Influence of Active Power Output and Control Parameters of Full-Converter Wind Farms on Sub-Synchronous Oscillation Characteristics in Weak Grids

Abstract

Active power outputs of a wind farm connected to a weak power grid greatly affect the stability of grid-connected voltage source converter (VSC) systems. This paper studies the impact of active power outputs and control parameters on the subsynchronous oscillation characteristics of full-converter wind farms connected weak power grids. Eigenvalue and participation factor analysis was performed to identify the dominant oscillation modes of the system under consideration. The impact of active power output and control parameters on the damping characteristics of subsynchronous oscillation is analysed with the eigenvalue method. The analysis shows that when the phase-locked loop (PLL) proportional gain is high, the subsynchronous oscillation damping characteristics are worsened as the active power output increases. On the contrary, when the PLL proportional gain is small, the subsynchronous oscillation damping characteristics are improved as the active power output increases. By adjusting the control parameters in the PLL and DC link voltage controllers, system stability can be improved. Time-domain results verify the analysis and the findings.

1. Introduction

In recent years, as a clean, renewable and relatively proven technology, wind power generation has grown significantly in order to tackle the climate change and replace fossil fuels generators. By the end of 2019, the cumulative installed capacity of wind power worldwide reached 650 GW, of which 60.4 GW was newly added [1]. With the development of wind power and high voltage direct current transmission system (HVDC), subsynchronous interaction (SSI) has attracted the attention of academia and industry. The SSI is generally classified into the following three types: subsynchronous resonance (SSR), subsynchronous control interaction (SSCI) and subsynchronous torsional interaction (SSTI) [2]. In 2009, an SSI incident occurred in southern Texas, USA. A doubly fed induction generator (DFIG)-based wind farm was integrated into the grids via a high-series compensation transmission line. This caused a subsynchronous control interaction, resulting in a large number of wind turbine trips [3,4]. In 2012, the Guyuan wind farm in China also experienced the interaction between the control of DFIG and series compensation devices, causing the SSI event.

With the increase of grid-connected wind power capacity and the use of long-distance transmission lines, the support from the grids for wind farm connection is weakened [5]. There is a different type of interaction observed recently between full-converter wind farms and weak AC networks. In 2015, the permanent magnet synchronous generators based wind farm in Xinjiang, China, suffered from a severe oscillation event without series compensation. The oscillation frequency coincided with the torsional vibration frequency of the nearby thermal power unit, which led to the torsional vibration protection action of the thermal power unit resulting in generator shut down [6]. This type of interaction between full-converter wind farms and weak AC networks is also called subsynchronous oscillation (SSO), which is the topic investigated in this paper.

The dominant elements that affect the subsynchronous interaction characteristics in different scenarios of wind farms connected to the grids [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. References [7,8,9] established DFIG-based wind farms interconnected with the grids and analysed the influence of the number of wind turbine generators (WTGs), wind speed, series compensation, line resistance, and outer and inner loop control parameters on subsynchronous interaction. For instance, reference [8] points out that when the DFIG-based wind farm is connected to series capacitive compensated transmission systems, the system damping decreases with the rise of series compensation or the decrease of total line resistance. Meanwhile, the variations of series compensation also affect the oscillation frequency of subsynchronous interaction. As for SSO in the full-converter wind farm or the VSC connected to AC grid system, the eigenvalue analysis, impedance-based analysis and the complex torque coefficient approach are conducted in [6,10,11,12,13,14,15,16,17,18,19,20] to research the dominant elements that affect the SSO characteristics. AC system strengths, WTGs number, wind speed, converter control parameters, PLL parameters and aggregation characteristics of wind farms are considered in these works. The work in [6] indicates the SSO will occur with the decrease of the AC system strengths. And control parameters also have great impacts on the SSO characteristics. In addition, SSO caused by the interconnection of direct-drive wind farms via voltage source converter based high voltage direct current (VSC-HVDC) transmission system has been studied in the references [21,22,23,24,25,26,27,28]. These elements, including wind farm control parameters, HVDC control parameters, PLL parameters and filter parameters are analysed and the coordinated controller is designed.

