Electric Power Systems Research

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1 Electrc Power Systems Research 81 (2011) Contents lsts avalable at ScenceDrect Electrc Power Systems Research jou rn al h om epage: Fully reference-ndependent LMP decomposton usng reference-ndependent loss factors Fangxng L Department of EECS, The Unversty of Tennessee, Knoxvlle, TN 37996, USA a r t c l e n f o Artcle hstory: Receved 1 March 2011 Receved n revsed form 1 May 2011 Accepted 3 May 2011 Avalable onlne 20 July 2011 Keywords: Power market LMP decomposton Reference-ndependent Loss factor DCOPF Loss dstrbuton factor (LDF) a b s t r a c t The decomposton of locatonal margnal prce (LMP) under the popular DCOPF framework generally depends on the choce of the reference bus. A prevous work has acheved reference ndependence for the overall LMP and LMP congeston component, but not all ndvdual LMP components. Ths paper proposes a method to obtan a truly reference-ndependent LMP decomposton such that all three components of LMP at each bus wll be nvarant w.r.t. the choce of the system reference bus. Ths s acheved wth loss factors based on a new AC-based dstrbuton factor model, whch depends on the network topology and the present operatng condton only, but not the system reference bus. Ths model gves referencendependent loss prces, whch can serve for a better loss hedgng fnancal transmsson rghts, snce the choce of reference bus wll not change the loss prces. Further, ths paper uses the fcttous nodal demand (FND) model to obtan loss dstrbuton factors (LDFs). FND gves more reasonable power flows snce losses should be dstrbuted n each ndvdual lne, rather than at load buses when the load-weghted LDFs are appled. Also, the proposed reference-ndependent dstrbuton factors and loss factors may have great potentals n other areas of power system analyss Elsever B.V. All rghts reserved. 1. Introducton The locatonal margnal prce (LMP) methodology has been mplemented or s under consderaton at a number of US RTOs or ISOs such as PJM, New York ISO, ISO-New England, Calforna ISO, ERCOT, and Mdwest ISO. LMP at a gven Bus B can be decomposed nto three components: margnal energy prce, margnal congeston prce, and margnal loss prce [1 5]. Ths can be wrtten as LMP B = LMP energy + LMP cong + LMP loss B B (1) A number of earler works [6 18] have reported LMP-related research results. In partcular, Refs. [6 13] dscussed the modellng of LMPs and ts decomposed components. Also, a dstrbuted reference bus model s dscussed n [14]. The senstvty of LMP s dscussed n [13,15]. The comparson of AC-based and DC-based results s dscussed n [9,13]. A modfcaton of LMP methodology s proposed n [16]. Forecastng of LMP consderng load varaton and uncertanty s presented n [17,18]. Loss hedgng rghts are dscussed n [19,20]. Ref. [10] presents an ACOPF-based decomposton whch s ndependent of the choce of energy reference bus. Ref. [8] presents an approach to calculate reference-ndependent LMP and ts conges- Correspondng author. Tel.: E-mal address: fl6@utk.edu ton component based on DCOPF usng lnear programmng (LP). Snce LP-based DCOPF s popularly employed n ndustral practces for real-tme LMP calculaton at a number of RTOs/ISOs, ths paper uses DC models to extend the prevous LMP research n [8]. In the decomposton model n [8], LMP congeston component at Bus B,.e., LMP cong, remans nvarant w.r.t. dfferent reference B buses, and the combnaton of the other two components,.e., LMP energy + LMP loss B, s also reference-ndependent. The LMP cong s B needed for fnancal transmsson rghts to hedge the transmsson congeston cost, whle LMP loss B s useful for the proposed loss prce hedgng n [19,20]. Snce the prevous works dd not provde a separaton of LMP energy and LMP loss B, ths can be controversal when loss prce hedgng s consdered. Based on ths motvaton, ths paper wll present a decomposton model that makes three ndvdual LMP components fully ndependent of the choce of reference. Ths s acheved by usng a loss factor model based on a proposed new AC-based dstrbuton factor model, whch depends on the network topology and the operatng condton only and does not requre a system reference bus. Also, the loss factor and the LMP loss component at the man-made reference bus wll not be zero, whle the lterature gves zero values. The non-zero values are more reasonable snce n realty there s no reference bus and every bus should have some contrbuton to losses. There are a few assumptons of the formulatons n ths paper that are lsted here to avod confuson: (1) each bus has one generator and one load for smplcty of dscusson; (2) each transmsson /$ see front matter 2011 Elsever B.V. All rghts reserved. do: /j.epsr

