Abstract
The dynamic coupling effect, which is introduced by the dual-sequence phase-locked loops (PLLs) used in doubly- fed induction generator (DFIG) based wind energy generation systems (WEGSs) during asymmetric low voltage ride-through (LVRT) in weak grid, needs attention. In order to study this new dynamic coupling effect, an equivalent two-degree-of-freedom (2-DOF) spring damper particle model is used in this paper to develop a small-signal model for the dual-sequence PLLs. The dynamic interaction between the positive-sequence (PS) and negative-sequence (NS) PLLs is unveiled. Moreover, the impact of the dynamic coupling between the dual-sequence PLLs on the dynamic stability during the steady-state stage of an asymmetric fault is analyzed. The analysis results show that the dynamic coupling between the dual-sequence PLLs will cause drift in the frequency and damping for the PS and NS PLL modes. This will change the instability modal of the system and introduce the risk of dynamic instability. Hence, the effectiveness of existing control strategies for enhancing the dynamic stability will be decreased. Finally, a novel PLL structure is designed to improve the dynamic stability of the system during the steady-state stage of an asymmetric fault. The effectiveness of the proposed strategy is verified by simulations and experiments.
WITH the development of the renewable energy power generation industry, doubly-fed induction generator (DFIG) based wind energy generation systems (WEGSs) have been widely applied in the market in recent years [
Some existing studies on DFIG-based WEGSs under asymmetric grid faults focus on control schemes for the positive-sequence (PS) and negative-sequence (NS) components. References [
The faults in a power system comprise two transient stages (fault-initiation and fault-clearing) and one steady-state stage [
However, the systems investigated in the above studies apply the single-sequence phase-locked loop (PLL) structure during asymmetric LVRT, which is different from the dual-sequence PLL structure adopted in [
This study is carried out for filling the study gap about the influence of applying dual-sequence PLLs structure on the dynamic stability of power system. The main contributions of this study are as follows.
1) Based on the developed equivalent two-degree-of-freedom (2-DOF) spring damper particle model for dual-sequence PLLs, the new dynamic coupling between the PS and NS PLLs introduced by the dual-sequence PLL structure is revealed.
2) The impact of the dynamic coupling effect on the stability of the system and the proposed control strategy for enhancing the dynamic stability during asymmetric LVRT are analyzed. Moreover, the risk of dynamic instability caused by the dynamic coupling is demonstrated.
3) A novel PLL structure with an adjustable damping shock absorber added to the NS PLL is designed to improve the dynamic stability of the system when dual-sequence PLLs are applied during asymmetric LVRT. Simulations and experiments verify the effectiveness of this structure.
The rest of this paper is organized as follows. The model of dual-sequence PLLs is developed, and the dynamic coupling between the PS and NS PLLs is discussed in Section II. In Section III, the influence of the new dynamic coupling introduced by the dual-sequence PLLs on the dynamic stability is analyzed. In Section IV, a novel PLL structure that can improve the dynamic stability of systems adopting dual-sequence PLLs is designed. The results of simulations and experiments are presented in Sections V and VI, respectively. Finally, Section VII concludes this paper.
The typical control structure of DFIG-based WEGS during asymmetric LVRT is shown in

Fig. 1 Typical control structure of DFIG-based WEGS during asymmetric LVRT.
During an asymmetric voltage dip, the system needs to inject the corresponding PS and NS reactive currents into the faulty grid in accordance with the severity of the fault [
In this paper, the command values of the PS and NS reactive currents that should be injected by the DFIG WEGS during asymmetric LVRT are expressed as [
(1) |
where .
The dynamic stability during the steady-state stage of a fault in which the transient fault component has completely attenuated is the main focus of this paper. The crowbar on the rotor side or the chopper on the DC side has been deactivated during the steady-state stage of a fault. Therefore, a small-signal model is obtained by linearizing the mathematical model at the stable operating point during steady-state stage of a grid fault.
The control structure of dual synchronous reference frames (SRFs) is normally used to control the PS and NS components [

