AN IMPROVED OPTIMIZATION ALGORITHM FOR NETWORK SKELETON RECONFIGURATION AFTER POWER SYSTEM BLACKOUT

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1 H. Lan obolšan alortam optmzace za prerazmešta sheme mreže nakon blokrana eneretsko sustava A IMROVED OTIMIZATIO ALORITHM FOR ETWORK SKELETO RECOFIURATIO AFTER OWER SSTEM BLACKOUT Hapn Lan ISS (rnt), ISS (Onlne) DOI: /TV Ornal scentfc paper etwork skeleton reconfuraton s an mportant task durn power system restoraton after blackout. The uncertanty of the restoraton tme and the restoraton successful rate durn the network skeleton reconfuraton are consdered n the paper. The restoraton tme of transmsson lnes and transformers, unts startup tme lmt and the restoraton successful rate are selected as trapezodal fuzzy varables. The optmal network skeleton reconfuraton model after power system blackout s constructed based on a fuzzy chance-constraned prorammn. An optmzaton alorthm combnn fuzzy smulaton wth SO s mplemented to solve the optmal model. The optmal network skeleton wth hher relablty and the restoraton sequence whch meet a certan confdence level are optmzed by the proposed method. The restoraton sequence can ensure the restoraton as quckly as possble. The effectveness of the proposed method s valdated by Matlab wth IEEE 30 bus system test. Keywords: blackout; chance-constraned prorammn; network skeleton reconfuraton; partcle swarm optmzaton; power system restoraton obolšan alortam optmzace za prerazmešta sheme mreže nakon blokrana eneretsko sustava Izvorn znanstven članka rerazmešta sheme mreže važan e zadatak tekom restaurace eneretsko sustava posle blokrana. U radu se razmatra nepredvdlvost vremena restaurace uspešnost brzne restaurace tekom prerazmešta sheme mreže. Vreme restaurace dalekovoda transformatora, rančno vreme pokretana ednca brzna restaurace zabran su kao trapezodne fuzzy varable. Optmaln model za prerazmešta sheme mreže posle blokrana eneretsko sustava zrađen e na osnovu fuzzy chance-constraned proramrana. Implementran e alortam optmzace ko kombnra fuzzy smulacu s SO kako b rešo optmaln model. ouzdan optmaln skelet mreže sled optmzace ko zadovolavau određen nvo pouzdanost optmzrau se predloženom metodom. Sled restaurace može osurat što brže provođene restaurace. Učnkovtost predložene metode proverena e pomoću Matlaba s testom sustava sabrnce IEEE 30. Klučne reč: restauraca eneretsko sustava; blokrane; prerazmešta sheme mreže; chance-constraned proramrane; optmzaca roa čestca Introducton ower system restoraton after blackout s a complex decson and control problem. Many electrc power companes have developed power system restoraton schemes to smplfy the operaton and shorten the restoraton procedure. ower system restoraton nvolves the follown three tasks: unt startup, network skeleton reconfuraton, and load pck-up. As far as network skeleton reconfuraton s concerned, ts scheme has reat effects on the whole restoraton process, and t has been researched wdely. Reasonable network skeleton can reduce the burden of power system restoraton effectvely. The optmal network skeleton based on network reconfuraton effcency and dscrete partcle-swarm optmzaton s researched n []. The network skeleton reconfuraton stratees based on assessment of node mportance and lne betweenness are proposed n [2 3]. A network skeleton reconfuraton method usn related node betweenness, topoloy prorty and path electrcal effects s proposed n [4]. But none of the above references nvolve n restoraton sequence of transmsson lnes and transformers. A network reconfuraton alorthm consdern both network skeleton optmzaton and restoraton sequence optmzaton s studed n [5 6]. The restoraton tme of every transmsson lne and transformer s rearded as a certan factor n the former research, but the actual experence demonstrates that t s uncertan. The uncertan property of the restoraton tme s more sutable to be expressed as a fuzzy varable. Meanwhle, the restoraton of transmsson lnes and transformers s not deal. Consdern the complex crcumstance, the restoraton maybe fals. Thus, the restoraton successful rate s also uncertan and s sutable to be rearded as a fuzzy varable. It