Until recently, there were very few papers specifically analysing the impact of the active power output of wind farms on the SSO characteristics. However, the change of active power output during the operation of wind farms will have a more significant impact on system stability. References [17,18,19,20] established the model of full-converter wind farm integrated into the grids or the VSC connected to AC grid. The SSO characteristics of the system are studied, and the impact of active output is analysed. The works in [17,18,19] pointed out that as the active power output of wind farms increases, the damping of the SSO mode decreases. However, it is revealed in [20] that increasing the active power output of wind farms will increase the damping of the SSO mode and increase system stability. When the active power output is too small, the system will result in diverging oscillation and loss of system stability.

In the view of the impact of active power output on SSO characteristics, some studies identified that the greater the active power output, the worse the SSO damping characteristics will be [17,18,19]. However, some studies that found that the higher the active output, the better the SSO damping characteristics will be [20]. Meanwhile, the existing researches are based on a certain set of control parameter without considering the influence of different control parameters. Therefore, it is necessary to study further the relationship between active power output and damping characteristics of SSO mode.

This paper investigates the impact of active power output on SSO characteristics by a small-signal analysis based on analytical models. The correlation between the active power output and the damping of the SSO mode with different control parameters is analysed through dynamic modelling and linear system analysis. First, the critical factor that determines the correlation is identified. Then, based on the eigenvalue analysis results, the strategy to increase the damping of SSO mode and improve system stability is proposed. Case studies and time-domain simulation verify the analysis result.

The rest of the paper is organized as follows: Section 2 builds the dynamic model of the system with full converter wind farm connected to the AC grids. In Section 3, both eigenvalue analysis and calculation of participation factors are carried to study the impact of active power output on SSO characteristics. The correlation between the active power output and the damping of the SSO mode is analysed with different control parameters and the critical factors that affect the SSO characteristics are presented. Meanwhile, the strategy to improve the stability of the system is proposed. Section 4 presents case studies and time-simulation results. Finally, the brief conclusions are given in Section 5.

2. System Modeling

A full-converter wind model including wind turbine, synchronous generator (SG), machine-side converter (MSC), DC link, grid-side converter (GSC), phase-locked loop (PLL), and converter control system is considered. It is assumed that wind farms usually consist of the same type of wind turbines with similar control parameters and operating conditions. Therefore, a wind farm is represented by an equivalent wind turbine. The schematic diagram of grid-connected wind power system structure is shown in Figure 1. L_f1 and R_f1 represent the filter inductance and filter resistance, respectively. C₁ represents the reactive power compensation parallel capacitor. R₂+jX₂ represents the equivalent impedance of both 25 kV line and 220 kV line. R₃+jX₃ represents the equivalent impedance of the transmission line near the grids. v_pcc denotes the voltage of point of common coupling (PCC). v_grid denotes the infinite grid voltage. i₁ and i₂ are the grid-side output current and transmission line current, respectively. Since the grid-connected dynamics of full-converter mainly depends on the control features of GSC, this paper ignores the machine-side dynamics. The wind turbine, SG and MSC are simplified as constant power sources [6].

The following section will establish a dynamic mathematical model of the grid-connected system. There are two dq reference frames in the dynamic mathematical model, namely the PLL-based dq frame and the grid-based dq frame. The PLL-based reference frame aligns its d-axis with the PCC voltage space vector v_pcc through the PLL output phase. Meanwhile, the grid-based reference frame has its d-axis aligned with the grid voltage space vector v_grid [10,17]. Superscripts ‘c’ and ‘g’ represent variables in the PLL-based reference frame and the grid-based reference frame, respectively. Phasor diagram of the component in different reference frames is shown in Figure 2.

2.1. Modeling of DC-Link

Since the machine-side dynamics are ignored, it is assumed that the active power output of the generator remains constant and is represented by P_wind. The dynamic mathematical model can be obtained from the DC link active power balance equation as Equation (1).

$$C_{\mathrm{dc}}{V}_{\mathrm{dc}}\frac{d{V}_{\mathrm{dc}}}{dt}=P_{\mathrm{wind}}-P_{\mathrm{g}}$$

(1)

$$\frac{C_{\mathrm{dc}}{V}_{\mathrm{dc},\mathrm{base}}^2}{2P_{\mathrm{base}}^2}\frac{d{(V_{\mathrm{dc}}^{\mathrm{pu}})}^2}{dt}=P_{\mathrm{wind}}^{\mathrm{pu}}-P_{\mathrm{g}}^{\mathrm{pu}}$$

