Measurement data based method for identifying wind turbine generators which cause sub-synchronous oscillations in complex power system

Abstract

A measurement data based method for identifying wind turbine generators which cause sub-synchronous oscillations in a complex power system has a theoretical foundation of the open-loop modal resonance and the parallel filter design. The advantages of the present invention are as follows. 1) The present invention can identify the wind turbine generators which cause the SSOs in the complex power system using measurement data instead of parametric model. Hence, it simplifies the computation and reduces the modeling cost effectively. 2) The present invention can identify the wind turbine generators which cause the SSOs in the complex power system precisely with reduced amount of measurement data, reducing the cost of hardware and data measurement effectively. 3) The present invention can identify the wind turbine generators which cause the SSOs in the complex power system based on the open-loop modal resonance theory.

Claims

1. A measurement data based method for identifying wind turbine generators which cause sub-synchronous oscillations (SSOs) in a complex power system, comprising following steps of: (1) obtaining a set of measurement data when the SSOs occur in the power system; reducing noise by applying a filtering algorithm; identifying an SSO frequency by using a signal processing method; wherein the identified SSO frequency is denoted as f.sub.s; (2) designing a parallel band-pass filter to ensure that only a signal at frequency f.sub.s passes through to earth and signals at other frequencies are suppressed; (3) for a power system with n wind turbine generators, installing the parallel band-pass filter designed in the step (2) on an interface between each of the wind turbine generators and the power system; wherein subsequently, dynamic interactions between the wind turbine generators and the power system at the SSO frequency f.sub.s are suppressed and filtered out; whilst those at other frequencies are not affected; (4) measuring a response from each of the wind turbine generators by adding a disturbance signal on a side of a wind turbine generator near the band-pass filter; (5) from the measurement data of the response of an i-th wind turbine generator, identifying an oscillation frequency, f.sub.ij, and residuing R.sub.ij by applying a signal processing method; wherein in notations above for f.sub.ij and R.sub.ij, subscript i refers to the i-th wind turbine generator and the subscript j refers to the j-th SSO frequency identified; afterwards, comparing f.sub.ij and f.sub.s; recording the f.sub.ij and R.sub.ij if a following condition is met:
1.05*f.sub.sf.sub.ij0.95*f.sub.s; wherein finally, if no f.sub.ij which can meet the above condition, it is concluded that the i-th wind turbine generator is not the wind turbine generator which causes the SSOs in the power system; (6) measuring the response from the power system by adding a disturbance signal on the side of the power system near the band-pass filter for the i-th wind turbine generator; (7) from the measurement data of the response, identifying an oscillation frequency f.sub.aij and a residue R.sub.aij by using the signal processing method; wherein in notations of f.sub.aij and R.sub.aij, subscript a indicates that the measurement data is about the power system, subscript i refers to the i-th wind turbine generator and subscript j refers to the j-th oscillation frequency and residue; afterwards, comparing f.sub.aij and f.sub.s; recording f.sub.aij and R.sub.aij if a following condition is met:
1.05*f.sub.sf.sub.aij0.95*f.sub.s and (8) Computing Z.sub.i=|{square root over (R.sub.ijR.sub.aij)}|, i=1, 2, L n based on recorded results in the step (5) and (7); wherein if Z.sub.k is the largest among Z.sub.i, i=1, 2, L n, the i-th wind turbine generator is identified to be the wind turbine generator causing the SSOs in the power system.

2. The measurement data based method for identifying the wind turbine generators which cause the sub-synchronous oscillations in the complex power system, as described in claim 1, wherein the interface in the step (3) adopts any point between the i-th wind turbine generator and the power system; at the point, the power system is divided to two subsystems; one subsystem consists of the i-th wind turbine generator; the other subsystem is comprised of remainder of the power system excluding the i-th wind turbine generator.

3. The measurement data based method for identifying the wind turbine generators which cause the sub-synchronous oscillations in the complex power system, as described in claim 1, wherein the disturbance signal in the step (4) excites an observable dynamic response of the i-th wind turbine generator.

4. The measurement data based method for identifying the wind turbine generators which cause the sub-synchronous oscillations in the complex power system, as described in claim 1, wherein the response of the wind turbine generator in the step (4) is an active power output from the wind turbine generator.

5. The measurement data based method for identifying the wind turbine generators which cause the sub-synchronous oscillations in the complex power system, as described in claim 1, wherein the disturbance signal in the step (4) excites an observable dynamic response of the power system.

6. The measurement data based method for identifying the wind turbine generators which cause the sub-synchronous oscillations in the complex power system, as described in claim 1, wherein the power system response in the step (6) is a magnitude of a terminal voltage at a node in the power system where the i-th wind turbine generator is connected to the power system.

7. The measurement data based method for identifying the wind turbine generators which cause the sub-synchronous oscillations in the complex power system, as described in claim 1, wherein the disturbance signal in the step (4) or step (6) is an impulse signal; a time duration and a magnitude of the signal are respectively 0.1 second and 2% of a magnitude of a terminal voltage at a node where the i-th wind turbine generator is connected with the power system.

