Transmission Cost Allocation by Cooperative Games and Coalition Formation Juan M. Zolezzi, Member, IEEE, and Hugh Rudnick, Fellow, IEEE

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1 1 Transmsson Cost Allocaton by Cooperatve Games and Coalton Formaton Juan M. Zolezz, Member, IEEE, and Hugh Rudnc, Fellow, IEEE Abstract-- The allocaton of costs of a transmsson system to ts users s stll a pendng problem n many electrc sector maret regulatons. Ths paper contrbutes wth a new allocaton method among the electrc maret partcpants. Both cooperaton and competton are defned as the leadng prncples to far solutons and effcent cost allocaton. The method s based manly on the responsblty of the agents n the physcal and economc use of the networ, ther ratonal behavor, the formaton of coaltons and cooperatve game theory resoluton mechansms. The desgned method s applcable to exstng networs or to ther expanson. Smulatons are made wth sample networs. Results conclude that adequate solutons are possble n a decentralzed envronment wth open access to networs. Comparsons wth tradtonal allocaton systems are shown, cooperatve game solutons compare better n economc and physcal terms. Index Terms Cooperatve Game Theory, Nucleolus, Open Access, Transmsson Cost Allocaton, Transmsson Expanson, Shapley Value. I. INTRODUCTION he deregulaton of the power ndustry, whch started n T South Amerca n the 1980 s and whch developed worldwde has orgnated a rch dscusson about the best way to manage, economcally and techncally, the dfferent marets and stuatons developed wth deregulaton. Wthout any doubt, the deregulaton process has faced dffcultes, successes and falures, wth many problematc areas stll beng explored and developed [1], [2]. Among them, one of the most complcated ones has been the non-dscrmnatory open access to transmsson and dstrbuton networs, and the cost allocaton among the dfferent maret agents utlzng those networs [3]. Dfferent authors have emphaszed the mportance of the transmsson system n the new deregulated marets [4], as facltator of generator competton, allowng generators to allocate ther producton n consumer centers and allowng consumers to beneft from that compettve envronment. Wthn that framewor, the transmsson tarff system and the user cost allocaton must preserve an adequate resource allocaton among maret agents. It s desred that transmsson prces and payment do not dsturb decsons for new generaton nvestment, for generator operaton and for Ths wor was supported by Fondecyt Project J. M. Zolezz s wth the Department of Electrcal Engneerng, Unversdad de Santago de Chle, Caslla 10233, Correo 2, Santago, Chle (e-mal: jzolezz@eee.org). H. Rudnc s wth the Department of Electrcal Engneerng, Unversdad Católca de Chle, Caslla 306, Correo 22, Santago, Chle (e-mal: h.rudnc@eee.org). consumer demand [5], [6]. At the same tme, ths must be acheved n a smple and far form. Each country has had to fnd ts soluton n agreement wth the characterstcs of ts transmsson system. Classc and modern solutons to transmsson cost allocaton have not been able to satsfy expectatons of regulators and maret agents. Many solutons have been formulated and many appled, ragng from wheelng blateral schemes to multlateral open access ones. Most of them face problems due to lac of economc foundatons [7]. Dfferent allocaton schemes have been formulated n recent years based on the natural economc use of the transmsson system [8]-[10]. They am at consderng the way the transmsson system s mpacted by generators and consumers by the smple fact of beng connected to the networ, rrespectve of ther commercal contracts wth other agents usng the same networ. In some countres ths framewor gves brth to the area of nfluence concept [7], [8]. Dfferent methods have been suggested to dentfy the mpact a generator or a consumer has on the flow of a transmsson lne wthn a transmsson networ, as well as how much of the generaton of a gven generator corresponds to a gven load [11], [12]. These methods consder topologcal studes, proportonalty prncples and networ flow equatons and determne factors that are used to allocate complementary transmsson charges or total transmsson payments, based on the use made by the dfferent agents. The benefcares method s another alternatve that has been proposed, wth sold economc bases, but t faces dffcultes n practcal applcatons [7]. Game theory provdes nterestng concepts, methods and models that may be used when assessng the nteracton of dfferent agents n compettve marets and n the soluton of conflcts that arse n that nteracton [13], such