Optimal Placement of Unified Power Flow Controllers : An Approach to Maximize the Loadability of Transmission Lines

Transcription

1 S. T. Jaya Chrsta Research scholar at Thagarajar College of Engneerng, Madura. Senor Lecturer, Department of Electrcal and Electroncs Engneerng, Mepco Schlenk Engneerng College, Svakas , Taml Nadu, Inda P. Venkatesh Assstant Professor, Department of Electrcal and Electroncs Engneerng, Thagarajar College of Engneerng, Madura , Taml Nadu, Inda. J. Electrcal Systems 2-2 (2006): Regular paper Optmal Placement of Unfed Power Flow Controllers : An Approach to Maxmze the Loadablty of Transmsson Lnes JES Journal of Electrcal Systems Ths paper deals wth the optmal locaton and parameters of Unfed Power Flow Controllers (UPFCs) n electrcal power systems, usng partcle swarm optmzaton (PSO). The objectve s to maxmze the transmsson system loadablty subject to the transmsson lne capacty lmts and specfed bus voltage levels. Usng the proposed method, the locaton of UPFCs and ther parameters are optmzed smultaneously. PSO s used to solve the above non-lnear programmng problem for better accuracy. The proposed approach s examned and tested on IEEE 30-bus system and IEEE 118-bus system. The results obtaned are qute promsng for the power system operaton envronment. Keywords: Unfed Power Flow Controller (UPFC), system loadablty maxmzaton, Partcle Swarm Optmzaton (PSO), Evolutonary Computaton, power flow. 1. INTRODUCTION In recent years, wth the deregulaton of the electrcty market, the tradtonal practces of power systems have been changed a lot. Years of under nvestment n the transmsson sector n many electrcty markets has drawn attenton to better utlze the exstng transmsson lnes. The advent of FACTS devces based on the advancement of semconductor technology opens up new opportuntes for ncreasng the capactes of the exstng transmsson systems [1, 2]. The UPFC s one of the most promsng FACTS devces n terms of ts ablty to control power system quanttes. It can ether smultaneously or selectvely control the actve and reactve power flow through the lnes and also bus voltages [3-7]. The above mentoned salent features rendered by UPFCs depend on the confguraton of UPFCs. Hence, for the practcal mplementaton of UPFCs n a power system, a systematc procedure s needed n fndng the optmal locaton and parameters of UPFCs. By optmally placng the UPFCs, t s possble to mnmze transmsson loss, mnmze power generaton cost, maxmze the loadablty of the transmsson system etc. Some papers have been publshed on solvng the optmal locaton of FACTS devces wth respect to dfferent purposes and methods [1, 8, 9]. In [1], Genetc Algorthm has been appled for the optmal placement of mult-type FACTS devces ncludng TCSC, TCPST, TCVR and SVC to maxmze the loadablty of transmsson lnes. In [8], an approach based on augmented Lagrange multpler method has been used to determne the optmal locaton of UPFCs to be nstalled. An mproved evolutonary programmng has been used to fnd the optmal locaton of UPFCs n [9] wth the purpose of ncreasng the system loadablty. A determnstc based method has been appled to evaluate the network losses. Ths research paper s an outcome of the AICTE sponsored research project AICTE / 8023 / RID / BOR / RPS 55/ Copyrght JES 2006 on-lne : journal.esrgroups.org/jes

