LMP in Single Auction Model of a Restructured Power System using Bat Algorithm

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1 LM n Sngle Aucon Model of a Resrucured ower Sysem usng a Algorhm Mural Macha, Maam Salaja Kumar, Member, IEEE, and 3 Maheswarapu Sydulu, Member, IEEE Deparmen of Elecrcal Engneerng,.I.S.T rahmapur, Odsha, Inda,3 Deparmen of Elecrcal Engneerng,.I.T Warangal, Telangana, Inda mural33.nw@gmal.com Absrac In resrucured elecrcy mares, an effecve ransmsson prcng mehod s requred o address ransmsson ssues and o generae correc economc sgnals. Transmsson lne consrans can resul n varaons n energy prces hroughou he newor. These prces depend on generaor bds, load levels and ransmsson newor lmaons. A congeson charge s ncurred when he sysem s consraned by physcal lms. Locaonal Margnal rcng (LM) has become popular mehod n resrucured power mares o address he congeson charges. Ths paper presens LM compuaon wh L, GA and proposed a algorhm (A) under hree dfferen cases n sngle aucon model. In sngle aucon model only supplers subm he bds and he load s assumed o be nelasc. xed and Lnear bds are consdered for generaors. All he mehods are esed on IEEE 4 bus sysem, ew England 39 bus sysem and Indan 75 bus sysem. Lnear bds wh A show mos opmal fuel cos compared o all oher mehods. Keywords Delvery facors; xed bd; Generaon shf facors; Genec Algorhm; Lnear bd; a algorhm I. ITRODUCTIO In Aprl 003 Whe aper [3] he U.S. ederal Energy Regulaory Commsson (ERC) proposed a mare desgn for common adopon by U.S. wholesale power mares. The elecrc power ndusry has undergone deregulaon around he world, a core ene of whch s o buld an open-access, unambguous and far elecrcy mares. roper and far prcng of real power s an mporan ssue n hs compeve mare. Core feaures of a mare desgn nclude; a wo selemen sysem conssng of a day-ahead mare suppored by a real-me mare o ensure connual balancng of supply and demand for power; and grd congeson managemen by means of nodal prcng. Under a deregulaed elecrcy mare envronmen, ransmsson newors hold a val role n supporng he ransacon beween producers and consumers. One drawbac of ransmsson consran s congeson. Congeson occurs when ransmsson lnes or ransformers operae a or above s hermal lms and hs prevens he sysem operaors from dspachng addonal power from a specfc generaor. Congeson can resul n an overall ncrease n he cos of power delvery. resenly here are wo prcng srucures [] ha are beng used n a compeve energy mare o accoun for congeson: he unform prcng mehod.e., mare clearng prce (MC) and he nonunform prcng mehod.e., LM. In he frs mehod, all generaors are pad he same prce (MC) based on he bd of he margnal generaor ha would be dspached n he absence of congeson. The second mehod /4/$ IEEE has been he basc approach n power mares o calculae he LMs and o manage ransmsson congeson. LM s he addonal cos for provdng one addonal MW a ceran node. uyers pay ISO, based on her nodal prce for he dspached energy. The ISO pays sellers, based on her respecve nodal prces. The dfference n LMs beween wo adjacen buses s he congeson cos whch arses when he energy s ransferred from one locaon (njecon) o he oher locaon (whdrawal). Margnal es represen ncremenal changes n sysem es due o ncremenal demand changes. Incremenal es yeld addonal coss whch are referred o as he cos of margnal es [5]. Thus LM s he summaon of he coss of margnal energy, margnal es and congeson. Therefore LM a bus s saed as follows: energy cong LM LM + LM + LM () In hs decomposon model, LM congeson componen a bus,.e., LM cong remans nvaran w.r.o dfferen reference buses, and he combnaon of he oher wo componens,.e., LM energy + LM, s also referencendependen. u each of LM energy or LM s sll reference-dependen []. In mare plannng and smulaon, DC model s desred due o s robusness and speed. DCO s broadly employed by a number of ndusral LM smulaors, such as A s Gd VewTM, GE s MASTM, Semens romod IVR, and ower World [4], []. Several papers have repored dfferen models for nodal prce calculaon. Reference [5] presened dfferen mehods and