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Optimal Placement of UPFC in Power Systems Using Genetic Algorithm D. Arabkhaburi, A. Kazemi, M.Yari, J. Aghaei Center of Excellence for Power System Automation and Operation, Department of Electrical Engineering Iran University of Science and Technology (IUST), Tehran, Iran Abstract- in this paper GA is used for determining optimal [7]. In the power system GA has been recently applied for location of UPFC in the power system. Optimal location in this optimization of generation expansion planning [8], paper is meant finding line number for UPFC location and its Economic dispatch [9], unit commitment [10] and capacitor parameters for specified number of UPFCs. Unlike other FACTS devices, UPFC has great flexibility that can control the Plaement [1 ]. active and reactive power flow and bus voltages, Power flow study of power system including UPFC with simultaneously. The system loadability is applied as measure of conventional methods such as Newton Raphson needs to use power system performance. Decoupled model of UPFC is used models for UPFC. These models don't must change Newton in simulation. The concept of simulation is the load flow Raphson method. One model with thesespecification s is incorporated with UPFC. Studies are applied on the 14-buses named Decoupled model. In the decoupled UPFC model, the IEEE network. The results have shown that steady state ends of UPFC may be assigned as PQ buses or the PV buses performance of power system can be effectively enhanced due with the desired values. In this model the power flow of the to the optimal location and parameters of UPFC. Index Terms- Genetic Algorithm (GA), UPFC, Power Flow power system with UPFC can be implemented without (PF), Loadability, Optimal Location, Load Increasing changing the conventional power flow program only with addition of two buses per one UPFC and the model can be used for power flow control easily. So, decoupled UPFC I. INTRODUCTION model is used in this paper. On the power system various types of studies are carried out. These studies are done in planning and operation with II. IMPLEMENTEDUPFC MODEL dynamic or steady state goals. Load increasing is one of In order to study the effects of UPFCs in the steady state conditions that happen in the power systems. Different types power system, it is essential to use a UPFC model for load of load increasing may be happed in reality and each of them flow calculation. Two types of UPFC model is represented in has particularly importance on power system studies. When the papers. One is a coupled model [12] and the other is load increasing is very high the power system may be faced decoupled model [13, 14]. In the first type a UPFC is with voltage dropping at buses and overloading at lines that modeled with a voltage source series with impedance in the result in instability on the power system. FACTS devices transmission line. In the second type a UPFC is model with will help to maintain the system stability or to restore the two separated buses. The first type is more difficult system to normal conditions. compared with the second one and the modification of the FACTS devices open a new opportunity for controlling the Jacobean matrix of system should be applied if it is used. On power flow and enhancing the loadability of power system. the other hand the decoupled model can be easily used in Among the FACTS devices, UPFC is a powerful device that conventional power flow methods without changing the can control both transmitted active and reactive power flow Jacobean matrix of system. In this paper the decoupled independently as well as bus voltages. model, as shown in Fig. 1, is used for power flow study The concept of UPFC and its characteristics has been established by L. Gyugyi for the first time [1]. On P1i + jQ1j + J implementation of UPFC in power system there is a practical | __ _ _ concern for finding optimal location of them. Optimal l location of FACTS devices allows controlling its power flow BUS i P +jQ +jQ BU [2] and thus increasing system loadability [3] and security margin [4]. Beyond a limited number of FACTS devices, the power system loadability can not be improved by this devices, this result has been observed at [5]. It is important to Fig. 1. The decoupled model of UPFC choose the suitable types of these devices in order to reach The UPFC controls the power flow in the transmission line the desired goals; UPFC is a device that will cover all where it is installed. In order to identify UPFC in load flow FACTS devices specifications. One method for solving study, it is represented by variables.Af the UPFC is assumed optimal placement problems is genetic algorithm. The to be lossless, the real power flow 'J that flows from busi Genetic algorithm was first proposed by Holland in early to bus j can be written as equation (1). 1970 [6] and put into practical applications in the late 1980s P.j PU (1) 1-4244-0726-5/06/$20.OO '2006 IEEE 1694 Authorized licensed use limited to: Jawaharlal Nehru Technological University. Downloaded on May 03,2010 at 03:19:47 UTC from IEEE Xplore. Restrictions apply.
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Page 1: GA Optimal Placement of UPFC in Power Systems Using GA

