Investigation of Hybrid Particle Swarm Optimization Methods for Solving Transient-Stability Constrained Optimal Power Flow Problems

Transcription

1 Investgaton of Hybrd Partcle Swarm Optmzaton Methods for Solvng Transent-Stablty Constraned Optmal Power Flow Problems K. Y. Chan, G. T. Y. Pong and K. W. Chan Abstract In ths paper, hybrd partcle swarm optmzaton (PSO) s proposed for solvng the challengng mult-contngency transent stablty constraned optmal power flow (MC-TSCOPF) problem. The objectve of ths nonlnear optmzaton problem s to mnmze the total fuel cost of the system and at the same tme fulfl the transent stablty requrements. The optmal power flow (OPF) wth transent stablty constrants consdered s re-formulated as an extended OPF wth addtonal rotor angle nequalty constrants, whch s sutable for hybrd PSO to solve. Comparson between varous exstng hybrd PSO technques s carred out by solvng the New England 39-bus system. Expermental results ndcate that the hybrd PSO ntegrated wth the mutaton operaton of genetc algorthms s better than the other exstng hybrd PSO methods n both soluton qualty and stablty. As a result, reasonable solutons can be reached wth faster convergence speeds and smaller computatonal efforts. Index Terms Partcle swarm optmzaton, genetc algorthms, transent stablty, optmal power flow, constraned optmzaton. I. INTRODUCTION The MC-TSCOPF ams to acheve an optmal soluton of a specfc objectve functon, such as fuel cost, network loss, by settng the system control varables, whle satsfyng the system to wthstand specfed contngences (dsturbances) and reach an acceptable steady-state operatng condton [1]. In solvng MC-TSCOPF, the dffculty manly comes from the non-convexty nature of OPF and the nonlnear dfferental-algebrac equatons whch descrbe the transent stablty constrants of the power system. Nonlnear and sem-nfnte programmng [,3] was proposed to solve the MC-TSCOPF. However, not only ther formulaton s complex and heavly ted to the system models, but also they rely on convexty to obtan the global optmum soluton and as such are forced to smplfy relatonshps n order to ensure convexty [4]. Partcle swarm optmzaton (PSO) s a recently proposed populaton based stochastc optmzaton algorthm whch was nspred by the socal behavors of anmals such as fsh The authors gratefully acknowledge the support by the Hong Kong Polytechnc Unversty (Project A-PA4Y and 4-ZV8). K.Y. Chan s wth the Department of Industral and Systems Engneerng, The Hong Kong Polytechnc Unversty, Hong Kong SAR. (rchanky@net.polyu.edu.hk) G.T.Y. Pong s wth the Department of Appled Mathematcs, The Hong Kong Polytechnc Unversty, Hong Kong SAR. (Glory.Pong@polyu.edu.hk). K. W. Chan s wth the Department of Electrcal Engneerng, The Hong Kong Polytechnc Unversty, Hong Kong SAR (eekwchan@polyu.edu.hk) schoolng and brd flockng [5]. Compared wth other stochastc optmzaton methods, PSO has comparable or even superor search performance for many hard optmzaton problems wth faster and stable convergence rates [6]. It requres only few parameters to be tuned and hence s attractve from an mplementaton vewpont. It has attracted broad attenton n the felds of evolutonary computng, optmzaton and many others. In recent years there have been a lot of reported works focused on the PSO. It has been appled wdely n the functon optmzaton, artfcal neural networks development, fuzzy control and some other felds. However, t can be notced that PSO performs well n the early state of the search, but the mprovement decreases gradually along the searchng stages. Its mprovement even termnates n the later stages of the search. It behaves lke the tradtonal local searchng methods that drop nto a local optma and cannot escape from t. Many mproved PSO algorthms have been proposed by ncorporatng wth other optmzaton methods, so as to explore better solutons. It can be found