Maintenance Scheduling of Thermal Power Units in Electricity Market

Transcription

1 ainenane Sheduling of Thermal Power Uni in Eleriiy arke ADNAN UHAREOVIC Independen Syem Operaor in Bonia and Herzegovina Hamdije Cemerlia 2, Sarajevo BOSNIA AND HERZEGOVINA SAJO BISANOVIC Publi Enerprie Elekroprivreda of Bonia and Herzegovina Vilonovo ealie 5, Sarajevo BOSNIA AND ENSUR HAJRO Fauly of Elerial Engineering Univeriy of Sarajevo Zmaja od Bone bb, Sarajevo BOSNIA AND HERZEGOVINA Abra: The objeive of hi paper i o preen a deiion-baed funion o deermine he opimal mainenane heduling raegy of hermal power uni aking ino aoun he pariular obligaion of Generaion Company, uh a bilaeral onra. Opporuniie for energy elling a he eleriiy marke a well a a deailed modeling of he power plan are onidered in he opimizaion problem. The deerminii onrained ombinaorial opimizaion problem ha onider maximizaion of profi i olved uing he inerior poin uing plane mehod. Thi mehod poee he advanage of boh, he inerior poin mehod and he uing plane mehod, and beome very promiing approah for he large-ale diree opimizaion problem. Furhermore, in order o improve he robune and effiieny of hi mehod, a new general opimal bae idenifiaion mehod i developed and inrodued o deal wih variou ype of opimal oluion. The implemenaion and performane of propoed oluion ehnique are preened. The effeivene of he approah i eed on he realii ize ae udy, and numerial reul are demonraed and diued. Key-Word: ainenane heduling; eleriiy marke; bilaeral onra; marke prie, inerior poin uing plane mehod, opimal bae idenifiaion mehod. Inroduion The mainenane aiviie for Generaion Company (GenCo) preen one of imporan ak ha have ignifian refleion on i profi and effiieny. I i pariularly emphaized in liberalized ambien where GenCo are faed wih numerou hallenge wih repe o enuring reliable eleriiy upply a profi-efeive rae. One of hee hallenge onern he planned prevenaive mainenane of ompany' power generaing uni. ainenane heduling a riial ehnial ak require arefully planning and analyi o guaranee yem reliabiliy and eonomi benefi for he GenCo. Beaue all power uni mu be mainained and inpeed, he planner in GenCo mu hedule planned ouage during he year. Several faor enering ino hi heduling analyi inlude: weekly (or daily) power profile (bilaeral onra), marke prie, amoun of mainenane o be done on all power uni, apaiy of uni, availabiliy of mainenane rew, elaped ime from he la mainenane aiviie, ehnologial reriion and eaon limi, obligaion oward Syem Operaor (SO) regarding o anillary ervie. All hee faor mu be inluded ino GenCo objeive for profi maximizaion. The mainenane heduling i hard, omplex ombinaorial opimizaion problem ha ha been udied widely in pa. Tradiional opimizaion ehnique uh a ineger programming [,2], E-ISSN: X 3 Volume 9, 204

2 deompoiion mehod [2,3,4], goal programming [5] have been ued o olve hi problem. odern evoluionary ehnique, a genei algorihm [6,7], imulaed annealing [7,8], memei algorihm [9], abu earh [7,0,] and fuzzy e heory [2,3] have been applied o he problem. The mainenane heduling of hermal power uni hould be opimized in erm of he objeive funion under erie of onrain. The eleion of objeive and onrain depend on he pariular need of mainenane heduling problem, he daa available, an auray o be ough and hoen mehodology for olving hi problem. There are generally wo aegorie of objeive in mainenane heduling problem: baed on o [,3,4,7,4] or on profi [,5,6,7] and baed on reliabiliy [8,0,2]. The mo ommon objeive baed on o i o minimize he oal operaing o over he planning period (horizon). Thi minimizaion ofen require many approximaion or ompuaionally inenive imulaion o yield a oluion. I wa repored in lieraure ha minimizaion of he oal operaing o (or produion o ha i he main par of he operaing o for hermal uni) i an ineniive he objeive for mainenane heduling problem [6,8,4]. A number of reliabiliy indie, uh a expeed lak (horage) of reerve, expeed energy no upplied and lo of load probabiliy, whih are baed on power yem meaure ha been ued a reliabiliy rieria for he formulaion of objeive funion [8,0,4]. The mainenane imeable hould aify e of onrain relaed o power uni (mainenane window onrain), preven he imulaneou mainenane of e of uni (exluion onrain), reriion he ar of mainenane on ome uni afer period of mainenane of oher uni (equene onrain), yem onrain (balane onrain, ranmiion onrain), rew onrain, e. In reen lieraure he mainenane heduling problem ha been oriened oward new relaion in eleri power eor. In a number of eleriiy marke, deregulaion of he power indury ha given GenCo he independene o mainain power uni in deenralized manner wih a minimum regulaory inervenion for yem euriy purpoe only. The mainenane period of ime for power uni are heduled eiher by profi-eeking GenCo only, or by oordinaion beween profi-eeking GenCo and reliabiliy-onerned SO, and he exen of oordinaion depend