An integrated approach for optimal placement and tuning of power system stabilizer in multi-machine systems

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An integrated approach for optimal placement and tuning of power system stabilizer in multi-machine systems

Vahid Keumarsi

^⇑

, Mohsen Simab, Ghazanfar Shahgholian

Department of Electrical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran

Department of Electrical Engineering, College of Engineering, Fars Science and Research Branch, Islamic Azad University, Fars, Iran

a r t i c l e i n f o

Article history:

Received 4 August 2013

Received in revised form 23 May 2014 Accepted 27 May 2014

Keywords:

Power system stabilizer Tuning and placement of PSS PSO algorithm

Fuzzy logic

Multi-machine system

a b s t r a c t

In this paper, a hybrid approach for tuning and placement of power system stabilizers (PSS) in multi-machine power systems is provided with the aim of reducing low-frequency oscillations (LFO) and improving power system dynamic stability with wide range of changes in system parameters and operating point of the system. PSS parameters are adjusted using Particle Swarm Optimization algorithm (PSO), and using Takagi–Sugeno (TS) fuzzy, optimal location for the PSS is determined. The employed fuzzy system has two inputs, the real part and the damping coefﬁcient of network eigenvalues. The results of test on a grid four machine-two area sample, show optimal performance of the proposed method in improving system stability and reducing low-frequency oscillations of local and inter-area modes.

1. Introduction

The power system is a complex and nonlinear system.

Mechanical nature of the power systems produces low-frequency oscillations (LFO) in the system, which will negatively affect the stability and performance of the system and will restrict the capac- ity of the transmission line[1]. In order to solve this problem, a supplementary controller is employed in the excitation system of the generators. This supplementary controller, known as PSS, is widely used to reduce the LFO and improve system stability[2].

Many new intelligent methods such as neural networks[3] and fuzzy logic [4,5] have been used in PSS design. Modern control methods such as adaptive control have also been used in PSS design[6]. In[7], an approach based on state feedback control is presented for tuning PSS parameters. Pole placement has also been employed to design PSS for multi-machine power systems in coordination with FACTS devices[8]. In[9], PSO was used to tune PSS for interconnected power systems. In [10], the use of adaptive neuro-fuzzy inference was proposed for PSS design to enhance the damping issue of the conventional PSSs. Bacterial swarm optimization is the other approach used for the design of PSS in coordination with thyristor controlled series capacitor (TCSC) for

multi-machine power systems [11]. A fuzzy PI Takagi–Sugeno stabilizer is presented in[12]. An optimal power system stabilizer (OPSS) based on second-order linear regulator with conventional lead-lag compensation structure is also proposed in[13]. The mod- iﬁed particle optimization (MPSO) is the other method proposed to adjust the stabilizer parameters[14].

In[15]presented a method to determine the optimal location and the number of multi-machine power system stabilizers (PSSs) using participation factor (PF) and genetic algorithm (GA). A type-2 fuzzy logic power system stabilizer with differential evolution algorithm presented in[16]. In[17], the design of a conventional power system stabilizer (CPSS) is carried out using the bat algorithm (BA). In[18]presents an enhanced indirect adaptive fuzzy sliding mode based power system stabilizer for damping local and inter-area modes of oscillations for multi-machine power systems. In [19] presented a new technique named cultural algorithms (CAs) to tune the PSS parameters. In [20] present the design and implementation of Power System Stabilizers in a multi-machine power system based on innovative evolutionary algorithm overtly as Breeder Genetic Algorithm with Adaptive Mutation.

This paper presents a hybrid approach to tune and place PSSs in multi-machine systems. PSS parameters are adjusted using PSO algorithm, according to the operating point of the network. The optimal location for PSS installation is then determined by fuzzy Takagi–Sugeno (TS) system. The fuzzy system has two inputs, the real part and the damping coefﬁcient of network eigenvalues.

⇑Corresponding author at: Department of Electrical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran. Tel.: +98 9179517691.

E-mail address:vahid_keumarsi@yahoo.com(V. Keumarsi).

Contents lists available atScienceDirect

Electrical Power and Energy Systems

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j e p e s

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Network eigenvalues are obtained by the linearization of differential–algebraic equations of the power system. The experimental results on a four machine-two area network show the good performance of the proposed method improving system stability and reducing low frequency oscillations of local and inter-area modes.

Power system modeling equations are presented in section

‘Power system modelling’ and the algorithm used for tuning and placement of PSSs is given in section ‘PSS tuning and placement’.

