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4568 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 4, NOVEMBER 2013
Transmission Network Cost Allocation Based onCircuit Theory and the Aumann-Shapley Method
Yuri P. Molina, Osvaldo R. Saavedra, Member, IEEE, and Hortensia Amarís, Member, IEEE
Abstract—This paper presents a newmethod to allocate the costsof the transmission system among generators and loads. The al-location is calculated for each branch of the transmission systemto identify and quantify the individual responsibility of generatorsand loads. A two-step method based on the perfect coupling of thecircuit theory with the Aumann-Shapley method is proposed here.First: to determine the participation of the generators in the costsof the transmission network, the generators aremodeled as currentinjections and the loads as impedances. Second: to determine theparticipation of the loads in the cost of the transmission network,the loads are modeled as current sources and the generators asimpedances. The Aumann-Shapley method and the circuit theoryare used to calculate the participation of each real and imaginarycurrent component in the “Allocation of the costs of the transmis-sion system” game by considering them as independent agents. Theproperties of the Aumann-Shapley method ensure equitable allo-cation and recovery of the total costs. Numerical results are pre-sented and discussed to demonstrate the applicability of the pro-posed method.
Index Terms—Aumann-Shapley method, circuit laws, currentcomponent, transmission network cost.
NOTATION
The notation used throughout this paper is stated below forquick reference.
A. Game Theory
Occurrence probability of coalition .
Group of agents who are in the game.
Cost due to coalition .
Cost allocated to agent .
Number of agents in the set .
Number of agents in coalition .
Infinitesimal value of .
Cost evaluated with equal to .
Cost allocated to agent .
Manuscript received January 23, 2013; revised May 06, 2013; accepted Au-gust 07, 2013. Date of publication August 23, 2013; date of current versionOctober 17, 2013. Paper no. TPWRS-00013-2013.Y. P. Molina is with the Department of Electrical Engineering, Federal
University of Paraíba, Joao Pessoa, PB 58051-900, Brazil (e-mail: [email protected]).O. R. Saavedra is with the Power System Group, Federal University of
Maranhão, São Luís, MA 65085 580, Brazil (e-mail: [email protected]).H. Amarís is with the Department of Electrical Engineering, University
Carlos III of Madrid, Madrid 28911, Spain.Digital Object Identifier 10.1109/TPWRS.2013.2278296
Quantity corresponding to agent .
Unitary cost in Aumann-Shapley method foragent .
B. Game Theory Applied to Transmission Network CostAllocation
Complex power flow through line kmcalculated at bus .
Complex nodal voltage at bus .
Current through line .
Element of the impedance matrixconsidering loads as impedances.
Series impedance of the equivalentcircuit of line .
Shunt admittance of the equivalentcircuit of line .
Complex current of the generator locatedat bus .
Real part of current provided by generatorlocated at bus .
Imaginary part of current provided bygenerator located at bus .
Number of generator buses.
Element of the impedance matrixconsidering generators as impedances.
Complex load current at bus .
Real part of load current at bus .
Imaginary part of load current at bus .
Number of load buses.
Unitary participation of player in thecomplex power flow through line .
Total participation of player in thecomplex power flow through line .
Participation of player in the activepower flow through line .
Unitary participation of player inthe complex power flow through line
.
0885-8950 © 2013 IEEE
MOLINA et al.: TRANSMISSION NETWORK COST ALLOCATION BASED ON CIRCUIT THEORY AND THE AUMANN-SHAPLEY METHOD 4569
Total participation of player in thecomplex power flow through line .
Participation of player in the activepower flow through line .
Participation of player in the activepower flow through line .
Unitary participation of player in thecomplex power flow through line .
Participation of player in the activepower flow through line .
Unitary participation of player inthe complex power flow through line
.
Participation of player in the activepower flow through line .
Participation of player in the activepower flow through line .
Usage of line .
Usage of line allocated to thegenerator located at bus .
Usage of line allocated to the loadlocated at bus .
