# Dynamic Modeling and Improved Control of a Grid-Connected DFIG Used in Wind Energy Conversion Systems

Authors: Abdeslam Jabal Laafou; Abdessalam Ait Madi; Adnane Addaim; Abdessamad Intidam
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The proposed work presented in this paper is mainly focused on the control of the active and reactive stator powers generated by a wind energy conversion system (WECS) based on the dual feed induction generator (DFIG). This control is achieved by acting on the rotor side converter (RSC) to extract the maximum power from the wind turbine (WT) while regulating the rotor currents. Furthermore, another control objective is achieved by acting on the grid side converter (GSC), in which the DC bus voltage is maintained constant and a unity power factor is ensured. To do that, a new robust control known as active disturbance rejection control (ADRC) has been proposed and applied to the WECS. This control is based on the extended state observer (ESO), which is the main core of this algorithm; it makes the estimation and cancellation of the total effect of various uncertainties (internal and external disturbances) possible in real time. To validate the effectiveness of the proposed approach, the system was modeled and simulated by using the Matlab/Simulink software. Two tests, namely, tracking and robustness tests, were performed to compare the proposed ADRC technique and classical PI controllers. The obtained results are promising and have shown that the proposed control strategy based on ADRC, especially when varying the mode parameters, is performant and very useful.


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## Body

## 1. Introduction

Both population growth and industrialization during these recent decades, especially in emerging countries, have led to a significant increase in global demand for electrical energy [1]. In addition, the cost of energy continues to increase due to the scarcity of nonrenewable resources used in the supply of power generation plants. Renewable energies are a particularly suitable response to the considerable energy needs of the planet, which could increase by 50% or more by 2030 [2].Among these renewable energies, we find green wind energy, which is one of the sustainable and renewable energy solutions for the production of electrical energy [3]. One major issue affecting these kinds of energy resources is the wind speed, which can change quickly, especially during gusts. These speed variations generate significant mechanical stresses on the system, which are more reduced with the use of an asynchronous machine than with a synchronous generator, which operates at fixed speed. That is why today the use of variable speed wind turbines is increasing compared to the fixed-speed wind turbines.The use of a dual feed induction generator (DFIG) is one solution proposed in the literature, for variable speed wind turbine structures. Several reasons have led to the use of this DFIG; the first one is that these machines are known by their robustness and their reduced efforts on mechanical parts, and the second one is their possibility for active and reactive control power. On the other hand, the use of the back-to-back converters to connect the generator rotor to the grid allows the transit of a fraction of the total system power. Consequently, the cost and losses in the components of power electronics converters are then reduced [4].In the literature, to obtain a decoupled control of the active and reactive powers of the DFIG, the authors [5] have proposed the use of the oriented stator tension vector control when the authors [6] have proposed the use of the oriented stator flux vector control. The control of active and reactive power is obtained with a controller named rotor current regulator [7], direct torque control (DTC) [8], or modified direct power control (MDPC) [9]. However, the wind turbine system (WTS) is a complex system with nonlinearities, strong coupling, multiple variables, signal wind energy, which are random, and time, which is varying, and disturbance of the system parameters caused by the disturbance of the external environment. These issues make it difficult to obtain the accurate mathematical model, which brought great challenges to the initial design [10]. Therefore, for these characteristics of the WTS, designing more detailed and comprehensive control methods to solve these disturbance and nonlinear problems is of great significance for the safe and reliable operation of the WTS and also achieving maximum power tracking. At present, few research studies have been presented [5–7] using proportional integral (PI) regulators and oriented stator flux control for controlling the rotor current. But the problem while using the PI regulator is the parameters tuning and its robustness to DFIG parameter variations. Some authors have looked for other power control options for DFIG using the rotor current vector loop like the functional predictive controller [11], the internal model controller [12], the model predictive control [13], the sliding mode control [14, 15], or feedback linearization control [16]. These controllers are designed using the DFIG model and have a satisfactory power response compared to the power response of the PI, although they are barely implemented due to the complex formulations of the controllers. Another possibility for power control can be done by using artificial intelligence (AI) approaches such as fuzzy logic (FL) and neural networks (NN) [17]. These strategies are designed by system knowledge and have a satisfactory power response, although they present expensive implementations in terms of high calculation, uncertainty due to knowledge of system parameters, and variations can lead to a degradation of system performance. In order to overcome the shortcomings of the classical linear PID control, Jingqing Han has proposed a new controller known as the active disturbance rejection control (ADRC) [18].The ADRC extracts the disturbance information directly from the input and output signals of the controlled object by the means of an extended state observer (ESO) and eliminates the disturbance by the final control amount; the disturbance signal is cancelled before it is applied to the output signal [18]. At present, the ADRC control strategy has been widely used in various fields such as machinery production, power systems, and process control, and it has been highly praised by researchers at worldwide [19–23]. This disruption rejection command allows the user to treat the system being studied as a simpler model, as the negative effects of external disturbances and modeling uncertainties are compensated in real time.In this context, this paper investigates the control ADRC of the grid connected wind conversion system based on DFIG, in order to deal with the system complexity and its uncertainties. Principally, the main contribution of this work lies on the design and development of the new robust control strategy by the use of the ADRC for the grid connected WECS based on DFIG. This approach ensures the maximum power capture available from the wind and controls the power delivery. Therefore, the ADRC is used to control the RSC, GSC, direct current (DC), and active and reactive powers.For this purpose, the present paper is organized as follows: the first section treats the general introduction; the second presents the overall architecture of the proposed system and the modeling of its components. The third section introduces the mathematical theory and design of the ADRC strategy. The fourth section illustrates the application of the proposed control by ADRC for the overall conversion system, and in the fifth section, the simulation results and discussion are presented.


## 2. Modeling of the Wind Energy Conversion System

In this section, the modeling of the elements of the WECS is shown, going from the conversion of wind kinetic energy into mechanical energy to the connection with the electrical grid. Based on physics laws and neglecting the mechanical and electrical losses that are assumed to have no effect on the dynamic behavior of the system, a dynamic model of each element of WECS is established. The WECS is illustrated in Figure1. Its components are as follows:(i)
Three-bladed, horizontal axis wind turbine(ii)
Gearbox system that adapts the turbine low rotational speed to the speed required by the generator(iii)
DFIG with nominal power of 1.5 MW(iv)
Two bidirectional converters (RSC and GSC) interconnected via a DC bus(v)
Three-phase filter and a step-up transformerFigure 1
Schematic diagram of wind turbine based on DFIG.