(2)

$$P_{\mathrm{g}}=v_{\mathrm{pcc},\mathrm{d}}^{\mathrm{c}}{i}_{1\mathrm{d}}^{\mathrm{c}}+v_{\mathrm{pcc},\mathrm{q}}^{\mathrm{c}}{i}_{1\mathrm{q}}^{\mathrm{c}}$$

(3)

P_g and P_base are the GSC power delivered to the grids and base power, respectively. V_dc and V_dc,base are expresses as DC voltage and rated DC voltage, respectively. Superscript ‘pu’ represents per unit variables. Subscripts ‘d’ and ‘q’ respectively notate the d-axis and q-axis components of variables. Hereafter the dc-link dynamic mathematical model is expressed by Equation (2). For convenience, the superscript ‘pu’ is omitted.

2.2. Outer and Inner Control Loop of GSC

The GSC control block diagram is shown in Figure 3. DC-link voltage control (DVC) and reactive power control are adopted for GSC, which contributes to balancing the power flow through DC link, maintaining DC-link voltage and operating at unit power factor for wind farm. The dynamic mathematical model of the outer and inner loop can be expressed as

$$\{\begin{array}{l}\frac{d{x}_1}{dt}=K_{\mathrm{idc}}(V_{\mathrm{dc}}^2-V_{\mathrm{dc},\mathrm{ref}}^2)\hfill \\ \frac{d{x}_2}{dt}=K_{\mathrm{ii}}(i_{1\mathrm{d},\mathrm{ref}}^{\mathrm{c}}-i_{1\mathrm{d}}^{\mathrm{c}})\hfill \\ \frac{d{x}_3}{dt}=K_{\mathrm{ii}}(i_{1\mathrm{q},\mathrm{ref}}^{\mathrm{c}}-i_{1\mathrm{q}}^{\mathrm{c}})\hfill \end{array}$$

(4)

$$\{\begin{array}{l}\begin{array}{c}{v}_{\mathrm{d}}^{\mathrm{c}}=K_{\mathrm{pdc}}{K}_{\mathrm{pi}}(V_{\mathrm{dc}}^2-V_{\mathrm{dc},\mathrm{ref}}^2)+K_{\mathrm{pi}}{x}_1-K_{\mathrm{pi}}{i}_{1\mathrm{d}}^{\mathrm{c}}+\\ x_2-\omega L_{\mathrm{f}1}{i}_{1\mathrm{q}}^{\mathrm{c}}+v_{\mathrm{pcc},\mathrm{d}}^{\mathrm{c}}\end{array}\hfill \\ v_{\mathrm{q}}^{\mathrm{c}}=K_{\mathrm{pi}}(i_{1\mathrm{q},\mathrm{ref}}^{\mathrm{c}}-i_{1\mathrm{q}}^{\mathrm{c}})+x_3+\omega L_{\mathrm{f}1}{i}_{1\mathrm{d}}^{\mathrm{c}}\hfill \end{array}$$

(5)

where x₁, x₂ and x₃ represent intermediate state variables. K_pdc and K_idc notate DVC proportional gain and integral gain, respectively. K_pi and K_ii are the proportional gain and integral gain of the inner current control loop, respectively. Subscript ‘ref’ denotes the system reference value.

2.3. Phase-Locked Loop Model

PLL uses the three-phase voltage at PCC bus as inputs to obtain the phase of the PCC voltage to achieve synchronization between the wind farm and the grids. The control block diagram of the PLL is illustrated in Figure 4. The PLL principle has been well documented [26] and will not be discussed here. ω₀ represents the rated angular frequency of the grids. Δω notates the frequency deviation. θ_pll is the voltage phase of the PLL output. K_ppll and K_ipll denote the PLL proportional gain and integral gain, respectively. PLL dynamic mathematical model can be expressed as

$$\{\begin{array}{l}\frac{d{x}_{\mathrm{pll}}}{dt}=K_{\mathrm{iPLL}}{v}_{\mathrm{pcc},\mathrm{q}}^{\mathrm{c}}\hfill \\ \frac{d\Delta {\theta }_{\mathrm{pll}}}{dt}={\omega }_0+\mathsf{\Delta }\omega \hfill \end{array}$$

(6)