s necessary to consder the uncertanty of the restoraton tme and the restoraton successful rate durn the network skeleton reconfuraton, n order to make the network skeleton more reasonable. The partcle swarm optmzaton (SO) s a computatonal method that optmzes a problem by teratvely tryn to mprove a canddate soluton wth reard to a ven measure of qualty. SO s ornally attrbuted to Kennedy, Eberhart and Sh [7 8]. It s frst ntended for smulatn socal behavour n [9], as a stylzed representaton of the movement of oransms n a brd flock or fsh school. The alorthm s smplfed and t s observed to be performn optmzaton. An extensve survey of SO applcatons s made n [0 ]. In ths paper, reardn the restoraton tme and restoraton successful rate as trapezodal fuzzy varables, a network skeleton optmal confuraton model based on the chance-contaned prorammn (CC) s constructed. Accordn to the model characterstcs, SO s used to et the optmal network skeleton and the restoraton sequence of the transmsson lnes and transformers. 2 The optmal network skeleton reconfuraton model consdern uncertanty A stable network skeleton of the power system should be rebult frstly, whch s the oal of the power system restoraton after blackout. And then the remann unserved loads are restored. The obectves need to be acheved safely, flexbly, smoothly and delberately n the Tehnčk vesnk 22, 6(205),

2 An mproved optmzaton alorthm for network skeleton reconfuraton after power system blackout H. Lan shortest tme. In order to fully consder the uncertantes the fuzzy chance-constraned prorammn combned optmzaton alorthm for network skeleton reconfuraton s proposed n the paper. 2. Chance-constraned prorammn In a stochastc envronment, n order to mze the optmstc return wth a ven confdence level subect to some chance constrants [2], the follown CC model s provded as: mαx mαx f x suβect to : os os { f ( x, ξ) f } β { ( x, ξ) 0, =, 2,..., p} α f, () where, x s a decson vector, ξ s a stochastc vector, f(x,ξ) s a return functon, and (x,ξ) are stochastc constrant functons, =, 2,, p. α and β are the ven confdence levels, and f s the β optmstc return. os{} s the possblty measure. 2.2 CC based skeleton network reconfuraton model 2.2. Selecton of fuzzy varables In the paper trapezodal fuzzy varables are used to represent the restoraton tme for transmsson lnes and transformers. The trapezodal fuzzy varable s shown n F.. A trapezodal fuzzy varable means the fuzzy varable fully determned by quadruplet (t, t 2, t 3, t 4 ) of crsp varables wth t < t 2 < t 3 < t 4, t s an optmstc restoraton tme, t 4 s a pessmstc restoraton tme, the tme between t 2 and t 3 s the most lkely restoraton tme. transformers. Assumn the restoraton successful rate of the transmsson lnes and transformers s ndependent. If a restoraton path ncludes several transmsson lnes and transformers, the relablty of the restoraton path s defned as follows: R = R =, (3) where R s the relablty of the restoraton path, R s the restoraton successful rate of the transmsson lne or transformer n restoraton path, s the number of transmsson lnes and transformers n the restoraton path. Defne the relablty of network skeleton s all the relablty of restoraton paths, whch s determned by the restoraton successful rate of the transmsson lnes and transformers. The restoraton relablty of the skeleton network s shown n Eq. (4). R R = = = R = =, (4) where R s the relablty of the skeleton network, s the number of the restoraton path The CC model Consdern the power system dspatchn rules constrant, the unts can only be allowed to be restored one by one. If there are more than one thermal unt, ratonal unts start-up sequence and loads restoraton sequence mht shorten the restoraton tme. The obectve functon s as follows: ( ) ux R f f. (5) t t 2 t t 3 4 Fure The trapezodal fuzzy varable Then membershp functon s ven as the follown. x t f t x t2 t 2 t f t2 x t3 u ( x) =. (2) x t4 f t3 x t4 t3 t4 0 otherwse Relablty ndex of the skeleton network In the stae of network skeleton reconfuraton, assumn that power statons are completely relable and then the restoraton relablty wll depend on the restoraton successful rate of transmsson lnes and Subect to: R os f β, T (6) os T t k α, k = (7) = S T, (8) t m m= where, Eq. (5) s the obectve functon, t ensures the shortest restoraton tme and the hhest relablty of network skeleton, the reater the better. Eq. (6) ensures the obectve functon value f at least at confdence level β. Eq. (7) ensures unts can start-up at least at confdence level α, T s the crtcal mum nterval of unt [3 4], t k s the fuzzy restoraton tme of transmsson lne or transformer k n restoraton path whch provdes power supply to unt, and t s a trapezodal fuzzy 360 Techncal azette 22, 6(205),

