Transmission cost allocation in pool systems.

Transcription

1 Tranmiin ct allcatin in pl ytem. Juli Uala Univeridad Carl III de Madrid Legané (Madrid), Spain Abtract The tranmiin ct allcatin prblem may be divided int three different ubprblem: the flw allcatin t tranactin, the definitin f thee tranactin, and the ct allcatin baed n the flw allcatin. The flw allcatin ubprblem i lved here uing a differential, lack invariant methd. The lat ne can be given many different lutin, but each chice ha different cnequence. Thee cnequence are analyzed fr the IEEE-RTS96 tet ytem, and can be eaily extraplated t real netwrk. Keywrd - tranmiin ct allcatin, differential methd. 1 Intrductin Tranmiin ct allcatin methd are f increaing imprtance in the deregulating prce that i taking place in the Eurpean Unin and elewhere. They are pecially intereting in the ncming Eurpean Internal Electricity Market, ince they mut be applied t quantify the cmpenatin fr Cr Brder Trade (CBT) f electricity frm the ue that a freign tranactin make f a natinal grid. Althugh thi paper de nt deal directly with thi prblem, the cncluin here exped ha direct applicatin t it. A cmmnly agreed feature that a tranmiin ct allcatin methd mut have i t prvide lcatinal ignal and incentive in rder t encurage efficient ue f the tranmiin facilitie. They al mut cmply with me cnditin, namely t avid cr-ubidie, t be tranparent and eay t implement, t enure ct recvery, t prvide adequate ecnmic ignal and t have cntinuity with time [1]. The ct allcatin methd prped in literature culd be claified a embedded ct methd and marginal ct methd. The latter, hwever, d nt guarantee ct recvering in real netwrk [2]. Embedded ct methd, n the ther hand, allcate the tranmiin ct accrding t the extent f ue f generatr and cnumer. Several methd f thi kind have been prped and are in ue in different ytem. They can be divided int rlled-in methd and lad flw baed methd. Rlled-in methd charge a fixed amunt per energy unit, and their main drawback i that they ignre actual netwrk ue and that they d nt end adequate ecnmic ignal t grid uer. Flw-baed methd, n the ther hand, charge the uer in prprtin t the ue they make f grid facilitie. Sme prped methd f thi kind may be claified a prprtinal [3], r differential methd [4]. The prprtinal methd ha everal advantage: it i imple t undertand and prvide everal reult uch a l allcatin, grid ue and lad haring amng generatr. Hwever, althugh beginning frm a lved lad flw, it de nt fllw the Vltage Kirchhff Law in the allcatin prce, ignre the cunterflw, and the reult eem t be t vlatile. Differential (r incremental) methd are, n the ther hand, well knwn in literature and are baed n the enitivitie f branch flw t pwer injectin in nde. Thee enitivitie, hwever, depend n the chice f the lack bu in the tudied cae, and, therefre, there i a part f arbitrarine in the allcatin. One methd fr vercming thi difficulty, i given in [4]. Thi methd make ue f the well knw prperty f the invariance f thee enitivitie t a tranactin when uing DC lad flw equatin. It allcate tranmiin ct t tranactin, btaining participatin factr in the tranmiin netwrk fr each ne f thee tranactin, and auming unnecearily that the generatr nde i a lack bu in each tranactin. Uually, the prped allcatin methd eem linked t a definitin f pwer exchange in the ytem. Fr intance, [4] prpe the Equivalent Bilateral Exchange principle, r the haring f the lad prprtinally t the pwer prvided by each generatr. Al, claical differential methd aume an exchange between each nde and the lack. In [3], the allcatin f ct and the definitin f the exchange are al made fllwing the Prprtinal Sharing Principle (PSP). In thi paper, hwever, the prblem f tranmiin grid ct allcatin t tranactin are divided int three ubprblem, that can be addreed independently. Firt, the definitin f tranactin. Secnd, the allcatin f