Optimal Reactive Power Dispatch Considering FACTS Devices

Transcription

1 Leonardo Journal of Scences ISSN Issue 18, January-June 2011 p Optmal Reactve Power Dspatch Consderng FACTS Devces Ismal MAROUANI *, Tawfk GUESMI, Hsan HADJ ABDALLAH and Abdarrazak OUALI Sfax Natonal Engneerng School, Electrcal Department. BP W, 3038 Sfax, Tunsa E-mal: * Correspondng author E-mal: smal.marouan@setks.rnu.tn Receved: 29 November 2010 / Accepted: 16 March 2011 / Publshed: 28 June 2011 Abstract Because ther capablty to change the network parameters wth a rapd response and enhanced flexblty, flexble AC transmsson system (FACTS) devces have taken more attenton n power systems operatons as mprovement of voltage profle and mnmzng system losses. In ths way, ths paper presents a mult-obectve evolutonary algorthm (MOEA) to solve optmal reactve power dspatch (ORPD) problem wth FACTS devces. Ths nonlnear mult-obectve problem (MOP) conssts to mnmze smultaneously real power loss n transmsson lnes and voltage devaton at load buses, by tunng parameters and searchng the locaton of FACTS devces. The constrants of ths MOP are dvded to equalty constrants represented by load flow equatons and nequalty constrants such as, generaton reactve power sources and securty lmts at load buses. Two types of FACTS devces, statc synchronous seres compensator (SSSC) and unfed power flow controller (UPFC) are consdered. A comparatve study regardng the effects of an SSSC and an UPFC on voltage devaton and total transmsson real losses s carred out. The desgn problem s tested on a 6-bus system. 97

2 Optmal Reactve Power Dspatch Consderng FACTS Devces Ismal MAROUANI, Tawfk GUESMI, Hsan HADJ ABDALLAH and Abdarrazak OUALI Keywords Mult-obectve optmzaton; Evolutonary algorthms; Power flow; statc synchronous seres compensator (SSSC); unfed power flow controller (UPFC); Voltage profle. Introducton The ORPD problem s consdered as a MOP. It conssts to mprove the voltage profle and mnmze the real power loss n transmsson lnes under several equalty and nequalty constrants. Such as, load flow equatons and securty lmts. To mantan the load buses voltage wthn ther permssble lmts many techncal methods are proposed [1, 2], such as, reallocatng reactve power generaton n the system adustng transformer taps, generator voltage and swtchable reactve power sources. To mnmze system losses, a redstrbuton of reactve power n the network can be used [3]. Because the recent progress of power electroncs, FACTS devces have taken more attenton n transmsson power systems. They have the capablty to change the network parameters wth a rapd response and enhanced flexblty, such as, mprovng voltage profle and mnmzng system losses. Some types of those devces, are, statc synchronous seres compensator (SSSC), statc synchronous compensator (STATCOM) and unfed power flow controller (UPFC). SSSC s consdered as a controllable voltage source nverter that s connected n seres wth transmsson lne. Ths nected voltage s almost n quadrature wth the lne current. Consequently, t provdes a varable reactance n seres wth the transmsson lne, whch, can be nductve or capactve. Ths reactance controls the power flow n the lne where t s ntroduced. The STATCOM s a shunt connected FACTS devces. It conssts of a voltage source converter lnked to the system va a shunt transformer. Its prncpal functon s to amelorate the voltage profle at the pont of connecton. To explot the benefts of those two devces, the UPFC can be a combnaton of SSSC and STATCOM. In a MOP, there sn t one soluton that s best wth respect to all obectves. Generally, the am s to determne the trade-off surface, whch s a set of non-domnated soluton ponts, known as Pareto optmal solutons. Every ndvdual of ths set s an acceptable soluton. 98

