Ramp Rate Constrained Unit Commitment by Improved Adaptive Lagrangian Relaxation

Transcription

1 Inernaonal Energy Journal: Vol. 6, o., ar 2, June Ramp Rae Consraned Un Commmen by Improved Adapve Lagrangan Relaxaon W. Ongsakul and. echaraks Energy Feld Of Sudy, School of Envronmen, Resources and Developmen, Asan Insue of Technology, ahumhan 220 THAILAD ABSTRACT Ths paper proposes an mproved adapve Lagrangan relaxaon (ILR) for ramp rae consraned un commmen problem. The proposed ILR mnmzes he oal supply cos subjec o he power balance, 5 mnue spnnng reserve response me consran, generaon ramp lm consrans, mnmum up and new down consrans. ILR s mproved by new mnmum down me o accoun for sarup and shu down ramp consrans, new nalzaon o oban a hgh qualy nal soluon, dynamc economc dspach o nclude he operang rampng lms, and adapve Lagrangan mulplers o speedup he convergence. If he 24 hour schedule s feasble, dynamc economc dspach by quadrac programmng s used o mnmze he producon cos subjec o he power balance and new generaon ramp operang frame lm. For hours wh nsuffcen 5 mnue response me spnnng reserves, reparng sraegy by quadrac programmng s used o mnmze he generaor fuel cos subjec o power balance, generaor operang ramp rae lm, and 5 mnue spnnng reserve response me consrans. The proposed ILR algorhm s esed on he 26 un IEEE relably es sysem. I s shown ha ILR could oban a hgher qualy soluon han dynamc economc dspach based on arfcal neural nework.. ITRODUCTIO Un commmen (UC) s used o schedule he generaors such ha he oal sysem producon cos over he scheduled me horzon s mnmzed under he spnnng reserve and operaonal consrans of generaor uns. UC problem s a nonlnear, mxed neger combnaoral opmzaon problem. The global opmal soluon can be obaned by complee enumeraon, whch s no applcable o large power sysems due o s excessve compuaonal me requremens []. Ramp rae consraned un commmen (RUC) was solved by mxed neger lnear programmng (MIL) [2], enhanced dynamc programmng [3], arfcal neural nework (A) [4], Lagrangan relaxaon (LR) [5-2], augmened Lagrangan relaxaon (ALR) [3-5], and dynamc prory ls [6]. In [4], a un commmen soluon was frs obaned whou rampng consrans. Thereafer, a dynamc adjusng process s adoped o oban un commmen schedule consderng he rampng consrans. To sasfy he sarup process, he consraned uns were adjused by changng her saus from 0 o for a few hours earler, leadng o overcommmen. In [7], he objecve funcon was augmened wh an addonal rampng cos, whch was relaed o he deprecaon n shaf lfe. LR s he mos effcen mehod o solve UC problem [7]. LR decomposes he man problem no un subproblems solved by dynamc programmng. Ths smplfes he problem sgnfcanly. However, rampng consrans n UC problem requres enlargng sae spaces dramacally [6], [8], [9], [4], for dynamc programmng o solve each un subproblem. The oal number of saes s he sum of number of down saes, number of ramp up saes, number of up saes, and number of ramp down saes [6].

