Prcng Energy and Ancllary Servces n Integrated Maret Systems by an Optmal Power Flow Tong Wu, Member, IEEE, Mar Rothleder, Member, IEEE, Zad Alaywan, Senor Member, IEEE, Alex D. Papalexopoulos, Fellow, IEEE Abstract A detaled AC OPF-based formulaton for procurng, prcng, and settlng energy and ancllary servce n smultaneous auctons by ntegrated maret systems s presented. The paper provdes clear defntons of Locatonal Margnal Prces for energy and Ancllary Servce Margnal Prces n terms of Lagrange multplers. The characterstcs of the prces are analyzed especally when economc substtuton among ancllary servces s requred. The paper also evaluates the condtons under whch opportunty costs are ncurred to unts that provde ancllary servces. It s partcularly shown that the ntutve belef that the provson of regulaton down servce does not ncur opportunty cost to the provder, n general, s not true. Index Terms Power system economcs, optmal power flow, deregulaton, locatonal margnal prce, ancllary servce, transmsson congeston, transmsson losses, opportunty cost. C I. ITRODUCTIO OMPETITIVE energy marets are nsttuted around the world and electrc supply ndustres are restructured to compete n the new emergng marets. In general, two extreme forms of maret auctons exst for tradng of varous energy products and servces. Ther dfference stems from choosng between tghter coordnaton and greater relance on prvate marets. Certan hybrd versons that clam to obtan the best of both maret forms are also begnnng to emerge. In the frst form of aucton used n maret systems, called unbundled systems, maret products are procured sequentally through central auctons managed by the ISO/RTO. The ntal maret s the energy maret, followed by a transmsson maret to manage congestons, followed by a maret for Ancllary Servces (A/S to conform to mandated relablty crtera. The forward marets (on a day-ahead and hour-ahead bass are followed by a real tme maret n whch the ISO/RTO uses A/S energy and supplemental energy offers to balance the system n real tme. Partcpaton n each maret s voluntary, so that traders can move freely from one maret to another to arbtrage prce dfferences between the marets. Proponents of these auctons clam that the voluntary nature of maret partcpaton allows the effcences provded by an Tong Wu s an Independent Consultant afflated wth ECCO Internatonal, Folsom, CA 95630 USA (e-mal: tongwu@eee.org. Mar Rothleder and Zad Alaywan are wth Calforna ISO, Folsom, CA 95630 USA (e-mals: mrothleder@caso.com, zalaywan@caso.com. Alex Papalexopoulos s wth ECCO Internatonal, San Francsco, CA 9404 USA (e-mal: alexp@eccontl.com. optmzed pool to be captured wthout havng to deal wth all the problems assocated wth complex optmzaton software tools. []. Examples of unbundled systems are n Australa, Scandnava, Calforna 998-2000, and Texas, as well as n Brtan s new system that began operaton n 200. In the second form of auctons used n maret systems, called ntegrated systems, maret products are procured smultaneously through central auctons. The ncentve for developng ntegrated systems s to realze gans from tght coordnaton n daly operatons, whle strengthenng system relablty. The basc argument for ntegrated systems s that optmzaton s necessary to mnmze the total costs of coordnatng generaton, transmsson and reserves to meet demand and ensure relablty. Proponents of ntegrated systems clam that the resultng prcng s superor n the sense that the shadow prces derved from the constraned optmzaton accurately reflects the system-wde opportunty costs of scarce resources, both nter-temporally and spatally. Examples of ntegrated systems are n Brtan 989-200 and n the US, n ew Yor, ew England and PJM. Current experence from operatng energy marets seems to gve credence to the clam that practcal unbundled systems, as currently mplemented, are crude and ntegrated systems mght be superor, at least n the ntal stage of maret evoluton. A thorough evaluaton of unbundled and ntegrated maret systems s gven n [2]. Ths paper analyzes the characterstcs of smultaneous auctons of ntegrated systems and the prcng mechansms for smultaneously procured energy and Ancllary Servces (A/S usng an AC Optmal Power Flow (OPF-based formulaton. The ew Yor ISO (YISO has mplemented the approach of smultaneous aucton by an AC OPF formulaton [3]. The Calforna ISO s also n the process of mplementng a smlar formulaton [4][5]. However, a detaled formulaton of the problem and a clear explanaton of the mplcatons of the smultaneous formulaton have not been forthcomng. Ths paper presents a detaled mathematcal formulaton of the smultaneous aucton of energy and A/S and a rgorous analyss of the characterstcs of the prces defned by the resultng Lagrange multplers. The theoretcal analyss provded by ths paper has helped us valdate ntutve belefs and nsghts ganed over the course of many years of desgnng, mplementng and runnng energy marets, and dentfy and dscard msconceptons that unfortunately are stll prevalent n the desgn of wholesale energy marets. For example, ths paper shows that the provson of Regulaton-