properes on LM calculaons based on DCO wh and whou. Refs. [7], [9] gves a sysemac descrpon on how he LM s are produced; also descrbed boh he modellng and mplemenaon challenges and soluons. Reference [8] descrbed ACO based LM calculaon consderng dsrbued. Reference [6] demonsraed an erave DCO based algorhm wh less model, consderng margnal es, and wh fcous nodal demand model o calculae nodal prce. All hese 3 models are solved wh lnear programmng. In hs paper a dealed explanaon abou whou, concenraed, and dsrbued cases for LM compuaon s gven. In whou case, sysem es are gnored. In concenraed case, oal sysem s suppled by he slac bus whch creaes a burden on he slac bus. To elmnae a large msmach a he slac bus, s dsrbued o all buses as an exra load n dsrbued case. In hs paper DCO s used for all he hree cases and s solved wh L approach, GA, and wh proposed a

2 algorhm. In [6] L approach s used for all he hree cases wh pecewse lnear cos curves bu does no gve acual margnal cos of generaon. To address hs drawbac, lnear bds are used under hree cases wh GA and A. The lnear bd curve nroduces non-lneares n he problem, however, s a more realsc represenaon of prce han ha of a fxed prce for power. The resuls of A are compared wh L and GA for all he 3 cases. Secon II presens sngle aucon model. Secon III dscusses he problem formulaon for LM calculaon usng delvery facors for 3 cases. Secon IV presens mplemenaon of GA and proposed a algorhm. Secon V gves a case sudy of Indan 75 bus sysem []. Secon VI presens resuls and dscusson. Secon VII concludes he paper. II. SIGLE AUCTIO MODEL In hs model, he consumer power demand s nsensve o varaon n prces and remans consan as shown n g.. The consumer s ready o accep a specfed amoun of power a he prevalng mare prce. The major par of he Dscom s demand bd s prce ang, whch s requred o mee s essenal daly servces o resdenal, domesc, and ndusral loads. g.. Sngle aucon model In he g., when here s no congeson n he sysem he aggregae suppler margnal cos curve and consumer nelasc demand nersecs a a pon λ whch s called Mare Clearng rce. Suppler surplus exss. When congeson occurs, due o addon of congeson prce and prce MC becomes LM leadng o merchandsng surplus or ISO surplus o ISO. Then he oal socal surplus s he sum of suppler surplus and ISO surplus. III. ROLEM ORMULATIO OR LM CALCULATIO In hs paper for compuaon of LMs, DCO problem s solved wh L fxed bds, GA fxed and lnear bds, and a algorhm lnear bds of generaors under all he hree cases. Acve power generaons of he generaors excep slac generaon are consdered as varables for opmzaon. The obaned G s are used n calculaon of LM for he congesed ransmsson sysem. Generaon Shf facors (GS) have been used for he calculaon of ransmsson lne flows. Delvery facors (D) a buses have been used o nclude he mpac of margnal es on nodal prce. A. Generaon Shf acor Generaon shf facor s he rao of change n power flow of lne o change n njecon of power a bus. GS coeffcen can be compued as ( ( a, ) ( b, ) ) GS () X where - nverse of (he magnary par of Y bus marx) X reacance of lne a s sendng bus and b s recevng bus of lne. Delvery acor The ey o consder margnal prce s he margnal facor, or jus facor (L) for smplcy, and he margnal delvery facor, or jus delvery facor (D). Mahemacally, hey can be wren as I s defned as D L (3) where L facor a bus M GS R where lne flow of lne, R ressance of lne, G - D njecon a bus GS - generaon shf facor o lne from bus. The facor (L ) a he h bus may be vewed as he change of oal sysem wh respec o a MW ncrease n njecon a ha bus. C. LM Calculaon wh Dfferen Loss Cases Case. Whou Losses usng DCO In hs mehod he objecve funcon s mnmzaon of oal margnal producon cos subjeced o power balance and lne flow consrans. LMs are calculaed from he obaned generaor power oupus. ISO paymens o generaors, Generaor prof, load paymen o ISO, ISO prof and sysem socal surplus are also calculaed. The objecve funcon s Mnmze J s.. G mn G G D C s) (4) (5) ( (6) lm,, M (8) max G,,, (9) where s number of buses, M s number of lnes, C (s) s cos funcon of he generaor.e. ( C ( S) b G + c ) n $/hr, G G s oupu power of generaor a bus (MW), D s he demand a bus (MW), s he lne flow of lne, lm s he hermal lm of lne. (7)

