Model Predictive Control on Transient Flux Linkage and Reactive Power Compensation of Doubly Fed Induction Wind Generator

For weak grid scenario with high new energy proportion, large ﬂ uctuations of load are prone to cause low-voltage ride through. Moreover, stator transient magnetic ﬂ ux will cause overvoltage and overcurrent problems in the rotor of doubly fed induction generator. Based on model predictive control, a control strategy for transient ﬂ ux linkage and reactive power compensation is proposed. Firstly, regarding the issue of reactive power allocation of grid side converters (GSC) and rotor side converters (RSC), an allocation strategy is derived under minimizing winding energy loss on case of low-voltage ride through, enabling the wind energy conversion system to provide reactive power support during grid voltage recovery process. Meanwhile, an improved mixed second-and third-order generalized integrator (MSTOGI) phase-locked loop (PLL) is used to extract the positive and negative sequence components of the power grid voltage, further for RSC control. Secondly, in response to power grid faults, considering the in ﬂ uence of stator DC transient ﬂ ux and negative sequence ﬂ ux components on rotor current, by injecting rotor transient compensation current and stator ﬂ ux feedforward compensation into RSC, the rotor impulse voltage and loop current are reduced. Moreover, combined with model predictive control algorithm, a control strategy of rotor current is designed. Finally, a simulation platform is built to validate the e ﬀ ectiveness of the proposed method based on comparing with several traditional vector control low-voltage ride through methods.


