Coupling DFIG-Based Wind Turbines with the Grid under Voltage Imbalance Conditions

: A smooth coupling is implemented between the grid and doubly fed induction generator-based wind turbines (DFIG-WTs) during grid voltage imbalance. The nonlinear characteristics of a grid-connected DFIG-WT system may increase stresses on the mechanical and electrical components of wind turbines. Such difﬁculties are greatly increased during periods of voltage imbalance. Consequently, in this paper, a new control scheme is proposed to regulate DFIGs in order to support a smooth connection to the grid during voltage imbalance. In synchronization mode, the positive sequence of the rotor dq -axes currents regulates the stator q -axis EMF that is to be synchronized with the q -axis voltage of the grid-side voltage. The phase difference between the grid and stator voltages is compensated by adjusting the stator d -axis EMF to zero. Under normal conditions, a PR controller is used to dampen the negative sequence of the rotor dq -axes currents. PI current controllers are tuned to control the positive sequence of the DFIG rotor currents, while PR current controllers are used to regulate the negative sequence of the rotor currents during synchronization and under normal operation conditions. Experiments are performed to verify the smooth synchronization of the DFIG and the robustness of the proposed control scheme during grid voltage imbalance.


Introduction
One of the major concerns of modern life is the issue of energy. Nowadays, power generation is primarily carried out by plants that use non-renewable resources as energy sources, which are mainly derived from fossil fuels. The limitation of reserves and the consequent increase in the price that is associated with fossil fuels point to the need to diversify energy sources. In addition, the environmental impact that is generated by the current scenario within the energy sector is considered high and needs to be continually rethought. The growing electricity demand, combined with the factors that were mentioned above, has motivated research into renewable sources that are environmentally friendly, which have low or no environmental impacts. Wind energy is a renewable alternative to the use of fossil fuels. This primary energy source is renewable and abundant in nature. The use of wind energy has high growth rates among renewable energy sources and is widely utilized worldwide [1]. The worldwide accumulated wind energy production has grown to 743 GW by incorporating new projects of 93 GW in 2020, as reported by the global wind energy council, which is sufficient for generating 7% of the world's electricity demand [2].
In the wind turbine marketplace, the doubly fed induction generator (DFIG) is considered to be the most prevalent wind energy generation system. The DFIG rotor windings are connected to the grid through a back-to-back converter, which consists of the grid-side converter (GSC), DC-link capacitor, and the rotor-side converter (RSC), as shown in Figure 1. In the normal generation regime, the GSC is responsible for controlling the DC-link voltage and the reactive power that flows from that bus into the network. However, the RSC aims to control the stator's active and reactive power that passes into the grid [3]. A DFIG is characterized by a low converter capacity (only 30% of the generator rating). Additionally, the control of the two converters allows an excursion of the machine speed that is limited to ±30% of the nominal speed [4].
process of a DFIG, the RSC aims to provide a smooth coupling between the stator windings and the grid [3]. Over the last decade, a limited number of publications have presented different control strategies for enhancing the coupling of DFIGs to the grid [5][6][7][8][9][10][11][12]. Vector control schemes with PI regulators have been commonly applied to achieve the synchronization of a DFIG and the grid [5,6]. However, these methods can affect the stability of the controlled DFIG or cause high-frequency oscillations. Recently, control schemes that were based on modern nonlinear controllers have been used for the enhancement of DFIG grid synchronization, in which the stator voltage of a DFIG was directly controlled to achieve a synchronization scheme that was based on sliding mode control (SMC) [7,8]. However, SMC suffers from the inadmissible chattering effect, which needs an additional control action to increase the robustness of the system. The model predictive control (MPC) of the DFIG rotor current has been employed to perform fast coupling to the grid [9]. Nevertheless, the main drawback of this technique is the necessary awareness of an accurate mathematical model of the rotor current. In [10,11], a robust grid synchronization scheme was proposed that involved applying a state feedback controller to both the RSC and GSC; however, this scheme is considered to be slightly complicated in application. The well-known fuzzy logic control (FLC) was also established for grid synchronization in [12]; however, the design and accuracy of the controller are dependent on the expertise of the user.  The most common arrangement of the controllers for synchronization and the normal operation of a DFIG is composed of two loops with two proportional-integral (PI) cascading controllers. The outer loops control the active and reactive power of the stator during normal operation and the stator EMF during coupling with the grid, while the internal loops control the rotor currents. The controlled bidirectional slip power of a DFIG system makes it act as if it were a synchronous generator. Therefore, during the start-up process of a DFIG, the RSC aims to provide a smooth coupling between the stator windings and the grid [3]. Over the last decade, a limited number of publications have presented different control strategies for enhancing the coupling of DFIGs to the grid [5][6][7][8][9][10][11][12]. Vector control schemes with PI regulators have been commonly applied to achieve the synchronization of a DFIG and the grid [5,6]. However, these methods can affect the stability of the controlled DFIG or cause high-frequency oscillations. Recently, control schemes that were based on modern nonlinear controllers have been used for the enhancement of DFIG grid synchronization, in which the stator voltage of a DFIG was directly controlled to achieve a synchronization scheme that was based on sliding mode control (SMC) [7,8]. However, SMC suffers from the inadmissible chattering effect, which needs an additional