A SURVEY ON REACTIVE POWER OPTIMIZATION AND VOLTAGE STABILITY IN POWER SYSTEMS

Transcription

1 Internatonal Journal on Techncal and Physcal Problems of Engneerng (IJTPE) Publshed by Internatonal Organzaton of IOTPE ISSN IJTPE Journal March 2014 Issue 18 Volume 6 Number 1 Pages A SURVEY ON REACTIVE POWER OPTIMIZATION AND VOLTAGE STABILITY IN POWER SYSTEMS N.M. Tabatabae 1,2 A. Jafar 1,2 N.S. Boushehr 1,2 1. Electrcal Engneerng Department, Seraj Hgher Educaton Insttute, Tabrz, Iran n.m.tabatabae@gmal.com, al.jafar.860@gmal.com, nargesboush@yahoo.com 2. Taba Elm Internatonal Insttute, Tabrz, Iran Abstract- Reactve power plays an mportant role n supportng the real power transfers by mantanng voltage stablty and system relablty. It s a crtcal element for a transmsson operator to ensure the relablty of an electrc system whle mnmzng the cost assocated wth t. The tradtonal objectves of reactve power dspatch are focused on the techncal sde of reactve support such as mnmzaton of transmsson losses. Reactve power cost compensaton to a generator s based on the ncurred cost of ts reactve power contrbuton less the cost of ts oblgaton to support the actve power delvery. In ths paper, we tred to ntroduce reactve power optmzaton problem, mscellaneous objectves, voltage stablty ndces types and formulatng of them, revewng recent studes n ths fled and comparson between them for checkng performance of them. Ths paper also ntroduces complete references n ths case wth a short notng to them n text body. Ths paper wll good reference for who they want begn to study n ths feld due to ths paper supports all ssues n reactve power optmzaton feld. Keywords: Reactve Power Optmzaton, Heurstc Algorthms Applcaton, Reactve Power Compensaton, Voltage Stablty Indces, Real Power Loss Mnmzaton. I. INTRODUCTION The reactve power optmzaton problem has a sgnfcant nfluence on secure and economc operaton of power systems. The reactve power generaton, although tself havng no producton cost, does however affect the overall generaton cost by the way of the transmsson loss. A procedure, whch allocates the reactve power generaton so as to mnmze the transmsson loss, wll consequently result on the lowest producton cost for whch the operaton constrants are satsfed [6]. The operaton constrants may nclude reactve power optmzaton problem. The conventonal gradent-based optmzaton algorthm has been wdely used to solve ths problem for decades. Obvously, ths problem s n nature a global optmzaton problem, whch may have several local mnma and the conventonal optmzaton methods easly lead to local optmum. On the other hand, n the conventonal optmzaton algorthms, many mathematcal assumptons, such as analytc and dfferental propertes of objectve functons and unque mnma exstng n problem domans, have to be gven to smplfy problem [6]. Otherwse, t s very dffcult to calculate the gradent varables n the conventonal methods. Further, n practcal power system operaton, the data acqured by the SCADA (Supervsory Control and Data Acquston) system are contamnated by nose. Such data may cause dffcultes n computaton of gradents. Consequently, the optmzaton could not be carred out n many occasons. In the last decade, many new stochastc search methods have been developed for the global optmzaton problems such as Smulated Annealng (SA), Genetc Algorthms (GA) and Evolutonary Programmng (EP) and etc. [6]. The man objectve of OPD s to consder and address the all the objectves of modern power systems. The man frst objectve of OPD s economy of the system, the economy of the system related to real power loss as a second objectve and reactve power dspatch s thrd objectve. The fourth objectve s voltage stablty enhancement and s related wth voltage profle optmzaton, relablty analyss and control of voltage devaton level when before, durng and post contngency condton. The fnal objectve s optmal locaton of FACTS devces and ts mportant objectve n modern power systems when dynamc loadng condton. The Man am of OPD problem s to optmze the all the objectves n smultaneously [3]. The smultaneously optmzaton not only consst of optmzaton and also satsfy the controls and ts related to optmzaton problem. The control strateges am s to avod some of the symptoms, voltage nstablty whch lead to voltage collapse lke heavy loadng, transmsson outages, or shortage of reactve power and the ts or constrants of OPD problem are real power generaton, reactve power generaton, bus voltages and settngs of transformer taps wth FACTS devces [3]. The ncreases of actve power loss s affects the economy of the power systems and systems need to reschedulng for proper operaton. The connecton of above the reactve power loss leads to devates the system voltage profle, fnally t dmnshes the relablty and stablty of the system [3]. 220

2 So, the OPD problem s one of the most mportant and challengng problems n de-regulated envronment and because, It s address to the optmal ponts of multobjectve functons of OPD problem as to determne the cost of operatng, mnmze the real power loss by Reactve power dspatch and t s by optmal locaton of the Flexble AC Transmsson Systems (FACTS) wth mnmum cost whle keepng an adequate voltage profle. Hence, the system n need of proper coordnaton between FACTS devces and transformer taps and stablty ndces wll leads the compensaton requrements, voltage stablty and coordnaton controls [3]. The man objectve of optmal reactve power dspatch (ORPD) of electrc power system s to mnmze an actve power loss va the optmal adjustment of the power system control varables, whle at the same tme satsfyng varous equalty/nequalty constrants. The equalty constrants are the power flow balance equatons, whle the nequalty constrants are the ts on the control varables and the operatng ts of the power system dependent varables. The problem control varables nclude the generator bus voltages, the transformer tap settngs, and the reactve power of shunt compensator, whle the problem dependent varables nclude the load bus voltages, generator reactve powers, and the power lne flows. Generally, the ORPD problem s a large-scale hghly constraned nonlnear non-convex and multmodal optmzaton problem [11]. Lnear programmng (LP), non-lnear programmng and gradent based technques have been proposed n the lterature [19-22] for solvng RPD problems. However, due to the approxmatons ntroduced by lnearzed models, the LP results may not represent the optmal soluton for nherently non-lnear objectve functons such as the one used n the reactve power dspatch problem. It s very dffcult to calculate gradent varables and a large volume of computatons s nvolved n ths approach [16]. Fgure 1. Flowchart of ABC algorthm for reactve power optmzaton [10] Also, these conventonal technques are known to converge to a local optmal soluton rather than the global one. Lately, expert system approach [23] has been proposed for the reactve power control computatons. Ths approach s based on If-then based producton rules. The constructon of such rules requres extensve help from sklled knowledge engneers [16]. II. REVIEWING SOME STUDIES A number of technques rangng from classcal technques lke gradent-based optmzaton algorthms to varous mathematcal programmng technques have been appled to solve ths problem [24-31]. Each of these has ndvdual merts n terms of computatonal tme and convergence propertes. However, mathematcal programmng technques suffer from ted modelng capabltes.e. they have severe tatons n handlng nonlnear, dscontnuous functons and constrants, and functons havng multple local mnma, as s normally the case wth the RPD problem. The development of Soft Computng and Evolutonary algorthms over the last decade has enabled researchers to consder these ssues n a better fashon. The advantages of Evolutonary algorthms n terms of modelng capablty and search power have encouraged ther applcaton to the RPD problem n power systems [32-38]. K. Iba [32] was probably the frst to apply GA to the reactve power dspatch problem. The method decomposes the system nto a number of subsystems and employs nterbreedng between the subsystems to generate new solutons. All the controller states, ncludng those wth a contnuous nature, are dscretzed and represented as nteger values. K.Y. Lee et al. [33] employed a modfed smple genetc algorthm for reactve power plannng. The populaton selecton and reproducton uses Benders cut n decomposed system and successve lnear programmng has been used to solve the operatonal optmzaton sub-problems. However, a bnary representaton of control varables ntroduces an element of approxmaton at the representaton stage tself. J.T. Ma and group [34-37] present an evolutonary programmng approach for solvng RPD. The technque uses a floatng pont representaton for control varables. Mutaton, used wth an adaptve probablty, s the only reproducton operator n the technque. An nner loop s used for functon mnmzaton wthout consderng constrants. Constrant satsfacton s carred out n an outer loop. Non-feasble solutons n the outer loop are rejected by attachng a penalty to ther ftness values. D.B. Das et al. have proposed two technques for the soluton of RPD. The frst, presented n D.B. Das et al. [39], s a Hybrd Stochastc Search technque that uses SA n selecton process of GA. The second s the Hybrd Evolutonary Strategy whch s an ES based technque wth a domnant mutaton operator and other mprovements presented n D.B. Das et al. [40]. Zhang et al. [41] have proposed a Mult-Agent Systems based approach for optmal reactve power dspatch. Jang et al. 221

