Analysis of Subsynchronous Torsional of Wind–Thermal Bundled System Transmitted via HVDC Based on Signal Injection Method

Abstract

With the development of large-scale new energy, the wind–thermal bundled system transmitted via high-voltage direct current (HVDC) has become the main method to solve the problem of wind power consumption. At the same time, the problem of subsynchronous oscillation among wind power generators, high-voltage direct current (HVDC), and synchronous generators (SGs) has become increasingly prominent. According to the dynamic interaction among doubly fed induction generators (DFIGs), HVDC, and SGs, a linearization model of DFIGs and SGs transmitted via HVDC is established, and the influence of the electromagnetic transient of wind turbines and HVDC on the electromechanical transient processes of SGs is studied. Using the method of additional excitation signal injection, the influence of the main factors of DFIG on the damping characteristics of each torsional mode of SG is analyzed, including control parameters and operation conditions when the capacity of HVDC is fixed. The mechanism of the negative damping torsional of SGs is identified. A time-domain simulation model is built in Electromagnetic Transients including DC/Power Systems Computer Aided Design (EMTDC/PSCAD) to verify the correctness and effectiveness of the theoretical analysis.

1. Introduction

In recent years, developing wind resources vigorously has been an important way of promoting energy transition and meeting the environmental challenges [1,2]. In the first half of 2020, China’s new grid-connected installed capacity was 6.32 million MW. By the end of June, the installed capacity of wind power was 217 million kW [3], and the installed wind power capacity has increased steadily. It is estimated that, by 2050, the installed wind power capacity will reach 2.4 billion kW in China [4].

Due to the uncoordinated distribution of wind power resources and load development in China, there is a problem of wind power accommodation. Wind power requires large-scale, long-distance, and stable transportation. Thus, the wind–thermal bundled system transmitted via high-voltage direct current (HVDC) is the key means to solve this problem, such as the Lugu HVDC project and Hazheng HVDC project. However, while solving the problem of accommodation, the security and stability of the sending network face hidden risks [5,6], example.g., the problem of power oscillation. In July 2015, a subsynchronous oscillation (SSO) occurred in a wind farm in Hami, Xinjiang, China, which caused the shaft torsional oscillation of the synchronous generator (SG) at a distance of 300 km, threatening the safe operation of the system [7,8].

Focusing on the power oscillation problem of the sending network, domestic and foreign scholars have investigated the following aspects using the eigenvalue method, complex torque coefficient approach, impedance analysis approach, etc: (1) the problem of SSO in SGs via HVDC transmission [9,10,11]; (2) the SSO of wind turbines transmitted via HVDC [12,13,14,15]; (3) the SSO of wind–thermal bundled systems transmitted via HVDC [16,17,18].

In the research of SGs via HVDC transmission, [9] pointed out that the HVDC transmission capacity, strength of grid, and current loop control strategy all affect the torsional vibration of the SG shafting and induce 11.5 Hz negative damping oscillation. The authors of [10,11] studied the stability mechanism of SSO caused by HVDC using the complex torque coefficient method, as well as pointed out that the nearby generators can produce SSO negative damping when the bandwidth of the constant current controller on the rectifier side is 10–20 Hz. In terms of wind turbines via HVDC transmission, [12,13] used the eigenvalue method and complex torque coefficient approach to study doubly fed induction generator (DFIG) and HVDC system wind farm damping characteristics, showing that their interaction is not obvious. The influence of DFIG controller parameters and wind speed on the torsional mode of SG was evaluated using eigenvalue analysis in [14]. The results showed that the frequency and damping of the torsional mode increased when the rotor-side converter power outer loop parameter increased from 1 to 5 or the current inner loop control parameter changed from 0.1 to 10; on the other hand, when the wind speed increased from 11 m/s to 15 m/s, the torsional mode first decreased and then increased. The authors of [15] pointed out that the damping of the SSO increases, and the coupling mechanism was deduced as a function of the dynamic process between voltage and current disturbance at the coupling point, when the proportional coefficient of the permanent magnetic synchronous generator (PMSG) outer loop and HVDC constant current controller increased or the integral coefficient decreased.

The research on the interaction between SGs or wind turbines and HVDC has been relatively complete, but the research on the interaction of wind turbines, SGs, and HVDC is still rare. The oscillation mechanism of the wind–thermal bundled system transmitted via HVDC was evaluated using a time-domain simulation and eigenvalue analysis in [16,17], pointing out that the connection of DFIG can alleviate the SSO of SGs caused by HVDC. The authors of [18] proposed an active disturbance rejection additional damping control using the least square to suppress the shaft torsional vibration of the wind–thermal bundled system effectively transmitted via HVDC. However, the dynamic interaction mechanism among the three has not been fully explained.

In wind–thermal bundled systems transmitted via HVDC, the torsional vibration of the SG caused by grid-connected wind turbines is manifested in the rotor speed oscillation. To be specific, the speed contains abundant electromechanical coupling information. If a certain controllable disturbance is applied to stimulate speed oscillation, we can observe the characteristics of speed response after the disturbance is removed, as well as judge the electromechanical coupling characteristics among wind turbines, HVDC, and SGs, and the contribution of various factors to torsional vibration can be identified.

