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Power system multi-parameter small signal stability analysis based on 2nd order perturbation theory
Electrical Power and Energy Systems 67 (2015) 409–416
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Power system multi-parameter small signal stability analysis based on 2nd order perturbation theory Jing Ma a,b,⇑, Shangxing Wang a, Yinan Li a, Yang Qiu a a b
State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China The Bradley Department of Electrical and Computer Engineering, Virginia Polytechnic Institute and State University, 24061, USA
a r t i c l e
i n f o
Article history: Received 31 December 2013 Received in revised form 24 November 2014 Accepted 5 December 2014 Available online 23 December 2014 Keywords: Low-frequency oscillation Mode analysis Sensitivity matrix 2nd order perturbation theory
a b s t r a c t A novel method based on multi-parameter 2nd order perturbation sensitivity is proposed to analyze the low-frequency oscillation modes in large-scale interconnected power system, since the low-frequency oscillation mode change is hard to determine due to the violent fluctuation of multiple parameters during operation. Firstly, the multi-parameter 2nd order perturbation sensitivity matrices of eigenvalues and eigenvectors are deduced. Then, their multi-parameter 2nd order estimated values are calculated. On the basis of this, the changing system oscillation modes under multiple parameters variation are estimated. The simulation results of WECC (Western Electricity Coordinating Council) system verify that this method is able to assess the small signal stability of the system relatively accurately even several parameters of the system change. Then it can adjust appropriate dispatching method accordingly to improve the damping of dominant oscillation mode. Also, this method makes the solving process direct and clear since it avoids the burdensome derivation calculation of 2nd order sensitivity, and it is time-saving by avoiding solving complicated high-dimensional state matrices. Ó 2014 Elsevier Ltd. All rights reserved.
Introduction Due to the increasing scale of the power system as well as the changing and complex operation conditions, the safety and stability of the system has been seriously threatened by the often occurrence of low-frequency oscillation and small signal stability [1–5]. Thus, a thorough and explicit method to analyze and control small signal stability is in urgent need. Currently, most small signal stability analysis methods are based on certainty theory. There are fewer researches considering uncertain parameters, mainly including the probability analysis method [6,7], the interval analysis method [8–10] and the sensitivity analysis method [11–13], etc. Probability analysis method uses probabilistic eigenvalue to demonstrate the influence of random factor on system stability by establishing a relationship between random parameters and state variables, and a relationship between the expectations and covariance of the state variables and eigenvalues. Interval analysis method uses interval distribution to model the small signal stability of the system under uncertain information
⇑ Corresponding author at: No. 52 Mailbox, North China Electric Power University, No. 2 Beinong Road, Changping District, Beijing 102206, China. Tel.: +86 10 80794899 (work), mobile: +86 15801659769. E-mail address: [email protected] (J. Ma). http://dx.doi.org/10.1016/j.ijepes.2014.12.021 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
by establishing a relationship between operation conditions and eigenvalues with uncertain information. The sensitivity analysis method reflects the relationship between the change of system parameters and the change of eigenvalues. It can not only be used to analyze the dominant parameters that affect small signal stability, but also guide power output and parameter adjustment, providing operators with relatively comprehensive information of small signal stability. With small fluctuation of the operation parameters, the system can be approximately considered operating in the linear region. Thus, a result that is accurate enough can be calculated by 1st order eigenvalue sensitivity [14,15]. But when the fluctuation of the operation parameters is relatively large, the non-linear characteristics of the eigenvalues and system parameters begin to be exhibited. And the accuracy of the calculation based on 1st order eigenvalue becomes limited [16]. Meanwhile, multi-parameter fluctuation in large-scale interconnected system leads to the mode change of low-frequency oscillation as well as the oscillation mode. So the change of the system oscillation mode under large multi-parameter fluctuation needs urgent study. Therefore, this paper uses matrix perturbation theory and Taylor expansion to deduce the multi-parameter 2nd order perturbation sensitivity matrix of the eigensolutions. Then, the multi-parameter 2nd order estimated value of eigensolutions is calculated. Then the mode change of system oscillation under
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J. Ma et al. / Electrical Power and Energy Systems 67 (2015) 409–416
Nomenclature A A0 DA
the state matrix of the system the state matrix of original system the incremental matrix caused by the parameters change As the perturbation sensitivity matrix of state matrix DA(a(s)) the increment of A in terms of a(s) ui the corresponding right eigenvector of the ith eigenvalue uji the jnd order perturbations of the right eigenvector ujik the kth row of the jst order increment matrix of ui(a) ki the ith eigenvalue kji the jnd perturbations of the eigenvalue a the vector composed of certain part of uncertain parameters Da the vector composed of the uncertain part of uncertain parameters L the dimension of a a0 the perturbation parameter’s original value GTki ða0 Þ
the gradient matrices of ki(a)
GTuik ða0 Þ
the gradient matrices of uik(a)
Hki ða0 Þ
the Hessian matrices of ki ðaÞ
~T G ki ~T G uik Gui ~T G uik h1 h2 h ~T H ki Huik ~ uik H f0 fi ferr n0 ni nerr
the 1st order perturbation sensitivity matrix of eigenvalue the kth row of matrix Gui the 1st order perturbation sensitivity matrix of the eigenvector the kth row of matrix Gui L L order square matrix L L order square matrix the 2nd order perturbation matrix of eigenvalue the 2nd order perturbation sensitivity matrix of eigenvalue the 2nd order perturbation matrix of the kth row of eigenvalue the 2nd order perturbation sensitivity matrix of eigenvector the real system frequency after perturbation the ith estimated value of the frequency the frequency error the real system damping ratio after perturbation the ith order estimated value of the damping ratio the damping ratio error
