Research on broadband oscillation characteristics and additional damping control of new energy grid-connected systems

The large-scale grid connection of new energy sources makes the power system highly electronic. The resulting broadband oscillation problem is the main challenge hindering the safe and stable operation of current high-proportion new energy power systems. To ensure the safe operation of the power grid, it is important to carry out systematic research on broadband oscillation suppression measures. At present, an additional damping controller method is often used in broadband oscillation suppression, but the parameter optimization problem for the additional controller of new energy units and flexible DC transmission systems must still be further explored. This article first establishes a state–space model of direct-drive permanent magnet wind farms connected to the AC and DC grid systems and uses eigenvalue analysis to analyze the broadband oscillation modes involved. Then, additional damping controllers are designed for wind turbines and flexible DC transmission systems, and a parameter optimization model is established with the maximum damping ratio of the coupled oscillation mode as the optimization goal. The Improved Plant Growth Simulation algorithm (I-PGSA), based on cloud model theory with good global search and a fast calculation speed, is used to optimize and solve the additional damping controller parameters. In addition, considering that adjusting the controller parameters during the optimization process may cause the target mode frequency to change too much, so it is impossible to continue to optimize it, this article further introduces mode tracking technology to track the target mode to ensure that no target offset occurs during the optimization process. Finally, through eigenvalue analysis and power systems computer-aided design/electromagnetic transients, including DC (PSCAD/EMTDC) time domain emulation, the system is compared before and after governance, and it is verified that the proposed controller parameter optimization strategy can effectively suppress the system broadband oscillation problem.

1 Introduction

In recent years, in the face of the depletion of traditional energy sources, climate warming, and other problems, the Chinese government has proposed to improve energy structure and fully promote the development of clean energy. Wind energy has the advantages of being clean and pollution-free with large reserves and is an extremely important component of new energy. With the continuous expansion of wind power grid-connected capacity, flexible DC transmission is widely used in long-distance and large-capacity wind power transmission problems, with its advantages of small transmission loss, independent control of active and reactive power, and no phase change failure (Wang et al., 2014). At the same time, wind farm power consumption in closer cities often uses AC transmission. Therefore, the wind farm power transmission processes will often form a mixed AC and DC grid structure. However, the large number of power electronic components brought by wind power grid integration to the source and network sides of the grid will cause wide-frequency oscillation problems, which will affect the safe and stable operation of the grid.

Since the beginning of the 21st century, wide-frequency oscillation problems have occurred frequently in global wind power grid-connected projects: in 2009, a wide-frequency oscillation problem of approximately 22 Hz occurred in a wind farm in Texas, United States, due to an excessively high fault-induced string complement (Xing et al., 2020). In 2015, a multi-band wide-frequency oscillation phenomenon of 20–90 Hz occurred in the direct-driven wind farm of the Hami region, Xinjiang, in a trans-ac transmission project, which caused a protection shutdown of a nearby thermal power unit (Jiang and Wang, 2020; Ma et al., 2020). In addition, wide-frequency oscillations of wind farms via flexible direct or AC feeder systems have also occurred in Guangdong, Shanghai, Fujian, and other places (Wei et al., 2015; Zhao et al., 2012).

These accidents have made scholars pay more attention to the wide-frequency oscillation caused by wind farms and power electronic equipment brought into the grid by flexible transmission. Many studies have been conducted on the above problems both within China and in other countries, but they mostly focus on a single transmission mode. For example, Zhang et al. (2014) only considered the wind turbine alone through the AC and DC when designing the damping controller, and the coupled oscillation mode between the AC and DC systems was not studied. Yang et al. (2020) established a research model for the series compensation system of doubly fed wind turbines and analyzed the new type of sub-synchronous resonance problem caused by the integration of power electronic equipment. Gao et al. (2020) established a small signal model for the line-commutated, converter-based high-voltage direct current (LCC-HVDC) transmission system of a doubly fed wind farm and analyzed the sub-synchronous oscillation characteristics of the system through eigenvalue analysis and participation factor analysis. Sun et al. (2018) established a system equivalent model for doubly fed wind farms to directly enter the AC power grid via flexible input and studied the sub-synchronous oscillation characteristics of the system based on the impedance stability analysis method. Bian et al. (2018) studied the sub-synchronous oscillation mechanism between offshore doubly fed wind farms and flexible DC transmission systems and proposed oscillation suppression measures based on additional damping controllers using signal testing methods. Wang et al. (2020) studied the problem of sub-synchronous oscillation in the grid-connected system of direct-drive wind turbines using the impedance analysis method.

The above literature models wind farms as an equivalent wind turbine connected to the power grid but does not consider the impact of system structural parameters on coupled oscillations during the research process. Lyu and Cai (2015) established a small signal impedance analysis model for three-phase MMC on the AC side of a modular multilevel converter-based high-voltage DC (MMC-HVDC) grid-connected system for offshore wind farms. The frequency domain analysis method was used to study the influence of controller parameters on system stability. Lyu et al. (2016) further considered circulating current control in the impedance model of MMC (Lyu et al. (2017). Lyu et al. (2018) established a small signal impedance model of MMC based on the harmonic state–space modeling method and studied the phenomenon of sub-synchronous oscillation in the system. Amin et al. (2015) and Amin and Molinas (2017) proposed an active damping scheme that can increase the stability of wind turbines connected to a flexible DC transmission grid system.

