Impedance Modeling Based Method for Sub/Supsynchronous Oscillation Analysis of D-PMSG Wind Farm

Subsynchronous oscillation (SSO) is a critical issue for the direct-drive permanent magnet synchronous generator (D-PMSG) based wind farm integrated to a weak onshore AC grid. To analyze the mechanism of the SSO phenomenon of D-PMSG based wind farm, widely used impedance-based stability analysis method is utilized in this paper. First, the impedance model based on the harmonic linearization theory of grid-connected D-PMSG is proposed, and the mechanism of sub/supsynchronous currents coupling is analyzed quantitatively for the first time. Then, based on the impedance model and relative stability criterion, the influence of wind farm operating parameters and grid impedance on stability is discussed. Simulations are carried out to verify the correctness of theoretical analysis.


Introduction
Recently, renewable energy generation has been developing rapidly all around the world due to the depletion of traditional energy and serious environmental pollution. Wind and solar energy with the benefits of cleanness, short construction cycle, and low running cost, have become the promising energy source. For practical power grids, the doubly fed induction generator (DFIG) and the direct-drive wind generator (direct drive permanent magnet synchronous generator (D-PMSG)) are the most applied generators for wind turbines due to the advantages of variable speed-based operation and independent control of active and reactive power [1]. Compared to DFIG, D-PMSG uses full power converters and has lower maintenance cost. The back-to-back pulse width modulation (PWM) converter isolates the generator from the grid, which improves the low voltage ride through ability. Because of this, D-PMSG based wind farms get a wide range of applications.
Due to the reverse distribution of the energy and power load in China, the renewable energy, mainly based on the large-scale of wind and solar power, long-distance high voltage direct current (HVDC) transmission system and series compensation transmission lines are widely utilized in nowadays power grid. However, previous researches pointed out that subsynchronous oscillation (SSO) is a potential problem in this series-compensated and HVDC transmission system [2,3]. On the other hand, with the increase of the wind energy, the short circuit ratio (SCR) at the point of common coupling (PCC) will decrease since the wind farm cannot provide enough support to the power grid, which will worsen the stability of power grid.
The SSO of grid-connected wind farm has been detected worldwide, for instance, in the wind farms of USA (Texas, 2009), China (GU yuan, Hebei Province, 2011) [4,5]. The above events are all due

Main Contribution
Most existing literature assumed that the PCC voltage perturbation at f p will only bring the same frequency error components in Park's transformation. Moreover, the output current harmonic component at coupled frequency 2f 1 − f p was neglected. However, these assumptions are obviously incorrect according to time-domain simulation results.
Based on the existing researches, this paper constructs the impedance model of grid-connected D-PMSG by using harmonic linearization small-signal modeling method. By reasonable simplification, the outer-loop control of converter is not considered [26], and the converter model is treated as voltage-controlled voltage source. It is worth mentioning that the frequency coupling of PLL and the relationship between sub/supsynchronous coupling currents are analyzed quantitatively. The simplified PWM converter and related theoretical analysis were all verified by simulation then. Furthermore, based on the correct impedance model and relative stability criterion, the impedance sensitivity of wind farm operation parameters and the influence on system stability are discussed. Finally, a time-domain simulation of SSO appearance when SCR decreases is carried out, which is theoretically explained.

Impedance Characteristics Analysis of D-PMSG
This section constructs the impedance model of grid-connected D-PMSG based wind farm. The structure diagram of the wind farm integrated in weak AC grid is presented first. Secondly, two assumptions are adopted to simplify the D-PMSG to a grid-connected converter. Then, the small-signal model of PLL considering frequency coupling is proposed, which is a key step to apply the harmonic linearization in different reference frame. Based on the PLL model, harmonic components in current control loop could be further obtained. Another simplification is about the PWM converter model. Finally, the impedance of D-PMSG is calculated and the relationship between coupled currents is analyzed.

Simplified Model of D-PMSG
The structure diagram of D-PMSGs integrated in a weak AC grid is shown in Figure 1. The wind farm in Xinjiang consists of hundreds of 1.5 MW D-PMSGs. Every unit is radially connected to 35 kV bus through a 0.69/35 kV transformer. Then, the electric power is collected at the 35-kV substation and next transmitted along higher voltage level transmission lines, to the 750-kV power grid 200 km away. The thermal synchronous power units are also connected to the common bus. Extra power is transmitted along HVDC to Eastern China. Since the transmission line has a long-distance and multi-stage transformers boost the voltage, the grid side equivalent impedance is as large as 0.4 pu [21]. The SCR is 2.5 (<3) and the grid wind farm integrate into belongs to weak AC grid [27].
Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 16 model of grid-connected converter. However, this model was too complicated and the relationship of sub/supsynchronous currents was not analyzed either.

