Applications of Cooperative Game Theory in Power System Allocation Problems
Electrical Engineering Department. Intelligent Control and Electrical Power System Laboratory (ICEPS), Djillali Liabes University, SidiBelAbbes, 22000, Algeria.
Emails: reseaukratima@live.fr; fzgherbi@gmail.com; flakdja@yahoo.fr
^{* }Corresponding author:^{* }Phone/Fax: 213 48 54 41 00
Abstract
This paper proposes a variant of cooperative game is derived and it has been proved that to determine the analytical solution to determine the nucleolus. The power generations or loads associated with the market are modeled as individual current injections based on a realtime solved AC power flow solution. Each load can be modeled as a current injection or equivalent constant impedance depending on whether it is required to be responsible for the system loss. Each current injection is then treated as an individual player of the transmission loss allocation game. The concept of Shapley value adopted from cooperative game theory is utilized to deal with the fairness of loss allocation. It has been applied to a 14bus system and the results are discuses.
Keywords
Power system planning; Cooperative game theory; Shapley Value; Coalition formation; Transmission loss allocation.
Introduction
In the deregulated power market one of the most important issues is the allocation of transmission losses among market participants since system losses can typically represent significant portion of the total generation. The main difficulty of loss allocation is caused by the highly nonlinear and nonseparable properties of the loss function. The electric power industry is undergoing a series of challenging changes due to deregulation and competition. One of the most important issues is the allocation of transmission losses among market participants since system losses can typically represent from five to ten percent of the total generation and costs millions of dollars per year. However, it is not a trivial task to “fairly” allocate a component of system losses to an individual participant of the market. The main difficulty of loss allocation is caused by the highly nonlinear and nonseparable properties of the loss function. To deal with the loss allocation problem, a number of allocation schemes have been proposed in the literature. These schemes fall into the following categories: Prorate, proportional haring, incremental transmission loss, loss formula, and circuit theory. Some approaches are based on DC power flow, while some use AC load flow for matching the calculation results and actual power flows. Some schemes are branchpowerflow based, while some focus on the branchcurrent based allocation techniques.
Game theory provides wellbehaved solution mechanisms with economic features for assessing the interaction of different participants in competitive markets and resolving the conflicts among players [1]. In particular, cooperative game theory is the most convenient tool to solve cost allocation problem [2, 3]. Some game theory based solutions have been proposed for power engineering problems, such as transmission cost allocation [1] and wheeling transactions [4]. The application of Shapley value concept arisen from the cooperative game theory was investigated to allocate losses and the work is extended in this paper. The transmission loss is derived as function individual current injections. Two basic formulations are presented to determine individual current injections. One basic model allocates losses only to the generators and the other allocates losses to both generators and loads. The main difference is that the former treats the load demands as equivalent constant admittances based on a realtime solved AC power flow solution and accordingly the bus admittance method impedance matrix (Ybus) is then modified, while the later formulates the load demands as equivalent current injections directly form bus impedance matrix.
Each current injection is then treated as an individual player of the transmission loss allocation game.
In the proposed approaches, the power generations and/or loads associated the market transactions are modeled as individual current injections [5]. Each current injection is then treated as an individual player of the transmission loss allocation game. The approaches are branchcurrent based, not branchpowerflow based. Without any approximations or assumptions like those made for a DC power flow or proportional sharing, the proposed approaches utilize the method of Shapley value [1] adopted from cooperative game theory to deal with the fairness issue of loss allocation. Some modified or alternative allocation approaches with or without a normalization procedure are also proposed to deal with the aggregated player of ancillary services and to speed up the computation when the number of players is large. The proposed approaches are consistent with the realtime AC power flow solution and recover the total system loss. The Kirchhoff’s laws and superposition principle are satisfied and both the network configuration and the voltagecurrent relationships are reflected. The interactions of players are naturally and fully considered. Moreover, the effect of reducing transmission loss can be identified from the negative loss allocation and the negative allocation can provide economic signals for the players.
Cooperative game will be described by the equations. These approaches (cooperative game) have been implemented and tested using the wellknown IEEE14bus system. A discussion about the significance, relevance, and usefulness of results obtained from these methods is presented,
Cooperative game
Consider an ‘n’ person balanced linear cooperative game described by the equations [6, 7]. The balanced game can be described as the solution vector satisfies all the collation constraints.
x(S) = ν(S) 
(1) 
x(N) = ν(N) 
(2) 
where x(S) is the set of possible coalitions and x(N) is the grand coalition.