3 Leonardo Journal of Scences ISSN Issue 18, January-June 2011 p In the lterature, several methods are used to solve the ORPD. In [2], a nonlnear programmng technque was used. Gradent-based optmzaton algorthms consst to lnearze the obectve functons and the system constrants around an operatng pont was presented n [3]. Those tradtonal technques consume an mportant computng tme and they are teratve methods. Also, they can converge to a local optmum. Recently, genetc algorthms (GA) are very much used to solve MOP. Many researchers have transformed the MOP to a sngle obectve problem usng approprate weghts. Then, GA was been appled [4]. Unfortunately, the obtaned soluton depends on the weght vector used n the process. Also, ths method requres a number of runs equal to the sze of the desred Pareto optmal solutons. MOEAs are used to elmnate most dffcultes of these methods. MOEAs are no teratve and they gve the Pareto optmal solutons n one run. Thus, n ths artcle a no conventonal technque based on MOEA s employed. It s based on non-domnated Sortng Genetc Algorthm II (NSGAII) approach. Whch s an eltst approach and t can mantan populaton dversty n the set of the non-domnated solutons. Therefore, the obectve of the present paper s to develop a power flow model for power system wth FACTS devces. The modfed Newton-Raphson power flow algorthm was used [5]. Then, a new ORPD problem s formulated. The solutons of ths problem are the FACTS parameters and locaton. Statc models of two FACTS devces consstng of UPFC and SSSC have been used n the present work. Materal As ndcated prevously, SSSC and UPFC are used and mathematcally modelled n ths paper. SSSC Mathematcal Modellng Fgure 1 shows the crcut model of an SSSC connected at poston d between two buses n and m. So, another two buses and are added to the total number of buses of the system. 99

4 Optmal Reactve Power Dspatch Consderng FACTS Devces Ismal MAROUANI, Tawfk GUESMI, Hsan HADJ ABDALLAH and Abdarrazak OUALI n d.x l (1-d).X l m Converter Fgure 1. SSSC locaton between buses n and m (where d s the porton of the mpedance of lne n-m, d [0,1] ) In ts equvalent crcut shown n Fgure 2, the SSSC s represented by a voltage source V se n seres wth the transformer mpedance X s [6, 7]. In practce, V se can be regulated to control the power flow of lne m-n and the voltage at buses and. Xs V I V Fgure 2. Voltage-source model of SSSC Ths last model can be developed by replacng voltage source V se by a current source I se parallel wth the transmsson lne as shown n Fgure 3. Bus V b se =1/Xse Bus V Ise Fgure 3. Replacement of seres voltage source by a current source I se = -b se V se (1) V se =rv e γ (2) where r and γ are respectvely the p.u. magntude and phase angle of seres voltage source. 0 r r max and 0 γ 2π The power necton model of the SSSC can be seen as two dependent power 100

5 Leonardo Journal of Scences ISSN Issue 18, January-June 2011 p nectons at auxlary buses and as shown n Fgure 4 [8]. V X s V P, Q P, Q Fgure 4. SSSC mathematcal model The apparent power suppled by the model s calculated as: S = V (-I se ) * (3) S = V (I se ) * (4) Then, actve and reactve power suppled by the SSSC can be deduced from Equatons (5) to (8). P,SSSC = -rb se V 2 sn(γ) (5) Q,SSSC = -rb se V 2 cos(γ) (6) P,SSSC = rb se V V sn(α -α +γ) (7) Q,SSSC = rb se V V cos(α -α +γ) (8) UPFC Mathematcal Modellng An UPFC can be represented as shown n Fgure 5. It conssts of, two voltage sources V se and V sh, and two transformer mpedances X se and X sh. Voltage sources V se and V sh are controllable n both ther magntude and phase angles [9, 10]. bus V Xsh Vse ' V Xse bus V Vsh Fgure 5. Two voltage-source model of UPFC (where V se s defned as n Equaton (2)) The voltage source V se n the last model can be replaced by a current source I se parallel wth the transmsson lne as shown n Fgure

6 Optmal Reactve Power Dspatch Consderng FACTS Devces Ismal MAROUANI, Tawfk GUESMI, Hsan HADJ ABDALLAH and Abdarrazak OUALI The shunt branch n UPFC s employed manly to provde all the losses n the UPFC and the actve power, P seres, whch s nected wth the system by the seres branch. If the total losses of the two converters are estmated to be about 2% of P seres, then, the provded power by shunt branch P shunt wll be expressed by [5]: P shunt = -1.02P sere (9) The functon of the reactve power delvered or absorbed by the shunt branch s to mantan the level of tenson to bus wthn acceptable lmts. Fnally, UPFC mathematcal model can be constructed by combnng the seres and shunt power nectons at both bus and as shown n Fgure 6 [5]. The elements of equvalent power necton n Fgure 6 are: P,ufc = 0.02rb se V 2 snγ-1.02rb se V V sn(α -α +γ) (10) P,upfc = rb se V V sn(α -α +γ) (11) Q,upfc = -rb se V V cos(γ) (12) Q,upfc = rb se V V cos(α -α +γ) (13) V V P,upfc+ Q,upfc P,upfc+ Q,upfc Fgure 6. UPFC mathematcal model Presentaton of the Test System The proposed procedure for solvng dspatch VAR ncludng FACTS devces s tested on the 6-bus system [11]. The one-lne dagram of ths system s shown n Fgure 7. The system conssts of three generators at buses 4, 5 and 6. Bus 6 s consdered at the slack bus. Buses 1, 2 and 3 are the load buses. 102

