Probabilistic eig enva I ue sensitivity indices for robust PSS site selection

K.W.Wang, C.Y.Chung, C.T.Tse and K.M.Tsang

Abstract: A probabilistic approach suitable for selecting locations of power system stabilisers (PSSs) is introduced in this paper. When multi-operating conditions of a power system are described by statistical attributes of nodal injections and considered in eigenvalue analysis, the probabilistic distribution of an eigenvalue will be expressed by its expectation and variance under the assumption of normal distribution. PSSs will improve both expectations and variances for all concerned eigenvalues. Sensitivities of eigenvalue expectations and variances are calculated by means of the second-order eigenvalue sensitivities with respect to nodal injections and PSS gains. Probabilistic sensitivity indices used for selecting the robust PSS sites are equivalent to the probabilistic representation of the conventional participation factor and the residue. The proposed probabilistic approach is validated on two test systems.

1 Introduction

Power system stabilisers have been widely used to solve dynainic instability problems. Many approaches or indices based on the open-loop system model have been proposed and successfully used for selecting the optimum PSS sites, such as the modal analysis approach [I], residue method [2] and different sensitivity coefficients [3]. Relationships between sensitivity, residue and participation have also been discussed in [3]. To consider more machines and state variables in a large-scale sjstem, some indices were derived from the reduced-order modal analysis [4, 51. A compara- tive study was presented in [5] and the most popular tech- niques or indices were found to be the residue method and the damping torque analysis. However, all of these tech- niques [l-51 are based on a deterministic condition with constant system parameters and a particular load level. These weaknesses can be overcome by considering a wide range of operating conditions and uncertainties using the probabilistic approach.

In an earlier probabilistic dynamic stability algorithm [6], the probabilistic property of an eigenvalue was deteimined from the known statistical attributes of variations of system parameters, such as the rotor angle and mechanical dainp- ing used in its two-machine test system. The uncertainties considered stemmed from measurement, estimation and forecast errors under a particular load level. The concept of stochastic stability was also employed in [7] to study the stability problem from dynamic stability limit curves based on a single-machine infinite-bus system. However, these probability studies are only applied to simple systems [6, 71. Varying the system operating conditions in a multimachine system was first considered in [8]. With nodal voltages used

0 IEE, 2001 IEE Pruceediigs online no. 20010638 DOL 10.1049/ip-gtd:20010638 Paper first received 9th April and in revised foiin 21st October 1999 The authors arc with the Departiiicnt ol' Electrical Engineering, The Hong Kong Polytechnic University, Hung Horn, Kowloon, Hong Kong

as basic random variables and determined by probabilistic load flow calculation, the probabilistic distribution of each eigenvalue was obtaiiied from the probabilistic attributes of the nodal voltages. Assuming a normal distribution, the random characteristic of each eigenvalue is described by its expectation and variance.

Based on the probabilistic analysis for system dynamic stability [SI, this paper extends the probabilistic approach to determine the robust PSS locations. Probabilistic sensitivity indices (PSIS) for damping and damping ratios are deter- mined from the second-order eigenvalue sensitivity repre- sentation [9], which extends the commonly used participation factor and residue index into the probabilistic environment. To validate the proposed PSI, PSS parame- ters are also determined to improve the system dynamic stability under multi-operating conditions.

2 Residue and sensitivity relationship

To describe the relationship between a residue and an eigenvalue sensitivity [3], the state space equation of a line- arised multimachine system can be simply described by

X = A X + B Z (14 Y = cx ( I b )

RI, = cuI,wp (2)

WpJI, = 1 (3)

2 = F(s, q)Y (4)

For a particular mode k, the residue matrix is the product of the controllability vector and the observability vector

where Wl' and Ulc are the lcth left and right eigenvectors of system matrix A , respectively, with

If a PSS with transfer matrix F(s, q) is regarded as the feed- back, such that

for the extended closed-loop system, the coefficient matrix becomes

A, = A + BF(s , q)C (5)

603 IEE Proc.-Getrer. T ~ ~ o u ~ ? r . Di.strih., Vol. 148, No. 6, Noveinher 2001

Consequently, the sensitivity of an eigenvalue AI< with respect to a feedback parameter q is derived as

Comparison of eqns. 2 and 6 gives [3]

(7)

For a monovariable feedback, the sensitivity is equal to res- idue, i.e.

From eqn. 8, if q stands for a PSS gain, the residue of the open-loop system (eqn. 1) can also be obtained from the eigenvalue sensitivity (dAkldq)l,=o of its closed-loop system.