The above literature examined the situation where wind farms are connected to the power grid separately through flexible DC or AC but did not consider the interactive effects between wind farms and AC/DC transmission modes under different connection methods. In summary, the current research on the parallel operation of different types of wind turbines in different transmission modes (e.g., AC and DC) is still insufficient, and the research on the interaction between AC and DC and the structural parameters of the system also must be in-depth.

In this article, we first construct a state–space model of multiple wind turbines connected to the power grid through AC and DC. The root-of-feature method is used to analyze the mechanism of the oscillatory modes that can easily destabilize the system and to determine the key factors that are affected. Then, an additional damping controller is designed for the corresponding oscillation modes. The parameters of the additional damping controller are used as the optimization variables, and the coupled oscillation modes are used as the improvement targets. The overall optimization model of the controller parameters is constructed by raising the damping ratio of the coupled oscillation modes over a certain threshold to achieve the optimization target of not affecting the stability of the system as much as possible. To keep the target from shifting during the optimization process of the algorithm, the mode tracking technique is introduced at the same time, and the I-PGSA is used to solve the model and calculate the optimal additional controller parameters. Finally, the optimization effect of control parameters is verified and compared by eigenvalue analysis and simulation.

2 Wind farm AC/DC grid-connected system modeling

Figure 1 shows the system structure of the wind farm connected to the grid via AC and DC, including the DC part and the AC part. According to the literature (Gao et al., 2018), the single-machine and multi-machine wind farm models show a high degree of similarity in the oscillation law. Therefore, in this article, the direct-drive permanent magnet synchronous generator (D-PMSG) aggregation model is used to simulate the whole wind farm. In the overall system, the permanent magnet direct-drive wind farm is connected to the flexible system through AC lines and transformers, and, finally, the power is delivered to the recipient grid.

Figure 1

Figure 1. Structural diagram of a wind farm connected to the power grid system through AC and DC.

2.1 Dynamic model of a permanent magnet direct-drive wind turbine

The D-PMSG consists of a wind turbine shaft system, a back-to-back converter, and a DC link. A dual-mass module is used in the modeling of the permanent magnet direct-drive wind turbine to describe the mechanical characteristics of the shaft system of the D-PMSG, and the linearized model of the shaft system is as follows:

$\left\{\begin{array}{l}p\Delta {\omega }_{\mathrm{a}}=T_{\text{Ja}}^{-1}\left[-\right.\left(D_{\text{aa}}+D_{\text{ab}}\right)\Delta {\omega }_{\mathrm{a}}+D_{\text{ab}}\Delta {\omega }_{\mathrm{b}}-K_{\text{ab}}\Delta {\delta }_{\mathrm{a}}+K_{12}\Delta {\delta }_{\mathrm{b}}+\Delta T_{\mathrm{a}}]\\ p\Delta {\omega }_{\mathrm{b}}=T_{\text{Jb}}^{-1}\left[-\right.\left(D_{\text{bb}}+D_{\text{ab}}\right)\Delta {\omega }_{\mathrm{b}}+D_{\text{ab}}\Delta {\omega }_{\mathrm{a}}+K_{\text{ab}}\Delta {\delta }_{\mathrm{a}}-K_{\text{ab}}\Delta {\delta }_{\mathrm{b}}+\Delta T_{\mathrm{b}}]\\ p\Delta {\delta }_{\mathrm{a}}={\omega }_{\mathrm{r}}\Delta {\omega }_{\mathrm{a}}\\ p\Delta {\delta }_{\mathrm{b}}={\omega }_{\mathrm{r}}\Delta {\omega }_{\mathrm{b}}\end{array}\right.(1)$

where p is the differential operation factor; subscripts a and b correspond to the wind turbine and generator, respectively; Δω_a, Δω_b, and T_Ja, T_Jb are the rotational speed and inertia time constant, respectively; ω_r is the rotational speed reference value; D_aa, D_bb, and D_ab are the self-damping coefficients as well as mutual damping coefficients; K_ab is the stiffness coefficient of the shaft system; ΔT_a and ΔT_b are the mechanical torque and electromagnetic torque, respectively; Δδ_a and Δδ_b are the electrical angular displacements of the wind turbine rotor and the generator rotor, respectively, relative to the synchronous rotating reference axis of the rated electrical speed.

The remaining parts of the dynamic model can be found in the literature (Chen et al., 2018).

2.2 Flexible dynamic modeling

The flexible model is composed of the rectifier inverter and its controller, the inverter and its controller, and the DC transmission link, and the specific structure is shown in Figure 2.

Figure 2

Figure 2. Flexible and straight topological structure.