### 2.1. Wind Turbine Modeling

The WT is a rotating device that converts a part of the wind kinetic power into a mechanical power at the turbine rotor [24], or the aerodynamic wind power available on a surface S swept by the turbine blades is expressed by the derivation of the kinetic energy of the air mass passing through this surface, as shown by(1)Pw=12ρSV3,where ρ: the air density (≃1.22kg/m3), S=πR2: the swept area by the blades of the radius turbineR, and V: the wind speed (m/s).According to the Betz law [13, 14], the turbine recovers only a fraction of this power Pw; or each wind turbine is defined by its own power coefficient Cpλ,β. This power coefficient represents the aerodynamic efficiency of the WT, which depends on the geometric characteristics of the blades, the pitch angle β and the speed ratio λ [12]. As a result, the captured aerodynamic power is given by(2)Pt=CPλ,βPw=12Cpλ,βρπR2V3.The mechanical torque appearing on the turbine rotor can therefore be subsequently represented by(3)Tt=12⋅ΩtCPλ,βρπR2V3,where Ωt is the turbine speed.The power coefficient is most often presented as a nonlinear function ofλ and β and whose theoretical upper limit is given by Betz's law [13, 14]. In this article, CPλ,β is expressed by the following function:(4)CPλ,β=0.5176116λ′−0.4β−5e−21/λ′+0.0068λ,with(5)1λi=1λ+0,08β+0,0351+β3,(6)λ=Ωt⋅RV.Figure2 shows the evolution of the power coefficient CPλ,β as a function of λ for different values of β. It can be observed that when the pitch angle β increases, the coefficient CP decreases, which results in a reduction in the wind kinetic energy captured by the turbine.Figure 2
Evolution of the power coefficient according toλ and β.


### 2.2. Gearbox and Mechanical Shaft Modeling

The gearbox, which connects the slow shaft of the turbine to the fast shaft that drives the generator, aims to adapt the turbine speed to that required by the generator [24]. The gearbox used generally includes two or three epicyclic gear trains to obtain high multiplication ratios (from ≈50 to ≈100). It is modeled using a torque and speed gain G corresponding to the multiplication ratio:(7)Ωt=1GΩm,Tm=1GTt.The total inertia of the system consisting of the WT and the generator is expressed by equation (8) (the inertia of the turbine is transferred to the generator rotor):(8)J=JtG2+Jg.Consequently, the mechanical shaft model is presented by(9)JdΩmdt=Tm−Tem−fΩm,where Ωm is the generator rotor speed, J is the total inertia, Jg the DFIG inertia, Jt is the turbine inertia, Tm the torque applied to the shaft of the generator, Tem is the electromagnetic torque produced by the generator, and fΩm represents the torque of viscous friction.


### 2.3. DFIG Modeling with Stator Flux Orientation

The DFIG consists of a three-phase stator directly connected to the grid and a rotor formed by the three-phase windings accessible by three rings equipped with sliding contacts and connected to the grid via a two-stage power electronics converter [14]. In the literature, several modeling methods have been discussed, although the most commonly used model for DFIG is the Park model [13], which allows to write a dynamic model in a direct quadrature (DQ) reference frame as follows:The electrical equations are given by(10)Vsd=Rsisd+dϕsddt−ωsϕsq,Vsq=Rsisq+dϕsqdt+ωsϕsd,Vrd=Rrird+dϕrddt−ωrϕrq,Vrq=Rrirq+dϕrqdt+ωrϕrd.The stator and rotor flux are shown by(11)ϕsd=Lsisd+Lmird,ϕsq=Lsisq+Lmirq,ϕrd=Lrird+Lmisd,ϕrd=Lrird+Lmisd.The electromagnetic torque Tem is written as(12)Tem=32pLmLsϕsqird−ϕsdirq.The active and reactive stator and rotor powers are given indq reference frame by(13)PS=32Vsdisd+Vsqisq,QS=32Vsqisd+Vsdisq,Pr=32Vrdird+Vrqirq,Qr=32Vrqird+Vrdirq,where vsd and vsq, vrd and vrq, isd and isq,ird and irq, ψsd and ψsq, ψrd and ψrq are the direct and quadrature components of stator voltages, rotor voltages, stator currents, rotor currents, stator flux, and rotor flux in the DQ reference frame. Rs and Rr are the stator and rotor resistances, Ls and Lr are the stator and rotor inductances, Lm is the mutual inductance, ωs and ωr are the stator and rotor pulsations speed, and p is the poles pair number [21].In order to simplify the control of the active and reactive powers, and to achieve the vector control of the DFIG, an orientation of the stator flux is required. In this article, the stator field vector is oriented along the d-axis; hence,ϕsd=ϕs and ϕsq=0 [25].Assuming that the stator resistanceRs is negligible and that the stator flux is constant, the DFIG equations become as follows:Rotor and stator voltages are(14)Vsd=Rsisd+dϕsddt=0,Vsq=ωsϕs,Vrd=Rrird+Lrσdirddt−gωsLrσirq,Vrq=Rrirq+Lrσdirqdt−gωsLrσird+gωsVsLmLsωs.Active and reactive stator powers are(15)PS=−32VsLmLsirq,QS=32VsϕsLs−VsLmLsird.Electromagnetic torque is(16)Tem=−32pLmLsϕsirq,where σ=1−Lm2/LrLs is the dispersion coefficient.


### 2.4. Back-to-Back Converters Modeling

The WECS studied in this work transfers a portion of its power to the grid by means of two-stage power converter coupled on a common DC bus; these converters are bidirectional and controlled by pulse width modulation (PWM). In this work, both converters are identical and can be used in both inverter and rectifier modes. Figure3 illustrates the simplified model of the GSC, filter, and grid. The converter consists of controllable components (e.g., insulated gate bipolar transistor (IGBT)) connected with antiparallel diodes allowing the bidirectional circulation of current [26].Figure 3
Simplified model of the GSC, filter, and grid.Each transistor-diode set is considered as a perfect switchTij, and every switch state is quantized by a connection function sij given by(17)sij=1,ifTijON,0,ifTijOFF,with i=1,2 and j=1,2,3.The switches states are complementary:(18)s1j+s2j=1,∀j∈1,2,3.The modulated voltages are expressed by(19)vf_a=2s11−s12+s133Udc,vf_b=2s12−s11+s133Udc,vf_c=2s13−s11+s123Udc.The modulated currentIinvis given by(20)Iinv=s11if_a+s12if_b+s13if_c.Applying Park’s transformation, equations (19) and (20) become as follows:(21)vfd=sdUdc,vfq=sqUdc,Iinv=sdifd+sqifq.where vfd and vfq are the inverter voltages, ifdandifq are the grid filter currents, Udc is the DC bus voltage, and sd and sq are the switches state in DQ reference frame.


### 2.5. DC Link and Filter Modeling

The connection with the electrical grid by means of rotor side is carried out via a filterLf,Rf [26]:(22)vs_a=vf_a−Rfifa−Lfdif_adt,vs_b=vf_b−Rfifb−Lfdif_bdt,vs_c=vf_c−Rfif_c−Lfdif_cdt.Equation (22) becomes in DQ frame as follows:(23)vsd=vfd−Rfifd−Lfdifddt+Lfωsifq,vsq=vfq−Rfifq−Lfdifqdt−Lfωsifd,where vsdandvsq are the grid voltages.The modeling of DC Link Voltage is given by(24)dUdcdt=1Cir−ig,with C being the DC link Capacitor and ir,ig being, respectively, the currents modulated by the RSC and the GSC.