2 Down A/S can ncur opportunty cost to the provder, under certan condtons, whch have been ntally perceved as counter-ntutve. Tradtonal OPF formulatons and ther soluton characterstcs are well descrbed n [6], [7], [8], [9] and [0]. The relatonshp between nodal prces and transmsson shadow prces s also well analyzed n [], [2] and [3]. The sequental aucton for A/S that s currently deployed by the Calforna ISO s descrbed n [4] and [5]. The optmzaton formulatons that form the bass of the smultaneous auctons at the ew Yor ISO and ew England ISO are descrbed n [6] and [7]. Few other papers, such as [8], are presentng the methodology of prcng energy and A/S usng OPF formulatons. However, these papers do not provde suffcent detals to allow a thorough analyss of the relatonshps among the prces for energy and A/S. In partcular, we could not fnd n the lterature the defnton and analyss of prces for A/S when economc substtuton of servces s requred. Although the Ratonal Buyer approach n [5] for procurng A/S allows economc substtuton of servces, t does not smultaneously optmze procurement of energy and A/S. Moreover, the ratonal buyer formulaton mnmzes total payment nstead of total cost; and the A/S prces so produced have exposed the Calforna ISO to fnancal neutralty problems. The rest of the paper s organzed as follows. Secton II presents the AC OPF formulaton used n ntegrated systems for the smultaneous aucton of energy and A/S. Secton III provdes nsghts on the characterstcs of locatonal margnal prces for energy and A/S. Secton IV llustrates the concepts by examples and Secton V concludes the paper. II. FORMULATIO FOR SIMULTAEOUS EERGY AD ACILLARY SERVICE AUCTIO The proposed maret desgn s based on Locatonal Margnal Prcng (LMP of energy and regonal Ancllary Servce Margnal Prcng (ASMP of Regulaton Up (Reg-Up, Regulaton Down (Reg-Down, Spnnng Reserve (Spn, and on-spnnng Reserves (on-spn. The Calforna ISO plans to operate such a maret n the future. Specfcally t wll run a Day-Ahead (DA maret and an Hour-Ahead (HA maret to aucton both energy and A/S. In both forward marets, the LMPs and the ASMPs are determned by an AC Optmal Power Flow functon that s part of a Securty Constraned Unt Commtment (SCUC program. The AC OPF functon optmally dspatches power and procures reserve capacty from the already commtted generaton, partcpatng nterchange, and dspatchable load whle satsfyng AC power flow equatons, A/S requrements, transmsson and operatng constrants. Contngences can be formulated by repeatng the power flow equatons and other voltage-dependent constrants for each contngency that needs to be consdered. However, wthout loss of generalty and n order to smplfy the exhbton of the paper, the formulaton for the normal operatng condton s used n ths paper. Although the Unt- Commtment (UC problem s an ntegral part of the new CAISO maret desgn, the descrpton of the UC problem s not wthn the scope of ths paper. The focus of ths paper s the AC OPF that determnes the fnal schedules and prces based on the UC results. Inter-temporal constrants are not modeled explctly n the formulaton. In the general case, the AC OPF functon needs to be ntegrated wth an AC OPF functon to resolve nter-temporal constrant volatons. Certan algorthms for modelng rampng constrants n the UC formulaton can mnmze the number of teratons at the expense of ncreased dmenson. However, the focus of ths paper s to defne and solve the problem for prcng assumng a set of scheduled unts rather than developng algorthms for solvng the general schedulng problem n ts entrety. A. Obectve The obectve of the AC OPF formulaton s to mnmze the sum of energy costs and A/S costs over a prescrbed settlement nterval. The settlement nterval s usually one hour for the ntegrated forward marets. Each energy cost functon s a pecewse lnear convex curve. Each A/S cost functon s a lnear functon represented by the product of the procured quantty and the bd prce. The energy cost curve for each resource s ether maret-based bds submtted by the resources or cost-based bds calculated from heat-rate, gas prce ndex, and operaton and mantenance costs. The mnmum load cost and the startup cost are not ncluded n the OPF formulaton because they have been consdered n the unt commtment stage of the SCUC