2.4. Grid Dynamics

The grid dynamics mainly include shunt capacitor dynamics, filter inductance dynamics, and transmission line equivalent inductance dynamics. The dynamic mathematical model of the grid is established in the grid-based reference frame. The grid dynamic mathematical model can be written as Equation group (7):

$$\{\begin{array}{l}\frac{\mathrm{d}{i}_{1\mathrm{d}}^{\mathrm{g}}}{\mathrm{d}t}=\frac{1}{L_{\mathrm{f}1}}(v_{\mathrm{d}}^{\mathrm{g}}-v_{\mathrm{pcc},\mathrm{d}}^{\mathrm{g}}-R_{\mathrm{f}1}{i}_{1\mathrm{d}}^{\mathrm{g}}+{\omega }_0{L}_{\mathrm{f}1}{i}_{1\mathrm{q}}^{\mathrm{g}})\hfill \\ \frac{\mathrm{d}{i}_{1\mathrm{q}}^{\mathrm{g}}}{\mathrm{d}t}=\frac{1}{L_{\mathrm{f}1}}(v_{\mathrm{q}}^{\mathrm{g}}-v_{\mathrm{pcc},\mathrm{q}}^{\mathrm{g}}-R_{\mathrm{f}1}{i}_{1\mathrm{q}}^{\mathrm{g}}-{\omega }_0{L}_{\mathrm{f}1}{i}_{1\mathrm{d}}^{\mathrm{g}})\hfill \\ \frac{\mathrm{d}{i}_{2\mathrm{d}}^{\mathrm{g}}}{\mathrm{d}t}=\frac{1}{L_{\mathrm{g}}}(v_{\mathrm{pcc},\mathrm{d}}^{\mathrm{g}}-v_{\mathrm{grid},\mathrm{d}}^{\mathrm{g}}-R_{\mathrm{g}}{i}_{2\mathrm{d}}^{\mathrm{g}}+{\omega }_0{L}_{\mathrm{g}}{i}_{2\mathrm{q}}^{\mathrm{g}})\hfill \\ \frac{\mathrm{d}{i}_{2\mathrm{q}}^{\mathrm{g}}}{\mathrm{d}t}=\frac{1}{L_{\mathrm{g}}}(v_{\mathrm{pcc},\mathrm{q}}^{\mathrm{g}}-v_{\mathrm{grid},\mathrm{q}}^{\mathrm{g}}-R_{\mathrm{g}}{i}_{2\mathrm{q}}^{\mathrm{g}}-{\omega }_0{L}_{\mathrm{g}}{i}_{2\mathrm{d}}^{\mathrm{g}})\hfill \\ \frac{\mathrm{d}{v}_{\mathrm{pcc},\mathrm{d}}^{\mathrm{g}}}{\mathrm{d}t}=\frac{1}{C_1}(i_{1\mathrm{d}}^{\mathrm{g}}-i_{2\mathrm{d}}^{\mathrm{g}}+{\omega }_0{C}_1{v}_{\mathrm{pcc},\mathrm{q}}^{\mathrm{g}})\hfill \\ \frac{\mathrm{d}{v}_{\mathrm{pcc},\mathrm{q}}^{\mathrm{g}}}{\mathrm{d}t}=\frac{1}{C_1}(i_{1\mathrm{q}}^{\mathrm{g}}-i_{2\mathrm{q}}^{\mathrm{g}}-{\omega }_0{C}_1{v}_{\mathrm{pcc},\mathrm{d}}^{\mathrm{g}})\hfill \end{array}$$

(7)

R_g and L_g denote the total equivalent resistance and inductance of the grid, including the transformers and the transmission lines. The impedance from the PCC to the grid can be represented as a single impedance R_g + jω₀L_g, R_g = R_T1 + R_T2 + R₂ + R₃, L_g = L_T1 + L_T2 + L₂ + L₃.

3. Eigenvalue Analysis

3.1. Analysis of the Dominant Oscillation Mode

In this paper, a wind farm consisting of fifty 2 MW wind turbines connected to the AC grid through long-distance transmission lines is used as the target test system. The parameters of the system are listed in

Table 1. Parameters of the grid-connected system.