Let the solution vector be
x = [x^{·}_{1}, x^{·}_{2}, ... x^{·}_{n1}, x^{·}_{n}] 
(3) 
Then second order (quadratic) cooperative game which is described as follows.
minε(S) 

Subject to
y(S) ≥ (ν(S))^{2} + ε(S) 
(4) 
y(N) = (ν(N))^{2} 
(5) 
If the solution to the game is
y = [y^{·}_{1}, y^{·}_{2}, ... y^{·}_{n1}, y^{·}_{n}] 
(6) 
Then the relation ship between the solution vectors is
y = ν(N)x 
(7) 
Multiply equations (1) by ν (N)
x(S)ν(N) = ν(S)ν(N) 
(8) 
In a balanced cooperative game it is understood that
x(S)Èx(S') = x(N) 
(9) 
ν(S) + ν(S') = ν(N) 
(10) 
where S' is the conjugate of coalition S
y(S) = (ν(S))^{2} + ν(S)ν(S') 
(11) 
y(N) = (ν(N))^{2} 
(12) 
By comparing equations (4&11) the minimum value of the lexicographical excess vector is determined.
e(S) = ν(S) ν(S') = e(S') 
(13) 
Hence, it is proved and the proof can be extended to all coalition values which are real as well as complex numbers, which exhibits balancing condition. The equations 4&5 are modified for complex numbers
y(S) ≥ ν(S)^{2} +ε(S) 
(14) 
y(N) = ν(N)^{2} 
(15) 
A. Generation and Load Models Based on a solved AC power flow solution for a pool based electric power market, let the complex power injection in to a generator bus i be _{} then the generation current injection is written as:
_{} 
(16) 
where V_{i }is its bus voltage. Similarly, let the complex power injection in to a load bus j be _{} we can then have load current injection
_{} 
(17) 
Or the equivalent load impedance
_{} 
(18) 
B. Transmission loss allocation problem
1. Loss allocation to generators only
For an ‘n’ node power network having ‘m’ generator buses the transmission loss of element ij connected between nodes ‘i’ and ‘j’ is derived in terms individual current contribution of each generator as
_{ } 
(19) 
where _{} is the current contribution of ’k’^{th} generator to the element ‘ij’ and it can be determined from modified Y bus method using converged load flow solution.
_{}is the resistance of line element ’ij’ connected between nodes ‘i’ and ‘j’. The individual voltage contribution of each generator is derived in terms of current injections.
_{} 
(20) 
_{}is a square matrix of size ‘n’ and the columns m+1 to n will be zero since they are load buses.
2. Loss allocation to generators and loads
In this formulation loss allocation is made for generator as well as load buses. The individual voltage contribution of each bus is derived.
_{} 
(21) 
Now the transmission loss of ‘ij’th element in terms of individual current contribution is given:
_{} 
(22) 
For both methods the current contributions of ‘k’th bus for ‘ij’th element is
_{} 
(23) 
z_{ij} is the transmission line impedance of element ‘ij’ (pi model for transmission line is considered).
Since transmission loss is real the effect of shunt admittances can be ignored.
Now the current contribution of ‘k’th generator to element ‘ij’ is given by:
_{} 
(24) 
where z_{ij} is the transmission line impedance of element ‘ij’ (pi model for transmission line is considered). It can be observed that the branch current flowing through is the algebraic sum of individual current contributions of each generator
_{} 
(25) 
For each element ‘ij’ the coalitions present a balancing condition because of Kirchoff’s current law. Let ‘S’ be set of possible coalitions:
_{} 
(26) 
_{} 
(27) 
Let the solution vector for this balanced cooperative game be:
_{} 
(28) 
Now the coalition values for the transmission loss allocation problem is derived as:
_{} 
(29) 
_{} 
(30) 
_{} 
(31) 
_{} 
(32) 
The transmission loss contribution of ‘k’th generator to ‘ij’th element is determined as:
_{} 
(33) 
Now the transmission loss contribution of ‘k’th generator is the summation of losses to every line element of that generator.