Introduction
The rapid development of the global economy has exacerbated the speed of traditional fossil fuel depletion and environmental degradation.It is urgent to continuously develop the new energy technologies [1,2].Now, the proportion of wind power in the total grid power generation is increasing rapidly and has exceeded 733 GW [3].Among which, the wind energy conversion system (WECS) based on doubly fed induction generators (DFIG) has been widely used due to high wind energy capture efficiency [4], mature wind power generation control technology, and small converter capacity [5,6].
However, due to the special linking structure that stator winding connects to the power grid directly, DFIG is very sensitive to grid disturbances [7].Once a power grid fault occurs, a large induced EMF (electromotive force) in the rotor circuit will be generated for the reason that stator flux cannot change suddenly with the stator voltage, so resulting in transient overvoltage and overcurrent of the rotor [8].Especially under asymmetric faults in the power grid, negative sequence voltage components in the stator would exacerbate the oscillation of rotor side overvoltage and overcurrent [9].
Heretofore, disturbances and faults in the power grid would greatly affect the reliable operation and self-protection of DFIG, a typical example is the ability to ride through low voltage (LVRT).The LVRT technologies involved in existing literature are mainly divided into two categories, hardware intervention and software control [10].Hardware protection technology depends on the speed of material science development, so the breakthrough speed is relatively slow.In software technology, the literature research results mainly present the control of back-to-back converters, and typical demagnetization methods are using the stator flux as a feedforward compensation for the RSC, effectively improving the LVRT capability of DFIG [11,12].By adding demagnetization current to RSC, the stator DC and negative sequence flux components of the stator can be reduced, and the LVRT capability of DFIG can be enhanced [13].Hence, wavelet packet transform has been used for transient stability analysis of DC bus voltage [14].In reference [15], Kumar and Chatterjee used the artificial bee colony algorithm to solve the maximum power point tracking problem by considering rotor speed as an optimization problem to improve system robustness.In addition, the combination use of supercapacitor energy storage components and passive series dynamic resistors can effectively control rotor overcurrent and overvoltage and suppress fluctuations in DC bus voltage [16].A virtual capacitor control strategy is introduced to reduce the impact of line impedance by compensating for the output angle deviation of the PLL and improves the stability of DFIG [17].
In terms of hardware protection measures, crowbar protection and chopping protection are commonly used in the LVRT process of DFIG [18].Followed by which, collaborative control of two types of protection is utilized to cope with electromagnetic changes in DFIG [19].A two-level threephase converter is proposed, which compensates for stator voltage drop by connecting the converter in series with the power grid [20].In addition, in the event of a low-voltage fault in the power grid, a fault current limiter based on a new type of iron core material reduces the peak fault current by increasing the line impedance [21].
Of course, the reliability of grid-connected power generation control must take into account the entire process of fault occurrence, troubleshooting, and system recovery.As a key technology of PLL, it is necessary to accurately extract the fundamental power grid voltage parameters accurately.In reference [22], Vijay et al. used a hybrid generalized integral filter to filter out the DC bias voltage and harmonics contained in the power grid voltage.Followed by which, a new control strategy for symmetric PLL was proposed [23].A virtual impedance design scheme was designed to improve the output current frequency stability of DFIG.Moreover, considering the shortcomings of the widely used proportional integral (PI) control in DFIG control systems [24,25], a model predictive control (MPC) scheme is proposed in RSC control design [26], which can predict future behavior based on the system model.Due to understandable and good dynamic performance, MPC has been widely used [27], where the switch state of the rotor side converter is taken as the control input and augmented decision variables are introduced, thereby reducing computational control time and improving DFIG control performance [28].The research did not pay more attention to the scenario of grid harmonics and LVRT.
Due to the uncertainty of weather conditions and the complexity of the model of DFIG, to prevent the disconnection problem of DFIG system in weak grid operation, as well consider strong robustness and good control performance of MPC under uncertainty process, innovative exploration of control methods is performed under multigrid fault scenarios based on previous research results.During the period of voltage drop in the power grid, the DFIG rotor side induces a large peak current, which in turn damages the inverter equipment.Therefore, this article first studies the impact of stator transient magnetic flux caused by voltage drop in the power grid on rotor current.A collaborative control is proposed by reactive power compensation and MPC for stator transient flux, to minimize the amplitude of rotor current and voltage during power grid faults.
Highlights of the work are the following: (1) By introducing a stator flux feedforward compensation strategy, the transient speeds of flux attenuation of the stator and the system recovery are all accelerated.(2) By injecting reverse rotor current to control the injection of rotor current and rotor terminal voltage angle, all are suppressed.(3) Considering the problem of inaccurate voltage phase detection in traditional PLL during power grid voltage faults, an improved MSTOGI-PLL is proposed to accurately lock the grid voltage and improve the stability of the wind power generation system.(4) Considering winding losses in DFIG, a reactive power allocation strategy of stator and rotor is proposed to reduce the total winding losses, which provides reactive power support for grid voltage recovery certainly, as well as control strategy to protect the safe and stable operation of DFIG.
The remainder of this paper is organized as follows: Section 2 describes the grid-connected power generation model for wind energy conversion systems.The electromagnetic transient analysis process of DFIG under LVRT is presented in Section 3. Section 4 provides LVRT control strategy based on MPC.Section 5 discusses the simulation results of the system in different cases.Finally, Section 6 concludes the study.

