# Power Extraction Control of Variable Speed Wind Turbine Systems Based on Direct Drive Synchronous Generator in All Operating Regimes

Authors: Youssef Errami; Abdellatif Obbadi; Smail Sahnoun; Mohammed Ouassaid; Mohamed Maaroufi
Journal: Journal of Electrical and Computer Engineering
 (2018)

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## Abstract

Due to the increased penetration of wind energy into the electrical power systems in recent years, the turbine controls are actively occupied in the research. This paper presents a nonlinear backstepping strategy to control the generators and the grid sides of a Wind Farm System (WFS) based Direct Drive Synchronous Generator (DDSG). The control objectives such as Tracking the Maximum Power (TMP) from the WFS, pitch control, regulation of dc-link voltage, and reactive and active power generation at varying wind velocity are included. To validate the proposed control strategy, simulation results for 6-MW-DDSG based Wind Farm System are carried out by MATLAB-Simulink. Performance comparison and evaluation with Vector Oriented Control (VOC) are provided under a wide range of functioning conditions, three-phase voltage dips, and the probable occurrence of uncertainties. The proposed control strategy offers remarkable characteristics such as excellent dynamic and steady state performance under varying wind speed and robustness to parametric variations in the WFS and under severe faults of grid voltage.


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## Body

## 1. Introduction

In the past decades, various renewable energy sources have received increasing attention as alternatives of fossil fuels [1, 2]. Among various modern renewable resources, wind power is considered the backbone of renewable power generation. It is regarded as a clean energy resource and it is easily accessible and cost effective. So, wind power penetration greatly increases in the electric power systems and it is anticipated to keep steady growth in the upcoming years [3, 4]. On the other hand, in modern wind power systems, three of the most promising types of wind turbine generators are the Doubly Fed Induction Generator (DFIG), the Direct Drive Synchronous Generator (DDSG) with permanent magnet, and the Squirrel Cage Induction Generator (SCIG) [5–12]. But, the main advantage of DDSG is the absence of gearbox coupling the wind turbine to the DDSG and the dc external excitation current. So, during the past few years direct driven DDSGs become very attractive in wind energy application because of their high power density, flexible magnet topologies, reduced maintenance, high effectiveness, high operating efficiency, excellent operation performance, and increased reliability [13–19]. In the area of wind power generation technology, Variable Speed Wind Power Generation System (VS-WPGS) has many advantages over fixed speed ones such as better power quality, higher overall efficiency, lower mechanical stress, and increased energy capture. So, it can be controlled to enable the turbine to operate at its maximum coefficient of power and to ensure the Maximum Power Tracking (MPT) ability [20–22]. To control the VS-WPGS based DDSG, full scale power electronic converter systems are generally used as the interface between the VS-WPGS and the electrical network for satisfying the new standards and grid connection requirements [23, 24]. Thus, they allow controlling the wind turbine system, decoupling the DDSG from the power grid, and the VS-WPGS does not need to synchronize its rotational speed with the electrical network frequency [25]. Thus, VS-WPGS is able to attain high efficacy and performance when connected to the electrical network, not only under normal working conditions but also under irregular and faulty network conditions. Figure 1 illustrates the general control structure for VS-WPGS. This consists of a DDSG side and a grid side inverter interconnected through a dc-link system. So, at the DDSG sides, the speed is controlled to ensure the MPT. The grid side converter is employed to regulate the dc-link voltage for a proper transfer of power [13, 19, 26, 27]. Also, the inverter system ought to have the ability to regulate reactive and active power that the VS-WPGS exchange with the electrical network and attain unity power factor (UPF) of total system [28]. Various power electronic device configurations were presented in the literature for VS-WPGS based DDSG [29–31]. On the other hand, to increase the annual energy yield of VS-WPGS, MPT control is essential under the rated wind velocity. Then, by changing the rotor velocity of DDSG according to the varying wind velocity, maximum power is extracted from available wind [21, 22, 32, 33].Figure 1
Configuration of a VS-WPGS based DDSG.Also, variable speed wind turbines frequently employ pitch control system, either active stall or pitch to feather, to reduce the aerodynamic power captured [34, 35].According to Fault Ride-Through (FRT) requirements, during faults of grid voltage, the WFS must remain connected to the electric network. As an example, Figure2 presents a diagram of the FRT requirements in some countries. So, these requirements present a considerable challenge to the well-established WFS technologies and their control [36]. Besides, developing effective control approaches WFS will improve the system availability to extract as much power from the wind as possible whereas it is available. On the basic aspect, Vector Oriented Control (VOC) is widely used because of its simple control configuration, ease of design, and economical cost. In [37, 38], control strategy has been developed for integrated control of wind turbine based PMSG. The works [39, 40] have presented Low Voltage Ride-Through (LVRT) of WECS. But, this technique cannot offer satisfactory control performance if the controlled system is highly nonlinear and uncertain. Moreover, it generally works at particular operating range and it is obligatory to retune the controller if the operating range is modified, whereas the WFS is usually affected by varying the operating condition and outside disturbance.Figure 2
National grid codes [36].To overcome the above problems, Direct Control Techniques (DCT) were proposed to substitute the VOC strategy for both rectifier systems and electrical network connection control. Direct power control (DPC) and Direct Torque Control (DTC) of a three-phase converter for fault tolerant DDSG drive were proposed in [29]. The works [19, 41] have proposed TMP algorithm using DTC for DDSG wind turbines. The most important advantages of DCT strategies include low sensitivity to the accuracy of WFS parameter estimation, rapid dynamic response, and easy implementation and they do not necessitate space vector modulation (SVM). But, they can suffer from high torque ripple and variable switching frequency of the power electronic converters and their performances deteriorate at low speed.On the other hand, modern WFS necessitate efficient and supple technologies that adapt to modifications in load and generation. Also, in order to increase the availability and reliability of WFS, effective strategies are considered crucial means to achieve these goals and to improve the electrical network connected operation capability. Recently, backstepping control approach has received worldwide attention due to its systematic and recursive design methodology for nonlinear control [42]. The stabilization of a virtual control state is the fundamental idea of the backstepping algorithm. This nonlinear control method is principally interesting due to its capacity to design adaptive controllers for nonlinear systems and to analyze the problem of stability. The advantages of backstepping control are related to the employment of the Lyapunov functions to ensure the robustness and stability of the system, consequently enhancing the performance of system [43]. Also, because of the flexibility to avoid cancellations of suitable nonlinearities and attain the objectives of tracking and stabilization, the backstepping approach has been generally applied in control systems. In the work of [44] backstepping control is developed to deal with the Tracking the Maximum Power (TMP) from the WFS. The work of [45] presents a backstepping control strategy to design a permanent magnet liner synchronous motor servo drive system with uncertainty. Wai et al. [46] proposed the backstepping design for a grid-connected inverter. Ruan et al. [47] proposed an adaptive control strategy using backstepping technique for the Voltage Source Converter (VSC) based High Voltage dc (HVDC) system to develop its dynamic performance. The idea of backstepping approach is to choose recursively some suitable functions of state variables like pseudo-control inputs for lower dimension subsystems of the entire system. The most attractive point of this technique is to employ the virtual control variable to create the original high order system simple. It decomposes a difficult control problem into simpler ones and it chooses recursively suitable functions of state variables which are the virtual control variables. Each backstepping step results in a novel virtual control to deal with a decomposed subsystem problem. So, this virtual control becomes a reference to the next algorithm step for another subsystem. Also, it produces a related error variable that can be stabilized by carefully choosing proper control inputs which can be determined using Lyapunov stability analysis [43]. When the procedure is finished, a true control input for the original control objectives can be generated by summing up the Lyapunov functions associated with each individual design step [48].This study presents a backstepping control design for the WFS using PMSG, connected to the electrical network. So, a backstepping scheme is developed in the sense of Lyapunov stability theorem to ensure the control objectives of Tracking the Maximum Power (TMP) from the WFS, pitch control, regulation of dc-link voltage, and reactive and active power generation at varying wind velocity. In the first place, mathematical model of WFS is described. In the meanwhile, design of the backstepping technique is provided. Then, simulation studies using MATLAB/Simulink are carried out in Section3. Finally, some conclusions are summarized in Section 4.