8. The measurement data based method for identifying the wind turbine generators which cause the sub-synchronous oscillations in the complex power system, as described in claim 2, wherein the disturbance signal in the step (4) or step (6) is an impulse signal; a time duration and a magnitude of the signal are respectively 0.1 second and 2% of a magnitude of a terminal voltage at a node where the i-th wind turbine generator is connected with the power system.

9. The measurement data based method for identifying the wind turbine generators which cause the sub-synchronous oscillations in the complex power system, as described in claim 3, wherein the disturbance signal in the step (4) or step (6) is an impulse signal, a time duration and a magnitude of the signal are respectively 0.1 second and 2% of a magnitude of a terminal voltage at a node where the i-th wind turbine generator is connected with the power system.

10. The measurement data based method for identifying the wind turbine generators which cause the sub-synchronous oscillations in the complex power system, as described in claim 4, wherein the disturbance signal in the step (4) or step (6) is an impulse signal; a time duration and a magnitude of the signal are respectively 0.1 second and 2% of a magnitude of a terminal voltage at a node where the i-th wind turbine generator is connected with the power system.

11. The measurement data based method for identifying the wind turbine generators which cause the sub-synchronous oscillations in the complex power system, as described in claim 5, wherein the disturbance signal in the step (4) or step (6) is an impulse signal; a time duration and a magnitude of the signal are respectively 0.1 second and 2% of a magnitude of a terminal voltage at a node where the i-th wind turbine generator is connected with the power system.

12. The measurement data based method for identifying the wind turbine generators which cause the sub-synchronous oscillations in the complex power system, as described in claim 6, wherein the disturbance signal in the step (4) or step (6) is an impulse signal, a time duration and a magnitude of the signal are respectively 0.1 second and 2% of the magnitude of the terminal voltage at the node where the i-th wind turbine generator is connected with the power system.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0076] FIG. 1 illustrates configuration of the power system integrated with multiple WTGs;

[0077] FIG. 2 illustrates closed-loop transform function of the power system integrated with multiple WTGs;

[0078] FIG. 3 illustrates expansion of power system by multiple oscillation modes and corresponding residues;

[0079] FIG. 4 illustrates magnitude-frequency characteristic of the ideal band-pass filter;

[0080] FIG. 5 illustrates expansion of the power system installed with the band-pass filter;

[0081] FIG. 6 illustrates magnitude-frequency characteristic of the ideal band-pass filter;

[0082] FIG. 7 illustrates points could be chosen to install filter in the step (3);

[0083] FIG. 8 illustrates points could be chosen to inject disturbance signal and monitor signal on the side of the i-th wind turbine generator in the step (4);

[0084] FIG. 9 illustrates the typical impulse signal in the step (4);

[0085] FIG. 10 illustrates points could be chosen to inject disturbance signal and measuring signal on the side of the power system in the step (6);

[0086] FIG. 11 illustrates technical schemes of the present invention;

[0087] FIG. 12 illustrates configuration of 4-machine 2-area power system integrated with four wind turbine generators;

[0088] FIG. 13(a) illustrates magnitude-frequency characteristic of the low-pass filter;

[0089] FIG. 13(b) illustrates active power output from PMSG-1 before filtering;

[0090] FIG. 13(c) illustrates active power output from PMSG-1 after filtering;

[0091] FIG. 14 illustrates FFT analysis results of the measurement data in FIG. 13;

[0092] FIG. 15 illustrates magnitude-frequency and phase-frequency characteristics of the high-pass filter;

[0093] FIG. 16(a) illustrates configuration of single-machine infinite power system integrated with a wind turbine generator;

[0094] FIG. 16(b) illustrates configuration of single-machine infinite power system integrated with a wind turbine generator and the parallel band-pass filter;

[0095] FIG. 17(a) illustrates active power output from wind turbine generator without the parallel band-pass filter;

[0096] FIG. 17(b) illustrates active power output from wind turbine generator with the parallel band-pass filter;

[0097] FIG. 18 illustrates installation location (1) of the high-pass filter;

[0098] FIG. 19 illustrates typical disturbance signal;

[0099] FIG. 20(a) illustrates active power output from the wind turbine generator before filtering;

[0100] FIG. 20(b) illustrates active power output from the wind turbine generator after filtering;

[0101] FIG. 21 illustrates point for injecting the disturbance signal and measuring signal;

[0102] FIG. 22(a) illustrates active power output from AC system before filtering;

[0103] FIG. 22(b) illustrates active power output from AC system after filtering;

[0104] FIG. 23 illustrates installation location (2) of the high-pass filter;

[0105] FIG. 24(a) illustrates active power output from the wind turbine generator before filtering;

[0106] FIG. 24(b) illustrates active power output from the wind turbine generator after filtering;

