In light of the preceding research, this paper proposes a new scheme for allocating DS losses to network participants in the deregulated energy market. The proposed method is based on a synthesis of electrical circuit theory and weighted Shapley values - a new notion in CGT.

To allocate the losses, existing techniques that cited in the given literature, take into account either the steady-state power rating of loads and DGs or their in-cooperative contribution w.r.t. to their location in the network. This led to bias and unfair solutions to passive participants who are connected away from the substations. However, on the other hand, in comparison to techniques cited in the given literature, the key contribution of this work is that the proposed method considers the average marginal contribution of each load and DG, as well as their weights, as a coefficient for sharing system losses. This not only benefits the loads but also grant maximum incentives to DGs for their contribution to loss reduction.

The management and handling of coalitions associated with CGT are the most tedious and time-consuming tasks. Coalitions can further grow to be very large in size depending on the number of participants in the system. To alleviate the stress associated with coalition computation, the paper proposes an algorithm named as majority rule game strategy for reducing the population of coalitions without jeopardizing the CGT axioms or the solution’s effectiveness. Novelty of this work is stated that proposed method can manage and reduce the number of coalitions effectively without compromission in fairness of loss sharing so that it may apply for any practical distribution systems which have thousands of nodes and participants.

The highlight of this paper exhibits that the proposed method produces efficient results by determining the significant contributions of consumers and DGs to overall system losses (including loss saving). The impacts of different market participants and their mutual contribution on system power losses are also considered. The proposed method fairly and rationally assigns system losses, and the allocation results are unaffected by the reduced number of coalitions constituted. Based on cooperative game theory, a weighted Shapley value-based power loss allocation approach is proposed and thoroughly compared with the Shapley value-based allocation method and the pro-rata method.

The remainder of the paper is structured as follows: The second section introduces the terminology and axioms of CGT; it also discusses the Weighted Shapley value, Shapley value, and majority rule game in CGT; the third section derives mathematical expressions for the proposed method; the fourth section discusses the obtained results; and the fifth section concludes the work.

For any transferable utility game N , υ , Harsanyi dividend can be expressed as, λ υ S = ∑ τ ⊆ S − 1 S − τ υ τ , ∀ S ⊆ N   (6)

Let ω = ω i i ∈ N ∈ R N be a vector. Allocation rules φ ω N , υ = φ 1 ω , φ 2 ω , . . . , φ N ω T ∈ R N are described by φ i ω N , υ = ∑ S ⊆ N : i ∈ S ω i ω S λ υ S , ∀ i ∈ N (7)

Such that ω S = ∑ i ∈ S ω i , are called Weighted Shapley values (Kalai and Samet, 1987; Nowak and Radzik, 1995).

For any transferable utility game N , υ , Shapley value allocation rules φ N , υ = φ 1 , φ 2 , . . . , φ N T ∈ R N are described by (Gilles, 2010b) φ i N , υ = ∑ S ⊆ N : i ∉ S S ! N − S − 1   ! N ! υ S ∪ i − υ S ∀ i ∈ N (8)

Where, the term S ! N − S − 1   ! N ! denotes the probability for player i to incorporate exactly to S. The term υ S ∪ i − υ S is known as the marginal contribution of player i, when it is incorporated into coalition S. Therefore, by using the Shapley value, the marginal contribution of each player is averaged for all possible network permutations that they can be part of.

Any transferable utility game N , υ is simple if for every coalition S ⊂ N , either υ S = 1 or υ S = 0 Typically, majority rule game is defined as: υ S = 1 i f   S = ⌈ N t ⌉   w h e r e 1 ≤ t ≤ 2 (9) or υ S = 0 otherwise.

Losses in distribution networks are caused by both consumer and DG. Therefore, responsibility for losses must be allocated among them based on their contributions.

This paper uses node and branch indexing techniques, as well as the power flow algorithm described in (Savier and Das, 2012). Following the execution of the power flow, system losses are calculated. The phasor value of current drawn by load (consumer current) at bus i, i.e., load i, may be calculated as follows: I L i = S L i * V i * = P L i − j Q L i V i * (10) where, S L i * is the conjugate of complex power of the load i; V i * is the conjugate of voltage of bus i; and P L i and Q L i are the active and reactive power of the load i, respectively.

Similarly, the generator current phasor value may be represented as: I G i = S G i * V i * = P G i − j Q G i V i * (11)

Where, S G i * is the conjugate of complex power of DG i; and P G i and Q G i are the active and reactive power of DG i, respectively.

Assumption for power sign convention of participants as follows: consumers draw power from the network, so their power and current are both assumed to be positive, whereas DGs inject power into the network, so their power and current are both assumed to be negative.