3 [42, 43] have proposed the mult-objectve approach for reactve power dspatch usng technques based on Evolutonary Programmng and Partcle Swarm Optmzaton respectvely. Zhao et al. [44] presented another mult-agent based PSO approach for optmal reactve power dspatch. ndvduals nstead the way of SGA. [14] Proposes a novel heurstc optmzaton algorthm namely the Mean Varance Mappng Optmzaton (MVMO) s proposed to handle the ORPD problem (see Fgure 3). [18] Presents optmal reactve power dspatch (ORPD) for mprovement of voltage stablty. Ths paper uses Dfferental Evoluton method (DE) as approach for solvng optmzaton ssues. The flowchart of ths algorthm has shown n Fgure 4. Fgure 2. Bg Bang-Bg Crunch (BB-BC) algorthm for reactve power optmzaton [8] Latest development n the feld of EAs s Quantum Evolutonary Algorthms (QEA) [45, 46], whch synergstcally combnes the prncples of Quantum Computng and EAs. QEA s a populaton-based probablstc Evolutonary Algorthm that ntegrates concepts from quantum computng for hgher representaton power and robust search. [1] Proposed an alternatve approach based on QEA s proposed for the frst tme for soluton of RPD. In [3] a novel bo-heurstc algorthm called Refned Bacteral Foragng Algorthm (RBFA) s proposed n the paper to solve the optmal power dspatch of deregulated electrc power systems. [8] Proposed the nature nspred Bg Bang-Bg Crunch (BB-BC) algorthm s mplemented to solve the mult constraned optmal reactve power flow problem n a power system. The flowchart of ths algorthm has shown n Fgure 2. [10] Presents Artfcal Bee Colony (ABC) based optmzaton technque s to handle RPO problem as a true mult-objectve optmzaton problem wth competng and non-commensurable objectves (Fgure 1 for related flowchart). [12] Proposes an Optmal Reactve Power Flow (ORPF) ncorporatng statc voltage stablty based on a mult-objectve adaptve mmune algorthm (MOAIA). [13] Proposed advanced an Improved Genetc Algorthm Combnng Senstvty Analyss (IGACSA) for reactve power optmzaton. The new algorthm combned senstvty analyss to generate ntal generaton of Fgure 3. The flowchart of ORPF based on MOAIA [12] III. PROBLEM DEFINITION The objectves of reactve power (VAR) optmzaton are to mprove the voltage profle, to mnmze system actve power losses, and to determne optmal VAR compensaton placement under varous operatng condtons. To acheve these objectves, power system operators utlze control optons such as adjustng generator exctaton, transformer tap changng, shunt capactors, and SVC [9]. However, the sze of power systems and prevalng constrants produce strenuous crcumstances for system operators to correct voltage problems at any gven tme. In such cases, there s certanly a need for decson-makng tools n predomnantly fluctuatng and uncertan computatonal envronments. There has been a growng nterest n VAR optmzaton problems over the last decade. Most conventonal methods used n VAR optmzaton are based on lnear programmng and nonlnear programmng. Some smplfed treatments n these methods may nduce local mnma. So, there s hghly need to fnd accurate and fast algorthms to use n reactve power optmzaton problem [9]. 222

4 IV. RELIABILITY ANALYSIS FOR CRITICAL LINES AND BUSES Many voltage stablty margn ndces have been proposed [47]. Ref. [48] proved that the statc voltage stablty margn could be measured by mnmal egenvalue of the non-sngular Jacoban matrx n a mult-generator system. Many artcles also have used ths ndex to mprove voltage stablty margn successfully [49-51]. Some of these ndexes are descrbed as follows [12]: A. Voltage Stablty Analyss and Fast Lne Flow Index The Fast Lne Flow Index (FLFI) method s to ensure the power flow control and stablty ndex between the recevng and sendng end power n the ntercommoned power system network. In ths method the set of power flow equatons s to coordnate the real and reactve power flow control over a transmsson lne n both the drectons of flow. The set of equatons were used to analyss and dentfcaton of crtcal lnes and weak buses [3]. The mum voltage devatons are ponted out n the partcular systems n the vew of voltage stablty analyss. The analyss of lne flow approach s gven for two bus system: 4XQj Lfl (1) V sn( ) 2 where, L fl s Fast Lne Flow Index, θ s angle n the mpedance angle from mpedance trangle, δ s Influence of the vector dagram, angle between sendng end and recevng end voltage, X s lne reactance, Q j s reactve power flow at the recevng, V s sendng end voltage [3]. Fgure 4. The flowchart of dfferental evoluton B. Voltage Stablty Approach (VSA) The Voltage Stablty Approach (VSA) s comprses a Voltage Stablty Index (VSI) aganst voltage collapse and lne stablty based on concept of mum power transferred through a transmsson lne flow. The optmal locaton and control varables of FACTS devces are based on voltage stablty ndex of each transmsson lne. The loadng of real or reactve powers are leads to dentfy the crtcal transmsson paths and va weak buses [3]. A voltage stablty ndex s deals the mum voltage devaton va power flow n transmsson, whch s leads to mantan the voltage profle aganst loadng condton. Therefore voltage stablty approach s gves the corrected voltage drop of a lne segment s defned as the projecton of the recevng end bus voltage of that segment on the voltage Phasor of the generator whch s the startng pont of that transmsson path. Ths ndex s gven by [3]: VSA hvact ΔV (2) where, V act s actual generator voltage, h s parameter for correct the desred constant value and ΔV s sum of corrected voltage drops by the sde of a transmsson path. The real power and reactve power flow n transmsson lne s defned as a sequence of connected buses wth declnng voltage magntudes agan startng from a generator bus [3]. The FLFI and VSA are analyss to carry out the real and reactve power loadng and wth address of crtcal lnes and weak buses. The voltage devaton and voltage stablty enhancement s happen for placng of FACTS devces. The optmal locaton FACTS devces, voltage control va reactve power support, the relablty analyss s carred out va stablty ndces. Further Q-V analyss s deals of voltage stablty analyss and reactve power compensaton desgn n FACTS devces [3]. C. Reactve Power Control and Voltage Stablty Index, Q-V Analyss The Q-V analyss encompass of voltage stablty analyss, reactve power control varables and VAR compensaton desgn s gven below the matrx: JP JPV P J J V Q (3) Q QV where, ΔP and ΔQ are ncremental real, reactve power, Δδ and ΔV are ncremental bus voltages and bus angles, J Pδ, J PV, J QV and J Qδ are sub matrxes of Jacobean n power flow equaton [3]. The Q-V analyss s method to dentfy FACTS devces for compensaton n partcular pont after dentfcaton of weak buses and crtcal lnes, by the way to mprove the voltage stablty and fnally provdes nformaton to enhance voltage stablty by takng necessty actons. Ths analyss gves a detal vew of stablty enhancement by modfcatons and reschedulng of control varables lke real and reactve power controls [3]. Power flow equatons after the ncrements n bus voltage magntude and angel, real and reactve power are can be wrtten as follows: J1 J2 JP JPV P J3 J 4 V JQ J QV V Q (4) The stablty pont of vew, accordng to pont of operaton keepng real power constant s. The ncremental relatonshp of Q-V analyss s gven below [3]: V J 1 Q (5) R 223

