Pricing System Security in Electricity Markets. latter might lead to high prices as a result of unrealistic

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1 1 Pro. Bulk Power Systems Dynams and Control{V, Onomh, Japan, August Prng System Seurty n Eletrty Markets Claudo A. Ca~nzares Hong Chen Wllam Rosehart UnverstyofWaterloo Unversty of Calgary Dept. Eletral & Computer Eng. Dept. Eletral & Computer Eng. Waterloo, ON, Canada N2L 3G1 Calgary, AB, Canada C.Canzares@ee.uwaterloo.a rosehart@eee.org Abstrat Ths paper proposes and desrbes n detal two dstnt and novel methodologes to try to address the ssue of a proper representaton of system seurty n the operaton of deentralzed and hybrd eletrty markets, so that adequate pre sgnals an be gven to all market partpants. Both methodologes are based on the use of voltage stablty theory to better aount for system seurty. For eah of the presented tehnques, a methodology to determne \nodal" margnal pres s proposed and desrbed n detal. A 6-bus system wth both supply and demand-sde bddng s desrbed and used to test and ompare the proposed tehnques. Keywords Eletrty markets, loatonal margnal pres (LMP), seurty, avalable transfer apablty (ATC). I. Introduton THE urrent rush to deregulate eletrty markets throughout the world has led to the proposal of several market strutures that ould be ategorzed nto three man groups, namely, pool or entralzed markets (e.g., the \old" U.K. market, Chle, and PJM), standard auton or deentralzed markets (e.g., Span and the former Calforna markets), and spot prng or hybrd markets (e.g., New Zealand, Ontaro, and the urrent U.K. and New England markets) [1], [2], [3]. A. Centralzed or Pool Markets Centralzed markets an be basally vewed as unt ommtment problems where a \entral" broker/operator takes are of \dspathng" the market partpants based on ther bds, whle aountng somewhat for the transmsson system and network seurty. Beause of the omplexty of the problem, whh s n tself a dsadvantage sne t s not \transparent" to the market partpants, the network s usually modeled usng only a d power ow model. Lmts on the transmsson lne power ows used to try to represent system seurty tend to be somewhat arbtrary, as these lmts are predetermned based on o-lne studes that do not neessarly orrespond to the atual system operatng ondtons. Ths model has the pereved advantage of produng an \optmal" soluton, takng nto onsderaton temporal restrtons suh as mnmum up and down tmes, but there s no guarantee that a unque soluton, or even a soluton may be found for ths problem [4]. The d model used to represent the network leads to solutons that most lkely do not reet the atual seurty level of the system, espeally f onservatvely low power ow lmts are used n the transmsson system, whh s usually the ase. The latter mght lead to hgh pres as a result of unrealst onstrants beomng atve. Another dsadvantage of ths methodology s that t fores the unbundlng of anllary serves, suh as reatve power, requrng teratve tehnques to obtan adequate solutons for energy and anllary serve markets [2]. Although some of these problems ould be resolved n theory by nludng the a power ow equatons, ths s not onsdered pratal due to the omplexty of the problem. Thus, ths problem, together wth the \entralzed" nature of ths type of market, gves partpants a sense of lak of \transpareny" n the handlng of transatons. Ths has led to the use of other market arhtetures n the mplementaton of most new eletrty markets. For all of these reasons, the ssue of proper onsderaton and prng of seurty nthstype of market struture s not pursued further n ths paper. B. Deentralzed or Smple Auton Markets Deentralzed markets are basally \transparent" markets run by a entral broker where only the partpants' bds are used to determne a market learng pre usng a smple auton mehansm, wthout onsderng system onstrants. The results of ths auton are then passed on to a market operator who may approve, modfy and/or rejet the transatons, dependng on the market rules. In these types of markets, anllary serves are basally unbundled and determned n sequental markets (e.g., reatve power and reserve markets). Ths approah has the problem of not onsderng the atual system ondtons durng the bdng proess, partularly system seurty, whhmay lead to nadequate pre sgnals that would make t dult for the partpants to develop approprate bdng strateges. The present paper proposes a methodology to properly ntrodue system seurty nthstype of market struture, so that pres somewhat reet the \level" of seurty of the real system. The proposed methodology s based on an teratve omputaton of the Avalable Transfer Capablty (ATC) and the \redspath" of generators and loads by determnng the mpat of the derent transatons through senstvty analyss. The ATC and requred senstvtes are all determned based on voltage stablty rtera. These