Grid-Connected Power Generation Model for WECS
Figure 1 shows that WECS includes modules such as a DFIG, energy storage battery, transformer, power grid, and load.The energy storage battery is connected to the DC bus capacitor through a DC/DC converter.Wind turbine is controlled in maximum power tracking mode.The MPC control models of the GSC and RSC are also shown in the figure.The goal is to reduce the total winding loss of the DFIG and provide reactive power support for grid restoration.
In the case of voltage drop, rotor overvoltage and overcurrent are reduced by injecting reverse rotor current.At the same time, by increasing stator flux feedforward compensation, the transient magnetic flux attenuation speed is accelerated, so rotor overcurrent is suppressed.Furthermore, to accurately detect the phase and frequency of nonideal grid voltage, an improved MSTOGI-PLL structure is proposed.Based on which, reactive power allocation strategies are adopted on the rotor side and grid side under the consideration of winding losses of DFIG, as well as the need for which to absorb reactive power during the voltage recovery process of the grid.The wind power system model and control strategy are analyzed as follows.By comparing with 2 International Journal of Energy Research traditional control strategies, the simulation results verify the superiority of the proposed control strategy, significantly enhancing the fault traversal ability of DFIG.
2.1.Wind Energy Capture Model.In the design of simulation parameters, consider the acceleration gear mechanism.A 5 kW wind turbine is designed at a rated wind speed of 10 m/s.The captured power of the wind turbine is P m [29].
where ρ is the air density, r is the turbine radius, v is the current wind speed, C p λ, β is the wind energy utilization coefficient, λ is the tip speed ratio, and β is the yaw angle.Relationship of C p λ, β and λ can be expressed as follows: where β = 0, ω t is the fan blade angular velocity, η is the gearbox ratio, and ω r is the DFIG rotor speed.For the used experimental prototype parameters of DFIG simulation platform, refer to literature [30].s −30% ∼ 30% , u rms = 380 V, where Q mc is the DFIG magnetization reactive power, f s is the grid frequency, ν l is the line voltage of the power grid, and L m is the mutual inductance between the stator and rotor.
Here, the DFIG excitation power provided by GSC is assumed as Q s = 1 − τ Q mc , τ 0 < τ < 1 , which is the reactive power distribution coefficient.The provided reactive power of RSC is Q r = τsQ mc .When the wind power generation system operates stably, the calculation for stator and rotor winding losses W all is as follows: In Eq. (4), W s and W r are the stator and rotor winding losses, I s and I r are stator and rotor current, and S s and S r are stator and rotor windings capacity.

International Journal of Energy Research
To realize W all minimum, namely, to find the minimum value of a quadratic function R s + R r τ 2 − 2R s τ + R s with respect to τ, the peak value is obtained when τ = R s / R s + R r .Substituting which into Eq.( 4) yields When the excitation power is fully provided by RSC, τ = 1.At this point, the rotor winding loss W all is as follows: Subtracting Eq. ( 5) from Eq. ( 6) yields According to Eq. ( 7), ΔW all > 0; therefore, the total copper loss of DFIG winding can be reduced.

2.3.
Grid-Connected PLL Model. Figure 2 shows the improved MSTOGI-PLL structure, which is used to decompose the positive and negative sequence components of the power grid voltage.A feedforward compensation link is added to the traditional MSTOGI structure to enhance the attenuation speed of the amplitude of the low-frequency and high-frequency harmonic components contained in the three-phase voltage.
Figure 2(a) shows the improved MSTOGI structure diagram.Among them, D 1 s and Q 1 s have the same amplitude frequency characteristics.The transfer function is as follows: where s is a Laplace transform operator (s not the slip rate), u 1 s and u 2 s are two mutually orthogonal output signals, u 1 s leads 90 degrees ahead of the phase u 2 s , u 3 s represents the DC voltage component contained in the three-phase voltage, u s is the input voltage signal, k is the damping coefficient, ω 0 is the resonant frequency, and ω 1 is the frequency of the compensation phase.
where sech x = 2/ e x + e −x is a hyperbolic secant function, k p is the value of the conventional PI regulator, a 1 is the value when it exceeds the frequency deviation threshold, a 2 is the value when e t is within the frequency deviation threshold range, sech a 3 e t is a hyperbolic function, and a 3 is the attenuation coefficient of which.According to Eq. ( 9), when the output frequency of the PLL is consistent with the three-phase voltage frequency, k p e t = k p .According to the polarity of the deviation, namely, e t is a positive or negative deviation, determine whether the value exceeds the deviation threshold, further to select an appropriate curve value k p e t that varies with the error.Construction is as follows: where k i is the value of the conventional PI regulator and c is the attenuation coefficient of the hyperbolic function.
In Eq. (10), when e t = 0, k i e t = k i .When e t increases, k i e t gradually decreases to prevent oscillation in the response and reduce overshoot.The specific parameter selection in the paper is shown in Table 1.
The results of single-phase voltage drop and two-phase voltage drop are shown in Figure 3, including improved mixed second-and third-order generalized integrator PLL (MSTOGI2 in the figure), delay signal cancellation PLL (DSC in the figure), double second-order generalized integrator PLL (DSOGI in the figure), and traditional mixed second-and third-order generalized integrator PLL (MSTOGI1 in the figure).It can be seen from which that the proposed MSTOGI2 PLL can more accurately extract the frequency and phase of grid voltage when a fault happens.In the case of asymmetric grid voltage faults, according to the symmetrical component method and superposition principle, the stator voltage space vector is represented as follows:

