Torsional Oscillation Damping Analysis and Suppression Strategy for PMSG-based Wind Generation System

Yizhuo Ma; Graduate; Jin Xu; Guojie Li; Keyou Wang

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Torsional Oscillation Damping Analysis and Suppression Strategy for PMSG-based Wind Generation System PDF

- ORCID：
Yizhuo Ma
✉
- ORCID：
Graduate (Student Member, IEEE)
- ORCID：
Jin Xu
✉
- ORCID：
Guojie Li (Senior Member, IEEE)
✉
- ORCID：
Keyou Wang (Member, IEEE)
✉

Key Laboratory of Control of Power Transmission and Conversion Ministry of Education, Shanghai Jiao Tong University, Shanghai 200240, China

Updated：2025-03-26

DOI：10.35833/MPCE.2024.000219

OUTLINE

Abstract

External disturbances can induce torsional oscillation with weak damping in the shaft system of permanent magnet synchronous generators (PMSGs) based wind generation system, thereby inducing low-frequency oscillations. However, the influence of electromagnetic torque on the shaft system damping and corresponding parameter laws have been scarcely explored. We define the electrical damping coefficient as a quantitative measure for the influence of electromagnetic torque on the shaft system damping. The torsional oscillation damping characteristics of the shaft system under vector control are analyzed, and the transfer function for electromagnetic torque and speed is derived. Additionally, we elucidate the mechanism by which the electromagnetic torque influences the shaft system damping. Simultaneously, laws describing the influence of wind speed, system parameters, and control parameters on the torsional oscillation damping are analyzed. Accordingly, the optimal damping angle of the shaft system a torsional oscillation suppression strategy is proposed to compensate for with uncertainty in the parameters affecting damping. The studied system is modeled using MATLAB/Simulink, and the simulation results validate the effectiveness of the theoretical analysis and proposed torsional oscillation suppression strategy.

Keywords

Permanent magnet synchronous generator (PMSG); wind generation system; torsional oscillation; shaft system model; damping; oscillation suppression

I. Introduction

MANY countries have recently established carbon neutrality targets. In line with these targets, the installed wind power capacity is projected to grow by 430 GW from 2022 to 2025, reflecting an upward trend in wind power adoption [

1], [2]. The permanent magnet synchronous generator (PMSG) based wind generation system has gained popularity owing to benefits such as gearbox-free operation and high energy conversion efficiency and performance [3], [4]. The integration of large-scale grid-connected wind power systems poses various challenges to power systems, including power quality, voltage and frequency control, security, and stability [5]. Therefore, the dynamic characteristics of PMSG-based wind generation systems must be examined along with their interactions with power grids [6]. Furthermore, analyzing and understanding the dynamic characteristics of the shaft system in PMSG-based wind generation systems can notably enhance the safe and stable operation of power systems [7].

Compared with the doubly-fed induction generator, the shaft system of PMSG is simpler and consists of only three main components: wind turbine (WT), low-speed transmission shaft, and generator. These components are directly connected without a gearbox through a low-speed transmission [

8], [9]. Owing to its multipole structure, the shaft system of PMSG is more flexible than that in a conventional power plant [10]. In addition, the PMSG has more pole pairs in addition to the flexible shaft system [11].

Although a single-mass model may explain the transient instability in a PMSG under drastic disturbances, a double-mass model is required to accurately represent the system dynamics [

12]. This is because the double-mass model accurately represents the flexibility and mechanical oscillations of the shaft system.

Under severe external disturbances such as short-circuit faults, the shaft system in a PMSG-based wind generation system experiences torsional oscillation, resulting in power oscillation in the grid-connected wind power system [

11]. Torsional oscillations not only induce fatigue in the transmission shaft, thereby reducing its service life, but also give rise to low-frequency oscillations, potentially compromising the safety and stability of the entire PMSG-based wind generation systems [12]. Hence, the torsional oscillations in PMSG-based wind generation systems must be investigated [13].

The torsional oscillation characteristics of PMSG-based wind generation systems are influenced by three factors: input mechanical torque, inherent torque of the shaft system, and input electromagnetic torque under external disturbances [

14]. The former two factors are determined by the intrinsic system characteristics, including the generator rotor [15]. The effect of the electromagnetic torque on the torsional oscillation and its potential for oscillation suppression have been analyzed in [16] and [17]. However, a research gap persists concerning the specific parameters and underlying laws that govern the influence of the electromagnetic torque on torsional oscillation characteristics. Therefore, the damping characteristics of PMSG transmission shafting should be unveiled, and its influence mechanism should be understood. This study provides insights into the torsional oscillation characteristics of PMSG-based wind generation systems. Bridging this research gap may have practical implications for designing and optimizing torsional oscillation damping controllers.

To enhance the stability of PMSG-based wind generation systems, various active damping control methods have been devised to mitigate torsional oscillation [

18]-[20]. In [18], a simple real power control strategy is introduced based on the rapid torque control for a direct-drive wind energy conversion system employing a PMSG. This study also outlines the parameter tuning procedure for the proposed control strategy. In [19], a sensorless active damping control strategy for direct-driven permanent magnet WT generators uses cascade observers to estimate the speeds of the WT and PMSG. In [20], the torsional oscillation damping is improved by adding a damping transfer function proportional to the speed difference between the WT and generator. Although previous studies have suggested methods to suppress torsional oscillations, to the best of our knowledge, few of them have thoroughly and quantitatively analyzed the mechanism of the influence of electromagnetic torque on the shaft system damping. In addition, an optimal damping angle compensation method has not been devised to improve the system stability.

We analyze the torsional oscillation damping characteristics under vector control using an electromagnetic damping torque method. The electrical damping coefficient is defined as a quantitative representation of the effect of the electromagnetic torque on the shaft system damping. Then, the transfer function for electromagnetic torque and speed is derived, and the influence of the electromagnetic torque on the shaft system damping is characterized. Further, the impact of wind speed, system parameters, and control parameters on torsional oscillation damping are examined in detail. Based on these findings, we propose a torsional oscillation suppression strategy for the active power control section of a machine-side converter (MSC). This strategy compensates for the damping angle of the shaft system and ensures the maximum damping.