The participation factor defined by

(9) is the sensitivity of h k with respect to the ith diagonal ele- ment of matrix A , and can be regarded as another measure of machine damping under special circumstances [5].

In this paper, residues and the participation factors obtained from eigenvalue sensitivities will be extended to multi-operating conditions.

3 Probabilistic eigenvalue sensitivity indices

Eigenvalues obtained from a probabilistic eigenvalue calcu- lation [8] are regarded as random variables described by their probabilistic density functions (PDFs) or numerical attributes. Assuming normal distribution, the statistic nature of a random variable can be determined by its expectation and variance. For a particular eigenvalue AI< = a k + jP,, the real part aIc with expectation and standard deviation oak (square root of the variance) will distribute within { Clc - 4oa/,, a/, + 40,~) with the probability 0.99993 which is very close to unity. (Although the acceptable coef- ficient of onk can be selected from the range 3 4 [8, lo], 4 is selected in this study.) The probabilistic distribution of a particular a is shown in Fig. 1. To ensure the stability of Ale, the PDF curve of a/, should be located on the left-hand side of ‘alc = 0’ Therefore, the upper limit

a; = E& + 4aac (10) can be regarded as an extended damping coefficient. If a\, is negative, hlc will be regarded as robust stable (under multi-operating conditions); otherwise robust unstable, as shown in Fig. 1. Sensitivities of a), with respect to PSS gains will indicate which generator using PSS is more effec- tive in a i movement or system damping enhancement.

~~~ a U = a

q F $ $ s * - ” “

stable unstable

- r , 0 0 0 ; 2 2 2 U

Fig. 1 5 = 4.364, U, = 0.2683, p { n < 0) = 0.9115 distributing range (-1.4372, 0.70923

Prohubilistic distribution o ja purticiilur. a

Therefore, the probabilistic sensitivity index adopted in this paper can be expressed as eqn. 11 for the real part of the Idh eigenvalue with respect to mth PSS gain, GI?,.

604

The imaginary part of the kth eigenvalue and the damping ratio [8] can be analysed similarly

4 Probabilistic sensitivity representation

From eqn. 11 it can be seen that the sensitivity of ak is the same as the common eigenvalue sensitivity discussed in many papers [3, 8, 9, 111, but the sensitivity of 0% has to be derived from the second-order eigenvalue sensitivities [9].

4.7 Sensitivities of real parts of eigenvalues With eigenvectors Wk, Ulc satisfying eqn. 3, K~ and 5 stand- ing for system parameters, the first- and second-order eigenvalue sensitivities, in general are

For a system of n eigenvalues, the derivative of the left eigenvector WlcT is a linear combination of all eigenvectors 191

m f k

In probabilistic eigenvalue analysis, the covariance matrix CV of the nodal voltages is solved from the covariances of the nodal injections by ‘probabilistic load flow’ [8], and covariances Cnk,al[, Cak,/3k, Cpk,, and Cpk,pk of an eigen- value AIc = a, + j& are calculated from C , by an analytical expression of eqn. 14 with (yk, qk) standing for the four combinations of aIc and PIC.

j=1

where N is the bus number of a power system, 5 and V J are nodal voltages and CviJ is the (i,j)th element of C ,

From eqn. 14, the sensitivity of the covariance with respect to the mth PSS gain Gni can be expressed as

(15) Considering oa: = C,,,, and y,< = qlc = at( in eqn. 15, the sensitivity of the standard deviation is simply

From eqn. 11 therefore, the calculation of PSIS will require the second-order eigenvalue sensitivities of eqn. 15.

4.2 Sensitivities of damping ratios The damping ratio & of a particular eigenvalue, Ak = aIc + ,jk?’k is 181

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From the hnearised expression of eqn. 17 at the expectation point ill< = El, + jDIt

with A t k = D A k A a k + D B ~ A ~ , (18)

-2 - 3 D A k - P k / l A k I , Dal; = - ( u k B k / I X k l 3 (19)

the variance of Ek is

' ( h = D?llccai. ,OK + D g k c f l ~ $1. f 2 D A k D B k c a i . $1.

(20) and the sensitivity of the variance with respect to the mth PSS gain will be

where

The sensitivity of a standard deviation qk is similar to eqn. 16 as

In eqns. 21-23, eigenvalue covariances and covariance sen- sitivities are determined by eqns. 14 and 15 respectively. The PSI for damping ratio will be expressed as

where d~lc/dG,, is similar to eqn. 18 as

4.3 Partial derivatives of matrix A The first- and second-order derivatives in eqns. 14, 15, 22 and 25 can be determined from eqn. 12 with K; and Ki in eqn. 12 representing the PSS gain or nodal voltages.

The first-order sensitivities of matrix A with respect to the arbitrary parameters in machine, control equipment and power network have been derived in [l I] and sensitivi- ties with respect to nodal voltages have been derived in [8] for the probabilistic eigenvalue analysis. These approaches can be extended to second-order partial derivatives of A with .respect to PSS gains and nodal voltages.

IEE Proc.-Gener. Transni. Disl~ih., Vol. 148, h'o. 6. November 2001

Matrices (except covariance matrices) included in eigen- value sensitivity expressions are very sparse, and the spar- sity technique is employed in this study, in which the matrix storage technique is the same as that in [12]. Because there are many matrix operations in the sensitivity computation, the sparsity technique is important to reduce the computation requirement and the accumulative error. Moreover, it becomes possible to store all first-order deriv- atives of A for the computation of the second-order sensi- tivities in eqn. 12b.

5 Case studies

The proposed method is tested with two different networks. In the 3-machine 9-bus system of Fig. 2, all machines are equipped with fast-acting static exciters and speed gover- nors. Block diagrams of all control systems are shown in [13]. The 8-machine 24-bus system of Fig. 3 is modified from the 36-bus test system [14] by omitting the DC links and eliminating some nodes which do not have injections. Details of network parameters, nodal powers and control system parameters are given in [14].

load B

113+j6.7 MVA L+ loo + j35 MVA . 8 5 ~ ~

I 1.025(pu) 0

load A 125 + j50 MVA

G3 @ V G ~ = 1.04L 0' (pu)

Fig. 2 Three-muchine system