control action to increase the robustness of the system. The model predictive control (MPC) of the DFIG rotor current has been employed to perform fast coupling to the grid [9]. Nevertheless, the main drawback of this technique is the necessary awareness of an accurate mathematical model of the rotor current. In [10,11], a robust grid synchronization scheme was proposed that involved applying a state feedback controller to both the RSC and GSC; however, this scheme is considered to be slightly complicated in application. The well-known fuzzy logic control (FLC) was also established for grid synchronization in [12]; however, the design and accuracy of the controller are dependent on the expertise of the user. In addition, DFIGs are typically very susceptible to grid disturbances, such as a grid voltage imbalance due to the direct connection of DFIG stator windings to the electrical grid, which may affect the quality of the synchronization process. A voltage imbalance causes oscillating active and reactive power components, which are delivered to the electrical grid. Furthermore, a grid voltage imbalance produces a high amount of current imbalance within the DFIG rotor and stator windings due to the low impedance that is associated with the negative sequence components [13]. These currents are responsible for uneven heating in the windings of the machine and fluctuations in its electromagnetic torque. Nowadays, this is quite undesirable due to the continuous rise in the participation of wind energy in electrical power systems [14]. Several control techniques for DFIGs have been proposed in the scientific literature on the compensation of voltage imbalance. The existing strategies can be divided into two categories: hardware-based solutions and software-based solutions. Regarding hardware-based solutions, they either have low costs and simple structures but less reliability, such as crowbars and series dynamic braking resistors, or they have high reliability but higher costs and more complex structures, such as fault current limiters and DC choppers. However, software-based solutions can achieve a trade-off between complexity and reliability without additional costs [15].
Therefore, it is preferable to apply nonlinear controllers for the compensation of an imbalance in the grid-side voltage. These controllers can reduce the oscillations of the stator active power, reduce the DFIG electromagnetic torque oscillations, reduce stator current imbalance and reduce rotor current imbalance. Various control techniques have been proposed to minimize the four above-mentioned quantities under voltage imbalance conditions [16][17][18][19][20][21][22][23][24][25][26][27][28]. The use of four cascaded PI controllers to control the positive and negative sequence components of the rotor currents provides an acceptable performance when a system is running in a steady-state condition [16,17]. However, systems experience a slow response when DFIGs are running in transient conditions. Multivariable state feedback (MSF) controllers have been used to control the rotor current and allow it to perform better in transient conditions [18]. However, this scheme has a low transient response and low control accuracy due to the change in DFIG parameters. In [19], the direct torque control (DTC) technique was performed for DFIGs under symmetrical voltage dips. The DTC method offers the advantage of a fast response, but the issue of ripples remains the main challenge. Direct power control (DPC) has been used to reduce power and torque ripples during grid voltage imbalance [20][21][22]. However, due to the major effects of the active and reactive power oscillations and generator speed, the DPC method experiences the problem of unfixed switching frequency. A powerful nonlinear auto disturbance rejection control (ADRC) technology was proposed to fulfil the control requirements of a DFIG under grid voltage imbalance [23,24]. However, this framework employs an offline tuning method for the ADRC parameters that may be not suitable for changes in generator parameters that are caused by grid disturbances. SMC [25,26], MPC [27] and FLC [28] have also been implemented to control DFIGs during grid imbalance. Even though these controllers provide robust control performance when dealing with grid disturbances, they have some flaws that make their implementation more complex in practice. These drawbacks include the high control gains of the SMC method due to nonlinear compensation, the dependency on the generator parameters in the MPC method, and the possible control errors arising from not regularly updating the rules of the FLC method.
A proportional resonance (PR) controller has been used to regulate the negative sequence of the rotor currents, while PI controllers have been used to control the positive sequence [17,[29][30][31][32]. This arrangement can eliminate the components of current imbalance from the rotor current by damping the resonant rotor current components at a frequency of 120 Hz. However, this arrangement has not yet been tested in the synchronization process. In this sense, the primary contributions of this article can be summarized as follows: 1.
The positive sequence of the rotor current is controlled using PI controllers, while the negative sequence is controlled using proportional resonant (PR) controllers for both synchronization and operational modes; The synchronization process of a DFIG starts when the generator speed reaches its minimum value. The positive and negative sequences of the q-axis stator EMF is regulated through the positive sequence of the dq-axes rotor current until the stator EMFs are equal to the grid voltages; 3.
To compensate for the phase difference between the stator EMFs and the grid voltages, the d-axis stator EMFs are controlled to match the reference value of the d-axis grid voltages, which is equal to zero; 4.
The stator EMF can be synchronized with the grid voltage by adding the calculated value of the compensation component to the slip angle; 5.
Directly after the connection process, the average active and reactive power is adjusted by controlling the positive sequence of the rotor current. Meanwhile, the PR controllers are tuned at 120 Hz to regulate the negative sequence of the rotor current and attenuate the influence of the grid voltage imbalance on the DFIG's torque; 6.
The effectiveness and robustness of the proposed control scheme for both synchronization and operational modes are proven experimentally during a grid imbalance.