I. INTRODUCTION
I N recent years, electric power systems worldwide havemoved from a regulated environment based upon tradi-
tional, vertically integrated utilities toward a decentralizedenvironment based on competitive markets. In this new sce-nario for electrical power systems, the costs of electricityservices should be properly identified. An important issue tobe addressed in the competitive environment is how to allocatethe total cost of the transmission system in an equitable wayamong generators and loads. That is, the allocated cost shouldreflect the real use they make of the transmission network andprovide efficient sitting signals to promote new investments atthe same time [1].Traditionally, the costs of the transmission system were re-
covered in a flat or pro rata way, in which the cost was allocatedto any agent in proportion to the MW connected to the network[2]. This method provides an easy calculation system; however,transmission system users are not differentiated by their respec-tive real usages. As a result, this method provides incorrect eco-nomic signals to transmission system users [3].Some flow-based methods were presented and revised in [4]
and [5]. Thesemethods usually combine power flow results withthe proportional sharing principle, and the assumption made isthat the network bus is a perfect combination of all incomingflows, so that it is impossible to determine the output element(line) of an electron that reaches into the bus. Despite the rea-sonable assumptions used in these methods, high volatility is
observed in allocating transmission network cost with respectto temporal load variations and to generator dispatch strategies[6].Recently, a method based on the min-max fairness algo-
rithm, using optimization-based real power tracing, has beenproposed in [3]. Subsequently, in [7], an optimal power flowtracing method has been proposed to “explicitly” model fair-ness constraints in the tracing framework of the fair min-maxtracing discussed in [3]. The proposed approach shows goodperformance and addresses concerns such as cycling and con-vergence to suboptimal solutions.In [6], an approach using the principle of equivalent bilat-
eral exchanges (EBE) is presented as an alternative method thatavoids some of the disadvantages of other methods. To buildthe EBEs, each demand is proportionally assigned a fraction ofeach generation, and conversely, each generation is proportion-ally assigned a fraction of each demand, in such a way that bothKirchhoff’s laws are satisfied.In [8], a marginal participation method is proposed where
the incremental line (or network) usage for a generator or aload when extra (MW) is injected into or drawn from the gridis evaluated. Then, the cost of the line (or network) is appor-tioned among the load and generator entities according to theirweighted marginal participation. Note that loads or generatorsthat reduce a line flow (or network usage) by their marginalusage are exempt from bearing the cost of the line because noentity should be paid for grid use. The limitations of this schemeare described in [3]. In fact, one of the main restrictions is itssensitivity to the choice of slack bus. In order to overcome thislimitation, the use of a dispersed slack bus to reduce price forloads and generators is discussed in [9].In [10], an approach to allocate the transmission network cost
based on the matrix is presented, which was previouslyapplied to loss allocation [11]. It should be emphasized thatall transmission lines must be modeled including shunt admit-tances. By doing so, the method provides an appropriatenumerical behavior.Lately, some methods based on the cooperative game theory
have been proposed to allocate the costs of the transmission net-work among generators and loads [1], [12], [13]. These over-come the drawbacks of the conventionally usedmethods and en-courage the economically optimal usage of transmission facili-ties [14]. Game theory provides interesting concepts, methods,andmodels that may be used to assess the interaction of differentagents in competitive markets and solve conflicts that arise inthat interaction [12]. The solution mechanisms of the cooper-ative game theory behave well in terms of fairness, efficiency,and stability, i.e., qualities required for the correct allocation oftransmission costs. Nevertheless, the proposals presented so farare still in the early stages of development, with contributionsbeing formulated in wheeling transactions [13], loss allocation[15]–[17], and others.In this particular application, the Aumann-Shapley method is
used to allocate the cost of the transmission network as a di-rect consequence of the successful application of the Aumann-Shapley method for solving the problem of loss allocation re-ported in [17]. This method calculates the average of the usagecontributions of each generator or load in each transmission line
4570 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 4, NOVEMBER 2013
by considering all relief orders; it can completely avoid the im-pact of the relief order on the allocation, ensuring equitable al-location of transmission network cost.The main contributions of this paper are as follows:1) it considers circuit laws and has desirable characteristics interms of economic coherence, because it is based on circuitlaws in combination with the Aumann-Shapley method;
2) it identifies and quantifies the active and reactive powerconsumed in each transmission line per user;
3) it fully allocates the transmission network cost (total costrecovery), due to the additive property of the Aumann-Shapley method;
4) it allocates the transmission network costs to generatorsand loads by considering the independence of agents, evenwhen they are in the same bus;
5) the proposed approach does not require much computa-tional effort because an analytical solution of the Aumann-Shapley method is obtained, avoiding the computationalburden and errors associated to iterative processing;
6) has little sensitivity to the power flow direction;7) by modeling the generators and loads (in different sce-narios) as constant admittances, the process maintains thepaths of active and reactive power in the transmission net-work with good approximation, identifying and separatingan appropriate network usage cost.