from the lteratures that PSO algorthms could be enhanced by ncorporaton wth GAs [7-10]. In these approaches, operatons of PSO and GAs are crossed over the search smultaneously. GAs operatons lke crossover, mutaton, selecton are ntegrated nto PSO. However, so far no conclusve concluson has been reached n whch hybrd PSO algorthm s better than the others. In ths paper, we re-formulated a smple transformaton of the mult-contngency-transent stablty constrants to the optmal power flow problem, whch s sutable for PSO algorthms to solve. A refned PSO s adopted as the man solver for ths challengng MC-TSCOPF problem. Among all the exstng hybrd PSO algorthms, four selected ones [7-10] have been mplemented and tested on the New England 39 bus system. Expermental result shows that the hybrd PSO algorthm whch ntegrates PSO wth the mutaton operaton of GAs s better than the other exstng hybrd PSO algorthms n both soluton qualty and soluton stablty n whch smaller computatonal effort s requred. II. MC-TSCOPF PROBLEM FORMULATION MC-TSCOPF s mathematcally defned as mn f ( xy, ) (1) st.. g( x, y) = 0 () H x, y 0 (3) ( ) () t ( ) U x, y 0, t T (4)

2 where x () t s a dependent vector whch ncludes actve and reactve power of the swng bus, voltage angle and reactve power of generator buses, and voltage angle and magntude of load buses. T = t, ) (, 0 tcl tcl te s the transent perod from the occurrence of the dsturbance at tme t to the clearng tme 0 t cl and then to the endng tme t e. x represents the ntal value of x () t at t = 0. y s a control whch ncludes vector actve power and voltage magntude of generator buses, voltage angle and magntude of the swng bus, and tap poston of LTCs. f can be expressed as the total generaton cost, total network loss, corrdor transfer power, total cost of compensaton, etc. g s the set of equalty constrants whch are usually the power flow constrants for a specfed operatng condton. H s nequalty constrants for the steady-state securty lmts lke bus voltage magntude lmts, generator power lmts, thermal lmts for transmsson lnes, etc. The dynamc securty constrants set U s nfnte n the functonal space. Further detals of the formulaton of MC-TSCOPF are avalable n [3,11]. Snce the equalty constrants g are mposed mplctly by the power flow calculaton ncorporated wthn the algorthm and also the nequalty constrants H s drectly satsfed by the PSO, the MC-TSCOPF can be formulated as a penalty functon problem: F x = mn f( x, y) + β [ U( x t, y ) ] (5) { } ( ) ( ) Generally, transent stablty constrants can be consdered as hard constrants that should not be volated whlst the statc constrants are soft n nature that slght volaton could be tolerant. Compared wth other constrant handlng approaches [1,13], penalty functon offers a smple and flexble strategy to effectvely deal wth mxed hard and soft constrants. In addton, there s no need to have separate penalty factors for each type of constrants. In (5), any transent nstablty would ntroduce a huge angle devaton and thus produce a large volaton and thus dscrmnaton even though the same penalty factor s used for all type of volatons. Typcally, β = 1000 works very well n most power systems [11]. III. TSCOPF USING PARTICLE SWARM OPTIMIZATION PSO s a novel optmzaton method developed by Kennedy and Eberhart [5,6]. Ths type of algorthms s modeled on processes of the socologcal behavour assocated wth brd flockng, and s one of the evolutonary computaton technques essentally. It uses a number of partcles that consttute a swarm. Each partcle traverses the search space lookng for the global mnmum (or mum). In a PSO system, partcles fly around n a multdmensonal search space. Durng flght, each partcle adjusts ts poston accordng to ts own experence, and the experence of neghbourng partcles, makng use of the best poston encountered by tself and ts neghbours. The swarm drecton of a partcle