II. Modeling and Control of PMSG-based Wind Generation System

The topology of a grid-connected PMSG-based wind generation system typically includes the shaft system, PMSG, back-to-back full-power converter, transformers, and the control sections of the grid-side converter (GSC) and MSC, as shown in Fig. 1, where MPPT is short for maximum power point tracking; PI is short for proportional and integral; PCC is short for point of common coupling; and PLL is short for phase-locked loop. This section presents a mathematical model of the considered system using the per-unit system.

Fig. 1 Typical topology of PMSG-based wind generation system.

A. PMSG Model

The PMSG is controlled in the d-q rotating coordinates with the d-axis aligned with magnetic flux linkage of PMSG rotor $ψ_{f}$ . The stator voltage of PMSG is given by [

21]:

\{\begin{array}{l} u_{s d} = - R_{s} i_{s d} - \frac{L_{d}}{ω_{e b}} \frac{d i_{s d}}{d t} + ω_{g} L_{q} i_{s q} \\ u_{s q} = - R_{s} i_{s q} - \frac{L_{d}}{ω_{e b}} \frac{d i_{s q}}{d t} - ω_{g} L_{d} i_{s d} + ω_{g} ψ_{f} \end{array}

(1)

where $u_{s d}$ and $u_{s q}$ are the d- and q-axis stator terminal voltages, respectively; $i_{s d}$ $a n d$ $i_{s q}$ are the d- and q-axis stator currents, respectively; $R_{s}$ is the stator resistance of PMSG; $ω_{e b}$ is the base value of stator angular frequency; $ω_{g}$ is the angular frequency of PMSG rotor; and $L_{d}$ and $L_{q}$ are the d- and q-axis self-inductances of PMSG stator, respectively.

The speed of a megawatt PMSG is relatively low, and most PMSG is nonsalient surface-mounted ( $L_{d} = L_{q}$ ). Therefore, the electromagnetic torque $T_{e}$ can be formulated as [

20]:

T_{e} = ψ_{f} i_{s q}

(2)

B. Double-mass Model

The double-mass model can be expressed as [

16]:

\{\begin{array}{l} 2 H_{t} \frac{d ω_{t}}{d t} = T_{m} - T_{s h} \\ 2 H_{g} \frac{d ω_{g}}{d t} = T_{s h} - T_{e} \\ \frac{d θ_{s h}}{d t} = ω_{e b} (ω_{t} - ω_{g}) \\ T_{s h} = K_{s h} θ_{s h} + D_{s h} (ω_{t} - ω_{g}) \end{array}

(3)

where $H_{t}$ and $H_{g}$ are the inertial time constants of WT and PMSG, respectively; $ω_{t}$ is the WT speed of generator rotor; $θ_{s h}$ is the torsion angle of WT relative to the generator rotor; $K_{s h}$ is the stiffness coefficient of shaft system; $D_{s h}$ is the damping coefficient of shaft system; and $T_{m}$ , $T_{e}$ , and $T_{s h}$ are the mechanical, electromagnetic, and shaft system torques, respectively.

C. MSC Model

The active power generated by PMSG $P_{e}$ is expressed as:

P_{e} = ψ_{f} i_{s q} ω_{g}

(4)

From (4), $T_{e}$ and $P_{e}$ can be accurately controlled by varying $i_{s q}$ . The active power control loop of MSC is shown in Fig. 2.

Fig. 2 Active power control loop of MSC.

In Fig. 2, the superscript * represents the reference value; K_p₁ and K_i₁ are the proportional and integral coefficients for the power outer loop of MSC, respectively; and K_p₂ and K_i₂ are the proportional and integral coefficients of the current inner loop on MSC, respectively. $P_{e}^{*}$ can be expressed as:

P_{e}^{*} = \{\begin{array}{l} k_{o p t} ω_{g}^{3} 0 < v < v_{r} \\ 1 v_{r} \leq v < v_{m a x} \end{array}

(5)

where $k_{o p t}$ is the MPPT curve coefficient; and $v$ , $v_{r}$ , and $v_{m a x}$ are the actual, rated, and maximum wind speeds, respectively.

Based on Fig. 2, the variable transfer of the active power control loop can be formulated as [

22]:

\{\begin{array}{l} i_{s q}^{*} = (P_{e}^{*} - P_{e}) (K_{p 1} + \frac{K_{i 1}}{s}) \\ u_{s q}^{} = - (i_{s q}^{*} - i_{s q}) (K_{p 2} + \frac{K_{i 2}}{s}) \\ i_{s q} = - \frac{u_{s q}^{}}{L_{q} s / ω_{e b} + R_{s}} \end{array}

(6)

III. Torsional Oscillation Damping Characteristics

This section presents the effect of electromagnetic torque $T_{e}$ on the torsional oscillation damping characteristics during MPPT. We derive the incremental transfer function of the electromagnetic torque and rotational speed difference, calculate the quantitative expression of electrical damping based on the electrical damping of synchronous generators (SGs), and analyze the effect of electrical damping on shaft system damping. Our findings provide a foundation for analyzing the influence of the PMSG parameters on the shaft system damping characteristics.

A. Electrical Damping Characteristics of PMSG

Using electromagnetic damping analysis, the linearized electromagnetic torque of PMSG $Δ T_{e}$ can be expressed as:

Δ T_{e} = D_{e} Δ ω_{Δ} + K_{e} Δ θ_{s h}

(7)

where $D_{e}$ and $K_{e}$ are the electrical damping and synchronization coefficients of the PMSG, respectively; and $Δ ω_{Δ}$ is the difference between $Δ ω_{g}$ and $Δ ω_{t}$ , i.e., $Δ ω_{Δ} = Δ ω_{t} - Δ ω_{g}$ .

We omit self-damping and the generator rotor. Using (3) and (7), the transfer function for torsional angle $θ_{s h}$ and mechanical torque $T_{m}$ is given by:

\frac{Δ θ_{s h}}{Δ T_{m}} = \frac{\frac{ω_{e b}}{2 H_{t}}}{s^{2} + (\frac{D_{s h}}{2 H_{t}} + \frac{D_{s h}}{2 H_{g}} - \frac{D_{e}}{2 H_{g}}) s + ω_{e b} (\frac{K_{s h}}{2 H_{t}} + \frac{K_{s h}}{2 H_{g}} - \frac{K_{e}}{2 H_{g}})}

(8)

The damping attenuation factor for torsional oscillation $ξ$ can be determined using (8) as:

ξ = \frac{D_{s h} (H_{t} + H_{g}) - D_{e} H_{t}}{4 H_{t} H_{g} ω_{o s c}}

(9)

where $ω_{o s c}$ is the natural oscillation angular frequency of the shaft system, and $ω_{o s c} = \sqrt[]{ω_{e b} [K_{s h} / (2 H_{t}) + (K_{s h} - K_{e}) / (2 H_{g})]}$ .