Huik ða0 Þ the Hessian matrices of uik(a)
multi-parameter fluctuation is estimated. The simulation results of 127-bus equivalent of the Western Electricity Coordinating Council (WECC) system verify the validity and practicability of this method. Sensitivity analysis based on matrix perturbation theory In the power system, the matrix eigenvalue is described as:
Aui ¼ ki ui
ð1Þ
where A is state matrix of the system, ki is ith eigenvalue, ui is corresponding right eigenvector of the ith eigenvalue. After the system parameter change, the state matrix of the system changes as follows:
A ¼ A 0 þ DA
ð2Þ
where A0 is the state matrix of original system, DA is the incremental matrix caused by the parameters change. From the Taylor expansion, the matrix increment caused by system parameters change is expressed as:
DA ¼
L X
ðsÞ
As Da
ð3Þ
s¼1
(
ðLÞ T
a ¼ ½að1Þ ; að2Þ ; ; a T Da ¼ ½Dað1Þ ; Dað2Þ ; ; DaðLÞ
ð4Þ
ðsÞ
a Þ where As ¼ DAð is perturbation sensitivity matrix of state matrix. DaðsÞ
DA(a(s)) is the increment of A in terms of a(s). a is the vector composed of certain part of uncertain parameters. Da is the vector composed of the uncertain part of uncertain parameters. L is the dimension of a. Perform 2nd order Taylor expansion of ki(a), which is the ith eigenvalue, at perturbation parameter’s original value a0:
1 ki ðaÞ ¼ ki ða0 Þ þ GTki ða0 ÞDa þ DaT Hki ða0 ÞDa; i ¼ 1; ; n 2
ð5Þ
Meanwhile, perform 2nd order Taylor expansion of uik(a), the kth parameters of eigenvector ui(a), which is the corresponding eigenvector of ki(a), at a0 as well:
1 uik ðaÞ ¼ uik ða0 Þ þ GTuik ða0 ÞDa þ DaT Huik ða0 ÞDa 2
ð6Þ
where GTki ða0 Þ and GTuik ða0 Þ are the gradient matrices of ki(a) and uik(a) expressed in terms of a respectively. Hki(a0) and Huik ða0 Þ are the Hessian matrices of ki(a) and uik(a) expressed in terms of a, respectively. From the direct derivation method, the gradient matrix of ki(a) and uik(a) expressed in terms of a is written as:
8 h i > i ðaÞ @ki ðaÞ i ðaÞ < GTki ða0 Þ ¼ @k @k @ að1Þ @ að2Þ @ aðLÞ h i ðaÞ @uik ðaÞ @uik ðaÞ > : GTu ða0 Þ ¼ @uikð1Þ ð2Þ @ aðLÞ @ @ a a ik
ð7Þ
From the direct derivation method, the Hessian matrix of ki(a) and uik(a) expressed in terms of a is written as:
8 2 2 @ ki ðaÞ > > > > 6 @að1Þ @ að1Þ > > 6 > .. .. > > Hki ða0 Þ ¼ 6 . . > 4 > > 2 > @ k ð a Þ i < ðLÞ ð1Þ 2@ a 2 @ a @ u ð a Þ > ik > > > 6 @ að1Þ @ að1Þ > > 6 > .. > Huik ða0 Þ ¼ 6 ... > > . 4 > > > @ 2 uik ðaÞ : @ aðLÞ @ að1Þ
@ 2 ki ðaÞ @ að1Þ @ aðLÞ
.. . @ 2 ki ðaÞ @ aðLÞ @ aðLÞ
3 7 7 7 5
@ 2 uik ðaÞ @ að1Þ @ aðLÞ
.. . @ 2 uik ðaÞ @ aðLÞ @ aðLÞ
3
ð8Þ
7 7 7 5
In the power system, the gradient matrix and Hessian matrix cannot be obtained by the derivation method directly since the eigenvalue and eigenvector are implicit functions of the system parameters. Therefore, the analysis of low-frequency oscillation modes of the power system by the matrix perturbation theory is proposed in this paper. From the matrix perturbation theory [17,18], the 2nd order estimated values of the eigenvalue and its corresponding eigenvector after system parameter change is written as:
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ki ¼ k0i þ k1i þ k2i
ð9Þ
ui ¼ u0i þ u1i þ u2i
ð10Þ
where k0i and u0i are the eigenvalue and right eigenvector of the original system, respectively. u1i and u2i are the 1st and 2nd order perturbations of the right eigenvector, respectively. k1i and k1i are the 1st and 2nd perturbations of the eigenvalue respectively. The 1st order perturbation of the eigenvalue k1i and 1st order perturbation of the eigenvector u1i, are expressed as follows, respectively:
k1i ¼ wT0i DAu0i n X
u1i ¼
i–j j¼1
ð11Þ
1 wT0j DAu0i u0j k0i k0j
ð12Þ
where w0i is the left eigenvector of the original system which meet following equations after normalization with right eigenvector:
wT0i u0i ¼ 1
ð13Þ
wT0i u0j
ð14Þ
¼ 0 ði–jÞ
Substitute Eq. (3) in Eq. (11), the 1st order perturbation of ki(a) expressed in terms of a is written as:
k1i ¼
wT0i
L L X X ðsÞ ~ T Da As DaðsÞ u0i ¼ k1i DaðsÞ ¼ G ki
~ T Da u1ik ¼ G uik
In Eqs. (9) and (10), k2i and u2i, the 2nd order perturbations of the eigenvalue and eigenvector, are given by the following equations respectively:
k2i ¼ wT0i DAu1i k1i wT0i u1i
i–j j¼1
k2i ¼
wT0i
s¼1
n X j¼1 j–i
¼
n X j¼1 j–i
1 k0i k0j
1 k0i k0j
wT0j
! L X As DaðsÞ u0i u0j
s¼1
where:
2
ðLÞ 3 wT0i A1 u1i 7 .. .. 7 5 . .
ð1Þ
wT0i A1 u1i 6 .. h1 ¼ 6 4 .
ð1Þ
wT0i AL u1i ð1Þ
ð1Þ
k1i wT0i u1i .. .
ðLÞ ð1Þ k1i wT0i u1i
ð21Þ
ðLÞ
wT0i AL u1i ð1Þ
k1i .. .
ðLÞ 3 wT0i u1i 7 .. 7 5 .
ð22Þ
ðLÞ ðLÞ k1i wT0i u1i
In Eq. (20), h1 and h2 are both L L order square matrix. The 2nd order perturbation matrix of eigenvalue can be obtained by summing these two up, which is h:
h ¼ h1 þ h2
s¼1
wT0j As u0i DaðsÞ uoj
ð20Þ
¼ DaT h1 Da þ DaT h2 Da
6 h2 ¼ 6 4
u1i ¼
" # L X ðsÞ As Da u1i k1i wT0i u1i
L X ¼ ðwT0i As u1i ÞDaðsÞ k1i wT0i u1i
2
ity matrix of eigenvalue. Substitute Eq. (3) in Eq. (12), the 1st order perturbation of ui(a) expressed in terms of a is:
ð19Þ
Substitute Eq. (3) in Eq. (18):
h i ~ T ¼ kð1Þ ; kð2Þ ; ; kðLÞ is the 1st order perturbation sensitivwhere G ki 1i 1i 1i
s¼1
ð18Þ
1 1 wT0j DAu1i k1i wT0j u1i u0j uT1j u1i u0i k0i k0j 2
n X
u2i ¼
ð15Þ
s¼1
ð17Þ
ð16Þ
h
i ð1Þ ðsÞ ðLÞ ¼ u1i ; ; u1i ; ; u1i Da ¼ Gui Da ðsÞ u1i
where is the 1st order perturbation increment of ui ðaÞ expressed in terms of parameter Da(s). Gui is the 1st order perturbation sensitivity matrix of the eigenvector. ~ T is the kth row of matrix Gui, then the kth row of the Suppose G uik 1st order increment matrix of ui(a) is written as:
ð23Þ
~ T ¼ h þ hT as the 2nd order perturbation sensitivity matrix Define H ki of eigenvalue, then the 2nd order perturbation of ki(a) expressed in terms of a is given by:
k2i ¼
1 T ~T Da Hki Da 2
ð24Þ
Substitute Eqs. (11) and (24) in Eq. (9), the 2nd order estimated values of system eigenvalue after multi-parameter change is obtained:
~ T Da þ 1 DaT H ~ T Da ki ¼ k0i þ G ki ki 2
ð25Þ
Substitute Eq. (3) in Eq. (19), the 2nd order perturbation of ui(a) expressed in terms of a is written as:
u2i ¼
n X i–j j¼1
" # L X 1 1 wT0j As DaðsÞ u1i k1i wT0j u1i u0j uT1j u1i u0i k0i k0j 2 s¼1 ð26Þ
Substitute Eqs. (15) and (16) in Eq. (26), then: ð1Þ
ðjÞ
ðnÞ
u2i ¼ U0 ½DaT h Da; ; DaT h Da; ; DaT h Da when j – i, h
(j)
2
Fig. 1. The calculation process of 2nd order perturbation method and direct numerical method.