Wind Turbine Modeling
A turbine's output power can be represented as: where P t is the turbine's output power, ρ is the density of air, v is the wind speed through the turbine, and R is the radius of the turbine blades. The function C p (β, λ), which is given by Equations (2) and (3), represents the portion of wind energy that is collected by the wind turbine and is characterized as a power coefficient [33]: The tip speed ratio is defined as: where ω t is the angular speed of the turbine rotor (rad/s). The behavior of the C p (β, λ) function for various angles β and velocity relationships λ can be seen in Figure 2. During the normal operation of a wind turbine, the maximum power delivery is sought, which is a condition that, according to Equation (1), happens over the curve with the greatest gradient of the C p (β, λ) function. Equation (2) applies to all three-bladed turbines with parameters C 1 , C 2 , C 3 , C 4 , C 5 , and C 6 , which are determined using a regression approach [34].

DFIG Modeling
Using the DFIG d−q equivalent circuit in Figure 3  where Rs is the stator resistance, Rr is the rotor resistance, ωe is the angul the slip angular velocity, vdqs is the stator dq-axes voltages, vdqr is the rotor d idqs is the stator dq-axes currents, λdqs is the stator dq-axes flux linkage, idq axes currents, λdqr is the rotor dq-axes flux linkage, Ls is the stator inductan inductance, and Lm is the magnetizing inductance. As shown in Figure 4 [34], the flux vector of the stator is aligned w follows:

DFIG Modeling
Using the DFIG d−q equivalent circuit in Figure 3 [34], the voltage, current, and fluxes under stator flux-oriented control can be represented as [35]: λ ds = L s i ds + L m i dr (9) λ qs = L s i qs + L m i qr (10) λ dr = L m i ds + L r i dr (11) λ qr = L m i qs + L r i qr (12) where R s is the stator resistance, R r is the rotor resistance, ω e is the angular velocity, ω sl is the slip angular velocity, v dqs is the stator dq-axes voltages, v dqr is the rotor dq-axes voltages, i dqs is the stator dq-axes currents, λ dqs is the stator dq-axes flux linkage, i dqr is the rotor dq-axes currents, λ dqr is the rotor dq-axes flux linkage, L s is the stator inductance, L r is the rotor inductance, and L m is the magnetizing inductance.
where Ps is the stator's active power, Qs is the stator's reactive power, and Te is the electromagnetic torque.

DFIGs under Grid Voltage Imbalance
The stator's voltages and active and reactive power during grid voltage imbalance can be stated as [36]: As shown in Figure 4 [34], the flux vector of the stator is aligned with the d-axis as follows: where Ps is the stator's active power, Qs is the stator's reactive power, and Te is the electromagnetic torque.

DFIGs under Grid Voltage Imbalance
The stator's voltages and active and reactive power during grid voltage imbalance can be stated as [36]: The active and reactive power of the rotor and stator, as well as the electromagnetic torque that is produced by the DFIG, in the condition of invariant power can be calculated as [34]: where P s is the stator's active power, Q s is the stator's reactive power, and T e is the electromagnetic torque.

DFIGs under Grid Voltage Imbalance
The stator's voltages and active and reactive power during grid voltage imbalance can be stated as [36]: whereˆis the superscript of the conjugated space vector, v p dqs is the positive components of the stator dq-axes voltages, i p dqs is the positive components of the stator dq-axes currents, and λ p dqs is the positive components of the stator dq-axes flux linkage. During grid voltage imbalance, the instantaneous DFIG stator active and reactive power in (19) can be expressed as [37]: where P s0 is the average component of the stator active power, Q s0 is the average component of the stator reactive power, P esin2 is the sin oscillating stator power component at twice the grid frequency, and P ecos2 is the cos oscillating stator power component at twice the grid frequency. The positive sequence of the d-axis is aligned with the positive sequence of the stator voltage in stator voltage orientation, as illustrated in Figure 5, which implies that v p qs = 0. The dq-axes sequence of the stator voltage can be defined as [36]: where ^ is the superscript of the conjugated space vector, is the positive components of the stator dq-axes voltages, is the positive components of the stator dq-axes currents, and is the positive components of the stator dq-axes flux linkage. During grid voltage imbalance, the instantaneous DFIG stator active and reactive power in (19) can be expressed as [37]: where Ps0 is the average component of the stator active power, Qs0 is the average component of the stator reactive power, Pesin2 is the sin oscillating stator power component at twice the grid frequency, and Pecos2 is the cos oscillating stator power component at twice the grid frequency. The positive sequence of the d-axis is aligned with the positive sequence of the stator voltage in stator voltage orientation, as illustrated in Figure 5, which implies that = 0. The dq-axes sequence of the stator voltage can be defined as [36]: Under grid voltage imbalance, the electromagnetic power of a DFIG is divided into its continuous elements and two pulsating parts as follows [13]: The electromagnetic torque is given as follows: where ωr is the rotor angular velocity. Accordingly, the electromagnetic torque matrix during grid voltage imbalance can be expressed as [13]: where Te0 is the average component of the electromagnetic torque, Tesin2 is the sin oscillating torque component at twice the grid frequency, Tecos2 is the cos oscillating torque component at twice the grid frequency, is the positive rotor components of the dq-axes currents, and is the negative rotor components of the dq-axes currents. Under grid voltage imbalance, the electromagnetic power of a DFIG is divided into its continuous elements and two pulsating parts as follows [13]: P e = P e0 + P ecos2 cos(2ω e t) + P esin2 sin(2ω e t) The electromagnetic torque is given as follows: where ω r is the rotor angular velocity. Accordingly, the electromagnetic torque matrix during grid voltage imbalance can be expressed as [13]: where T e0 is the average component of the electromagnetic torque, T esin2 is the sin oscillating torque component at twice the grid frequency, T ecos2 is the cos oscillating torque component at twice the grid frequency, i p dqr is the positive rotor components of the dq-axes currents, and i n dqr is the negative rotor components of the dq-axes currents.
The dq-axes negative reference components of a DFIG rotor can be calculated by eliminating the torque ripple components T ecos2 and T esin2 in Equation (25) where i n * dqr is the rotor negative components of the dq-axes currents. The RSC controllers are implemented under grid voltage imbalance situations in the normal operating mode, as shown in Figure 6. In this system and when using two control loops, the currents in the rotor are controlled along the dq reference axes. The generator's reactive and active power (or electromagnetic torque) are controlled by the dq-axes current loops. The bottom part of the block diagram shows the control of the negative sequence by controlling the 120-Hz torque components. The extraction of the positive and negative sequences of the dq-axes rotor is shown in Figure 7. The dq-axes negative reference components of a DFIG rotor can be calculated by eliminating the torque ripple components Tecos2 and Tesin2 in Equation (25), such that [13]: where * is the rotor negative components of the dq-axes currents.
The RSC controllers are implemented under grid voltage imbalance situations in the normal operating mode, as shown in Figure 6. In this system and when using two control loops, the currents in the rotor are controlled along the dq reference axes. The generator's reactive and active power (or electromagnetic torque) are controlled by the dq-axes current loops. The bottom part of the block diagram shows the control of the negative sequence by controlling the 120-Hz torque components. The extraction of the positive and negative sequences of the dq-axes rotor is shown in Figure 7.
To obtain the proportional and integral gain of the PI controller for the internal current loop of the grid-side converter, the Naslin polynomial technique was used in this study [38]. The proportional gain Kp and the integral gain Ki for the positive dq-axes current controllers are listed in Table 1.