Fig. 2 Diagram of dual-sequence PLL structure.
The PCC voltage is first decomposed into PS and NS reference voltages by the PS and NS separation modules, respectively. The PS and NS separation algorithm can be expressed as:
(2) |
where
Subsequently, the q-axis components of the PS and NS voltages, i.e., and , will enter the proportional-integral (PI) controller in independent PS and NS PLLs to obtain the angular velocities for PS and NS SRFs, and then obtain the PS and NS angles for the PS and NS SRFs through the integral operation, respectively.
The small-signal model for the dual-sequence PLLs is expressed as:
(3) |
Due to the PLL dynamics, the system will have four rotating synchronous frames during asymmetric grid faults: d

Fig. 3 Control vector F in αβ, d
Considering the PLL dynamics, we have:
(4) |
The relationships of the control vector F in these four SRFs are expressed as:
(5) |

Fig. 4 Structure of power grid under two typical asymmetric grid faults.
Because the DFIG-based WEGS generally adopts the three-phase three-wire system to connect an AC grid, there is no zero-sequence (ZS) component path in the system [
(6) |
During asymmetric grid faults, there exist dynamic coupling phenomena in the PS and NS networks [
(7) |
where and are the grid sequence impedances; and and are the inter-sequence coupling impedances of the faulty network. The output current dynamics of the DFIG-based WEGS can be expressed as:
(8) |
Combining (3)-(8), the small-signal model of dual-sequence PLLs is shown in

Fig. 5 Small-signal model of dual-sequence PLLs.
Owing to the inter-sequence coupling effect, there is dynamic interaction between the PS and NS PLLs. These interaction is not investigated in previous studies. From
(9) |
As shown in (9), the interactively coupled dual-sequence PLLs are similar to two coupled harmonic oscillators. Therefore, the Newton’s equations of motion can be written for the system as:
(10) |
where M, D, and K are the mass, damping, and spring matrices of the system, respectively, and M is a unit matrix indicating that the masses of both blocks are 1. In the damping and spring matrices, the diagonal elements and represent the damping and spring coefficients of an independent single-sequence PLL, respectively, whereas the nondiagonal elements and () represent the damping and spring coupling terms between the dual-sequence PLLs, respectively.
The equivalent 2-DOF spring damper particle model with the force analysis of particles is shown in