Unlike previous studies, this paper proposes using the signal injection method to analyze the torsional vibration of a high-proportion wind power system [19], which involves applying an alternating current (AC) voltage signal on the excitation winding to stimulate speed oscillation. This paper observes the difference of the free response characteristics of the speed disturbance, which is caused by the change in wind turbine scale, and it identifies the coupling mechanism of DFIGs, HVDC, and SG shafting. The main contributions of this paper are as follows:

• The linearization models of DFIGs and SGs transmitted via HVDC are established. Thus, the influence of the electromagnetic transient process of wind turbines and HVDC on the electromechanical transient process of SGs is studied. Moreover, this paper reveals the interaction mechanism of the system;
• The influences of DFIG control parameters and operating conditions on the damping characteristics of each torsional mode of SGs under the condition of a fixed capacity of HVDC is analyzed on the basis of injection signal identification. Then, the negative damping torsional vibration mechanism of the SG shafting system is identified;
• Considering the differences in operation conditions and control parameters of different DFIGs in the wind farm, taking two-machine parallel systems as an example, the influence of DFIGs on the torsional vibration of SG shafting is further expounded.

2. System Model

The topological structure of a wind–thermal bundled system transmitted via HVDC is shown in Figure 1, which mainly includes DFIGs, SGs, and HVDC. Considering that the DFIG in the actual system consists of hundreds and even thousands of wind turbines, the model order is high; therefore, it is difficult to establish a detailed model. Therefore, this paper establishes a single-machine model for DFIGs using the principle of similar transformation, while the model of the SG is taken from the first sub-synchronous resonance (SSR) benchmark [20], and the CIGRE Benchmark Model is used for HVDC [21].

2.1. Model of DFIG

The DFIG topology structure and its control strategy are shown in Figure 2, consisting of wind turbines, an induction generator, a rotor-side converter (RSC), and a grid-side converter (GSC). The definitions of variables in Figure 2 are shown in

Table A1. The definitions of variables for the DFIG.

Variable	Notation
Reactive power of stator	Qs
Active power of stator	Ps
Reactive power of grid	Qg
Voltage of rotor	ur
Current of rotor	ir
Voltage of grid	ug
Current of grid	ig
Voltage of stator	us
DC capacitor voltage	udc
Direct current capacitor	Cdc
Connection reactance	Lc
Output phase of phase-locked loop	θPLL
Wind speed	v
Rotor speed	ωr
The flux linkage of stator	ψs
The flux linkage of rotor	ψr
Output voltage of GSC	ucg
Output current of GSC	icg
Inductance of GSC	L1
Inductance of rotor	L’
Rated angular speed of the stator	ω0
Slip	s
Mutual reactance	xm
Reactance of the stator	xs
State variables in the control system	x1~x8

(Appendix A). The stator is directly connected to the network, and the rotor is connected to the grid through the RSC and GSC. The RSC and GSC usually use voltage vector-oriented control [22,23]. In order to maximize the utilization of wind energy, the RSC adopts the maximum power point tracking (MPPT) control strategy, whereas the GSC takes DC voltage and reactive power as the control targets.

The mathematical model of the DFIG, as shown in [24], is linearized at the equilibrium point to obtain the linearized model shown in Equation (1).

$$\{\begin{array}{l}\Delta {\dot{x}}_{\mathrm{DFIGs}}=A_{\mathrm{DFIGs}}\Delta x_{\mathrm{DFIGs}}+B_{\mathrm{DFIGs}}\Delta u_{\mathrm{DFIGs}}\\ \Delta y_{\mathrm{DFIGs}}=C_{\mathrm{DFIGs}}\Delta x_{\mathrm{DFIGs}}+D_{\mathrm{DFIGs}}\Delta u_{\mathrm{DFIGs}}\end{array},$$

(1)

where A_DFIGs, B_DFIGs, C_DFIGs, and D_DFIGs are the state matrix, input matrix, and output matrix of DFIGs; the state variables are Δx_DFIGs = [Δω_r, Δψ_sd, Δψ_sq, Δψ_rd, Δψ_rq, Δi_gd, Δi_gq, Δu_dc, Δx₁~Δx₈], and Δx_DFIGs is a 16 × 1 matrix; the input and output variables are Δu_DFIGs = [Δu_sd, Δu_sq]^T and Δy_DFIGs = [Δi_sd, Δi_sq]^T, and Δu_DFIGs and Δy_DFIGs are 2 × 1 matrices. The transfer function of DFIGs is denoted in Equation (2).

$$\Delta Y_{\mathrm{DFIGs}}=C_{\mathrm{DFIGs}}\cdot {\left(sI-A_{\mathrm{DFIGs}}\right)}^{-1}\cdot B_{\mathrm{DFIGs}}+D_{\mathrm{DFIGs}}.$$

(2)

2.2. Model of SG

The topological structure of an SG was described in [20], consisting of an electrical part and shafting part. The electrical part includes stator/rotor excitation and voltage, and the shafting part is equivalent to six mass blocks, namely, a high-pressure cylinder (HP), intermediate-pressure cylinder (IP), low-pressure cylinder A (LPA), low-pressure cylinder B (LPB), generator, and exciter. There are five natural oscillation frequencies in the shaft system, which are 15.71 Hz (torsional mode 1, TM1), 20.21 Hz (TM2), 25.55 Hz (TM3), 32.28 Hz (TM4), and 47.46 Hz (TM5), respectively, of which TM5 is not considered in this paper.