Optimal Placement of UPFC in Power SystemsUsing Genetic Algorithm

D. Arabkhaburi, A. Kazemi, M.Yari, J. AghaeiCenter of Excellence for Power System Automation and Operation, Department of Electrical Engineering

Iran University of Science and Technology (IUST), Tehran, Iran

Abstract- in this paper GA is used for determining optimal [7]. In the power system GA has been recently applied forlocation of UPFC in the power system. Optimal location in this optimization of generation expansion planning [8],paper is meant finding line number for UPFC location and its Economic dispatch [9], unit commitment [10] and capacitorparameters for specified number of UPFCs. Unlike otherFACTS devices, UPFC has great flexibility that can control the Plaement [1].active and reactive power flow and bus voltages, Power flow study of power system including UPFC withsimultaneously. The system loadability is applied as measure of conventional methods such as Newton Raphson needs to usepower system performance. Decoupled model of UPFC is used models for UPFC. These models don't must change Newtonin simulation. The concept of simulation is the load flow Raphson method. One model with thesespecification s isincorporated with UPFC. Studies are applied on the 14-buses named Decoupled model. In the decoupled UPFC model, theIEEE network. The results have shown that steady state ends of UPFC may be assigned as PQ buses or the PV busesperformance of power system can be effectively enhanced due with the desired values. In this model the power flow of theto the optimal location and parameters of UPFC.Index Terms- Genetic Algorithm (GA), UPFC, Power Flow power system with UPFC can be implemented without(PF), Loadability, Optimal Location, Load Increasing changing the conventional power flow program only with

addition of two buses per one UPFC and the model can beused for power flow control easily. So, decoupled UPFC

I. INTRODUCTION model is used in this paper.

On the power system various types of studies are carriedout. These studies are done in planning and operation with II. IMPLEMENTEDUPFC MODELdynamic or steady state goals. Load increasing is one of In order to study the effects of UPFCs in the steady stateconditions that happen in the power systems. Different types power system, it is essential to use a UPFC model for loadof load increasing may be happed in reality and each of them flow calculation. Two types ofUPFC model is represented inhas particularly importance on power system studies. When the papers. One is a coupled model [12] and the other isload increasing is very high the power system may be faced decoupled model [13, 14]. In the first type a UPFC iswith voltage dropping at buses and overloading at lines that modeled with a voltage source series with impedance in theresult in instability on the power system. FACTS devices transmission line. In the second type a UPFC is model withwill help to maintain the system stability or to restore the two separated buses. The first type is more difficultsystem to normal conditions. compared with the second one and the modification of theFACTS devices open a new opportunity for controlling the Jacobean matrix of system should be applied if it is used. On

power flow and enhancing the loadability of power system. the other hand the decoupled model can be easily used inAmong the FACTS devices, UPFC is a powerful device that conventional power flow methods without changing thecan control both transmitted active and reactive power flow Jacobean matrix of system. In this paper the decoupledindependently as well as bus voltages. model, as shown in Fig. 1, is used for power flow studyThe concept of UPFC and its characteristics has been

established by L. Gyugyi for the first time [1]. On P1i + jQ1j + Jimplementation ofUPFC in power system there is a practical | __ _ _concern for finding optimal location of them. Optimal llocation of FACTS devices allows controlling its power flow BUS i P +jQ +jQ BU[2] and thus increasing system loadability [3] and securitymargin [4]. Beyond a limited number of FACTS devices, thepower system loadability can not be improved by thisdevices, this result has been observed at [5]. It is important to Fig. 1. The decoupled model of UPFCchoose the suitable types of these devices in order to reach The UPFC controls the power flow in the transmission linethe desired goals; UPFC is a device that will cover all where it is installed. In order to identify UPFC in load flowFACTS devices specifications. One method for solving study, it is represented by variables.Af the UPFC is assumedoptimal placement problems is genetic algorithm. The to be lossless, the real power flow 'J that flows from busiGenetic algorithm was first proposed by Holland in early to bus j can be written as equation (1).1970 [6] and put into practical applications in the late 1980s P.j PU (1)

1-4244-0726-5/06/$20.OO '2006 IEEE 1694

Authorized licensed use limited to: Jawaharlal Nehru Technological University. Downloaded on May 03,2010 at 03:19:47 UTC from IEEE Xplore. Restrictions apply.