## 2.1. Wind Turbine Modeling

The WT is a rotating device that converts a part of the wind kinetic power into a mechanical power at the turbine rotor [24], or the aerodynamic wind power available on a surface S swept by the turbine blades is expressed by the derivation of the kinetic energy of the air mass passing through this surface, as shown by(1)Pw=12ρSV3,where ρ: the air density (≃1.22kg/m3), S=πR2: the swept area by the blades of the radius turbineR, and V: the wind speed (m/s).According to the Betz law [13, 14], the turbine recovers only a fraction of this power Pw; or each wind turbine is defined by its own power coefficient Cpλ,β. This power coefficient represents the aerodynamic efficiency of the WT, which depends on the geometric characteristics of the blades, the pitch angle β and the speed ratio λ [12]. As a result, the captured aerodynamic power is given by(2)Pt=CPλ,βPw=12Cpλ,βρπR2V3.The mechanical torque appearing on the turbine rotor can therefore be subsequently represented by(3)Tt=12⋅ΩtCPλ,βρπR2V3,where Ωt is the turbine speed.The power coefficient is most often presented as a nonlinear function ofλ and β and whose theoretical upper limit is given by Betz's law [13, 14]. In this article, CPλ,β is expressed by the following function:(4)CPλ,β=0.5176116λ′−0.4β−5e−21/λ′+0.0068λ,with(5)1λi=1λ+0,08β+0,0351+β3,(6)λ=Ωt⋅RV.Figure2 shows the evolution of the power coefficient CPλ,β as a function of λ for different values of β. It can be observed that when the pitch angle β increases, the coefficient CP decreases, which results in a reduction in the wind kinetic energy captured by the turbine.Figure 2
Evolution of the power coefficient according toλ and β.


## 2.2. Gearbox and Mechanical Shaft Modeling

The gearbox, which connects the slow shaft of the turbine to the fast shaft that drives the generator, aims to adapt the turbine speed to that required by the generator [24]. The gearbox used generally includes two or three epicyclic gear trains to obtain high multiplication ratios (from ≈50 to ≈100). It is modeled using a torque and speed gain G corresponding to the multiplication ratio:(7)Ωt=1GΩm,Tm=1GTt.The total inertia of the system consisting of the WT and the generator is expressed by equation (8) (the inertia of the turbine is transferred to the generator rotor):(8)J=JtG2+Jg.Consequently, the mechanical shaft model is presented by(9)JdΩmdt=Tm−Tem−fΩm,where Ωm is the generator rotor speed, J is the total inertia, Jg the DFIG inertia, Jt is the turbine inertia, Tm the torque applied to the shaft of the generator, Tem is the electromagnetic torque produced by the generator, and fΩm represents the torque of viscous friction.


## 2.3. DFIG Modeling with Stator Flux Orientation

The DFIG consists of a three-phase stator directly connected to the grid and a rotor formed by the three-phase windings accessible by three rings equipped with sliding contacts and connected to the grid via a two-stage power electronics converter [14]. In the literature, several modeling methods have been discussed, although the most commonly used model for DFIG is the Park model [13], which allows to write a dynamic model in a direct quadrature (DQ) reference frame as follows:The electrical equations are given by(10)Vsd=Rsisd+dϕsddt−ωsϕsq,Vsq=Rsisq+dϕsqdt+ωsϕsd,Vrd=Rrird+dϕrddt−ωrϕrq,Vrq=Rrirq+dϕrqdt+ωrϕrd.The stator and rotor flux are shown by(11)ϕsd=Lsisd+Lmird,ϕsq=Lsisq+Lmirq,ϕrd=Lrird+Lmisd,ϕrd=Lrird+Lmisd.The electromagnetic torque Tem is written as(12)Tem=32pLmLsϕsqird−ϕsdirq.The active and reactive stator and rotor powers are given indq reference frame by(13)PS=32Vsdisd+Vsqisq,QS=32Vsqisd+Vsdisq,Pr=32Vrdird+Vrqirq,Qr=32Vrqird+Vrdirq,where vsd and vsq, vrd and vrq, isd and isq,ird and irq, ψsd and ψsq, ψrd and ψrq are the direct and quadrature components of stator voltages, rotor voltages, stator currents, rotor currents, stator flux, and rotor flux in the DQ reference frame. Rs and Rr are the stator and rotor resistances, Ls and Lr are the stator and rotor inductances, Lm is the mutual inductance, ωs and ωr are the stator and rotor pulsations speed, and p is the poles pair number [21].In order to simplify the control of the active and reactive powers, and to achieve the vector control of the DFIG, an orientation of the stator flux is required. In this article, the stator field vector is oriented along the d-axis; hence,ϕsd=ϕs and ϕsq=0 [25].Assuming that the stator resistanceRs is negligible and that the stator flux is constant, the DFIG equations become as follows:Rotor and stator voltages are(14)Vsd=Rsisd+dϕsddt=0,Vsq=ωsϕs,Vrd=Rrird+Lrσdirddt−gωsLrσirq,Vrq=Rrirq+Lrσdirqdt−gωsLrσird+gωsVsLmLsωs.Active and reactive stator powers are(15)PS=−32VsLmLsirq,QS=32VsϕsLs−VsLmLsird.Electromagnetic torque is(16)Tem=−32pLmLsϕsirq,where σ=1−Lm2/LrLs is the dispersion coefficient.


## 2.4. Back-to-Back Converters Modeling

The WECS studied in this work transfers a portion of its power to the grid by means of two-stage power converter coupled on a common DC bus; these converters are bidirectional and controlled by pulse width modulation (PWM). In this work, both converters are identical and can be used in both inverter and rectifier modes. Figure3 illustrates the simplified model of the GSC, filter, and grid. The converter consists of controllable components (e.g., insulated gate bipolar transistor (IGBT)) connected with antiparallel diodes allowing the bidirectional circulation of current [26].Figure 3
Simplified model of the GSC, filter, and grid.Each transistor-diode set is considered as a perfect switchTij, and every switch state is quantized by a connection function sij given by(17)sij=1,ifTijON,0,ifTijOFF,with i=1,2 and j=1,2,3.The switches states are complementary:(18)s1j+s2j=1,∀j∈1,2,3.The modulated voltages are expressed by(19)vf_a=2s11−s12+s133Udc,vf_b=2s12−s11+s133Udc,vf_c=2s13−s11+s123Udc.The modulated currentIinvis given by(20)Iinv=s11if_a+s12if_b+s13if_c.Applying Park’s transformation, equations (19) and (20) become as follows:(21)vfd=sdUdc,vfq=sqUdc,Iinv=sdifd+sqifq.where vfd and vfq are the inverter voltages, ifdandifq are the grid filter currents, Udc is the DC bus voltage, and sd and sq are the switches state in DQ reference frame.