pwer flw thrugh branche t each tranactin, and finally, the ct allcatin t the already allcated flw. Different lutin can be given t all f them. Thi lat prblem i given pecial imprtance in thi paper, becaue it ha a great influence in the reult. The allcatin i finally lved auming an equal hare f the grid ct fr generatr and lad. The definitin f the tranactin i traightfrward fr market baed n bilateral cntract, but in pl rganized market a definitin f tranactin mut be made. Thi definitin ha a great influence n the reult, and thi tpic i duly addreed in the paper. Tw principle fr defining thee tranactin in a pl baed ytem that have been prped in literature, the Prprtinal Sharing Principle (PSP) and the Equivalent Bilateral Exchange (EBE), are examined in thi paper and the cnequence f thi chice analyzed. It mut be recalled that when uing differential 15th PSCC, Liege, Augut 25 Sein 4, Paper 1, Page 1

2 methd applied directly t nde, a tranactin i tacitly aumed between each nde and the lack nde. The allcatin f flw t tranactin i made frm a lved lad flw f the tudied grid. Thi lad flw may be repreentative f a certain lad and generatin pattern. The methd culd be applied t all the hur f the year in rder t quantify the amunt f ue that a tranactin ha made f the tranmiin grid. The allcatin i made t tranactin, lking fr the invariance prperty f the allcatin t the lack bu chice made in the initial lad flw. Unlike in [4], the AC lad flw equatin are ued. Thi require me additinal cnideratin fr achieving thi invariance. The ct allcatin prblem ha al been addreed, and different lutin have been cnidered. A further deciin mut be taken abut the percentage f ct allcated t generatin and lad. In nn lack invariant methd, thi chice i taken by ching a lack nde cle t the generatin r lad center. Hwever, the fact that the ct are allcated t tranactin, and frm thi t uer, allw mre flexibility. The main cntributin f thi paper are intended t be the fllwing The plit f the whle prblem int three ubprblem: tranactin definitin, ue f grid allcatin, and ct allcatin, with different pible lutin fr each f them. Thi diviin allw t make chice fr each ubprblem being aware f their cnequence. The prpal f a flw allcatin methd t tranactin that i differential and lack invariant (DSI methd). Thi methd make ue f the AC lad flw equatin. A tudy abut the cnequence f the definitin f tranactin in pl ytem. A tudy abut the cnequence f the ct allcatin chice. The paper i rganized a fllw. Sectin 2 expe the differential, lack-invariant (DSI) methd f flw and le allcatin t tranactin. Sectin 3 deal with the ubject f tranactin definitin chice, while in Sectin 4 the prblem f ct allcatin t tranactin i addreed. In Sectin 5 the numerical reult f the applicatin f the methd t the IEEE 24 nde Reliability Tet Sytem [6] ytem under different hypthei are given and cmmented. 2 Flw allcatin The methd begin frm the reult f a lved AC lad flw in a ytem with a given lad and generatin. Thi lutin i ued a a tart pint. If P k m and P m k are the active pwer in branch r between nde k and m, injected, repectively, in nde k, and m, the average pwer flw and le in branch r are given by equatin (1). F r = P k m P m k 2 (1) Let u define T ij a the tranactin t f a pwer P ij between the generatin nde j and the demand nde i. Differential methd allcate thee pwer flw t tranactin frm the value f enitivitie f average flw and le t a differential increae in each tranactin. Thee enitivitie f average pwer in branch r t a differential variatin f the tranactin t, T ij, are given by (2). φ rt = F r F r P j F r P i (2) The value f thee enitivitie may be btained by mean f incremental lad flw. The cnditin under which thi enitivitie are lack invariant are given in the Appendix. Hence, the increment f average flw f branch r due t the differential tranactin t, df rt, culd be fund a (3). df rt = φ rt