Research Article Solving Unit Commitment Problem Using Modified Subgradient Method Combined with Simulated Annealing Algorithm

Transcription

1 Hndaw Publshng Corporaon Mahemacal Problems n Engneerng Volume 2010, Arcle ID , 15 pages do: /2010/ Research Arcle Solvng Un Commmen Problem Usng Modfed Subgraden Mehod Combned wh Smulaed Annealng Algorhm Ümmühan Başaran Flk and Mehme Kurban Deparmen of Elecrcal and Elecroncs Engneerng, Anadolu Unversy, Esksehr, Turkey Correspondence should be addressed o Ümmühan Başaran Flk, ubasaran@anadolu.edu.r Receved 13 December 2009; Revsed 7 Aprl 2010; Acceped 14 May 2010 Academc Edor: Jyh Horng Chou Copyrgh q 2010 Ü. Başaran Flk and M. Kurban. Ths s an open access arcle dsrbued under he Creave Commons Arbuon Lcense, whch perms unresrced use, dsrbuon, and reproducon n any medum, provded he orgnal work s properly ced. Ths paper presens he solvng un commmen UC problem usng Modfed Subgraden Mehod MSG mehod combned wh Smulaed Annealng SA algorhm. UC problem s one of he mporan power sysem engneerng hard-solvng problems. The Lagrangan relaxaon LR based mehods are commonly used o solve he UC problem. The man dsadvanage of hs group of mehods s he dfference beween he dual and he prmal soluon whch gves some sgnfcan problems on he qualy of he feasble soluon. In hs paper, MSG mehod whch does no requre any convexy and dfferenably assumpons s used for solvng he UC problem. MSG mehod dependng on he nal value reaches zero dualy gap. SA algorhm s used n order o assgn he approprae nal value for MSG mehod. The major advanage of he proposed approach s ha guaranees he zero dualy gap ndependenly from he sze of he problem. In order o show he advanages of hs proposed approach, he four-un Tuncblek hermal plan and en-un hermal plan whch s usually used n leraure are chosen as es sysems. Penaly funcon PF mehod s also used o compare wh our proposed mehod n erms of oal cos and UC schedule. 1. Inroducon UC s very mporan problem for power sysem engneerng. The problem can be descrbed as a nonlnear, mxed-neger, and nonconvex and s consdered o be a nondeermnsc polynomal-me hard NP-hard problem 1. Therealdffculy n solvng he problem s he hgh dmenson of he possble soluon space. Mea-heursc and mahemacal based mehods have been developed for solvng he hermal and hydrohermal UC problem n he leraure. The mos used meaheursc mehods are SA 2 5, exper sysems 6,abusearch 7 9, evoluonary programmng 10, 11, genec algorhms 12 15, memec algorhm

2 2 Mahemacal Problems n Engneerng 16, parcle swarm opmzaon 17, neror pon mehod 18, neural nework 19, 20, and greedy randomzed adapve search procedure 21. The mahemacal-based mehods depend on he dualy heory The oher mehods used for solvng he UC problem are dynamc programmng 29, 30, mxed-neger programmng 31, benders decomposon 32, and some hybrd mehods Mea-heursc mehods are used for solvng dffcul combnaoral opmzaon problems. To solve UC problem by usng hese mehods, prory ls s precalculaed and all he consrans are heurscally ncorporaed. The LR mehod subsequenly res o fnd he values of he Lagrange mulplers ha maxmze he dual objecve funcon based on he dualy heory. The dualy gap, a major problem n he nonlnear programmng, has been long recognzed as an nheren dsadvanage of hese mehods. If he LR-based mehods are used for solvng he UC problem, dual soluon may be far away from he opmal soluon. The dualy gap for he problem of UC s an mporan measure of he qualy of he soluon. When he gap s smaller, he soluon s beer 36.In 37,dfferen mahemacal-based mehods LR, penaly funcon, and augmened Lagrangan penaly funcon are compared o each oher accordng o feasble cos, dual cos, dualy gap, number of eraons, and duraon me. Accordng o 37, s seen ha here are dfferences beween prmal value and dual value. MSG mehod does no requre any convexy and dfferenably assumpons. In he nonlnear NP, he dualy gap has been nvesgaed and he heorecal ools for zero dualy gap condon have been mproved exensvely n In hs paper, one of he mehods based on dual opmzaon echnque, MSG mehod, whch has he bes performance n elmnang he dualy gap n he leraure, s used for solvng he UC problem. A dual problem wh respec o he sharp augmened Lagrangan s consruced for UC problem. The dsadvanage of he MSG mehod s ha he zero-dualy gap value depends on he nal value of he upper dual value. Ths dual value s found by usng SA algorhm. In hs proposal approach consrucs he dual problem and solves whou any dualy gap for large class of nonconvex consraned problems. Ths proposed approach s compared o PF mehod because he MSG mehod removes some of he problems occurred n hs mehod. The remanng secons are oulned as follows. Secon 2 provdes a descrpon of he UC problem formulaon. Ths secon ncludes an objecve funcon and he consrans of he problem. In Secon 3, MSG mehod wh SA algorhm Explaned n dealed. In Secon 4, applcaons and numercal resuls are presened and dscussed. Fnally he concluson s gven. 2. Un Commmen Problem Formulaon UC has been used o plan over a gven me horzon he mos economcal schedule of commng and dspachng generang uns o mee forecased demand levels and spnnng reserve requremens whle all generang un consrans are sasfed. The objecve funcon can be represened mahemacally as F ( P ) T,U N [ 1 1 F ( P ) ( SC 1 U 1 )] T N ( U SD 1 U ) U