5 where, J R s known as reduced Jacobean and s gven as follows: 1 J J J J J (6) R QV Q PV P The voltage stablty analyss s further wth help of sub matrx Jacoban s gven n the followng equaton: Qk qdk Lk BkkVk (7) Vk Vk where, q dk s reactve power demand at nth bus, L k s voltage stablty ndex at nth bus, B kk s magnary part of admttance matrx [3]. Usng the reduced Jacoban matrx, the senstvty of voltage stablty ndex wth respect to VAR njecton at kth bus can be wrtten as: Qk VSI V Vk (8) V J 1 R Qnj Voltage stablty ndex depends upon the followng parameters voltage profle mprovement, reactve power demand, voltage at kth bus and connectvty of the bus,.e. B kk Generally the product B kk V k s mportant and domnant. If B kk s large then relatvely lesser voltage magntude may be suffcent to gve requred voltage stablty margn [3]. NC f k k kj k k 1 L L A C (9) where, ΔC k s kth bus change n reactve power control varables, NC s total number of reactve power control varables whch ncludes PV buses, tap changers and swtchable shunt reactors, A kj s the senstvty coeffcent of VSI wth respect to the change n reactve power control varables. In order to mprove the voltage stablty and mantan the voltage profle end results of Q-V analyss, t s requred to nject reactve power at the crtcal and weak buses [3]. D. Voltage Stablty Index, L-Index For voltage stablty bus evaluaton uses L-ndex [52], [53], the ndcator value vares n the range between zero (the no load condton) and one (voltage collapse) whch corresponds to [18]: Ibus YbusV bus (10) By segregatng the load buses from generator buses, can wrte as: IL Y1 Y2 VL I G Y3 Y 4 V (11) G VL IL H1 H2 IL H I G V G H3 H 4 V (12) G where, V L and I L are voltages and currents at the load buses, V G, I G are voltages and currents at the generator buses, H 1, H 2, H 3, H 4 are sub-matrces of the hybrd matrx H, generated from bus Y partal nverson. From Equatons (11) and (12), we can wrte as [18]: 1 1 V H I H V Y I Y Y V (13) L 1 L 2 G 1 L 1 2 G H Y Y (14) The no load condton, currents at the load buses (I L ) are zero, can be wrtten as: V H V (15) 0j 2j G where, V 0j s voltages at bus j for no load condton. Ths representaton can then be used to defne a voltage stablty ndcator at the load bus, whch s gven by [18]: L 1 V / V (16) j 0 j j where, L j s L-ndex voltage stablty ndcator for bus j, V j s voltage for bus j. The L-ndex approaches the numercal value 1.0, when a load bus approaches a steady state voltage collapse stuaton. So f the ndex evaluated at any bus s less than unty, the system can keep voltage stablty [18]. V. PROBLEM FORMULATION The ORPF formulaton ncludes the objectve functons, the varable constrant condtons and the load flow constrant equatons [12]. A. Objectve Functon The mult-objectve functons of the power system ORPF nclude the techncal goal and the economc goal. The economc goal s manly to mnmze the system actve power transmsson loss. The techncal goals are to mnmze the load buses voltage devaton from the deal voltage and to mprove the voltage stablty margn (VSM). Therefore, mult-objectve functons for both the techncal and economc goals are consdered n ths paper as follows [12]: mn( PL ) f ( x) mn( Vb ) (17) ( VSM ) where, P L s total real power losses, V b s voltage devaton, VSM s the voltage stablty margn. B. Voltage Devaton Objectve Functon The voltage devaton objectve functon can be wrtten as the mnmum of the total sum of each load bus voltage devaton [12]: deal B Vb Vb V b mn Vb mn (18) b1 V b where, V b s the actual voltage of the system load bus b, deal Vb s the deal voltage of the load bus b and δv b s the mum permtted voltage devaton of the load bus b. In ths paper, V s 1 pu and δv b s -5% to +5%. When deal b deal b V b < V, δv b = -5%, otherwse δv b = +5%. The functon ϕ(x) s: 0 f x 0 ( x) (19) x f else x 0 In addton, B s the total number of system load buses. When the voltage V b of load bus b s runnng at deal deal V V, V V b b b b, deal Vb Vb Vb 0 C. System Voltage Stablty Margn. 224