2 1996 F. L / Electrc Power Systems Research 81 (2011) constrant (thermal, contngency, nomogram, etc.) may have bdrectonal lmts n realty, but t s modelled as f a undrectonal lmt for smple formulaton; (3) a sngle-block generaton cost or bd model s assumed, whle n realty a monotoncally ncreasng mult-block model s commonly used; and (4) the demand elastcty s not explctly modelled snce unserved loads can be smply vewed as generaton resources. 2. Revew of DCOPF formulatons for LMP calculaton 2.1. Model 1 (lossless) Earler studes of LMP calculaton wth lnearzed OPF gnore lne losses. Thus, the energy prce and the congeston prce follow a perfect lnear model gven by: Mn s.t. c G (2) G = D (3) GSF k (G D ) Lmt k, for k = 1 M (4) G mn G G max, for = 1 N (5) The LMP decomposton of ths model s straghtforward and gnored here. It s well known that the actual GSF values depend on the choce of reference bus. However, the lne flow models n (4) based on the reference-dependent GSF are reference-ndependent. So, ths model produces the same power flow results and hence the same LMP regardless of the choce of reference bus. Ths model can be acceptable for estmaton purposes or as a startng pont for market-related research. However, t may not be preferred n operaton because of the lack of the loss component. Also, the energy and congeston components wll vary w.r.t. dfferent choces of reference Model 2 When losses are consdered, the key to consder margnal loss prce s margnal loss factor (LF) and the margnal delvery factor (DF), defned as: DF = 1 LF = 1 P Loss (6) LFs and DFs wll be one of the man topcs n ths paper. For now, they are assumed avalable. Then, we can formulate dfferent DC-based OPF models. A straghtforward approach presented n the past s to multply DF by nodal njectons to account for losses n the energy balance equaton whle keepng (2) (5) unchanged. Mn s.t. c G (7) DF G DF D + offset = 0 (8) GSF k (G D ) Lmt k, for k = 1 M (9) G mn G G max, for = 1 N (10) The LMP decomposton of ths model s straghtforward and can be found n [8,13]. It has been observed that the margnal DF may produce doubled losses n Eq. (8) wthout offset. Ref. [13] rgorously proves that n a fully DC-based model the value of offset should be the estmated total system losses. Wthout offset, generators may output doubled losses. Also, offset may consder errors n ntal loss estmaton Model 3 Ths s the optmzaton model named LP2 n [8] for LMP calculaton consderng losses. It can be wrtten as: Mn s.t. P Loss c G (11) G D P Loss = 0 (12) LF G + LF D + offset = 0 (13) GSF k (G D LDF P Loss ) Lmt k, for k = 1 M (14) G mn G G max, for = 1 N (15) As prevously mentoned, LMP cong s reference-ndependent, B and the combnaton of the other two components,.e., LMP energy + LMP loss B, s also reference-ndependent. But each of LMP energy or LMP loss B s stll reference-dependent Model 4 Another model of LMP wth losses s proposed n [13] usng an teratve approach. It can be wrtten as: Mn s.t. c G (16) DF est G DF est D + P est Loss = 0 (17) GSF k (G D E est ) Lmt k, for k = 1 M (18) G mn G G max, for = 1 N (19) Here an teratve approach s used. Intally, DF est = 1, E est = 0, and P est Loss = 0; and essentally a lossless DCOPF s performed. Then, DF, E and P Loss are updated to start a new DCOPF. Ths s repeated tll convergence. It s also dscussed that a two-teraton smplfcaton,.e., lossless model n (2) (5) for the frst teraton and then (16) (19) for the second teraton, can produce good enough results. The LMP decomposton of ths model s the same as Model 2, as shown n [13]. An mportant feature of ths formulaton s the fcttous nodal demand (FND) model to mmc lne losses. For each lne, 50% of the lne loss s assgned to each connected bus as an extra demand, represented by E. Hence, the losses are dstrbuted n each lne. As shown n Secton 5, ths FND model can be used to gve farer and more reasonable loss dstrbuton factors (LDFs) to mprove Model 3.