3 Mahemacal Problems n Engneerng 3 In 2.1, T s perod, N s he number of generaors, P s he generaon power oupu of un a hour, F P s quadrac fuel cos funcon of generang un F P a b P c P 2, U s saus of un a hour on 1, off 0, SC s sarup cos of un a hour, SD s shudown cos of un a hour 43. The mnmzaon of he objecve funcon s provded o he followng consrans. Sysem Consrans Power Balance Consrans. For sasfyng he load balance n each sage, he forecased load demand should be equal o he oal power of he generaed power for feasble combnaon N U P P d In 2.2, P s sysem load demand a hour. d Un Consrans Generaon Lms. Each un mus sasfy he generaon range P,mn U P P,max U, 1, 2,...,N. 2.3 In 2.3, P,mn s mnmum power oupu of un, andp,max s maxmum power oupu of un. Ramp Up and Ramp Down Consrans. For each un, oupu s lmed by ramp up/down rae a each hour as follows: P 1 P RD, f U 1&U 1 1, P P 1 RU, f U 1&U 1 1, 2.4 RD s ramp down rae lm of un,andru s ramp up rae lm of un. The problem s nonconvex because s srucured bnary varables. These varables cause a grea deal of rouble and dffculy n solvng he UC. Load balance s couplng consran for he UC problem. The couplng consrans across he un so ha one un affecs wha wll happen on oher uns f he couplng consrans are me. 3. Modfed Subgraden Mehod Combned wh Smulaed Annealng Algorhm 3.1. Smulaed Annealng Algorhm for Un Commmen SA algorhm s a srong echnque for solvng hard combnaoral opmzaon problems whou specfc srucure. Ths mehod has he ably of escapng local mnma by performng uphll moves. The man advanage of SA algorhm s ha does no need large compuer memory. SA s based on he erave mehod, orgnally proposed by Meropols e al. 44, whch smulaes he ranson of aoms n equlbrum a a gven

4 4 Mahemacal Problems n Engneerng emperaure. The man dsadvanage of hs mehod s very greedy regardng compuaon me requremens, due o he large number of eraons needed for he convergence of he mehod. All he generang and accepance dsrbuons depend on he emperaure 3. One of he measure seps n he SA s he coolng schedule. Ths schedule has hree componens: he nal emperaure seng, he emperaure decreasng scheme, and a fne number of nal value of emperaure should be large enough o allow all ransens o be acceped for unconsraned opmzaon problem. On he oher hand, he emperaure s lowered based on mulplyng he emperaure n whch ypcal values le beween 0.8 and If he accepance rao s low or he sampled mean and varance of cos values a curren emperaure have bg drops, hen he facor s adjused o a hgher value n order o avod geng suck a a local opmal confguraon. Oherwse, he facor s adjused o a lower value o ncrease he convergence speed. The number of moves a each emperaure s based on he requremen ha a each emperaure quasequlbrum s o be resored. The fnal emperaure s obaned f, a fve consecuve emperaures, eher he sampled mean values of cos funcon do no change or he accepance rao s small enough 2, 3. In solvng he UC problem by usng SA algorhm, wo ypes of varables need o be deermned: he un saus bnary varables U and he uns oupu power connuous varables P. Then, hs problem can be consdered no wo subproblems, a combnaoral opmzaon problem n U and a nonlnear opmzaon problem n P. The flow dagram of SA algorhm for UC problem s gven n Fgure 1 3. The economc dspach problem s solved by usng lambda-eraon mehod n he SA Modfed Subgraden Mehod for Un Commmen The MSG mehod proposed by Gasmov has a noable performance n havng zero dualy gaps for large class of nonconvex problems 38, 39. In he sandard form he nonlnear programmng problem can be expressed as follows: mn K f K 3.1 h K 0, subjec o K Ω, 3.2 where h K h 1 K h 2 K h n K s he consran vecor. The prmal problem s 3.1. Sharp augmened Lagrangan s defned as follows: L K, v, c f K c h K v T h K, 3.3 where v R 3 and c R. Defne he dual funcon as H v, c mn L K, v, c. 3.4 K Ω