This paper is organized as follows. In Section II, the Aumann-Shapley method is revised and the cause of the problem of trans-mission network cost allocation is identified as being highly re-lated to game theory. In Section III, the proposed method is pre-sented. In Section IV, illustrative numerical results with a 4-bustest system are reported, as well as the results with the IEEE30-bus test system. Conclusions, comments, and final consider-ations are presented in Section V.
II. COST ALLOCATION BY GAME THEORY
Game theory has been widely recognized as an importanttool in many fields, with practical applications to social, eco-nomic, political, biological, and other problems. In the elec-tricity sector, game theory has been extensively applied to sev-eral situations, especially allocation problems. In this section,two cost allocation methods are revised.
A. Shapley Value Method
The Shapley value method finds an expected marginal con-tribution to each player in the game with respect to a uniformdistribution on the set of all permutations in the entry order ofplayers to analyze all possible combinations in the game. Thecost, the profit, the benefits, or the participation (for now, thecost) of each agent is calculated. The average value of the in-cremental costs calculated in each permutation determines thecost that corresponds to each agent. Thus, the influence of theplayer entry order on the cost allocation is eliminated [1], [17].The Shapley value can be interpreted as being the average
value of the incremental costs of including the agent; it con-siders all sub-coalitions that do not contain this particular agent,including the empty sub-coalition. Assuming that the occur-rence probabilities of sub-coalitions of several sizes are the
same, the allocation is defined formally through the followinganalytical expression:
(1)
Based on probability concepts
(2)
With the application of the Shapley value, plausible resultsare obtained and the solution is intuitively considered “equi-table”, because all agents have the same opportunity to be inthe best and worst order positions. However, due to its combi-natorial nature, the problem size grows exponentially with thenumber of agents, and the method becomes computationallyunfeasible. For example, the total number of permutations forthe case with agents is equal to (there are approximately3 628 800 possible permutations for ten agents).
B. Aumann-Shapley Method
The Aumann-Shapley method is a natural consequence ofthe Shapley value method. It is based on the premise that eachagent has to be sub-divided into infinitesimal sub-agents, and theShapley method is applied to each one as if each sub-agent werean individual. The Aumann-Shapley value reflects the averagemarginal contributions to the coalition for all agents, makingthe problem insensitive to order of entry. Additionally, desirablecharacteristics in terms of economic coherence and isonomy areintroduced, making it more complete than the Shapley valuemethod according to [1], [15], [18], and [19].At first glance, the computational effort would be even greater
than in the Shapley value method due to the significant increasein the number of combinations. However, there is an analyticalsolution when agents are divided into infinitesimal sub-agents.This method is the only method that satisfies fundamental ax-ioms for fair allocation among agents:• Symmetry—The cost assigned to each player does not de-pend on the order of the players;
• Effectiveness—The sum of costs assigned to an individualplayer is equal to the total cost (full cost recovery);
• Additivity—the sum of costs assigned to a player who de-cides to play two games separately is equal to the cost as-signed to this player when the two games are played to-gether;
• Isonomy—the players who have the same cost functioninfluence should have the same unitary participation.
