Abstract
The brushless doubly-fed induction generator (BDFIG) presents significant potential for application in wind power systems, primarily due to the elimination of slip rings and brushes. The application of virtual synchronous control (VSynC) has been demonstrated to effectively augment the inertia of BDFIG systems. However, the dynamic characteristics and stability of BDFIG under weak grid conditions remain largely unexplored. The critical stabilizing factors for BDFIG-based wind turbines (WTs) are methodically investigated, and an enhanced VSynC method based on linear active disturbance rejection control (LADRC) is proposed. The stability analysis reveals that the proposed method can virtually enhance the stability of the grid-connected system under weak grid conditions. The accuracy of the theoretical analysis and the effectiveness of the proposed method are affirmed through extensive simulations and detailed experiments.
AMONG the various types of grid-connected wind turbines (WTs), the doubly-fed induction generator (DFIG) is often preferred [
However, with the increasing integration of power electronics-based renewable energy sources, the inertia and damping characteristics of power systems are progressively diminishing [
For instance, the VSynC approaches have been employed in WTs to enhance the inertia support capability [
To tackle this challenge, a linear active disturbance rejection control (LADRC) is employed for a standard three-phase grid-connected converter, as demonstrated in [
Nevertheless, the stability analysis of the BDFIG system is more challenging, owing to its complex machine structure. Most of the previous studies on control methods of BDFIG system generally fall into two categories, which are the field-/voltage-oriented vector control (VC) [
Addressing these challenges, the critical stabilizing factors crucial for BDFIG-based WTs are methodically investigated, and an enhanced VSynC method based on LADRC is proposed to enhance the stability of the BDFIG system. The accuracy of the theoretical analysis and the effectiveness of the proposed method are affirmed through extensive simulations and detailed experiments.
The topology and control structure of BDFIG system are shown in

Fig. 1 Topology and control structure of BDFIG system.
When employing the VSynC, the BDFIG system acquires inertia and damping properties, enabling it to emulate the output characteristics of traditional SGs. The power outer control loop includes the active power control loop and reactive power control loop.
The active power control loop can be expressed as:
(1) |
Pm contains two components, namely the reference active power and the output of the virtual functionary, which can be expressed as:
(2) |
Similarly, the reactive power control can be expressed as:
(3) |
In the synchronous coordinate system, the voltage and magnetic equations of BDFIG system can be obtained as [
(4) |
where , is the mechanical angular velocity; and is the differential operator.
PW and CW voltages are taken as input variables, and the currents are taken as output variables. The open-loop input admittance model of BDFIG system is established, as shown in
(5) |

Fig. 2 Open-loop input admittance model of BDFIG system.
Simplifying the transfer functions in
(6) |
From the mathematical model of the BDFIG system, the relation between the CW and PW currents can be obtained. Under the d-axis orientation of the PW magnetic flux, the relation can be re-written as:
(7) |
where .
It can be observed from (7) that the PW magnetic flux basically keeps constant under the stable condition. Besides, the changing rates of both the PW current and voltage are approximately first-order linear. Neglecting the coupling term, the equation of voltage control can be derived as:
(8) |
Thus, the voltage control loop can be disclosed as:
(9) |
Typically, when the PI regulator is used in the current control loop, and can be expressed as:
(10) |
Thus, the current control loop can be obtained as:
(11) |
In the synchronous coordinate system, the active and veactive power references of the PWs under steady state can be denoted as Pp,ref and , respectively. The calculation of Pp,ref and Qp,ref are given by (B1) in Appendix B. Injecting a small-signal perturbation under the steady-state working point yields (refer to Appendix B (B1) for detailed derivation process):
(12) |
Then, the above equation is expanded and the steady-state values are eliminated, which can be expressed as:
(13) |
The simplified structure of VSynC loop is illustrated in

Fig. 3 Simplified structure of VSynC loop.
Setting the small-signal perturbation of the active and reactive power as the input, the phase and amplitude can be derived as:
(14) |
Then, the PW voltage to the synchronous coordinate system is converted, which can be expressed as:
(15) |
Linearizing (15) via the small signal, we can obtain:
(16) |
where subscript 0 is the corresponding small-signal variable.
Combining (14) and (16), the vector of PW voltage disturbance reference can be expressed as:
(17) |
Combining (5)-(17), the complete input admittance model of BDFIG system with VSynC can be summarized as in
(18) |

