In the context of modern energy systems, each prosumer is treated as an individual subject with the primary objective of minimizing the total cost while fulfilling the energy demand of customers. This total cost encompasses various components, including the operational costs of generation, conversion, and storage equipment, the cost of energy trading with external municipal networks, the cost of energy sharing with other prosumers, management fees paid to the coordinator (SO), costs for demand response, and the cost associated with carbon dioxide emissions. The objective function of each prosumer is expressed as Equation 1: C pro i = ∑ t ∑ k ∈ K 1 c o m , k i x k , t i + ∑ k ∈ K 2 c o m , k i x k , t i + ∑ k ∈ K 3 c o m , k i s c k , t i + s d k , t i + x p e , t i p t p e − x s e , t i p t s e + x n g , t i p t n g + x p h , t i p t p h + x e , t s i − x e , t i s p e , t sis + x h , t s i − x h , t i s p h , t sis + p t s o x e , t s i + x e , t i s + p t s o x h , t s i + x h , t i s + ∑ j ∈ J x j , d r u p , t i + x j , d r d o w n , t i p j , d r , t i + c co2 i x n g , t i δ n g + x p e . t i δ p e + x p h . t i δ p h (1) where C pro i represents the overall cost of prosumer i , c o m , k i denotes the operational cost of technology k , x k , t i is the energy generation from devices, s c k , t i and s d k , t i indicate the energy charging and discharging quantities of storage units, respectively. K 1 , K 2 , and K 3 represent the sets of generation, conversion, and storage technologies, respectively. x n g , t i , x p e , t i , x s e , t i , and x p h , t i denote the natural gas purchasing, electricity purchasing, electricity selling, and heat purchasing quantities by prosumer i , p t n g , p t p e , p t s e , p t p h represent the corresponding trading prices. x e , t s i , x e , t i s , x h , t s i , and x h , t i s are the internal electricity purchasing and selling quantities from or to other prosumers, p e , t sis and p h , t sis denote the corresponding energy sharing prices. p t ems is the unit management fee paid to the SO. x j , d r u p , t i , x j , d r d o w n , t i , and p j , d r , t i are the energy shifting in and out quantities and unit price for energy shifting, respectively. δ n g , δ p e , and δ p h are the emission factors for natural gas, utility grid electricity, and municipal network heat, respectively.

The electricity output from photovoltaic (PV) systems and wind power plants (WPPs) is modeled as Equations 2, 3: x e , p v , t i = η p v , P I , t i θ P V , t i ξ P V , t i S r P V , t i S r t i 1 + 0.005 × T t − 25 ∀ i ∈ I , t ∈ T (2) x e , w p p , t i = 0 0 ≤ v t ≤ v cin Q r i × v t − v cin v r − v cin v cin ≤ v t ≤ v r Q r i v r ≤ v t ≤ v c o 0 v c o ≤ v t ∀ i ∈ I , t ∈ T (3) where x e , p v , t i and x e , w p p , t i are the electricity generation of PV and WPP, η p v , P I , t i is the conversion efficiency, θ P V , t i represents panel number, and ξ P V , t i denotes rated output under standard test conditions, S r P V , t i and S r t i denote the standard and actual solar irradiance intensity. v t is the wind speed at period t, Q r i is the rated capacity of wind turbine, v cin , v r , and v c o are cut in speed, rated speed, and cut out speed, respectively.