Fig. 6 Equivalent small-signal model of dual-sequence PLLs. (a) 2-DOF spring damper particle model. (b) Force analysis of particles.
The two blocks in
(11) |
(12) |
(13) |
There is a coupling effect between the two blocks, which will affect the damping and oscillation frequencies of these two modes.
According to the established model, the application of dual-sequence PLLs has introduced a new dynamic coupling effect to the system (the dynamic coupling between the PS and NS PLLs). The influence of this new dynamic coupling effect on the system stability is still unclear.
In order to study the influence of this new dynamic coupling effect, a small-signal state-space equation is established for the DFIG-based WEGS during asymmetric LVRT, which includes the dynamics of the RSC current control loops (CCLs), grid-side converter (GSC) CCLs, and dual-sequence PLLs. During asymmetric faults, the small-signal model of a DFIG in the PS and NS dq frames is expressed as:
(14) |
(15) |
where Lm is the mutual inductance; and . The small-signal model of the RSC control system in the PS and NS dq frames is defined as:
(16) |
Similarly, the small-signal model of the GSC and its control system in the PS and NS dq frames can be obtained as:
(17) |
(18) |
(19) |
By combining the above equations and the model of the faulty grid, the small-signal model of a DFIG during the steady-state stage of a weak-grid asymmetric fault can be obtained. By solving the small-signal state-space equation, the dominant oscillation model of the DFIG-based WEGS and its corresponding participation factors can be studied during asymmetric LVRT. In addition, the severity of the asymmetric faults is measured by the voltage unbalance factor (VUF) defined as .
When and , there is a pair of poles entering the right half of the complex plane. The dominant oscillation mode of the system and its analysis results are summarized in
Condition | Eigenvalue λ3,4 | Damping ratio |
---|---|---|
Without considering coupling between CCLs and NS PLL | -0.009 | |
Without considering coupling between dual-sequence PLLs | 0.039 |
Eigenvalue λ3,4 | Damping ratio | Frequency (Hz) | Dominated state variable | Participation factor |
---|---|---|---|---|
-0.01 | 23.16 | 0.47 | ||
0.22 | ||||
0.02, 0.02 | ||||
0.05, 0.05 | ||||
0.02, 0.02 | ||||
0.05, 0.05 |
From the analysis, the dynamic stability of a DFIG-based WEGS adopting the dual-sequence PLL structure is dominated by the NS PLL during asymmetric LVRT. In contrast, the PS PLL affects the dynamic stability of the system by the dynamic coupling between the dual-sequence PLLs during asymmetric LVRT.
Considering different coupling effects, an eigenvalue analysis of the dominant poles is summarized in
Because of the influence of this new dynamic coupling introduced by dual-sequence PLLs, the improved control strategies proposed in the existing studies may be less effective for WEGSs that apply the dual-sequence PLL structure. Therefore, this paper will focus on this influence on the dynamic stability of the system and strategies for addressing it to improve the stability of the system.
The damping ratios and natural oscillation frequencies (eigenfrequencies) for the decoupled PLL modes can be calculated using (20).
(20) |
According to [
It is assumed that the 2-DOF system is viscous proportional damping. Substituting its natural exponential solution into (10) and dividing both sides by the nonzero term
(21) |
According to Cramer’s Rule, when the coefficient determinant of an n-variable homogeneous linear equation system composed of n equations is nonzero, the homogeneous equation system has only trivial solutions, i.e., . Thus, if (21) has a nontrivial solution, the determinant of the coefficient matrix must be zero.
(22) |
Substituting into (22), the solution to (22) can be obtained as:
(23) |
where and are the approximate modal frequencies of the PS and NS PLLs considering the dynamic coupling between the dual-sequence PLLs, respectively. Intuitively, owing to kpnknp.
In accordance with (23), the dynamic coupling between the dual-sequence PLLs increases the frequency of the PS PLL mode and decreases the frequency of the NS PLL mode. In addition, the magnitude of the frequency shift is related to kpnknp.
From (10), the dynamic coupling between the dual-sequence PLLs also affects the damping of the system. Using (10), we can obtain:
(24) |
To find a nontrivial solution, we have:
(25) |
(26) |
(27) |
(28) |
(29) |
In (25), the PS and NS PLLs are equated to coupled complex torque models that have symmetric structures. For example, the model of NS PLL can be depicted as shown in

Fig. 7 Model of NS PLL. (a) Spring damper particle model of coupled NS PLL. (b) Equivalent complex torque model of NS control subsystem.
According to complex torque theory, the magnitude of the imaginary part of Gnp(s) can be used to measure the influence of the dynamic coupling between the dual-sequence PLLs on the damping of the NS PLL. Substituting into Gnp(s), the damping offset for the NS PLL mode can be obtained as:
(30) |
where ; and .
It can be observed from (30) that because , i.e., the dynamic coupling between the dual-sequence PLLs will provide negative damping of the NS PLL mode. Owing to the symmetry of the PS and NS PLLs, the dynamic coupling between the dual-sequence PLLs will provide positive damping of the PS PLL mode. The damping offset of the PS PLL mode caused by dynamic coupling is expressed as:
(31) |
It can be observed from (31) that because .
Similar to the frequency offset, the magnitude of the damping offset is related to kpnknp. The magnitude of kpnknp is related to the coupling impedances and and the PS and NS reactive currents and . The more serious the asymmetrical faults, the more obvious the inter-sequence coupling impedance [
The above small-signal state-space model will be used for a modal analysis to obtain the dynamic stability of the dominant poles of the system during the steady-state stage of an asymmetric fault: the poles dominated by the PS and NS PLL modes.