PI Gains
Kp K i Positive sequence controllers 3.64 123.55

Synchronization with the Grid
Initially, during the start-up process of the system, which is represented by Fi switch S1 is open and the synchronization process starts when the generator speed r its minimum value. The grid voltages eabc, the stator voltages vabcs, and the rotor cu iabcr are obtained experimentally during the synchronization process. The positive an ative sequences of the q-axis stator EMF are regulated through the positive seque the dq-axes rotor current until the stator EMFs are equal to the grid voltages. The p θs is obtained by integrating the speed signal, which the PI controller permanently lates in order to maintain the d-axis grid voltage at zero, as shown in Figure 9a. T pensate for the phase difference between the stator EMFs and grid voltages, the stator EMFs are controlled to match the reference value of the d-axis grid voltages, is equal to zero. Figure 9b depicts the phase difference compensation process and pr the compensation component δθsl. Therefore, the stator EMF can be synchronize the grid voltage by adding the calculated value of δθsl to the slip angle. The synch tion is then concluded with the closing of the switch, which connects the machine's to the grid. Directly after the connection process, the DFIG control mode is change synchronization to operational mode. The average active and reactive power are lated to follow their command targets and minimize the response of the voltage imb on the generator's torque. The synchronization sequence is depicted in the flowcha is shown in Figure 10. To obtain the proportional and integral gain of the PI controller for the internal current loop of the grid-side converter, the Naslin polynomial technique was used in this study [38]. The proportional gain K p and the integral gain K i for the positive dq-axes current controllers are listed in Table 1.