## 2.5. DC Link and Filter Modeling

The connection with the electrical grid by means of rotor side is carried out via a filterLf,Rf [26]:(22)vs_a=vf_a−Rfifa−Lfdif_adt,vs_b=vf_b−Rfifb−Lfdif_bdt,vs_c=vf_c−Rfif_c−Lfdif_cdt.Equation (22) becomes in DQ frame as follows:(23)vsd=vfd−Rfifd−Lfdifddt+Lfωsifq,vsq=vfq−Rfifq−Lfdifqdt−Lfωsifd,where vsdandvsq are the grid voltages.The modeling of DC Link Voltage is given by(24)dUdcdt=1Cir−ig,with C being the DC link Capacitor and ir,ig being, respectively, the currents modulated by the RSC and the GSC.


## 3. Structure and Principle of ADRC

The ADRC control strategy is robust control method proposed by Jinging HAN [18] to overcome the deficiencies of the conventional control by PID [21]. To illustrate the principle of the ADRC technique, let us consider a single-input, single-output nonlinear time-varying controlled object [24]:(25)xn=fx,x˙,…,xn−1,w,t+b⋅u,x=y,where: x,x˙,…,xn respectively represents the object state and its various order dynamics, w is the external disturbances, fx,x˙,…,xn−1,w,t represents all internal and external (total) disturbances affecting the system to be controlled, u and y are the system input and output, respectively, and b is the control gain.In engineering practice, it is often difficult to accurately establish and determine the system dynamic model and control gain. There are various uncertainties; because of this, model-based control theory and methods have encountered great difficulties and challenges in engineering practice [19]. The advantage of ADRC is that even if the dynamic model of the system is not clear, and there is a large uncertainty in the control gain, good control performance can still be obtained. The basic structure of the ADRC controller is shown by the block diagram in Figure 4. It includes three parts: the tracking differentiator (TD), the ESO, and the state error feedback (SEF) control law. These three parts can have many different forms. For the difference, if each part of the ADRC contains a nonlinear link, it is called nonlinear ADRC; otherwise, if they are designed as linear links, they are called linear ADRCs [19]. The principles of the three parts are introduced below. Among them, b0 can be an approximate estimated constant value of b, which can also be adjusted according to the control needs and can also be adaptive online.Figure 4
Block diagram of n order nonlinear ADRC.


### 3.1. Tracking Differentiator (TD)

The purpose of the TD is to arrange the transition process and then reduce the initial error, which affects the system in the initial stage, by effectively resolving the contradiction between overshoot and fastness. It can be achieved by TD or an appropriate function generator. TD was originally used to track the input signal as quickly as possible, while giving an approximate differential signal. At present, TD is often used to arrange the transition process, so the form of TD is mainly introduced here.The general form of a continuous nonlinear tracking differentiator is shown in [19] and given by(26)v1˙=v2,v2˙=v3,⋮vn˙=rnfv1−v,v2r,…,vnrn−1,where v is the TD input signal, vii=1,2,…,n is the TD output signal, v1 is the tracking signal of the input v, and vii=1,2,…,n is the i−1 order differential of v1, which can be approximated as (i − 1) order differentiation; r is called the speed factor, and the larger r is, the faster v1 tracks the input signal v.


### 3.2. Extended State Observer (ESO)

In this stage, it is assumed that the dynamic model of the system is completely unknown, and the “total disturbance”f· is estimated online and in real time through the extended state assuming that x1=x,…,xn=xn−1; let xn+1=f· be the extended state variables of the system, then the general form of designing a continuous extended state observer is shown in [19] and expressed by(27)e=z1−y,z1˙=z2−β01·g1e,z2˙=z3−β02·g2e,⋮zn˙=zn+1−β0n·gne+b0·u,zn+1˙=−β0n+1·gn+1e,where zii=1,2,…,n+1 are the estimated values of states xii=1,2,…,n and xn+1=f· the total disturbances; β0ii=1,2,…,n+1is the observer adjustable gains, and gnei=1,2,…,n+1 is a nonlinear constructed function.Under certain conditions, ESO can estimate the state of the object and the total disturbance of the system with a certain accuracy; that is,(28)z1⟶x1,…,zn⟶xn,zn+1⟶f·.For the specific design of ESO, a large number of existing observer and filter design techniques can be used for the specific form of ESO. Han Jingqing [18] chose gie as the specific nonlinear function, which has the form mentioned in [21] by(29)gie=fale,ai,δ=eaisgnε,e>δ,eδ1−ai,e≤δ,where 0<ai<1, and δ>0 are adjustable parameters; when ai=1, gie=e, that is, the traditional Luenberger observer, also known as linear ESO.


### 3.3. State Error Feedback Control Law

ESO obtains the estimated value of the total disturbance in real time. If it is compensated in the control law, the function of active disturbance rejection can be realized. Therefore, the control law is taken as shown in [20], given by(30)u=u0−zn+1b0,where u0 is the initial control component. If the estimation error of zn+1 on the unknown total disturbance “f·+b−b0u” is ignored, then the object defined by the equation (25) is converted into a “series of integrators” as follows:(31)xn=f·+b−b0u−zn+1+u0≈u0.In this way, the controlled objects that are full of disturbances, uncertainties, and nonlinearities are uniformly converted into standard integrators types, which turns the design of the control system from complex to simple, from abstract to intuitive, and having broad applicability.The control componentu0 has multiple implementations [21], and here is a general Nonlinear SEF (NLSEF) control law for controlled objects of any order:(32)u0=∑i=1nkifalei,ai′,δ′,where ei=vi−zi, ki is the gain coefficient, and ai′andδ′ are undetermined constants, usually chosen as 0<ai<1<ai′i=1,2,…,n. In this way, the differential effect will become smaller when it is close to the steady state, which will help improve the performance of the control system [21].Whenai′=1i=1,2,…,n, the control law becomes linear. The advantage of linear control law is that the parameter tuning is simple and the control effect is relatively smooth.


### 3.4. Linear ADRC Design

In practice, the nonlinear ADRC has a large number of parameters that need to be adjusted, and adjusting them is a very hard and complicated task. As a result, in order to reduce the model complexity and the controller computational, a linear ADRC design method is proposed. In the proposed structure, the TD block is omitted and the standard Luenberger observer with state expansion is used as the linear ESO to estimate the system states and the generalized disturbance; the NLSEF controller is replaced by a proportional controller that drives the tracking error between the system output and reference signal to zero. Figure5 shows the block diagram of a first-order linear ADRC [26].Figure 5
First-order linear ADRC block diagram.Consider a first-order plant where the plant dynamics is given by(33)x˙=−1Tx+b⋅u,x=y.An external disturbanced is added to the system and modeling errors are taken into account by means of Δbb=b0+Δb. Then the system is rewritten by equation (34) as shown in [26]:(34)y˙=−1Ty+w+Δb⋅u+b0⋅u=fy,w,t+b0u=f·+b0u.Letx1=y, x2=f, and f˙=h.The system state-space model is given then by(35)x˙=Ax+Bu+Eh,y=Cx.where(36)A=0100,B=b0,C=10,E=01,x=x1x2.The linear ESO (LESO), presented in Figure6, is designed as ai=1; therefore, the corresponding observer is presented by(37)z˙=Az+Bu+Ky−y^,y^=Cz,where z is the observed states vector, z=z1z2T (z1 is the estimation of x1 and z2 is the estimation of x2), y^ is the estimated output, and K is the observer gain vector.Figure 6
First-order linear extended state observer.VectorK is defined by(38)K=β01β02T=2ω0ω02T,where ω0 is determined by the poles placement in closed-loop to ensure both fast observer dynamics and minimal perturbations sensitivity [26].As mentioned in the previous section, ifai′=1, the control law by ADRC is set as a linear controller. According to [26], first-order plant equation (33) becomes(39)u0=kp.As a result, a proportional controller is used to control system (34):(40)u0=kpv−z1=kpv−y^.The controller tuning is chosen ask1=ωc=4/Tsettle where Tsettle is the desired closed loop settling time. The dynamics of the ESO must be fast compared to that of the controller. Therefore, the poles of the observer are placed to the left of the system closed-loop poles. In this case, ω0 is selected as ω0=3∼10ωc.