~~~ 12 15

23 A h L9 -- 20

,., O G 4 G7

Fig. 3 Eight-muchine sy~stem

The voltage dependency of the load characteristics has been represented by the exponential model: PL = PLOP, QL = QLoV/'. Four additional equivalent admittances are derived from the linearisation of PL and Q L for each load bus [l I] and are taken into account in the eigenvalue and sensitivity analysis.

In these two systems, the noma1 operation values of the nodal powers and PV voltages are regarded as their expec- tations. Each nodal power or PV voltage is assigned with a standardised daily operating curve [8]. From these curves, 480 system operating samples are created and the covari- ances of the nodal injections are determined.

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Table 1: Eigenvalues and damping ratios without PSS

No. E s acl n pa s 9 E* pc

16 -1.000 0.00 0.0000 999.00 1.000 17 -0.999 0.00 0.0001 999.00 1.000 19 -0.689 0.71 0.0732 9.42 1.000 0.6946 0.0493 12.07 1.0 21 -0.534 10.47 0.1811 2.95 0.998 0.0509 0.0106 -4.61 0.0 23 -0.403 7.19 0.0675 5.96 1.000 0.0559 0.0058 -7.66 0.0

24 -0.203 0.00 0.0000 999.00 1.000 25 -0.017 0.00 0,0002 82.35 1.000 Expectations: x = 5 Standardised expectations: a* =-E/u,, E* = (6- O.l)/q. Probabilities: P, = P{cr < 0) and PE = P{E > 0.1).

a (rad/s) and $. Standard deviation: 0.

Table 2: PSls corresponding to participation factor

State variables

AEbl Awl A61 AEb2 A C U ~ A62 AEb3 Am3

a 2 1 -0.0024 0.0701 0.0541 -0.0012 0.4138 0.4442 0.0002 0.0144 0.0122 0.3114 0.4277 -0.0021 0.0393 0.0703 -0.0045 0.1470

Table 3 PSls corresponding to residue

Input signals

Awl APe1 Amp A P g A w ~ APe3

0.0001 1.9461 0.0119 8.0335 0.0003 -0.7001 &y 0.0000 -0.2476 -0.0007 -0.9480 0.0000 0.0537 a 2 3 0.0001 6.5837 -0.0056 0.4553 0.0044 -1.4436 E23 0.0007 -1.0116 0.0009 -0.0644 -0.0007 0.1548