## 2. Description of the Proposed WFS

This section describes the proposed wind energy conversion system using DDSG and presents its modelling and control strategies.


### 2.1. System Description

The schematic diagram of the grid-connected DDSG-WTS is shown in Figure3. It is composed of three wind turbines directly linked to DDSGs connected to the grid throughout a converter, which consists of a grid side inverter (GSI) and a generator-side rectifier (MSR). The MSR control the DDSGs to extract the maximum power from the wind [49]. The GSI sustains a constant dc-link voltage and controls the active and reactive power that the WTS exchanges with the electrical power.Figure 3
Configuration of the VS-WFS.So, the converters allow adapting frequencies and voltages between the DDSG terminals and the grid.


### 2.2. Wind Turbine Characteristics

According to aerodynamics, the maximum power extracted by the rotor blades is expressed as [50]:(1)Pw-max=12ρwπArw2Cw-maxλw-m,βwvw3,where ρw is the air density, Arw is the rotator radius, vw is the speed of wind, λw-m is the optimal tip speed ratio (TSR), Cw-max(λm,βw) is the maximum value of the performance coefficient of the turbine, and βwis the blade pitch angle. The tip speed ratio λw is determined by the rotor speed ωr and the wind speed as(2)λw=ωrArwvw.So, an efficient TMP is fundamental for WFS to develop the energy capture effectiveness from wind and the pitch angle βw is set as zero during the TMP operation. The optimal TSR strategy is realized by directly regulating the tip speed ratio, which is calculated by real time using wind velocity and generator speed, to follow a preobtained λw-m. Therefore, one way to modify the captured wind power is to adjust the DDSG speed, which is depicted in Figure 4 [51]. As seen, different turbine velocities are associated with different captured wind power. Consequently, the generator-side rectifiers in Figure 3 regulate the speeds of DDSGs to attain TMP in combination with the pitch control system. Figure 4 depicts the typical wind turbine power-rotor velocity characteristic curves for different wind velocities, where the optimal power curve illustrates the TMP at different conditions of wind speed. The maximum power extracted Pw-max can be obtained as a function of the shaft velocity:(3)Pw-max=ρwπArw5Cw-max2λw-m3ωr3=Kmax⁡ωr3,where Kmax⁡ is the coefficient of rotational velocity for the maximum extracted power through Cw-max at the wind generator. It should be noted that when the wind velocity is high, the generator velocity and power would exceed the maximum value if the MPPT control is still employed. The pitch control is provided to limit the captured power and the generator speed. So, the pitch angle is set to zero to obtain TMP if the speed is below the maximum speed. But, if the velocity exceeds its rated value, the pitch angle is controlled to limit the captured power to the maximum rated value and the WFS would lead to deviating the TMP searching strategy to prevent overspeeding. The pitch control diagram is shown in Figure 5.Figure 4
Wind turbine power curves at different values of wind speed.Figure 5
Pitch control diagram for generator speed limiting.


### 2.3. DDSG Dynamic Modelling

If the rotating reference frame is aligned with the magnetic axis of the rotor, the electric dynamics equations of a three-phase DDSG can be written in a synchronously rotatingdq reference frame as [52](4)vsd⁡vsq=Rs+p·Lsd⁡-ωeLsqωeLsd⁡Rs+p·Lsd⁡isd⁡isq+0ωeψs,where vsq, vsd⁡ are stator terminal voltages in the dq frame (V); isq, isd⁡ are direct and quadrature currents components in the stator windings (A); Lsq, Lsd⁡ are inductances of the DDSG in the dq frame (H); Rs is resistance of the stator windings (Ω); ψs is flux linkage generated by the permanent magnets (Wb).The electrical angular velocity of the rotor,ωe, is defined as(5)ωe=pnωr,where pn is the number of pole pairs of the generator; ωr is the mechanical angular velocity (rad/sec).If the DDSG is assumed to have equalq-axis, d-axis in inductances (Lsq=Lsd⁡=Lgs), the torque developed by the DDSG is(6)Tsg=32pnψsisq.The generator velocity in the one-mass model of the wind turbine systems is given as follows:(7)dωrdt=1JsTsg-Twt-Fsωr,where Js is the moment of inertia (Kg·m2), Fs is the viscous friction coefficient (Nm/rad·sec−1), and Twt is the mechanical torque developed by the wind turbine (WT) (Nm).


