damping of low frequency oscillations of multi-machine multi-upfc power systems, based on adaptive...

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 27, NO. 4, NOVEMBER 2012 1831

Damping of Low Frequency Oscillationsof Multi-Machine Multi-UPFC Power Systems,Based on Adaptive Input-Output Feedback

Linearization ControlShahrokh Shojaeian, Jafar Soltani, Member, IEEE, and Gholamreza Arab Markadeh

Abstract—In this paper, damping of the low frequency oscilla-tions of multi-machine multi-UPFC power systems is investigatedbased on adaptive input-output feedback linearization control(AIFLC) approach. Considering a three-phase symmetrical fault,ignoring the subtransient states of the synchronous machines, thenonlinear state equations of the system are derived in order toobtain the UPFC reference control signals as well as the systemparameters estimation laws. The stability of the system controlleris proved by Lyapunov theory. Moreover using the six reducedorder model of synchronous machine, some simulation results arepresented in order to verify the validity and effectiveness of theproposed control approach.

Index Terms—Low frequency oscillations, nonlinear control,power system stability, UPFC.

I. INTRODUCTION

I N recent years, the fast progress in the field of power elec-tronics has opened new opportunities for the power industry

via utilization of the controllable FACTS devices such as unifiedpower flow controller (UPFC), thyristor controlled series capac-itor (TCSC), and static VAR compensator (SVC) as alternativemeans to mitigate power system oscillations [1]–[3]. Becauseof the extremely fast control action associated with FACTS-de-vice operations, they are promising candidates for mitigationpower system oscillation and improving power system steadystate performance [4], [5]. UPFC, regarded as one of the mostversatile ones in the FACTS device family [6], [7], has the ca-pabilities of controlling power flow in the transmission line, im-proving the transient stability, mitigating system oscillation, andproviding voltage support. It performs this through the controlof the in-phase voltage, quadrate voltage, and shunts compen-sation due to its main control strategy [7], [8].

Manuscript received May 20, 2011; revised September 09, 2011, December12, 2011, January 25, 2012 and March 03, 2012; accepted April 01, 2012. Dateof publication May 15, 2012; date of current version October 17, 2012. Paperno. TPWRS-00471-2011.S. Shojaeian is with the Department of Electrical Engineering, Science

and Research Branch, Islamic Azad University, Tehran, Iran (e-mail: [email protected]).J. Soltani is with the Department of Electrical Engineering, Islamic Azad

University, Khomeinishahr Branch, and also with the Faculty of Electrical andComputer Engineering, Isfahan University of Technology, Isfahan, Iran (e-mail:[email protected]; [email protected]).G. Arab Markadeh is with the Department of Engineering, Shahrekord Uni-

versity, Shahrekord, Iran (e-mail: [email protected]).Digital Object Identifier 10.1109/TPWRS.2012.2194313

Several investigations on UPFC main control effects showthat UPFC can improve system transient stability and enhancethe system transfer limit as well. The application of UPFC tothe modern power system can therefore lead to a more flexible,secure, and economic operation [9].FACTS devices and their power system applications are de-

scribed in [10]–[12]. From the view point of power system dy-namics, the essential problem is how to control specific FACTSdevices, and in particular UPFC. For example, one approach isto apply optimal control [13]–[16]. The problem with standardoptimal control is that it tends to use a linearized system model,which is valid only for a given operating point. This raises thequestion of robustness, as control based on linearized systemmodel is valid only when the system is in the vicinity of thechosen operating point and one can never be sure whether ornot the control will still be satisfactory when the system oper-ating conditions change or when the system model changes dueto line or generator outages. Moreover, stressed power systemsare known to exhibit nonlinear behavior. Hence the motivationfor the work reported so far is to derive a state-variable controlusing a nonlinear system model in order to take into account theinfluence of changing operating conditions and changes in thenetwork parameters.In the past two decades, in order to solve the low frequency

oscillation problems of the modern power systems which in-clude the FACTS devices, some authors have tried to apply thenonlinear control methods to these systems [16]–[26].In [16], a nonlinear coordinated generator excitation is pro-

posed to enhance the transient stability of a simple power systemwhich is connected to an infinite bus via two parallel transmis-sion lines. For this system, the input-output feedback lineariza-tion control (IFLC) approach is used first in order to achievethe nonlinear offline stability estimator as well as to obtain thesystem fourth order linearized model.According to the linear quadratic (LQ) optimal control tech-

nique, a linear Luenberger observer is used to detect the di-rect axis transient reactance and the inertia constant of the syn-chronous machine. These estimated parameters are used by theIFLC controller. In control method of [16], the control and pa-rameters estimation laws are obtained independently from twodifferent approaches. It means that at each step of time inseconds, the system parameters are detected by LQ control tech-nique, and are used by IFL controller. Such parallel proceduremay cause the overall system to become unstable especiallyduring the system faults and load disturbances. One may note