The sender-side converter model and the receiver-side converter model set up the positive sequence d-q coordinate system with the grid voltage u_s1 and u_s2 as the reference, respectively. Then, the dynamic model of the voltage source converter-based high-voltage direct current transmission (VSC-HVDC) system is as follows:

$\left\{\begin{array}{l}{\omega }_{\mathrm{s}1\mathrm{b}}^{-1}{L}_{1}p{i}_{\mathrm{s}1\mathrm{d}}=-R_1{i}_{\mathrm{s}1\mathrm{d}}+{\omega }_{\mathrm{s}1}{L}_1{i}_{\mathrm{s}1\mathrm{q}}+u_{\mathrm{s}1\mathrm{d}}-u_{\mathrm{c}1\mathrm{d}}\\ {\omega }_{\mathrm{s}1\mathrm{b}}^{-1}{L}_{1}p{i}_{\mathrm{s}1\mathrm{q}}=-R_1{i}_{\mathrm{s}1\mathrm{q}}{-\omega }_{\mathrm{s}1}{L}_{\mathrm{s}1\mathrm{d}}+u_{\mathrm{s}1\mathrm{q}}-u_{\mathrm{c}1\mathrm{q}}\\ C{u}_{\text{dc}1}p{u}_{\text{dc}1}=\left(u_{\mathrm{c}1\mathrm{d}}{i}_{\mathrm{s}1\mathrm{d}}+u_{\mathrm{c}1\mathrm{q}}{i}_{\mathrm{s}1\mathrm{q}}\right)-i_{\text{dc}}{u}_{\text{dc}1}\\ L_{\text{dc}}p{i}_{\text{dc}}=u_{\text{dc}1}-u_{\text{dc}2}-R_{\text{dc}}{i}_{\text{dc}}\\ C{u}_{\text{dc}2}p{u}_{\text{dc}2}=-\left(u_{\mathrm{c}2\mathrm{d}}{i}_{\mathrm{s}2\mathrm{d}}+u_{\mathrm{c}2\mathrm{q}}{i}_{\mathrm{s}2\mathrm{q}}\right)-i_{\text{dc}}{u}_{\text{dc}2}\\ {\omega }_{\mathrm{s}2\mathrm{b}}^{-1}{L}_{2}p{i}_{\mathrm{s}2\mathrm{d}}=-R_2{i}_{\mathrm{s}2\mathrm{d}}+{\omega }_{\mathrm{s}2}{L}_2{i}_{\mathrm{s}2\mathrm{q}}+u_{\mathrm{s}2\mathrm{d}}-u_{\mathrm{c}2\mathrm{d}}\\ {\omega }_{\mathrm{s}2\mathrm{b}}^{-1}{L}_{2}p{i}_{\mathrm{s}2\mathrm{q}}=-R_2{i}_{\mathrm{s}2\mathrm{q}}-{\omega }_{\mathrm{s}2}{L}_2{i}_{\mathrm{s}2\mathrm{d}}+u_{\mathrm{s}2\mathrm{q}}-u_{\mathrm{c}2\mathrm{q}}\end{array}\right.(2)$

where i_s1d and i_s2d are the sending end current and the receiving end current; i_dc is the direct current; u_dc1 and u_dc2 are the sending end DC voltage and the receiving end DC voltage; ω_s1b and ω_s2b are the grid angular frequencies on the AC side of the sending end and the receiving end; R₁ and L₁ are the equivalent resistance and inductance of the outlet transformer of the sending end converter; R₂ and L₂ are the equivalent resistance and inductance of the outlet transformer of the receiving end converter, respectively.

Among them, the sender end controller adopts the fixed active and fixed reactive power control strategy, and the receiver end controller adopts the fixed DC voltage and fixed reactive power control strategy.

2.3 Interface equation establishment

In building the unified state–space model, the coordinate system of each module must be unified using a coordinate transformation because the reference coordinate system of each module is different. In the grid-side converter model of D-PMSG, the d₁-q₁ rotational coordinate system is set based on the grid voltage u_s, while in the feeder converter model of VSC-HVDC, the d₂-q₂ rotational coordinate system is set based on the voltage u_s1.

The interface dynamic equations, that is, the dynamic equations of the AC transmission line, are obtained by transforming the physical quantities in the VSC-HVDC feeder converter and its controller model based on the d₁-q₁ rotational coordinate system to the d₂-q₂ rotational coordinate system:

$\left\{\begin{array}{c}{u}_{\text{sd}}-u_{\mathrm{s}1\mathrm{d}}^{\prime }=R_1{i}_{\text{sd}}+{\omega }_{\mathrm{s}1\mathrm{b}}^{-1}{L}_{1}p{i}_{\text{sd}}-{\omega }_{\mathrm{s}1}{L}_1{i}_{\text{sq}}\\ u_{\text{sq}}-u_{\mathrm{s}1\mathrm{q}}^{\prime }=R_1{i}_{\text{sq}}+{\omega }_{\mathrm{s}1\mathrm{b}}^{-1}{L}_{1}p{i}_{\text{sq}}+{\omega }_{\mathrm{s}1}{L}_1{i}_{\text{sd}}\end{array}\right.(3)$

Among them,

$\left[\begin{array}{c}{u}_{\mathrm{s}1\mathrm{d}}^{\prime }\\ u_{\mathrm{s}1\mathrm{q}}^{\prime }\end{array}\right]=T\left[\begin{array}{c}{u}_{\mathrm{s}1\mathrm{d}}\\ u_{\mathrm{s}1\mathrm{q}}\end{array}\right](4)$

2.4 System-wide equation of state

The non-linear mathematical model of each module is converted into a state–space model by means of linearization as follows:

$\left\{\begin{array}{l}p\mathit{X}={\mathit{A}}_1\mathit{X}+{\mathit{B}}_1\mathit{Y}\\ \mathit{Y}={\mathit{C}}_1\mathit{X}\end{array}\right.(5)$

where X and Y are the system state variables and algebraic variables, respectively; A₁, B₁, and C₁ are the state matrix, the state and algebraic variable relationship matrix, and the algebraic and state variable relationship matrix, respectively.