2 2-76 Inernaonal Energy Journal: Vol. 6, o., ar 2, June 2005 The addonal Lagrangan mulplers corresponds o rampng consrans for each hour and each un were proposed n [8], [9], and []. These exra Lagrangan mulplers cause addonal compuaonal burden, leadng o a slower convergence rae. Furhermore, backward economc dspach s requred o reduce he generaon oupu from s consraned maxmum o zero under he ramp down lm whn one hour [4]. Ths paper proposes an mproved adapve Lagrangan relaxaon (ILR) for ramp rae consraned un commmen problem. The proposed ILR mnmzes he oal supply cos subjec o he power balance, 5 mnue spnnng reserve response me consrans, generaon ramp lm consrans, mnmum up and new down consrans whou enlargng sae spaces for dynamc programmng o solve un subproblems. ILR s mproved by new mnmum down me o accoun for sarup and shu down ramp consrans, new nalzaon o oban a hgh qualy nal soluon, dynamc economc dspach o nclude he operang rampng lms, and adapve Lagrangan mulplers o speedup he convergence. If he 24 hour schedule s feasble, dynamc economc dspach by quadrac programmng s used o mnmze he producon cos subjec o he power balance and new generaon ramp operang frame lm. For hours wh nsuffcen 5 mnue response me spnnng reserves, reparng sraegy by quadrac programmng s used o mnmze he generaor fuel cos subjec o power balance, generaor operang ramp rae lm, and 5 mnue spnnng reserve response me consrans. The proposed ILR algorhm s esed on he 26 un IEEE relably es sysem. The organzaon of hs paper s as follows. Secon 2 addresses he RUC problem formulaon. The ILR algorhm s descrbed n Secon 3. umercal resuls are presened n Secon 4. Lasly, he concluson s gven. 2. RAM RATE COSTRAIED UIT COMMITMET ROBLEM FORMULATIO In hs paper, hree ypes of rampng consrans for each un are consdered. Sarup ramp consrans: when an off-lne un s urned on, akes T, SR mnues o ncrease s generaon oupu from zero o s mnmum level as shown n Fg. a. Durng hs perod, hs un s off-lne bu consumng fuel. Shu down ramp consrans: when an onlne un s urned off, akes some mes o decrease s generaon oupu from he curren generaon oupu o zero. However, decommng he onlne un wh generaon oupu hgher han mnmum level s no allowed unless s forced shu down (generaon sheddng). Ths s because before he un saus s changed from a hour o 0 a hour +, hs un s operaed a he equlbrum pon where mechancal npu power equals o elecrcal oupu power. A hour + when hs un s dsconneced from he sysem, he elecrcal power of hs un s changed from s curren generaon o zero. Ths resuls n accelerang power on he roor and causes ransen nsably, whch may furher affec he generaor s lfe me and s performance. Hence, he generaon oupu of hs un should be a he mnmum level a he las commed hour [9]. I wll ake T, SDR mnues o decrease s generaon oupu from s mnmum level o zero as shown n Fg. b.

3 Inernaonal Energy Journal: Vol. 6, o., ar 2, June ,max,max,mn,mn T,SR UH DH Fg. Sarup and shu down ramp consrans T,SDR a) b) Operang ramp consrans: The generaon oupu of curren hour mus be lmed by he up ramp lm and down ramp lm, UR 60, as generaon ncreases () DR 60, as generaon decreases (2) Un ramp rae consrans requre modfcaon of operang consrans (un bass) and spnnng reserve consrans (sysem bass). On he un bass, he change of he generaon level of each un on any wo successve perods mus be whn ramp rae lmaon. The sum of ramp lms of commed uns mus be a leas suffcen o mee he change n he sysem load from one hour o he nex hour. On he sysem bass, he spnnng reserve amoun conrbued by each un mus be calculaed by consderng ramp rae consrans. The spnnng reserve calculaed from he dfference beween maxmum commed power and acual load can be large enough o mee he reserve requremen, bu rampng lmaons may cause he acual avalable spnnng nsuffcen. In hs paper, generaon ramp lm, new mnmum down me, and un reserve conrbuon are used. Generaon ramp lm: To sasfy he generaon operang lm consran n () and (2),,hgh low, generaon ramp lm, and are used as shown n Fg. 2. ew Mnmum down me: To sasfy he shu down and sarup rampng consrans, he generaon oupu of he shu down hour and he sarup hour s lmed o s mnmum level. The new mnmum down me s exended by he me for shu down process (from mnmum generaon level o zero) and me for sarup process (from zero o mnmum generaon level) as shown n Fg. 3. T = T + T + T. (3),down,SDR,down,SR Un reserve conrbuon: Due o he un ramp up lms, he spnnng reserve conrbued by each un whn τ mnues s r = mn [, τ UR ]. (4),max

4 2-78 Inernaonal Energy Journal: Vol. 6, o., ar 2, June 2005,max,hgh,low Hour Hour + UR 60 DR 60 +,hgh + +,low UR 60 DR 60 +2,hgh +2,low,mn Fg. 2 Generaon ramp lm for un,max,max,mn,mn T,SDR T,down T,SR DH UH T,down Fg. 3 ew mnmum down me The objecve of UC problem s o mnmze he producon cos over he scheduled me horzon (e.g. 24 hours) under he generaor operaonal, power balance, and spnnng reserve consrans. The objecve funcon o be mnmzed s: Subjec o: T F (,U, ) = [ F ( ) + ST, (-U,- )] U, (5) = = (a) ower balance consran load = U (b) 5 mnue spnnng reserve response me consran, = 0, (6) R r U, 0, (7) =