2 J. Electrcal Systems 2-2 (2006): The present objectve of ths paper s to analyze, once more, the problem of system loadablty maxmzaton. However, the tool of analyss employed s partcle swarm optmzaton whch s a new evolutonary computatonal stochastc technque. The man advantages of usng PSO are that, t can generate hgh qualty solutons wthn shorter calculaton tme and has more stable convergence characterstc compared to other stochastc methods [11]. The applcaton of ths tool to power system problems has been found n some papers [10-12]. For example, [10] focuses on the problem of fuel cost mnmzaton. The PSO method s very recent n the lterature. The man dea s based on the food searchng behavor of brds. Each ndvdual n PSO fles n the search space wth a velocty. It s ths velocty whch s dynamcally adjusted accordng to ts own flyng experence and ts companons flyng experence. Here, the global and local best postons are computed at each teraton and the output s the new drecton of search. Makng use of ths PSO technque, the proposed method can fnd hgh-qualty solutons relably wth faster convergence characterstcs n a reasonably good computaton tme. Fndng the optmal placement and parameters of UPFCs for maxmzng the system loadablty s a large scale non-lnear optmzaton problem. In ths paper, a novel technque s proposed to ncrease the system loadablty. PSO appled for optmal placement of UPFCs s evaluated on IEEE 30-bus and IEEE 118-bus power systems. Smulaton results show that the proposed approach converges to better solutons wth lesser computatonal burden. 2. UPFC EQUIVALENT CIRCUIT AND POWER EQUATIONS 2.1 UPFC Equvalent Crcut In ths paper, a smplfed equvalent crcut of UPFC gven n [9] s used and s shown n Fgure 1. Fgure 1: Equvalent crcut of UPFC. The three controllable parameters of UPFC are V T, φ T and I q. V T denotes the magntude of the voltage njected n seres wth the transmsson lne through the seres transformer. φ T s the phase angle of ths voltage. I q s the shunt reactve current of UPFC. The UPFC parameters lmtatons. V T, φ T and VT V T mn, V T max, φt 0,2π I q are chosen wthn a range due to physcal and economc Iq Iqmax, I qmax The lmts of UPFC parameters are taken from [13]. 83

3 S. T. Jaya Chrsta & P. Venkatesh: Optmal Placement of Unfed Power Flow Controllers UPFC Power equatons The equvalent crcut of UPFC embedded n transmsson lne -j s shown n Fgure 2. The two power njectons ( Pnj ( ), Q nj ( )) and ( Pj ( nj), Q j( nj) ) of the UPFC are calculated accordng to the followng expressons [13]: P = G e e + f f + B f e e f + G ee + f f (1) ( ) ( ) ( ) 2 '( ) nj j j T j T j j T j T T T Q ( ) = G' ( fe e f ) B' ( ee + f f ) VI (2) nj T T T T q P ( ) = G ( e e + f f ) B ( f e e f ) (3) j nj j j T j T j j T j T Q ( ) = G ( e f f e ) + B ( e e + f f ) (4) j nj j j T j T j j T j T Where: P, P : the actve power njectons at bus and j, respectvely; ( nj) j( nj) Q, Q : the reactve power njectons at bus and j, respectvely; ( nj) j( nj) e, f : real part and magnary part of voltage at bus ; e j, f j : real part and magnary part of voltage at bus j; e T, f : real part and magnary part of voltage of seres voltage source, respectvely and T e = V cos( φ ), f = V sn( φ ) T T T T T T V : the voltage magntude of bus ; Gj, Bj, gj, b j : the parameters of lne - j G' = G + g, B' = B + b j j j j Fgure 2: Equvalent crcut of UPFC embedded branch. Accordng to equatons (1) (4), the addtonal elements of Jacoban Matrx at the bus and j are: For bus, when j P Δ H = = G f + B e nj ( ) j fj j T j T P Δ N = = G e B f nj ( ) j ej j T j T (5) (6) 84

4 J. Electrcal Systems 2-2 (2006): Qnj ( ) Δ Mj = = 0 f Qnj ( ) Δ Lj = = 0 e j j (7) (8) when = j Pnj ( ) Δ H = = 2 G' f f Pnj ( ) Δ N = = 2 G' e e Q Δ M = = G f B e e I V T T nj ( ) ' T ' T q / f Q Δ L = = G e B f fi V nj ( ) ' T ' T q / e For bus j, when j Pj( nj) Δ H j = = 0 f Pj( nj) Δ N j = = 0 e Qj( nj) Δ M j = = 0 f Qj( nj) Δ Lj = = 0 e when j = P Δ H = = G f B e j( nj) jj fj j T j T P Δ N = = G e + B f j( nj) jj ej j T j T Q Δ M = = G e + B f j( nj) jj fj j T j T Q Δ L = = G f + B e j( nj) jj ej j T j T (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) 85