Main Contribution
Most existing literature assumed that the PCC voltage perturbation at fp will only bring the same frequency error components in Park's transformation. Moreover, the output current harmonic component at coupled frequency 2f1 − fp was neglected. However, these assumptions are obviously incorrect according to time-domain simulation results.
Based on the existing researches, this paper constructs the impedance model of grid-connected D-PMSG by using harmonic linearization small-signal modeling method. By reasonable simplification, the outer-loop control of converter is not considered [26], and the converter model is treated as voltage-controlled voltage source. It is worth mentioning that the frequency coupling of PLL and the relationship between sub/supsynchronous coupling currents are analyzed quantitatively. The simplified PWM converter and related theoretical analysis were all verified by simulation then. Furthermore, based on the correct impedance model and relative stability criterion, the impedance sensitivity of wind farm operation parameters and the influence on system stability are discussed. Finally, a time-domain simulation of SSO appearance when SCR decreases is carried out, which is theoretically explained.

Impedance Characteristics Analysis of D-PMSG
This section constructs the impedance model of grid-connected D-PMSG based wind farm. The structure diagram of the wind farm integrated in weak AC grid is presented first. Secondly, two assumptions are adopted to simplify the D-PMSG to a grid-connected converter. Then, the smallsignal model of PLL considering frequency coupling is proposed, which is a key step to apply the harmonic linearization in different reference frame. Based on the PLL model, harmonic components in current control loop could be further obtained. Another simplification is about the PWM converter model. Finally, the impedance of D-PMSG is calculated and the relationship between coupled currents is analyzed.

Simplified Model of D-PMSG
The structure diagram of D-PMSGs integrated in a weak AC grid is shown in Figure 1. The wind farm in Xinjiang consists of hundreds of 1.5 MW D-PMSGs. Every unit is radially connected to 35 kV bus through a 0.69/35 kV transformer. Then, the electric power is collected at the 35-kV substation and next transmitted along higher voltage level transmission lines, to the 750-kV power grid 200 km away. The thermal synchronous power units are also connected to the common bus. Extra power is transmitted along HVDC to Eastern China. Since the transmission line has a long-distance and multistage transformers boost the voltage, the grid side equivalent impedance is as large as 0.4 pu [21]. The SCR is 2.5 (<3) and the grid wind farm integrate into belongs to weak AC grid [28]. As shown in Figure 2, grid-connected D-PMSG unit consists of wind turbine, permanent magnet synchronous generator, machine side converter (MSC), and grid side converter (GSC). The MSC controls the rotor speed to track the maximum wind power. The GSC is controlled to maintain the DC-link voltage and the output power factor.  As shown in Figure 2, grid-connected D-PMSG unit consists of wind turbine, permanent magnet synchronous generator, machine side converter (MSC), and grid side converter (GSC). The MSC controls the rotor speed to track the maximum wind power. The GSC is controlled to maintain the DC-link voltage and the output power factor. It is well understood that the small signal disturbance from the GSC cannot transmit to the MSC since the DC capacitor decouples the interactions between two converters. Therefore, the generator and the MSC is usually simplified as controlled current source when applying the small signal stability analysis. On the other hand, the outer loops of VSC control, DC voltage control, and reactive power control are proved not to take part in the dynamics under frequency band of SSO, 10~100 Hz approximately [26]. Consequently, the impedance model constructed in this paper mainly concerns the influence of GSC control parameters, operating condition. The structure diagram of simplified grid-connected D-PMSG system is shown in Figure 3. For the reasons above, Vdc is assumed constant in this study and the active and reactive current references (Idref and Iqref) are assumed constant.
, ...... Phase voltages are denoted as va, vb, and vc, while phase currents as ia, ib, and ic. In d-q axis, recall the current as id, iq. The ABC/dq transformation reference angle θpll is the output of PLL, whose function is obtaining the phase angle of AC voltage. The block diagram of PLL is depicted in Figure  4, where HPLL(s) is the PLL regulator which uses a PI controller and an integrator. Current errors are adjusted by the current controller and added with feedforward compensation. The results vdref, vqref are the dq-domain voltage references, which are transformed to varef, vbref, and vcref for PWM. It is well understood that the small signal disturbance from the GSC cannot transmit to the MSC since the DC capacitor decouples the interactions between two converters. Therefore, the generator and the MSC is usually simplified as controlled current source when applying the small signal stability analysis. On the other hand, the outer loops of VSC control, DC voltage control, and reactive power control are proved not to take part in the dynamics under frequency band of SSO, 10~100 Hz approximately [26]. Consequently, the impedance model constructed in this paper mainly concerns the influence of GSC control parameters, operating condition. The structure diagram of simplified grid-connected D-PMSG system is shown in Figure 3. For the reasons above, V dc is assumed constant in this study and the active and reactive current references (I dref and I qref ) are assumed constant. As shown in Figure 2, grid-connected D-PMSG unit consists of wind turbine, permanent magnet synchronous generator, machine side converter (MSC), and grid side converter (GSC). The MSC controls the rotor speed to track the maximum wind power. The GSC is controlled to maintain the DC-link voltage and the output power factor. It is well understood that the small signal disturbance from the GSC cannot transmit to the MSC since the DC capacitor decouples the interactions between two converters. Therefore, the generator and the MSC is usually simplified as controlled current source when applying the small signal stability analysis. On the other hand, the outer loops of VSC control, DC voltage control, and reactive power control are proved not to take part in the dynamics under frequency band of SSO, 10~100 Hz approximately [26]. Consequently, the impedance model constructed in this paper mainly concerns the influence of GSC control parameters, operating condition. The structure diagram of simplified grid-connected D-PMSG system is shown in Figure 3. For the reasons above, Vdc is assumed constant in this study and the active and reactive current references (Idref and Iqref) are assumed constant. Phase voltages are denoted as va, vb, and vc, while phase currents as ia, ib, and ic. In d-q axis, recall the current as id, iq. The ABC/dq transformation reference angle θpll is the output of PLL, whose function is obtaining the phase angle of AC voltage. The block diagram of PLL is depicted in Figure  4, where HPLL(s) is the PLL regulator which uses a PI controller and an integrator. Current errors are adjusted by the current controller and added with feedforward compensation. The results vdref, vqref are the dq-domain voltage references, which are transformed to varef, vbref, and vcref for PWM. Phase voltages are denoted as v a , v b, and v c , while phase currents as i a , i b , and i c . In d-q axis, recall the current as i d , i q . The ABC/dq transformation reference angle θ pll is the output of PLL, whose function is obtaining the phase angle of AC voltage. The block diagram of PLL is depicted in Figure 4, where H PLL (s) is the PLL regulator which uses a PI controller and an integrator.