7 Leonardo Journal of Scences ISSN Issue 18, January-June 2011 p G3 Load Load 5 4 Load G2 G1 Fgure 7. Test system Method Problem Formulaton The ORPD problem s to optmse the steady performance of a power system n terms of one or more obectve functons whle satsfyng several equalty and nequalty constrants. In ths part, we suppose that the extremtes FACTS devces are referred by bus and. Obectve functons In ths paper two obectve functons are used: Real power loss: Ths obectve conssts to mnmse the real power loss P L n transmsson lnes that can be expressed as [1, 11]: P = L N b P (14) k k=1 Nb ( P,SSSC or P,UPFC ) + VV hyhcos( α - αh - θ h );f k= h=1 Nb P k = ( P,SSSC or P,UPFC ) + VV hyhcos( α - αh - θ h );f k= h=1 Nb VVY k h khcos( αk - αh - θ kh),f k¹, h=1 (15) where: N b : number of buses; V k <α k and V h <a h : respectvely voltages at bus k and h; Y kh and θ kh : respectvely modulus and argument of the kh-th element of the nodal admttance matrx Y. 103

8 Optmal Reactve Power Dspatch Consderng FACTS Devces Ismal MAROUANI, Tawfk GUESMI, Hsan HADJ ABDALLAH and Abdarrazak OUALI Voltage devaton: Ths obectve s to mnmze the devaton n voltage magntude at load buses that can be expressed as: N L ref D =1 V = V -V (16) where: N L: number of load buses; at the -th load bus; ref V ref V s usually set to be 1.0 p.u. : prespecfed reference value of the voltage magntude Problem Constrants The problem constrants are dvded to equalty and nequalty constrants. Equalty constrants These represent typcal load flow equatons as follows: N b ( ) ( ) P -P - V G cos α - α +B sn α - α = 0 (17) G D =1 N b ( ) ( ) Q -Q - V G sn α - α -B cos α - α = 0 G D =1 (18) where: P G and Q G : generator real and reactve power at -th bus, respectvely; P D and Q G : load real and reactve power at -th bus, respectvely; G and B : transfer conductance and susceptance between buses and, respectvely. Inequalty constrants These constrants can be summarzed by: Securty constrants: These nclude the constrants of voltage at the -th load buses V L as follows: V V V,=1,,...,N (19) mn max L L L L Parameters FACTS constrants: r mn r r max (20) γ mn γ γ max (21) d mn d d max (22) 104

9 Leonardo Journal of Scences ISSN Issue 18, January-June 2011 p We should note that the vector of decson varables s U = [r, γ, d]. Mult-Obectve Optmzaton In a MOP, there may not be one soluton that s best wth respect to all obectves. Usually, the am s to determne the trade-off surface, whch s a set of nondomnated soluton ponts, known as Pareto optmal solutons. Every ndvdual n ths set s an acceptable soluton. For any two X 1 and X 2, we can have one of two possbltes: one domnates the other or none domnates the other. In a mnmzaton problem, we say that the soluton X 1 domnates X 2, f the followng two condtons are satsfed [13]: { } { } 1, 2,..., N ob, f (X 1) f (X 2) 1, 2,..., N ob,f (X 1) < f (X 2 ) where N ob : Number of obectve functons and f : -th obectve functon. The goal of a mult-obectve optmzaton algorthm s not only to gude the search towards the Pareto optmal front, but, also to mantan populaton dversty n the set of the nondomnated solutons. In the rest of ths secton, we wll present the eltst MOEA NSGAII. So, we must be start wth a presentaton of the NSGA approach. NSGA approach: The basc dea behnd NSGA s the rankng process executed before the selecton operaton. The rankng procedure conssts to fnd the nondomnated solutons n the current populaton P. These solutons represent the frst front F 1. Afterwards, ths frst front s elmnated from the populaton and the rest s processed n the same way to dentfy nondomnated solutons for the second front F 2. Ths process contnues untl the populaton s properly ranked. So, we can wrte [14]: (23) U r F = 1 P = (24) where r s the number of fronts. The same ftness value f k s assgned to all of ndvduals of the same front F k. Ths ftness value decreases whle passng from the front F k to the F k+1. To mantan dversty n the populaton, a sharng method s used. Let consder d the varable dstance (Eucldean norm) between two solutons X and X. 105