Fig. 8 Movement trajectories of dominant poles of PLLs as δ varies from 0.41 to 0.63.
δ | Dynamic coupling | Dominant mode | Eigenvalue | Damping ratio | Frequency (Hz) |
---|---|---|---|---|---|
0.55 | Decoupled | PS PLL | 0.040 | 30.67 | |
Coupled | 0.068 | 32.35 | |||
Decoupled | NS PLL | 0.038 | 24.54 | ||
Coupled | -0.001 | 23.30 | |||
0.63 | Decoupled | PS PLL | 0.038 | 29.16 | |
Coupled | 0.070 | 31.37 | |||
Decoupled | NS PLL | 0.039 | 24.84 | ||
Coupled | -0.010 | 23.16 |
As can be observed from
From the perspective of vibration mechanics, greater dynamic coupling between the dual-sequence PLLs will increase the nondiagonal elements of D, and D will change from a positive-definite matrix to a non-positive-definite one, which will affect the dynamic stability of the system during the steady-state stage of a fault with asymmetric LVRT.
According to the above analysis, the adoption of the dual-sequence PLL structure has changed the form of the dynamic stability of the system during the steady-state stage of a fault when there is an asymmetric voltage dip. On one hand, the dynamic coupling between the dual-sequence PLLs has introduced negative and positive damping to the NS and PS PLL modes, respectively. On the other hand, the dynamic coupling between the dual-sequence PLLs becomes stronger as the severity of the asymmetric fault increases. Consequently, this dynamic coupling effect due to the dual-sequence PLL structure changes the dominant mode of the instability of the system.
There are two control strategies proposed in the existing studies to enhance the dynamic stability during the steady-state stage of asymmetric faults: the algorithm that optimizes the current control reference in [
Reference [

Fig. 9 Control structure of compensated control strategy in [

Fig. 10 Motion trajectories of dominant poles of PS and NS PLLs when compensated control strategy in [
From
When the dual-sequence PLL structure is applied, the improvement in the dynamic stability using the existing improved control strategies during asymmetric LVRT will be reduced or even invalidated. Hence, a more effective control strategy needs to be designed to improve the dynamic stability of systems that apply the dual-sequence PLL structure during asymmetric LVRT.
The negative damping of the NS PLL mode is the main cause of the dynamic instability of the system during the steady-state stage of asymmetric faults. Therefore, a method for enhancing the damping of the NS PLL mode is the key to enhancing the dynamic stability during the steady-state stage of an asymmetric voltage dip.
This paper proposes an improved NS PLL structure that negatively feeds the deviation between and back to the q-axis input voltage of the NS PLL through the PI controller. The control structure of improved NS PLL is shown in

Fig. 11 Control structure of improved NS PLL.
After adopting this improved NS PLL structure, the damping ratio of the NS PLL mode during asymmetric LVRT is revised as:
(32) |
From (32), the actual damping of the NS PLL mode during asymmetric LVRT can be adjusted according to the additional PI controller parameters kcp and kci.

Fig. 12 Damping ratio of NS PLL mode as kcp and kci change.

Fig. 13 Motion trajectories of dominant poles of dual-sequence PLLs after adopting improved PLL structure as δ varies from 0.41 to 0.63.
The improved control strategy proposed in this paper is physically equivalent to the adjustable damping shock absorber widely used in engineering mechanics (such as vehicle engineering), as shown in