In order to provide an intuitive example of the Au-mann-Shapley method, a given amount of service isconsidered to be requested by all agents. At this point, thecost of using this service will be equal to . If a given agentrequests a greater amount of service equal to , there willbe an increase in the cost of service to . Theincremental cost caused by this agent is .As discussed in the previous subsection, the order of entry
affects cost allocation. Likewise, the order in which the agentsrequesting extra amounts of service should influence cost al-location. The problem can be solved by dividing each agent
MOLINA et al.: TRANSMISSION NETWORK COST ALLOCATION BASED ON CIRCUIT THEORY AND THE AUMANN-SHAPLEY METHOD 4571
into infinitesimal parts . Thus, the effect in the newsub-agent in the cost function is
(3)
where is the cost evaluated with equal to , and isthe infinitesimal value of . That is, the incremental cost due to
is approximately equal to the marginal cost. In some math-ematical conditions [20], it can be verified that when the valueof increases from zero to its maximum value, all marginalcost averages tend to a value known as the “Aumann-Shapleyunitary cost”, which is mathematically expressed as
(4)
Equation (4) should be analytically or iteratively solved. Insome problems, it is possible to obtain analytical solutions asthose presented in this paper.Finally, the cost allocated to each agent is
(5)
III. GAME THEORY APPLIED TO TRANSMISSIONNETWORK COST ALLOCATION
Given a set of agents (generators and loads) that use the trans-mission system to generate energy from production centers tothe consumption centers, the total cost of this service (includinginvestment in circuits, O&M, etc.) should be allocated to allusers in an equitable way, so that the allocation reflects the ef-fective use that each agent makes of the transmission system.The question posed here is how to allocate the costs of trans-mission.Clearly, the transmission system is used by generators and
loads as a direct result of the energy supplied by the generatorsand the power required by the loads. The first question to beanswered is as follows: which agents should pay for the use ofthe transmission system? The most accepted answer is that bothgenerators and loads are responsible for the use of the transmis-sion network. Therefore, it is assumed that distribution shouldbe equal: 50% for generators and 50% for loads.The 50/50 rule has been advocated by the authors in previous
works. Game theory has demonstrated the existence of a fair andequitable allocation [15]. Furthermore, from a physical point ofview, in a simple two-bus system, with one generation and oneload, it is clear that the line flow is the result of the power sup-plied by the generator and the power demanded by load. Theycoexist in a mutually advantageous cooperation. Without load,there is no need for generation, and without generation, there isno load [17].The proposed method in this paper consists of two steps: 1)
generators are modeled as current injections and the loads asimpedances to determine the responsibilities of network usageby generators. 2) the loads are modeled as current sources and
Fig. 1. equivalent of line .
the generators as impedances to determine the responsibilitiesof network usage by loads.
A. Background
In order to determine the responsibility of the agents (gen-erators and loads) in the use of the transmission system, it isnecessary to determine first their responsibility in a transmis-sion line. Subsequently, the costs of each transmission line areadded and, finally, the costs associated with the use of the trans-mission system for each agent system.The expression for the power flow in the transmission line in
Fig. 1 can be expressed as follows:
(6)
Both and can be expressed as a function of the equiva-lent current sources of the generators. The voltage at node isgiven by
(7)
The current through the line is obtained as
(8)
Substituting (7) into (8) and rearranging gives
(9)
Analogously, to determine the responsibility of the loads inthe use of the transmission system, the responsibility of loads ina transmission line is determined first.The terms of the expression (6) can be rewritten as a function
of the equivalent current sources of the loads:
(10)
The current through the line is
(11)
4572 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 4, NOVEMBER 2013
Substituting (10) into (11) and rearranging gives
(12)
B. Allocating Cost to Generators by Aumann-Shapley Method
To allocate the responsibility of the generators in the trans-mission system, the generators are modeled as current sourcesand the loads as impedances. Then, Aumann-Shapley methodis applied in the “transmission system cost allocation” game,with players who may be expandedinto players by considering their real and imaginarycomponents as independent parts. Therefore, all players are
. Equation (4)is used here to obtain the participation of each player.1) Unitary Participation of in the Complex Power Flow
Through Line Is:
(13)
where
(14)
Developing (14), and substituting it into (13) the unitary par-ticipation of gives
(15)