6 As mentoned before, there are many ndces for ensurng voltage stablty ssue. Therefore, we can choose one of them regardng to our problem formulaton proporton. In ths paper we wll use statc voltage stablty margn can be measured by the mnmal egenvalue of the non-sngular Jacoban matrx n a mult-generator system. So, enhancng the mnmal egenvalue of the non-sngular Jacoban matrx can be wrtten as [12]: ( VSM ) mn eg( Jacob) (20) where, Jacob s the Jacoban matrx of the power flow, eg(jacob) s all the egenvalues of the Jacoban matrx, mn(eg(jacob)) s the mnmum of the egenvalues n the Jacoban matrx and (mn(eg(jacob))) s mzng the mnmal egenvalue n the Jacoban matrx. Thus, the objectve functon of the ORPF s [12]: mn( F) mn P,, ( ) T L Vb VSM (21) D. System Varable Constrant Condtons Varable constrant condtons nclude the control and the state varable constrant condtons. The control varable constrant condtons nclude the transformer tap changer settng T, the compensatng capactance capacty C and the generator bus voltage U. The state varables nclude each load bus voltage and each generator bus output reactve power Q. Thus, the varable constrant condtons may be wrtten as [12]: Vgk mn Vgk Vgk T mn T T (22) C j mn C j C j Qgk mn Qgk Qgk (23) Vl mn Ul Vl where, V gk mn (V gk ), T mn (T ), C j mn (C j ), Q gk mn (Q gk ) and V l mn (V l ) are the lower (upper) t values of the generator bus voltage, transformer rato, capacty of compensaton capactor, generator bus reactve power and each load bus voltage, respectvely [12]. E. System Power Flow Constrant Equatons The ORPF must satsfy the system power flow equatons, whch are wrtten as: n G L j j j j j j1 P P P V V G cos B sn 0 (24) r Q QG QC QL 1 n V V j Gj snj Bj cosj 0 j1 The system actve power loss s: n (25) cos sn (26) P V V G B L j j j j j 1 jh where, n s the total number of nodes, P G, Q G are the bus generator actve power and reactve power, respectvely, P L, Q L are the bus load actve power and reactve power, respectvely, V, V j are the buses and j voltages, respectvely, and G j, B j, δ j are the conductance and phase angle between bus and j, respectvely, h s the number of buses connectng wth bus. At the same tme, the system transmsson power s ted by the upper capacty of the branch (transformer and transmsson lne) [12]. In addton, consder that, the mentoned problem formulaton can use for all optmzaton algorthms by a lttle changes n equaton forms. At the followng we wll note to mscellaneous algorthms test results and comparson between them. VI. AN EXAMPLE OF REACTIVE POWER OPTIMIZATION USING A HEURISTIC ALGORITHM - GENETIC ALGORITHM A. Introducton to Genetc Algorthm A.1. Representaton of Desgn Varables In GAs, the desgn varables are represented as strngs of bnary numbers, 0 and 1. For example, f a desgn varable x s denoted by a strng of length four (or a fourbt strng) as ( ), ts nteger (decmal equvalent) value wll be =5. If each desgn varable x, =1, 2 n s coded n a strng of length q, a desgn vector s represented usng a strng of total length nq. For example, f a strng of length 5 s used to represent each varable, a total strng of length 20 descrbes a desgn vector wth n=4. The followng strng of 20 bnary dgts denote the vector (x 1 =18, x 2 =3, x 3 =1, x 4 =4) [54-58]: Fgure 5. Example of strng length In general, f a bnary number s gven by b q b q-1 b 2 b 1 b 0, where b k =0 or 1, k=1, 2, q then ts equvalent decmal number y (nteger) s gven by: q y 2 b (27) k0 k k Ths ndcates that a contnuous desgn varable x can only be represented by a set of dscrete values f bnary representaton s used. If a varable x (whose bounds are gven by x l and x u ) s represented by a strng of q bnary numbers, as shown n Equaton (27), ts decmal value can be computed as [54-58]: u l q l x x k x x 2 b q k (28) 2 1 k0 Thus f a contnuous varable s to be represented wth hgh accuracy, we need to use a large value of q n ts bnary representaton. In fact, the number of bnary dgts needed (q) to represent a contnuous varable n steps (accuracy) of x can be computed from relaton [54-58]: u l q x x 2 1 (29) x For example, f a contnuous varable x wth bounds 1 and 5 s to be represented wth an accuracy of 0.01, we need to 225

7 use a bnary representaton wth q dgts where q or. Equaton (28) shows why GAs are naturally suted for solvng dscrete optmzaton problems [54-58]. q 9 A.2. Representaton of Objectve Functon and Constrants Because Genetc Algorthms are based on the survval of the fttest prncple of nature, they try to mze a functon called the ftness functon. Thus GAs are naturally sutable for solvng unconstraned mzaton problems. The ftness functon, F(X), can be taken to be same as the objectve functon f(x) of an unconstraned mzaton problem so that F(X) = f(x). A mnmzaton problem can be transformed nto a mzaton problem before applyng the GAs. Usually the ftness functon s chosen to be nonnegatve. The commonly used transformaton to convert an unconstraned mnmzaton problem to a ftness functon s gven by [54-58]: 1 F( X) (30) 1 f( X) It can be seen that Equaton (30) does not alter the locaton of the mnmum of f(x) but converts the mnmzaton problem nto an equvalent mzaton problem. A general constraned mnmzaton problem can be stated as: Mnmze f(x) subject to g (X) 0; =1, 2,, m and h j (X) 0; j=1, 2,, p. Ths problem can be converted nto an equvalent unconstraned mnmzaton problem by usng concept of penalty functon as [54-58]: mnmze ( X ) f ( X ) r g ( X ) p j1 R h ( X ) j j 2 m 1 2 (31) where r and R j are the penalty parameters assocated wth the constrants g (X) and h j (X), whose values are usually kept constant throughout soluton process. In Equaton (5), the functon g (X), called the bracket functon, s defned as [54-58]: g( X ) f g( X ) 0 g ( X) (32) 0 f g ( X ) 0 In most cases, the penalty parameters assocated wth all the nequalty and equalty constrants are assumed to be the same constants as: r =r; =1, 2,, m and R j =R; j=1, 2,, p, where r and R are constants. The ftness functon, F(X), to be mzed n the GAs can be obtaned, smlar to Equaton (30), as [54-58]: 1 F( X) (33) 1 ( X ) Equatons (31) and (32) show that the penalty wll be proportonal to the square of the amount of volaton of the nequalty and equalty constrants at the desgn vector X, whle there wll be no penalty added to f(x) f all the constrants are satsfed at the desgn vector X [54-58]. The soluton of an optmzaton problem by GAs starts wth a populaton of random strngs denotng several (populaton of) desgn vectors. The populaton sze n GAs (n) s usually fxed. Each strng (or desgn vector) s evaluated to fnd ts ftness value. The populaton (of desgns) s operated by three operators reproducton, crossover, and mutaton to produce a new populaton of ponts (desgns). The new populaton s further evaluated to fnd the ftness values and tested for the convergence of the process [54-58]. One cycle of reproducton, crossover, and mutaton and the evaluaton of the ftness values s known as a generaton n GAs. If the convergence crteron s not satsfed, the populaton s teratvely operated by the three operators and the resultng new populaton s evaluated for the ftness values. The procedure s contnued through several generatons untl the convergence crteron s satsfed and the process s termnated. The detals of the three operatons of GAs are gven below [54-58]. A.4. Reproducton Reproducton s the frst operaton appled to the populaton to select good strngs (desgns) of the populaton to form a matng pool. The reproducton operator s also called the selecton operator because t selects good strngs of the populaton. The reproducton operator s used to pck above average strngs from the current populaton and nsert ther multple copes n the matng pool based on a probablstc procedure. In a commonly used reproducton operator, a strng s selected from the matng pool wth a probablty proportonal to ts ftness. Thus f F denotes the ftness of the strng n the populaton of sze n, the probablty for selectng the th strng for the matng pool (p ) s gven by [54-58]: F p 1,...,n (34) n j1 F j Note that Equaton (34) mples that the sum of the probabltes of the strngs of the populaton beng selected for the matng pool s one. The mplementaton of the selecton process gven by Equaton (34) can be understood by magnng a roulette wheel wth ts crcumference dvded nto segments, one for each strng of the populaton, wth the segment lengths proportonal to the ftness of the strngs as shown n Fgure (5) [54-58]. By spnnng the roulette wheel n tmes (n beng the populaton sze) and selectng, each tme, the strng chosen by the roulette-wheel ponter, we obtan a matng pool of sze n. Snce the segments of the crcumference of the wheel are marked accordng to the ftness of the varous strngs of the orgnal populaton, the roulette-wheel process s expected to select F/F copes of the th strng for the matng pool, where F denotes the average ftness of the populaton [54-58]: 1 n Fj n j 1 F (35) A.3. Genetc Operators 226