5.1 3-machine 9-bus system With all generators represented by the third-order machine model, the probabilistic eigenvalue analysis for this system results in 25 eigenvalues with the last 10 eigenvalues (4 real and 3 complex) in the ascending order of a as shown in Table 1. Standardised expectations a" = -doa and r = (5 - O.l)loE, listed in the 5th and 9th colunm, respectively, provide a direct measure for stability probabilities under the assumption of normal distribution. Referring to the dis- cussion of the eigenvalue distribution in Section 3, if a standardised expectation a* is less than 4, the correspond- ing mode is regarded as robust unstable, otherwise it is robust stable [8, 101. In the 9th column of Table 1, the limit for the damping ratio E applied in this paper is 0.1 [8]. Because aZl*, K1* and &* are 2.95, 4 .61 and -7.66, respectively, which are less than 4, i.e. inadequate distribu- tion probabilities of Pa or PE the 21st and 23rd eigenvalues (the two electromechanical modes) may be regarded as crit- ical under the probabilistic notion. Although a25 = -0.017 is very close to zero, its distribution probability is adequate.

With reference to Section 4, the probabilistic sensitivity indices of eqns. 11 and 24 have been calculated for the two critical eigenvalues and are listed in Tables 2 and 3 with respect to the participation factor and residue, respectively. The indices in those tables arrive at the same conclusion: the best PSS site for the 21st mode is machine G2 and for the 23rd mode is G1. Modal analysis [ 151 based on eigen- value expectation (regardless of the effect of eigenvalue var- iance) shows that the 21st mode is the oscillation between G2 and GI + G3, whilst the 23rd mode is between G1 and G2 + G3.

It should be noted that the selected coefficient value of oak in eqn. 10 affects the contribution of the variance sensi- tivity to the corresponding PSI in eqn. 11. If a different

606

coefficient value, say 3, is used in eqn. 10, the PSIs in Tables 2 and 3 will change slightly, but it will not affect the PSS location.

For this simple 3-machine system, it is possible to con- sider l j 2 , and ,& simultaneously. The criterion for PSS input signal selection is that PSIs should have (a) the opposite sign for a and Zj (b) the same sign for ql and ~ 1 2 3 , same sign for lj21 and e2:23 (c) relatively larger magnitudes Therefore, the power deviation AP, of GI or G2 may be selected as the input signal.

It should be noted that PSI denotes the eigenvalue sensi- tivity under multi-operating conditions. The solution obtained from PSIs can be validated from Table 4. The probabilistic distributions of a and 5 are described by their standardised expectations a* and r . respectively, in Table 4. With one PSS on G2 only, and if the PSS gain magnitude is increased from zero, aZl* is increased (i.e. a2, is improved) due to its positive PSI 8.0335 in Table 3. €j2, and c223 are also improved due to their negative PSIs -0.9480 and -0.0644. aZl and arrive at the desired prob- ability (with ql* 2 4 and nl* 2 4) when the gain changes from 0 to -0.024 and -1.05, respectively. However, E2:23 remains unsatisfied even if the gain value is changed to - I .39 which has made two other real eigenvalues unstable (not shown in Table 4). Similarly, when a PSS on G1 is used alone, a21, K1 and lj23 cannot be improved to desired values. Therefore, if only the real part of the eigenvalues is involved, one PSS is adequate; otherwise, two PSSs will be required to ensure satisfactory damping ratios.

Table 4 Eigenvalue shift against G2 PSS gain

Gain 0.00 -0.01 -0.024 -0.10 -1.05 -1.39

a;, 2.95 3.36 4.01 9.77 4.96 4.04 -4.61 -4.76 -4.95 -0.62 4.01 14.06

E;3 -7.66 -7.74 -7.87 -8.58 -3.43 -3.33

see Table 1 for symbols

The power-input PSS on G1 and G2 have the same transfer function as eqn. 26. The PSS gain Kpss, time con- stants T , T, and T2 are repeatedly adjusted under the guidance of the sensitivities of eigenvalue expectations and

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variances. Finally, all eigenvalues have adequate probabili- ties in terms of damping and damping ratio (the results of two electromechanical models are listed in Table 5). There- fore, the system performance under all the 480 operating samples can be guaranteed by employing the probabilistic technique.

where Tw = lOs, T, = O.lSs, T2 = O.OSs, Kpssl = 4 . 5 , Kpssz = 4 . 7 (KpssI and Kpss2 are p.u. on the system base 100 MVA).

5.2 8-machine 24-bus system With all generators represented by the fifth-order machine model [I 11, the solution gives 88 eigenvalues (38 real and 25 complex). Probabilistic results for the seven electromechan- ical modes are shown in Tables 6 9 (other eigenvalues with

adequate stability probabilities are not shown for simplic- ity).