3 F. L / Electrc Power Systems Research 81 (2011) Reference-ndependent dstrbuton factors and loss factors 3.1. Motvaton for reference-ndependent LF When losses are consdered, LF s commonly employed as shown n Models 2 4. However, LF n the prevous works depends on the reference choce. In general, ths leads to reference-dependent decomposton. Model 3 s the present state of the art n terms of achevng reference-ndependence LMP cong and (LMP energy + LMP loss B B ) usng DC model. The LMP decomposton s gven by: LMP energy = (20) LMP loss B = LF B (21) LMP cong B = + + k GSF k B = ( k LDF GSF k ( k GSF k B) (22) From (20) to (21), the combnaton of LMP energy and loss components can be wrtten as LMP energy + LMP loss B = LF B = (1 LF B ) (23) There are only two varables n (23). Hence, f we can fnd a way to make one varable (such as LF B ) reference-ndependent, the other varable,, wll be reference-ndependent as well. Therefore, a fully reference-ndependent LMP decomposton can be acheved Basc model for loss factors As shown n [13], LF can be modelled as: P Loss = F 2 k R k (24) LF = P Loss = ( M ) F 2 k R k = ) R k 2F k F k (25) In the above equatons, the lne flow F k should be obtaned from state estmaton results n operaton. If we assume a perfect data measurement and state estmaton to smplfy our dscusson, F k wll be the same as the results from economc dspatch by solvng ACOPF or a close approxmaton (a lossless DCOPF as an extreme smplfcaton). Although ether ACOPF or lossless DCOPF needs a voltage zero-angle reference, the lne flow F k should be always reference-ndependent. Hence, seekng reference-ndependent loss factors (LF ) s converted to seekng a reference-ndependent F k /, the dstrbuton factor (a.k.a. senstvty factor) of lne flow w.r.t. bus njecton. Please note F k / nvolves real power only. The real-power-only F k / naturally makes one to consder a lnear lossless DC network, n whch a lne flow s usually consdered as the aggregaton of the contrbuton from all power sources (generaton as postve and load as negatve) based on superposton. Ths can be wrtten as F k = GSF k j (G j D j ) = GSF k j P j (26) From (26), we have F k = GSF k (27) Hence, the conventonal LF n (25) s reference-dependent because the above DC-based dstrbuton factor, namely GSF n ths paper, s reference-dependent. For ths reason, t s not lkely to have DC-based reference-ndependent dstrbuton factors. Thus, we may have to explore other approaches lke AC-based model, whch wll be ntensvely studed next. To avod confuson, we frst defne the generalzed AC-based dstrbuton factor of lne flow n MVA wth respect to nodal njecton n MVA, namely, S k / S or k for notatonal convenence. Also, we need to defne the real-power dstrbuton factor, k,re, as: k,re = F k = ( S k) RE ( S ) RE (28) It should be noted that the above defnton takes out the mpact of reactve components because the magnary components n S k and S wll partly contrbute to the real part of k. Hence, (28) wll be truly the real power senstvty. It should be noted that n general we have k,re /= ( S k / S ) RE. Apparently, the challenge here s to fnd a reference-ndependent k,re,.e., AC-based dstrbuton factor ( S k ) RE /( S ) RE. The remanng part of ths secton wll show a prevous model of AC-based senstvty, pont out an unjustfed assumpton n ts dervaton, and then gve a mathematcally rgorous model wth numercal verfcaton Prevous model of reference-ndependent k Ref. [21] shows a well-known dervaton of k, whch s reference-ndependent. Ths s shown as follows: k = S k = (V k1i k ) S (V I ) = (V k1 ((V k1 V k2 ) /z k )) (V I ) ((V k1 V k2 ) /z k ) (I ) = Z k1, Z k2, z k = 1 z k ( V k1 I V k2 I ) (29) Regardless of the dfference between k and k,re that Eq. (29) does not address, the above dervaton mples njecton as current sources, because t assumes that all bus voltages are held nvarant and close enough n both magntude and angle (.e., V k1 = V and they are nvarant). The voltage magntudes may be close to constant, but the angle dfference can be consderable. In addton, from the vewpont of crcut analyss, f there s a current njecton change at Bus, t should change the voltage at every bus as well. Otherwse, f voltages do not change, the current and power njecton wll not change at all snce we have V = Z bus I. So, generally speakng, voltages are not ndependent varables f we consder the network and do not consder some voltage control actons. Hence, the assumpton of constant voltage magntude and angle s not justfable and should be relaxed Reference-ndependent k,re V B = The bus voltage at any Bus B can be wrtten as: (Z B,j I j ) (30)