3 H. Lan obolšan alortam optmzace za prerazmešta sheme mreže nakon blokrana eneretsko sustava varable, s the number of unts needed to be restored. In Eq. (8), T represents the total restoraton tme of the network skeleton, t m s the restoraton tme of the transmsson lne or transformer m, whch s a trapezodal fuzzy varable, S s the number of transmsson lnes or transformers n the skeleton network. In order to meet branches power and nodes voltae lmts, power flow must be calculated. The network skeleton should obey the follown constrants: = U Q = U Q mn mn mn U (3) lm = = U ( U ( cos δ + B sn δ ), (9) cos δ B sn δ ), (0), () Q Q, (2) U lm U V, m, (4) S where, Eq. (9) and Eq. (0) are the power balance equatons. Eq. () and Eq. (2) are the nequalty constrants of unts. Eq. (3) s the nequalty of buses voltae. Eq. (4) s the actve power flow constrant n the transmsson lnes and transformers. 3 The optmzaton solvn alorthm of the network skeleton reconfuraton model In the paper, fuzzy smulaton s used to check the fuzzy constrants and compute the fuzzy obectve functon. Accordn to destnaton nodes and the power rd characterstcs, the partcle swarm optmzaton alorthm combned wth fuzzy smulaton and Floyd alorthm s used to et the restoraton sequence of the nodes and the correspondn paths. 3. The Floyd alorthm Floyd alorthm n the paper s used to fnd the shortest restoraton paths of network. The Floyd alorthm compares all possble paths throuh the raph between each par of vertces and can fnd the shortest path between every par of vertces of a raph. As a result, a matrx S denotes the shortest dstance between vertces and, and a matrx denotes the next vertex k on the path from vertex to vertex. 3.2 Fuzzy smulaton 3.2. The check of fuzzy constrants Fuzzy smulaton s often used n fuzzy prorammn [5]. After the restoraton paths whch are from blackstart power source to the unts to be restored were found throuh Floyd alorthm, the fuzzy constrants should be checked by fuzzy smulaton. The fuzzy smulaton steps are as follows. Step : enerate crsp varables unformly from fuzzy varables t k, T at confdence level of α. Step 2: Check the constrants. If the crsp varables satsfy the chance constrants represented n Eq. (7), the restoraton path s feasble. Step 3: If the crsp varables are not satsfy the chance constrant represented n Eq. (7), enerate the crsp varables aan. Step 4: If the chance constrants represented n Eq. (7) cannot be satsfed after tmes fuzzy smulaton, the unts cannot be restored usn the current restoraton sequence Computaton of the fuzzy obectve functon After checkn the fuzzy constrants, the fuzzy obectve functon should be computed. The steps are shown as follows. Step : Set f = ; Step 2: enerate crsp varables unformly from fuzzy varables R, t m at confdence level β. Step 3: Substtute the crsp varables nto equatons (4), (6) and (8). Calculate the obectve functon value f. If f < f, then set f = f. Step 4: o to step 2, repeat the fuzzy smulaton for tmes. Step 5: Return the mum obectve functon value f. 3.3 The partcle swarm optmzaton alorthm SO s ntalzed wth a roup of random partcles and then searches for optma by updatn eneratons. In every teraton, each partcle s updated by the follown two best values. The frst one s the local best ftness called pbest. Another best value s a lobal best called best. The alorthm has the follown procedure: Step : Random eneraton of an ntal partcle populaton. Step 2: Reckon the ftness value, t wll drectly depend on the dstance to the optmum. Step 3: Modfy best, pbest and partcle velocty. Step 4: Move each partcle to a new poston. Step 5: o to step 2, and repeat untl converence or a stoppn condton s satsfed. In the paper, SO s used to calculate the optmal restoraton sequence of the unts and loads to be restored. 