8 Fgure 6. Roulette-Wheel selecton scheme In Fgure (5), the populaton sze s assumed to be 6 wth ftness values of the strngs 1, 2, 3, 4, 5, and 6 gven by 12, 4, 16, 8, 36, and 24, respectvely. Snce the ffth strng (ndvdual) has the hghest value, t s expected to be selected most of the tme (36% of the tme, probablstcally) when the roulette wheel s spun n tmes (n=6 n Fgure (5)). The selecton scheme, based on the spnnng of the roulette wheel, can be mplemented numercally durng computatons as follows [54-58]. The probabltes of selectng dfferent strngs based on ther ftness values are calculated usng Equaton (34). These probabltes are used to determne the cumulatve probablty of strng beng coped to matng pool, p by addng ndvdual probabltes of strngs 1 through as: P p (36) j1 j Thus the roulette-wheel selecton process can be mplemented by assocatng the cumulatve probablty range P -1 -P to the th strng. To generate the matng pool of sze n durng numercal computatons, n random numbers, each n the range of zero to one, are generated (or chosen). By treatng each random number as the cumulatve probablty of the strng to be coped to the matng pool, n strngs correspondng to the n random numbers are selected as members of matng pool [54-58]. By ths process, the strng wth a hgher (lower) ftness value wll be selected more (less) frequently to the matng pool because t has a larger (smaller) range of cumulatve probablty. Thus strngs wth hgh ftness values n the populaton, probablstcally, get more copes n the matng pool. It s to be noted that no new strngs are formed n the reproducton stage; only the exstng strngs n the populaton get coped to the matng pool. The reproducton stage ensures that hghly ft ndvduals (strngs) lve and reproduce, and less ft ndvduals (strngs) de. Thus the GAs smulate the prncple of survval-of-the-fttest of nature [54-58]. A.5. Crossover After reproducton, the crossover operator s mplemented. The purpose of crossover s to create new strngs by exchangng nformaton among strngs of the matng pool. Many crossover operators have been used n the lterature of GAs. In most crossover operators, two ndvdual strngs (desgns) are pcked (or selected) at random from the matng pool generated by the reproducton operator and some portons of the strngs are exchanged between the strngs [54-58]. In the commonly used process, known as a sngle-pont crossover operator, a crossover ste s selected at random along the strng length, and the bnary dgts (alleles) lyng on the rght sde of the crossover ste are swapped (exchanged) between the two strngs. The two strngs selected for partcpaton n the crossover operators are known as parent strngs and the strngs generated by the crossover operator are known as chld strngs. For example, f two desgn vectors (parents), each wth a strng length of 10, are gven by [54-58]: (Parent1) X { } 1 (Parent 2) X { } 2 The result of crossover, when the crossover ste s 3, s gven by: ( Offsprng1) X { } 3 ( Offsprng2) X { } 4 Snce the crossover operator combnes substrngs from parent strngs (whch have good ftness values), the resultng chld strngs created are expected to have better ftness values provded an approprate (sutable) crossover ste s selected. However, the sutable or approprate crossover ste s not known beforehand. Hence the crossover ste s usually chosen randomly. The chld strngs generated usng a random crossover ste may or may not be as good as or better than ther parent strngs n terms of ther ftness values [54-58]. If they are good or better than ther parents, they wll contrbute to a faster mprovement of the average ftness value of the new populaton. On the other hand, f the chld strngs created are worse than ther parent strngs, t should not be of much concern to the success of the GAs because the bad chld strngs wll not survve very long as they are less lkely to be selected n the next reproducton stage (because of survval-of-the-fttest strategy used) [54-58]. As ndcated above, the effect of crossover may be useful or detrmental. Hence t s desrable not to use all the strngs of the matng pool n crossover but to preserve some of the good strngs of the matng pool as part of the populaton n the next generaton. In practce, a crossover probablty, p c s used n selectng the parents for crossover. Thus only 100p c percent of the strngs n the matng pool wll be used n the crossover operator whle 100(1-p c ) percent of the strngs wll be retaned as they are n the new generaton (of populaton) [54-58]. A.6. Mutaton The crossover s the man operator by whch new strngs wth better ftness values are created for the new generatons. The mutaton operator s appled to the new strngs wth a specfc small mutaton probablty, p m. The mutaton operator changes the bnary dgt (allele s value) 1 to 0 and vce versa. Several methods can be used for mplementng the mutaton operator [54-58]. In the sngle-pont mutaton, a mutaton ste s selected at random along the strng length and the bnary dgt at that ste s then changed from 1 to 0 or 0 to 1 wth a probablty of p m. In the bt-wse mutaton, each bt (bnary dgt) n the strng s consdered one at a tme n sequence, and the dgt s changed from 1 to 0 or 0 to 1 wth a 227

9 probablty p m. Numercally, the process can be mplemented as follows. A random number between 0 and 1 s generated/chosen [54-58]. If the random number s smaller than p m, then the bnary dgt s changed. Otherwse, the bnary dgt s not changed. The purpose of mutaton s 1- to generate a strng (desgn pont) n neghborhood of current strng, thereby accomplshng a local search around the current soluton, 2- to safeguard aganst a premature loss of mportant genetc materal at a partcular poston, and 3- to mantan dversty n the populaton [54-58]. As an example, consder the followng populaton of sze n = 5 wth a strng length 10: Here all the fve strngs have a 1 n the poston of the frst bt. The true optmum soluton of the problem requres a 0 as the frst bt. The requred 0 cannot be created by ether the reproducton or the crossover operators. However, when the mutaton operator s used, the bnary number wll be changed from 1 to 0 n the locaton of the frst bt wth a probablty of np m [54-58]. Fgure 7. Flowchart of GA based RPD algorthm Note that the three operator s reproducton, crossover, and mutaton are smple to mplement. The reproducton operator selects good strngs for the matng pool, the crossover operator recombnes the substrngs of good strngs of the matng pool to create strngs (next generaton of populaton), and the mutaton operator alters the strng locally. The use of these three operators successvely yelds new generatons wth mproved values of average ftness of the populaton [54-58]. Although, the mprovement of the ftness of the strngs n successve generatons cannot be proved mathematcally, the process has been found to converge to the optmum ftness value of the objectve functon. Note that f any bad strngs are created at any stage n the process, they wll be enated by the reproducton operator n the next generaton. The GAs have been successfully used to solve a varety of optmzaton problems n the lterature [54-58]. B. Ftness Functon for Implementaton of Genetc Algorthm to Reactve Power Optmzaton Problem In the RPD problem under consderaton the objectve s to mnmze the total power loss satsfyng the constrants n Equatons (22) to (25). For each ndvdual, the equalty constrants (24) and (25) are satsfed by runnng Newton-Raphson algorthm and the constrants on the state varables are taken nto consderaton by addng a quadratc penalty functon to the objectve functon. Wth the ncluson of penalty functon, the new objectve functon then becomes [16]: mnmze f ( V,Q,T,C )= P + NPQ N 2 2 g + k V V k Q Q v q g g 1 1 NT 2 N 2 C f l 1 1 NC g loss k T T k C C 2 N 2 C k S S k L L s h 1 1 (37) where, k v, k q, k f, k l, ks, and k h are penalty factors, V s generator bus voltages, Q g s reactve power generaton va generators, T s tap changer transformers tap poston, C s capactors reactve power generaton, S s transmsson lnes ts, L s voltage stablty ndex. In the above objectve functon V and Q are defned as [16]: V Q g T V V V V V V mn mn f f g f g g g (38) mn mn Qg f Qg Qg (39) Q Q Q T T T T T T mn mn f f (40) mn mn C f C C C (41) C f C C The value of the penalty factor should be large so that there s no volaton for unt output at the fnal soluton. Snce GA s desgned for the soluton of mzaton problems, the GA ftness functon s defned as the nverse of Equaton (37) [58]. 1 F ftness (42) f Therefore, we should optmze Equaton (37) then reactve power n the power system wll be optmzed. The flowchart and steps of reactve power optmzaton by genetc algorthm s shown n Fgure