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1832 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 27, NO. 4, NOVEMBER 2012

that in the control method described in [16], the parameters es-timation law is obtained by a linear observer, which does nottake into account the system nonlinearities.In [24], a nonlinear variable-gain fuzzy controller is applied

to a typical two-area power system, with a UPFC in order to en-hance the transient stability and damp the low frequency oscilla-tions of this network. The fuzzy controller uses a numerical con-sequent rule base of Takagi-Sugeno type, which can be eitherlinear or nonlinear producing control gain variation over a verywide range. This type of fuzzy control could be more robust andeffective compared to conventional PI regulators. Although thefuzzy controller of [24] could be robust against the slow changesthat occur in the system steady state operating points, it cannotguarantee the robustness and stability of the system subjectinguncertain parameters and load disturbances.In [25], considering the second order classical model of the

synchronous machine, the IFLC method is used for a powersystem which includes the series and shunt FACTS devices.These devices are interfaced with the bulk power systemthrough injection buses. Having controlled the system busvoltage vector, the low frequency oscillations of the systemcan be effectively damped. In [25], the coherency identifica-tion method has been achieved based on modal analysis andGaussian elimination with full pivoting on the selected eigen-vectors of the system applied in order to obtain the referencegenerators and their group members. In addition, in [25], theimpact of the system parameters uncertainties and the systemload disturbances has not been taken into account.In [26], the IFLC technique is applied to a multi-machine

power system with a UPFC in order to regulate the active powerdemands in the system steady state operating conditions.According to our little search and effort, no more reported pa-

pers were found which have shown the application of adaptivenonlinear control technique for damping the low frequencyoscil-lations of themodern power systems including FACTS devices.The main contribution of this paper is to use the well-known

adaptive input-output feedback linearization control (AIFLC)technique in order to damp the low frequency oscillations ofpractical multi-machine multi-UPFC power systems. The con-trol method proposed in this paper is shown to be very well ro-bust and stable when subjeced to system abnormal conditionsand system parameters uncertainties. The overall system sta-bility is proved by Lyapunov theory. In this method, the thirdorder model of the synchronous machine is used for developingthe power system state equations which are required to designthe AIFL controller first using the sixth reduced order of the syn-chronous machines. Some digital computer simulation resultsare presented to support the validity and effectiveness of theproposed control approach. The static fourth order Runge-Kuttamethod is used to solve the system nonlinear differential equa-tions using MATLAB code with a time step of .

II. AIFL CONTROLLER DESIGN

A. Machine Model

AnAIFL nonlinear controller is designed in order to damp thelow frequency oscillations of the practical multi-machine multi-UPFC power systems. This controller is not expected to enhancethe transientstabilityof theselectedpowersystembefore therotor

single swings of the synchronous machines. One may note thatthe transient stability of the system can be achieved by a highperformanceprotection systemusinghigh speedcircuit breakers.Thatwill result inchoosingthepropersystemfaultclearance.Thismeans that for designing the AIFL controller, there is no need totake into account the synchronous machines subtransient states.Notice that the UPFCs used in themodern power systems shouldhave high capacity PWM inverters with low frequency powerswitches. As a result, these types of FACTS devices practicallyare not able to have any contribution for improving the systemtransient stability in the first few cycles after the fault occurring.The UPFCs used in power systems could be for two differentpurposes or both. The first is to regulate the active and reactivepowers which are decided to flow in some selected transmissionlines in thesystemsteadystatecondition.Thesecondpurpose is touse these FACTS in order to damp the low frequency oscillationsof the power system. Itmeans that for this aim, theUPFCshave tobe switched on after the fault clearance and switched off when areasonable damp is achieved for the power system.Asmentionedabove, havingused these converters during the fault is practicallyuseless.Based on above descriptions, the third order model of the

synchronous machine [27] is used for designing the AIFL con-troller:

(1)

where is the th turbine injected power, is the th syn-chronous machine active power, is the th transient inducedvoltage (behind the direct axis transient inductance), andis the rotor excitation voltage of the th machine.In addition, a simple machine rotor excitation system control

is used as

(2)

where is the th machine terminal voltage magnitude.and are the th machine AVR gain and time constant, re-spectively.