### 2.4. Backstepping Control Design for the MSR

To attain optimum power effectiveness regardless of wind speed variations, the MSR is used to control the DDSG speeds. Control of the MSR permits the generators to regulate the rotational velocities according to the wind variation. On the other hand, the TMP algorithm adopted in this work is the tip speed ratio control. The backstepping control is employed to perform the TMP control of the wind turbine systems. So, according to (2), the reference of the mechanical DDSG speed can be derived as(8)ωr-ref=λwvwArw.For each MSR, the major control objective is to track the velocity reference as(9)αω=ωr-ref-ωr.The dynamic of the velocity tracking error is expressed as(10)dαωdt=dωr-refdt-dωrdt.From (6)-(7) and (10), we have(11)dαωdt=dωr-refdt-1Js32pnψsisq-Twt-Fsωr.So, a first Lyapunov function candidate is defined as [53](12)Γ1=12αω2.Differentiating (12) gives us(13)dΓ1dt=αωdαωdt=αωdωr-refdt-αωJs32pnψsisq-Twt-Fsωr.Equation (13) can be reorganized as(14)dΓ1dt=-δωαω2+αωJsJsδωαω+Twt-32pnψsisq+Fsωr+Jsdωr_refdt,where δω is the positive closed-loop feedback constant. According to (14), the backstepping law can be designed as follows:(15)isd-ref=0,(16)isq-ref=23pnψsJsδωαω+Jsdωr_refdt+Fsωr+Twt.If the speed error can be made to zero by selecting proper control input, (14) can be simplified to [54](17)dΓ1dt=-δωαω2≺0.Consequently, by Lyapunov stability analysis, the velocity controller is asymptotically stable. To stabilize the currents components isd⁡ and isq, define the following current errors:(18)αsd⁡=isd-ref-isd⁡αsq=isq-ref-isq,where isq-ref, isd-ref are the current references. Taking the derivative of αsd⁡ and αsq with respect to time and using (4), one can obtain (19)αsd⁡dαsd⁡dt=αsd⁡-1Lgsvsd⁡+Lgsωeisq-Rsisd⁡=-λsd⁡αd2+αsd⁡Lgs-vsd⁡-Lgsωeisq+Rsisd⁡+Lgsλsd⁡αsd⁡αsqdαsqdt=αsqdisq-refdt-disqdt=-λsqαsq2+αsqLgsLgsdisq-refdt-vsd⁡+ωeψs+Rsisq+Lgsωeisd⁡+λsqLgsαsq,where λsd and λsq are the positive closed-loop feedback constants. The second Lyapunov function candidate is defined as (20)Γ2=Γ1+12αsq2+12αsd2=12αω2+12αsq2+12αsd2.Differentiating (20) gives us(21)dΓ2dt=αωdαωdt+αsddαsddt+αsqdαsqdt.From (14) and (18)–(19), (21) can be reorganized as(22)dΓ2dt=-δωαω2+αωJsJsδωαω+Twt-32pnψsisq+Fsωr+Jsdωrrefdt-λsqαsq2+αsqLgsLgsdisq-refdt-vsd⁡+ωeψs+Rsisq+Lgsωeisd⁡+λsqLgsαsq-λsd⁡αsd⁡2+αsd⁡Lgs-vsd⁡-Lgsωeisq+Rsisd⁡+Lgsλsd⁡αsd⁡.The commands vsq-ref and vsd-ref are carried out from (22) as(23)vsq-ref=ωeψs+Rsisq+Lgsωeisd⁡+Lgsdisq-refdt+Lgsλsqαsqvsd-ref=Lgsλsd⁡αsd⁡-Lgsωeisq+Rsisd⁡.Using (16) and (23), (22) can be simplified to(24)dΓ2dt=-δωαω2-λsdαsd2-λsqαsq2≤0.Therefore, by Lyapunov stability analysis, the backstepping regulator is asymptotically stable and the control of the DDSG speeds is attained. The control diagram for individual MSR is depicted in Figure 6.Figure 6
Control block diagram of MSR using backstepping approach.


