Effects of various power system stabilizers on improving power system dynamic performance

Electrical Power and Energy Systems 46 (2013) 175–183

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Effects of various power system stabilizers on improving power system dynamic performance Ping He a,b, Fushuan Wen c,d,⇑, Gerard Ledwich c, Yusheng Xue e, Kewen Wang f a

School of Electrical Engineering, South China University of Technology, Guangzhou 510640, China College of Electrical and Information Engineering, Zhengzhou University of Light Industry, Zhengzhou 450002, China c School of Electrical Engineering and Computer Science, Queensland University of Technology, Brisbane, Queensland 4001, Australia d School of Electrical Engineering, Zhejiang University, Hangzhou 310027, China e State Grid Electric Power Research Institute, Nanjing 210003, China f School of Electrical Engineering, Zhengzhou University, Zhengzhou 450001, China b

a r t i c l e

i n f o

Article history: Received 14 June 2012 Received in revised form 27 September 2012 Accepted 9 October 2012 Available online 23 November 2012 Keywords: Power system stabilizer Eigenvalue analysis Small/large-signal stability Electromechanical modes

a b s t r a c t To ensure the small-signal stability of a power system, power system stabilizers (PSSs) are extensively applied for damping low frequency power oscillations through modulating the excitation supplied to synchronous machines, and increasing interest has been focused on developing different PSS schemes to tackle the threat of damping oscillations to power system stability. This paper examines four different PSS models and investigates their performances on damping power system dynamics using both smallsignal eigenvalue analysis and large-signal dynamic simulations. The four kinds of PSSs examined include the Conventional PSS (CPSS), Single Neuron based PSS (SNPSS), Adaptive PSS (APSS) and Multi-band PSS (MBPSS). A steep descent parameter optimization algorithm is employed to seek the optimal PSS design parameters. To evaluate the effects of these PSSs on improving power system dynamic behaviors, case studies are carried out on an 8-unit 24-bus power system through both small-signal eigenvalue analysis and large-signal time-domain simulations. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Power systems worldwide have been continuously expanding in scale and evolving into more complicated structures in past decades, and have to operate more frequently close to their limits as the results of geographical and physical limitations as well as the power industry restructuring. The secure operation of power systems, therefore, has become a major concern, and the applications of power system stabilizers (PSSs) for dynamic stability enhancement have drawn more attention than ever before [1–10]. Conventional lead-lag PSSs (CPSSs) have been widely used by electric utilities for this purpose. A PSS is used to provide some supplemental damping to rotor oscillations via an electric torque which is in phase with the speed deviation [1]. In view of the fact that power systems are highly nonlinear and operating conditions can vary over a wide range, various kinds of PSSs have been developed in the past decades, such as the fixed parameter decentralized PSS, the adaptive PSS, and the fuzzy logic based PSS, to name a few [2–6].

⇑ Corresponding author at: School of Electrical Engineering, Zhejiang University, Hangzhou 310027, China. Mobile: +86 13968105384; fax: +86 571 87952014. E-mail address: [email protected] (F. Wen). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.10.026

In order to take into account more system operating conditions, the probabilistic eigenvalue analysis method was proposed for designing power system damping controllers [11–13]. With this approach, the system stability is reinforced by shifting the distribution ranges of the critical eigenvalues toward the left side of the complex plane. Coordination of the controller parameters was achieved through solving a non-linear programming problem, in which the objective function is composed of all unsatisfactory eigenvalues. The objective function is minimized by using optimization approaches such as the steepest descent (SD) method so that the overall performance of the controller could be optimized under the given system states. Up to now, various PSS design methods have been proposed and some applied to different degrees in actual power systems [3– 8,11,12]. CPSSs are generally based on fixed parameters, and it is hence not yet possible to achieve the optimal behavior for various operating conditions of a power system. The adaptive power system stabilizers, as reported in [3–6], could track the changes of system dynamics in real time, and hence could perform well for various operating conditions in principle. In 2003, a novel PSS architecture was proposed in [7] and later included in the revised IEEE Std-421.5 as PSS4B in [15]. Up to now, several kinds of PSSs have been proposed, and it is not clear about their relative performances. With the development of large-scale power systems, a