The generation of electricity and heat from natural gas-fired CHP systems and boilers is modeled in Equations 4–10. For CHP systems, low operation load leads to the low electric generation efficiency. To avoid the low efficiency, a threshold for load ratio χ chp,t i is set. Moreover, the ramp rate for the power generation unit is considered. As an alternative source of heat, the natural gas-fired boilers are taken into account. x e , c h p , t i =   x ngchp,t i η chp,t e ∀ i ∈ I , t ∈ T (4) x h , c h p , t i = x ngchp,t i − x e , c h p , t i η rec ∀ i ∈ I , t ∈ T (5) x e , c h p , m i n i ≤ x e , c h p , t i ≤ x e , c h p , m a x i ∀ i ∈ I , t ∈ T (6) χ chp,t i α chp,t i x e , c h p , m a x i ≤ x e , c h p , t i ≤ α chp,t i x e , c h p , m a x i ∀ i ∈ I , t ∈ T (7) x e , c h p , r r , m i n i ≤ x e , c h p , t i − x e , c h p , t − 1 i ≤ x e , c h p , r r , m a x i ∀ i ∈ I , t ∈ T − , T + (8) x h , n g b o , t i = x ngbo,t i η ngbo,t i ∀ i ∈ I , t ∈ T (9) x h , n g b o , m i n i ≤ x h , n g b o , t i ≤ x h , n g b o , m a x i ∀ i ∈ I , t ∈ T (10) where x ngchp,t i is the natural gas consumption, x e , c h p , t i and x h , c h p , t i are the electricity and heat generated from the CHP system, η rec and η chp,t e are the efficiencies of the heat recovery system and the power generation unit, x e , c h p , m i n i and x e , c h p , m a x i represent the minimum and maximum electricity output, α chp,t i is a binary variable indicating the on-off state. x e , c h p , r r , m i n i and x e , c h p , r r , m a x denote the ramp rate limit. x h , n g b o , t i represents the heat generation of boilers, x ngbo,t i is the natural gas consumption, η ngbo,t i is the thermal efficiency, x h , n g b o , m i n i and x h , n g b o , m a x i denote the minimum and maximum heat output.

The power-to-heat technology converts low-grade heat into high-grade heat, modeled as Equations 11,12: x h , P 2 H , t i = x e , P 2 H , t i η P2H,t i ∀ i ∈ I , t ∈ T (11) x h , P 2 H , m i n i ≤ x h , P 2 H , t i ≤ x h , P 2 H , max i ∀ i ∈ I , t ∈ T (12) where x h , P 2 H , t i and x e , P 2 H , t i are the heat generation and electricity consumption, e t a P2H,t i is the conversion efficiency.

The charging and discharging processes for electrical and heat storage are given by Equations 13–17: z k , t i = z k , t − 1 i + s c k , t i η k , s c , t i − s d k , t i / η k , s d , t i ∀ i ∈ I , k = e s , h s , t ∈ T − , T + (13) 0 ≤ s c k , t i ≤ ε k , s c , t i s c k , m a x i ∀ i ∈ I , k = e s , h s , t ∈ T − , T + (14) 0 ≤ s d k , t i ≤ ε k , s d , t i s d k , m a x i ∀ i ∈ I , k = e s , h s , t ∈ T − , T + (15) ε k , s c , t i + ε k , s d , t i ≤ 1 ∀ i ∈ I , k = e s , h s , t ∈ T − , T + (16) 1 − d e k , t i z k , m a x i ≤ z k , t i ≤ z k , m a x i ∀ i ∈ I , k = e s , h s , t ∈ T (17) where z k , t i is the energy stored at time t, s c k , t i and s d k , t i are the charging and discharging quantities, η k , s c , t i and η k , s d , t i are the charging and discharging efficiencies, s c k , m a x i and s d k , m a x i denote the rated charging and discharging rates, ε k , s c , t i and ε k , s d , t i are binary variables indicating the charging and discharging states, z k , m a x i is the storage capacity, d e k , t i represents the depth of discharge.

The demand response program adjusts energy consumption to align demand with supply. Load shifting-based incentive demand response is considered, modeled as Equations 18–21: 0 ≤ x j , d r u p , t i ≤ β j , u p , t i d j , t i α j , u p , t i ∀ i ∈ I , t ∈ T , j ∈ J (18) 0 ≤ x j , d r d o w n , t i ≤ β j , d o w n , t i d j , t i α j , d o w n , t i ∀ i ∈ I , t ∈ T , j ∈ J (19) α j , u p , t i + α j , d o w n , t i ≤ 1 ∀ i ∈ I , t ∈ T , j ∈ J (20) ∑ t x j , d r u p , t i = ∑ t x j , d r d o w n , t i ∀ i ∈ I , j ∈ J (21) where d j , t i represents the electricity and heat demand, j is the type of energy, β j , u p , t i and β j , d o w n , t i are the maximum percentages of energy shifting up and down, α j , u p , t i and α j , d o w n , t i are binary variables.

