1
Transmission Cost Allocation by Cooperative
Games and Coalition Formation
Juan M. Zolezzi, Member, IEEE, and Hugh Rudnick, Fellow, IEEE
Abstract-- The allocation of costs of a transmission system to its
users is still a pending problem in many electric sector market
regulations. This paper contributes with a new allocation method
among the electric market participants. Both cooperation and
competition are defined as the leading principles to fair solutions
and efficient cost allocation. The method is based mainly on the
responsibility of the agents in the physical and economic use of
the network, their rational behavior, the formation of coalitions
and cooperative game theory resolution mechanisms. The
designed method is applicable to existing networks or to their
expansion. Simulations are made with sample networks. Results
conclude that adequate solutions are possible in a decentralized
environment with open access to networks. Comparisons with
traditional allocation systems are shown, cooperative game
solutions compare better in economic and physical terms.
Index Terms— Cooperative Game Theory, Nucleolus, Open
Access, Transmission Cost Allocation, Transmission Expansion,
Shapley Value.
I. INTRODUCTION
T
he deregulation of the power industry, which started in
South America in the 1980’s and which developed
worldwide has originated a rich discussion about the best way
to manage, economically and technically, the different markets
and situations developed with deregulation. Without any
doubt, the deregulation process has faced difficulties,
successes and failures, with many problematic areas still being
explored and developed [1], [2]. Among them, one of the most
complicated ones has been the non-discriminatory open access
to transmission and distribution networks, and the cost
allocation among the different market agents utilizing those
networks [3].
Different authors have emphasized the importance of the
transmission system in the new deregulated markets [4], as
facilitator of generator competition, allowing generators to
allocate their production in consumer centers and allowing
consumers to benefit from that competitive environment.
Within that framework, the transmission tariff system and
the user cost allocation must preserve an adequate resource
allocation among market agents. It is desired that transmission
prices and payment do not disturb decisions for new
generation investment, for generator operation and for
This work was supported by Fondecyt Project 1000517.
J. M. Zolezzi is with the Department of Electrical Engineering,
Universidad de Santiago de Chile, Casilla 10233, Correo 2, Santiago, Chile
(e-mail: jzolezzi@ieee.org).
H. Rudnick is with the Department of Electrical Engineering, Universidad
Católica de Chile, Casilla 306, Correo 22, Santiago, Chile (e-mail:
h.rudnick@ieee.org).
consumer demand [5], [6]. At the same time, this must be
achieved in a simple and fair form. Each country has had to
find its solution in agreement with the characteristics of its
transmission system.
Classic and modern solutions to transmission cost allocation
have not been able to satisfy expectations of regulators and
market agents. Many solutions have been formulated and many
applied, raging from wheeling bilateral schemes to multilateral
open access ones. Most of them face problems due to lack of
economic foundations [7].
Different allocation schemes have been formulated in recent
years based on the “natural economic use” of the transmission
system [8]-[10]. They aim at considering the way the
transmission system is impacted by generators and consumers
by the simple fact of being connected to the network,
irrespective of their commercial contracts with other agents
using the same network. In some countries this framework
gives birth to the “area of influence” concept [7], [8]. Different
methods have been suggested to identify the impact a
generator or a consumer has on the flow of a transmission line
within a transmission network, as well as how much of the
generation of a given generator corresponds to a given load
[11], [12]. These methods consider topological studies,
proportionality principles and network flow equations and
determine factors that are used to allocate complementary
transmission charges or total transmission payments, based on
the use made by the different agents.
The beneficiaries’ method is another alternative that has
been proposed, with solid economic bases, but it faces
difficulties in practical applications [7].
Game theory provides interesting concepts, methods and
models that may be used when assessing the interaction of
different agents in competitive markets and in the solution of
conflicts that arise in that interaction [13], such as those of the
electricity markets. In particular, cooperative game theory
arises as a most convenient tool to solve cost allocation
problems [14]. The solution mechanisms of cooperative game
theory behave well in terms of fairness, efficiency and
stability, qualities required for the correct allocation of
transmission costs. Nevertheless, proposals to date are still in a
developing stage, with contributions been formulated in
wheeling transactions [15], in the allocation of expansion costs
[16], [17] and other [18].