formulaton. The obectve for the AC OPF s to mnmze the sum of the followng components: ( Energy Costs, ( Reg-Up Costs, ( Spn Costs, (v on-spn Costs, and (v Reg-Down Costs. Mathematcally the obectve s to mnmze: C Total (. C ( P C [ P ( x ] I ( I C I ( I C C ( C ( ( where the symbols are defned as follows: C Total (. Total cost of energy and A/S C (P Energy cost functon at node C ( on-spn cost functon at node C ( Reg-Down cost functon at node C ( Reg-Up cost functon at node C ( Spn cost functon at node C [P (x] Energy cost functon of reference node I Set of nodes provdng non-spn reserve I Set of nodes provdng Reg-Down I Set of nodes provdng Reg-Up I Set of nodes provdng Spn on-spn reserve provded by node Reg-down capacty provded by node Reg-Up capacty provded by node Spn capacty provded by node B. Power Balance Constrant The power balance constrants are descrbed by the AC power flow equatons. The demand s the scheduled quantty n the
3 forward energy maret. To smplfy the presentaton, t s assumed n ths paper that each node has at maxmum one resource. In a practcal mplementaton when there are multple resources connected to the same bus, each resource necton s modeled as a separate varable and the bus necton s consdered to be the sum of all the resource nectons. Gven a power system wth nodes, we number the nodes as follows for convenence of reference: PQ nodes (.e., load or generator operatng at reactve power lmt are numbered from to d. PV nodes (.e., generator or load wth voltage control are numbered from d to d g. Slac node (.e., the reference node s numbered as the last bus,. The set of AC power flow equatons generally conssts of: d equatons that descrbe the actve power balance at the PQ nodes. g equatons that descrbe the actve power balance at the PV nodes, d equatons that descrbe the reactve power balance at the PQ nodes. Mathematcally, the equatons are descrbed as follows: P ( x, P P ( x P 0 for, 2,, - (2 Q ( x, Q Q ( x Q 0 for, 2,, d (3 where x[θ, θ 2,..., θ Ν V, V 2,, V d ] T representng the voltage phase angles θ and magntudes V. Equaton (2 represents actve power balancng equatons at all nodes except the reference node and P denotes actve power necton at node. Equaton (3 represents reactve power balancng equatons at the PQ nodes and Q denotes reactve necton at node. The actve power loss of the system s determned by (4 P ( x P 0 (4 loss where P loss denotes the actve power transmsson loss of the system. C. Capacty Reserve Constrants Reg-Up, Reg-Down, Spn on-spn are procured optmally to mnmze the total cost of energy and reserves. Voltage Support and Blac Start servces are procured by resource specfc agreements between the ISO and the supplers, whch are not part of the optmzaton process. The capacty reserve constrants are nequalty constrants to ensure the rght amount of capacty s procured accordng to the prescrbed A/S requrements by A/S regons that are defned off-lne based on the ISO load forecast and other operatng system condtons consstent wth ERC standards. The resources wthn the same A/S regon must meet a prescrbed porton of the regonal A/S requrements. Moreover, the followng substtutons are allowed: ( Reg-Up can meet Spn and on-spn requrements; ( Spn can meet on-spn requrements. Regulaton Up Requrement Equaton (5 specfes the amount of Reg-Up that needs to be procured from generators n each regon : I Z R 0 (5 where the symbols are defned as follows: R Requrement of Reg-Up n regon Set of nodes n regon Z 2 Spnnng Reserve Requrement Equaton (6 specfes the total amount of Reg-Up and Spn that needs to be procured from resources n each regon : I Z R R 0 (6 I Z where R denotes the requrement of Spn n regon 3 on-spnnng Reserve Requrement Equaton (7 specfes the total amount of Reg-Up, Spn and on-spn that needs to be procured from resources n each regon : R R R 0 (7 I Z I Z I Z where R denotes the requrement of on-spn n regon 4 Regulaton Down Requrement Equaton (8 specfes the amount of Reg-Down, R, that needs to be procured from generators n each regon : R 0 (8 I Z 5 Regulaton Up Bd Lmt The awarded quantty for Reg-Up for each generator must be non-negatve and may not be greater than an upper lmt,, whch represents the bd lmt or physcal lmts such as ramp rates. 0 (9 6 Spnnng Bd Lmt Smlarly, the awarded quantty for Spn s non-negatve and lmted by an upper lmt,, as follows: 0 (0 7 on Spnnng Bd Lmt The awarded quantty for on-spn s also non-negatve and lmted by an upper lmt,, as follows: 0 ( 8 Regulaton Down Bd Lmt The awarded quantty for Reg-Down s also non-negatve and lmted by an upper lmt,, as follows: 0 (2 D. Supply Constrants Actve power supples are lmted by avalable capacty and rampng capablty as follows.