Simulation results and conclusions are described in sections

‘Experimental results and Conclusion’, respectively.

Power system modelling

In order to analyze the power system, linearization of Differen- tial–Algebraic Equations (DAE) around the operating point is used [21].

Generator equations

For each generator, the following fourth-order model has been considered:

ddi

dt ¼

x

i

x

s ð1Þ

d

x

i

dt ¼TMi

Mi ðE⁰_qiX⁰_diIdiÞIqi

Mi ðE⁰_diþX⁰_qiIqiÞIdi

Mi Dið

x

i

x

sÞ Mi

ð2Þ

dE⁰_qi dt ¼ E⁰_qi

T⁰_doiðXdiX⁰_diÞIdi

T⁰_doi þEfdi

T⁰_doi ð3Þ

dE⁰_di dt ¼ E⁰_di

T⁰_qoi_iþ Iqi

T⁰_qoi_iðXqiX⁰_qiÞ ð4Þ

Exciter equation

Exciter equation is:

dEfdi

dt ¼ Efdi

T⁰_doiðXdiX⁰_diÞ T⁰_doi IdiþEfdi

T⁰_doi ð5Þ

A detailed description of all symbols and quantities can be found in[21]. By the linearization of the power system equation, explained in[21], and by the addition of PSS equations, the power system model is:

Dx^_¼AxþBu ð6Þ

Dy¼CxþDu ð7Þ

whereAis the state variables matrix,Bis the input matrix,Cis the output matrix,Dis the feed-forward matrix,xis the vector of state variables,uis the vector of control inputs, andyis the output. Here, the gole of PSS design is to place the eigenvalues of matrixAin the left half of the complex plane. Eigenvalues of the system can be evaluated from matrixA:

ki¼

r

ij

x

i ð8Þ

wherei= 1, 2, 3,. . .,nandndenotes the total number of eigenvalues.

The eigenvalues may be real or complex. The imaginary part of the complex eigenvalue (x) is the radian frequency of the oscillations and the real part (

r

) is the decrement rate. Then, the damping ratio (nj) of thejth eigenvalue is deﬁned with the following equation:

ni¼

r

i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

r

²_i þ

x

²_i

q ð9Þ

PSS structure

The PSSs used here possess a conventional lead-lag structure and have speed deviation input [22–24]. The Gain of stabilizer (KPSS) determines the amount of damping created by PSS. Like a high-pass ﬁlter with time constant Tw, the Filtering block (washout) passes the cross speed deviations and blocks steady- state values of the speed. The two-level compensation blocks pro- vide an appropriate lead characteristic, in order to compensate the lag characteristic between the input excitation control and the electrical torque of the generator. Five parameters of PSS, including T1–T4(time constant) andKPSS, are optimized by PSO algorithm and Twis considered to be constant.Fig. 1shows the PSS structure.

PSS tuning and placement

The classical PSS tuning methods are usually used for speciﬁc frequency and operating point, and hence, any change in system operating point degrades the performance of PSS. In this paper, Particle Swarm Optimization algorithm (PSO) is used to adjust PSS parameters.

Particle Swarm Optimization (PSO) algorithm

PSO is an evolutionary computation algorithm, inspired from the nature and based on repetition[25]. The PSO algorithm is com- posed of ﬁxed number of particles taking random initial values.

Two values are assigned to each particle, position (X^k_i) and velocity (V^k_i). These particles repetitively move in then-dimensional space (corresponding to the number of parameters) to look for new possible answer choices by calculating the optimality of each particle based on the objective function. At each step, the best position of each particle (pbest^k_i) and the best position among all particles (gbest^k_i) are stored. New speed (V^kþ1_i ) and new position (X^kþ1_i ) of each particle will be updated using following equations[26]:

V^kþ1_i ¼wV^k_i þc1r1pbest^k_i X^k_i

þc2r2

gbest^k_i X^k_i

ð10Þ

X^kþ1_i ¼X^k_i þV^kþ1_i ð11Þ

wherec1andc2are positive numbers illustrating the weight of the acceleration of each term, guiding each particle toward the best individual (p_best) and the best swarm (g_best) positions,r₁andr₂are two random number in the range [0 1], andwis the inertia calculated by the following equation[26]:

w¼wmax wmaxwmin

kmax

k ð12Þ

wherew_maxandw_minare the maximum and minimum values ofw, kmaxis the maximum number of iterations andkis the current iter- ation number.