10 Optmal Reactve Power Dspatch Consderng FACTS Devces Ismal MAROUANI, Tawfk GUESMI, Hsan HADJ ABDALLAH and Abdarrazak OUALI S () () Xk -Xk max mn k=1 Xk -Xk d = 2 (25) where S s the number of varables n the MOP. The parameters are the upper and lower bounds of varable X k. () () () ( 1 2 S ) max k mn X and X respectvely X = X,X,,X (26) The sharng procedure s as follows: Step 1: Fx the nche radus σ share and a small postve number, ε. Step 2: Intate f mn = N pop +ε and the counter of front =1. Step 3: From the r nondomnated fronts F whch consttute P. P = U r F = 1 Step 4: For each ndvdual Xq F : assocate the dummy ftness f (q) = f mn - ε; calculate the nche count n cq as gven n [13]; k calculate the shared ftness (q) f '(q) f = n. cq '(q) Step 5: F = mn(f ; q P ) and = + 1. mn Step 6: If r, then, return to step 4. Else, the process s fnshed. The MOEAs usng nondomnated sortng and sharng have been crtczed manly for ther O(MN 3 ) computatonal complexty (M s the number of obectves and N s the populaton sze). Also, these algorthms are not eltst approaches and they need to specfy the sharng parameter. To avod these dffcultes, we present n the followng an eltst MOEA whch s called Nondomnated Sortng Genetc Algorthm II (NSGAII). NSGAII approach: In ths approach, the sharng functon approach s replaced wth a crowded comparson. Intally, an offsprng populaton Q t s created from the parent populaton P t at the t th generaton. After, a combned populaton R t s formed [14]. R t = P Q R t s sorted nto dfferent no domnaton levels F as shown n the NSGA approach. So, we can wrte: t t 106

11 Leonardo Journal of Scences ISSN Issue 18, January-June 2011 p where r s the number of fronts. U r = 1 R t = F Fnally, one teraton of the NSGAII procedure s as follows: Step 1: Create the offsprng populaton Q t from the current populaton P t. Step 2: Combne the two populatons Q t and P t to form R t. Step 3: Fnd the all nondomnated fronts F and R t. Step 4: Intate the new populaton P t+1 = φ and the counter of front for ncluson = 1. Step 5: Whle P t+1 + F N pop, do: P t + 1 Pt F Step 6: Sort the last front F usng the crowdng dstance n descendng order and choose the frst (N pop - P t+1 ) elements of F. Step 7: Use selecton, crossover and mutaton operators to create the new offsprng populaton Q t+1 of sze N ob. To estmate the densty of soluton surroundng a partcular soluton X n a nondomnated set F, we calculate the crowdng dstance as follows: Step 1: Let s suppose q = F. For each soluton X n F, set d = 0. Intate m = 1. Step 2: Sort F n the descendng order accordng to the obectve functon of rank m. Let s m consder I =sort[ ]( F ) the vector of ndces,.e. I m s the ndex of the soluton X n the f m > sorted lst accordng to the obectve functon of rank m. Step 3: For each soluton m+1 m-1 I X whch verfes 2 I m (q-1), update the value of d as follows: I fm -fm (27) max mn m m d d + f -f Then, the boundary solutons n the sorted lst (solutons wth smallest and largest functon) are assgned an nfnte dstance value,.e. f, I m = 1 or I m = q, d =. Step 4: If m = M, the procedure s fnshed. Else, m = (m+1), and return to step 2. Implementaton of the NSGAII: The proposed NSGAII has been mplemented usng real-coded genetc algorthm (RCGA) [14]. So, a chromosome X correspondng to a decson varable s represented as a strng of real values x,.e. X = x 1 x 2 x lchrom. lchrom s the 107