Electromagnetic Transient Analysis and Control
In Eq. ( 12), u + m , u − m , and u m0 are the amplitude of the positive sequence, negative sequence, and zero sequence voltage components, respectively.Once asymmetric grid voltage drop occurs, originating from positive sequence voltage and negative sequence voltage, the steady-state magnetic flux component of the stator is generated.
So, during asymmetric faults, the stator magnetic flux φ s is as follows [32].
where φ + sf and φ − sf are the steady-state magnetic flux generated by the positive and negative sequence voltages of the stator.
According to Eq. ( 13), including the positive sequence and transient flux components, the influence of negative sequence flux is also necessary to be considered in the case of asymmetric faults.
According to the DFIG state space model, the stator flux state equation in the dq synchronous rotating coordinate system is as follows [33].
In Eq. ( 14), u ds and u qs are the stator voltage, φ ds and φ qs are the stator flux, and i dr and i qr are the rotor current.
Solving the system state equations in Eq. ( 14), characteristic roots of which are λ s = −R s /L s ± jω s .Due to small R s , the transient DC flux decay rate of the stator is very slow.To restore stable operation as soon as possible, a feedforward stator flux fluctuation compensation term is introduced in the rotor current loop.The improved stator flux state equation is as following.
where k d is the coefficient of stator flux compensation term, φ ds0 = 0 is the steady-state initial value of the stator flux d-axis, and φ qs0 = u m /ω s is the steady-state initial value of the stator flux q-axis.Solve the system state equations

Transient Compensation of Rotor Current.
Considering the capacity limitation of RSC, the induced EMF of the rotor is prone to exceed the controllable voltage range, which will weaken the current limiting ability of RSC and cause overcurrent on the rotor side.Therefore, it is necessary to analyze the rotor side voltage.
When operating in a stable state, the rotor voltage equation u r is as follows: According to Eq. ( 11), in the case of grid symmetrical faults, the stator flux includes steady-state flux φ sf and transient DC flux φ sn .Corresponding voltage components are induced at the rotor end, namely, u rf and u rn .Considering Substitute the steady-state flux φ sf and transient DC flux φ sn in Eq. ( 11) into the first term of Eq. ( 16), ignoring the smaller term 1/τ s ; Eq. ( 18) is obtained.
In Eq. ( 18), s = ω s − ω r /ω s is the slip rate.In general, the DFIG slip s is within 0.3, so the steady-state induced electromotive force u rf of the rotor is relatively small.However, the transient DC flux-induced EMF u rn of the rotor is proportional to k s 1 − s du m , and its amplitude is several times higher than that under steady-state conditions.Therefore, there is an excessive peak current in the rotor circuit, which is not conducive to the stability of DFIG.
Multiply Eq. ( 18) by e −jω r t and convert it to the rotor coordinate system to obtain the following equation: The superscript r in the equation represents the parameters under the rotor shaft system.The voltage equation in the rotor coordinate system can be expressed as follows: When the voltage drops deeply, u r rf is relatively small.Substitute u r rn and the transient DC flux current i r rn into Eq.( 20), and convert which to time domain to obtain the transient DC flux-induced voltage component of the rotor as follows: According to Eq. ( 21), in order to avoid transient surge voltage at the rotor end, the voltage can be suppressed by accelerating the decay rate of the stator transient DC flux component φ r sn , injecting rotor current i r rn opposite to the stator transient DC flux, and increasing rotor impedance.
When the rotor current is not zero and φ r sf is eliminated, the differential equation of φ r sn is obtained as follows: Substitute Eq. ( 22) into Eq.( 23), and the transient DC flux φ r sn is solved by differential equation, resulting as follows: φ r sn = c r sn0 e −t/τ , 24 where c r sn0 is the initial value of the transient DC flux φ r sn , and τ 1 = τ s / 1 + L m k s / σL r is the new stator time constant.According to Eq. ( 24), compensating current on the rotor side can accelerate the stator flux decay rate.
Under asymmetric faults, the rotor terminal voltage can be expressed as follows [34].
According to Eq. ( 25), due to that rotor side overvoltage under asymmetric faults caused by stator transient attenuation DC component and negative sequence flux component, the rotor terminal voltage induced by the negative sequence component of the magnetic flux and corresponding transient current can be analyzed based on similar principles as the transient DC component of the magnetic flux.

