents, DC-bus voltage, and grid currents, respectively, which are(57)Cs=Kp+Kis;Irs=kr1+γrs;Vs=1Cs;Ifs=kf1+γfs,where: kr=1/Rr, kf=1/Rf, γr=σLr/Rr, and γf=Lf/Rf.The rotor-side converter PI controller parameters are(58)Kpir=3γrtrckr;Kiir=3trckr,where trc is the settling time for the rotor currents.The grid-side converter PI controller parameters:(i)
For the grid currents:(59)Kpif=3γftrgkf;Kiif=3trfkf.(ii)
For the DC-link voltage:(60)Kpvdc=2ζCω0;Kivdc=Cω02;ω0=3trvζ,where trf and trv are the settling time for the grid currents and the DC-link voltage, respectively and ζ is the damping coefficient; it is generally between 0 < ζ < 1 in terms of stability. Thus, in regulation loops of the grid side, we will take into consideration that the outer loop (vdc) has a slower response time than the dynamics of the inner loop (if) [25, 26].


## 5.1.1. Rotor-Side Converter Control

To control the rotor-side converter, we have applied the indirect stator field oriented control (ISFOC), in which the stator field vector Ψs is oriented along the axis d. Then, the stator field components are Ψsq = 0; Ψsd = Ψs [21].it is assumed for high power machine that the resistanceRs in negligible and that the stator field Ψs is constant when the grid is stable.The DFIG model can be written as(43)dirddt=−c1ird+ωrirq+c2vrd,(44)dirqdt=−c1irq−ωrird−ωrc3Ψs+c2vrq,where c1=Rr/σLr; c2=1/σLr; c3=Lm/σLsLr; ωr = ωg − pΩmecThe expression of the rotor angle is(45)θr=∫ωg−ωmecdt.The compensation and the coupling terms are(46)erd=1c2ωrirq,erq=1c2ωrird+c3c2ωrΨs.The electromagnetic torque equation is then(47)Tem=−32pLmLsΨsirq.The active and reactive powers can be described by(48)Ps=−32LmLsvgirq,(49)Qs=32−LmLsvgird+ΨsLsvg.From equations equations (47) and (49), we can derive the reference currents irqref and irdref to regulate the active power produced by the DFIG and to control the reactive power supplied or absorbed by the generator:(50)irqref=−23LspΨsLmPaeroopt,(51)irdref=23ΨsLm−LsvgLmQsref,where the torque Paeroopt is generated by MPPT and Qsref is the reference reactive power; it is given as zero in normal operations Qsref=0 to obtain a unit power factor [22].


## 5.1.2. Grid-Side Converter Control

Among the most used techniques to control the grid-side converter, we can find the voltage oriented control (VOC) or the direct power control (DPC) techniques. However, as described in [3, 9–23], the VOC is considered to be more efficient due to lower energy losses and lower current distortion compared to direct power control (DPC). Hence, vgd = 0 and vgq = vg. Then, the power Pf and Qf are expressed in following equations:(52)Pf=32vgifq,Qf=32vgifd,and the filter currents are(53)difddt=−c4ifd+ωgifq+c5vid,(54)difqdt=−c4ifq−ωgifd+c5viq−vg,where c4=Rf/Lf and c5=1/Lf.The compensation and the coupling terms are(55)efd=−1c5ωgifq,efq=vg+1c5ωgifd.The direct reference currentifdref is(56)ifdref=23vgQfref,where Qfref is the reference reactive power; it is given as zero to obtain a unit power factor in grid side.Figure6 shows the block diagram of the DFIG wind energy conversion system by PI control strategy.Finally, there are many methods to tune the PI controller parameters (gainsKp and Ki) of rotor-side control and grid-side control. As described in [3], the compensation method for the closed loop system is not complex and it is effective, which is shown in Figure 7.Figure 7
The method to tune the PI controller parameters [24].C(s) is the PI controller. The transfer functions are Ir(s), V(s), and If(s) of the rotor currents, DC-bus voltage, and grid currents, respectively, which are(57)Cs=Kp+Kis;Irs=kr1+γrs;Vs=1Cs;Ifs=kf1+γfs,where: kr=1/Rr, kf=1/Rf, γr=σLr/Rr, and γf=Lf/Rf.The rotor-side converter PI controller parameters are(58)Kpir=3γrtrckr;Kiir=3trckr,where trc is the settling time for the rotor currents.The grid-side converter PI controller parameters:(i)
For the grid currents:(59)Kpif=3γftrgkf;Kiif=3trfkf.(ii)
For the DC-link voltage:(60)Kpvdc=2ζCω0;Kivdc=Cω02;ω0=3trvζ,where trf and trv are the settling time for the grid currents and the DC-link voltage, respectively and ζ is the damping coefficient; it is generally between 0 < ζ < 1 in terms of stability. Thus, in regulation loops of the grid side, we will take into consideration that the outer loop (vdc) has a slower response time than the dynamics of the inner loop (if) [25, 26].