Objective function

The Stability of a power system can be determined based on its eigenvalues. Eigenvalues with large negative real parts ensure system stability. the overshoot and oscillations values are determined

ω ω sT sT +

1 1 ₂¹

1 sT sT +

+ output

input

KPSS

4 1 3 1

sT sT + +

Fig. 1.Structure of power system stabilizer.

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by the damping coefficient of the system, where any increase in the damping coefficient reduces oscillations and improves the stability of the system. In this paper, the objective function is a combination of the two indices; the damping coefficient and the real part of eigenvalues, which are considered as follows[27]:

Minimize J¼J₁þ

a

J₂ ð

a

¼10Þ ð13Þ

J₁¼Xⁿ

i¼1

ð

r

i

r

0Þ²for

r

i

r

0; ð

r

0¼ 3Þ ð14Þ

J2¼Xⁿ

i¼1

ðn0niÞ²fornin0; ðn0¼0:3Þ ð15Þ

ndenotes the total number of eigenvalues. termJ₁in the objective function controls the real part (

r

) of eigenvalues and generally leads the eigenvalues of the system to the left side of imaginary axis in an area less than

r

0. The termJ2in the objective function brings the damping (n) of eigenvalues to the desired damping (n0) and controls the overshoot of the system. value of

a

obtained by experimenta- tion and considered constant equal to 10[27].

Optimal PSS placement using TS fuzzy

Optimal PSS Locating in multi-machine systems is the impor- tant issues which can improve PSS performance and increase dynamic stability of the power system. The TS fuzzy system is sui- ted for mathematical analysis such as nonlinear systems modeling [28]. The output of a TS fuzzy system is deﬁned as a function of inputs:

If x1isMandx2isLthen output¼ax1þbx2þc ð16Þ whereLandMare fuzzy sets;x₁andx₂are fuzzy system inputs; and a,bandcare constants. In this paper, the TS fuzzy system determines the optimal location for PSS installation by proper design of membership functions and fuzzy rules. The zero-order TS fuzzy system (a,b= 0) is used, and thus the output of each rule is a constant.

The TS fuzzy system inputs are the damping coefficient and the real part of power system eigenvalues. Five linguistic variables were considered for each input. The membership functions of the damping coefficient and the real part inputs can be seen inFigs. 2 and 3, respectively. Also, the 25 rules defined for fuzzy output are shown inTable 1. In this paper, it is assumed that a fixed number of PSSs, k, are installed among the m generators of the network. Various cases of PSS location in network, defined as position index variable (PlaceIndex), are obtained by enumeration ofkstabilizers amongm generators. The maximum possible value forPlaceIndexis obtained by Eq.(17):

PlaceIndexmax¼ k m

¼ m!

k! ðmkÞ! ð17Þ

1 PlaceIndex PlaceIndexmax ð18Þ

In order to determine the best location for PSS, all possible locations are determined by(17) and (18)and are stored inPlaceIndex variable. Then, according to the algorithm shown inFig. 5, enumeration of variablePlaceIndex is performed for each PlaceIndex. In each enumeration step, after tuning PSS parameters, the eigenvalues of the system are calculated and sorted based on the least damping. The first eight eigenvalues (corresponding to the quan- tity of mechanical modes), having the least damping coefficients are sent to the fuzzy system. Receiving the damping coefficient and the real part of eigenvalues by the fuzzy system as inputs, it calculates theout(i), output corresponding to theith eigenvalue.

Also FuzzyPlace(k), corresponding to the kth placeIndex at each enumeration step, can be calculated using Eq.(19):

FuzzyPlaceðkÞ ¼10outðiÞ þ8outðiþ1Þþ

6outðiþ2Þ þ. . .þoutð8Þ fori¼1;2;3;. . .;8 ð19Þ The variableFuzzyPlace(k) indicates the ﬁtness of thekth for installing PSS and a largeFuzzyPlace(k) shows a better place for PSS installation. Finally, the best place for PSS installation is determined based on the maximum value ofFuzzyPlace(k). The algorithm used PSS tuning and ﬁnding its location can be observed in Fig. 5.

Experimental results

In order to study the performance of the proposed method, the four machine-two area network shown inFig. 4is simulated. This benchmark system is introduced to study inter-area oscillations and their effects on the system dynamic stability. It consists of two areas, each having two generators. The two areas are con- nected by a double communication line. About 400 MW of power is transferred through the communication lines between the two areas. All the simulations have been carried out by Matlab soft- ware. In this system, there are two generation areas, which have two generators, and two loads interconnected by transmission lines. details of the power system can be seen in reference[29].

