Abstract
The emergence of prosumers in distribution systems has enabled competitive electricity markets to transition from traditional hierarchical structures to more decentralized models such as peer-to-peer (P2P) and community-based (CB) energy transaction markets. However, the network usage charge (NUC) that prosumers pay to the electric power utility for network services is not adjusted to suit these energy transactions, which causes a reduction in revenue streams of the utility. In this study, we propose an NUC calculation method for P2P and CB transactions to address holistically economic and technical issues in transactive energy markets and distribution system operations, respectively. Based on the Nash bargaining (NB) theory, we formulate an NB problem for P2P and CB transactions to solve the conflicts of interest among prosumers, where the problem is further decomposed into two convex subproblems of social welfare maximization and payment bargaining. We then build the NUC calculation model by coupling the NB model and AC optimal power flow model. We also employ the Shapley value to allocate the NUC to consumers fairly for the NUC model of CB transactions. Finally, numerical studies on IEEE 15-bus and 123-bus distribution systems demonstrate the effectiveness of the proposed NUC calculation method for P2P and CB transactions.
IN recent years, with the gradual maturity of renewable energy generation technologies and continual reduction in installation costs, prosumers with small-scale distributed generation in distribution energy networks play an increasingly critical role in transactive energy markets [
Currently, the common methods used to determine the NUC include: ① the postage stamp method [
The NUC design based on a unique cost allocation method requires prosumers to share the incurred costs equally or proportionally. In [
As a solution, dynamic energy pricing and DLMP methods with dynamic characteristics have been introduced to calculate the NUC to balance and maximize the interests of multiple parties. Time-of-use (ToU) tariffs with dynamic change characteristics are considered in the NUC as an efficient pricing mechanism to improve social welfare and facilitate energy utilization. In [
The aforementioned NUC calculation methods mostly focus on the design of the pricing mechanism, which is not only related to pricing but also to the quantity of transactive energy transmitted over the network. The energy management problem for P2P transactions is viewed as a Nash bargaining (NB) game in which the agents mutually negotiate to achieve fair profit allocation [
To these ends, this paper proposes a calculation method for the distribution of NUCs for P2P and community-based (CB) market structures, which allocates the NUC efficiently to consumers while considering the operational characteristics of the distribution network in the context of the prosumer era. The design of the NUC considers the effects of both the energy pricing mechanism based on the DLMP method and the transaction behavior of users based on NB theory in the transactive energy market. First, based on NB theory, we formulate the NB problem for P2P and CB transactions and then decompose the problem into two convex subproblems: the social welfare maximization problem (P1 and S1) and the payment bargaining problem (P2 and S2), which can determine the optimal transaction quantity and price. Second, the NUC calculation model is constructed by coupling the NB model and AC optimal power flow (OPF) model, in which the DLMPs are composed of the dual multipliers in the AC OPF model. In addition, the Shapley value is employed to allocate the NUC to consumers fairly for the NUC model of CB transactions. In the simulation, the NUC, welfare, and network operational characteristics of P2P and CB transactions are compared. The main contributions of this study are as follows.
1) Compared with the existing NUC models, a holistic NUC calculation model is built for the trading optimization scheduling problems of P2P and CB for prosumers. The NUC pricing method and optimal trading scheduling are innovatively incorporated into the NUC analytic framework, which has not been previously studied.
2) The energy trading problem in the transactive energy market is formulated based on NB theory, which captures the transaction behavior of prosumers and improves their mutual benefits. The formulated non-convex bargaining problem is decomposed into the two convex subproblems of social welfare maximization and payment bargaining.
3) A cooperative NUC allocation strategy is proposed for CB trading of prosumers based on the Shapley value. Compared with the NUC of P2P trading, which is derived from the differences among DLMPs, the NUC of CB trading is fairly allocated to consumers based on the marginal contribution of the coalition, which is obtained from the Shapley value.
The remainder of this paper is organized as follows. Section II presents the energy trading model in energy transaction markets. The NUC model is discussed in Section III. The case study is presented in Section IV. Conclusions are given in Section V.
Different prosumer markets in the prosumer era are proposed in [

Fig. 1 Structural attributes of P2P and CB markets. (a) P2P market. (b) CB market.