on he marke environmen and aual legilaive. Alhough he oordinaion proedure how SO adju individual GenCo' mainenane hedule and how eah GenCo repond o adjued hedule i imporan, i i no a main onern of hi paper and one an inveigae more abou hi ubje. An appliable proedure ha oniliae objeive for GenCo, o hedule heir uni for mainenane in order o maximize heir profi, and SO requiremen ha enure adequae euriy hroughou he week of year, i deermined hrough muliple ineraion beween GenCo and SO and given in [5,6]. In hi paper he mainenane heduling problem i analyzed from he GenCo poin of view. In order o enure adequae level of euriy, in hi paper we aume imple ineraion of he SO oward he GenCo aking ino aoun minimal level of reerve requiremen. Thi requiremen an be a par of SO' oal poliy, onained in i plan of anillary ervie. For minimal level of reerve GenCo will have benefi hrough prie of apaiy in reerve (hi revenue i no analyzed in hi paper). The mainenane heduling i an aive reearh area in power yem opimizaion. The omplexiy inrodued by planning onep uh a muliple and onradiory objeive, aoiaed wih he ombinaorial naure of he problem, lead o he perepion of limiaion of radiional mehod. Rounding off baed mehod ha quik ompuaion peed, bu i ignifianly degrade opimaliy and may be impoible o obain a feaible oluion. Sandard ineger (binary) programming mehod, a branh and bound algorihm, are non-polynomial. Conequenly, i i low and inraable for largeale problem. Heurii algorihm, a genei algorihm, imulaed annealing, abu earh and fuzzy e heory, are very ompuaionally imeonuming provided ha he ize of he earh pae i huge. In reen year, inerior poin mehod (IP) ha been widely ued for olving opimizaion problem in eleri power yem, for i fa onvergene haraerii and dealing wih inequaliie onvenienly. Generally, IP deal wih problem in whih all variable are oninuou. Inerior poin uing plane mehod (IPCP) appeared a a powerful ool o deal wih mixed ineger programming ha enlarge he appliaion area of IP. In 992, IPCP wa propoed by ihell and Todd o olve he perfe mah problem [8]. ihell and Borher olve linear ordering problem by IPCP [9]. The omparion wih implex uing plane mehod (SCP) [20,2,22] how ha i ha remarkable advanage a problem ize inreae. Ding, e al. ue IPCP o olve large-ale, diree and nonlinear mixed ineger opimal power flow problem [23,24]. If he opimal oluion of he relaxaion problem olved by IP i a degenerae oluion or onvex E-ISSN: X 32 Volume 9, 204

3 ombinaion oluion, he uing plane will fail o be generaed, and he IPCP will fail. A new bae idenifiaion mehod i preened. The improved algorihm an find opimal bae for variou ype of opimal oluion. Effiieny of bae idenifiaion proedure i improved. Large ompuaion ime may be onumed in marix rank alulaion and rowolumn ranformaion. The perurbaion mehod and ome linear algebra ehnique are inrodued o IPCP ha an ignifianly improve ompuaion effiieny. The fou of hi paper i developmen of omprehenive model for mainenane heduling raegy of hermal power uni aking ino aoun he pariular obligaion of he GenCo from long-erm bilaeral onra, a well a deerminaion of power profile for elling on eleriiy marke baed on foreaed prie. The main onribuion of hi paper are a follow:. preenaion of he approah ha i flexible and robu o be ued in he mainenane heduling of hermal power uni; 2. developmen of a hybrid model ha ombine energy ale hrough bilaeral onra and energy ale on he marke for he maximizaion of GenCo profi wih maller unerainy from marke prie volailiy; 3. appliaion of he inerior poin uing plane mehod for mixed-ineger linear programming approah ha guaranee onvergene o he opimal oluion and ompuaional effiieny in large-ale ae udie. Thi paper i organized a follow. Seion 2 provide he noaion ued hroughou he paper. In Seion 3 opimal mainenane heduling problem i modeled a deerminii programming problem. Seion 4 deail he priniple of IPCP and i appliaion for he problem. In Seion 5 reul from a realii ize ae udy are preened and diued. Seion 6 ae all of he onluion of hi paper. 2 Noaion The noaion ued hroughou he paper i aed below: Indexe: i hermal uni index k hermal power plan index m bilaeral onra index ime period (week) index Conan: θ number of hour in week ( θ=68 ) π m () prie of bilaeral onra m in period [$/Wh] π () marke prie of energy in period [$/Wh] a ik, fixed operaing o of uni i in plan k [$/h] b ik, linear o erm in o haraerii of uni i in plan k [$/Wh] ik, quadrai o erm in o haraerii of uni i in plan k [$/W 2 h] d ik, variable O& o of uni i in plan k [$/Wh] C ik, mainenane o of uni i in plan k [$/W] ET ik, earlie mainenane ar of uni i in plan k LT ik, lae mainenane ar of uni i in plan k ik, duraion of mainenane for uni i in plan k N k number of uni in plan k ha an be mainained imulaneouly O ab, number of period during ha mainenane of uni a and b hould overlap pm () power from bilaeral onra m in period [W] P ik, apaiy