From Tables 7-9, the PSls, for a and 6 for different sig- nals, reach the same conclusion for the first five modes, have a slight difference for the 6th mode, but a relatively large difference for the last mode. (The large difference may be attributed to the fact that inode seven (interarea) is asso- ciated with more machines and has a more complex sensi- tivity relationship than the other modes.) From Tables 8 and 9, both power and speed deviations can be selected as PSS input signals, though larger PSS gains will be required for speed signals. Power signal stabilisers are again employed for this system.

From Table 8, modes 2-6 can be improved by using PSS located on G2, G6, GS (or G4), G3 and G7, respec- tively. Mode 7 cannot rely on the PSS of G6, because this PSS has been used for mode 3 and the PSI signs between mode 7 and mode 3 with respect to APe6 are opposite.

Table 5 Eigenvalues and damping ratios with PSSs

No. Tx P 0, CY p, E 9 E* Pc 13 * -3.030 6.77 0.3205 9.45 1.0 0.4085 0.0637 4.84 1.0

18" -1.522 5.43 0.0927 16.42 1.0 0.2698 0.0225 7.54 1.0"

13th and 18th eigenvalues corresponding to the 21st and 23rd in Table 1, respectively. See Table 1 for symbols

Table 6: Electromechanical modes without PSS

CY No. Z 7j 0, Pa F 9 E* PE 1 -1.731 15.85 0.0439 39.47 1.000 0.1085 0.0023 3.64 0,9999

2 -0.757 11.11 0.0827 9.16 1.000 0.0680 0.0077 -4.14 0.0000

3 -0.565 9.84 0.0491 11.50 1.000 0.0573 0.0062 -6.87 0.0000

4 -0.637 7.90 0.0172 37.09 1.000 0.0803 0.0007 -26.38 0.0000

5 -0.650 7.53 0.0985 6.60 1.000 0.0860 0.0148 -0.95 0.1711

6 -0.421 6.48 0.0306 13.75 1.000 0.0647 0.0047 -7.54 0.0000

7 -0.010 3.82 0.0117 0.86 0.802 0.0026 0.0031 -31.84 0.0000

See Table 1 for symbols

Table 7: PSls corresponding to the participation factor

~i~~~~~~~~ State variables

CY AUjl A 0 2 A 0 3 A w ~ A w ~ A* A Q 3

a1 0.5065 0.0044 0.0045 0.0006 0.0008 0.0016 0.0008 0,0000

a 2 0.001 1 0.4387 -0.0026 0.0029 0.0037 0.0680 -0.0002 0.0000

a3 0.0002 0.0034 0.0059 0.0026 -0.0458 0.4559 0.0603 0.0060

a 4 0.0000 0.0000 -0.0042 0.2947 0.2078 0.0000 -0.0002 0.0000

a5 0.0007 -0.0008 0.4201 0.0532 0.0529 4.0002 0.0105 -0.0004

0.0003 0.0002 0.01 15 0.0997 0.2475 -0.0325 0.2418 0.0000

CY1 0.0096 0.0120 0.0319 0.0493 5.8438 0.2970 0.0480 -0.5703

Table 8: PSIS corresponding to residue

Input signals

Ape1 Ape2

E1 0.1510 0.0007

0.0006 0.1009

E3 0.0001 -0.0059

5 4 0.0000 0.0000

-0.0007 0.0000

0.0002 0.0002

Ape3 Ape4 APe5

-0.0007 -0.0001 -0.0002

-0.001 1 -0.0004 -0.0006

-0.0017 0.0000 0.0000

0.0000 -0.0882 -0.0994

-0.4380 -0.0682 -0.0108

-0.01 60 -0.0368 -0.0784

AP&i Ape7 Apes

0.0004 -0.0003 0.0000

0.0383 0.0014 0.0000

0.1719 0.0004 -0.0010

0.0000 0.0003 0.0000

-0.0006 0.0055 -0.0001

-0.0027 -0.0991 -0.0007

67 -0.0565 -0.0147 -0.5844 -0.1097 -0.2272 -0.9304 -0.6181 -0.3626

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Table 9: PSIS corresponding to residue (x I O )

E1 -0.0554 0.0000 -0.0003 0.0000 0.0000 0.0000 0.0000 0.0000

-0.0010 -0.0721 0.0005 0.0000 0.0001 -0.0023 0.0002 0.0000

63 -0.0005 0.0000 0.0014 0.0000 0.0001 -0.0699 0.0026 -0.0003

0.0000 0.0000 0.0036 -0.0418 -0.0304 0.0000 0.0000 0.0000 -0.0018 0.0010 -0.1892 -0.0080 -0.0025 0.0004 0.0017 0.0000 -0.0033 -0.0002 -0.0179 -0.0140 -0.0215 -0.0030 -0.0204 0.0000

h -0.3305 -0.0801 -0.3225 -0.0724 -0.1041 -0.0008 -0.1404 -0.0619

Thus mode 7 could be improved by PSS on G3 and G7. In mode 1, P ( 5 > 0.13 is 0.9999 in Table 6 and extra PSS may not be necessary. Therefore, a total of five PSS is adequate.