Parameter	Value (pu, SB = 100 MVA)
Transformer T1(575 V/25 kV)	XT1 = 0.06, RT1 = 0.006
Transformer T2(25 kV/220 kV)	XT2 = 0.065, RT2 = 0.0065
Long-distance transmission line	X2 = 0.525, R2 = 0.0525
Short-distance transmission line	X3 = 0.01, R3 = 0.001

. The short circuit ratio (SCR) of this system is 1.53, which indicates that the wind farm is connected to a very weak AC grid [27]. The parameters of the wind generator are shown in

Table 2. The parameters of a single wind generator.

Parameter	Value (pu, SB = 2 MVA)
Rated power	2 MW
Rated frequency	50 Hz
GSC filter	Xf1 = 0.15, Rf1 = 0.003, yc1 = 0.25
DC capacitor	0.09 F
Rated DC voltage	1100 V
DVC	Kpdc = 1.1, Kidc = 137.5
Current control	Kpi = 0.4758, Kii = 3.28
PLL	Kppll = 314, Kipll = 24,700

.

In the system dynamic mathematical model established in this paper, the state variables are x = [ i^g_1d, i^g_1q, i^g_2d, i^g_2q, v^g_pcc,d, v^g_pcc,q, x_pll, Δθ_pll, V_dc, x₁, x₂, x₃]. By linearizing the dynamic mathematical model at an operating condition x₀, the small signal model of the system can be established as Equation (8) shows.

$$\frac{d\mathsf{\Delta }\mathit{x}}{dt}=\mathit{A}\mathsf{\Delta }\mathit{x}$$

(8)

In Equation (8), A represents the eigenmatrix of the small signal model as shown in Appendix A and Δx denotes incremental state vector.

When the active power output of the wind farm is maintained at 0.8 pu, the eigenmatrix is used to calculate the eigenvalues of the system as shown in

Table 3. Eigenvalues of the weak grids-connected wind power system.

Mode	Eigenvalue
λ1,2	−569.33 ± j1764.69
λ3,4	−87.53 ± j836.15
λ5	−976.21
λ6,7	2.62 ± j199.47
λ8	−91.51
λ9,10	−15.89 ± j55.99
λ11	−6.90
λ12	−6.89

. It can be observed that there are four oscillation modes in the target system, of which λ_6,7 and λ_9,10 belong to the SSO mode. However, the real parts of the eigenvalues λ_6,7 are positive, which indicates that the mode exhibits negative damping and the system is unstable. For this mode, the participation factors of state variables are shown in Figure 5. In Figure 5, the first six state variables represent the dynamics of the grids and the last six state variables represent the dynamics of the wind farm. Therefore, this mode is related to both the grid dynamics and the wind farm dynamics and reflects the subsynchronous interaction between the AC grids and the wind farm. As far as the control loops are concerned, the participation factors of these state variables (Δθ_pll, x_pll, V²_dc, x₁) are higher. That is, PLL and DVC have a greater impact on this mode.

3.2. Impacts of the Active Power Outputs of the Wind Farm on Subsynchronous Oscillation Characteristics with Different Control Parameters

There are two main factors that affect the eigenvalues in the weak grids: one is active power output (operating condition), and the other is the control structure and control parameters. By calculating the participation factors, it can be seen that the PLL and the DVC loop have a greater impact on the dominant oscillation mode. In this section, the eigenvalue method will be used to analyse the impact of active power output on SSO characteristics with different control parameters. For convenience of expression, the following sections will use comparative gain to express the control parameters. The comparative gain represents a multiple of the pre-set value of the parameters given in Table 2.

3.2.1. Impacts of Active Power Outputs with Different PLL Proportional Gains

To evaluate this case, K_ppll is selected between 0.1 and 1.2 times of its pre-set value. When the active power output increases from 0.6 pu to 1.0 pu, the variations of the dominant eigenvalues with different K_ppll are shown in Figure 6 (only those parts are shown where the imaginary part is positive). When the value of K_ppll is large (e.g., when the factors are larger than 0.3 times), the eigenvalues move toward the right half plane (RHP) with the increase of the active power output, the mode damping decreases, and the system stability decreases. The active power output is negatively related to the mode damping. When the value of K_ppll is small (e.g., when the factors are smaller than 0.3 times), the eigenvalues move towards the left half plane (LHP) as the active power output increases. The active power output is positively correlated with the mode damping. There are only slight changes of the frequency of the SSO modes with different active power outputs.

