 Original research
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Drivetrain torsional vibration suppression of large scale PMSGbased WECS
Protection and Control of Modern Power Systems volume 7, Article number: 37 (2022)
Abstract
This paper provides a systematic analysis of the large scale PMSG (permanent magnet synchronous generator)based WECS (wind energy conversion system) torsional vibration problem under MPPT (maximum power point tracking) control and constant power control. This is from the perspective of SSO (subsynchronous oscillation), SSH (subsynchronous harmonics) and forced torsional vibration. The cause of SSO is the negative total system damping, weakened by the constant power control. The system is susceptible to inducing SSH in the grid current and voltage in the underdamped condition. To effectively suppress the torsional vibration of PMSGbased WECS, a stiffness compensation control strategy based on adaptive damping is proposed. The results show that SSO, SSH and the forced torsional vibration can be suppressed at the source using the proposed suppression strategy.
Introduction
Torsional vibration can cause severe damage to a drivetrain and can also lead to problems of SSO (subsynchronous oscillation) and SSH (subsynchronous harmonics) in a power system. In December 1970 and October 1971, two serious torsional vibration accidents occurred in the steam turbine units’ shaft in the Mohave power plant in the United States, causing damage to the generator shaft [1]. The cause of the two accidents was SSR (subsynchronous resonance) produced by the interaction between the torsional vibration of the steam turbine shaft and the series compensation capacitance of the transmission system. Subsequently, in 1977, it was found that an HVDC transmission system could also cause strong steam turbine torsional vibration [2, 3]. Compared with the torsional vibration caused by series compensation capacitance, the torsional vibration caused by HVDC cannot form a resonant loop. Therefore, the mechanisms of these two torsional vibrations are different, though they have been collectively referred to as SSO. Compared with SSR, SSO has a broader meaning and there have been many forms of SSO, although SSO involved in torsional vibration is still the major form. It can be seen that torsional vibration is a historical issue, and the early concern of SSO was caused by steam turbine unit torsional vibration.
In recent years, the development of renewable energy, e.g., wind energy, has made great progress [4, 5] and wind turbines are becoming larger. However, with larger wind turbine size and capacity, the torsional vibration problem is becoming increasingly prominent.
Harm of torsional vibration
Torsional vibration can seriously harm equipment and the connected power system as follows:

1)
Torsional vibration can aggravate the fatigue damage of a shaft. From the mechanics of materials, when torsional vibration occurs on a shaft, it is equivalent to imposing an alternating periodic stress on the shaft. Both the amplitude and frequency of the stress can affect the fatigue life of the shaft. The torsional vibration of highspeed PMSGbased WECS (Wind Energy Conversion System) is very serious, because its drivetrain contains some flexible components such as gearbox and flexible coupling. In many field accidents, gearbox damage and coupling skidding account for a large proportion of the damage.

2)
Torsional vibration can cause SSO and potentially make the system lose stability. As in a steam turbine unit, a DFIG (doubly fed induction generator)based WECS also adopts a direct grid connection mode. Therefore, the SSO mechanisms of DFIGbased WECS and a steam turbine unit are similar. There is currently a lot of research on the SSO problem of DFIGbased WECS [6,7,8,9]. Because a PMSGbased WECS uses an inverter gridconnected mode it removes the direct connection between generators and power grid. Therefore, the SSO problem of PMSGbased WECS may depend only on the generator control mode. There has also been research showing that power control can weaken system damping and even induce SSO in a PMSGbased WECS.

