Electrical Power and Energy Systemsrms.scu.ac.ir/Files/Articles/Journals/Abstract/1-s2.0-S... · ture of the lead-lag power system stabilizer is shown in Fig. 1 [14]. 2.2. STATCOM

Electrical Power and Energy Systems 53 (2013) 1003–1017

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Electrical Power and Energy Systems

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Analysis of the simultaneous coordinated design of STATCOM-baseddamping stabilizers and PSS in a multi-machine power system usingthe seeker optimization algorithm

0142-0615/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijepes.2013.06.021

⇑ Corresponding author.E-mail addresses: [email protected] (E. Afzalan), [email protected]

(M. Joorabian).

Ehsan Afzalan, Mahmood Joorabian ⇑Shahid Chamran University of Ahvaz, Ahvaz, Iran

a r t i c l e i n f o

Article history:Received 10 December 2012Received in revised form 10 June 2013Accepted 28 June 2013

Keywords:Seeker optimization algorithm (SOA)STATCOMPSSSimultaneous coordinated design

a b s t r a c t

Facts devices are increasingly being applied to increase the damping of the power system. Uncoordinateddesign of these devices with power system stabilizers (PSSs), however, may have a detriment effect onthe system performance. This paper presents seeker optimization algorithm (SOA) to design the param-eters of PSS and STATCOM coordinately to improve more stability of power system. The SOA is used toachieve the best optimal results to minimize the objective function of the optimization problem. Simu-lations are carried out on a two-area Kundur and 39-bus New England power systems, and simulationresults confirm the efficiency of the proposed method to stabilize power system oscillations. ComparingSOA with other intelligent method (GA) shows that a better performance is achieved. In addition, theeffectiveness of the coordinated design using the proposed method is confirmed through eigenvalueanalysis.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Dynamic stability is based on small amplitude oscillations thatcan be initiated by sudden changes in the load or the network. Sev-eral modes of inter-area oscillation have appeared in the dynamicresponse of the power system due to the complexity and increas-ing dimensions associated with modern power systems. Therefore,power system stabilizers (PSSs) are employed to enhance thedynamic and transient performance of power systems, commonlyinstalled as supplementary control devices to provide extra damp-ing. PSSs are very effective controllers in enhancing the damping oflow-frequency oscillations because the damping torque of local-area modes can be increased [1]. In recent decades, the complexity,inter-connections and nonlinearities of power systems are consid-erably being increased. It has been made designers determine PSSparameters simultaneously to safe and secure operation of multi-machine power system [2–4].

Evolutionary algorithms have been used to solve optimizationproblems in power system studies. The PSS parameters design isone of the most interesting optimization problems today, whichconsidered in many works by using evolutionary algorithms. Parti-cle swarm optimization (PSO) has been used to tune PSS parame-ters [2]. The performance of genetic algorithm (GA) [3] and

simulated annealing (SA) [4] has been investigated to design PSSparameters. One of the most serious problems of design of PSSparameters is related to the fact that the PSS parameters are setfor unique operating point and special faults [2–4]. As a result,the design of PSS parameters are not robust over a wide rang ofoperating points and different faults. Accordingly to resolve theproblem, a bacterial foraging algorithm (BFA) and PSO have beenemployed to design robust PSS parameters based on differentoperating points and various faults in multi-machine powersystems [5].

Recently, the need to expand modern power systems is increas-ing due to world economic growth. However, because of environ-mental problems and urbanization, the construction of newtransmission lines remains a challenge. Therefore, the use of newequipment, such as FACTS devices, has been proposed. These de-vices can improve the transfer power limit, further improving thesteady state, transient stability performance and damping the in-ter-area modes [6]. Nevertheless, interactions between the powersystem stabilizers and the FACTS controllers may cause dynamicinstabilities, if they are designed based on uncoordinated controlstrategies. Finally, the simultaneous coordinated design of PSSand FACTS device controllers is necessary, so the simultaneouscoordinated design of FACTS device controllers and PSS has beenstudied in some works [7–11]. Coordinated design of both PSSand TCSC has been performed with PSO algorithms [7] and a non-linear programming technique [8]. The coordinated design of SSSCand PSS has been studied by using a hybrid BFOA-PSO algorithm

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Fig. 1. Lead-lag structure of power system stabilizer.

1004 E. Afzalan, M. Joorabian / Electrical Power and Energy Systems 53 (2013) 1003–1017

[9]. Coordination design of SVC and PSS to enhance the transientstability of power systems has been examined by using probabilis-tic theory in Ref. [10]. The PSO algorithm [11] has been used tocoordinate the design of STATCOM and PSS controllers to improvethe performance of transient and steady state stability in multi-machine power systems. The main disadvantage of the particleswarm optimization algorithm is that it easily suffers from the par-tial optimism. It causes the less exact at the regulation of its speedand the direction. The 1th, 2th and 3th feature of SOA cover thisproblem of PSO.

The seeker optimization algorithm (SOA) is a population-basednew heuristic method. The algorithm has been used to search foroptimal solutions in optimization problems [12]. It has beenfounded on human comprehension and searching abilities to findthe optimal solution. In the SOA, the optimal solution is found bythe seeker population. The fundamental concept of the SOA is thecomparatively simple-model and easy implementation versusother optimization techniques. The most important features ofthe algorithm are given as follows; [12].

(a) The position of each particle (seeker) is updated based onthe search direction and the step length.

(b) A proportional selection rule is used to implement thesearch direction. It enhances the population diversity toincrease the global search capability, and the number of con-trol parameters is decreased, making it simpler toimplement.

(c) Fuzzy reasoning is used to create the step length withrespect to the uncertain reasoning of human searching,which can be represented by natural linguistic variables,and a simple if-else control rule.

In this paper, a supplementary damping controller with a lead-lag structure, which is the STATCOM-based controller to improvethe damping and the transient stability of the power system, isadded to the AC voltage controller of STATCOM. The difference inthe speed deviations is taken as the input signal for the proposedSTATCOM-based controller. In this paper, both AC and DC STAT-COM PI-controllers are considered in the coordination design. Fur-ther, the SOA method is used to coordinate the design of STATCOMcontrollers and PSS.