5 S. T. Jaya Chrsta & P. Venkatesh: Optmal Placement of Unfed Power Flow Controllers PSO 3.1 Overvew The PSO s a populaton-based optmzaton tool frst proposed by Kennedy and Eberhart [14]. PSO s bascally developed through smulaton of brd flockng or fsh schoolng n two-dmensonal space. The advantages of PSO compared to other evolutonary computatonal technques are: 1. PSO s easy to mplement. 2. There are few parameters to be adjusted n PSO. 3. All the partcles tend to converge to the best soluton quckly. 3.2 PSO Algorthm The PSO algorthm s presented below. 1. The technque s ntalzed wth a populaton of random solutons or partcles and then searches the optma by updatng generatons. Each ndvdual partcle has the followng three propertes: a current poston n search space x, a current velocty v and a personal best poston n search space y. 2. In every teraton, each partcle s updated by the followng two best values. The frst one s the personal best poston y whch s the poston of partcle n the search space, where t has acheved the best soluton so far. The second one s the global best poston y whch s the poston yeldng the best soluton among all the y s. The personal and global best values are updated at tme t usng equatons (21) and (22) respectvely. Here t s assumed that the swarm has s partcles. Thus, 1... s and assumng mnmzaton of the objectve functon f, y ( t 1), ( ) ( ( 1) ) ( 1, ) ( ( )) ( ( 1) ) y ( t) f f y ( t) f x t + + = x t + f f y t > f x t+ y t y t y t f y t = {,..., } ( ( )) mn { f ( y1 ( t) ),... f ( ys ( t ) )} ( ) 1 ( ) s ( ) (21) (22) 3. After fndng the two best values, each partcle updates ts velocty and current poston. The velocty of the partcle s updated accordng to ts own prevous best poston and the prevous best poston of ts companons and s gven by equaton (23). For all dmensons j 1 n, vj, ( t 1) wvj, ( t) c1r1, j( t) yj, ( t) xj, ( t) + = + + c2r2, j( t) [ yj( t) xj, ( t) ] Two pseudorandom sequences 1 ~ (0,1) r U and 2 ~ (0,1) r U are used to affect the stochastc nature of the algorthm. Ths new velocty s then added to the current poston of the partcle to obtan ts next poston. (23) 86

6 ( 1) ( ) ( 1) J. Electrcal Systems 2-2 (2006): x t + = x t + v t + (24) Partcle veloctes on each dmenson are clamped to a maxmum velocty v max whch s gven by, vmax = k xmax, where 0.1 k 1.0 (25) Where x max s the doman of the search space. The acceleraton coeffcents c 1 and c 2 control the dstance moved by a partcle n an teraton. The nerta weght w n (23) controls the convergence behavor of PSO. Usually the value of w s lnearly decreased from 1 to near 0 over the executon. The nerta weght w s set accordng to the followng equaton: wmax wmn w = wmax ter (26) termax where ter max s the maxmum number of teratons, and ter s the current teraton number. Reference [10] presents an useful dctonary about PSO. 3.3 PSO Parameter Control The parameters to be controlled whle usng PSO are lsted below. A. Number of partcles The typcal range of number of partcles to be used s 20 to 60. For most of the problems, 10 partcles are suffcent to obtan good results. For some dffcult or specal problems, we can go for 100 to 200 partcles. B. Dmenson of partcles Ths s determned by the problem to be optmzed. C. Range of partcles Ths also depends on the problem to be optmzed. We can specfy dfferent ranges for dfferent dmenson of partcles. D. Maxmum velocty v max v max determnes the maxmum change a partcle can take n an teraton. Usually the range of the partcle s taken as v max. E. Acceleraton coeffcents Usually the acceleraton coeffcents c 1 and c 2 are taken as 2. However, we can also use other values. But usually c 1 equals to c 2 and ranges from 0 to 4. F. Stop condton The stop condton s based on the maxmum number of teratons to be executed or the mnmum error requred n the problem. Ths s also determned by the problem to be optmzed. 4. PROBLEM FORMULATION The man goal of optmzaton s to ncrease the power transmtted by a transmsson 87