The electrical damping coefficient $D_{e}$ has a negative correlation with the torsional oscillation damping coefficient $ξ$ . When $D_{e} < 0$ , the phase difference between $Δ T_{e}$ and $Δ ω_{Δ}$ lies in $(90 °, 270 °$ ), resulting in an increase in $ξ$ . This in turn leads to a positive damping of the shaft system, enhancing its stability. When $D_{e} > 0$ , the phase difference between $Δ T_{e}$ and $Δ ω_{Δ}$ falls in $(0,90 °)$ , causing $ξ$ to decrease. Consequently, the shaft system experiences negative damping and thus instability.

B. Shaft System Damping Characteristics of PMSG

To obtain the small-signal output power of the PMSG, the MPPT curve can be approximated by a linear function around the steady-state operating point as:

Δ P_{e}^{*} = 3 k_{o p t} ω_{g 0}^{2} Δ ω_{g}

(10)

where $Δ P_{e}^{*}$ is the linearized increment of the reference generator output power; $Δ ω_{g}$ is the linearized increment of the generator speed; and $ω_{g 0}$ is the generator speed at the stable running point.

The linearized increment of reference electromagnetic torque on the MPPT curve can be formulated as:

Δ T_{e}^{*} = 2 k_{o p t} ω_{g 0}^{} Δ ω_{g}

(11)

By combining (2) and (4)-(6), the electromagnetic torque can be expressed as:

T_{e} = ψ_{f} i_{s q} = ψ_{f} \frac{k_{o p t} n_{1} n_{2} ω_{g}^{3} - n_{1} n_{2} T_{e} ω_{g}}{R_{s} + L_{q} s / ω_{e b} + n_{2}}

(12)

where $n_{1} = K_{p 1} + K_{i 1} / s$ ; and $n_{2} = K_{p 2} + K_{i 2} / s$ .

Linearizing both sides of the equation simultaneously allows to formulate the incremental transfer function for the linearized electromagnetic torque $Δ T_{e}$ and $Δ ω_{Δ}$ as (13).

\begin{array}{l} \frac{Δ T_{e}}{Δ ω_{Δ}} = - ψ_{f} \frac{H_{t}}{H_{t} + H_{g}} \frac{2 k_{o p t} ω_{g 0}^{2} [K_{p 1} K_{p 2} s^{2} +}{L_{q} s^{3} / ω_{e b} + (R_{s} + K_{p 2} + ω_{g 0} ψ_{f} K_{p 1} K_{p 2}) s^{2} +} \to \\ \leftarrow \frac{(K_{p 1} K_{i 2} + K_{i 1} K_{p 2}) s + K_{i 1} K_{i 2}]}{[K_{i 2} + ω_{g 0} ψ_{f} (K_{p 1} K_{i 2} + K_{i 1} K_{p 2})] s + ω_{g 0} ψ_{f} K_{i 1} K_{i 2}} \end{array}

(13)

Further, considering (13), we have (14).

\{\begin{array}{l} m = 2 k_{o p t} ω_{g 0}^{2} \frac{H_{t}}{H_{t} + H_{g}} \\ G (s) = - ψ_{f} \frac{K_{p 1} K_{p 2} s^{2} + (K_{p 1} K_{i 2} + K_{i 1} K_{p 2}) s +}{L_{q} s^{3} / ω_{e b} + (R_{s} + K_{p 2} + ω_{g 0} ψ_{f} K_{p 1} K_{p 2}) s^{2} +} \to \\ \leftarrow \frac{K_{i 1} K_{i 2}}{[K_{i 2} + ω_{g 0} ψ_{f} (K_{p 1} K_{i 2} + K_{i 1} K_{p 2})] s + ω_{g 0} ψ_{f} K_{i 1} K_{i 2}} \end{array}

(14)

The derivation process is detailed in Supplementary Material A.

The linearized electromagnetic torque can be expressed as $Δ T_{e} = m G (s) Δ ω_{Δ}$ .

According to (7) and (13), the real part of the incremental transfer function for $Δ T_{e}$ and $Δ ω_{Δ}$ corresponds to the electrical damping coefficient $D_{e}$ . Moreover, as indicated by (9), $D_{e}$ is the sole controllable parameter influencing shaft system damping. Hence, investigating the impact of system parameters on $D_{e}$ is essential for enhancing the damping.

IV. Influence of System Parameters on Torsional Oscillation Damping Characteristics

In this section, we examine the correlation between $D_{e}$ and various system parameters: $ω_{g 0}$ , $R_{s}$ , K_p₁, K_i₁, K_p₂, K_i₂, and $v$ . Specifically, we investigate the effects of wind speed $v$ , the PMSG parameters, and control parameters on $D_{e}$ and the oscillation characteristics of shaft system. Accordingly, we establish a theoretical basis for the proposed torsional oscillation suppression strategy outlined in Section V.

The main parameters of the PMSG-based wind generation system provided by China Wind Power Group Limited are listed in Table I. In addition, the control parameters are tuned using Simulink to achieve the optimal performance.