The first step is to determine the PSS gain KGj (for i = 2, 3, 5, 6, 7) and the adjusting direction is guided by the sensi- tivities of the eigenvalue expectations. After repeated adjustment and eigenvalue sensitivity calculation, the PSS gain KG s with the minimum absolute value are

.KG~ = 0.61 (for Az), K G ~ = 0.33 (for As),

K G ~ = -0.31 (for A,) I<G~ = -0.32 (for As), K G ~ = -0.35 (for A,) K G ~ = -0.32 and I I G ~ = -0.75 (for A,) (27)

In eqn. 27, KG2 and KG6 are positive and other gains are negative, whch can be simply explained as follows. The phase lag 8 due to an excitation system (EXC) increases with oscillation frequency U. For a typical EXC with exciter time constant T, = OSs, 0 > 90” when o > 8 (radh). As modes 2 and 3 have a larger oscillation frequency (Table 6), the corresponding PSS gains KG2 and &6 will be of different sign from other PSSs. This also agrees with Table 8 which shows that the dominant PSIS namely eel, APe2 and APe6 for the first three modes with values 0.1510, 0.1009 and 0.1719, respectively, are all positive.

Table I O : PSS parameters

PSSS PSS3 PSSS PSS6 PSS7

Kpss 0.353 -0.330 -1.453 0.492 -0.753

TW 10s 10s 10s 10s 10s

TI 0.151s 0.160s 0.151 s 0.160s 0.149s

T, 0.150s 0.149s 0.150s 0.148s 0.150s

iteration of (a) computing sensitivities of expectations and variances of concerned eigenvalues with respect to all PSS parameters; (b) adjusting PSS parameters which have the largest sensitivities to improve the probabilities of con- cerned eigenvalues.

The parameters of the five power-input PSSs equipped on G2, G3, G5, G6 and G7 are shown in Table 10. All Kpsss are p.u. on a system base of 100MVA. Distributions of damping and damping ratios of the seven electrome- chanical modes have all been improved (see Table 11). Therefore, PSS locations determined by the probabilistic approach are effective for considering multi-operating con- ditions.

6 Conclusions

This paper introduces the probabilistic analysis approach for selecting the robust PSS locations. Assuming normal distribution, the proposed probabilistic sensitivity index is derived and calculated based on the system’s multi-operat- ing conditions. Not only the sensitivities of eigenvalue expectations, but also the sensitivities of eigenvalue vari- ances are calculated by means of the second-order eigen- value sensitivities. Case studies show that the proposed probabilistic sensitivity index can provide the best PSS locations under all operating conditions considered.

7 Acknowledgment

The authors gratefully acknowledge the research funding provided by Research Grants Council of Hong Kong for this project.

8 References

The second step is to coordinate PSS gains and time con- stants in which the initial time constants are selected as: Tw = lOs, TI = T2 = 0.15 s for all PSSs and the initial gains are the five KGj values (for i = 2, 3, 5, 6, 7) in eqn. 27. The coordination of PSS parameters is accomplished by

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Table 11: Electromechanical modes with PSSs

1 -1.749 15.88 0.0429 40.73 1.0 0.1095 0.0023 4.11 1.0

2 -1.254 10.86 0.0179 69.98 1.0 0.1148 0.0036 4.09 1.0

3 -1.544 9.07 0.1463 10.55 1.0 0.1678 0.0167 4.06 1.0

4 -2.941 9.99 0.0460 62.24 1.0 0.2647 0.0066 25.04 1.0

5 -1.384 9.02 0.4263 10.96 1.0 0.1517 0.0129 4.00 1.0

6 -0.876 7.36 0.0310 28.23 1.0 0.1182 0.0045 4.04 1.0

7 -1.534 4.46 0.2313 6.63 1.0 0.3250 0.0558 4.03 1.0

See Table 1 for symbols

608 IEE Proc.-Genes. Trmsm. Distrih, Vol. 148. No. 6, November 2001

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