In the case where both the consumer and the DG are connected to a common bus i the resultant value of current at bus i is given by: I R i = I L i − I G i (12) where, I R i is resultant value of current at same bus i.

The sign of I R i is determined by the penetration of DG power at a common bus i and is stated as: I R i = I R i − I R i 0 if   I L i > I G i if   I L i < I G i if   I L i = I G i (13)

The structural design of the DS is radial, and the resultant power will flow unidirectional, i.e., from the reference/root node to the end nodes. Thus, in this paper, current flowing through any branch is calculated using the bus injection to branch current (BIBC) matrix and is stated as: I b 1 I b 2 ⋮ I b n − 1 = E 1,2 E 1,3 … E 1 , n E 2,2 E 2,3 … E 2 , n ⋮ ⋮ ⋮ ⋮ E n − 1,2 E n − 1,3 ⋯ E n − 1 , n × I R 2 I R 3 ⋮ I R n (14) Where, n is the total number of buses in the system; I b is the column vector of branch current where its subscript is varying from index 1 to n-1 and its dimension is n-1 × 1; I R is the column vector of bus resultant current where its subscript is varying from index 2 to n and its dimension is n-1 × 1.

Elements in the row matrix are defined as:

E_ij = 1 0 if   I R j ∈ N B i if   I R j ∉ N B i NB _i is set of buses connected downstream of upstream branch i where, E i , j is the element whose value is either 1 or 0, and subscript i belongs to branch number which is varying from 1 to n-1 whereas j belongs to bus number which is varying from 2 to n.

Eq. 14 can be expressed in its canonical form as: I b = B I B C × I R (15) where, BIBC is the bus injection to branch current matrix whose dimension is n − 1 × n − 1 .

Bus 1 serves as the root node/reference bus, and it is presumed that no load is connected to it, hence it is excluded from the BIBC matrix.

From (14), the equation for current in any branch i may be expressed as, I b i = ∑ j ∈ N B i E i , j I R j (16)

The real power loss of the system is stated as: P L S = ∑ i ∈ B r I b i 2 × R i (17) where, P L S is the real power loss of the system; R i is the resistance of branch i; and Br is the set of all the branches in the system.

Using (16) and (17), the real power loss of the system can also be expressed as, P L S = ∑ i ∈ B r ∑ j ∈ N B i E i , j I R j 2 × R i (18)

To yield maximum advantage to DGs in terms of their contributions in loss saving without affecting the economic efficiency of the system, or in other words, to eliminate consumer cross-subsidies, this article uniquely distinguishes the changes in losses caused by consumers and DGs separately. Hence, following Subsection 3.2.1 describes the allocation of system losses to load (i.e., base case or without DG), and thus (12) can be modified I R i = I L i , while on the other hand; subsection 3.2.2 describes the allocation of loss saving to DG (i.e., system with DG), and thus (12) remains intact.

A hypothetical 4-bus DS is used to illustrate the CGT model for system loss allocation and to evaluate the proposed method. The circuit diagram of the test system is depicted in Figure 1, and its actual line and load data are given in figure. As shown in Figure 1, bus 1 acts as a substation and it is assumed that no load is connected on it and voltage is regulated at specified value, whereas loads are connected to buses 2, 3, and 4. To analyze and comprehend the proposed mechanism, allocation of system losses using base case has been considered. The number of coalitions that participated in establishing a fair allocation of losses is 7, and they are shown in Table 1. Due to a smaller number of coalitions, it may be managed without incurring a computational execution burden. So, for the given network, no need to use the majority rule game for allocation.

Base values of test system are 11 kV and 1 MVA. After execution of power flow, voltages on system buses are V ₁=(1 + j0) pu; V ₂=(0.91533-j0.00572) pu; V ₃=(0.88127-j0.0096) pu; and V ₄=(0.86939-j0.00947) pu. The total real losses in the system are 0.335621 pu. The first stage in determining the participation of consumer to system losses is to ascertain the coalitions and their worth. Table 1 illustrates the coalitions that have been formed, and worth corresponding to each coalition.