4 1998 F. L / Electrc Power Systems Research 81 (2011) And the bus njecton power at Bus s gven by S = V I = I (Z,j I j ) (31) The exstng lne flow through Lne k at the sendng end,.e., Bus k1, s gven by S k = V k1 I k = V k1 (V k1 V k2 ) N = (Z k1,j I j ) z k z k (Z k1,j I j ) (Z k2,j I j ) [ N ] [ N ] (Z k1,j I j ) ((Z k1,j Z k2,j ) I j ) = (32) z k As we can see from (31) and (32), both bus njecton and lne flow are related to the system topology and the ntal njectons. The former s constant and the latter s ndependent varables. Hence, bus njecton S and lne flow S k can be expressed usng ndependent varables I, and there s no ntermedate varable lke voltages nvolved n (31) and (32). Ths wll make the dervaton below very clear wthout any possble confuson. Now we need to consder a small change of I, say, x. Although x can be any complex number n theory, here we can consder that x s appled to the magntude I only to keep the same power factor, roughly speakng. To facltate our dervaton, we can take angular shft to make the angle of I the reference angle,.e., 0 degrees for I. Ths s because angular shft (or changng voltage/current reference angle) does not affect bus njecton power and lne flows, snce phase angles are ndeed relatve measures whle power s not. Wthout losng generalty, we contnue to use the symbol I for smplcty after the angular shft. Snce I IM = 0, we have (I + x) I = x = (I + x) I (33) The change of bus njecton power at Bus s gven by S = x (Z,j I j ) + Z, I x = (Z,j I j ) + Z, I x (34) Smlar to (31) and (32), (36) nvolves only the network topology and the ntal condton of current njectons. There s no ntermedate varable, and there s no reference bus nvolved. Hence, the power dstrbuton factor k,re does not requre a reference bus. N Further, f we consder V B = (Z B,j I j ), we can smplfy (36) as follows: [(Z k1, /z k ) (V V ) + ((Z Z )/z k1 k2 k1,j k2,j k ) V k1 ] RE k,re(k1) = (37) [V + Z, I ]RE Ths smplfed equaton shows the nvolvement of ntermedate state varables of bus voltages, whch are essentally determned by the ntal condton and the network topology. So, t s stll reference-ndependent. It s not dffcult to observe that f we N assume voltages are all close to unty, and V (= (Z,j I j )) s much greater than Z, I, then (37) s smplfed back to (29). The recevng end senstvty s smlar to (37) except that Bus k2 s treated as k1. Ths s gven by ] RE [(Z k2, /z k ) (V k1 V k2 ) + ((Z k1,j Z k2,j )/z k ) V k2 k,re(k2) = (38) [V + Z, I ]RE The senstvty based on the flow at the center of Lne k s the average of (37) and (38). Thus, we have k,re = k,re(k1) + k,re(k2) 2 [((Z k1, + Z k2, ) (V V ) + (Z Z ) (V k1 k2 k1,j k2,j k1 + V k2 ))/2z k ] RE = (39) [V + Z, I ]RE The above analytcal dervaton of (39) s rgorous and does not need the assumpton of nvarant voltages mpled by (29). Hence, ths s the frst major contrbuton of ths paper. Ths can be summarzed as below: The complex power dstrbuton (senstvty) factor, k, and the correspondng real power dstrbuton factor, k,re, are determned by the network topology and the present operatng condton only, and do not requre a system reference. It should be noted that the dervaton of analytcal Eq. (39) assumes the bus njectons at the present operatng pont are current sources. Ths s also mpled n Ref. [21] for Eq. (29). Ths should be a reasonable assumpton because of two reasons: some loads behave as current sources ndeed as evdenced by many power system dynamc studes; and more mportant, the voltage or power control cannot have nstantaneously effect when a very tny current perturbaton s nstantaneously appled to a selected bus Numercal verfcaton of reference-ndependent k,re The change of the kth lne flow at the k1 end s gven by S k = Z k1, x (Z z k1,j I j ) (Z k2,j I j ) k N + (Z k1,j I j ) (Z z k1, Z k2, ) x (35) k To obtan k,re we need to combne (34) and (35). Snce the perturbaton x s a real-number scalar (magntude only), whch can be elmnated, so we have the senstvty k,re at the sendng end Bus k1 gven by: [ ( N ) k,re(k1) = ( S k) RE (Z k1, /z k ) ( S ) RE = (Z k1,j I j ) N (Z k2,j I j ) + [ N (Z,j I j ) + Z, I Numercal tests are presented n ths subsecton to verfy Eq. (39). The test system s a modfed PJM 5-bus system wth detals shown n Secton 5 (see Fg. 3 and related data). The test procedure s descrbed as follows: Frst, the current njectons for a base case are obtaned. Ths s done wth an ACOPF run to fnd dfferent voltage magntudes and angles, whch should represent the results from economc dspatch. The voltage phasors from Buses A to E are 1.1 0, , , , and , respectvely. Then, solvng I = Z 1 V gves the bus current njectons. The njectons wll be consdered as the ( N ) ] RE (Z k1,j I j )/z k (Z k1, Z k2, ) ] RE (36)

