Allocation of Static Var Compensator Using Gravitational Search Algorithm

Transcription

1 Frst Jont Congress on Fuzzy an Intellgent Systems Ferows Unversty of Mashha, Iran 9-3 Aug 007 Intellgent Systems Scentfc Socety of Iran Allocaton of Statc Var Compensator Usng Gravtatonal Search Algorthm Esmat Rashe, Hossen Nezamaba-pour, Sae Saryaz, Malhe M. Farsang Electrcal Engneerng Department of Shah Bahonar Unversty, Kerman Abstract: Gravtatonal optmzaton (GO) s use to place the Statc Var Compensator (SVC) n a large power system base on ts prmary functon, where the optmzaton s mae on two parameters: ts locaton an sze. The prmary functon of a SVC s to mprove transmsson system voltage, thereby enhancng the maxmum power transfer lmt. To valate the results, Partcle Swarm Optmzaton (PSO) Algorthm s apple an the performances of GO an PSO are compare. The results show GO quckly fns the optmal soluton n fnng the locaton an sze of SVC. Keywors: Gravtatonal optmzaton, partcle swarm optmzaton, voltage stablty, FACTs evces, SVC. Introucton In the last ecaes, efforts have been mae to fn the ways to assure the securty of the system n terms of voltage stablty. It s foun that flexble AC transmsson system (FACTS) evces are goo choces to mprove the voltage profle n power systems that operate near ther steay-state stablty lmts an may result n voltage nstablty. Many stues have been carre out on the use of FACTS evces n voltage an angle stablty. Takng avantages of the FACTS evces epens greatly on how these evces are place n the power system, namely on ther locaton an sze. Over the last ecaes there has been a growng nterest n algorthms nspre from the observaton of natural phenomena. It has been shown by many researches that these algorthms are goo replacement as tools to solve complex computatonal problems such as optmzaton of obectve functons, tranng neural networks, tunng fuzzy membershp functons, machne learnng, system entfcaton, control, etc. Varous heurstc approaches have been aopte by researches nclung genetc algorthm, tabu search, smulate annealng, ant colony an partcle swarm optmzaton. These algorthms have been proven to be very effectve for statc an ynamc optmzaton problems. Stuy on the use of heurstc approaches to seek the optmal locaton of FACTS evces n a power system s carre out by the researches aroun the worl [-0]. In ths paper a new optmzaton algorthm s use known as Gravtatonal Optmzaton (GO). The propose optmzaton algorthm s base on gravtatonal law an laws of moton base on followng efnton by Englsh mathematcan Sr Isaac Newton n 687: every partcle n the unverse attracts every other partcle wth a force that s rectly proportonal to the prouct of ther masses an nversely proportonal to the square of the stance between them. GO algorthm was ntrouce by Rashe n 007 []. Now n ths paper the ablty of the propose algorthm n power system for Var plannng by SVC s nvestgate. To valate the results obtane by GO, PSO s apple an the