1 6 6 ðjÞ h ¼ 6 k0i k0j 4
T
ð27Þ
is given by: ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ðLÞ
wT0j A1 u1i k1i wT0j u1i wT0j A1 u1i k1i wT0j u1i .. . ð1Þ
.. . ð1Þ
ð1Þ
wT0j AL u1i k1i wT0j u1i
ðLÞ
ðLÞ
ðLÞ
wT0j AL u1i k1i wT0j u1i
3 7 7 7 5 ð28Þ
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J. Ma et al. / Electrical Power and Energy Systems 67 (2015) 409–416
Fig. 2. WECC system.
when j = i, h(j) is given by:
2
h
ðjÞ
ð1Þ
ð1Þ
u0i u1i 16 ¼ 6 ... 24
ð1Þ ð1Þ u1i u1i
ðLÞT ð1Þ 3 u1i u1i 7 7 ... 5
~ uik ¼ Huik þ HT as Make matrix Huik summarization, and define H uik the 2nd order perturbation sensitivity matrix of eigenvector. Then the 2nd order perturbation of uik(a) expressed in terms of a is:
ð29Þ
ðLÞ ðLÞ u1i u1i
From Eqs. (28) and (29), u2ik, the 2nd order increment of uik(a), is given by:
u2ik ¼
n X ðjÞ u0jk DaT h Da ¼ DaT Huik Da
ð30Þ
j¼1
where Huik = u01kh(1) + u01kh(2) + + u01kh(n) is the 2nd order perturbation matrix of the kth row of eigenvalue.
u2ik ¼
1 T~ Da Huik Da 2
ð31Þ
Substitute Eqs. (16) and (31) in Eq. (10), the 2nd order estimated value of the kth row of the system eigenvector after multi-parameter change is as follows:
~ T Da þ 1 Da T H ~ uik Da uik ¼ u0ik þ G uik 2
ð32Þ
The method that calculates 2nd order perturbation sensitivity not only makes the solving process direct and clear since it avoids the burdensome derivation calculation, but also saves calculating time
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J. Ma et al. / Electrical Power and Energy Systems 67 (2015) 409–416
by avoiding solving complicated high-dimensional state matrix, only using variation of perturbation parameter and perturbation sensitivity matrix to obtain the system characteristic quantities after the perturbation (The variation of loads and generators outputs is able to be obtained by the PMUs installed on the external bus of generators and substations. The inertias and transient reactance are able to be obtained by the parameters of nameplates). To prove the effectiveness of the method, comparison of calculation process between proposed method and true value analysis is shown in Fig. 1. The main difference between the two methods is that it takes a great amount of time for the direct method to form a new matrix A after the disturbance; while for the 2nd order perturbation method, simple matrix multiplication of the disturbance variable is all that is needed for small perturbation analysis after the disturbance. Therefore, the 2nd order perturbation method is more efficient and could be attractive for on-line monitoring or control purposes. Testing results and analysis 127-bus equivalent of the Western Electricity Coordinating Council (WECC) systems introduced to verify the validity and practicability of the proposed method [19]. Table 1 Dominant modes for low-frequency oscillation. Mode
Eigenvalues
Freq (Hz)
Damp ratios (%)
Dominant generators
1 2 3 4
0.2925 ± 4.4842i 0.1333 ± 5.0381i 0.3162 ± 5.1638i 0.2602 ± 5.9488i
0.7137 0.8018 0.8218 0.9468
6.51 2.65 6.11 4.37
G18 G20 G1G2 G29 G1 G2 G29 G3 G5 G19 G22
The system is a multi-area inter-connected large scale power system, which has a generating capacity of 61.4 GW with a load of 60.8 GW, as shown in Fig. 2. The system model includes 29 generators, 127 buses, and 215 lines. Generators adopt the six-order detailed model. The excitation model is IEEE-DC1 model, and the load model adopts the WECC model, where 80% is constant active load, 80% is constant reactive load, 20% is motor load. The dominant mode of low-frequency oscillation is obtained by using mode analysis method to analyze the system, as shown in Table 1. When a certain or several generators in the equivalent system are put into or out of operation, parameters such as the inertia constant and d-axis transient reactance will change. A perturbation analysis of the related system parameters is conducted, taking the following two kinds of perturbation as examples: (1) The power output of equivalent generator 1, 2, and 3 increase by 5% and the active load on bus 9, 10, and 11 all increase by 5%, where c = 5% is the perturbation parameter in example 1. (2) The inertia time constant of equivalent generator 13, 14 and 15 increase by 5% and the d-axis transient reactance of generator 14, 15 and 16 increase by 5%, where b = 5% is the perturbation parameter in example 2. Define ferr as the frequency error, f0 as the real system frequency after perturbation, and fi as the ith estimated value, where i is 1 or 2. Define nerr as the damping ratio error, n0 as the real system damping ratio after perturbation, and ni as the ith order estimated value of the damping ratio, where i is 1 or 2.
f err ¼
jf i f 0 j f0
ð33Þ
Table 2 Changes of oscillation mode after perturbation 1. Mode 1
Mode 2
Mode 3
Mode 4
True value
k0 f0 n0(%)
0.2925 ± 4.4851i 0.7138 6.51
0.1273 ± 5.0539i 0.8043 2.52
0.3481 ± 5.1980i 0.8273 6.68
0.2591 ± 5.9552i 0.9478 4.35
1st order perturbation estimated value
k1 f1 n1(%) ferr1 nerr1
0.2920 ± 4.4856i 0.7139 6.49 0.0001 0.0031
0.0908 ± 5.0347i 0.8013 1.80 0.0037 0.2659
0.2393 ± 5.2974i 0.8431 4.51 0.0191 0.4940
0.2114 ± 5.9568i 0.9481 3.55 0.0003 0.2345
2nd order perturbation estimated value
k2 f2 n2(%) ferr2 nerr2
0.2924 ± 4.4853i 0.7139 6.51 0.0001 0.0007
0.1152 ± 5.0423i 0.8025 2.28 0.0022 0.0936
0.2991 ± 5.2367i 0.8334 5.70 0.0074 0.1464
0.2431 ± 5.9557i 0.9479 4.08 0.0001 0.0624
Table 3 Changes of oscillation mode after perturbation 2. Mode 1
Mode 2
Mode 3
Mode 4
True value
k0 f0 n0(%)
0.3175 ± 4.4368i 0.7061 7.14
0.1358 ± 5.0368i 0.8016 2.69
0.3158 ± 5.1492i 0.8195 6.12
0.2549 ± 5.9074i 0.9402 4.31
1st order perturbation estimated value
k1 f1 n1(%) ferr1 nerr1
0.3543 ± 4.4878i 0.7143 7.87 0.0116 0.1022
0.1280 ± 5.0456i 0.8030 2.54 0.0017 0.0558
0.3502 ± 5.2123i 0.8296 6.70 0.0123 0.0948
0.2421 ± 5.9118i 0.9409 4.09 0.0007 0.0510
2nd order perturbation estimated value
k2 f2 n2(%) ferr2 nerr2
0.3321 ± 4.4637i 0.7104 7.42 0.0061 0.0392
0.1327 ± 5.0392i 0.8020 2.63 0.0005 0.0223
0.3328 ± 5.1765i 0.8239 6.42 0.0054 0.0490
0.2478 ± 5.9087i 0.9404 4.19 0.0002 0.0278
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nerr ¼
J. Ma et al. / Electrical Power and Energy Systems 67 (2015) 409–416
jni n0 j n0
ð34Þ
In the two examples mentioned above, the estimated values of oscillation modes after perturbation calculated by 2nd order perturbation sensitivity matrix are shown in Tables 2 and 3. In example 1, the maximum error of the estimated 1st order value of frequency and damping ratio are 1.91% and 49.4% respectively, and the maximum error of the estimated 2nd order value of frequency and damping ratio are 0.74% and 14.64% respectively, as shown in Table 2. In example 2, the maximum error of the estimated 1st order value of frequency and damping ratio are 1.23% and 10.22% respectively, and the maximum error of the estimated 2nd order value of frequency and damping ratio are 0.61% and 4.9% respectively, as shown in Table 3. The accuracy advantage of the estimated 2nd order value calculated by the 2nd order perturbation sensitivity of the eigenvalues in analyzing low-frequency oscillation mode with multiple uncertain parameters can be seen by comparing Table 2 with Table 3.