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comparative analysis of some major kinds of PSSs representing the key equipment for mitigating low frequency oscillations is very demanding. Given this background, four major PSS design approaches are examined in this work. They are the CPSS, the Single Neuron Model based PSS (SNPSS) [3], the adaptive PSS (APSS) [4,6], and the multi-band PSS (MBPSS) [7,8]. The major consideration of selecting these four kinds of PSSs for comparisons lies in that CPSS and MBPSS are used most widely, while SNPSS and APSS are receiving extensive concerns. The performance of each PSS model is compared against that of others with regards to efficiency and robustness. An 8-unit 24-bus power system is used to perform the small-signal eigenvalue analysis and large-signal time domain simulations on each of the four PSS models. To make the comparisons equitable, the parameters of the four kinds of PSSs are all determined by the steepest descent method [11]. This paper is organized as follows. Section 2 briefly explains the probabilistic eigenvalue analysis and optimization method. The models of the four kinds of PSSs are presented in Section 3 and the small-signal eigenvalue analysis is carried out in Section 4. In Section 5, the large-signal time domain simulation is done, and simulation results obtained. The paper is concluded in Section 6. 2. System model and optimization The power system dynamic model can be linearized around the system operating point, thus the state space equation set of a power system can be expressed with the state variable vector X and the system matrix A as shown below.

(

n0k ¼ nk  4rnk P nc n ¼ ðnk  nc Þ=rn P 4 k

ð4Þ

k

where nc is the acceptable threshold of the damping ratio; in order to ensure the system dynamic performance, nc should not be less than a specified value (in this study, nc is specified to be 0.1) [14]. 2.2. The optimization method The sensitivities of a0k and n0k against the PSS gain reflect which generator’s PSS is more effective to stabilize the power system concerned, the probabilistic sensitivity indexes (PSIs) which are a0

n0

Skmk ¼ @ a0k =@K m and Skkm ¼ @n0k =@K m (K m is a parameter of the mth PSS) presented in [12] are thus used to determine the PSS siting and the parameters under probabilistic conditions. For the parameter tuning problem, an optimization model from [11] as shown in Eq. (5), is employed here, in which only the ‘‘weak’’ eigenvalues (ak < 4 or nk < 4) are included

X

Minimize FðKÞ ¼

ðak  4Þ2 þ

ak <4

X

ðnk  4Þ2

ð5Þ

nk <4

where FðKÞ is the objective function, K denotes the PSS parameter vector whose practical limits depend on the hardware restrictions, ak and nk are the functions of K. The optimization model represented by Eq. (5) can be solved by the steepest descent (SD) approach. Starting from an initial point K(0), the iteration proceeds as below ðiÞ

X_ ¼ AX

ð1Þ

By considering multiple operating conditions of a power system, all nodal injection power can be regarded as random variables with the statistical attributions being determined by the probabilistic load flow. The corresponding eigenvalues can also be random variables, even though their distribution characteristics are different. The probabilistic distributions and the stability probabilities of all the eigenvalues can be obtained by means of the probabilistic eigenvalue analysis [14].

K ðiþ1Þ ¼ K ðiÞ  l1 rFðK ðiÞ Þ

ð6Þ (i)

where i denotes the iteration number, rF(K ) is the gradient of ðiÞ ðiÞ F(K) at point K(i), i.e. rFðK ðiÞ Þ ¼ @FðKÞ=@KjK¼K , and l1 is the optimal step size along the direction rF(K(i)) and is obtained by running the one-dimensional search algorithm [12]. 3. PSS moldeling This section presents the models of the four PSSs to be evaluated.

