Applications of Cooperative Game Theory in Power System Allocation Problems

Transcription

1 Leonardo Journal of Sciences ISSN Issue 23, July-ecember 2013 p Applications of Cooperative ame Theory in Power System Allocation Problems Fayçal ELATRECH KRATIMA *, Fatima Zohra HERBI, and Fatiha LAKJA Electrical Engineering epartment. Intelligent Control and Electrical Power System Laboratory (ICEPS), illali Liabes University, Sidi-Bel-Abbes, 22000, Algeria. s: * Corresponding author: * Phone/Fax: 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 maret are modeled as individual current inections based on a real-time solved AC power flow solution. Each load can be modeled as a current inection or equivalent constant impedance depending on whether it is required to be responsible for the system loss. Each current inection 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 14-bus 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 maret one of the most important issues is the allocation of transmission losses among maret 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 non-separable properties of the loss function. The electric power industry is undergoing a series of challenging changes due to deregulation and competition

2 Applications of Cooperative ame Theory in Power System Allocation Problems Fayçal ELATRECH KRATIMA, Fatima Zohra HERBI, Fatiha LAKJA One of the most important issues is the allocation of transmission losses among maret 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 tas to fairly allocate a component of system losses to an individual participant of the maret. The main difficulty of loss allocation is caused by the highly nonlinear and non-separable 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 C power flow, while some use AC load flow for matching the calculation results and actual power flows. Some schemes are branch-power-flow based, while some focus on the branch-current based allocation techniques. ame theory provides well-behaved solution mechanisms with economic features for assessing the interaction of different participants in competitive marets 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 wor is extended in this paper. The transmission loss is derived as function individual current inections. Two basic formulations are presented to determine individual current inections. 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 real-time 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 inections directly form bus impedance matrix. Each current inection 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 maret transactions are modeled as individual current inections [5]. Each current inection is then treated as an individual player of the transmission loss allocation game. The approaches are branch-current based, not branch-power-flow based. Without any approximations or assumptions lie those made for a C power flow or proportional sharing, the proposed 126

3 Leonardo Journal of Sciences ISSN Issue 23, July-ecember 2013 p 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 real-time AC power flow solution and recover the total system loss. The Kirchhoff s laws and superposition principle are satisfied and both the networ configuration and the voltage-current 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 well-nown IEEE14-bus 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 n-1, x n] (3) Then second order (quadratic) cooperative game which is described as follows. minε(s) Subect to y(s) (ν(s)) 2 + ε(s) (4) y(n) = (ν(n)) 2 (5) If the solution to the game is 127

4 Applications of Cooperative ame Theory in Power System Allocation Problems Fayçal ELATRECH KRATIMA, Fatima Zohra HERBI, Fatiha LAKJA y = [y 1, y 2,... y n-1, 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) x(s') = x(n) (9) ν(s) + ν(s') = ν(n) (10) where S' is the conugate 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. eneration and Load Models Based on a solved AC power flow solution for a pool based electric power maret, let the complex power inection in to a generator bus i be i S = P + Q then the generation current inection is written as: i i i [ ] S V = [( P + Q ) V ] I (16) = i i where V i is its bus voltage. Similarly, let the complex power inection in to a load bus be ( P Q ) S = + we can then have load current inection S P + Q I = = (17) V V Or the equivalent load impedance 128

5 Leonardo Journal of Sciences ISSN Issue 23, July-ecember 2013 p V V i Z = = (18) I P Q B. Transmission loss allocation problem 1. Loss allocation to generators only For an n node power networ having m generator buses the transmission loss of element connected between nodes i and is derived in terms individual current contribution of each generator as loss where m * m P = I I. R = 1 = 1 I is the current contribution of th generator to the element and it can be determined from modified Y bus method using converged load flow solution. R (19) is the resistance of line element connected between nodes i and. The individual voltage contribution of each generator is derived in terms of current inections. v nn 1 Y Y L I =.diag (20) Y Y 0 L LL v nn buses. is a square matrix of size n and the columns m+1 to n will be zero since they are load 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. v = Z diag I (21) nn bus bus Now the transmission loss of th element in terms of individual current contribution is given: loss m * m P = I I. R = 1 = 1 For both methods the current contributions of th bus for th element is (22) 129