Using the input–output relationship between modules, that is, the above interface equations, combined with the method of coordinate transformation, the state equations of different modules are converted to a unified coordinate system to form the state–space model of the whole system.

$\left\{\begin{array}{c}p\mathit{X}=\mathit{A}\mathit{X}+{\mathit{B}}_1\mathit{Y}\\ \mathit{A}=\left({\mathit{A}}_1+{\mathit{B}}_1{\mathit{C}}_1\right)\end{array}\right.(6)$

The dynamic equation of the whole system is as follows:

$\left\{\begin{array}{l}\mathit{X}=\left[{\mathit{X}}_{\text{PMSG}1},{\mathit{X}}_{\text{plls}0},{\mathit{X}}_{\text{PMSG}1-\text{Ctrl}},{\mathit{X}}_{\text{plls}1},{\mathit{X}}_{\text{HVDC}},{\mathit{X}}_{\text{plls}2},\right.\\ \left.\times {\mathit{X}}_{\text{HVDC}-\text{Ctrl}},{\mathit{X}}_{\text{PMSG}2},{\mathit{X}}_{\text{plls}3},{\mathit{X}}_{\text{PMSG}2-\text{Ctrl}}\right]\\ {\mathit{X}}_{\text{PMSG}1}={\left[\Delta {\omega }_3,\Delta {\omega }_4,\Delta {\delta }_3,\Delta {\delta }_4,\Delta i_{\mathrm{d}},\Delta i_{\mathrm{q}},\Delta u_{\text{dc}},\Delta i_{\text{sld}},\Delta i_{\text{slq}}\right]}^T\\ {\mathit{X}}_{\text{plls}0}={\left[\Delta {\theta }_{\text{plls}0},\Delta Z_{\text{plls}0}\right]}^T\\ {\mathit{X}}_{\text{PMSG}1-\text{Ctrl}}={\left[\Delta w_4,\Delta w_5,\Delta w_6,\Delta x_4,\Delta x_5,\Delta x_6\right]}^T\\ {\mathit{X}}_{\text{plls}1}={\left[\Delta {\theta }_{\text{plls}1},\Delta Z_{\text{plls}1}\right]}^T\\ {\mathit{X}}_{\text{HVDC}}={\left[\Delta u_{\text{dc}1},\Delta i_{\text{dc}},\Delta u_{\text{dc}2},\Delta i_{\mathrm{s}2\mathrm{d}},\Delta i_{\mathrm{s}2\mathrm{q}}\right]}^T\\ {\mathit{X}}_{\text{plls}2}={\left[\Delta {\theta }_{\text{plls}2,}\Delta Z_{\text{plls}2}\right]}^T\\ {\mathit{X}}_{\text{HVDC}-\text{Ctrl}}={\left[\Delta y_1,\Delta y_2,\Delta y_3,\Delta y_4,\Delta z_1,\Delta z_2,\Delta z_3,\Delta z_4\right]}^T\\ {\mathit{X}}_{\text{PMSG}2}={\left[\Delta {\omega }_1,\Delta {\omega }_2,\Delta {\delta }_1,\Delta {\delta }_2,\Delta i_{2\mathrm{d}},\Delta i_{2\mathrm{q}},\Delta u_{2\text{dc}},\Delta i_{2\text{sd}},\Delta i_{2\text{sq}}\right]}^T\\ {\mathit{X}}_{\text{plls}3}={\left[\Delta {\theta }_{\text{plls}3},\Delta Z_{\text{plls}3}\right]}^T\\ {\mathit{X}}_{\text{PMSG}2-\text{Ctr}\mathrm{l}}={\left[\Delta w_1,\Delta w_2,\Delta w_3,\Delta x_1,\Delta x_2,\Delta x_3\right]}^T\end{array}\right.(7)$