The mathematical model of an SG was shown in [20], linearized at the equilibrium point, and the linearized mode of the SG was obtained, as represented by Equation (3).

$$\{\begin{array}{l}\Delta {\dot{x}}_{\mathrm{SG}}=A_{\mathrm{SG}}\Delta x_{\mathrm{SG}}+B_{\mathrm{SG}}\Delta u_{\mathrm{SG}}\\ \Delta y_{\mathrm{SG}}=C_{\mathrm{SG}}\Delta x_{\mathrm{SG}}+D_{\mathrm{SG}}\Delta u_{\mathrm{SG}}\end{array},$$

(3)

where A_SG, B_SG, C_SG, and D_SG are the state matrix, input matrix, and output matrix of SG; the state variables are Δx_SG = [Δω₁~Δω₆, Δδ₁~Δδ₆, ΔT₁~ΔT₃, Δα, Δμ, Δi_f, Δi_D, Δi_g, Δi_Q, ΔE_fd], and Δx_SG is a 22 × 1 matrix; the input and output variables are Δu_SG = [Δu_d, Δu_q]^T and Δy_SG = [Δi_d, Δi_q]^T, and Δu_SG and Δy_SG are 2 × 1 matrices. The definitions of variables are shown in

Table A2. The definitions of variables for the SG.

Variable	Notation
Stator voltage	Udq
Stator current	idq
Stator linkage	ψdq
Current of excitation winding	if
Current of damping winding in the d-axis	iD
Current of damping winding in the g-axis	ig
Current of damping winding in the Q-axis	iQ
Voltage of exciter	Efd
Location of speed relay	α
Opening angle of valve	μ
Motive power of steam turbine	T1~T3
Rotational speed of 6-mass	ω1~ω6
Rotor phase of 6-mass	δ1~δ6

(Appendix A).

The transfer function of SG is further obtained as shown in Equation (4).

$$\Delta Y_{\mathrm{SG}}=C_{\mathrm{SG}}\cdot {\left(sI-A_{\mathrm{SG}}\right)}^{-1}\cdot B_{\mathrm{SG}}+D_{\mathrm{SG}}.$$

(4)

2.3. Model of HVDC

The topology and mathematical model of HVDC were shown in [21]. The rectifier of HVDC adopts constant current control and the inverter-side controller adopts constant turn-off angle control. The linearized model structure can be given as

$$\{\begin{array}{l}\Delta {\dot{x}}_{\mathrm{HVDC}}=A_{\mathrm{HVDC}}\Delta x_{\mathrm{HVDC}}+B_{\mathrm{HVDC}}\Delta u_{\mathrm{HVDC}}\\ \Delta y_{\mathrm{HVDC}}=C_{\mathrm{HVDC}}\Delta x_{\mathrm{HVDC}}+D_{\mathrm{HVDC}}\Delta u_{\mathrm{HVDC}}\end{array},$$

(5)

where A_HVDC, B_HVDC, C_HVDC, and D_HVDC are the state matrix, input matrix and output matrix of HVDC; the state variables are Δx_HVDC = [ΔI_rd, ΔI_rq, ΔI_dc1, ΔI_dc2, ΔV_dc, ΔV_pccd, ΔV_pccq, ΔV_cr2d, ΔV_cr2q, ΔI_Lr1d, ΔI_Lr1q, ΔV_cr3d, ΔV_cr3q, ΔV_cr4d, ΔV_cr4q, ΔI_Lr2d, ΔI_Lr2q, Δx₉~Δx₁₂], and Δx_HVDC is a 21 × 1 matrix; the input and output variables are Δu_HVDC = [ΔV_rd, ΔV_rq, ΔI_dc]^T and Δy_HVDC = [ΔI_rd, ΔI_rq]^T, and Δu_HVDC is a 3 × 1 matrix and Δy_HVDC is a 2 × 1 matrix. The definitions of variables are shown in

Table A3. The definitions of variables for HVDC.

Variable	Notation
AC system voltage	Vr
AC system current	Ir
DC line current	Idc
DC line capacitance voltage	Vdc
AC bus voltage	VPCC
Filter capacitance voltage	Vcr
Filter capacitance current	ILr
State variables in the phase-locked loop controller	x9~x10
State variables in the current loop controller	x11~x12

(Appendix A).

The transfer function of HVDC can be obtained as shown in Equation (6):

$$\Delta Y_{\mathrm{HVDC}}=C_{\mathrm{HVDC}}\cdot {\left(sI-A_{\mathrm{HVDC}}\right)}^{-1}\cdot B_{\mathrm{HVDC}}+D_{\mathrm{HVDC}}.$$

(6)

3. Electromechanical Coupling Characteristics of Equipment

According to the SG linearized model, the electromagnetic torque disturbance ΔT_e can be expressed as shown in Equation (7).