as those of the electrcty marets. In partcular, cooperatve game theory arses as a most convenent tool to solve cost allocaton problems [14]. The soluton mechansms of cooperatve game theory behave well n terms of farness, effcency and stablty, qualtes requred for the correct allocaton of transmsson costs. Nevertheless, proposals to date are stll n a developng stage, wth contrbutons been formulated n wheelng transactons [15], n the allocaton of expanson costs [16], [17] and other [18]. Ths paper presents a method to allocate charges among users of a transmsson system, ether n an exstng networ or an expandng one. The method s based n a model that ntegrates cooperaton and coordnaton among the agents as

2 2 basc prncples. It also consders the physcal and economc use of the networ by the dfferent agents. It assumes a ratonal behavor of the agents, the formaton of coaltons among agents, along wth cooperatve game soluton mechansms. The proposed method ams at achevng the transmsson prcng prncples defned n [5]. It s smple to apply and t has an adequate physcal and economc understandng of the electrcty transmsson problem. It provdes adequate sgnals to agents mang nvestment decsons, ensurng farness, effcency and stablty n the resultant allocaton. The method s appled for cost allocaton to consumers n a networ, but t s equally applcable to generators or combnatons of both. II. METHODOLOGY A. Formulaton The proposed method analyses each transmsson lne at a tme, as a problem wth partcular characterstcs, wth consumers beng the agents that nteract n a cooperatve game. Each lne s understood as an element wthn the transmsson networ, tang nto account all ts nteractons. In an electrc maret, competton taes place essentally at the generaton level. Under these condtons, generators am at obtanng normal rents for the nvested captal (ncluded the radal lnes to reach the networ). Any addtonal charges for transmsson, for an nelastc demand, wll be passed over to consumers by means of a hgher generaton cost [7]. That means at the end that consumers are the ones who pay for the transmsson system that s not mplct n the generaton nvestment. The partcpaton of an agent n coverng the cost of a partcular transmsson lne s determned by playng a cooperatve game (NA, C), where NA s the set of the N players and C ts characterstc functon [13]. For basc concepts of cooperatve game theory n cost allocaton, refer to the appendx. The method consders the lne power flow as a measure of lne use, mportant for cost allocaton purposes. Further, t s assumed that maxmum lne flow condtons lne capacty and nvestment costs. Therefore, t s requred to determne the system operatonal condtons that produce maxmum flow n each lne, one at a tme. Ths means dentfyng the loads L and generators g for NB dfferent buses that condton max maxmum lne flows F,.e., for a lne between buses l and m (see appendx A for argmax concept): ( L, g ) = arg max F ( L, g ); = 1,... N (1), The methodology s applcable to exstng networs as well as to expansons. In the frst case there wll be abundant data on flows and operatonal condtons, for demand and for generaton, for all possble condtons, for example as many as 8760 scenaros n a gven year. From them the operatonal condton (generaton, demand and margnal generaton costs) can be obtaned, n whch the maxmum flow taes place n a gven transmsson lne. Thus, the characterstc functon of the cooperatve game for such lne, based on the stand-alone flow requrements for each consumer agent or possble coalton of agents. In the second case, an expanson of a transmsson system, the recommendaton of the authors s to consder a large quantty of possble operaton smulatons. The followng should be taen nto account: demand curves, hydrologcal scenaros for hydroelectrc plants (dry years, ntermedate, wet), mantenance of generatng unts and transmsson lnes. The need s to dentfy the scenaro that mples the greater flow through a lne. Each player or agent evaluates ts transmsson requrement for a gven system operaton condton, ether actng separately or cooperatng n a coalton wth other agents also requrng the transmsson networ. To perform ths evaluaton, each agent must assess ts use of each transmsson lne for a maxmum flow condton. The values of L obtaned n (1) dmenson the requrements of the game agents. The game characterstc functon s defned by the standalone requrement f s for each agent and for each potental coalton S. Thus, the game characterstc functon C S, for lne and each potental coalton S, wll be: C s s = f ; S NA (2) The usefulness for each agent or