Fig. 4 Model of BDFIG system with VSynC.
Accordingly, the frequency response curves of the BDFIG are obtained. The bode diagram of output admittance of the BDFIG system is shown in

Fig. 5 Bode diagram of output admittance of BDFIG system. (a) d-axis component. (b) d- and q-axis coupled components. (c) q- and d-axis coupled components. (d) q-axis component.
According to the developed input model, the BDFIG system behaves as multi-input multi-output characteristic in the dq coordinate. According to the generalized Nyquist criterion theorem, the stability of the BDFIG system can be estimated by the total time number of the loop matrix that wraps around (-1, 0).
For the BDFIG system, the loop matrix can be expressed as .
(19) |
As shown in

Fig. 6 Structure diagram of BDFIG system connected to grid.
(20) |
Considering that the value of the equivalent resistance R is much smaller than that of the equivalent reactance X, the grid impedance angle can be assumed to be °. Then, the output power is approximated as:
(21) |
According to (21), the active power-frequency loop can be conveyed as:
(22) |
From (22), the damping ratio and the natural oscillation frequency are obtained as:
(23) |
The three-dimensional relationship among J, D, and is shown in

Fig. 7 Three-dimensional relationship among J, D, and .
According to (22), the open-loop transfer function can be demonstrated as:
(24) |
Setting , the root trajectories of the grid-connected system can be derived as:
(25) |
Based on (25), the root trajectories of the system under different J and D can be obtained, as shown in

Fig. 8 Root trajectories under different J and D. (a) kg·
To analyze the effect of control parameters on the stability of the BDFIG system, the generalized Nyquist curves under different VSynC parameters are given in

Fig. 9 Generalized Nyquist curves under different VSynC parameters. (a) kg·
The power response of the BDFIG system under different VSynC parameters is shown in

Fig. 10 Power response under different VSynC parameters. (a) kg·
The generalized Nyquist curves under different SCRs are shown in

Fig. 11 Generalized Nyquist curves under different SCRs.
To validate the truth of the theoretical analysis, simulations under different SCRs are carried out. The results of electromagnetic transient simulation waveforms of BDFIG under different SCRs are presented in

Fig. 12 Electromagnetic transient simulation waveforms of BDFIG system under different SCRs.
At the beginning, , at which time the BDFIG system can operate stably. Nevertheless, when SCR drops to 2.0 at 1.0 s, the voltage and current waveforms get to oscillate and the BDFIG system is destabilized. Further, when SCR drops to 1.5 at 1.3 s, the BDFIG system oscillates more obviously, which validates the theoretical analysis results in
The LADRC can significantly enhance the stability of the BDFIG system compared with the traditional PI control under the same bandwidth condition [

Fig. 13 Control topology of BDFIG system based on LADRC.
The structure of first-order LADRC is presented in

Fig. 14 Structure of first-order LADRC.
The LESO enables real-time observation of the actual BDFIG system, which can be deduced as:
(26) |
The LSEF can increase the feedback amount based on the proportional control, which is conducive to enhancing the transient response of BDFIG system. The LSEF can be expressed as:
(27) |
The structure of single-parameter LADRC is applied, with symbolizing its bandwidth. Then, , , and kp are exhibited as:
(28) |
Further, based on (26) and (27), the equivalent topology of first-order LADRC is deduced, as shown in
(29) |

Fig. 15 Equivalent topology of first-order LADRC.
From (29), it can be concluded that C1(s) is equivalent to the combination of “a PI controller” plus “a first-order low-pass filter”. Note that the numerator order of the equivalent of is higher than that of the denominator, implying the achievement of phase compensation. This is why the LADRC behaves better in enhancing the stability of theBDFIG system than that of the traditional PI control.
The design for the voltage and current control loops is given as follows. Considering symmetry, only the d-axis design process is presented.
The input reference of current control loop is the d-axis CW current icd,ref, while the feedback is the d-axis CW current icd. And the output is the d-axis CW voltage . The relationship can be expressed as:
(30) |
According to (26), (27), and (30), the current loop can be designed as:
(31) |
Under the PW magnetic flux orientation, the changing rates of PW current and voltage can be approximated as a first-order linear relationship. Thus, the relationships between the PW and CW currents are exhibited as:
(32) |
According to (26), (27), and (31), the voltage control loop can be designed as:
(33) |
The generalized Nyquist curves of the BDFIG system under different SCRs and control methods are demonstrated in