$$\begin{array}{ll}\Delta T_e& =-\left\{\left[\begin{array}{cc}{i}_{\mathrm{q}0}& -i_{\mathrm{d}0}\end{array}\right]+\left[\begin{array}{cc}-{\psi }_{\mathrm{q}0}& {\psi }_{\mathrm{d}0}\end{array}\right]\cdot \left[\begin{array}{cc}{L}_1(s)& 0\\ 0& L_2(s)\end{array}\right]\right\}\cdot {\left[\begin{array}{cc}s/{\omega }_{\mathrm{base}}& -{\omega }_0\\ {\omega }_0& s/{\omega }_{base}\end{array}\right]}^{-1}\cdot \left[\begin{array}{c}-{\psi }_{\mathrm{q}0}\\ {\psi }_{\mathrm{d}0}\end{array}\right]\cdot \Delta {\omega }_r\\ & +\left\{\begin{array}{l}\left[\begin{array}{cc}{i}_{\mathrm{q}0}& -i_{\mathrm{d}0}\end{array}\right]\\ +\left[\begin{array}{cc}-{\psi }_{\mathrm{q}0}& {\psi }_{\mathrm{d}0}\end{array}\right]\cdot \left[\begin{array}{cc}{L}_1(s)& 0\\ 0& L_2(s)\end{array}\right]\end{array}\right\}\cdot {\left[\begin{array}{cc}s/{\omega }_{base}& -{\omega }_0\\ {\omega }_0& s/{\omega }_{base}\end{array}\right]}^{-1}\cdot \left[\begin{array}{c}\Delta u_d\\ \Delta u_{\mathrm{q}}\end{array}\right]\end{array},$$

(7)

where ω₀ is the SG rotor speed initial value, ω_base is the reference value of the SG, and ω_base = 60 Hz.

L₁(s) and L₂(s) represent the transfer function relationship between current and flux, and the expressions are denoted in Equation (8).

$$\{\begin{array}{l}{L}_1(s)=\frac{\left({\tau }_{\mathrm{d}0}^{″}s+1\right)\left({\tau }_{\mathrm{d}0}^{\prime }s+1\right)}{\left({\tau }_{\mathrm{q}}^{″}s+1\right)\left({\tau }_{\mathrm{d}}^{\prime }s+1\right)L_{\mathrm{d}}}\\ L_2(s)=\frac{\left({\tau }_{\mathrm{q}0}^{″}s+1\right)\left({\tau }_{\mathrm{q}0}^{\prime }s+1\right)}{\left({\tau }_{\mathrm{q}}^{″}s+1\right)\left({\tau }_{\mathrm{q}}^{\prime }s+1\right)L_{\mathrm{q}}}\end{array}.$$

(8)

According to Equation (7), the SG electromagnetic torque disturbance is related to the rotor speed disturbance and voltage disturbance, and the voltage disturbance is correlated with the current disturbance and grid structure. The expression of voltage disturbance is considered as shown in Equation (9).

$$\left[\begin{array}{c}\Delta u_{\mathrm{d}}\\ \Delta u_{\mathrm{q}}\end{array}\right]=\left[\begin{array}{cc}{Z}_{11}& Z_{12}\\ Z_{21}& Z_{22}\end{array}\right]\cdot \left[\begin{array}{c}\Delta i_d\\ \Delta i_{\mathrm{q}}\end{array}\right],$$

(9)

where Z₁₁, Z₁₂, Z₂₁, and Z₂₂ are the grid impedance matrix elements, and the current disturbance expression is shown in Equation (10).

$$\begin{array}{ll}\left[\begin{array}{c}\Delta i_{\mathrm{d}}\\ \Delta i_q\end{array}\right]=& -\left[\begin{array}{cc}{L}_1(s)& 0\\ 0& L_2(s)\end{array}\right]\cdot {\left[\begin{array}{cc}s/{\omega }_{base}& -{\omega }_0\\ {\omega }_0& s/{\omega }_{base}\end{array}\right]}^{-1}\cdot \left[\begin{array}{c}-{\psi }_{\mathrm{q}0}\\ {\psi }_{\mathrm{d}0}\end{array}\right]\cdot \Delta {\omega }_r\\ & +\left[\begin{array}{cc}{L}_1(s)& 0\\ 0& L_2(s)\end{array}\right]\cdot {\left[\begin{array}{cc}s/{\omega }_{\mathrm{base}}& -{\omega }_0\\ {\omega }_0& s/{\omega }_{base}\end{array}\right]}^{-1}\cdot \left[\begin{array}{c}\Delta u_{\mathrm{d}}\\ \Delta u_{\mathrm{q}}\end{array}\right]\end{array}.$$

(10)

Equations (7)–(10) reflect the formation mechanism of SG electromagnetic torque, thus giving the formation mechanism diagram shown in Figure 3a. The SG speed disturbance is the input signal, and the electromagnetic torque is composed of two branches, of which the first branch is mainly determined by the SG electrical part and the second branch is codetermined by the SG electrical part and the grid. The expressions of G₁(s), G₂(s), G₃(s), G₄(s), and G₅(s) in Figure 3b are shown in Equation (A1) (Appendix A).

It can be seen from Figure 3 that the SG speed disturbance transfer function can be obtained.

$$\Delta T_e=\left\{G_1(s)+(G_2(s)\cdot G_3(s)\cdot {\left[1-G_3(s)\cdot G_5(s)\right]}^{-1}\cdot G_4(s))\right\}\cdot \Delta {\omega }_r.$$

(11)

When the DFIG and HVDC are connected to the system, the mechanism of interaction with the SG is shown in Figure 4. It can be seen that the rotor disturbance Δω_r of the SG is the input disturbance of the electrical system. The output current disturbance Δi_sdq is formed through the electromagnetic coupling relationship of the electrical part. On the one hand, Δi_sdq directly forms the electromagnetic torque disturbance ΔT_e and acts on the rotor; on the other hand, the terminal voltage disturbance Δu_dq of the SG, wind turbines, and HVDC is formed by the grid coupling. The SG terminal voltage disturbance through its electromagnetic coupling relationship intensifies ΔT_e and Δi_sdq; then, Δi_sdq interacts with the terminal voltage disturbances of wind turbines and HVDC to produce current disturbances Δi_rdq and Δi_hdq. Furthermore, Δi_rdq and Δi_hdq are superimposed to continue the grid function and form a new terminal voltage disturbance Δu_dq, which further aggravates ΔT_e until the system rebalances or loses stability.