5 F. L / Electrc Power Systems Research 81 (2011) Table 1 Verfcaton of k,re of each lne w.r.t. Buses 1 and 4. Lne Analytcal: usng the proposed model n Eq. (39) Numercal: 1% perturbaton of njectng current Real part of k usng the prevous work shown n Eq. (29) w.r.t. Bus 1 w.r.t. Bus 4 w.r.t. Bus 1 w.r.t. Bus 4 w.r.t. Bus 1 w.r.t. Bus 4 AB AD AE BC CD DE Ref. Bus (a) No current can flow back to the ground wthout a reference bus to absorb the njecton. (b) Current can flow to the reference bus through the network. Fg. 1. Necessty of the reference bus for fndng GSF usng DC model. S (a) AC network wth shunts (b) Equvalent to have multple snks determned by network parameters Fg. 2. Usng AC network to fnd loss factors. ntal condton here. Note: The voltage angle reference for ACOPF s Bus 1. Changng t to another bus wll change the angle values at each ndvdual bus, but not the relatve dfferences between any two phasors. Hence, generaton dspatch and lne flow reman the same, because angle s a relatve measure, but power s not. Hence, generaton output and lne flows from ACOPF are ndeed reference ndependent. Second, a perturbaton s performed by applyng a 1% current change at a gven bus. The system s solved usng V = Z I (prme means the perturbed case), snce current njectons are the ndependent varables. Fnally, the rato of real-power lne-flow change versus realpower bus-njecton change s calculated usng power flows from the base case obtaned wth ACOPF and the perturbed case. The values of ths numercally calculated senstvty,.e., ( S k ) RE /( S ) RE, are compared wth the analytcally calculated values,.e., ( S k ) RE /( S ) RE, usng (39). The comparson s shown n the second and thrd columns n Table 1. Apparently, the analytcally calculated k,re usng (39) s accurate as verfed by the numercally calculated values,.e., ( S k ) RE ( S ( S ) RE k ) RE ( S ) RE The dstrbuton factors usng the real part of k from the prevous work shown n Eq. (29) s lsted n the last column n Table 1. If compared wth Eq. (39), Eq. (29) leads to sgnfcant dfference. Ths s reasonable because of the nonlnearty of power systems, whch means that the senstvty wth the present operatng condton gnored,.e., senstvty wthout no source, s certanly dfferent from the condton wth loads. Ths justfes the necessty of usng a more complcated analytcal model (39) that ncludes the ntal system condton and consders voltages as varables rather than constant. And, more numercal tests, not shown here smply due to space lmt, all support Eq. (39).

6 2000 F. L / Electrc Power Systems Research 81 (2011) Combnng (39) wth (25), the reference-ndependent loss factor s gven as follows: LF = P Loss = R k 2F k k,re (40) Brghton $20 600MW E Lmt = 240MW D Sundance 400 $40 200MW where k,re s defned n (39). Snce both F k and k,re are reference-ndependent, the loss factor LF s referencendependent Some llustratve comments $14 110MW $15 100MW Alta Park Cty A B C Soltude $30 500MW The foundaton of the DC power flow model s that the lne flow can be smplfed to F k = ı k1 ı k2 (41) x k Hence, only lne reactances are needed n the model. Lne resstances and shunt capactances are all gnored. To calculate the GSF of Bus to Lne k, t s assumed that there s a small njecton at Bus. Here can be vewed as a current njecton as well snce the voltage s held at 1.0 p.u. at every bus. Then the resultant lne flow at Lne k dvded by s the GSF of Bus to Lne k. Ths s shown n Fg. 1 wth the upper-rght bus as the reference. As shown n the fgure, snce we have an njecton, we must specfy a reference bus to absorb the njecton. Otherwse, there s no path to let the njecton at Bus to flow back to ground. Ths s why a reference/slack bus must be specfed. In case we have a dstrbuted reference bus, then the njecton wll flow back at dfferent buses to ground wth the amount decded by reference-bus weghtng factors. As shown n Fg. 1, when the reference bus changes, the GSF of each bus to dfferent lnes may change. Ths s the reason that LF from (25) and (27) s reference-dependent. In contrast, the AC network model has shunt capactances that are typcally represented by the p model shown n Fg. 2. The dstrbuton factor k means the change of MVA flow through Lne k when there s a per unt change of an njecton at Bus. As we assume the njectons are all current sources, a small change of I leads to voltage changes at all buses and then leads to changes of njecton power at all buses wth njecton sources, postve (generaton) or negatve (load). The changes at other buses are essentally equvalent to multple absorpton snks correspondng to the perturbaton ( S ) at Bus, as llustrated n Fg. 2. The absorpton amounts at dfferent buses are dfferent, and they are objectvely determned by the system topology and the ntal condton (operatng pont). It s not determned by any user-defned or man-made reference. Ths s the sgnfcant dfference between the proposed model and conventonal approaches. For nstance, the generc Power Transfer Dstrbuton Factor (PTDF) n [22 24] requres a par of user-defned njecton and snk. Also, the tradtonal (dstrbuted) reference bus s essentally a user-defned snk. The user-defned or man-made