agent coalton of utlzng lne s mplct n equaton 2, through the lne flow requrement. It s assumed that all agents act ratonally so that they prefer to earn more money, or prefer to reduce ther costs [13]. Thus, usng a monetary unt to represent the usefulness of the lne, t s possble to assocate to each MW flow a monetary unt, wthout producng dstortons n the game soluton. It s equvalent to solve the game consderng the MW usefulness for each agent or agent coaltons. Therefore, the characterstc functon wll be expressed n generc monetary unts, beng of no mportance what monetary unt s used. What s mportant s the responsblty or partcpaton each agent has n the fnancng of the lne, not the money mpled. It must be ept n mnd that there are no restrctons to the formaton of coaltons among the agents, understandng that they are ntellgent and ratonal,.e. that they create coaltons, whch mnmze ther cost partcpaton for gven transmsson lnes. In so far as lne cost functons are sub-addtve,.e. that the jont cost s lower than the sum of ndvdual costs, t would be convenent to create coaltons for groups of agents. Gven ths sub-addtve condton, coaltons wll certanly be formed, and further all maret agents wll partcpate. Ths s the typcal grand coalton of cost allocaton problems. The power flow for lne from bus l to bus m for coalton S s calculated usng a DC model, as: s 1 s s f = ( Θ l Θ m ) ; S NA (3) x l, m and wth phase angles Θ gven by

3 3 θ s 1 s = B P (4) where P S s the vector of njected powers (generaton mnus load) for each coalton S. To determne the njected generatons, an economc dspatch s performed, tang nto account the generaton varable costs cv and generaton producton level g. NB mn cv ( g ) g (5) = 1 subject to total generaton equal to total demand for coalton S. NB g L = 0 (6) = 1 L S g mn g g (7) max Thus, the economc dspatch for coalton S wll determne lne flows and these wll determne the game characterstc functon value for that coalton. Ths s done for all potental coaltons. Wth the obtaned characterstc functon values, the soluton of the game s obtaned usng any cooperatve game soluton mechansm, such as Nucleolus, Shapley Value, Kernel and other [13]. As a result, the payoff confguraton PC (payoff s the word for cost allocaton n game theory) for lne s obtaned. PC = ( x ; δ ) (8) where x s the payment vector and δ s the coalton confguraton for lne, see appendx B. Ths payoff confguraton allows to determne the percentage cost allocaton CA of lne to agent. x ( + ) CA, = N (9) x ( + ) = 1 where x (+) are the postve values of cost allocaton. The method does not consder compensaton for negatve values; these only reduce the magntude of the postve assgnments. As ndcated before, a DC flow algorthm s used. For the cooperatve game solutons, the Mathematca software s employed. To determne the allocaton of the total cost of a transmsson system to each agent, t s necessary to add ts allocatons for each lne,.e.: NL C = CA * CL (10) = 1, C : total cost allocaton for the transmsson system, correspondng to agent. CL : real total cost of lne n monetary unts NL : number of transmsson lnes. B. Electrc maret consderatons The proposed method solves the transmsson cost allocaton, gven an exstng networ. However, t also allows solvng the cost allocaton of an expanson, wth no changes n the method. In the same way, the method may be appled to total transmsson costs, as well as to complementary charges n a margnal cost scheme [8]. In the later arrangement, the margnal scheme couples well wth the economc foundatons of the game theory mechansms. Agents must cooperate and coordnate to share the costs of the transmsson networ that nterconnects them. That networ s assumed to be the property of a thrd party, not nvolved n the energy busness, but s nterested to recover nvestments, operaton and mantenance costs drectly, or through an ndependent system operator. Gven that there s open access, consumers may establsh commercal contracts wth any generators, wthout restrctons, other than generator capactes. Therefore, t s possble for each agent to consder dfferent supply condtons, for example wth one load demandng supply and absorbng all the transmsson cost and another condton where several consumers cooperate, sharng supply and transmsson costs. The assumpton s made that the networ has been adequately planned, n terms of transmsson capacty, qualty, securty and bacup. No congeston s assumed. The assumpton s made that agents partcpatng n the maret are ntellgent, ndependent and ratonal, that s, they prefer to maxmze ther profts, or they prefer to reduce