where X_PMSG1 and X_PMSG2 are the wind turbine generators (WTG) state variables; X_plls is the phase-locked loop state variable; X_PMSG1-Ctrl and X_PMSG2-Ctrl are the WTG control system state variables; and X_HVDC is the VSC-HVDC state variable. Δθ_pllsi and ΔZ_plli are the phase-locked loop i (i = 0, 1, 2, 3), respectively, negative state variables of the output phase and angle instantaneous change values of the controller; Δω₃ and Δω₄ are the electrical rotational speeds of the rotor of the PMSG1 wind turbine and generator, respectively; Δω₁ and Δω₂ are the electrical rotational speeds of the rotor of the PMSG2 wind turbine and generator, respectively; Δδ₃ and Δδ₄ are the electrical angular displacements of the rotor of the PMSG1 wind turbine and generator relative to the reference axis for synchronous rotation at rated electrical speed; Δδ₁ and Δδ₂ are the electrical angular displacement of PMSG2 wind turbine and generator rotor relative to the rated electrical speed synchronous rotation reference axis; Δi_d and Δi_q are the d- and q-axis components of the grid-connected current of PMSG1, respectively; Δi_2d and Δi_2q are the d- and q-axis components of the grid-connected current of PMSG2, respectively; Δu_dc and Δu_2dc are the DC voltages of PMSG1 and PMSG2, respectively; Δi_2sd and Δi_2sq are the d-axis and q-axis components of the PMSG2 outlet AC current, respectively; Δw₄ and Δw₆ are the d- and q-axis state variables of the PMSG1 machine-side current controller, respectively; Δw₁ and Δw₃ are the d- and q-axis state variables of the PMSG2 machine-side current controller, respectively; Δw₅ is the state variable of the PMSG1 machine-side speed controller; Δw₂ is the state variable of the PMSG2 machine-side speed controller; Δx₄ and Δx₆ are the d-axis state variables of PMSG1 grid-side voltage and the q-axis state variables of current controller, respectively; Δx₁ and Δx₃ are the d-axis state variables of PMSG2 grid-side voltage and the q-axis state variables of current controller, respectively; Δx₅ is the d-axis state variable of the PMSG1 grid-side current controller; Δx₂ is the d-axis state variable of the PMSG2 grid-side current controller; Δy₁ and Δy₃ are the d- and q-axis state variables of the VSC-HVDC external loop active and reactive power controller, respectively; Δy₂ is the d-axis state variable of the VSC-HVDC inner loop circuit controller; Δy₄ sends the d-axis state variable of the inner loop current controller at the end; Δz₁ and Δz₃ are the external loop voltage and reactive power controller state variables at the VSC-HVDC receiving end, respectively; and Δz₂ and Δz₄ are the d- and q-axis state variables of the VSC-HVDC receiver loop current controller, respectively.

3 Eigenvalue calculation and analysis

3.1 Eigenvalue calculation and participation factor analysis

The initial conditions of the system are set in Supplementary Table A1 in Supplementary Appendix A. To distinguish the intrinsic oscillation modes of the two wind turbines to avoid the resonance of the two oscillation modes with similar frequencies, the unit parameters of the two direct-driven wind turbines are slightly different. The specific parameter settings of each part are shown in Supplementary Tables A2–A6 in Supplementary Appendix A. The results of the analysis of the eigenvalues obtained under the initial conditions of the system are shown in Table 1.

Table 1

Table 1. Eigenvalue analysis results.

The damping ratio of the LFO-1 oscillatory mode is 2.93%, a value below the 3% damping ratio threshold, and the mode is weakly damped with a high risk of instability. In contrast, the damping ratio of the LFO-2 oscillation mode is 4.94%, which exceeds the damping ratio threshold, indicating that this mode is relatively stable. In the sub-synchronous frequency band, except for the SSO-1 mode with a lower damping ratio, the damping ratios of the SSO-2 to SSO-5 modes are all greater than the threshold value of 3%, thus possessing better stability. The damping ratio of the SupSO oscillation mode is 7.49%, which exceeds the threshold value of the damping ratio and, therefore, the mode has a high degree of stability.

In summary, the LFO-1 and SSO-1 modes have a greater risk of instability. Most of the remaining oscillation modes show high damping ratios and relatively stable performances. The participation factors of each oscillation mode are obtained by the eigenvalue analysis method, as shown in Table 2.

Table 2

Table 2. Ranking of participation factors for each oscillation mode.

In Table 2, Δi_s1dq, Δi_s2dq, and Δi_2sdq represent the communication network between D-PMSG and VSC-HVDC, the VSC-HVDC receiving end communication network, and the variables transmitted by D-PMSG through the communication network, respectively; Δi_s3dq is the sum of the VSC-HVDC output current and the D-PMSG AC line current; Δi_sq is the q-axis state variable of the AC network current component between D-PMSG and VSC-HVDC.

Analyses using Table 2 data show that

(1) The two sets of oscillation modes of LFO are caused by the shaft systems of D-PMSG1 and D-PMSG2, respectively, and the subsystems involved in the oscillations are their respective shaft systems.

(2) The SSO-3 and SSO-4 oscillation modes are caused by D-PMSG1 and D-PMSG2 themselves, which are mainly affected by the machine-side controllers of the permanent magnet direct-drive (PMD) wind turbine, and the subsystems involved in the oscillations are D-PMSG1 and D-PMSG2.

(3) The two SSO-1, SSO-2 oscillation modes are caused by the interaction between D-PMSG1 and the soft direct, and the subsystems involved in the oscillations are VSC-HVDC and the D-PMSG1 controller.

(4) The two sets of oscillation modes of the SSO-5 and SupSO are caused by the interaction between the flexible DC and the permanent magnet direct-drive feeder AC transmission line, and the subsystems involved in the oscillations are D-PMSG1 and VSC-HVDC.

3.2 Time domain simulation verification

The model shown in Figure 1 was built in the PSCAD/EMTDC environment, with a simulation duration of 10.0 s. When the system runs for 2.0 s, a single-phase ground short-circuit fault with a duration of 0.2 s is applied at the point of common coupling (PCC) of the receiving grid. The change in the active power output at the PCC is observed. The waveform of the output active power is shown in Figure 3a The output active power of the established system was analyzed using the Prony analysis method, and the results are shown in Figure 3b.

Figure 3

Figure 3. Active power waveform diagram and Prony analysis chart. (a) Active power oscillation waveform. (b) Prony analysis chart.