3)
Torsional vibration is the main source of SSH. In fact, for power systems containing wind power generation, it is worth noting that all disturbances, including wind, can cause a WECS to produce torsional vibrations in the subsynchronous frequency range. Lowintensity torsional vibration may only have a small impact on the unit itself, but it may still cause subsynchronous frequency oscillation of electrical parameters such as grid current or voltage. IEEE generally uses "subsynchronous frequency components" or "SSH" to describe this electrical oscillation. As the wind power penetration increases, a large amount of SSH is transmitted between the power grid sources. This may cause other forms of SSO. On July 1, 2015, a wind farm in Hami, Xinjiang, China produced a large amount of SSH. After 5 levels of transformation, the SSH was transmitted to the thermal power unit 300 km away and triggered the torsional vibration protection action of three sets of the 660 MW units of the garden power plant. The units were successively tripped and the power plant was completely shut down. On examination, there was a large directdriven wind farm connected to the adjacent power grid. Many SSH close to the natural frequencies of the thermal power units shafts were generated by the PMSGbased WECS, and these were transferred to the thermal power plant through the grid and subsequently caused the severe torsional vibration of the thermal power units. There have been some similar accidents in Texas, USA and Guyuan County in Hebei Province, China. The above accidents indicate that the monitoring of SSH in the power grid has been insufficient and generally SSH in wind farms has been neglected.
Causes and characteristics of torsional vibration
The causes and characteristics of torsional vibration are shown in Fig. 1. As shown, torsional vibration is mainly caused by wind disturbance and the influence of control on system damping. As the external disturbance source, persistent wind disturbance can excite the system and cause forced torsional vibration. The frequency of this forced torsional vibration depends on the wind disturbance frequency, while tower shadow effect, wind shear and turbulence should also be taken into account. The spectrum of wind disturbance is wide, and the frequency of wind disturbance is lower than the drivetrain natural frequency. In addition, the control function may change the system damping. If the total damping of the system becomes negative, the system will exhibit unstable torsional oscillation, indicating SSO in the system. Moreover, the system is susceptible to external disturbances in underdamped conditions. This can induce torsional vibration near the drivetrain natural frequency, while the underdamped torsional vibration can cause SSH. It is clear that the frequencies of both SSO and SSH are close to the drivetrain natural frequency and the spectrum of oscillations is concentrated.
Research status of torsional vibration suppression
For torsional vibration suppression, damping control is considered an effective strategy. Steam turbine units and DFIGbased WECS can usually add active damping control to excitation links to improve system damping. However, PMSGbased WECS cannot perform excitation control because of its permanent magnet rotor. For this reason, a torsional vibration suppression strategy was proposed in [10, 11]. This added active damping control to the generator torque control loop, and this method has been widely used in practice. The method needs to extract the torsional vibration information from the generator speed by a BPF (bandpass filter), and then multiply it by a magnification factor (or damping coefficient) as the damping compensation torque into the torque control loop [10,11,12,13,14,15,16,17]. However, the effectiveness of the BPF is affected by parameter uncertainties. In order to restore the performance of the damper, it is usually necessary to repeatedly adjust the parameters during the testing phase of the wind turbine to achieve a good matching effect. To overcome this problem, a damper based on an adaptive Kalman filter was proposed in [18], while a method for reconstructing DCLink current to extract torsional vibration signals was proposed in [19,20,21]. The method was further simplified in [22, 23], which also pointed out that the torsional vibration signal could be extracted from the fluctuation of the DCLink current. The effectiveness of the scheme was verified by experiment. In addition, there have been many studies on the torsional vibration suppression scheme of a WECS drivetrain. SMC (sliding mode control) was applied to the suppression of torsional vibration in [24] and a good suppression effect was obtained. Reference [25] proposed a virtual inertia control method to suppress drivetrain torsional vibration, while [26] proposed a SDDC (speed difference damping controller) to reduce the torsional vibration of the driveline. However, the methods in [24,25,26] were also based on BPF for extracting the torsional vibration information.
Existing problems of torsional vibration
Although SSO may occur in a PMSGbased WECS under power control, its mechanism is not clear. It is not conclusively determined whether PMSGbased WECS has the possibility to generate SSO under other control modes such MPPT control. Furthermore, the mechanism of SSH generated by torsional vibration also has not been understood. Thus, further research and discussion are required on effective suppression methods for forced torsional vibration caused by wind disturbance.
Main contributions and highlighted innovations
This paper focuses on the torsional vibration of a PMSGbased WECS in the two main control modes, i.e., MPPT control mode and constant power control mode, and provides an indepth analysis of SSO, SSH and forced torsional vibration problems. The main contributions and innovations are as follows:

1)
The mechanism of SSO is thoroughly explored. The study finds that power control can weaken system damping, and when the total damping becomes negative, SSO will occur. However, an MPPT strategy with OTC (optimal torque control) can increase the system damping and thus, SSO under MPPT control does not occur. Therefore, damping control must be adopted in the constant power mode. In addition, for conventional damping control, if the added damping is insufficient, the system will be in the underdamped state and SSH can be generated.