dt ij (3) Thee equatin are nly valid fr differential variatin in tranactin. In rder t btain the allcatin f the flw t a tranactin, it wuld be neceary t integrate them mehw. The integratin prce require an initial pint, t define an integratin path P, and t knw the value f the enitivitie alng thi path, φ rt (T ij ). T take a deciin fr bth initial pint and integratin path i difficult and metime unrealitic, and the required cmputatin time i much higher, ince thi prce mut be numerical. Fr thi rean a recnciliatin prce i fllwed here. Thi methd i equivalent t the integratin, when cnidering nly ne integratin tep, being the integratin made with the implicit Euler rule and auming that the integratin path begin frm zer value f all the tranactin, and then, that all are raied at the ame rate t their actual value. Under thee aumptin, the allcatin f flw t each tranactin i hwn in (4). F rt = φ rt (T ij )T ij (4) Thee cefficient F rt d nt lead t the crrect flw f each line, N T F r t=1 F rt and a recnciliatin prce i neceary, finding the crrected flw allcatin by (5). F c rt = Sme pint mut be highlighted: F r NT t=1 F F rt (5) rt 15th PSCC, Liege, Augut 25 Sein 4, Paper 1, Page 2

3 The fact that the enitivitie are fund with repect t tranactin make them practically independent f the lack bu given. A demntratin f thi fact i given in the Appendix. Once the allcatin t a tranactin i made, it may be neceary t hare it between generatin and lad. Thi may be made in everal way, by agreement between agent, in the cae f a bilateral cntract, r by dividing it between them mehw. In thi way, a great flexibility i pible fr the allcatin amng uer, becaue it might be even different fr each tranactin. The direct allcatin t nde, and the chice f an adequate lack bu cler t the generatin r the demand, in rder t allcate t them a part f the pwer flw, i al pible and ha been prped in literature. Hwever, thi lat lutin i le flexible than the prpal made here. Beide, in ytem with n clear patial eparatin between generatin and lad i nt traightfrward t che the adequate lack bu fr thi purpe. In the next paragraph, the tranactin definitin and the ct allcatin prcedure are examined in detail. 3 Tranactin When the market i baed n bilateral cntract, the tranactin are clearly defined. Hwever, in pl perated ytem, it i neceary t define them in a nndicriminatry way. Thi definitin f tranactin i a key iue becaue if affect deciively the reult. Tw principle are examined here, the Equivalent Bilateral Exchange (EBE) principle and the Prprtinal Sharing Principle (PSP). The firt ne aume that all the lad are upplied by all the generatr prprtinally t the pwer prvided by them. Thi principle ha been prped in [4] and me apect have been tudied in [5]. The ecnd ne aume that the utging pwer f each branch frm a nde cme frm the incming flw prprtinally t the amunt f thee incming flw. Thi principle ha been prped in [3], where it i al ued fr flw and le allcatin. In the prped methd the PSP principle will be nly ued fr the definitin f the tranactin. The applicatin f thi principle t the ytem hwn in figure 1, where the arrw hw the directin f the active pwer thrugh the branche, wuld ay that lad 3 i upplied by generatr 1 and 2, while lad 2 and 4 are upplied nly by generatr 4. Unlike in [3], the principle i nly ued t define the tranactin, but the flw in all the line are enitive t a variatin f every tranactin. Therefre, each tranactin participate in the flw f every line. 1 2 The EBE principle ha the prperty f maller variability with pace and time [4]. Beide, a all the generatin i upped t upply t all the lad, the peripheral generatin and lad make mre ue f the grid than the lcated in the center f the ytem. They are, cnequently, charged mre. Anther characteritic f thi principle i that if it were applied t very large ytem, it wuld lead t unrealitic reult, ince the