2.1. Probabilistic eigenvalue analysis 3.1. The conventional PSS (CPSS) The statistical characteristics of a random variable can be described by its corresponding numerical characteristics such as the mathematical expectation (mean value), variance and covariance. Similarly, the statistical characteristics of an eigenvalue can be represented similarly by these characteristics [14]. Under the assumption of the normal distribution, for a particular eigenvalue kk ¼ ak þ jbk ; under the 4r criterion the real part  k and the standard deviation rak will distribwith the expectation a  k  4rak ; a  k þ 4rak g with the probability 0.99993). To ute within fa  k  4rak ; a  k þ 4rak g should be located ensure the stability of kk ; fa on the left side of the complex plane, which can also be described by the upper limit a0k and the standardized real part expectation ak as shown below [11].



a0k ¼ a k þ 4rak 6 0 ak ¼ a k =rak P 4

ð2Þ

Therefore, a0k and ak can be regarded as two extended damping coefficients from which the robust stability of kk can be estimated. The damping ratio for an eigenvalue kk is defined as

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nk ¼ ak = a2k þ b2k

ð3Þ

Similar to the calculation of a0k and ak in Eq. (2), the expected damping ratios can be calculated with the expectation  nk and the standard deviation rnk as

For simplicity, the CPSS is modeled by two identical lead/lag networks represented by a gain KPSS, four time constants Tl, T2, T3 and T4, and a washout circuit taking a time constant Tw [15]. The transfer function of CPSS, G(s), can be expressed as Eq. (7).

GðsÞ ¼ K PSS

sT w 1 þ sT 1 1 þ sT 3 1 þ sT w 1 þ sT 2 1 þ sT 4

ð7Þ

3.2. The single neuron based PSS (SNPSS) Due to its strong ability to self-study and self-adapt, a single neuron [16] has been applied in power system controllers for achieving a better performance. With its simple algorithm and working principle, the single neuron has the structure as illustrated in Fig. 1. In Fig. 1, yðtÞ is the system output; rðtÞ is the desired or reference value; kðtÞ stands for the gain; xi ðtÞ stands for the ith input signal; wi ðtÞ is the connection weight corresponding to xi ðtÞ; A is the converting device; B is the controlled object, and stand for a power system with PSSs installed. The learning algorithm of the single neuron can be found in [3], and will not be detailed in this paper. The control signal of a PSS, such as the generator speed deviation Dx, or the power deviation DP, can be used as the input signal

P. He et al. / Electrical Power and Energy Systems 46 (2013) 175–183

177

and KN are the gains of different frequency subbands; TA, TB and TN are the corresponding time constants; Tw is the washout time constant. 4. Small-signal eigenvalue analysis This section performs small-signal stability analysis for each of the four kinds of PSSs. 4.1. The test system

Fig. 1. The configuration of the single neuron control system.

Fig. 2. The structure of the SNPSS.

of the signal neuron. In this study, DP is employed, and DP is zero under the normal operating conditions. The input signal x1 ðtÞ; x2 ðtÞ and x3 ðtÞ of the converting device A can be represented as

8 > < x1 ðtÞ ¼ rðtÞ ¼ 0 x2 ðtÞ ¼ rðtÞ  yðtÞ ¼ DP > : x3 ðtÞ ¼ x2 ðtÞ  x2 ðt  1Þ ¼ Dx2 ðtÞ ¼ @ DP=@t

ð8Þ

Under small disturbances, the equation set describing the system dynamic states can be linearized around a given operating point. An algorithm such as Eq. (8) can be represented by the form of a first-order transfer function block. Together with the transfer function G(s) of a PSS, the combined model is shown in Fig. 2. An auxiliary PSS gain kðtÞ can be obtained by multiplying the gain of the single neuron and G(s). For the convenience of presentation, in this work, kðtÞ is considered the function of the generator active power PðtÞ, which will be self-adjusted with the changes of the system’s operating conditions. 3.3. The adaptive PSS (APSS) Fig. 3 shows the configuration of APSS [4–6] using three design points. The weighing-coefficients W1, W2 and W3 are employed to simulate the actual operating conditions. 3.4. The multi-band PSS (MBPSS)