This paper presents a method to allocate charges among
users of a transmission system, either in an existing network or
an expanding one. The method is based in a model that
integrates cooperation and coordination among the agents as
2
basic principles. It also considers the physical and economic
use of the network by the different agents. It assumes a rational
behavior of the agents, the formation of coalitions among
agents, along with cooperative game solution mechanisms.
The proposed method aims at achieving the transmission
pricing principles defined in [5]. It is simple to apply and it has
an adequate physical and economic understanding of the
electricity transmission problem. It provides adequate signals
to agents making investment decisions, ensuring fairness,
efficiency and stability in the resultant allocation.
The method is applied for cost allocation to consumers in a
network, but it is equally applicable to generators or
combinations of both.
II. METHODOLOGY
A. Formulation
The proposed method analyses each transmission line at a
time, as a problem with particular characteristics, with
consumers being the agents that interact in a cooperative game.
Each line is understood as an element within the transmission
network, taking into account all its interactions.
In an electric market, competition takes place essentially at
the generation level. Under these conditions, generators aim at
obtaining normal rents for the invested capital (included the
radial lines to reach the network). Any additional charges for
transmission, for an inelastic demand, will be passed over to
consumers by means of a higher generation cost [7]. That
means at the end that consumers are the ones who pay for the
transmission system that is not implicit in the generation
investment.
The participation of an agent in covering the cost of a
particular transmission line is determined by playing a
cooperative game (NA, C), where NA is the set of the N
players and C its characteristic function [13]. For basic
concepts of cooperative game theory in cost allocation, refer to
the appendix.
The method considers the line power flow as a measure of
line use, important for cost allocation purposes. Further, it is
assumed that maximum line flow conditions line capacity and
investment costs. Therefore, it is required to determine the
system operational conditions that produce maximum flow in
each line, one at a time. This means identifying the loads Li
and generators gi for NB different buses that condition
maximum line flows Fkmax , i.e., for a line k between buses l
and m (see appendix A for argmax concept):
( L i , g i ) = arg max Fk ( L i , g i ); i = 1,..., N
(1)
The methodology is applicable to existing networks as well
as to expansions. In the first case there will be abundant data
on flows and operational conditions, for demand and for
generation, for all possible conditions, for example as many as
8760 scenarios in a given year. From them the operational
condition (generation, demand and marginal generation costs)
can be obtained, in which the maximum flow takes place in a
given transmission line. Thus, the characteristic function of the
cooperative game for such line, based on the stand-alone flow
requirements for each consumer agent or possible coalition of
agents.
In the second case, an expansion of a transmission system,
the recommendation of the authors is to consider a large
quantity of possible operation simulations. The following
should be taken into account: demand curves, hydrological
scenarios for hydroelectric plants (dry years, intermediate,
wet), maintenance of generating units and transmission lines.
The need is to identify the scenario that implies the greater
flow through a line.
Each player or agent evaluates its transmission requirement
for a given system operation condition, either acting separately
or cooperating in a coalition with other agents also requiring
the transmission network. To perform this evaluation, each
agent must assess its use of each transmission line for a
maximum flow condition. The values of Li obtained in (1)
dimension the requirements of the game agents.
The game characteristic function is defined by the standalone requirement fks for each agent and for each potential
coalition S. Thus, the game characteristic function CkS, for line
k and each potential coalition S, will be:
C
s
k
= f ks
; ∀ S ⊆ NA
(2)
The usefulness for each agent or agent coalition of utilizing
line k is implicit in equation 2, through the line flow
requirement. It is assumed that all agents act rationally so that
they prefer to earn more money, or prefer to reduce their costs
[13]. Thus, using a monetary unit to represent the usefulness of
the line, it is possible to associate to each MW flow a
monetary unit, without producing distortions in the game
solution. It is equivalent to solve the game considering the
MW usefulness for each agent or agent coalitions.
Therefore, the characteristic function will be expressed in
generic monetary units, being of no importance what monetary
unit is used. What is important is the responsibility or
participation each agent has in the financing of the line, not the
money implied.