### 2.5. Nonlinear dc-link Voltage Regulation

The GSI controls the dc-link voltage and transfers the power from turbine-DDSG to the grid. Also, it contributes to the reactive and active power control of the overall DDSG system [55]. If the reference frame is rotating synchronously with the electrical grid voltage vector, the dynamic model voltage equations of the electrical network connection in grid voltage oriented reference frame (d,q) are represented as follows:(25)did-griddt=1Lgfegd-Rgfid-grid+ωgLgfiq-grid-Usdiq-griddt=1Lgfegq-Rgfiq-grid-ωgLgfid-grid,where egd, egq are the grid side inverter voltage (d-q) components (V); Us is the amplitude of the grid voltage (V); Rgf (Ω) and Lgf (H) are the resistance and inductance of the filter; ωg is the angular frequency of grid; and id-grid and iq-grid are the d-axis current and q-axis current (A) of the electrical grid, respectively. The active Pg and reactive power Qg can be calculated as follows [56]:(26)Pg=32Usid-gridQg=32Usiq-grid.Equations (26) illustrate that the active and reactive powers are regulated to achieve the required voltage by changing the respective current of thed-q-axis. The dc-link voltage equation can be formulated as in the following equation:(27)CdVdcdt=ig-igrid,where C is the capacitance of the dc-link capacitor (F); Vdc is dc-link voltage (V); igrid is the current between the dc-link and the grid; and ig is the current between the dc-link and the DDSG stators (A). So, the active power along the side of the GSI is(28)Vdcigrid=32Usid-grid.From (27)-(28), the dc-link voltage dynamics can be calculated as(29)dVdcdt=ig-1C32UsVdcid-grid.Consequently, to transfer all of the active power generated in turbine-DDSG, the dc-link voltage Vdc must be maintained at a constant value using the current control of id-grid. Equation (29) can be reorganized as(30)VdcCdVdcdt=PTotal-32Usid-grid,where PTotal is the generated power of the DDSGs. Equation (30) can be expressed as(31)C2dVdc2dt=PTotal-32Usid-grid.The tracking error of the dc-bus voltage is defined as (32)εdc-link=Vdc-r2-Vdc2.The dc-link voltage tracking error has the following dynamic:(33)dεdc-linkdt=dVdc-r2dt-2CPTotal-32Usid-grid-r,where id-grid-r is the current reference. Consider a third Lyapunov function candidate:(34)Φdc-link=12εdc-link2.From (33), differentiating (34) gives us(35)dΦdc-linkdt=εdc-linkdεdc-linkdt=εdc-linkdVdc-r2dt-2CPTotal-32Usid-grid-r.So, (35) can be reorganized as(36)dΦdc-linkdt=-λbusεdc-link2+εdc-linkdVdc-r2dt-2CPTotal-32Usid-grid-r+λbusεdc-link,where λbus is the positive closed-loop feedback constant. According to (36), the backstepping law can be designed as follows: (37)id-grid-r=1Us23PTotal-C3dVdc-r2dt+λbusεdc-link.From (37), (38) can be simplified to(38)dΦdc-linkdt=-λbusεdc-link2≤0.Consequently, by Lyapunov stability analysis, dc-link voltage controller is asymptotically stable. To stabilize the currents components id-grid and iq-grid, define the following current errors:(39)ξd-grid=id-grid-r-id-grid(40)ξq-grid=iq-grid-r-iq-grid.The reference iq-grid-r is calculated from the desired power factor. Based on (37) and (39), (33) can be reorganized as(41)dεdc-linkdt=-λdc-linkεdc-link+3CUsξd-grid.Besides, from (25) and (39)-(40), one can obtain (42)ξd-griddξd-griddt=ξd-griddid-grid-rdt-did-griddt=-λd-gridξd-grid2+ξd-gridLgfLgfdid-grid-rdt-egd+Us+Rgfid-grid-Lgfωgiq-grid+λd-gridLgfξd-gridξq-griddξq-griddt=-diq-griddt=-λq-gridξq-grid2+ξq-gridLgf-egq+Rgfiq-grid+Lgfωgid-grid+λq-gridLgfλq-grid,where λd-grid and λq-grid are the positive closed-loop feedback constants. Consider the fourth Lyapunov function candidate: (43)ΦGSI=12εdc-link2+12ξq-grid2+12ξd-grid2.Differentiating (43) gives us(44)dΦGSIdt=εdc-linkdεdc-linkdt+ξd-griddξd-griddt+ξq-griddξq-griddt.From (41) and (42), (44) can be reorganized as(45)dΦGSIdt=-λdc-linkεdc-link2-λd-gridξd-grid2-λq-gridξq-grid2+ξd-gridLgf3LgfCUsεdc-link+Lgfdid-grid-rdt-egd+Rgfid-grid-Lgfωgiq-grid+Us+λd-gridLgfξd-grid+ξq-gridLgf-egq+Rgfiq-grid+Lgfωgid-grid+λq-gridLgfξq-grid.According to (45), the backstepping law can be designed as follows: (46)vd-GSI=3LgfCUsεdc-link+Lgfdid-grid-rdt+Rgfid-grid-Lgfωgiq-grid+Us+Lgfλd-gridξd-gridvq-GSI=Rgfiq-grid+Lgfωgid-grid+Lgfλq-gridξq-grid.Using (37) and (46), (45) can be simplified to(47)dΦGSIdt=-λdc-linkεdc-link2-λd-gridξd-grid2-λq-gridξq-grid2≤0.Therefore, by Lyapunov stability analysis, the backstepping regulator is asymptotically stable and the control of the GSI is attained. The control diagram is depicted in Figure 7.Figure 7
Control block diagram of GSI.


## 2.1. System Description

The schematic diagram of the grid-connected DDSG-WTS is shown in Figure3. It is composed of three wind turbines directly linked to DDSGs connected to the grid throughout a converter, which consists of a grid side inverter (GSI) and a generator-side rectifier (MSR). The MSR control the DDSGs to extract the maximum power from the wind [49]. The GSI sustains a constant dc-link voltage and controls the active and reactive power that the WTS exchanges with the electrical power.Figure 3
Configuration of the VS-WFS.So, the converters allow adapting frequencies and voltages between the DDSG terminals and the grid.


## 2.2. Wind Turbine Characteristics

According to aerodynamics, the maximum power extracted by the rotor blades is expressed as [50]:(1)Pw-max=12ρwπArw2Cw-maxλw-m,βwvw3,where ρw is the air density, Arw is the rotator radius, vw is the speed of wind, λw-m is the optimal tip speed ratio (TSR), Cw-max(λm,βw) is the maximum value of the performance coefficient of the turbine, and βwis the blade pitch angle. The tip speed ratio λw is determined by the rotor speed ωr and the wind speed as(2)λw=ωrArwvw.So, an efficient TMP is fundamental for WFS to develop the energy capture effectiveness from wind and the pitch angle βw is set as zero during the TMP operation. The optimal TSR strategy is realized by directly regulating the tip speed ratio, which is calculated by real time using wind velocity and generator speed, to follow a preobtained λw-m. Therefore, one way to modify the captured wind power is to adjust the DDSG speed, which is depicted in Figure 4 [51]. As seen, different turbine velocities are associated with different captured wind power. Consequently, the generator-side rectifiers in Figure 3 regulate the speeds of DDSGs to attain TMP in combination with the pitch control system. Figure 4 depicts the typical wind turbine power-rotor velocity characteristic curves for different wind velocities, where the optimal power curve illustrates the TMP at different conditions of wind speed. The maximum power extracted Pw-max can be obtained as a function of the shaft velocity:(3)Pw-max=ρwπArw5Cw-max2λw-m3ωr3=Kmax⁡ωr3,where Kmax⁡ is the coefficient of rotational velocity for the maximum extracted power through Cw-max at the wind generator. It should be noted that when the wind velocity is high, the generator velocity and power would exceed the maximum value if the MPPT control is still employed. The pitch control is provided to limit the captured power and the generator speed. So, the pitch angle is set to zero to obtain TMP if the speed is below the maximum speed. But, if the velocity exceeds its rated value, the pitch angle is controlled to limit the captured power to the maximum rated value and the WFS would lead to deviating the TMP searching strategy to prevent overspeeding. The pitch control diagram is shown in Figure 5.Figure 4
Wind turbine power curves at different values of wind speed.Figure 5
Pitch control diagram for generator speed limiting.