7 S. T. Jaya Chrsta & P. Venkatesh: Optmal Placement of Unfed Power Flow Controllers... network as much as possble, by optmally placng the UPFCs and also keepng the power system n a secure state n terms of branch loadng and bus voltage levels. In most of the optmzaton problems, the constrants are consdered by usng penalty terms n the objectve functon. Here also, the objectve functon used, penalzes the confguraton of UPFCs whch cause overloaded transmsson lnes and over or under voltages at buses [1]. Only, the techncal benefts of UPFCs, n terms of loadablty, are consdered n ths paper. The cost of UPFCs s not accounted n ths research. Ths mult-crtera constraned optmzaton problem s converted nto a sngle objectve optmzaton problem wth the objectve functon as a sum of two terms wth ndvdual crtera. The frst term, Ovl s related to branch loadng and penalzes overloads n the lnes. The second term, Vt s related to bus voltage levels and penalzes for bus voltages whch are not between 0.9 and 1.1 p.u. Therefore, for a confguraton of UPFCs, the objectve functon, f s gven by, Mnmze f = Ovllne + Vtbus (27) lne where, 1; f Ppq P Ovl = Pf ; f P > P pq max 1 pq pq max bus (28) Vt 1; f 0.9 V b 1.1 = Pf2; otherwse (29) P pq : Lne flow between buses p and q P pq max : Lne flow lmt for lne between buses p and q Pf 1 : Penalty factor for penalzng the overloaded lnes. V b : Voltage at bus b Pf 2 : Penalty factor for penalzng the voltage devaton at the buses. For a confguraton of UPFCs, f the constrants are satsfed, the value of the objectve functon s equal to 2. Also, equal value s gven for both the penalty factors. 5. PSO IMPLEMENTATION FOR OPTIMAL PLACEMENT OF UPFCS 5.1 Constructon of Partcle The confguraton of NU number of UPFCs s defned wth two parameters namely the locaton of UPFCs and ther correspondng controllable parameters such as V T, φ T and I q. The constructon of partcle for PSO mplementaton s shown n Fgure 3. L 1.. L NU V T1.. V TNU φ T1.. φ TNU I q1.. I qnu Fgure 3: Constructon of partcle. 88

8 J. Electrcal Systems 2-2 (2006): In Fgure 3, L n (where n = 1, 2, NU ) ndcates the locaton of UPFCs. Ths gves the numbers of the transmsson lnes where the UPFCs are to be located. The condton here s only one UPFC can be nstalled n a lne. Hence, a lne could appear at the maxmum of once n L n., φ Tn and I qn [where n = 1, 2, NU ] ndcate the controllable parameters of UPFC to be nstalled n lne L n. It can be seen that the total length of the partcle s 4 NU. V Tn For a gven power system, the ntal populaton of partcles s selected randomly based on the number of UPFCs to be nstalled and the ranges of the UPFC controllable parameters. 5.2 Methodology The step by step algorthm for solvng the proposed optmzaton problem s gven below. Step 1: The number of UPFCs to be placed and the ntal load factor are declared. Step 2: The ntal populaton of ndvduals s created satsfyng the UPFC constrants and also t s verfed that only one devce s placed n each lne. Step 3: For each ndvdual n the populaton, the ftness functon gven by equaton (27) s evaluated after runnng load flow. Step 4: The velocty s updated by equaton (23) and new populaton s created by equaton (24). Step 5: Steps 3 and 4 are repeated tll maxmum number of teraton s reached. Step 6: If the fnal best ndvdual obtaned satsfes all the constrants n the problem, then ncrement the load factor and go to step 2. Else, go to next step. Step 7: Prnt the prevous best ndvdual whch contans the locaton and parameters of UPFCs wth the correspondng load factor. Step 8: Stop the procedure. Actually n PSO technque, the ntal populaton of partcles for a gven load factor and number of UPFCs s generated randomly. But, t s found that for some systems, whle ncreasng the number of devces, f the results obtaned prevously are taken nto account, quck convergence wth better solutons s obtaned. Ths optmzaton strategy has been adopted n ths paper and t reduces the computatonal tme whch s sgnfcant especally n large systems. In ths paper, all loads are ncreased n the same proporton and t s assumed that the ncrease n real power generaton due to ths ncrease n load s met by the generator connected to slack bus. 6. NUMERICAL RESULTS AND DISCUSSION To verfy the effectveness and effcency of the proposed PSO based loadablty maxmzaton approach, the IEEE 30-bus power system and the IEEE 118-bus power system are used as the test systems. The numercal data for IEEE 30-bus and IEEE 118-bus systems are taken from [15]. The smulaton studes are carred out on a Pentum - IV, 3.0 GHz system n MATLAB envronment. 89