4 Actve Power mum Lmt The total power output plus the capacty reserves for Reg-Up, Spn and on-spn from each resource should not exceed ts maxmum operatng lmt,.e., P P 0 (3 where P s the maxmum operatng lmt of the resource at node for the partcular hour. 2 Actve Power Mnmum Lmt A generator once commtted must mantan a mnmum output. In addton, f a generator provdes Reg-Down, t must produce addtonal power to mae room for Reg-Down capacty. Such constrants are descrbed as follows: Mn P P 0 (4 where P Mn s the mnmum load of the resource at node. 3 Ten-mnute Ramp Lmt The total amount of provson for Reg-Up, Spn and on-spn from a resource,, s lmted by ts rampng capablty wthn 0 mnutes. 0 0 (5 OP RR RR where RR and RR OP are the ramp rates of resource n MW/mnute for provdng Reg-Up and operatng reserves. E. etwor Constrants etwor constrants n ths paper nclude the followng types: Reactve power supply lmts Voltage magntude and phase angle lmts Branch flow lmts, and Other networ lmts such as nomograms [9] Any networ constrant except te-lne constrants can be represented n the followng form: F F ( x 0 (6 where F (x s the quantty that s lmted by constrants ; and F s the upper lmt of the quantty descrbed by constrant. Specal examples of (6 nclude reactve power supply lmts and voltage lmts as follows: Q Q ( x 0 for d, d 2,, d g (7 Mn Q Q ( x 0 for d, d 2,, d g (8 where Q s the upper lmt of reactve power necton at node ; and Q Mn s the lower lmt of the reactve power necton at node. ote Q (x at PV nodes are functons of voltage varables. V Mn V V 0 for, 2,, d (9 V 0 for, 2,, d (20 where V s the upper lmt of voltage magntude at node ; and V Mn s the lower lmt of the voltage magntude at node. θ Mn θ θ 0 for, 2,, - (2 θ 0 for, 2,, - (22 where θ s the upper lmt of voltage phase angle at node ; and θ Mn s the lower lmt of the voltage phase angle at node. Equaton (6 also descrbes transmsson lmts n the followng forms: Branch or Branch Group Lmt: Power flow lmt on an ndvdual transmsson branch or a branch group represented by a constant. omogam: Power flow lmt on a transmsson nterface represented by a functon of other varables such as output of a certan group of generators, load of a certan area, or flows on other transmsson nterfaces. Such functons are modeled by pece-wse lnear functons. Reg-Up, Spn and on-spn provded from resources outsde of the ISO control area can compete wth energy schedules for transmsson usage on te lnes based on ther capacty bds. When A/S compete wth energy schedules for the use of te-lne, the constrant s descrbed as follows: ( F 0 F ( x (23 T where, and are quanttes of Reg-Up, Spn and on-spn capacty from resource provded across te lne ; T denotes the set of resources that compete for the use of telne. III. DEFIITIO, CHARACTERISTICS AD COMPOETS OF EERGY AD ACILLARY SERVICE PRICES A. Lagrange Functon The Lagrange functon n our formulaton s as follows. L C ( P C [ P ( x ] (Energy Cost C ( (Reg-Up Cost I C ( (Spn Cost I C ( (on-spn Cost I C ( (Reg-Down Cost I [ P ( x P ] d λ (Actve Power Balance [ Q ( x Q ] γ (Reactve power balance at PQ nodes I Z λ R (Reg-Up Requrement λ I Z I Z (Spn Requrement R R
λ R R R I Z I Z I Z (on-spn Requrement λ R (Reg-Down Requrement I Z π ( P P ( MW Lmt Mn Mn π ( P P (Mn MW Lmt ( I α ( Reg-Up Lmt ( I β (Mn Reg-Up Lmt ( I α ( Spn Lmt ( I β (Mn Spn Lmt ( I α ( on-spn Lmt ( I β (Mn on-spn Lmt ( I α ( Reg-Down Lmt ( I β (Mn Reg-Down Lmt OP α ( 0RR (0-Mn Ramp Lmt µ F (x ( F (etwor Lmt T The Gree symbols represent the Lagrange multplers assocated wth the correspondng constrants. The LMPs and the ASMPs are defned based on these multplers. B. Locatonal Margnal Prces for Energy The LMP for settlng energy at node equals to the ncremental cost of supplyng an addtonal MW of power at node. Suppose the power balance equaton at node s perturbed by P P (x P. The ncremental cost for the perturbaton at the optmal pont s as follows: P [ P (x P ] λ (24 where λ s the LMP for actve power at node. As s shown n the lterature [3][4], each nodal prce can be decomposed nto three components: ( ncremental cost at the reference bus, ( ncremental cost of thermal transmsson losses, and ( ncremental cost of networ constrants whch nclude transmsson constrants, power supply constrants, voltage constrants and phase angle constrants,.e., λ λ λ L µ S (25 where: C λ P System margnal cost of energy at the reference node Ploss L P The -th Loss Contrbuton Factor. F S P The senstvty of the quantty lmted by constrant wth respect to actve power nected nto node and wthdrawn at the reference node. The LMPs are not affected by the choce of the reference node because the losses are optmally dstrbuted accordng to the supply bds by the OPF. However, the Loss Contrbuton Factors are affected by the choce of the reference node. To avod the controversy