Significantly, some main features of paper can be explained asfollow:

� The SOA method is a novel optimization algorithm that recentlyit is applied to solve optimization problems.� The implicit concept of the SOA method is very easy to model

and relatively easier.� The SOA method has the ability to converge to a better solution

than other methods [12,16–18].� Performance of SOA shows ability of proposed method to

improve power system stability.� Different operating points and various faults are considered to

select parameters of controllers.

Consequently, coordinated design of STATCOM controllers andPSS parameters is done by using the SOA method for differentoperating points and various faults which is tested on two-areaKundur and 39-bus New England power systems.

The rest of paper is constructed as follows: The nonlinear equa-tions of STATCOM are presented in Section 2. The SOA and GAmethods are thoroughly described in Section 3, and the proposedobjective function is shown in Section 4. Implementation of theSOA method for coordinated design of the STATCOM controllersand PSS is presented in Section 5. The time-domain simulation re-sults of the simultaneous coordinated design between the

STATCOM controllers and PSS for two-area Kundur and 39-busNew England power systems are discussed in Section 6. Mean-while, the simulation results are compared to obtained simulationresults of GA method and uncoordinated design. Section 7 endsthis work with the conclusion.

2. Power system modeling

2.1. Synchronous generator

Consider an n-generator power system. Each generator can bemodeled with a five-order dynamic system using an IEEE type1(DC type) exciter. The equations, according to the dynamic modelof the ith generator, can be found in Ref. [13]. In addition, the struc-ture of the lead-lag power system stabilizer is shown in Fig. 1 [14].

2.2. STATCOM

Fig. 2 shows the basic structure of a multi-machine power sys-tem with a STATCOM, which is located in the midpoint of thetransmission line. In Fig. 2, Rp and Lp represent the ‘ON’ state resis-tance of the switches, including the transformer leakage resistanceand the transformer leakage inductance, respectively. A voltagesource inverter (VSI) is used to control the injected reactive powerof STATCOM, generating a balanced three-phase voltage Vp. Voltagecontrol is achieved by firing the angle control of the VSI. When theAC-bus voltage Vs is greater than Vp, the VSI draws reactive powerfrom the AC-bus, whereas if |Vs| < |Vp|, then it will supply reactivepower to the AC-system. The dynamic components of the currentcan be written as follows [15];

dipd0

dt¼ �Rp

Lpipd0 þxipq0 þ

1LpðVs � Vpd0 Þ ð1Þ

dipq0

dt¼ �xipd0 �

Rp

Lpipq0 þ

1Lpð�Vpq0 Þ ð2Þ

The d–q frame is now transformed into another frame, the d0–q0

frame in which the d0-axis coincides with the AC-bus voltage (VS),as shown in Fig. 3. The following equations are used to determinethe q- and d-axis components of the VSI voltage (Vpq0 and Vpd0 ).

Vpq0 ¼ �ðxLpipd0 þ Lpuq0 Þ ð3Þ

Vpd0 ¼ xLpipq0 þ Vs � Lpud0 ð4Þ

x ¼ xo þdds

dtð5Þ

where the ud0 and uq0 are virtual control signals which are deter-mined by linear PI controllers to make main control signals Vpq0

and Vpd0 .The STATCOM current can be modeled using a first order differ-

ential equation. By substituting Eqs. (3) and (4) into Eqs. (1) and(2), the differential equations of the STATCOM current are deter-mined as follows:

dipd0

dt¼ �Rp

Lpipd0 þ ud0 ð6Þ

Fig. 2. Kundur two-area multi machine power system with STATCOM.

Fig. 3. Phasor diagram.

E. Afzalan, M. Joorabian / Electrical Power and Energy Systems 53 (2013) 1003–1017 1005

dipq0

dt¼ �Rp

Lpipq0 þ uq0 ð7Þ

As shown in Fig. 4, the control signals ud0 and uq0 can be easilydetermined by linear PI controllers. The following equation repre-sents the dynamic model of the DC bus voltage.

dVdc

dt¼ 3

2CVdcVsipd0 � ði

2pd0 þ i2

pq0 ÞRp

h i� Vdc

CRdcð8Þ

where Vdc shows the voltage of the DC capacitor. In this model, theswitching losses are taken into consideration using a shunt DC-sideresistance Rdc.

The structure of the cascade controller of STATCOM is shown inFig. 4 in which there are two AC and DC voltage control loops [15].In Fig. 2, Kpt, Kit, Kpc, Kic, Kpq, Kiq, Kpd, and Kid are the parameters ofthe PI controllers. In addition, the STATCOM-based stabilizer isconsidered in this paper to improve the transient. The differencein the speed deviations are taken as the input signal for the pro-posed STATCOM stabilizer. The K1S, K2S, K3S and K4S are time con-stants of the STATCOM stabilizer block which are shown in Fig. 4.

Fig. 4. Cascade control architecture and

3. Optimization algorithms

3.1. Seeker optimization algorithm (SOA)

Seeker optimization algorithm (SOA) is a population-based newheuristic method. It has been used to search for optimal solutionsto optimization problems [12,16–18]. A seeker population is estab-lished in which each member is called a seeker, this populationperforms the optimization process to achieve an optimal solution.Searching over several different domains of the search space is car-ried out by classifying the entire population into same subpopula-tions. The total population is randomly categorized into Ksubpopulations which every subpopulation has the same size. Allthe seekers in the same subpopulation constitute a neighborhood.

3.1.1. Calculation of the seeker algorithmFor the ith seeker on each jth variable at each time step t, the

search direction dij(t) and the step length aij(t) are calculated sothat aij(t) P 0 and dij(t) e {�1,0,1} Moreover, i and j illustrate thepopulation number and the optimizing variable number,respectively.

3.1.2. Search directionsSwarms are used to make a reciprocal action in a cooperative

way, based on the innate ability to execute their needs and goals.Generally, with respect to swarm dynamics theory, there are twoextreme types of cooperative behavior:

(a) Egotistic behavior is entirely pro-self.(b) Altruistic behavior is entirely pro-group.

Egotistic behavior: every single sophisticated seeker is uni-formly egotistic, and he should go towards his historical best

STATCOM-based damping stabilizer.

u

δ-2δ-3δ 3δ2δδ0 αij

μij

Bell function curve

Fig. 5. The action part of the fuzzy reasoning.


position according to his own opinion. An empirical direction vec-tor di,ego(t), which is presented as follows, may implement thistreatment for the ith seeker

di;egoðtÞ ¼ signðpi;bestðtÞ � xiðtÞÞ ð9Þ

where the pi,best(t), xi(t) and sign(�) are Historical best position, posi-tion for each ith seeker at each time step t and a signum function.