Small-Signal Modeling of the PLL
As described in Figure 4, phase voltages are transformed to dq-domain voltage vd and vq by Park's transformation. When there is a harmonic frequency fp component in PCC voltage, the output angle θpll will be θ1 + Δθ. The Park's transformation could be broken into two parts as follows: In the time domain, the phase voltage with a small-signal perturbation can be written as: where V1 corresponds to the magnitude of the fundamental voltage at frequency f1, Vp, with ϕup correspond to the magnitude and phase of the positive-sequence perturbation at frequency fp. Similar to phasor analysis, in the frequency domain, the voltage (1) could be written as: According to (1), considering Δθ, vq is easily found to be: For voltage perturbation at fp, remove terms in Δθ(t) proportional to second and higher orders by the harmonic linearization principle, then Δθ(t) is given by Applying convolution to (4), (5), and (7), and neglecting terms in vq proportional to the second order, the result of (6) in frequency domain is as follows:

Small-Signal Modeling of the PLL
As described in Figure 4, phase voltages are transformed to dq-domain voltage v d and v q by Park's transformation. When there is a harmonic frequency f p component in PCC voltage, the output angle θ pll will be θ 1 + ∆θ. The Park's transformation could be broken into two parts as follows: where T(θ 1 ) is the Park's transformation when θ pll = θ 1 = ω 1 t.
In the time domain, the phase voltage with a small-signal perturbation can be written as: where V 1 corresponds to the magnitude of the fundamental voltage at frequency f 1 , V p, with φ up correspond to the magnitude and phase of the positive-sequence perturbation at frequency f p . Similar to phasor analysis, in the frequency domain, the voltage (1) could be written as: According to (1), considering ∆θ, v q is easily found to be: For voltage perturbation at f p , remove terms in ∆θ(t) proportional to second and higher orders by the harmonic linearization principle, then ∆θ(t) is given by Applying convolution to (4), (5), and (7), and neglecting terms in v q proportional to the second order, the result of (6) in frequency domain is as follows: Note that ∆θ = H pll (s) V q , then G p (s) can be solved from (8): To get the frequency errors of Park's transformation, the final step is to obtain the response of cos(θ pll (t)). Since cos(θ pll (t)) = cos(θ 1 (t) + ∆θ(t)) = cosω 1 tsin(ω 1 t) ∆θ(t), apply convolution again: From above we can find that a voltage perturbation at f p will result in both f p and 2f 1 − f p errors in Park's transformation, which gives a primary explanation of why sub/supsynchronous components usually exist together.
A simulation experiment was conducted to verify the correctness of (10). Appendix A gives the setup parameters of the experiment. The voltages at PCC are V 1 = 566 V, V p = V p e jφ up = 20·e jπ/4 . The response of cos(θ pll (t)) is depicted in Figure 5, compared with theoretical calculated results.
Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 16 Note that Δθ = Hpll(s) Vq, then Gp(s) can be solved from (8): To get the frequency errors of Park's transformation, the final step is to obtain the response of cos(θpll(t)). Since cos(θpll(t)) = cos(θ1(t) + Δθ(t)) = cosω1t -sin(ω1t) Δθ(t), apply convolution again: From above we can find that a voltage perturbation at fp will result in both fp and 2f1 − fp errors in Park's transformation, which gives a primary explanation of why sub/supsynchronous components usually exist together.
A simulation experiment was conducted to verify the correctness of (10). Appendix A gives the setup parameters of the experiment. The voltages at PCC are V1 = 566 V, Vp = Vp e jϕ up = 20•e jπ/4 . The response of cos(θpll(t)) is depicted in Figure 5, compared with theoretical calculated results.

Harmonic Components in Control Loop
It is easily to get from Figure 3 that the modulating signals, current, and PCC voltage satisfy: In order to solve (11) for impedance in the frequency domain, the sequence components in the modulating signals should be found as functions of the voltage and current perturbations. In the previous section, we have initially reached a conclusion that sub/supsynchronous always appear together. Therefore, for the voltage of (2), the current can be assumed as: where the annotations similar to (2). Other phase currents can be inferred.