5 Mahemacal Problems n Engneerng 5 Inalze all varables Fnd, randomly an nal feasble soluon. Solve he economc dspach problem. Calculae oal cos, F P,U Deermne he nal emperaure ha resuls n a hgh probably of accepng any soluon. Equlbrum s acheved? No Yes Sop Fnd a ral soluon. Perform he accepance es; accep or rejec he ral soluon. Soppng creron s sasfed No Yes Sop Decrease he emperaure. Fgure 1: SA algorhm for UC problem. The dual problem P s max v,c R 3 R H v, c. 3.5

6 6 Mahemacal Problems n Engneerng Dual funcon H v,c s no a defne funcon; however s a convex funcon for connuous f K, h K and compac se S. Inequaly consrans can be convered o sandard form by addng nonnegave slack varables o lef hand sdes of nequales; one of he advanages of he MSG mehod s ha he fac ha he soluon convergence s proved. Thus, n each eraon a beer soluon can be found. Ths mehod dffers from he classcal LR funcons, scans he soluon wh concs, and hence obans zero dual gap value. In oher words, here s no dfference beween prmal and dual problems. Thus gves an opmal soluon. Anoher advanage of hs mehod s ha does no requre any convexy and dfferenably assumpons. Usng he defnons, he MSG mehod s as follows. Inalzaon Choose a par v 1,c 1 wh v 1 R 3 and c R, c 1 0andlej 1, and go o Sep 1. Sep 1. Gven c j,v j, solve he followng subproblem: mn K f K c j h K v j h K H ( v j,c j ) 3.6 subjec o k Ω. In 3.6 c j,v j s mulpler of sharp augmened Lagrangan, f K s objecve funcon, h K s norm of he consran vecor, and H v j,c j s dual funcon of he problem. Le K j be a soluon of 3.6.Ifh K j 0, hen sop; v j,c j s an opmal soluon o he dual problem and K j s a soluon o 3.1 ;sof K j s he opmal value of problem 3.1. Oherwse, go o Sep 2. Sep 2. Updae v j,c j by v j 1 v j z j h ( K j ), c j 1 c j ( z j ε j ) h ( K j ), 3.7 where z j and ε j are posve scalar sep szes defned below. Replace j by j 1andgooSep 1. Sep Sze Calculaon Le us consder he par v j,c j and calculae H ( ) v j,c j mn L K, v, c, 3.8 K Ω

7 Mahemacal Problems n Engneerng 7 and le H K j / 0 for he correspondng K j, whch means ha K j s no opmal. Then he sep sze parameer z j can be calculaed as 0 <z j < 2 (H H ( ) ) v j,c j 5 h Kj, <ε j <z j, where H s an upper bound for he dual funcon. Consderng he dual funcon formed by usng he sharp Lagrangan, s value a any feasble pon s no larger han prmal problems objecve funcon value. The equaly occurs a a pon when boh prmal and dual problems acheve her opmal values. I has been proven n 38 ha f h K 0for any K obaned from 3.6, hen s he soluon of he prmal problem. If h K / 0, hen he value of H calculaed from 3.6 s srcly less han he opmal value of 3.1. In hs case we updae dual varables usng Sep 2, whch leads o an ncrease n he value of he dual funcon. Soluon of 3.6 correspondng o he updaed v, c s always greaer han he value obaned n he prevous sep. Noe ha hs propery s no guaraneed by he mulpler and penaly mehods 43, 45. Accordng o MSG mehod for solvng he UC problem, he objecve funcon and consrans are defned as follows. Objecve Funcon One has mn F ( P ) T,U N [ 1 1 F ( P ) ( SC 1 U 1 )] T N ( ) U SD 1 U U Slack varables are added o nequaly consrans. Equaly and nequaly consrans can be defned as follows. Generaon Lm Consrans One has h 1 N, T U P,mn P h 0, 3.11 h 2 N, T P U P,max h 0. Load Balance Consrans One has h 3 T N U P P d