As for example 1, in the process of the perturbation parameter varying from 0% to 5%, the variation curve of frequency and damping ratio of each oscillation mode is as shown in Fig. 3, where the dashed line represents the estimated 1st order value calculated by 1st order sensitivity matrix, the solid line represents the estimated 2nd order value calculated by 2nd order sensitivity matrix, and the asterisks represents the true value of system frequency and damping ratio after perturbation. Fig. 3 indicates that except for mode 1, which has nearly no change after perturbation, both the system oscillation frequency and damping ratio increase or decrease in a nonlinear way With the increase of perturbation, the estimated linear 1st order value calculated by 1st order sensitivity matrix becomes not accurate enough, even largely deviated. So the advantage of 2nd order sensitivity matrix becomes more obvious since it keeps relatively high calculation accuracy even under large perturbation. As for example 2, in the process of the perturbation parameter varying from 0% to 5%, the variation curve of frequency and damping ratio of each oscillation mode is as shown in Fig. 4.
Fig. 3. Frequencies and damping ratios in the case of perturbation No. 1.
J. Ma et al. / Electrical Power and Energy Systems 67 (2015) 409–416
By comparing Fig. 3(h) with 4(h), it can be seen that as the perturbation increases, the deviation of the estimated 1st order value from the actual value increases as well. Although the deviation of the estimated 2nd order value also increases, such deviation is still acceptable. This indicates that compared with the estimated 1st order value of frequency and damping ratio of the oscillation mode, its estimated 2nd order value has higher accuracy and is more close to the actual value. It needs to be noted that generators are often put into or out of operation in the equivalent system. The method proposed is able to obtain eigenvalues of the entire system only by taking the changed parameter into consideration. It avoids the complicated calculation progress of high-dimensional state matrix and saves calculation time. Participation factor is the product of corresponding elements of left and right eigenvectors. It is a comprehensive index to reflect the influence of system state parameters on the controllability and observability of a certain oscillation mode, and is a highly
415
important guidance on the choice of installation location of controllers. After the system parameter change, left and right eigenvectors corresponding to the system oscillation mode change accordingly, and the corresponding participation factors change as well. As shown in Table 1, the generators involved in mode 2 is from two area, making it a classical inter-area low-frequency oscillation mode. Thus, the participation factor of mode 2 under small perturbation is studied. As shown in Fig. 5, the participation factor changes accordingly in example 2. Fig. 5 illustrates that the participation factor of generator 1 is the largest before and after perturbation, followed by equivalent system 2 and 29. It can be concluded from the figure that generators involve in mode 2 does not change and 2nd order perturbation estimated value is almost the same with the true value after perturbation 2. To prove the rapidity of the proposed method, comparison of calculation speed between 2nd order perturbation method and direct computation method is shown in Table 4. It can be seen that
Fig. 4. Frequencies and damping ratios in the case of perturbation No. 2.
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J. Ma et al. / Electrical Power and Energy Systems 67 (2015) 409–416
Acknowledgement Many faculties and students contribute greatly to this research. The authors would like to thank Dr. Arun G. Phadke and DrYilu Liu. This work was supported by the National Natural Science Foundation of China (51277193), the Chinese University Scientific Fund Project (2014ZZD02), the 111 project (B08013), the Scientific Research Foundation for the Returned Overseas Chinese Scholars of State Education Ministry ([2011] No. 1139), Hebei Natural Science Foundation (E2012502034), Electric Power Youth Science and Technology Creativity Foundation of CSEE ([2012] No. 46), the New-Star of Science and Technology supported by Beijing Metropolis Beijing Nova program (Z141101001814012), the Excellent talents in Beijing City (2013B009005000001), the Fund of Fok Ying Tung Education Foundation (141057). References
Fig. 5. Participators of Mode 2 using 2st sensitivity after perturbation 2.
Table 4 Calculation time of two methods applying to WECC system. Method
CPU time consuming (s)
Direct computation method 2nd order perturbation method
22.236152 0.305780
the 2nd order perturbation method is more efficient than the direct computation method. The CPU used is the Intel core i3 processor 330 M (2.13 GHz). Conclusion A novel method to analyze low-frequency oscillation modes by multi-parameter 2nd order perturbation sensitivity is proposed in this paper. Firstly, the multi-parameter 2nd order perturbation sensitivity matrices of eigenvalue and eigenvector are deduced. Then, their multi-parameter 2nd order estimated values are calculated. Based on this, the change of system oscillation modes under multiple parameter variation is assessed. This method has following characteristics. (1) It is able to estimate the small signal stability of the system relatively accurately even several uncertain parameters in the system change largely, thus provide the operators with relatively comprehensive information. (2) It makes the solving process direct and clear since it avoids the burdensome derivation calculation of 2nd order sensitivity, and it saves calculating time by avoiding solving complicated high-dimensional state matrix. (3) Compared with the 1st order perturbation theory, it is more accurate in analyzing low-frequency oscillation mode.