10 VII. SIMULATION RESULTS The IEEE 57-bus system are used as the test case to examne the performance of several algorthm and compare them wth other heurstc algorthms. For test system, the lower and upper ts of load bus voltages are 0.95 p.u. and 1.05 p.u., respectvely. Generator voltages at the hgh voltage termnal are defned as contnuous varables. The lower and upper ts are set to 0.94 p.u. and 1.06 p.u., respectvely [14]. Dscrete control varables consst of transformer tap postons and the susceptance of shut compensators. All Under-Load Tap Changng (ULTC) transformers are assumed to have 21 dscrete taps wthn ±10% of the nomnal voltage (1% for each tap). Each transformer tap s defned by an nteger between -10 to 10. These ULTC data are fcttous values. The number of taps and the voltage range n practcal cases can be dfferent. All shunt compensators have 11 dscrete steps of dfferent ratngs (defned by an nteger between 0 and 10). The performance of MVMO s compared wth followng algorthms [14]. 1- PSO: A standard PSO verson 2007 [59] 2- DE: A basc DE namely DE/current-to-best/1 [60, 61] 3- JADE: An adaptve DE algorthm [62] 4- JADE-vPS: A modfed JADE algorthm [63] The IEEE 57-bus system conssts of seven generators, 80 lnes where 15 of whch are equpped wth ULTC transformers. Shunt reactve power compensators are connected to buses 18, 25 and 53. The t of these susceptances s [0, 0.2], [0, 0.18] and [0, 0.18], respectvely. Therefore, the ORPD search space has 25 dmensons. The populaton sze PS of PSO, DE and JADE and the ntal value of PS n JADE-vPS s set to 50 [14]. Fgure 9. Load bus voltage profles Fgure 10. Convergence of generator voltages Table 1. Statstcal results for actve power loss n MW MVMP PSO DE JADE JADE-vPS Mn Ave Max Standard devaton Fgure 11. Convergence of selected transformer taps Fgure 8. Average convergence characterstcs To farly compare each algorthms wth others, every algorthm s ndependently run for 50 tmes. Then statstcal values consstng of mnmum, average, mum and standard devaton of actve power losses are computed as lsted n Table 1 [14]. Fgure 12. Convergence of shunt compensator 229

11 The average convergence of actve power losses found by each algorthm s plotted after the frst 100 FEs as shown n Fgure 8. It s clearly shown that the convergence of MVMO s the fastest. In ths test case, the statstcal results of MVMO n Table 1 are not outstandng the other algorthms. However, the MVMO results are on average very close to the other technques. An nterestng observaton made from Fgure 8 s that MVMO s very fast n the global search capablty because the lowest power loss has been found after the frst 100 Fes [14]. As mentoned n [64] that there are fve buses (buses 25, 30, 31, 32 and 33) n ths network that the voltages are outsde the ts. After the ORPD result gven by each method, power flow s calculated to determne bus voltages as shown n Fgure 9. It s shown that all bus voltages can be mantaned wthn the ts. These voltage profles confrm the merts of ORPD n achevng both reduced power losses and voltage securty [14]. The convergence of optmzed control varables are shown n Fgures 10 to 12. From these fgures, the control varables change abruptly at the early searchng stage. Then, they settle to a steady state at the later stage. At ths phase, an optmum has been dscovered. The CPU tme of all methods s approxmately 5 mnutes [14]. VIII. CONCLUSIONS In ths paper, reactve power optmzaton s fully ntroduced and some studes n ths fled ntroduced too. In recent years, reactve power control problem has been concerned wth scences and researchers due to reactve power hghly affect n power system operaton and control. Reactve power has drect and non-drect relatonshp wth all parameters of power system, so reactve power optmzatons wll be a nonlnear and non-convex optmzaton problem. In addton, power system stablty and relablty ndexng have drect relatonshp by reactve power balance n power system. Reactve power can change and also control voltages of system buses drectly and keep them n deal ranges. Therefore, reactve power balance s very mportant to satsfyng preferred ranges n bus voltages. For checkng voltage stablty of power system many ndces are proposed and each of them have advantages and dsadvantages n problem formulaton and optmzaton process. Classc algorthms cannot reach to global optmum of optmzaton problems as reactve power optmzaton due to gradent-based optmzaton algorthms nature. Therefore, n recent years, efforts to reach to relable and accurate algorthm to solve ths problem have been done and advantages and dsadvantages of them ntroduced. Fnally, the IEEE 57-bus test system optmzaton results have been presented and comparsons between some of them have been done n smulaton result secton. REFERENCES [1] G.S. Salesh Babu, D. Bhagwan Das, C. Patvardhan, Quantum Evolutonary Algorthm Soluton of Real Valued Functon Optmzaton - Reactve Power Dspatch Example, XXXII Natonal Systems Conference, NSC, pp , December [2] D. Taleb Khanmr, et al., Reactve Power Optmal Dspatch Usng Genetc Algorthm to Reduce Real Power Losses n Power System, 11th Conference on Power Dspatch Networks, pp , Mazandaran, Iran. [3] S. Jaganathan, S. Palansamy, Foragng Algorthm for Optmal Reactve Power Dspatch wth Voltage Stablty and Relablty Analyss, Journal of Theoretcal and Appled Informaton Technology, Vol. 47, No. 1, pp , Jan [4] A.H. Mantawy, M.S. Al-Ghamd, A New Reactve Power Optmzaton Algorthm, IEEE Bologna Power Tech Conference, Bologna, Italy, June [5] P.R. Sujn, T.R.D. Prakash, M.M. Lnda, Partcle Swarm Optmzaton Based Reactve Power Optmzaton, Journal of Computng, Vol. 2, Issue 1, pp , Jan [6] K. Lenn, M.R. Mohan, Ant Colony Search Algorthm for Optmal Reactve Power Optmzaton, Serban Journal of Electrcal Engneerng, Vol. 3, No. 1, pp , June [7] R. Ouddr, M. Rahl, L. Abdelhakem Kordak, Economc Dspatch Usng a Genetc Algorthm - Applcaton to Western Algera s Electrcal Power Network, Journal of Informaton Scence and Engneerng, Vol. 21, pp , [8] S. Sakthvel, M. Gayathr, V. Manmozh, A Nature Inspred Optmzaton Algorthm for Reactve Power Control n a Power System, Internatonal Journal of Recent Technology and Engneerng (IJRTE), Vol. 2, Issue 1, pp , March [9] J. Zhu, Optmzaton of Power System Operaton, John Wley & Sons Inc. Publcaton, Hoboken, New Jersey, [10] B. Sngh Prajapat, L. Srvastava, Mult-Objectve Reactve Power Optmzaton Usng Artfcal Bee Colony Algorthm, Internatonal Journal of Engneerng and Innovatve Technology (IJEIT), Vol. 2, Issue 1, pp , July [11] M. Abdelmoumene, B. Mohamed, Optmal Reactve Power Dspatch Usng Effcent Partcle Swarm Optmzaton Algorthm, frames/artcles/ix/ix-2_messaoudi_djelfa.pdf. [12] X. Hugang, Ch. Haozhong, L. Hayu, Optmal Reactve Power Flow Incorporatng Statc Voltage Stablty Based on Mult-Objectve Adaptve Immune Algorthm, Elsever, Energy Converson and Management, Vol. 49, pp , [13] Y.Q. Chen, Y. Zhang, Y. We, Applcaton of Improved Genetc Algorthm Combnng Senstvty Analyss to Reactve Power Optmzaton for Power System, DRPT, pp , Nanjng, Chna, Aprl [14] W. Nakawro, I. Erlch, J.L. Rueda, A Novel Optmzaton Algorthm for Optmal Reactve Power Dspatch - A Comparatve Study, IEEE, pp ,