of uni i in plan k [W] P ik, minimum oupu of uni i in plan k [W] R0( ) minimum reerve level aigned o GenCo from he SO in period [W] S ab, number of period required beween he end of mainenane of uni a and he beginning of mainenane of uni b Variable: Pik, () power generaed by uni i in plan k in period [W] p () power for bid on marke in period [W] vik, () 0/ variable, equal o one if uni i in plan k i online in period, oherwie zero yik, () 0/ variable, equal o one if uni i in plan k i being mainained in period, oherwie zero Number: I k number of hermal uni in plan k K number of hermal power plan number of bilaeral onra number of period of he planning horizon. T 3 Problem Formulaion 3. Objeive funion Imporan poin in mainenane heduling of hermal power uni preen eleion of objeive funion. I depend of long-erm GenCo raegi parameer, i obligaion oward SO, regulaory agreemen and e. Beaue of ha, mainenane heduling i eenially muli-objeive ak wih E-ISSN: X 33 Volume 9, 204

4 onfliing objeive. In hi paper he objeive i o maximize profi for he GenCo. The expeed profi for he GenCo i alulaed a a differene beween expeed revenue and operaing o. Operaing o inlude o of energy produion and mainenane o. The bilaeral ale onra wih pariular energy paern and prie profile are inluded in hi objeive. Alo, in objeive () he marke learing prie for eah period are known. The objeive funion for GenCo i expreed a profi maximizaion and formulaed a follow: T θ π () () + θπ max m pm p() Co() = m = () K Ik 2 Co() = θ aik, vik, () + bik, Pik, () + ik, Pik, () + k= i= K Ik K Ik +θ dik, Pik, () + Cik, Pik, yik, () k= i= k= i= (2) In equaion () he fir erm i relaed o revenue from bilaeral onra beween he GenCo and oher marke player (load erving eniie, rader, diribuion ompanie). The amoun of power ha he GenCo ha agreed o erve in period a reul of bilaeral onra m i pm () and he prie ha he GenCo will be paid i π m (). Wih hi onra, he GenCo revenue inreae for π m() pm(). The eond erm repreen expeed revenue from elling power p () on marke wih foreaed prie π () in period. The hird erm repreen he oal o Co() oni of produion o (fuel o) FCik, (), variable O& o d ik, and mainenane o C, a aed in (2). The fuel o are ik, repreened by quadrai funion: FCik, () = aik, + bik, Pik, () + ik, Pik, () (3) In order o ue ae-of-he-ar ehnique for linear programming, quadrai fuel o funion are modelled by by pieewie linear approximaion [25]. Thi enure formulaion of problem a mixedineger linear programming model ha an be olved faer han original mixed-ineger nonlinear programming model. The repreenaion of hi approximaion i aed in Seion Bilaeral onra and energy for marke In he newly reruured eleriiy marke, he GenCo and oher marke player (load erving eniie, diribuion ompanie) an ign long-erm 2 bilaeral onra o over player need, whih are derived from he demand of heir uomer. Thee bilaeral onra over he real phyial delivery of elerial energy. The aor agree on differen prie, quaniie, or differen qualiie of elerial energy. Alo, duraion of he onra may differ, from medium-erm (weekly, monhly) o long-erm (yearly, few year). How muh of heir apaiy and demand GenCo and player will onra hrough bilaeral onra, and how muh hey will leave open for marke ranaion, i heir raegi and fundamenal queion. Baially, heir reaon for onraing bilaeral onra are follow. Beaue of prie volailiy, marke power rik and poible onrain in ranmiion nework, he GenCo will eimae how muh of i apaiy will be onraed hrough bilaeral onra, and how muh of apaiy will be offered on he marke. Bilaeral onra redue rik for he GenCo beaue i apaiie may go unued a a reul of no finding buyer or ranporaion apaiy on he marke. Alo, load-erving eniie, diribuion ompanie, a oher pary in bilaeral onra wih he GenCo, fae wih rik on he marke beaue of prie volailiy. Addiionally, for large onumer whoe load need high reliable eleri energy, he bilaeral onra give guaranee ha heir load will be alway upplied. The bilaeral onra define ha erain amoun of energy during number of hour will be delivered a given ime in he fuure, a agreed prie and a defined loaion. The GenCo mu ake hee bilaeral onra ino onideraion when heduling i uni [25]. Uually, bilaeral onra have a diree power paern during erain number of period a well a orreponding prie paern. Power p () and prie m π m () in period are onan. Thi implie ha revenue from all bilaeral onra i onan. Aording o foreaed weekly prie on marke, GenCo ha poibiliy o ell a par of i remaining produion on he marke. Level of power for bid on he marke in period, p (), depend of marke prie in period, π (). The revenue from elling power on he marke i π p(). The variable p () are opimizaion variable. Prie and power quaniie relevan for he bilaeral onra an be obained by yemai negoiaion heme [26] hroughou he GenCo and i onra parner an reah a muually benefi and olerable rik. Negoiaion for prie and power quaniie will onverge only if boh ide an find prie mix ha provide an aepable ompromie beween he rik and benefi (uually, par of he porfolio managemen). E-ISSN: X 34 Volume 9, 204