3 Case. Wh Concenraed Loss usng DCO In hs mehod also he man objecve s mnmzaon of oal margnal producon cos subjeced o energy balance and lne flow consrans. However, n LM based elecrcy mares, sysem margnal es have sgnfcan mpac on he economcs of power sysem operaon. So, sysem margnal es have o be aen no accoun for obanng accurae prces. In hs model s assumed ha oal sysem s suppled by slac bus generaor. The problem s o Mnmze J C ( s) (0) s.. D ( ) + 0 () mn G G lm,, M (),,, (3) max G where s he oal sysem. n () s used o offse he doubled average sysem caused by he margnal facor (L) and he margnal delvery facor (D). Afer geng power oupus of generaors for he above dspach, slac bus power s calculaed usng (7) or () and he prce a he reference (slac) bus needs o be calculaed by subsung slac bus power eher n fxed bds or lnear bds. A he reference bus, boh prce and congeson prce are always zero. Therefore, he prce a he reference bus s equal o he energy componen. The decomposon of LM s shown here LM prce a he reference bus (4) energy λ M cong LM GS μ (5) where μ s he consran cos of lne, defned as: change n oal cos μ change n consra n' s flow LM λ ( D ) (6) (LM 0 for less power sysem) Afer calculang he ndvdual componens of LM usng (4-6), oal LM a each bus s calculaed usng (). Case 3. Wh Dsrbued Loss usng DCO Concenraed model addresses he margnal prce hrough he delvery facors. However, he lne flow consran n () sll assumes a less newor. u he equaly consran n () nforms ha oal generaon s greaer han he oal demand by he average sysem. Ths causes a msmach a slac bus and hs msmach s absorbed by he sysem slac bus. If sysem demand s huge e.g., a few GW hen he sysem may be of he order of hundreds of MW and may no be possble o add all he es o slac bus. To address hs ssue, s necessary ha he lne es are o be shared among he buses as exra loads. Ths paper employs he concep of dsrbued, o represen he es of he lnes conneced o a bus. In hs mehod sysem es are dsrbued among all he buses and elmnae he large msmach a he reference bus. In hs approach, n each ransmsson lne s dvded no wo equal halves, and each half s added o respecve end buses of ha lne as an exra load. So for each bus, he oal exra load s he sum of halves of lne es whch are conneced o ha bus. The exra load a bus s assumed as E, and s defned as follows: E M R where M s number of lnes conneced o bus. The lne flow for hs case s calculaed as n (8). GS ( G D E ) (8) The problem formulaon for hs case s same as n case. Afer geng power oupus of generaors, LMs a all buses are compued. Wh hs approach, he fuel cos can be furher reduced han he concenraed case and he burden on he slac bus can be elmnaed. IV. IMLEMETATIO O GA AD AT ALGORITHM A. a Algorhm (A) The basc ba algorhm (A) developed by Xn-She Yang [3]. Ths algorhm s he combnaon of SO and Harmony search algorhms. or he bas n smulaons, he followng equaons defne how her posons x and veloces v n a d- dmensonal search space are updaed. The new soluons x and veloces v a me sep are gven by f + f + ( f fmn )β (9) mn max * v v + ( x x ) f x x + v + (0) () where f s he h ba frequency, β [0,] s a random vecor drawn from a unform dsrbuon. Here x * s he curren global bes locaon (soluon) whch s locaed afer comparng all he soluons among all he n bas a each eraon. As he produc λ f s he velocy ncremen, adjus he velocy change by usng f or (λ ) whle fxng he oher facor λ (or f ), dependng on he ype of he problem of neres. The presen wor uses f mn 0 and f max (dependng on he doman sze of he problem of neres). Inally, each ba s randomly assgned a frequency whch s drawn unformly from [f mn, f max ]. or he local search par, once a soluon s seleced from among he curren bes soluons, a new soluon for each ba s generaed locally usng random wal. * x x + (0.0 rand) where x s he curren poson of soluon, (7) () * x s he curren bes soluon. The n deph deals and pseudo code of a algorhm are gven n [3].