## 3.1. Tracking Differentiator (TD)

The purpose of the TD is to arrange the transition process and then reduce the initial error, which affects the system in the initial stage, by effectively resolving the contradiction between overshoot and fastness. It can be achieved by TD or an appropriate function generator. TD was originally used to track the input signal as quickly as possible, while giving an approximate differential signal. At present, TD is often used to arrange the transition process, so the form of TD is mainly introduced here.The general form of a continuous nonlinear tracking differentiator is shown in [19] and given by(26)v1˙=v2,v2˙=v3,⋮vn˙=rnfv1−v,v2r,…,vnrn−1,where v is the TD input signal, vii=1,2,…,n is the TD output signal, v1 is the tracking signal of the input v, and vii=1,2,…,n is the i−1 order differential of v1, which can be approximated as (i − 1) order differentiation; r is called the speed factor, and the larger r is, the faster v1 tracks the input signal v.


## 3.2. Extended State Observer (ESO)

In this stage, it is assumed that the dynamic model of the system is completely unknown, and the “total disturbance”f· is estimated online and in real time through the extended state assuming that x1=x,…,xn=xn−1; let xn+1=f· be the extended state variables of the system, then the general form of designing a continuous extended state observer is shown in [19] and expressed by(27)e=z1−y,z1˙=z2−β01·g1e,z2˙=z3−β02·g2e,⋮zn˙=zn+1−β0n·gne+b0·u,zn+1˙=−β0n+1·gn+1e,where zii=1,2,…,n+1 are the estimated values of states xii=1,2,…,n and xn+1=f· the total disturbances; β0ii=1,2,…,n+1is the observer adjustable gains, and gnei=1,2,…,n+1 is a nonlinear constructed function.Under certain conditions, ESO can estimate the state of the object and the total disturbance of the system with a certain accuracy; that is,(28)z1⟶x1,…,zn⟶xn,zn+1⟶f·.For the specific design of ESO, a large number of existing observer and filter design techniques can be used for the specific form of ESO. Han Jingqing [18] chose gie as the specific nonlinear function, which has the form mentioned in [21] by(29)gie=fale,ai,δ=eaisgnε,e>δ,eδ1−ai,e≤δ,where 0<ai<1, and δ>0 are adjustable parameters; when ai=1, gie=e, that is, the traditional Luenberger observer, also known as linear ESO.


## 3.3. State Error Feedback Control Law

ESO obtains the estimated value of the total disturbance in real time. If it is compensated in the control law, the function of active disturbance rejection can be realized. Therefore, the control law is taken as shown in [20], given by(30)u=u0−zn+1b0,where u0 is the initial control component. If the estimation error of zn+1 on the unknown total disturbance “f·+b−b0u” is ignored, then the object defined by the equation (25) is converted into a “series of integrators” as follows:(31)xn=f·+b−b0u−zn+1+u0≈u0.In this way, the controlled objects that are full of disturbances, uncertainties, and nonlinearities are uniformly converted into standard integrators types, which turns the design of the control system from complex to simple, from abstract to intuitive, and having broad applicability.The control componentu0 has multiple implementations [21], and here is a general Nonlinear SEF (NLSEF) control law for controlled objects of any order:(32)u0=∑i=1nkifalei,ai′,δ′,where ei=vi−zi, ki is the gain coefficient, and ai′andδ′ are undetermined constants, usually chosen as 0<ai<1<ai′i=1,2,…,n. In this way, the differential effect will become smaller when it is close to the steady state, which will help improve the performance of the control system [21].Whenai′=1i=1,2,…,n, the control law becomes linear. The advantage of linear control law is that the parameter tuning is simple and the control effect is relatively smooth.


## 3.4. Linear ADRC Design

In practice, the nonlinear ADRC has a large number of parameters that need to be adjusted, and adjusting them is a very hard and complicated task. As a result, in order to reduce the model complexity and the controller computational, a linear ADRC design method is proposed. In the proposed structure, the TD block is omitted and the standard Luenberger observer with state expansion is used as the linear ESO to estimate the system states and the generalized disturbance; the NLSEF controller is replaced by a proportional controller that drives the tracking error between the system output and reference signal to zero. Figure5 shows the block diagram of a first-order linear ADRC [26].Figure 5
First-order linear ADRC block diagram.Consider a first-order plant where the plant dynamics is given by(33)x˙=−1Tx+b⋅u,x=y.An external disturbanced is added to the system and modeling errors are taken into account by means of Δbb=b0+Δb. Then the system is rewritten by equation (34) as shown in [26]:(34)y˙=−1Ty+w+Δb⋅u+b0⋅u=fy,w,t+b0u=f·+b0u.Letx1=y, x2=f, and f˙=h.The system state-space model is given then by(35)x˙=Ax+Bu+Eh,y=Cx.where(36)A=0100,B=b0,C=10,E=01,x=x1x2.The linear ESO (LESO), presented in Figure6, is designed as ai=1; therefore, the corresponding observer is presented by(37)z˙=Az+Bu+Ky−y^,y^=Cz,where z is the observed states vector, z=z1z2T (z1 is the estimation of x1 and z2 is the estimation of x2), y^ is the estimated output, and K is the observer gain vector.Figure 6
First-order linear extended state observer.VectorK is defined by(38)K=β01β02T=2ω0ω02T,where ω0 is determined by the poles placement in closed-loop to ensure both fast observer dynamics and minimal perturbations sensitivity [26].As mentioned in the previous section, ifai′=1, the control law by ADRC is set as a linear controller. According to [26], first-order plant equation (33) becomes(39)u0=kp.As a result, a proportional controller is used to control system (34):(40)u0=kpv−z1=kpv−y^.The controller tuning is chosen ask1=ωc=4/Tsettle where Tsettle is the desired closed loop settling time. The dynamics of the ESO must be fast compared to that of the controller. Therefore, the poles of the observer are placed to the left of the system closed-loop poles. In this case, ω0 is selected as ω0=3∼10ωc.