relatin between very ditant nde wuld ignre grid cngetin r ther cntraint and nnlinearitie. The Prprtinal Sharing Principle ha advantage and diadvantage cmplementary t the EBE principle. Firt, it ha mre variability with time and pace. Althugh thi may be a diadvantage, it mut be taken int accunt that in me cae, uch a the cmpenatin fr CBT, the charge fr grid ue are cmputed thrughut a lng perid- fr intance a year- and charged annually. Under thee cnditin, the vlatility with time i le imprtant. Under thi principle, the generatin and demand lcated in the center f the ytem wuld likely upply r receive pwer t r frm mre nde, and they wuld have greater relative charge than if EBE principle i fllwed. On the ther hand, peripheral generatin and demand are mre cnfined and their charge tend t be maller than under the EBE principle. If thi principle i applied t very large ytem, it wuld plit them int maller balanced ne, there are le interference with ytem nnlinearitie and cngetin. 4.1 Different methd. 4 Ct allcatin Once the flw ha been allcated t the different tranactin, it till remain the prblem f allcating the ct t them. Thrughut thi ectin, it will be aumed that the ct recvery principle i applied, and therefre, the ct are allcated t each tranactin accrding t the ue that it make f the grid. It mut be al remarked that the flw allcatin t a tranactin may be pitive r negative. If it i pitive, it mean that thi tranactin cntribute t the dminant flw f the branch. The ct f each branch can be allcated in different way t the branch uer. If the negative term are allwed, the flw pped t the dminant flw huld be paid. The allcatin f the ct a branch r t a tranactin t, S rt wuld be given by (6). S rt = C r F c rt F r (6) 3 Figure 1: Oriented graph. 4 Where C r i the branch ct, F c rt i the flw allcatin f branch r t tranactin t, and F r the flw thrugh branch r. The ttal grid ct allcatin t a tranactin t wuld be given by (7). 15th PSCC, Liege, Augut 25 Sein 4, Paper 1, Page 3

4 N R S t = S rt (7) r=1 Anther methd i t allcate the ct accrding t the ablute value f the flw allcatin t a tranactin. Thi i given by equatin (8). U rt = C r F c rt NT t=1 F c rt (8) where U rt i the ct allcatin f branch r t the tranactin t and C r i the ct f branch r. With thi allcatin, there are n incentive t the cunterflw, but the payment fr the ue f ne branch i prprtinal t the ue that the tranactin make f it in a given ituatin. The ttal flw allcatin t a tranactin i given by equatin (9). L Figure 2: Fur nde ytem. L1 Pwer in nde (MW) Cae I II III-A III-B III-C L3 Table 1: Lad and generatin pattern in the 4 nde ytem (MW). 31 N R U t = U rt (9) r=1 In the next paragraph thee chice are dicued. 4.2 Cnequence f the ct allcatin methd chice. Let u cnider the mall ytem in figure 2, with the different lad and generatin pattern hwn in table 1. Tw ituatin are cnidered in rder t illutrate al the imprtance f the tranactin definitin. The firt ne aume that all the nde belng t a ingle ytem (Dmetic ytem). The ecnd ne uppe that the nde 11 and 12 are ne ytem (Sytem 1) urrunded by ther tw that exchange the pwer thrugh the line L1 f Sytem 1 (Multiarea ytem). Fr implicity, all the line have the ame ct and the grid ct are hared equally by generatin and lad. Reult are given fr the tw aumptin,dmetic (table 2) and Multiarea (table 3, and fr the tw ct chice, Flw prprtinal ( Flw ) and Ablute value prprtinal ( Ab ). Value in bth table repreent the ttal amunt f the grid ct in p.u. paid by each nde. The fllwing cncluin may be drawn frm the reult: If the ct are allcated prprtinally t the branch flw, an incentive fr dicharging the line i prvided, a hwn fr ituatin I and II. When the flw in line L1 i very mall, the allcatin change abruptly with mall difference in flw. In ituatin III-B, nde 11 and 12 are paid mre than 12 time 1 the ct f the whle grid, while in ituatin