The electricity, heat, and natural gas balances for prosumer i are constrained by Equations 22–24: x e , p v , t i + x e , w p p , t i + x e , n g c h p , t i + x e , t s i + x e , p e , t i + s d e s , t i + x e , d r d o w n , t i ≥ d e , t i + x e , t i s + x e , s e , t i + x e , p 2 h , t i + s c e s , t i + x e , d r u p , t i ∀ i ∈ I , t ∈ T (22) x h , n g c h p , t i + x h , n g b o , t i + x h , t s i + x h , p h , t i + x h , p 2 h , t i + s d h s , t i + x h , d r d o w n , t i ≥ d h , t i + x h , t i s + s c h s , t i + x h , d r u p , t i ∀ i ∈ I , t ∈ T (23) x ngchp,t i + x ngbo,t i = x n g , t i ∀ i ∈ I , t ∈ T (24)

In the multi-energy sharing framework, the SO plays a pivotal role in orchestrating the schedules of each prosumer, facilitating energy sharing among prosumers, and maintaining the balance of supply and demand within the district. To efficiently manage these tasks, the SO imposes a management fee, which is predetermined prior to participation in the energy-sharing program. The objective function of the SO can be expressed as Equation 25: C s o = − ∑ t ∑ i ∑ j p t s o x j , t s i + x j , t i s (25) where C s o represents the cost incurred by the SO, with a negative value indicating a benefit. x j , t s i and x j , t i s denote the energy sharing quantities for prosumer i in terms of purchasing and selling, respectively. During the internal sharing process, each prosumer declares its intended sharing quantity without knowledge of other prosumers’ strategies. Consequently, the SO must ensure that the internal supply and demand are balanced. This balance can be calculated as Equation 26: ∑ i x j , t s i = ∑ i x j , t i s ∀ j ∈ J , t ∈ T (26)

To maximize the social welfare of all stakeholders, the resources of all sharing participants should be effectively coordinated. Typically, these coordinated prosumers represent different interest groups. Without a SO, prosumers can only trade with external energy entities and cannot share energy with their neighbors. Therefore, we propose a centralized coordination scheme, introducing an independent system operator (ISO) as the supervisor. The ISO will integrate all objectives and constraints into a centralized optimization problem. Under contracts signed by the sharing participants, this centralized scheme can leverage all available resources to maximize the social welfare of the entire system. However, the centralized scheme requires all participants to disclose accurate information to the ISO, making it suitable for scenarios where all participants are rational and willing to cooperate. The ISO optimizes dispatch strategies to maximize profit based on the provided supply and demand information, and the cooperative surplus is then allocated according to a predefined payment rule.

The multi-energy sharing problem aims to minimize the total cost for all stakeholders. The coordination model for all participants is defined in Equation 27: C cen = min ∑ i C pro i + C s o (27) E q s . 2 − 24 , E q . 26 ∀ i ∈ I (28)

Equation 28 encompasses all the constraints for prosumers and the SO, detailed in Section 3.1 and Section 3.2.

The multi-energy sharing framework generates a cooperative surplus, which must be allocated to each participant. Let N = 1,2...,i,...,n represent the set of participants, with S ∈ N as a subset. The payoff vector y = { x 1 , x 2 , … , x n } corresponds to the allocation schemes, and f denotes the profit for different subsets.

A core solution in cooperative games must satisfy group, subgroup, and individual rationality. Hence, the following constraints are considered: ∑ i = 1 n y i = f N (29) ∑ i = 1 s y i ≥ f S ∀ S ∈ N (30) y i ≥ f i ∀ i ∈ N (31)

Equation 29 guarantees that the aggregate benefit for the coalition is equivalent to the sum of benefits accrued by each individual member. The benefit for any subset of participants should not exceed the allocation for each member within that subset, as stipulated by Equation 30. Equation 31 ensures that members derive greater benefits through collaboration than they would by operating independently. Satisfying these core conditions is crucial for the efficacy and fairness of the benefit distribution mechanism.

A fair and judicious benefit distribution mechanism is essential for sustaining a robust energy sharing alliance. In this study, three cost allocation schemes will be taken into account: the Shapley value, the Nucleolus, and the N-H solution. In the following, a short description of the three main existing approaches is provided.

The Shapley value, a widely recognized approach, effectively captures the average marginal impact and significance of participants within the coalition (Shapley, 1953; Jin et al., 2018). In the context of multi-energy sharing, a prosumer’s marginal contribution, contingent upon their involvement in the coalition, serves as the basis for estimating their individual costs or benefits. For an n -participant sharing scenario, the Shapley value for prosumer i is mathematically defined as: ψ i f = ∑ s ∈ N n − s ! s − 1 ! n ! f S − f S − i ∀ i = 1,2 , … , n (32) where f S is the profit of coalition S , and f S − i is the profit of coalition S except individual i .