## 4. DFIG Wind Turbine Control by ADRC

After having modeled the different elements of the wind conversion chain based on the DFIG, this section discusses the control strategy by ADRC of the DFIG through the two power electronics converters with three main objectives:(i)
Maximum power extraction by optimal torque control(ii)
A decoupled control of the active and reactive stator powers by controlling the rotor side converter with ADRC approach(iii)
DC bus voltage control and unit power factor ensuring by acting on the grid-side converter using the ADRC approach


### 4.1. Maximum Power Point Tracking (MPPT)

The most successful control strategy for WTs directly connected to the power grid is based on the MPPT technique [27]. The principle of this technique is to rotate the turbine over a certain wind speed range in order to maintain the tip speed ratio (TSR) λ at its optimal value λopt which makes the turbine operating at CP=CPmax.Consider equation (6). The captured power from the wind is expressed as a function of the rotational speed ωm:(41)Pt=0.5ρπR5CPλ,βλ3ωm3.Replacingλ by λopt and replacing CPλ,β=CPmax, the maximum power to be captured is expressed by(42)Ptmax=Koptωm3,where Kopt is a coefficient:(43)Kopt=0.5ρπR5CPmaxλopt,βλopt3.Therefore, the reference electromagnetic torque for adjusting the rotational speed in order to extract the maximum power from the wind is given by(44)Tem_ref=Koptωm2.


### 4.2. Rotor Side Converter Control by ADRC

The main objective in this stage is to regulate the transfer of the stator active and reactive powers to the power grid [25]. The control is ensured by the regulation of rotor currents, where the direct component Ird ensures the control of the reactive power, and the quadrature component Irq controls either the active power to its desired reference or the electromagnetic torque to ensure the MPPT. The corresponding control is illustrated in Figure 7.Figure 7
Control of the RSC by ADRC.From equation (16), we deduce the reference current irq_ref given by equation (45) allowing the electromagnetic torque produced by the DFIG to be regulated to its reference value imposed by the MPPT strategy:(45)irq_ref=−23LspLmψsTemref=−23LspLmψsKoptωm2.Similarly, the reference rotor currentird_ref shown by equation (46) is determined, from equation (15), in order to control the reactive power supplied or absorbed by the generator:(46)ird_ref=23ψsLm−LsvsLmQs_ref.The rotor currentsird and irq are controlled by two ADRC type regulators as shown in Figure 7.The knowledge of the stator fluxψs is necessary for the design of this control; it can be estimated from measurements of the direct component of the stator and rotor currents (equation (11)). For the synthesis of the ADRC controllers, the last two equations from equation (14) are rewritten to express the dynamics of the rotor currents:(47)dirddt=−RrσLrird+ωrirq+1σLrvrd,(48)dirqdt=−RrσLrirq−ωrird−ωrLmσLsLrψs+1σLrvrq.These expressions can be written in the canonical form of an ADRC controller given by(49)dirddt=fd·+br0ud,where(50)fd·=−RrσLrird+ωrirq+1σLr−br0vrd,br0=1σLr;ud=vrd.In addition,(51)dirqdt=fq·+br0uq,where(52)fq·=−RrσLrirq−ωrird−ωrLmσLsLrψs+1σLr−br0vrq,br0=1σLr;uq=vrq,where fd· and fq·represent the total disturbances affecting the currents ird and irq, respectively. br0 is the known part of the generator parameters.


### 4.3. Grid Side Converter Control by ADRC

The control of this converter makes it possible to control the currents flowing in the filterRf,Lf in order to keep the DC bus voltage constant regardless of the power exchanged between the DFIG and the power grid, and to ensure the control the active and reactive powers flowing through the connection point. The power factor can be kept unitary by imposing a zero reactive power reference. The block diagram of the adopted control strategy is shown in Figure 8. It involves a dual-loop control structure: an outer dc-link voltage control loop and an inner current control loop.Figure 8
Control of GSC by ADRC.As mentioned before the grid voltages are oriented such thatvsd=0 and vsq=Vs, As a result, equations 24 are rewritten by using(53)difddt=−RfLfifd−ωsifq−1Lfvfd,(54)difqdt=−RfLfifq+1LfVs−ωsifd−1Lfvfq.The active and reactive powers exchanged with the power grid through the filter are expressed by(55)Pf=32Vsifq,(56)Qf=32Vsifd.The power across the DC link Capacitor C can be expressed by(57)Pdc=Udcig−ir.And we have(58)dUdcdt=1Cig−ir,or(59)Pdc=CUdcdUdcdt.By neglecting all the losses in filter, power electronics converters, and the capacitor, the exchanged power on the DC bus is given by(60)Pdc=Pf−Pr,where PrPf are the generator rotor side and filter side powers, respectively.By taking into account equations (55)–(60), the DC bus voltage is expressed by using(61)CUdcdUdcdt=32Vsifq−Udcir,or(62)dUdc2dt=3VsCifq−2UdcCir.We putX=Udc2; then, we have(63)dXdt=3VsCifq−2XCir.Therefore, we obtain the canonical form of the ADRC controller such that(64)dXdt=fX·+bc0uX,or(65)fX·=−2XCir+3VsC−bc0ifq,bc0=3VsC;uX=ifq.The external voltage regulation loop makes it possible to maintain the voltage across the capacitorUdc constant and to generate the reference current ifq_ref for the current inner loop [25]. The reference current ifd_ref, given by equation (66), is calculated by the desired delivery of reactive power:(66)ifd_ref=23VsQf_ref.For the inner current control loop, equations (53) and (54) are adapted with the canonical forms of ADRC and then we obtain equations (67) and (69):(67)difddt=ffd·+bf0ufd,with(68)ffd·=RfLfifd−ωsifq+1Lf−bf0Vfd,bf0=−1Lf;ufd=Vfd.In addition,(69)difqdt=ffq·+bf0ufq,with(70)ffq·=1LfVs−RfLfifq−ωsifd+1Lf−bf0Vfq,bf0=−1Lf;ufq=Vfq.


## 4.1. Maximum Power Point Tracking (MPPT)

The most successful control strategy for WTs directly connected to the power grid is based on the MPPT technique [27]. The principle of this technique is to rotate the turbine over a certain wind speed range in order to maintain the tip speed ratio (TSR) λ at its optimal value λopt which makes the turbine operating at CP=CPmax.Consider equation (6). The captured power from the wind is expressed as a function of the rotational speed ωm:(41)Pt=0.5ρπR5CPλ,βλ3ωm3.Replacingλ by λopt and replacing CPλ,β=CPmax, the maximum power to be captured is expressed by(42)Ptmax=Koptωm3,where Kopt is a coefficient:(43)Kopt=0.5ρπR5CPmaxλopt,βλopt3.Therefore, the reference electromagnetic torque for adjusting the rotational speed in order to extract the maximum power from the wind is given by(44)Tem_ref=Koptωm2.