It can be seen from Figure 4 that the rotational speed disturbance transfer function among the DFIG, HVDC, and SG is considered as shown in Equation (12).

$$\Delta T_{\mathrm{e}}=\left\{G_1(s)+(G_2(s)\cdot G_3(s)\cdot {\left[1-G_3(s)\cdot (G_5(s)+G_{\mathrm{DFIG}\mathrm{s}}(s)+G_{\mathrm{HVDC}}(s))\right]}^{-1}\cdot G_4(s))\right\}\cdot \Delta {\omega }_r.$$

(12)

The above analysis reflects the influence of the electromechanical transient process of power electronic equipment on the electromechanical transient process of current magnetization equipment. The electromechanical coupling process among wind turbines, HVDC, and SGs is revealed from the mechanism. At present, eigenvalue analysis, complex torque analysis, impedance analysis, and the time-domain simulation method are commonly used to analyze system SSO problems. Among them, eigenvalue analysis can obtain the inherent mode of the system and judge the stability of the system at one operating point, but it cannot explain the subsynchronous interaction mechanism clearly. Whether the complex torque analysis is applicable to multimachines and multi-power electronics systems is still inconclusive, and impedance analysis can only determine the stability of the subsystem but not the entire system. The above methods are difficult to ensure calculation accuracy when studying the SSO problem of complex systems including DFIGs, SGs, and HVDC, while the time-domain simulation method can guarantee model integrity and calculation accuracy to the maximum extent. Therefore, this paper studies the influence of various equipment on SG shafting using the identification signal injection method.

4. Influence of Various Online Equipment on the Damping Characteristics of SGs

4.1. Additional Excitation Signal Injection Method

Common SG identification methods include the stator-side additional current injection method and rotor-side additional excitation signal injection method. The latter is used in this paper. The advantages of this method are that the frequency, amplitude, and time length of the excitation signal can be changed according to needs, and all concerned modal attenuation coefficients can be identified; in online application, due to the controllable amplitude, it can ensure the safety of the shafting system without affecting the normal operation of the system. From the dynamic coupling mechanism of various equipment, the torsional vibration of the SG shafting system is caused by the transmission of wind turbines via HVDC, which is manifested in the rotor speed oscillation; that is, the speed contains abundant electromechanical coupling information. If a controllable disturbance is applied to stimulate the rotation speed oscillation and to observe the response characteristics of the rotation speed after the disturbance is removed, the electromechanical coupling characteristics among the wind turbines, HVDC, and SGs can be judged, and then the contribution of each equipment to the shaft torsional vibration of the unit can be identified.

Taking the rotor motion equation as an example, the variation rules of rotor angle and its frequency are as follows when there is external disturbance:

$$\Delta {\ddot{\delta }}_k+2{\sigma }_k\Delta {\dot{\delta }}_k+{\omega }_{nk}^2\Delta {\delta }_k=-\Delta T_{ek},$$

(13)

where the electromagnetic torque ΔT_ek = −Acos(ω_st).

Assuming the angle increment Δδ_k(t = 0) = 0, the torsional vibration of SG can be expressed as

$$\{\begin{array}{l}\Delta {\delta }_k=\Delta {\delta }_{k,t}\left(t\right)+\Delta {\delta }_{k,s}\left(t\right)\\ \Delta {\delta }_{k,t}\left(t\right)=-B{e}^{-{\sigma }_{k}t}\left(\cos \varphi \cos {\omega }_{d}t+\frac{{\omega }_s\sin \varphi +{\sigma }_k\cos \varphi }{{\omega }_d}\sin {\omega }_{d}t\right)\\ \Delta {\delta }_{k,s}\left(t\right)=B\cos \left({\omega }_{s}t-\varphi \right)\end{array},$$

(14)

$$\{\begin{array}{l}B=\frac{A}{\sqrt{{\left({\omega }_{nk}^2-{\omega }_{\mathrm{s}}^2\right)}^2+4{\sigma }_k^2}}\\ \varphi =\arctan \frac{2{\sigma }_k/{\omega }_{nk}}{1-{\left({\omega }_{\mathrm{s}}/{\omega }_{nk}\right)}^2}\\ {\omega }_d=\sqrt{{\omega }_{nk}^2-{\sigma }_k^2}\end{array},$$

(15)

where Δδ_k,t(t) is the transient response that only exists for a period of time, and Δδ_k,s(t) is the forced response under the excitation of the injected signal. It can be seen that the SG rotor dynamic process contains attenuation components and forced excitation, and the attenuation coefficient can characterize the modal damping of the system.