5 Inernaonal Energy Journal: Vol. 6, o., ar 2, June (c) Generaon ramp lm consrans,low U,,hgh U, =,,, (8), where, hgh, = mn[, + UR 60], fu = U =,,max,mn,, f U = 0, U, (9),, =, low, = max[, DR 60], f U = U =,,mn,mn,, f U = 0, U. (0),, =, (d) Mnmum up and new down me consrans (e) Sarup cos where, χ, δ, and γ are sarup cos parameers., on, f T < T,up, U, = 0, f T,off < T,down, () 0 or, oherwse, T,off ST = [ χ + δ ( exp( )], (2) γ 3. A IMROVED ADATIVE LAGRAGIA RELAXATIO FOR RUC The LR procedure solves he UC problem by decomposng he man problem no subproblems whch are coupled by he Lagrangan mulplers. The LR procedure solves he UC problem by relaxng or emporarly gnorng he couplng consrans and solvng he problem as f hey dd no exs. Ths s done hrough he dual opmzaon procedure aempng o reach he consraned opmum by maxmzng he Lagrangan, T L (,U, ) = F(,U, ) + ( load U, ) + ( R r U, ), (3) = Wh respec o nonnegave λ and μ whereas mnmzng wh respec o he oher conrol varables n he problem, ha s: = T = = q* ( λ, μ) = Max q( λ, μ), (4), λ μ

6 2-80 where, Inernaonal Energy Journal: Vol. 6, o., ar 2, June 2005 q( λ, μ) = Mn L(,U, λ, μ). (5) U,, Eqs. (6) and (7) are he couplng consrans across he uns. In parcular, wha s done o one un affecs he oher uns. The Lagrangan funcon s rewren as: L = T = = {[ F ( ) + ST, ( -U, - )] U, - U T The erm { [ F ( +,, ],,, } ) ST ( U )U U μ r U =, T r U } + ( + R )., = load (6) λ can be mnmzed separaely for each generang un, when he couplng consrans are emporarly gnored. Then he mnmum of he Lagrangan funcon s solved for each generang un over he me horzon, ha s Mn L(,U, U, μ) T = mn {[ F ( ) + ST, = = ( -U,- )] U, U, r U, }, Subjec o U,,low, U for =,,T, and he consrans n ()., hgh 3. Dynamc rogrammng In he convenonal Lagrangan relaxaon mehod, he dual soluon s obaned by usng dynamc programmng for each un separaely. Snce generaon ramp lm and new mnmum down me are used o accoun for he rampng consrans, he dynamc programmng does no requre enlargng sae spaces. Ths can be vsualzed n Fg. 4 showng he only wo possble saes for un (.e., U, = 0 or ): U = U = 0 ST ST ST ST = 0 = = 2 = 3 = T- = T Fg. 4 Search pahs of each un n dynamc programmng A he U, = 0 sae, he value of he funcon o be mnmzed s rval (.e., equals zero), a he sae where U, =, he funcon o be mnmzed s (he sar-up cos s dropped here snce he mnmzaon s wh respec o ) mn[ F ( ) - μ r ]. To fnd he dual power, he erm mn[ F ( ) - μ r ] wll be mnmzed by he opmaly condon,

7 Inernaonal Energy Journal: Vol. 6, o., ar 2, June The soluon o hs equaon s d [ F ( ) λ - μ r ] = 0. (7) d The dual power s obaned, df ( d,op,op ) = λ μ. (8) μ - b =. (9) 2c There are hree cases o check, op agans s lms:,op. If <, =., low,op, low 2. If,,op =.,low,op, hgh,hgh 3. If >, =., hgh Dynamc programmng s used o deermne he opmal schedule of each un over he scheduled me perod. More specfcally, for each sae n each hour, he on/off decson makng s needed o selec he lower cos by comparng he combnaon of he sar up cos and he accumulaed coss from wo hsorcal roues. A hour, he dual power calculaed by (9) and whn he lm n (8), wll be subsued n [ F ( ) + ST, ( -U, - )] U, U, μ r U, (20) Then, dynamc programmng searches for he opmal schedulng for each un o oban he lowes value of he erm T + {[ F ( ) ST, ( - U, - )] U, U, μ r U, } (2) = Subjec o mnmum up and down me consrans and generaon ramp lm n (8). 3.2 Inalzaon The nal values of Lagragan mulplers are very crcal o he LR soluon snce hey may preven LR from reachng he opmal soluon or requre a longer compuaonal me o reach one [8]. Dfferen nal values may also lead LR o dfferen soluons. In [4], he nal mulpler λ was se o he hourly sysem margnal cos of he schedule o sasfy he power balance consran and he nal mulpler μ was se o zero, leadng o an nfeasble nal soluon. On he oher hand, he