12 Optmal Reactve Power Dspatch Consderng FACTS Devces Ismal MAROUANI, Tawfk GUESMI, Hsan HADJ ABDALLAH and Abdarrazak OUALI chromosome sze and x s a real number wthn ts lower lmt a and upper lmt b..e. x [a, b ]. Thus, for two ndvduals havng as chromosomes respectvely X and Y and after generatng a random number α [ 0,1], the crossover operator can provde two chromosomes X and Y wth a probablty P c as follows [14]: ( ) ( ) X' = αx+ 1-α Y Y' = 1- α X+αY (28) In ths study, the non-unform mutaton operator has been employed. So, at the t th generaton, a parameter x of the chromosome X wll be transformed to other parameter x ' wth a probablty P m as follows: ( ) ( ) ' x +Δ t,b - x, f τ =0 x = x -Δ t,x - a, f τ =1 ( 1-t/g ) Δ( t, y ) = y( 1- ε ) β max where τ s a random bnary number, r s a random number ε [ 01, ] and g max s the maxmum number of generatons. β s a postve constant chosen arbtrarly. (29) (30) Results and Dscusson The characterstcs of lnes and buses are marked n the Tables 1 and 2 respectvely. These values are gven n p.u. consderng a base power of 100 MVA for the overall system and base voltages of 100 KV. The lower voltage magntude lmts at all buses are 0.9 p.u and the upper lmts are 1.1 p.u. Three cases of power analyss are consdered. Case 1 assumes the study wthout any compensaton. Case 2 assumes an SSSC between two load buses. In case 3, an UPFC s assumed too between two load buses. Table 1. Data lnes Lne Impedance Bus Bus R X

13 Leonardo Journal of Scences ISSN Issue 18, January-June 2011 p Table 2. Data buses Bus V P G Q G P D Q D Fgure 8 shows the convergence characterstcs of the load flow program after 6 teratons wth a tolerance of Fgure 8. Convergence of the load flow For the system wthout FACTS controllers, the voltage magntude of load buses gven by the load flow program, are not mantaned wthn ther permssble lmts (Table 3). The correspondng values of voltage devaton and real power loss are respectvely: V D = p.u. and P L = p.u. Table 3. Voltage magntudes and phase angles for the system wthout FACTS controllers Bus V α [rad] Resoluton of the ORPD After executon of the optmzaton program, the values of the voltage devaton and the real power losses for the varous postons of elements FACTS, are gven n Fgure

14 Optmal Reactve Power Dspatch Consderng FACTS Devces Ismal MAROUANI, Tawfk GUESMI, Hsan HADJ ABDALLAH and Abdarrazak OUALI 0,45 0,4 0,35 0,3 0,25 0,2 0,15 0,1 0,05 0 Voltage devaton [p.u.] Power loss [p.u.] Lne 1-6 Lne 1-2 Lne 2-4 Lne 4-5 Lne 2-3 Lne 3-6 Lne 3-5 (a) UPFC 0,45 0,4 0,35 0,3 0,25 0,2 0,15 0,1 0,05 0 Voltage devaton [p.u.] Power loss [p.u.] Lne 1-6 Lne 1-2 Lne 2-4 Lne 4-5 Lne 2-3 Lne 3-6 Lne 3-5 (b) SSSC Fgure 9. The best poston for system FACTS Fgure 9 shows that the best poston of UPFC and SSSC for mnmum voltage devaton and mnmum real power loss, s n lne 1-2. Therefore, we wll gve only the results correspondng to ths optmal poston. Optmsaton Mono-Obectve To get convergence of power loss and voltage devaton functons whch are shown n Fgures 10 and 11, these two obectve functons are optmzed ndvdually. Fgure 10 shows the convergence of power losses functon versus number of generatons. From ths Fgure 10, we can see that a power losses correspondng to the two 110

15 Leonardo Journal of Scences ISSN Issue 18, January-June 2011 p cases converge to p.u. and p.u., respectvely, for the system wth SSSC and wth UPFC. And t shows that the UPFC corresponds to the best soluton. Fgure 10. Convergence of power losses wth generatons From Fgure 11, we can see that voltage devaton correspondng to the two cases converges to p.u. and p.u., respectvely, for the system wth SSSC and wth UPFC. Fgure 11. Convergence of voltage devaton wth generatons Optmzaton B-Obectve In ths secton, the two obectve functons are optmzed smultaneously. 111