## 5.2. Wind Turbine System Control by ADRC

As mentioned previously in the Section3, the active disturbance rejection control (ADRC) is a novel and robust control strategy. Similar to the control by PI, we elaborate the rotor-side converter and the grid-side converter controls by linear ADRC. Figure 8 shows the block diagram of the DFIG wind turbine system by ADRC control strategy connected into grid.Figure 8
The ADRC control strategy of the DFIG used in WECS.


### 5.2.1. Linear ADRC Design for Rotor-Side Converter

To control rotor-side converter by ADRC, we have used two linear ADRC controllers which control the wind turbine system powers to extract the available maximum power from the wind turbine. Ultimately, to design the ADRC regulator on rotor side, we have achieved the rotor current regulations by two linear ADRC controllers, wherein equations (43) and (44) are adapted to the canonical form of linear ARDC:(61)dirdqdt=ftirdqt+b0irdquirdqt.(i)
The design of the linear ADRC controller ford-axis rotor current loop ird:(1)
Total (external and internal) disturbances affecting the rotor currentird:(62)ftird=−c1irq+ωrird+c2−b0irqvrq.(2)
Known parts of the controlled system parameters:(63)b0ird=c2.(3)
Control inputs of the rotor currentird:(64)uird=vrd.(ii)
The design of the linear ADRC controller forq-axis rotor current loop irq:(1)
Total (external and internal) disturbances affecting the rotor currentirq:(65)ftirq=−c1irq+ωrird−ωrc3Ψs+c2−b0isqvrq.(2)
Known parts of the controlled system parameters:(66)b0irq=c2.(3)
Control inputs of the rotor currentirq:(67)uirq=vrq.


### 5.2.2. Linear ADRC Design for Grid-Side Converter

The grid-side converter is used to control the active power injected into the utility grid through the regulation of the DC-link voltage and to control the reactive power to achieve a unity power factor for wind variations. The voltage oriented control (VOC) based linear ADRC is used here and the controllers are designed as follows:(i)
The filter current controllers:Similar to the rotor-side control, the filter currents are adapted to canonical form of linear ADRC. Replacing equations (53) and (54) into equation (68):(68)difdqdt=ftifdqt+b0ifdquifdqt.(1)
Total (external and internal) disturbances affecting the rotor current (ird and irq):(69)ftifd=−c4ifd+ωgifq+c5−b0ifdvid,ftifq=−c4ifq+ωgifd−c5vg+c5−b0ifqviq.(2)
Known parts of the controlled system parameters:(70)b0ifd=−c5,b0ifq=−c5.(3)
Control inputs of the rotor currents (ird and irq):(71)uifd=vid,uifq=viq.(ii)
The DC-link voltage controller:In equation (19), motivated by the work in [3, 9–27], we neglect all the losses in converters, wherein, the powers Pr and Pf can be written as follows: Pr = irvdc and Pf = ifvdc.The DC-link voltage equation:(72)dvdc2dt=2CPr−3Cvgifq.Furthermore, we substituteU=vdc2:(73)dUdt=c6Pr−c7ifq,wherec6=2/C and c7=3/Cvg.After adaptation of the DC-Link voltage equation to the canonical form, Linear-ADRC design for DC-Link voltage regulation is as follows.(1)
Total (external and internal) disturbances affecting the DC-link voltagevdc:(74)ftvdc=−c6Pr+−k7−b0vdcifq.(2)
Known parts of the controlled system parameters:(75)b0vdc=c7.(3)
Control inputs of the DC-link voltagevdc:(76)uvdc=ifq.


## 5.2.1. Linear ADRC Design for Rotor-Side Converter

To control rotor-side converter by ADRC, we have used two linear ADRC controllers which control the wind turbine system powers to extract the available maximum power from the wind turbine. Ultimately, to design the ADRC regulator on rotor side, we have achieved the rotor current regulations by two linear ADRC controllers, wherein equations (43) and (44) are adapted to the canonical form of linear ARDC:(61)dirdqdt=ftirdqt+b0irdquirdqt.(i)
The design of the linear ADRC controller ford-axis rotor current loop ird:(1)
Total (external and internal) disturbances affecting the rotor currentird:(62)ftird=−c1irq+ωrird+c2−b0irqvrq.(2)
Known parts of the controlled system parameters:(63)b0ird=c2.(3)
Control inputs of the rotor currentird:(64)uird=vrd.(ii)
The design of the linear ADRC controller forq-axis rotor current loop irq:(1)
Total (external and internal) disturbances affecting the rotor currentirq:(65)ftirq=−c1irq+ωrird−ωrc3Ψs+c2−b0isqvrq.(2)
Known parts of the controlled system parameters:(66)b0irq=c2.(3)
Control inputs of the rotor currentirq:(67)uirq=vrq.