Fig. 14 Adjustable damping shock absorber applied to an automobile. (a) Coupling between vehicle body and tire. (b) Equivalent 2-DOF system in vehicle. (c) DFIG-based WEGS with proposed novel PLL structure.
While driving a vehicle, the small-signal disturbances are inevitable owing to uneven ground. The tires and vehicle body are equivalent to a coupled 2-DOF system. The adjustable damping shock absorber can adjust the coupled damping between the tires and the vehicle body according to the magnitude of the disturbance, thereby reducing the vibration of the vehicle body and improving the driving performance and the comfort of passengers [
When an asymmetric voltage dip occurs in the grid, the DFIG-based WEGS activates the NS control system and injects NS current into the faulty grid. The PS and NS PLL modes of the system during the steady-state stage of an asymmetric fault are similar to a coupled 2-DOF mass damping spring system, similar to the vehicle body and tires. The severity of the asymmetric fault is analogous to the magnitude of the disturbance on the road while driving a vehicle. The coupled damping and coupling springs between the PS and NS PLL modes are related to the magnitude of the disturbance (severity of the fault). When the system enters the steady-state stage of an asymmetric fault after the transient process ends, the proposed adjustable damping shock absorber (the novel PLL structure) is activated. The parameters of the adjustable damper are set according to the magnitude of the disturbance (severity of the asymmetric fault).
When a fault is detected, the fault type is first determined. The fault type, fault location, and fault resistance can be obtained through the fault detection technology, which has been studied in [
The above analysis and the effectiveness of the novel PLL structure for improving the dynamic stability during asymmetric LVRT are verified by simulation using MATLAB/Simulink. In the simulation model, the DFIG-based WEGS is connected to the power grid through step-up transformers. An asymmetric ground fault is triggered at the fault point.
The simulation results as the severity of the asymmetric fault changes without applying any improved control are shown in Figs.

Fig. 15 Simulation results when without any improved control. (a) . (b) . (c) . (d) . (e) . (d) THD.

Fig. 16 Simulation results when δ=0.63 without any improved control. (a) . (b) . (c) . (d) . (e) . (d) THD.
A small-signal instability phenomenon occurs during the steady-state stage of an asymmetric voltage dip without any improved control strategy. The stator-side current Isabc and voltage , rotor-side current Irabc, DC-side voltage Udc, and GSC current Icabc all have oscillations. The total harmonic distortion (THD) of Usa during asymmetric LVRT are 17.73% and 25.54% when and , respectively. Comparing the simulation results in Figs.
Figures

Fig. 17 Simulation results when with compensated control strategy in [

Fig. 18 Simulation results when with novel PLL structure proposed in this paper. (a) . (b) . (c) . (d) . (e) . (f) THD.
Figures

Fig. 19 Simulation results when with compensated control strategy in [

Fig. 20 Simulation results when with novel PLL structure proposed in this paper. (a) . (b) . (c) . (d) . (e) . (f) THD.
According to the analysis results in
Figures

Fig. 21 Simulation results under phase-to-phase short-circuit fault without improved control strategy. (a) . (b) . (c) . (d) . (e) . (f) THD.

Fig. 22 Simulation results under phase-to-phase short-circuit fault with proposed novel PLL structure. (a) . (b) . (c) . (d) . (e) . (f) THD.
The control strategies for improving the dynamic stability during asymmetric LVRT proposed in the existing studies have limited effect for the systems adopting dual-sequence PLLs. In particular, as the severity of the asymmetric fault increases, the enhancement in the dynamic stability provided by the existing improved control strategies will decrease or become invalidated. In contrast, the novel PLL structure proposed in this paper can improve the dynamic stability of the systems adopting dual-sequence PLLs during asymmetric LVRT, regardless of the severity of the asymmetric fault.
In this paper, the experimental platform shown in

Fig. 23 Structure of experimental platform.

Fig. 24 Experimental results under a two-phase asymmetrical voltage dip when . (a) , , , without improved control strategy. (b) , , , with novel PLL structure. (c) FFT analysis of Usab without improved control strategy. (d) FFT analysis of Usab with novel PLL structure.