5 3.3 ainenane onrain The following relaion repreen e of onrain ha mu be aified in mainenane heduling problem. Alo, minimal reque on reerve level deermined by he SO i here aken in onideraion a obligaion for he GenCo. a) inimum and maximum power oupu: The power oupu for eah online uni mu be wihin delared range repreened by i minimum and maximum power oupu: Pik, vik, () Pik, () Pik, vik, () i, k, (4) The uni anno be online if i i in mainenane ha enured by onrain: v y ik, () + ik, () i, k, (5) If he uni undergo mainenane in period, y ik, () =, onrain (5) enure ha v ik, () = 0, beaue of ha onrain (4) enure he oupu of he uni i e o zero during mainenane. The power oupu of he uni an be equal o zero if he uni i no online and i no undergo mainenane. b) Conraed arrangemen and power for marke: The oal power generaed in hermal uni mu be enough o over he onraed load paern and power deermined for he marke for eah period: K Ik P, () () ik = pm + p () k= i= m= (6) ) Requiremen on minimum of reerve: Available apaiy of uni mu aified requiremen on minimal level of reerve impoed by he SO for eah period: K Ik P, (, ()) () ik yik pm p () R0() k= i= m= (7) d) ainenane duraion: For eah uni mu be enured he neeary number of ime period for i mainenane during he horizon. The onrain (8) enure hi reque: T yik, () = ik, i, k = (8) e) Coninuou mainenane period: Thi onrain enure ha he mainenane for eah uni mu be finihed one when begin: y y y ik, ( ) ik, ( ) ik, ( + ik, ) i, k, (9) f) Earlie and lae mainenane ar ime: Planner in he GenCo deermine earlie and lae mainenane ar ime for eah hermal power uni aking ino onideraion peifi uni mainenane requiremen, appropriae eaon limi (heaing, working feaibiliy, rew availabiliy). Suppoe Tik, T i he e of period when mainenane uni i in plan k may ar, o: { } Tik, = T : ETik, LTik, i, k (0) g) Number of uni in he plan ha an be mainain imulaneouly: The nex onrain limi he number of uni in one plan ha an be mainained a he ame ime: Ik yik, () Nk() k, i = () h) Inompaible pair of uni: The requiremen ha ome uni anno be mainained a he ame ime i eaily aed by binary onrain (2). If uni a and b (in he ame plan or in oher plan) anno undergo mainenane during he ame period, hi i aed a follow: y y ak, () + bk, () (2) i) ainenane prioriy: If power uni a mu be mainained before uni b, ha following onrain mu be aified: yak, ( τ ) ybk, ( ) τ=, ( ) = 0, ( τ ) 0 (3) { yak, for } j) Separaion among oneuive mainenane ouage: If beween finih of mainenane of uni a and begin of mainenane of uni b (in he ame plan or in differen plan) i needed eparaion of S ab, period, han following onrain mu be aified [5]: yak, ( τ ak, Sab, ) ybk, () τ= (4) E-ISSN: X 35 Volume 9, 204

6 min max abk,, yak, ( τ ak, Sab, ) abk,, ybk, () τ τ= τ=, ( ) = 0, ( τ S ) 0 min abk,, { yak, for ak, ab, } (5) = min { ak,, bk, }, max = max { ak,, bk, } abk,,. k) Overlap in mainenane ouage: If during period in ha uni a finihe he mainenane before uni b and if duraion he mainenane of uni b mu overlap peified number of period O ab,, han following onrain mu be aified [5]: i neeary o modify he laial ehnique for generaing uing plane from he opimal ableau [23]. Seondly, how o idenify he bae variable in IPCP? The bae idenifiaion in IPCP i a imporan a implex ableau generaion in SCP. In [23] hown how i obained bae informaion T 2 T from he marix DA (AD A ) AD under nondegenerae hypoheize. Unforunaely, mo of linear programming problem are degenerae and many problem have muliple opimum, whih limi he IPCP appliaion. iniializaion k = 0 yak, ( τ ak, + Oab, ) ybk, () (6) τ= min max abk,, yak, ( τ ak, + Oab, ) abk,, ybk, () τ τ= τ=, { yak, ( ) = 0, for ( τ ak, + Oab, ) 0} (7) Uni a and uni b an be in he ame power plan, or in differen plan. If i Oab, = bk,, ha uni a and uni b finih mainenane imulaneouly. k = k + relax he mainenane heduling problem o a linear programming LP(k) by deleing he inegral onrain olve LP(k) by uing IP LP(k) i feaible? Y ineger value are all inegral feaible? N idenify he opimal bai N Y he mainenane heduling problem i infeaible. Sop! he opimum i found. Sop! 4 ehodology and Algorihm 4. IPCP priniple and i implemenaion In radiional uing plane mehod (CP), he linear programming relaxaion have been olved uing he implex mehod. Simplex mehod ha an exponenial-ime haraerii, whih reri i real-world appliaion. Conrarily, IP earhe opimum inide he feaible region, i ieraion number do no obviouly hange a he ale of yem inreae, o i i uperior o he implex in onvergene and alulaion peed [27]. Generally, uing plane mehod for (0/) mixed-ineger linear programming require olving a large number of linear programming relaxaion, o i i obviou ha replaing implex algorihm wih IP will improve alulaion effiieny. Baed on hi idea, IPCP in many heoreial udie and praial appliaion were hown a very promiing ool for olving large-ale diree opimal problem. The main ompuaional priniple of IPCP and i implemenaion for he mainenane heduling problem i given in Fig.. Obviouly, wo poin are very imporan for IPCP ue. Firly, how o generae uing plane wihou implex ableau? I ge uing plane by he informaion of opimal bai add he uing plane ino IP(k), IP(k+) i obained ig. : The alulaion flowhar of IPCP for he mainenane heduling problem. 