ther costs. Thus, they wll be eager to cooperate n a ratonal economc envronment. Ths means that no agent or coalton of agents wll have a cost greater than ts stand-alone cost, and the resultant game cost allocaton wll cover all transmsson costs. Ths condton s nown as a brea-even condton or Pareto optmum. C. A smple example To better understand the proposal, a smple example wth two buses and two generators s developed, based on the system ndcated n Fg. 1. It s assumed that the varable cost of generator 1 (G1) s lower than that of generator 2 (G2). The maxmum demand condton and the maxmum lne flow are gven by P1 = 220 MW; P2 = 50 MW; L1 = 120 MW; L2 = 150 MW; F = 100 MW. P1 G1 L1 F, C Fg. 1. Example system. P2 G2 Only consumers L1 and L2 are to pay the transmsson lne. Those consumers are suppled at mnmum cost, wthn the restrctons of the maret where they are located. L2

4 4 If consumer L1 would ndependently requre supply at mnmum cost, t would request t from G1, havng a lower cost and enough capacty to supply. Its stand-alone transmsson cost s zero. For consumer L2, the stuaton s dfferent, t needs to use the transmsson lne to connect to the cheapest generator, but t has the alternatve of supply from G2 n bus 2, wth a hgher cost. If t chooses the frst alternatve, ts stand-alone transmsson requrement s 150 MW. The characterstc functon of each game s determned based on the flow requrements for each possble coalton. Thus, the coalton where only L2 partcpates s possble and t has a flow requrement of 150 MW. If L2 wants to obtan energy from the most economc source, t would be nterested n buldng ts own transmsson lne wth that capacty, or cooperatng wth other agents to reduce ts costs. The 150 MW represents, n monetary terms, a cost of 150 monetary unts for agent L2. That s why the characterstc game functon for the coalton formed just by L2 has a monetary value of 150. If both consumers cooperate and coordnate to obtan supply together, they would face a common requrement of 100 MW for the transmsson lne, whch corresponds to the necessary flow to obtan supply from the generators, consderng ther varable costs and generaton capactes. Therefore, both consumers would be nterested to cooperate n the payment of the transmsson system (t would be ndfferent for L1 and t would be attractve for L2. It s assumed that agents are nterested n cooperaton only f t mples a reducton of costs for each or at least, not an ncrease of costs). A coalton would be acheved. The coalton would be mantaned over tme, dependng on how convenent t s for the agents and ther transmsson payments. From the above analyss, the followng characterstc functon of the cooperatve game s obtaned: C1=0; C2 = 150; C12 = 100. The result of the game, usng dfferent cooperatve game soluton mechansms, gves the followng payment confguraton: PC = {0,100; 12}, whch ndcates that the total cost of transmsson lne 1-2 s to be pad by consumer L2. The soluton ndcates that the consumer located n the bus wth lower cost generaton must not pay for the transmsson lnes t does not use, gven enough generatng capacty to supply t. The sun demand concept s supported under ths analyss, but only for the load at the margnal bus, when there s enough local generaton to supply t. Other methods formulated n the lterature (area of nfluence of consumers, margnal partcpaton, average partcpaton, generalzed load dstrbuton factors, generaton dsplacement dstrbuton factors [9], [10]) gve the same answer to the cost allocaton when consumers pay for the networs. III. TEST SIMULATIONS To test and better understand the method, two systems are assessed, a radal networ and a meshed networ. A. Radal system Many South Amercan transmsson systems have a radal character, ncluded the Chlean central nterconnected networ. Thus, the method s tested frst n a radal networ (Fg. 2), smlar n structure to the Chlean one, wth data provded n tables I and II. Economc dspatches for three operatonal condtons (maxmum, medum and low demand) are gven n table III G1 G2 G3 G L1 L2 Fg. 2. Radal system. TABLE I LINE DATA Lne Impedance Maxmum power j MW j MW j MW TABLE II GENERATOR DATA Maxmum power Varable cost Generator Type (MW) (US$/MWh) G1 Desel G2 Coal G3 Natural gas G4 Hydroelectrc TABLE III ECONOMIC DISPATCHES (MW) Demand G1 L1 G2 L2 G3 L3 G4 L4 Pea Medum Low TABLE IV LINE FLOWS (MW) Demand F21 F32 F43 Pea Medum Low Consumers are the nteractng agents, whch may act ndvdually or formng coaltons, f t s benefcal for each one of them. The analyss s made for every transmsson lne n the system, consderng the maxmum flow condton for each lne at a tme. In real systems, and n the example, the maxmum flow for each lne does not necessarly tae place at maxmum demand. Table IV shows the resultant flows n the three lnes for the three operatonal condtons. For lne 1-2, the maxmum flow taes place for the maxmum demand condton, for lne 2-3 t taes place for medum demand whle for lne 3-4 t s at low demand. The use of each lne by the dfferent agents s assessed as well as L3 L4