III-C, very imilar t the previu ne, it i them wh mut pay and the ther receive. The nly cnditin i that the amunt paid by the fur nde i 1. The limit i ituatin III-A, where the amunt t pay r being paid i infinite, a may be een in equatin 6. Nde Cae Chice I Flw Ab II Flw Ab III-A Flw ± ± Ab III-B Flw Ab III-C Flw Ab Table 2: Allcatin fr Dmetic ytem. Nde Cae Chice I Flw Ab II Flw Ab III-A Flw ± ± Ab III-B Flw Ab III-C Flw Ab Table 3: Allcatin fr Multiarea ytem. The allcatin prprtinal t the ablute value reult in a negative incentive fr nde 11 and 12, becaue they pay mre althugh they have reduced the flw in L1. Hwever, the behavir i much mre teady in lw flw ituatin in line L1. 1 Net flw in L2 i 1 MW due t flw f 75,75 MW frm 21 t 31 and 74,75 MW frm 11 t 12. Nde 21 and 31 wuld have t pay 37,875 time the ct f L2. Ct f L2 i a third part f ytem ttal ct. 15th PSCC, Liege, Augut 25 Sein 4, Paper 1, Page 4

5 Tranactin definitin ha a key rle in the final allcatin. In the Dmetic example, the generatr are upped t upply every lad prprtinally t the generated and demanded pwer. There are, therefre, fur different tranactin. In the Multiarea ituatin, hwever, there are nly tw exchange, between nde 11 and 12, and between nde 21 and 31. In thi cae, the difference between the allcatin prprtinal t flw allcatin, and t the ablute value, are even greater. Therefre, it eem that the benefit f the incentive fr a mre efficient ue f the line i cunterbalanced by the unteadine in ituatin f lw flw due t ppite exchange. Thi i nt an unuual ituatin. It ha been claimed that a 2% f the flw in the line in 14 cuntrie f the UCTE change their directin becaue f the freign tranit. It eem, hence, mre adequate t allcate prprtinally t the ablute value, if the ct recvery principle i aumed. 5 Applicatin example. Thi methd ha been applied t the IEEE-RTS96 ytem [6]. Lad flw have been run uing the Matlab Pwer Sytem Tlbx [7]. The IEEE-RTS96 1-area ytem i hwn in figure 3. In thi ytem, mt f the generatin i ituated in the upper part f the ytem, while the lad i mtly cncentrated in the lwer part. Therefre, the main pwer flw cme dwnward. The ct f the grid element are prprtinal t the reactance f the branch Single area ytem. The IEEE-RTS96 ytem ct i allcated amng it uer. Reult are given in figure 4 t 7. In all f them, the amunt f grid ct allcated t each nde, under different hypthei, i hwn. Thee hypthei are the tranactin definitin, (EBE and PSP principle are cnidered), and the ct allcatin (prprtinal t the flw allcatin, r t it ablute value). Ue f grid (p.u.),18,16,14,12,1,8,6,4,2 Generatin. Prprtinal t ab. value nde Figure 4: Charge fr ue f grid amng generatr. Charge prprtinal t ablute value. Ue f grid (p.u.),8,7,6,5,4,3,2,1 Demand. Prprtinal t ab. value nde Figure 5: Charge fr ue f grid amng demand. Charge prprtinal t ablute value. Generatin. Prprtinal t flw ,35, Ue f grid (p.u.),25,2,15,1,5 -,5 -, nde Figure 6: Charge fr ue f grid amng generatr. Charge prprtinal t flw. Demand. Prprtinal t flw 4 5 8, Figure 3: IEEE-RTS96 ytem. Several reult are hwn. Firt f all, different tudie fr a ingle area ytem have been run under different definitin f tranactin and ct allcatin methd. Then, the IEEE-RTS96 multiarea ytem i tudied in rder t draw cncluin abut the tranactin definitin in thee multiarea ytem, with pible applicatin t CBT cmpenatin evaluatin. 7 Ue f grid (p.u.),1,5 -,5 -,1 -, nde Figure 7: Charge fr ue f grid amng demand. Charge prprtinal t flw. 15th PSCC, Liege, Augut 25 Sein 4, Paper 1, Page 5