PSS tuning

The results of using PSO algorithm to tune the stabilizer parameters (PSOPSS) are compared with those of using classical methods (CPSS) and Genetic Algorithm (GAPSS). PSO and GA parameters

-1 -0.7 -0.4 -0.1 0.2 0.5 0.8 1

0 0.2 0.4 0.6 0.8 1

Damping Ratio

Degree of membership

NS ZE PS PM PB

Fig. 2.Membership functions for damping coefﬁcient.

-15 -12 -9 -6 -3 0 3 5

0 0.2 0.4 0.6 0.8 1

Real Part

Degree of membership

NB NM NS ZE PS

Fig. 3.Membership functions for realpart.

Table 1

Fuzzy system rules.

r n

PB PM PS ZE NS

NB Mf1 Mf1 Mf2 Mf3 Mf3

NM Mf1 Mf1 Mf2 Mf3 Mf3

NS Mf1 Mf2 Mf2 Mf3 Mf4

ZE Mf2 Mf3 Mf3 Mf4 Mf5

PS Mf3 Mf4 Mf4 Mf5 Mf5

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considered in this comparison can be seen inTable 2. Eigenvalues and the damping coefﬁcients of network modes for cases of no stabilizers (No PSS), CPSS, PSOPSS and GAPSS are illustrated in Table 3.

It can be seen that in NoPSS case, some of the network modes have very weak damping. Moving up to the CPSS case, relative

improvement in the weak modes of network is observed, suggesting the positive impact of PSS on network stability. As the results show, optimizing PSS parameters, signiﬁcantly improves the damping of network modes. It can also be observed that using PSO algorithm results in more favorable results than using other methods to tune PSS parameters.

Fig. 4.Network four machine-two areas.

Fig. 5.Simultaneous PSS tuning and locating algorithm.

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PSS locating

As a common rule, not all the generators in a network are equipped by PSSs, but the minimum number of PSSs used in a network equals to half of the total generators used in it[30]. In this paper, one, two and three PSSs are considered for the power system. The optimal location for installing PSSs in each case is presented in Table 4, obtained using the TS fuzzy system. The 3rd column of the table shows the ﬁtness value of eachPlaceIndex, with larger values showing better locations for PSS. As observed in Table 4, generators G1, G3, G4 are the best PSS locations for the case of using three generators, while generators G1, G3 and generator G1 are the best places for PSS installation for the cases of using two and one generators, respectively. The tuned PSS parameters in each case, best installation position, and the least network damping coefﬁcient are summarized inTable 5.

As observed inTable 5, the minimum damping coefﬁcients in cases of using one and two PSSs in network are 0.047 and 0.067, respectively, suggesting a network with weakly damping modes.

These values increase to 0.151, a suitable value, for the case of using three PSSs and 0.352 when using four PSSs. As a conclusion, installing three PSSs in the network yields to acceptable damping of oscillatory and weak network modes.

Time domain simulations

In order to study the performance of the proposed algorithm in setting and placing PSSs in power systems, some time domain simulations have been performed on the network under study. Mean- while, to investigate the proposed method performance when the operating point of the system changes, two different operating cases are considered for the system (Table 6).

PSS tuning

All generators of the studied network are equipped by PSSs.

Dynamic behavior ﬂuctuations of inter-area modes are depicted

inFigs. 6 and 7with different methods of setting PSS parameters, including CPSS, GAPSS, PSOPSS and without PSS simulations.

According toFigs. 6 and 7tuning PSS parameters by PSO algorithm provides suitable results and results in more favorable damping for network compared to the use of CPSS and GAPSS. Ana- lyzing system eigenvalues, shown inTable 3, in combination with the dynamical simulation performed, suggests the good performance of the proposed PSOPSS method tuning PSS, improving damping and reducing local and inter-area network oscillations.

PSS locating

In order to study the performance of the proposed method ﬁnd- ing the best location for PSS, some time domain simulations has been applied to the network under study. The experiments were performed for the cases of installing two and three PSSs in the network.

Investigating the speeds of generators.Figs. 8–15, show the speeds of generators with three and two PSSs in the network for different installation locations and load conditions in the network.

According toFigs. 8–15, the best locations for installing three and PSSs in the network are generators G1, G3, G4.

Table 2

PSO and GA parameters.