[0107] FIG. 25 illustrates points for injecting the disturbance signal and measuring signal;

[0108] FIG. 26(a) illustrates active power output from AC system before filtering;

[0109] FIG. 26(b) illustrates active power output from AC system after filtering;

[0110] FIG. 27 illustrates installation location (3) of the high-pass filter;

[0111] FIG. 28(a) illustrates active power output from the wind turbine generator before filtering;

[0112] FIG. 28(b) illustrates active power output from the wind turbine generator after filtering;

[0113] FIG. 29 illustrates points for injecting the disturbance signal and measuring signal;

[0114] FIG. 30(a) illustrates active power output from AC system before filtering;

[0115] FIG. 30(b) illustrates active power output from AC system after filtering;

[0116] FIG. 31 illustrates installation location (4) of the high-pass filter;

[0117] FIG. 32(a) illustrates active power output from the wind turbine generator before filtering;

[0118] FIG. 32(b) illustrates active power output from the wind turbine generator after filtering;

[0119] FIG. 33 illustrates points for injecting the disturbance signal and measuring signal;

[0120] FIG. 34(a) illustrates active power output from AC system before filtering;

[0121] FIG. 34(b) illustrates active power output from AC system after filtering;

[0122] FIG. 35 illustrates configuration of the power system without DFIG-1

[0123] FIG. 36 illustrates active power output from SG-1 without DFIG-1;

[0124] FIG. 37 illustrates active power output from SG-2 without DFIG-1;

[0125] FIG. 38 illustrates active power output from SG-3 without DFIG-1;

[0126] FIG. 39 illustrates active power output from SG-4 without DFIG-1;

[0127] FIG. 40 illustrates configuration of the complex power system integrated with four wind turbine generators;

[0128] FIG. 41(a) illustrates magnitude-frequency characteristic of the low-pass filter;

[0129] FIG. 41(b) illustrates frequency response of the power system before filtering;

[0130] FIG. 41(c) illustrates frequency response of the power system after filtering;

[0131] FIG. 42 illustrates FFT analysis results of the measurement data in FIG. 41;

[0132] FIG. 43 illustrates magnitude-frequency and phase-frequency characteristics of the high-pass filter;

[0133] FIG. 44 illustrates installation location (1) of the high-pass filter;

[0134] FIG. 45 illustrates the typical disturbance signal;

[0135] FIG. 46(a) illustrates active power output from PMSG-1 before filtering;

[0136] FIG. 46(b) illustrates active power output from PMSG-1 after filtering;

[0137] FIG. 47 illustrates points for injecting the disturbance signal and measuring signal;

[0138] FIG. 48(a) illustrates active power output from SG-1 before filtering;

[0139] FIG. 48(b) illustrates active power output from SG-2 after filtering

[0140] FIG. 49 illustrates installation location (2) of the high-pass filter;

[0141] FIG. 50(a) illustrates active power output from PMSG-2 before filtering;

[0142] FIG. 50(b) illustrates active power output from PMSG-2 after filtering;

[0143] FIG. 51 illustrates points for injecting the disturbance signal and measuring signal;

[0144] FIG. 52(a) illustrates active power output from SG-8 before filtering;

[0145] FIG. 52(b) illustrates active power output from SG-8 after filtering;

[0146] FIG. 53 illustrates installation location (3) of the high-pass filter;

[0147] FIG. 54(a) illustrates active power output from PMSG-3 before filtering;

[0148] FIG. 54(b) illustrates active power output from PMSG-3 after filtering;

[0149] FIG. 55 illustrates points for injecting the disturbance signal and measuring signal;

[0150] FIG. 56(a) illustrates active power output from SG-10 before filtering;

[0151] FIG. 56(b) illustrates active power output from SG-10 after filtering;

[0152] FIG. 57 illustrates installation location (4) of the high-pass filter;

[0153] FIG. 58(a) illustrates active power output from PMSG-4 before filtering;

[0154] FIG. 58(b) illustrates active power output from PMSG-4 after filtering;

[0155] FIG. 59 illustrates points for injecting the disturbance signal and measuring signal;

[0156] FIG. 60(a) illustrates active power output from SG-7 before filtering;

[0157] FIG. 60(b) illustrates active power output from SG-7 after filtering;

[0158] FIG. 61 illustrates configuration of the power system without PMSG-1;

[0159] FIG. 62(a) illustrates active power output from PMSG-2 without PMSG-1;

[0160] FIG. 62(b) illustrates active power output from PMSG-3 without PMSG-1;

[0161] FIG. 62(c) illustrates active power output from PMSG-4 without PMSG-1.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

[0162] The flowchart of the present invention shown in FIG. 11 is demonstrated by following two examples.

EXAMPLE 1

[0163] FIG. 12 shows a 4-machine 2-area power system integrated with two PMSGs and two DFIGs, which represent four grid-connected wind farms. When the SSOs occur, the wind turbine generators which cause the SSOs in the power system are identified by the present invention without using the parametric model of the power system. Detailed steps of identification are as follows.