8 2-82 Inernaonal Energy Journal: Vol. 6, o., ar 2, June 2005 nal mulpler λ was se o he hourly sysem margnal cos of he schedule o sasfy boh he power balance and spnnng reserve consran, whereas he nal mulpler, μ was se o zero whch was generally lower han he opmal value μ [9]. Our nalzaon procedure nends o creae a hgh qualy feasble schedule n he frs eraon. The generang uns are sored n he ascendng order of full load average producon cos, FL avg (, max ). For each of he 24 hours, he un wh he leas FL avg (, max ) wll be commed one by one unl he power balance consran s sasfed. Subsequenly, economc dspach n each hour s (0) carred ou o oban he hourly equal lambda whch s nally se o Lagrangan mulplers λ. For he hours wh nsuffcen spnnng reserve requremen consderng he maxmum capacy consran load + R =,max U, 0, (22) More un(s) are needed o be commed o gve he nal feasble soluon. Ths s obaned by commng a un wh he leas FL avg (, max ) one by one unl he spnnng reserve s sasfed. ( 0 ) For each of he 24 hours, each nonnegave μ s deermned by he upper bound of zero on/off decson crera of he commed un as follows: μ χ + δ ( 0 ) = max[ [ F ( ) + λ ],0]. (23) T ( 0 ),max,up ( ) The nal μ 0 s deermned by he hghes μ ( 0 ) among he commed uns as: 0 (0) m ( ) (0) μ = max[ μ,...,μ ] (24) where, m s he margnal un wh he hghes FL avg (, max ) gvng he suffcen spnnng reserve a hour. 3.3 Adapve Updang of he Lagrangan mulpler In general, adjusng Lagrangan mulpler by subgraden mehod s no effcen n he presence of he spnnng reserve consran [20]. The LR performance s heavly dependen on he mehod used o updae he mulplers. In hs paper, he Lagrangan mulpler updae rule s o desgn he large sep sze a he begnnng of eraons and smaller as he eraon grows as proposed n [8]. The values of α and β are deermned heurscally [2]. Each nonnegave λ and μ are adapvely updaed by: ( k) ( k-) pdf = max[ +,0], (25) ( + k) norm( pdf )

9 Inernaonal Energy Journal: Vol. 6, o., ar 2, June where, pdf = load U,, (26) = T 2 norm ( pdf ) = ( pdf ) + ( pdf ) ( pdf ), (27) ( k) ( k-) rdf μ = max[ μ +,0], (28) ( + k) norm( rdf ) where, rdf = max( load + R,maxU,, R τ UR U, ), (29) = = T 2 = ( rdf ) + ( rdf ) +... ( rdf (30) norm ( rdf ) + ) α and β are dvded no hree cases dependng on he sgns of pdf and rdf as follows: Case : pdf 0 and rdf 0 : updang boh λ and μ by usng α = 0.02 and β = 0.05 Case 2: pdf < 0 and rdf 0 : updang boh λ and μ by usng α = 0.6, β = 0.3 Case 3: pdf < 0 and rdf > 0 : updang only μ by usng α = 0.02, β = 0.05 In fac, updang hese wo mulplers λ and μ n hour mus move hem n he same drecon. In hour, f pdf and rdf have he same sgns, eher posve or negave, λ and μ wll be updaed (ncrease or decrease) by (25) and (28) respecvely. When he oal dual generaon oupu s larger han he load n ha hour (pdf < 0) bu he spnnng reserve s nsuffcen (rdf > 0), more commed un(s) are requred o sasfy he spnnng reserve consrans. However, updang λ by (25), wll decrease s value, resulng n commng less uns. Therefore, when pdf < 0 and rdf >0, only μ wll be updaed. 3.4 Dynamc Economc Dspach The un rampng consran lnks he generaon oupu of he prevous hour o ha of he presen hour, hus nroducng a dynamc characersc n he economc dspach procedure known as dynamc economc dspach (DED). Snce forward economc dspach sars from he frs hour o he las hour, he resulng generaon level a he las commed hour may be hgher han he mnmum level before decommng hs un (shu down generaon level consran). Smlarly, backward economc dspach, sarng from he las hour o he frs hour may resul n he generaon level a he frs commed hour hgher han he mnmum level (sarup generaon level consran). If he economc dspach s nally performed a he maxmum demand hour as n [5], performng forward economc dspach for he subsequen hours and backward economc dspach for he prevous hours does no guaranee ha he sarup and shu down generaon level consrans wll be sasfed. Therefore, n hs paper, he new lm frame conssng of down and up frame s proposed for dynamc economc dspach. The down frame lm s used o guaranee ha he generaon oupu a he las commed hour wll be a s mnmum power oupu before decommng. On he oher hand, he up frame lm s used o guaranee ha he generaon oupu a he frs commed hour wll be a s mnmum power oupu.