5 5 the use by dfferent coaltons. Applyng the proposed method, the resultant characterstc game functon s gven n table V. The analyss to determne the characterstc functon for each of the agents and for each possble coalton s done as follows. For lne 1-2 the assessment s made that the only demand s that of consumer L1, and t must be suppled by exstng generators, accordng to a mert order. Lne 1-2 usage wll be 120. Under that approach, other consumers, f alone, wll not use lne 1-2. If consumers cooperate, for example L1 and L2, they wll share use of lne 1-2, wth a requrement of 120. The same would happen for coaltons L1&L3 and L1&L4. For that lne, any coalton nvolvng demand L1 wll mpose a requrement of 120. Any coalton not nvolvng L1 wll mply a null requrement of that lne. In the same form, characterstc game functons wll be determned for the other lnes. TABLE V GAME CHARACTERISTIC FUNCTION Coalton Lne 1-2 Lne 2-3 Lne 3-4 C C C C C C C C C C C C C C C Table VI fnally gves the percentage cost allocaton to the dfferent agents usng Shapley Value and Nucleolus cooperatve game mechansms. The table also provdes the allocaton resultant from the margnal partcpaton method (area of nfluence approach) [10] and the generalzed load dstrbuton factors (GLDF) [9]. Although several solutons are close, dfferences arse dependng on the lne. All methods assgn full responsblty of lne 1-2 to consumer L1, whch seems logc and predctable. The same happens for lne 2-3, although allocatons vary. For lne 3-4, all methods, except Nucleolus, gve equal results, wth more responsblty for load L2, less for L3 and even less for L1. The dfferent results gven by the Nucleolus mechansm s explaned by ts sensblty to the game features [13]. It s possble to demonstrate (see appendx) that the results obtaned wth the proposed method comply wth the ndvdual ratonalty prncple, n the sense that nobody pays more than ts stand-alone cost. They also comply wth group ratonalty, n that the sum of all agent contrbutons s equvalent to the total lne cost. The collectve ratonalty also apples; the sum of the payments of any two possble coaltons s lower than the sum of ndvdual payments. The solutons are part of the game core, whch allows assgnment stablty as well as coalton stablty. None of the agents has the ncentve to leave the coalton and group n a dfferent way. There s no alternatve coalton that would mprove the assgnment acheved by the agents. TABLE VI GAME RESULTS FOR DIFFERENT LINES AND COMPARISSONS WITH ALTERNATIVE METHODS Lne 1-2 Load L1 Load L2 Load L3 Load L4 [%] [%] [%] [%] Shapley Value Nucleolus GLDF Margnal Partcpatons Lne 2-3 Load L1 Load L2 Load L3 Load L4 [%] [%] [%] [%] Shapley Value Nucleolus GLDF Margnal Partcpatons Lne 3-4 Load L1 Load L2 Load L3 Load L4 [%] [%] [%] [%] Shapley Value Nucleolus GLDF Margnal Partcpatons The mportance of cooperatve game soluton mechansms, and that of the proposed method, s that these ratonalty, stablty and farness consderatons are part of those methods and soluton mechansms. The obtaned soluton s based on those economc ratonal and nteracton prncples, tang nto account the transmsson networ physcal (electrc) characterstcs as well as the electrcal maret economc and fnancal condtons. Solutons are shown based on two cooperatve game soluton mechansms: Shapley Value and Nucleolus. They are compared wth other cost allocatons methods, such as the GGDF and the margnal partcpaton [9, 10]. Those methods are tradtonally used for transmsson cost allocaton, but they do not consder the economc nteracton among agents for cost allocaton, and base themselves on consderaton of physcal networ flows, based on DC modelng. It s mportant to remember that the proposed method, usng game models, consder that networ techncal aspects, as well as economc ones, nfluence decsons by the agents, whch act ratonally. Smlarly, these models consder other nformaton, such as nstalled capacty per bus, varable generaton costs and consumer szes. B. Sx bus Garver system The classc sx-bus system (Fg 3), orgnally proposed by Garver [19], has been consdered a good test networ to assess transmsson expanson methods and expanson cost allocaton [16], [17]. The expanson decsons are assumed nown for