B. Power System Model

A simplifiedUPFCmodel is shown in Fig. 1(a) and (b), whereis the admittance of the transmission line containing UPFC.

With reference toFig.1(b), theanglesofUPFCinjectedseriesandshunt space vectors are assumed to be, respectively, in90 degrees lagging and leading with respect to transmission linecurrent andUPFCbusvoltage .Suchanassump-tion is necessary to make, in order to modulate the instantaneousactive power that flows in the transmission line including UPFC,withmaintheobjectiveofdampingthelowfrequencyoscillationsof the power system in the abnormal conditions [27], [28].Noticethat if inFig.1(b), theangleof isadjustedforaphaseangledif-ferent from90degreeswithrespect to thephaseangleof the ,both the instantaneous active and reactive powers of the UPFC

SHOJAEIAN et al.: DAMPING OF LOW FREQUENCY OSCILLATIONS OF MULTI-MACHINE MULTI-UPFC POWER SYSTEMS 1833

Fig. 1. UPFC model. (a) With voltage sources. (b) With current sources.

line are modulated [10], [28], which is not necessary for the pur-pose described in this paper.Referring to Fig. 1(b), when the instantaneous active power of

the UPFC transmission line tends to increase, the UPFC seriesinjected voltage acts as an inductive series reactance. Re-versely, when the instantaneous active power of this line tendsto decrease, the mentioned signal acts as a capacitive series re-actance. Such behavior of signal causes to damp the systemlow frequency oscillations. It is not necessary to say that at eachstep of time in seconds, the phase angle of andspace vectors are obtained by simulation and in practice by mea-surement. Having obtained these angles, the angles of the UPFCinjected signals and ( and ) are calculated as

Consider a practical multi-machine multi-UPFC modernpower system, numbering machines terminal buses from 1 toN, the UPFC buses from to and remaining busesfrom to . Injected currents to the lastbuses are zero. equations of this network can be derivedas

(3)

where vectors and are the machines terminals injectedcurrents and UPFCs injected currents, into the network, respec-tively. Vectors , , and are the machines terminals volt-ages, UPFCs buses voltages, and the last buses voltages, respec-tively. With reference to Fig. 2, vector can be obtained as

(4)

where to are UPFCs injected series voltages,to are UPFCs injected shunt currents, and toare the transmission lines admittances which include UPFCs.Using (3) and (4), one can obtain that

(5)

Fig. 2. Two area multi-machine power system which has been used for simu-lation.

where is a unity matrix and is azero matrixes. Defining

(6)

(7)

From (4)

(8)

Combining (5) and (8) gives

(9)

where

(10)

(11)

Notice that in transient state condition of the power systemunder consideration, the first machine is assumed to be as slackmachine, same as the steady state condition. Considering this as-sumption, the machines sixth reduced order equations parallelwith (9) are solved, in such a way that the terminal voltage ofslackmachine is forced not to change from its steady state value.As a result, at each step of time in seconds, the injected cur-rent of this machine (into the network defined by ) can be cal-culated from (9). That is because its terminal voltage is a knownconstant value. Under this condition, it can be assumed that the-axis of common synchronous rotating reference frame is to

be in the direction of the slack machine constant voltage vector,which rotates with nominal power system frequency.It is not necessary to say that the rotor axis of the slack

machine also rotates with respect to common reference frameso that at each step of time in seconds, the angle of rotationcan be obtained by solving the mechanical state equations of


this machine. Therefore, there is no need to solve the electricalstate equations of the slack machine in our system simulation.Using vector (9), the th machine injected current vector is

transferred to common reference as

(12)

Transferring the th machine injected current vector to itsrotor reference frame by multiplying both sides of (12) bygives

(13)

where is the rotor angle of this machine measured from the-axis of the common reference frame and is themagnitude

of the th component of the control vector given in (6). Using(13), one can obtain that