[1] Paserba J. Analysis and control of power system oscillation, in CIGRE Special Pub. 38.01.07; 1996. [2] Kundur P. Power system stability and control. New York: McGraw-Hill; 1994. [3] Ramos Rodrigo A, Alberto Luís FC, Bretas Newton G. A new methodology for the coordinated design of robust decentralized Power system damping controllers. IEEE Trans Power Syst 2004;19(1):444–54. [4] Kamwa I, Trudel G, Gérin-Lajoie L. Robust design and coordination of multiple damping controllers using nonlinear constrained optimization. IEEE Trans Power Syst 2000;15:1084–92. [5] Chaudhuri B, Pal BC. Robust damping of multiple swing modes employing global stabilizing signals with a TCSC. IEEE Trans Power Syst 2004;19(1):499–506. [6] Jin Ma, Dong Han, Ren-mu He, Zhao-Yang Dong, Hill DJ. Reducing identified parameters of measurement-based composite load model. IEEE Trans Power Syst 2008;23(1):76–83. [7] Chung CY, Wang KW, Tse CT, Niu R. Power-systemstabilizer(PSS) design by probabilistic sensitivity indexes (PSIs). IEEE Trans Power Syst 2002;17(3):688–93. [8] Jing Ma, Tong Wang, Zengping Wang, Thorp J.S. Interval eigenvalue analysis of closed-loop control for power system oscillation with interval parameters. IEEE Power and Energy Society General Meeting; 22–26 July 2012. p. 1–6. [9] Jing Ma, Mingfa Peng, Tong Wang. Low frequency oscillation analysis of the topology variation network based on matrix perturbation. Asia-pacific power and energy engineering conference (APPEEC); 27–29, March 2012. p. 1–4. [10] Chen SH, Qiu ZP. A new method for computing the upper and lower bounds on frequencies of structures with interval parameters. Mech Res Commun 1994;21(6):583–92. [11] Tse CT, Tso SK. Design optimisation of power system stabilisers based on model and eigenvalue-sensitivity analyses. IEE Proc Gener Trans Distribut 1988;135(5):406–15. [12] Skogestad S, Postlethwaite I. Multivariable feedback control. New York: Wiley; 2001. [13] Pal B, Chaudhuri B. Robust control in power systems. New York: Springer; 2005. [14] Wang KW, Chung CY, Tse CT, Tsang KM. Probabilistic eigenvalue sensitivity indices for robust PSS site selection. IEE Proc-Gener Trans Distribut 2001;148(6):603–9. [15] Yu W-S, Ko H-J. Improved eigenvalue sensitivity for finite-precision digital controller realisations via orthogonal Hermitian transform. IEE Proc-Control Theory Appl 2003;150(4):365–75. [16] Nam HK, Kim YK, Shim KS, Lee KY. A new Eigen-sensitivity theory of augmented matrix and its applications to power system stability analysis. IEEE Trans Power Syst 2000;15(1):363–9. [17] Moore RE. Interval analysis. New York: Prentice-Hall; 1966. [18] Chen SH. Matrix perturbation theory in structural dynamics designs. Beijing: Science Press; 1999. [19] Peng Zhang, Thorp JS, Xiaoru Wang, Xing Wei. The effect of electromechanical wave controllers on inter-area modes. IEEE Power Energy Soc General Meet 2012:1–8.
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Power system multi-parameter small signal stability analysis based on 2nd order perturbation theory Jing Ma a,b,⇑, Shangxing Wang a, Yinan Li a, Yang Qiu a a b
State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China The Bradley Department of Electrical and Computer Engineering, Virginia Polytechnic Institute and State University, 24061, USA
a r t i c l e
i n f o
Article history: Received 31 December 2013 Received in revised form 24 November 2014 Accepted 5 December 2014 Available online 23 December 2014 Keywords: Low-frequency oscillation Mode analysis Sensitivity matrix 2nd order perturbation theory
a b s t r a c t A novel method based on multi-parameter 2nd order perturbation sensitivity is proposed to analyze the low-frequency oscillation modes in large-scale interconnected power system, since the low-frequency oscillation mode change is hard to determine due to the violent fluctuation of multiple parameters during operation. Firstly, the multi-parameter 2nd order perturbation sensitivity matrices of eigenvalues and eigenvectors are deduced. Then, their multi-parameter 2nd order estimated values are calculated. On the basis of this, the changing system oscillation modes under multiple parameters variation are estimated. The simulation results of WECC (Western Electricity Coordinating Council) system verify that this method is able to assess the small signal stability of the system relatively accurately even several parameters of the system change. Then it can adjust appropriate dispatching method accordingly to improve the damping of dominant oscillation mode. Also, this method makes the solving process direct and clear since it avoids the burdensome derivation calculation of 2nd order sensitivity, and it is time-saving by avoiding solving complicated high-dimensional state matrices. Ó 2014 Elsevier Ltd. All rights reserved.
Introduction Due to the increasing scale of the power system as well as the changing and complex operation conditions, the safety and stability of the system has been seriously threatened by the often occurrence of low-frequency oscillation and small signal stability [1–5]. Thus, a thorough and explicit method to analyze and control small signal stability is in urgent need. Currently, most small signal stability analysis methods are based on certainty theory. There are fewer researches considering uncertain parameters, mainly including the probability analysis method [6,7], the interval analysis method [8–10] and the sensitivity analysis method [11–13], etc. Probability analysis method uses probabilistic eigenvalue to demonstrate the influence of random factor on system stability by establishing a relationship between random parameters and state variables, and a relationship between the expectations and covariance of the state variables and eigenvalues. Interval analysis method uses interval distribution to model the small signal stability of the system under uncertain information
⇑ Corresponding author at: No. 52 Mailbox, North China Electric Power University, No. 2 Beinong Road, Changping District, Beijing 102206, China. Tel.: +86 10 80794899 (work), mobile: +86 15801659769. E-mail address: [email protected] (J. Ma). http://dx.doi.org/10.1016/j.ijepes.2014.12.021 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.
by establishing a relationship between operation conditions and eigenvalues with uncertain information. The sensitivity analysis method reflects the relationship between the change of system parameters and the change of eigenvalues. It can not only be used to analyze the dominant parameters that affect small signal stability, but also guide power output and parameter adjustment, providing operators with relatively comprehensive information of small signal stability. With small fluctuation of the operation parameters, the system can be approximately considered operating in the linear region. Thus, a result that is accurate enough can be calculated by 1st order eigenvalue sensitivity [14,15]. But when the fluctuation of the operation parameters is relatively large, the non-linear characteristics of the eigenvalues and system parameters begin to be exhibited. And the accuracy of the calculation based on 1st order eigenvalue becomes limited [16]. Meanwhile, multi-parameter fluctuation in large-scale interconnected system leads to the mode change of low-frequency oscillation as well as the oscillation mode. So the change of the system oscillation mode under large multi-parameter fluctuation needs urgent study. Therefore, this paper uses matrix perturbation theory and Taylor expansion to deduce the multi-parameter 2nd order perturbation sensitivity matrix of the eigensolutions. Then, the multi-parameter 2nd order estimated value of eigensolutions is calculated. Then the mode change of system oscillation under