12 [15] B. Radbratovc, Reactve Optmzaton of Transmsson and Dstrbuton Networks, A Thess Presented to the Academc Faculty, Georga Insttute of Technology, May [16] S. Duraraj, P.S. Kannan, D. Devaraj, Applcaton of Genetc Algorthm to Optmal Reactve Power Dspatch Includng Voltage Stablty Constrant, Journal of Energy & Envronment, Vol. 4, pp , [17] H. Sharfzade, N. Amjad, Reactve Power Optmzaton Usng Partcle Swarm Optmzaton, Journal of Smulatons n Engneerng, No. 18, pp , Iran, [18] C. Thammasrrat, B. Marungsr, R. Oonsvla, A. Oonsvla, Optmal Reactve Power Dspatch Usng Dfferental Evoluton, World Academy of Scence, Engneerng and Technology, Vol. 60, pp , [19] H.W. Dommel, W.F. Tnny, Optmal Power Flow Solutons, IEEE Trans. on Power App & Systems, Vol. 87, pp , [20] K.Y. Lee, Y.M. Park, J.L. Ortz, A Unted Approach to Optmal Real and Reactve Power Dspatch, IEEE Transactons on Power Apparatus and Systems (PAS), Vol. 104, pp , [21] G.R. M. Da Costa, Optmal Reactve Dspatch Through Prmal-Dual Method, IEEE Trans. on Power Systems, Vol. 12, No. 2, pp , May [22] L.D.B. Terra, M.J. Short Securty Constraned Reactve Power Dspatch, IEEE Trans. on Power Systems, Vol. 6, No. 1, February [23] K.H.A. Rahman, S.M. Shahdehpour, Applcaton of Fuzzy Set Theory to Optmal Reactve Power Plannng wth Securty Constrants, IEEE Trans. on Power Systems, Vol. 9, No.2, pp , May [24] K.R.C. Mamandur, R.D. Chenoweth, Optmal Control of Reactve Power Flow for Improvements n Voltage Profles and for Real Power Loss Mnmzaton, IEEE Transactons on Power Systems, Vol. PAS-100, No. 7, pp ,1981. [25] J.M. Brght, K.D. Demaree, J.P. Brtton, Reactve Securty and Optmalty n Real Tme, IFAC Power Systems and Power Plant Control, pp , Bejng, [26] J. Goossens, Reactve Power and System Operaton - Incpent Rsk of Generator Constrants and Voltage Collapse, Invted Paper, IFAC Power Systems and Power Plant Control, pp. 1-10, Seoul, Korea, [27] L.D.B. Terra, M.J. Short, A Global Approach for VAR/Voltage Management, IFAC Power Systems and Power Plant Control, pp , Seoul, Korea, [28] N. Deeb, S.M. Shadepour, Lnear Reactve Power Optmzaton n a Large Power Network Usng the Decomposton Approach, IEEE Transactons on Power Systems, Vol. 5, No. 2, pp , [29] S. Granvlle, Optmal Reactve Dspatch through Interor Pont Methods, IEEE Transactons on Power Systems, Vol. 9, No. 1, pp , [30] B. Cova, N. Losgnore, P. Marannno, M. Montagna, Contngency Constraned Optmal Reactve Power Flow Procedures for Voltage Control n Plannng and Operaton, IEEE Transactons on Power Systems, Vol. 10, No. 2, pp , [31] J.R.S. Manlovan, A.V. Garca, A Heurstc Method for Reactve Power Plannng, IEEE Transactons on Power Systems, Vol. 11, No. 1, pp , [32] K. Iba, Reactve Power Optmzaton by Genetc Algorthm, IEEE Transactons on Power Systems, Vol. 9, No. 2, [33] K.Y. Lee, Y.M. Park, Optmzaton Method for Reactve Power Plannng by Usng a Modfed Smple Genetc Algorthm, IEEE Transactons on Power Systems, Vol. 10, No. 4, pp , [34] Q.H. Wu, J.T. Ma, Power System Optmal Reactve Power Dspatch Usng Evolutonary Programmng, IEEE Transactons on Power Systems, Vol. 10, No. 3, pp , [35] J.T. Ma, L.L. La, Evolutonary Programmng Approach to Reactve Power Plannng, IEE proceedngs on Generaton, Transmsson and Dstrbuton, Vol. 143, No. 4, pp , [36] L.L. La, J.T. Ma, Applcaton of Evolutonary Programmng to Reactve Power Plannng - Comparson wth Nonlnear Programmng Approach, IEEE Transactons on Power Systems, Vol. 12, No. 1, pp , [37] L.L. La, J.T. Ma, Practcal Applcaton of Evolutonary Computng to Reactve Power Plannng, IEE proceedngs on Generaton, Transmsson and Dstrbuton, Vol. 145, No. 6, pp , [38] Q.H. Wu, Y.J. Cao, J.Y. Wen, Optmal Reactve Power Dspatch Usng an Adaptve Genetc Algorthm, Electrcal Power and Energy Systems, Vol. 20, No. 8, pp , [39] D.B. Das, C. Patvardhan, Reactve Power Dspatch wth Hybrd Stochastc Search Technque, Electrcal Power and Energy Systems, Vol. 24, pp , [40] D.B. Das, C. Patvardhan, A New Hybrd Evolutonary Strategy for Reactve Power Dspatch, Electrc Power Systems Research, Vol. 65, pp , [41] Y. Zhang, Z. Ren, Real-Tme Optmal Reactve Power Dspatch Usng Mult-Agent Technque, Electrc Power Systems Research, Vol. 69, pp , [42] C. Jang, C. Wang, Improved Evolutonary Programmng wth Dynamc Mutaton and Metropols Crtera for Mult-Objectve Reactve Power Optmzaton, IEE Proceedngs on Generaton, Transmsson and Dstrbuton, Vol. 152, No. 2, pp , March [43] C. Jang, E. Bompard, A Hybrd Method of Chaotc Partcle Swarm Optmzaton and Lnear Interor for Reactve Power Optmzaton, Mathematcs and Computers n Smulaton, Vol. 68, pp ,