## 2.3. DDSG Dynamic Modelling

If the rotating reference frame is aligned with the magnetic axis of the rotor, the electric dynamics equations of a three-phase DDSG can be written in a synchronously rotatingdq reference frame as [52](4)vsd⁡vsq=Rs+p·Lsd⁡-ωeLsqωeLsd⁡Rs+p·Lsd⁡isd⁡isq+0ωeψs,where vsq, vsd⁡ are stator terminal voltages in the dq frame (V); isq, isd⁡ are direct and quadrature currents components in the stator windings (A); Lsq, Lsd⁡ are inductances of the DDSG in the dq frame (H); Rs is resistance of the stator windings (Ω); ψs is flux linkage generated by the permanent magnets (Wb).The electrical angular velocity of the rotor,ωe, is defined as(5)ωe=pnωr,where pn is the number of pole pairs of the generator; ωr is the mechanical angular velocity (rad/sec).If the DDSG is assumed to have equalq-axis, d-axis in inductances (Lsq=Lsd⁡=Lgs), the torque developed by the DDSG is(6)Tsg=32pnψsisq.The generator velocity in the one-mass model of the wind turbine systems is given as follows:(7)dωrdt=1JsTsg-Twt-Fsωr,where Js is the moment of inertia (Kg·m2), Fs is the viscous friction coefficient (Nm/rad·sec−1), and Twt is the mechanical torque developed by the wind turbine (WT) (Nm).


## 2.4. Backstepping Control Design for the MSR

To attain optimum power effectiveness regardless of wind speed variations, the MSR is used to control the DDSG speeds. Control of the MSR permits the generators to regulate the rotational velocities according to the wind variation. On the other hand, the TMP algorithm adopted in this work is the tip speed ratio control. The backstepping control is employed to perform the TMP control of the wind turbine systems. So, according to (2), the reference of the mechanical DDSG speed can be derived as(8)ωr-ref=λwvwArw.For each MSR, the major control objective is to track the velocity reference as(9)αω=ωr-ref-ωr.The dynamic of the velocity tracking error is expressed as(10)dαωdt=dωr-refdt-dωrdt.From (6)-(7) and (10), we have(11)dαωdt=dωr-refdt-1Js32pnψsisq-Twt-Fsωr.So, a first Lyapunov function candidate is defined as [53](12)Γ1=12αω2.Differentiating (12) gives us(13)dΓ1dt=αωdαωdt=αωdωr-refdt-αωJs32pnψsisq-Twt-Fsωr.Equation (13) can be reorganized as(14)dΓ1dt=-δωαω2+αωJsJsδωαω+Twt-32pnψsisq+Fsωr+Jsdωr_refdt,where δω is the positive closed-loop feedback constant. According to (14), the backstepping law can be designed as follows:(15)isd-ref=0,(16)isq-ref=23pnψsJsδωαω+Jsdωr_refdt+Fsωr+Twt.If the speed error can be made to zero by selecting proper control input, (14) can be simplified to [54](17)dΓ1dt=-δωαω2≺0.Consequently, by Lyapunov stability analysis, the velocity controller is asymptotically stable. To stabilize the currents components isd⁡ and isq, define the following current errors:(18)αsd⁡=isd-ref-isd⁡αsq=isq-ref-isq,where isq-ref, isd-ref are the current references. Taking the derivative of αsd⁡ and αsq with respect to time and using (4), one can obtain (19)αsd⁡dαsd⁡dt=αsd⁡-1Lgsvsd⁡+Lgsωeisq-Rsisd⁡=-λsd⁡αd2+αsd⁡Lgs-vsd⁡-Lgsωeisq+Rsisd⁡+Lgsλsd⁡αsd⁡αsqdαsqdt=αsqdisq-refdt-disqdt=-λsqαsq2+αsqLgsLgsdisq-refdt-vsd⁡+ωeψs+Rsisq+Lgsωeisd⁡+λsqLgsαsq,where λsd and λsq are the positive closed-loop feedback constants. The second Lyapunov function candidate is defined as (20)Γ2=Γ1+12αsq2+12αsd2=12αω2+12αsq2+12αsd2.Differentiating (20) gives us(21)dΓ2dt=αωdαωdt+αsddαsddt+αsqdαsqdt.From (14) and (18)–(19), (21) can be reorganized as(22)dΓ2dt=-δωαω2+αωJsJsδωαω+Twt-32pnψsisq+Fsωr+Jsdωrrefdt-λsqαsq2+αsqLgsLgsdisq-refdt-vsd⁡+ωeψs+Rsisq+Lgsωeisd⁡+λsqLgsαsq-λsd⁡αsd⁡2+αsd⁡Lgs-vsd⁡-Lgsωeisq+Rsisd⁡+Lgsλsd⁡αsd⁡.The commands vsq-ref and vsd-ref are carried out from (22) as(23)vsq-ref=ωeψs+Rsisq+Lgsωeisd⁡+Lgsdisq-refdt+Lgsλsqαsqvsd-ref=Lgsλsd⁡αsd⁡-Lgsωeisq+Rsisd⁡.Using (16) and (23), (22) can be simplified to(24)dΓ2dt=-δωαω2-λsdαsd2-λsqαsq2≤0.Therefore, by Lyapunov stability analysis, the backstepping regulator is asymptotically stable and the control of the DDSG speeds is attained. The control diagram for individual MSR is depicted in Figure 6.Figure 6
Control block diagram of MSR using backstepping approach.