10 2-84 Inernaonal Energy Journal: Vol. 6, o., ar 2, June 2005 A he las commed hour, he generaon oupu s se a he mnmum level. Thus, a he b hour pror o hs hour, he upper operang lm s se by a down frame lm, l DF,hgh, as shown n Fg. 5(a). mn[,max,,mn + DR 60 b ], f l = T,,2, b 0,andU, l b =, l b,hgh DF =, max, f U =, =,,T,, 0, f U = 0,, l b (3) Aferward, a lm frame s consruced sarng from he frs scheduled hour o he las scheduled hour. A he frs commed hour, he generaon oupu s se a he mnmum level. Thus, f a he subsequen hours, he upper operang lm s se by an up frame lm, F a,hgh, as shown n Fg. 5(b).,hgh f F a = mn[ DF,max f a,hgh,,mn + UR 60, f U =, =,,T,, a ], f U, =, f =,,T, and 0, f a a (32) A whole frame for un s shown n Fg. 6, he doed block represenng he unconsraned capacy, regardless ramp rae lm, whereas he bold lne represenng ramp rae lm frame. Afer he complee frame for each un s obaned, he economc dspach s performed nally a he hour ha corresponds o he maxmum sysem demand. Then, he dspachng process proceeds forward and backward n he oher me nervals. DF = l b,hgh,max DF l b,hgh = 0 b = 3 b = 2 b = b = 0,mn DF l b,hgh Fg. Fg. 5a. 5(a) A down An down frame frame for for un un l,max f a F,hgh,mn a = 0 a = a = 2 a = 3 DF l b,hgh f Fg. 5(b) An up frame for un

11 Inernaonal Energy Journal: Vol. 6, o., ar 2, June Unconsraned capacy,max F,hgh Fg. 6 A whole frame for un In hs paper, he un commmen schedule obaned from dynamc programmng descrbed n Secon 4. s checked for he feasbly for each hour by consderng he maxmum capacy consran (22), reserve rampng capacy consran and commed un consran R τ UR U, 0, (33) = =, low U, load. (34) The 5 mnue spnnng reserve response me consran (7) s needed o guaranee ha he sum of he reserve conrbuon by each un deermned by boh he dfference beween s capacy and curren generaon, and rampng capably s a leas suffcen o mee he spnnng reserve requremen whn 5 mnues. The maxmum capacy consran (22) forces commmen of suffcen generaon o mee he demand and reserve requremen. Whereas he reserve rampng capacy consran (33) s necessary o assure ha he commed generaors have suffcen reserve rampng capacy o sasfy he reserve requremens. To sasfy he 5 mnue spnnng reserve response me consran (7), boh maxmum capacy consran (22) and reserve rampng capacy consran (33) mus be sasfed. Because sasfyng only maxmum capacy consran (22), he spnnng reserve may be lmed by he rampng capably whereas sasfyng only reserve rampng capacy consran (33), he spnnng reserve may be lmed by he generaon capacy. The commed un consran (34) assures ha he demand can be me wh all dspach generaors loaded above her respecve lower lms. If he 24 hour schedule does no volae (22) and (33), hourly dynamc economc dspach by quadrac programmng s used o mnmze he oal producon cos subjec o power balance equaon and operang lm. Oherwse, curren Lagragan mulplers are no suable o oban a feasble schedule. Subjec o: F,ed ), ed = Mnmz e (, (35) (a) ower balance consran load =,ed U, = 0, (36)

12 2-86 Inernaonal Energy Journal: Vol. 6, o., ar 2, June 2005 (b) ew generaon ramp operang frame lm,ed low U,,ed,ed hgh U, =,,, (37), where,,ed hgh =, hgh, ed mn[ F, + UR 60], fu = U =,,mn,, f U = 0, U or U, U = 0, (38),, =,, =, +,ed low =, ed max[, DR 60], f U = U =,,mn,mn,, f U = 0, U or U, U = 0. (39),, =,, =, +,ed hgh,hgh n (38) and,ed low n (39) are dfferen from n (9) and n (0).,ed hgh uses he lm frame as he upper lm nsead of he maxmum real power generaon,, max whch s used for, hgh. In addon, he new generaon ramp operang frame lm s se o he mnmum real power generaon, whenever he un saus s changed eher from 0 o or from o 0., mn 3.5 Reparng Sraegy To mee he sysem load whou volang ramp up rae, backward economc dspach was used n [4]. However, backward economc dspach generally may end up wh an nfeasble soluon, volang 5 mnue response me spnnng reserve consran. Thus, forward and backward economc dspach sraeges were used o resolve he problem n []. Inally, he forward economc dspach s performed. If a feasble soluon can no be found from he forward pah, a backward economc dspach s needed o sasfy he 0 mnue response me spnnng reserve consran. Consrucng boh forward and backward pahs requre excessve compung requremen. Sarng a he maxmum sysem demand hour as n [5], performng forward economc dspach for he subsequen hours and backward economc dspach for he prevous hours does no guaranee ha he 0 mnue spnnng reserve consran wll be sasfed, especally for he sharp ncrease or decrease loadng level. In hs paper, reparng sraegy s proposed o releve he 5 mnue response me spnnng reserve consran. For each hour, afer he dynamc economc dspach n Secon 4.4 s performed and he generaon ed, power oupus, are obaned, he 5 mnue spnnng reserve response me consran n (7) s checked. If s volaed, he generaon power oupus wll be redspached. The commed uns are classfed no wo caegores, shown n Fg. 7. low, Caegory : Uns whose generaons can ncrease whou reducng her reserve conrbuon and her generaons sasfy condon,,max, ed τ UR. (40) Ther redspached oupu power mus no be less han her generaon dspached n Secon 4.4. Therefore, her new lower lm s se by her dspached power oupu,