Altruistic behavior is divided into two parts:

(1) Pro-group behavior: in the behavior, seekers want to com-municate with each other, cooperate and adjust their behav-iors in response to other seeker in the same neighborhoodregion to achieve the desired goal.

(2) Self-organized aggregation behavior: In the behavior, posi-tive feedback usually takes the form of attraction toward agiven signal source.

Both optional altruistic behaviors may be implemented asfollows:

di;alt1ðtÞ ¼ signðgbestðtÞ � xiðtÞÞ ð10Þ

di;alt2ðtÞ ¼ signðIbestðtÞ � xiðtÞÞ ð11Þ

where gbest(t) and Ibest(t) show the neighbors’ historical best positionand the neighbors’ current best position, respectively.

Pro-activeness behavior: seekers enjoy the properties of pro-activeness; seekers do not simply act in response to their environ-ment; they are able to exhibit goal-directed behavior. In addition,the future behavior can be predicted and guided by the past behav-ior. As a result, the seeker may be pro-active to change his searchdirection and exhibit goal-directed behavior according to his pastbehavior. Hence, each seeker is associated with an empirical direc-tion called as pro-activeness direction as follows:

di;proðtÞ ¼ signðxiðt1Þ � xiðt2ÞÞ ð12Þ

where t1, t2 e {t, t � 1, t � 2}, and it is supposed that xi(t1) is betterthan xi(t2) according to a cost function value. The four mentionedempirical directions are presented in Eqs. (9)–(12) direct seekersto execute rational decisions in their respective search directions.

If the jth variable of the ith seeker lies to either the right or leftdirection of the coordinate axis, the value of dij(t) is taken as +1 or�1, respectively. If the jth variable of the ith seeker is placed at thecurrent position, the value of dij(t) becomes 0. A proportional selec-tion rule is applied to select each variable j of di(t) as follows

diðtÞ ¼

0 if rj 6 pð0Þj

þ1 if pð0Þj 6 rj 6 pð0Þj þ pðþ1Þj

�1 if pð0Þj þ pðþ1Þj 6 rj 6 1

8>><>>: ð13Þ

where rj is a uniform random number in [0,1] andpðmÞj ðm 2 0;þ1;�1Þ is the percent of the numbers of ‘m’ from theset {dij,ego,dij,alt1,dij,alt2,dij,pro} for each variable j of all four empiricaldirections; i.e. pðmÞj = (the number of m)/4. For example, value of mis considered equal 0, and numbers of 0 from the set {dij,ego = 0,dij, alt1 = 1,dij,alt2 = �1,dij,pro = 1} is calculated. It indicates that mequal 2. Finally, pð0Þj ¼ ð2=4Þ ¼ 0:5.

3.1.3. Calculation of step lengthDifferent optimization problems often have different ranges of

fitness values. To design a Fuzzy system to be applicable to a widerange of optimization problems, the fitness values of all the seekersare descendingly sorted and turned into the sequence numbersfrom 1 to S as the inputs of Fuzzy reasoning. The linear member-ship function is used in the conditional part since the universe of

discourse is a given set of numbers. The expression is presentedas follows:

l ¼ lmax �S� Ii

S� 1ðlmax � lminÞ ð14Þ

where Ii and lmax are the sequence number of xi(t) after sorting thefitness values and the maximum membership degree value that isequal to or a slightly less than 1.0, respectively. In this paper the va-lue of lmax is equal 0.95. The Bell membership function, consideredas lðxÞ ¼ e�x2=2d2

(shown in Fig. 5), is applied to present the actionpart of the fuzzy system. The membership degree values of the in-put variables beyond [�3d, +3d] are less than 0.0111(l(±3d) = 0.0111).

The minimum value of l is tuned 0.0111. Moreover, the param-eter d of the Bell membership function shown as follows:

d ¼ x� absðxbest � xrandÞ ð15Þ

where x is used to decrease the step length while increasing thetime step to gradually improve the search accuracy, as it linearlydeclines from 0.9 to 0.1 during the run. In Eq. (15), both the bestseeker xbest and a randomly selected seeker xrand can be selectedfrom the same subpopulation, which is related to the ith seeker.The parameter d is shared by all seekers within the same subpopu-lation. In order to introduce randomness into each variable and toenhance the local search ability, the following equation convertsli into a vector li with elements given by:

lij ¼ RANDðli;1Þ ð16Þ

where RAND(li, 1) presents a uniformly random real number within[li,1].

The action part of the fuzzy reasoning approach can be repre-sented by Eq. (17), and the step length (aij) for each variable jcan be calculated as follows:

aij ¼ dj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� lnðlijÞ

qð17Þ

3.1.4. Updating the position of a seekerThe position of the ith seeker (1 6 i 6 S) for the jth variable is

updated as follows:

xijðt þ 1Þ ¼ xijðtÞ þ aijðtÞ � dijðtÞ ð18Þ

3.1.5. Subpopulations learn from each otherEach subpopulation is searching for the optimal solution using

its own information. It hints that the subpopulation may trap intolocal optima yielding a premature convergence. Subpopulationsmust learn from each other about the optimum information sofar they have acquired in their respective domain. Thus, the posi-tion of the worst seeker of each subpopulation is combined withthe best one in each of the other subpopulations using the follow-ing binomial crossover operator as expressed

xknj;worst ¼xlj;bestt if randj 6 0:5xknj;worst else

�ð19Þ

Specify the parameters for GA

start


where randj is a uniformly random real number within [0,1],xknj;worst is denoted as the jth variable of the nth worst position inthe kth subpopulation, and xlj,best is the jth variable of the bestposition in the lth subpopulation so that, n; k; l ¼ 1;2; . . . ;

K � 1; and k–l. To increase the diversity in the population, thesuitable information obtained by each subpopulation is sharedamong the other subpopulations. The flowchart of the algorithmis shown in Fig. 6.