The details of the development to achieve (15) from (13) isshown in the Appendix.Finally, to determine the total participation of player in
the complex power flow of the line , the unitary partici-pation is multiplied by the amount of the player:
(16)
Thus, a portion of the complex power flow through lineis directly associated with . Then, the participation in theactive power flow at line of the player is
(17)
2) Unitary Participation of in the Complex Power FlowThrough Line Is:
(18)
where
(19)
By developing (19), and substituting it into (18), the unitaryparticipation of in the complex power flow through line
is
(20)
Therefore, to determine the total participation of playerin the complex power flow through line , the unitary par-ticipation is multiplied by the “amount” of the player:
(21)
Analogously, the participation in the active power of lineof the player is
(22)
The total participation of the generator located at bus on theactive power flow of line is the sum of (17) and (22):
(23)
C. Allocating Cost to Loads by the Aumann-Shapley Method
To allocate the responsibility of loads on the transmissionsystem, the loads are modeled as current sources (negativevalues) and the generators as impedances (negative values). Inthis case, it is possible to obtain a nonlinear function of the linein the same way used for the generators. This strategy is onlyvalid for the operation point under analysis (a snapshot of thesystem operation).The procedure to allocate the costs for transmission system
usage is similar to that for generators, as shown in the previous
MOLINA et al.: TRANSMISSION NETWORK COST ALLOCATION BASED ON CIRCUIT THEORY AND THE AUMANN-SHAPLEY METHOD 4573
section. Thus, the unitary participation of in active powerflow of the line at the bus is
(24)
The participation of player in active power flow at lineis
(25)
The unitary participation of (bus ) in the active powerflow at line is
(26)
The participation of player in the active power flow atline is
(27)
The total participation of the load located at bus in the activepower flow of line is the sum of (25) and (27):
(28)
D. Quantification of the Line Usage
The most accepted criterion, though not the only, to quantifythe use of line , is based on the contribution to the activepower flow module [10], [6]:
(29)
because both dominant flows and counter flows make use of theline. Therefore, the total usage of the line is
(30)
E. Power Flow Direction
As discussed in [10], the method is sensitive to thepower flow direction, i.e., the cost allocation obtained by con-sidering a power flow from to is quite different than when
considering a power flow from to . This difference also leadsto the conclusion that the difference between power flowsand does not represent the losses in this line. To overcomethis problem, in [10], the authors proposed the method,which uses the average of the allocated costs considering powerflows and .In this paper, the cost allocations using the input and output
flows are slightly different, and this difference can be attributedto effect of the losses. Typically, losses are small in high voltagelines. To maintain the applicability of the proposed method tomedium and low voltage power circuits, we also adopted theaverage of power flows and , as criterion to quantifythe line usage:
(31)
(32)
Therefore, the allocations are more balanced among load andgenerating buses.
F. Allocation to Loads and Generators Plugged at the SameBus
The loads and the generators are considered independentagents. Thus, their responsibility for the transmission costs iscomputed independently, which is a very important point toconsider and an additional contribution with respect to othermethods that consider the equivalent power injection at thistype of bus.
IV. NUMERICAL RESULTS
In this section, the proposed method is compared with otherexisting methods using two test systems; the first one is a di-dactic four-bus system, and the second one is the IEEE 30-bussystem. For two systems, it is assumed that maximum trans-mission system stress occurs under peak demand conditions. Itis emphasized that the proposed method can be applied to verylarge systems because it uses an analytical solution based on theAumann-Shapley method and does not require significant com-putational effort.
A. Four-Bus System
The four-bus system (see Fig. 2) used in [10] is considered. Ithas five transmission lines with identical values of resistance, re-actance, and shunt susceptance: 0.1275, 0.097, and 0.4611 p.u.,respectively. It is also assumed that the cost of each line is pro-portional to their series reactances: .Four methods of cost allocation are considered: pro-rata (PR),
and [10], and proportional sharing (PS) [4], whichare compared with the proposed method.Table I shows that the proposed method allocates the cost
of the lines as 50% to loads and 50% to generators, as in thePS and PR methods. This principle is based on the fact thatpower will flow through a line if there is at least one generatorand one load at their respective terminal nodes. Therefore, it isreasonable to expect an allocation in the same proportion among
4574 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 4, NOVEMBER 2013
Fig. 2. Four-bus system.