When K_ppll takes these intermediate values, the correlation between the active power output and the mode damping will change from negative correlation to positive correlation with the decrease of K_ppll. When K_ppll takes the critical value, the real part of the dominant eigenvalues changes with the active power output as shown in Figure 7. Moreover, as depicted in Figure 7, the real part of the dominant eigenvalues gradually increases when the active power output increases from 0.6 pu to 0.75 pu, while the real part of the dominant eigenvalues decreases when the active power output increases from 0.75 pu to 1.0 pu. It can be found that when K_ppll takes the critical value, the mode damping decreases first and then increases as the active power output increases.

In addition, it can also be seen from Figure 6 that when the active power output is negatively correlated with the mode damping, the larger the value of K_ppll, the greater the variation of the mode damping with the active power output will be. That is, the stability of the system is more affected by the active power output. Conversely, when the active power output is positively correlated with the mode damping, the smaller the value of K_ppll, the stability of the system is more affected by the active power output.

From the results above, a conclusion can be drawn that when selecting a larger K_ppll, the active power output is negatively correlated with the damping of this SSO mode, while when selecting a smaller K_ppll, the active power output is positively correlated with the damping of this SSO mode. Moreover, there is a critical value K_ppll for correlation. Meanwhile, the closer K_ppll is to the critical value, the less the system stability is affected by the active power output.

3.2.2. Impacts of Active Power Outputs with Different PLL Integral Gain

In the two cases where K_ppll is selected to be larger (negative correlation) and smaller (positive correlation), the impact of the active power outputs on the mode damping with different K_ipll is observed. When the active power output increases from 0.6 pu to 1.0 pu, the dominant eigenvalue is plotted as shown in Figure 8. As shown in Figure 8a, with different K_ipll, the dominant eigenvalues move towards the RHP as the active power output increases and in effect decreasing the mode damping. At the same time, Figure 8b shows response with smaller K_ppll value. With different K_ipll, the dominant eigenvalues move towards the LHP as the active power output increases and the mode damping increases. It can be observed that adjusting K_ipll does not affect the correlation between the active power output and the damping of this SSO mode. However, under the same active power output condition, the damping of the SSO mode increases when K_ipll decreases. This is because the typical control parameters of a PLL are designed to ensure good phase tracking responses. However, in a weak grid, a fast PLL response will enlarge the interaction between the weak grid and the wind turbine converter, which will reduce the system stability. Therefore, a smaller integral gain is selected to improve the stability by compromising the PLL response characteristics.

3.2.3. Impacts of the Active Power Outputs with Different DVC Proportional Gain.

According to the above analysis on PLL parameters, four representative PLL parameters are selected as shown in

Table 4. Four different PLL parameters.

	Kppll	Kipll
Case 1	314 (the pre-set value)	24,700 (the pre-set value)
Case 2	314	24,700 × 0.8
Case 3	314 × 0.2	24,700
Case 4	314 × 0.2	24,700 × 0.8

. The impact of K_pdc on the correlation between the active power output and the damping of the dominant SSO mode is analysed with the four different PLL parameters. When the active power output increases from 0.6 pu to 1.0 pu, the variations of the dominant eigenvalues with different K_pdc are presented in Figure 9.

The K_ppll of PLL is selected to be larger in Figure 9a,b. Figure 9a,b show that with different K_pdc, the dominant eigenvalues move towards the RHP as the active power output increases, and the mode damping decreases. The K_ppll of PLL is selected to be smaller in Figure 9c,d and shows that with different K_pdc, the dominant eigenvalues move towards the LHP as the active power output increases, and the mode damping increases. Therefore, adjusting K_pdc does not change the correlation between the active power output and the mode damping. However, when the active power output is negatively correlated with the mode damping, the smaller the value of K_pdc, the greater the variation of the mode damping with the active power output. That is, system stability is more affected by the active power output (as shown in Figure 9a,b). Conversely, when the active power output is positively correlated with the mode damping, the greater the value of K_pdc, the greater the system stability affected by the active power output (as shown in Figure 9c,d).

Meanwhile, it can be found that the increase in K_pdc leads to increase the mode damping under the same active power output condition. When the damping of the SSO mode is small, the stability can be improved by increasing K_pdc. Comparing Figure 9a,c with Figure 9b,d, it can be seen that better and improved system stability can be achieved by simultaneously decreasing K_ipll and increasing K_pdc.