## 2.5. Nonlinear dc-link Voltage Regulation

The GSI controls the dc-link voltage and transfers the power from turbine-DDSG to the grid. Also, it contributes to the reactive and active power control of the overall DDSG system [55]. If the reference frame is rotating synchronously with the electrical grid voltage vector, the dynamic model voltage equations of the electrical network connection in grid voltage oriented reference frame (d,q) are represented as follows:(25)did-griddt=1Lgfegd-Rgfid-grid+ωgLgfiq-grid-Usdiq-griddt=1Lgfegq-Rgfiq-grid-ωgLgfid-grid,where egd, egq are the grid side inverter voltage (d-q) components (V); Us is the amplitude of the grid voltage (V); Rgf (Ω) and Lgf (H) are the resistance and inductance of the filter; ωg is the angular frequency of grid; and id-grid and iq-grid are the d-axis current and q-axis current (A) of the electrical grid, respectively. The active Pg and reactive power Qg can be calculated as follows [56]:(26)Pg=32Usid-gridQg=32Usiq-grid.Equations (26) illustrate that the active and reactive powers are regulated to achieve the required voltage by changing the respective current of thed-q-axis. The dc-link voltage equation can be formulated as in the following equation:(27)CdVdcdt=ig-igrid,where C is the capacitance of the dc-link capacitor (F); Vdc is dc-link voltage (V); igrid is the current between the dc-link and the grid; and ig is the current between the dc-link and the DDSG stators (A). So, the active power along the side of the GSI is(28)Vdcigrid=32Usid-grid.From (27)-(28), the dc-link voltage dynamics can be calculated as(29)dVdcdt=ig-1C32UsVdcid-grid.Consequently, to transfer all of the active power generated in turbine-DDSG, the dc-link voltage Vdc must be maintained at a constant value using the current control of id-grid. Equation (29) can be reorganized as(30)VdcCdVdcdt=PTotal-32Usid-grid,where PTotal is the generated power of the DDSGs. Equation (30) can be expressed as(31)C2dVdc2dt=PTotal-32Usid-grid.The tracking error of the dc-bus voltage is defined as (32)εdc-link=Vdc-r2-Vdc2.The dc-link voltage tracking error has the following dynamic:(33)dεdc-linkdt=dVdc-r2dt-2CPTotal-32Usid-grid-r,where id-grid-r is the current reference. Consider a third Lyapunov function candidate:(34)Φdc-link=12εdc-link2.From (33), differentiating (34) gives us(35)dΦdc-linkdt=εdc-linkdεdc-linkdt=εdc-linkdVdc-r2dt-2CPTotal-32Usid-grid-r.So, (35) can be reorganized as(36)dΦdc-linkdt=-λbusεdc-link2+εdc-linkdVdc-r2dt-2CPTotal-32Usid-grid-r+λbusεdc-link,where λbus is the positive closed-loop feedback constant. According to (36), the backstepping law can be designed as follows: (37)id-grid-r=1Us23PTotal-C3dVdc-r2dt+λbusεdc-link.From (37), (38) can be simplified to(38)dΦdc-linkdt=-λbusεdc-link2≤0.Consequently, by Lyapunov stability analysis, dc-link voltage controller is asymptotically stable. To stabilize the currents components id-grid and iq-grid, define the following current errors:(39)ξd-grid=id-grid-r-id-grid(40)ξq-grid=iq-grid-r-iq-grid.The reference iq-grid-r is calculated from the desired power factor. Based on (37) and (39), (33) can be reorganized as(41)dεdc-linkdt=-λdc-linkεdc-link+3CUsξd-grid.Besides, from (25) and (39)-(40), one can obtain (42)ξd-griddξd-griddt=ξd-griddid-grid-rdt-did-griddt=-λd-gridξd-grid2+ξd-gridLgfLgfdid-grid-rdt-egd+Us+Rgfid-grid-Lgfωgiq-grid+λd-gridLgfξd-gridξq-griddξq-griddt=-diq-griddt=-λq-gridξq-grid2+ξq-gridLgf-egq+Rgfiq-grid+Lgfωgid-grid+λq-gridLgfλq-grid,where λd-grid and λq-grid are the positive closed-loop feedback constants. Consider the fourth Lyapunov function candidate: (43)ΦGSI=12εdc-link2+12ξq-grid2+12ξd-grid2.Differentiating (43) gives us(44)dΦGSIdt=εdc-linkdεdc-linkdt+ξd-griddξd-griddt+ξq-griddξq-griddt.From (41) and (42), (44) can be reorganized as(45)dΦGSIdt=-λdc-linkεdc-link2-λd-gridξd-grid2-λq-gridξq-grid2+ξd-gridLgf3LgfCUsεdc-link+Lgfdid-grid-rdt-egd+Rgfid-grid-Lgfωgiq-grid+Us+λd-gridLgfξd-grid+ξq-gridLgf-egq+Rgfiq-grid+Lgfωgid-grid+λq-gridLgfξq-grid.According to (45), the backstepping law can be designed as follows: (46)vd-GSI=3LgfCUsεdc-link+Lgfdid-grid-rdt+Rgfid-grid-Lgfωgiq-grid+Us+Lgfλd-gridξd-gridvq-GSI=Rgfiq-grid+Lgfωgid-grid+Lgfλq-gridξq-grid.Using (37) and (46), (45) can be simplified to(47)dΦGSIdt=-λdc-linkεdc-link2-λd-gridξd-grid2-λq-gridξq-grid2≤0.Therefore, by Lyapunov stability analysis, the backstepping regulator is asymptotically stable and the control of the GSI is attained. The control diagram is depicted in Figure 7.Figure 7
Control block diagram of GSI.


## 3. Simulation Result Analysis

The performance of the proposed nonlinear control strategy is demonstrated in this section. The controller is tested on the WFS control strategy found in Section2. The schematic of the overall control scheme of the WFS is illustrated in Figure 7. Numerical simulations are carried out using MATLAB/Simulink software to verify WFS performance. System data used are listed in Tables 1, 2, and 3. The performance coefficients of the turbines are evaluated as the following form:(48)Cwλw,βw=12116λwc-0.4βw-5e-21/λwc1λwc=1λw+0.08βw-0.035βw3+1,where Cw reaches the maximum value Cw-max=0.4104 when λw is λw-m=8.1.Table 1
Parameters of the DDSG.
Parameter Value P r: rated power of DDSG 2 (MW) ω m: rated mechanical speed 2.57 (rd/s) R: stator resistance 0.008 (Ω) L s: stator d-axis inductance 0.0003 (H) ψ f: permanent magnet flux 3.86 (wb) p n: pole pairs 60Table 2
Parameters of the turbine.
Parameter Value Blade number 3 ρ: the air density 1.08 kg/m3 A: area swept by blades 4775.94 m2 v w - n: rated wind speed 12.4 m/sTable 3
System parameters.
Parameter Value dc-link voltage reference 2100 V Grid frequency 50 Hz Grid phase voltage 660 V RatioXGrid/RGrid 5 dc-link capacitor 38000μFBesides, the simulation will be performed in three steps with different objectives. The first scenario consists in operation under changing wind velocities, the second is robustness of proposed control strategy against electrical and mechanical parameter changes, and the third is an AC voltage sag.