2 2 senstvtes are also used to determne a margnal pre for eah of the market partpants, so that ther eet on system seurty s aounted for n the nal pres. C. Hybrd or OPF-based Markets Hybrd markets are based on lassal spot prng theory [1], [2], and allow for an adequate bundlng of energy and anllary serve markets, so that proper pre sgnals an be gven to all market partpants from a global system operaton perspetve through the use of loatonal or \nodal" margnal pres [5]. These markets an be vewed basally as Optmal Power Flow (OPF) \seurty onstraned" problems, wth the objetve of maxmzng soal welfare as opposed to mnmzng osts,.e., the objetve s to mnmze the derenes between supply and demand bds. The unt ommtment problem s assumed to be resolved by \preproessng" of the bds for example, temporal restrtons suh asramp rates may be nluded as lmts on the power bds (e.g., [6]). An advantage of these types of markets s that there s extensve experene wth the use of OPF n power system operaton, wth mature soluton methodologes that have been proven to work n prate and that an be readly appled to eletrty markets. For these reasons, many eletrty markets are beng mplemented or moded to operate based on these tehnques. A problem wth these types of markets, as wth the lassal OPF problem, s that system seurty s somewhat arbtrarly represented through lmts n power transfers that most lkely do not reet the atual seurty levels assoated wth a gven operatng ondton, thus yeldng napproprate pre sgnals [7]. In ths paper, a tehnque to better aount for system seurty than just usng somewhat \arbtrary" lmts on transmsson lne power ows s proposed for hybrd markets. The proposed methodology s based on a voltage stablty onstraned OPF proposed n [8]. Although n ths method t s not possble to dretly represent the (N-1) ontngeny rtera used n the omputaton of the ATC, the eets of ontngenes an be ndretly represented through an \adequate" hoe of the value of the stablty margn n the optmzaton proedure, as explaned below. D. Content The methodologes presented n ths paper try to address the ssue of a proper representaton of system seurty n the operaton of a deentralzed and a hybrd market, so that adequate pre sgnals an be gven to all market partpants. Both methodologes are based on the use of voltage stablty theory to better aount for system seurty [9] although ths approah does not fully aount for the atual dynamal seurty of the system, t does represent system seurty namuh more aurate way than the smple use of predetermned lmts on power ows. The paper s thus strutured as follows: Seton II desrbes n detal the proposed auton system wth redspath methodology and the assoated ongeston prng mehansms. Seton III dsusses the ntroduton of voltage stablty onstrants n an OPF-based eletrty market, nludng a dsusson on the eet of these onstrants on the margnal pres. Seton IV omments on the results obtaned for a 6 bus test system wth demand-sde bddng, onentratng espeally n omparng the derent proposed methodologes. Fnally, Seton V summarzes the man deas and results obtaned n ths paper, as well as the advantages and dsadvantages of the proposed tehnques. II. Smple Auton System wth Redspath The redspath and prng methodology proposed here s basally an mplementaton of a smple auton system where ATC omputatons and senstvty analyses, based on voltage stablty rtera, are arred out for the gven bddng ondtons. Ths method s based on senstvty formulas developed for the stablty lmt ondtons dened by the ATC the redspath osts are then dstrbuted among the market partpants based on the senstvty fators obtaned from these formulas. Ths seton onentrates on desrbng the tehnques used for the ATC and senstvty alulatons, and ther use n the proposed redspath and seurty prng algorthm. A. ATC Calulatons and Senstvty Formulas The ATC, as dened bynerc, s a measure of the transfer apablty remanng n the physal transmsson network for further ommeral atvty over and above already ommtted uses [10]. Thus, mathematally t s de- ned as ATC = TTC ; ETC ; TRM (1) where TTC = mn(p maxilm P maxvlm P maxslm ) represents the Total Transfer Capablty,.e., the maxmum power that the system an delver gven the seurty onstrants dened by thermal lmts (I lm ), voltage lmts (V lm ) and stablty lmts (S lm ), and an (N-1) ontngeny rteron (worst sngle ontngeny of the transmsson system). ETC stands for the Exstng Transmsson Commtments, and TRM orresponds to the Transmsson Relablty Margn, whh nludes a Capaty Benet Margn (CBM) and s meant to aount for unertantes n system operaton. From the ATC denton (1), t s lear that ts value hanges wth the system operatng ondtons, and hene has to be omputed for every bddng ondton. However, n most of the urrent mplementatons of eletrty markets, the ATC s determned n o-lne studes and represented n the bddng proess as rather onservatve lmts on the power owng through the transmsson lnes or man transmsson \orrdors". Ths s obvously unrealst and ould easly lead to ether ttous ongeston problems, wth the orrespondng prng mplatons, or unseured operatng ondtons, whh may lead to ollapse problems.