s defned by the set of partcles neghbourng the partcle and ts hstory experence. The best prevous poston of a partcle s recorded and represented as pbest. The poston of the best partcle among all the partcles s represented as gbest. The velocty and poston of each partcle can be calculated usng the followng formulas [14]: v = k ( w v + ϕ rand() ( pbest x ) ϕ rand () ( gbest x )) (6) x = x + v where x and v are the current poston and velocty of partcle at the th generaton respectvely, w s nerta weght factor, ϕ 1 and ϕ are acceleraton constants, rand() returns a unform random number n the range of [0,1], k s constrcton factor derved from the stablty analyss of equaton (6) to ensure the system to be converged but not prematurely [15]. Mathematcally, k s a functon of ϕ 1 and ϕ as reflected n the followng equaton: k = ϕ ϕ ϕ 4 where ϕ = ϕ + ϕ 1 and ϕ > 4. PSO utlzes pbest and gbest to modfy the current search pont to avod the partcles movng n the same drecton, but to converge gradually toward pbest and gbest. Sutable selecton of nerta weght w provdes a balance between global and local exploratons. Generally, w can be dynamcally set wth the followng equaton [6]: w w mn w = w (8) where s the mum number of teratons, s the current number of teratons, w and w mn are the upper and lower lmts of the nerta weght, and are set to 1. and 0.1, respectvely, n the case studes. In the above procedures, the partcle velocty s lmted by a mum value v. The parameter v determnes the resoluton, or ftness, wth whch regons are to be searched between the present poston and the target poston. Ths lmt enhances the local exploraton of the problem space and t realstcally smulates the ncremental changes of human learnng. If v s too hgh, partcles mght fly past good solutons. If v s too small, partcles may not explore suffcently beyond local solutons. Based on our experences wth PSO, v s often set at 10% to 0% of the dynamc range of the varable on each dmenson. In ths paper, 0% of the varable dynamc range s adopted as the lmt of v. The followng descrbes the ncorporaton of PSO algorthm () nto the mult-contngency transent stablty constraned optmal power flow. Step 1: Input system data, contngency set, PSO parameters and specfy the lower and upper boundares of each varable. Control varables nclude the actve power and termnal voltage of each generator, voltage angle and magntude of the swng bus, and tap poston of each LTC. Step : Each partcle n the swarm represents a feasble canddate soluton to the optmzaton problem and s ntalzed randomly wth all control varables satsfed ther practcal operaton constrants. (7)

3 Step 3: For each partcle, an unconstraned Newton- Raphson power flow calculaton s used to determne the power flow soluton, whch ncludes all the dependent varables, for a gven set of control varables. Step 4: Evaluate the ftness of each partcle usng the evaluaton functon descrbed n (5). Power flow soluton obtaned n Step 3 s used to evaluate the objectve functon (1) and the statc volatons ()-(3). For transent stablty volaton evaluaton (4), transent stablty smulaton s used to produce the generator rotor responses. The mum rotor angle devaton from the COI, among all generators and contngences, s then used to compute a transent stablty penalty usng (5). Step 5: Fnd the best poston of the swarm gbest and the best poston of each partcle pbest by comparng the evaluaton value F ( x) of each partcle wth the one n pbest. If F ( x) s better, then set pbest to the correspondng x. The best among pbest s denoted as gbest. Step 6: If there s any stoppng crtera beng satsfed, go to Step 11; otherwse, ncrement the teraton number. Step 7: Update the nerta weght w accordng to equaton (8). Step 8: Update the velocty v of each partcle accordng to equaton (6). If v > v, v = v. If v < v, v = v. Step 9: Update the poston of each partcle. If a partcle volates