Fig. 2 Transactive energy trading framework in a distribution system. (a) P2P market. (b) CB market.
The P2P market is more scalable due to the lack of a central coordinator required for the negotiation process. However, in a transactive energy market with many prosumers, the P2P transaction model generates a heavy computational burden because each prosumer must interact both directly and simultaneously. Compared with the P2P market, the CB market can form an independent community based on the preferences of prosumers, e.g., trading with other closest prosumers, which can effectively improve the operational efficiency of systems and reduce the computational burden. However, the CB market generally does not have good scalability.
We assume that the energy declarations of all participants for P2P and CB transactions meet the minimum entry requirements for market transactions to avoid wasting resources and all participants can benefit from the P2P and CB markets. For modeling convenience, each prosumer in the transactive energy market is defined as a producer with the index or a consumer with the index , where , , and and are the numbers of producers and consumers, respectively. The set of prosumers is and .
Energy is traded in a P2P market consisting of np producers and nc consumers. Bilateral transactions between producers and consumers can be described by the supply and demand matrix D given in (1). Each decision variable pij of matrix D represents the energy supply of producer i to consumer j or the energy demand of consumer j from producer i.
(1) |
The total demand of consumer j is expressed as:
(2) |
Similarly, the total supply of producer i is:
(3) |
The energy transaction in the P2P market is modeled as:
(4) |
(5) |
(6) |
(7) |
(8) |
(9) |
where and are the utility function of consumer j and cost function of producer i, respectively; is the set of decision variables; and are the lower and upper limits of energy supply for producer i, respectively; and and are the lower and upper limits of energy demand for consumer j, respectively. The model objective function in (4) is the social welfare maximization for all prosumers. If the energy demand of the consumer is inelastic, the objective function is equivalent to minimizing the cost of the producer. In this study, the cost function is set as a general linear function. Constraints (5) and (6) indicate the total energy sold by producer i and purchased by consumer j in the P2P market, respectively. Constraints (7) and (8) represent the limits of the energy that can be sold by producer i and purchased from consumer j, respectively. Constraint (9) sets the energy offered by producer i to consumer j as a non-negative value.
Inspired by the method of decomposition and coordination, producers and consumers in the CB market can choose to participate in energy transactions within or outside the community according to their preferences, and thus the scope of communities is divided. By matching energy transactions based on the transaction preferences of producers and consumers, the CM submits the matching information to the DSO, which completes the energy dispatch and clears the CB market. The CM can be considered a nonprofit local data-sharing center or a small-scale aggregator [
(10) |
(11) |
(12) |
(13) |
(14) |
(15) |
(16) |
(17) |
(18) |
where and are the sets of producers and consumers in community m, respectively; is the set of decision variables; and are the utility function of producer j and cost function of consumer i in community m, respectively; is the transaction function of CM in community m; and are the energy sold by producer i of community m to consumers in community m and other communities, respectively; and are the energy purchased by consumer j of community m from producers in community m and other communities, respectively; and and are the import and export energy of community m, respectively. The objective function of the CB market is the social welfare maximization in (10). Constraint (11) represents the energy offered by producer i of community m. Constraint (12) represents the energy demanded by consumer j of community m. Constraints (13) and (14) represent the energy transactions handled centrally by the CM among the communities. For each producer and consumer, we consider the energy balance in community m and CB market of (15) and (16). Constraints (17) and (18) represent the limits of the energy that can be sold by producer i and purchased from consumer j in community m, respectively.
For the CB model, the utility function of consumer j in community m is given by:
(19) |
where the constant is the satisfaction parameter of consumer j, which indicates the utilization value of the energy demand; and and are the favorable transaction costs for internal energy transactions and the additional penalty factor for external energy trading, respectively, which can effectively encourage energy transactions within the community.
The cost function of producer i in community m with energy trading is defined as:
(20) |
where is the generation cost of producer i in community m; and is the additional penalization for producers, which reflects the will of producer i to improve community autonomy in the context of the CB market.
In the negotiation process for CM, the focus of the CB market operation is to achieve a common agreement on prosumer preferences in matching energy inside and outside the community. We address the critical role of CM and define as the transaction function, which represents the cost for community m to conduct a common agreement with other communities. The transaction function of the CM in community m is given by [
(21) |
Prosumer transactions in the transactive energy market have self-interest properties. These include the cases in which prosumers in the P2P market share energy transactions with the aim of maximizing individual benefits, and prosumers in the CB market rely on transaction preferences to divide communities based on transactions to maximize individual satisfaction and enhance transaction value. We assume that prosumers who trade in the transactive energy market are independent and rational. As independent rational individuals, each prosumer expects to achieve a consensus through negotiation and seeks a balanced strategy to maximize their benefits; otherwise, a disagreement is the result. Reference [
(22) |
s.t.
(23) |
where K is the set of negotiators; and and are the benefits of negotiator k before and after the bargaining process, respectively. Constraint (23) indicates that the benefits of negotiator k in bargaining transactions are enhanced.
Previous studies have considered the application of NBS in market-clearing models of P2P and CB markets [
(24) |
(25) |
where is the transaction price for producer i and consumer j.
The formulation of our proposed energy trading problem based on NB in the P2P market is given as:
(26) |
s.t.
(27) |
(28) |
(29) |
(30) |
(31) |
The objective function (26) corresponds to the welfare improvement of producer i and consumer j. Constraint (27) summarizes the local energy-scheduling constraints for each prosumer in the P2P market. Constraints (28) and (29) denote the balance of energy payments and the balance of the quantity of energy transactions, respectively. Constraints (30) and (31) correspond to (23), which indicates that prosumers increase their benefits through bargaining transactions.
The NB model in (26)-(31) for P2P transactions is a nonconvex optimization problem that is difficult to solve directly. To address this issue, the solution of model in (26)-(31) is decomposed into two convex subproblems P1 and P2 as follows. The detailed proof is presented in [
(34) |
s.t.
(35) |
(36) |
where , , and are the optimal solutions for P1; and and are the disagreement points obtained from optimization model given in (4)-(9).
Similarly, in a CB market with bargaining transactions, the energy exchange payments for producer and consumer are given by (37) and (38), respectively. Note that the bargaining transactions between producers and consumers are handled centrally by CMs, and CMs in different communities can also trade the energy with each other.
(37) |
(38) |
where and are the transaction prices of producer i inside and outside community m through bargaining transactions, respectively; and and are the transaction prices of producer j inside and outside community m through bargaining transactions, respectively.