TABLE Ⅰ Main Parameters of PMSG-based Wind Generation System

Parameter	Value
Base power value $S_{b}$	2 MW
Base value of AC phase voltage $U_{b}$	575 V
Base value of DC-link voltage $U_{d c b}$	1150 V
Base value of stator angular frequency $ω_{e b}$	377 rad/s
Base value of PMSG rotor speed $ω_{m b}$	7.85 rad/s
Base value of grid angular frequency $ω_{b}$	377 rad/s
Rated frequency of PMSG $f_{n}$	60 Hz
Pole pair of PMSG $n_{p}$	48
Magnetic flux linkage of PMSG rotor $ψ_{f}$	1.18842 p.u.
d-axis self-inductance of PMSG stator $L_{d}$	0.5131 p.u.
q-axis self-inductance of PMSG stator $L_{q}$	0.5131 p.u.
Stator resistance of PMSG $R_{s}$	0.0001 p.u.
Inertia time constant of WT $H_{t}$	6.69 s
Inertia time constant of PMSG $H_{g}$	1 s
Damping coefficient of shaft system $D_{s h}$	1 p.u.
Stiffness coefficient of shaft system $K_{s h}$	1.6 p.u.
Proportional coefficient of power outer loop on MSC $K_{p 1}$	1 p.u.
Integral coefficient of power outer loop on MSC $K_{i 1}$	20 $s^{- 1}$
Proportional coefficient of current inner loop on MSC $K_{p 2}$	1 p.u.
Integral coefficient of current inner loop on MSC $K_{i 2}$	10 $s^{- 1}$
Capacitance of DC-link voltage $C$	0.6232 p.u.
Proportional coefficient of voltage outer loop on GSC $K_{p 3}$	5 p.u.
Integral coefficient of voltage outer loop on GSC $K_{i 3}$	300 $s^{- 1}$
Proportional coefficient of current inner loop on GSC $K_{p 4}$	0.83 p.u.
Integral coefficient of current inner loop on GSC $K_{i 4}$	5 $s^{- 1}$
Grid inductance $L_{c g}$	0.55 p.u.
Grid resistance $R_{c g}$	0.006 p.u.
Lowpass filter for $P_{e}$ $ω_{P}$	100 rad/s
PLL filter $ω_{f}$	100 rad/s
Proportional coefficient of PLL $K_{p, p l l}$	4.1 p.u
Integral coefficient of PLL $K_{i, p l l}$	200 $s^{- 1}$

A. Influence of Wind Speed and Stator Resistance on $D_{e}$

Figure 3 shows the Bode diagram of $Δ T_{e} / Δ ω_{Δ}$ with various $ω_{g 0}$ ranging from 0.6 to 1.0 p.u., where $θ_{o s c}$ is the phase angle of transfer function $G (s)$ at $ω_{s o c}$ . The phase lag of $Δ T_{e}$ with respect to $Δ ω_{Δ}$ falls between 90° and 180° near $ω_{o s c}$ .

Fig. 3 Bode diagram of $Δ T_{e} / Δ ω_{Δ}$ with various $ω_{g 0}$ .

Hence, $D_{e}$ is negative, resulting in positive damping provided by $Δ T_{e}$ for the shaft system, thus improving the system stability. The $D_{e}$ values corresponding to various $ω_{g 0}$ are presented in Table II. $D_{e}$ decreases as $ω_{g 0}$ increases, indicating an enhanced effect of the electromagnetic torque on the shaft system damping.

TABLE Ⅱ

D_{e}

Values Corresponding to Various

ω_{g 0}

$ω_{g 0}$ (p.u.)	$D_{e}$
0.6	-0.248
0.7	-0.334
0.8	-0.428
0.9	-0.527
1.0	-0.626

The variation range of the stator resistance should be moderated when investigating the impact of temperature changes on $R_{s}$ . The Bode diagram of $Δ T_{e} / Δ ω_{Δ}$ and the corresponding $D_{e}$ are shown in Fig. 4 and Table III, respectively, with various $R_{s}$ . The curves in the diagram are very similar, suggesting that changes in $R_{s}$ have a small influence on $D_{e}$ . Thus, the effect of $Δ T_{e}$ on the shaft system damping is negligible and does not affect the stability of the shaft system under MPPT.

Fig. 4 Bode diagram of $Δ T_{e} / Δ ω_{Δ}$ with various $R_{s}$ .

TABLE Ⅲ

D_{e}

Values Corresponding to Various

R_{s}

R_s (p.u.)	$D_{e}$ (p.u.)
0.0001	-0.6260
0.0003	-0.6260
0.0005	-0.6260
0.0007	-0.6260
0.0009	-0.6259

B. Influence of Control System Parameters on $D_{e}$

The active power control loop comprises two PI controllers and four control parameters: $K_{p 1}$ , $K_{i 1}$ , $K_{p 2}$ , and $K_{i 2}$ . We set $K_{p 1} = 1$ p.u., $K_{i 1} = 20 s^{- 1}$ , $K_{p 2} = 1$ p.u., and $K_{i 2} = 10 s^{- 1}$ as default control parameters. Subsequently, we select parameter within the ranges of 1-3 p.u. for $K_{p 1}$ , 5-25 $s^{- 1}$ for $K_{i 1}$ , 1-3 p.u. for $K_{p 2}$ , and 10-30 $s^{- 1}$ for $K_{i 2}$ .

Using (13), the Bode diagram of $Δ T_{e} / Δ ω_{Δ}$ with various $K_{p 1}$ ranging from 1 to 3 p.u. is shown in Fig. 5. The phase lag of $Δ T_{e}$ with respect to $Δ ω_{Δ}$ falls between 90° and 180° near $ω_{o s c}$ for $K_{p 1}$ of 1.0, 1.5, 2.0, 2.5, and 3.0 p.u.. In such cases, $D_{e}$ is negative, indicating that $T_{e}$ contributes to positive damping of the shaft system, thus enhancing its stability. When $K_{p 1} = 3.0$ p.u., the phase lag of $T_{e}$ with respect to $Δ ω_{Δ}$ is between 0° and 90° near $ω_{o s c}$ . In this case, the negative value of $D_{e}$ suggests that $T_{e}$ still contributes to positive damping of the shaft system, but its stability deteriorates. Table IV lists the $D_{e}$ values corresponding to various $K_{p 1}$ . As the PMSG operates under the MPPT, $D_{e}$ transitions from negative to positive as $K_{p 1}$ increases. This indicates a weakening effect of the electromagnetic torque on the shaft system damping, which is unfavorable to stability.

Fig. 5 Bode diagram of $Δ T_{e} / Δ ω_{Δ}$ with various $K_{p 1}$ .