A conclusion can be drawn from the analysis that the correlation between the active power output and the damping of the dominant SSO mode mainly depends on K_ppll. When K_ppll is large, the active power output is negatively correlated with the damping of this SSO mode. When K_ppll is small, the active power output is positively correlated with the damping of the dominant SSO mode. Moreover, there is a critical range for K_ppll, in which SSO damping is near consistent irrespective to the change of active power variation. Meanwhile, the system stability can be improved by appropriately decreasing K_ipll or increasing K_pdc.

4. Case Study and Simulation Verifications

To validate the effectiveness of the conclusions in Section 3, the impact of the active power output on the eigenvalues of the system is analysed with different control parameters shown in Figure 1. At the same time, the detailed simulation model of the studied system is developed in Matlab/Simulink (2018a, MathWorks, Natick, MA, USA) for validation.

4.1. Verification of the Negative Correlation when the PLL Proportional Gain is Large

When the active power output is 0.6pu, the system has good stability through trial-and-error and adjustment of control parameters. The control parameters in this case are called the based-case as shown in

Table 5. Four different control parameters

Parameters		Based-Case	Group 1	Group 2	Group 3
PLL	Kppll	314	314	314 × 0.2	314 × 0.2
	Kipll	24,700	24,700 × 0.8	24,700 × 0.8	24,700 × 0.7
DVC	Kpdc	1.1	1.1 × 1.6	1.1 × 1.6	1.1 × 2
	Kidc	137.5	137.5	137.5	137.5
Inner current control loop		Kpi = 0.4758		Kii = 3.28

. When the control parameters of the based-case in Table 5 are used (with the larger K_ppll selected), the eigenvalue locus of the two SSO modes with the increase in active power output are plotted in Figure 10a. It is found that under the control parameters of the based-case, the eigenvalues λ_6,7 move to the RHP with the increase of active power output. The mode damping decreases continuously, and the system stability is weakened. When the active power output reaches 0.75 pu, λ_6,7 first crosses the imaginary axis and enters the RHP. The system becomes unstable. That is, there is a negative correlation between the active power output and the damping of the λ_6,7 mode. The results proved that when K_ppll is large, the active power output is negatively correlated with the damping of this SSO mode.

When the control parameters of group 1 in Table 5 are used, the impact of the active power output on the eigenvalues of the SSO modes is shown in Figure 10b. Clearly, the eigenvalues are always in the LHP during the variations of active power output. The damping of the λ_6,7 mode is always positive, and the system remains stable. Therefore, it is proved that the system stability can be improved by decreasing K_ipll and increasing K_pdc.

In order to verify the above analysis, an electromagnetic transient simulation model of the grid-connected wind farm system in Figure 1 is built in MATLAB/Simulink. The studied system adopts the control parameters of the based-case and the group 1 control parameters, respectively. At t = 3 s, the active power output of the wind farm increases from 0.7 pu to 0.75 pu. Responses of active power output, DC voltage and phase-a voltage of the PCC are observed and analysed. The corresponding time-domain simulation results are presented in Figure 11. It can be seen that when the active power output increases from 0.7 pu to 0.75 pu, the system with the control parameters of the based-case oscillates and becomes unstable. As shown in Figure 11a, the wind power has 31Hz oscillation. This further confirms the conclusion in Section 3 that the active power output is negatively correlated to the damping of this λ_6,7 mode with a lager K_ppll. Furthermore, the system with the group 1 control parameters is able to keep stable after disturbance, indicating that the damping of the SSO mode increased after adjusting K_ipll and K_pdc. The simulation results are consistent with the analysis results above.

4.2. Verification of the Positive Correlation when the PLL Proportional Gain Is Small

When the control parameters of group 2 in Table 5 are used (the smaller K_ppll is selected), the eigenvalue locus with varied active power output is depicted in Figure 12a. It can be seen that the eigenvalues λ_6,7 move to the LHP as the active power output increases. The damping of this SSO mode increases and the system stability is enhanced. Moreover, when the active power output of the wind farm is too small (less than 0.75 pu), the eigenvalue λ_6,7 will be in the RHP. The system will result in diverging oscillation and become unstable. That is, the active power output is positively correlated to the damping of the λ_6,7 mode. It is proved that when K_ppll is small, the active power output is positively correlated with the damping of this SSO mode.