The mathematical model of the proposed energy-trading problem based on NB in the CB market is given as:
(39) |
s.t.
(40) |
(41) |
(42) |
(43) |
The objective function (39) corresponds to the increment in welfare for all bargaining transactions. Constraint (40) is the energy scheduling for prosumers in a CB market, and constraint (41) denotes the payment balance for bargaining transactions among prosumers. Constraints (42) and (43) indicate that the benefits to consumers and prosumers, respectively, are improved by bargaining.
We equivalently transform bargaining model (39)-(43) into two easily solvable convex subproblems, namely, social welfare maximization subproblem S1 and payment bargaining subproblem S2. The solution to the original NB model (39)-(43) can be determined by successively solving S1 and S2.
The optimal energy transactions for the P2P and CB markets presented in Section II have no guarantee that their results will satisfy the operational constraints of distribution network, which may otherwise cause line congestion, voltage instability, and power quality degradation. To alleviate these problems, the NUC that links the energy bargaining transactions for P2P and CB markets to the secure operating conditions of the distribution network must be analyzed. The following four subsections present the modeling mechanism of NUC and the modeling process when NUC is used to coordinate transactive energy market transactions with energy dispatch in the distribution network. We formulate the DLMPs as a general function of the dual multipliers of the AC OPF to capture resource scarcity, and we use the DLMPs to derive the NUC to incentivize or penalize the transaction behaviors of the prosumer. Considering the transferability of DLMPs, the NUC is calculated by coupling the AC OPF model and the energy transaction model of transactive energy markets and is allocated using the Shapley value for CB transactions. Note that the objective of the NUC model is not to minimize the NUC, which is based on post-processing.
Consider a radial distribution network (N, L), where node set N consists of substation node {0} and other nodes , and each node (indexed as n) in contains a parent node and a set of child nodes . The branch flow model of the radial distribution is given as [
(49) |
(50) |
(51) |
(52) |
(53) |
(54) |
(55) |
(56) |
(57) |
(58) |
(59) |
(60) |
where represents the set of decision variables; the constants and indicate the marginal cost of generators and the marginal benefit of consumers, respectively; and are the active power generation and consumption, respectively; and are the current and voltage magnitudes squared, respectively; and are the active and reactive power, respectively; and are the conductance and susceptance, respectively; is the complex power flow limit; and and are the resistance and reactance, respectively. The objective of the model in (49) is to maximize social welfare. Constraints (50) and (51) denote the active and reactive power balances of each node, respectively. Constraints (52) and (53) correspond to the complex power flow constraints of each line. Constraint (54) corresponds to a relaxation of the power flow equations and is derived in [
The dual multipliers , , , and on the left of the corresponding constraints constitute the DLMPs, which are derived using the Karush-Kuhn-Tucker (KKT) condition through the AC OPF model given in (61) [
(61) |
where is the nonlinear function given in [
DLMPs that offer incentive price signals are used to derive NUCs with bargaining transactions in different markets. In a P2P market, to reflect the effects of bargaining transactions on DLMPs, constraint (50) must be modified to realize an interconnection between the bargaining model with P2P transactions (26)-(31) and the branch flow model (49)-(60). The modified constraint is formulated as:
(62) |
The NUC model for the P2P market is given as:
(63) |
s.t.
(64) |
The optimal solutions and DLMPs are obtained by solving the NUC optimization model in (63) and (64). The NUC is paid by consumers according to the principle of “who benefits and who bears”, and the NUC of consumer j with bargaining transactions in the P2P market can be calculated by:
(65) |
where and are the DLMPs for producer i and consumer j, respectively.
Unlike the P2P market, the CB market not only simplifies the complexity of the energy transaction process but also reduces the computational burden of the NUC by introducing the role of the CM and fully considering the self-organization of the community. Correspondingly, constraint (50) must be adjusted so that the bargaining transaction model given in (39)-(43) for the CB market and the branch flow model (49)-(60) are coupled. The adjusted constraint is formulated as:
(66) |
The NUC model for the CB market is given by:
(67) |
s.t.
(68) |
The community is a self-organized structure with weakly centralized transactions, and all energy transactions are matched by the CM, which simplifies the calculation process of the NUC and reflects the advantage of the decomposition and coordination approach. The NUC of community m is determined based on the average energy selling and purchasing prices, which are given by:
(69) |
where and are the DLMPs for producer i and consumer j within community m, respectively; and are the numbers of consumers and producers within community m, respectively; and is the energy purchased by community m from community f. The first and second items of (69) represent the NUC generated by trading within community m and with other communities, respectively. Note that, in contrast to the NUC modeling mechanism of P2P transactions, the CB market considers the participation of third-party agents, i.e., CMs, in energy bargaining transactions, making it difficult to allocate the NUC directly to consumers. Therefore, a feasible allocation scheme must be designed for each consumer to ensure the NUC of community m obtained from DLMPs.
Because prosumers build communities based on transaction preferences and the NUC is allocated accordingly, the relationship between prosumers in the CB market is typically both cooperative and competitive. In coalitional games with self-interested characteristics, the worth of the coalition must be distributed among the players of the coalitional group using a fairness rule. The rational allocation of the NUC among community prosumers is a critical problem for CB market energy transactions. For a coalition composed of multiple participants, the Shapley value is used to design an allocation scheme that allocates coalition benefits based on the average marginal contribution of each participant. The Shapley value has been studied and widely applied in the fields of transmission network construction cost allocation [
In general, we denote as the contribution of coalition for any of the proposed coalition benefit allocation problems, where is derived from (69). We denote as the Shapley value, which is assigned to each game contribution represented by function , i.e., a vector of NUC {} for each consumer k of coalition N such that . Let be any non-empty subset of consumers from the set of N that may build a coalitional group, and the number of consumers in set Q is . The marginal contribution, i.e., NUC, of a consumer k with respect to the overall NUC savings obtained by a subset of Q is defined by the value . Then, the Shapley value assigned to each consumer k is given by (17). This sum is calculated over all possible subsets Q of N without containing consumer k.
(70) |
where and are obtained by (69), i.e., and .
The modified IEEE 15-bus and IEEE 123-bus distribution test systems are used to validate the effectiveness of the proposed NUC calculation model. The optimization problems of P2P and CB transaction involve the use of decentralized or distributed optimization techniques, e.g., the alternating direction method of multipliers, to guarantee the privacy of prosumers. However, because the distributed optimization techniques have the problem of slow convergence or non-convergence within limited iterations, the centralized optimization method with fast convergence is applied in this study to obtain global optimal results. The program is developed in MATLAB 2017b, where the commercial solver IPOPT is used to solve the optimization problems of NUC and to acquire the DLMPs.