Parameters PSO GA

Npop 20 20

Iter 300 300

c1 2 –

c2 2 –

xmin 0.45 –

xmax 0.95 –

Pcrossover – 0.7

Pmutation – 0.05

Table 3

Eigenvalues and damping coefﬁcients.

Method Eigenvalues Damping

NoPSS 0.0907 ± 4.3183i 0.0210

0.2851 ± 6.1351i 0.0464

0.4054 ± 6.2126i 0.0651

CPSS 0.4708 ± 4.4266i 0.1058

0.7694 ± 6.3661i 0.1200

0.7954 ± 6.3383i 0.1245

GAPSS 2.5545 ± 8.6985i 0.2818

2.5440 ± 7.8648i 0.3078

3.0023 ± 8.1775i 0.3446

PSOPSS 2.9973 ± 7.9819i 0.3515

2.9764 ± 7.1223i 0.3856

3.5894 ± 8.1883i 0.4015

Table 4 Best place of PSS.

No. PSS Best place () Fuzzy place

3PSS G1, G3, G4 349.5003

G2, G3, G4 332.9920

G1, G2, G3 320.7537

G1, G2, G4 318.4545

2PSS G1, G3 263.7264

G1, G4 259.143

G2, G3 254.7809

G2, G4 247.361

G1, G2 247.2790

G3, G4 236.9329

1 PSS G1 201.8627

G2 194.7218

G3 183.9345

G4 180.6046

Table 5

PSS parameters and damping.

No Place KPSS T1 T2 T3 T4 nmin

1 PSS G1 21.25 0.184 0.041 1.384 0.037 0.047

2 PSS G3 36.23 0.276 0.042 0.522 0.014 0.067

G1 31.38 0.805 0.013 0.241 0.077

3 PSS G1 17.97 0.333 0.038 0.957 0.084 0.151

G3 19.02 1.175 0.047 0.235 0.017

G4 6.783 1.066 0.011 0.563 0.077

4PSS G1 10.62 0.911 0.034 0.710 0.200 0.352

G2 20.07 0.309 0.068 0.822 0.020

G3 6.706 0.559 0.102 1.399 0.013

G4 6.312 1.451 0.036 0.436 0.049

Table 6

Two different operating cases.

Case Number

Description

Case 1 Single line between 8 and 9 out of service, and a three-phase short circuit is applied to bus 8 for 0.1 s

Case 2 Single line between 7 and 9 out of service, and a three-phase short circuit is applied to bus 9 for 0.15 s

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Inter-area modes. Figs. 16–21, show inter-area oscillations (w1–w3) and (w2–w3), with three and two PSSs in the network with different installation locations and loading conditions.

0 2 4 6 8 10 12

-2 0 2

x 10^-3

Time (sec)

w1-w3 (pu)

w1- w3 - 4pss (case1)

NoPSS CPSS GAPSS PSOPSS

Fig. 6.Inter-area modesw1–w3, case 1, 4 PSSs installed.

0 2 4 6 8 10 12

-2 0 2

x 10^-3

Time (sec)

w2-w3 (pu)

w2-w3 - 4pss (case1)

NoPSS CPSS GAPSS PSOPSS

0 2 4 6 8 10 12

1 1.002 1.004 1.006 1.008

Time (sec)

Speed (pu)

speed G1 - 3pss (case1)

No pss pss in G1,G2,G3 pss in G1,G2,G4 pss in G1,G3,G4 pss in G2,G3,G4

Fig. 8.G1 speed, case1, 3 PSSs installed.

0 2 4 6 8 10 12

1 1.002 1.004 1.006 1.008

Time (sec)

Speed (pu)

Fig. 9.G1 speed, case 2, 3 PSSs installed.

0 2 4 6 8 10 12

1 1.002 1.004 1.006 1.008

Time (sec)

Speed (pu)

0 2 4 6 8 10 12

1 1.002 1.004 1.006 1.008

Time (sec)

Speed (pu)

0 2 4 6 8 10 12

1 1.002 1.004 1.006 1.008 1.01

Time (sec)

Speed (pu)

0 2 4 6 8 10 12

1 1.002 1.004 1.006 1.008 1.01

Time (sec)

Speed (pu)

0 2 4 6 8 10 12

1 1.002 1.004 1.006 1.008 1.01

Time (sec)

Speed (pu)

0 2 4 6 8 10 12

1 1.002 1.004 1.006 1.008 1.01

Time (sec)

Speed (pu)