6 Frm thee reult, the fllwing cncluin culd be made: In general, peripheral generatin and lad are charged mre under EBE aumptin than under PSP principle when allcatin i made fllwing the ablute value criteria. Thi i becaue with Equivalent Bilateral Exchange the exchange between remte nde are cnidered. Under PSP aumptin, hwever, a mre lcal allcatin f lad t generatr i made, reulting in greater charge t centrally lcated nde, becaue they are included in mre tranactin. When the allcatin i prprtinal t the flw allcatin, generatr 1, 2 and 3 are allcated a negative amunt with Equivalent Bilateral Exchange due t the fact that there are exchange that are ppite t the main pwer flw frm the upper t the lwer part. Thi effect, hwever, de nt appear when applying the PSP principle, becaue the exchange are limited t cler nde. Therefre, thi cunter flw de nt exit. It culd al be added that all the generatr with lcal lad are allcated a maller amunt under PSP aumptin that under EBE principle, becaue the PSP principle make thee generatr t upply mtly their lcal lad, and cnequently they make le ue f the tranmiin grid. Nde 3 i allcated a negative amunt in bth cae in figure 7 becaue thi lad i upplied, accrding t the PSP principle, by generatr 15, 21 and 22. An increae in thee tranactin wuld lead t a decreae in the flw f line (3-1) and (3-9), al upplied by thee generatr, and that have a very high ct. A imilar thing happen under EBE aumptin. It can be eaily cncluded that great difference can be btained with different aumptin. Fr the ake f tability it culd be recmmended t allcate ct accrding t the ablute value f the flw allcatin, but f cure thi culd be a ubject f endle dicuin. 5.2 Multiarea ytem. A a final example, let u cnider the cmplete (3 area) IEEE-RTS96 ytem. In thi example the three ytem are almt exactly the ame, and they are all balanced in generatin and lad. Only exchange lp flw take place between them. Thee exchange (active pwer in p.u.) may be een in figure 8, where the number f the nde in the ytem ha been preceded by anther number indicating it area. In thi ituatin, if the EBE principle i ued fr defining the tranactin, the part f the ct f each grid and le that wuld have t pay the uer f each area (rw) t the ther area (clumn) i given in table 4. Table 5 hw the ame cncept, but when the PSP i ued fr defining the tranactin. It can be berved that PSP plit the whle grid int it balanced ubet, while the EBE principle divide the verall ct amng all uer. It i dubiu whether the allcatin fllwing EBE principle i adequate fr large ytem, like the real pwer netwrk. Thi i pecially pertinent if the whle UCTE grid were tudied. In thi cae, the PSP principle wuld plit it int balanced et f generatin and lad. On the ther hand, it de nt eem adequate t ue the PSP principle in mall ytem, becaue there culd be aburd ituatin like a generatr nt upplying a cle lad due t the ene f the pwer flw in the ytem. What i the right ize fr the applicatin f each methd i till an pen quetin ,152 1, ,9825 1, , Area 1 Area 2 Area 3 Figure 8: Active pwer exchange in the IEEE-RTS96 3-area ytem. Active pwer flw in p.u. Area Area 1 Area 2 Area 3 1,5292,2355,2355 2,2436,5513,1983 3,2272,2132,5996 Table 4: EBE tranactin. Payment frm area in rw t the area in clumn. Area Area 1 Area 2 Area 3 1,822,984,541 2,1242,8482,333 3,538,534,9126 Table 5: PSP tranactin. Payment frm area in rw t the area in clumn. 6 Cncluin Frm the preented reult, it may be cncluded that: Differential methd are adequate fr lving the grid ct allcatin prblem. They fllw the netwrk phyical rule and are eay t undertand and implement. The prblem f grid ct allcatin may be divided in three ubprblem: flw allcatin t tranactin, the definitin f thee tranactin and the ct allcatin baed n the flw allcatin. Thee ubprblem may be dealt with independently. The flw allcatin prblem t tranactin may be eaily lved uing a differential technique running differential lad flw. The invariance f the reult t the tranactin i ttal if DC lad flw equatin are ued. With AC lad flw equatin, thi invariance i nly apprximate, and me additinal meaure mut be taken t enure thi invariance. 15th PSCC, Liege, Augut 25 Sein 4, Paper 1, Page 6