### 3.1. Case 1: Performance Evaluation under Normal Operation of Proposed Control

Figure8 shows the wind velocities for the different wind turbine systems (a), the individual pitch angle for each wind turbine (b), the performance coefficients of the turbines (c), the speeds of DDSG (d), and the total power extracted by the wind farm (f). It has been noted that the speeds of DDSG are adapted relative to the variation of wind speed and the Cw can be preserved at a maximum value which could yield to TMP by working the turbines at the speed references. The pitch angle controls are activated for velocities above the rated wind velocity to control the rated power of the DDSGs. If the wind turbines receive wind velocities which are lower than the rated speed, the pitch angles are set as 0° to generate the maximum extracted power. The wind turbine systems operated at theirs maximum performance coefficients and the pitch angle were kept at their optimal values. The controller performs the power maximization according to the wind velocity variation. But, pitch controls are activated to restrict the rotational velocities to below the maximum speed and to limit the powers of turbines when the wind velocities are above the rated speed. In this study, the DDSG rated speed is 2.57 rad/s. So, in high wind velocity regions, the DDSG velocities and extracted power would not exceed the maximum. The generated power of the WFS is shown in Figure 8(f). Figure 8(e) shows wind DDSG1 speed follows the optimal speed provided by the TMP operation as the wind velocity varies. The DDSG1 velocity is able to track its optimal values precisely. Figures 8(g)–8(h) show the currents of DDSG1, where the generator currents are able to track their optimal values precisely. Figure 9 illustrates the dc-link voltage response. The dc-link voltage controller tried to maintain the constant value so that the dc-link voltage had an approximately constant value of 2100 V. Figure 10 shows the simulation result of reactive power which is controlled to be zero. Figure 11 illustrates a detailed view of the grid currents and phase voltage during the wind variation where the proposed WFS produces sinusoidal currents at electrical network with unity power factor. The simulation results show that the proposed backstepping controllers have good performance and can guarantee acceptable global regulation and tracking performance to extract and convert power under varying wind speeds.Figure 8
Simulation of the WFS in normal operation and using backstepping strategy.
(a)
Wind speeds (m/s) (b)
Pitch anglesβw (in degree) (c)
Coefficient of power,Cw (d)
Speed of DDSGs (rd/s) (e)
Speed of DDS1 (rd/s) (f)
Total power extracted (W)  (g)
Quadrature current componentisq of DDSG1 (A) (h)
Direct current componentisd⁡ of DDSG1 (A)Figure 9
dc-link voltage (V).Figure 10
Reactive power (VAR).Figure 11
Simulation of the three-phase current and voltage of electrical network.


### 3.2. Case 2: WFS Response Subject to Parameter Uncertainties

In order to verify the robustness of the proposed control strategy against the variation of parameters, it is supposed that the stator inductanceLgs, the stator resistor Rs, and the total moment of inertia Js of the system values are increased by 50% under the condition of nominal values. In Figures 12–15, robustness against WFS parameter variations is tested for the proposed strategy. In the simulation, Lgs, Rs, and Js are increased to 150% of their nominal values, and the other simulation parameters are the same as those in first scenario. The simulation results of the backstepping control and the VOC strategy with and without the mentioned parameter variation are shown in Figures 12–15. The results with and without the parameter variations are marked by “A” and “B,” respectively. It can be seen that the backstepping strategy has a fast dynamic response, a reduced settling time, and a lower overshoot. The simulation results confirm that the proposed control approach is robust against uncertainties in the WFS compared with conventional VOC.Figure 12
DDSG1 speed (rd/s).
(a)
Backstepping control (b)
VOC strategyFigure 13
Coefficient of power.
(a)
Backstepping control (b)
VOC strategyFigure 14
dc-link voltage (V) (with nominal values).
(a)
Backstepping control (b)
VOC strategyFigure 15
dc-link voltage (V) (with parameter variations).
(a)
Backstepping control (b)
VOC strategy


### 3.3. Case 3: Simulation under Grid Fault Condition

This simulation example deals with the WFS behaviour during a Three-Phase Grid Fault (3PGF) produced att = 5.5 s and characterised by a voltage drop of 200 ms at Bus 3 (Figure 3). Figures 16 and 17 illustrate the voltage of the WFS at Bus 4 in normal and close zoom views, respectively, for backstepping and VOC strategies. It can be observed that, by applying the backstepping controller, the transient stability of the WFS has been enhanced and the oscillations of the three-phase voltage have been damped quickly when the fault is cleared. Besides, Figure 17 shows that the transient duration with the proposed control is faster than that with the VOC and the three-phase voltage oscillation amplitude at VOC is evidently larger than that at backstepping control. The results illustrated that the backstepping control strategy can not only reduce the peak value of three-phase voltage but also shorten the transient response time compared with the VOC strategy. Compared to the results depicted in Figure 18, it can be seen that the peak value of frequency deviations can be eliminated during grid voltage drop with the proposed control strategy. Compared with VOC strategy, the peak value of frequency deviations is reduced from 1.9 Hz to less than 0.25 Hz with backstepping strategy. Figure 19 illustrates the dc-link voltage response under proposed control and VOC strategy. The voltage drop will not be transmitted to the dc-link system voltage and the converter dc-link voltage is controlled invariable at the reference value 2100 V as in Figure 19.Figure 16
Waveforms of three-phase voltage at Bus 4.
(a)
Backstepping control (b)
VOC strategyFigure 17
Zoom-in view of three-phase voltage at Bus 4.
(a)
Backstepping control (b)
VOC strategyFigure 18
Frequency response of the VS-WFS under grid faults.
(a)
Backstepping control (b)
VOC strategyFigure 19
dc-link voltage (V).
(a)
Backstepping control (b)
VOC strategy