regardng the selecton of the reference node, we recommend usng the entre LMP value at each node rather than ts ndvdual components for settlement purposes. If the loss component of the LMP value at each bus has to be settled separately from energy for commercal, regulatory, or other reasons, a Load Center Penalty Factor [20] approach can be used. C. Ancllary Servce Margnal Prces (ASMP Defntons The ASMP for an A/S n regon s the ncremental cost for meetng an addtonal MW of the requrement for the A/S n regon as follows: R R R R λ λ λ λ λ λ λ 5 (Reg-Up Prce (26 (Spn Prce (27 (on-spn Prce (28 (Reg-Down Prce (29 2 Propertes for eutralty It s shown next that when a hgher qualty servce s procured to meet the requrement of a lower qualty servce, e.g., Reg- Up s procured to meet the Spn requrement, the ASMPs for the two servces are equal. Suppose Reg-Up s used to meet the Spn requrement n regon ; that s:
R < 0 (30 I Z Accordng to the Kuhn-Tucer condtons, the followng holds true: λ R 0 λ 0 (3 I Z Consequently, the ASMPs for Reg-Up and Spn n regon are the same,.e., λ λ (32 Smlar analyss can be carred out for substtuton among other A/S. Ths mportant property allows A/S costs ncurred to the ISO be allocated to maret partcpants wthout the neutralty problem that occurs to the ratonal buyer approach [5]. To llustrate ths pont, let s contnue wth the above scenaro but further assume that there s only one A/S regon and that the partcpants are oblgated to pay exactly the requrements for Reg-Up and Spn. The total payment from the ISO to the provders s: Payment I Z I The total charge to the maret partcpants s: Charge R R Z (33 (34 When Reg-Up s used to meet the requrement for Spn only, we have: I Z > R I Z, < R and (35 R R 0 (36 I Z I Z The neutralty mbalance for the ISO s the dfference between the payment and the charge as shown n (37; t s zero f and only f the prces for Reg-Up and Spn are equal. Payment Charge D. Opportunty Costs I R Z Opportunty Costs for Provson of Regulaton Up, Spnnng Reserve and on-spnnng Reserve Reg-Up, Spn and on-spn are referred to as upward reserves. A suppler of upward reserves may sell less energy n a forward maret (.e., DA or HA than t would have been economc for t to sell because of the provson for upward reserves. Ths happens f and only f the resource s constraned by ts maxmum capacty. Suppose resource partcpates n the energy and only n the on-spn marets. Accordng to the Kuhn-Tucer condton, P C condton, (37 C C P λ π Mn α β α OP µ 0 6 (38 λ π π 0 (39 In order to demonstrate the opportunty costs, let s assume, wthout loss of generalty, that resource s an nternal resource (.e., elmnatng µ from (38, provdng a postve amount of on-spn (.e, β 0, not constraned by ts bd quantty (.e., α 0, not constraned by ramp rate (.e., α OP 0, and not constraned by ts mnmum load lmt (.e., π Mn 0. Under these assumptons, C λ π (40 C π λ (4 P ow f the resource s not lmted by ts maxmum capacty (.e., π 0, the ASMP for on-spn n regon and the LMP for node are determned to be the margnal cost of on- Spn and the margnal cost of energy ndependently. However, f the margnal resource for on-spn s lmted by ts maxmum capacty (.e., π > 0, we have, C C λ λ (42 P Equaton (42 shows that the ASMP for on-spn consst of two components: ( the margnal on-spn bd prce, and ( a component that represents the opportunty cost to resource for the provson of on-spn nstead of energy. Ths analyss can be done for other upward A/S or under more complex condtons. The concluson, however, remans the same. 2 Opportunty Costs for Provson of Regulaton Down A suppler of Reg-Down may have to sell more energy n a forward maret (.e., DA or HA than t would have been economc for t to sell because of the provson for Reg-Down. Ths happens f and only f the resource s constraned by ts mnmum load lmt. Suppose resource partcpates n the Reg-Down maret only. Accordng to the Kuhn-Tucer Mn λ π α β 0 (43 In order to derve the opportunty costs, assume that resource s provdng a postve amount of Reg-Down (.e, β 0, not constraned by ts upper lmt (.e., α 0, and not constraned by ts maxmum capacty (.e., π 0. Under these assumptons, C Mn λ π (44
