Using Lagrangian relaxation in optimisation of unit commitment and planning

Transcription

1 Faklea za lekroehniko va horin Heike Brand Chrioph Weber Uing Lagrangian relaxaion in opimiaion of ni commimen and planning SCGN Dicion aper No. 3 Conrac No. NK5-C--94 rojec co-fnded by he ropean Commniy nder he 5 h Framework rogramme 998- Conrac No. BBW.67 rojec co-fnded by he Swi Federal Agency for dcaion and Science Febrary Conrac No. CU-BS-/ rojec co-fnded by ermoelekrarna oplarna Ljbljana d.o.o.

2 INRDUCIN... BASIC RY... 3 LAGRANGIAN RLAXAIN FR IMISAIN F UNI CMMIMN AND DISACHING RBLMS DSCRIBD IN LIRAUR FRMULAIN FR A SIMLIFID CH-RBLM ASCS N SLVING LAGRANGIAN RLAXD RBLM CNCLUSINS LIRAUR... 9

3 IR Sgar v.. Inrodcion Lagrangian relaxaion algorihm eem o be efficien in peeding p he olving of opimiaion problem and here are everal example in he lierare where hey have been ed for opimiaion of ni commimen and dipaching /Aoki e al 989/ /Cheng e al / /Dozaer / /Honker / /Sern e al / /Virmani e al 989/. Wihin he SCGN projec we have problem o reach aifacory olion wihin reaonable ime wih he deeriic model for he long-erm planning. Uing he Lagrangian relaxaion can be one way o enhance he compaional performance. Baic heory he baic concep of Lagrangian relaxaion i illraed wih he imiaion problem below. he conrain are divided ino wo ype he eqaliie g j x j m and ineqaliie h j x j.p. X. { f x } g x h j j x X j... m j... p he idea i o decompoe he problem called he primal problem o everal maller problem ha are eaier o olve. hi can hen be done hrogh creaion of he relaxed problem by inclding he conrain ino he objecive fncion. By mliplying he conrain wih he Lagrange mliplier λ and repecively and inclding hem in he objecive fncion he primal problem i ranformed ino an nconrained opimiaion problem called he relaxed problem.. x X x { f x λ g x h x } Φ λ Φλ i he dal objecive fncion. By maximiing Φ wih repec o λ and we ge a lower bond for he feaible olion of he primal problem. a high vale of he mliplier indicae ha he conrain are conidered o a higher degree. max λ { Φ λ }.

4 IR Sgar v.. * * Φ λ Φ λ f x * indicae he opimal olion * f x he difference beween he olion of he dal and primal problem i called he daliy gap δ δ f x Φ λ From he opimal dal variable we ge he opimal olion of he relaxed problem and if he daliy gap i zero hi i alo he olion of he primal problem. If a daliy gap exi beween he opimal olion he olion of he relaxed problem doe no need o be primal feaible and hen a primal feaible olion ha o be conrced from he rel of he relaxed problem. 3 Lagrangian Relaxaion for opimiaion of ni commimen and dipaching 3. roblem decribed in he lierare In /Dozaer / hor-erm planning of power and heaing yem wih algorihm baed on Lagrangian relaxaion i conidered. wo baic model one for he power prodcion problem and one for he hea prodcion problem are ed. he oal yem co i imied and he model inclde ar-p co. For he hea prodcion problem hea orage i inclded. Minimm operaion and h-down ime are alo conidered. L. L L L { } L L DMAND imm operaion and h L RS L L - down ime

5 IR Sgar 3 v... o { } o o DMAND o o imm operaion and h - down ime he elecric power prodced by ni a ime L variable indicaing he on-off a for he ni k a ime i he pinning reerve a ime L RS L DMAND DMNAD o he hea prodced by ni a ime conan in he prodcion co eqaion for ni a ime ar-p co for ni a ime he elecric power demand a ime he hea demand a ime energy conen of orage a ime hea aken from orage a ime For he power prodcion problem he relaxed problem i conrced by inclding he rericion for he elecric demand and he reerve reqiremen ino he objecive fncion of he primal problem. L L Φ λ L λ L DMAND L L RS. L { } L imm operaion and h L - down ime L For given Lagrange mliplier he relaxed problem hen decompoe ino one b-problem for each prodcion ni.