13 Inernaonal Energy Journal: Vol. 6, o., ar 2, June Whereas he new upper lm s se by,,ed low =,ed, Ω. (4) > τ UR, = τ UR, Ω. (42) If,ed hgh (,max ),ed hgh (,max ) Caegory 2: Uns whose reserve conrbuon can be ncreased f her generaons are decreased and her generaons sasfy condon,,max, ed < τ UR. (43) Ther redspached oupu power mus no be more han her generaon dspached n Secon 4.4. Therefore, her new upper lm s se by her dspached power oupu, Whereas he new lower lm s se by,,ed hgh =,ed, Ω 2. (44) If,ed low < (,max UR),,ed low = (,max τ UR ), Ω 2 τ. (45),ed hgh,max,ed hgh,ed, max τ UR r,ed r =τ UR,ed low,ed low Caegory Caegory Caegory Caegory 2 2 Fg. 7 Classfcaon commed uns no caegory and 2 In hs paper, he reparng sraegy by quadrac programmng s proposed o mnmze he producon cos n (35) subjec o balance consran (36), operang lm (37), 5 mnue spnnng reserve response me consran (7) and addonal consrans,,ed = TΩ Ω,ed = TΩ 2 Ω 2 + ΔSR, (46) ΔSR, (47)

14 2-88 Inernaonal Energy Journal: Vol. 6, o., ar 2, June 2005 where, ΔSR = R = r U,. (48) 3.6 Soppng Crera The relave dualy gap s G J ([ U, ]) L(,U, λ, μ ) = (k), (49) L(,U, λ, μ ) where, from (6). T, ]) = [ F (,ed ) + ST, ( - U,- U, = = J ([ U )] and L (,U, λ, μ ) s calculaed The relave dualy gap s used o measure he soluon qualy, by checkng agans he soppng crera. The eraon process sops when eher he relave dualy gap s less han he specfed olerance or he eraon couner exceeds he maxmum allowable number of eraons. 3.7 Overall rocedure Sep : Inalze λ and μ descrbed n Secon 4.2. Sep 2: Inalze he ILR eraon couner, k =, and J B = $0 7. Sep 3: Solve he un subproblems by usng dynamc programmng descrbed n Secon 4.. Sep 4: If he dual soluon does no sasfy maxmum capacy consran (22), reserve rampng capacy consran (33), and commed un consran (34), go o Sep 0. Sep 5: Carry ou he dynamc ED by quadrac programmng descrbed n Secon 4.4. If 5 mnue spnnng reserve response me consran (7) s sasfed, go o Sep 7. Sep 6: erform reparng sraegy by quadrac programmng descrbed n Secon 4.5. (k ) Sep 7: Calculae he prmal cos J ([ U, ]), he dual cos L (,U, λ, μ ) and he relave dual gap G (k) as descrbed n Secon 4.6. (k ) (k ) Sep 8: If J ([ U, ]) < J B, J B = J ([ U, ]) and [ U, ] = [ U, ]. Sep 9: If he relave dual gap G (k) < ε, go o Sep. Sep 0: If k < K max, k = k +, updae Lagrangan mulpler adapvely as descrbed n Secon 4.3 and reurn o Sep 3. Sep : Termnae ILR. B k 4. UMERICAL RESULTS The ILR algorhm s esed on he IEEE relably sysem conssng of 26 generang uns [4], [22], and [23]. In he smulaon, he daly load, un daa are shown n Appendx. There are wo cases:

15 Inernaonal Energy Journal: Vol. 6, o., ar 2, June Case : Case 2: To compare he resuls wh hose n [4], he spnnng reserve s calculaed from he dfference beween he maxmum capacy and he curren generaon oupu of each un. The spnnng reserve requremen s se o he larges commed un wh respec o he consran n (22), neglecng he 5 mnue spnnng reserve response me consran n (7). The spnnng reserve s calculaed from un reserve conrbuon whn 5 mnues. The spnnng reserve s se o 4% of he oal load demand. In boh cases, here are wo load levels, load A and B. Load B s smaller han load A, hus here are more medum sze uns o sar up and shu down. ILR compuaonal mes are obaned from a penum IV,.6 GHz personal compuer. As shown n Table, he oal cos obaned from ILR s lower han A [4] n boh load levels. The CU mes may no be drecly comparable because he compuer used n [4] was no specfed. In addon, ILR could be used wh 5 mnue spnnng reserve response me consrans as shown n Table 2. Table roducon Cos n Case Load Mehod CU me (s) Cos($) Load A A [4] ,326.5 ILR ,996.9 Load B A [4] ,653.6 ILR ,6.5 Table 2 roducon Cos n Case 2 Load CU me (S) Cos($) Load A ,64.9 Load B , COCLUSIOS In hs paper, he proposed ILR s effcenly and effecvely mplemened o solve he RUC problem. ILR oal producon coss over he scheduled me horzon are less expensve han dynamc economc dspach based on arfcal neural nework. ILR s suable for RUC due o he subsanal producon cos savngs. 6. ACKOWLEDGEMETS Ths work s suppored n par by he Energy Conservaon romoon Fund under conrac 029/2545, Energy olcy and lannng Offce of Thaland. 7. OMECLATURE DH = he shu down me of un o decrease s generaon from, max o zero (h), DR = he ramp down rae lm of un (MW/mn),

16 2-90 Inernaonal Energy Journal: Vol. 6, o., ar 2, June 2005 l DF b,hgh = he down frame of un a b hours pror o he las commed hour l (MW), F ( ) = he generaor fuel cos funcon n a quadrac form, F ( ) = a ( ) 2 + b + c ($/h), f F a,hgh = he lm frame of un a a hours afer he frs commed hour f (MW), FL avg (, max ) = he full load average producon cos of un, F (,max ) /, max ($/MWh), G (k) = he relave dualy gap a eraon k, J B = he bes oal economc dspach producon cos reached ($), ( k ) J([U, ]) = he oal economc dspach producon cos a eraon k ($), k = he ILR eraon couner, K max = he maxmum allowable number of eraons, = he oal number of generaor uns, T,down = he new mnmum down me of un (h),,hgh = he hghes possble power oupu of un a hour (MW),, low = he lowes possble power oupu of un a hour (MW),,ed hgh = he hghes possble dspached power oupu of un a hour (MW),,ed low = he lowes possble dspached power oupu of un a hour (MW),,mn = he mnmum real power generaon of un (MW),,max = he maxmum real power generaon of un (MW) = he generaon oupu power of un a hour (MW),, ed = he economc dspach generaon oupu of un a hour (MW), load = he load demand a hour (MW), r = he un reserve conrbuon a hour, mn[, τ UR ] (MW), R = he spnnng reserve a hour (MW), ST, = he sarup cos of un a hour ($), T = he oal number of hours, T,down = he mnmum down me of un (h),, on,max T = he connuously on me of un up o hour -, T,off = he connuously off me of un up o hour -, T,SR = he me for un o ncrease s generaon from zero o, mn (h), T,SDR = he me for un o decrease s generaon from, mn o zero (h), T,up = he mnmum up me of un (h), TΩ = he oal generaon oupu of uns n caegory a hour (MW),

17 Inernaonal Energy Journal: Vol. 6, o., ar 2, June TΩ 2 = he oal generaon oupu of uns n caegory 2 a hour (MW), a = he number of hours afer he frs commed hour, b = he number of hours pror o he las commed hour, f = he frs commed hour, l = he las commed hour, U, = he saus of un a hour (on =, off = 0), UH = he sarup me of un o ncrease s generaon from zero o, max (h), UR = he ramp up rae lm of un (MW/mn), B, [ U ] = he bes feasble soluon reached, Δ SR = he defc spnnng reserve a hour (MW), ε = he relave dualy gap olerance, Ω = he se of commed un classfed n caegory, Ω 2 = he se of commed un classfed n caegory 2, τ = he reserve response me frame (.e., 0-5 mn), ( 0 ) ( 0 ) λ, ( k ) ( k ) λ, μ = he nal Lagrange mulplers a hour ($/MWh, $/MW), μ = he Lagrange mulplers a hour a eraon k ($/MWh, $/MW), and (0) μ = he nal Lagrangan mulpler of un a hour ($/MW), 9. REFERECES [] Wood, A. J. and Wollenberg, B. F ower generaon, operaon & conrol. 2 nd ed., ew York: John Wley&Sons. [2] Dllon, T. S.; Edwn, K. W.; Kochs, H. D.; and Taud, R. J Ineger programmng approach o he problem of opmal un commmen wh probablsc reserve deermnaon. IEEE Trans. ower Apparaus and Sysems AS-97(6): [3] Hobbs, W. J.; Hermon, G.; Warner, S.; and Sheble, G. B An enhanced dynamc programmng approach for un commmen. IEEE Trans. ower Sysems 3(3): [4] Wang, C. and Shahdehpour, S. M Effecs of ramp rae lms on un commmen and economc dspach. IEEE Trans. ower Sysems 8(3): [5] Ruzc, S. and Rajakovc,. 99. A new approach for solvng exended un commmen problem. IEEE Trans. ower Sysems 6(): [6] Wang, C. and Shahdehpour, S. M Ramp rae lms n un commmen and economc dspach ncorporang roor fague effec. IEEE Trans. ower Sysems 9(3): [7] Wang, C. and Shahdehpour, S. M Opmal generaon schedulng wh rampng coss. IEEE Trans. ower Sysems 0(): [8] eerson, W. L. and Brammer, S. R A capacy based Lagrangan relaxaon un commmen wh ramp rae consrans. IEEE Trans. on ower Sysems 0(2):