[0164] 1. Determine the SSO Frequency

[0165] In order to identify a wind turbine generator which causes the SSOs, the SSO frequency in the power system needs to be determined firstly. Afterwards, the band-pass filter can be designed near this frequency.

[0166] Step 1: reducing the noise in the measurement data.

[0167] A low-pass filter depicted in FIG. 13(a) is designed to reduce the harmonic components in the measurement data.

[0168] Step 2: determining the SSO frequency by the signal processing method.

[0169] The SSOs are observed in the active power output from wind turbine generators, for example that from PMSG-1 shown in FIG. 13 (c). The FFT analysis results are displayed in FIG. 14. From FIG. 14, it can be concluded that the SSO frequency in the power system is 15 Hz, i.e. f.sub.s=15 Hz

[0170] 2. Design a Parallel Band-Pass Filter

[0171] Step 3: designing a parallel band-pass filter.

[0172] Design a parallel band-pass filter to ensure that the signal at frequency 15 Hz can pass through to earth, the signals at other frequencies are suppressed. Subsequently, dynamic interactions between the wind turbine generators and the power system at the SSO frequency f.sub.s are suppressed and filtered out; whilst at other frequencies are not affected. A 4th order band-pass filter is designed by Besself method as shown by Eq. (22), where the cut-off frequencies are 14 Hz and 16 Hz.

[00011]$\begin{array}{cc}{G}_T\left(s\right)=\frac{4.Math..Math.s^2}{s^4+3.4641.Math.s^3+452.Math.s^2+775.96.Math.s+50176}& \left(22\right)\end{array}$

[0173] Results of magnitude-frequency and phase-frequency characteristics of the filter designed by Eq. (22) are shown in FIG. 15. From FIG. 15, it can be concluded that the filter designed by Eq. (22) can meet the requirement.

[0174] Step 4: verifying the effectiveness of the parallel band-pass filter designed in the step 3.

[0175] A single-machine infinite-bus power system installed with PMSG-1 is shown by FIG. 16(a). It is used to verify the effectiveness of the parallel band-pass filter. FIG. 16(b) shows that the parallel band-pass filter is installed at the interface between the PMSG-1 and remainder of the power system (ROPS).

[0176] When the SSOs occur in the power system of FIG. 16(a), the SSO frequency is calculated to be 15 Hz from the measurement data of active power output from PMSG-1. After the parallel band-pass filter is installed, the active power output from PMSG-1 is measured which is shown in FIG. 17(b). By comparing FIG. 17(b) and FIG. 17(a), it can be seen that the 15 Hz component in the variation of active power output from PMSG-1 is reduced effectively. Therefore, the effectiveness of the parallel band-pass filter is verified.

[0177] 3. Measure Open-Loop Residues

[0178] Open-loop residues of each wind turbine generator represented by either PMSG or DFIG and the ROPS can be measured by installing a parallel band-pass filter at the PCC of each wind turbine generator.

[0179] Step 5: installing a parallel band-pass filter on the side of PMSG-1 and measure the residues.

[0180] The installing location of the band-pass filter, the place to inject the disturbance signal and to gain the measurement data are indicated in FIG. 18. The disturbance signal is 2% increase of magnitude of PCC voltage for 0.1 second as shown by FIG. 19. The measurement data gained on the side of PMSG-1 is shown by FIG. 20(a). With the noise being reduced, the same measurement data is shown in FIG. 20(b).

[0181] Prony analysis is applied to the measurement data shown by FIG. 20(b) to obtain the residue of PMSG-1 to be 0.0001+0.0003j.

[0182] Similarly, in order to measure the residue of the ROPS subsystem, the parallel band-pass filter is installed on the side of power system near PMSG-1. The installing location of the band-pass filter, the place to inject the disturbance signal and to gain the measurement data are indicated in FIG. 21. The disturbance signal is the same to that shown by FIG. 19. The measurement data gained on the side of power system and noise reduced data are shown in FIG. 22(a) and FIG. 22(b), respectively.

[0183] From the measurement data displayed in FIG. 22(b), the residue of the ROPS subsystem is estimated by the Prony analysis to be 0.0441+0.0473j.

[0184] Step 6: installing a parallel band-pass filter on the side of PMSG-2 and measure the residues.

[0185] The procedure to measure the residues for PMSG-2 is as same as that presented above in the step 5 for PMSG-1. The measurement data on the side of PMSG-2 and the noise reduced data are shown in FIG. 24(a) and FIG. 24(b). The residue of PMSG-1 is obtained to be 0.0001+0.0003j.

[0186] The measurement data and noise reduced data on the side of power system are shown by FIG. 26(a) and FIG. 26(b) respectively. The residue of the ROPS subsystem is obtained to be 0.0820+0.0405j.