source-snk par or reference bus(es) wll gve dfferent ncremental lne flow wth a dfferent choce of snk or reference. However, the equvalent snks shown n Fg. 2(b) are objectvely determned by the network topology and the present operatng pont, both of whch are referencendependent. Therefore, the AC-based senstvty factor k (or k,re ) and the loss factor LF do not need a reference bus defned by users. Here s another explanaton from the mathematcal vewpont. The Z bus matrx n Eq. (39) s the nverson of the Y bus matrx. If all lne resstances and shunt branches are gnored, the AC model s smplfed to the DC model. That s, k = k,re = GSF k. Then, we cannot obtan the Z bus matrx because the N N Y bus matrx s sngular wth a rank of N 1 due to the gnored shunt branches. Hence, to make the Y bus matrx nvertble, we have to specfy a reference Generaton Center Load Center Fg. 3. The base case for smulaton test. bus. Ths means to delete the row and column assocated wth the reference bus such that the rank of the new (N 1) (N 1) Y bus matrx s N 1 (.e., nonsngular). Apparently, wth shunt branches and the AC model, the orgnal N N Y bus matrx s nonsngular and can be nversed to obtan Z bus. 4. Proposed LMP model 4.1. Applcaton of FND model for LDF Besde the proposed reference-ndependent loss factors expressed n (40) and (39), another proposed mprovement of LMP model les n the representaton of loss dstrbuton factors (LDFs). In the prevous work [8], LDFs are smply modelled wth bus loads as the weghtng factors as follows: LDF = D D = N (42) D D j The proposed mproved model apples fcttous nodal demand (FND) to represent LDFs. Ths can be done by obtanng lne flows frst va an ntal ACOPF or DCOPF. Then, the kth lne losses can be calculated as F 2 k R k and then equally allocated to each of the two connected buses [13]. The accumulated FND at each bus wll be used as the weghts to calculate LDF n Eq. (14). The new LDF model at Bus can be wrtten as: M 1 E = 2 F 2 k R k (43) LDF = E N E j (44) where M = the number of lnes connected to Bus. If compared wth the LDF model usng bus loads as weghtng factors n (42), ths FND-based model s a more reasonable approach gvng better power flow results wth very lttle extra computng and modellng effort, because there s no new varable ntroduced nto the optmzaton model n (11) (15) Proposed models Here are two models that wll be used n the smulaton test n ths paper. Model 5 (for comparson purpose): Eqs. (11) (15) for DCOPF-based dspatch; Eq. (40) and (39) for loss factors; Eq. (42) to obtan LDFs usng bus loads as weghts.

7 F. L / Electrc Power Systems Research 81 (2011) Table 2 Lne parameters. Lne AB AD AE BC CD DE R (%) X (%) B/2 (% 10 3 ) Model 6 (fnal model): Eqs. (11) (15) for DCOPF-based dspatch; Eqs. (40) and (39) for loss factors; Eqs. (43) and (44) to obtan LDFs usng the FND model. Both models use (20) (22) for LMP decomposton. It should be noted that the DC-based GSF s stll needed n the dspatch model,.e., (11) (15), to model lne flows under lnear programmng because of the nature and advantage of DCOPF n handng transmsson constrants and achevng a straghtforward LMP decomposton. Otherwse, the fnal lne flows from dspatch model cannot perfectly produce the same power flow results, and the reference-ndependent LMP decomposton cannot be acheved. The above models requre ntal values of system status, such as lne flows, generaton outputs, and so on, to obtan E n (43), LDF n (44), offset n (13), and LF n (40). Snce they need to be reference-ndependent, the ntal values of system status need to be reference-ndependent. Apparently, runnng an ntal ACOPF or lossless DCOPF should ft ths need. Then, the DC-based Model 5 or 6 can be appled for LMP calculaton wth fully referencendependent decomposton Role of the ntal OPF It s mportant to be noted that runnng an ntal ACOPF s algned wth the popular ndustral practce of Ex Post LMP model. In operaton, there are three typcal steps: 1. An ACOPF or a close approxmaton, lossless DCOPF as an extreme, s performed for Ex Ante economc dspatch. 2. State estmaton s performed to smooth measurement error and to provde nput (lne flows, generaton outputs, etc.) for LMP and other real-tme applcatons. 3. A DCOPF-wth-loss model s performed for Ex Post LMP. If we reasonably assume measurement s perfect to skp the state estmaton such that we can focus on LMP models, the above process s bascally an ACOPF (or a lossless DCOPF) for dspatch and a DCOPF-wth-loss for LMP calculaton. Snce ACOPF (or lossless DCOPF) gves reference-ndependent lne flows, generaton outputs, etc., t s justfable for the proposed process to use the ntal ACOPF results for reference-ndependent LF and then DC-based Model 5 or 6 for reference-ndependent LMP decomposton. 