18 2-92 Inernaonal Energy Journal: Vol. 6, o., ar 2, June 2005 [9] Svoboda, A.J.; Tseng, C. L.; L, C.; and Johnson, R. B Shor erm resource schedulng wh ramp consrans. IEEE Trans. ower Sysems 2(): [0] La, S. Y. and Baldck, R Un commmen wh ramp mulplers. IEEE Trans. ower Sysems 4(): [] Wu, H. and Goo, H. B Opmal schedulng of spnnng reserve wh ramp consrans. In roceedng of IEEE ower Engneerng Socey Wner Meeng. 2: ew York, U.S.A., 3 January - 4 February. [2] Goo, H. B.; Mendes, D..; Bell, K. R. W.; and Krchen, D. S Opmal schedulng of spnnng reserve. IEEE Trans. ower Sysems 4(4): [3] Bau, J. and Renaud, A Daly generaon schedulng opmzaon wh ransmsson consrans: a new class of algorhms. IEEE Trans. ower Sysems 7(3): [4] Abdul-Rahman, K. H.; Shahdehpour, S. M.; Aganac, M.; and Mokhar, S A praccal resource schedulng wh OF consrans. IEEE Trans. ower Sysems (): [5] Ma, H. and Shahdehpour, S. M Un commmen wh ransmsson secury and volage consrans. IEEE Trans. ower Sysems 4(2): [6] Wang, M.; Zhang, B.; and Deng, Y A novel un commmen mehod consderng varous operaon consrans. In roceedngs of IEEE ower Engneerng Socey Wner Meeng January, Vol. 3. [7] Bhaacharya, K.; Bollen, M. H. J.; and Daalder, J. E Operaon of resrucured power sysems: Kluwer Academc ublshers. [8] Dekrajangpech, S.; Sheble, G. B.; and Conejo, A. J Aucon mplemenaon problems usng Lagrangan relaxaon. IEEE Trans. ower Sysems 4(): [9] Yan, H.; Luh,. B.; Guan, X.; and Rogan,.M Schedulng of hydrohermal power sysems. IEEE Trans. ower Sysems 8(3): [20] Merln, A. and Sandrn, A new mehod for un commmen a Elecrce De France. IEEE Trans. ower Apparaus and Sysems as-02(5): [2] Fsher, M. L. 98. The Lagrangan relaxaon mehod for solvng neger programmng problems. Managemen Scence 27: -8. [22] IEEE RTS Task Force of AM Subcommee 979. IEEE Relably Tes Sysem. IEEE Trans. ower Apparaus Sysems as-98(6): [23] IEEE relably es sysem ask force The IEEE Relably Tes Sysem IEEE Trans. ower Sysems 4(3):

19 Inernaonal Energy Journal: Vol. 6, o., ar 2, June AEDIX The daa were obaned from [4], [22], and [23]. The load and un daa are summarzed n Tables 3, 4 and 5. Table 3 The Daly Load A Hour Load (MW) Hour Load (MW) Hour Load (MW) Hour Load (MW) Table 4 The Daly Load B Hour Load (MW) Hour Load (MW) Hour Load (MW) Hour Load (MW)

20 2-94 Inernaonal Energy Journal: Vol. 6, o., ar 2, June 2005 Un, mn (MW),max (MW) c (k$) b (k$/mw) a (k$/mw 2 ) Table 5 Un Daa T,up (h) T,down (h) In. Sa. (h) UH (h) DH (h) UR (MW/ mn) DR (MW/ mn) T,down (h) χ ($) δ ($) γ (h)