Fig. 3 IEEE 15-bus distribution test system.
Prosumer | π ($/MWh) | γcom ($/MWh) | γimp ($/MWh) | γexp ($/MWh) |
---|---|---|---|---|
0 | 50 | |||
1 | 0.2000 | 1.0 | ||
2 | 10 | 0.1859 | 0.90 | |
3 | 10 | 0.9045 | 0.90 | |
4 | 0.2000 | 1.0 | ||
5 | 0.2000 | 1.0 | ||
6 | 0.2000 | 1.0 | ||
7 | 10 | 0.9045 | 0.95 | |
8 | 0.2000 | 1.1 | ||
9 | 0.2000 | 1.1 | ||
10 | 0.2000 | 1.1 | ||
11 | 0.2000 | 1.1 | ||
12 | 0.2000 | 1.0 | ||
13 | 0.2000 | 1.0 | ||
14 | 0.2000 | 1.0 |

Fig. 4 Energy transactions with NUCs in IEEE 15-bus distribution test system. (a) P2P market. (b) CB market.
Note that some consumers pay a negative NUC. For example, the NUC for consumer 4 is -$0.2036. This is because a bidirectional power flow exists in the active distribution network, and a consumer with a negative NUC can relieve the line congestion by purchasing the power from the producer. Thus, the electric power utility subsidizes the consumer.
Index of communities | Index of consumers | NUC ($) |
---|---|---|
1 | 1 | 4.6746 |
12 | 0.1029 | |
13 | 0.0240 | |
14 | 0.1766 | |
2 | 4 | 0.0004 |
5 | 0.0013 | |
6 | 0.0014 | |
3 | 8 | 0.0020 |
9 | 0.0023 | |
10 | 0.0026 | |
11 | 0.0015 |