Page 2: GA Optimal Placement of UPFC in Power Systems Using GA

Although a UPFC can control the power flow but cannot voltage levels. The objective function is made in order togenerate the real power. So equation (2) should be penalize configurations of UPFC which lead to overloadconsidered in the model. transmissions lines and over or under voltage at busses.

Pu1 + Pu2 = 0 (2) Only technical benefits of UPFC, in terms of loadability,are taken into account. Other criteria such as costs of

The values of Q.1 Qu2 , can be set to an arbitrary value installing and maintaining of devices are not considered.within the capacity of UPFC to maintain the bus voltage. In As it is commonly done for multi criteria constrainedthe same way if multiple UPFCs are installed in the power optimization problems, this problem is transformed intosystem, the control variables for the kth installed UPFC are some separate objective optimization problems. Objectiveshown as follows: function is defined as the sum of two terms. The first one is

UPFCkh K' Q" P" Q"2 ] (3) related to branch loading which penalizes overloads in lines.This term is called LF and is computed for all lines of the

So that: power system, if branch loading is less than 100% its value is

Pk +Pkj =0 (4) equal to 1 otherwise it decreases exponantionally withrespect to the overload. To accelerate convergence, product

It is assumed that the UPFCs are installed in the middle of of values for all objective functions is calculated. The secondlines. part of the objective function is for voltage levels that are

named BF. This function is calculated for all buses of power

III. THE PROPOSED GENETIC ALGORITHM system. For voltage levels between 0.95 and 1.05 values ofobjective functions is equal to 1. Outside of this range value

In a simple genetic algorithm, individuals are coded to a it decreases exponantionally with the voltage deviations.chromosome that contains variables of the problem. The Therefore for a configuration of UPFCs, objective function isConfiguration of chromosome in order to optimal location of as below:the UPFC consists of two types of parameters: location of LF t BL <100 (5)UPFC and Pi, Qi, Pj and Qj as decoupled model Lexp[O.0461(100- BL)] BL .100parameters of UPFC. In the Fig. 1, the chromosome for theproposed algorithm has been shown.

1 .0.95 < VL < 1.05

Objective Location BF (6)Function ~Of I Q1 P. ~r 1.05.<VL.<1.25| Function U ... f|PFpi Qi | Qi exp............[-23.0259|11 VL| 05]I°0V<5UPFC exp1..~~~~~~~~~~~~~~~~ ~or 0.75.< VL . 0.95

Fig. 2. The Chromosome of Proposed GA

The total number of UPFCs inserted in the power system Function= fl LF + (7)is limited, due to the cost of devices and the influences on Oblective =nthe operating characteristics of the power system. A genetic i-Lines j=Busesalgorithm is governed by three factors: mutation rate, where, LF is the line flows index and BL is the Branchcrossover rate and population size. The GA is a search Loading (Percentage of line flow respect to line capacityprocess which can be applied to constrained problems; the rate). BF is bus voltage index and VL is per unit value of busconstraints may be included into the fitness function as voltages.added penalty terms.

In this algorithm issues for optimization that must be noted V. OPTIMIZATION FLOWCHARTare as follows:Location of UPFC: No more than one UPFC can be installed As explained previously, the aim is to find maximumin one branch power flow computations. value of the power that the system is able to supply withoutControl parameters: The performance of the GA depends on overloading lines and disallowed voltage level.control parameters, such as population size, crossover It is searched to locate a given number of UPFCs toprobability and mutation probability. Therefore selection of increase as much as possible the capacity of the powerproper value of GA parameters has high effect for gain system. In the search for the best location with best values anoptimum solution. optimization flowchart has been implemented as shown in

fig. 3. Starting from an initial load the GA described in

IV. OBJECTIVE FUNCTION OF OPTIMIZATION section III is applied recursively. The stop criterion is eithera maximum number of generations or solutions with

The aim of optimization is to perform a best utilization of objective function equal to 1. In the first case, the algorithmthe existing transmission lines. In this respect, UPFC device is stopped otherwise load is increased and a newis located in order to maximize the system loadability while optimization starts again. Loads are increased in according toobserving thermal and voltage constraints. In other words, It considered case studies. Additional losses because ofwas tried to increase as much as possible the power increasing of the power transmitted are shared among alltransmitted by power system to costumers with holding generators proportion to their power.power system in security state in terms of branch loading and