8 8 Mahemacal Problems n Engneerng Ramp Up and Ramp Down Consrans One has h 4 N, T P 1 P RD, h, f U 1&U 1 1, h 5 N, T P P 1 RU h, f U 1&U h shows slack varables from 3.11 o H h 1 N, T h 2 N, T h 3 T h 4 N, T h 5 N, T. Accordng o 3.3, sharp augmened Lagrangan L SAL s defned as follows: L SAL T N [ ( ) ( F P SC 1 1 ( (U c sqr 1 U 1 P,mn P h T N )]U SD ( 1 U 1 1 ) 2 ( P U P,max h ) 2 ) U 1 ( N 2 U P d) P 1 ( P 1 ( ) 2 P P 1 RU h P RD h ) v [ (U P,mn P ) ( h P U P,max h ) ( ) N U P P d 1 ( P 1 P ( P P 1 RU h) ]. RD h ) The epslon value s chosen as 0.95z j n he UC problem soluon. Afer 3.14 s consruced usng he objecve funcon and consrans, all he seps of he MSG mehod are appled o he UC problem. In hs sudy, SA algorhm s used n order o assgn he approprae nal value H for MSG mehod. Then, he UC problem s solved usng hs proposed mehod for fndng he opmal soluon. 4. Applcaons and Numercal Resuls The UC problem for four-un Tuncblek hermal plan s solved by usng he MSG mehod combned wh SA algorhm. Frs he SA algorhm s appled o he UC problem and hen he cos funcon value found from he SA algorhm s used as upper dual value n MSG mehod for four-un Tuncblek hermal plan and en-un hermal plan. SA algorhm s

9 Mahemacal Problems n Engneerng 9 Table 1: Un characerscs for four-un Tuncblek hermal plan. Un P mn P max SC a b C SD RD RU No MW MW $ $/h $/MWh $/MW 2 h $ MW/h MW/h Table 2: Load daa for four-un Tuncblek hermal plan MW. Sage Load Sage Load coded n MATLAB PF and MSG mehods are coded n GAMS whch s a hgh-level modelng sysem for mahemacal programmng problems 46. The daa for four-un Tuncblek hermal plan s aken from Turksh Elecrc Power Company and Elecrcy Generaon Company. The un characerscs for four-un Tuncblek hermal plan are gven n Table 1. In hs sudy, a 24-hour day s subdvded no 8 dscree sages for four-un Tuncblek hermal plan. The load demands for he sages are gven n Table 2. The un characerscs for en-un hermal plan are gven n Table 3. The load demands for each perod are gven n Table 4 for en un hermal plan. UC schedule for he SA algorhm for four-un Tuncblek hermal plan s gven n Table 5. UC schedule for he SA algorhm for en-un hermal plan s gven n Table 6. The cos value $ found from he SA algorhm s used for upper dual value n MSG mehod for four-un Tuncblek hermal plan. The cos value $ found from he SA algorhm for en-un hermal plan. Opmal soluon of he dual model se up by he augmened Lagrange funcon, whch s proposed by Azmov and Gasmov for nonconvex problems, s equal o he prmal soluon of he sysem CONOPT2 s used as a GAMS solver for MSG and PF mehods. The MSG mehod s run unl he norm s equal o for four-un Tuncblek hermal plan; he norm s equal o for en-un hermal plan. Then all he consrans reduce o zero and feasble soluon s obaned. Prmal value s equal o he dual value yeldng zero dualy gap value for boh of hese sysems. The value for he c parameer s for four-un Tuncblek hermal plan; c parameer s for en-un hermal plan. The UC schedule for MSG mehod combned wh SA algorhm s gven for four-un Tuncblek hermal plan and en-un hermal plan n Tables 7 and 8, respecvely. In Table 7, he oal cos value s found $ and n Table 8 oal cos value s found $ for MSG mehod combned wh SA algorhm. I can be seen from Tables 7 and 8 ha MSG reaches o zero dualy gap value prmal value dual value for UC problem. To show he advanages of hs mehod, PF mehod s used. UC schedule, prmal-dual values for PF mehod are gven n Tables 9 and 10 for four-un Tuncblek hermal plan and for en-un hermal plan, respecvely. I s seen ha here are dfferences beween prmal value and dual value n he PF mehod. The qualy of he soluon s mproved when he dualy gap s decreased. In hs