4 . Algorhm forlm Compuaon usng GA and A ) Usng GA Sep : Read no. of buses, no. of lnes, slac bus number, and us daa. Read GA parameers le populaon sze, chromosome lengh, no. of uns, maxmum no. of generaons, elsm probably, crossover probably, muaon probably, and epslon. Read a, b, c coeffcens; mn. and max. lms of generaors. Read lne daa ncludng lne hermal lms. Sep : Generae randomly power generaons of all generaors excep slac generaor and decode hem. Sep 3: Calculae Generaon shf facors usng (). Sep 4: Calculae nal lne flows usng (5). Sep 5: Calculae he sysem.e., n each lne usng (4) for case and case3. Sep 6: or case 3, calculae he exra load a each bus usng (7) from nal lne flows, and hen calculae new lne flows usng (8). Sep 7: Calculae facors and hen delvery facors a each bus usng (3). Sep 8: Calculae gen of slac bus usng (7) of case or () for cases and 3. Sep 9: Chec for lne flow lms usng (8). If he lne lms are volaed add penales o objecve funcon. Sep 0: Chec for slac bus power lms usng (9). If volaes he lms add penales o objecve funcon. Sep : Calculae he margnal fuel coss of all uns wh he randomly generaed G s; calculae he oal fuel cos usng (6) for all cases and hen calculae he fness funcon 00/ (+ objecve funcon+ penales). Sep : Sor he chromosomes n he descendng order of fness. Sep 3: Is eraon max. no. of eraons. If yes go o sep 8 else go o sep 4. Sep 4: If fness () fness (psze) problem converged. If no go o sep 7. Calculae he energy prce of he reference bus eher wh fxed bds or wh lnear bds and hen calculae he decomposon of LM usng (4), (5) and (6). Sep 5: Calculae ISO paymen o generaors (generaon a bus * LM a bus ) and hen calculae suppler surplus (ISO paymen o generaor - fuel cos of generaor). Sep 6: Calculae load paymens o ISO (demand a bus j * LM a bus j), and ISO surplus (oal load paymen o ISO oal ISO paymen o generaors) for all he 3 cases, and hen fnally calculae socal surplus (suppler surplus + ISO surplus) n 3 cases & STO. Sep 7: Use selecon, crossover and muaon operaors. Generae new populaon. eraon eraon +; Go o sep 4. Sep 8: STO and prn maxmum number of eraons reached. ) Usng A Usng A, he algorhm mahemacally s smlar o GA excep he A operaors used. Due o space consran a algorhm for LM compuaon canno be presened here. V. CASE STUDY O IDIA 75 US SYSTEM Tes sysems used are IEEE 4 bus, ew England 39 bus and Indan 75 bus sysems o compue LMs under all he 3 cases wh L, GA and A. Only Indan 75 bus sysem [] (Uar radesh Sae Elecrcy oard daa.e., 400V, 0V and 3V grd daa) resuls are presened n hs paper whch has 5 generaors, 7 lnes and 4 ransformers. GA parameers used are opulaon sze: 40, umber of bs for each generaor n he chromosome:, Elsm probably: 0.5, Crossover probably: 0.85, Muaon probably: 0.0, Tolerance: A parameers used are opulaon sze 5 (0-5), Loudness parameer (A) 0.5, Maxmum eraons 500, ulse rae (r) 0.5, requency mn 0, requency max 0.0, Alpha 0.9, Gamma 0.9. In hs wor for L approach, OWER WORLD SIMULATOR s used. or base case loadng, 9h lne connecng 4-8 buses