5. Test results 5.1. Test case The PJM 5-bus system [1] s used for smulaton n ths secton. The system confguraton, generaton bds, generaton lmts, and loads are shown n Fg. 3. Only the Lne DE s assumed to have a thermal lmt of 240 MW. The lne parameters are gven n Table 2, where the reactances are from the orgnal case n [1] and the resstances are assumed to be 10% of correspondng lne reactances. Each of the two shunt capactances of a p model transmsson lne s assumed to have a reactance value of 100 tmes the lne reactance. And the reactve generaton lmts are smply set to 150 MVar, from leadng to laggng. The above data are from [8] except the assumed Table 3 Intal dspatch results from ACOPF. Gen Alta Park Cty Soltude Sundance Brghton Dspatch (MW) Table 4 Intal MW lne flows from ACOPF. LINE AB AD AE BC CD DE Sendng end Recevng end Lne center Table 5 Results from Model 5 usng Bus A as the reference bus. Bus A B C D E Bus gen Bus load Bus loss Loss factor GSF LDF LMP LMP energy LMP loss LMP cong lne resstances, shunt capactances, and the reactve generaton lmts, whch are not avalable from [8]. An ACOPF s appled to obtan the ntal economc dspatch soluton used for the follow-up LMP calculaton. For nstance, the results can be used to obtan nodal AC currents and voltages and then the reference-ndependent k,re, as exactly shown n Secton 3.5. Then, the reference-ndependent loss factor, LF, can be calculated usng Eq. (40). The estmated system losses can be used to set the offset n Eq. (13), and the lne flows can be used to obtan E n (43) and LDF n (44). Table 3 shows the generaton output. Table 4 shows the MW lne flows at the sendng end, at the recevng end, and at the lne center. If the results from an ntal lossless DCOPF are appled to fnd reference-ndependent LF, the fnal LF s extremely close to (less than 3% error) the one obtaned usng results from ACOPF. Ths s reasonable because lnearzed DC model n hgh voltage AC system s usually consdered effcent enough Results from Model 5 (bus loads as weghts for LDFs) In ths test, LDFs are calculated usng bus loads as the weghtng factors as specfed n Model 5. So, the LDFs are 0, 0.3, 0.3, 0.4, and 0 from Buses A to E, respectvely. Here two cases are studed: (1) Bus A as the reference; and (2) dstrbuted reference buses of B, C and D wth bus loads as the weghts (.e., 0.3, 0.3, and 0.4). As shown n Tables 5 and 6, the GSFs are dfferent w.r.t. dfferent references. But the dspatches are the same. Each LMP component s also dentcal. The LMPs at the margnal unt buses (C and E) are equal to the correspondng margnal unt cost. Ths meets the prncple of LMP modellng. Fg. 4 also shows the lne flows. It can be easly verfed that the values of bus loss or bus msmatch (=ncomng lne flows + generaton outgong flows load) accountng for losses are 0, , , , and 0 at Buses A E, respectvely. There s no loss balance at Buses A and E because the LDFs are 0 at these two buses. Ths s dfferent from realty because losses are dstrbuted n each lne, and every bus should balance some losses. The next subsecton wll show that each bus wll absorb some losses

8 2002 F. L / Electrc Power Systems Research 81 (2011) Table 6 Results from Model 5 usng dstrbuted reference at B, C, and D. Bus A B C D E Bus gen Bus load Bus loss Loss factor GSF LDF LMP LMP energy LMP loss LMP cong Table 8 Results from Model 6 usng dstrbuted reference at B, C, and D. Bus A B C D E Bus gen Bus load Bus loss Loss factor GSF LDF LMP LMP energy LMP loss LMP cong Fg. 5 shows that the losses are dstrbuted n each lne and eventually balanced at every bus as FNDs (.e., E ). The values of bus losses are , , , , and at Buses A E, respectvely. It can be easly verfed that each bus loss equals to ts loss factor multpled by the system total losses. Hence, ths s a farer and more reasonable model for obtanng LDFs Comparson wth Model 3 Fg. 4. Lne flow results usng load-weghted LDF (Model 5). Fg. 5. Lne flow results usng FND-based LDF (Model 6). Table 7 Results from Model 6 usng Bus A as the reference bus. Bus A B C D E Bus gen Bus load Bus loss Loss factor GSF LDF LMP LMP energy LMP loss LMP cong correspondng to ts connectng lne losses wth the FND model for LDFs Results from Model 6 (FND-based LDFs) Here the proposed fnal model (Model 6) s used for another smulaton run, n whch the FND model s appled for calculatng LDFs. Results are shown n Fg. 5 and Tables 7 and 8. Agan, the GSFs are dfferent w.r.t. dfferent reference buses; but the dspatch results and the LMP decomposton are dentcal. Smlar to Model 5, the LMPs at margnal unt buses (C and E) are equal to local margnal unt cost. Models 5 and 6 use the same LDF model as Model 3. The loss factor model s the only dfference between Models 5 and 3. The loss factors of Model 3 are reference-dependent; therefore, LF at the reference n Model 3 should be 0. Hence, LMP loss at the reference should be 0 as well. Ths s shown n