## 3.1. Case 1: Performance Evaluation under Normal Operation of Proposed Control

Figure8 shows the wind velocities for the different wind turbine systems (a), the individual pitch angle for each wind turbine (b), the performance coefficients of the turbines (c), the speeds of DDSG (d), and the total power extracted by the wind farm (f). It has been noted that the speeds of DDSG are adapted relative to the variation of wind speed and the Cw can be preserved at a maximum value which could yield to TMP by working the turbines at the speed references. The pitch angle controls are activated for velocities above the rated wind velocity to control the rated power of the DDSGs. If the wind turbines receive wind velocities which are lower than the rated speed, the pitch angles are set as 0° to generate the maximum extracted power. The wind turbine systems operated at theirs maximum performance coefficients and the pitch angle were kept at their optimal values. The controller performs the power maximization according to the wind velocity variation. But, pitch controls are activated to restrict the rotational velocities to below the maximum speed and to limit the powers of turbines when the wind velocities are above the rated speed. In this study, the DDSG rated speed is 2.57 rad/s. So, in high wind velocity regions, the DDSG velocities and extracted power would not exceed the maximum. The generated power of the WFS is shown in Figure 8(f). Figure 8(e) shows wind DDSG1 speed follows the optimal speed provided by the TMP operation as the wind velocity varies. The DDSG1 velocity is able to track its optimal values precisely. Figures 8(g)–8(h) show the currents of DDSG1, where the generator currents are able to track their optimal values precisely. Figure 9 illustrates the dc-link voltage response. The dc-link voltage controller tried to maintain the constant value so that the dc-link voltage had an approximately constant value of 2100 V. Figure 10 shows the simulation result of reactive power which is controlled to be zero. Figure 11 illustrates a detailed view of the grid currents and phase voltage during the wind variation where the proposed WFS produces sinusoidal currents at electrical network with unity power factor. The simulation results show that the proposed backstepping controllers have good performance and can guarantee acceptable global regulation and tracking performance to extract and convert power under varying wind speeds.Figure 8
Simulation of the WFS in normal operation and using backstepping strategy.
(a)
Wind speeds (m/s) (b)
Pitch anglesβw (in degree) (c)
Coefficient of power,Cw (d)
Speed of DDSGs (rd/s) (e)
Speed of DDS1 (rd/s) (f)
Total power extracted (W)  (g)
Quadrature current componentisq of DDSG1 (A) (h)
Direct current componentisd⁡ of DDSG1 (A)Figure 9
dc-link voltage (V).Figure 10
Reactive power (VAR).Figure 11
Simulation of the three-phase current and voltage of electrical network.


## 3.2. Case 2: WFS Response Subject to Parameter Uncertainties

In order to verify the robustness of the proposed control strategy against the variation of parameters, it is supposed that the stator inductanceLgs, the stator resistor Rs, and the total moment of inertia Js of the system values are increased by 50% under the condition of nominal values. In Figures 12–15, robustness against WFS parameter variations is tested for the proposed strategy. In the simulation, Lgs, Rs, and Js are increased to 150% of their nominal values, and the other simulation parameters are the same as those in first scenario. The simulation results of the backstepping control and the VOC strategy with and without the mentioned parameter variation are shown in Figures 12–15. The results with and without the parameter variations are marked by “A” and “B,” respectively. It can be seen that the backstepping strategy has a fast dynamic response, a reduced settling time, and a lower overshoot. The simulation results confirm that the proposed control approach is robust against uncertainties in the WFS compared with conventional VOC.Figure 12
DDSG1 speed (rd/s).
(a)
Backstepping control (b)
VOC strategyFigure 13
Coefficient of power.
(a)
Backstepping control (b)
VOC strategyFigure 14
dc-link voltage (V) (with nominal values).
(a)
Backstepping control (b)
VOC strategyFigure 15
dc-link voltage (V) (with parameter variations).
(a)
Backstepping control (b)
VOC strategy


## 3.3. Case 3: Simulation under Grid Fault Condition

This simulation example deals with the WFS behaviour during a Three-Phase Grid Fault (3PGF) produced att = 5.5 s and characterised by a voltage drop of 200 ms at Bus 3 (Figure 3). Figures 16 and 17 illustrate the voltage of the WFS at Bus 4 in normal and close zoom views, respectively, for backstepping and VOC strategies. It can be observed that, by applying the backstepping controller, the transient stability of the WFS has been enhanced and the oscillations of the three-phase voltage have been damped quickly when the fault is cleared. Besides, Figure 17 shows that the transient duration with the proposed control is faster than that with the VOC and the three-phase voltage oscillation amplitude at VOC is evidently larger than that at backstepping control. The results illustrated that the backstepping control strategy can not only reduce the peak value of three-phase voltage but also shorten the transient response time compared with the VOC strategy. Compared to the results depicted in Figure 18, it can be seen that the peak value of frequency deviations can be eliminated during grid voltage drop with the proposed control strategy. Compared with VOC strategy, the peak value of frequency deviations is reduced from 1.9 Hz to less than 0.25 Hz with backstepping strategy. Figure 19 illustrates the dc-link voltage response under proposed control and VOC strategy. The voltage drop will not be transmitted to the dc-link system voltage and the converter dc-link voltage is controlled invariable at the reference value 2100 V as in Figure 19.Figure 16
Waveforms of three-phase voltage at Bus 4.
(a)
Backstepping control (b)
VOC strategyFigure 17
Zoom-in view of three-phase voltage at Bus 4.
(a)
Backstepping control (b)
VOC strategyFigure 18
Frequency response of the VS-WFS under grid faults.
(a)
Backstepping control (b)
VOC strategyFigure 19
dc-link voltage (V).
(a)
Backstepping control (b)
VOC strategy


## 4. Conclusion

This paper has presented a nonlinear backstepping approach of WFS driven DDSG operating under normal and voltage drop situations. A proposed control has been investigated to deal with problems of simultaneous control of the DDSG speeds and the dc-link voltage to achieve Maximum Power Tracking and pitch control. Simulation research on a 6-MW-DDSG Wind Farm System validates the proposed control strategy. Performance comparison and evaluation with Vector Oriented Control (VOC) are provided under a wide range of functioning conditions, three-phase voltage dips, and the probable occurrence of uncertainties. The main conclusions are summarized as follows:(1) The proposed backstepping strategy can guarantee acceptable global regulation and tracking performance. (2) The proposed control strategy offers remarkable characteristics such as excellent dynamic and steady state performance under varying wind speed and robustness to parametric variations in the WFS and under severe faults of grid voltage.



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*Source: 3837959-2018-05-20.xml*