_{} 
(34) 
The system parameters are shown in Tables 1, 2, 3
Table 1. Bus data of 14 bus system
Bus i 
type 
Pd 
Qd 
Gs 
Bs 
area 
Vm 
Va 
baseKV 
zone 
Vmax 
Vmin 
1 
3 
0 
0 
0 
0 
1 
1.06 
0 
0 
1 
1.06 
0.94 
2 
2 
21.7 
12.7 
0 
0 
1 
1.045 
4.98 
0 
1 
1.06 
0.94 
3 
2 
94.2 
19 
0 
0 
1 
1.01 
12.72 
0 
1 
1.06 
0.94 
4 
1 
47.8 
3.9 
0 
0 
1 
1.019 
10.33 
0 
1 
1.06 
0.94 
5 
1 
7.6 
1.6 
0 
0 
1 
1.02 
8.78 
0 
1 
1.06 
0.94 
6 
2 
11.2 
7.5 
0 
0 
1 
1.07 
14.22 
0 
1 
1.06 
0.94 
7 
1 
0 
0 
0 
0 
1 
1.062 
13.37 
0 
1 
1.06 
0.94 
8 
2 
0 
0 
0 
0 
1 
1.09 
13.36 
0 
1 
1.06 
0.94 
9 
1 
29.5 
16.6 
0 
19 
1 
1.056 
14.94 
0 
1 
1.06 
0.94 
10 
1 
9 
5.8 
0 
0 
1 
1.051 
15.1 
0 
1 
1.06 
0.94 
11 
1 
3.5 
1.8 
0 
0 
1 
1.057 
14.79 
0 
1 
1.06 
0.94 
12 
1 
6.1 
1.6 
0 
0 
1 
1.055 
15.07 
0 
1 
1.06 
0.94 
13 
1 
13.5 
5.8 
0 
0 
1 
1.05 
15.16 
0 
1 
1.06 
0.94 
14 
1 
14.9 
5 
0 
0 
1 
1.036 
16.04 
0 
1 
1.06 
0.94 
Table 2. Generator data of 14 bus system
bus 
Pg 
Qg 
Qmax 
Qmin 
Vg 
mBase 
status 
Pmax 
Pmin 
1 
232.4 
16.9 
10 
0 
1.06 
100 
1 
332.4 
0 
2 
40 
42.4 
50 
40 
1.045 
100 
1 
140 
0 
3 
0 
23.4 
40 
0 
1.01 
100 
1 
100 
0 
6 
0 
12.2 
24 
6 
1.07 
100 
1 
100 
0 
8 
0 
17.4 
24 
6 
1.09 
100 
1 
100 
0 
Table 3. Branch data of 14 bus system
fbus 
tbus 
r 
X 
b 
rateA 
rateB 
rateC 
ratio 
angle 
status 
1 
2 
0.01938 
0.05917 
0.0528 
9900 
1 
1 
1 
0 
1 
1 
5 
0.05403 
0.22304 
0.0492 
9900 
1 
1 
1 
0 
1 
2 
3 
0.04699 
0.19797 
0.0438 
9900 
1 
1 
1 
0 
1 
2 
4 
0.05811 
0.17632 
0.034 
9900 
1 
1 
1 
0 
1 
2 
5 
0.05695 
0.17388 
0.0346 
9900 
1 
1 
1 
0 
1 
3 
4 
0.06701 
0.17103 
0.0128 
9900 
1 
1 
1 
0 
1 
4 
5 
0.01335 
0.04211 
0 
9900 
0 
1 
1 
1 
1 
4 
7 
0 
0.20912 
0 
9900 
0 
0 
0.978 
0 
1 
4 
9 
0 
0.55618 
0 
9900 
0 
0 
0.969 
0 
1 
5 
6 
0 
0.25202 
0 
9900 
0 
0 
0.932 
0 
1 
6 
11 
0.09498 
0.1989 
0 
9900 
0 
1 
1 
0 
1 
6 
12 
0.12291 
0.25581 
0 
9900 
0 
1 
1 
0 
1 
6 
13 
0.03181 
0.13027 
0 
9900 
0 
1 
1 
0 
1 
7 
8 
0 
0.17615 
0 
9900 
0 
0 
1 
1 
1 
7 
9 
0 
0.11001 
0 
9900 
0 
0 
1 
1 
1 
9 
10 
0.03181 
0.0845 
0 
9900 
0 
1 
1 
0 
1 
9 
14 
0.12711 
0.27038 
0 
9900 
0 
0 
1 
0 
1 
10 
11 
0.08205 
0.19207 
0 
9900 
0 
0 
1 
0 
1 
12 
13 
0.22092 
0.19988 
0 
9900 
0 
0 
1 
0 
1 
13 
14 
0.17093 
0.34802 
0 
9900 
0 
0 
1 
0 
1 
Results and Discussions
Several systems have been used to test the proposed method. In this paper, the test results of a 14bus system are presented and discussed.
The oneline diagram of a 14 bus system with 2 generation buses, 12 load buses, and 20 transmission lines is shown in Fig. 1. A solved power flow solution is shown in Table 1. The players of the loss allocation game are defined as the bus injected complex powers according to the solution listed in Table 4. The losses allocated to only generators and only loads for each transmission line and the total system loss allocations are listed in Tables 5 and 6, respectively.