16 Optmal Reactve Power Dspatch Consderng FACTS Devces Ismal MAROUANI, Tawfk GUESMI, Hsan HADJ ABDALLAH and Abdarrazak OUALI Fgure 12. Pareto-optmal front of the proposed approach Fgure 12 gves the Pareto-optmal front correspondng to the two cases. And t shows that the UPFC corresponds to the best results. Table 4. The best soluton for voltage devaton R γ d V D Corresp. P L SSSC UPFC Table 5. The best soluton for real power loss R γ d P L Corresp. V D SSSC UPFC After optmzaton by the NSGAII approach, we have obtaned the Tables 4 and 5 gvng the best solutons for mnmum voltage devaton and mnmum real power loss respectvely. The values of r, d, V D and P L are gven n power unts and γ are n radans. Table 6. For mnmum voltage devaton V 1 [pu] V 2 [pu] V 3 [pu] SSSC UPFC Table 7. For mnmum real power loss V 1 [pu] V 1 [pu] V 1 [pu] SSSC UPFC

17 Leonardo Journal of Scences ISSN Issue 18, January-June 2011 p The profle voltage n load buses correspondng to the best solutons gven n Tables 4 and 5 are shown respectvely n Tables 6 and 7 and V s n power unts. From tables 6 and 7, we can see that the voltage magntude of load buses gven wtn FACTS are mantaned wthn ther permssble lmts (0.9 pu<v<1.1 pu). Concluson In ths paper, a procedure based on MOEA to solve ORPD problem wth FACTS was presented. Ths problem conssts to mnmze smultaneously real power loss n transmsson lnes and voltage devaton at load buses by usng FACTS devces. The decson varables are parameters and locaton of FACTS devces. The NSGAII approach s opted to solve ths nonlnear MOP. UPFC and SSSC are used n ths work. The resoluton of the MOP shows that UPFC and SSSC have a postve effect on ORPD. Knowng that, UPFC gves the best results. The presented procedure was been tested on a 6-bus system. References 1. Abdo M.A., Bakhashwan J.M., Optmal VAR dspatchng usng a multobectve evolutonary algorthm, Electrcal Power and Energy System, 2005, 27(1), p Ben Arba H., Had Abdallah H., Mult Obectves Reactve Dspatch Optmsaton, Leonardo Journal of Scences (LJS), January-June, 2007, 6(10), p Mamandur K.R.C., Chnoweth R.D., Optmal control of reactve power flow for mprovement n voltage profles and for real power loss mnmzaton, IEEE Transactons on Power Apparatus and Systems, 1981, PAS-100(7), p Mshra S., Dash P.K., Hota P.K., Trpathy M., Genetcally optmzed neuro-fuzzy PFC for dampng modal oscllatons of power system, IEEE Transactons on Power Systems, 2002, 17(4), p Mete Vural A., Tumay M., Mathematcal modellng and analyss of unfed power flow 113

18 Optmal Reactve Power Dspatch Consderng FACTS Devces Ismal MAROUANI, Tawfk GUESMI, Hsan HADJ ABDALLAH and Abdarrazak OUALI controller: A comparson of two approaches n power flow studes and effects of UPFC locaton, Electrcal Power and Energy Systems, 2007, 29(8), p Ghadr R., Reshma S.R., Power flow model/calculaton for power systems wth multple FACTS controllers, Electrc Power Systems Research, 2007, 77(12), p Sen K.K., SSSC-statc synchronous seres compensator: Theory, modellng, and applcaton, IEEE Transactons on Power Delevery, 1998, 13(1), p Zhang X.P., Advanced modellng of multcontrol functonal statc synchronous seres compensator (SSSC) n Newton-Raphson power flow, IEEE Transactons on Power Systems, 2003, 18(4), p Fuerte-Esquvel C.R., Acha E., Unfed power flow controller: a crtcal comparason of Newton-Raphson UPFC algorthms n power flow studes, IEE Proceedngs Generaton Transmsson and Dstrbuton, 1997, 144(5), p Noroozan M., Angqust L., Ghandhar M., Andersson G., Use of UPFC for optmal power flow control, IEEE Transactons on Power Delvery, 1997, 12(4), p Rubaa A., Vllaseca F.E., Transent stablty herarchcal control n multmachne power systems, IEEE Transactons on Power Systems, 1989, 4(4), p Abdo M.A., A novel multobectve evolutonary algorthm for envronmental/economc power dspatch, Electrc Power System Research, 2003, 65(1), p Das A.H.F., de Vasconcelos J.A., Mult-obectve genetc algorthms appled to solve optmzaton problems, IEEE Transactons On Magnetcs, 2002, 38(2), p Herrera F., Lozano M., Verdegay J.L., Tacklng real-coded genetc algorthm: operators and tools for behavoural analyss, Artfcal Intellgence Revew, 1998, 12(4), p