2 convergence characterstcs of two algorthms are compare. In the next secton a bref overvew of gravtatonal force s gven to prove a proper backgroun an followe by explanaton of GO an PSO. OVERVIEW ON GRAVITATIONAL FORCE In physcs, gravtaton s the tenency of obects wth mass to accelerate towar each other. Gravtaton s one of the four funamental nteractons n nature, the other three beng the electromagnetc force, the weak nuclear force, an the strong nuclear force. Gravtaton s the weakest of these nteractons, but acts over great stances an s always attractve. Gravty s a force, pullng together all matter. Every boy n the unverse attracts every other boy. Gravty s every where. In the Newton gravtatonal law, each partcle attracts every other partcle wth a force whch s 'gravtatonal force'. The way Newton's gravtatonal force behave was calle 'acton at a stance'. Ths means gravty acts between separate boes wthout any ntermeary an wthout any elay. The gravtatonal force between tow boes s rectly proportonal to the prouct of ther masses an nversely proportonal to the square of the stance between them. Newton's law of gravtaton s efne as follows: mm F = G () r where F s the magntue of the gravtatonal force between the two pont masses, G s the gravtatonal constant, m the mass of the frst pont mass, m s the mass of the secon pont mass, r s the stance between the two pont masses. Also Newton ntrouce three laws for moton. The frst law s that, once a boy s set n moton, t wll reman movng n constant spee n a straght lne unless a force acts on t. Newton's secon law s that when a force s apple to a boy, ts acceleraton epens only on the force an on the mass of boy. Newton's thr law efnes that acton s equal to reacton. The epenence of the acceleraton ' a ' an the force ' F ' an the mass ' m ' can be wrtten as follows: F a = () m Base on ()-(), there s an attractng force of gravty among all partcles of the unverse where the effects of bgger an the closer mass s more. A ecreasng gravty wll mean an ncreasng stance between them. Also n physcs three fferent masses are efne as follows: -Inertal mass. It s a measure of an obect's resstance to changng ts state of moton when a force s apple. An obect wth small nertal mass changes ts moton more realy, an an obect wth large nertal mass oes so less realy. -Passve gravtatonal mass. It s a measure of the strength of an obect's nteracton wth the gravtatonal fel. Wthn the same gravtatonal fel, an obect wth a smaller passve gravtatonal mass experences a smaller force than an obect wth a larger passve gravtatonal mass. -Actve gravtatonal mass. It s a measure of the strength of the gravtatonal fel ue to a partcular obect. For example, the gravtatonal fel that one experences on the Moon s weaker than that of the Earth because the Moon has less actve gravtatonal mass. It s shown that there s no any fference between these three masses. Wth the above escrptons, the GO algorthm s explane n the next secton. 3 Gravtatonal Optmzaton an Partcle swarm optmzaton. GO: In GO a set of agents calle masses are ntrouce to fn the optmum soluton. Each mass has three specfcatons: the poston of the mass, nertal mass an gravtatonal mass. The poston of the mass s representng a soluton for the problem. By gravtatonal mass, we mean actve gravtatonal mass an passve gravtatonal mass s consere to be. The values of gravtatonal an nertal masses are etermne base on the ftness functon calculate for each mass. For solvng a problem usng GO, an artfcal system s set up. It s assume that only gravtatonal an moton laws are governerng. The general vews of such laws are smlar to that one n the nature, whch can be escrbe as follows: Gravtatonal law: each partcle attracts every other partcle an the gravtatonal force between tow boes s rectly proportonal to the prouct of ther masses an nversely proportonal to the square of the stance between them.

3 Moton law: the current velocty of any mass s equal to the sum of a fracton of prevous velocty of mass an varaton n the velocty. Varaton n the velocty or acceleraton of any mass s equal to the force acte on the system ve by mass of nerta. Now conser a system wth m masses. The poston of the th partcle s efne as follows: X = x,..., x,..., x ) (3) ( n x s the poston of th th mass n the menson. In the tme t a force acts on mass from mass. Ths force s gven as follows: G( * Mg F = ( x x ) R + ε (4) where Mg s gravtatonal mass relate to mass, G ( s gravtatonal constant n tme t, ε s a small value an R ( s the Euclan stance between two masses an efne as follows: R = X, X (5) The total force actng on mass n the th menson n tme t s gven as follows: F = r F ( ) (6) t kbest, where r s a ranom number n the range [0,]. Kbest s the frst K masses wth the best ftness functon. Base on the Newton's secon law, the acceleraton relate to mass n tme t n the th menson s gven as follows: F a = (7) M where M s the nertal mass of th mass. The next velocty of each mass s the sum of a fracton of current velocty of mass an ts acceleraton. The poston an the velocty of each mass are efne as follows: V ( t + ) = r * V + a (8) x ( t + ) = x + V ( t + ) (9) where r an r are two ranom numbers n the range [0,]. In GO, gravtatonal constant wll take an ntal value an t wll be reuce by tme as follows: t T β α G = (0) where T s the number of teraton. The gravtatonal mass an the nertal mass are upate by followng equatons: ft worst( Mg = best( worst( () M = + Mg () where ft ( represent the ftness relate to mass n tme t, worst ( an best ( efne as follows: best( = mn ft (3) {,.., m} worst( = max ft (4) {,.., m} The prncple of GO s shown by Fgure. Generate ntal populaton Evaluate the ftness for each mass Calculate the G, best, worst, M an M g of the populaton No Upate poston an velocty Meetng en of crteron? Yes Return best soluton Fgure : General prncple of GO algorthm.. PSO s as an optmzaton tool that proves a populaton-base search proceure n whch nvuals, calle partcles, change ther postons wth tme [-3]. In a PSO system, partcles fly aroun n a multmensonal search space. Durng flght, each partcle austs ts poston accorng to ts own experence, an the experence of neghbourng partcles, makng use of the best poston encountere by tself an ts neghbours. Partcles n the PSO are efne by two varables:

4 x an v n whch x s the poston of the partcle representng a canate soluton to the problem an v escrbes the velocty. In the PSO, two fferent efntons are use as: the nvual best an the global best. As a partcle moves through the search space, t compares ts ftness value at the current poston to the best ftness value t has ever attane prevously. The best poston that s assocate wth the best ftness encountere so far s calle the nvual best known or pbest. The global best, or gbest, s the best poston among all of the nvual s best postons acheve so far. Usng the gbest an pbest, the th partcle velocty n the th menson s upate accorng to the followng equaton: v t + ) = w. v + c. A c B (5) ( +. where, A = ran ( pbest x ), B = Ran ( gbest x ), w s nerta weght factor, c an c are acceleraton constant, ran () an Ran () are ranom number between 0 an. Base on the upate veloctes, each partcle changes ts poston accorng to the followng equaton: x ( t + ) = x + v ( t + ) (6) 4 Voltage Stablty Analyss Voltage stablty s the ablty of a power system to mantan acceptable voltages at all buses n the system not only uner normal operaton, but also after followng sturbances. Voltage stablty can be categorze as large-sturbance an smallsturbance voltage stabltes. Large sturbance voltage stablty s the ablty of the system to control the voltage after beng subecte to large sturbances such as system faults, an loss of loa or generaton. Small sgnal voltage stablty s the ablty of the system to control voltage after beng subecte to small perturbatons, such as graual changes n loas [4]. In ths paper GO an PSO are use for analyss of voltage stablty of a power system. 5 Stuy System A 5-area-6-machne system: The stuy system s shown n Fgure, consstng of 6 machnes an 68 buses. Ths s a reuce orer moel of the New Englan (NE) New York (NY) nterconnecte system. The frst nne machnes are the smple representaton of the New Englan system generaton. Machnes 0 to 3 represent the New York power system. The last three machnes are the ynamc equvalents of the three large neghborng areas nterconnecte to the New York power system. GO an PSO are use to locate SVC optmally n the power system shown n Fgure. Implementatons of the two fferent technques are presente below. 4 G G G G 6 63 G G G G G G Fgure : Sngle lne agram of a 5-area stuy system. Placng of SVC usng GO an PSO starts from an ntal loa. All loas are ncrease graually near to the pont of voltage collapse, all at once. Bus voltage magntue profle when system s heavly stresse s shown n Fgure 3. voltage n pu nternal bus number Fgure 3: Bus voltage magntue profle when system s heavly stresse. The goal of the optmzaton s to fn the best locaton of SVC where the optmzaton s mae on 5 G G4 9 6 G9 58 G G7

5 two parameters: ther locaton an sze. In the PSO an GO algorthm, n partcles are generate ranomly where n s selecte to be 50. Snce optmzatons are mae on two parameters: ther locaton an sze, therefore, each partcle s a -mensonal vector n whch =. The ntalzaton s mae on the poston ranomly for each partcle. Also, the mensons are normalze between [0-]. The number of teraton s consere to be 70. In Go, α s set to be 0 an β s set to an ncreasng lnearly to 3. K s consere to be 00 percent of total masses an ecreasng lnearly to percent. In PSO, the parameter n (5) must be tune. These parameters control the mpact of the prevous veloctes on the current velocty where, n ths paper, c = c = an w s ecreasng lnearly from 0.9 to 0.. It means that at the frst teraton w s 0.9 an n the last teraton w s 0.. Each partcle n the populaton s evaluate usng the obectve functon efne as follows, searchng for the partcle assocate wth ob : best voltage n pu nternal bus number Fgure 4: Bus voltage magntue profle of the stresse system after placng 546 MVar SVC at bus. ob = max V V (7) k Ω k ref k where Ω s the set of all loa buses, V s the k voltage magntue at loa bus k an V refk s the nomnal or reference voltage at bus k. To locate SVC by GO, sutable buses are selecte base on 0 nepenent runs, uner fferent ranom sees. Both GO an PSO entfy bus as the bus vulnerable to the voltage collapse. By applyng the GO an PSO, the level of compensaton s foun to be 546 MVar. Fgure 4 shows bus voltage magntue profle of the stresse system after placng a 546 MVar SVC at bus. The results obtane by GO an PSO are average over nepenent runs. The average best-so-far an average level of compensaton of each run are recore an average over 0 nepenent runs. To have a better clarty, the convergence characterstcs n fnng the locaton an sze of a SVC s gven n Fgure 5-6. These fgures show that GSO has a better feature to fn optmal soluton. Fgure 5: Convergence characterstcs of GO an PSO on the average best-so- far n fnng the soluton, 546 MVar SVC at bus. Fgure 6: Convergence characterstcs of GO an PSO on the average mean Ftness n fnng the soluton, 546 MVar SVC at bus. 6 Concluson In ths paper, a new algorthm known as GO s