ts poston lmts (.e. lmts of the control varables) n any dmenson, set ts poston at the proper lmt. Step 10: Return to Step 4 to repeat the evaluaton process wth updated poston, untl the termnaton condton s reached. Step 11: The partcle that generates the latest gbest s the optmal value. IV. HYBRID PARTICLE SWARM OPTIMIZATION METHOD Ths secton presents the operaton of the four selected hybrd PSO methods [7-10] for solvng the MS-TSCOPF problem: A. Ahmed et al s hybrd PSO () Ahmed et al [7] observed that PSO performs well n the early teratons, but t usually presents problems reachng a near-optmal soluton. The behavor of the PSO n the model presents some mportant aspects related wth the velocty update. If a partcle s current poston concdes wth the global best poston, the partcle wll only move away from ths pont f ts nerta wegh w and velocty v are dfferent from zero. If ther veloctes are very close to zero, then all the partcles wll stop movng once they catch up wth the global best partcle, whch may lead to a premature convergence to the PSO. In fact, ths does not even guarantee that the PSO has converged on a local mnmum t merely means that all the partcles have converged to the best poston dscovered so far by the swarm. Ths phenomenon s known as stagnaton [15]. To prevent t, Ahmed et al proposed to ntegrate the mutaton of GAs nto the PSO. Ths approach allows the search to escape from local optma and search n dfferent zones of the search space. It starts wth the random choce of a partcle n the swarm and moves to dfferent postons nsde the search area. Ahmed et al employed the mutaton operaton by the followng equaton: mut p k = p k 1 + ω (9) ( [ ]) ([ ] ) where p[ k ] s the random choce partcle from the swarm, and ω s randomly generated wthn the range [ 0, 0.1 ( x x )], mn representng 0.1 tmes the length of the search space. The procedures of Ahmed et al [7] s PSO () are shown: Step 1: Step 1 to Step 9 of Step : Perform the mutaton operaton based on (9) Step 3: Step 10 to Step 11 of B. Juang s hybrd PSO () Juang [8] observed that GA and PSO work wth a populaton of solutons. Orgnally, PSO works based on socal adaptaton of knowledge, and all partcles are consdered to be n the same teraton. On the contrary, GA works based on evoluton from teraton to teraton, and the changes of partcles (or chromosomes n GA s termnology) n a sngle teraton are not consdered. In the reproducton and crossover operaton of GAs, partcles are reproduced or selected as parents drectly to the next generaton wthout any enhancement. However, n nature, partcles wll grow up and become more sutable to the envronment before producng offsprng. To ncorporate ths phenomenon, PSO that s nspred by socal nteracton of knowledge s adopted to enhance the top-rankng partcles on each teraton. It enhances partcles by both sharng nformaton between each other and ther ndvdually learned knowledge. Then, these enhanced partcles are reproduced and selected as parents for crossover operaton and mutaton operaton as n genetc algorthms. Offsprng produced by the enhanced partcles are expected to perform better than some of those partcles n orgnal teraton, and the poor-performed partcles wll be weeded out from teraton to teraton. The procedures of Juang s hybrd PSO () are shown: Step 1: Step 1 to Step 4 of Step : Select top-half best performng partcles as eltes. Step 3: Perform the PSO operaton (same as the one shown n Step 5 to Step 9 of ) on the selected eltes rather than all partcles. Step 4: Select two enhanced eltes n Step 3 as two parents by tournament selecton, n whch two enhanced eltes are selected randomly, and ther ftness values are compared to select the elte wth better ftness values as one parent, and then the other parent s selected n the same way. Step 5: Produce two offsprng by performng two-pont crossover on the two parents selected n Step 4.