3.4 The prncple of the optmzaton solvn alorthm The flowchart of the network skeleton reconfuraton alorthm prncple s shown n F. 2. The steps are shown as follows: () Start and set the black-start power source, the unts and loads to be restored. (2) Calculate the shortest paths by Floyd alorthm, store the shortest dstance between vertces wth matrx S, store the next vertex on the path wth matrx. From matrx S and matrx we can et the shortest path between any par of vertces. (3) Reardn the restoraton sequence of unts and loads to be restored as partcles and ntally the partcle wth random sequence. Tehnčk vesnk 22, 6(205),

4 An mproved optmzaton alorthm for network skeleton reconfuraton after power system blackout H. Lan (4) Select the th partcle and search matrces S and to et the restoraton paths from black-start power source and the unts or loads to be restored. (5) Check the chance constrants throuh fuzzy smulaton. (6) If the chance constrants cannot be satsfed, then update the partcles by mutaton. (7) If the chance constrants can be satsfed, then calculate the power flow and check whether the lmts of nodes voltae and branches power can be satsfed. (8) If the teraton tmes have reached p, then calculate the ftness. (9) Updatn the partcles by cross. (0) If the teraton tmes do not reach, then repeat aan, otherwse output the optmal partcles that mean the optmal restoraton sequence. () Accordn to the optmal restoraton sequence, the network skeleton can be otten. Start Set the black-start power source and the unts to be restored Calculate shortest paths by Floyd alorthm and et the matrx S and Form and ntal partcles Set = : Select the th partcle and calculate the shortest paths by matrx S and Check chance constrants wth fuzzy smulaton Satsfy the constrant? ower flow checkn = p? Calculate the ftness by fuzzy smulaton Update the partcles by cross Reach the mal teraton Output the optmal partcle and the correspondn restoraton path Update the partcle by mutaton End Fure 2 The flowchart of the network skeleton reconfuraton alorthm prncple 4 Applcaton example and analyss The optmzaton solvn alorthm proposed n the paper s mplemented by Matlab and tested wth IEEE30 bus system. In ths system, there are 6 unts and 40 transmsson lnes and transformers. Suppose the unt at bus s the black-start unt, the unts to be restored at bus [ ] and the load to be restored at bus [ ]. Suppose the trapezodal fuzzy restoraton tme of unt at bus 27 s (0, 2, 8, 20) mnutes, the other unts trapezodal fuzzy restoraton tme s (20, 25, 30, 35) mnutes. The trapezodal fuzzy restoraton tme of transmsson lnes s (2; 2,2; 2,8; 3) mnutes. The restoraton successful rates of transmsson lnes are shown n Tab.. Durn the network skeleton reconfuraton based on CC, the number of partcles s p = 0, the mal teraton = 00, the tmes of fuzzy smulaton = 000, the confdence level α = 0,96, β = 0,96. Durn the power flow checkn the mal actve power output of unts to be restored s 30 %. Table The restoraton successful rates of transmsson lnes The transmsson lne The restoraton successful rate 3-4 (0,9; 0,92; 0,98; ) 4-6 (0,9; 0,92; 0,98; ) 8-28 (0,9; 0,92; 0,98; ) 0-20 (0,9; 0,92; 0,98; ) 0-7 (0,9; 0,92; 0,98; ) (0,9; 0,92; 0,98; ) Others (0,85; 0,9; 0,95; ) After calculaton by the proposed alorthm, the optmal network skeleton s shown n F. 3, n whch the sold lne shows the optmal network skeleton. Tab. 2 shows the restoraton sequence and the correspondn restoraton paths. Table 2 The restoraton sequence and the restoraton path Sequence Bus to be Transmsson lnes to be restored restored , , , , Fure 3 The network skeleton of IEEE 30 bus system After calculaton, the obectve functon value s 0,0204, and the relablty of the skeleton network s 0,908. The total restoraton tme of the network skeleton Techncal azette 22, 6(205),