Fig. 25 Experiment results under an asymmetric voltage dip when . (a) , , , without improved control strategy. (b) , , , with novel PLL structure. (c) FFT analysis of Usab without improved control strategy. (d) FFT analysis of Usab with novel PLL structure.
A comparison of the experimental results in Fig.
The influence of the new dynamic coupling effect introduced by dual-sequence PLLs (the dynamic coupling between the PS and NS PLLs) on the dynamic stability of DFIG-based WEGSs during the steady-state stage of an asymmetric fault is studied. The conclusions are as follows.
1) The dynamic coupling between the dual-sequence PLLs will increase the frequency of the PS PLL mode and decrease the frequency of the NS PLL mode. Moreover, the damping of the PS PLL mode will increase, and the damping of the NS PLL mode will decrease. Further, the dynamic coupling between the dual-sequence PLLs will increase as the severity of the asymmetric fault increases, and the damping offset will increase. Hence, this dynamic coupling will change the mode of instability and introduce a risk of dynamic instability to the system.
2) The dynamic coupling introduced by the dual-sequence PLLs will decrease the effectiveness of the existing control strategies for enhancing the dynamic stability. An adjustable damping shock absorber from the field of engineering mechanics was used in this paper to design a novel PLL structure to enhance the dynamic stability of DFIG-based WEGSs that adopt dual-sequence PLLs during the steady-state stage of an asymmetric fault. Simulations and experiments verify the limitations of the existing control strategies and the effectiveness of the proposed novel PLL structure towards improving the dynamic stability during the steady-state stage of an asymmetric fault.
Nomenclature
Symbol | —— | Definition |
---|---|---|
A. | —— | Parameters and Variables |
ω, θ | —— | Angular velocity and angle |
I, U | —— | Current and voltage |
Pg, Qg | —— | Active power and reactive power |
R, L, C | —— | Resistance, inductance, and capacitance |
Kp1,2, Ki1,2 | —— | Proportional and integral coefficients in positive-sequence (PS) and negative-sequence (NS) phase-locked PLLs |
Kp3,4, Ki3,4 | —— | Proportional and integral coefficients in PS current control loop (CCL) of rotor-side converter (RSC) |
Kp5,6, Ki5,6 | —— | Proportional and integral coefficients in NS CCL in RSC |
Kp7,8, Ki7,8 | —— | Proportional and integral coefficients in PS CCL of grid-side converter (GSC) |
Kp9,10, Ki9,10 | —— | Proportional and integral coefficients in NS CCL in GSC |
Kp11, Ki11 | —— | Proportional and integral coefficients in DC voltage control loop |
Udc | —— | DC-side voltage |
Zg, Zg1, Zg2 | —— | Grid impedances |
B. | —— | Subscripts and Superscripts |
α, β | —— | Components in αβ coordinates |
+, - | —— | PS and NS components |
0 | —— | Initial value |
a, b, c | —— | Phases a, b, and c |
d, q | —— | Components in synchronous dq coordinates |
f | —— | Components in faulted branch |
pll | —— | PLL-detected components |
ref | —— | Reference value |
s, r, c, g | —— | Stator, rotor, GSC, and grid sides |
p, n | —— | Components in PS and NS synchronous reference frames (SRFs) |
Appendix
Parameter | Value | Parameter | Value |
---|---|---|---|
Pbase | 2 MW | Ubase | 690 V |
Udc | 1200 V | Rs, Ls | 0.02, 2.9 p.u. |
Rr, Lr | 0.03, 2.95 p.u. | Lm | 2.8 p.u. |
NS PLL bandwidth | 25 Hz | PS PLL bandwidth | 29 Hz |
Rg, Lg | 0.1, 0.5 p.u. |
Parameter | Value | Parameter | Value |
---|---|---|---|
Pbase | 3 kW | Ubase | 220 V |
Udc | 400 V | Rs, Ls | 0.07, 1.3 p.u. |
Rr, Lr | 0.1, 1.35 p.u. | Lm | 1.26 p.u. |