(14)

with

(15)

(16)

Based on synchronous machine third order model, the th ma-chine terminal voltage equations are

(17)

(18)

where , , and are the th machine stator resistance,direct axis transient reactance, and quadrature axis reactance,respectively. From (17) and (18), and can be obtainedas

(19)

(20)

with

(21)

Neglecting the stator copper losses, the thmachine generatedactive power is

(22)

Combining (19), (20), and (22) gives

(23)

where

(24)

(25)

Using (23), the third equation of (1) can be rewritten as

(26)

with

(27)

(28)

(29)

Using the last two equations of (1)

(30)

General form of (30) is rewritten as

(31)

where

and

Choosing the system output variable as

(32)

Applying the Lie theory [29] gives

(33)


(34)

where

(35)

(36)

(37)

Assuming

(38)

Combining (38) and (34) gives

(39)

Using (39) according to IFLC method, is chosen to be

(40)

Substituting for (40) in (39) results

(41)

where is a real positive number. Rewriting (38) for togives

(42)

is an matrix with components of and isinvertible when . It means, for example, in a six-machine three-UPFC power system, one of the UPFCs shouldbe operated like a static synchronous compensator (STATCOM)or static synchronous series compensator (SSSC). If ,the state equations of the machines should be subtracted fromeach other so that the number of resultant equations becomeequal to . In (42), is

(43)

It should be noted that each component of vector given in(43) is the magnitude of its corresponding component of vectorcontrol described by (6). For example, is the magnitudeof . Using (43)

(44)

C. AIFL Controller Design

Considering (34) with constant unknown parameters gives

(45)

and with the estimated parameters

(46)

Obviously can stabilize (45). Combining (45) and(46) results in

(47)

where

(48)

(49)

(50)

Lyapunov function can be candidate as

(51)

where , , and are the positive estimation weighing co-efficients. Derivating function with respect to the time in sec-onds gives

(52)

Then the system parameter estimation laws are obtained as

(53)

(54)

(55)

As a result, (54) is reduced to

(56)

Having obtained , , and from (53)–(55), can becalculated from(42) as

(57)

where in (42) is

(58)


Fig. 3. Block diagram of the proposed control approach.

III. POWER SYSTEM SIMULATION

The reduced sixth order model of synchronous machine (thatis obtained by ignoring the derivative terms in the stator d and qaxis voltage equations with assumption of 1 pu of rotor angularspeed in electric radian per second for speed voltage terms ofthese equations) is used for power system simulations [30]. Con-sidering (9)–(58) for any three-phase symmetrical fault that oc-curs in the power system, the system nonlinear differential equa-tions are solved by applying the numerical fourth order staticRunge-Kutta method using MATLAB code with a time stepof .Referring to (58), it can be seen that the magnitudes of the

UPFCs reference signals ( and for to M) are linearfunctions of the system state variables described by (38). No-tice that , where is the error of the thmachine generated active power. As a result, it is obvious thatthe UPFCs injected signals may not be in synchronism and inphase with UPFCs transmission lines active powers. In this case,the AIFL controller has failed to damp the low frequency oscil-lations of these lines (and then, to damp the whole power systemlow frequency oscillations). To solve this problem, we found outthat for each UPFC, we need to pass its injected signals at leastthrough a single stage lead-lag phase compensator before in-jecting to the corresponding transmission lines (see Fig. 3). Thetime constants of these phase compensators and AIFL controllergains has to be obtained by trial-and-error method so that theoutput of the phase compensators are to be roughly in synchro-nism and in phase with UPFCs transmission lines active powersignals.

A. Case Study 1

We consider the test network IEEE 30-bus system with sixsynchronous machines and three UPFCs as shown in Fig. 4. Thesynchronous machines specifications are given in Table I. As-suming the initial steady state condition of this network obtainedby AC load flow analysis, suppose that a three-phase symmet-rical fault occurs at exactly at the middle of the linewhich connects the buses 17 and 24, clearing at ac-complished with line interruption and reclosing at . Ac-cording to the flowchart of Fig. 5, simulation results obtained forthis fault are shown in Figs. 6 and 7. Notice that the UPFCs areswitched on when the fault cleared and are switched off when areasonable damping is achieved for the power system.To test the proposed control approach robustness, an initial

error of 20% in the power system resistances, as well asan initial error of 20% in the synchronous machines directaxis transient reactance, is considered. It can be seen that theAIFL controller joined with the lead-lag phase compensatorsis capable of damping the power system low frequency os-cillations in spite of parameters uncertainties. In addition to

Fig. 4. Power system configuration for case study 1.