TABLE Ⅳ D_e Values Corresponding to Various

K_{p 1}

$K_{p 1}$ (p.u.)	$D_{e}$ (p.u.)
1.0	-0.63
1.5	-0.51
2.0	-0.30
2.5	0.01
3.0	0.31

We also use (13) to obtain the Bode diagram of $Δ T_{e} / Δ ω_{Δ}$ with various $K_{i 1}$ ranging from 5 to 25 $s^{- 1}$ in steady states. As shown in Fig. 6, for $K_{i 1}$ of 5, 10, 15, 20, and 25 $s^{- 1}$ , the phase lag of $Δ T_{e}$ with respect to $Δ ω_{Δ}$ falls between 90° and 180° near $ω_{o s c}$ . In such cases, $D_{e}$ is negative, indicating that $T_{e}$ contributes to the positive damping of shaft systems, thereby enhancing its stability. The $D_{e}$ values corresponding to various $K_{i 1}$ are listed in Table V. When the PMSG operates under the MPPT, $D_{e}$ decreases as $K_{i 1}$ increases, thereby augmenting the damping effect of the electromagnetic torque on the shaft system and improving its stability.

Fig. 6 Bode diagram of $Δ T_{e} / Δ ω_{Δ}$ with $K_{i 1}$ .

TABLE V

D_{e}

Values Corresponding to Various

K_{i 1}

$K_{i 1}$ (s^-1)	$D_{e}$ (p.u.)
5	-0.36
10	-0.63
15	-0.96
20	-1.37
25	-1.91

The analysis regarding $K_{p 2}$ and $K_{i 2}$ and their impacts on the shaft stability is analogous to the analysis conducted for $K_{p 1}$ and $K_{i 1}$ , respectively, as given in Figs. 7 and 8 and Tables VI and VII, which is thus not discussed further in this subsection.

Fig. 7 Bode diagram of $Δ T_{e} / Δ ω_{Δ}$ with $K_{p 2}$ .

Fig. 8 Bode diagram of $Δ T_{e} / Δ ω_{Δ}$ with various $K_{i 2}$ .

TABLE VI

D_{e}

Values Corresponding to Various

K_{p 2}

$K_{p 2}$ (p.u.)	$D_{e}$ (p.u.)
1.0	-0.63
1.5	-0.41
2.0	-0.15
2.5	0.10
3.0	0.33

TABLE VII

D_{e}

Values Corresponding to Various

K_{i 2}

$K_{i 2}$ (s^-1)	$D_{e}$ (p.u.)
10	-0.41
15	-0.63
20	-0.88
25	-1.20
30	-1.61

V. Torsional Oscillation Suppression Strategy

We present a system comprising MPPT, MSC, and GSC control strategies. The MSC aims to regulate the output active power $P_{e}$ and the d-axis stator current.

An additional damping controller is introduced into the active power control loop to mitigate torsional oscillations in a PMSG-based wind generation system.

A. Proposed Torsional Oscillation Suppression Strategy

To enhance the effect of the electromagnetic torque on the shaft system damping, D_e should be reduced, as discussed in Section IV. Therefore, we introduce an additional damping controller into the active power control loop. Figure 9 shows the linearized model of the transfer relationship between $Δ T_{m}$ and $Δ ω_{Δ}$ , which is obtained using (1) and (13). The dotted line in Fig. 9 represents the addition of damping controller $H (s)$ . The transitive relation between $Δ T_{e}$ and $Δ ω_{Δ}$ can be derived as:

\begin{array}{l} Δ T_{e} = (m + H (s)) G (s) Δ ω_{Δ} = (m + H (j ω)) G (j ω) Δ ω_{Δ} = \\ m |G (j ω)| e^{j θ} Δ ω_{Δ} + |H (j ω)| |G (j ω)| e^{j θ + φ} Δ ω_{Δ} \end{array}

(15)

Fig. 9 Linearized model of transfer relationship between $Δ T_{m}$ and $Δ ω_{Δ}$ .

where $θ$ is the phase angle of transfer function $G (s)$ ; and $φ$ is the phase angle of transfer function $H (s)$ .

With the additional damping controller in the active power control loop, $D_{e}$ near $ω_{o s c}$ can be expressed as:

\{\begin{array}{l} D_{e} = D_{e 0} + D_{e H} \\ D_{e 0} = m \cdot R e (| G (j ω_{o s c}) | e^{j θ_{o s c}}) \\ D_{e H} = R e (| H (j ω_{o s c}) | | G (j ω_{o s c}) | e^{j φ_{o s c}}) \end{array}

(16)

where D_e₀ is the electrical damping coefficient without the additional damping controller; $D_{e H}$ is the additional electrical damping coefficient with the additional damping controller; and $φ_{o s c}$ is the phase angle of transfer function $H (s)$ at $ω_{o s c}$ .

From (16) and the expression of $G (s)$ , the system reaches stability, with both the amplitude and phase of $G (s)$ remaining stable once the operating conditions, control parameters, and system parameters are determined. Therefore, based on $G (s)$ , we can adjust the amplitude and phase of $| H (j ω_{o s c}) |$ to compensate for phase $φ_{o s c}$ by modifying additional damping controller $H (s)$ and the internal parameters. This adjustment aims to decrease $D_{e}$ , enhance the damping effect of $T_{e}$ on the shaft system, and improve the stability of the shaft system.

The additional damping controller $H (s)$ comprises two primary components: bandpass filter controller $H_{f} (s)$ and phase compensation controller $H_{i} (s)$ . The bandpass filter controller $H_{f} (s)$ employs a second-order bandpass filter expressed as:

H_{f} (s) = \frac{2 ξ_{f} ω_{n} s}{s^{2} + 2 ξ_{f} ω_{n} s + ω_{n}^{2}}

(17)

where $ξ_{f}$ is the damping ratio of $H_{f} (s)$ ; and $ω_{n}$ is the center angular frequency of $H_{f} (s)$ .

We set the damping ratio of $H_{f} (s)$ to be $ξ_{f} = 0.15$ . In addition, the center angular frequency of the bandpass filter, $ω_{n} =$ 12.748 rad/s, corresponds to the inherent oscillation angular frequency of the shaft system.

The phase compensation controller $H_{i} (s)$ enhances the controller performance, and its transfer function is given by:

H_{i} (s) = K_{t o d} {(\frac{1 + s T_{1}}{1 + s T_{2}})}^{2}

(18)

where K_tod is the damping controller gain; and T₁ and T₂ are the lead and lag correction time constants, respectively.