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J. Ma et al. / Electrical Power and Energy Systems 67 (2015) 409–416
Nomenclature A A0 DA
the state matrix of the system the state matrix of original system the incremental matrix caused by the parameters change As the perturbation sensitivity matrix of state matrix DA(a(s)) the increment of A in terms of a(s) ui the corresponding right eigenvector of the ith eigenvalue uji the jnd order perturbations of the right eigenvector ujik the kth row of the jst order increment matrix of ui(a) ki the ith eigenvalue kji the jnd perturbations of the eigenvalue a the vector composed of certain part of uncertain parameters Da the vector composed of the uncertain part of uncertain parameters L the dimension of a a0 the perturbation parameter’s original value GTki ða0 Þ
the gradient matrices of ki(a)
GTuik ða0 Þ
the gradient matrices of uik(a)
Hki ða0 Þ
the Hessian matrices of ki ðaÞ
~T G ki ~T G uik Gui ~T G uik h1 h2 h ~T H ki Huik ~ uik H f0 fi ferr n0 ni nerr
the 1st order perturbation sensitivity matrix of eigenvalue the kth row of matrix Gui the 1st order perturbation sensitivity matrix of the eigenvector the kth row of matrix Gui L L order square matrix L L order square matrix the 2nd order perturbation matrix of eigenvalue the 2nd order perturbation sensitivity matrix of eigenvalue the 2nd order perturbation matrix of the kth row of eigenvalue the 2nd order perturbation sensitivity matrix of eigenvector the real system frequency after perturbation the ith estimated value of the frequency the frequency error the real system damping ratio after perturbation the ith order estimated value of the damping ratio the damping ratio error
Huik ða0 Þ the Hessian matrices of uik(a)
multi-parameter fluctuation is estimated. The simulation results of 127-bus equivalent of the Western Electricity Coordinating Council (WECC) system verify the validity and practicability of this method. Sensitivity analysis based on matrix perturbation theory In the power system, the matrix eigenvalue is described as:
Aui ¼ ki ui
ð1Þ
where A is state matrix of the system, ki is ith eigenvalue, ui is corresponding right eigenvector of the ith eigenvalue. After the system parameter change, the state matrix of the system changes as follows:
A ¼ A 0 þ DA
ð2Þ
where A0 is the state matrix of original system, DA is the incremental matrix caused by the parameters change. From the Taylor expansion, the matrix increment caused by system parameters change is expressed as:
DA ¼
L X
ðsÞ
As Da
ð3Þ
s¼1
(
ðLÞ T
a ¼ ½að1Þ ; að2Þ ; ; a T Da ¼ ½Dað1Þ ; Dað2Þ ; ; DaðLÞ
ð4Þ
ðsÞ
a Þ where As ¼ DAð is perturbation sensitivity matrix of state matrix. DaðsÞ
DA(a(s)) is the increment of A in terms of a(s). a is the vector composed of certain part of uncertain parameters. Da is the vector composed of the uncertain part of uncertain parameters. L is the dimension of a. Perform 2nd order Taylor expansion of ki(a), which is the ith eigenvalue, at perturbation parameter’s original value a0:
1 ki ðaÞ ¼ ki ða0 Þ þ GTki ða0 ÞDa þ DaT Hki ða0 ÞDa; i ¼ 1; ; n 2
ð5Þ
Meanwhile, perform 2nd order Taylor expansion of uik(a), the kth parameters of eigenvector ui(a), which is the corresponding eigenvector of ki(a), at a0 as well:
1 uik ðaÞ ¼ uik ða0 Þ þ GTuik ða0 ÞDa þ DaT Huik ða0 ÞDa 2
ð6Þ
where GTki ða0 Þ and GTuik ða0 Þ are the gradient matrices of ki(a) and uik(a) expressed in terms of a respectively. Hki(a0) and Huik ða0 Þ are the Hessian matrices of ki(a) and uik(a) expressed in terms of a, respectively. From the direct derivation method, the gradient matrix of ki(a) and uik(a) expressed in terms of a is written as:
8 h i > i ðaÞ @ki ðaÞ i ðaÞ < GTki ða0 Þ ¼ @k @k @ að1Þ @ að2Þ @ aðLÞ h i ðaÞ @uik ðaÞ @uik ðaÞ > : GTu ða0 Þ ¼ @uikð1Þ ð2Þ @ aðLÞ @ @ a a ik
ð7Þ
From the direct derivation method, the Hessian matrix of ki(a) and uik(a) expressed in terms of a is written as:
8 2 2 @ ki ðaÞ > > > > 6 @að1Þ @ að1Þ > > 6 > .. .. > > Hki ða0 Þ ¼ 6 . . > 4 > > 2 > @ k ð a Þ i < ðLÞ ð1Þ 2@ a 2 @ a @ u ð a Þ > ik > > > 6 @ að1Þ @ að1Þ > > 6 > .. > Huik ða0 Þ ¼ 6 ... > > . 4 > > > @ 2 uik ðaÞ : @ aðLÞ @ að1Þ
@ 2 ki ðaÞ @ að1Þ @ aðLÞ
.. . @ 2 ki ðaÞ @ aðLÞ @ aðLÞ
3 7 7 7 5
@ 2 uik ðaÞ @ að1Þ @ aðLÞ
.. . @ 2 uik ðaÞ @ aðLÞ @ aðLÞ
3
ð8Þ
7 7 7 5
In the power system, the gradient matrix and Hessian matrix cannot be obtained by the derivation method directly since the eigenvalue and eigenvector are implicit functions of the system parameters. Therefore, the analysis of low-frequency oscillation modes of the power system by the matrix perturbation theory is proposed in this paper. From the matrix perturbation theory [17,18], the 2nd order estimated values of the eigenvalue and its corresponding eigenvector after system parameter change is written as:
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J. Ma et al. / Electrical Power and Energy Systems 67 (2015) 409–416
ki ¼ k0i þ k1i þ k2i
ð9Þ
ui ¼ u0i þ u1i þ u2i
ð10Þ
where k0i and u0i are the eigenvalue and right eigenvector of the original system, respectively. u1i and u2i are the 1st and 2nd order perturbations of the right eigenvector, respectively. k1i and k1i are the 1st and 2nd perturbations of the eigenvalue respectively. The 1st order perturbation of the eigenvalue k1i and 1st order perturbation of the eigenvector u1i, are expressed as follows, respectively:
k1i ¼ wT0i DAu0i n X
u1i ¼
i–j j¼1
ð11Þ
1 wT0j DAu0i u0j k0i k0j
ð12Þ
where w0i is the left eigenvector of the original system which meet following equations after normalization with right eigenvector:
wT0i u0i ¼ 1
ð13Þ
wT0i u0j
ð14Þ
¼ 0 ði–jÞ
Substitute Eq. (3) in Eq. (11), the 1st order perturbation of ki(a) expressed in terms of a is written as:
k1i ¼
wT0i
L L X X ðsÞ ~ T Da As DaðsÞ u0i ¼ k1i DaðsÞ ¼ G ki
~ T Da u1ik ¼ G uik
In Eqs. (9) and (10), k2i and u2i, the 2nd order perturbations of the eigenvalue and eigenvector, are given by the following equations respectively:
k2i ¼ wT0i DAu1i k1i wT0i u1i
i–j j¼1
k2i ¼
wT0i
s¼1
n X j¼1 j–i
¼
n X j¼1 j–i
1 k0i k0j
1 k0i k0j
wT0j
! L X As DaðsÞ u0i u0j
s¼1
where:
2
ðLÞ 3 wT0i A1 u1i 7 .. .. 7 5 . .
ð1Þ
wT0i A1 u1i 6 .. h1 ¼ 6 4 .
ð1Þ
wT0i AL u1i ð1Þ
ð1Þ
k1i wT0i u1i .. .
ðLÞ ð1Þ k1i wT0i u1i
ð21Þ
ðLÞ
wT0i AL u1i ð1Þ
k1i .. .
ðLÞ 3 wT0i u1i 7 .. 7 5 .