Generate initial population

Find the fitness of each individual

Time-domain simulation

Gen=1
3.2. Genetic algorithm (GA)
The GA has been used for optimizing the parameters of controlsystem that are complex and difficult to solve by conventionaloptimization methods. GA maintains a set of candidate solutionscalled population and repeatedly modifies them. At each step, the

Decide the personal best position, neighborhood best position and population best position

Start

End

Output the results

t = t+1

Initialize s seekers positions

Divide the population into K subpopulation randomly

Calculate the objective function value of each seeker

Subpopulations learn from each other using (19)

Update the position of each seeker

Yes

No

t = 0

Compute search direction for each seeker using (13)

Compute step length for each seeker using (17)

Calculate the objective function value of each seeker

Update the personal best position, neighborhood best position and population best position

Is stopping condition satisfied?

Fig. 6. The flowchart of SOA method.

in the current population

Yes

Gen.>Max.Gen.? Stop

No

Apply GA operators: Selection, Crossover and Mutation

Gen = Gen+1

Fig. 7. The flowchart of Genetic algorithm.

GA selects individuals from the current population to be parentsand uses them produce the children for the next generation. Candi-date solutions are usually represented as strings of fixed length,called chromosomes. An objective function is used to reflect thegoodness of each member of population. Given a random initialpopulation GA operates in cycles called generations, as follows [19]:

(1) Each member of the population is evaluated using a fitnessfunction.

(2) The population undergoes reproduction in a number of iter-ations. One or more parents are chosen stochastically, butstrings with higher fitness values have higher probabilityof contributing an offspring.

(3) Genetic operators, such as crossover and mutation areapplied to parents to produce offspring.

(4) The offspring are inserted into the population and the pro-cess is repeated.

The computational flow chart of GA is shown in Fig. 7.

4. Proposed objective function

The parameters of the STATCOM controllers and PSS are opti-mized so that both the speed deviation of the generators and theSTATCOM bus voltage deviation are reduced to enhance the powersystem performance. For this purpose, the coordinated design ofthe STATCOM controllers and PSS parameters can be convertedinto an optimization problem. Various objective functions havebeen considered in the previous studies [2,20,11] on the coordi-nated design of FACTS devices controllers and PSS. In this paper,The Integral of Time multiplied Absolute Error (ITAE) criterion isused as the objective function for coordinated design [21,22]. Theoverall goal is to minimize the objective function subject withrespect to several constraints. These constraints illustrate theupper and lower limits of each parameter. The constrained optimi-zation problem is formulated as follows [20]:

Set the SOA parameters and limits of each variable

Start

t = t+1

Initialize the positions of the seekers and t=0

Run the simulation file for different loading condition

Calculate the objective function based on Eq. (20)

Determine search direction, step length, and the neighbors for each seeker

Update the position of each seeker


Min : F ¼XNP

l¼1

XNF

j¼1

Al �Z t2¼tsim

t1¼0al

Xn

i¼1

jDxij þ bljDVSj" #

t dt

subject to :

KminPSS;i 6 KPSS;i 6 Kmax

PSS;i; Tmin1;i 6 T1;i 6 Tmax

1;i ð20Þ

Tmin2;i 6 T2;i 6 Tmax

2;i ; Tmin3;i 6 T3;i 6 Tmin

3;i

Tmin4;i 6 T4;i 6 Tmax

4;i ; Kminpt 6 Kpt 6 Kmax

pt

Kminit 6 Kit 6 Kmax

it ; Kminpc 6 Kpc 6 Kmax

pc

Kminic 6 Kic 6 Kmax

ic ; KminS 6 KS 6 Kmax

S

Kmin1S 6 K1S 6 Kmax

1S ; Kmin3S 6 K3S 6 Kmax

3S

where NP and NF are the number of operating points and faults forwhich the optimization is carried out, respectively. n is the numberof generators of the multi-machine power system. Dxi and DVS arethe speed deviation of the ith machine and the voltage deviation ofthe STATCOM bus, respectively. a and b are weighting factors. t1 andt2 are the start and end time of the simulation, respectively, and t isthe simulation time in seconds. Limits are placed on the STATCOM-based stabilizer and the PI-controllers of the AC and DC voltageloops and PSS parameters to maintain the system within the stabil-ity margin.

The optimization of coordination design between the STATCOMcontrollers and PSS parameters is done by assessing the objectivefunction as given in Eq. (20) that a multiple of the operating condi-tions and faults are considered.

End

Show the optimized parameters of coordinated design

Calculate the fitness value of the new positions

Meet the stopping criterion ?

Yes

No

Fig. 8. The coordinated design flowchart based on the proposed algorithm.

Table 1Cost function determined by coordinating design problem using SOA method byconsidering variation of parameters.

5. Implementation of SOA for the coordinated design

The coordinated design optimization based on the SOA methodis presented in the follow.

Step (1) Read the parameters of the proposed algorithm, andspecify the lower and upper limits of each variable.

Step (2) Randomly and uniformly initialize the positions of theseekers in the search space. Set the time step t = 0.

Step (3) Calculate the fitness values of the initial positions usingthe objective function in (20), which is based on theresults of the time-domain analysis. The time-domainsimulation results are computed for large and smallfaults and different operating points. The initial histor-ical best position in the population is attained. Set thehistorical best position of each seeker to his currentposition.

Step (4) Let t = t + 1.Step (5) Determine the search direction (Section 3.1.1), the step

length (Section 3.1.2) and the neighbors for eachseeker.

Step (6) Update the position of each seeker using Eq. (18).Step (7) Calculate the fitness values of the new positions using

the objective function based on the results of the time-domain analysis. Update the historical best positionamong the population and the historical best positionof each seeker.

Step (8) Go to Step 4 until a stopping criterion is satisfied.

Fig. 8 illustrates a flowchart of the proposed coordinated designof the STATCOM controllers and PSS by the SOA method.

Parameter Different sizes of parameters

S Value 20 40 50 70 90Cost function (F) 45.09 35.53 30.42 30.40 30.35

K Value 1 2 3 4 5Cost function (F) 34.02 32.17 30.42 30.42 30.42

6. Simulation results

To validate the proposed method, the simulations are carriedout on a two-area Kundur and 39-bus New England power

systems. The parameters of the excitation system and the STAT-COM model are given in the appendix. The ranges considered forthe PSS parameters of the multi-machine system are given in [5].The SOA method parameters are S = 50 and Itr = 80. The GA methodparameters are Pop = 50 and Itr = 80, Crossover rate = 0.6 andMutation rate = 0.01. In this study, trial and error method is usedto determine parameter settings of the SOA method. Furthermore,robustness of the results of the SOA method against change in the

Table 4Parameters of STATCOM controllers and PSS by SOA method for two-area powersystem.