4.2 Failure reaon analyi Cuing plane an be generaed one he opimal bae i obained. For radiional SCP, he opimal oluion onverge o he verex poin of onvex hull. Opimal bae an be obained baed on he implex ableau and hen he uing plane are generaed. However, for IPCP, if relaxed linear program i degenerae or ha muliple oluion, he bae anno be orrely idenified. There are wo ae in ha he opimal bae anno be idenified: (i) he opimal oluion i degenerae and (ii) he linear relaxaion programming ha muliple opimal oluion. Aume ha he linear programming problem ha T he andard form: min{ x : Ax = b, x 0}, and i T T dual problem form: max{ b y: A y+ =, 0}, n m where i (, x, ) R, (, by ) R, A R m n, and A i aumed o have full row rank. E-ISSN: X 36 Volume 9, 204

7 A imple problem i ued o hown he failure reaon of IPCP aued by he above wo ae: max 2x + 4x2 (a).. x + 2x2 + x3 = 8 (b) x + x4 = 8 () x2 + x5 = 3 (d) x, x2, x3, x4, x5 0 The problem (a) (d) i equivalen o: max 2x + 4x2 (e).. x + 2x2 8 (f) x 8 (g) x2 3 (h) x, x2 0 I i a muliple oluion for linear programming problem. There are hree ype of oluion: ) non-degenerae oluion x =(2,3,0,6,0), number of nonzero elemen i equal o 3 (i equal o number of equaliy onrain); 2) degenerae oluion x =(8,0,0,0,3), number of nonzero elemen i le han 3; 3) onvex ombinaion oluion x =αx +(-α)x and for α (0,), e.g. x =(4.5,.75,0,3.5,.25) when α=7/2, he amoun of nonzero elemen i greaer han 3. The geomeri meaning of he above opimal oluion ype are hown in Fig. 2. Line AC repreen (h); line CD repreen (g); line BD repreen (f); he region area (onvex polyope) enloed by ABD0 i he feaible region onrued by onrain (f), (g) and (h). Beaue he line BD parallel he line repreened by objeive funion, any poin on line BD i he opimal oluion A B(2,3) 2 3 opimizaion rajeory of IPCP opimizaion rajeory of implex mehod opimal fae 4 P(4.5,.75) Fig. 2: Shemai diagram of opimal oluion C 8 D A he implex mehod earhe for opimal oluion hrough verie, he oluion i definiely verex poin of he opimal fae. The verex oluion i orreponding o he non-degenerae oluion (poin B) or degenerae oluion (poin D) hown in Fig. 2. Gomory uing plane an be generaed from he final implex ableau. Obviouly, a implex ableau i no available when he inerior poin mehod i ued. Inerior poin mehod earhe for opimal oluion hrough he inerior of feaible region, and all hree ype of oluion may be obained. A hown in Fig. 2, he opimal oluion i more likely o onverge o any poin peraining o line BD. In oher word, he opimum i onvex ombinaion oluion. In hi iuaion, IPCP would fail a he opimal bae anno be idenified. 4.3 The improvemen of IPCP An opimal bae idenifiaion mehod i developed [28] and i proedure an be deribed a in Fig. 3. The index e {,,n} i pariioned ino ube I, I 2 and I 3, and defined a follow: I = { j x j > 0, j = 0}, I 2 = { j x j = 0, j = 0}, I 3 = { j x j = 0, j > 0}, ( dual lak variable). j nondegenerae oluion nondegenerae oluion he orreponding olumn of nonzero elemen oniue bae marix Fig. 3: Diagram of opimal bae idenifiaion. The olumn of marix A and elemen of veor are laified a hree par: A = { A j I }, = { j I } * j j A = { A j I }, = { j I } 2 * j 2 2 j 2 A = { A j I }, = { j I } 3 * j 3 laify oluion ype move he oluion o verex poin of opimal fae and idenify oluion ype op proedure onvex ombinaion oluion 3 j 3 degenerae oluion expand he orreponding olumn of nonzero elemen from bae marix degenerae oluion where A, A 2 and A 3 are ompoed of he orreponding olumn of index e I, I 2 and I 3 E-ISSN: X 37 Volume 9, 204

8 of marix A eparaely;, 2 and 3 are ompoed of he orreponding elemen of index e I, I 2 and I 3 of veor eparaely. The proedure i a follow: Claify he ype of oluion: non-degenerae oluion, degenerae oluion or onvex ombinaion oluion. The index e I, I 2 and I 3 are obained: i) if i i a non-degenerae oluion, he index e of nonzero elemen i he index e of opimal bae, le B = A, go o 0; ii) if i i a degenerae oluion, le B = A, go o 6; iii) if i i onvex ombinaion oluion, go o 2. 2The movemen of primal opimal oluion: i) olve equaion Az = 0. A A i olumn linearly dependen, z may have muliple oluion. Chooe one of nonzero veor z; ii) olve x, = x + z 0, alulae he range of alar : min max. 3Chooe anyone of min and max, le x, = x + z. For erain x, j = 0 ( x, j repreen he j h elemen of,, x ), x j i removed from x, and added ino x 2, he olumn orreponding o x, j i removed from A and added ino A 2, le x =, x, x = ( x,, x2, x 3). 