It must be kept in mind that there are no restrictions to the
formation of coalitions among the agents, understanding that
they are intelligent and rational, i.e. that they create coalitions,
which minimize their cost participation for given transmission
lines. In so far as line cost functions are sub-additive, i.e. that
the joint cost is lower than the sum of individual costs, it
would be convenient to create coalitions for groups of agents.
Given this sub-additive condition, coalitions will certainly be
formed, and further all market agents will participate. This is
the typical “grand coalition” of cost allocation problems.
The power flow for line k from bus l to bus m for coalition S
is calculated using a DC model, as:
1
(3)
f ks =
( Θ ls − Θ ms ) ; ∀ S ⊆ NA
x l ,m
and with phase angles Θ given by
3
−1
θ = B
s
P
s
(4)
where PS is the vector of injected powers (generation minus
load) for each coalition S. To determine the injected
generations, an economic dispatch is performed, taking into
account the generation variable costs cvi and generation
production level gi.
NB
∑
min
i =1
(5)
cv i ( g i ) g i
subject to total generation equal to total demand for coalition
S.
NB
∑g − ∑L
i
i =1
Li ∈S
i
=0
(6)
g imin ≤ g i ≤ g imax
(7)
Thus, the economic dispatch for coalition S will determine
line flows and these will determine the game characteristic
function value for that coalition. This is done for all potential
coalitions. With the obtained characteristic function values, the
solution of the game is obtained using any cooperative game
solution mechanism, such as Nucleolus, Shapley Value, Kernel
and other [13]. As a result, the payoff configuration PC
(payoff is the word for cost allocation in game theory) for line
k is obtained.
PC k = ( x k ; δ k )
(8)
where x is the payment vector and δ is the coalition
configuration for line k, see appendix B. This payoff
configuration allows to determine the percentage cost
allocation CA of line k to agent i.
k
CAi , k =
k
xik ( + )
N
∑x
i =1
k
i
(9)
(+)
k
i
where x (+ ) are the positive values of cost allocation.
The method does not consider compensation for negative
values; these only reduce the magnitude of the positive
assignments.
As indicated before, a DC flow algorithm is used. For the
cooperative game solutions, the Mathematica software is
employed.
To determine the allocation of the total cost of a
transmission system to each agent, it is necessary to add its
allocations for each line, i.e.:
Ci =
B. Electric market considerations
The proposed method solves the transmission cost
allocation, given an existing network. However, it also allows
solving the cost allocation of an expansion, with no changes in
the method.
In the same way, the method may be applied to total
transmission costs, as well as to complementary charges in a
marginal cost scheme [8]. In the later arrangement, the
marginal scheme couples well with the economic foundations
of the game theory mechanisms.
Agents must cooperate and coordinate to share the costs of
the transmission network that interconnects them. That
network is assumed to be the property of a third party, not
involved in the energy business, but is interested to recover
investments, operation and maintenance costs directly, or
through an independent system operator.
Given that there is open access, consumers may establish
commercial contracts with any generators, without restrictions,
other than generator capacities. Therefore, it is possible for
each agent to consider different supply conditions, for example
with one load demanding supply and absorbing all the
transmission cost and another condition where several
consumers cooperate, sharing supply and transmission costs.
The assumption is made that the network has been adequately
planned, in terms of transmission capacity, quality, security
and backup. No congestion is assumed.
The assumption is made that agents participating in the
market are intelligent, independent and rational, that is, they
prefer to maximize their profits, or they prefer to reduce their
costs. Thus, they will be eager to cooperate in a rational
economic environment. This means that no agent or coalition
of agents will have a cost greater than its stand-alone cost, and
the resultant game cost allocation will cover all transmission
costs. This condition is known as a break-even condition or
“Pareto optimum”.
C. A simple example
To better understand the proposal, a simple example with
two buses and two generators is developed, based on the
system indicated in Fig. 1. It is assumed that the variable cost
of generator 1 (G1) is lower than that of generator 2 (G2). The
maximum demand condition and the maximum line flow are
given by P1 = 220 MW; P2 = 50 MW; L1 = 120 MW; L2 =
150 MW; F = 100 MW.
P1
P2
F, C
G1
G2
NL
∑ CA
k =1
i ,k
* CL
k
(10)
Ci: total cost allocation for the transmission system,
corresponding to agent i.