Similarly, for each coalition, compute the Harsanyi dividend based on the worth of the characteristic function. Thus, λ υ 2 = − 1 0 × 0.030825 = 0.030825 ; λ υ 3 = − 1 0 × 0.034049 = 0.034049 ; λ υ 4 = − 1 0 × 0.068331 = 0.068331 ; λ υ 2,3 = − 1 2 − 1 × 0.030825 + − 1 2 − 1 × 0.034049 + − 1 2 − 2 × 0.116096 = 0.051222 ; λ υ 2,4 = − 1 2 − 1 × 0.030825 + − 1 2 − 1 × 0.068331 + − 1 2 − 2 × 0.164059 = 0.064903 ; λ υ 3,4 = − 1 2 − 1 × 0.034049 + − 1 2 − 1 × 0.068331 + − 1 2 − 2 × 0.188664 = 0.086284 ; λ υ 2,3,4 = − 1 3 − 1 × 0.0308245 + − 1 3 − 1 × 0.0340485 + − 1 3 − 1 × 0.068331 + − 1 3 − 2 × 0.116096 + − 1 3 − 2 × 0.164059 + − 1 3 − 2 × 0.188664 + − 1 3 − 3 × 0.335615 = 0

Now, the share of losses of consumers connected to bus 2 is determined as x 2 ω = 1 2 + 0 . 5 2 1 2 + 0 . 5 2 × 0.030825 + 1 2 + 0 . 5 2 1 2 + 0 . 5 2 + 0 . 8 2 + 0 . 4 2 × 0.051222 + 1 2 + 0 . 5 2 1 2 + 0 . 5 2 + 1 2 + 0 . 5 2 × 0.064903 + 1 2 + 0 . 5 2 1 2 + 0 . 5 2 + 0 . 8 2 + 0 . 4 2 + 1 2 + 0 . 5 2 × 0 = 0.091734 ; the share of losses of consumers connected to bus 3 is determined as x 3 ω = 0 . 8 2 + 0 . 4 2 0 . 8 2 + 0 . 4 2 × 0.068331 + 0 . 8 2 + 0 . 4 2 1 2 + 0 . 5 2 + 0 . 8 2 + 0 . 4 2 × 0.051222 + 0 . 8 2 + 0 . 4 2 0 . 8 2 + 0 . 4 2 + 1 2 + 0 . 5 2 × 0.086284 + 0 . 8 2 + 0 . 4 2 1 2 + 0 . 5 2 + 0 . 8 2 + 0 . 4 2 + 1 2 + 0 . 5 2 × 0 = 0.095163 ; and the share of losses of consumers connected to bus 4 is determined as x 4 ω = 1 2 + 0 . 5 2 1 2 + 0 . 5 2 × 0.068331 + 1 2 + 0 . 5 2 1 2 + 0 . 5 2 + 1 2 + 0 . 5 2 × 0.064903 + 1 2 + 0 . 5 2 1 2 + 0 . 5 2 + 0 . 8 2 + 0 . 4 2 × 0.086284 + 1 2 + 0 . 5 2 1 2 + 0 . 5 2 + 0 . 8 2 + 0 . 4 2 + 1 2 + 0 . 5 2 × 0 = 0.148724 .

After estimating the shares of consumers in system losses, it is required to determine whether or not they are fairly allocated by establishing axioms.

The losses shared by the proposed method fulfil all of the key axioms for fair game as specified in Section 2, as follows: By following axiom 1, the proposed method has penalized the consumers for the cause of system losses of a given DS in the game N , υ i.e., x 2 ω = 0.091734 > 0 ;   x 3 ω = 0.095163 > 0 ; a n d   x 4 ω = 0.148724 > 0 . By following axiom 2, the proposed method assigns individual worth of losses such that the shared losses of consumers are non-negative in the game N , υ as given in Table 1. By following axiom 3, the proposed method obeys individual rationality in which the shared losses of each consumer are greater than the losses created by the individual consumers, i.e., x 2 ω = 0.091734 ≥ υ 2 = 0.030825 ;   x 3 ω = 0.095163 ≥ υ 3 = 0.034049 ; a n d   x 4 ω = 0.148724 ≥ υ 4 = 0.068331 . Also, the proposed method follows the coalition rationality, described by axiom 4, in which the sum of shared losses of consumers in a coalition is greater than the loss when only those consumers are presented in the same coalition, i.e., x 2 ω + x 3 ω = 0.186897 ≥ υ 2,3 = 0.116096 ;   x 2 ω + x 4 ω = 0.240458 ≥ υ 2,4 = 0.164059 ; a n d   x 3 ω + x 4 ω = 0.243887 ≥ υ 3,4 = 0.188664 . By following axiom 5, the proposed method shares system losses calculated by the power flow algorithm among consumers who participated in the game N , υ efficiently, i.e., x 2 ω + x 3 ω + x 4 ω = 0.335615 .