An 8-unit 24-bus power system as shown in Fig. 5 is used as the study system [13]. Each generator is represented by a sixth-order model, and the models of the excitation system as well as the turbine governor are given in the Appendix. Table 1 lists the electromechanical modes for the situation without PSS installed (NPSS). 720 samples are taken from the standardized daily operating curves of this system to simulate multiple system operating conditions (Fig. 6). The probabilistic sensitivity indices as described in Section 2 are listed in Table 2. Six PSSs are installed on G1, G2, G3, G5, G6 and G7 of this 8-unit system. The results in Table 2 shows that PSSs at G3, G5, G6 and G7, especially G7, are more sensitive in the electromechanical modes. Hence, the four kinds of PSSs are separately applied to PSS7 to test their effectiveness. The PSS parameters obtained by the preceding probabilistic method in [11] are used as the initial values of the succeeding method, and optimized by the steepest descent approach as described in Section 2.2. Table 3 lists the results of seven electromechanical modes with the CPSS. 4.2. SNPSS As mentioned before, 720 samples are taken from the standardized daily operating curves of Fig. 6. In order to identify the auxiliary PSS gain kðtÞ, the PSS parameters are optimized under 720 operating conditions, where each sample of kðtÞ is the optimum value under each of the 720 system conditions by minimizing the objective function of Eq. (5). 720 PSS7 samples are then obtained, including the auxiliary gain k7(t) and the corresponding generator’s active power. To better verify the self-regulation of the auxiliary PSS gain, P3(t) (the power generation of G3) is selected because of its relatively large fluctuation. From the 720 samples, the relationship between k7(t) and P3(t) can be obtained by curve fitting using the least squares method. By numerous simulation comparisons, the four-order fitting function produces the least error compared to the linear, second-order, cubic and exponential fitting functions. The four-order fitting function thus obtained is

k7 ðtÞ ¼ 15222:9921 þ 14203:580  P 3 ðtÞ  4950:9448  P3 ðtÞ2 The main characteristics of the MBPSS model [7,8] examined in this work are shown in Fig. 4. K is the main gain of the PSS; KA, KB

þ 764:59  P3 ðtÞ3  44:1415  P 3 ðtÞ4

Fig. 3. The APSS block diagram.

ð9Þ

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3

L2 L7

Power (p.u.)

2.5 2 1.5 1 0.5 0

2

4

6

8

10

12

14

16

18

20

22

24

time (h) Fig. 4. The MBPSS block diagram.

Fig. 6. The standard daily operating curves.

Table 2 Probabilistic sensitivity indices of the critical damping ratios to PSS gains.

n1 n2 n3 n4 n5 n6 n7

K1

K2

K3

K4

K5

K6

K7

.1378 .0007 .0003 .0000 .0006 .0004 .0244

.0005 .1235 .0031 .0000 .0001 .0003 .0057

.0005 .0017 .0024 .0017 .3370 .0094 .2473

.0001 .0003 .0000 .0636 .0164 .0331 .0462

.0002 .0005 .0001 .0762 .0107 .0701 .0959

.0004 .0422 .2262 .0000 .0007 .0010 .9789

.0005 .0012 .0033 .0002 .0021 .1046 .2549

By applying Eq. (10) in PSS7, seven electromechanical modes and the eigenvalues with ak < 4 or nk < 4 are obtained, as listed in Table 5.

4.4. MBPSS Fig. 5. The 8-unit 24-bus power system.

Applying Eq. (9) in PSS7 and performing the probabilistic eigenvalue computation, seven electromechanical modes are obtained as given in Table 4. 4.3. APSS Three typical time points, i.e. 0 am, 9 am and 15 pm are employed, and the weighting coefficients in Fig. 3 are specified as W1 = 0.3625, W2 = 0.4142 and W3 = 0.2234 [4]. The output of the APSS can be expressed as

GPSS7 ¼ 0:987

5s 1 þ 1:153s 1 þ 0:052s 1 þ 5s 1 þ 0:9s 1 þ 0:05s

ð10Þ

In Fig. 4, the washout time constant Tw usually takes 3–10 s. In this work, Tw is specified to be 5 s, and the other time constants can be calculated by Eq. (11).