ð22Þ
ðLÞ ðLÞ k1i wT0i u1i
In Eq. (20), h1 and h2 are both L L order square matrix. The 2nd order perturbation matrix of eigenvalue can be obtained by summing these two up, which is h:
h ¼ h1 þ h2
s¼1
wT0j As u0i DaðsÞ uoj
ð20Þ
¼ DaT h1 Da þ DaT h2 Da
6 h2 ¼ 6 4
u1i ¼
" # L X ðsÞ As Da u1i k1i wT0i u1i
L X ¼ ðwT0i As u1i ÞDaðsÞ k1i wT0i u1i
2
ity matrix of eigenvalue. Substitute Eq. (3) in Eq. (12), the 1st order perturbation of ui(a) expressed in terms of a is:
ð19Þ
Substitute Eq. (3) in Eq. (18):
h i ~ T ¼ kð1Þ ; kð2Þ ; ; kðLÞ is the 1st order perturbation sensitivwhere G ki 1i 1i 1i
s¼1
ð18Þ
1 1 wT0j DAu1i k1i wT0j u1i u0j uT1j u1i u0i k0i k0j 2
n X
u2i ¼
ð15Þ
s¼1
ð17Þ
ð16Þ
h
i ð1Þ ðsÞ ðLÞ ¼ u1i ; ; u1i ; ; u1i Da ¼ Gui Da ðsÞ u1i
where is the 1st order perturbation increment of ui ðaÞ expressed in terms of parameter Da(s). Gui is the 1st order perturbation sensitivity matrix of the eigenvector. ~ T is the kth row of matrix Gui, then the kth row of the Suppose G uik 1st order increment matrix of ui(a) is written as:
ð23Þ
~ T ¼ h þ hT as the 2nd order perturbation sensitivity matrix Define H ki of eigenvalue, then the 2nd order perturbation of ki(a) expressed in terms of a is given by:
k2i ¼
1 T ~T Da Hki Da 2
ð24Þ
Substitute Eqs. (11) and (24) in Eq. (9), the 2nd order estimated values of system eigenvalue after multi-parameter change is obtained:
~ T Da þ 1 DaT H ~ T Da ki ¼ k0i þ G ki ki 2
ð25Þ
Substitute Eq. (3) in Eq. (19), the 2nd order perturbation of ui(a) expressed in terms of a is written as:
u2i ¼
n X i–j j¼1
" # L X 1 1 wT0j As DaðsÞ u1i k1i wT0j u1i u0j uT1j u1i u0i k0i k0j 2 s¼1 ð26Þ
Substitute Eqs. (15) and (16) in Eq. (26), then: ð1Þ
ðjÞ
ðnÞ
u2i ¼ U0 ½DaT h Da; ; DaT h Da; ; DaT h Da when j – i, h
(j)
2
Fig. 1. The calculation process of 2nd order perturbation method and direct numerical method.
1 6 6 ðjÞ h ¼ 6 k0i k0j 4
T
ð27Þ
is given by: ð1Þ
ð1Þ
ð1Þ
ð1Þ
ð1Þ
ðLÞ
wT0j A1 u1i k1i wT0j u1i wT0j A1 u1i k1i wT0j u1i .. . ð1Þ
.. . ð1Þ
ð1Þ
wT0j AL u1i k1i wT0j u1i
ðLÞ
ðLÞ
ðLÞ
wT0j AL u1i k1i wT0j u1i
3 7 7 7 5 ð28Þ
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J. Ma et al. / Electrical Power and Energy Systems 67 (2015) 409–416
Fig. 2. WECC system.
when j = i, h(j) is given by:
2
h
ðjÞ
ð1Þ
ð1Þ
u0i u1i 16 ¼ 6 ... 24
ð1Þ ð1Þ u1i u1i
ðLÞT ð1Þ 3 u1i u1i 7 7 ... 5
~ uik ¼ Huik þ HT as Make matrix Huik summarization, and define H uik the 2nd order perturbation sensitivity matrix of eigenvector. Then the 2nd order perturbation of uik(a) expressed in terms of a is:
ð29Þ
ðLÞ ðLÞ u1i u1i
From Eqs. (28) and (29), u2ik, the 2nd order increment of uik(a), is given by:
u2ik ¼
n X ðjÞ u0jk DaT h Da ¼ DaT Huik Da
ð30Þ
j¼1
where Huik = u01kh(1) + u01kh(2) + + u01kh(n) is the 2nd order perturbation matrix of the kth row of eigenvalue.
u2ik ¼
1 T~ Da Huik Da 2
ð31Þ
Substitute Eqs. (16) and (31) in Eq. (10), the 2nd order estimated value of the kth row of the system eigenvector after multi-parameter change is as follows:
~ T Da þ 1 Da T H ~ uik Da uik ¼ u0ik þ G uik 2
ð32Þ
The method that calculates 2nd order perturbation sensitivity not only makes the solving process direct and clear since it avoids the burdensome derivation calculation, but also saves calculating time
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J. Ma et al. / Electrical Power and Energy Systems 67 (2015) 409–416
by avoiding solving complicated high-dimensional state matrix, only using variation of perturbation parameter and perturbation sensitivity matrix to obtain the system characteristic quantities after the perturbation (The variation of loads and generators outputs is able to be obtained by the PMUs installed on the external bus of generators and substations. The inertias and transient reactance are able to be obtained by the parameters of nameplates). To prove the effectiveness of the method, comparison of calculation process between proposed method and true value analysis is shown in Fig. 1. The main difference between the two methods is that it takes a great amount of time for the direct method to form a new matrix A after the disturbance; while for the 2nd order perturbation method, simple matrix multiplication of the disturbance variable is all that is needed for small perturbation analysis after the disturbance. Therefore, the 2nd order perturbation method is more efficient and could be attractive for on-line monitoring or control purposes. Testing results and analysis 127-bus equivalent of the Western Electricity Coordinating Council (WECC) systems introduced to verify the validity and practicability of the proposed method [19]. Table 1 Dominant modes for low-frequency oscillation. Mode
Eigenvalues
Freq (Hz)
Damp ratios (%)
Dominant generators
1 2 3 4
0.2925 ± 4.4842i 0.1333 ± 5.0381i 0.3162 ± 5.1638i 0.2602 ± 5.9488i
0.7137 0.8018 0.8218 0.9468
6.51 2.65 6.11 4.37
G18 G20 G1G2 G29 G1 G2 G29 G3 G5 G19 G22
The system is a multi-area inter-connected large scale power system, which has a generating capacity of 61.4 GW with a load of 60.8 GW, as shown in Fig. 2. The system model includes 29 generators, 127 buses, and 215 lines. Generators adopt the six-order detailed model. The excitation model is IEEE-DC1 model, and the load model adopts the WECC model, where 80% is constant active load, 80% is constant reactive load, 20% is motor load. The dominant mode of low-frequency oscillation is obtained by using mode analysis method to analyze the system, as shown in Table 1. When a certain or several generators in the equivalent system are put into or out of operation, parameters such as the inertia constant and d-axis transient reactance will change. A perturbation analysis of the related system parameters is conducted, taking the following two kinds of perturbation as examples: (1) The power output of equivalent generator 1, 2, and 3 increase by 5% and the active load on bus 9, 10, and 11 all increase by 5%, where c = 5% is the perturbation parameter in example 1. (2) The inertia time constant of equivalent generator 13, 14 and 15 increase by 5% and the d-axis transient reactance of generator 14, 15 and 16 increase by 5%, where b = 5% is the perturbation parameter in example 2. Define ferr as the frequency error, f0 as the real system frequency after perturbation, and fi as the ith estimated value, where i is 1 or 2. Define nerr as the damping ratio error, n0 as the real system damping ratio after perturbation, and ni as the ith order estimated value of the damping ratio, where i is 1 or 2.