Synchronization with the Grid
Initially, during the start-up process of the system, which is represented by Figure 8, switch S1 is open and the synchronization process starts when the generator speed reaches its minimum value. The grid voltages e abc , the stator voltages v abcs , and the rotor currents i abcr are obtained experimentally during the synchronization process. The positive and negative sequences of the q-axis stator EMF are regulated through the positive sequence of the dq-axes rotor current until the stator EMFs are equal to the grid voltages. The position θ s is obtained by integrating the speed signal, which the PI controller permanently regulates in order to maintain the d-axis grid voltage at zero, as shown in Figure 9a. To compensate for the phase difference between the stator EMFs and grid voltages, the d-axis stator EMFs are controlled to match the reference value of the d-axis grid voltages, which is equal to zero. Figure 9b depicts the phase difference compensation process and provides the compensation component δθ sl . Therefore, the stator EMF can be synchronized with the grid voltage by adding the calculated value of δθ sl to the slip angle. The synchronization is then concluded with the closing of the switch, which connects the machine's stator to the grid. Directly after the connection process, the DFIG control mode is changed from synchronization to operational mode. The average active and reactive power are regulated to follow their command targets and minimize the response of the voltage imbalance on the generator's torque. The synchronization sequence is depicted in the flowchart that is shown in Figure 10. the grid voltage by adding the calculated value of δθsl to the slip angle. The synchronization is then concluded with the closing of the switch, which connects the machine's stator to the grid. Directly after the connection process, the DFIG control mode is changed from synchronization to operational mode. The average active and reactive power are regulated to follow their command targets and minimize the response of the voltage imbalance on the generator's torque. The synchronization sequence is depicted in the flowchart that is shown in Figure 10.

Experimental Results and Discussion
This section presents the experimental results for a wind system that was based on the 3 kW DFIG from LabVolt and the wind turbine simulator from Labvolt, as shown in Figure 11. The DSP board, data acquisition board, and sensors were utilized, as well as the laboratory bench, to acquire the experimental results. The experimental results of the