4 If A i olumn dependen, go o 2, ele le B = A, go o nex ep. 5If rank( B) < rank([ A, A 2]) go o 8, ele go o nex ep. 6 Add he olumn of A 2 whih are linearly independen of B ino B. 7If rank( B ) = m, go o 0, ele go o nex ep (m i number of linear independen onrain). 8The movemen of dual opimal oluion: T i) olve equaion B z = 0 and hooe anyone of nonzero veor z ; T ii) aording o A 3 y' 3, alulaing he range of : min max. 9Chooe anyone of min and max, le y' = y z erainly j aifie a T 3jy' = 3j ( a 3j i j h olumn of 3, i j h elemen of 3 ), a 3j i removed from A 3j 3 A and added ino A 2 and B, meanwhile B hould be kep olumn linearly independen, go o 7. 0Sop proedure, B i opimal bae marix. 5 Cae Sudy To illurae he effeivene of he propoed model we have preened an illuraive ae udy. The model ha been implemened and olved wih C++ language on PC baed plaform wih GenuineInel proeor loking a 3.20 GHz wih 3 GB of RA. 5. Inpu daa Inpu daa from realii ae udy are preened in hi eion. The GenCo generaion yem oni of five hermal power plan wih oal 20 power uni. Table and Table 2 how li of hermal uni wih i apaiie, mainenane parameer, fuel o oeffiien, O& and mainenane o. The lengh of he planning horizon i 52 week, and mainenane hedule for eah uni will our ju one during he planning horizon. Table : The parameer of hermal power uni plan k # #2 #3 #4 #5 uni i no. Pmin Pmax ET LT Table 2: Fuel o oeffiien, O& o and mainenane o of hermal power uni plan uni a b d C # #2 #3 #4 # E-ISSN: X 38 Volume 9, 204

9 Uing of pieewie linear approximaion for quadrai fuel o haraerii given in Eq. (3), omplee model i hen preened a mixed-ineger linear programming (ILP) formulaion of he mainenane heduling problem ha enure an effiien oluion uing IPCP. The pieewie linear approximaion of o haraerii (variable o) i formulaed a follow [25,29]: N VCik, () = Fn(, ikd ) n(, ik, ), i, k, n = N Pik, () = Pik, v ik, () + dn(, ik, ), i, k, n = 0 d n( ik,, ) d n ( ik, ), i, k,, n =,2,..., N where N i he number of blok of he pieewie linear variable o funion, Fn (, ik ) repreen he lope of blok n of he variable o of hermal uni i, dn (, ik, ) repreen he power produed by uni i in period uing n h power blok, dn (, ik ) i ize of he n h power blok for uni i. Variable o have been modelled uing he pieewie linear approximaion wih hree blok a hown in Table 3. Table 3: Pieewie linear approximaion of fuel o haraerii of hermal power uni plan uni # #2 #3 #4 #5 T T 2 F ($/Wh) F 2 ($/Wh) F 3 ($/Wh) In Table 3 onan T and T 2 mean upper limi of blok and 2 of hermal uni variable o. Table 4 how foreaed weekly prie on he marke. I hould be noed ha prie profile hould be obained by appropriae foreaing proedure. Table 4: Weekly foreaed marke prie π () π () π () π () The reque on minimum reerve deermined by he SO, ha GenCo mu aify a an obligaion, aumed o be value of 250 for eah week. The GenCo ha wo yearly bilaeral onra, for example, wih large onumer. Fir onra ha power paern ha i onan during erain number of week wih orreponding prie paern and eond onra wih onan power during year wih fixed prie. The onraed power profile and prie are preened in Table 5. Table 5: Bilaeral onra wih power profile and prie profile ($/Wh) CONTRACT # p () π () CONTRACT #2: p2 () = 250 (BASE LOAD); π 2 () = 45.8 The reul of following e ae are analyed: ae #: only onrain () (); ae #2: ae # plu inompaible pair of uni (uni 4 and 5 in # and uni 7 and 8 in #2 anno be mainained a he ame ime); ae #3: ae #2 plu mainenane prioriy (in #3, uni 9 mu be mainained before uni 3 in #4); ae #4: ae #3 plu eparaion among oneuive mainenane ouage (afer finihing mainenane of uni 6 in #4 and beginning mainenane of uni 20 in #5, eparaion of 5 week i needed); E-ISSN: X 39 Volume 9, 204

10 ae #5: ae #4 plu overlap in mainenane ouage (mainenane of uni 4 in #4 mu begin 3 week before uni 9 in #3 finih i mainenane). 5.2 Te reul and analyi For peified e ae, Table 6 how oal o, profi and oal energy for marke. aximum profi and he bigge energy amoun for he marke are obained in e ae # ha onider only bai onrain (4) (). The lowe o and he malle energy amoun for he marke are obained in e ae #5 haraerized wih he malle value of profi ompared wih oher ae. I an be een from Table 6 how differen e of onrain aigned o mainenane heduling problem affe oal o of GenCo generaion yem, and how affe i profi. In all ae, oal energy from boh bilaeral onra i equal 29,4,400 (Wh). Table 6: The global reul for differen e ae wih bilaeral onra e ae oal o ($) profi ($) oal energy for marke (Wh) # 872,372, ,634,84.3 7,0,600 #2 875,677, ,948, ,73,533 #3 873,235, ,893, ,09,390 #4 875,29, ,636, ,67,367 #5 866,444, ,087,69.8 6,96,560 I i illuraive o onider he influene of bilaeral onra on GenCo profi for he above e ae. Table 7 how oal o, profi, and he oal generaed energy for marke wihou bilaeral onra. Inroduion of bilaeral