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Loss Reduction Allocation to Distributed Generation Units in Distribution Systems N. Madinehi, H. Askarian Abyaneh, Senior Member, IEEE, M. Mousavi Agah, H. Nafisi, and K. Shaludegi
Abstract-- This paper proposes a novel loss reduction allocation method in distribution systems in the presence of distributed generation (DG) units. The proposed method aims to procure the participation of each DG unit in the reduced amount of energy losses in distribution systems brought about by participation of all DG units in power market. The method is based on the assumption that DG units have private owners; hence, it provides a proper solution for distribution companies to have a fair incentive to specify which DG units are more beneficial to them. Two cooperative game theory approaches namely Shapley Value and t Value methods are applied in order to provide fair and stable models for loss reduction allocation in distribution networks. Consistent and stable results qualify the equity and validity of the method. Index Terms-- Loss allocation, Distributed Generation, Shapley Value, t -Value
I. INTRODUCTION Distributed Generations (DGs) have become an economic solution for electrical generation in distribution networks. Integration of DG units in distribution network provides many potential benefits to distribution companies (DISCOs). The benefits include loss reduction, reliability improvement, voltage support, improved power quality, capacity release, as well as deferment of upgrading distribution infrastructures [1]. As DISCOs have no control on placement of DG units, it is important for them to provide incentives for DG owners to place DG units in critical points of distribution network. In order to have a fair incentive, DISCOs should specify which DG units are more beneficial to them. Hence, they should consider the parameters by which DG units can be selected as the most economical solution. Among them, loss reduction is a key consideration to select the most beneficial DG units. N. Madinehi is with the Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran (email:
[email protected]). H. Askarian Abyaneh is with the Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran (email:
[email protected]). M. Mousavi Agah is with the Department of Electrical Engineering, Amirkabir University of Technology (email:
[email protected]). H. Nafisi is with the Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran (email:
[email protected]). K. Shaudegi is with the Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran (email:
[email protected]).
Several studies have been conducted to assess losses variations in distribution networks in the presence of DG units. Reference [1] described the importance of losses variations for DG connection costs and proposed a method to compute such loss variations. However, the proposed method is only valid for small DG penetration variations. Allocation of energy losses to consumers connected to distribution networks was proposed in [2]. A comparison was then made between the allocation methods of energy loss in distribution network. A method was proposed in [3] for optimally allocating various types of DG technologies in distribution network to minimize energy loss. In [4], the impact of DG on distribution losses was analyzed. It was shown that energy losses variation, as a function of the DG penetration, presents a U-shape trajectory. The authors of [4] have considered the overall impact of DG units on loss reduction. However, it was not specified the share of each DG unit. This brief review of the literature shows that considerable work has been performed to assess the effect of DG units in loss reduction of distribution networks. The problem of specifying the share of each DG in the reduced losses can be regarded as a mathematical equation with equilibrium constraints. The problem is solved in this paper by using Cooperative game approaches based on Shapley Value and t Value which provide fair and stable models. Cooperative game theory is concerned primarily with coalitions groups of players who coordinate their actions and pool their winnings [5]. Consequently, one of the problems here is how to divide the reduced losses among the DG units participate in power market. Game theory approaches has been used in many power system studies. Some studies have proposed fair schemes for the transmission loss allocation under pool-based or bilateral contracts in electricity markets. Hrieh have implemented Shapley Value to allocate transmission losses under poolbased electricity market [6, 7]. The impact of bilateral transactions on system losses in order to allocate a corresponding loss component to each individual transaction for improving economic efficiency has been considered in other studies [8]-[11]. In [11], two cooperative game theory methods named Nucleolus-based Method and Shapley Value Method have been applied for loss allocation of bilateral transactions in transmission systems. Some researches have