[0187] Step 7: installing a parallel band-pass filter on the side of DFIG-2 and measure residues.

[0188] The procedure to measure the residues for DFIG-2 is as same as that presented above in the step 5 for PMSG-1. The measurement data on the side of DFIG-2 and the noise reduced data are shown in FIG. 28(a) and FIG. 28(b). The residue of DFIG-2 is obtained to be 0.0003+0.0006j.

[0189] The measurement data and noise reduced data on the side of power system are shown by FIG. 30(a) and FIG. 30(b) respectively. The residue of the ROPS subsystem is obtained to be 0.0502+0.0413j.

[0190] Step 8: installing a parallel band-pass filter on the side of DFIG-1 and measure residues.

[0191] The procedure to measure the residues for DFIG-1 is as same as that presented above in the step 5 for PMSG-1. The measurement data on the side of DFIG-1 and the noise reduced data are shown in FIG. 32(a) and FIG. 32(b). The residue of DFIG-1 is obtained to be 0.0078+0.0050j.

[0192] The measurement data and noise reduced data on the side of power system are shown by FIG. 34(a) and FIG. 34(b) respectively. The residue of the ROPS subsystem is obtained to be 0.0976+0.0805j.

[0193] 4. Identify the Wind Turbine Generators which Cause the SSOs According to the Index

[0194] Step 9: calculating the indexes.

[0195] The indexes Z.sub.i=|{square root over (R.sub.iR.sub.ai)}|, i=1, 2, 3, 4 are computed based on the residues obtained above, where R.sub.i and R.sub.ai, i=1, 2, 3, 4 are the residues of each wind turbine generator and the corresponding ROPS subsystem, respectively. The computational results are presented in Tab. 1-1.

TABLE-US-00001 TABLE 1-1 Results of residues and corresponding index obtained by using the present invention Index Value The cause of the SSOs Z4 (DFIG-1) 0.0291 0.0016i Yes Z3 (DFIG-2) 0.0054 0.0038i No Z1 (PMSG-1) 0.0036 + 0.0052i No Z2 (PMSG-2) 0.0055 + 0.0031i No

[0196] Step 10: identifying the wind turbine generator which causes the SSOs.

[0197] According to the open-loop modal resonance theory, the wind turbine generator corresponding to the maximum index among all Zi is identified to be the wind turbine generator causing the SSOs. Therefore, DFIG-1 is identified to be the cause of the SSOs in the four-machine two-area power system.

[0198] 5. Verify the Correctness of Result Obtained by the Present Invention

[0199] The linearized model of FIG. 9 is used in this section to verify the correctness of the above identification made by the present invention.

[0200] Step 11: establishing the parametric model of the power system integrated with the wind turbine generators.

[0201] The open-loop linearized models of each wind turbine generator subsystem and the corresponding ROPS subsystem described by Eq. (1) and Eq. (2) are respectively derived. Following notations are used to denote various results of derived parametric models and computational results.

[0202] A.sub.dfi, b.sub.dfi, c.sub.dfi, i=1, 2 are the state matrix, control vector and output vector of open-loop state-space models of DFIG-1 and DFIG-2.

[0203] p.sub.dfi, r.sub.dfi, i=1, 2 are the left and right eigenvectors corresponding to the open-loop SSO modes of DFIG-1 and DFIG-2.

[0204] A.sub.dfact, b.sub.dfact, c.sub.dfact, i=1, 2 are the state matrix, control vector and output vector of open-loop state-space models of the ROPS subsystem without DFIG-1 and DFIG-2 being included, respectively;

[0205] p.sub.dfact, r.sub.dfact, i=1.2 are the left and right eigenvectors corresponding to the open-loop SSO modes of ROPS subsystem without DFIG-1 and DFIG-2 being included, respectively;

[0206] A.sub.pmi, b.sub.pmi, c.sub.pmi, i=1, 2 are the state matrix, control vector and output vector of open-loop state-space models of PMSG-1 and PMSG-2.

[0207] p.sub.pmi, r.sub.pmi, i=1, 2 are the left and right eigenvectors corresponding to the open-loop SSO modes of PMSG-1 and PMSG-2.

[0208] A.sub.pmaci, b.sub.pmaci, c.sub.pmaci, i=1, 2 are the state matrix, control vector and output vector of the ROPS subsystem without PMSG-1 and PMSG-2 being included, respectively.

[0209] p.sub.pmaci, r.sub.pmaci, i=1, 2 are the left and right eigenvectors corresponding to the open-loop SSO modes of ROPS subsystem without PMSG-1 and PMSG-2 being included, respectively.

[0210] Step 12: calculating the residues of each open-loop wind turbine generator subsystem and corresponding open-loop ROPS subsystems.