6 Applications of Cooperative ame Theory in Power System Allocation Problems Fayçal ELATRECH KRATIMA, Fatima Zohra HERBI, Fatiha LAKJA I = ( v v ) / z (23) i z is the transmission line impedance of element (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 th generator to element is given by: I = ( v v ) / z (24) i where z is the transmission line impedance of element (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 n = 1 I = I (25) For each element the coalitions present a balancing condition because of Kirchoff s current law. Let S be set of possible coalitions: x (S ) = I(S ) (26) x (N) = I (27) Let the solution vector for this balanced cooperative game be: [ I I... I ] x = (28) 1 2 n Now the coalition values for the transmission loss allocation problem is derived as: min e(s) (29) * y ( S) ( I( S) I ( S)) + e( S) (30) y (N) = 2 I (31) y = real(i. * x) (32) The transmission loss contribution of th generator to th element is determined as: P = y R (33) Now the transmission loss contribution of th generator is the summation of losses to every line element of that generator. 130

7 Leonardo Journal of Sciences ISSN Issue 23, July-ecember 2013 p loss P = P The system parameters are shown in Tables 1, 2, 3 Table 1. Bus data of 14 bus system Bus i type Pd Qd s Bs area Vm Va basekv zone Vmax Vmin Table 2. enerator data of 14 bus system bus Pg Qg Qmax Qmin Vg mbase status Pmax Pmin Table 3. Branch data of 14 bus system fbus tbus r X b ratea rateb ratec ratio angle status (34) 131

8 Applications of Cooperative ame Theory in Power System Allocation Problems Fayçal ELATRECH KRATIMA, Fatima Zohra HERBI, Fatiha LAKJA Results and iscussions Several systems have been used to test the proposed method. In this paper, the test results of a 14-bus system are presented and discussed. The one-line 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 inected 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 Figure bus system The total allocated loss is consistent with the power flow solution and can reasonably reflect the amounts of transactions inected complex powers according to the solution listed in Table 7. Since the networ configuration and the location of each player are taen 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. 132

9 Leonardo Journal of Sciences ISSN Issue 23, July-ecember 2013 p Table 4. Converged load flow solution of 14 bus system Voltage Mag Voltage Angle Real Power Reactive Power Bus no pu egrees P Q Transmission loss Table 5. Transmission loss allocation (only generator buses) Line N0: 1 2 Loss Total

10 Applications of Cooperative ame Theory in Power System Allocation Problems Fayçal ELATRECH KRATIMA, Fatima Zohra HERBI, Fatiha LAKJA Table 6. Transmission loss allocation (only load buses) Line No L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L Total Line No 1 2 L3 Table 7. Transmission loss allocation (generator and load buses) L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L Total

11 Leonardo Journal of Sciences ISSN Issue 23, July-ecember 2013 p Conclusions The loss impacts between one player and any other coalitions of players are taen 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 References 1. ross., Tao S., A physical-flow-based approach to allocating transmission losses in a transaction framewor, IEEE Trans. PowerSyst, 2000, 15, p ing Q., Abur A., Transmission loss allocation in a multiple transaction framewor, IEEE Trans. Power Syst, 2004, 19, p Leite da Silva A. M., uilherme de Carvalho Costa J., Transmission loss allocation: part I single energy maret, IEEE Trans. Power Syst, 2003, 18, p Zolezzi J. M., Rudnic H., Transmission cost allocation by cooperative games and coalition formation, IEEE Trans. Power Syst, 2002, 17, p Hsieh S. C., Wang H. M., Allocation of transmission losses based on cooperative game theory and current inection models, in Proc.IEEE Int. Conf. Industrial Tech., Bango, Thailand, 2002, 11-14,p ross., Tao S., A physical-flow-based approach to allocating transmission losses in a transaction framewor, IEEE Trans. PowerSyst, 2000, 15, p Saloman anara, Shanarappa Kodad F., Tulsi Ram as., Analytical solution to balanced quadratic cooperative game and its application to transmission loss allocation, Indian Journal of Science and Technology, 2007, Vol.1 No Krishna eeshit., Poornachandra Rao N., Transmission loss allocation in a multiple transaction framewor, International Journal of Engineering Research and Applications (IJERA), May-Jun 2012, 2 (3), p

12 Applications of Cooperative ame Theory in Power System Allocation Problems Fayçal ELATRECH KRATIMA, Fatima Zohra HERBI, Fatiha LAKJA 9. aliana F.., Phelan M., Allocation of transmission losses to bilateral contracts in a competive environment, IEEE Trans. PowerSyst, 2000, 15, p Coneo A. J., Arroyo J. M., Alguacil N., uarro A. L., Transmission loss allocation: a comparison of different practical algorithms, IEEE Trans. Power Syst, 2002, 17, p omez Exposito A., Riquelme Santos J. M., onzalez arcia T., Ruiz Velasco E.A., Fair allocation of transmission power losses, IEEE Trans. Power Syst, 2000, 15, p aniel J. S., Salgado R. S., Irving M. R., Transmission loss allocation through a modified Ybus, IEE Proceedings- eneration, Transmission and istribution, 2005, 152, p