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considered the development of a fair pricing model as a contentious issue [12]-[14]. These papers have proposed cooperative game theory for fair allocation of transmission loss cost among generators and loads in the transmission grids. In none of the above studies, the effect of each DG unit on loss reduction of distribution network has considered. However, there is a knowledge gap in partitioning the share of each DG unit in loss reduction. This problem refers to procuring the participation of each DG unit in the reduced amount of power losses in distribution systems brought about by all DG units participating in power market. With this paper, we precisely intend to fill the gap of calculating the share of each DG unit in reduced amount of losses by proposing a method for loss reduction allocation between various DG units connected to distribution network. By applying the proposed method on a sample distribution network, share of each DG unit is specified in annual loss reduction of the sample network. DG models used in this paper are adopted from the models proposed in [15]. The novelties of this study are as follows: 1. Presenting a fair and rational loss reduction allocation method for DG units. 2. Applying t -Value Method as the mathematical solution to the problem and comparing the results with those of Shapley Value Method. II. REVIEW OF RECENT EXISTING APPROACHES This section is devoted to provide a short review of the two main existing approaches based on cooperative game theory. A. t -Value Method The t -Value Method, first introduced by Tijs in 1981 [16], was defined for quasi-balanced games. The goal of using the t value concept is to fairly allocate the energy losses that are jointly reduced by all DG participants. The solution is based on the upper vector M ( N , v ) and the lower vector m( v ) of a game v . A cooperative game in characteristic function form is an order pair N , v consisting of the player set N = {1,2 ,...,n} and the characteristic function v. Each subset S Ì N is referred to as coalition. i Î N means the player number. v(S ) is the reduced amount of losses related to coalition S, which means the subtract of losses in the system with and without participation of the players in that coalition. The t value is defined by:
t (v ) = am(v ) + ( 1 - a )M (N , v ) where
(1)
M i (N ,v ) = v (N ) - v(N - {} i)
(2)
é ù mi (v ) = maxêv(s ) M j (N , v )ú êë úû jÎ{s}- i
(3)
å
and a Î[0,1] is uniquely determined by:
åt (v ) = v(N )
(4)
i
iÎ N
B. Shapley Value Method The Shapley Value is the basic method to solve the problem of cooperation games. The importance of every player in the coalition is considered in this method. Shapley Value of a game v is the average of the marginal vectors of the game which is shown below:
fi (v ) =
åW ( S )´ [v(s ) - v(s - i )] iÎS
W(S )=
(n - S )!´( S
- 1)!
n!
(5) (6)
where i represents the DG player which takes part in loss reduction allocation. S is the number of members within each coalition S. v(s - i ) is the reduced amount of losses related to coalition S when player i is not participating and W ( S ) is the weight factor which represents the percentage of marginal reduced losses that should be allocated to each player. The following conditions should take into consideration in a cooperative game in order to be fair and justifiable: 1) The amount of reduced losses in the presence of DG participants in each coalition should be more than the reduced losses that each DG unit participates independently; otherwise the players won't accept the loss reduction allocation plan. 2) The total amount of reduced losses due to the existence of all DGs should be completely allocated among all participants. III. PROBLEM FORMULATION This section presents formulation of the problem of loss reduction allocation from perspective of DISCO. The objective of the proposed formulation is encouraging DG units which have more proportion in reduction of power system losses. In this work, DISCO is considered to purchase power from wholesale market and DG units in its territory and is responsible to supply all demands in the area. Reduction in power losses of the distribution system has this benefit that DISCO purchases less power from wholesale market. The IEEE-30 bus distribution system is considered as case study in this work. The purchased power from the wholesale market is transferred from transmission network to distribution grid through main substation which is located in bus 1. The scheme of IEEE 30-bus distribution system is shown in Fig.1. The system parameters can be found in [17].
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Fig. 1. 30 bus IEEE Distribution Test System [19]
IV. THEORETICAL APPROACHES Three DG units have been considered as participants in the market. Each unit can be regarded as one player. The set of players in the game is represented as N = {1,2 ,3} , in which 1 implies the DG connected to bus 14 with 12.4 MW power capacity, 2 implies the DG in bus 19 with 9.5 MW power capacity and 3 implies the DG in bus 26 with 8.4 MW power capacities. There are seven nonempty subsets as follows: {1} , {2} , {3} , {1,2} , {1,3} , {2 ,3} , {1,2,3} . Each subset forms a coalition where different combinations of DG units take part in the power market. The empty subset is the situation in which there is no DG participant in the system and all the demand is supplied through wholesale market. The solution introduced in this paper employs MATLAB thorough an algorithm to calculate system loss in each coalition and implement Shapely Value and t Value methods to determine the share of each DG unit in reducing the losses. The sample network is simulated in DIgSILENT software and the load flow analysis is run for each state. Total load in the system is 102.9 MW and the system loss without DG units is 10431.8 kW. Suppose the three independent DG participate in wholesale market simultaneously. Table I shows the reduced amount of distribution losses when each coalition acts individually. The negative values show the reduction of loss. As shown in the table, the amount of reduced losses in the presence of DG participants in each coalition is more than the reduced losses that each DG unit participate provides independently. The following subsection describes the problem solution by using each method.