3 3 The problem n the omputaton of the ATC s the atual determnaton of the stablty lmts, whh requre ostly tme doman smulatons. Hene, n the present paper, the stablty lmts are approxmately represented usng voltage stablty margns, whh an be readly and qukly omputed, gvng a good dea of the \relatve" stablty of the network [9]. In ths ase, a ontnuaton power ow approah s used to determne the maxmum system loadng or TTC, as dened n (1) [11]. Sne voltage stablty rtera are used to determne the ATC value, one an also readly determne the senstvtes of the ATC wth respet to varous systems parameters, espeally wth respet to the partpants' bds. The requred senstvty formulas an be obtaned from the defnton of the TTC and the use of bas voltage stablty onepts [9]. Thus, onsder that the system an be represented n steady state wth the followng set of nonlnear equatons: f(x p) =0 (2) where x 2 < n stands for the state and algebra system varables, suh asbusvoltage magntudes and angles 2<s the \bfuraton" or loadng parameter used to represent the system loadng level, as the load powers are modeled as P L = P Lo + P D, wth P Lo representng the load power levels at the ntal loadng ondtons and P D orrespondng to the demand power bds (all loads are assumed to have onstant power fators) and p 2 < m orresponds to \ontrollable" system parameters, suh as the supply and demand power bds P S and P D, respetvely. In ths analyss, generator powers are modeled as P G = P Go +(+k G )P S, where P Go s the ntal generaton loadng ondtons and k G savarable used to represent a dstrbuted slak bus. Equatons (2) typally orrespond to a set of \moded" power ow equatons, whh basally result from modelng system ontrols and lmts n greater detal than n the typal power ow equatons. The voltage stablty lmts for system (2) are basally assoated wth saddle-node and lmt-ndued bfuratons of the orrespondng set of nonlnear equatons [9] at these bfuraton ponts, the system ollapses. Thermal and voltage lmts, on the other hand, an be treated mathematally, for the purpose of senstvty analyses, n a smlar way as lmt-ndued bfuratons, although the system does not ollapse when these lmts are reahed. Hene, for (2), the ATC an then be dened as ATC = (3) f the TRM s negleted, and where s the \rtal" (\maxmum") loadng level at whh the system s at a lmt ondton, as per denton (1). In ths paper, all ATC and requred senstvtyvalues are omputed based on the results generated by UWPFLOW [12], whh s a ontnuaton power ow program apable of representng varous power system elements usng \detaled" steady state models. A.1 Saddle-Node Bfuratons Saddle-node bfuratons (SNB) are haraterzed by a par of equlbrum ponts oalesng and dsappearng as the bfuraton parameter \slowly" hanges. Mathematally, the SNB pont s an equlbrum pont (x p ) wth a sngular Jaoban D x fj and assoated unque rght and left \sngular" egenvetors v and w, respetvely,.e., D x fj v = D T x fj w =0. By takng the dervatves of (2), one has at the SNB pont that D x fj dx + D fj d + D p fj dp = 0 ) w T D x fj dx + w T D fj d + w T D p fj dp = 0 Hene, from these equatons and as proposed n [13], one has that the senstvtes of the system loadng wth respet to hanges n the parameters p at the SNB pont an be determned by usng A.2 Lmts d 1 dp = ;! T! T D p fj (4) D fj Lmt-ndued bfuratons (LIB) are equlbrum ponts where a system ontrol lmt s reahed, whh n some ases may lead to a system ollapse haraterzed by a par equlbrum ponts oalesng and dsappearng for slowhanges of the bfuraton parameter. At a LIB,as opposed to a SNB, the system Jaoban s not sngular at the bfuraton pont (x p ) hene, equaton (4) does not apply at ths pont. In general, a system reahng a lmt at an equlbrum pont (x p ) an be haraterzed by two derent sets of equatons,.e., f 1 (x p ) = 0 (5) f 2 (x p ) = 0 where the rst set f 1 () orresponds to the \orgnal" system equatons, whereas the seond set f 2 () orresponds to a moded set of equatons where the lmt s atve. For example, when a reatve power generator lmt s reahed at a bus, a generator voltage ontrol equaton, say V ;V =0, may be replaed by Q G ; Q Glm = 0 at the lmt ondton. Hene, takng the dervatves of (5) at the equlbrum pont where the lmt beomes atve, D x f 1 j dx + D f 1 j d + D p f 1 j dp = 0 D x f 2 j dx + D f 2 j d + D p f 2 j dp = 0 Elmnatng dx from these equatons, leads to d dp = 1 ; T T D x f 2 j D x f 1 j ;1 D p f 1 j ; D p f 2 j (6) where = D f 2 j ; D x f 2 j D x f 1 j ;1 D f 1 j Observe that the senstvty formula (6) apples to any lmt ondton, ndependent of whether t orresponds to