10 10 Mahemacal Problems n Engneerng Table 3: Un characerscs for en-un hermal plan. Un P mn P max SC SC a b C RD RU No MW MW ho $ cold $ $/h $/MWh $/MW 2 h MW/h MW/h Table 4: Load daa for en-un hermal plan MW. Sage Load Sage Load Sage Load Table 5: UC schedule for SA algorhm for four-un Tuncblek hermal plan. Sage Un Combnaon TC $ paper, zero dualy gap s acheved and feasble soluon s aaned by usng oal cos value of he SA algorhm as an upper dual value for he MSG mehod. 5. Concluson The qualy of he soluon of UC problem s mproved when he dualy gap s decreased. In hs paper zero dualy gap s acheved and feasble soluon s aaned by applyng he novel proposed mehod, MSG mehod combned wh SA algorhm, o solvng he UC problem. The cos funcon found from he SA algorhm s used for upper dual value n MSG mehod.

11 Mahemacal Problems n Engneerng 11 Table 6: UC schedule for SA algorhm for en-un hermal plan. Perod Un Combnaon Perod Un Combnaon TC $ Table 7: UC schedule for MSG mehod combned wh SA algorhm for four-un Tuncblek hermal plan. Sage Un Combnaon TC $ prmal value dual value. Table 8: UC schedule for MSG mehod combned wh SA algorhm for en-un hermal plan. Perod Un Combnaon Perod Un Combnaon TC $ prmal value dual value. The mos aracve feaure of he proposed approach s ha he dualy gap value of MSG mehod over he scheduled me horzon s zero. Noe ha here s a general accepance ha whenever he sysem sze ges smaller, he dualy gap value ges bgger for UC problem.

12 12 Mahemacal Problems n Engneerng Table 9: UC schedule for PF mehod for four-un Tuncblek hermal plan. Sage Un Combnaon Prmal value $ , Dual value $ Table 10: UC schedule for PF mehod for en-un hermal plan. Perod Un Combnaon Perod Un Combnaon Prmal value $ , Dual value $ However, s shown ha hs s no he case wh he MSG mehod for a small sze sysem. The resuls of he proposed mehod for solvng he UC problem are very mpressve, and he qualy of feasble soluon s sgnfcanly mproved. Ths approach can be appled o UC problem for any sze of sysems o oban he feasble schedule. Ls of Symbols c: Mulpler of sharp augmened Lagrangan f K : Objecve funcon F P : Generaor fuel cos funcon n a quadrac form, F P a b P c P 2 $/h H: Upper bound of he dual funcon H v, c : Dual funcon h x : Equaly consran