4 Step 6: Repeat Step 4 and Step 5 untl the reproduced offsprng occupes the whole populaton of the eltes. Step 7: Unform mutaton s adopted on offsprng reproduced on Step 6, that the mutated gene s drawn randomly, unformly from the correspondng search nterval. A constant mutaton probablty p = 0.1 s used. m Step 8: Replace the bottom-half worst performng partcles by the offsprng produced n Step 7. Step 9: Step 10 to Step 11 of. C. Noel and Jannett s hybrd PSO () Noel and Jannett [9] ntended to ncrease the convergence speed of the PSO by ntegratng the dervatve nformaton of gradent nto the formulaton of the velocty of each partcle as n equaton (6). The classcal gradent descent s assumed as: ( ) x = x η C x + 1 (10) where C s a cost functon (n our case the cost functon s defned as (5)), η s the learnng rate, and x s the current poston of partcle at the th generaton. An updated equaton was proposed by combnng (6) and (10): v = k w v + k ϕ rand () ( pbest x ) rand() + ( C( x + ε E ) C( x )) ε (11) x = x + v where w s defned as n (8) and k s defned as n (7), ε s a th n small constant, E s the standard bass vector for R, and n s the number of varables of the cost functon. The frst term n (11) s the nertal term, the second term moves the partcle towards the global best soluton, and the thrd term moves the partcle n the drecton opposte the gradent. Noel and Jannett s hybrd PSO (we call t ) s dentcal to except that the update equaton (11) s used n Step 8 nstead of usng equaton (6). D. Sh s hybrd PSO (ShPSO) The man dea of Sh s hybrd PSO [10] s to run PSO and GA methods alternatvely n seres. It performs a pre-defned number of PSO teratons smultaneously at frst. After the PSO teratons, the fnal partcles are consttuted the frst populaton of GA. Then the populaton s evolved usng GA-operators untl the pre-defned number of teratons of GA reached. After runnng wth the pre-defned number of teratons of GA, the reproduced populaton of GA s transmtted back to PSO as the frst populaton of partcles. Then the PSO operaton performs untl the pre-defned number of teratons of the second PSO termnaton condton reached. The procedures of Sh s hybrd PSO are shown: Step 1: Step 1 to Step 9 of Step : Return to Step 1 to repeat the PSO operatons for partcles updatng untl pre-defned number of PSO teratons s reached. /* Step 1 to Step are the steps of PSO */ Step 3: Pass the fnal populaton of partcles of the PSO to GA as ts frst populaton. Step 4: Select the parents n the populaton based on the roulette-wheel selecton. Step 5: Product offsprng by performng dscrete crossover on the selected parents wth the crossover rate p = 0.8. c Step 6: Mutate the produced offsprng by performng mutaton operator of Gaussan perturbaton wth the mutaton rate p = 1/ n, where n s the number of m varables of the cost functon. Step 7: Return to Step 4 to repeat the evaluaton process untl the pre-defned number of teratons of GA s reached. /* Step 3 to Step 7 are the steps of GA */ Step 8: Pass the fnal populaton of GA to the second PSO as ts frst partcles populaton. Step 9: Step 1 to Step 9 of Step 10: Return to Step 9 to repeat the PSO operatons for partcles updatng untl the termnaton condton s reached. /* Step 8 to Step 10 are the steps of PSO */ V. CASE STUDY A case study of solvng the optmal power flow problems wth stablty constrants on the New England 39-bus system s used to demonstrate the effectveness and robustness of the hybrd PSO based approaches (,,, NeolPSO and ShPSO) for solvng MC-TSOCPF problems. All the hybrd PSO based approaches are coded n Matlab. The system data of the power system s collected n [16,17]. The New England 39-bus test system comprses 10-generator, 39-bus, and 46-lne. Power System Toolbox [16] s employed to perform tme-doman transent stablty smulatons for determnng generator rotor trajectores. The tme step adopted s 0.01s and the ntegraton tme nterval s fxed to 1.5s. The total load for the operatng condton consdered s 6,098 MW and 1,409 MVAr. There are three onload tap changers connected buses 11-1, 