4 4 a LIB or a thermal or voltage lmt. Hene, ths equaton together wth (4) are used to determne the senstvty of the ATC wth respet to the system parameters p, whh for the purpose of ths paper orrespond to the supply and demand bds P S and P D, respetvely. B. Redspath and Seurty Prng Algorthm The redspath tehnque proposed n ths paper follows smlar general rtera as the one urrently used n the New England eletrty market [14], where redspath s used to address the ssue of a smple auton mehansm, whh neglets system losses, yeldng a market learng ondton that volates ertan ongeston rtera, as de- ned by power ow lmts on the transmsson system. The osts resultng from dspathng a more expensve unt than the market learng pre (MCP) to solve the ongeston problem are then \dstrbuted" among the derent partpants. The man mprovements of the tehnque proposed here wth respet to known methodologes are: The ATC s omputed \on-lne",.e., takng nto onsderaton the atual system ondtons, as opposed to usng power ow lmts determned o-lne that do not neessarly represent the system seurty level. The ongeston osts are dstrbuted among the market partpants based on the atual mpat that eah one of them has on the ATC value. The suggested redspath tehnque to address the problem of market learng ondtons that do not meet the atual ATC requrements, as dened n the prevous seton, s summarzed n the ow-dagram shown n Fg. 1. Ths methodology determnes the transaton osts for the derent market partpants as follows: N k=k+1 Y Avalable generator for reshedulng or urtalable load? N Intalzaton Compute transaton mpat (senstvtes) Pk generators and j for reshedulng or urtalable load λ (k+1) Calulate Adjust step length Transaton Contrbuton Fator and reshedulng ost λ (k+1) >= T Y Update last reshedulng amount and ost B.1 Step 1 Usng a smple auton mehansm, the MCP and assoated total transaton power level T are determned, together wth the load and generator power levels that lear the market, namely, P D and P S and for all loads and generators, respetvely. The values of P D and P S are used n the determnaton of the ATC, as these dene the load and generator dreton used for the omputaton of n UWPFLOW [12]. B.2 Step 2 If the ATC s volated,.e., f < T, the mpat of eah possble system transaton s determned usng (4) or (6), dependng on the lmtng fator that denes the ATC. Thus, the generator wth the most postve mpat on the ATC that has not been fully dspathed n the bdng proess, and the generator j dspathed n step 1 wth the most negatve or least postve mpat n the ATC are hosen for reshedulng. Thus, the orrespondng nrease and derease n generaton s dened as S = ; S j = S where k s the number of teraton n the redspath proess, and S s hosen dependng on the value that one wants Total reshedulng osts Fg. 1. Redspath tehnque for a smple auton system. The ATC at eah teraton s represented by (k+1), whereas T represents the transaton level that lears the market. for the ATC, sne the new value of the ATC may beapproxmated by (k+1) + d dp S S + d It s assumed here that d=dp S j dp Sj S j (7) > d=dp Sj j, otherwse no ATC mprovements an be attaned by redspathng generaton. Sne the whole proess s based on a lnearzaton, one annot make large hanges n generated power, otherwse ths mghthave a large eet on the atual ATC value, whh hanges nonlnearly as the parameters hange hene the need for an teratve proess. The amount of generaton hosen for redspath S may be readjusted when de- usng the full nonlnear termnng the atual value of (k+1) system, as explaned below. Observe that system losses are not onsdered here, as these are just assgned to a \slak" bus. Ths basally

5 5 means that other market mehansms are used to aount for losses n the proess, whh stypally the ase n deentralzed market strutures. Only f there are no adequate generators avalable for redspath, s the load onsdered for urtalment. Ths approah s to be expeted when the load s nelast, as these types of loads requre that the foreasted load be dspathed, gven the hgh \osts" of load urtalment. In the ase of demand-sde bddng, however, the load ould be onsdered for redspathng n the same way as the generators,.e., the load wth the most negatve mpat on the ATC, say, may be redued by anamount that has a \sgnant" mpat on the ATC value, as per approxmaton (k+1) ; d dp D D (8) Ths equaton an be used for reshedulng nelast loads as well. In ths ase, there s no seurty ost for load urtalment, sne one assumes that the loads are ntrnsally runnng a rsk of not beng dspathed by partpatng n the market. However, ths means that overall system revenues are lost, as loads that ould be served by redspathng avalable generaton would not be dspathed, even f these loads are wllng to pay the hgher osts of reshedulng generated by the proposed market proess. Hene, the load redspath opton, whether the loads are nelast or not, s only onsdered f no generaton bds are avalable to solve the ATC volaton. Observe that n ths ase, the transaton level T s aeted by the load reduton thus, T (k+1) = T ; D Ths s not the ase for generaton redspath n ths ase, T (k+1) = T. B.3 Step 3 The step hanges n generaton or load are readjusted by omputng the atual value of (k+1) and omparng t to the approxmated value omputed usng (7) or (8). If the derene s greater than a hosen tolerane, the prevous step and ths one are repeated wth smaller hanges n the supply and demand untl the desred tolerane s met. B.4 Step 4 The redspathng osts for the gven teraton k are then determned based on a Transaton Contrbuton Fator (TCF) as dened by = d=dp j Pj d=dp jj p p j where stands for the bus number, and p orresponds to the value of the orrespondng parameter,.e., the value of S or D. Only buses wth negatve mpat on the ATC,.e., buses