the results n [8]. For nstance, Tables III and IV n [8] show LF = 0 and LMP loss = 0 at the reference bus. Note that the weghted average values for LF and LMP loss should be used for the dstrbuted reference bus n the case of Table IV n [8]. In contrast, Models 5 and 6 gve non-zero loss factors and nonzero LMP loss at all buses. Ths should be more reasonable than Model 3 because n realty there s no reference (slack) bus and every bus should have some contrbuton to losses. Hence, LF and LMP loss should not be zero at a gven reference whch s purely man-made or user-defned. As prevously mentoned, Model 6 further mproves Model 5 by usng the FND model for a better power flow representaton such that losses are dstrbuted nto each lne, rather than load buses Comparson wth ACOPF-based LMP Although true ACOPF s not commonly used n ndustral practces due to the convergence ssue, t s a good tool for benchmark purpose because ACOPF gves the exact dspatch results consderng all transmsson and generaton constrants n full AC model. Thus, t gves the accurate LMP at each bus. Therefore, a good approxmate, wth-loss, DCOPF-based model should produce results close to that from ACOPF. However, t should be noted that decomposton of the exact ACOPF-based LMP nto three LMP components has to take some approxmaton for lnearzaton because ACOPF only gves the total LMP at each bus, whch s the Lagrange multpler of the correspondng AC power flow constrants [13]. Usually, the generaton shft factors and/or the loss factors are nvolved durng the approxmate decomposton of LMP. Hence, ths leads back to the orgnal queston of a far loss allocaton such as beng referencendependent. In other words, ACOPF gves the accurate and unque results of the generaton dspatch and the total LMP at each bus, but there s no accurate or unque LMP decomposton. An mportant goal of the LMP research works s to dentfy more reasonable LMP decompostons such as the proposed decomposton method n ths paper.

9 F. L / Electrc Power Systems Research 81 (2011) The ACOPF-based LMPs for the test system are $23.410, $28.272, $30.000, $ and $ per MWh, from Buses A to E, respectvely. The generaton dspatches are MW from Alta, 100 MW from Park Cty, MW from Soltude, 0 MW from Sundance, and MW from Brghton. If compared wth results from Model 5 or 6, the numbers are very close. As a matter of fact, the FND-based wth-loss model usually produces the dspatch and LMP results very close to that from ACOPF, as evdenced n [13]. Hence, t s reasonable that the proposed method, whch ncorporates the FND model, produces the results very close to ACOPF whle gvng a fully reference ndependent decomposton. 6. Conclusons The man contrbutons of ths paper are as follows: Frst, t presents new analytcal equatons to calculate the ACbased dstrbuton factors and then loss factors that only depend on the system topology and the present operatng pont. Hence, the proposed new model of dstrbuton factors and loss factors s reference-ndependent. The rgorous dervaton consders the change of bus voltages when there s a perturbaton of bus current njecton. Ths leads to more reasonable dstrbuton factors, f compared wth (29) from [21] that gnores the nodal voltage changes and the present operaton pont. The referencendependent LMP loss component can serve for a better loss hedgng FTR proposed n [19,20], snce t gves LMP loss prces nvarable to the energy reference bus. Next, ths paper plugs the proposed loss factor model nto the orgnal LMP Model 3 to acheve a fully reference-ndependent LMP decomposton. In addton, t also combnes the FND model n [13] nto Model 3 to obtan new loss dstrbuton factors (LDFs) such that losses are dstrbuted at each lne to acheve a farer and more reasonable model. Therefore, the fnal model of LMP decomposton usng the proposed Model 6 s fully referencendependent, and the system losses are dstrbuted at each ndvdual lne gvng a better power flow results. Future works may nclude the nvestgaton of possble applcatons of the proposed reference-ndependent dstrbuton factor and loss factor to other areas n power system analyses. Acknowledgement The author would lke to thank Eugene Ltvnov and Tongxn Zheng for many useful dscussons related to LMP. Appendx A. Lst of symbols Lagrangan multpler of Eq. (12) Lagrangan multpler of Eq. (13) k Lagrangan multpler of the kth trans. constrant n Eq. (14) k AC-based dstrbuton factors from Bus to Lne k k,re real power dstrbuton factor from Bus to Lne k obtaned from AC-based model ı, ı voltage angles at Bus and Bus j c generaton cost at Bus D demand at Bus DF margnal delvery factor at Bus G, G max, and G mn generaton output, mnmum lmt, and maxmum lmt at Bus GSF k reference-dependent, DC-based generaton shft factor (or just shft factor) of Lne k w.r.t. 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