TABLE IGENERATORS PARAMETERS FOR CASE STUDY 1

these results, as a sample, the simulation results correspondingto the UPFC2 injected signals magnitudes and phase angles,and machine2 estimated lumped parameters are, respectively,shown in Figs. 8–10. As an example, synchronous machine 3damper windings currents for case 1 are shown in Fig. 11.Comparing these results to those presented in Fig. 6 achieved

by AIFL controller, the superiority and capability of the pro-posed AIFL controller with respect to PI controller is very ob-vious.

B. Case Study 2

For this case, a typical power system with four synchronousgenerators and one UPFC as shown in Fig. 12 was selected.The machines specifications are given in [27]. Considering thesteady state condition of this network, suppose that a three-phase symmetrical fault occurs at exactly at the middleof the line 1, clearing at accomplished with line inter-ruption and reclosing at . Assuming an initial error of20% in the power system resistances as well as an initial error

of 20% in the synchronous machines direct axis transient re-actance, simulation results for this test are shown in Fig. 13. Forthis power system, since we have only one UPFC, the AIFL con-troller must produce only the magnitudes of two control signals;


Fig. 5. Flowchart of the computer simulation program.

therefore, two state equations are needed to design the AIFLC.Considering (39) for to 4, subtracting equation corre-sponding to G2 from those of related to G3 and G4, two stateequations are achieved which are used to design the AIFL con-troller. Referring to Fig. 12, G1 is forced to remain as a slackbus similar to that described in case study 1.At the end of this section, it may be useful to compare the

proposed AIFL controller behavior with conventional PI con-trollers behaviors in damping the low frequency oscillations ofthe multi-machine multi-UPFC power system under the faultconditions.Considering the IEEE 30-bus power system configuration

(Fig. 4), suppose that the injected series voltage and shuntcurrent of each UPFC are to be obtained by two PI regula-tors (say, series and shunt PIs). The controllers are tuned (bytrial-and-error method) in order to adjust the active powerof UPFC transmission line and the UPFC output bus voltagemagnitude equal to those of values obtained by steady stateload flow analysis of the power system. The input of the firstPI controller is the error of UPFC transmission line activepower and its output is the magnitude of the UPFC injectedseries voltage. The input of the second PI controller is the errorof UPFC bus voltage magnitude and its output is the injectedUPFC shunt current. The phase angles of these signals areonline obtained by the same method that was described for

Fig. 6. Machines rotor angles with and without AIFL controller for the powersystem of case study 1.

AIFL controller. Notice that the reference values of the PIcontrollers are chosen to be equal to those of values obtainedby steady state load flow analysis.In the normal condition of the power system, the mentioned

regulators are able to do their jobs properly. However, comparedto AIFL controller, they are not able to damp the low frequencyoscillations of the power system when any electrical fault oc-curs.That is because a conventional proportional integral type con-

troller (PI) is in fact an adjusting controller and is not reallya tracking controller; it means that at each step of time inseconds, its output signal depends on the value of input errorsignal at this step and the previous steps. As a result, the UPFCsinjected series voltages and shunt currents produced by theseregulators cannot be in synchronism and in phase with the errorsignals of the active powers that flow in the UPFC transmissionlines or the signal errors of the UPFCs output bus voltages. Onemore point is that even in the normal conditions of the powersystem, the PI controllers may not be robust and stable againstthe system parameters uncertainties and small variations that


Fig. 7. Machines rotor speeds with and without AIFL controller for the powersystem of case study 1.

Fig. 8 UPFC2 reference control signals using the AIFLC approach assuminga 1 kHz sampling frequency for the power system of case study 1.

usually occur in the steady state conditions. For these cases, thePI regulators should be online retuned for the new conditions.This is not really so easy and even is impractical.Considering the same fault that has been assumed for case

study 1 with AIFL controller, this controller is replaced by anumber of PI controllers.

Fig. 9. Phase angle of UPFC2 reference control signals and for thepower system of case study 1

Fig. 10. Estimation of using the AILF control approach for the powersystem of case study 1.