6 IR Sgar 4 v.. { } down ime - imm operaion and h. L L L I i L L L L λ. Φ o DMAND λ λ For he hea prodcion problem he rericion for he hea demand and he hea orage are inclded in he objecive fncion of he primal problem. { } imm operaion and h down ime. o o o o. o o o DMAND o λ λ { } down ime - imm operaion and h. λ For given Lagrange mliplier he relaxed problem decompoe ino one b-problem for each prodcion ni and one problem for he hea orage.

7 IR Sgar 5 v.. he hea orage problem can hen be frher eparaed ino one problem for each ime inerval. he power prodcion problem decribed above ha alo been died in /Cheng e al /. /Virmani e al 989/ decribe a imilar power prodcion problem. hey have inclded a more complicaed reerve rericion. In [Aoki e al 989] alo a imilar power prodcion problem i decribed. hey however conider long-erm opimiaion one week p o one monh. A pmp-orage hydro ni a well a an aigned fel rericion are alo inclded in heir problem. 3. Formlaion for a implified CH-problem In or long-erm model we conider boh he power prodcion and he hea prodcion problem imlaneoly. Frher we maximie he profi inead of imiing he co ince we alo inclde he poibiliy o ell elecric power a he po marke. ake-or-pay conrac for bying elecric power and fel are alo conidered. Here we conider a yem of CH rbine wih given hea and elecric power demand and he poibiliy o by elecric power from a ake-or-pay conrac and o by and ell elecric power from he po marke. he problem can hen be decribed a follow: max L. L L LBUY S LSLLS LBUY Sx S L LBUY LSLLS { } ε ε L SLL S price DMAND co LBUY S L LBUY S L LBUY L co LBUY r co LDMAND β β L elecric power old a he po marke a ime β LSLLS

8 IR Sgar 6 v.. price S he price for he elecric power when yo ell i a he po marke a ime co r L he elecric power prodced by ni a ime β ε conan for ni in eqaion decribing he Q-char variable indicaing he on-off a for he ni a ime elecric power bogh a he po marke a ime LBUY S co S he co for bying elecric power a he po marke a ime LBUY elecric power bogh from ake-or-pay conrac a ime co he co for bying elecric power from he ake-or-pay conrac DMAND L DMAND he co for fel r he hea prodced by ni a ime conan for ni in fel conmpion eqaion he elecric power demand a ime he hea demand a ime he fel conmpion i decribed wih a linear eqaion wih he conan and. he relaion beween he power and hea prodcion i decribed wih plan-characeriic map o called Q-char. hey coni of hree raigh line wih he conan β and β ε and ε and and repecively. By inclding he fir wo rericion we ge he following relaxed problem:

9 IR Sgar 7 v.. Φ λ. L L max L L BUY S L BUY L SLL S { } ε ε L BUY S S L SLL S λ L L co L DMAND price L DMAND L L BUY S L SLL S L BUY S β β co L BUY r co L BUY L β he dal problem hen become: λ { Φ λ } For given λ and he relaxed problem can be decompoed o one problem for each CH rbine and one problem for he erm ha are no rbine-dependen. co β λ max r L. L L { } L L ε ε L L L β β