## 4.2. Rotor Side Converter Control by ADRC

The main objective in this stage is to regulate the transfer of the stator active and reactive powers to the power grid [25]. The control is ensured by the regulation of rotor currents, where the direct component Ird ensures the control of the reactive power, and the quadrature component Irq controls either the active power to its desired reference or the electromagnetic torque to ensure the MPPT. The corresponding control is illustrated in Figure 7.Figure 7
Control of the RSC by ADRC.From equation (16), we deduce the reference current irq_ref given by equation (45) allowing the electromagnetic torque produced by the DFIG to be regulated to its reference value imposed by the MPPT strategy:(45)irq_ref=−23LspLmψsTemref=−23LspLmψsKoptωm2.Similarly, the reference rotor currentird_ref shown by equation (46) is determined, from equation (15), in order to control the reactive power supplied or absorbed by the generator:(46)ird_ref=23ψsLm−LsvsLmQs_ref.The rotor currentsird and irq are controlled by two ADRC type regulators as shown in Figure 7.The knowledge of the stator fluxψs is necessary for the design of this control; it can be estimated from measurements of the direct component of the stator and rotor currents (equation (11)). For the synthesis of the ADRC controllers, the last two equations from equation (14) are rewritten to express the dynamics of the rotor currents:(47)dirddt=−RrσLrird+ωrirq+1σLrvrd,(48)dirqdt=−RrσLrirq−ωrird−ωrLmσLsLrψs+1σLrvrq.These expressions can be written in the canonical form of an ADRC controller given by(49)dirddt=fd·+br0ud,where(50)fd·=−RrσLrird+ωrirq+1σLr−br0vrd,br0=1σLr;ud=vrd.In addition,(51)dirqdt=fq·+br0uq,where(52)fq·=−RrσLrirq−ωrird−ωrLmσLsLrψs+1σLr−br0vrq,br0=1σLr;uq=vrq,where fd· and fq·represent the total disturbances affecting the currents ird and irq, respectively. br0 is the known part of the generator parameters.


## 4.3. Grid Side Converter Control by ADRC

The control of this converter makes it possible to control the currents flowing in the filterRf,Lf in order to keep the DC bus voltage constant regardless of the power exchanged between the DFIG and the power grid, and to ensure the control the active and reactive powers flowing through the connection point. The power factor can be kept unitary by imposing a zero reactive power reference. The block diagram of the adopted control strategy is shown in Figure 8. It involves a dual-loop control structure: an outer dc-link voltage control loop and an inner current control loop.Figure 8
Control of GSC by ADRC.As mentioned before the grid voltages are oriented such thatvsd=0 and vsq=Vs, As a result, equations 24 are rewritten by using(53)difddt=−RfLfifd−ωsifq−1Lfvfd,(54)difqdt=−RfLfifq+1LfVs−ωsifd−1Lfvfq.The active and reactive powers exchanged with the power grid through the filter are expressed by(55)Pf=32Vsifq,(56)Qf=32Vsifd.The power across the DC link Capacitor C can be expressed by(57)Pdc=Udcig−ir.And we have(58)dUdcdt=1Cig−ir,or(59)Pdc=CUdcdUdcdt.By neglecting all the losses in filter, power electronics converters, and the capacitor, the exchanged power on the DC bus is given by(60)Pdc=Pf−Pr,where PrPf are the generator rotor side and filter side powers, respectively.By taking into account equations (55)–(60), the DC bus voltage is expressed by using(61)CUdcdUdcdt=32Vsifq−Udcir,or(62)dUdc2dt=3VsCifq−2UdcCir.We putX=Udc2; then, we have(63)dXdt=3VsCifq−2XCir.Therefore, we obtain the canonical form of the ADRC controller such that(64)dXdt=fX·+bc0uX,or(65)fX·=−2XCir+3VsC−bc0ifq,bc0=3VsC;uX=ifq.The external voltage regulation loop makes it possible to maintain the voltage across the capacitorUdc constant and to generate the reference current ifq_ref for the current inner loop [25]. The reference current ifd_ref, given by equation (66), is calculated by the desired delivery of reactive power:(66)ifd_ref=23VsQf_ref.For the inner current control loop, equations (53) and (54) are adapted with the canonical forms of ADRC and then we obtain equations (67) and (69):(67)difddt=ffd·+bf0ufd,with(68)ffd·=RfLfifd−ωsifq+1Lf−bf0Vfd,bf0=−1Lf;ufd=Vfd.In addition,(69)difqdt=ffq·+bf0ufq,with(70)ffq·=1LfVs−RfLfifq−ωsifd+1Lf−bf0Vfq,bf0=−1Lf;ufq=Vfq.


## 5. Simulations and Results

The DFIG based wind conversion system was modeled and simulated under the Matlab/Simulink environment. The WECS parameters used for the simulation are given in the Appendix section.To illustrate the performance of the ADRC control strategy, two tests, namely, tracking and robustness tests, have been conducted under different conditions. It should be noted that, throughout the first test, the reference reactive powersQs_ref and Qf_ref are set at zero, in order to ensure a unit power factor at the grid connection point, and in the second test, the stator reactive power reference has been set to Qs_ref=−1e5VAR.


### 5.1. Tracking Test

In this test, a variable speed wind profile is applied to the WT blades, as illustrated in Figure9, to validate tracking effectiveness of the proposed control strategy. The following wind speed profile corresponds, as shown in Figure 10, to a DFIG mechanical speed in MPPT control varying between 1200 rpm and 1750 rpm. The generator therefore operates in both hyposynchronous and hypersynchronous modes with slip varying from -16, 6% to about 20%.Figure 9
Applied variable wind speed profile.Figure 10
DFIG Mechanical speed (rpm).It can be observed from Figure11 that the power coefficient Cp was maintained at its optimum value, which corresponds to the MPPT. The corresponding extracted power by the WT is shown in Figure 12.Figure 11
Turbine power coefficient.Figure 12
Extracted wind turbine mechanical power.Figures13 and 14 show the regulation of the rotor currents ird and irq by PI and ADRC; it can be seen that both of them follow their references; however, the ADRC shows a better dynamic response and follows its references perfectly. Current irq, which controls the electromagnetic torque Tem of the generator, varies according to the wind speed profile so as to extract maximum power; this can be seen in Figure 15. Current ird, which controls the reactive power Qs, is kept constant to have a unit power factor on the stator side as highlighted in Figure 16.Figure 13
Regulation of the currentird by PI and ADRC.Figure 14
Regulation of the currentirq by PI and ADRC.Figure 15
Active power regulation by PI and ADRC.Figure 16
Reactive power regulation by PI and ADRC.Figures17 and 18 show the waveforms of the stator and rotor currents generated by the DFIG; it can be seen that stator currents follow the variation of the wind; on the other hand, the rotor currents show that the DFIG works in hyposynchronous when the wind speed is less than 1500 rpm and in hypersynchronous when the wind speed is higher than 1500 rpm (2 s to 3.7 s).Figure 17
Waveforms of generated stator currents.Figure 18
Waveforms of generated rotor currents.Figure19 shows that the DC bus voltage Udc was kept constant by both regulators (ADRC and PI); however, the PI controllers present an overshoot of 250 V compared to the ADRC controller.Figure 19
DC bus voltage regulation by ADRC and PI.Figures20–23 represent the simulation results of currents ifd and ifq of the (Rf, Lf) filter and the active and reactive powers exchanged with power via the rotor side.Figure 20
Regulation of the currentifd by PI and ADRC.Figure 21
Regulation of the currentifq by PI and ADRC.Figure 22
Reactive power regulation by PI and ADRC.Figure 23
Active power regulation by PI and ADRC.Currentifd tracks perfectly its zero setpoint, which allows having a reactive power zero on the grid side. Current ifq was maintained at its set value, which allows the exchange of active power between the DFIG Rotor and the grid according to the wind profile.Figure24 illustrates the waveforms of phase stator current isa and grid voltage vsa; it can be observed that the current and voltage are in phase opposition, which means zero reactive power injection and active power delivery to the grid.Figure 24
Waveforms of phase a statorisa current and grid voltage vsa.