Fig. 11. Synchronous machine 3 damper windings currents for the powersystem of case study 1.

Fig. 12. Power system configuration for the power system of case study 2.

Simulation results obtained by PI controllers are shown inFig. 14. Comparing these results with similar simulation resultsobtained by AIFL controllers as shown in Fig. 6, it can be easilyrecognized that the PI regulators are nearly failed to damp thelow frequency oscillations of the power system. Hereby, the su-periority of the AILF controller with respect to PI controllershas been demonstrated.


Fig. 13. Rotor angles of the generators for the power system of case study 2.

IV. CONCLUSION

In this paper, the low frequency oscillations damping ofmulti-machine multi-UPFC power systems has been inves-tigated by using the AIFLC approach. The AIFL controllergenerates the magnitudes of UPFCs injected series voltage andinjected shunt current vectors. For each UPFC, at each stepof time in seconds, the phase angle of these vectors are obtainedto be in 90 degrees lagging and leading, respectively, with re-spect to UPFC transmission line current and its bus voltage. Inorder to make the mentioned vectors roughly in synchronismand in phase with UPFCs transmission lines active powerssignals, a lead-lag phase compensator has been used in joinwith AIFL controller, for each UPFC. The AIFL controllers’gains, time constants of the lead-lag phase compensators, andthe estimation weighting factors, all have been achieved bytrial-and-error method.The UPFCs have been arranged to be automatically switched

on just after clearing the fault and automatically switched offwhen a reasonable damp is achieved for the power system. As aresult, the subtransient states of the synchronous machines havebeen neglected in designing the AIFLC. It means that the thirdorder model of synchronous machines can be used for this pur-pose. In addition, the sixth reduced order model of synchronousmachines have been used in our power system simulation.Since the uncertain parameters of the power system have

been chosen to be the lumped uncertain functions, therefore anynumber of system parameters, even all, can be assumed to be un-certain.Simulation results presented in this paper have been obtained

for an IEEE 30-bus power system with six generators and threeUPFCs and also for a typical power system with four machinesand one UPFC. For each power system, a three-phase symmet-rical fault which accomplished with line interruption and line

Fig. 14. Machines rotor angles with and without using PI controller for thepower system of case study 1.

reclosing has been tested. These results very well support theeffectiveness and capability of the proposed control approach.A comparison has been made between the behaviors of the

proposed AIFL controller and conventional PI controllers indamping the low frequency oscillations of the multi-machinemulti-UPFC power system. From simulation results obtained byPI regulators, it has been shown that these controllers are not ca-pable of damping the low frequency oscillations of the faultedpower system under consideration.

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Shahrokh Shojaeian received the B.Sc. and M.Sc.degrees in electrical engineering from IsfahanUniversity of Technology, Isfahan, Iran, in 1997 and2001 respectively. He is currently pursuing the Ph.D.degree in the Faculty of Electrical Engineering, Sci-ence and Research Branch, Islamic Azad University,Tehran, Iran.His research interests are nonlinear control, power

system control and stability, and power system relia-bility.

Jafar Soltani (M’93) received the B.Sc. degree fromTabriz University, Tabriz, Iran, in 1974 and the M.Sc.and Ph.D. degrees from the University of ManchesterInstitute of Science and Technology (UMIST), Man-chester, U.K., in 1983 and 1987, respectively.He is currently a Professor in the Department

of Electrical Engineering, Khomeinishar Branch,Islamic Azad University, Isfahan, Iran. He is alsoan Emeritus Professor with the Faculty of Electricaland Computer Engineering, Isfahan University ofTechnology, Isfahan, Iran. His main area of research

is electrical machines and drive, power electronics, and power system control.He has published many international journal and conference papers and is theholder of a U.K. patent.Dr. Soltani is a member of IET. In addition, he is the reviewer of some inter-

national journals in European and IEEE Transactions.

Gholamreza Arab Markadeh received the B.Sc.,M.Sc., and Ph.D. degrees, all from Isfahan Univer-sity of Technology, Isfahan, Iran, in 1996, 1998, and2005, respectively.He is currently an Associate Professor in the

Department of Engineering, Shahrekord University,Shahrekord, Iran. His main areas of research arenonlinear control, power electronics, and variablespeed ac drives.

damping of low frequency oscillations of multi-machine multi-upfc power systems, based on adaptive...

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