10 IR Sgar 8 v.. max L. LSLLS LBUY S LBUY LSLLS price λ co LDMAND S L λ LBUY λ LBUY LDMAND LBUY LBUY S λ cos DMAND he amon of bogh elecriciy from he po marke can never be higher han he elecriciy demand. he amon of elecriciy old o he po marke can never be higher han wha can be prodced by he rbine and wha can be bogh from conrac. he co for bying elecric power a he po marke price for elling i alway higher han he. herefore he opimally bogh amon of elecric power from he po marke will alway be zero when he old amon i non-zero. he oppoie i alo re. price S co S Since we do no have any ime-dependen rericion he relaxed problem a well a he primal problem can be frher decompoed o one problem for each ime inerval. If imal operaion and h-down ime are inclded we will ge a dependency beween he ime inerval for he par of he relaxed problem relaed o he CH rbine. If a limi for he amon bogh from he ake-or pay conrac i inclded we will alo ge a ime dependency for he b-problem inclding he conrac erm. 4 Apec on olving he Lagrangian relaxed problem he dal problem of a Lagrangian relaxed problem i olved ieraively where he Lagrange mliplier are pdaed beween each ieraion. In /Dozaer / he procedre i decribed a follow:. Chooe aring vale for he mliplier.. Solve he relaxed problem for he crren vale of he mliplier. 3. Conrc a primal feaible olion from he olion of he relaxed problem. 4. Sop if he convergence crierion i aified he daliy gap i fficienly mall. 5. Updae he mliplier and go o. he difficly i hen o find good aring vale for he mliplier and a iable algorihm o pdae he mliplier. In boh /Dozaer / and /Honker / everal previoly ed o-called b-gradien mehod are menioned. In /Cheng e al / geneic algorihm are ed o pdae he mliplier. In /Aoki e al 989/ an algorihm baed on he variable meric mehod for dal maximiaion i preened.

11 IR Sgar 9 v.. n n n For he b-gradien mehod λ λ n where n i he nmber of he ieraion ep λ i he Lagrange mliplier i he ep lengh and i he b gradien. he b-gradien eqal he vale of he rericion inclded in he relaxed problem. here are everal mehod o deere and in /Dozaer / ome differen mehod are compared for he hea prodcion problem decribed nder 3.. Aoki e al /Aoki e al 989/ ay ha he advanage of he variable meric mehod hey e for pdaing Lagrange mliplier over he radiional b-gradien mehod i ha heir mehod preven he olion from ocillaing near he dal maximm. Alo in hi mehod he bgradien are ed. he eqaion for pdaing he mliplier i however differen from he one hown above for he b-gradien mehod. A mehod o olve he dal and primal problem m alo be choen. Dynamic programg eem o be common o e in connecion wih Lagrangian relaxaion. For example in /Cheng e al / /Dozaer / /Serner e al / and /Virmani e al 989/ hi mehod i ed. However in /Dozaer / combinaion wih branch-and-bond mehod are alo menioned and preened. In one example he e a branch-and-bond framework for olving he primal problem. 5 Conclion o e he Lagrangian relaxaion cold be one way o improve he compaional performance of he long-erm SCGN opimiaion model. o ar wih we have o look more ino how he algorihm for pdaing he Lagrange mliplier are be formlaed for or problem and alo how o be chooe he aring vale of he mliplier. hen we can ry o olve a implified problem in he form of he problem preened in Chaper 3.. Frher we have o conider he be way o olve he primal and dal problem. We will look more ino he poibiliie o e dynamic programg. 6 Lierare /Aoki e al 989/ Aoki K; Ioh M.; Saoh.; Nara K.; Kanezahi M.: pimal Long-erm Uni Commimen in Large Scale Syem Inclding Fel Conrained hermal and mped-sorage Hydro In: I ranacion on ower Syem Vol. 4 No /Cheng e al /

12 IR Sgar v.. Cheng Chan-ing; Li Chih-Wen; Li Chn-Chang: Uni Commimen by Lagrangian Relaxaion and Geneic Algorihm In: I ranacion on ower Syem Vol. 5 No /Dozaer / Dozaer rik: nergy Syem peraion by Lagrangian Relaxaion Linköping Sdie in Science and echnology Dieraion No. 665 Diviion of pimizaion Deparmen of Mahemaic Linköping niverie Sweden /Honker / Honker U.: Krafwerkeinazplanng mi dem Lagrange pimizaion Syem In: pimerng in der nergiverorgng VDI-Beriche Nr 67 /Sern e al / Sern B.; Habrich H.-J.; wer A.: Sochaiche pimerng von Krafwerkeinaz nd Sromhandel zr Berückichng von lanngicherheien In: pimerng in der nergiverorgng VDI-Beriche Nr. 67 /Virmani e al 989/ Virmani S.; Imhof K.; Mkherjee S.: Implemenaion of a Lagrangian Relaxaion Baed Uni Commimen roblem In: I ranacion on ower Syem Vol. 4 No