wth d=dp j < 0, are onsdered n ths omputaton buses wth postve mpat are gvenazero TCF value, so that market partpants that do not reate (9) the seurty problem are not harged for the ost of keepng the system seure. The parameter values p are nluded n ths \normalzaton" proess to aount for the \sze" of the orrespondng transatons n the seurty ost. The total generator reshedulng seurty ost of the k th teraton maybedenedas SC k =(C S ; MCP) S (10) where s the generator hosen n Step 2, wth a margnal ost or bd C S,andMCP s the market learng pre obtaned from the smple bddng proess n Step 1. In the ase of nelast loads, one an argue that there should be a ost assoated wth urtalng the load that should be onsdered as part of the ost of keepng the system seure thus, one would have n ths ase that SC k = A D D (11) where A D s the \ost" of urtalng the load at the hosen bus. B.5 Step 5 If the ATC requrements are met,.e., f (k+1) <T (k+1), then the teratve proess stops, say atk = m. At ths pont, the nal generator or load reshedules are adjusted based on (7) or (8), respetvely, sothattheatc and nal transaton level are the same,.e., (m+1) T (m) Thus, or P (m) S B.6 Step 6 = ;P (m) S j = P (m) D d=dp S j (m) T (m) ; (m) = T (m) ; (m) d=dp D j (m) ; d=dp Sj j (m) The nal transaton levels and margnal osts for eah node are readly determned as follows: Generators: Loads P S = P (0) S + mx k=1 S = MCP ; 1 P S P D = P (0) D ; mx k=1 D = MCP + 1 P D S (12) mx k=1 D mx k=1 SC k SC k

6 6 B.7 Observatons It s mportant to hghlght the fat that the proposed tehnque to ompute \nodal" margnal pres assumes that all the osts of redspathng are fully dstrbuted among the market partpants, as s the ase of the urrent New England eletrty markets [14]. Hene, sne a \zero sum" approah s used n ths ase, there are no \leftovers" to be used by the ISO for possble upgrades of the transmsson system. Furthermore, osts assoated wth losses and reatve power dspath, as well as reserves for frequeny ontrol, are assumed to be resolved n other ndependent auton markets or by other market mehansms, whh may n turn yeld system operatng ondtons that ould have a sgnant eet on the margnal osts generated by eah one of the derent markets. Although the latter mght be resolved by usng other teratve proesses, somewhat smlar to the one proposed here for the ntegraton of the ATC nto the auton system, ths s ertanly a dsadvantage of these types of markets. Hybrd markets, on the other hand, produe solutons that aount for several of the market osts through an \ntegrated" soluton approah, thus resolvng most of these problems. However, these market strutures present several other dsadvantages assoated wth a \blak box" approah, suh as lak of \transpareny" and onvergene dultes, as dsussed n the next seton. III. Spot Pre of Seurty Hybrd market strutures are based on the determnaton of a spot pre of eletrty obtaned from an OPF soluton and ts assoated Lagrangan multplers [1], [2], [4]. Ths tehnque basally onssts on solvng the followng OPF problem [5]: Mn. C T S P S ; C T DP D (13) s.t. f( V Q G P S P D )=0 (PF eqs.) 0 P S P S max (supply bd bloks) 0 P D P D max (demand bd bloks) jp j ( V)j P j max Q G mn Q G Q G max V mn V V max (\seurty" lmts) (gen. Q lmts) (bus V lmts) where C S and C D are vetors of supply and demand bds n $/MWh, respetvely Q G stands for the generator reatve powers V and represent the bus phasor voltages and P j represent the powers owng through the lnes, whh are typally used to \ndretly" represent system seurty. P S and P D are the vetors of supply and demand bds of power n MW, provded n bloks of P S max and P D max, and are assoated wth generator and load powers, sne P G = P Go + P S,and P L = P Lo + P D, where P Go and P Lo basally dene the ntal system operatng ondtons (loads are assumed to have onstant power fators). The OPF problem (13) an be transformed nto the followng optmzaton problem based on the Lagrangan that results from a logarthm barrer Interor Pont approah: Mn. L = C T S P S ; C T D P D (14) where 2< n and ; T f( V Q G P S P D ) ; T P S max (P S max ; P S ; s PS max ) ; T P D max (P D max ; P D ; s PD max ) ; T P j max (P j max ;jp j ( V)j;s Pj max ) ; T Q G mn (Q G ; Q G mn ; s QG mn ) ; T Q G max (Q G max ; Q G ; s QG max ) ; T V mn (V ; V mn ; s Vmn ) ; T V max (V max ; V ; s Vmax )! X ; s ln s = P S max PD max Pj max QG mn QG max Vmn Vmax s T >0, are the Lagrangan multplers, and s = s P S max s PD max s Pj max s QG mn s QG max s Vmn s Vmax T s 0, are the slak varables (s =[s ]). The margnal osts for the supply and demand an be dened from S = C S ; S + PS max D = ;C D + D + =0 PD max =@P S =1and@f =@P D = ;1, from the power ow equatons. Thus, from (15), the margnal or shadow pre for eah market partpant,.e., the Loatonal Margnal Pre (LMP), s gven by the orrespondng Lagrangan multpler [5],.e., LMP = (16) In (13), the lmts on the lne power ows are used to somehow represent the system seurty level,.e., ndretly represent the system's ATC. These lmts are determned o-lne, assumng ertan \typal" system ondtons and ontngenes, and hene do not represent the atual operatng ondtons and the eet of the bddng proess on system seurty. Ths s obvously one of the major dsadvantages of ths method, and hene s resolved here by nludng addtonal onstrants that dene a mnmum voltage stablty margn, as orgnally proposed for the standard OPF problem n [8]. Thus, the followng modatons

