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

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.


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.
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.

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. Lf1 and Rf1 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. vpcc denotes the voltage of point of common coupling (PCC). vgrid 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 vpcc through the PLL output phase. Meanwhile, the grid-based reference frame has its d-axis aligned with the grid voltage space vector vgrid [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. 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.

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 Pwind. The dynamic mathematical model can be obtained from the DC link active power balance equation as Equation (1) Pg and Pbase are the GSC power delivered to the grids and base power, respectively. Vdc and Vdc,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.

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  Figure 2. Phasor diagram of component in different reference frame.

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).
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.

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

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. θpll is the voltage phase of the PLL output. Kppll and Kipll denote the PLL proportional gain and integral gain, respectively. PLL dynamic mathematical model can be expressed as

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):

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. ω 0 represents the rated angular frequency of the grids. ∆ω 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

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. θpll is the voltage phase of the PLL output. Kppll and Kipll denote the PLL proportional gain and integral gain, respectively. PLL dynamic mathematical model can be expressed as

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):

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): 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 + jω 0 L g , R g = R T1 + R T2 + R 2 + R 3 , L g = L T1 + L T2 + L 2 + L 3 .

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. 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. Table 1. Parameters of the grid-connected system.

Parameter
Value (pu, S B = 100 MVA) Transformer T1(575 V/25 kV) X T1 = 0.06, R T1 = 0.006 Transformer T2(25 kV/220 kV) X T2 = 0.065, R T2 = 0.0065 Long-distance transmission line 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 , ∆θ pll , V dc , x 1 , x 2 , x 3 ]. By linearizing the dynamic mathematical model at an operating condition x 0 , the small signal model of the system can be established as Equation (8) shows.
In Equation (8), A represents the eigenmatrix of the small signal model as shown in Appendix A and ∆x 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. 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 (∆θ pll , x pll , V 2 dc , x 1 ) are higher. That is, PLL and DVC have a greater impact on this mode. 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 (Δθpll, xpll, V 2 dc, x1) are higher. That is, PLL and DVC have a greater impact on this mode.

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

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.

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.
Energies 2020, 13, 5225 8 of 17 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 Kppll 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 Kppll 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 Kppll. When Kppll 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 Kppll takes the critical value, the mode damping decreases first and then increases as the active power output increases. 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. 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 Kppll 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 Kppll 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 Kppll. When Kppll 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 Kppll 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.

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

Impacts of Active Power Outputs with Different PLL Integral Gain
In the two cases where Kppll is selected to be larger (negative correlation) and smaller (positive correlation), the impact of the active power outputs on the mode damping with different Kipll 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 Kipll, 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 Kppll value. With different Kipll, the dominant eigenvalues move towards the LHP as the active power output increases and the mode damping increases. It can be observed that adjusting Kipll 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 Kipll 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. According to the above analysis on PLL parameters, four representative PLL parameters are selected as shown in Table 4. The impact of Kpdc 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 Kpdc are presented in Figure 9. According to the above analysis on PLL parameters, four representative PLL parameters are selected as shown in Table 4. 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 Kppll of PLL is selected to be larger in Figure 9a,b. Figure 9a,b show that with different Kpdc, the dominant eigenvalues move towards the RHP as the active power output increases, and the mode damping decreases. The Kppll of PLL is selected to be smaller in Figure 9c,d and shows that with different Kpdc, the dominant eigenvalues move towards the LHP as the active power output increases, and the mode damping increases. Therefore, adjusting Kpdc 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 Kpdc, 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 Kpdc, 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 Kpdc 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 Kpdc. Comparing Figure 9a,c with Figure 9b,d, it can be seen that better and improved system stability can be achieved by simultaneously decreasing Kipll and increasing Kpdc.  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 .

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.

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. 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. active power output is negatively correlated with the damping of this SSO mode. When Kppll is small, the active power output is positively correlated with the damping of the dominant SSO mode. Moreover, there is a critical range for Kppll, 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 Kipll or increasing Kpdc.

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.

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. When the control parameters of the based-case in Table 5 are used (with the larger Kppll 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 λ6,7 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, λ6,7 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 λ6,7 mode.
The results proved that when Kppll 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.
Energies 2020, 13, x 12 of 18 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 λ6,7 mode is always positive, and the system remains stable. Therefore, it is proved that the system stability can be improved by decreasing Kipll and increasing Kpdc.
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 λ6,7 mode with a lager Kppll. 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 Kipll and Kpdc. The simulation results are consistent with the analysis results above.

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 Kppll is selected), the eigenvalue locus with varied active power output is depicted in Figure 12a. It can be seen that the eigenvalues λ6,7 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 λ6,7 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 λ6,7 mode. It is proved that when Kppll is small, the active power output is positively correlated with the damping of this SSO mode.

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 . 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 λ6,7 mode is always positive, and the system remains stable. This analysis indicates again that the stability of the system can be enhanced by decreasing Kipll and increasing Kpdc.
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 λ6,7 mode with a smaller Kppll. 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.

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 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. 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 λ6,7 mode is always positive, and the system remains stable. This analysis indicates again that the stability of the system can be enhanced by decreasing Kipll and increasing Kpdc.
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 λ6,7 mode with a smaller Kppll. 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.

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

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. The simulation results are given in Figures 15 and 16. Energies 2020, 13, x 14 of 18 parameters are given in Figure 14. The control parameters are given in x L x R x R x R x R x R  Figure 14. The diagram of a complex system. In Figure 15, a large Kppll 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 Kipll and increasing Kpdc, as in Group 5, the system can be maintained stable, as shown in Figure 15b. In Figure 16, a small Kppll 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 Kipll and increasing Kpdc, as in Group 7, the system can be maintained as stable, as shown in Figure 16b.  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.
Energies 2020, 13, x 14 of 18 parameters are given in Figure 14. The control parameters are given in x L x R x R x R x R x R  Figure 14. The diagram of a complex system. In Figure 15, a large Kppll 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 Kipll and increasing Kpdc, as in Group 5, the system can be maintained stable, as shown in Figure 15b. In Figure 16, a small Kppll 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 Kipll and increasing Kpdc, as in Group 7, the system can be maintained as stable, as shown in Figure 16b. 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.
Energies 2020, 13, x 15 of 18 The simulation of the complex system further verifies the proposed theoretical analysis.

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

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.

Conflicts of Interest:
The authors declare no conflict of interest.