Machine Kpss T1 T2 T3 T4

G1 8.1384 1.555 0.3382 3.0124 8.25G2 17.7442 0.005 0.4518 3.0524 4.7097G3 26.6819 0.4176 0.419 4.7109 3.1388G4 25.6080 0.0984 0.2056 7.7185 5.287

STATCOM PI-controller Kpt Kit Kpc Kic –23.198 2328.379 1.8e�5 0.0182 –

Stabilizer KS T1S T2S T3S T4S

33.7662 0.8615 0.5 0.7942 0.5


values of each parameter should be evaluated to determine howrobust the algorithm is with respect each parameter. For this pur-pose, the value of cost function for two-area power system deter-mined. The simulation results of power system for increment anddecrement of each parameter is presented in Table 1. As shownin this table, the performance of the SOA method is robust respectto select the mentioned parameters of SOA method.

where K is the number of subpopulations.In majority of previous literatures, T1 and T3 are usually vari-

ables to be optimized. In this paper, T2 and T4 are considered inthe optimization problem to increase complexity of optimizationproblem. Limits of PSS parameters are shown in Table 2.

Table 5Commutation time for SOA and GA for the two-area power system.

Time (s)

SOA GA

204.25 406.39

6.1. Example 1: Two-area 11-bus Kundur power system

Fig. 2 shows a two-area four-machine power system with aSTATCOM of 100 MVA installed at bus 8 [11]. The nominal systemparameters are given in [1]. In simulations, three loading condi-tions are considered which are listed in Table 3.

To obtain the parameters of STATCOM controllers and PSSs,optimization is performed by subjecting the power systems tosmall and large disturbances. In this paper, a temporary transmis-sion line outage measuring 200 ms in duration is placed (on one ofthe tie lines), and when the system returns to steady state, a three-phase short circuit of 200 ms in duration is applied at the middle ofthe tie lines. Consequently, the parameters for the STATCOM con-trollers and PSSs are obtained such that the value of F is minimizedand the transient and dynamic stabilities of the power system areimproved. The obtained parameters of the STATCOM controllersand PSS using by SOA method are illustrated in Table 4. It is impor-tant to note that computation time of the two-area power systemshow in Table 5. It can be clearly seen that the SOA method takesleast amount of time in its row to finish iterations. Thus, systemsemploying SOA method can save considerable amount of timeand therefore are feasible for online optimization with high speedprocessors.

To show simulation results, the application of a 200 ms three-phase short circuit at t = 1 s at bus 8 is followed immediately bya 200 ms line outage between buses 8 and 9, as shown in Fig. 2.The performance of the SOA method is compared with simulationresults of GA method and the uncoordinated design. The simula-tion results for three operating points are shown as follows:

6.1.1. Case 1Performance of the proposed simultaneous coordinated design

approach is examined for the first operating point in the Table 2.The inter-area and the local oscillation modes using the coordi-

Table 2Parameter limits of PSS used in optimization.

Two-area power system 39-bus new England

5 6 K 6 30 10 6 K 6 2000.005 6 T1 6 2 0.005 6 T1 6 10.001 6 T2 6 1 0.001 6 T2 6 10.01 6 T3 6 10 0.01 6 T3 6 100.005 6 T4 6 15 0.05 6 T4 6 1

Table 3Operating condition of two area power system (pu).

Operating condition P1 Q1 P2

Case 1 0.7767 0.1524 0.7659Case 2 0.7323 0.1711 0.7183Case 3 0.8268 0.1297 0.8199

nated and uncoordinated designs of the controllers are shown inFigs. 9a and b and 10a and b, respectively. It is clear from these fig-ures that simultaneous design of the STATCOM controllers and PSSusing the proposed approach significantly improves the stabilityperformance of the example power system, and the low frequencyoscillations are significantly damped in comparison to the resultsof the other methods. In other words, the coordination design im-proved the damping in the system oscillations. The response ofpower system clearly shows that the responses of SOA and GAare comparable. SOA optimized STATCOM controllers and PSSs ex-hibit better damping than GA uncoordinated design, in turn, exhi-bit better damping than uncoordinated design. Fig. 11a is the rotorangle deviation of machine-1 with respect to reference machine-3,and Fig. 11b is the rotor angle deviation of machine-1 with respectto reference machine-2. The results show that the system isswitching back to steady state from the state at which the systemmoves during violent swings.

6.1.2. Case 2The coordination design between the STATCOM controllers and

PSS to evaluate the performance of the multi-machine power sys-tem is tested for the second operating point. The inter-area and lo-cal modes of the oscillations using the proposed method areillustrated in Figs. 12a and b and 13a and b, respectively. It canbe clearly seen that the simultaneous coordination design betweenthe STATCOM controllers and PSS using by SOA method signifi-cantly enhances the stability performance of the power systemwith respect to either that produced for coordinated design usingthe GA method or that of the uncoordinated design. Fig. 14a andb are the rotor angle deviations of machine-1 with respect to refer-ences machine-3 and machine-2, respectively.

6.1.3. Case 3Performance of the power system for the third operating point

is verified for the coordinated design. The inter-area and localmodes of the oscillations using the proposed method are shown

Q2 P3 Q3 P4 Q4

0.1799 0.8163 0.1090 0.7912 0.11190.1888 0.7679 0.1337 0.7404 0.12660.1689 0.8712 0.0793 0.8492 0.0936

0 1 2 3 4 5 6 7 8 9 10-6

-4

-2

0

2

4

6

8 x 10-3

Time (sec)Spee

d D

evia

tion

of W

1-W

3 (r

ad/s

ec)

Co-ordinated STATCOM and PSS by SOACo-ordinated STATCOM and PSS by GAUn-coordinated STATCOM and PSS

0 1 2 3 4 5 6 7 8 9 10-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5 x 10-3

Time (sec)

Spee

d D

evia

tion

of W

1-W

2 (r

ad/s

ec)


(a) (b)Fig. 9. (a) Inter-area mode (W1 �W3) of oscillations for case 1. (b) Local mode (W1 �W2) of oscillations for case 1.