5 H. Lan obolšan alortam optmzace za prerazmešta sheme mreže nakon blokrana eneretsko sustava s 44,8 mnutes. Durn the tme all the unts and the loads can be restored successfully. As shown n Tab. 2 and F. 3, the network skeleton ncludes part of the transmsson lnes, whch are 4-6, 0-20 and wth hh restoraton successful rate. Althouh the transmsson lnes 3-4, 0-7 and 8-28 are also wth hh relablty, they do not need to be restored for ther adacency. The results above have verfed the effectveness and correctness of the proposed method. 5 Concluson An mproved optmzaton alorthm for the skeleton network reconfuraton after blackout s put forward n the paper. The restoraton tme of transmsson lnes and transformers, unts start-up tme lmt and the restoraton successful rate are selected as trapezodal fuzzy varables. The network skeleton optmal model s constructed based on CC. An optmzaton alorthm combnn fuzzy smulaton wth SO s mplemented to solve the network skeleton reconfuraton optmal model. The network skeleton and restoraton sequence meet a certan confdence level. Wth the network skeleton havn hher restoraton relablty, the start-up power to unts s provded as much as possble. The network skeleton and unts restoraton sequence are obtaned and optmzed by the proposed method. The effectveness of the proposed method s valdated by Matlab wth IEEE 30 bus system test. Acknowledements [8] Sh,.; Eberhart, R. C. A Modfed artcle Swarm ptmzer. // roceedns of IEEE Internatonal Conference on Evolutonary Computaton /Florda, 998, pp [9] Kennedy, J. The partcle swarm: socal adaptaton of knowlede. // roceedns of IEEE Internatonal Conference on Evolutonary Computaton / Indanapols, 997, pp DOI: 0.09/cec [0] ol, R. An Analyss of ublcatons on artcle Swarm Optmzaton Applcatons. // Techncal Report CSM-469, [] ol, R. Analyss of the publcatons on the applcatons of partcle swarm optmzaton. // Journal of Artfcal Evoluton and Applcatons, (2008), pp. -0. [2] Lu, B. D. Theory and ractce of Uncertan rorammn. Sprner-Verla Berln and Hedelber mbh, hyscaverl, DOI: 0.007/ [3] Adb, M. M.; Fnk, L. H. Specal Consderaton n ower System Restoraton. // IEEE Transacton on ower System. 2, (992), pp DOI: 0.09/mper [4] Lu, C. C.; Lou, K. L.; Chu, R. F. eneraton Capablty Dspatch for ower System Restoraton: A Knowlede- Based Approach. // IEEE Transacton on ower Systems. 8, (993), pp DOI: 0.09/ [5] Eberhart, R.; Kennedy, J. A ew Optmzer Usn artcles Swarm Theory. // roceedns of the 6th Internatonal Symposum on Mcro Machne and Human Scence / aoya, 995, pp DOI: 0.09/MHS Author s addresses Hapn Lan, Lecturer School of Electrcal and Electronc Enneern orth Chna Electrc ower Unversty, Baodn 07003, Chna E-mal: lanhapn@alyun.com Ths paper s supported by "the Fundamental Research Funds for the Central Unverstes (3MS72)". 6 References [] Lu,.; u, X.. Reconfuraton of etwork Skeleton based on Dscrete artcle-swarm Optmzaton for Black- Start Restoraton. // IEEE ower Enneern Socety eneral Meetn / Montreal, 2006, pp. -7. [2] Lu,.; u, X.. ode Importance Assessment based Skeleton etwork Reconfuraton. // roceedns of the CSEE. 27, 0(2007), pp [3] Wan, L.; Lu,.; u, X.. Skeleton-etwork Reconfuraton based on ode Importance and Lne Betweenness. // Automaton of Electrc ower Systems. 34, 2(200), pp [4] Zhan,. S.; Lu, J..; We, Z. B. Skeleton-etwork Reconfuraton based on Topoloy rorty and ath Electrcal Effects. // ower System rotecton and Control. 39, 7(20), pp. -6. [5] Ln, Z. Z.; Wen, F. S. A ew Optmzaton Method for Determnn Restoraton aths based on Wehted Complex etwork Model. // Automaton of Electrc ower Systems. 33, 6(2009), pp. -5. [6] Zhan, C.; Ln, Z. Z.; Wen, F. S. A Two Stae Stratey for etwork Reconfuraton based on Concept of Reret. // Automaton of Electrc ower Systems. 37, 8(203), pp , 75. [7] Kennedy, J.; Eberhart, R. artcle Swarm Optmzaton. // roceedns of IEEE Internatonal Conference on eural etworks IV /erth, 995, pp DOI: 0.09/IC Tehnčk vesnk 22, 6(205),

Affine Invariant Matching of Broken Boundaries Based on Differential Evolution

Affine Invariant Matching of Broken Boundaries Based on Differential Evolution Affne Invarant Matchn of Broken Boundares Based on Dfferental Evoluton Wuchao tu P.W.M. Tsan Department of Electronc Enneern Cty Unversty of Hon Kon Hon Kon Chna Abstract - Affne nvarant matchn of a par