Fig. 5 Comparison of line loading ratio and nodal voltage magnitude under P2P and CB markets. (a) Line loading ratio. (b) Nodal voltage magnitude.
In a pool-based distribution market, i.e., the proposed benchmark market, the objective is to maximize social welfare by collecting bids and offers from all market participants through the DSO and then clearing the market in a centralized manner. Simultaneously, specific operating constraints must be met to ensure the reliable operation of the power system. The centralized trading model can be modeled as model I in [
Compared with the P2P and CB markets, no line congestion occurs in the benchmark market, and the voltage profile of the nodes is relatively stable because all energy transactions are scheduled uniformly by the DSO. Furthermore, the line loading ratio and nodal voltage magnitude of P2P transactions are higher than those of CB transactions. Consumers in communities 2 and 3 complete the nearby energy trading, and the energy is not transmitted through lines 3 and 8, as shown in
The prosumer payments and NUC with or without NB are presented in
Market | Transaction behavior | Revenue of producer ($) | Payment of consumer ($) | NUC ($) |
---|---|---|---|---|
P2P | Without NB | 72.2987 | 84.5360 | 12.2373 |
With NB | 73.5621 | 83.7618 | 10.1997 | |
CB | Without NB | 75.8217 | 81.5949 | 5.7732 |
With NB | 76.1426 | 81.1322 | 4.9896 |

Fig. 6 IEEE 123-bus distribution test system.
Index of producers | Capacity (MW) | Incremental cost ($/MWh) |
---|---|---|
0 | 0.8 | 20 |
2 | 0.5 | 10 |
12 | 0.5 | 10 |
20 | 0.5 | 9 |
60 | 0.5 | 11 |
71 | 0.5 | 10 |
77 | 0.5 | 9 |
107 | 0.5 | 10 |
120 | 0.5 | 11 |

Fig. 7 Quantities of different energy transactions. (a) P2P market. (b) CB market.

Fig. 8 Comparison of line loading ratios of different energy transactions. (a) Line loading ratio under P2P and CB markets. (b) Line loading ratio of each community under CB market.

Fig. 9 Comparison of numerical distributions of nodal voltage magnitudes for different energy transactions. (a) Nodal voltage magnitude under P2P and CB markets. (b) Nodal voltage magnitude of each community under CB market.
The NUCs for P2P and CB transactions are $81.5976 and $34.9272, respectively, which represent the grid asset utilization of different transaction markets. The line loading ratio and the voltage magnitude profile for P2P transactions are higher than those for CB transactions. The NUCs for each community are listed in
Index of community | NUC ($) | Index of community | NUC ($) |
---|---|---|---|
1 | 4.7014 | 5 | 3.4853 |
2 | 4.6185 | 6 | 5.0821 |
3 | 3.1625 | 7 | 5.6079 |
4 | 5.3824 | 8 | 2.8871 |
In this study, we propose the methods to calculate the NUCs for P2P and CB markets with the aim of improving the revenue of network services provided by the electric power utility and effectively allocating the NUC to consumers in the prosumer era. In particular, the NB method is employed to formulate energy transaction models for the P2P and CB markets to capture the transaction behaviors of prosumers with self-interest characteristics. For both energy transaction markets, we use the DLMPs to coordinate the distribution system operation and energy trading and combine them with the Shapley value to allocate the NUC that consumers paid for the electric power utility. Case studies on modified IEEE 15-bus and 123-bus distribution test systems demonstrate the effectiveness of our proposed NUC calculation methods for P2P and CB transactions. The simulation results show that the line loading ratio and voltage magnitude fluctuation of CB transactions are lower than those of P2P transactions, indicating that the grid asset utilization and energy loss for prosumers with CB transactions are relatively low, and the NUC is less than the P2P transactions. In addition, the prosumers could improve their welfares by bargaining transactions and reduce the NUC for P2P and CB transactions.
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