Harmonic Components in Control Loop
It is easily to get from Figure 3 that the modulating signals, current, and PCC voltage satisfy: In order to solve (11) for impedance in the frequency domain, the sequence components in the modulating signals should be found as functions of the voltage and current perturbations. In the previous section, we have initially reached a conclusion that sub/supsynchronous always appear together. Therefore, for the voltage of (2), the current can be assumed as: where the annotations similar to (2). Other phase currents can be inferred.
Recall that currents id and iq are outputs of a dq-domain transformation, which in the frequency domain involves a convolution of the frequency components in the phase currents, with the frequency components in Park's transformation. Currents in frequency domain can be defined as (3), and frequency Appl. Sci. 2019, 9, 2831 7 of 16 components of Park's transformation are described in (10). Table 1 shows the possible combinations to consider in the convolution, it is worth noting that second and higher orders small signals are neglected. The dc components are not listed here, because id and iq would be exactly equal to I dref and I qref at the stable operating point.
Combine the frequency components in Table 1 noting that : Figure 3 depicts a dq-domain current controller, according to its diagram, the dq reference voltages are given by: It is easy to find that the dq reference voltages have dc and (fp − f 1 ) components. Moreover, (14) can be further described as: These signals are convoluted with inverse Park's transformation to generate their phase-domain counterparts. Counting equations (10) and (13~16) together, the harmonics of phase-domain voltage references are as followings:

Simplified Model of the Converter
The converter depicted in Figure 3 uses pulse width modulation techniques. The amplitude of output phase voltage could reach as large as V dc /2, correspond modulation factor is 1. Therefore, to get good control performance, it is necessary to take account in this characteristic when designing both system setup parameters and control strategies.
On one hand, in the design of the grid-connected converter parameters under the unit power factor control, it is worth mentioning that the output voltage, current, PCC voltage, and power grid voltage meet the relationship shown in Figure 6. get good control performance, it is necessary to take account in this characteristic when designing both system setup parameters and control strategies.
On one hand, in the design of the grid-connected converter parameters under the unit power factor control, it is worth mentioning that the output voltage, current, PCC voltage, and power grid voltage meet the relationship shown in Figure 6. From the following figure, we can conclude that usually Upwm < Us since the soothing inductance usually smaller than line inductance Lline. As a result, it will be praised if the dc side voltage Vdc is set to be twice as large as Vsa, at least.
On the other hand, modulation factor in PWM should not be larger than 1 the carrier usually uses (−1, 1) triangular wave. Therefore, in control strategy design, the three-phase voltage reference should be divided by Vdc/2, amplitude limiter should be added respectively.
If the grid-connected GSC of D-PMSG system satisfied the two conditions above, the VSC could be supposed as a voltage-controlled voltage source, and vga, vgb, and vgc would follow varef, vbref, and vcref.
To verify the correctness of the assumption, a simulation experiment was conducted. The structure diagram of experimental circuit is shown in Figure 3. The converter side parameters are listed in Appendix A. The grid side parameters used in this simulation are as follows: Vsa = 650 V, no resistance, and Lline increases from 0.3 mH to 0.46 mH at 3 s. The converter output voltage compared with reference voltage is depicted in Figure 7. So, in this paper, the switching dynamics process of converter will not be detailed described. The simplified converter model is proposed and verified in this section, which satisfies: From the following figure, we can conclude that usually U pwm < U s since the soothing inductance usually smaller than line inductance L line . As a result, it will be praised if the dc side voltage V dc is set to be twice as large as V sa , at least.
On the other hand, modulation factor in PWM should not be larger than 1 the carrier usually uses (−1, 1) triangular wave. Therefore, in control strategy design, the three-phase voltage reference should be divided by V dc /2, amplitude limiter should be added respectively.
If the grid-connected GSC of D-PMSG system satisfied the two conditions above, the VSC could be supposed as a voltage-controlled voltage source, and v ga , v gb, and v gc would follow v aref , v bref, and v cref . To verify the correctness of the assumption, a simulation experiment was conducted. The structure diagram of experimental circuit is shown in Figure 3. The converter side parameters are listed in Appendix A. The grid side parameters used in this simulation are as follows: V sa = 650 V, no resistance, and L line increases from 0.3 mH to 0.46 mH at 3 s. The converter output voltage compared with reference voltage is depicted in Figure 7. From the following figure, we can conclude that usually Upwm < Us since the soothing inductance usually smaller than line inductance Lline. As a result, it will be praised if the dc side voltage Vdc is set to be twice as large as Vsa, at least.
On the other hand, modulation factor in PWM should not be larger than 1 the carrier usually uses (−1, 1) triangular wave. Therefore, in control strategy design, the three-phase voltage reference should be divided by Vdc/2, amplitude limiter should be added respectively.
If the grid-connected GSC of D-PMSG system satisfied the two conditions above, the VSC could be supposed as a voltage-controlled voltage source, and vga, vgb, and vgc would follow varef, vbref, and vcref.
To verify the correctness of the assumption, a simulation experiment was conducted. The structure diagram of experimental circuit is shown in Figure 3. The converter side parameters are listed in Appendix A. The grid side parameters used in this simulation are as follows: Vsa = 650 V, no resistance, and Lline increases from 0.3 mH to 0.46 mH at 3 s. The converter output voltage compared with reference voltage is depicted in Figure 7. So, in this paper, the switching dynamics process of converter will not be detailed described. The simplified converter model is proposed and verified in this section, which satisfies: So, in this paper, the switching dynamics process of converter will not be detailed described. The simplified converter model is proposed and verified in this section, which satisfies: (19) Appl. Sci. 2019, 9, 2831 9 of 16