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Figs. 20 and 21, show inter-area oscillations (w1–w3) and (w2–w3), with two PSSs installed:

Figs. 16–21conﬁrm the previously obtained optimized locations for installing three and two PSSs in the network. Time domain simulations, shown inFigs. 8–21, introduce generators G1, G3, G4,

and G1, G3 as the best installing locations in the network for the cases of using three and two PSSs, respectively. By comparing these results toTable 4, which shows the installing locations determined using fuzzy systems, the optimal performance of the proposed fuzzy method to determine the best places for PSSs in the system

0 2 4 6 8 10 12

-4 -2 0 2 4^{x 10}

-3

Time (sec)

w1-w3 (pu)

w1- w3 - 3pss (case1)

No PSS pss in G1,G2,G3 pss in G1,G2,G4 pss in G1,G3,G4 pss in G2,G3,G4

0 2 4 6 8 10 12

-4 -2 0 2 4^{x 10}

-3

Time (sec)

w1-w3 (pu)

0 2 4 6 8 10 12

-4 -2 0 2 4^{x 10}

-3

Time (sec)

w2-w3 (pu)

0 2 4 6 8 10 12

-4 -2 0 2 4^{x 10}

-3

Time (sec)

w2-w3 (pu)

0 2 4 6 8 10 12

-3 -2 -1 0 1 2

x 10^-3

Time (sec)

w1-w3 (pu)

pss in G1,G2 pss in G1,G3 pss in G1,G4 pss in G2,G3 pss in G2,G4 pss in G3,G4

0 2 4 6 8 10 12

-2 -1 0 1 2

x 10^-3

Time (sec)

w2-w3 (pu)

pss in G1,G2 pss in G1,G3 pss in G1,G4 pss in G2,G3 pss in G2,G4 pss in G3,G4

0 2 4 6 8 10 12

0.998 1 1.002 1.004 1.006 1.008 1.01

Time (sec)

Speed (pu)

generators speed - No pss

G1 G2 G3 G4

Fig. 22.Generators speed, No PSS.

0 2 4 6 8 10 12

0.998 1 1.002 1.004 1.006 1.008 1.01

Time (sec)

Speed (pu)

generators speed - 3pss (case2)

G1 G2 G3 G4

Fig. 23.Generators speed, case 2, 3 PSSs installed.

0 2 4 6 8 10 12

-2 0 2

x 10^-3

Time (sec)

Speed (pu)

Inter-Area Mods - No pss

w1-w3 w1-w4 w2-w3 w2-w4

Fig. 24.Inter-area modes, No PSS installed.

0 2 4 6 8 10 12

-2 0 2

x 10^-3

Time (sec)

Speed (pu)

Inter-Area Mods - 3pss (case2)

w1-w3 w1-w4 w2-w3 w2-w4

Fig. 25.Inter-area modes, case 2, 3 PSSs installed.

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is conﬁrmed. In order to study the performance of the proposed method improving system stability, reducing local and inter-area oscillations of the system, a simulation assuming three PSSs installed in the network is performed. PSS parameters and optimized installation locations are determined by the proposed algorithm, and PSSs are installed on the generators G1, G3, G4.Figs. 22–

25show the speeds of generators and inter-area oscillations for cases of no PSS and three PSSs installed.

Figs. 22–25 show the favorable performance of the proposed method setting and placing PSSs in the network with the aim of improving the system dynamic stability.

Conclusions

In this paper, a hybrid approach for simultaneous PSS setting and placing in multi-machine systems is proposed. PSS parameters are adjusted using PSO optimization algorithm and the optimal PSS installation locations are determined by Takagi–Sugeno fuzzy system. The TS fuzzy system employs damping coefﬁcients and real parts of network eigenvalues as its two inputs. A sample four- machine, two-area network is employed to investigate the effectiveness of the proposed method, in different operating point conditions. Analysis of the system eigenvalues and dynamic simulation of the network, conﬁrmed optimal performance of the proposed method (PSOPSS) setting PSS parameters and improving damping modes (particularly mechanical oscillating modes), compared to the cases of use of classical methods (CPSS) and Genetic Algorithm (GAPSS). The proposed fuzzy method was applied to determine the PSS installation locations, and the obtained results showed optimal performance of the proposed method to simulta- neously determine the best PSS adjustments and installation locations, providing appropriate damping for the system. The proposed method is robust against changes in network operating point, improves dynamic performance of the network, reduces local and inter-area low-frequency oscillations of the system, and is suitable for a wide range of power systems.

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