6 apple to place SVC n a power system. Also, to valate the obtane results by GO, PSO s apple. Both GO an PSO gve the same bus an the same level of compensaton for the SVC placement. Although the results obtane by GO an PSO are the same but GO quckly fns the hgh-qualty optmal soluton n fnng the locaton an sze of SVC. GO has a great potental n solvng complex power system problems. To have an optmal placement for SVC, mult-obectve VAr plannng shoul be consere whch s the future work of the authors. References [] P. Patern, S. Vtet, M. Bena an A. Yokoyama, Optmal locaton of phase shfters n the French network by genetc algorthm, IEEE Trans. Power Systems., vol. 4, pp. 37 4, 999. [] H.C. Leung an T.S. Chung, Optmal placement of FACTS controller n power system by a genetc base algorthm n Proc. 999 IEEE Power Electroncs an Drve Systems Conference, Vol., pp [3] S. Gerbex, R. Cherkaou an A.J. Germon, Optmal locaton of multtype FACTS evces n a power system by means of genetc algorthms, IEEE Trans. Power Syst., vol. 6, No. 3, pp , August 00. [4] F. Wang an G.B. Shrestha, Allocaton of TCSC evces to optmze total transmsson capacty n a compettve power market n Proc. 00 IEEE Wnter Meetng, Vol., pp [5] E.E. El-Araby, N. Yorno an H. Sasak, A comprehensve approach for FACTS evces optmal allocaton to mtgate voltage collapse n Proc. 00 IEEE Transmsson an Dstrbuton Conference, pp [6] L.J Ca, I. Erlch an G. Stamtss, Optmal choce an allocaton of FACTS evces n eregulate electrcty market usng genetc algorthms n Proc. 004 IEEE PES General Meetng, pp [7] N.P. Pahy, M.A. Abel-Moamen an B.J. Praveen Kumar, Optmal locaton an ntal parameter settngs of multple TCSCs for reactve power plannng usng genetc algorthms n Proc. 004 IEEE PES General Meetng, pp. 0 4, [8] L. Ippolto an P. Sano, Selecton of optmal number an locaton of thyrstorcontrolle phase shfters usng genetc base algorthms IEE Proc.-Gener., Transm. Dstrb., Vol. 5, No. 5, pp , September 004. [9] S. Ebrahm, M. M. Farsang, H. Nezamaba-Pour an K. Y. Lee, Optmal Allocaton of STATIC VAR COMPENSATORS usng Moal analyss, Smulate annealng an Tabu search, Proc. 006 IFAC Symposum on Power Plants an Power Systems, Calgary, Canaa, July, 006. [0] M. M. Farsang, H. Nezamaba-Pour an K. Y. Lee, Mult-obectve VAr Plannng wth SVC for a Large Power System Usng PSO an GA, Proc. 006 IEEE PES Power Systems Conference an Exposton (PSCE), Atlanta, USA,. 9Oct-Nov, 006. [] ع. راشدي ح. نظام آبادي "الگ وريتم ج ستجوی گران شی" پ انزدهمين کنفرانس مهندسی برق ايران ارديبهشت ١٣٨٦. [] J. Kenney an R. Eberhart, Partcle swarm optmzaton, n Proc. 995 IEEE Int. Conf. Neural Networks (ICNN 95), vol. IV, pp [3] J. Kenney, The partcle swarm: Socal aaptaton of knowlege, n Proc. 997 IEEE Int. Conf. Evol. Comput., pp [4] P. Kunur, Power System Stablty an Control, McGraw-Hll: New York, 994.