Fig. 16 Generalized Nyquist curves of BDFIG system under different SCRs and control methods. (a) . (b) . (c) .
As demonstrated in

Fig. 17 Simulation waveforms of BDFIG system under different SCRs and control methods. (a) . (b) .
Under weak grid condition, when Method I is used, the voltage and current of BDFIG system are destabilized. However, when Method II is utilized, the instability phenomenon gets enhanced to some extent, but instability phenomenon still exists when SCR goes down to 1.5. When Method III is triggered, the BDFIG system maintains stable even when , which is consistent with
To further verify the accuracy of the analysis, a 3 kW BDFIG experimental platform is built, as depicted in

Fig. 18 Experimental platform of BDFIG.
The experimental results of power step response of the BDFIG system under different VSynC parameters are shown in

Fig. 19 Experimental results of BDFIG system power step response under different VSynC parameters. (a) N·m·s/rad, kg·
Experimental results with traditional PI control under different SCRs are shown in

Fig. 20 Experimental results with traditional PI control method under different SCRs. (a) . (b) . (c) .
The experimental results with enhanced VSynC method under different SCRs are shown in

Fig. 21 Experimental results with enhanced VSynC method under different SCRs. (a) . (b) . (c) .
The experimental results under different speeds are shown in

Fig. 22 Experimental results under different speeds. (a) PW voltage under sub-synchronous operation mode. (b) PW current under sub-synchronous operation mode. (c) PW voltage under super-synchronous operation mode. (d) PW current under super-synchronous operation mode.
Ⅵ. Conclusion
In this paper, the admittance model of BDFIG with VSynC is constructed. Furthermore, an enhanced VSynC method based on LADRC is proposed to enhance the stability of the BDFIG system under weak grid condition. The specific conclusions are summarized as follows.
1) Based on the established BDFIG model, the VSynC loop mainly changes the impedance characteristics of the BDFIG in low frequency band, while it has almost no effect on the middle and high frequency bands.
2) J and D have significant influence on the stability of the BDFIG system, i.e., increasing D can enhance the stability margin, while increasing J can reduce the stability margin.
3) The proposed method can significantly enhance the stability of the BDFIG system with a simple structure.
The influence of the LADRC on the BDFIG transient characteristics remains to be evaluated, which will be studied in the near future.
Nomenclature
Symbol | —— | Definition |
---|---|---|
ψp, ψc, ψr | —— | Vectors of power winding (PW), control winding (CW) and rotor winding (RW) magnetic flux |
—— | Overshoot value | |
ω | —— | Angular velocity of grid |
ω0 | —— | Synchronous angular velocity of grid |
ωp | —— | PW angular velocity |
—— | RW angular velocity | |
—— | CW angular velocity | |
—— | Grid voltage angle perturbation | |
—— | Grid voltage angle | |
—— | RW angle | |
—— | Angular aberration | |
, | —— | d- and q-axis CW current compensations |
ΔPout | —— | Deviation value of outturn active power |
ΔPref | —— | Deviation value of reference active power |
, | —— | Observer gains |
b0 | —— | Known part of control gain b |
b0,i | —— | Current of b0 |
C1(s) | —— | Closed-loop feedback controller |
Cf(s) | —— | Closed-loop feedforward controller |
D | —— | Virtual damping coefficient |
E0 | —— | Rated grid voltage value |
E1 | —— | Identity matrix |
E | —— | Reference grid voltage amplitude |
ftotal | —— | Sum of system perturbations |
Gipp | —— | Transfer function matrix between PW voltage and current |
Gp | —— | Control object |