CLk : real total cost of line in monetary units
NL : number of transmission lines.
L1
L2
Fig. 1. Example system.
Only consumers L1 and L2 are to pay the transmission line.
Those consumers are supplied at minimum cost, within the
restrictions of the market where they are located.
4
If consumer L1 would independently require supply at
minimum cost, it would request it from G1, having a lower
cost and enough capacity to supply. Its stand-alone
transmission cost is zero. For consumer L2, the situation is
different, it needs to use the transmission line to connect to the
cheapest generator, but it has the alternative of supply from G2
in bus 2, with a higher cost. If it chooses the first alternative,
its stand-alone transmission requirement is 150 MW. The
characteristic function of each game is determined based on
the flow requirements for each possible coalition. Thus, the
coalition where only L2 participates is possible and it has a
flow requirement of 150 MW. If L2 wants to obtain energy
from the most economic source, it would be interested in
building its own transmission line with that capacity, or
cooperating with other agents to reduce its costs.
The 150 MW represents, in monetary terms, a cost of 150
monetary units for agent L2. That is why the characteristic
game function for the coalition formed just by L2 has a
monetary value of 150. If both consumers cooperate and
coordinate to obtain supply together, they would face a
common requirement of 100 MW for the transmission line,
which corresponds to the necessary flow to obtain supply from
the generators, considering their variable costs and generation
capacities. Therefore, both consumers would be interested to
cooperate in the payment of the transmission system (it would
be indifferent for L1 and it would be attractive for L2. It is
assumed that agents are interested in cooperation only if it
implies a reduction of costs for each or at least, not an increase
of costs). A coalition would be achieved. The coalition would
be maintained over time, depending on how convenient it is
for the agents and their transmission payments.
From the above analysis, the following characteristic
function of the cooperative game is obtained: C1=0; C2 = 150;
C12 = 100.
The result of the game, using different cooperative game
solution mechanisms, gives the following payment
configuration: PC = {0,100; 12}, which indicates that the total
cost of transmission line 1-2 is to be paid by consumer L2. The
solution indicates that the consumer located in the bus with
lower cost generation must not pay for the transmission lines it
does not use, given enough generating capacity to supply it.
The “sunk demand” concept is supported under this analysis,
but only for the load at the marginal bus, when there is enough
local generation to supply it.
Other methods formulated in the literature (area of influence
of consumers, marginal participation, average participation,
generalized load distribution factors, generation displacement
distribution factors [9], [10]) give the same answer to the cost
allocation when consumers pay for the networks.
III. TEST SIMULATIONS
To test and better understand the method, two systems are
assessed, a radial network and a meshed network.
A. Radial system
Many South American transmission systems have a radial
character, included the Chilean central interconnected
network. Thus, the method is tested first in a radial network
(Fig. 2), similar in structure to the Chilean one, with data
provided in tables I and II. Economic dispatches for three
operational conditions (maximum, medium and low demand)
are given in table III
G1
G2
1
G3
2
G4
3
L1
4
L2
L3
L4
Fig. 2. Radial system.
TABLE I
LINE DATA
Line
Impedance
1-2
2-3
3-4
0.00 + j0.15
0.00 + j0.05
0.00 + j0.14
Maximum
power
200 MW
500 MW
200 MW
TABLE II
GENERATOR DATA
Generator
Type
G1
G2
G3
G4
Diesel
Coal
Natural gas
Hydroelectric
Maximum power
(MW)
50
600
400
200
Variable cost
(US$/MWh)
70
22
12
0
TABLE III
ECONOMIC DISPATCHES (MW)
Demand
Peak
Medium
Low
G1
0
0
0
L1
120
84
48
G2
400
100
0
L2
500
350
200
G3
400
400
200
L3
300
210
120
G4
200
200
200
L4
80
56
32
TABLE IV
LINE FLOWS (MW)
Demand
Peak
Medium
Low
F21
120
84
48
F32
220
334
248
F43
120
144
168
Consumers are the interacting agents, which may act
individually or forming coalitions, if it is beneficial for each
one of them. The analysis is made for every transmission line
in the system, considering the maximum flow condition for
each line at a time. In real systems, and in the example, the
maximum flow for each line does not necessarily take place at
maximum demand. Table IV shows the resultant flows in the
three lines for the three operational conditions.