TABLE I DISTRIBUTION LOSS REDUCTION OF EACH COALITION Distributed Generation Amount of Reduced Losses Combination
{1} {2 } {3 } {1 ,2 } {1,3} {2 ,3} {1,2 ,3}
-3497.62 -2051.69 -1132.56 -4738.64 -4236.79 -2994.46 -5395.09
A. t -Value Solution for Loss Reduction Allocation
In this part, the allocating process of loss reduction in t Value Method is described. The upper vector M ( N , v ) and the lower vector m( v ) of the case study are as follows: M 1 (N ,v ) = v({1,2 ,3}) - v({2 ,3}) = -2400.63 M 2 (N , v ) = v ({1,2 ,3}) - v({1,3}) = -1158.3 M 3 (N , v ) = v ({1,2 ,3}) - v({1,2}) = -656.45
m1 (v ) = max[ -3580.3,-3580.3,-3580.3,-3497.6 ] = -3497.6 m2 (v ) = max[ -2338.1,-2338.1,-2338.1,-2051.69 ] = -2051.69 m1 (v ) = max[ -1836.2 ,-1132.5 ,-1836.2 ,-1836.2 ] = -1132.5
The amount of a will be equal to 0.4783 according to equations 1 and 4. The proportion of each DG in the reduction of losses will be calculated as: tv1 = .4783* -3497.6 + .522* -2400.63 = -2924.981 tv2 = .4783* -2051.69 + .522* -1158.3 = -1585.94 tv3 = .4783* -1132.5 + .522* -656.45 = -884.169
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B. Shapley Value Solution for Loss Reduction Allocation The allocating process of loss reduction in Shapley Value is shown in table II. This process is performed only for DG 1 for the sake of simplicity. TABLE II LOSS REDUCTION ALLOCATION IN SHAPLEY VALUE METHOD 1
1,2
1,3
1,2,3
C(s)
-3497.62
-4738.64
-4236.79
-5395.09
C(s-1)
0
-2051.69
-1132.56
-2994.46
Fig. 3. Loss Reduction Allocation in power system with
C(s)-C(s-1)
-3497.62
-2686.95
-3104.23
-2400.63
s
1
2
2
3
Compared to the reduced amount of losses by each DG unit individually, the allocated losses are relatively less for all the three players, which is consistent with the basic conditions of the cooperative game model.
w(s)
0.33
0.17
0.17
0.33
w*C(s)-C(s-1)
-1165.873
-447.825
-517.3717
-800.21
Total
-2931.28
By applying the two methods of cooperative game theory to the case study, the share of each DG unit in reduced amount of losses is shown as Table III. The amounts of losses allocated to players in both methods are approximately similar. The sum of each player's portion is equal to the total amount of reduced losses due to the existence of all DGs. The results reveal that the total reduced losses has been completely allocated among all participants and the allocation procedure is fair and rational. TABLE III LOSS REDUCTION ALLOCATION IN IEEE-30 BUS TEST SYSTEM DG Number
Shapley Value
t
value
Active loss (kW)
Active loss (kW)
1
-2931.28
-2924.981
2
-1587.15
-1585.94
3
-876.66
-884.169
Sum
-5395.09
-5395.09
By comparing the results of Table III, it can be observed that DG1 located in bus 1 is more than other DG units, which means that this unit has more partnership in loss reduction of the system and is more beneficial from the view point of DISCO; hence, there should be more incentive for the owner of this DG to generate more power in order to have more proportion in loss reduction of the system. Figs. 2 and 3 show the cylinderical diagram of the loss reduction allocation calculated with Shapley Value and t Value methods respectively. The reduced losses allocated to each DG unit in both of the methods are approximately the same.