8 pffiffiffiffiffi pffiffiffiffiffi > < T A2 ¼ T A3 ¼ 1=2pfA RA ¼ 1=xA RA T A1 ¼ TRA2A > : T A4 ¼ T A2 RA

ð11Þ

where xA is the A-band central angular frequency, and RA contains the compensation coefficients of the A-band [7,8]. In this work, PSS7 is designed with the two-band structure, with xM ¼ 4:5; xH ¼ 50; K M ¼ 5; K H ¼ 25; RM ¼ 1:18 and RH ¼ 1:14. The seven electromechanical modes are listed in Table 6.

Table 1 Electromechanical modes without PSS (NPSS). Mode

a

 b

ra

a⁄

Pa

 n

rn

n⁄

Pn

1 2 3 4 5 6 7

1.731 0.757 0.565 0.637 0.650 0.421 0.010

15.854 11.113 9.840 7.908 7.530 6.482 3.823

0.0439 0.0827 0.0491 0.0172 0.0985 0.0306 0.0117

38.33 8.55 10.48 34.18 6.09 12.11 3.43

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0003

0.1085 0.0680 0.0573 0.0803 0.0860 0.0647 0.0026

0.0033 0.0072 0.0037 0.0036 0.0112 0.0047 0.0030

2.57 4.46 11.44 5.55 1.26 7.50 31.96

0.9949 0.0000 0.0000 0.0000 0.1057 0.0000 0.0000

  a ; b; n are the expectations of the real part, the imaginary part and the damping ratio of eigenvalue; ra ; rn are the standard deviation of k; a ; n are the standardized  Þ=ra ; n ¼ ð expectation of k; a ¼ ðac  a n  nc Þ=rn , and aC ¼ 0:01; nC ¼ 0:1; P a ¼ Pða < aC Þ and P n ¼ Pðn > nC Þ are the stability probabilities of k.

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P. He et al. / Electrical Power and Energy Systems 46 (2013) 175–183 Table 3 Electromechanical modes with CPSS installed. Mode

a

 b

ra

a

Pa

 n

rn

n

Pn

1 2 3 4 5 6 7

2.417 1.434 1.213 1.558 0.838 1.766 1.013

15.030 10.672 9.682 9.122 7.743 7.334 4.198

0.0473 0.0336 0.0472 0.1337 0.0145 0.1755 0.1588

50.04 41.20 24.64 11.27 54.23 9.78 6.06

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.1587 0.1331 0.1243 0.1683 0.1076 0.2341 0.2345

0.0048 0.0052 0.0028 0.0102 0.0009 0.0180 0.0334

12.15 6.34 8.69 6.67 8.95 7.44 4.02

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Table 4 Electromechanical modes with SNPSS installed. Mode

a

 b

ra

a

Pa

 n

rn

n

Pn

1 2 3 4 5 6 7

2.418 1.445 1.211 2.174 1.750 0.873 1.276

15.030 10.670 9.671 9.314 8.152 7.759 4.586

0.0464 0.0257 0.0466 0.2005 0.0887 0.0115 0.1028

51.08 54.19 24.92 10.59 19.16 71.38 11.93

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.1589 0.1342 0.1242 0.2273 0.2099 0.1119 0.2681

0.0047 0.0045 0.0029 0.0334 0.0199 0.0013 0.0274

12.40 7.67 8.38 3.81 5.52 9.10 6.14

1.0000 1.0000 1.0000 0.9999 1.0000 1.0000 1.0000

Table 5 Electromechanical modes with APSS installed. Mode

a

 b

ra

a⁄

Pa

 n

rn

n

Pn

1 2 3 4 5 6 7

2.420 1.453 2.325 1.208 1.520 0.889 1.138

15.031 10.667 9.745 9.667 8.206 7.765 4.669

0.0456 0.0222 0.2451 0.0414 0.1256 0.0095 0.0816

51.95 63.26 9.28 27.99 11.71 88.15 13.34

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.1589 0.1350 0.2321 0.1239 0.1822 0.1137 0.2369