According to Figure 3, the frequency components of the output active power include sub-synchronous oscillation components at 1.99 Hz and 4.69 Hz. The amplitude of 1.99 Hz is relatively large, indicating strong oscillation. The results in Figure 3 are consistent with those in Table 2, verifying the correctness of the model established in the previous section.

4 Damping controller parameter optimization, model building, and solving

Given that the coupled oscillation mode is often influenced by multiple factors, it is difficult to adjust a controller parameter alone to achieve the effect of suppressing the oscillation, so more attention is needed for the governance of the coupled oscillation mode. Additional damping control has better results for the suppression of broadband oscillations, but when dealing with multimodal oscillation problems, the corresponding suppression links must be designed separately for the frequency characteristics of each modal oscillation, and the calculation process is more complicated. Therefore, this article further improves the suppression effect of the system’s broadband oscillations by optimizing the parameters of additional controllers on the basis of existing studies.

4.1 Damping controller design

A damping controller uses system electrical quantities (average unit speed, active power, reactive power, etc.) as the input signal. Different frequencies of the oscillating components extracted will be filtered through the bandpass filter. Then, the amplitude and the phase of this group of signals can be moved to form a signal that can offset or compensate for part of the original oscillating components of the damping to achieve the damping control.

4.1.1 Direct-drive turbine network side damping controller design

The design structure of the single-group damping controller is shown in Figure 4.

Figure 4

Figure 4. Design structure of a single-group damping controller.

The damping controller state equation is

$\left\{\begin{array}{l}p{X}_1=X_2\\ p{X}_2={-\omega }_{\mathrm{c}}^2{X}_1-2\xi {\omega }_{\mathrm{c}}{X}_2+{\omega }_{\mathrm{c}}p\Delta \omega \\ p{X}_3=\frac{K}{T_2}{X}_1+\frac{K{T}_1}{T_2}{X}_2-\frac{1}{T_1}{X}_3\\ p{X}_4=\frac{K{T}_3}{T_1{T}_4}{X}_1-\frac{K{T}_2{T}_3}{T_1{T}_4}{X}_2+\frac{T_2{-T}_3}{T_1{T}_4}{X}_3-\frac{T_2}{T_1{T}_4}{X}_4\end{array}\right.(8)$

$\left\{\begin{array}{l}1\le K\le 50,\\ 0\le T_i\le 5,i=1,2,3,4\end{array}\right.(9)$

where p = d/dt is the differential operator; ω_c is the cut-off frequency; ξ is the damping ratio; Δω is the rotational speed deviation signal; K is the gain coefficient; X₁ and X₂ are the output variables of the filters, respectively; X₃ and X₄ are the output variables of the phase compensation link, respectively; and T_i is the time constant.

Meanwhile, the damping controller designed to be multimodal has multiple sets of bandpass filters for being multimodal. The specific structure is shown in Figure 5.

Figure 5

Figure 5. Additional damping control for a permanent magnet direct-drive network side controller.

4.1.2 Design of a damping controller for flexible straightening

For flex-straight, the sender end controller triggers more complex broadband oscillations, that is, SSO-1 and SSO-2. It can be governed by additional damping control of the VSC-HVDC sender end controller. The control strategy is shown in Figure 6.

Figure 6

Figure 6. Flexible and straight additional damping control.

As seen in Figure 6, the additional damping control is to achieve the decoupling control of active and reactive power by adding damping controllers in the active and reactive power control loops, respectively, which helps to quickly calm the power oscillation phenomenon in the system. The parameter optimization for the additional damping of the VSC-HVDC is mainly for K_p, T_1p, T_2p, K_q, T_1q, and T_2q, and the other parameters can be set as T_wf = 10; T_R = 10; p = 2, respectively. In Figure 6, K_p and K_q are gain coefficients; T_1p, T_2p, T_1q, T_2q, T_wf, and T_R are time constants; and p is the power.

4.2 Damping controller parameter optimization model

To govern the coupled oscillation modes generated by the system with the additional damping controller, the parameter optimization model of the additional damping controller is established by taking the additional damping controller parameters as the optimization variables, setting the coupled oscillation modes as the improvement objective, and taking the damping ratio of the coupled oscillation modes as high as possible as the optimization objective.

$F=\mathit{max}\left({\displaystyle \sum _{i=1}^K}{\eta }_i{\xi }_i\right)(10)$

where K is the number of target modes; η is the weight coefficient of the target mode; ξ is the damping ratio of the target mode; and F is the objective function.

(1) The damping ratio of the mode to be improved should be greater than a certain threshold. The current study shows that a damping ratio below 3% can be regarded as a low-damping mode, but considering a certain margin, this article takes 5% as the threshold.

(2) The damping ratio of the remaining oscillation modes cannot be smaller than the pre-optimization damping ratio.