1-13 and After a complete scan of all possble sngle lne fault contngences, the followng two conflctng contngences were dentfed. Contngency 1: A three phase fault occurred at the end of lne 6-7 near bus 6. The fault was cleared by trppng the lne at bus 6 after 110 ms and at bus 7 after 10 ms. Contngency : A three phase fault occurred at the end of lne near bus 16. The fault was cleared by trppng the lne at bus 16 after 80 ms and at bus 17 after 100 ms. Wth the above two contngences, the followng 4 cases were bult. Case 1: conventonal OPF wthout any transent stablty constrants Case : transent stablty constraned OPF wth contngency 1 consdered only

5 Case 3: transent stablty constraned OPF wth contngency consdered only Case 4: transent stablty constraned OPF wth contngency 1 and consdered The parameters used n all the hybrd PSO based approaches are followngs: swarm sze = 30, ntal nerta weght w = 1., acceleraton constants ϕ 1 = ϕ =.05, penalty factor β = 1000, pre-defned number of teratons = test runs were performed to collect the four statstcs for the average, varance, best and worst results among the 50 test runs. Table I gves the average optmzaton results of the 50 runs for the above four cases. The number n bracket s ther poston rankng. Table I. Mean Cost n 50 Test Runs Methods Case 1 Case Case 3 Case (3) (4) (3) (4) (1) (1) (1) (1) (4) () () () (5) (5) (5) (5) ShPSO () (3) (4) (3) It s observed that algorthm acheves the best mean cost among the fve PSO algorthms. In fact, the obtans the lowest s n all cases. Table II shows the varance of the 50 runs. The smaller the varance means the closer the values cluster around the mean. Snce three out of four of the varances of are the smallest, t demonstrates that the algorthm s capable to approach and keep searchng around the optmal mean closer. Table II. Varance n 50 Runs Methods Case 1 Case Case 3 Case ShPSO Therefore these results ndcate that algorthm s better than the other hybrd PSO based approaches n both soluton qualty and soluton stablty n solvng the MC-TSOCPF problem. It demonstrates that PSO ntegrated wth the mutaton operaton of GAs can make enhancement for searchng better solutons. The convergence plots of all the PSO methods for cases 1-4 are shown n Fg 1-4, respectvely. They show the progresses of each PSO method through the searches for the frst 50 teratons. It can be observed clearly from the fgures that the convergence speeds of wth ntegraton of mutaton operaton are faster than the other four methods whlst ts soluton s also among the best. In other words, s more lkely to reach better solutons whlst pre-mature convergence s more unlkely to be happened n than the other four PSO methods. For accessng the computatonal efforts requred for each hybrd method to reach reasonable solutons, the solutons obtaned from the standard PSO,.e. are used as the acceptable benchmark solutons of the MC-TSCOPF problems. Table III shows the number of teratons needed for each hybrd PSO method to reach the solutons found by wth 50 teratons. For the hybrd methods, the mum number of teratons was set to 50. Any methods whch cannot reach an acceptable soluton as found by would have a Nl n the table. Based on the number of teratons as well as the computaton tme, as detaled n Table III and IV, needed to reach an acceptable soluton, the computatonal efforts of each hybrd method could be accessed and compared. It can be found from Table III and IV that can reach the acceptable solutons wth smallest numbers of computatonal teratons and shortest computatonal tmes than the rest four hybrd PSO methods. Table III. Number of teratons performed n the PSO methods untl the acceptable soluton reached Methods Case 1 Case Case 3 Case Nl Nl Nl Nl Nl ShPSO Nl Nl Table IV. Computatonal tme (n seconds) performed n the PSO methods untl the acceptable soluton reached Methods Case 1 Case Case 3 Case Nl Nl Nl Nl Nl ShPSO Nl Nl VI. CONCLUSION In ths paper, four hybrd partcle swarm optmzaton algorthms have been selected from the exstng hybrd methods publshed n recent years for solvng the challengng mult-contngency transent stablty constraned optmal power flow (MC-TSCOPF) problem. The feasblty and robustness of each hybrd PSO method for the TSCOPF problem are demonstrated on the New England 39-bus system. Expermental results ndcate that the superorty of the hybrd PSO method, namely, whch ntegrate the PSO wth mutaton operaton of GA for solvng mult-contngency TSCOPF n both soluton qualty and stablty wth smaller