0.0047 0.0039 0.0323 0.0026 0.0222 0.0015 0.0217

12.59 9.08 4.09 9.39 3.70 9.08 6.30

1.0000 1.0000 1.0000 1.0000 0.9999 1.0000 1.0000

Table 6 Electromechanical modes with MBPSS installed. Mode

a

 b

ra

a

Pa

 n

rn

n

Pn

1 2 3 4 5 6 7

1.880 1.626 1.346 0.888 1.192 1.700 1.483

15.995 11.437 9.966 7.825 7.690 5.930 4.170

0.0547 0.0831 0.0947 0.0040 0.1101 0.1899 0.1175

34.38 19.57 14.22 219.39 10.82 8.95 12.63

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.1167 0.1408 0.1339 0.1127 0.1531 0.2756 0.3352

0.0041 0.0078 0.0080 0.0017 0.0117 0.0248 0.0227

4.04 5.19 4.26 7.48 4.52 7.08 10.36

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

4.5. Performance comparisons By comparing Tables 3–6 with Table 1, it is found that the damping ratios of most electromechanical modes are significantly enhanced. Tables 3 and 4 give the probabilistic results of the seven electromechanical modes with the CPSS and the SNPSS, respectively. In general, the SNPSS performs better except for a comparatively large drop of  n6 which decreases from 0.2341 to 0.1119. It should be pointed out, nevertheless, that  n6 ¼ 0:1119 >  nC ¼ 0:1 does satisfy the acceptable threshold of the damping ratio, and that more importantly,  n4 ;  n5 and  n7 have been significantly enhanced. In practice, it is not easy to receive a timely remote signal due to communication network delays and some other difficulties. The APSS is designed to adapt to these working conditions. Table 5 shows that the damping ratios of modes 3, 5, and 7 are enhanced with the APSS. Similarly, according to the results in Table 6, all electromechanical modes have sufficient stability margins, though some are slightly reduced, such as  n1 and  n4 , but still satisfy the accept-

able threshold. More importantly, the damping ratio of mode 7 is increased by 43% compared with that of the CPSS. Assuming a normal distribution, the probabilistic density functions (PDFs) of the real part and the damping ratio in modes 5 and 7 with different PSSs are shown in Figs. 7 and 8. As revealed by Fig. 7, the performances of mode 5 with SNPSS is better than that of other PSSs. Fig. 8 illustrates that the MBPSS is more robust as the corresponding eigenvalues fall into the region with larger stability margin. Table 7 outlines the degrees of improvement of real parts and damping ratios when using SNPSS, APSS and MBPSS over CPSS, in which the negative values (shown in bold) indicate the degrees of decline. It can be seen from the data of Table 7 that mode 6 is deteriorated with SNPSS installed, modes 4 and 6 worsened with APSS installed, and the damping ratios of modes 1 and 4 are reduced with MBPSS installed. As shown in Table 2, the sensitivities of modes 6 and 7 against the PSS7 are 0.1046 and 0.2549, respectively, and this means that PSS7 influences modes 6 and 7 more than the other modes.

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Fig. 7. The PDF curves of mode 5 with different PSSs installed on G7.

Fig. 8. The PDF curves of mode 7 with different PSSs installed on G7.

Table 7 The improvement degrees of real parts and damping ratios when using SNPSS, APSS and MBPSS over CPSS. Mode

1 2 3 4 5 6 7

Real part improvement (%)

Damping ratio improvement (%)

SNPSS

APSS

MBPSS

SNPSS

APSS

MBPSS

0.04 0.77 0.16 39.5 109 50.1 25.9

0.13 1.32 91.7 22.5 81.3 49.6 12.3

22.0 13.4 11.0 43.0 42.2 3.74 46.4

0.13 0.83 0 35.1 95.1 52.2 14.3

0.13 1.43 86.7 26.4 69.3 51.4 1.02

26.5 5.79 7.73 33.1 42.3 17.7 42.9

Fig. 9. The relative rotor angle curves between G1 and G8 under a three-phase grounding short-circuit fault with different kinds of PSSs respectively installed.

P. He et al. / Electrical Power and Energy Systems 46 (2013) 175–183

181

Fig. 10. The relative rotor angle curves between G6 and G8 under a three-phase grounding short-circuit fault with different kinds of PSSs respectively installed.

Fig. 11. The voltage on bus 1 under a three-phase grounding short-circuit fault with different kinds of PSSs respectively installed.