(3) Each controller must satisfy its specific parameter constraints, that is, keep its proportional and integral gains within the limited constraints. Because the additional damping controller strategies are different for a permanent magnet direct drive and a flexible direct drive, the parameter values are also different. For the flexible additional damping controller, K_pmax takes the value of 300, K_imax takes the value of 1, and K_pmin and K_imin take the value of 0. For the permanent magnet direct-drive additional damping controller, K_pmax takes the value of 50, K_imax takes the value of 5, and K_pmin and K_imin take the value of 0. Then, we can get the constraints on the parameter solution model of the damping controller, as shown in the following equation:

$\left\{\begin{array}{l}{k}_{\mathrm{p}-a.b-\mathit{min}}\le k_{\mathrm{p}-a.b}\le k_{\mathrm{p}-a.b-\mathit{max}}\\ k_{\mathrm{i}-a.b-\mathit{min}}\le k_{\mathrm{i}-a.b}\le k_{\mathrm{i}-a.b-\mathit{max}}\\ {\xi }_j\ge \mathit{max}\left({\lambda }_{j1},{\lambda }_{j2}\right)\\ {\xi }_k>{\lambda }_k\end{array}\right.(11)$

where k_p-a.b, k_p-a.b-max, and k_p-a.b-min are the bth proportional gain coefficients and their upper and lower limit values of the ath additional damping controller, respectively; k_i-a.b, k_i-a.b-max, and k_i-a.b-min are the bth integral gain coefficients and their upper and lower limit values of the ath additional damping controller, respectively; ξ_j, λ_j1, and λ_j2 are the damping ratio, damping ratio threshold, and initial damping ratio of the target oscillation mode j, respectively; and ξ_k and λ_k are the damping ratio and damping ratio threshold of the remaining oscillation mode k, respectively.

4.3 Mode tracking

Considering that the optimization objective is to maximize the damping ratio of the target modes, additional damping controllers are applied to the grid-side controller of the permanent magnet direct-drive wind turbine and the flexible and straight receiver-end controller. When solving the parameters of the additional damping controller, adjusting the controller parameters leads to changes in the eigenvalues of the system, which may cause significant changes in the frequency of the target oscillation mode. If the frequency of the target mode changes too much, it may lead to the loss of tracking of the optimization target in the subsequent optimization iterations, and thus, the optimization of this target mode cannot be continued. For this reason, a mode tracking technique is introduced in the optimization strategy proposed in this article to ensure that the optimization process always improves on the pre-selected oscillatory modes. The target mode is tracked through the information of the first five participant orderings of the mode, the left and right eigenvectors, and the oscillation frequency to ensure that the optimization is always performed for the given target mode.

For an oscillatory pattern m, the oscillatory pattern similarity is defined by drawing on the paradigm similarity search as

$S_m\left(R_a,R_b\right)=\sqrt{{\displaystyle \sum _{i=1}^K}{w}_i{\left[1-\left(\frac{a_i-b_i}{F_i}\right)\right]}^2}(12)$

where R_a and R_b are the sets of corresponding feature information (including the eigenvalues of the mode, the ordering of the participation factors, the frequency of oscillation, and the left and right eigenvectors and the angle between them) of the oscillatory mode m in the optimization process and the initial conditions, respectively; a_i and b_i are the ith feature information of R_a and R_b, respectively; w_i and F_i are the weights and the ranges of values of the ith feature information of the oscillatory mode m, respectively; S_m is the similarity of oscillation modes.

4.4 Optimal model solution based on the I-PGSA algorithm

PGSA is an algorithm to simulate the phototropism (optimal growth) of plants, but the conventional PGSA can only be used for problems in integer programming. Therefore, the cloud model theory is incorporated into the conventional method. The cloud model, with its ability to find certainty in uncertainty and to achieve dynamic changes in stability, simulates the fundamental mechanism of plant growth in nature well and can change the accuracy of the results of each calculation, targeting the problem that can only be dealt with in integer planning. At the same time, it adopts the method of varying the step size to improve the convergence speed and to shorten the computation time. The flow of optimizing the damping controller parameters using the I-PGSA algorithm is shown in Figure 7.

Figure 7

Figure 7. Damping controller parameter optimization flowchart.

5 Simulation example

5.1 Eigenvalue analysis and comparison

For SSO-1, SSO-2 (coupled oscillation mode between flexible direct and permanent magnet direct-drive wind turbines), its related controllers (grid-side converter (GSC), sending-end controller (SEC)), and additional damping controllers are governed, and its calculation results are compared with those of the undamped controller to determine the feasibility of optimization. The initial parameters of the additional damping controller are shown in Table 3. In Table 3, K_s1 and K_s2 are the gain coefficients of the GSC additional damping controller; T_s11, T_s12, T_s13, T_s14, T_s21, T_s22, T_s23, and T_s24 are the time constants of the GSC additional damping controller.

Table 3

Table 3. Initial parameters of the GSC additional controller.

I-PGSA is used to solve the above controller parameters, with the maximum number of iterations being 20; the other parameters of the system remain unchanged. The final iteration results are shown in Table 4, while the eigenvalue calculations are performed under this parameter, and the results are shown in Table 5 and Figure 8.

Table 4

Table 4. Additional controller optimized parameters.

Table 5

Table 5. Initial parameters of the GSC additional controller.

Figure 8

Figure 8. Comparison of the damping ratios of various oscillation modes before and after governance.

From Table 5 and Figure 8, it can be seen that, in the absence of the additional damping controller, SSO-1, SSO-2 oscillation mode damping is relatively low. With the additional damping controller parameters gained through the optimization, the SSO-1, SSO-2 oscillation mode damping ratio has a considerable increase. The impact of the remaining oscillation mode is smaller, and it is possible to accurately sort out some of the oscillation modes. Therefore, it has better feasibility.