Gpr | —— | Transfer function matrix between PW and RW currents |
Grc | —— | Transfer function matrix between RW and CW currents |
Gcr | —— | Transfer function matrix between CW and RW currents |
Gcu | —— | Transfer function matrix of voltage control loop |
Gci | —— | Transfer function matrix of current control loop |
Girp | —— | Transfer function matrix between RW voltage and PW current |
Gicc | —— | Transfer function matrix between CW voltage and current |
, | —— | Transfer function matrices of q-axis voltage and current coordinate transformation |
—— | Transfer function matrix of d-axis virtual synchronization control | |
—— | Transformer transfer function matrix of d-axis PW voltage | |
, | —— | d- and q-axis PW current disturbances |
—— | Vectors of PW and RW current disturbances | |
, | —— | Vectors of CW current disturbance and its reference |
ip, ic, ir | —— | Vectors of PW, CW, and RW currents |
, | —— | d- and q-axis CW currents |
, | —— | d- and q-axis CW current discrepancies |
, | —— | d- and q-axis PW currents |
, | —— | Steady-state values of d- and q-axis PW currents |
J | —— | Virtual inertia coefficient |
kp | —— | Error feedback coefficient |
Kω | —— | Primary frequency modulation coefficient |
Kpu, Kiu | —— | Proportional and integral coefficients of voltage loop |
Kpi, Kii | —— | Proportional and integral coefficients of current loop |
Lp, Lc, Lr | —— | PW, CW, and RW self-inductances |
—— | Mechanical inductance | |
Mpr | —— | Mutual inductance between PW and RW |
Mcr | —— | Mutual inductance between CW and RW |
n | —— | Droop coefficient |
P | —— | Output active power |
Pm | —— | Input virtual mechanical power |
Pe | —— | Output electromagnetic active power |
Pref | —— | Reference active power |
—— | PW reference active power | |
—— | PW active power disturbance | |
Qref | —— | Reference reactive power |
—— | PW reference reactive power | |
Q | —— | Output electromagnetic reactive power |
—— | PW reactive power disturbance | |
Rp, Rc, Rr | —— | PW, CW, and RW resistances |
Rg, Lg | —— | Resistance and inductance of grid |
—— | Control input quantity | |
—— | Output of linear state error feedback (LSEF) | |
, | —— | d- and q-axis PW voltages |
, | —— | d- and q-axis CW voltages |
, | —— | d- and q-axis PW voltage discrepancies |
up, uc, ur | —— | PW, CW, and RW voltage vectors |
, | —— | Vectors of PW voltage disturbance and its reference |
, | —— | d- and q-axis PW voltage disturbances |
, | —— | Vectors of CW voltage disturbance and its reference |
, | —— | Steady-state values of d- and q-axis PW voltages |
, | —— | PW voltage and grid voltage amplitudes |
—— | Stator voltage | |
—— | PW reference voltage | |
—— | PW voltage amplitude disturbance | |
—— | Reference of LSEF input | |
—— | Small-signal disturbance matrix of brushless doubly-fed induction generator (BDFIG) system | |
—— | d- and q-axis disturbances | |
X, R | —— | Equivalent reactance and resistance |
—— | Matrix of BDFIG system admittance | |
—— | Transfer function matrix of open-loop admittance of BDFIG system | |
—— | Transfer function matrix of admittance of BDFIG system | |
—— | Real-time observation of output quantity of BDFIG system | |
—— | Estimate of real-time observation of internal and external disturbances | |
—— | Matrix of grid impedance |
Appendix
(A1) |
(A2) |
(A3) |
(A4) |
(A5) |
(A6) |
(A7) |
(A8) |
(A9) |
In calculating open-loop input admittance model of BDFIG, Fig. 2 is simplified to the following form and (6) can be obtained from Fig. A1.

Fig. A1 Simplified open-loop input admittance model of BDFIG system.
In the synchronous coordinate system, the active and reactive power references of PWs during steady state are and , respectively.
(B1) |