For line 1-2, the maximum flow takes place for the
maximum demand condition, for line 2-3 it takes place for
medium demand while for line 3-4 it is at low demand. The
use of each line by the different agents is assessed as well as
5
the use by different coalitions. Applying the proposed method,
the resultant characteristic game function is given in table V.
The analysis to determine the characteristic function for
each of the agents and for each possible coalition is done as
follows. For line 1-2 the assessment is made that the only
demand is that of consumer L1, and it must be supplied by
existing generators, according to a merit order. Line 1-2 usage
will be 120. Under that approach, other consumers, if alone,
will not use line 1-2. If consumers cooperate, for example L1
and L2, they will share use of line 1-2, with a requirement of
120. The same would happen for coalitions L1&L3 and
L1&L4. For that line, any coalition involving demand L1 will
impose a requirement of 120. Any coalition not involving L1
will imply a null requirement of that line. In the same form,
characteristic game functions will be determined for the other
lines.
Line 1-2
120
0
0
0
120
120
120
0
0
0
120
120
120
0
120
Line 2-3
84
350
0
0
434
84
84
350
350
0
390
434
84
334
334
TABLE VI
GAME RESULTS FOR DIFFERENT LINES AND COMPARISSONS WITH ALTERNATIVE
METHODS
Line 1-2
Shapley Value
Nucleolus
GLDF
Marginal
Participations
Line 2-3
TABLE V
GAME CHARACTERISTIC FUNCTION
Coalition
C1
C2
C3
C4
C12
C13
C14
C23
C24
C34
C123
C124
C134
C234
C1234
The solutions are part of the game core, which allows
assignment stability as well as coalition stability. None of the
agents has the incentive to leave the coalition and group in a
different way. There is no alternative coalition that would
improve the assignment achieved by the agents.
Line 3-4
48
200
120
0
200
168
48
200
168
120
200
168
168
168
168
Table VI finally gives the percentage cost allocation to the
different agents using Shapley Value and Nucleolus
cooperative game mechanisms. The table also provides the
allocation resultant from the marginal participation method
(area of influence approach) [10] and the generalized load
distribution factors (GLDF) [9].
Although several solutions are close, differences arise
depending on the line. All methods assign full responsibility of
line 1-2 to consumer L1, which seems logic and predictable.
The same happens for line 2-3, although allocations vary.
For line 3-4, all methods, except Nucleolus, give equal
results, with more responsibility for load L2, less for L3 and
even less for L1. The different results given by the Nucleolus
mechanism is explained by its sensibility to the game features
[13].
It is possible to demonstrate (see appendix) that the results
obtained with the proposed method comply with the individual
rationality principle, in the sense that nobody pays more than
its stand-alone cost. They also comply with group rationality,
in that the sum of all agent contributions is equivalent to the
total line cost. The collective rationality also applies; the sum
of the payments of any two possible coalitions is lower than
the sum of individual payments.
Shapley Value
Nucleolus
GLDF
Marginal
Participations
Line 3-4
Shapley Value
Nucleolus
GLDF
Marginal
Participations
Load L1
[%]
100
100
100
Load L2
[%]
0
0
0
Load L3
[%]
0
0
0
Load L4
[%]
0
0
0
100
0
0
0
Load L1
[%]
15.64
25.15
19.35
Load L2
[%]
84.36
74.85
80.65
Load L3
[%]
0
0
0
Load L4
[%]
0
0
0
19.35
80.65
0
0
Load L1
[%]
13.04
0
13.04
Load L2
[%]
54.35
100
54.35
Load L3
[%]
32.61
0
32.61
Load L4
[%]
0
0
0
13.04
54.35
32.61
0
The importance of cooperative game solution mechanisms,
and that of the proposed method, is that these rationality,
stability and fairness considerations are part of those methods
and solution mechanisms. The obtained solution is based on
those economic rational and interaction principles, taking into
account the transmission network physical (electric)
characteristics as well as the electrical market economic and
financial conditions.