2)
The mechanism of SSH is also analyzed in detail. The study finds that conventional control can usually result in a underdamped system which is susceptible to external disturbances to produce SSH.

3)
A new stiffness compensation control strategy based on adaptive damping is proposed to reduce forced torsional vibration. Detailed analysis, design and simulation experiments are provided to verify the correctness of the strategy.
Mathematical model of PMSGbased WECS
A typical configuration of a generic 2 MW PMSGbased WECS is shown in Fig. 2. It mainly consists of a wind turbine, a flexible drivetrain and an electrical system including a PMSG, two backtoback connected VSCs (voltage source converters) comprised of a MSC (machine side converter), a DCLink and a GSC (grid side converter), and the control system. The system components are implemented in Simulink and the parameters are given in the Appendix.
Wind turbine model
According to the Betz theory, the mechanical power absorbed by a wind turbine [27,28,29,30,31] is given by:
where the subscript tur refers to wind turbine. ρ is air density, R is radius of wind turbine, v is wind speed, β is pitch angle, λ is TSR (tipspeed ratio), C_{P} is defined as the wind energy conversion coefficient. Usually, C_{P} is a function of β and λ, and λ is given by:
where ω_{tur} is rotor speed. From (1) and (2), the aerodynamic torque of wind turbine is:
Flexible drivetrain model
Because of the heavy load on a largescale wind turbine drivetrain, its resonance mode must be considered. In general, there are two methods for flexible drivetrain modeling:

1)
Continuous mass spring damping model. This model can accurately reflect the drivetrain dynamics, but the solution process is very complicated.