Fig. B1 Experimental results of power step response without VSynC (). (a) Waveforms of active power dynamic response. (b) Waveforms of PW voltage. (c) Waveforms of PW current.
Parameter | Value |
---|---|
Ug (V) | 380 |
Rp (p.u.) | 0.067 |
Rc (p.u.) | 0.073 |
Rr (p.u.) | 0.130 |
Lp (p.u.) | 24.510 |
Lc (p.u.) | 2.660 |
Lr (p.u.) | 27.250 |
Mpr (p.u.) | 24.280 |
Mcr (p.u.) | 2.540 |
Parameter | Value |
---|---|
Pref (kW) | 3 |
Ug (V) | 380 |
Pp | 3 |
Pc | 1 |
Rp (Ω) | 3.200 |
Rc (Ω) | 5.320 |
Rr (mΩ) | 0.130 |
Lp (H) | 0.292 |
Lc (H) | 0.642 |
Lr (mH) | 0.048 |
Mpr (mH) | 2.160 |
Mcr (mH) | 4.000 |
References
R. McMahon, P. Tavner, E. Abdi et al., “Characterizing brushless doubly fed machine rotors,” IET Electric Power Applications, vol. 7, no. 7, pp. 535-876, Aug. 2013. [Baidu Scholar]
P. Mukherjee and V. V. Rao, “Superconducting magnetic energy storage for stabilizing grid integrated with wind power generation systems,” Journal of Modern Power Systems and Clean Energy, vol. 7, no. 2, pp. 400-411, Mar. 2019. [Baidu Scholar]
T. Long, S. Shao, P. Malliband et al., “Crowbar-less fault ride-through of the brushless doubly fed induction generator in a wind turbine under symmetrical voltage dips,” IEEE Transactions on Industrial Electronics, vol. 60, no. 7, pp. 2833-2841, Jul. 2013. [Baidu Scholar]
Y. Wang, H. Xu, P. Ge et al., “Stability estimation and enhanced control of BDFIG-driven wind turbines under weak grid,” in Proceedings of 2023 3rd New Energy and Energy Storage System Control Summit Forum (NEESSC), Mianyang, China, Sept. 2023, pp. 243-249. [Baidu Scholar]
C. Wang, H. Xu, P. Ge et al., “Transient stability analysis and enhancement for BDFIG based virtual synchronous control,” in Proceedings of 2023 3rd New Energy and Energy Storage System Control Summit Forum (NEESSC), Mianyang, China, Sept. 2023, pp. 259-263. [Baidu Scholar]
A. Nair, S. Kamalasadan, J. Geis-Schroer et al., “An investigation of grid stability and a new design of adaptive phase-locked loop for wind-integrated weak power grid,” IEEE Transactions on Industry Applications, vol. 58, no. 5, pp. 5871-5884, Sept. 2022. [Baidu Scholar]
L. Xiong, P. Li, F. Wu et al., “Stability enhancement of power systems with high DFIG-wind turbine penetration via virtual inertia planning,” IEEE Transactions on Power Systems, vol. 34, no. 2, pp. 1352-1361, Mar. 2019. [Baidu Scholar]
J. Liu, W. Yao, J. Wen et al., “Impact of power grid strength and PLL parameters on stability of grid-connected DFIG wind farm,” IEEE Transactions on Sustainable Energy, vol. 11, no. 1, pp. 545-557, Jan. 2020. [Baidu Scholar]
P. Chen, C. Qi, and X. Chen, “Virtual inertia estimation method of DFIG-based wind farm with additional frequency control,” Journal of Modern Power Systems and Clean Energy, vol. 9, no. 5, pp. 1076-1087, Sept. 2021. [Baidu Scholar]
V. Nayanar, N. Kumaresan, and N. A. Gounden, “A single-sensor-based MPPT controller for wind-driven induction generators supplying DC microgrid,” IEEE Transactions on Power Electronics, vol. 31, no. 2, pp. 1161-1172, Feb. 2016. [Baidu Scholar]
R. Peña-Alzola, D. Campos-Gaona, P. F. Ksiazek et al., “DC-link control filtering options for torque ripple reduction in low-power wind Turbines,” IEEE Transactions on Power Electronics, vol. 32, no. 6, pp. 4812-4826, Jun. 2017. [Baidu Scholar]
W. Du, W. Dong, Y. Wang et al., “Small-disturbance stability of a wind farm with virtual synchronous generators under the condition of weak grid connection,” IEEE Transactions on Power Systems, vol. 36, no. 6, pp. 5500-5511, Nov. 2021. [Baidu Scholar]
N. Mohammed, M. H. Ravanji, W. Zhou et al., “Online grid impedance estimation-based adaptive control of virtual synchronous generators considering strong and weak grid conditions,” IEEE Transactions on Sustainable Energy, vol. 14, no. 1, pp. 673-687, Jan. 2023. [Baidu Scholar]