13 [44] B. Zhao, C.X. Guo, Y.J. Cao, A Mult-Agent Based Partcle Swarm Optmzaton Approach for Optmal Reactve Power Dspatch, IEEE Transactons on Power Systems, Vol. 20, No. 2, pp , [45] K.H. Han, J.H. Km, Quantum-Inspred Evolutonary Algorthms wth a New Termnaton Crteron, HE Gate, and Two-Phase Scheme, IEEE Transactons on Evolutonary Computaton, Vol. 8, No. 2, pp , [46] K.H. Han, J.H. Km, Quantum-Inspred Evolutonary Algorthms wth a New Termnaton Crteron, HI Gate, and Two-Phase Scheme, IEEE Transactons on Evolutonary Computaton, Vol. 8, No. 2, pp , [47] R. Raghunathaa, R. Ramanujama, K. Parthasarathyb, D. Thukaramb, Optmal Statc Voltage Stablty Improvement Usng a Numercally Stable SLP Algorthm for Real Tme Applcatons, Electr. Power Energy Syst., Vol. 21, pp , [48] F. Zhhong, L. Qu, N. Yxn, et al, Analyss Steady- Stale Voltage Stablty n Mult-Machne Power Systems by Sngular Value Decomposton Method, CSEE, Vol. 12, pp. 8-10, Chnese, [49] E.E. Souza Lma, L. Flomeno, Assessng Egenvalue Senstvtes, IEEE Trans Power Syst., Vol. 15, No. 1, pp , [50] H.C. Nallan, P. Rastgoufard, Computatonal Voltage Stablty Assessment of Large-Scale Power Systems, Electrcal Power Systems Research, Vol. 38, pp , [51] A. Berzz, P. Bresest, P. Marannno, G.P. Granell, M. Montagna, System-Area Operatng Margn Assessment and Securty Enhancement Aganst Voltage Collapse, IEEE Trans. Power Syst., Vol. 11, No. 3, pp , [52] K. Vasakh, P.K. Rao, Dfferental Evoluton Based Optmal Reactve Power Dspatch for Voltage Stablty Enhancement, Journal of Theoretcal and Appled Informaton Technology, pp , [53] C.A. Belhadj, M.A. Abdo, An Optmzaton Fast Voltage Stablty Indcator, IEEE Power Tech 99 Conference, Budapest, Hungary, [54] S.R. Sngresu, Engneerng Optmzaton - Theory and Practce, Fourth Edton, July [55] R.L. Haupt, S.E. Haupt, Practcal Genetc Algorthms Second Edton, John Wley and Sons, New York, [55] F.M. Bayat, Optmzaton Algorthms Inspred by Nature, Unversty of Zanjan, Zanjan, Iran. [56] M. Melane, An Introducton to Genetc Algorthms, Cambrdge, UK, [57] N.M. Tabatabae, A. Jafar, N.S. Boushehr, K. Dursun, Genetc Algorthm Applcaton n Economc Load Dstrbuton, Internatonal Journal on Techncal and Physcal Problems of Engneerng (IJTPE), Issue 17, Vol. 5, No. 4, pp , December [58] V.Ya. Lyubchenko, D.A. Pavlyuchenko, Reactve Power and Voltage Control by Genetc Algorthm and Artfcal Neural Network, Internatonal Journal on Techncal and Physcal Problems of Engneerng (IJTPE), Issue. 1, Vol. 1, No. 1, pp , December [59] Standard PSO 2007 (SPSO-2007) on the Partcle Swarm Central, [Onlne] [60] R. Storn, K. Prce, Dfferental Evoluton - A Smple and Effcent Heurstc for Global Optmzaton Over Contnuous Spaces, Journal of Global Optmzaton Vol. 11, pp , [61] K. Prce, R. Storn, J. Lampnen, Dfferental Evoluton - A Practcal Approach to Global Optmzaton, Sprnger Verlag, [62] J. Zhang, A.C. Sanderson, JADE - Adaptve Dfferental Evoluton wth Optonal External Archve, IEEE Trans. Evol. Compt., Vol. 13, No. 5, pp , Oct [63] W. Nakawro, Voltage Stablty Assessment and Control of Power Systems usng Computatonal Intellgence, Ph.D. Thess, Unversty of Dusburg-Essen, [64] C. Da, W. Chen, Y. Zhu, X. Zhang, Seeker Optmzaton Algorthm for Optmal Reactve Power Dspatch, IEEE Trans. Power Syst., Vol. 24, No. 3, pp , Aug BIOGRAPHIES Naser Mahdav Tabatabae was born n Tehran, Iran, He receved the B.Sc. and M.Sc. degrees from Unversty of Tabrz (Tabrz, Iran) and the Ph.D. degree from Iran Unversty of Scence and Technology (Tehran, Iran), all n Power Electrcal Engneerng, n 1989, 1992, and 1997, respectvely. Currently, he s a Professor n Internatonal Organzaton of IOTPE. He s also an academc member of Power Electrcal Engneerng at Seraj Hgher Educaton Insttute (Tabrz, Iran) and teaches power system analyss, power system operaton, and reactve power control. He s the General Secretary of Internatonal Conference of ICTPE, Edtor-n-Chef of Internatonal Journal of IJTPE and Charman of Internatonal Enterprse of IETPE all supported by the IOTPE ( He has authored and co-authored of sx books and book chapters n Electrcal Engneerng area n nternatonal publshers and more than 130 papers n nternatonal journals and conference proceedngs. Hs research nterests are n the area of power qualty, energy management systems, ICT n power engneerng and vrtual e-learnng educatonal systems. He s a member of the Iranan Assocaton of Electrcal and Electronc Engneers (IAEEE). 232

14 Al Jafar was born n Zanjan, Iran n He receved the B.Sc. degree n Power Electrcal Engneerng from Abhar Branch, Islamc Azad Unversty, Abhar, Iran n He s currently the M.Sc. student n Seraj Hgher Educaton Insttute, Tabrz, Iran. He s the Member of Scentfc and Executve Commttees of Internatonal Conference of ICTPE and also the Scentfc and Executve Secretary of Internatonal Journal of IJTPE supported by Internatonal Organzaton of IOTPE ( Hs research felds are ntellgent algorthms applcaton n power systems, power system dynamcs and control, power system analyss and operaton, and reactve power control. Narges Sadat Boushehr was born n Iran. She receved her B.Sc. degree n Control Engneerng from Sharf Unversty of Technology (Tehran, Iran), and Electronc Engneerng from Central Tehran Branch, Islamc Azad Unversty, (Tehran, Iran), n 1991 and 1996, respectvely. She receved the M.Sc. degree n Electronc Engneerng from Internatonal Ecocenergy Academy (Baku, Azerbajan), n She s the Member of Scentfc and Executve Commttees of Internatonal Conference of ICTPE and also the Scentfc and Executve Secretary of Internatonal Journal of IJTPE supported by Internatonal Organzaton of IOTPE ( Her research nterests are n the area of power system control and artfcal ntellgent algorthms. 233