onra furher ompliae finding opimal mainenane hedule. However, alway i enure an addiional profi for he GenCo. Furhermore, plan generaion beome far le eniive o fluuaion of marke prie, imply beaue bilaeral onra ipulae energy produion and plaemen. Table 7: The global reul for differen e ae wihou bilaeral onra e ae oal o ($) profi ($) oal energy for marke (Wh) # 865,54, ,259, ,987,77 #2 864,898, ,97, ,980,997 #3 865,54, ,259, ,987,77 #4 864,55, ,58, ,956,082 #5 864,580, ,363, ,972,597 Le aume ha marke prie have deviaion in he range of ±5%, a reaon, for example, error in forea prie. The inremen of hi deviaion i % (dereae prie), i.e. +% (inreae prie). The elemen from wo bilaeral onra remain he ame and hey are given in Table 5. Nex, we fou on e ae #5. Table 8 how profi for every inremenal prie hange on he marke, a well a profi deviaion for a given prie hange. Reul in Table 8 lead o he onluion ha exiene of bilaeral onra play he role of hok aborber in he profi funion, meaning ha hange of profi happen lowly han hange of marke prie. Eenially, profi deviaion are aenuaed when ompared o he marke prie hange. For inane, 5% prie dereae aue only 2.04% dereae in he profi. The eniiviy i omewha larger in he ae of prie inreae, where 5% marke prie inreae aue 2.25% inreae in he profi. Table 8: The profi deviaion wih hange in he marke prie marke prie deviaion 5% 4% 3% 2% % 0% profi (mil. $) profi deviaion (%) marke prie deviaion % 2% 3% 4% 5% profi (mil. $) profi deviaion (%) Obained mainenane hedule of he hermal power uni for e ae # and #5 an be een in Table 9 and 0. Table 9 and 0 how oal produion of hermal uni P T (), power for marke p S (), oal reerve R() and power in mainenane P () for eah week. Shown in Table 9 and 0, reuling hedule during he horizon aified all peified onrain and enured profi maximizaion obained by IPCP deribed in Seion 4. Shedule for e ae #2, #3 and #4 are obained in imilar manner, where given onrain are aified for eah e ae. In analyzed e ae mainenane onrain, weekly power profile from bilaeral onra and weekly foreaed marke prie are dominan faor for high number of onemporaneou power plan in mainenane ondiion. The effe of onrain in he mainenane heduling problem ignifianly impa boh oal produion, a well a he power offered o he marke. E-ISSN: X 40 Volume 9, 204

11 Table 9: The mainenane hedule for ae # Table 0: The mainenane hedule for ae #5 week uni in mainenane P T () p S () R() P () week uni in mainenane P T () p S () R() P () ,4, ,4,5, ,4,5,6, ,4,5,6, ,0, ,0,4, ,0,4, ,2,0,4, ,8,4,5,8, ,8,5, ,8,9,5, ,8,9,5, ,8,9,3,5, ,9,3, ,9,3,6,7, ,9,,6,7, ,9,,6,7, ,6,7, ,6,7, , ,6, ,9,6, ,9,6, ,9,4, ,9,0,2, ,9,0,2, ,7,0,2, ,7,0,2,4, ,5,7,2,9, ,5,7,2,5,9, ,5,7,5,9, ,5,8,5,9, ,5, ,8,3,5, ,8,,3,5, ,6,8,,3, ,6,, ,, E-ISSN: X 4 Volume 9, 204

12 Table : odel dimenion for ae #5 # of binary variable # of real variable # of onrain Table 2: Evoluion uing plane hrough ieraion progre for ae #5 ieraion ID uing plane ieraion ID uing plane ieraion ID uing plane Table ummarize alulaion dimenion for ae #5. The preened mainenane heduling problem ha many binary and oninuou variable ha are ypial for ILP. Beaue he omplexiy of he preened formulaion, i anno be olved by original IPCP. In fa, he main reaon i failing o find he uing plane, a i i analyzed in eion 4.2. The opimal oluion i obained by uing improved IPCP and number of uing plane during 5 ieraion i lied in Table 2. 6 Conluion In reruured power yem and liberalized marke, mainenane heduling problem ha new haraerii differen from hoe in radiional environmen. In preened mainenane heduling model, GenCo inere i o maximize profi, ombining energy ale hrough bilaeral onra and energy ale on marke. GenCo i reponible for performing neeary mainenane of i power uni in order o uain i poiion on he marke. The mainenane period of ime for power uni are heduled eiher by profi-eeking GenCo only, or by oordinaion beween profi-eeking GenCo and reliabiliy-onerned Syem Operaor and exen of oordinaion depend on he marke haraerii. Alhough he oordinaion proedure how Syem Operaor adju he individual GenCo' mainenane hedule and how eah GenCo repond o he adjued hedule i imporan, i i no a main onern of hi paper and one an inveigae more abou hi ubje. In hi paper he mainenane heduling problem i analyzed from he GenCo poin of view. To olve hi mainenane heduling problem, he inerior poin uing plane mehod i ued. Thi mehod poee advanage of boh, inerior poin mehod and uing plane mehod, and beome very promiing approah for large-ale and diree opimizaion problem. The new bae idenifiaion mehod i preened o olve problem of degenerae oluion and onvex ombinaion oluion. The improved algorihm an olve diffiulie brough by muliple oluion. The preened model ha been uefully eed on he realii ize ae udy. Numerial reul have revealed he auray and ompuaionally effiien performane of he preened formulaion. Aknowledgemen The auhor