2)
Segmented concentrated mass spring damping model. This can be regarded as a simplified model. If the mass block is appropriately selected, the drivetrain dynamics can also be well reflected. Therefore, this method is widely used.
For a PMSGbased WECS, both the wind turbine and generator can be considered as a single mass block. Thus, its flexible drivetrain is usually described by the twomass spring damping model shown in Fig. 3.
The flexible drivetrain dynamic model [30] is expressed as:
where the subscript gen refers to generator. J_{tur} is wind turbine rotational inertia, J_{gen} is generator rotational inertia, ω_{gen} is generator speed, T_{gen} is electromagnetic torque, θ is drivetrain twist angle, K is drivetrain stiffness coefficient, D is the drivetrain damping.
Electrical system model
Figure 4 shows the control block of a full power converter. The purpose of MSC control is to realize the control of generator torque, while GSC control stabilizes the DCLink voltage U_{dc} at its reference value U_{dc_ref} and achieves the necessary reactive power support for grid.
As described in [32], the PMSG model and MSC dynamics can be regarded as a firstorder inertia link after innerloop current feedforward decoupling control and firstorder tuning for the innerloop current PI parameters, while the inertia delay of torque control can be adjusted by the PI parameters shown in Fig. 4(a). If the PI parameters are large enough, the dynamic delay of the torque loop can be neglected. Because MPPT control is generally used below rated wind speed v_{rate} and constant power control is adopted above v_{rate}, the electromagnetic torque T_{gen} is:
where T_{ref} is the torque command, P_{rate} is the rated power, C_{P_max} is the maximum C_{P} and λ_{opt} is OTSR (optimal tipspeed ratio). In this paper, C_{P_max} = 0.48 and λ_{opt} = 8.
Torsional vibration analyses
SSO problem discussion
As is wellknown, PMSGbased WECS torsional vibration is mainly caused by wind disturbance and generator control, and is less affected by the power grid because of its inverter gridconnected mode. Thus, its SSO problem may be limited to MSC control. By the assumptions of:
Equation (4) can be rewritten as:
where the superscript “ ~ ” and “—” refer to small signal values and steady state values, respectively. Equation (7a) characterizes the rigid motion mode while (7b) reflects the resonance mode. From (7b), the flexible drivetrain natural frequency is expressed as:
Under the drivetrain parameters considered in this paper, the drivetrain natural frequency is ω_{n} = 97.28 rad/s (f_{n} = 15.48 Hz). Clearly, the natural frequency falls into the frequency range of the SSO (10–50 Hz). Rewrite (3) and (5) as:
where a, b and c are given in Appendix, and \(\tilde{T}_{gen}\) is:
Usually, the inertia moment has the ability to suppress speed or frequency mutation. Due to J_{tur} > > J_{gen}, \(\tilde{\omega }_{tur} \ll \tilde{\omega }_{gen}\). This means that \(\tilde{\omega }_{tur}\) can be ignored when analyzing the torsional vibration. If ω_{tur} is used for the pitch control, \(\tilde{\beta }\) can also be neglected. Therefore, there are \(\dot{\tilde{\theta }} \approx  \tilde{\omega }_{gen}\), \(\tilde{T}_{tur} \approx c\tilde{v}\) and
The simplified small signal model is shown in Fig. 5. The small signal model includes the dynamics of wind turbine, drivetrain, generator and its control, but does not involve the power grid and GSC control (For DFIG, it may be necessary to consider the power grid impact). In order to verify the correctness of the simplified small signal model shown in Fig. 5, a comparison between the simplified model and simulation test is conducted with the variable wind speed v = 9 + sin(10t). From the results in Fig. 6, it proves the correctness of the simplified model as the output results of the simplified model are very close to the simulation outputs in steady state.
Damping has a certain physical meaning and is expressed as the ability to block certain movements, e.g., vibration or oscillation. As Fig. 5 shows, torque control can change the system damping. If \({{\partial T_{gen} } \mathord{\left/ {\vphantom {{\partial T_{gen} } {\partial \omega_{gen} }}} \right. \kern\nulldelimiterspace} {\partial \omega_{gen} }} > 0\), the total damping can be enhanced. On the other hand, system damping is weakened if \({{\partial T_{gen} } \mathord{\left/ {\vphantom {{\partial T_{gen} } {\partial \omega_{gen} }}} \right. \kern\nulldelimiterspace} {\partial \omega_{gen} }} < 0\). From (10), the MPPT strategy with OTC can increase system damping while constant power control can weaken system damping. Therefore, SSO can occur under constant power control when the total damping is negative. Thus, an additional active damping control strategy must be adopted under constant power control. The mechanism of damping control method is shown in Fig. 7.
As Fig. 8 illustrates, when the system controller switched from MPPT control mode to constant power control mode at 2.2 s, SSO occurrs without damping control and the main variables such as generator torque, DCLink voltage, generator speed and drivetrain twist angle oscillate. In contrast, after the damping control is used, SSO is completely suppressed. Figure 9 shows that the oscillation frequency (about 13.33 Hz) is close to the drivetrain natural frequency.
The system power curves are shown in Fig. 10. It is evident that gridconnected power oscillation can also occur under constant power control. However, the electromagnetic power does not change much and is stable at 2 MW. The difference between the electromagnetic power and actual gridconnected power reflects the system power loss (mainly copper loss).
SSH problem analysis
From Figs. 8–10, although the system tends to be stable after adding electrical damping, the system variables have convergent oscillations in the first few cycles. This means that the underdamped system is susceptible to disturbance (e.g., wind disturbance or controller switching). Figure 11 shows the system responses to sustained excitation of periodical wind disturbance. The wind disturbance excites the oscillation of the underdamped system in the form of pulsation. This oscillation presents two frequency characteristics including underdamped oscillation (13.33 Hz) and forced oscillation caused by wind disturbance (about 3.33 Hz). By slowly increasing damping to critical damping condition, the underdamped oscillation can be completely suppressed.
From the above analyses, it can be concluded that negative damping can cause SSO and the under damping may produce SSH. In the following sections, the SSH problem will be further elaborated. It is assumed that the system is underdamped, and the disturbance produced the torsional vibration is near the drivetrain natural frequency, as:
where A is oscillation amplitude and ω_{0} is oscillation frequency. In such a condition, the generator speed contains this frequency component, as:
As the electromagnetic power P_{gen} can be maintained at rated power P_{rate} (see Fig. 10), T_{gen} = P_{gen} /ω_{gen} = P_{rate} /ω_{gen}. From (10), (13), and the relationship between electromagnetic torque and qaxis stator current i_{sq}, there is:
where n_{p} is a generator pole logarithm and ψ is the permanent magnet flux magnitude. From the power flow in Fig. 2, there is:
where V_{gd} is the voltage at the gridconnected point. In general, the system loss mainly refers to generator copper loss, because the resistance of the coupling reactor on the gridside is very small (assumed to be zero in this paper). Thus, the grid connection loss can be neglected and P_{Out} = P_{grid}. As Fig. 4(a) shows, if the qaxis stator current i_{sq} is fully tracking its reference i_{sq_ref}, the qaxis stator voltage u_{sq} will not contain this frequency component. Thus, there is:
Three scenarios in gridside control are considered as follows:

The control function for the DCLink voltage is very strong, and the DCLink voltage does not fluctuate. From Fig. 4(b), when there is no fluctuation in the DCLink voltage, \(\tilde{i}_{gd} = 0\) and there is:
From (17), there is:
Equation (18) means that the voltages at the gridconnected point contain the same subsynchronous component as the torsional vibration frequency.

The capacitor completely filters out the active power fluctuation caused by the torsional vibration., i.e.:
$$\left\{ \begin{gathered} \tilde{P}_{Out} = 0 \hfill \\ \tilde{P}_{dc} \approx \overline{U}_{dc} \tilde{i}_{dc} + \tilde{U}_{dc} \overline{i}_{dc} \approx C\overline{U}_{dc} \dot{\tilde{U}}_{dc} = \tilde{P}_{In} = \frac{{2P_{rate} A\omega_{0} \overline{u}_{sq} }}{{3n_{p} \psi \omega^{2} }}\cos \omega_{0} t \hfill \\ \end{gathered} \right.$$(19)Thus
$$\tilde{U}_{dc} = \frac{{2P_{rate} A\overline{u}_{sq} }}{{3n_{p} \psi \omega^{2} C\overline{U}_{dc} }}\sin \omega_{0} t$$(20)From Fig. 4(b), if the lag phase of the innercurrent loop is neglected, the following expression can be obtained:
$$\tilde{i}_{gd} = \left( {K_{P} + \frac{{K_{I} }}{s}} \right)\tilde{U}_{dc} = \frac{{2P_{rate} A\overline{u}_{sq} }}{{3n_{p} \psi \omega^{2} C\overline{U}_{dc} }}\left( {K_{P} \sin \omega_{0} t  \frac{{K_{I} }}{{\omega_{0} }}\cos \omega_{0} t} \right)$$(21)Equation (21) indicates that the grid currents contain the same subsynchronous component as the torsional vibration frequency.
The final case is between case 1) and case 2), i.e., both DCLink voltage and gridconnected power fluctuate. The actual system conforms to the third situation (see Figs. 8–11), and both grid voltage and current contain SSH of this frequency.
Forced torsional vibration reduction
The study has found that increasing damping can suppress SSO and SSH, but has limited effect on suppressing the forced torsional vibration. Therefore, this paper proposes a new method by adding appropriate stiffness to reduce the forced torsional vibration. The method consists of superimposing the stiffness compensation control based on the original damping control shown in Fig. 12.
From Fig. 12, the closedloop transfer functions can be derived as:
Equation (22b) indicates that K_{s} can reduce the static gain and increase the system resonance frequency. By reducing the static gain, the forced torsional vibration caused by wind disturbance can be reduced. However, the system critical damping condition will also change:
The proposed control schematic to suppress forced torsional vibration is shown in Fig. 13. Because the system needs different injection damping at different working points, the damping gain K_{D} should be adjusted by (23).
Simulation results
In order to verify the effectiveness of the proposed algorithm in this paper, a detailed simulation test platform is developed in MATLAB/SIMULINK SimPowerSysterms environment shown in Fig. 14. The transient response of PMSGbased WECS can be obtained with this simulation test platform. In this section, the focus is on the results obtained from the case study of a 2 MW PMSGbased WECS and the injected stiffness coefficient meeting K_{s} = 4 K.
The simulation of this section consists of two tests. The first test is carried out below the rated wind speed and the wind speed curve is shown in Fig. 15(a). Thus it aims to test torsional vibration suppression under MPPT control. The system responses in these conditions are illustrated in Fig. 15. It is evident that both the conventional damping control and proposed control can effectively suppress SSH. However, compared with the conventional damping control, the proposed control method can further reduce the forced torsional vibration caused by wind disturbance, as described below.