Experimental Results and Discussion
This section presents the experimental results for a wind system that was based on the 3 kW DFIG from LabVolt and the wind turbine simulator from Labvolt, as shown in Figure 11. The DSP board, data acquisition board, and sensors were utilized, as well as the laboratory bench, to acquire the experimental results. The experimental results of the synchronization process between the stator and grid are shown during the system's start-up process. The DFIG rotor current dynamics are presented with the results that were obtained experimentally. The experimental results were based on a 10 kHz switching frequency, a 2200 µF DC bus capacitance, and a 100 µs sampling time. The DFIG and turbine simulator data are presented in Tables A1 and A2. Sustainability 2022, 14, x FOR PEER REVIEW 11 of 20 synchronization process between the stator and grid are shown during the system's startup process. The DFIG rotor current dynamics are presented with the results that were obtained experimentally. The experimental results were based on a 10 kHz switching frequency, a 2200 μF DC bus capacitance, and a 100 μs sampling time. The DFIG and turbine simulator data are presented in Tables A1 and A2. Figure 11. Experimental set-up of the DFIG system.
As depicted in Figure 12, the amplitudes of two phases of the 3-phase grid voltages were reduced to generate the grid voltage imbalance conditions. As depicted in Figure 12, the amplitudes of two phases of the 3-phase grid voltages were reduced to generate the grid voltage imbalance conditions. Figure 11. Experimental set-up of the DFIG system. As depicted in Figure 12, the amplitudes of two phases of the 3-phase grid voltages were reduced to generate the grid voltage imbalance conditions. The stator voltage and grid voltage during the synchronization process are shown in Figure 13, which shows the start and end waveforms of the synchronization. The synchronization process was carried out in approximately 20 ms, which indicates that the synchronization process was achieved smoothly and without causing any disturbances in the grid voltage. The stator voltage and grid voltage during the synchronization process are shown in Figure 13, which shows the start and end waveforms of the synchronization. The synchronization process was carried out in approximately 20 ms, which indicates that the synchronization process was achieved smoothly and without causing any disturbances in the grid voltage. After closing the switch, the reactive and active power controllers were activated to control the average values by regulating the positive sequence of the dq-axes rotor currents. When the negative sequence of the rotor currents was not controlled, the reactive and active power oscillated around the average values, which were 0 VAR and 1000 W, respectively. Figure 14a,b show the power oscillations around the average values with a frequency of 120 Hz, which represents the negative current sequence. After connecting to the grid, the stator reactive power oscillation amplitude was approximately 120 VAR. Similarly, the stator active power oscillation amplitude was about 100 W around the mean power.
Due to the existence of the 120 Hz components within the stator currents, the torque oscillated around the average value with an amplitude of 2 Nm when there was no control of the torque ripple, as illustrated in Figure 15a. When torque ripple control was applied, the amplitude of the ripple reduced to about 0.3 Nm, as shown in Figure 15b. The proposed controller was robust and minimized the torque ripple significantly. Figure 16a shows the spectrum of the generator torque when using a conventional control method, while Figure 16b shows the spectrum when the proposed controller was used. The generator's 120 Hz torque oscillation component was reduced significantly. In comparison, the components 240 Hz and 360 Hz were canceled by regulating the negative sequence of the rotor dq-axes currents, which reduced the mechanical stress on the turbine shaft. The harmonic spectrum of the stator dq-axes currents is illustrated in Figure 17. The After closing the switch, the reactive and active power controllers were activated to control the average values by regulating the positive sequence of the dq-axes rotor currents. When the negative sequence of the rotor currents was not controlled, the reactive and active power oscillated around the average values, which were 0 VAR and 1000 W, respectively. Figure 14a,b show the power oscillations around the average values with a frequency of 120 Hz, which represents the negative current sequence. After connecting to the grid, the stator reactive power oscillation amplitude was approximately 120 VAR. Similarly, the stator active power oscillation amplitude was about 100 W around the mean power.
Due to the existence of the 120 Hz components within the stator currents, the torque oscillated around the average value with an amplitude of 2 Nm when there was no control of the torque ripple, as illustrated in Figure 15a. When torque ripple control was applied, the amplitude of the ripple reduced to about 0.3 Nm, as shown in Figure 15b. The proposed controller was robust and minimized the torque ripple significantly. Sustainability 2022, 14, x FOR PEER REVIEW 13 of 20      Figure 16a shows the spectrum of the generator torque when using a conventional control method, while Figure 16b shows the spectrum when the proposed controller was used. The generator's 120 Hz torque oscillation component was reduced significantly. In comparison, the components 240 Hz and 360 Hz were canceled by regulating the negative sequence of the rotor dq-axes currents, which reduced the mechanical stress on the turbine shaft. The harmonic spectrum of the stator dq-axes currents is illustrated in Figure 17. The double frequency component (120 Hz) was dominant, especially in the d-axis current. These components were eliminated and, consequently, the torque ripple decreased.  When the conventional control method was utilized to regulate the stator's active and reactive power, both included the 120 Hz components, as illustrated in Figure 18a,b. The amplitude of the reactive power ripple was 120 VAR when the mean reactive power was controlled to be zero, as in Figure 18a. Similarly, the active power oscillated with an  When the conventional control method was utilized to regulate the stator's active and reactive power, both included the 120 Hz components, as illustrated in Figure 18a,b. The amplitude of the reactive power ripple was 120 VAR when the mean reactive power was controlled to be zero, as in Figure 18a. Similarly, the active power oscillated with an When the conventional control method was utilized to regulate the stator's active and reactive power, both included the 120 Hz components, as illustrated in Figure 18a,b. The amplitude of the reactive power ripple was 120 VAR when the mean reactive power was controlled to be zero, as in Figure 18a. Similarly, the active power oscillated with an amplitude of 100 W, as in Figure 18b. The stator reactive power ripple was suppressed when the generator torque ripple was controlled, as illustrated in Figure 19a. However, controlling the torque ripple by regulating the negative sequence of the rotor current did not reduce the stator active power ripple, as seen in Figure 19b. This was because it is impossible to reduce both the stator active and reactive power ripples simultaneously.  The stator reactive power ripple was suppressed when the generator torque ripple was controlled, as illustrated in Figure 19a. However, controlling the torque ripple by regulating the negative sequence of the rotor current did not reduce the stator active power ripple, as seen in Figure 19b. This was because it is impossible to reduce both the stator active and reactive power ripples simultaneously. Figure 20a,b show the grid voltage and current waveforms when the compensation technique for grid voltage imbalance was implemented. By controlling the negative component of the grid current, the ripple component of the grid current was minimized. There was a difference in the amplitudes of the 3-phase currents due to a small negative q-axis current, as shown in the circled peaks. The validity of the proposed controller was presented clearly when the generator torque, stator reactive power, stator currents, and grid current ripples were controlled and reduced.
From the aforementioned results, it is evident that the synchronization process was achieved smoothly and without causing any disturbances in the grid voltage by using the proposed control scheme. In addition, the generator's 120 Hz torque oscillation component was suppressed significantly, while the 240 Hz and 360 Hz components were canceled by regulating the negative sequence of the rotor dq-axes currents. As a result, the torque oscillations of the DFIG were reduced from 2 Nm to only 0.3 Nm, which indicates the robustness and effectiveness of the proposed control method. Furthermore, it is well noted that the suggested torque elimination technique was effective in terms of reducing the stator reactive power ripples. Finally, the grid current ripple component was suppressed by controlling the negative component of the grid current.  Figure 20a,b show the grid voltage and current waveforms when the compensation technique for grid voltage imbalance was implemented. By controlling the negative component of the grid current, the ripple component of the grid current was minimized. There was a difference in the amplitudes of the 3-phase currents due to a small negative q-axis current, as shown in the circled peaks. The validity of the proposed controller was presented clearly when the generator torque, stator reactive power, stator currents, and grid current ripples were controlled and reduced. From the aforementioned results, it is evident that the synchronization process was achieved smoothly and without causing any disturbances in the grid voltage by using the proposed control scheme. In addition, the generator's 120 Hz torque oscillation component was suppressed significantly, while the 240 Hz and 360 Hz components were canceled by regulating the negative sequence of the rotor dq-axes currents. As a result, the torque oscillations of the DFIG were reduced from 2 Nm to only 0.3 Nm, which indicates the robustness and effectiveness of the proposed control method. Furthermore, it is well noted that the suggested torque elimination technique was effective in terms of reducing