grealy aknowledge he onribuion of Xiaoying Ding from Shool of Elerial Engineering, Xi an Jiaoong Univeriy, Republi of China for help in he developmen of hi work. Referene: [] K. W. Edwin, F. Curiu, New mainenane heduling mehod wih produion o minimizaion via ineger linear programming, Inernaional Journal of Eleri Power and Energy Syem, Vol. 2, 990, pp [2] L. Chen, J. Toyoda, Opimal generaing uni mainenane heduling for muli-area yem wih nework onrain, IEEE Tranaion on Power Syem, Vol. 6, No. 3, 99, pp [3]. K. C. arwali, S.. Shahidehpour, Longerm ranmiion and generaion mainenane heduling wih nework, fuel and emiion onrain, IEEE Tranaion on Power Syem, Vol. 4, No. 3, 999, pp [4]. K. C. arwali, S.. Shahidehpour, Inegraed generaion and ranmiion mainenane heduling wih nework onrain, IEEE Tranaion on Power Syem, Vol. 3, No. 3, 998, pp [5] L.. oro, A. Ramo, Goal programming approah o mainenane heduling of generaing uni in large ale power yem, IEEE Tranaion on Power Syem, Vol. 4, No. 3, 999, pp [6] K. P. Dahal, C. J. Aldridge, J. R. Donald, Generaor mainenane heduling uing a genei algorihm wih a fuzzy evaluaion funion, ELSEVIER, Fuzzy Se and Syem, No. 02, 999, pp [7] E. K. Burke, J. A. Clarke, A. J. Smih, Four mehod for mainenane heduling, in G. D. Smih, N. C. Seele and R. Albreh (ed.) Proeeding of Third Inernaional Conferene Arifial Neural Ne and Genei Algorihm E-ISSN: X 42 Volume 9, 204

13 (ICANNGA '97), Springer-Varlag, Vienna, 998, pp [8] K. P. Dahal, G.. Bur, J. R. Donald, S. J. Galloway, GA/SA baed hybrid ehnique for he heduling of generaor mainenane in power yem, Proeeding of Congre of Evoluionary Compuaion (CEC2000), San Diego, 2000, pp [9] E. K. Burke, A. J. Smih, Hybrid evoluionary ehnique for he mainenane heduling problem, IEEE Tranaion on Power Syem, Vol. 5, No., 2000, pp [0] I. El-Amin, S. Duffuaa,. Abba, A abu earh algorihm for mainenane heduling of generaing uni, ELSEVIER, Eleri Power Syem Reearh, Vol. 54, No. 2, 2000, pp [] J. Sugimoo, H. Tajima, S. ahi, R. Yokoyama, V. Silva, Profi-baed hermal uni mainenane heduling under prie volailiy in ompeiive environmen, Inernaional Conferene on Inelligen Syem and Conrol ISC, Cambridge, [2] H. S. Kim, S. P. oon, J. S. Choi, S. Y. Lee, D. H. Do,.. Gupa, Generaor mainenane heduling onidering air polluion baed on he fuzzy heory, IEEE Inernaional Fuzzy Syem Conferene Proeeding, Vol. III, Seoul, 999, pp [3] H. S. Kim, J. S. Choi, Developmen of a mehod for flexibile generaor mainenane heduling uing he fuzzy heory, Proeeding of Aia Fuzzy Syem Sympoium (AFSS2000), Tukuba, [4] H. H. Zurn, V. H. Quinana, Several objeive rieria for opimal generaor prevenive mainenane, IEEE Tranaion on Power Apparau and Syem, Vol. PAS-96, No. 3, 977, pp [5] A. J. Conejo, R. Berrand,. D. Salazar, Generaion mainenane heduling in reruured power yem, IEEE Tranaion on Power Syem, Vol. 20, No. 2, 2005, pp [6] R. Ehraghnia,. H.. Shanehi, H. R. ahhadi, A new approah for mainenane heduling of generaing uni in power marke, 9h Inernaional Conferene on Probabilii ehod Applied o Power Syem, KTH, Sokholm, [7]. Shahidehpour, H. Yamin, Z. Li, arke operaion in eleri power yem: Foreaing, heduling and rik managemen, fir ed., New York, John Wiley & Son, [8] J. E. ihell,. Todd, Solving ombinaorial opimizaion problem uing Karmarkar algorihm, ahemaial Programming, Vol. 56, 992, pp [9] J. E. ihell, B. Borher, Solving real-world linear ordering problem uing a primal-dual inerior poin uing plane mehod, Annal of Operaion Reearh, Vol. 62, 996, pp [20] R. J. Vanderbei, Linear programming: Foundaion and exenion, eond ed., Boon: Kluwer Aademi Publiher, 200. [2] H. Taha, Operaion reearh: An inroduion, Sixh ed., New Jerey: Prenie Hall, 997. [22] F. S. Hillier, G. J. Lieberman, Inroduion o operaion reearh, ixh ed., New York, NY: Graw-Hill, 995. [23] X. Y. Ding, X. F. Wang, Y. H. Song, An inerior poin uing plane mehod for opimal power flow, Inernaional Journal of anagemen ahemai, Vol. 5, No. 4, 2004, pp [24] L. Lin, X. Wang, X. Ding, H. Chen, A robu approah o opimal power flow wih diree variable, IEEE Tranaion on Power Syem, Vol. 24, No. 3, 2009, pp [25] S. Bianovi,. Hajro,. Dlaki, Hydrohermal elf-heduling problem in a day-ahead eleriiy marke, ELSEVIER, Eleri Power Syem Reearh, Vol. 78, No. 9, 2008, pp [26] S. El Khaib, F. D. Galiana, Negoiaing bilaeral onra in eleriiy marke, IEEE Tranaion on Power Syem, Vol. 22, No. 2, 2007, pp [27] I. J. Luig, R. E. aren, D. F. Shanno, Inerior poin mehod for linear programming: Compuaional ae of he ar, ORSA Journal on Compuing, Vol. 6, No., 994, pp. -4. [28] L. Lin, W. Xi-fan, X. Ding, Z. Qin, Robu approah of inerior poin uing plane mehod in opimal power flow, Auomaion of Eleri Power Syem, Vol. 3, No. 9, 2007, pp. -5. [29] A. B. Keha, I. R. Faria, G. L. Nemhauer, odel for repreening pieewie linear o funion, Operaion Reearh Leer, No. 32, 2004, pp E-ISSN: X 43 Volume 9, 204