1)
Figures 15(b–d) present the curves of generator torque, DCLink voltage and drivetrain twist angle. Their fluctuations under the presented control strategy have clearly been further reduced. As is well known, drivetrain twist angle fluctuation can severely damage the drivetrain, and its multiplication with the stiffness coefficient has been used in many studies to characterize the shaft resonant loading. Thus, the reduction of twist angle fluctuation can help to reduce the fatigue of the drivetrain.

2)
Figure 15(e) shows the system power curve. As seen, the proposed control method also has the ability to reduce the fluctuations of electromagnetic power and grid connected power.

3)
Although the proposed control increases speed fluctuation, the generator speed becomes closer to the rotor speed as shown in Fig. 15(f). In addition, the MPPT controller can provide positive damping to the torsional vibration. From (23), when the generator is above 1.37 rad/s, the damping gain K_{D} will be zero. In fact, the system only performs stiffness compensation under MPPT control and the compensation torque is shown in Fig. 15(g).
The second test is implemented above the rated wind speed and the wind speed curve is shown in Fig. 16(a), while the system responses are presented in Figs. 16(b)(g). From Figs. 16(b)(g), the proposed control strategy is also able to reduce the forced torsional vibration under constant power control and has low fluctuations of generator torque, DCLink voltage, drivetrain twist angle and system power.
Conclusions
In this paper, the torsional vibration problem using MPPT control and constant power control has been studied. The hazards, causes, characteristics, conventional damping control methods and existing problems of torsional vibration are discussed. The model of the whole system and basic control of a wind turbine is presented, while the mechanisms of SSO, SSH and forced torsional vibration by wind disturbance are analyzed. A stiffness compensation control method based on adaptive damping is presented to further mitigate the forced torsional vibration. The conclusions are as follows:

1)
An MPPT controller can increase system damping. However, the power controller can weaken system damping and if the total damping becomes negative, SSO will occur.

2)
Both MPPT control at low rotor speed ( or low wind speed) and power control with insufficient damping can put the system into an underdamped state, while an underdamped system is susceptible to external disturbances to produce SSH.

3)
It is found that the conventional damping control has little effect on the forced torsional vibration, but appropriate addition of stiffness can mitigate the forced torsional vibration. Therefore, the proposed new stiffness compensation control strategy based on adaptive damping can suppress forced torsional vibration.
Availability of data and materials
The data used to support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
Thanks to the editorial department for timely feedback, the senior duty editor and the assistant editor for handling the paper and the reviewers for the valuable comments.
Funding
This work has been partially supported by Shaanxi Provincial Department of Education Project (17JK0691).
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Authors and Affiliations
Contributions
The author systematically studied and analyzed the torsional vibration problem of PMSGbased WECS and proposed a new way to suppress the torsional vibration. The author read and approved by the final manuscript.
Authors' information
Feihang Zhou (198910), male, PHD. He graduated from Xi'an University of Technology in China and received the PHD degree. He is a lecturer of Xi`an University of Posts & Telecommunications now. His research focuses on the suppression of wind turbine vibration.
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Appendix
Appendix
PMSGbased WECS parameters: Wind turbine inertia moment J_{tur} is 2 × 10^{4} kg·m^{2}; Air density ρ is 1.225 kg/m^{3}; Blade length R is 31 m; Rated wind speed v_{rate} is 13.1; Generator inertia moment J_{gen} is 700 kg·m^{2}; Rated power P_{rate} is 2 MW; Rated torque T_{rate} is 4 × 10^{2} kN·m; Rate rotor speed ω_{rate} is 5 rad/s; Generator pole logarithm n_{p} is 102; Permanent flux φ is 1.25 Wb; Statorresistance is 0.11 Ω; Statorreluctance L is 8.35 mH; DCLink voltage U_{dc} 1800 V; Gridside filter inductor L_{c} is 6 mH; Drivetrain stiffness K is 6.4 × 10^{6} N·m·s/rad; Drivetrain damping D is 1.58 × 10^{5}.
Aerodynamic factors a, b, c are as follows:
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Zhou, F. Drivetrain torsional vibration suppression of large scale PMSGbased WECS. Prot Control Mod Power Syst 7, 37 (2022). https://doi.org/10.1186/s41601022002578
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DOI: https://doi.org/10.1186/s41601022002578
Keywords
 Adaptive damping
 Constant power control
 MPPT
 PMSGbased WECS
 SSH
 SSO
 Stiffness compensation
 Torsional vibration