congesed n he drecon 4 8 for all he hree cases wh L, GA and A. The correspondng resuls are lsed n able I. The LMs a all buses are also calculaed. The fuel cos comparson graphs for cases, and 3 s shown n g.. gs. 3, 4 and 5 show he LMs comparson for cases, and 3. LM comparson graphs mply ha due o congeson n he lne 4 8, LM a 4h bus (.e., source node of congesed lne and also havng generaor) undergoes a large varaon n all he mehods for all he 3 cases. g. 6 shows he convergence characerscs of GA and A. Ths case sudy hghlghed ha A approach wh lnear bds of generaors mnmzes he oal fuel cos of he sysem and mproves he socal surplus of he sysem. Slac generaor power n dsrbued model s comparavely reduced han n concenraed model. The consran cos or shadow prce of congesed lne s 3.35 $/MWh for case, 3.67 $/MWh for case and 3.7 $/MWh for case 3, whch are obaned from ower World Smulaon. rom he able I can be observed ha n case, wh ba lnear bds, slac bus generaon volaed he maxmum lm whch s he major drawbac of case poned ou n hs paper. In case 3, wh ba lnear bds, slac bus generaon ses whn s lms. In g. 3 wh all he mehods, he LMs a all he buses excep a he 4h bus are same due o he absence of prce n he LM componens. In gs. 4 and 5 usng all he mehods, LMs slghly vary a he buses excep a 4h bus due o he presence of prce n LM. g.. uel cos comparson graph for all he hree cases of Indan 75 bus sysem

5 TALE I. ACTIVE OWER GEERATIOS O GEERATORS OR IDIA 75 US SYSTEM I 3 CASES Generaor bus o. (slac bus) L- CASE CASE CASE 3 GA approach GA approach GA approach L A-L L- L A-L L- L VI. RESULTS AD DISCUSSIOS rom able I can be observed ha, ISO profs are negave n some cases. Ths s because, he objecve funcon for hs problem s fuel cos mnmzaon and n he process of mnmzng he fuel cos alone, he oher parameers le generaor prof, ISO prof and socal surplus are found o be negave. These negave values can be removed, when consumers are modeled as elasc consumers whle modfyng he objecve funcon as socal welfare maxmzaon for LM calculaon and can be consdered as a fuure scope for hs wor. rom g. s observed ha, A-L gves he mos opmal fuel cos. rom able I s observed ha, wh A-L socal surplus s mproved han compared o oher (lm volaed) A-L Sysem Loss (MW) uel cos ISO paymen o Generaors Generaor prof Load paymen o ISO ISO prof Socal surplus eraons CU me (sec) mehods under all he 3 cases. rom able I can also be observed ha, number of eraons aen by A s less han GA o converge he problem and hence he CU me wh A s also reduced han wh GA. The advanage of usng lnear bds over fxed bds s ha, recovers rue margnal cos of slac generaor or energy prce for deermnng LM. As menoned n case 3, n concenraed case s no feasble o add oal sysem o slac generaor whch may creae a burden on slac generaor or slac generaor may volae s lm. So dsrbued case s suggesed, n whch oal sysem s dsrbued among all he generaors, and hence burden on slac generaor s removed. I was observed ha, slac bus generaon n dsrbued case s reduced han n concenraed case for he suded es sysem.