f err ¼
jf i f 0 j f0
ð33Þ
Table 2 Changes of oscillation mode after perturbation 1. Mode 1
Mode 2
Mode 3
Mode 4
True value
k0 f0 n0(%)
0.2925 ± 4.4851i 0.7138 6.51
0.1273 ± 5.0539i 0.8043 2.52
0.3481 ± 5.1980i 0.8273 6.68
0.2591 ± 5.9552i 0.9478 4.35
1st order perturbation estimated value
k1 f1 n1(%) ferr1 nerr1
0.2920 ± 4.4856i 0.7139 6.49 0.0001 0.0031
0.0908 ± 5.0347i 0.8013 1.80 0.0037 0.2659
0.2393 ± 5.2974i 0.8431 4.51 0.0191 0.4940
0.2114 ± 5.9568i 0.9481 3.55 0.0003 0.2345
2nd order perturbation estimated value
k2 f2 n2(%) ferr2 nerr2
0.2924 ± 4.4853i 0.7139 6.51 0.0001 0.0007
0.1152 ± 5.0423i 0.8025 2.28 0.0022 0.0936
0.2991 ± 5.2367i 0.8334 5.70 0.0074 0.1464
0.2431 ± 5.9557i 0.9479 4.08 0.0001 0.0624
Table 3 Changes of oscillation mode after perturbation 2. Mode 1
Mode 2
Mode 3
Mode 4
True value
k0 f0 n0(%)
0.3175 ± 4.4368i 0.7061 7.14
0.1358 ± 5.0368i 0.8016 2.69
0.3158 ± 5.1492i 0.8195 6.12
0.2549 ± 5.9074i 0.9402 4.31
1st order perturbation estimated value
k1 f1 n1(%) ferr1 nerr1
0.3543 ± 4.4878i 0.7143 7.87 0.0116 0.1022
0.1280 ± 5.0456i 0.8030 2.54 0.0017 0.0558
0.3502 ± 5.2123i 0.8296 6.70 0.0123 0.0948
0.2421 ± 5.9118i 0.9409 4.09 0.0007 0.0510
2nd order perturbation estimated value
k2 f2 n2(%) ferr2 nerr2
0.3321 ± 4.4637i 0.7104 7.42 0.0061 0.0392
0.1327 ± 5.0392i 0.8020 2.63 0.0005 0.0223
0.3328 ± 5.1765i 0.8239 6.42 0.0054 0.0490
0.2478 ± 5.9087i 0.9404 4.19 0.0002 0.0278
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nerr ¼
J. Ma et al. / Electrical Power and Energy Systems 67 (2015) 409–416
jni n0 j n0
ð34Þ
In the two examples mentioned above, the estimated values of oscillation modes after perturbation calculated by 2nd order perturbation sensitivity matrix are shown in Tables 2 and 3. In example 1, the maximum error of the estimated 1st order value of frequency and damping ratio are 1.91% and 49.4% respectively, and the maximum error of the estimated 2nd order value of frequency and damping ratio are 0.74% and 14.64% respectively, as shown in Table 2. In example 2, the maximum error of the estimated 1st order value of frequency and damping ratio are 1.23% and 10.22% respectively, and the maximum error of the estimated 2nd order value of frequency and damping ratio are 0.61% and 4.9% respectively, as shown in Table 3. The accuracy advantage of the estimated 2nd order value calculated by the 2nd order perturbation sensitivity of the eigenvalues in analyzing low-frequency oscillation mode with multiple uncertain parameters can be seen by comparing Table 2 with Table 3.
As for example 1, in the process of the perturbation parameter varying from 0% to 5%, the variation curve of frequency and damping ratio of each oscillation mode is as shown in Fig. 3, where the dashed line represents the estimated 1st order value calculated by 1st order sensitivity matrix, the solid line represents the estimated 2nd order value calculated by 2nd order sensitivity matrix, and the asterisks represents the true value of system frequency and damping ratio after perturbation. Fig. 3 indicates that except for mode 1, which has nearly no change after perturbation, both the system oscillation frequency and damping ratio increase or decrease in a nonlinear way With the increase of perturbation, the estimated linear 1st order value calculated by 1st order sensitivity matrix becomes not accurate enough, even largely deviated. So the advantage of 2nd order sensitivity matrix becomes more obvious since it keeps relatively high calculation accuracy even under large perturbation. As for example 2, in the process of the perturbation parameter varying from 0% to 5%, the variation curve of frequency and damping ratio of each oscillation mode is as shown in Fig. 4.
Fig. 3. Frequencies and damping ratios in the case of perturbation No. 1.
J. Ma et al. / Electrical Power and Energy Systems 67 (2015) 409–416
By comparing Fig. 3(h) with 4(h), it can be seen that as the perturbation increases, the deviation of the estimated 1st order value from the actual value increases as well. Although the deviation of the estimated 2nd order value also increases, such deviation is still acceptable. This indicates that compared with the estimated 1st order value of frequency and damping ratio of the oscillation mode, its estimated 2nd order value has higher accuracy and is more close to the actual value. It needs to be noted that generators are often put into or out of operation in the equivalent system. The method proposed is able to obtain eigenvalues of the entire system only by taking the changed parameter into consideration. It avoids the complicated calculation progress of high-dimensional state matrix and saves calculation time. Participation factor is the product of corresponding elements of left and right eigenvectors. It is a comprehensive index to reflect the influence of system state parameters on the controllability and observability of a certain oscillation mode, and is a highly
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important guidance on the choice of installation location of controllers. After the system parameter change, left and right eigenvectors corresponding to the system oscillation mode change accordingly, and the corresponding participation factors change as well. As shown in Table 1, the generators involved in mode 2 is from two area, making it a classical inter-area low-frequency oscillation mode. Thus, the participation factor of mode 2 under small perturbation is studied. As shown in Fig. 5, the participation factor changes accordingly in example 2. Fig. 5 illustrates that the participation factor of generator 1 is the largest before and after perturbation, followed by equivalent system 2 and 29. It can be concluded from the figure that generators involve in mode 2 does not change and 2nd order perturbation estimated value is almost the same with the true value after perturbation 2. To prove the rapidity of the proposed method, comparison of calculation speed between 2nd order perturbation method and direct computation method is shown in Table 4. It can be seen that
Fig. 4. Frequencies and damping ratios in the case of perturbation No. 2.
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Acknowledgement Many faculties and students contribute greatly to this research. The authors would like to thank Dr. Arun G. Phadke and DrYilu Liu. This work was supported by the National Natural Science Foundation of China (51277193), the Chinese University Scientific Fund Project (2014ZZD02), the 111 project (B08013), the Scientific Research Foundation for the Returned Overseas Chinese Scholars of State Education Ministry ([2011] No. 1139), Hebei Natural Science Foundation (E2012502034), Electric Power Youth Science and Technology Creativity Foundation of CSEE ([2012] No. 46), the New-Star of Science and Technology supported by Beijing Metropolis Beijing Nova program (Z141101001814012), the Excellent talents in Beijing City (2013B009005000001), the Fund of Fok Ying Tung Education Foundation (141057). References
Fig. 5. Participators of Mode 2 using 2st sensitivity after perturbation 2.
Table 4 Calculation time of two methods applying to WECC system. Method
CPU time consuming (s)
Direct computation method 2nd order perturbation method
22.236152 0.305780
the 2nd order perturbation method is more efficient than the direct computation method. The CPU used is the Intel core i3 processor 330 M (2.13 GHz). Conclusion A novel method to analyze low-frequency oscillation modes by multi-parameter 2nd order perturbation sensitivity is proposed in this paper. Firstly, the multi-parameter 2nd order perturbation sensitivity matrices of eigenvalue and eigenvector are deduced. Then, their multi-parameter 2nd order estimated values are calculated. Based on this, the change of system oscillation modes under multiple parameter variation is assessed. This method has following characteristics. (1) It is able to estimate the small signal stability of the system relatively accurately even several uncertain parameters in the system change largely, thus provide the operators with relatively comprehensive information. (2) It makes the solving process direct and clear since it avoids the burdensome derivation calculation of 2nd order sensitivity, and it saves calculating time by avoiding solving complicated high-dimensional state matrix. (3) Compared with the 1st order perturbation theory, it is more accurate in analyzing low-frequency oscillation mode.
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