LVRT Prediction Model Based on MPC for Rotor Side.
Compared to conventional PI control methods, MPC is a prior control method, with good ability to predict the future behavior and select control variables.The mathematical model of DFIG in stator voltage-oriented synchronous rotating coordinate system is as follows [35].
where u ds , u qs , u dr , and u qr are the dq axis components of the stator and rotor voltage; i ds , i qs , i dr , and i qr are the dq axis components of the stator and rotor current; and φ ds , φ qs , φ dr , and φ qr are the dq axis components of the stator and rotor flux linkage.Using the forward Euler method, Eq. ( 28) is done discretize, where  29), (30) Storage cost function J r (j) When DFIG is connected to the grid, the control objective of the rotor current loop is to follow the reference value to the cost function, as follows: where J r is the cost function of rotor current.

Structure Diagram of System Control Strategy.
Figure 4 shows the flowchart of the DFIG three-vector MPC algorithm.Firstly, set parameters such as wind speed, stator, and rotor windings of the doubly fed generator, self-inductance, and mutual inductance.Then, collect the three-phase voltage and current of the power grid for calculating the transient magnetic flux φ sn , negative sequence magnetic flux φ − sf , and magnetic flux φ ds and φ qs of the doubly fed generator stator in the dq rotating synchronous coordinate system.Secondly, calculate the rotor compensation current i r rn and the stator flux feedforward compensation amount φ r sn in the event of a power grid fault.Design a three-vector MPC control algorithm.
Equations ( 28)-( 30) are a discrete mathematical model of a doubly fed generator in the dq coordinate system.Consider the relationship between transient DC flux and tran-sient DC flux compensation current angle θ, so that θ = ±5 ∘ changes (θ = 0 ∘ in traditional control schemes) to seek the optimal rotor terminal voltage.The objective function of the rotor side converter is shown in Eq. (31).Substitute i ref dr , i ref qr , i dr k + 2 , and i qr k + 2 into the objective function, calculate the first effective vector u i , and store the switch function number i of the vector.Then, obtain the second effective vector u j from the switch functions and store the switch function number j of the vector.Finally, calculate the effective action time t i , t j , and t z of the three vectors, and use the seven-segment zero vector distribution SVPWM implementation method to generate pulse output signals to control the DFIG operation.

Simulation Case Analysis
To verify the accuracy of the proposed control method, a WECS simulation platform is built on the MATLAB/Simulink system.The reactive power distribution strategy is adopted on the grid side and rotor side to reduce the copper loss of DFIG windings.Secondly, study the changes in rotor circuit current, stator transient magnetic flux, and DC bus voltage under three different operating conditions of symmetrical and asymmetric voltage faults in the power grid.Further verify that the control strategy can improve the LVRT performance of wind power generation systems.

Reactive Power Distribution of WECS under Variable
Wind Speed. Figure 5 shows the power curve of the wind power generation system as the wind speed changes.The wind speed increased from 7 m/s to 9 m/s.The rotor firstly absorbs power from the grid and then outputs power.The excess energy output by DFIG is absorbed by the battery.At t = 2 s, reactive power allocation strategies are executed at the GSC and RSC sides.From the partially enlarged image in Figure 5, it can be seen that as the rotor current increases, and the stator current decreases, while the total winding loss of DFIG decreases, which prove the effectiveness of the control strategy.Table 2 provides an analysis of stator and rotor winding losses at a wind speed of 9 m/s, and the total winding losses are reduced.