Impedance Analysis of D-PMSG
Transforming (11) to frequency domain, at the voltage perturbation frequency f p : Positive-sequence impedance is defined as: Introducing (17), frequency-domain version of (19) and (20) in (21), the positive-sequence impedance of the grid-connected D-PMSG can be easily obtained: Of course, the impedance characteristics have also been verified by simulation. Directly apply voltage v a (t) = 565.3cos(2πf 1 t) + 20cos(2πf p t + π/4), f p =(0, 100 Hz) at PCC, and the other set up parameters remained the same as Appendix A. Voltage and current signals collected and analyzed by FFT tools, the calculated simulation results compared with impedance prediction are shown in Figure 8. From the phase characteristics we can find that the D-PMSG's equivalent inductance X PMSG is positive at subsynchronous frequency while negative (capacitive) at supsynchronous frequency.  (20) Positive-sequence impedance is defined as: Introducing (17), frequency-domain version of (19) and (20) in (21), the positive-sequence impedance of the grid-connected D-PMSG can be easily obtained: Of course, the impedance characteristics have also been verified by simulation. Directly apply voltage va(t) = 565.3cos(2πf1t) + 20cos(2πfpt + π/4), fp =(0, 100 Hz) at PCC, and the other set up parameters remained the same as Appendix A. Voltage and current signals collected and analyzed by FFT tools, the calculated simulation results compared with impedance prediction are shown in Figure 8. From the phase characteristics we can find that the D-PMSG's equivalent inductance XPMSG is positive at subsynchronous frequency while negative (capacitive) at supsynchronous frequency.

Frequency Coupling of the Sub/Supsynchronous Currents
Furthermore, the relationship between sub/supsynchronous currents could be analyzed. Since there is no (2f1 − fp) harmonic voltage at PCC, phase-domain voltage reference at (2f1 − fp) satisfies: Introducing (18) into (23), and eliminating Vp by (22), the current response is given by: For the system parameters setup in 2.2 section, it can be calculated that Ip2 = 0.7754•e j3 . 063 •(−Ip). In the simulation experiment conducted in 2.2, the comparison of sub/supsynchronous FFT results is shown in the following Figure 9. It is worth mentioning that for clearly observing, phases of the insignificant frequency components except for 20, 50, and 80 Hz are neglected. The simulation results in Figure 9 verified the correctness of (24) at 20 Hz. By changing the frequency of voltage perturbation, the correctness at (0, 100 Hz) could be validated, too.

Frequency Coupling of the Sub/Supsynchronous Currents
Furthermore, the relationship between sub/supsynchronous currents could be analyzed. Since there is no (2f 1 − f p ) harmonic voltage at PCC, phase-domain voltage reference at (2f 1 − f p ) satisfies: Introducing (18) into (23), and eliminating V p by (22), the current response is given by: For the system parameters setup in 2.2 section, it can be calculated that I p2 = 0.7754·e j3 . 063 ·(−I p ). In the simulation experiment conducted in 2.2, the comparison of sub/supsynchronous FFT results is shown in the following Figure 9. It is worth mentioning that for clearly observing, phases of the insignificant frequency components except for 20, 50, and 80 Hz are neglected. The simulation results in Figure 9 verified the correctness of (24) at 20 Hz. By changing the frequency of voltage perturbation, the correctness at (0, 100 Hz) could be validated, too.

Impedance-Based Stability Analysis of SSO
The impedance-based stability analysis was widely used and developed in recent years. It was firstly proposed to analyze the stability of inverter-grid system in Reference [19]. Under the assumption that the inverter is controlled as a current source parallel with an output impedance, it will remain stable with a nonideal grid if the ratio of the grid impedance to the inverter output impedance satisfies the Nyquist criterion. It was developed by proposing a simplified method based on aggregated RLC circuit model to intuitively explain and quantitatively evaluate this type of SSO [5]. This method has a clear physical meaning and is easy to carry out as follows: at the resonant frequency fr decided by negative XPMSG and positive XGrid, if the total resistance RPMSG + RGrid is negative, the grid-connected D-PMSG system will be unstable. This paper chose this method to analyze gridconnected D-PMSG stability, and simulation is conducted to verify the correctness.

Impedance Sensitivity Analysis
Based on the correct and effective impedance model, influence of control and circuit parameters can be further discussed. The impedance dynamics effects on SSO stability would be explained, too. It can be inferred from (22) that the impedance characteristics are mainly influenced by initial operating conditions which are decided by power flow, also affected by control parameters of PLL and dq-current controller.