6 6 ths assessment and the problem s to determne how to assgn expanson costs among agents partcpatng n the system. Besdes the system data provded n [19], requred data for generators has been ntroduced and s gven n table VII. Bus 6 has been consdered the slac bar (needed for the margnal partcpaton method). 3 G3 5 L3 (40 MW) L5 (240 MW) G1 1 L1 (80MW) The characterstc functon values for lne 1-4 have been determned followng the proposed method. The operaton condton (L, g ), equaton 1, that condtons maxmum flow for lne 1-4, s frst determned. Usng a DC load flow, the flow requrements mposed by each load coalton on that lne, accordng to equatons 3 and 4, are determned. Mnmum cost generaton s consdered, for each coalton S. To determne the njected generatons, an economc dspatch s performed, followng equatons 5, 6 and 7. Table VIII s obtaned, accordng to equaton 2. The game soluton, usng Shapley Value and Nucleolus, for lne 1-4 s gven n Table IX. TABLE IX TRANSMISSION NETWORK COST ALLOCATION RESULTS FOR LINE 1-4 G6 2 L2 (240 MW) Method Load L1 Load L2 Load L3 Load L4 Load L5 [%] [%] [%] [%] [%] Shapley Value 30,81 0,00 4,15 0,00 65,04 Nucleolus 16,72 1,65 0,00 0,00 81,63 GLDF 33,15 0,37 6,67 0,00 59,81 Margnal Partcpatons 30,61 0,00 7,75 0,00 61,64 6 Fg. 3. Sx bus Garver system dagram. TABLE VII MAXIMUM GENERATION CAPACITY DATA AND PRODUCTION VARIABLE COSTS Generator Generaton Maxmum Capacty (MW) Generaton Level Because of space restrctons, the characterstc functons of each of the cooperatve games for each transmsson lne are not detaled, but only an example s gven n Table VIII, whch provdes the data for the game functon for lne 1-4. TABLE VIII GAME CHARACTERISTIC FUNCTION FOR LINE 1-4 Coalton Value Coalton Value { f } 0 {1,2,3} {1} {1,2,4} 18,3993 {2} {1,2,5} {3} {1,3,4} {4} {1,3,5} {5} {1,4,5} {1,2} {2,3,4} {1,3} {2,3,5} {1,4} {2,4,5} {1,5} {3,4,5} {2,3} {1,2,3,4} {2,4} {1,2,3,5} {2,5} {1,2,4,5} {3,4} {1,3,4,5} {3,5} {2,3,4,5} {4,5} {1,2,3,4,5} L4 (160 MW) Varable Cost (US$/MWh) G G G Results obtaned by the cooperatve game soluton mechansms, Shapley Value and Nucleolus, dffer among them, because of the fundamental dfferent nature of the agent nteracton they model. Shapley Value loos for a far dstrbuton based on margnal contrbutons of agents when randomly jon a gven coalton. Instead, Nucleolus mnmzes the maxmum dscomfort of the fnal resultant allocaton. Both methods have advantages and dsadvantages. If they compare wth tradtonal allocaton costs, such as GGDF and margnal partcpaton, the Shapley Value method provdes closer results. On the other hand, all methods concde n assgnng the largest responsblty of the cost of lne 1-4 to agents L1 and L5. The proposed methods wor wth games that are convex, whch ensures that the Shapley Value wll be at the center of the core. Besdes, results comply wth the ndvdual ratonalty prncple (no one pays more than ts stand-alone above cost) and group ratonalty prncple (all costs are covered). The collectve ratonalty prncple also apples. Solutons are part of the game core, whch mples allocaton stablty and coalton stablty. None of the agents has the ncentve to leave the coalton or group n a dfferent manner, as no alternatve coalton may mprove the allocaton. Once the games, for each of the lnes of the sx-bus Garver system, are solved, t s possble to obtan the aggregated results for the whole system, as shown n Table X. TABLE X TRANSMISSION NETWORK TOTAL COST ALLOCATION RESULTS Method Load L1 Load L2 Load L3 Load L4 Load L5 [%] [%] [%] [%] [%] Shapley Value 15,4 19,63 4,75 9,51 50,72 Nucleolus 13,22 24,92 4,75 9,04 48,05 GLDF 15,43 14,66 5,18 19,91 44,83 Margnal Partcpatons 14,98 13,99 5,37 19,81 45,84 Those results nclude new nvestments, whch correspond to one addtonal crcut for 3-5, two for 4-6 and 4 for 2-6 [19].