Solutions are shown based on two cooperative game
solution mechanisms: Shapley Value and Nucleolus. They are
compared with other cost allocations methods, such as the
GGDF and the marginal participation [9, 10]. Those methods
are traditionally used for transmission cost allocation, but they
do not consider the economic interaction among agents for
cost allocation, and base themselves on consideration of
physical network flows, based on DC modeling.
It is important to remember that the proposed method, using
game models, consider that network technical aspects, as well
as economic ones, influence decisions by the agents, which act
rationally. Similarly, these models consider other information,
such as installed capacity per bus, variable generation costs
and consumer sizes.
B. Six bus Garver system
The classic six-bus system (Fig 3), originally proposed by
Garver [19], has been considered a good test network to assess
transmission expansion methods and expansion cost allocation
[16], [17]. The expansion decisions are assumed known for
6
this assessment and the problem is to determine how to assign
expansion costs among agents participating in the system.
Besides the system data provided in [19], required data for
generators has been introduced and is given in table VII. Bus 6
has been considered the slack bar (needed for the marginal
participation method).
G1
L5 (240 MW)
1
5
L1 (80MW)
G3
3
The characteristic function values for line 1-4 have been
determined following the proposed method. The operation
condition (Li, gi), equation 1, that conditions maximum flow
for line 1-4, is first determined. Using a DC load flow, the
flow requirements imposed by each load coalition on that line,
according to equations 3 and 4, are determined. Minimum
cost generation is considered, for each coalition S. To
determine the injected generations, an economic dispatch is
performed, following equations 5, 6 and 7. Table VIII is
obtained, according to equation 2. The game solution, using
Shapley Value and Nucleolus, for line 1-4 is given in Table
IX.
TABLE IX
TRANSMISSION NETWORK COST ALLOCATION RESULTS FOR LINE 1-4
L3 (40 MW)
Load L1
[%]
30,81
Method
2
Shapley Value
L2 (240 MW)
G6
6
4
L4 (160 MW)
Fig. 3. Six bus Garver system diagram.
TABLE VII
MAXIMUM GENERATION CAPACITY DATA AND PRODUCTION VARIABLE COSTS
Generator
Generation Maximum
Capacity (MW)
Generation
Level
Variable Cost
(US$/MWh)
G1
G3
G6
150
165
545
50
165
545
70
22
12
Because of space restrictions, the characteristic functions of
each of the cooperative games for each transmission line are
not detailed, but only an example is given in Table VIII, which
provides the data for the game function for line 1-4.
TABLE VIII
GAME CHARACTERISTIC FUNCTION FOR LINE 1-4
Coalition
{f}
{1}
{2}
{3}
{4}
{5}
{1,2}
{1,3}
{1,4}
{1,5}
{2,3}
{2,4}
{2,5}
{3,4}
{3,5}
{4,5}
Value
0
22.375
11.9227
5.66697
15.897
45.0414
34.2962
28.0405
6.47654
67.4149
17.5897
3.97424
56.9641
10.23
50.784
29.1444
Coalition
{1,2,3}
{1,2,4}
{1,2,5}
{1,3,4}
{1,3,5}
{1,4,5}
{2,3,4}
{2,3,5}
{2,4,5}
{3,4,5}
{1,2,3,4}
{1,2,3,5}
{1,2,4,5}
{1,3,4,5}
{2,3,4,5}
{1,2,3,4,5}
Value
39.9632
18,3993
77.2125
12.1435
73.0819
51.5179
1.69273
62.6311
27.6081
34.8114
24.0662
77.2125
37.2677
57.1849
27.6081
31.7479
Load L2
[%]
0,00
Load L3
[%]
4,15
Load L4
[%]
0,00
Load L5
[%]
65,04
Nucleolus
16,72
1,65
0,00
0,00
81,63
GLDF
33,15
0,37
6,67
0,00
59,81
Marginal Participations
30,61
0,00
7,75
0,00
61,64
Results obtained by the cooperative game solution
mechanisms, Shapley Value and Nucleolus, differ among
them, because of the fundamental different nature of the agent
interaction they model. Shapley Value looks for a fair
distribution based on marginal contributions of agents when
randomly join a given coalition. Instead, Nucleolus minimizes
the maximum discomfort of the final resultant allocation. Both
methods have advantages and disadvantages. If they compare
with traditional allocation costs, such as GGDF and marginal
participation, the Shapley Value method provides closer
results.