6 g.3. LMs comparson for case of Indan 75 bus sysem g.4. LMs comparson for case of Indan 75 bus sysem g.5. LMs comparson for case 3 of Indan 75 bus sysem g.6. Convergence characerscs comparson of GA and A for Indan 75 bus sysem rom g. 6 can be observed ha, a algorhm aes less eraons o converge han GA. Smlar nd of resuls are obaned for IEEE 4 bus and ew England 39 bus sysems also. u due o space consran he resuls of 4 and 39 bus sysems are no presened n hs paper. VII. COCLUSIOS Ths paper presens a smple ransmsson prcng scheme.e., Locaonal Margnal rcng usng DCO o calculae he ransmsson varable cos consderng hree dfferen cases n a pool ype resrucured elecrcy mare. Under all he 3 cases, LMs are evaluaed a all he buses n a power sysem. To solve he DCO problem under each case, L, GA and a algorhm are used. xed bds of generaors are used wh L and GA. Lnear bds of generaors are used wh GA and A. Inally comparson s made beween L and GA. I s observed ha GA-L gves he mos opmal fuel cos n hs comparson. ex for furher opmzaon of fuel cos A-L s used. or all he used es sysems, he slac bus generaor power wh dsrbued approach s reduced han wh he concenraed approach and burden on he slac generaor s removed. I s observed ha consderable savngs n oal fuel cos of generaors s acheved wh A- L under all he 3 cases. LMs wh lnear bds are calculaed o avod he non smooh naure of bd curve n fxed bds. In all he used es sysems, he dsrbued wh A-L approach for LM calculaon shows relable resuls wh opmzed fuel cos values. REERECES [] R.E.Clayon, R.Muerj, Sysem plannng ools for he compeve mares, IEEE Compu. Appl. ower, vol.9, no.3, pp50-55, Jul.996. [] H.Sngh, S.Hao, A.apalexopoulos, Transmsson congeson managemen n compeve elecrcy mares, IEEE Trans. ower Sys., vol. 3, pp , May 998. [3] U.S. ederal Energy Regulaory Commsson, oce of Whe aper, Aprl 8, 003. [4] J.Yang,.L, L.A.A.reeman, A mare smulaon program for he sandard mare desgn and generaon/ransmsson plannng, n roc, IEEE ower Eng. Soc. General Meeng, 003, pp [5] Yong u, Zuy L, Dfferen models and properes on LM calculaons, IEEE ower Eng. Soc. General Meeng, 006. [6] angxng L, Ru o, DCO-based LM Smulaon: Algorhm, Comparson wh ACO, and Sensvy, IEEE Trans. ower Sys., vol., o.4, ovember 007. [7] E.Lvnov, Desgn and operaon of he locaonal margnal prcesbased elecrcy mares, IET Gener. Transm. Dsrb., 00, vol.4, Iss., pp [8] Zechun Hu, Haozhong cheng, urong L, An erave LM calculaon mehod consderng dsrbuons, IEEE Trans. ower. Syem., vol.5, o.3, Augus 00. [9]. C. Schweppe, M. C. Caramans, and R. D. Tabors and R.E.ohn, Spo rcng of Elecrcy. oson, MA: Kluwer, 988. [0]. J. Rng, "Dspach based prcng n decenralsed power sysems," h.d. dsseraon, Unv. Canerbury, Chrschurch, ew Zealand, 995. [] Indan 75 bus Sysem daa (Uar radesh Sae Grd daa) obaned from IIT Kanpur". [] angxng L, ully reference-ndependen LM decomposon usng reference-ndependen facors, Elecrc ower Sysems Research, ELSEVIER, vol.8, pp , 0. [3] Xn-She Yang, aure-inspred Meaheursc Algorhms, Uned Kngdom: nd ed, Lunver ress, 00. [4] M.Mural, M.Salaja Kumar, M.Sydulu, A Genec Algorhm based Secury Consraned Economc dspach approach for LM calculaon, Inernaonal Journal of Energy Scence, vol.3, no., pp.6-6, 03. [5] M.Mural, M.Salaja Kumar, M.Sydulu, Esmaon of Locaonal Margnal rce n a Resrucured Elecrcy Mare wh Dfferen Loss Cases usng Seed Genec Algorhm, Araban Journal for Scence and Engneerng, SRIGER, vol.39, no., pp , 04.