[0211] From the results obtained in the step 11, the open-loop residues of DFIG-1 and the corresponding ROPS subsystem are calculated to be R.sub.df1=p.sub.df1.sup.Tb.sub.df1c.sub.df1r.sub.df1=1.28+164.55i and R.sub.dfac1=p.sub.dfac1.sup.Tb.sub.dfac1c.sub.dfac1r.sub.dfac1=0.019+0.009i, respectively. Thus, the index associated with DFIG-1 is calculated to be Z.sub.r4={square root over (R.sub.df1R.sub.dfac1)}=1.01-1.57i.

[0212] Similarly, the open-loop residues of DFIG-2 and the corresponding ROPS subsystem are calculated to be R.sub.df2=p.sub.df2.sup.Tb.sub.df2c.sub.df2r.sub.df2=1.38+130.01i and R.sub.dfac2=p.sub.dfac2.sup.Tb.sub.dfac2c.sub.dfac2r.sub.dfac2=0.0061-0.002i, respectively. The index associated with DFIG-2 is calculated to be Z.sub.r3={square root over (R.sub.df2R.sub.dfac2)}=0.27-0.79i.

[0213] The open-loop residues of PMSG-1 and the corresponding ROPS subsystem are calculated to be R.sub.pm1=p.sub.pm1.sup.Tb.sub.pm1c.sub.pm1r.sub.pm1=3.03+17.2i and R.sub.pmac1=p.sub.pmac1.sup.Tb.sub.pmac1c.sub.pmac1r.sub.pmac1=0.011+j0.006, respectively. The index associated with PMSG-1 is calculated to be Z.sub.r1={square root over (R.sub.pm1R.sub.pmac1)}=0.38+0.273i.

[0214] The open-loop residues of PMSG-2 and the corresponding ROPS subsystem are calculated to be R.sub.pm1=p.sub.pm2.sup.Tb.sub.pm2c.sub.pm2r.sub.pm2=25.78+23.33i and R.sub.pmac2=p.sub.pmac2.sup.Tb.sub.pmac2c.sub.pmac2r.sub.pmac2=0.009+j0.002, respectively. The index associated with PMSG-2 is calculated to be Z.sub.r2={square root over (R.sub.pm2R.sub.pmac2)}=0.26+0.503i

[0215] Step 13: verifying the correctness of result obtained in the step 10 by the present invention.

[0216] Computational results of indexes above by using the parametric model are listed in Table 1-2. It can be seen that DFIG-1 causes the SSOs, confirming the correctness of identification made previously from Table 1-1 by using the present invention.

TABLE-US-00002 TABLE 1-2 Results of residues and index obtained using the parametric model for confirmation Method in Parametric The cause the present The cause of model of Index invention the SSOs method the SSOs Z4 (DFIG-1) Z.sub.b4 = 1 Yes Z.sub.br4 = 1 Yes Z3 (DFIG-2) Z.sub.b3 = 0.2268 No Z.sub.br3 = 0.4521 No Z1 (PMSG-1) Z.sub.b1 = 0.2165 No Z.sub.br1 = 0.2550 No Z2 (PMSG-2) Z.sub.b2 = 0.2165 No Z.sub.br2 = 0.3063 No

[0217] DFIG-1 is disconnected from the example power system as shown by FIG. 35. Simulation results are shown from FIG. 36 to FIG. 39. It can be seen that no SSOs occur in the example power system, confirming that DFIG-1 causes the SSOs as identified by the present invention.

EXAMPLE 2

[0218] FIG. 40 shows a 10-machine 39-node New England power system integrated with four PMSGs, which represent four grid-connected wind farms. When the SSOs occur, the wind turbine generators which cause the SSOs in the power system are identified by the present invention without using the parametric model. Detailed steps of identification are as follows.

[0219] 1. Determine the SSO Frequency

[0220] In order to identify the wind turbine generators which cause the SSOs, the SSO frequency in the power system needs to be determined firstly. Afterwards, the band-pass filter can be designed near this frequency.

[0221] Step 1: reducing the noise in the measurement data.

[0222] A low-pass filter depicted in FIG. 41 is designed to reduce the harmonic components in the measurement data.

[0223] Step 2: determining the SSO frequency by the signal processing method.

[0224] The SSOs can be observed in frequency response curve in FIG. 41(a)-(c). The FFT analysis results are summarized in FIG. 42. From FIG. 42, it can be concluded that the SSO frequency in the power system is 14.5 Hz, i.e. f=14.5 Hz

[0225] 2. Design a Parallel Band-Pass Filter

[0226] Step 3: designing a parallel band-pass filter.

[0227] Design a parallel band-pass filter to ensure that the signal at frequency 14.5 Hz can pass through to earth, the signals at other frequencies are suppressed. Subsequently, dynamic interactions between the wind turbine generators and the power system at the SSO frequency f.sub.s are suppressed and filtered out; whilst at other frequencies are not affected. A 4th order band-pass filter is designed by Besself method, where the cut-off frequencies are 14 Hz and 16 Hz. The analysis results of magnitude-frequency characteristics of the filter are shown in FIG. 43. From FIG. 43, it can be concluded that the designed filter can meet the requirement.