On the other hand, all methods coincide in assigning the
largest responsibility of the cost of line 1-4 to agents L1 and
L5. The proposed methods work with games that are convex,
which ensures that the Shapley Value will be at the center of
the core.
Besides, results comply with the individual
rationality principle (no one pays more than its stand-alone
above cost) and group rationality principle (all costs are
covered). The collective rationality principle also applies.
Solutions are part of the game core, which implies
allocation stability and coalition stability. None of the agents
has the incentive to leave the coalition or group in a different
manner, as no alternative coalition may improve the allocation.
Once the games, for each of the lines of the six-bus Garver
system, are solved, it is possible to obtain the aggregated
results for the whole system, as shown in Table X.
TABLE X
TRANSMISSION NETWORK TOTAL COST ALLOCATION RESULTS
Method
Shapley Value
Nucleolus
GLDF
Marginal
Participations
Load L1
[%]
15,4
13,22
15,43
Load L2
[%]
19,63
24,92
14,66
Load L3
[%]
4,75
4,75
5,18
Load L4
[%]
9,51
9,04
19,91
Load L5
[%]
50,72
48,05
44,83
14,98
13,99
5,37
19,81
45,84
Those results include new investments, which correspond to
one additional circuit for 3-5, two for 4-6 and 4 for 2-6 [19].
7
Allocations of costs for only the new investments are given in
table XI.
Results show that larger loads have a larger cost allocation
for the existing network as well as for the expansions. There
are no published studies on the Garver system that could be
used for comparison of cost allocation to consumers.
References [16], [17] focus their assessment on cost allocation
to buses. The authors are developing further research to also
use the method to determine generator payments.
TABLE XI
ESPANSION COST ALLOCATION RESULTS FOR LINES 2-6, 3-5 AND 4-6
Method
Shapley Value
Nucleolus
GLDF
Marginal
Participations
Load L1
[%]
10,52
5,01
10,95
Load L2
[%]
29,76
36,79
28,98
Load L3
[%]
4,66
3,34
4,77
Load L4
[%]
18,33
17,19
19,12
Load L5
[%]
36,74
37,67
36,18
11,01
27,98
4,69
19,62
36,7
All methods give similar results, assigning more
responsibility to the most important loads in the system.
IV. CONCLUSION
A new method has been proposed to allocate transmission
costs to agents in a network. It is simple and transparent in its
application and it is based on the real operation of the
transmission system. If used as a complement to marginal
pricing, it provides adequate economic signals to the agents for
their optimal behavior in terms of efficiency.
Cooperative game theory is used, allowing to incorporate
efficiency and fairness principles. As a result, there is no
alternative better allocation assignment and two agents in
identical conditions are allocated equal payments. Thus, its
applicability to competitive markets becomes most attractive.
The presented method allows solving the transmission cost
allocation problem, taking into account existent conditions of a
given electric network. But it also allows solving the allocation
of expansion costs, without any methodological change.
Results are compared and validated with existing methods.
A potential problem of the method is the handling of the
dimension of the games, which grows with the ratio 2N, with N
being the number of agents. The authors are developing an
application process for systems with many buses, to be applied
to the Chilean electrical market.
The authors are also continuing research in algorithms for
independent coalition formation, stability of coalitions and
further applications in network expansion.
V. APPENDIX
A. ARGMAX CONCEPT
For any set Z and any function f: ZR, argmax y e Z f(y) denotes
the set of points in Z that maximize the function f, so
arg max f ( y ) = { y ∈ Z | f ( y) = max f ( z )}
y∈Z
z∈Z
(A.1)
B. CONCEPTS OF COOPERATIVE GAME THEORY IN COST
ALLOCATION
A cost allocation cooperative game is given by a couple
(NA, C), where NA is the set of the N agents and C is the cost
function.
The agents can group in many different ways according to
their interests and convenience. The way in which N players
group in m mutually exclusive and excluding coalitions S, is
the coalition configuration.
δ = {S1 , S 2 ,....S m }
(B.1)
d is a partition of NA that fulfills three conditions.
S j ≠ Φ; j = 1,2,.., m
(B.2)
Si S j = Φ ; ∀i ≠ j
S
j
= NA
S j∈δ
where Φ is the empty set.