9 S. T. Jaya Chrsta & P. Venkatesh: Optmal Placement of Unfed Power Flow Controllers... For the gven optmzaton problem, t s found that, by settng the acceleraton coeffcents c 1 and c 2 both equal to 2.5, better solutons are got n a reasonable tme. Although PSO s senstve to the tunng of parameters, ths paper has proved ts potental n solvng complex power system problems. It has been found from the results that, PSO quckly fnds the hgh qualty optmal soluton. 6.1 IEEE 30-bus system The envronment parameters of PSO chosen for IEEE 30- bus system are: Number of teratons = 50 Number of partcles = 20 The results obtaned usng the proposed method for the IEEE 30-bus system are summarzed n table I. Fgure 4 shows the maxmum possble loadablty wth the gven number of UPFCs for IEEE 30-bus system. It s observed that the optmal locaton and parameters of UPFCs ncrease the system loadablty. Table I gves the optmal locaton and parameters of UPFCs for dfferent loadng factors for IEEE 30-bus system. From the results, t s observed that the loadablty has been ncreased to 112% by nstallng an UPFC n lne 4 whch connects buses 3 and 4. The maxmum loadablty wth 2 UPFCs wthout volatng the thermal and voltage constrants s 154%. For ths load factor, the UPFCs are embedded n lnes connectng buses 1,2 and 6,7. From Fgure 4, t s evdent that, there s a maxmum number of devces beyond whch the effcency of the network cannot be further mproved. Accordng to the used optmzaton crteron, for IEEE 30 bus system, the maxmum number of UPFCs beyond whch the loadablty cannot be ncreased s 3. To demonstrate the superorty of the proposed PSO based approach, smulaton results have been compared wth the results avalable n lterature usng Evolutonary Programmng (EP) method presented n [9]. In [9], a determnstc based method has been appled to evaluate the network losses. The network losses are set to 10% of the total losses. The objectve functon framed n [9] ncludes the loadablty term also whch s slghtly dfferent from the objectve functon used n ths paper. In ths paper, losses are calculated accurately by conductng Newton Raphson method of load flow soluton. Table I: Optmal locaton and parameters of UPFCs for dfferent load factors for IEEE 30-bus system No. of UPFCs Loadng factor Branches embedded wth UPFCs V T ( pu..) UPFC parameters φ T (deg) I q ( pu..)