The additional damping controller H(s) is formulated as:

H (s) = H_{f} (s) H_{i} (s) = K_{t o d} {(\frac{1 + s T_{1}}{1 + s T_{2}})}^{2} \frac{2 ξ_{f} ω_{n} s}{s^{2} + 2 ξ_{f} ω_{n} s + ω_{n}^{2}}

(19)

Figure 10 shows a model of a PMSG-based wind farm (WF) connected to a four-machine two-area system. In this model, the WF comprises two PMSG-based wind generation systems with different parameters, which are equivalent to two 350 MW WFs, denoted as WF1 and WF2, respectively [

23]. The parameters for the AC grid and SGs are shown in Supplementary Material B and have some modifications. The parameters for the PMSG-based wind generation system of WF1 are listed in Table I, whereas those for WF2 are provided in Supplementary Material C.

Fig. 10 Model of PMSG-based WF connected to four-machine two-area system.

This model is employed to evaluate the torsional oscillation damping characteristics of PMSG-based wind generation systems with various parameters. We also investigate the independence of the torsional oscillation suppression strategies when multiple PMSG-based wind generation systems with different parameters are connected to power grid. Additionally, the coupling characteristics of the damping controller and torsional oscillation are examined.

Increasing K_tod enhances the torsional oscillation suppression ability of the damping controller but introduces a new oscillation mode. Excessive K_tod values can lead to positive real roots, thereby impacting the system stability. Therefore, the impact of K_tod on the system stability should be analyzed. Supplementary Material D presents a detailed linearization model of a PMSG-based WF connected to the four-machine two-area system.

The main oscillations are categorized into eight modes (modes 1-8): ① local oscillation mode dominated by the SGs in area 1; ② local oscillation mode dominated by the SGs in area 2; ③ torsional oscillation mode dominated by shaft system of WF1; ④ oscillation mode dominated by the PLL; ⑤ interarea oscillation mode dominated by all the SGs; ⑥ oscillation mode dominated by the proposed additional damping controller in WF1; ⑦ torsional oscillation mode dominated by shaft system of WF2; and ⑧ oscillation mode dominated by the proposed additional damping controller in WF2.

First, we vary K_tod₁from 0 to 24 with an increment step of 0.08. The root loci of oscillation modes are shown in Fig. 11(a). As shown in Fig. 11(b), when K_tod₁ increases, modes 3 and 6 initially converge and then gradually diverge. The absolute value for the damping ratio of mode 3 first increases and then decreases, and when K_tod₁ reaches 0.48, it reaches the maximum value. Additionally, the variation in K_tod₁ has no effect on modes 7 and 8. Concurrently, we increase K_tod₂ from 0 to 24 with an increment step of 0.08. As shown in Fig. 12, with an increase in K_tod₂, the root loci of modes 7 and 8 follow the same trend as those of modes 3 and 6 when $K_{t o d 1}$ increases, respectively. Furthermore, the changes in the damping characteristics of shaft system for WF2 do not affect those for WF1. Therefore, when dealing with multiple WTs connected to AC power systems, a shaft system damping controller should be designed considering the characteristics of each PMSG-based wind generation system.

Fig. 11 Root loci and damping ratio of oscillation modes 1-8 with various K_tod₁. (a) Root loci. (b) Damping ratio.

Fig. 12 Root loci and damping ratio of oscillation modes 1-8 with various K_tod₂. (a) Root loci. (b) Damping ratio.

B. Implementation of Proposed Torsional Oscillation Suppression Strategy

The proposed oscillation suppression strategy is summarized in the following steps.

Step 1: determine the system operating and control parameters such as wind speed, inertial time constants, stator resistance, $K_{p 1}$ , $K_{i 1}$ , $K_{p 2}$ , and $K_{i 2}$ .

Step 2: calculate the torsional oscillation frequency.

Step 3: obtain the Bode diagram of $Δ T_{e} / Δ ω_{Δ}$ using (13).

Step 4: calculate the phase angle of the Bode diagram at the torsional oscillation frequency and obtain the compensation angle.

Step 5: calculate the parameters of the phase compensation and bandpass filter controllers based on the compensation angle.

Step 6: obtain the optimal damper controller gain $K_{t o d}$ based on eigenvalue analysis.

Based on the parameters listed in Table I, when the system reaches steady state, the phase-frequency characteristics of $G (s)$ are depicted in Fig. 13 with various $ω_{g 0}$ ranging from 0.7 to 1.1 p.u.. Figure 13 shows that $θ_{o s c}$ ranges from -132° to -122° at $ω_{o s c}$ of 12.748 rad/s. For instance, taking $ω_{g 0}$ =1.1 p.u. as a reference, the phase angle at $ω_{o s c}$ in the shaft system stabilizes at approximately -122° during steady-state operation.

Fig. 13 Phase-frequency characteristic of $G (s)$ with various $ω_{g 0}$ .

Based on (9) and (13), when $Δ T_{e}$ and $Δ ω_{Δ}$ are in antiphase, $D_{e}$ reaches its minimum value, and the damping provided by $Δ T_{e}$ is maximized, considerably benefiting the shafting stability. Hence, $H (s)$ is used to adjust the phase angle of $G (s)$ at $ω_{o s c}$ to -180°. The phase angle that $H_{i} (s)$ must compensate is $φ_{o s c} = - 130 ° - θ_{o s c} = - 53 °$ . Time constants T₁ and T₂ are computed as 0.56 and 0.2 s, respectively. The phase-frequency characteristic of $G (s)$ after incorporating $H (s)$ is shown in Fig. 14. By comparing Figs. 13 and 14, the phase angle of $G (s)$ at $ω_{o s c}$ has been adjusted to -180°, maximizing the damping effect of $Δ T_{e}$ .

Fig. 14 Phase-frequency characteristic of $G (s)$ with various $ω_{g 0}$ after incorporating $H (s)$ .

VI. Case Study

We conduct simulations using the MATLAB/Simulink platform to evaluate the effectiveness of the proposed torsional oscillation suppression strategy. The simulations are executed on a personal computer equipped with a 3.3 GHz processor and 16 GB of RAM to implement the model depicted in Fig. 1. The converter is modeled as a digital system with a switching frequency of 2 kHz and sampling time of 5×10^-6 s.

The parameters of the PMSG-based wind generation system listed in Table I are obtained from China Wind Power Group Limited based on an actual system. In addition, the control parameters are fine-tuned using Simulink to achieve optimal performance.