S. Wang, J. Hu, and X. Yuan, “Virtual synchronous control for grid-connected DFIG-based wind turbines,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 3, no. 4, pp. 932-944, Dec. 2015. [Baidu Scholar]
L. Shang, J. Hu, X. Yuan et al., “Improved virtual synchronous control for grid-connected VSCs under grid voltage unbalanced conditions,” Journal of Modern Power Systems and Clean Energy, vol. 7, no. 1, pp. 174-185, Jan. 2019. [Baidu Scholar]
S. Wang, J. Hu, X. Yuan et al., “On inertial dynamics of virtual-synchronous-controlled DFIG-based wind turbines,” IEEE Transactions on Energy Conversion, vol. 30, no. 4, pp. 1691-1702, Dec. 2015. [Baidu Scholar]
H. Nian and Y. Jiao, “Improved virtual synchronous generator control of DFIG to ride-through symmetrical voltage fault,” IEEE Transactions on Energy Conversion, vol. 35, no. 2, pp. 672-683, Jun. 2020. [Baidu Scholar]
H. Shao, X. Cai, Z. Li et al., “Stability enhancement and direct speed control of DFIG inertia emulation control strategy,” IEEE Access, vol. 7, pp. 120089-120105, Aug. 2019. [Baidu Scholar]
C. Li, Y. Yang, Y. Cao et al., “Frequency and voltage stability analysis of grid-forming virtual synchronous generator attached to weak grid,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 10, no. 3, pp. 2662-2671, Jun. 2022. [Baidu Scholar]
H. Xu, F. Nie, Z. Wang et al., “Impedance modeling and stability factor assessment of grid-connected converters based on linear active disturbance rejection control,” Journal of Modern Power Systems and Clean Energy, vol. 9, no. 6, pp. 1327-1338, Nov. 2021. [Baidu Scholar]
Z. Xie, Y. Chen, W. Wu et al., “Admittance modeling and stability analysis of grid-connected inverter with LADRC-PLL,” IEEE Transactions on Industrial Electronics, vol. 68, no. 12, pp. 12272-12284, Dec. 2021. [Baidu Scholar]
H. Liu, F. Bu, W. Huang et al., “Linear active disturbance rejection control for dual-stator winding induction generator AC power system,” IEEE Transactions on Industrial Electronics, vol. 70, no. 7, pp. 6597-6607, Jul. 2023. [Baidu Scholar]
S. Shao, E. Abdi, F. Barati et al., “Stator-flux-oriented vector control for brushless doubly fed induction generator,” IEEE Transactions on Industrial Electronics, vol. 56, no. 10, pp. 4220-4228, Oct. 2009. [Baidu Scholar]
X. Yan and M. Cheng, “Backstepping-based direct power control for dual-cage rotor brushless doubly fed induction generator,” IEEE Transactions on Power Electronics, vol. 38, no. 2, pp. 2668-2680, Feb. 2023. [Baidu Scholar]
M. Lu, Y. Chen, D. Zhang et al., “Virtual synchronous control based on control winding orientation for brushless doubly fed induction generator (BDFIG) wind turbines under symmetrical grid faults,” Energies, vol. 12, no. 2, pp. 319-319, Jan. 2019. [Baidu Scholar]
P. Lin, Z. Wu, Z. Fei et al., “A generalized PID interpretation for high-order LADRC and cascade LADRC for servo systems,” IEEE Transactions Industrial Electronics, vol. 69, no. 5, pp. 5207-5214, May 2022. [Baidu Scholar]
R. Zhou, C. Fu, and W. Tan, “Implementation of linear controllers via active disturbance rejection control structure,” IEEE Transactions on Industrial Electronics, vol. 68, no. 7, pp. 6217-6226, Jul. 2021. [Baidu Scholar]
W. Ma, Y. Guan, and B. Zhang, “Active disturbance rejection control based control strategy for virtual synchronous generators,” IEEE Transactions on Energy Conversion, vol. 35, no. 4, pp. 1747-1761, Dec. 2020. [Baidu Scholar]
M. Zhu, Y. Ye, Y. Xiong et al., “Parameter robustness improvement for repetitive control in grid-tied inverters using an IIR filter,” IEEE Transactions on Power Electronics, vol. 36, no. 7, pp. 8454-8463, Jul. 2021. [Baidu Scholar]
W. Xu, O. M. E. Mohammed, Y. Liu et al., “Negative sequence voltage compensating for unbalanced standalone brushless doubly-fed induction generator,” IEEE Transactions on Power Electronics, vol. 35, no. 1, pp. 667-680, Jan. 2020. [Baidu Scholar]