Each agent belongs to one and only one of the m coalitions
and the members of a certain coalition are related to each other
but not with other agents that belong to other coalitions.
The costs assigned to each agent as a result of the game
correspond to the vector of assignments or payments
x = {x 1 , x 2 ,....., x N } , where xi is the agent i payoff. The
result of a game is the payoff configuration PC:
PC = {x; δ) = ( x 1 , x 2 ,....., x N ; S1 , S 2 ,....Sm )
(B.3)
The solution of the game should fulfill the conditions of
rationality (individual, collective and global) and stability.
x (i) ≤ C (i ); ∀i ∈ NA
(B.4)
x ( S ) ≤ C ( S ); ∀S ∉ δ
x ( NA) = C ( NA)
with:
x( S ) = ∑ x i
(B.5)
i∈S
Any agent or coalition of agents should not have a bigger
cost that their alternative cost or stand-alone cost. The
resulting assignment of costs of the game to all the players
must be identical to the total costs to be covered. This last
condition is known as break-even condition or Paretooptimum.
The payoffs that are Pareto-optimum and individually
rational are denominated imputations. Besides, if it is
collectively rational, the core of the game is obtained.
The core of the game ( NA, C ) is the set of all the solutions
PC (x; d) such:
x (T ) ≤ C (T ); ∀T ⊂ NA , with: x(T ) = ∑ x i
i∈T
(B.6)
8
The core is, therefore, a subset of the group of imputations.
This core concept is the simplest of all the solution concepts of
cooperative games. It corresponds to a group of imputations
that leaves no space for a better allocation to its players and
does not allow subsidies among coalitions. Unfortunately,
there are many games with too large cores (many solutions),
no core at all or empty core. The empty core takes place when
the collective rational is not achieved.
Starting from the core it is possible to establish different
theories that outline solutions to this type of games. Solutions
could be: extensions of the core, stable set and bargaining set.
Other solutions can be achieved with the excess theory,
defined as the difference between the coalition stand-alone
cost and the payoff for that coalition as a result of the game.
Thus, the excess of coalition R with respect to the payoff
vector x of the PC (x; d) is defined as:
e( R , x ) = C ( R ) − x ( R )
[4]
[5]
[6]
[7]
[8]
[9]
[10]
(B.7)
[11]
and represents the total amount that potential members of
the R coalition collectively win or lose if they withdraw from
R.
From the excess definition it is possible to incorporate the
Nucleolus and Kernel solution mechanisms.
The nucleolus solution corresponds to imputations for which
the maximal excess is minimized:
max{min e( R, x)}; ∀R
(B.8)
Other traditional solution is the Shapley Value, given by:
φi =
∑
(N − s )! (s − 1)![C(S ) − C(S − {i})]
N!
S ⊆ NA
where:
N: total number of players
S: coalition of players
s = |S|: number of players in coalition S
i: player
i ∈ NA
(B.9)
VI. ACKNOWLEDGMENT
The authors gratefully acknowledge the support from
Universidad de Santiago de Chile and Universidad Católica de
Chile. We also acknowledge the helpful comments received
from three anonymous reviewers
VII. REFERENCES
[1]
[2]
[3]
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VIII. BIOGRAPHIES
Juan Zolezzi was born in Valdivia, Chile, and graduated as an Electrical
Engineer from Technical State University of Chile, later obtaining his M.Sc.
from University of Chile and presently is a Ph.D. candidate at Catholic
University of Chile. He is a professor at Santiago University of Chile. His
research activities focus on power economics, economic transmission issues,
planning and regulation of electric markets. He has been a consultant with
utilities and regulators in Chile.
Hugh Rudnick was born in Santiago, Chile, and graduated as an Electrical
Engineer from University of Chile, later obtaining his M.Sc. and Ph.D.
degrees from the Victoria University of Manchester, UK. He is a professor at
Catholic University of Chile. His research activities focus on the economic
operation, planning and regulation of electric power systems. He has been a
consultant with the World Bank and with utilities and regulators in Argentina,
Bolivia, Brazil, Canada, Central America, Chile, Colombia, Peru and
Venezuela, as well as in Europe.