TABLE ITRANSMISSION COST ALLOCATION TO BUSES
generators and loads. However, and do not followthis proportion because its allocation depends basically of thenetwork parameters [21].The proposed method allocates the transmission cost by con-
sidering the amount, the location, and the effective use of theline. The generator at bus 1 uses little of the line 1–2 becausethe dominant direction of the power flows along the direction2–1. Therefore, the cost allocation to the generator 1 is small,because the counterflow is considered. However, andallocate a significant amount of the cost of each line to the agents(loads and generators) directly connected to the lines, regardlessof the effective use of each line by the agent, which can be ob-served lines 1–2, 1–3, 1–4, and 3–4, in which the allocation toagents located at the ends of these lines have higher cost withrespect to other agents.
Fig. 3. IEEE 30-bus test system.
The proposed method can be considered less sensitive tothe flow direction, that is, the allocation cost considering theincoming flow is slightly higher than when outgoingflow is considered. For example, the cost allocationto generator 1 (bus 1) due to line 1-4 when considering theincoming flow is 30.88 /h; when considering the outflow, weobtain 30.78 /h (the difference is due to the line losses). Byallocating the average value, the result is practically the same:(30.78 /h /h)/ /h. However, themethod is highly sensitive to the active power flow direction;that is, the allocation considering the incoming flow on the lineis very different from the allocation considering the outgoingflow of the line. For example, the allocation to generator 1 (bus1) in line 1–4 is 49.69 /h by considering the incoming flow andapplying the method, and is 21.57 /h by considering theoutflow and applying the method (this difference cannotbe attributed to line losses). The method allocates theaverage .
B. IEEE 30-Bus System
Tovalidate theproposedmethod, the IEEE30-bussystem[17],shown in Fig. 3 is used. The same methods used in the previouscase study are considered again. It is assumed that the analyzedoperation point corresponds to the peak load of the system.According to the proposed method, both generators and loads
are equally responsible for using each transmission line. Giventhe large number of lines, only the participation of the generatorsand loads in the total costs of the transmission system are shown,that is, the total contribution at each line, and one line (5–7) isselected for the analysis of the behavior of the proposed method.Table II shows that all the methods allocate a significant
amount of the total cost of the transmission system to thegenerator located at bus 1, which is responsible for the largestactive power injection. Thus, all of the methods implicitlyreflect the order of magnitude of the generation sources.This table also shows that the allocation to the generator lo-
cated at bus 5 by the PS method is null because the load is largerthan the generation at this bus, and, consequently, it is handledas a load bus. Therefore, only the load is responsible for the al-located costs at this bus, and the generator has null allocation.
MOLINA et al.: TRANSMISSION NETWORK COST ALLOCATION BASED ON CIRCUIT THEORY AND THE AUMANN-SHAPLEY METHOD 4575
TABLE IITRANSMISSION COST ALLOCATION TO GENERATORS
TABLE IIITRANSMISSION COST ALLOCATION TO LOADS
However, for the and methods, the generation andthe load at the same bus share the responsibility for the use ofthe grid. The proposed method allocates 399.79 ( /h) to the gen-erator located at bus 5 and 317.43 ( /h) to the load located at thesame bus (see Table III). This independence in the results is be-cause the proposed method handles both the generator and theload as independent players. This result is in accordance withcircuits and game theory.Table III shows that the cost allocated to the load buses due to
lines without shunt admittances by the and methodsare much larger than in proposed method, which is most evidentfor buses 29 and 30. This difference is observed because allo-cation by the method is very sensitive to small variationsin the admittance matrix as mentioned in [21].Table IV illustrates the cost allocation of line 5–7 to the gener-
ators. The allocation provided by the PS method indicates thatgenerators 5, 11, and 13 are not responsible for the use of theline, which is not reasonable from an intuitive analysis.