10 J. Electrcal Systems 2-2 (2006): Loadng Factor Number of UPFCs Fgure 4: Maxmum loadng factor wth respect to gven number of UPFCs for IEEE 30-bus system Table II summarzes the results as obtaned by the two methods for the IEEE 30-bus system usng ther proposed methodologes. The results show that the optmal solutons determned by PSO lead to ncreased loadablty of the lnes wth less number of UPFCs, whch confrms that PSO based present approach s capable of determnng global optmal or near global optmal soluton. Table II also shows that PSO s faster than EP n speed because of lesser number of generatons or teratons and smaller populaton sze used by PSO to obtan the optmal soluton. Table II: Comparson of smulaton results of IEEE 30-bus system Compared tem EP based method PSO based method Maxmum possble loadablty Number of UPFCs requred for obtanng the maxmum loadablty 4 3 Total number of generatons Populaton Sze IEEE 118- bus system The envronment parameters of PSO chosen for IEEE 118- bus system are: Number of teratons = 100 Number of partcles = 100 The results obtaned usng the proposed method for the IEEE 118-bus system are summarzed n table III. Fgure 5 shows the maxmum possble loadablty wth the gven number of UPFCs for IEEE 118-bus system. Table III gves the optmal locaton and parameters of UPFCs for dfferent loadng factors for IEEE-118 bus system. From the results, t s evdent that the maxmum possble loadablty for the system s 119% wth 6 UPFCs. Beyond ths lmt, the loadablty cannot be mproved wth ncrease n UPFCs. It can also be noted from the results that, for ths system, always one UPFC s placed n lne number 163 connectng buses 100 and 103 for achevng maxmum loadablty of the transmsson system. So, when ths locaton of UPFC s nserted n any one of the partcles of the ntal populaton, whle gong for hgher number of UPFCs, convergence s acheved quckly and global optmal soluton s obtaned. 91

11 S. T. Jaya Chrsta & P. Venkatesh: Optmal Placement of Unfed Power Flow Controllers... Table III: Optmal locaton and parameters of UPFCs for dfferent load factors for IEEE 118-bus system No. of UPFCs Loadng factor Branches embedded wth UPFCs V T ( pu..) UPFC parameters φ T (deg) I q ( pu..) Loadng Factor Number of UPFCs Fgure 5: Maxmum loadng factor wth respect to gven number of UPFCs for IEEE-118 bus system 92

12 J. Electrcal Systems 2-2 (2006): Table IV compares the results as obtaned by the two methods for the IEEE 118-bus system usng ther proposed methodologes. Results obtaned by the proposed PSO based method are better than the one observed by the EP based method avalable n [9] n terms of loadablty and computaton tme. Table IV: Comparson of smulaton results of IEEE 118-bus system Compared tem EP based method PSO based method Maxmum possble loadablty Total number of generatons Populaton Sze Due to the randomness n the PSO technque, the algorthm s executed for 60 trals when appled to the test systems. It s found from the results that, PSO shows good consstency n obtanng the optmal solutons. 7. CONCLUSION Ths paper nvestgates one of the most promsng FACTS devces, namely UPFC as control agent n a power system. Here, partcle swarm optmzaton s used to determne the optmal locaton and parameters of UPFCs. The system loadablty was employed as a measure of power system performance. Smulaton results valdate the effcency of ths new approach n maxmzng the loadablty of the system. Furthermore, the locaton of UPFCs and ther parameters are optmzed smultaneously. Results have shown that, as the number of UPFCs s ncreased, the system loadablty also ncreases up to a lmt. It s also observed that for a gven system, there s a maxmum number of UPFCs beyond whch the loadablty cannot be mproved. The performance of the proposed method demonstrated through ts evaluaton on the IEEE 30-bus power system and the IEEE 118-bus power system shows that PSO s able to undertake global search wth a fast convergence rate and a feature of robust computaton. The proposed algorthm s an effectve and practcal method for the allocaton of UPFCs n large power systems. References [1] S. Gerbex, R. Cherkaon and A. J. Germond, Optmal locaton of mult type FACTS devces n a power system by means of genetc algorthms, IEEE Trans. Power Systems, vol.16, no.3, pp , August [2] F. D. Galana, K. Almeda, M. Toussant, J. Grffn and D. Atanckov, Assessment and control of the mpact of FACTS devces on power system performance, IEEE Trans. Power Systems, vol.11, no.4, November [3] L. Gyugy, C. D. Schauder, S. L. Wllams, T. R. Retman and D. R. Torgerson, The unfed power flow controllers: a new approach to power transmsson control, IEEE Trans. Power Delv., vol.10, no.2, pp , [4] A. Nabav-Nak and M. R. Iravan, Steady -state and dynamc models of unfed power flow controller (UPFC) for power system studes, IEEE Trans. Power Systems, vol.11, no.4, pp , November

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