[0228] Step 4: verifying the effectiveness of the parallel band-pass filter designed in the step 3.

[0229] The procedure to verify the effectiveness of the parallel band-pass filter is the same as that presented above in the step 4 of example 1.

[0230] 3. Measure Open-Loop Residues

[0231] Open-loop residues of each wind turbine generator represented by PMSG and the remainder of the ROPS can be measured by installing a parallel band-pass filter at the PCC of each wind turbine generator.

[0232] Step 5: installing a parallel band-pass filter on the side of PMSG-1 and measure the residues.

[0233] The installing location of the band-pass filter, the place to inject the disturbance signal and to gain the measurement data are indicated in FIG. 44. The disturbance signal is 10% increase of magnitude of PCC voltage for 0.1 second as shown by FIG. 45. The measurement data gained on the side of PMSG-1 is shown by FIG. 46(a). With the noise being reduced, the same measurement data is shown in FIG. 46(b).

[0234] Prony analysis is applied to the measurement data shown by FIG. 46(b) to obtain the residue of PMSG-1 to be 0.0183+0.0497.

[0235] Similarly, in order to measure the residue of the ROPS subsystem, the parallel band-pass filter is installed on the side of power system near PMSG-1. The installing location of the band-pass filter, the place to inject the disturbance signal and to gain the measurement data are indicated in FIG. 47. The disturbance signal is the same as that shown by FIG. 45. The measurement data gained on the side of power system and noise reduced data are shown in FIG. 48(a) and FIG. 48(b) respectively.

[0236] From the measurement data displayed in FIG. 48(b), the residue of the ROPS subsystem is estimated by the Prony analysis to be 0.0020+0.0026j.

[0237] Step 6: installing a parallel band-pass filter on the side of PMSG-2 and measure the residues.

[0238] The procedure to measure the residues for PMSG-2 is the same as that presented above in the step 5 for PMSG-1. The measurement data on the side of PMSG-2 and the noise reduced data are shown in FIG. 50(a) and FIG. 50(b). The residue of PMSG-2 is obtained to be 0.0003+0.0000j.

[0239] The measurement data and noise reduced data on the side of power system are shown by FIG. 52(a) and FIG. 52(b) respectively. The residue of the ROPS subsystem is obtained to be 0.0000+0.0007j.

[0240] Step 7: installing a parallel band-pass filter on the side of PMSG-3 and measure the residues.

[0241] The procedure to measure the residues for PMSG-3 is the same as that presented above in the step 5 for PMSG-1. The measurement data on the side of PMSG-3 and the noise reduced data are shown in FIG. 54(a) and FIG. 54(b). The residue of PMSG-3 is obtained to be 0.0001+0.0000j.

[0242] The measurement data and noise reduced data on the side of power system are shown by FIG. 56(a) and FIG. 56(b) respectively. The residue of the ROPS subsystem is obtained to be 0.0000+0.0000j.

[0243] Step 8: installing a parallel band-pass filter on the side of PMSG-4 and measure the residues.

[0244] The procedure to measure the residues for PMSG-4 is the same as that presented above in the step 5 for PMSG-1. The measurement data on the side of PMSG-4 and the noise reduced data are shown in FIG. 58(a) and FIG. 58(b). The residue of PMSG-4 is obtained to be 0.0006+0.0005j.

[0245] The measurement data and noise reduced data on the side of power system are shown by FIG. 60(a) and FIG. 60(b) respectively. The residue of the ROPS subsystem is obtained to be 0.0000+0.0007j.

[0246] 4. Identify the Wind Turbine Generators which Cause the SSOs According to the Indexes

[0247] Step 9: calculating the indexes.

[0248] The indexes Z.sub.i=|{square root over (R.sub.iR.sub.ai)}|, i=1, 2, 3, 4 are computed based on the residues obtained above, where R.sub.i and R.sub.ai, i=1, 2, 3, 4 are the residues of each wind turbine generator and the corresponding ROPS subsystem, respectively. The computational results are presented in Tab. 2-1.

TABLE-US-00003 TABLE 2-1 Results of residues and corresponding index obtained by using the present invention Index Value The cause of the SSOs Z1 (PMSG-1) 0.0020 + 0.0130i Yes Z2 (PMSG-2) 0.0003 + 0.0003i No Z3 (PMSG-3) 0 No Z4 (PMSG-4) 0 No

[0249] Step 10: identifying the wind turbine generator which causes the SSOs.

[0250] According to the open-loop modal resonance theory, the wind turbine generator corresponding to the maximum index among all Zi is identified to be the wind turbine generator causing the SSOs. Therefore, PMSG-1 is identified to be the cause of the SSOs in the 10-machine 39-node power system.

[0251] 5. Verify the Correctness of Result Made by the Present Invention

[0252] The linearized model of FIG. 40 is used in this section to verify the correctness of the above identification made by the present invention.