7 7 Allocatons of costs for only the new nvestments are gven n table XI. Results show that larger loads have a larger cost allocaton for the exstng networ as well as for the expansons. There are no publshed studes on the Garver system that could be used for comparson of cost allocaton to consumers. References [16], [17] focus ther assessment on cost allocaton to buses. The authors are developng further research to also use the method to determne generator payments. TABLE XI ESPANSION COST ALLOCATION RESULTS FOR LINES 2-6, 3-5 AND 4-6 Load L1 Load L2 Load L3 Load L4 Load L5 Method [%] [%] [%] [%] [%] Shapley Value 10,52 29,76 4,66 18,33 36,74 Nucleolus 5,01 36,79 3,34 17,19 37,67 GLDF 10,95 28,98 4,77 19,12 36,18 Margnal 11,01 27,98 4,69 19,62 36,7 Partcpatons All methods gve smlar results, assgnng more responsblty to the most mportant loads n the system. IV. CONCLUSION A new method has been proposed to allocate transmsson costs to agents n a networ. It s smple and transparent n ts applcaton and t s based on the real operaton of the transmsson system. If used as a complement to margnal prcng, t provdes adequate economc sgnals to the agents for ther optmal behavor n terms of effcency. Cooperatve game theory s used, allowng to ncorporate effcency and farness prncples. As a result, there s no alternatve better allocaton assgnment and two agents n dentcal condtons are allocated equal payments. Thus, ts applcablty to compettve marets becomes most attractve. The presented method allows solvng the transmsson cost allocaton problem, tang nto account exstent condtons of a gven electrc networ. But t also allows solvng the allocaton of expanson costs, wthout any methodologcal change. Results are compared and valdated wth exstng methods. A potental problem of the method s the handlng of the dmenson of the games, whch grows wth the rato 2 N, wth N beng the number of agents. The authors are developng an applcaton process for systems wth many buses, to be appled to the Chlean electrcal maret. The authors are also contnung research n algorthms for ndependent coalton formaton, stablty of coaltons and further applcatons n networ expanson. V. APPENDIX A. ARGMAX CONCEPT For any set Z and any functon f: Z R, argmax y e Z f(y) denotes the set of ponts n Z that maxmze the functon f, so argmax f ( y) = { y Z f ( y) = max f ( z)} (A.1) y Z z Z B. CONCEPTS OF COOPERATIVE GAME THEORY IN COST ALLOCATION A cost allocaton cooperatve game s gven by a couple (NA, C), where NA s the set of the N agents and C s the cost functon. The agents can group n many dfferent ways accordng to ther nterests and convenence. The way n whch N players group n m mutually exclusve and excludng coaltons S, s the coalton confguraton. δ = S,S,...S } (B.1) { 1 2 m d s a partton of NA that fulflls three condtons. Sj Φ; j = 1,2,..,m S S = Φ ; j j S j δ j S = NA (B.2) where Φ s the empty set. Each agent belongs to one and only one of the m coaltons and the members of a certan coalton are related to each other but not wth other agents that belong to other coaltons. The costs assgned to each agent as a result of the game correspond to the vector of assgnments or payments x = {x1, x 2,..., x N}, where x s the agent payoff. The result of a game s the payoff confguraton PC: PC = { x ; δ) = (x, x,...,x ;S,S,...S ) (B.3) 1 2 N The soluton of the game should fulfll the condtons of ratonalty (ndvdual, collectve and global) and stablty. x( ) C( ); NA x( S) C( S); S δ x( NA) = C( NA) 1 2 m (B.4) wth: x ( S) = (B.5) x S Any agent or coalton of agents should not have a bgger cost that ther alternatve cost or stand-alone cost. The resultng assgnment of costs of the game to all the players must be dentcal to the total costs to be covered. Ths last condton s nown as brea-even condton or Paretooptmum. The payoffs that are Pareto-optmum and ndvdually ratonal are denomnated mputatons. Besdes, f t s collectvely ratonal, the core of the game s obtaned. The core of the game ( NA, C) s the set of all the solutons PC (x; d) such: x( T ) C( T ); T NA, wth: x ( T ) = x (B.6) T

8 8 The core s, therefore, a subset of the group of mputatons. Ths core concept s the smplest of all the soluton concepts of cooperatve games. It corresponds to a group of mputatons that leaves no space for a better allocaton to ts players and does not allow subsdes among coaltons. Unfortunately, there are many games wth too large cores (many solutons), no core at all or empty core. The empty core taes place when the collectve ratonal s not acheved. Startng from the core t s possble to establsh dfferent theores that outlne solutons to ths type of games. Solutons could be: extensons of the core, stable set and barganng set. Other solutons can be acheved wth the excess theory, defned as the dfference between the coalton stand-alone cost and the payoff for that coalton as a result of the game. Thus, the excess of coalton R wth respect to the payoff vector x of the PC (x; d) s defned as: e( R, x ) = C( R) x( R) (B.7) and represents the total amount that potental members of the R coalton collectvely wn or lose f they wthdraw from R. From the excess defnton t s possble to ncorporate the Nucleolus and Kernel soluton mechansms. The nucleolus soluton corresponds to mputatons for