WECS Case Analysis under Grid Fault.
To verify the effectiveness of the control strategy proposed, comparisons for waveforms of rotor current and stator flux are done among traditional vector control (TVC), traditional demagnetization control (TDC), MPC+demagnetization control (MPC+DC), and the proposed control strategy.Three scenarios are simulated as following.
Scenario A: analysis of LVRT performance when the three-phase grid voltage drops symmetrically to 20% Scenario B: analysis of LVRT performance when the two-phase grid voltage drops to 20% Scenario C: analysis of LVRT performance when the single-phase grid voltage drops to 20% Figure 6 shows a symmetrical drop in three-phase voltage between t = 1 1 s and 1.7 s under scenario A. It can be seen that the rotor speed gradually increases during the drop period, and a maximum rotor current of 2.6 pu occurs by the TVC method, while a rotor current of 1.9 pu occurs by the control strategy proposed, and reactive power support is provided during the voltage recovery period.Compared with traditional vector control strategies, the reactive power absorbed is reduced by 49%, and the minimum fluctuation rate of DC bus voltage is controlled under 2.0%.
Figure 7 shows operating results under scenario B. In the early stage of voltages drop, the rotor speed rapidly increases, and the TVC scheme increases the rotor speed by 37.5%, not conducive to the stability of wind turbines.Among the four control strategies, under the TVC and TDC schemes, the peak rotor current is numerically similar.Applying MPC control technology to DFIG control strategies can reduce the overshoot of DC bus voltage.The rotor current of 2.2 pu occurs by control strategy proposed.Figure 8 shows operating results under scenario C, the single-phase voltage drops between t = 1 1 s and 1.7 s.During the voltage drop period, the proposed method slowly increases the rotor speed and gradually decreases after the voltage is restored, with a speed overshoot of only 7.1%.The TVC scheme increases the rotor speed by 17.8%.Among the four control strategies, under the TVC and TDC schemes, the rotor current peak values are 4.1 pu and 3.7 pu, which seriously affect the stability of DFIG operation.
The method proposed has a significant effect on suppressing rotor overcurrent by injecting reverse current into the rotor through RSC, with a peak rotor current of 1.3 pu.DC bus voltage overshoot by the method is only 0.8%.The reactive power absorbed during the restoration of the grid is 1.5 kW, which is 40% less than that under the TVC scheme.
The transient magnetic flux attenuation speed of the stator is accelerated, and the reactive power of the stator is restored to a stable state more quickly.Table 3 lists the rotor speed, current, and stator active and reactive power indicators of DFIG under three scenarios.During power grid faults, the rotor current of TVC is high, absorbing a large amount of reactive power, which is not conducive to the recovery of grid voltage.The use of TDC can accelerate the decay rate of stator flux, but the effect on suppressing rotor current is limited.The control strategy proposed can not only reduce rotor overcurrent but also accelerate the decay rate of stator magnetic flux, provide reactive power to the grid, reduce stator and rotor winding losses, and accelerate the recovery speed of power grid faults.

Conclusion
A transient magnetic flux and reactive power compensation control strategy based on MPC for DFIG is proposed.In the RSC control strategy, reverse rotor current compensation is introduced to suppress rotor overcurrent during LVRT.Reactive power support for grid voltage recovery is provided by combining GSC and RSC reactive power allocation strategies, as well as reducing the total loss of DFIG windings.Compared with traditional vector control strategies, the stator flux feedforward control accelerates the transient flux decay rate of the stator, and LVRT capability of DFIG is improved.Specially, by the proposed control strategy under scenario C, the peak rotor current is reduced by 69%, the overshoot of the DC bus voltage by 50%, and the reactive power absorption of the stator by 40%.In addition, during the LVRT process, the rotor speed overshoot is controlled, which increases the stability of the wind power generation system.Therefore, the control strategy proposed would own a good application in this field.