7 7 are proposed to better aount for system seurty: Mn. C T S P S ; C T DP D (17) s.t. f( V Q G P S P D )=0 (PF base ase) f( V Q G P S P D )=0 o (mn. stab. margn) 0 P S P S max (supply bd bloks) (PF max. load) 0 P D P D max (demand bd bloks) jp j ( V)j P jmax jp j ( V )jp jmax Q G mn Q G Q G max Q G mn Q G Q G max V mn V V max V mn V V max (thermal lmts) (gen. Q lmts) (bus V lmts) where s a salar that represents a loadng parameter that drves the system to a \rtal" pont, and thus denes the generaton and load powers at the rtal pont as follows: P G = P Go +(1+ + k G ) P S,and P L = P Lo +(1+ ) P D,wthk G representng a varable used to dstrbute the system losses among the generators n proporton to the generator bds (dstrbuted slak bus model). The loadng parameter s used to guarantee a mnmum voltage stablty margn o, and thus ndretly represents the system ATC however, ontngenes annot be dretly nluded n ths problem as opposed to the prevously proposed tehnque. The latter an somewhat be resolved wth the hoe of the value of o,as the larger ths value, the larger the ontngeny that the system an wthstand, gven that ths ndretly represents the system stablty margn. However, one has to be areful when hoosng ths margn, as a \ large" value mght lead to onvergene dultes or a lak of soluton all together, whh s one of the man problems of ths approah. Note that, n ths ase, the lmts n the transmsson lne ows are only used to represent atual thermal lmts as opposed to the \seurty" onstrants n (13). The LMPs an also be determned usng the orrespondng Lagrangan multplers for (17). Thus, t an be readly S = C S ; S ; S (1 + + k G ) (18) + =0 D = ;C D + D + D (1 + ) + PD max =0 where orresponds to the Lagrangan multpler of the power ow equatons at the rtal pont. Hene, the LMP for eah market partpant s also gven by equaton (16),.e., LMP =. Observe that n (18), the LMP value aounts for the atual bd (C ), the bd lmts ( Pmax ), as well as the stablty margn( and ). As wth any spot prng methodology, the bndng onstrants have a dret eet on the soluton of optmzaton problem (17), and hene are ndretly represented n the pre through the value of the orrespondng Lagrangan multplers [5]. For example, generator Q lmts do have an eet on the optmal soluton and thus aet the LMPs, even though reatve power prng s not dretly onsdered n the optmzaton problem. For nelast loads, all the prevous equatons should be hanged to assume a xed demand value. In ths ase, the load s not treated as a varable parameter wthn the optmzaton proess. However, one may stll use the Lagrangan multplers at eah load bus as the LMPs for the load, sne an argument an be made that these represent the margnal osts of supplyng power at the load buses [5]. Observe thatlak of soluton or onvergene dultes maybeenountered n ths ase, dependng on the value of the nelast demand and other system onstrants n other words, the system mght not be able to supply the requred demand under the gven onstrants. The possble onvergene dultes or lak of soluton, as well as the ntegrated approah that somewhat \hdes" the eet of the derent market omponents and onstrants nto a nal overall soluton, are ertanly dsadvantages of OPF-based approahes when ompared to the use of smple autons. Furthermore, as opposed to the auton mehansm proposed n ths paper, an adequate representaton of the eet of ontngenes on system seurty, and the nluson of a mehansm to allow and aount for nelast load urtalment due to seurty reasons are not feasble n the OPF methodology proposed n ths paper. However, OPF-based tehnques yeld pre values that dretly or ndretly aount for the eet that the derent system onstrants have n the market. Furthermore, these methodologes yeld a relatvely smple mehansm to generate revenues for the ISO to be used n possble system mprovements, as demonstrated on the example dsussed n the next seton. IV. Example The two tehnques proposed n the prevous seton are mplemented and ompared to standard smple auton and spot prng tehnques based on the results obtaned for the test system of Fg. 2 [2]. Ths s a smple 6 bus test system wth 3 generaton ompanes (GENCOs) that provde supply bds, and three energy supply ompanes (ESCOs) that provde demand bds, as shown n Table I. The data for ths system s extrated from [2] and s shown n Tables II and III. The maxmum transmsson lne power ows were determned