Similarly, when the control parameters of group 3 in Table 5 are adopted, the impact of active power output on the eigenvalues of the SSO modes is shown in Figure 12b. In the process of active power output change, the eigenvalues are always in the LHP. The damping of the λ_6,7 mode is always positive, and the system remains stable. This analysis indicates again that the stability of the system can be enhanced by decreasing K_ipll and increasing K_pdc.

To validate the above analysis, group 2 and group 3 were selected as the control parameters of the system respectively. At t = 3 s, the active power output of the wind farm decreases from 0.8 pu to 0.7 pu. Figure 13 presents the corresponding time-domain simulation results. It can be observed that when the active power output decreases from 0.8 pu to 0.7 pu, the system using group 2 of control parameters is unstable and the oscillation frequency of the wind power is 21 Hz. This result matches the conclusion in Section 3 well, which demonstrates that there is a positive correlation between the active power output and the damping of this λ_6,7 mode with a smaller K_ppll. Meanwhile, the system using the control parameters of group 3 can remain stable after disturbance. This indicates that the damping of the SSO mode increases after adjusting the parameters. The simulation results are in accordance with the analysis results above.

4.3. Simulation Verification for a Complex System

To verify the analyses, simulation has been carried out for a complex system with different wind farm ratings, grid configurations and grid voltage levels, as shown in Figure 14. The system parameters are given in Figure 14. The control parameters are given in

Table 6. Four different control parameters.

Parameters		Group 4	Group 5	Group 6	Group 7
PLL	Kppll	314	314	314 × 0.2	314 × 0.2
	Kipll	24,700	24,700 × 0.8	24,700 × 0.8	24,700 × 0.7
DVC	Kpdc	1.1 × 1.2	1.1 × 1.8	1.1×1.8	1.1 × 2.0
	Kidc	137.5	137.5	137.5	137.5
Inner current control loop		Kpi = 0.4758		Kii = 3.28

. The simulation results are given in Figure 15 and Figure 16.

In Figure 15, a large K_ppll is used. Figure 15a gives the simulation results when the Group 4 control parameters are used. When the wind power increases, system tends to be unstable. If the control parameters are adjusted properly, by reducing K_ipll and increasing K_pdc, as in Group 5, the system can be maintained stable, as shown in Figure 15b.

In Figure 16, a small K_ppll is used. Figure 16a gives the simulation results when the Group 6 control parameters are used. When the wind power decreases, system tends to be unstable. If the control parameters are adjusted properly, by reducing K_ipll and increasing K_pdc, as in Group 7, the system can be maintained as stable, as shown in Figure 16b.

The simulation of the complex system further verifies the proposed theoretical analysis.

5. Conclusions

This paper investigates the influence of active power output on subsynchronous oscillation characteristics in weak grids. Compared to the research in the literature, this is the first of its kind to investigate the distinctive correlations between active power output and the damping of the SSO mode. The reasons for the different correlation between active power output and SSO mode damping have been explained. The findings and contributions of the study include:

The change of active power output in one direction can either improve or reduce SSO mode damping. This work identifies that the correlation between active power variation and damping mainly depends on the proportional gain of the phase-locked loop (PLL).

• When the PLL proportional gain is large, the active power output is negatively correlated with the damping of the SSO mode. When the PLL proportional gain is small, the active power output is positively correlated with the damping of the SSO mode. This clarifies the confusions in the understanding of the correlation between active power output and SSO damping.
• The PLL integral gain and the DC voltage control proportional gain have little influence on the correlation between the active power output and SSO damping. However, the system stability can be improved by appropriately retuning the PLL integral gain and the DC voltage control proportional gain.
• There is a critical range for the PLL proportional gain, in which SSO damping is near consistent irrespective to the change of active power variation. The influence of active power output on the stability can be minimized by selecting proper the PLL proportional gain first when the damping variation is at the critical range. Then adjustment of other parameters will improve the stability. This is valuable for engineering applications in designing PLL parameters.

For full-converter wind power systems, the grid-connected dynamics mainly depend on the control of GSC and are not affected by the wind turbine types. The conclusions of this paper are applicable to full-converter wind farms with induction generators or permanent magnet synchronous generators. DFIG is not covered in the study, and the analysis of the DFIG-based wind farms and the auxiliary control design will be undertaken in future research.