### 5.2. Robustness Test

The robustness test consists in varying the DFIG model parameters. Indeed, the calculations of the regulators are based on functions whose parameters are assumed to be fixed. However, in a real system, these parameters are subject to variations caused by different physical phenomena. Figures25–28 show the evolution of the active and reactive power injected by stator side or the image of the rotor currents regulation. During this test, the same wind speed profile was kept, and the reference of the stator reactive power was changed from zero to Qs_ref=−1e5VAR at t = 2.5 and then returned to zero at t = 4. Two robustness tests were made; in the first one, the rotor resistance Rr was increased by 100%, and in the second one, the inductance Lr was changed by 30%.Figure 25
Stator Active power regulation for 2∗ Rr.Figure 26
Stator Reactive power regulation for 2∗ Rr.Figure 27
Stator Active power regulation for 1.3∗ Lr.Figure 28
Stator reactive power regulation for 1.3∗ Lr.As it can be concluded from the figures shown, these variations of Rr and Lr have almost no influence on the ADRC regulators, due to its capability in automatically estimating and compensating the disturbances by means of the ESO, which is not the case for the classical PI controller; its tuning gains are designed according to the model parameters, and this variation affects its performance compared to the ADRC.


## 5.1. Tracking Test

In this test, a variable speed wind profile is applied to the WT blades, as illustrated in Figure9, to validate tracking effectiveness of the proposed control strategy. The following wind speed profile corresponds, as shown in Figure 10, to a DFIG mechanical speed in MPPT control varying between 1200 rpm and 1750 rpm. The generator therefore operates in both hyposynchronous and hypersynchronous modes with slip varying from -16, 6% to about 20%.Figure 9
Applied variable wind speed profile.Figure 10
DFIG Mechanical speed (rpm).It can be observed from Figure11 that the power coefficient Cp was maintained at its optimum value, which corresponds to the MPPT. The corresponding extracted power by the WT is shown in Figure 12.Figure 11
Turbine power coefficient.Figure 12
Extracted wind turbine mechanical power.Figures13 and 14 show the regulation of the rotor currents ird and irq by PI and ADRC; it can be seen that both of them follow their references; however, the ADRC shows a better dynamic response and follows its references perfectly. Current irq, which controls the electromagnetic torque Tem of the generator, varies according to the wind speed profile so as to extract maximum power; this can be seen in Figure 15. Current ird, which controls the reactive power Qs, is kept constant to have a unit power factor on the stator side as highlighted in Figure 16.Figure 13
Regulation of the currentird by PI and ADRC.Figure 14
Regulation of the currentirq by PI and ADRC.Figure 15
Active power regulation by PI and ADRC.Figure 16
Reactive power regulation by PI and ADRC.Figures17 and 18 show the waveforms of the stator and rotor currents generated by the DFIG; it can be seen that stator currents follow the variation of the wind; on the other hand, the rotor currents show that the DFIG works in hyposynchronous when the wind speed is less than 1500 rpm and in hypersynchronous when the wind speed is higher than 1500 rpm (2 s to 3.7 s).Figure 17
Waveforms of generated stator currents.Figure 18
Waveforms of generated rotor currents.Figure19 shows that the DC bus voltage Udc was kept constant by both regulators (ADRC and PI); however, the PI controllers present an overshoot of 250 V compared to the ADRC controller.Figure 19
DC bus voltage regulation by ADRC and PI.Figures20–23 represent the simulation results of currents ifd and ifq of the (Rf, Lf) filter and the active and reactive powers exchanged with power via the rotor side.Figure 20
Regulation of the currentifd by PI and ADRC.Figure 21
Regulation of the currentifq by PI and ADRC.Figure 22
Reactive power regulation by PI and ADRC.Figure 23
Active power regulation by PI and ADRC.Currentifd tracks perfectly its zero setpoint, which allows having a reactive power zero on the grid side. Current ifq was maintained at its set value, which allows the exchange of active power between the DFIG Rotor and the grid according to the wind profile.Figure24 illustrates the waveforms of phase stator current isa and grid voltage vsa; it can be observed that the current and voltage are in phase opposition, which means zero reactive power injection and active power delivery to the grid.Figure 24
Waveforms of phase a statorisa current and grid voltage vsa.


## 5.2. Robustness Test

The robustness test consists in varying the DFIG model parameters. Indeed, the calculations of the regulators are based on functions whose parameters are assumed to be fixed. However, in a real system, these parameters are subject to variations caused by different physical phenomena. Figures25–28 show the evolution of the active and reactive power injected by stator side or the image of the rotor currents regulation. During this test, the same wind speed profile was kept, and the reference of the stator reactive power was changed from zero to Qs_ref=−1e5VAR at t = 2.5 and then returned to zero at t = 4. Two robustness tests were made; in the first one, the rotor resistance Rr was increased by 100%, and in the second one, the inductance Lr was changed by 30%.Figure 25
Stator Active power regulation for 2∗ Rr.Figure 26
Stator Reactive power regulation for 2∗ Rr.Figure 27
Stator Active power regulation for 1.3∗ Lr.Figure 28
Stator reactive power regulation for 1.3∗ Lr.As it can be concluded from the figures shown, these variations of Rr and Lr have almost no influence on the ADRC regulators, due to its capability in automatically estimating and compensating the disturbances by means of the ESO, which is not the case for the classical PI controller; its tuning gains are designed according to the model parameters, and this variation affects its performance compared to the ADRC.


## 6. Conclusion

This paper was devoted to the modeling and control of the variable speed wind energy conversion system based on the DFIG, where the main objective is to operate the WT so as to extract the maximum power from the wind energy. To reach this goal a control strategy based on ADRC regulators has been proposed and applied to control the RSC via the rotor currents, the GSC and DC bus voltage through the currents flowing in filter, and the active and reactive power transits between the WT and the electrical power grid.To validate the effectiveness of the proposed system, the system was modeled and simulated under the Matlab/Simulink environment. Two tests, namely, tracking and robustness, were performed to compare the proposed ADRC technique and classical PI controllers. The obtained results are promising and have shown that the proposed control strategy based on ADRC, especially when varying the mode parameters, is very performant.



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*Source: 1651648-2020-07-23.xml*