13 Mahemacal Problems n Engneerng 13 g x : Inequaly consran LR: Lagrangan Relaxaon N: Number of generang uns P: Prmal problem P : Dual problem PF: Penaly funcon mehod P d : Nomnal demand a hour MW P : Generaon oupu of un a hour MW P max, : Maxmum avalable capacy of un a hour MW P mn, : Mnmum avalable capacy of un a hour MW RU : Ramp up rae of un MW/h RD : Ramp down rae of un MW/h SC : Sar up cos of un $ SD : Shu down cos of un $ UC: Un commmen U : Saus value of un a me T: Tme horzon for UC h TC: Toal cos v: Mulpler of sharp augmened Lagrangan. Acknowledgmen The suppor and gudance on MSG mehod by Professor Rafal N. Gasmov s graefully acknowledged. References 1 A. J. Wood and B. F. Wollenberg, Power Generaon Operaon and Conrol, Wley-Inerscence, New York, NY, USA, 2nd edon, F. Zhuang and F. D. Galana, Un commmen by smulaed annealng, IEEE Transacons on Power Sysems, vol. 5, no. 1, pp , A. H. Manawy, Y. L. Abdel-Magd, and S. Z. Selm, A Smulaed annealng mehod for un commmen, IEEE Transacons on Power Sysems, vol. 13, no. 1, pp , A. Vana, J. P. Sousa, and M. Maos, Smulaed annealng for he un commmen problem, n Proceedngs of IEEE Poro Power Tech Conference, Poro, Porugal, Sepember C. C. Asr Rajan, M. R. Mohan, and K. Manvannan, Refned smulaed annealng mehod for solvng un commmen problem, n Proceedngs of he Inernaonal Jon Conference on Neural Neworks (IJCNN 02), pp , May S. L, S. M. Shahdehpour, and C. Wang, Promong he applcaon of exper sysems n shor-erm un commmen, IEEE Transacons on Power Sysems, vol. 8, no. 1, pp , H. Mor and T. Usam, Un commmen usng Tabu search wh resrced neghborhood, n Proceedngs of he Inernaonal Conference on Inellgen Sysems Applcaons o Power Sysems, pp , February B. Xaomn, S. M. Shahdehpour, and Y. Erkeng, Consraned un commmen by usng abu search algorhm, n Proceedngs of he Inernaonal Conference on Elecrcal Engneerng, vol. 2, pp , A. Rajan, C. C. Mohan, and M. R. Manvannan, Neural based abu search mehod for solvng un commmen problem, n Proceedngs of he 5h Inernaonal Conference on Power Sysem Managemen and Conrol, vol. 488, pp , K. A. Juse, H. Ka, E. Tanaka, and J. Hasegawa, An evoluonary programmng soluon o he un commmen problem, IEEE Transacons on Power Sysems, vol. 14, no. 4, pp , 1999.

14 14 Mahemacal Problems n Engneerng 11 C. C. A. Rajan and M. R. Mohan, An evoluonary programmng-based abu search mehod for solvng he un commmen problem, IEEE Transacons on Power Sysems, vol. 19, no. 1, pp , P.-C. Yang, H.-T. Yang, and C.-L. Huang, Solvng he un commmen problem wh a genec algorhm hrough a consran sasfacon echnque, Elecrc Power Sysems Research, vol. 37, no. 1, pp , S. A. Kazarls, A. G. Bakrzs, and V. Perds, A genec algorhm soluon o he un commmen problem, IEEE Transacons on Power Sysems, vol. 11, no. 1, pp , S. O. Orero and M. R. Irvng, A genec algorhm modellng framework and soluon echnque for shor erm opmal hydrohermal schedulng, IEEE Transacons on Power Sysems, vol. 13, no. 2, pp , T. Senjyu, H. Yamashro, K. Shmabukuro, K. Uezao, and T. Funabash, A un commmen problem by usng genec mehod based on characersc classfcaon, n Proceedngs of IEEE Power Engneerng Socey Wner Meeng, vol. 1, pp , J. Valenzuela and A. E. Smh, A seeded memec algorhm for large un commmen problems, Journal of Heurscs, vol. 8, no. 2, pp , P. Sryanyong and Y. H. Song, Un commmen usng parcle swarm opmzaon combned wh lagrange relaxaon, n Proceedngs of IEEE Power Engneerng Socey General Meeng, pp , June L. M. Kmball, K. A. Clemens, P. W. Davs, and I. Nejdaw, Mulperod hydrohermal economc dspach by an neror pon mehod, Mahemacal Problems n Engneerng, vol. 8, no. 1, pp , H. Sasak, M. Waanabe, J. Kubokawa, N. Yorno, and R. Yokoyama, A soluon mehod of un commmen by arfcal neural neworks, IEEE Transacons on Power Sysems, vol. 7, no. 3, pp , V. N. Deu and W. Ongsakul, Improved mer order and augmened Lagrange Hopfeld nework for un commmen, IET Generaon, Transmsson and Dsrbuon, vol. 1, no. 4, pp , A. Vana, J. P. de Sousa, and M. Maos, A new meaheursc approach o he un commmen problem, n Proceedngs of he 14h Power Sysems Compuaon Conference, Sevlla, Span, June 2002, Sesson 05, Paper A. Merln and P. Sandrn, New mehod for un commmen a elecrce de France, IEEE Transacons on Power Apparaus and Sysems, vol. 102, no. 5, pp , R. Neva, A. Inda, and I. Gullen, Lagrangan reducon of search-range for large-scale un commmen, IEEE Transacons on Power Sysems, vol. 2, no. 2, pp , F. Zhuang and F. D. Galana, Towards a more rgorous and praccal un commmen by Lagrangan relaxaon, IEEE Transacons on Power Sysems, vol. 3, no. 2, pp , S. Vrman, E. C. Adran, K. Imhof, and S. Mukherjee, Implemenaon of a Lagrangan relaxaon based un commmen problem, IEEE Transacons on Power Sysems, vol. 4, no. 4, pp , N. J. Redondo and A. J. Conejo, Shor-erm hydro-hermal coordnaon by lagrangan relaxaon: soluon of he dual problem, IEEE Transacons on Power Sysems, vol. 14, no. 1, pp , Q. Zha, X. Guan, and J. Cu, Un commmen wh dencal uns: successve subproblem solvng mehod based on Lagrangan relaxaon, IEEE Transacons on Power Sysems, vol. 17, no. 4, pp , W. Ongsakul and N. Pecharaks, Un commmen by enhanced adapve lagrangan relaxaon, IEEE Transacons on Power Sysems, vol. 19, no. 1, pp , A. K. Ayoub and A. D. Paon, Opmal hermal generang un commmen, IEEE Trans Power App Sys, vol. 90, no. 4, pp , W. L. Snyder Jr., H. D. Powell Jr., and J. C. Rayburn, Dynamc programmng approach o un commmen, IEEE Transacons on Power Sysems, vol. 2, no. 2, pp , M. Carrón and J. M. Arroyo, A compuaonally effcen mxed-neger lnear formulaon for he hermal un commmen problem, IEEE Transacons on Power Sysems, vol. 21, no. 3, pp , H. Ma and S. M. Shahdehpour, Transmsson-consraned un commmen based on Benders decomposon, Inernaonal Journal of Elecrcal Power and Energy Sysems, vol. 20, no. 4, pp , H. Y. Yamn and S. M. Shahdehpour, Un commmen usng a hybrd model beween Lagrangan relaxaon and genec algorhm n compeve elecrcy markes, Elecrc Power Sysems Research, vol. 68, no. 2, pp , 2004.