Conclusions
This paper presented a coupling of a grid scheme for DFIG-based wind turbines under grid voltage imbalance conditions. The complete system design and control were implemented experimentally and the implementation methodology was presented. The proposed controller was validated in the laboratory with a 3 kW system. The results of implementing the proposed algorithm were presented and discussed in the discussion section for symmetrical grid voltage imbalance during synchronization and operational conditions. The implementation of PI controllers to control the positive components of the rotor currents and PR controllers to control the negative sequence components of the rotor currents minimized the generator's torque ripples. At the same time, the stator reactive power ripple was also reduced significantly. The stator active power ripple, on the other hand, stayed similar to that generated by the standard control technique. Finally, the validity of the suggested method guaranteed smooth and fast synchronization and torque ripple elimination. During the start-up process, the proposed algorithm took about one cycle to adjust the generator EMF to the q-axis component of the grid voltage. The proposed structure and applied algorithm control of the positive and negative components of the rotor current significantly improved basic control performance, particularly when grid voltage imbalance was investigated.  Average component of the stator active power Q s0 Average component of the stator reactive power T e0 Average component of the electromagnetic torque P esin2 Sin oscillating stator power component at twice the grid frequency P ecos2 Cos oscillating stator power component at twice the grid frequency Q esin2 Sin oscillating stator reactive power component at twice the grid frequency P ecos2 Cos oscillating grid reactive power component at twice the grid frequency T esin2 Sin oscillating torque component at twice the grid frequency T ecos2 Cos oscillating torque component at twice the grid frequency v Wind speed * A superscript refers to the reference value of the variable

Appendix A
The specifications for the induction machine that was used for testing were as follows: three-phase, four poles, 230 V, 50 Hz, and 3 kW.