7 Mn C π λ (45 P ow f the resource s not lmted by ts mnmum capacty (.e., π Mn 0, the ASMP for Reg-Down n regon and the LMP for node are determned to be the margnal cost of Reg- Down and margnal cost of energy ndependently. However, f the resource s lmted by ts mnmum capacty (.e., π Mn > 0, we have, C C λ λ (46 P Equaton (46 shows that the ASMP for Reg-Down conssts of two components: ( the margnal Reg-Down bd prce, and ( a component that represents the opportunty cost to resource due to sellng energy below ts bd prce as a consequence of provdng Reg-Down. E. Prce for Transmsson Defnton of Shadow Prce The shadow prce for transmsson constrant s determned to be the margnal cost of constrant as follows: F µ (47 2 Prce for Pont-To-Pont Transmsson The prce for usng the transmsson system to delver one MW from node to node s defned as follows: λ λ λ L L µ ( S S (48 ( The frst term on the rght hand sde of (48 represents the cost of losses attrbutable to the transacton between node and node ; the second term represents the cost of transmsson constrants ncludng thermal lmts on transmsson branches (groups, reactve power lmts, voltage lmts and other general nomogram constrants. 3 Prce for etwor Servce Transmsson To avod double subscrpts n notatons, any networ servce rght can be descrbed as the rght of sendng (p, p 2,, p s % of one MW at nodes (,2,, s and recevng (p s, p s2,, p sr % of one MW at nodes (s, s2,, sr. Usng ths notaton, the prce for payng any networ servce rght s descrbed as follows: s r s λ λ p s L p s λ p s r s L p µ s S p s r S s (49 p The frst term on the rght hand sde of (49 represents the cost of losses attrbutable to the transactons assocated wth the transmsson servce; the second term represents the cost of transmsson constrants ncludng thermal lmts on transmsson branches (groups, reactve power lmts, voltage lmts and general nomogram constrants. 4 Total Congeston Revenue from Energy Settlement The total congeston revenue collected by the ISO through LMP settlements for energy has the followng relatonshp: CR λ P µ F λ L P Ploss (50 The left hand sde of (50 ndcates that the total amount of congeston revenue s the leftover from the energy settlement at all the nodes. Replacng the λ n (50 by the expresson n (25, one can obtan, after some manpulaton, the rght hand sde of (50. On the rght hand sde of (50, the frst term represents the revenue assocated wth transmsson congeston and resource lmt volatons. The second term represents an over collecton from the compensaton of margnal losses. 5 Transmsson Prce for Importng Ancllary Servces Snce A/S requrements are determned accordng to load and export quanttes of the nternal control area, A/S provded from outsde of the control area partcpate n the A/S aucton n the A/S regons wthn the control area. A/S mports are settled by the same regonal ASMPs that are used by the nternal resources. However, mportng Reg-Up, Spn and on-spn through congested te lnes ncurs congeston charges to the mporters. Ths charge s prced by the shadow prce on the te lne. Suppose an external resource s provdng on-spn over te lne. Consder the same Kuhn Tucer condton shown n (38. Snce mports are not lmted by capacty or ramp rates, α OP π π Mn 0 and snce the mport of on-spn s postve, β 0. The only constrants that are potentally actve are the on-spn max bd lmt and the te-lne flow lmt. Therefore, C λ α µ (5 After applyng the congeston charge to resource, the actual prce receved by resource for provdng on-spn s as follows: C λ µ α (52 On the rght hand sde of (52, the frst term represents the bd prce of resource for on-spn, and the second term represents the suppler s surplus, whch s zero f resource s the margnal suppler for on-spn n regon. IV. EXAMPLES The examples are desgned to facltate understandng of the paper rather than presentng smulaton results. In order to focus on the ey ssues, a smple 3-node DC networ as shown n Fg. s used. The networ has 3 dentcal branches; each branch s rated at 50 MW n both drectons. The load L 3 has a fxed schedule of 50 MW. All the 3 generators G, G 2 and G 3 can operate between 0 and 00 MW wth nfnte rampng
8 capablty. It s further assumed that all the generators are located wthn the same A/S regon nsde the control area; and therefore the Spn reserves do not compete wth energy for the congested transmsson networ. The Spn requrement, R, s 0 MW n Case and 30 MW n Case 2, whch are used to llustrate ASMPs wthout and wth opportunty costs. G G 2 ode #2 ode # ode #3 Fg.. A 3-bus DC networ wth 3 dentcal branches Case : R 0 MW As s shown n Table I, all generators offer both energy and Spn reserve; G s the most economc energy and Spn provder. Snce the three branches are dentcal, to supply each MW of power from G to L 3, 2/3 MW goes through Branch ##3; and /3 MW goes through Branches ##2 and #2#3. The followng smplfed DC OPF problem s formulated: Mnmze: 0P 30P 2 45P 3 5 5 2 40 3 (53 Subect to the constrants: Power Balance: P P 2 P 3 50 (54 Spn Requred: 2 3 R (55 Flow (# #2: 50 (/3 P (/3 P 2 50 (56 Flow (# #3: 50 (2/3 P (/3 P 2 50 (57 Flow (#2 #3: 50 (/3 P (2/3 P 2 50 (58 Capacty lmts: 0 P 00 for, 2, 3 (59 Lower Bounds: 0 P and 0 for, 2, 3 (60 Resources TABLE I BIDS AD RESULTS FOR THE 3-ODE EXAMPLE I CASE Energy Bd Prce ($/MWh Spn Bd Prce ($/MW Total Capacty (MW Energy Schedule (MWh LMP ($/MWh L 3 Spn Award (MW G 3 Spn ASMP ($/MW Opportunty Cost ($/MW G 0 5 00 75 0 0 5 0 G 2 30 5 00 0 27.5 0 /A /A G 3 45 40 00 75 45 0 /A /A L 3 Fxed /A 