15 Mahemacal Problems n Engneerng P. Aavryanupap, H. Ka, E. Tanaka, and J. Hasegawa, A hybrd evoluonary programmng for solvng hermal un commmen problem, n Proceedngs of he 12h Annual Conference Power and Energy Socey, G. K. Purushohama and L. Jenkns, Smulaed annealng wh local search a hybrd algorhm for un commmen, IEEE Transacons on Power Sysems, vol. 18, no. 1, pp , L. A. F. M. Ferrera, On he dualy gap for hermal un commmen problems, n Proceedngs of IEEE Inernaonal Symposum on Crcus and Sysems, vol. 4, pp , May M. Kurban and U. B. Flk, A comparave sudy of hree dfferen mahemacal mehods for solvng he un commmen problem, Mahemacal Problems n Engneerng, vol. 2009, Arcle ID , 13 pages, R. N. Gasmov and A. M. Rubnov, On augmened lagrangans for opmzaon problems wh a sngle consran, Journal of Global Opmzaon, vol. 28, no. 2, pp , R. N. Gasmov, Augmened Lagrangan dualy and nondfferenable opmzaon mehods n nonconvex programmng, Journal of Global Opmzaon, vol. 24, no. 2, pp , A. Y. Azmov and R. N. Kasmov, On weak conjugacy, weak subdfferenals and dualy wh zero gap n nonconvex opmzaon, Inernaonal Journal of Appled Mahemacs, vol. 1, no. 2, pp , R. T. Rockafellar and R. J. B. Wes, Varaonal Analyss, vol. 317 of Grundlehren der Mahemaschen Wssenschafen, Sprnger, Berln, Germany, A. M. Rubnov and R. N. Gasmov, Srcly ncreasng posvely homogeneous funcons wh applcaon o exac penalzaon, Opmzaon, vol. 52, no. 1, pp. 1 28, M. S. Bazaraa, H. D. Sheral, and C. M. Shey, Nonlnear Programmng: Theory and Algorhm, John Wley & Sons, Hoboken, NJ, USA, 3rd edon, N. Meropols, A. Rosenbluh, M. Rosenbluh, and E. Teller, Equaons of sae calculaons by fas compung machnes, Journal of Chemcal Physcs, vol. 21, pp , D. P. Bersekas, Nonlnear Programmng, Ahena Scenfc, Belmon, Mass, USA, A. Brooke, D. Kendrck, A. Meeraus, and R. Raman, GAMS: a user s gude, GAMS Developmen Corporaon, 1998, hp://