0 1 2 3 4 5 6 7 8 9 10-8

-6

-4

-2

0

2

4

6

8 x 10-3

Time (sec)

Spee

d D

evia

tion

of W

1-W

4 (r

ad/s

ec)


0 1 2 3 4 5 6 7 8 9 10-1.5

-1

-0.5

0

0.5

1

1.5

2 x 10-3

Time (sec)

Spee

d D

evia

tion

W3-

W4

(rad

/sec

)Co-ordinated STATCOM and PSS by SOACo-ordinated STATCOM and PSS by GAUn-coordinated STATCOM and PSS


0 2 4 6 8 10-10

0

10

20

30

40

50

60

70

Time (sec)

Rot

or a

ngle

of G

1-G

3

0 2 4 6 8 100

5

10

15

20

Time (sec)

Rot

or a

ngle

of G

1-G

2

(a) (b)Fig. 11. (a) Rotor angle difference between G1 and G3. (b) G1 and G2 for case 1 (SOA method).


in Figs. 15a and b and 16a and b, respectively. The simultaneouscoordinated design of the STATCOM controllers and PSS using theSOA method improves the transient and dynamic stability perfor-mance of the power system. It goes without saying that, the coor-dinated design improve the damping in the system; a systemhaving coordinated design by SOA and GA methods show betterdamping than the uncoordinated design. Damping is best with sys-tems having coordinated design by SOA method followed by GAmethod and uncoordinated design. The speed responses for SOA

method have a settling time faster than the GA method. In addi-tion, Fig. 17a and b are the rotor angle deviations of machine-1with respect to references machine-3 and machine-2, respectively.

6.1.4. Analysis of the damping performanceThe performance and robustness of the proposed method are

evaluated using two performance indices, including the ITAE andFigure of Demerit (FD), which are defined based on the system per-formance characteristics as follows [23]:

0 1 2 3 4 5 6 7 8 9 10-4

-2

0

2

4

6

8 x 10-3

Time (sec)

Spee

d de

viat

ion

of W

1-W

3 (r

ad/s

ec)


0 1 2 3 4 5 6 7 8 9 10-2.5

-2-1.5

-1-0.5

00.5

11.5

22.5 x 10-3

Time (sec)

Spee

d D

evia

tion

of W

1-W

2 (r

ad/s

ec)



0 1 2 3 4 5 6 7 8 9 10-8

-6

-4

-2

0

2

4

6

8 x 10-3

Time (sec)

Spee

d de

viat

ion

of W

1-W

4 (r

ad/s

ec)


0 1 2 3 4 5 6 7 8 9 10-2

-1.5

-1

-0.5

0

0.5

1

1.5x 10-3

Time (sec)

Spee

d de

viat

ion

of W

3-W

4 (r

ad/s

ec)



0 2 4 6 8 10-10

0

10

20

30

40

50

60

Time (sec)

Rot

or a

ngle

of G

1-G

3

0 2 4 6 8 10-5

0

5

10

15

20

Time (sec)

Rot

or a

ngle

of G

1-G

2



ITAE ¼ 1000Z t2¼tsim

t1¼0t � ðjw1 �w2j þ jw1 �w3j þ jw1 �w4j

þ jw3 �w4j þ jDVsjÞdt ð21Þ

FD ¼X4

i¼1

ðOSi � 4000Þ2 þ ðUSi � 4000Þ2 þ TS2i ð22Þ

where w, t, OS, S and TS are the rotor speed, simulation time, over-shoot, undershoot and settling time to evaluate the ITAE and FDindices. Lower values of these indices illustrate enhanced systemresponse in terms of the time-domain characteristics. The numeri-cal results of the performance and robustness for all system loadingcases and the ITAE and FD indices are shown in Figs. 18 and 19,respectively. Application of the proposed simultaneous coordinated

0 1 2 3 4 5 6 7 8 9 10-6

-4

-2

0

2

4

6

8 x 10-3

Time (sec)

Spee

d D

evia

tion

of W

1-W

3 (r

ad/s

ec)


0 1 2 3 4 5 6 7 8 9 10-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5 x 10-3

Time (sec)

Spee

d D

evia

tion

of W

1-W

2 (r

ad/s

ec)


(a) (b)Fig. 15. (a) Inter-area mode (W1 �W3) of oscillations for case 3. (b). Local mode (W1 �W2) of oscillations for case 3.

0 1 2 3 4 5 6 7 8 9 10-8

-6

-4

-2

0

2

4

6

8 x 10-3

Time (sec)

Spee

d D

evia

tion

of W

1-W

4 (r

ad/s

ec)


0 1 2 3 4 5 6 7 8 9 10-2

-1.5

-1

-0.5

0

0.5

1

1.5

2 x 10-3

Time (sec)

Spee

d D

evia

tion

of W

3-W

4 (r

ad/s

ec)



0 2 4 6 8 100

10

20

30

40

50

60

Time (sec)

Rot

or a

ngle

of G

1-G

3

0 2 4 6 8 10

0

5

10

15

Time (sec)

Rot

or a

ngle

of G

1-G

2



design of the STATCOM controllers and PSS achieves the best re-sponse in terms of speed deviation, overshoot, undershoot and set-tling time in comparison to the coordinated design with the GAmethod and the uncoordinated design. These figures claim thatthe coordinated design of the STATCOM controllers and PSS usingby SOA method enhances the transient and dynamic stability per-formance of the power system in terms of ITAE and FD indices.

6.1.5. Eigenvalue analysisThe eigenvalue analysis investigates the dynamic behavior of a

power system under different characteristic frequencies (‘‘modes’’)[24]. The eigenvalues indicate the system stability and how is tobecoming unstable. In addition, it is required that all modes arestable in a power system and it is desired that all electromechan-ical oscillations are damped out as quick as possible. In this respect,

0

50

100

150

200

250

Case 1 Case2 Case3

Coordinated by SOA Coordinated by GA Uncoordinated

Fig. 18. Values of performance index ITAE for different operating point.

0

500

1000

1500

2000

2500

3000

3500

4000

4500

Case1 Case2 Case3

Coordinated by SOA Coordinated by GA Uncoordinated

Fig. 19. Values of performance index FD for different operating point.


in the paper, eigenvalue analysis is used to show robustness andimprovement stability of power system using proposed coordi-nated design. The two-area power system has nine electromechan-ical modes. Table 6 list the complex eigenvalues and their dampingratios of all the generators for different load conditions in the two-area power system to coordinated design using by SOA and GAmethods. The eigenvalues generated by a system using coordi-nated design of STATCOM and PSS have the highest negative realpart in that row and thus improve system stability, the best eigen-value of each method is highlighted in all the table. SOA and GAoptimized parameters of STATCOM controllers and PSSs exhibitbest results for the inter-area in different areas for the two-areapower system. Consequently, coordination design of STATCOMand PSSs improve the system dynamic and transient stability ofpower system. It goes without saying, eigenvalue analysis point

Table 6Electromechanical modes and damping ratio of two-area power system.