6 computatonal effort over other exstng hybrd PSO methods, have been tested. Snce tme-doman smulaton s adopted for transent stablty evaluaton, the computaton task of the OPF wth transent stablty constrants consdered s farly tmeconsumng. However, the proposed method shows the potental for on-lne and off-lne applcatons n a parallel computng envronment. Ths s an area for the future work. Also standard mutaton operaton wth constant mutaton space s currently n use, whch can be mproved by replacng wth an enhanced mutaton operaton wth dynamc mutaton space. The results wll be reported n the near future. REFERENCES [1] J. A. Momoh, R. J. Koessler and M. S. Bond: Challenges to Optmal Power Flow, IEEE Trans. Power Syst., vol. 1, no. 1, pp , Feb [] D. Gan, R. J. Thomas and R.D. Zmmerman: Stablty-constraned optmal power flow, IEEE Transactons on Power Systems, vol. 15, pp , 000. [3] Y. Xa, K.W. Chan and M. Lu: Drect nonlnear prmal-dual nteror-pont method for transent stablty constraned optmal power flow, IEE Proceedngs: Generaton, Transmsson and Dstrbuton, vol. 15, pp , 005. [4] J. A. Momoh and J. Z. Zhu: Improved nteror pont method for OPF problems, IEEE Trans. Power Syst., vol. 14, no.3, pp , Aug [5] J. Kennedy and R. Eberhart: Partcle swarm optmzaton, Proceedngs of IEEE Internatonal Conference on Neural Networks, vol. 4, pp , [6] J. Kennedy and R. Eberhart: Swarm Intellgence, Morgan Kaufmann Publshers, 001. [7] A. A. E. Ahmed, L. T. Germano and Z. C. Antono: A hybrd partcle swarm optmzaton appled to loss power mnmzaton, IEEE Transactons on Power Systems, Vol. 0, No., pp , May 005. [8] C. F. Juang: A hybrd genetc algorthm and partcle swarm optmzaton for recurrent network desgn, IEEE Transactons on Systems, Man and Cybernetcs Part B: Cybernetcs, vol. 34, no., pp , 004. [9] M. M. Noel and T. C. Jannett: Smulaton of a new hybrd partcle swarm optmzaton algorthm, Proceedngs of the Thrty-Sxth Southeastern Symposum on System Theory, pp , 004. [10] X. H. Sh, Y. H. Lu, C. G. Zhou, H. P. Lee, W. Z. Ln and Y. C. Lang: Hybrd evolutonary algorthms based on PSO and GA, IEEE Congress on Evolutonary Computaton, Vol. 4, pp , 003. [11] N. Mo, G. T. Y. Pong, K. W. Chan, S. W. Me: Mult-contngency Transent Stablty Constraned Optmal Power Flow by Genetc Algorthm, Proc. 7 th Conf. on Advances n Power System Control, Operaton and Management, CDROM, 006. [1] R. Farman and J. A. Wrght: Self-Adaptve Ftness Formulaton for Constraned Optmzaton, IEEE Transactons on Evolutonary Computaton, volume 7, pp , 003. [13] T. P. Runarsson and X. Yao: Stochastc rankng for constraned evolutonary optmzaton, IEEE Transactons on Evolutonary Computaton, vol. 4, pp , 000. [14] B. Zhao, C. X. Guo, and Y. J. Cao: A multagent-based partcle swarm optmzaton approach for optmal reactve power dspatch, IEEE Trans. Power Syst., vol. 0, no., May 005. [15] R. C. Eberhart and Y. Sh: Comparson between genetc algorthms and partcle swarm optmzaton, Evolutonary Programmng VII. New York: Sprnger-Verlag, 1998, vol. 1447, Lecture Notes n Computer Scence, pp [16] J. H. Chow: Power System Toolbox Verson.0, Cherry Tree Scentfc Software, 000. [17] R. Zmmerman and D. Gan: MATPOWER: A Matlab power system smulaton package, x 104 Case NeolPSO 3.66 ShPSO generaton numbers Fg. 1 Convergence curves of varous PSO methods for Case 1

7 4 x 104 Case ShPSO generaton numbers Fg. Convergence curves of varous PSO methods for Case 3.9 x 104 Case ShPSO generaton numbers Fg. 3 Convergence curves of varous PSO methods for Case 3 4. x 104 Case ShPSO Iteraton numbers Fg. 4 Convergence curves of varous PSO methods for Case 4