The additional damping controller initial parameters for the SSO-5, SupSO (coupled oscillation mode between AC and DC subsystems) and its related control system’a (flexible direct receiver control system) additional damping controller for governance are shown in Table 6. The final iteration results are shown in Table 7. The eigenvalue calculations are carried out under this parameter, and the results are shown in Table 8 and Figure 9.

Table 6

Table 6. Additional controller initial parameters.

Table 7

Table 7. Additional controller optimized parameters.

Table 8

Table 8. Comparison of eigenvalues before and after adding an additional damping controller.

Figure 9

Figure 9. Comparison of damping ratios of various oscillation modes before and after governance.

As can be seen from Table 8 and Figure 9, when there is no additional damping controller, the damping ratio of the SSO-5 and SupSO oscillation modes is relatively low. After the optimization of the parameters of the additional damping controller, the damping ratio of the SSO-5 and SupSO oscillation modes has risen considerably. The impact on the rest of the oscillation modes is small, so that it is feasible to accurately sort out certain oscillation modes.

5.2 Time domain simulation verification

The model shown in Figure 1 was built in PSCAD/EMTDC software, and a damping controller was attached to the grid-side controller of the permanent magnet direct-drive wind turbine. The PCC of the grid was set at the receiving end to cause a single-phase ground short-circuit fault at 2–2.2 s. The active power output of the system was observed. The comparative analysis chart of the obtained active power is shown in Figure 10.

Figure 10

Figure 10. Comparison analysis chart of active power.

From Figure 10, it can be seen that the system is oscillating, and it is not easy to converge when a small disturbance is added before the damping controller is put into use. However, after the damping controller is put into use and the parameters are optimized, the oscillation of the system is quickly calmed after a small disturbance is added, which verifies the feasibility of the damping controller design method in this article.

The damping controller is attached to the flexible direct receiver controller, and the same simulation conditions as above are set. The resulting active power comparison analysis is shown in Figure 10.

From Figure 11, it can be seen that after the damping controller is put into operation and the parameters are optimized, adding small disturbances does not flatten the oscillation of the system. Considering that the damping ratios of the coupled oscillation modes related to the receiving end of the VSC-HVDC are affected by various factors, adjusting only the parameters of the VSC-HVDC controller may improve a single oscillation mode, but it may also deteriorate other oscillation modes, thus obtaining the simulation results. However, the previous verification showed that adjusting the parameters of the additional damping controller can effectively improve both sets of coupled oscillation modes simultaneously. It also proved that the damping controller design method proposed in this article has excellent governance capabilities for coupled oscillation modes.

Figure 11

Figure 11. Comparison analysis chart of active power.

6 Conclusion

In this article, the state–space model of a wind farm with AC/DC access to a grid system is established, and the mechanism analysis of the oscillatory modes, which are more likely to destabilize the system, is carried out by using the eigenroot method to determine the key factors affected by it. Then, an additional damping controller is designed, and its parameters are used as the optimization variables. The coupled oscillation mode is used as the improvement objective to achieve a faster optimization model solution through the I-PGSA algorithm. To keep the objective from shifting during the optimization process of the algorithm, a mode tracking technique is also introduced to track the target mode. Finally, the control parameter optimization effect is verified and compared by eigenvalue analysis and time domain simulation. The conclusion is as follows:

(1) When governing the oscillation modes of SSO-1 and SSO-2, the optimized additional controller parameters K_s1, T_s11, T_s12, T_s13, and T_s14 are 23.66, 0.0211, 0.0157, 0.0215, and 0.0137, respectively. The parameters K_s2, T_s21, T_s22, T_s23, and T_s24 are 27.43, 0.0187, 0.0253, 0.0176, and 0.0235, respectively. After parameter optimization, the damping ratio of the SSO-1 oscillation mode increased from 0.25% to 3.25%, and the damping ratio of the SSO-2 oscillation mode increased from 3.67% to 4.78%.

(2) When governing the oscillation modes, the optimized parameters K_p, T_1p, and T_2p of the additional controller are 35.26, 0.0385, and 0.0083, respectively, and the parameters K_q, T_1q, and T_2q are 125.23, 0.0074, and 0.0281, respectively. After parameter optimization, the damping ratio of the SSO-5 oscillation mode increased from 7.45% to 10.76%, and the damping ratio of the SupSO oscillation mode increased from 7.49% to 13.85%.

The simulation results show that this method can calculate the optimal damping controller parameters to improve the damping and has a good ability to suppress the system-wideband oscillation.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

Author contributions

QF: Formal Analysis, Methodology, Resources, Writing – original draft. ZD: Conceptualization, Resources, Visualization, Writing – original draft. DZ: Software, Validation, Writing – original draft. XGY: Data curation, Validation, Writing – original draft. AP: Investigation, Project administration, Writing – review and editing. JZ: Data curation, Writing – review and editing. XY: Supervision, Writing – review and editing.

Funding

The author(s) declare that financial support was received for the research and/or publication of this article. This research was funded by the project “State Grid Corporation of China Science and Technology Project Funding” (No. 520940230036).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors declare that this study received funding from State Grid Corporation of China. The funder had the following involvement in the study: study design, collection, analysis, interpretation of data, the writing of this article, and the decision to submit it for publication.

Generative AI statement

The author(s) declare that no Generative AI was used in the creation of this manuscript.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fenrg.2025.1531338/full#supplementary-material

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