Impedance Sensitivity of Voltage Drops
When the fundamental voltage drops from 1.0 pu (636.4 V) to 0.6 pu while the other parameters in (22) keep the same as Appendix A, the impedance characteristics dynamics is shown in Figure 10. The reactance XPMSG and resistance RPMSG are listed below separately. It can be concluded that the XPMSG changes from inductive to capacitive, crossing zero at 50 Hz. RPMSG are negative around 50 Hz. To be noticed that XPMSG gets larger along with voltage drops lower, while the resistance curve moves to the positive direction. Larger XPMSG means lower resonant frequency but larger RPMSG brings bigger total resistance, which means the influence on stability depends on which one is dominant.

Impedance-Based Stability Analysis of SSO
The impedance-based stability analysis was widely used and developed in recent years. It was firstly proposed to analyze the stability of inverter-grid system in Reference [19]. Under the assumption that the inverter is controlled as a current source parallel with an output impedance, it will remain stable with a nonideal grid if the ratio of the grid impedance to the inverter output impedance satisfies the Nyquist criterion. It was developed by proposing a simplified method based on aggregated RLC circuit model to intuitively explain and quantitatively evaluate this type of SSO [5]. This method has a clear physical meaning and is easy to carry out as follows: at the resonant frequency f r decided by negative X PMSG and positive X Grid , if the total resistance R PMSG + R Grid is negative, the grid-connected D-PMSG system will be unstable. This paper chose this method to analyze grid-connected D-PMSG stability, and simulation is conducted to verify the correctness.

Impedance Sensitivity Analysis
Based on the correct and effective impedance model, influence of control and circuit parameters can be further discussed. The impedance dynamics effects on SSO stability would be explained, too. It can be inferred from (22) that the impedance characteristics are mainly influenced by initial operating conditions which are decided by power flow, also affected by control parameters of PLL and dq-current controller.

Impedance Sensitivity of Voltage Drops
When the fundamental voltage drops from 1.0 pu (636.4 V) to 0.6 pu while the other parameters in (22) keep the same as Appendix A, the impedance characteristics dynamics is shown in Figure 10. The reactance X PMSG and resistance R PMSG are listed below separately. It can be concluded that the X PMSG changes from inductive to capacitive, crossing zero at 50 Hz. R PMSG are negative around 50 Hz. To be noticed that X PMSG gets larger along with voltage drops lower, while the resistance curve moves to the positive direction. Larger X PMSG means lower resonant frequency but larger R PMSG brings bigger total resistance, which means the influence on stability depends on which one is dominant.

Impedance-Based Stability Analysis of SSO
The impedance-based stability analysis was widely used and developed in recent years. It was firstly proposed to analyze the stability of inverter-grid system in Reference [19]. Under the assumption that the inverter is controlled as a current source parallel with an output impedance, it will remain stable with a nonideal grid if the ratio of the grid impedance to the inverter output impedance satisfies the Nyquist criterion. It was developed by proposing a simplified method based on aggregated RLC circuit model to intuitively explain and quantitatively evaluate this type of SSO [5]. This method has a clear physical meaning and is easy to carry out as follows: at the resonant frequency fr decided by negative XPMSG and positive XGrid, if the total resistance RPMSG + RGrid is negative, the grid-connected D-PMSG system will be unstable. This paper chose this method to analyze gridconnected D-PMSG stability, and simulation is conducted to verify the correctness.

Impedance Sensitivity Analysis
Based on the correct and effective impedance model, influence of control and circuit parameters can be further discussed. The impedance dynamics effects on SSO stability would be explained, too. It can be inferred from (22) that the impedance characteristics are mainly influenced by initial operating conditions which are decided by power flow, also affected by control parameters of PLL and dq-current controller.

Impedance Sensitivity of Voltage Drops
When the fundamental voltage drops from 1.0 pu (636.4 V) to 0.6 pu while the other parameters in (22) keep the same as Appendix A, the impedance characteristics dynamics is shown in Figure 10. The reactance XPMSG and resistance RPMSG are listed below separately. It can be concluded that the XPMSG changes from inductive to capacitive, crossing zero at 50 Hz. RPMSG are negative around 50 Hz. To be noticed that XPMSG gets larger along with voltage drops lower, while the resistance curve moves to the positive direction. Larger XPMSG means lower resonant frequency but larger RPMSG brings bigger total resistance, which means the influence on stability depends on which one is dominant.

Impedance Sensitivity of Current Drops
Voltage remaining no changed at 1.0 pu, the impedance characteristics dynamics when the current reference I dref drops from 1.0 pu (1847 A) to 0.6 pu is shown in Figure 11. It is worth noting that the output fundamental current equals to I dref under the control strategy. X PMSG gets lager and R PMSG decreases while the output current drops. It can be inferred that this system will be more unstable.

Impedance Sensitivity of Current Drops
Voltage remaining no changed at 1.0 pu, the impedance characteristics dynamics when the current reference Idref drops from 1.0 pu (1847 A) to 0.6 pu is shown in Figure 11. It is worth noting that the output fundamental current equals to Idref under the control strategy. XPMSG gets lager and RPMSG decreases while the output current drops. It can be inferred that this system will be more unstable.