When the excitation signal is removed, the rotational speed of the torsional vibration mode decays exponentially. Solving Equation (16) can obtain the expression of shafting torsional vibration, which can be expressed as shown in Equations (17) and (18).

$$\{\begin{array}{l}△{\ddot{\delta }}_k+2\left({\sigma }_k+{\sigma }_{\mathrm{e}k}\right)△{\dot{\delta }}_k+{\omega }_{nk}^2△{\delta }_k=0\\ △{\delta }_k\left(t_0\right)=△{\delta }_{k,0}\end{array},$$

(16)

$$△{\delta }_k\left(t\right)=C_{\theta }{e}^{-{\displaystyle \sum {\sigma }_k\left(t-t_0\right)}}\sin \left({\omega }_d\left(t-t_0\right)+\varphi \right),\hspace{0.17em}\hspace{0.17em}t\ge t_0,$$

(17)

$$△{\omega }_k\left(t\right)=C_{\omega }{e}^{-{\displaystyle \sum {\sigma }_k\left(t-t_0\right)}}\cos \left({\omega }_d\left(t-t_0\right)+\varphi \right),\hspace{0.17em}\hspace{0.17em}t\ge t_0,$$

(18)

where C_θ, C_ω are coefficients, and t₀ is the time to remove the excitation. It can be seen from Equation (18) that, after the excitation is withdrawn, the modal speed changes in the form of attenuated oscillation, and the attenuation rate is the sum of mechanical damping and electrical damping. The total mode can be obtained by identifying the attenuation rate of the torsional vibration mode speed after removing the excitation.

The basic principle of the additional excitation signal injection method is shown in Figure 5 [25], which shows that the modal signal generator superimposes the excitation test signal u_s with controllable angular velocity, amplitude, and duration to the output of automatic voltage regulator (AVR). The u_s can generate the electromagnetic torque disturbance ΔT_e, and it excites a safe and controllable SSO in shafting system. After removing u_s, the change in generator speed depends on the result of the interaction between the mechanical damping and electrical damping of the system, and the damping can be identified using the spectrum analysis method on the basis of discrete Fourier transform. Since the influence of mechanical damping is not considered in this paper, the mechanical damping was set as 0. The main process of the injection method is depicted in Figure 6.

4.2. Influence of Access to Various Equipment on SG Damping Characteristics

In order to illustrate the feasibility of the above method, the system shown in Figure 1 was taken as an example. The transmission capacity of HVDC was 800 MW, and the output power of the SG was 600 MW. The installed capacity of a single DFIG was 1.5 MW, and the grid-connected number and wind speed of DFIG were 1500 and 4 m/s.

When t = 0 s, the excitation signal disturbance was added, and then the signal was removed at 15 s. The rotational speed curve of SG shafting in torsional vibration mode is shown in Figure 7. The logarithm of the ω_r amplitude was taken and curve-fitting was performed. The fitting results are shown in Figure 8, and the attenuation coefficients of TM1–TM4 were −0.1235, −0.007909, −0.02113, and −0.06042, respectively.

In order to further analyze the influence of wind speed and control parameters on the torsional vibration of the SG shafting system under the condition of fixed HVDC transmission capacity, the evolution law of the influence of various factors on the torsional vibration was studied by taking a single-DFIG model and a two-machine parallel model as examples.

4.2.1. Influence of Access to Single Equivalent DFIG on SG Shafting

The wind speed was 4 m/s, the transmission capacity of HVDC was 800 MW, the output power of the SG was 600 MW, and the installed capacity of a single DFIG was 1.5 MW. When 1500 units were connected in parallel, the attenuation coefficient of each SG torsional vibration mode was as shown in Figure 9 under the control parameters of different DFIG rotor converter current inner loops.

It can be seen from Figure 9 that, when the number of grid-connected DFIGs was given, and when K_p changed from 1 p.u. to 10 p.u., the attenuation coefficients of the torsional vibration modes of SG shafting generally showed a trend of decrease, whereby the attenuation coefficient of TM1 decreased 1.2-fold, the attenuation coefficients of TM2 and TM3 decreased 1.7-fold and 1.9-fold, and the attenuation coefficient of TM4 decreased 1.8-fold. The damping characteristics of the modes were enhanced, and, within the same time range, TM1 changed more obviously. As the integral coefficient of the current inner loop increased, the attenuation coefficients of the torsional modes of the SG did not change significantly.

The installed capacity of a single DFIG was 1.5 MW, the output power of the SG was 600 MW, and the output power of HVDC remained at 800 MW. Under different operating conditions of grid-connected DFIGs, the attenuation coefficients of the torsional mode of the SG were as shown in Figure 10.

It can be seen from Figure 10 that, as the wind speed increased from 4 m/s to 7 m/s, the attenuation coefficients of the torsional modes of SG shafts increases, and the mode damping characteristics weakened. Specifically, the attenuation coefficients of TM1–TM4 increased 1.6-fold, 1.8-fold, 1.5-fold, and 1.3-fold, respectively.

From all the above results, it can be seen that the K_i of DFIGs, as well as the operating conditions of the wind turbines, can affect the damping characteristics of the SG, while the setting of K_i has little effect on the SG damping characteristic.

4.2.2. Influence of Parallel Connection of Two DFIGs on SG Shafting

Considering that the actual wind turbines are often distributed in different regions, the wind speed and the number of grid-connected units in different regions may be quite different. Therefore, this section takes two DFIGs in parallel as an example, considering the geographical distribution of wind turbines; furthermore, it studies the influence of the control parameters and the change in operating conditions on SG shafting in a wind–thermal bundled system transmitted via HVDC.

When the operating condition of DFIGs was 4 m/s, the number of grid-connected units was 1500, and the output power of HVDC and SG was 800 MW and 600 MW, respectively, the K_p of the two DFIGs increased from 1 p.u to 10 p.u. The attenuation coefficient of TM1 of the SG shafting system is shown in

Table 1. Damping coefficient for TM1 of SG under different K_p.