A. Evaluation of Proposed Torsional Oscillation Suppression Strategy

Throughout the simulation, the wind speed is maintained constant at 12.1 m/s, which corresponds to the rated wind speed. The time-domain simulation incorporates a three-phase short-circuit fault on node B12 of the studied system at $t = 50$ s. The fault persists for 0.2 s before being cleared at $t = 50.2$ s, after which the system returns to steady state.

The time-domain simulation is aimed to investigate two scenarios with and without applying the proposed torsional oscillation suppression strategy, i.e., scenarios 1 and 2, respectively.

The simulation results of studied system in scenarios 1 and 2 are shown in Fig. 15. The application of the double-mass model and proposed torsional oscillation suppression strategy results in a stabilization of $ω_{g}$ after the short-circuit fault is cleared. In addition, the system attains transient stability, with $P_{g}$ , $U_{d c}$ , and voltage at PCC $U_{o}$ returning to steady-state levels, accompanied by oscillations following the disturbance. Implementing the proposed torsional oscillation suppression strategy mitigates transient instability and reduces the likelihood of low-frequency oscillations during the short-circuit fault. Consequently, the oscillations in $ω_{g}$ , $U_{d c}$ , $U_{o}$ , and $P_{g}$ are attenuated, enhancing the dynamic and steady-state operation of the studied system and improving the power grid stability compared with the results without the additional damping controller.

Fig. 15 Simulation results of studied system in scenarios 1 and 2. (a) ω_g.(b) P_g.(c) U_dc.(d) U_o.

Then, the fundamental parameters affecting the damping of the shaft system, including wind speed, system parameters, and control parameters, are analyzed. The simulation curves of $ω_{g}$ and $P_{g}$ with different wind speeds under MPPT are shown in Fig. 16. Figure 17 shows the simulation curves of $ω_{g}$ and $P_{g}$ with various $R_{s}$ under MPPT. Additionally, Fig. 18 illustrates the simulation curves of $ω_{g}$ and $P_{g}$ with various $K_{p 1}$ and $K_{i 1}$ under MPPT. The results for $K_{p 2}$ and $K_{i 2}$ are ignored because they are similar to those for $K_{p 1} a n d K_{i 1}$ , respectively.

Fig. 16 Simulation curves of $ω_{g}$ and $P_{g}$ with different wind speeds under MPPT. (a) $ω_{g}$ with $v = 1.0$ p.u.. (b) $ω_{g}$ with $v = 0.8$ p.u.. (c) $ω_{g}$ with $v = 0.6$ p.u.. (d) $P_{g}$ with $v = 0.8$ p.u.. (e) $P_{g}$ with $v = 1.0$ p.u.. (f) $P_{g}$ with $v = 0.6$ p.u..

Fig. 17 Simulation curves of $ω_{g}$ and $P_{g}$ with various $R_{s}$ under MPPT. (a) $ω_{g}$ . (b) $P_{g}$ .

Fig. 18 Simulation curves of $ω_{g}$ and $P_{g}$ with various $K_{p 1}$ and $K_{i 1}$ . (a) $ω_{g}$ with various $K_{p 1}$ . (b) $P_{g}$ with various $K_{p 1}$ . (c) $ω_{g}$ with various $K_{i 1}$ . (d) $P_{g}$ with various $K_{i 1}$ .

Figures 16-18 show that the oscillation amplitudes of $ω_{g}$ and $P_{g}$ are smaller and the oscillation attenuation is faster with increase of $K_{i 1}$ and $K_{i 2}$ , indicating the shaft system damping improves. However, the oscillation amplitude of $ω_{g}$ and $P_{g}$ is larger and the oscillation attenuation is slower with increase of $K_{p 1}$ and $K_{p 2}$ , indicating that the shaft system damping deteriorates. Additionally, the simulation results are nearly insensitive to variations in stator resistance R_s within a reasonable range influenced by temperature.

B. Comparison of Proposed Torsional Oscillation Suppression Strategy with Other Strategies

To evaluate the performance of the proposed torsional oscillation suppression strategy, we compared it with common strategies, including speed-feedback-based damping and active damping based on torque estimation [

19].

We consider two cases that illustrate the effectiveness of the proposed torsional oscillation suppression strategy. In case 1, wind speed v remains constant at 12.1 m/s, as in Section VI-A. A three-phase short-circuit fault occurs at node B12 of the system and $t = 50$ s in the time-domain simulation. The fault persists for 0.2 s before being cleared at $t = 50.2$ s, after which the system returns to steady state. In case 2, the initial wind speed is 12.1 m/s and abruptly drops to 7.3 m/s at $t = 50 s$ .

Figure 19 shows the transient response curves of $ω_{g}$ , $P_{g}$ , $U_{d c}$ , $U_{o}$ , $T_{e}$ , and $θ_{s h}$ in response to the three-phase short-circuit fault using the proposed torsional oscillation suppression strategy, active damping strategy considering torque estimation, and damping strategy based on speed feedback [

3], [19].

Fig. 19 Transient response curves of $ω_{g}$ , $P_{g}$ , $U_{d c}$ , $U_{o}$ , $T_{e}$ , and $θ_{s h}$ in response to three-phase short-circuit fault using different control strategies. (a) ω_g. (b) P_g. (c) U_dc. (d) U_o. (e) $T_{e}$ . (f) $θ_{s h}$ .

The proposed torsional oscillation suppression strategy and other strategies improve the shaft system damping and suppress the oscillatory instability of the PMSG-based wind generation system during the three-phase short-circuit fault. The proposed torsional oscillation suppression strategy is more effective in suppressing torsional oscillations than the comparison strategies. Additionally, the proposed torsional oscillation suppression strategy attenuates the oscillations of $ω_{g}$ , $P_{g}$ , $U_{d c}$ , $T_{e}$ , and $θ_{s h}$ faster than other strategies. These results indicate that the proposed torsional oscillation suppression strategy outperforms other strategies in suppressing torsional oscillations.

Figure 20 shows that following a drop in wind speed, P_g, $T_{e}$ , and $ω_{g}$ initially decrease, and then stabilize after oscillation attenuation.

Fig. 20 Transient response curves of $ω_{g}$ , $P_{g}$ , $U_{d c}$ , $U_{o}$ , $T_{e}$ , and $θ_{s h}$ following a drop in wind speed. (a) ω_g.(b) P_g.(c) U_dc.(d) U_o.(e) $T_{e}$ .(f) $θ_{s h}$ .