Fig. 2. Loss Reduction Allocation in power system with Shapley Value method
t
-Value method
V. CONCLUSION In this paper, new loss reduction allocation method has been proposed from the perspective of DISCO’s benefit. The proposed method considers the disco as the purchaser of power energy from wholesale market and DG units. In order to have a fair incentive, DISCO should specify which DG units are more beneficial; to achieve this objective, the share of each DG unit in the reduced amount of losses in the presence of all the units has been calculated. The problem of specifying the share of each DG in the reduced losses is regarded as a mathematical equation with equilibrium constraints. Cooperative game approaches based on Shapley Value and t Value Methods have been adopted to provide fair and stable models for our purpose. The results reveal that the allocation results can be easily understood and accepted by market players. The allocation solutions would not be affected by the sequence that each distributed generation unit participate and generate power. VI. REFERENCES [1]
L. Soder, "Estimation of reduced electrical distribution losses depending on dispersed small scale energy production", 12th Power Sys. Comp. Conf., Zurich, Switzerland, 1996. [2] J. S. Savier, D. Das, "Energy loss allocation in radial distribution systems: A comparison of practical algorithms", IEEE Trans. Power Del., vol. 24, no. 4, pp. 260-267, Nov. 2009. [3] Y. M. Atwa, E. F. El-Saadany, "Optimal renewable resources mix for distribution system energy loss minimization ", IEEE Trans. on Power Sys., vol. 25, p.p. 360-370, 2009. [4] V. H. M. Quezada, J. R. Abbad, "Assessment of energy distribution losses for increasing penetration of distributed generation", IEEE Trans. On Power Del., vol. 21, p.p. 533-540, 2006. [5] R. Branzei, D. Dimitrov,"Models in Cooperative Game Theory", Springer, 2nd Edition, 2008. [6] S.C. Hsieh, "Fair Transmission Loss Allocation Based on Equivalent Current Injection and Shapley Value", IEEE Conf., 2006. [7] S.C. Hrieh, f.M. Wang, "Allocation of Transmission Losses Based on Cooperative Game Theory and Current Injection Models", IEEE Conf., 2002. [8] F. D. Galiana, ZEEE, M. Phelan," Allocation of Transmission Losses to Bilateral Contracts in a Competitive Environment", IEEE Trans. on Power Sys., Vol. 15. NO. I, FEB. 2000 [9] B. Xin-zhong, C. Wei, "Allocation of Transmission losses in electricity System Based on Cooperative Game Theory", IEEE Conf., p.p. 1-4, 2010. [10] D. Songhuai, Z. Xinghua, M. Lu, X. Hui," A Novel Nucleolus-Based Loss Allocation Method in Bilateral Electricity Markets", IEEE Trans. on Power Sys., Vol. 21, NO. 1, Feb. 2006
5 [11] D. A. Lima, J. Contreras, A. P. Feltrin, "A cooperative game theory analysis for transmission loss allocation", Science Direct, Electric Power Systems Research, p.p. 264–275, 2008. [12] M. Bockarjova, M. Zima, G. Andersson," On Allocation of the Transmission Network Losses using Game Theory", IEEE Conf., 2008 [13] Y. P. Molina, R. B. Prada, O. R. Saavedra," Allocation of transmission loss cost using game theory", IEEE Conf., 2007 [14] R. Bhakar, V. S. Sriram, N. P. Padhy, H. O. Gupta," Probabilistic Game Approaches for Network Cost Allocation", IEEE Trans. on Power Sys., Vol. 25, NO. 1, Feb. 2010 [15] J.H. Teng, "Modeling distributed generations in three-phase distribution load flow", IEEE Trans. Power Del., Vol.2, NO. 3, p.p. 330-340, 2008. [16] S. Tijs, “Bounds for the core and the τ –value”, Game Theory and Mathematical Economics, North-Holland, Amsterdam, p.p. 123-132, 1981. [17] www.ee.washington.edu/research/pstca/