Impedance Sensitivity of PLL Parameters
Both voltage and current reference set at 1.0 pu increase the Kp and Ki and capture the impedance curve dynamics. The results are shown in Figure 12. From (a) we can know that when Kp increases alone, XPMSG curve changes lightly while the equivalent RPMSG curve becomes "wider", which means the frequency range of negative resistance is larger. Thus, larger Kp is not conducive to system stability. Whereas, the variation of Ki does not have obvious impact on stability.

Impedance Sensitivity of Current Controller Parameters
Similarly, the impedance curve dynamics while increasing the current controller parameters Kpi and Kii are shown in Figure 13. It can be concluded that adjustments of Kpi and Kii hardly affect the stability because the zero crossing frequency of resistance is nearly not changed.

Impedance Sensitivity of PLL Parameters
Both voltage and current reference set at 1.0 pu increase the K p and K i and capture the impedance curve dynamics. The results are shown in Figure 12. From (a) we can know that when K p increases alone, X PMSG curve changes lightly while the equivalent R PMSG curve becomes "wider", which means the frequency range of negative resistance is larger. Thus, larger K p is not conducive to system stability. Whereas, the variation of K i does not have obvious impact on stability.

Impedance Sensitivity of Current Drops
Voltage remaining no changed at 1.0 pu, the impedance characteristics dynamics when the current reference Idref drops from 1.0 pu (1847 A) to 0.6 pu is shown in Figure 11. It is worth noting that the output fundamental current equals to Idref under the control strategy. XPMSG gets lager and RPMSG decreases while the output current drops. It can be inferred that this system will be more unstable.

Impedance Sensitivity of PLL Parameters
Both voltage and current reference set at 1.0 pu increase the Kp and Ki and capture the impedance curve dynamics. The results are shown in Figure 12. From (a) we can know that when Kp increases alone, XPMSG curve changes lightly while the equivalent RPMSG curve becomes "wider", which means the frequency range of negative resistance is larger. Thus, larger Kp is not conducive to system stability. Whereas, the variation of Ki does not have obvious impact on stability.

Impedance Sensitivity of Current Controller Parameters
Similarly, the impedance curve dynamics while increasing the current controller parameters Kpi and Kii are shown in Figure 13. It can be concluded that adjustments of Kpi and Kii hardly affect the stability because the zero crossing frequency of resistance is nearly not changed.

Impedance Sensitivity of Current Controller Parameters
Similarly, the impedance curve dynamics while increasing the current controller parameters K pi and K ii are shown in Figure 13. It can be concluded that adjustments of K pi and K ii hardly affect the stability because the zero crossing frequency of resistance is nearly not changed.

SSO Simulation and Theoretical Explanation
The structure diagram of the studied system is shown in Figure 14. The converter side parameters are set as Appendix A, the grid side parameters set in simulation are as Table 2. The SCR decreases from 5.37 to 2.13 at 3 s, and decreases to 1.73 at 4 s.  Furthermore, the power waveform and current spectrum analysis are shown in Figure 15. From (a) we can find that the output power of D-PMSG oscillates twice at the SCR changing moments. The first oscillation quickly stabilized, but the second oscillation at 24 Hz gradually diverges until the system crashes. The magnitude of harmonic components at 5.5 s in output current is shown in Figure  15b, the 26/74 Hz sub/supsynchronous currents exist simultaneously.

SSO Simulation and Theoretical Explanation
The structure diagram of the studied system is shown in Figure 14. The converter side parameters are set as Appendix A, the grid side parameters set in simulation are as Table 2. The SCR decreases from 5.37 to 2.13 at 3 s, and decreases to 1.73 at 4 s.