Fig. 12. The voltage on bus 11 under a three-phase grounding short-circuit fault with different kinds of PSSs respectively installed.

From Tables 3 and 7, it is known that the performance of mode 7 has more or less improved with four kinds of PSSs. However, only MBPSS improves mode 6 relative to CPSS. It can be seen from Table 7 and Fig. 8 that the improvement of the damping characteristics with MBPSS is better than that of other PSSs, namely SNPSS and CPSS. 5. Large-signal stability assessment All simulation results presented in this section are obtained using the same power system as shown in Fig. 5. The parameters

of all generators as well as those of the various kinds of PSSs remain the same as in the situation of the small-signal stability analysis. A particular system operating condition from the 720 samples is chosen to perform the large-signal stability time domain simulation, and bus 24 is specified to be the slack bus. A three-phase grounding short-circuit fault occurred between buses 9 and 11 near bus 11 with the fault time tf = 1.0 s and clearing time tc = 1.2 s. The behaviors of the four types of PSSs at G7 are simulated in the time domain. The relative rotor angle curves between G1, G6 and G8 with different PSSs respectively installed under a three-phase grounding

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Fig. 13. The active power curve of G3 under a three-phase grounding short-circuit fault with different kinds of PSSs respectively installed.

Fig. A1. The excitation system of generators G1, G2, G3, G6, G7 and G8.

Fig. A2. The excitation system of generators G4 and G5.

Fig. A3. The turbine governor of all generators.

short-circuit fault are given in Figs. 9 and 10, the voltages on buses 1 and 11 are shown in Figs. 11 and 12, and the active power curve of G3 is shown in Fig. 13. It can be observed from Figs. 9–13 that under the three-phase grounding short-circuit fault, the relative rotor angle curves are divergent in the case without PSSs installed, the bus voltage at the short-circuit point fall down to zero, and finally the system loses the stability at around 2 s. When the four kinds of PSSs are respectively installed, the system tends to stabilize at around 5s. From Figs. 9 and 10, it can be seen that the amplitudes of the power angle curves with CPSS and APSS respectively installed are larger than those with MBPSS and SNPSS respectively installed, and the swing characteristics of the system with MBPSS and SNPSS respectively installed are similar. Similar situations are observed in the profile of the active power curve of G3, as shown in Fig. 13. The

bus voltage curves with four kinds of PSSs respectively installed exhibit similar features, as shown in Figs. 11 and 12.

6. Conclusions This paper examined the effects of four different PSS design approaches, namely CPSS, SNPSS, APSS and MBPSS, on improving power system dynamic performances. The steepest descent method is used to obtain the optimal PSS parameters. An 8-unit 24-bus sample power system is used to evaluate the system behaviors under small and large disturbances with four types of PSSs respectively installed. The eigenvalue analysis for the small disturbance scenario shows that the damping ratios of most electromechanical modes are significantly enhanced with four kinds of PSSs respectively

P. He et al. / Electrical Power and Energy Systems 46 (2013) 175–183

installed. Compared with CPSS, the performance of mode 7 with MBPSS is more robust as the corresponding eigenvalues all fall into the region with a larger stability margin, and the performance of mode 5 with SNPSS is better than those of other PSSs. On the other hand, it is shown by time-domain simulations for the large disturbance scenario that the amplitudes of the curves with CPSS and APSS respectively installed are larger when a fault occurs, while the swing characteristics with MBPSS and SNPSS respectively installed are similar. These results suggest that the performances of MBPSS and SNPSS are superior to CPSS and APSS in improving the amplitude oscillations of the system. Acknowledgement This work is supported by a discovery project of Australia Research Council (DP120101345).

Appendix A. Block diagram of the excitation systems and the turbine governors See Figs. A1–A3. References [1] Kundur P, Klein M, Rogers GJ, Zywno MS. Application of power system stabilizers for enhancement of overall system stability. IEEE Trans Power Syst 1989;4(2):614–26. [2] Nechadi E, Harmas MN, Hamzaoui A, Essounbouli N. A new robust adaptive fuzzy sliding mode power system stabilizer. Int J Electr Power Energy Syst 2012;42(1):1–7.

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