	Group 1	1 p.u	3 p.u	6 p.u	10 p.u
Group 2
1 p.u		−0.1137	−0.125	−0.1291	−0.1262
3 p.u		−0.125	−0.1339	−0.1371	−0.1373
6 p.u		−0.1291	−0.1371	−0.1412	−0.1418
10 p.u		−0.1262	−0.1373	−0.1418	−0.1375

, and the other modes are shown in

Table A4. Damping coefficient for TM2 of SG under different K_p.

	Group 1	0.01	0.03	0.06	0.10
Group 2
0.01		−0.007161	−0.008144	−0.008696	−0.008758
0.03		−0.008144	−0.009209	−0.01047	−0.00998
0.06		−0.008696	−0.01047	−0.01127	−0.01153
0.10		−0.008758	−0.00998	−0.01153	−0.01187

,

Table A5. Damping coefficient for TM3 of SG under different K_p.

	Group 1	0.01	0.03	0.06	0.10
Group 2
0.01		−0.01933	−0.02213	−0.02481	−0.0267
0.03		−0.02213	−0.02492	−0.02751	−0.03029
0.06		−0.02481	−0.02751	−0.03117	−0.03423
0.10		−0.0267	−0.03029	−0.03423	−0.03682

and

Table A6. Damping coefficient for TM4 of SG under different K_p.

	Group 1	0.01	0.03	0.06	0.10
Group 2
0.01		−0.05832	−0.06379	−0.06968	−0.07934
0.03		−0.06379	−0.06961	−0.07486	−0.0854
0.06		−0.06968	−0.07486	−0.08635	−0.09565
0.10		−0.07934	−0.0854	−0.09565	−0.1061

(Appendix B); in the same way, the current inner loop control integral coefficients of the two DFIGs increased from 0.5 p.u to 3 p.u. The attenuation coefficient of TM1 is shown in

Table 2. Damping coefficient for TM1 of SG under different K_i.

	Group 1	0.5 p.u	1.5 p.u	2 p.u	3 p.u
Group 2
0.5 p.u		−0.1412	−0.1427	−0.1425	−0.1427
1.5 p.u		−0.1427	−0.1438	−0.1427	−0.1432
2 p.u		−0.1425	−0.1427	−0.1436	−0.143
3 p.u		−0.1427	−0.1432	−0.143	−0.1428

. Table 1 and Table 2 show that (1) when the K_p of group1 was constant, with the increase in K_p of the current inner loop of group2, the damping characteristics of the system were enhanced and the oscillation frequency decreased, (2) when the K_i of group1 was constant, with the increase in K_i of group2, the damping characteristics of the SG shafting system changed little, and (3) in the two-DFIG parallel system, the increase in proportional coefficient of the DFIG inner loop was conducive to the stability of TM1, and the change trend was basically the same as the conclusion of the single system.

When K_p was 3 p.u, the capacity of HVDC was 800 MW, and the SG output power was 600 MW, the operating condition of the DFIGs increased from 4 m/s to 7 m/s, and the attenuation coefficient of SG shafting torsional (TM1) was as shown in

Table 3. Damping coefficient for TM1 of SG under different operating conditions.

	Group 1	4 m/s	5 m/s	6 m/s	7 m/s
Group 2
4 m/s		−0.1339	−0.1376	−0.1219	−0.1071
5 m/s		−0.1376	−0.1345	−0.1224	−0.1085
6 m/s		−0.1219	−0.1224	−0.107	−0.09429
7 m/s		−0.1071	−0.1085	−0.09429	−0.08924

. It can be seen from Table 3 that with the increase in wind speed of Group 2, the damping characteristics of the system weakened and the oscillation frequency increased as the wind speed of group1 was constant. Therefore, in the two-machine parallel system, the increase in DFIG operating conditions was harmful to the stability of TM1, and the change trend was consistent with the conclusion of the single system.

4.2.3. Influence of SG Parameters on SG Shafting

The influence of SG parameters on the torsional vibration of SG shafting were not considered in the above analysis. This section mainly analyzes the influence of AVR and mechanical damping coefficient of SG on the torsional modes of each shafting. The AC exciter adopts the IEEE AC exciter module AC1A, and the transfer function block diagram is shown in Figure A1 (Appendix C).

When the wind speed of the DFIGs was 4 m/s, the number of grid-connected units was 1500, the HVDC transmission power was 800 MW, and the output power of the SG was 600 MW, under different AVR control parameters (K_A and T_A), the attenuation coefficients of each torsional mode of SG were as shown in

Table 4. Damping coefficient for TM1–TM4 of SG under different K_A (T_A = 0.02).

KA	TM1	TM2	TM3	TM4
50	−0.1206	−0.008813	−0.0217	−0.02252
150	−0.1207	−0.008823	−0.02172	−0.02255

and

Table 5. Damping coefficient for TM1–TM4 of SG under different T_A (K_A = 50).

TA	TM1	TM2	TM3	TM4
0.02	−0.1206	−0.008813	−0.0217	−0.02252
0.2	−0.1206	−0.008816	−0.0217	−0.02253

. The attenuation coefficients of each torsional mode of SG are shown in

Table 6. Damping coefficient for TM1–TM4 of SG under different T_A (K_A = 50).