SSO Simulation and Theoretical Explanation
The structure diagram of the studied system is shown in Figure 14. The converter side parameters are set as Appendix A, the grid side parameters set in simulation are as Table 2. The SCR decreases from 5.37 to 2.13 at 3 s, and decreases to 1.73 at 4 s.  Furthermore, the power waveform and current spectrum analysis are shown in Figure 15. From (a) we can find that the output power of D-PMSG oscillates twice at the SCR changing moments. The first oscillation quickly stabilized, but the second oscillation at 24 Hz gradually diverges until the system crashes. The magnitude of harmonic components at 5.5 s in output current is shown in Figure  15b, the 26/74 Hz sub/supsynchronous currents exist simultaneously.  Furthermore, the power waveform and current spectrum analysis are shown in Figure 15. From (a) we can find that the output power of D-PMSG oscillates twice at the SCR changing moments. The first oscillation quickly stabilized, but the second oscillation at 24 Hz gradually diverges until the system crashes. The magnitude of harmonic components at 5.5 s in output current is shown in Figure 15b Theoretical analysis is conducted to explain this simulation result. Since the current reference does not change, from Figure 6 we can know that the voltage at the PCC will drop along while the grid inductance Lg is increasing. So, the output power is also decreasing. Quantitative stability analysis should be done to figure out why the second oscillation is unstable. In these two SCR condition, calculate the power flow, then the total impedance of D-PMSG and grid ZPMSG + ZGRID could be obtained as follows: Then, according to the developed RLC circuit-based stability criterion mentioned in the beginning of this section, the system will be unstable if: (26) This criterion is easier to implement by plotting them in the same figure and the direction of Rtotal at the resonant frequency is obviously to see. The analysis shown in Figure 16 gives the impedance comparison of D-PMSG and grid when SCR = 2.13 and 1.73.  Theoretical analysis is conducted to explain this simulation result. Since the current reference does not change, from Figure 6 we can know that the voltage at the PCC will drop along while the grid inductance L g is increasing. So, the output power is also decreasing. Quantitative stability analysis should be done to figure out why the second oscillation is unstable. In these two SCR condition, calculate the power flow, then the total impedance of D-PMSG and grid Z PMSG + Z GRID could be obtained as follows: Then, according to the developed RLC circuit-based stability criterion mentioned in the beginning of this section, the system will be unstable if: R total ( f r ) < 0, when X total ( f r ) = 0 (26) This criterion is easier to implement by plotting them in the same figure and the direction of R total at the resonant frequency is obviously to see. The analysis shown in Figure 16 gives the impedance comparison of D-PMSG and grid when SCR = 2.13 and 1.73. Theoretical analysis is conducted to explain this simulation result. Since the current reference does not change, from Figure 6 we can know that the voltage at the PCC will drop along while the grid inductance Lg is increasing. So, the output power is also decreasing. Quantitative stability analysis should be done to figure out why the second oscillation is unstable. In these two SCR condition, calculate the power flow, then the total impedance of D-PMSG and grid ZPMSG + ZGRID could be obtained as follows: Then, according to the developed RLC circuit-based stability criterion mentioned in the beginning of this section, the system will be unstable if:   ( ) 0, ( ) 0 total r total r R f when X f (26) This criterion is easier to implement by plotting them in the same figure and the direction of Rtotal at the resonant frequency is obviously to see. The analysis shown in Figure 16 gives the impedance comparison of D-PMSG and grid when SCR = 2.13 and 1.73. Figure 16. Impedance comparison of ZPMSG and ZGrid. Figure 16. Impedance comparison of Z PMSG and Z Grid.
The green vertical line indicates that the total resistance at the resonant frequency is positive when SCR = 2.13, which means this system is stable. This is consistent with the power waveform after 3 s in Figure 15a.
When SCR decreases to 1.73, the total resistance at the resonant frequency 72.77 Hz is −0.04, and this system is judged unstable under this operating condition. The f p supsynchronous oscillation current will be captured by the GSC controller and introduces a (2f 1 − f p ) subsynchronous current in return, as discussed before, about 27 Hz and 73 Hz, respectively. Moreover, the power will oscillate at a frequency of 23 Hz (|f p − f 1 |), which is very close to the simulation results after 4 s, considering the calculating and modeling errors. It can be concluded that the mechanism of SSO when SCR decreases is theoretically revealed.

Discussion
This paper focus on the modeling and analysis of sub/supsynchronous oscillation in D-PMSG based wind farm integrated to a weak AC grid. The small-signal harmonic linearization method is utilized to develop the positive sequence of the GSC. In this main step, the dynamics of PLL under harmonic voltage perturbation is analyzed. Moreover, the mechanism of sub/supsynchronous currents frequency coupling is revealed quantitatively. Based on the impedance model, the influence of converter parameters is further discussed. Finally, the influence of the SCR of AC system on SSO is well explained by impedance-based stability analysis. Simulation experiments are carried out to verify these theoretical discussions. Following conclusions can be made: 1.
The D-PMSG's equivalent inductance X PMSG is positive at subsynchronous frequency while negative (capacitive) at supsynchronous frequency, which means the resonance between D-PMSG and weak grid occurs at the supsynchronous frequency. The relationship of coupled sub/supsynchronous currents could be quantitatively calculated.

2.
The D-PMSG's impedance is sensitive to the PCC voltage and output current, which are decided by power flow. Further proof that when the output current is relatively small as a result of low wind speed, it is very likely to observe oscillations in this system. The influence of control parameters is also discussed. No obvious impact of current controller is found.

3.
A simulation experiment was conducted to study the mechanism of SSO when SCR decreases. Both simulation results and theoretical analysis proved that when the grid side impedance increases, the grid-connected D-PMSG system will be more unstable.
Future work based on the work in this paper will discuss the following several issues: The first issue is about how to obtain the reduced-order equivalent model of wind farm which contains hundreds of wind turbine generators. The second issue focus on oscillation suppression, proposing advice related to parameters adjustment or additional damping control strategy. The last issue will discuss the interaction with synchronous generator, further analyzing the accident in Xinjiang, China.

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

Appendix A
The PSCAD-EMTDC simulation experimental parameters setup in this paper are as following: Fundamental frequency: f 1 = 50 Hz; dc bus voltage: V dc = 1300 V; dq current controller parameters: H i (s) = (K pi + K ii /s) where K pi = 0.25 and K ii = 355; PLL parameters: H pll (s) = (K p + K i /s)/s where K p = 0.085 and K i = 32; current reference: I dref = 1847, I qref = 0; converter circuit parameters: L = 0.15 mH.