[0253] Step 11: establishing the parametric model of the power system integrated with wind turbine generators.

[0254] The open-loop linearized models of each wind turbine generator subsystem and the corresponding ROPS subsystem described by Eq. (1) and Eq. (2) are respectively derived. Following notations are used to denote various results of derived parametric models and computational results

[0255] A.sub.pmi, b.sub.pmi, c.sub.pmi, i=1, 2, 3, 4 are the state matrix, control vector and output vector of open-loop state-space models of PMSG-1, PMSG-2, PMSG-3, and PMSG-4, respectively;

[0256] p.sub.pmi, r.sub.pmi, i=1, 2, 3, 4 are the left and right eigenvectors corresponding to the open-loop SSO modes of PMSG-1, PMSG-2, PMSG-3, and PMSG-4.

[0257] A.sub.pmaci, b.sub.pmaci, c.sub.pmaci, i=1, 2, 3, 4 are the state matrix, control vector and output vector of open-loop state-space models of the ROPS subsystem without PMSG-1, PMSG-2, PMSG-3, and PMSG-4 being included, respectively;

[0258] p.sub.pmaci, r.sub.pmaci, i=1, 2, 3, 4 are the left and right eigenvectors corresponding to the open-loop SSO modes of ROPS subsystem without PMSG-1, PMSG-2, PMSG-3, and PMSG-4 being included, respectively;

[0259] Step 12: calculating the residues of each open-loop wind turbine generator subsystem and corresponding open-loop ROPS subsystems.

[0260] From the results obtained in the step 11, the open-loop residues of PMSG-1 and the corresponding ROPS subsystem are calculated to be R.sub.pm1=p.sub.pm1.sup.Tb.sub.pm1c.sub.pm1r.sub.pm1=3.02+134.24i and R.sub.pmac1=p.sub.pmac1.sup.Tb.sub.pmac1c.sub.pmac1r.sub.pmac1=0.01+j0.002, respectively. Thus, the index associated with PMSG-1 is calculated to be Z.sub.r1={square root over (R.sub.pm1R.sub.pmac1)}=0.63+0.64i

[0261] Similarly, the open-loop residues of PMSG-2 and the corresponding ROPS subsystem are calculated to be R.sub.pm2=p.sub.pm2.sup.Tb.sub.pm2c.sub.pm2r.sub.pm2=0 and R.sub.pmac2=p.sub.pmac2.sup.Tb.sub.pmac2c.sub.pmac2r.sub.pmac2=0.006+j0.004, respectively. The index associated with PMSG-2 is calculated to be Z.sub.r2={square root over (R.sub.pm2R.sub.pmac2)}=0.

[0262] The open-loop residues of PMSG-3 and the corresponding ROPS subsystem are calculated to be R.sub.pm3=p.sub.pm3.sup.Tb.sub.pm3c.sub.pm3r.sub.pm3=0 and R.sub.pmac3=p.sub.pmac3.sup.Tb.sub.pmac3c.sub.pmac3r.sub.pmac3=0.011+j0.001, respectively. The index associated with PMSG-3 is calculated to be Z.sub.r3={square root over (R.sub.pm3R.sub.pmac3)}=0.

[0263] The open-loop residues of PMSG-4 and the corresponding ROPS subsystem are calculated to be R.sub.pm4=p.sub.pm4.sup.Tb.sub.pm4c.sub.pm4r.sub.pm4=0 and R.sub.pmac4=p.sub.pmac4.sup.Tb.sub.pmac4c.sub.pmac4r.sub.pmac4=0.008+j0.002, respectively. The index associated with PMSG-4 is calculated to be Z.sub.r4={square root over (R.sub.pm4R.sub.pmac4)}=0.

[0264] Step 13: verifying the correctness of result obtained in the step 10 by the present invention.

[0265] Computational results of indexes above by using the parametric model are listed in Table 2-2. It can be seen that PMSG-1 causes the SSOs, confirming the correctness of identification made previously from Table 2-1 by using the present invention.

TABLE-US-00004 TABLE 2-2 Results of residues and index obtained using the parametric model for confirmation Method in Parametric The cause the present The cause of model of Index invention the SSOs method the SSOs Z1 (PMSG-1) Z.sub.b1 = 1 Yes Z.sub.br1 = 1 Yes Z2 (PMSG-2) Z.sub.b2 = 0.0321 No Z.sub.br2 = 0.0000 No Z3 (PMSG-3) Z.sub.b3 = 0.0000 No Z.sub.br3 = 0.0000 No Z4 (PMSG-4) Z.sub.b4 = 0.0000 No Z.sub.br4 = 0.0000 No

[0266] PMSG-1 is disconnected from the example power system as shown by FIG. 61. Simulation results are shown from FIG. 62(a) to FIG. 62(c). It can be seen that no SSOs occur in the example power system, confirming that PMSG-1 causes the SSOs as identified by the present invention.