Data Availability
There are no data available.
13 International Journal of Energy Research

Figure 1 :
Figure 1: Detailed model structure diagram of doubly fed wind motor.

Figure 2 (
Figure 2(b) shows a nonlinear PI regulator model; construction is as follows:

Figure 3 :
Figure 3: Results of PLL during single-phase and two-phase voltage drop.

7
International Journal of Energy Research Substitute the flux equation in Eq. (27) into the voltage equation in Eq. (26), and perform dq decomposition on DFIG rotor current, and the rotor current equation is organized as follows.
s is the discrete period.Predict the rotor current at time k + 1, and take into account that the DFIG discretization model has a long delay of one cycle from the step; therefore, u dr k and u qr k in the rotor current equation are changed to u dr k − 1 and u qr k − 1 , and multistep prediction is used to obtain the optimal control quantity through the model prediction algorithm.The prediction equation for rotor current at time k + 1 is as follows:Storage cost function J r () Collect signals U abc , i sabc , i rabc Calculate  sf -,  sn ,  ds and  qs

29 8
International Journal of Energy ResearchAccording to the model predictive control system rolling optimization, eliminating PWM update delay requires a second step prediction of the rotor current.The prediction equation for the rotor current at time k and time k + 2 is as follows:

Figure 6 :
Figure 6: System waveform diagram of three-phase power grid voltage drop.

Figure 7 :
Figure 7: System waveform diagram of two-phase power grid voltage drop.

Figure 8 :
Figure 8: System waveform diagram of single-phase power grid voltage drop.
Assuming that both the stator and rotor adopt the motor convention, and the rotor side parameters are converted to the stator side, the grid voltage experienced a symmetrical fault drop with a depth of d at time t 0 .Ignoring the stator resistance R s , the stator flux can be represented as the steady-state flux and transient DC flux components due to that the stator [31]StatorFlux Feedforward Compensation.fluxcannotchangesuddenlybefore and after the grid voltage drop.The differential equation can be expressed as follows[31].φs= φ sf + φ sn = 1 − du m jω s e jω s t + du m jω s e −t/τ s e −t/τ s , 11 4 International Journal of Energy Research where τ s = L s /R s is the stator time constant.According to Eq. (11), only when the transient DC flux φ sn gradually decays to zero can the system enter a new stable state.

Table 2 :
Analysis of DFIG winding losses at a wind speed of 9 m/s.

Table 3 :
Comparison of relevant control performance indicators.
Wind speed, tip speed ratio, yaw angle ρ, r, η: Air density, turbine radius, gearbox ratio ω t , ω r : Fan blade angular velocity, DFIG rotor speed ω s : Synchronous angular velocity of the power grid P s , Q s , S s : Active and reactive power on the stator side and stator side capacity P r , Q r , S r : Active and reactive power on the rotor side and rotor side capacity Q mc , v l , f s : DFIG magnetization reactive power, grid frequency, grid line voltage I s , I r : E ffective value of stator and rotor line current R s , R r : Stator and rotor winding resistance τ: Reactive power distribution coefficient on the GSC and RSC W s , W r , W all : Loss of stator and rotor windings and total loss D 1 s , Q 1 s : D i fferential operator, voltage drop depth φ sf , φ sn : Steady-state flux and DC transient flux in stator coordinate system u + m , u − m , u m0 , τ s : Amplitude of positive sequence, negative sequence, and zero sequence voltage components, stator time constant φ + sf , φ − sf : Steady-state magnetic flux generated by stator positive and negative sequence voltages k s , σ: Stator coupling coefficient, motor leakage coefficient u rk , u rf , u rn : Open circuit voltage, steady-state fluxand transient flux-induced electromotive force of rotor in stator coordinate system s : Characteristic roots of stator flux state equation.