whch the maxmal excess s mnmzed: max{mne( R, x )}; R (B.8) Other tradtonal soluton s the Shapley Value, gven by: ( N s)(! s 1 )! [ C( S) C( S {} )] NA φ = (B.9) N! S NA where: N: total number of players S: coalton of players s = S : number of players n coalton S : player VI. ACKNOWLEDGMENT The authors gratefully acnowledge the support from Unversdad de Santago de Chle and Unversdad Católca de Chle. We also acnowledge the helpful comments receved from three anonymous revewers VII. REFERENCES [1] H. Rudnc, The Electrc Maret Restructurng n South Amerca: Success And Falures on Maret Desgn, Plenary Sesson, Harvard Electrcty Polcy Group, San Dego, Calforna, January [2] H. Rudnc and J. Zolezz, "Electrc Sector Deregulaton and Restructurng n Latn Amerca: Lessons to be Learnt and Possble Ways Forward", IEE Proceedngs Generaton, Transmsson and Dstrbuton, vol 148, no. 2, pp , March [3] I. Pérez-Arraga, H. Rudnc and W. Stadln, Internatonal Power System Transmsson Open Access Experence, IEEE Transactons on Power Systems, vol. 10, no. 1, pp , February [4] R. Tabors, Transmsson System Management and Prcng: New Paradgms and Internatonal Comparsons, IEEE Transactons on Power Systems, vol. 9, no. 1, pp , February [5] R. Green, Electrcty Transmsson Prcng: An Internatonal Comparson, Utltes Polcy, vol. 6, no. 3, pp , [6] H. Rudnc, paper dscusson "Margnal prcng of Transmsson Servces: a Comparatve Analyss of Networ Cost Allocaton Methods", IEEE Transactons on Power Systems, vol. 15, no. 4, pp , November [7] F. Rubo, Metodología de Asgnacón de la Red de Transporte en un Contexto de Regulacón Aberto a la Competenca, Unversdad Pontfca Comllas de Madrd, Escuela Técnca Superor de Ingenería, Doctoral Thess, pp , [8] H. Rudnc, R. Palma and J. Fernández, Margnal Prcng and Supplement Cost Allocaton n Transmsson Open Access, IEEE Transactons on Power Systems, vol. 10, no. 2, pp , May [9] H. Rudnc, M. Soto and R. Palma, Use of System Approaches for Transmsson Open Access Prcng, Internatonal Journal of Electrcal Power and Energy Systems, vol. 21, no. 2, pp , [10] F. Rubo and I. Perez, Margnal Prcng of Transmsson Servces: a Comparatve Analyss of Networ Cost Allocaton Methods, IEEE Transactons on Power Systems, vol. 15, no. 1, pp , February [11] J. Bale, Topologcal Generaton and Load Dstrbuton Factors for Supplement Charge Allocaton n Transmsson Open Access, IEEE Transactons on Power Systems, vol. 12, no. 3, pp , August [12] D. Krschen, R. Allan and G. Strbac, Contrbutons of Indvdual Generators to Loads and Flows, IEEE Transactons on Power Systems, vol. 12, no. 1, pp , February [13] J. Kahan, A. Rapoport, Theores of Coalton Formaton, London, Lawrence Erlbaum Assocates, [14] H. Young, Cost Allocaton, Handboo of Game Theory, vol. 2, pp , [15] Y. Tsuamoto and I. Iyoda, Allocaton of Fxed Transmsson Cost to Wheelng Transactons by Cooperatve Game Theory, IEEE Transactons on Power Systems, vol. 11, no. 2, pp , May [16] J. Contreras and F. Wu, Coalton Formaton n Transmsson Expanson Plannng, IEEE Transactons on Power Systems, vol. 14, no. 3, pp , August [17] J. Contreras and F. Wu, A Kernel-Orented Coalton Algorthm for Transmsson Expanson Plannng, IEEE Transactons on Power Systems, vol. 15, no. 4, pp , November [18] C. Yeung, A. Poon and F. Wu, Game Theoretcal Mult-agent Modelng of Coalton Formaton for Multlateral Trades, IEEE Transactons on Power Systems, vol. 14, no. 3, pp , August [19] L. Garver, Transmsson Networ Estmaton Usng Lnear Programmng, IEEE Transactons on Power Apparatus and Systems, vol. PAS-89, no. 7, pp , September/October [20] H. Saadat, Power System Analyss, McGraw-Hll, pp , VIII. BIOGRAPHIES Juan Zolezz was born n Valdva, Chle, and graduated as an Electrcal Engneer from Techncal State Unversty of Chle, later obtanng hs M.Sc. from Unversty of Chle and presently s a Ph.D. canddate at Catholc Unversty of Chle. He s a professor at Santago Unversty of Chle. Hs research actvtes focus on power economcs, economc transmsson ssues, plannng and regulaton of electrc marets. He has been a consultant wth utltes and regulators n Chle. Hugh Rudnc was born n Santago, Chle, and graduated as an Electrcal Engneer from Unversty of Chle, later obtanng hs M.Sc. and Ph.D. degrees from the Vctora Unversty of Manchester, UK. He s a professor at Catholc Unversty of Chle. Hs research actvtes focus on the economc operaton, plannng and regulaton of electrc power systems. He has been a consultant wth the World Ban and wth utltes and regulators n Argentna, Bolva, Brazl, Canada, Central Amerca, Chle, Colomba, Peru and Venezuela, as well as n Europe.