From the preceding results, it is clear that no consumer is expected to agree to receive less than the consumer obtained acting alone, as indicated by individual rationality, and additionally, the payoff vector of the total system losses received by the consumer, with the understanding that consumer i must receive x i ω , as indicated by efficiency. Imputation occurs when a game fulfils both individual and coalitional rationalities. The stable imputation is referred to as the core, or in other words, the core of the game will remain stable if no coalition has the motivation or ability to undermine the agreement on equitable distribution among players. In given DS, imputation of the payoff vector x i ω has been stable through coalition S because shared losses have satisfied coalition rationality. Moreover, this is an essential game because ∑ i = 1 n υ i < υ N has been satisfied.

Because coalitional rationality is not a criterion for defining the weighted Shapley value, the solution produced using the proposed method does not necessarily belong to the core. However, depending on the network architecture, the number of consumers, and the manner in which power is distributed, the solution of the proposed method may belong to the core.

Table 2 depicts the allocated losses (in kW) by the proposed method compared to those of the other two methods. Because the power consumption of buses 2 and 4 is identical, the pro-rata method assigns identical losses. However, the proposed method and the Shapley value method differentiate consumers based on their marginal contributions to coalitions and thus assign different losses than the pro-rata method does. However, in addition to marginal contribution, the proposed method takes the weight of the consumer into account, and hence the resulting value differs from the Shapley value. Thus, as shown in Table 2, neither a consumer located far from the substation nor a consumer located close to the substation experiences economic overload as a result of the proposed method.

As a result of the aforementioned analysis, it is clear that the proposed method fulfils the agreement on unbiased system loss allocation. Additionally, while the proposed method satisfies all of the axioms necessary for equitable allocation, it may not fulfil coalition rationality for specific networks. Apart from this, the proposed method has an advantage over the Shapley value method in that it bases loss allocation on the marginal contribution of consumers to system losses as well as the probability of the consumer drawing power in coalitions of their marginal contributions.

In this test system, 33-bus and 69-bus DSs have been considered. Line and load data of 33-bus and 69-bus DSs were obtained from (Atanasovski and Taleski, 2012) and (Savier and Das, 2012), respectively. To reduce the system losses and improve the voltage profile, DGs are optimally located and sized in a 33-bus test system at bus numbers 7,17 and 32 with power injection of 240 + j 96 k V A , 400 + j 160 k V A , and 400 + j 100 k V A , respectively. Base values for the test system are taken as 12.66 kV and 1 MVA. System real losses with the base case are 202.6768 kW whereas loss saving achieved is 119.4674 kW. Similarly, for the 69-bus test system, DGs are optimally located and sized at bus numbers 50, 61 and 64 with power injection of 180 + j 160 k V A , 990 + j 720 k V A , and 210 + j 140 k V A , respectively. Base values for the test system are taken as 12.66 kV and 1 MVA. System real losses with the base case are 226.5886 kW whereas loss saving achieved is 191.1672 kW.

Table 3 depicts the loss allocated to consumers and the loss saving to DGs of a 33-bus DS using all three methods. In the CGT model, the total number of coalitions required for a 33-bus (excluding bus-1) DS is 4.295 × 10 9 . It is self-evident that a computer system will take a long time to execute and will require significantly more memory space in order to deal with this massive number of coalitions for loss allocation. This paper has proposed a way of resolving these intractable problems by employing a majority rule game. By employing a majority rule game, the proposed method includes only coalitions with 88% of all players, which indicates that system losses associated with each coalition exceed 50% of system losses associated with all participants. The results obtained by the proposed method in Table 3 have accounted for 41,449 coalitions with a value of ‘t’ (in the majority rule game) is 1.14. While it is true that the proposed method requires some computing time to process 41,449 coalitions, this time will undoubtedly be far less than the time required to execute 4.295 × 10 9 coalitions. Apart from the time savings associated with computing, the proposed method incorporates the principles of both the Shapley value and pro-rata methods.

As shown in Table 3, the proposed method did not assign the higher losses to the consumers with the large power rating nor did it assign equal losses to consumers with the same power rating in various buses. Additionally, Table 3 details the allocation of loss savings to DGs. Loss savings are used to reward DGs for their contributions to system performance improvement. Each of the three techniques assigned nearly identical loss savings to all connected DGs.

Table 4 shows the loss allocated to consumers and loss saving allocated to DGs of a 69-bus DS using all three methods. The total number of coalitions required for 69-bus DS is 2.951 × 10 20 , while the proposed method requires 52,463 coalitions with a value of ‘t’ is 1.04. Using the majority rule game, the proposed method accounted for only coalitions with at least 96% of total participants. The remainder of the observations made about the 33-bus DS have been followed by those about the 69-bus DS.

As a result of the foregoing test systems and discussion, it is concluded that the proposed method may be used to any size of realistic DSs without jeopardizing the consistency and fairness of loss allocation.