TABLE IVCOST ALLOCATION OF THE LINE 5–7 TO GENERATORS
TABLE VCOST ALLOCATION OF THE LINE 5–7 TO LOADS
The proposed method allocates a considerable amount to theagents who are near line 5–7. For instance, the value allocated tothe generator at bus 5 is much greater than the value allocated tothe other generators because the generator at bus 5 is close to line5–7. For this reason, in Table V, it is shown that the cost alloca-tion for the use of the line 5–7 by load 5 is considerably high. Asimilar behavior is obtained using the and methods.The proposed method is based on the law of circuits com-
bined with the Aumman-Shapley value. It presents good per-formance in terms of the location of an agent in the network,the distance, and the amount of power injected (generators) orextracted (loads) from the network. The expression obtained tocalculate the use of a line by an agent is less sensitive to theside of the line used as reference. Furthermore, the allocation isperformed for each generator/load individually. The proposedmethod naturally inherits the properties of the Aumann-Shapleymethod, such as equality, effectiveness, additivity, and sym-metry, and thus a fair allocation among agents is calculated.
4576 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 4, NOVEMBER 2013
V. CONCLUSION
This paper presented a new approach to determine the re-sponsibility of the generators and the loads to the transmissionsystem cost based on circuit theory in combination with gametheory, specifically the Aumann-Shapley method. In this ap-proach, it is possible to determine the participation of the realand imaginary components of the currents in the use of the trans-mission system.The analytical solution obtained by applying the Aumann-
Shapley method does not require significant computational ef-fort, and it is free from numeric mistakes that can result fromiterative processing.The transmission system costs are attributed to the genera-
tors and loads by considering the independence of the agents,although they are at the same bus. The obtained allocation re-flects the order of magnitude of the generators and the loads, aswell as their location in the grid.
APPENDIXUNITARY PARTICIPATION OF IN THE COMPLEX
POWER FLOW THROUGH LINE
The unitary participation of in the complex power flowthrough line is
(33)
where
(34)
developing the partial derivative of (34) is obtained:
(35)
substituting (35) into (33) and rearranging gives
(36)
by developing (36) is obtained
(37)
In order to check the total allocation in each step, the termof the (37) is removed. Therefore, the unitary participation
of in the complex power flow through line is
(38)
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Yuri P. Molina was born in 1978. He received theDiploma in electrical engineering in 2003 from theNational University of Engineering, Lima, Peru, theM.Sc. degree in power system in 2005 from FederalUniversity of Maranhão, and the Ph.D. degreein electrical engineering in 2009 from CatholicUniversity of Rio de Janeiro, Brazil.He joined the Electrical Engineering Department
of Federal University of Paraiba, Paraiba, Brazil, in2012. His research interests include control, opera-tions, planning, and economics of electric energy sys-
tems, as well as development of new electricity markets.
Osvaldo R. Saavedra (S’88–M’94) received theM.Sc. and Ph.D. degrees from the State Universityof Campinas, Campinas, Brazil, in 1988 and 1993,respectively.From 1983 to 1986, he was with Inecom Engineers
Ltd., Arica, Chile. From 1994 to 1997, he was a Vis-iting Lecturer at the Federal University of Maranhão(UFMA), Maranhão, Brazil. Since 1997, he has beena Professor in the Electrical Engineering Departmentat UFMA. He co-founded Renewable Energy Labo-ratory and the Electrical Energy Institute in this in-
stitution. From 2010 to 2012, he was vice-president of the Brazilian Society ofAutomatics. Since 2011, he has been with the state government of Maranhão,Brazil, as sub-secretary of science and technology. His main research interestsare in renewable energies and power system operations.
Hortensia Amarís (M’00) received the ElectricalEngineer degree and the Ph.D. degree from theTechnical University of Madrid (UPM), Madrid,Spain, in 1990 and 1995, respectively.She joined the Department of Electrical Engi-
neering of the University Carlos III of Madrid in1996, where she has been working mainly on powerquality, power-electronic converters, custom powersystems, renewable energy sources and smart grids.