Type of methods Case 1 Case 2

Eigenvalues Damping ratio (%) Eigenvalu

Co-ordinated by SOA �1.76 ± j 3.59 38.70 �1.56�1.43 ± j 5.53 25.03 3.49–1.73�2.29 ± j 7.51 29.16 5.73–2.19

Co-ordinated by GA �0.82 ± j 4.25 18.94 �1.01�0.95 ± j 6.14 15.29 �1.03�2.16 ± j 7.22 28.66 �2.10

Un-coordinated �0.68 ± j 4.25 15.79 �0.88�0.55 ± j 6.58 8.32 �0.75�0.96 ± j 8.52 11.19 �1.06

out that the SOA method has the better performance in the caseof coordinating STATCOM and PSS than the GA method.

6.2. Example 2: 39-bus New England power system

In this part of the study, the 10-machine 39-bus power systemshown in Fig. 20 is considered further demonstrate the versatilityof the suggested technique. Details of the system data are givenin [14,25]. Generator G1 is an equivalent power source represent-ing parts of the US–Canadian interconnection system. Withoutpower system stabilizers, the system damping is poor and the sys-tem exhibits highly oscillatory response [25]. It is therefore neces-sary to install one or more PSS to improve the dynamicperformance. To identify the optimum locations of PSS’s, the par-ticipation factor method [26] and the sensitivity of PSS effect meth-od [27] were used. The results indicate that machines G5, G7 and G9

are the optimum locations for installing PSS’s to damp out the localand inter-areas modes of oscillations. In addition, location of STAT-COM is selected at bus 28 [28]. To obtain the parameters of STAT-COM controllers and PSSs, optimization is performed by subjectingthe power systems to small and large disturbances. In this paper, atemporary transmission line outage measuring 200 ms in durationis placed (line 4–5), and when the system returns to steady state, athree-phase short circuit of 200 ms in duration is applied at the bus26. The coordination design of STATCOM and PSSs (machines G5, G7

and G9) is demonstrated by considering two operating pointslabeled as nominal and heavy loading conditions which show inTable 7. The parameters of the STATCOM controllers and PSSs usingwith the SOA method are illustrated in Table 8. The computationtime of the 39-bus New England power system shows in Table 9.It can be clearly seen that the SOA method takes least amount oftime in its row to finish iterations.

6.2.1. Case 1Performance of the proposed simultaneous design approach is

investigated for the nominal condition in Table 6. A three-phasefault on bus 26 at t = 1 s to t = 1.2 s is considered. Speed deviationof generators G5, G7 and G9 are shown in the Figs. 21–23 usingcoordinated design of STATCOM and PSS by SOA and GA methods,respectively. It can be clear seen that simultaneous design of theSTATCOM controllers and PSS using the proposed approach mean-ingfully enhances the stability performance of the power system incomparison to the results of the GA method.

6.2.2. Case 2A three-phase short circuit of 100 ms at t = 1 is applied at bus 26

for heavy load condition. The speed responses of generators G5, G7

and G9 for the aforementioned cases are shown in Figs. 24–26,respectively. It is clear from these figures, once again, that the coor-dinated design of STATCOM and PSSs improves the damping in the

Case 3

es Damping ratio (%) Eigenvalues Damping ratio (%)

± j 40.80 �1.86 ± j 3.97 42.42± j 28.90 �1.23 ± j 5.13 23.31± j 7.11 29.43 �2.79 ± j 7.24 35.95± j 3.56 27.29 �1.21 ± j 3.26 34.79± j 5.43 18.63 �0.98 ± j 4.97 19.38± j 7.78 26.05 �2.40 ± j 7.03 32.30± j 4.13 20.83 �0.78 ± j 4.24 18.09± j 6.19 12.02 �0.95 ± j 6.69 14.05± j 8.27 12.71 �1.26 ± j 8.87 14.06

Fig. 20. Ten-generator 39-bus new-England power system.

Table 7Operating condition of 39-bus power system.

Operating condition Active power Reactive power

Case 1 (nominalcondition)

Nominal active power Nominal reactive power

Case 2 (heavycondition)

Total active powerincreasing by 50%

Total active powerincreasing by 40%

Table 8Parameters of STATCOM controllers and PSS by SOA method.

Machine Kpss T1 T2 T3 T4

G5 178.023 0.0801 0.0217 0.0740 0.0292G7 190.324 0.0894 0.0228 0.2653 0.0248G9 169.6610 0.0783 0.0247 0.0794 0.0223

STATCOM PI-controller Kpt Kit Kpc Kic –14.452 2235.573 10.9e�5 0.0391 –

Stabilizer KS T1S T2S T3S T4S

43.4831 0.7640 0.5 0.6784 0.5

Table 9Commutation time for SOA and GA for the 39-bus NewEngland power system.

Time (s)

SOA GA

404.25 706.39

0 5 10 15-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3 x 10-3

Time (sec)

Spee

d D

evia

tion

of W

5 (r

ad/s

ec) Co-ordinated STATCOM and PSS by SOA

Co-ordinated STATCOM and PSS by GA

Fig. 21. Speed deviation of G5 for case 1.


system. In addition, damping of power system using by SOA meth-od is better than the GA method.

6.2.3. Analysis of the damping performanceTwo performance indices, including the ITAE and Figure of De-

merit (FD), are defined based on the system performance charac-teristics as follows:

ITAE ¼ 1000Z t2¼tsim

t1¼0t �X10

i¼1

jwi � 1jdt ð23Þ
FD ¼X10
i¼1

ðOSi � 4000Þ2 þ ðUSi � 4000Þ2 þ TS2i ð24Þ

0 5 10 15-4

-2

0

2

4

6

8 x 10-3

Time (sec)

Spee

d D

evia

tion

of w

7 (r

ad/s




0 2 4 6 8 10 12 14 16 18 20-3

-2

-1

0

1

2

3

4 x 10-3

Time (sec)

Spee

d D

evia

tion

of W

9 (r

ad/s




0 5 10 15-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5 x 10-3

Time (sec)

Spee

d D

evia

tion

of W

5 (r

ad/s




0 5 10 15-3

-2

-1

0

1

2

3 x 10-3

Time (sec)

Spee

d D

evia

tion

of W

7 (r

ad/s




0 5 10 15-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2 x 10-3

Time (sec)

Spee

d D

evia

tion

of W

9 (r

ad/s




0

100

200

300

400

500

600

Case1 Case2

Coordinated by SOA Coordinated by GA

Fig. 27. Values of performance index ITAE for different operating point.