The total allocated loss is consistent with the power flow solution and can reasonably reflect the amounts of transactions injected complex powers according to the solution listed in Table 7.
Since the network configuration and the location of each player are taken into account by the proposed schemes, the system loss is not evenly allocated to the supply side and the demand side. Thus, there is no need to specify the sharing factors of losses to be allocated to the supply side and demand side.
Table 4. Converged load flow solution of 14 bus system
Bus no 
Voltage Mag pu 
Voltage Angle Degrees 
Real Power P(MW) 
Reactive Power Q(MW) 
1 
1.0500 
0 
232.7115 
37.6883 
2 
1.0450 
5.2189 
18.3000 
51.0159 
3 
1.0100 
12.9697 
94.2000 
6.7984 
4 
1.0165 
10.5354 
47.8000 
3.9000 
5 
1.0175 
8.9813 
7.6000 
1.6000 
6 
1.0700 
14.4469 
11.2000 
6.3182 
7 
1.0610 
13.5833 
0.0000 
0.0000 
8 
1.0900 
13.5833 
0.0000 
17.9653 
9 
1.0554 
15.1618 
29.5000 
16.6000 
10 
1.0505 
15.3209 
9.0000 
5.8000 
11 
1.0567 
15.0151 
3.5000 
1.8000 
12 
1.0551 
15.3017 
6.1000 
1.6000 
13 
1.0503 
15.3817 
13.5000 
5.8000 
14 
1.0352 
16.2584 
14.9000 
5.0000 
Transmission loss (MW) 
13.71 
Table 5. Transmission loss allocation (only generator buses)
Line N0: 
G1 (MW) 
G2 (MW) 
Loss 
1 
4.5720 
0.0116 
4.5604 
2 
2.7478 
0.0481 
2.7959 
3 
2.1419 
0.1860 
2.3279 
4 
1.5186 
0.1596 
1.6782 
5 
0.7953 
0.1127 
0.9080 
6 
0.3574 
0.0189 
0.3763 
7 
0.4964 
0.0249 
0.5213 
8 
0 
0 
0 
9 
0 
0 
0 
10 
0 
0 
0 
11 
0.0536 
0.0029 
0.0559 
12 
0.0674 
0.0046 
0.0720 
13 
0.1995 
0.0135 
0.2130 
14 
0 
0 
0 
15 
0 
0 
0 
16 
0.0114 
0.0011 
0.0125 
17 
0.1065 
0.0086 
0.1151 
18 
0.0126 
0.0005 
0.0126 
19 
0.0060 
0.0004 
0.0064 
20 
0.0520 
0.0030 
0.0550 
Total 
13.1384 
0.5731 
13.7115 
Table 6. Transmission loss allocation (only load buses)
Line No 
L3 (MW) 
L4 (MW) 
L5 (MW) 
L6 (MW) 
L7 (MW) 
L8 (MW) 
L9 (MW) 
L10 (MW) 
L11 (MW) 
L12 (MW) 
L13 (MW) 
L14 (MW) 
1 
1.949 
0.876 
0.131 
0.194 
0.000 
0.010 
0.550 
0.168 
0.065 
0.111 
0.249 
0.279 
2 
0.890 
0.595 
0.112 
0.162 
0.000 
0.000 
0.392 
0.120 
0.048 
0.086 
0.188 
0.205 
3 
1.678 
0.236 
0.027 
0.042 
0.000 
0.004 
0.140 
0.042 
0.016 
0.026 
0.058 
0.068 
4 
0.410 
0.467 
0.052 
0.088 
0.000 
0.006 
0.261 
0.078 
0.029 
0.049 
0.109 
0.129 
5 
0.149 
0.216 
0.051 
0.068 
0.000 
0.003 
0.157 
0.049 
0.020 
0.037 
0.080 
0.084 
6 
0.677 
0.115 
0.012 
0.024 
0.000 
0.009 
0.055 
0.016 
0.006 
0.011 
0.024 
0.029 
7 
0.235 
0.195 
0.016 
0.007 
0.000 
0.011 
0.055 
0.014 
0.004 
0.002 
0.008 
0.020 
8 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
9 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
10 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
11 
0.007 
0.006 
0.001 
0.008 
0.000 
0.008 
0.042 
0.021 
0.015 
0.007 
0.015 
0.004 
12 
0.001 
0.001 
0.000 
0.001 
0.000 
0.002 
0.007 
0.001 
0.000 
0.032 
0.021 
0.012 
13 
0.006 
0.005 
0.001 
0.007 
0.000 
0.006 
0.035 
0.007 
0.001 
0.017 
0.101 
0.057 
14 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
15 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
16 
0.002 
0.001 