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Page 3: GA Optimal Placement of UPFC in Power Systems Using GA

hni romuberOf TJPFC5

X.VrLre =1d1POpU1Wt1

fc1rec uomogonme rmu

Foad| Floart ofptim

Vp.E Sas oa si s b o t tLoainreainge rearsyo e Ladi N re te

Yes

r

Eds

Fig.3: Flowchart of optimization

VI. CASE STUDY Case]: Load increasing at some buses on the network

Load increasing studies on the real power system are done Load increasing at some area on the system is one of cases

for different aims in planning and operation process of the that power system may be faced. For simulation of this casesystem. The implemented test system is IEEE 14-buses the area that was faced with load increasing modeled withnetwork that is showed in Fig. 4. Here the studies are one bus on the power system. Load increased bus at this caseassessed in five cases, then buses voltages and lines flows is bus#4 and power flow results have shown that some busresults are presented before and after using UPFC. In all voltages and lines flows violated of their allowed levels,these cases has been observed that more than allowed level therefore system is unstable. For restoring the system toof load increasing, voltage at some buses dropped and some stable conditions, Genetic algorithm has proposed that it islines overloaded except at one of cases that system was faced necessary to use one UPFC at line#9. All the Results ofwith bus voltage dropping but not line overloading. power flow before and after of using UPFC have been shown

in tables (I) to (IV).

1213 14 ,Case2: Active load increasing at all busesX0E}g < 1020 Real example for this case is electrical peak load of energy

1 1- 91tconsumption. At this period of consumption, mainly active\L](iLIa1S 113 l load consumption is increased and reactive load consumption

\;)L14 @ r.lo L o 9 S is constant. By active load increasing in compared with base6X) 1Lg g 1 | system, test system will be unstable such that one UPFC is

not able to restore system to security condition and the GAQ Q t9t = | ~~~~~~~~hasproposed that with2UPFCs in the lines #9 andl16.

1 4_>X 17 4 < ~~~~~~~Case3.Reactive load increasing at all buses|/ T I;lr ~~~~~~~~~~Thereis not real example for only reactive load increasing

1 < r.s_ __>~~~~~ gn at power system. In this case, it is observed that only* 1 >/- t ~~~~~~~~voltages of buses dropped and any lines are not overloaded.2lJ 2L 3 LL ~~~~~~~~Insuch case that only buses are under allowed level of

Fig. 4. IEEE 14-buses Network voltage it is proposed to use SVC or STATCOM instead of

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Page 4: GA Optimal Placement of UPFC in Power Systems Using GA

UPFC. The reason of this study is to show ability of UPFC STATCOM in the bus #14 and two UPFCs in the lines #6implementation in the system for optimal reactive power and 16.flow controlling. The GA has proposed that with two UPFCsin the lines #3 and 8. The results of power flow before and VII. CONCLUSIONSafter using UPFC on the system is shown in the tables (I) to(IV). In this paper, the simulation results of optimal UPFC

placement on an unstable power system because of loadCase 4: Active and reactive load increasing increasing have been presented. Load increasing studies onFor long-term studies of power system, it is necessary to the system can be applied at different cases and aspects. Itconsider both active and reactive load increasing. In this has been shown that all load increasing cases may be result

case, the GA has proposed that with 2 UPFCs in the lines #6 in instability on the system and optimal placement of UPFCand 8. Similar to other cases the result has been shown at can restore the system to stable condition. In other hand

tables (I) to (IV). loadability and security level of power system has increased.The advantage of the GA as optimal placement method is the

CaseActive and reactive load and generation increasin ability of solving multi-objective problem. Moreover, it canCase 5. A an e vl a t increasing also be applied to constrained problem which the constraintswith together can be included into the fitness function as added penaltyTo Complete study of a power system for loadability, it terms and when the solutions are multi-variable, the GA haveneeds that both loads and power generation increasing to be been shown that is very powerful. Also from the case studiesconsidered. Through this study ability of topology and it can be observed that the security level of system increasestructure of existing system for loadability can be more as the number of installed UPFC in the system becomeinvestigated. It is observed at this case using 2 UPFC more. This situation should be study from the economicsystems can not restore its stable state and bus# 14 is under point of view and at such state combination of UPFC andallowed limit. With use 3 UPFC, the system can come back another FACTS device for example SVC or STATCOM areto stable state but it is not economic to use 3 UPFCs on 14 proposed.buses network but also it is better to use one SVC or