Rg, Lg | 0.1, 0.42 p.u. | Synchronous speed | 1500 r/min |
PS PLL bandwidth | 29 Hz | NS PLL bandwidth | 25 Hz |
References
Z. Xie, X. Gao, S. Yang et al., “Improved fractional-order damping method for voltage-controlled DFIG system under weak grid,” Journal of Modern Power Systems and Clean Energy, vol. 10, no. 6, pp. 1531-1541. [Baidu Scholar]
S. S. Bernal-Perez, S. Añó-Villalba, and R. Blasco-Gimenez, “Stability analysis of multi-terminal HVDC with diode rectifier connected off-shore wind power plants,” International Journal of Electrical Power & Energy Systems, vol. 124, p. 106231, Jan. 2021. [Baidu Scholar]
S. Behzadirafi and F. de León, “Closed-form determination of the impedance locus plot of fault current limiters: asymmetrical faults,” IEEE Transactions on Power Delivery, vol. 35, no. 2, pp. 754-762, Apr. 2020. [Baidu Scholar]
Y. Zhu and H. Peng, “Multiple random forests based intelligent location of single-phase grounding fault in power lines of DFIG-based wind farm,” Journal of Modern Power Systems and Clean Energy, vol. 10, no. 5, pp. 1152-1163, Sept. 2022. [Baidu Scholar]
H. Xu, Y. Zhang, Z. Li et al., “Reactive current constraints and coordinated control of DFIG’s RSC and GSC during asymmetric grid condition,” IEEE Access, vol. 8, pp. 184339-184349, Oct. 2020. [Baidu Scholar]
Z. Rafiee, R. Heydari, M. Rafiee et al., “Enhancement of the LVRT capability for DFIG-based wind farms based on short-circuit capacity,” IEEE Systems Journal, vol. 16, no. 2, pp. 3237-3248, Jun. 2022. [Baidu Scholar]
Y. Chang, I. Kocar, J. Hu et al., “Coordinated control of DFIG converters to comply with reactive current requirements in emerging grid codes,” Journal of Modern Power Systems and Clean Energy, vol. 10, no. 2, pp. 502-514, Mar. 2022. [Baidu Scholar]
C. Ding, Y. Chen, and T. Nie, “LVRT control strategy for asymmetric faults of DFIG based on improved MPCC method,” IEEE Access, vol. 9, pp. 165207-165218, Dec. 2021. [Baidu Scholar]
G. Pannell, D. J. Atkinson, and B. Zahawi, “Minimum-threshold crowbar for a fault-ride-through grid-code-compliant DFIG wind turbine,” IEEE Transactions on Energy Conversion, vol. 25, no. 3, pp. 750-759, Sept. 2010. [Baidu Scholar]
S. Mortazavian and Y. A. I. Mohamed, “Dynamic analysis and improved LVRT performance of multiple DG units equipped with grid-support functions under unbalanced faults and weak grid conditions,” IEEE Transactions on Power Electronics, vol. 33, no. 10, pp. 9017-9032, Oct. 2018. [Baidu Scholar]
J. Hu, Q. Hu, B. Wang et al., “Small signal instability of PLL-synchronized Type-4 wind turbines connected to high-impedance AC grid during LVRT,” IEEE Transactions on Energy Conversion, vol. 31, no. 4, pp. 1676-1687, Dec. 2016. [Baidu Scholar]
J. Hu, B. Wang, W. Wang et al., “Small signal dynamics of DFIG-based wind turbines during riding through symmetrical faults in weak AC grid,” IEEE Transactions on Energy Conversion, vol. 32, no. 2, pp. 720-730, Jun. 2017. [Baidu Scholar]
B. Wang, S. Wang, and J. Hu, “Dynamic modeling of asymmetrical-faulted grid by decomposing coupled sequences via complex vector,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 9, no. 2, pp. 2452-2464, Apr. 2021. [Baidu Scholar]
L. Guan and J. Yao, “Small-signal stability analysis of weak-grid-connected DFIG-based WT during asymmetric faults,” in Proceedings of 2021 International Conference on Power System Technology (POWERCON), Haikou, China, Dec. 2021, pp. 1601-1605. [Baidu Scholar]
X. Fang, J. Yao, R. Liu et al., “Small-signal stability analysis and current control reference optimization algorithm of DFIG-based WT during asymmetric grid faults,” IEEE Transactions on Power Electronics, vol. 36, no. 7, pp. 7750-7768, Jul. 2021. [Baidu Scholar]