## 5.2.2. Linear ADRC Design for Grid-Side Converter

The grid-side converter is used to control the active power injected into the utility grid through the regulation of the DC-link voltage and to control the reactive power to achieve a unity power factor for wind variations. The voltage oriented control (VOC) based linear ADRC is used here and the controllers are designed as follows:(i)
The filter current controllers:Similar to the rotor-side control, the filter currents are adapted to canonical form of linear ADRC. Replacing equations (53) and (54) into equation (68):(68)difdqdt=ftifdqt+b0ifdquifdqt.(1)
Total (external and internal) disturbances affecting the rotor current (ird and irq):(69)ftifd=−c4ifd+ωgifq+c5−b0ifdvid,ftifq=−c4ifq+ωgifd−c5vg+c5−b0ifqviq.(2)
Known parts of the controlled system parameters:(70)b0ifd=−c5,b0ifq=−c5.(3)
Control inputs of the rotor currents (ird and irq):(71)uifd=vid,uifq=viq.(ii)
The DC-link voltage controller:In equation (19), motivated by the work in [3, 9–27], we neglect all the losses in converters, wherein, the powers Pr and Pf can be written as follows: Pr = irvdc and Pf = ifvdc.The DC-link voltage equation:(72)dvdc2dt=2CPr−3Cvgifq.Furthermore, we substituteU=vdc2:(73)dUdt=c6Pr−c7ifq,wherec6=2/C and c7=3/Cvg.After adaptation of the DC-Link voltage equation to the canonical form, Linear-ADRC design for DC-Link voltage regulation is as follows.(1)
Total (external and internal) disturbances affecting the DC-link voltagevdc:(74)ftvdc=−c6Pr+−k7−b0vdcifq.(2)
Known parts of the controlled system parameters:(75)b0vdc=c7.(3)
Control inputs of the DC-link voltagevdc:(76)uvdc=ifq.


## 6. Simulation Results and Discussion

In this work, we mainly aim to develop and compare two strategies of wind turbine system control based on a DFIG to evaluate the dynamical performances and sensitivity to DFIG parameter changes. The system parameters of turbine, DFIG, filter, and grid are given in Appendix. Furthermore, the proposed wind turbine system control has been simulated using the Matlab/Simulink environment.Figure9 illustrates the variable wind speed profile applied on wind turbine system.Figure 9
The wind speed profile.Figure10 illustrates the mechanical characteristic of the turbine, as the power coefficient and the tip speed ratio. Figure 11 shows the generator mechanical speed and the angle of blade orientation system.Figure 10
(a) The aerodynamic yieldCp and (b) the tip speed ratio λ.
(a)
(b)Figure 11
(a) The mechanical speed of generatorωmec and (b) the blades’ pitch angle β.
(a)
(b)As illustrated in previous figures, due to the pitch control, the mechanical speed of the generator does not exceed the rated value when the wind speed exceeds its rated value. Moreover, once the pitch angle increases, the tip speed ratio and the power coefficient of turbine decrease.Figure12 depicts the active and reactive power generated by the DFIG. Moreover, Figure 13 depicts the amount of error between reference powers and measured powers for linear ADRC approach and PI approach.Figure 12
(a) The stator active powerPs and (b) the stator reactive power Qs.
(a)
(b)Figure 13
(a) The error betweenPs and Psref and (b) the error between Qs and Qsref.
(a)
(b)Three phase winding stator currents and three phase winding rotor currents generated by DFIG are shown in Figure14.Figure 14
(a) The currents generated through stator and (b) the currents generated through rotor.
(a)
(b)It can be noticed in Figure14 that the three phase currents generated through rotor and stator of the DFIG depend on the wind speed. More importantly, for rotor currents, it can be seen that there is a power change between the grid and generator (subsynchronous mode and supersynchronous mode).Figure15 presents the DC-link voltage and the total power injected into the utility grid (stator power Ps and rotor power Pr).Figure 15
(a) The DC-link voltageVdc and (b) the total power injected into the grid Pgt.
(a)
(b)To evaluate the dynamical performances of the two controls strategies when changing DFIG parameters, it is necessary to change by considering parametric variation of the rotor inductanceLr and stator inductance Ls by an increase of 20% to their nominal value. The results of this test are depicted in Figure 16.Figure 16
(a) Error betweenQs and Qsref for 20% Ls and (b) error between Qs and Qsref for 20% Lr.
(a)
(b)Finally, from the previous simulation results of the control applied on the wind turbine system based on DFIG, Table1 summarizes the performance comparison of the control Methods.Table 1
Comparison of the control methods.
Technique
Precision
Rapidity
Efficiency
Robustness
PI control
Yes
Fast
Yes
Less
ADRC control
Yes
More fast
Yes
More


## 7. Conclusion

This paper presents a dynamic modeling and control of a wind turbine based on DFIG; we have been interested in the internal behaviour of the variable speed wind system while taking into account the internal disturbances. Moreover, we have proposed and detailed the classical indirect stator flux oriented method (ISFOC) based on the PI controller and the new method by linear ADRC.Due to wind fluctuation, the optimal torque algorithm is used to extract the maximum wind power in one hand, and in the other hand, to control the active and reactive powers exchanged with the utility grid independently, we have used the rotor side converter (for the stator powers control) and the grid side converter (for controls the filtre powers and DC bus voltage).Ultimately, the simulation results demonstrate that the ADRC control strategy gives very good dynamical performances and it is not sensitive to DFIG parameter changes unlike the classic control strategy by the proportional integral.



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