TABLE I

THE RESULTS OF BUSES VOLTAGES OF POWER FLOW WITHOUT UPFC FOR THE ALL CASESB3s# Case I Case 2 Case 3 Case 4 Case 5

1 1.0500 1.0500 1.0500 1.0500 1.05002 1.0500 1.0500 1.0500 1.0500 1.05003 1.0321 1.0316 L.0583 1.0358 1.05154 0.9683 0.9777 1.0239 0.9841 0.99515 0.9940 0.9906 1.0335 0.9981 1.01246 1.0500 1.0500 1.0500 1.0500 1.05007 0.9802 0.9715 0.9388 0.9555 0.92678 1.0500 1.0500 1.0500 1.0500 1.05009 0.9570 0.9406 0.8409 0.9013 0.851110 0.9543 0.9342 0.8113 0.8871 0.830111 0.9955 0.9817 0.9185 0.9566 0.925912 1.0016 0.9783 0.8985 0.9467 0.913513 1.0020 0.9723 0.9399 0.9532 0.929414 0.9479 0.8957 0.8327 0.8584 0.8080

TABLE II

THE RESULTS OF BUSES VOLTAGES OF POWER FLOW WITH UPFC FOR THE ALL CASES

Bus# Case I Case 2 Case 3 Casc 4 Casc 5(one UPFC) (2 UPFC's) (2 UPFC's) (2 UPFC's) ( UPFC's)1 1.0500 1.0500 1.0500 1.0500 1.05002 1.0500 1.0500 1.0500 1.0500 1.05003 1.0100 1.0100 1.0100 1.0100 1.01004 1.0225 0.9999 1.0226 1.0056 1.02115 1.0316 1.012 1.0329 1.0155 1.03026 1.0500 1.0500 1.0500 1.0500 1.0500

______ 70.9809 1.0030 1.0027 1.0038 1.00218____ 1.0500 1.0500 1.0500 1.0500 1.0500_____9 0.9642 0.9862 0.963 1 0.9747 0.9669

10 0.9622 0.9845 0.9501 0.9705 0.956911 1.0003 1.0112 0.9944 1.0044 0.996512 1.0094 1.0110 0.9812 0.9943 0.981313 1.0113 1.0092 1.0011 1.0022 0.991514 0.9647 0.9694 0.9572 0.9584 0.9451

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Page 5: GA Optimal Placement of UPFC in Power Systems Using GA

TABLE IIITHE RESULTS OF LINE FLOWS OF POWER FLOW WITHOUT UPFC FOR THE ALL CASES

Line # Case I Casc 2 Case 3 Casc 4 Casc 51 1.5475 1.7721 0.9992 1.3681 0.11732 0.1443 0.1487 0.4771 0.0618 0.30873 L.0958 0.8595 0.1607 0.7180 0.43334 1.3226 1.4202 0.6150 1.1421 0.34435 0.6511 0.6338 0.0776 0.5387 0.36276 0.9361 0.6955 0.3557 0.6500 0.77237 1.7726 0.9212 0.3629 0.7458 0.31668 0.0207 0.7964 0.3041 0.6865 0.26179 0.4062 0.2590 0.5888 0.1393 0.308210 0.9997 0.9988 0.9990 0.9985 1.497011 0.0335 0.3420 0.1104 0.2793 0.145212 0.5920 1.2518 0.4096 l.1313 1L180513 0.0307 0.1505 0.0213 0.1187 0.084614 0.2535 0.3264 0.1903 0.3221 0.361015 0.1469 0.3086 0.0936 0.2793 0.282116 0.3929 0.7552 0.3134 0.7205 0.757717 0.0840 0.3381 0.0405 0.2915 0.262618 0.1826 0.1826 0.1109 0.1825 0.210519 0.0353 0.0592 0.0089 0.0459 0.041520 0.1789 0.2530 0.1189 0.2466 0.2720

TABLE IVTHE RESULTS OF LINE FLOWS OF POWER FLOW WITH UPFC FOR THE ALL CASES

Line Casel1 Case 2 Case 3 Casc 4 Casc 5(one UPFC) (2 UPFCs) (2 UPECs) (2 UPFCs) ( UPFCs)