0.000 
0.001 
0.000 
0.003 
0.011 
0.015 
0.003 
0.002 
0.004 
0.001 
17 
0.008 
0.006 
0.001 
0.009 
0.000 
0.008 
0.045 
0.009 
0.001 
0.013 
0.040 
0.111 
18 
0.003 
0.003 
0.000 
0.004 
0.000 
0.004 
0.019 
0.010 
0.006 
0.003 
0.007 
0.002 
19 
0.000 
0.000 
0.000 
0.000 
0.000 
0.001 
0.003 
0.001 
0.000 
0.009 
0.008 
0.005 
20 
0.006 
0.005 
0.001 
0.008 
0.000 
0.006 
0.036 
0.007 
0.001 
0.011 
0.032 
0.058 
Total 
6.002 
2.483 
0.343 
0.506 
0.000 
0.025 
1.585 
0.508 
0.186 
0.333 
0.787 
1.005 
Table 7. Transmission loss allocation (generator and load buses)
Line No 
G1 
G2 
L3 (MW) 
L4 (MW) 
L5 (MW) 
L6 (MW) 
L7 (MW) 
L8 (MW) 
L9 (MW) 
L10 (MW) 
L11 (MW) 
L12 (MW) 
L13 (MW) 
L14 (MW) 
(MW) 
(MW) 

1 
4.139 
0.005 
0.320 
0.061 
0.002 
0.002 
0.000 
0.007 
0.017 
0.005 
0.002 
0.002 
0.005 
0.008 
2 
2.290 
0.049 
0.028 
0.135 
0.036 
0.055 
0.000 
0.013 
0.087 
0.027 
0.012 
0.023 
0.048 
0.051 
3 
0.856 
0.102 
1.302 
0.047 
0.004 
0.002 
0.000 
0.001 
0.016 
0.004 
0.001 
0.000 
0.001 
0.005 
4 
1.264 
0.148 
0.145 
0.188 
0.006 
0.021 
0.000 
0.009 
0.081 
0.023 
0.008 
0.012 
0.027 
0.038 
5 
0.572 
0.104 
0.116 
0.083 
0.029 
0.037 
0.000 
0.000 
0.069 
0.022 
0.010 
0.019 
0.040 
0.040 
6 
0.200 
0.014 
0.763 
0.071 
0.005 
0.012 
0.000 
0.006 
0.031 
0.009 
0.003 
0.006 
0.013 
0.016 
7 
0.542 
0.015 
0.013 
0.083 
0.034 
0.035 
0.000 
0.008 
0.011 
0.006 
0.004 
0.012 
0.023 
0.014 
8 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
9 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
10 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
11 
0.004 
0.002 
0.006 
0.005 
0.001 
0.008 
0.000 
0.009 
0.042 
0.021 
0.015 
0.007 
0.015 
0.004 
12 
0.001 
0.000 
0.001 
0.001 
0.000 
0.001 
0.000 
0.002 
0.007 
0.001 
0.000 
0.032 
0.021 
0.012 
13 
0.003 
0.001 
0.005 
0.005 
0.001 
0.008 
0.000 
0.007 
0.035 
0.007 
0.001 
0.017 
0.101 
0.057 
14 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
15 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
16 
0.001 
0.000 
0.001 
0.001 
0.000 
0.001 
0.000 
0.003 
0.011 
0.015 
0.003 
0.002 
0.004 
0.001 
17 
0.004 
0.002 
0.007 
0.006 
0.001 
0.009 
0.000 
0.009 
0.045 
0.009 
0.001 
0.013 
0.040 
0.111 
18 
0.002 
0.001 
0.003 
0.002 
0.000 
0.004 
0.000 
0.004 
0.019 
0.010 
0.006 
0.003 
0.007 
0.002 
19 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.000 
0.001 
0.003 
0.001 
0.000 
0.009 
0.008 
0.004 
20 
0.003 
0.001 
0.005 
0.005 
0.001 
0.008 
0.000 
0.006 
0.036 
0.007 
0.001 
0.011 
0.032 
0.058 
Total 
9.471 
0.396 
2.120 
0.535 
0.025 
0.048 
0.000 
0.015 
0.314 
0.118 
0.035 
0.072 
0.206 
0.357 
Conclusions
The loss impacts between one player and any other coalitions of players are taken into account and the choice of cross term sharing factors is not uniform or arbitrary. Also, there is no need to specify the sharing factors of losses to be allocated to the supply side and demand side.
The branch with negative loss allocation may provide one interesting application on congestion management, which is currently under investigation
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