7 The tranactin definitin ha a key rle n the allcatin reult. Tw principle fr thi definitin have been examined, the Equivalent Bilateral Exchange principle, and the Prprtinal Sharing Principle. If the EBE principle i fllwed, the allcatin i mre unifrm thrughut the ytem, and i mre like a flat rate methd. It eem le adequate fr large netwrk. The applicatin f PSP, n the cntrary, lead t a mre irregular allcatin pattern, while it de nt eem gd fr mall grid. What i a large and a mall grid i a ubject under dicuin. The ct allcatin criteria prprtinal t the flw allcatin, and allwing negative allcatin, may lead t unteady ituatin in line lightly charged due t great cunterflw. Thi effect may check the incentive that prvide fr a mre extenive ue f the tranmiin facilitie, when the ct recvery principle i aumed. Appendix. Branch flw enitivitie t tranactin with different lack bue In figure 9 a chematic electric ytem i hwn. The differential tranactin dt ij cnit in an active pwer injected in nde j and delivered in nde i. j r dp l Figure 9: Electric ytem When the lack bu i nde, the additinal le prduced by thi tranactin, dp l, are prvided by thi nde. The enitivity f the flw f branch r t thi differential F r increae f the tranactin, i given by (1): F r = F r P j F r P i (1) If, hwever, the lack bu i the nde, the variatin f flw df r when the ame tranactin dt ij take place, will have the ame value nly if the le prduced by thi tranactin are al upplied by the bu. In thi cae, thi enitivity culd be written a (11) i F r = F r P j F r P i + F r P dp l (11) Equatin (1) and (11) give the ame reult, and the difference between them i given by the term Fr P dp l. There are different actin that culd be taken t cnider it, namely: 1. T neglect thi term, ince it i a ecnd rder differential term. 2. T include the le in the generating nde j f tranactin T ij. Thi wuld be imilar t the methd prped in [4] fr the lle cae. Thi pibility, hwever, mut be cherent with the definitin f tranactin in ly ytem, ince the le prduced by each tranactin huld be included in it. It mut be recalled, hwever, that fr lle ytem thi term de nt exit, and therefre, in thee ytem, the difference f enitivitie are lack invariant. 3. Althugh the enitivity term culd be calculated inverting the given jacbian matrix, they can al be btained numerically running incremental lad flw, where the enitivitie, tgether with the term dp l, culd be fund. A cmplete invariance culd be achieved by running thee incremental lad flw with a ditributed lack bu, where the le wuld be aumed by all the nde, except the generating nde in the tranactin. The prcedure ued in thi paper i the here numbered a 2, althugh it ha been checked that the difference with methd 3 are negligibly mall. Acknwledgment Thank are due t Red Eléctrica de Epaña fr their upprt t thi wrk. Jé Lui Fernández Gnzález huld be al gratefully mentined fr ueful cmment. Reference [1] J.W. Marangn Lima, Allcatin f tranmiin fixed charge: an verview. IEEE Tran. n Pwer Sytem. Vl. 11 n. 3, pp: , Aug [2] J.I. Pérez Arriaga, F.J. Rubi, J.F. Puerta et al. Marginal pricing f tranmiin ervice: an analyi f ct recvery. IEEE Tran. n Pwer Sytem. Vl. 1 n. 1,pp , Feb [3] J. Bialek, Tracing the flw f electricity. IEE Prc.- Generatin, Tranmiin and Ditributin. Vl. 143, n. 4, pp , Jul [4] F.D. Galiana, A.J. Cnej, H.Gil, Tranmiin Netwrk Ct Allcatin Baed n Equivalent Bilateral Exchange, IEEE Tran. n Pwer Sytem, Vl. 18, n. 4, pp , Nv. 23. [5] C. Vázquez, Lui Olm, Ignaci J. Pérez-Arriaga On the Selectin f the Slack Bu in Mechanim fr Tranmiin Netwrk Ct Allcatin that are baed n Netwrk Utilizatin. Prc. f the 15th Pwer Sytem Cmputing Cnference. PSCC. Seville, 22. [6] The IEEE Reliability Tet Sytem, IEEE Tran. n Pwer Sytem. Vl 14, n. 3, pp , Aug [7] Pwer Sytem Tlbx. Verin 2.. Cherry Tree Scientific Sftware. 15th PSCC, Liege, Augut 25 Sein 4, Paper 1, Page 7