Dm	TM1	TM2	TM3	TM4
0.05 p.u	−0.1762	−0.0154	−0.1021	−0.08473
0.1 p.u	−0.2128	−0.02149	−0.1182	−0.1009
0.2 p.u	−0.2487	−0.03367	−0.1311	−0.1332

under different mechanical damping coefficients (D_m).

From Table 4, Table 5 and Table 6, it can be seen that the change in AVR control parameters had little influence on each torsional mode of the SG. The mechanical damping coefficient had a great influence on the torsional mode of the SG, whereby larger coefficients were more conducive to the stability of each torsional mode.

5. Time-Domain Simulation Verification

In order to verify the correctness of the theoretical analysis, a simulation model of DFIGs, SGs, and HVDC connected to a grid system was built in EMTDC/PSCAD. Among them, the SG adopted the IEEE SSR first standard model, and the simulation parameters of SG were as shown in [17]. The HVDC adopted the CIGRE Benchmark Model, whose rated transmission power was 1000 MW. The simulation parameters of HVDC and DFIG are shown in

Table A10. Main parameters of the DFIG.

Parameter	Unit/p.u.
Nominal capacity of DFIG PDFIG	1.5
Voltage of grid ug	0.69
Resistance of the stator rs	0.0164
Reactance of the stator xs	0.255
Resistance of the rotor rr	0.0183
Reactance of the rotor xr	0.222
Mutual reactance xm	13.68
DC capacitor voltage udc	1.5
Direct current capacity Cdc	0.09
Inductance of rotor L’	0.005
Inductance of GSC L1	0.005

and

Table A11. Parameters of the HVDC controller.

Parameter	Unit/p.u.
Proportional coefficient of constant current controller KPr	1.0989
Integral coefficient of constant current controller Kir	1/0.01092
Proportional coefficient of turn-off angle controller KPi	0.7506
Integral coefficient of fixed turnoff angle controller Kii	1/0.0544
Proportional coefficient of measuring link Kmr	0.5
Integral coefficient of measuring link Tmr	0.0012
proportional coefficient of PLL KpPLL	10
Integral coefficient of PLL KiPLL	50
Proportional coefficient of constant current controller KPr	1.0989
Integral coefficient of constant current controller Kir	1/0.01092
Proportional coefficient of turn-off angle controller KPi	0.7506

(Appendix C).

5.1. Influence of Single Machine on Torsional Vibration of SG Shafting

5.1.1. Different Current Inner Loop Control Parameters

The wind speed of the DFIGs was 4 m/s, the number of grid-connected units was 1500, the HVDC transmission power was 800 MW, and the output power of the SG was 600 MW. A three-phase short circuit fault occurred in the system at t = 6 s, and the duration was 75 ms. The SG rotor speed response curve under the K_p of DFIGs is shown in Figure 11a. Time–frequency analysis was used to analyze the rotor speed data, and the time–frequency analysis results are shown in Figure 11b,c.

As can be seen from Figure 11, with the increase in K_p, the attenuation trend of TM1 changed most significantly compared with other modes, and the mode damping was enhanced. The time-domain simulation results were consistent with the analysis results in Section 4.2.1.

5.1.2. Different Operation Conditions

When the control parameters of the DFIGs were constant, the number of grid-connected units was 1500, and the output powers of the HVDC and the SG were fixed, the system had a three-phase short circuit fault at t = 6 s with a duration of 75 ms. The results of the rotor speed response curve and time–frequency analysis under different operating conditions of DFIGs are shown in Figure 12.

Comparing Figure 12b with Figure 12c, it can be concluded that, when the wind speed of DFIGs increased from 4 m/s to 7 m/s, the change trend of TM1 was more obvious than the other three modes and TM1 damping became weaker, which was not conducive to system stability. The change trend was consistent with the identification results in Figure 10.

5.2. Influence of Two-Machine Parallel System on Torsional Vibration of SG Shafting

The system parameters were the same as in Section 5.1. In this section, the two-machine parallel system was simulated and verified. The torsional vibration of the SG transmitted via an HVDC system was analyzed, accounting for the regional differences in wind turbines, caused by different current inner loop control parameters, operating conditions, and the number of grid-connected units. The results of the SG rotor speed response curve and the time–frequency analysis are shown in Figure 13 and Figure 14.

From the time-domain analysis, it can be seen that, when the K_p of the RSC in the DFIG increased and the operating conditions decreased, it was conducive to the stability of the system.

6. Conclusions

This paper proposed a torsional vibration analysis method for a system containing HVDC, DFIGs, and SGs, using the additional excitation signal injection method. It revealed the torsional modes of the SG shafting caused by DFIGs connected to the grid. The main conclusions are as follows:

(1) In a typical wind–thermal bundled system transmitted via HVDC, the electromagnetic transient process among wind turbines, SGs, and HVDC influences the electromechanical transient process of the system. There are some factors, such as the types of grid-connected wind turbines, main control parameters, operating conditions, and; the transmission power and control parameters of HVDC that affect the torsional modes of the SG to some extent.

(2) For the DFIG and SG system transmitted via HVDC, under the given HVDC transmission capacity, as the current inner loop control parameters of the RSC of the DFIG decreased (K_p = 10 p.u–1 p.u), the damping characteristics of each torsional mode of the SG decreased; as the operating conditions of the DFIG decreased (v = 7–4 m/s), the damping characteristics of each torsional mode gradually increased. With the increase in mechanical damping coefficient (D_m = 0.05 p.u–2 p.u), the damping of each torsional mode increased, which was beneficial to the stability of the system.