1 0.9743 0.8919 0.5377 0.7680 0.06572 0.4815 0.0853 0.3331 0.2331 0.01303 0.1098 0.4816 0.9237 0.3981 0.23184 0.6981 0.7806 0.4336 0.6021 0.19785 0.0187 0.3880 0.1662 0.2481 0.20796 0.3831 0.3566 0.0984 0.9259 0.18637 0.5404 0.4055 0.3165 0.6180 0.11358 0.2487 0.5796 0.8796 0.8756 0.15949 0.8501 0.8091 0.1457 0.3777 0.029110 0.8879 0.2873 0.4440 0.3066 0.559911 0.0945 0.3392 0.0078 0.2732 0.096412 0.0521 0.2702 0.2982 0.6838 0.530713 0.1263 0.0090 0.0799 0.2100 0.050314 0.2601 0.2518 0.0057 0.0096 0.170115 0.0948 0.1016 0.0243 0.1114 0.064416 0.2863 0.8840 0.0900 0.2680 0.925417 0.0207 0.0614 0.0787 0.2376 0.077918 0.2136 0.1744 0.0366 0.0605 0.099419 0.0319 0.0150 0.0137 0.0047 0.041220 0.1749 0.2291 0.0012 0.0258 0.1802

VIII. REFERENCES of UPFC", Proceeding of the 2000 IEEE/PES summer

[1] L. Gyugyi, "Unified Power Flow Concept for Flexible meeting, pp.30-43.AC Transmission system", IEE proc. Generation, [5] S. Gerbex, R. Cherkaoui, A. J. Germond, "OptimalTransmission and Distribution, Vol.139, No.4, July Location Of Multi type FACTS devices in a Power1991, pp.323-332. System by Means of Genetic Algorithm", IEEE Trans.

[2] D. J. Gotham and G. T. Heydt, "Power Flow Control and Power Systems, vol.16, No. 3, pp.537-544, Aug.2001.Power studies for System with FACTS devices", IEEE [6] J. H. Holland, "Adaptation in natural and artificialTrans. Power Systems, vol.13, No.1, pp. 60-65, systems", The University of Ann Arbor, 1975.Feb.1998. [7] E. Goldberg, Genetic algorithms in search,

[3] F. DGaliana, K. Almeida, M. Toussaint, J. Griffin, D optimization& machine learning, Addison-WesleyAtanackovice, "Assessment and Control of the FACTS Publishing Company, Inc, 1989.Devices on Power System Performance", IEEE Trans. [8] Y. Fukuyama and Y- Ueki, "Application of ParallelPower Systems, vol.11, no.4, pp.193 1-1936, Nov.1996. Genetic Algorithm to Generation Expansion Planning

[4] 5. h. Kim, J. U. Lim, s. 1. Moon, "Enhancement of Power Using Parallel Processors", Electrical Engineering inSystem Security Level through the Power Flow Control Japan, 1995, Vol.115, No6, pp.71-78.

[9] D. C. Walters and G. B. Sheble, "Genetic Algorithm toOptimal Reactive Power Dispatch Including Voltage-

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Page 6: GA Optimal Placement of UPFC in Power Systems Using GA

dependent Load Models", IFAC International [14]T. H. Kim, G. Ch. Seo, G. U. Lim, S. Moon, J. P. Park,Symposium on Large Scale Systems, London. July 1995. B. M. Han, "A Decoupled unified power flow controller

[10]S. A. Kazarlis, A. G. Bakirtzis and V. Petridis, "A model for power flow considering limit resolution', IEEEGenetic Algorithm Solution to the Unit Commitment 1998, pp.1190-1195.Problem", IEEE Transaction on Power system, 1993,Vol.8, No3, PP.1325-1332.

[11 ]Gary Boone and HSIAO-Dong Chang, "OptimalCapacitor Placement in Distribution System by GeneticAlgorithm", Electric Power & Energy System 1993,Vol.15, No.3, pp.155-162.

[12]M. Noroozian, L. Angquist, M. Ghandhari andGAndertson, "use of UPFC for Optimal Power FlowControl", IEEE Trans. on Power Delivery, Vol 12, No.4,1997.

[13]Nabavi-Niaki, M. R, Iravani, "Steady state and dynamicmodels of unified power flow controller (UPFC) forpower system studies", IEEE Trans. on power delivery,Vol. 11, November 1996.

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