The numerical results of the performance and robustness for allsystem loading cases and the ITAE and FD indices are shown in Figs.27 and 28, respectively. It can be clearly seen that application ofthe proposed simultaneous coordination design of the STATCOMcontrollers and PSS achieves the best response in comparison withGA method. The indexes ITAE and FD for different operating loadingcondition using SOA method achieved less values in comparisonGA method. Consequently, overshoot, undershoot and settling timeof generators speed gained less amount by proposed method.

6.2.4. Eigenvalue analysisThe Eigenvalue analysis investigates the dynamic behavior of a

39-bus New England power system under different loadingconditions. The 39-bus New England power system has eighteenelectromechanical modes. Tables 10 and 11 present the complexeigenvalues and their damping ratios of all the generators for dif-ferent load conditions for coordination design using by SOA andGA methods. The eigenvalues of power system using coordinateddesign of STATCOM and PSS by SOA method have the highest neg-

10200

10400

10600

10800

11000

11200

11400

11600

11800

12000

Case1 Case2

Coordinated by SOA Coordinated by GA

Fig. 28. Values of performance index FD for different operating point.

Table 10Electromechanical modes and damping ratio 39-bus New England power system fornominal condition.

Co-ordinated by SOA Co-ordinated by GA

Eigenvalues Dampingratio (%)


�0.221 ± j 9.643 2.29 �0.146 ± j 9.934 1.46�0.522 ± j 9.235 5.64 �0.198 ± j 9.832 2.01�0.562 ± j 8.56 6.55 �0.259 ± j 9.476 2.73�0.654 ± j 8.143 8.00 �0.306 ± j 8.453 3.61�0.371 ± j 8.245 4.49 �0.164 ± j 8.295 1.97�0.502 ± j 7.120 7.03 �0.206 ± j 7.623 2.70�0.715 ± j 5.238 13.52 �0.321 ± j 6.867 4.66�3.046 ± j 3.340 67.38 �1.047 ± j 6.542 15.80�0.552 ± j 3.219 16.90 �0.1342 ± j 4.243 3.16

Table 11Electromechanical modes and damping ratio 39-bus New England power system forheavy condition.

Co-ordinated by SOA Co-ordinated by GA



�0.2423 ± j 9.789 2.47 �0.147 ± j 9.842 1.49�0.7012 ± j 9.234 7.57 �0.227 ± j 9.853 2.30�0.4523 ± j 8.472 5.33 �0.231 ± j 9.983 2.31�0.3834 ± j 8.385 4.56 �0.197 ± j 8.528 2.30�0.6356 ± j 8.197 7.73 �0.296 ± j 8.372 3.53�0.7023 ± j 7.309 9.56 �0.573 ± j 7.521 7.59�0.6124 ± j 6.043 10.08 �0.492 ± j 6.502 7.54�2.5721 ± j 3.990 54.18 �1.045 ± j 6.322 16.30�0.5523 ± j 3.452 15.79 �0.295 ± j 4.493 6.55


ative real part in that row and thus improve system stability thanthe GA method. In addition, the best eigenvalue of each method ishighlighted in all the tables. Consequently, coordinated design ofSTATCOM and PSSs improve the system dynamic and transient sta-bility of power system. It is obvious that, eigenvalue analysis indi-cate that the SOA method has the better performance in the case ofcoordinating STATCOM and PSS than the GA method.

Generally, inter-area oscillation is a typical phenomenon inlarge power systems. As systems are getting larger either by grow-ing or by interconnecting weakly damped inter-area oscillationsmay emerge. System dynamic and stability can be improved by(a) modulating power at the power plant by means of PSS and(b) affecting the power flow in the system by FACTS devices suchas STATCOM with their ability to change the power flow directlyor indirectly to make better use of the installed capacity. This paper

shows that the performance of FACTS devices and PSS simulta-neously to improve stability of power system.

7. Conclusions

In this paper, coordinated design of STATCOM controllers andPSSs is performed to enhance power system stability in a multi-machine power system. In this coordination design, a STATCOM-based damping stabilizer is added to the AC voltage loop. Mean-while, PI-controllers for the AC and DC voltage loops of STATCOMare considered. The problems associated with tuning the STATCOMcontrollers and PSS for a given set of operating points are convertedto an optimization problem. It is solved with a new intelligentalgorithm called the SOA method. The SOA method is successfullyapplied to find the optimal parameters of the coordinated design.The productivity of the proposed method is examined for a two-area and 39-bus power systems through the use of time-domainsimulation studies under different operating conditions. The simu-lation results illustrate the effectiveness of coordinated design toincrease the damping of low frequency oscillations. Additionally,the performance characteristics of the power systems are investi-gated in terms of the ITAE and FD indices and the eigenvalue anal-ysis. These demonstrate that the simultaneous coordinated designof STATCOM controllers and PSSs using the SOA method is superiorto the coordinated design with the GA method and the uncoordi-nated design.

Appendix A.

A.1. STATCOM data

500 kV; Rp ¼ 0:071; Lp ¼ 0:22; VDC ¼ 40 kV;CDC ¼ 3757 Micro-F; V ref ¼ 1:0 pu

A.2. Excitation system two-area power system

TA ¼ 0:05; KA ¼ 200; TW ¼ 10:

uSmax ¼ 0:15 pu; uSmin ¼ �0:15 pu; sensor time constant

¼ 0:015:

A.3. Excitation system 39-bus power system

TA ¼ 0:05; KA ¼ 400; TW ¼ 10:

uSmax ¼ 0:15 pu; uSmin ¼ �0:15 pu; sensor time constant

¼ 0:015:

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Electrical Power and Energy Systemsrms.scu.ac.ir/Files/Articles/Journals/Abstract/1-s2.0-S... · ture of the lead-lag power system stabilizer is shown in Fig. 1 [14]. 2.2. STATCOM