8 8 Bus 2 (GENCO 2) Bus 3 (GENCO 3) TABLE II Bus Data at Base Loadng Bus 1 (GENCO 1) Bus 6 (ESCO 3) Bus P Lo Q Lo P Go Q Gmax Q Gmn [MW] [MVar] [MW] [MVar] [MVar] Bus 4 (ESCO 1) Fg. 2. Bus 5 (ESCO 2) Test system. Bus V V max V mn [p.u.] [p.u.] [p.u.] TABLE I GENCO and ESCO Bds Bus Partpant C P max [$/MWh] [MW] 1 GENCO 1 (S 1 ) GENCO 2 (S 2 ) GENCO 3 (S 3 ) ESCO 1 (D 1 ) ESCO 2 (D 2 ) ESCO 3 (D 3 ) \o-lne" through a seres of maxmum loadng studes usng UWPFLOW [12], and based on (N-1) ontngeny rtera these lmts represent the maxmum power ow on the derent lnes for a maxmum loadng ondton alulated usng the load and generaton pattern dened by the bds, for a worst ontngeny senaro. Thermal lmts on the lnes were assumed to be large, and hene are negleted n ths ase. Results for nelast demand were not obtaned for ths system, to smplfy the analyss of the results as well as to shorten the length of the paper. Nevertheless, as dsussed n the prevous setons, ths problem an be vewed as a speal ase of demand-sde bdng. A. Smple Autons A smple auton mehansm, negletng losses, yelds the results depted n Fg. 3 and Table IV, whh nluded the voltages, total transaton level (T ), losses, total ost, and the leftover payment for the ISO for omparson purposes. Observe that the results onsder that one of the generators (GENCO 2), pks up the losses, thus leavngaost that has to be overed by the ISO usng another auton or market mehansm. The ATC omputatons, however, TABLE III Lne Data Lne R j X j B =2 P jmax -j [p.u.] [p.u.] [p.u.] [MW] yeld a value of =38:23 MW orrespondng to a mnmum load bus voltage lmt for a worst ontngeny on Lne 2-4. Hene, sne <T, generator redspath s neessary to make the transaton feasble. The generaton redspath proedure of Fg. 1 yelds the results depted on Tables V and VI. Ths proedure s based on the senstvty vetor d=dpj, as prevously explaned for example, at the ntal teraton k = 1 s om-

9 9 $/MWh TABLE V Reshedulng and Seurty Costs MCP=9.5 ESCO 1 ESCO 2 ATC ESCO 3 GENCO 1 k [MW] [MW] [$/h] G 1 G 2 =T SC k 8.8 GENCO 2 TABLE VI Transaton Contrbuton Fators 7 GENCO Fg. 3. Soluton of smple auton for 6-bus test system. TABLE IV Smple Auton Results MW k Bus Part. V = MCP P Pay [p.u.] [$/MWh] [MW] [$/h] 1 S S S D D D TOTALS T =45 MW Cost= $/h Losses=3.55 MW ISO= $/h ATC=38 MW puted usng (6) and s equal to d dp (1) = d=dp S1 j (1) d=dp S2 j (1) d=dp S3 j (1) d=dp D1 j (1) d=dp D2 j (1) d=dp D3 j (1) = :95 ;0:03 0:03 ;1:55 ;0:49 ;0:09 Table VII shows the nal results of the proposed redspath proedure. Observe the derene between these results and the orgnal smple auton results of Table IV although the total transaton level s the same, the power levels, margnal pres, and/or transaton osts for eah market partpant are derent B. OPF-based Prng The results of applyng the bas OPF-based proedure (13), whh does not properly aount for system seurty, to determne the partpants' margnal osts and power levels are depted n Table VIII. These results were obtaned n MATLAB based on an Interor Pont optmzaton approah. In ths ase, only the upper voltage lmts on generator buses play a role, as these voltages are fored to ther maxmum values by the optmzaton proedure, whh s to be expeted, sne ths would redue losses and n general mprove system seurty. The latter s demonstrated by the hgher value of ATC obtaned n ths ase, whh was alulated \o-lne" to determne the feasblty of the gven transaton. However, observe that the transaton level T s smaller, even though the ATC s larger there s no lear reason for ths reduton, whh hghlghts TABLE VII Smple Auton wth Redspath Results Bus Part. V P Pay [p.u.] [$/MWh] [MW] [$/h] 1 S S S D D D TOTALS T =45 MW Cost= $/h Losses=3.56 MW ISO= $/h ATC=45 MW

10 10 TABLE VIII OPF-based Results Bus Part. V P Pay [p.u.] [$/MWh] [MW] [$/h] 1 S S S D D D TOTALS T =42.4 MW Cost= $/h Losses=1.6 MW ISO=6.54 $/h ATC=69.37 MW TABLE IX OPF-based Results Inludng Seurty ( o =5:5 p.u.) Bus Part. V P Pay [p.u.] [$/MWh] [MW] [$/h] 1 S S S D D D TOTALS T =40.5 MW Cost= $/h Losses=1.4 MW ISO=7.02 $/h ATC=66.59 MW the fat that, as prevously mentoned, t s not easy to learly pnpont all the fators that nuene the nal soluton n an ntegrated OPF-based tehnque. Note that there s some money left from the transaton, whh the ISO an use to over for the transmsson system losses. When system seurty s dretly nluded n the OPFbased approah, the soluton obtaned from solvng (17) for a mnmum \stablty" margn value of o =0:1 p.u. s exatly the same as the one obtaned wthout onsderng system seurty. The reason for ths s that the requred mnmum margn s muh smaller than the atual system ATC, and hene the orrespondng system onstrants have no nuene n the soluton of the optmzaton problem. However, when ths value s sgnantly nreased to o =5:5 p.u., the soluton hanges to that shown n Table IX, as the stablty margn beomes a bndng onstrant. Observe that the margnal pres are smaller for all market partpants n ths ase, as one would expet from equatons (18), and that the transaton level has dereased slghtly, whh s also to be expeted, sne the seurty onstrants now play a role on the soluton proess, redung the amount of power that an be transated through the transmsson system. On the other hand, the amount of money left over has nreased, whh basally means that the ISO an use these funds to take are of losses as well as ongeston problems n the system. V. Conlusons Two dstnt tehnques to manage and pre system seurty n derent eletrty market strutures are proposed, explaned n detal, and brey tested n ths paper. Both tehnques are based on bas onepts of voltage stabltynpower systems, and an be vewed as mprovements of