X. He, H. Geng, R. Li et al., “Transient stability analysis and enhancement of renewable energy conversion system during LVRT,” IEEE Transactions on Sustainable Energy, vol. 11, no. 3, pp. 1612-1623, Jul. 2020. [Baidu Scholar]
L. B. Larumbe, Z. Qin, L. Wang et al., “Impedance modeling for three-phase inverters with double synchronous reference frame current controller in the presence of imbalance,” IEEE Transactions on Power Electronics, vol. 37, no. 2, pp. 1461-1475, Feb. 2022. [Baidu Scholar]
L. Guan, J. Yao, R. Liu et al., “Small-signal stability analysis and enhanced control strategy for VSC system during weak-grid asymmetric faults,” IEEE Transactions on Sustainable Energy, vol. 12, no. 4, pp. 2074-2085, Oct. 2021. [Baidu Scholar]
D. Wang, L. Liang, L. Shi et al., “Analysis of modal resonance between PLL and DC-link voltage control in weak-grid tied VSCs,” IEEE Transactions on Power Systems, vol. 34, no. 2, pp. 1127-1138, Mar. 2019. [Baidu Scholar]
M. G. Taul, X. Wang, P. Davari et al., “Current reference generation based on next-generation grid code requirements of grid-tied converters during asymmetrical faults,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 8, no. 4, pp. 3784-3797, Dec. 2020. [Baidu Scholar]
H. Pan, Z. Li, and T. Wei, “A novel phase-locked loop with improved-dual adaptive notch filter and multi-variable filter,” IEEE Access, vol. 7, pp. 176578-176586, Dec. 2019. [Baidu Scholar]
S. Das and B. Singh, “Enhanced control of DFIG based wind energy conversion system under unbalanced grid voltages using mixed generalized integrator,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 3, no. 2, pp. 308-320, Apr. 2022. [Baidu Scholar]
M. Darabian and A. Bagheri, “Design of adaptive wide-area damping controller based on delay scheduling for improving small-signal oscillations,” International Journal of Electrical Power & Energy Systems, vol. 133, p. 107224, Dec. 2021. [Baidu Scholar]
R. E. D Bishop and D. C. Johnson, The Mechanics of Vibration, Cambridge: Cambridge University Press, 2011. [Baidu Scholar]
S. Tan, H. Geng, and G. Yang, “Phillips-Heffron model for current-controlled power electronic generation unit,” Journal of Modern Power Systems and Clean Energy, vol. 6, no. 3, pp. 582-594, May 2018. [Baidu Scholar]
P. Liu, D. Ning, L. Luo et al., “An electromagnetic variable inertance and damping seat suspension with controllable circuits,” IEEE Transactions on Industrial Electronics, vol. 69, no. 3, pp. 2811-2821, Mar. 2022. [Baidu Scholar]
C. Yang, L. Huang, H. Xin et al., “Placing grid-forming converters to enhance small signal stability of PLL-integrated power systems,” IEEE Transactions on Power Systems, vol. 36, no. 4, pp. 3563-3573, Jul. 2021. [Baidu Scholar]
M. Pignati, L. Zanni, P. Romano et al., “Fault detection and faulted line identification in active distribution networks using synchrophasors-based real-time state estimation,” IEEE Transactions on Power Delivery, vol. 32, no. 1, pp. 381-392, Feb. 2017. [Baidu Scholar]
C. A. Apostolopoulos, C. G. Arsoniadis, P. S. Georgilakis et al., “Unsynchronized measurements based fault location algorithm for active distribution systems without requiring source impedances,” IEEE Transactions on Power Delivery, vol. 37, no. 3, pp. 2071-2082, Jun. 2022. [Baidu Scholar]
K. Kalita, S. Anand, and S. K. Parida, “A closed form solution for line parameter-less fault location with unsynchronized measurements,” IEEE Transactions on Power Delivery, vol. 37, no. 3, pp. 1997-2006, Jun. 2022. [Baidu Scholar]