50 50 45 /A /A /A The soluton n ths case s obtaned by solvng a smple LP problem; the results are gven n Table I. Snce G and G 2 compete for the use of Branch # #3 and G s far more compettve than G 2 n terms of provdng energy, G s awarded a 75 MWh energy schedule to fully utlze the 50 MW capacty of Branch # #3; G 3 pcs up the other 75 MW of load. The resultng LMPs at ode # and ode #3 are set by G and G 3 at $0/MWh and $45/MWh, respectvely. The LMP for ode #2 s calculated usng (25 dsregardng the loss component,.e., λ 2 λ 3 (/3µ 3 where µ 3 s obtaned from λ λ 3 (2/3µ 3. Snce λ $0/MWh and λ 3 $45/MWh, µ 3 $52.5/MWh and λ 2 45 (/3*52.5 $27.5/MWh. The ASMP for Spn s set by G at $5/MW. There s no opportunty cost for G n ths case because G stll has unused capacty after provdng energy and the Spn. Case 2: R 30 MW The soluton n ths case s also obtaned by solvng the LP problem; the results are gven n Table II. Snce G has reached ts full capacty by provdng 70 MW of energy and 30 MW of Spn, t s not a margnal unt and cannot set the margnal prce. The resultng LMPs at ode #2 and ode #3 are set by G 2 and G 3 at $30/MWh and $45/MWh, respectvely. The LMP for ode # s calculated usng (25 dsregardng the loss component,.e., λ λ 3 (2/3µ 3 where µ 3 s obtaned from λ 2 λ 3 (/3µ 3. Snce λ 2 $30/MWh and λ 3 $45/MWh, µ 3 $45/MWh and λ 45 (2/3*45 $5/MWh. The ASMP for Spn s set by G at $0/MW that equals to G s Spn bd of $5/MW plus the opportunty cost of $5/MW that equals to the LMP at ode # mnus G s energy bd. TABLE II BIDS AD RESULTS FOR THE 3-ODE EXAMPLE I CASE 2 Resources Energy Bd Prce ($/MWh Spn Bd Prce ($/MW Total Capacty (MW Energy Schedule (MWh LMP ($/MWh Spn Award (MW Spn ASMP ($/MW Opportunty Cost ($/MW G 0 5 00 70 5 30 0 5 G 2 30 5 00 0 30 0 /A /A G 3 45 40 00 70 45 0 /A /A L 3 Fxed /A 50 50 45 /A /A /A V. COCLUSIO A detaled formulaton of smultaneous energy and A/S auctons for ntegrated maret systems s presented. Rgorous defntons are gven for the Locatonal Margnal Prces (LMP for energy and the Ancllary Servce Margnal Prces (ASMP when economc substtuton among A/S s requred. The paper analyzes and provdes nsghts on the propertes of the prces and the relatonshps among the prces that are determned by ths optmal power flow formulaton. The followng fndngs resulted from the analyss of the prces resultng from our formulaton are not at all ntutve, and n some cases are counter-ntutve: Provson of Reg-Down ncurs opportunty cost to the provder f the unt s constraned by a mnmum schedule lmt n order to mae room for the provson of Reg- Down and therefore has to provde energy at a prce below ts bd. Provson of upward A/S (.e., Reg-Up, Spn and on- Spn ncurs opportunty cost f the unt s operatng aganst ts maxmum operatng lmt and therefore has to provde less energy than t s economc for t to provde n
9 order to leave room for the provson of the upward A/S. The ASMPs nclude compensaton for opportunty costs f there s any; no addtonal payment s necessary for opportunty costs ncurred to A/S provders. The ASMP for Reg-Up s never less than the ASMP for Spn. The ASMP for Spn s never less than the ASMP for on-spn. When one type of A/S capacty s procured to meet the requrement of another type of A/S capacty, the ASMPs for the two types of A/S are equal. The congeston charge to the A/S mport across a congested nterface s prced by the shadow prce of the nterface, whch s determned by the energy bds alone. The total congeston revenue collected by the ISO through LMP for energy based on an AC OPF ncludes not only congeston charges but also an over collecton of compensaton for losses. VI. ACKOWLEDGMET The authors gratefully acnowledge Drs. George Angelds and Roger Trenen for ther nsghts on maret desgn ssues related to ths subect. VII. REFERECES [] S. S. Oren, Authorty and Responsblty of the ISO: Obectves, Optons and Tradeoffs, n Desgnng Compettve Electrcty Marets, Hung-po Chao and Hllard G.Huntngton, Ed. Boston: Kluwer Academc Publshers, 998 [2] Robert Wlson, Archtecture of Power Marets, 999 Presdental Address to the Econometrc Socety presented at the orth Amercan Far Eastern, Australasan and European regonal meetngs. [3] YISO Tarff, Avalable: http://www.nyso.com/servces/oatt.html [4] CAISO 28-Jun MD02 Tarff Flng of Clean Tarff Sheets & Errata. Avalable: http://www.caso.com/ [5] July 7, 2002 FERC Order on the Calforna Comprehensve Maret Redesgn Proposal n Docet os. ER02-656-000, et al.. Avalable: http://www.caso.com/pubnfo/ferc/rulngs/ [6] H. W. Dommel, and W. F. Tnney, "Optmal Power Flow Solutons," IEEE Trans. Power Apparatus and Systems, vol. pas-87, pp. 866-876, Oct. 968. [7] A. D. Papalexopoulos, C. F. Imparato and F. F. Wu, Large-Scale Optmal Power Flow: Effects of Intalzaton, Decouplng and Dscretzaton, IEEE Transactons on Power Apparatus and Systems, Vol. PWR5 4, pp. 