methodologes urrently used on-lne to determne eletrty pres n atual markets. From the tests and analyses, the followng man onlusons an be drawn: The system ATC an be adequately aounted for n smple auton markets through the teratve tehnque proposed here. In the ase of OPF-based markets, although the urrent system stabltylevel an be aounted for n the soluton proess, the ssue of ontngenes annot be dretly nluded. In both tehnques, a value for the pre of seurty s generated. However, the OPF-based approah has the advantage of dretly generatng addtonal funds for the ISO to take are of system upgrades assoated wth ts losses and seurty other mehansms must be devsed and used n smple autons for ths spe purpose. The auton-based mehansm s robust and does not present majornumeral dultes. Furthermore, osts of load urtalment for nelast loads an be readly nluded n the proess. Ths s not the ase for the OPF-based approah, whh may present onvergene problems and, n some ases, even a lak of soluton, espeally for nelast loads the nluson of load urtalment osts does not appear to be a smple matter n ths ase. In the OPF-based tehnque, t s not smple to readly dstngushed the derent fators that nuene the nal results, as these are somewhat hdden n the soluton. In the ase of the auton-based tehnque, ths s relatvely smple to do, as the derent parts and transatons that make up the market are handled ndependently of eah other. However, teratve proedures are requred n ths ase to be able to aount for all the derent fators that aet the nal soluton. The authors are urrently workng on testng further the proposed tehnques to better understand ther benets and drawbaks, and to determne possble mprovements. Referenes [1] M. Il, F. Galana, and edtors L. Fnk, Power Systems Restruturng: Engneerng and Eonoms. Kluwer Aadem Publshers, Boston, [2] G. B. Sheble, Computatonal Auton Mehansms for Restrutured Power Industry Operaton. KluwerAadem Publshers, USA, [3] R. Wlson, \Market Arhteture," tehnal report, Graduate Shool of Busness, Stanford Unversty, [4] M. Madrgal-Martnez, Optmzaton Models and Tehnques for Implementaton and Prng of Eletrty Markest, PhD thess, Unversty ofwaterloo, Waterloo, Canada, [5] K. Xe, Y.-H. Song, J. Stonham, E. Yu, and G. Lu, \Deomposton Model and Interor Pont Methods for Optmal Spot Prng of Eletrty n Deregulaton Envronments," IEEE Trans. Power Systems, vol. 15, no. 1, February 2000, pp. 39{50.

11 11 [6] Independent Market Operator, \Market Rules of the Ontaro Eletrty Market," doument MDPGDE 0002, avalable at January [7] M. A. Bolton Zammt, D. J. Hll, and R. J. Kaye, \Desgnng Anllary Serves Markets for Power System Seurty," IEEE Trans. Power Systems, vol. 15, no. 2, May 2000, pp. 675{680. [8] W. Rosehart, C. Ca~nzares, and V.H. Quntana, \Optmal Power Flow Inorporatng Voltage Collapse Constrants," Pro IEEE/PES Summer Meetng, Edmonton, Alberta, July [9] C. A. Ca~nzares, edtor, \Voltage Stablty Assessment, Proedures and Gudes," tehnal report, IEEE/PES Power System Stablty Subommttee, January Fnal Draft avalable at [10] \Avalable Transfer Capablty Dentons and Determnaton," tehnal report, NERC, USA, [11] C.A. Ca~nzares, A. Berzz, and P. Marannno, \Usng FACTS Controllers to Maxmze Avalable Transfer Capablty," Pro. Bulk Power Systems Dynams and Control-IV, IREP, Santorn, Geere, August 1998, pp. 633{641. [12] C. A. Ca~nzares et al., \UWPFLOW Program," Unversty of Waterloo, avalable at [13] S. Greene, I. Dobson, and F. L. Alvarado, \Senstvtyofthe Loadng Margn to Voltage Collapse Wth Respet to Arbtrary Parameters," IEEE Trans. Power Systems, vol. 12, no. 1, February 1997, pp. 262{272. [14] D. Gan and Q. Chen, \Loatonal Margnal Prng{New England Perspetve," Pro IEEE/PES Wnter Meetng, Columbus, Oho, January Claudo A. Ca~nzares reeved n Aprl 1984 the Eletral Engneer dploma from the Esuela Poltena Naonal (EPN), Quto- Euador, where he held derent teahng and admnstratve postons from 1983 to Hs MS (1988) and PhD (1991) degrees n Eletral Engneerng are from the Unversty of Wsonsn-Madson. Dr. Ca~nzares s urrently an Assoate Professor and the Assoate Char for Graduate Studes at the E&CE Department of the Unversty of Waterloo, and hs researh atvtes onentrate mostly n studyng stablty, modelng, smulaton, ontrol, and omputatonal ssues n a/d/facts systems. Hong Chen reeved her Bahelors (1992) and Masters (1995) degree n Eletral Engneerng from Southeast Unversty n Chna. She worked n NARI (Nanjng Automaton Researh Insttute), P.R. Chna, from 1995 to 1998, and was engaged n the developmentofsoftware tools for EMS applatons. She s urrently a Ph.D. anddate n the E&CE Department at the Unversty of Waterloo, pursung researh on eletrty markets and omputer applatons n power systems. Wllam D. Rosehart reeved hs Bahelors, Masters and Ph.D. degrees n Eletral Engneerng from the Unversty of Waterloo n 1996, 1997 and 2001, respetvely. From 1991 to 1995, through the ooperatve eduaton program at the Unversty of Waterloo, he worked n the Power Industry n Canada, nludng GE Canada, Hammond Manufaturng, and Waterloo North Hydro. He s urrently an Assstant Professor at the E&CE Department of the Unversty of Calgary.