Likewise, U_dc, $θ_{s h}$ , and U_o oscillate before reaching steady state. The proposed torsional oscillation suppression strategy demonstrates a superior performance in attenuating the oscillations of $ω_{g}$ , $θ_{s h}$ , and P_g more rapidly compared with other strategies. This superiority is evident in Fig. 20(f), where the oscillation of $θ_{s h}$ is promptly suppressed. The simulation results validate the superiority of the proposed torsional oscillation suppression strategy in suppressing torsional oscillations.

C. Verification of Effect of $K_{t o d}$ on System Stability

In this subsection, we analyze the influence of $K_{t o d}$ on the system stability. Figure 21 shows the transient response curves of $ω_{g}$ , $P_{g}$ , $U_{d c}$ , $T_{e}$ , and $θ_{s h}$ with various $K_{t o d}$ . Figure 21(a) and (e) shows that $ω_{g}$ and $θ_{s h}$ decrease while the shaft system damping increases with increasing $K_{t o d}$ , reaching its peak when $K_{t o d} = 0.48$ . When $K_{t o d}$ increases to 2, the shaft system damping decreases, and the decay rates of $ω_{g}$ and $θ_{s h}$ decline.

Fig. 21 Transient response curves of $ω_{g}$ , $P_{g}$ , $U_{d c}$ , $T_{e}$ , and $θ_{s h}$ with various K_tod. (a) ω_g.(b) P_g.(c) U_dc.(d) T_e. (e) $θ_{s h}$ .

In Fig. 21(b)-(d), an increase in $K_{t o d}$ results in an increase in the active power reference value, leading to higher oscillation amplitudes in $P_{g}$ , $U_{d c}$ , and $T_{e}$ . Therefore, the proper selection of $K_{t o d}$ is crucial for effective oscillation suppression. If $K_{t o d}$ is excessively small, the suppression effect becomes prominent. By contrast, if $K_{t o d}$ is very large, the suppression effect diminishes, increasing the instantaneous values of P_g, U_dc,and $T_{e}$ , and potentially causing system instability.

VII. Conclusion

We analyze the torsional oscillation damping characteristics based on the electromagnetic damping torque method. Subsequently, the transfer function for electromagnetic torque and speed is derived, and the influence of electromagnetic torque on the shaft system damping is explained.

The electrical damping coefficient $D_{e}$ is the only controllable parameter that affects the shaft system damping and is negatively correlated with damping coefficient $ξ$ of the torsional oscillation. Additionally, the real part of the incremental transfer function of $Δ T_{e}$ and $Δ ω_{Δ}$ is equivalent to electrical damping coefficient $D_{e}$ .

The influence of wind speed and system control parameters on torsional oscillation damping is determined. We find that $v$ , $K_{i 1}$ , and $K_{i 2}$ are positively correlated with the shaft system damping, while $K_{p 1}$ and $K_{p 2}$ are negatively correlated with shaft system damping, and a change in stator resistance $R_{s}$ within a reasonable range has a negligible effect on damping. Therefore, to ensure the stable operation of a PMSG-based wind generation system, the high-speed operation is recommended. In addition, while ensuring an active power tracking rate and stability in the high-frequency range, we recommend minimizing $K_{p 1}$ and $K_{p 2}$ , as well as increasing $K_{i 1}$ and $K_{i 2}$ .

We propose an oscillation suppression strategy on the active power control loop to compensate for the damping angle between $Δ T_{e}$ and $Δ ω_{Δ}$ and thus improve the stability of the shaft system based on the previous analysis. The studied system is modeled using MATLAB/Simulink, and the simulation results show that the proposed torsional oscillation suppression strategy outperforms other strategies in eliminating torsional oscillations. The simulation results verifies the effectiveness of our theoretical analysis and control strategy.

In future work, we will investigate an adaptive compensation method for the damping angle of a PMSG-based wind generation system considering the frequency offset.

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Introduction

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Torsional Oscillation Damping Analysis and Suppression Strategy for PMSG-based Wind Generation System PDF

Abstract

Keywords

I. Introduction

II. Modeling and Control of PMSG-based Wind Generation System

A. PMSG Model

B. Double-mass Model

C. MSC Model

III. Torsional Oscillation Damping Characteristics

A. Electrical Damping Characteristics of PMSG

B. Shaft System Damping Characteristics of PMSG

IV. Influence of System Parameters on Torsional Oscillation Damping Characteristics

A. Influence of Wind Speed and Stator Resistance on $D_{e}$

B. Influence of Control System Parameters on $D_{e}$

V. Torsional Oscillation Suppression Strategy

A. Proposed Torsional Oscillation Suppression Strategy

B. Implementation of Proposed Torsional Oscillation Suppression Strategy

VI. Case Study

A. Evaluation of Proposed Torsional Oscillation Suppression Strategy

B. Comparison of Proposed Torsional Oscillation Suppression Strategy with Other Strategies

C. Verification of Effect of $K_{t o d}$ on System Stability

VII. Conclusion

References

Home

Introduction

Editorial Board

For Author

Call For Papers

APC

Sponsor & Publisher

Torsional Oscillation Damping Analysis and Suppression Strategy for PMSG-based Wind Generation System PDF

Abstract

Keywords

I. Introduction

II. Modeling and Control of PMSG-based Wind Generation System

A. PMSG Model

B. Double-mass Model

C. MSC Model

III. Torsional Oscillation Damping Characteristics

A. Electrical Damping Characteristics of PMSG

B. Shaft System Damping Characteristics of PMSG

IV. Influence of System Parameters on Torsional Oscillation Damping Characteristics

A. Influence of Wind Speed and Stator Resistance on De

B. Influence of Control System Parameters on De

V. Torsional Oscillation Suppression Strategy

A. Proposed Torsional Oscillation Suppression Strategy

B. Implementation of Proposed Torsional Oscillation Suppression Strategy

VI. Case Study

A. Evaluation of Proposed Torsional Oscillation Suppression Strategy

B. Comparison of Proposed Torsional Oscillation Suppression Strategy with Other Strategies

C. Verification of Effect of Ktod on System Stability

VII. Conclusion

References

A. Influence of Wind Speed and Stator Resistance on $D_{e}$

B. Influence of Control System Parameters on $D_{e}$

C. Verification of Effect of $K_{t o d}$ on System Stability