748 759, May 989. [8] A.D. Papalexopoulos, et. al. Challenges to Optmal Power Flow, 96 WM 32-9, PWRS, presented at the 96 IEEE Wnter Power Meetng, Baltmore, February 996. [9] B. Stott, O. Alsac, and A. J. Montcell, Securty Analyss and Optmzaton, Proceedngs of the IEEE, vol. 75, pp. 623-644, Dec. 987. [0] S. M. Shahdehpour, and V. C. Ramesh, "onlnear Programmng Algorthms and Decomposton Strateges for OPF," n IEEE/PES Tutoral Course, Optmal Power Flow: Soluton Technques, Requrements, and Challenges, pp. 0-25, 996. [] S. V. Venatesh, W-H. Edwn Lu and A.D. Papalexopoulos A Least Squares Soluton for Post Optmal Power Flow Senstvty Calculaton, IEEE Transactons on Power Systems, Vol. 7, pp. 394 40, August 992. [2] F. Wu, P. Varaya, P. Spller, and S. Oren, Fol Theorems on Transmsson Access: Proofs and Counterexamples, Journal of Regulatory Economcs; 0:5-23 (996, Kluwer Academc Publshers. [3] P. R. Grb, G. A. Angelds, and R. R. Kovacs, Transmsson Access and Prcng wth Multple Separate Energy Forward Marets, IEEE Trans. Power Systems, vol. 4, pp. 865-876, Aug. 999. [4] H. Sngh, A. D. Papalexopoulos, Compettve Procurement of Ancllary Servces by an Independent System Operator, IEEE Trans. Power Systems, vol. 4, o. 2, pp. 498-504, May. 999. [5] Y. Lu, et. al. A Ratonal Buyer s Algorthm Used for Ancllary Servce Procurement, presented at 2000 IEEE/PES Wnter Meetng, Sngapore, Jan. 23-27, 2000 [6] A. I. Cohen, V. Brandwan, and S. K. Chang, Securty Constraned Unt Commtment for Open Marets, Proc. 2 st Internatonal Conference on Power Industry Computer Applcatons, pp. 39-44. [7] K. W. Cheung, P. Shamsollah, and D. Sun, Energy and Ancllary Servce Dspatch for the Interm ISO ew England Electrcty Maret, Proc. 2 st Internatonal Conference on Power Industry Computer Applcatons, pp. 47-53. [8] J. Kumar, and G. Sheble, Framewor for Energy Broerage System wth Reserve Margn and Transmsson Losses, IEEE Trans. Power Systems, vol., pp. 763-769, ov. 996. [9] J. D. McCalley, S. Wang, R. T. Trenen, and A. D. Papalexopoulos, "Securty Boundary Vsualzaton For Systems Operaton," presented at 996 IEEE/PES Summer Meetng, Denver Colorado, July 28 Aug., 996 [20] A. J. Wood, and B. F. Wollenberg, Power Generaton, Operaton and Control, 2 nd Ed., John Wley & Sons, 996 VIII. BIOGRAPHIES Dr. Tong Wu (M 995 receved a Ph.D. degree n Electrcal Engneerng from Drexel Unversty n 995. He s an Independent Consultant afflated wth ECCO Internatonal, Inc, worng prmarly for the Calforna ISO. Pror to startng hs consultng career n 2000, he was a Prncpal Regulatory Analyst wth PG&E. Pror to onng PG&E n 997, he taught Electrcal Engneerng at the Unversty of Hong Kong. Hs specal felds of nterest nclude desgn of electrc power marets and related computer applcatons. Mar Rothleder (M 989 receved hs B.S. n E.E. from Calforna State Unversty Sacramento, n 989. He oned PG&E and held varous postons n Substaton Engneerng, Power Control Operatons and Transmsson Plannng. Mar oned the Calforna ISO n 997 and later became the Manager of Maret Applcaton overseeng the development, support and testng of all maret applcatons. Currently, Mr. Rothleder s on a specal assgnment managng the mplementaton of the Maret Desgn 2002 effort. Mr. Rothleder was the Char of the San Francsco Chapter of the Power Engneerng Socety. Zad Alaywan (M 987 receved hs B.S. and M.S. degrees n Electrcal Engneerng from Montana State Unversty n 987. He s currently Drector of Maret Operatons at the Calforna ISO. He has drected the desgn, development and mplementaton of the bddng, prcng, and schedulng systems at the CAISO. Pror to onng the CAISO, he wored at Pacfc Gas & Electrc Company n varous postons n system operatons, power plant operaton, and transmsson plannng. Mr. Alaywan s a certfed PE n the State of Calforna. Dr. Alex Papalexopoulos (M 980, SM 985, F 200 s presdent and founder of ECCO Internatonal, an Energy Consultng Company that provdes consultng servces on electrcty maret desgn and software ssues wthn and outsde the U.S. to a wde range of clents such as Regulators, Governments, Utltes, Independent System Operators, Power Exchanges, Mareters, Broers and Software vendors. ECCO Internatonal s currently nvolved n varous energy restructurng proects around the world ncludng orth Amerca, Europe, and Asa. Pror to formng ECCO Internatonal n 998, Alex was a drector of the PG&E s Electrc Industry Restructurng Group n San Francsco, Calforna. He receved the Electrcal and Mechancal Engneerng Dploma form the Unversty of Athens, Greece n 980 and the M.S. and Ph.D. degrees n Electrcal Engneerng from the Georga Insttute of Technology, Atlanta, Georga n 982 and 985, respectvely. He has publshed numerous scentfc papers n IEEE and other Journals and he s the 996 recpent of IEEE s PES Prze Paper Award. Dr. Papalexopoulos s a Fellow of IEEE.