While benefit distribution strategies derived from the three distribution mechanisms are encompassed within the core, certain participants may perceive these allocations as inequitable. Consequently, it is imperative to assess the fairness of the distribution mechanism. To assess the fairness of the distribution program from multiple perspectives, we introduce Shapley-Shubik Power index, Jain’s fairness index, and Gini coefficient in assessing the fairness of the distribution scheme (Wu et al., 2017; Soares et al., 2024).

For the Shapley-Shubik Power index, the degree of equity in the allocation outcomes is inversely proportional to the value of the Fairness Index F I α . In other words, a lower Fairness Index value indicates a more equitable distribution, and vice versa. The index can be mathematically expressed as: F I α = δ α α ̄ (35) α i = y i − f j ∑ i y i − f j ∀ i ∈ I (36) where 0 ≤ F I α ≤ 1 , α i represents the Shapley-Shubik power index with ∑ i α i = 1 , δ α denotes the standard variance, and α ̄ represents the average value.

Jain’s fairness index (JFI) is also one of the most widely used criterion to measure fairness. JFI is calculated by dividing the square of the total sum of individual gains by the sum of the squares of individual gains, then multiplying by the number of users. In resource allocation contexts, “individual gains” typically refer to the amount of resources or benefits each user receives. The index ranges from 1 N to 1 (where N is the number of participants), with values closer to one indicating a more equal distribution of gains among participants. By using JFI, it is possible to ensure that all users receive a fair portion of shared resources, preventing any single user from dominating the distribution. The formula for Jain’s fairness index is given by: J F I = ∑ i = 1 N E B i 2 N ∑ i = 1 N E B i 2 (37) where E B i represents the economic benefits of each user i participating in energy sharing, and is calculated by the difference with costs with C e i and without C r e f i participation in shared economy.

Another widely used fairness index is the F fairness index, which evaluates the distribution of gains or cost from a particular method by comparing it to those obtained using Shapley Value distributions. The F index is calculated by measuring the distance between a normalized billing vector from any trading mechanism and the normalized billing vector produced by the Shapley Value. This index operates under the assumption that the Shapley Value distribution is optimal, with lower values indicating that other distribution methods closely resemble the Shapley Value distribution. The formula for the F index is given by: F = ∑ i = 1 N B i ∑ i = 1 N B i − S i ∗ ∑ i = 1 N S i ∗ (38) where B i is the bill or income of prosumer i when using other trading mechanisms and S i ∗ is the bill or income of prosumer i when using the Shapley Value.

The decision-making process for multi-energy sharing and cost allocation among sharing participants is illustrated in Figure 2. To solve the optimization problem presented in our multi-energy sharing framework, we employ a two-stage approach:

1. Social Welfare Maximization: We use a Mixed Integer Linear Programming (MILP) solver to maximize the social welfare function (Equations (27) and (28)) subject to the constraints outlined in sections 3.1 and 3.2. Specifically, we utilize the Matlab and Gurobi optimizer due to its efficiency in handling large-scale MILP problems.

This two-stage approach allows us to efficiently solve the complex optimization problem while ensuring a fair distribution of benefits among participants.

To assess the efficacy of the proposed multi-energy sharing model, we examined the strategic decision-making processes of residential, office, and commercial prosumers under different ownership structures in Dalian, China. Situated in a temperate climate zone, Dalian’s electricity and heating demand profiles are illustrated in Figure 3 (Li, 2017). The study encompasses two residential clusters, one office cluster, and one commercial cluster, with respective floor areas of 100,000, 50,000, and 50,000 square meters. Table 1 delineates the equipment capacities for each prosumer.

Table 2 outlines the time-of-use (TOU) tariffs for electricity and heat supplied by municipal energy entities. The electricity feed-in tariff is fixed at $0.04/kWh, while natural gas is priced at $0.032/kWh. Operation and maintenance costs are derived from (Li and Zhang, 2021b). The combined heat and power generation efficiency is set at 0.38. Storage systems operate with 0.95 charging and discharging efficiencies, power-to-heat systems achieve an efficiency of 3, and gas boilers function at 0.85 efficiency (Nielsen et al., 2016). The model incorporates demand response strategies, including both electrical and thermal load shifting. Buildings are equipped with various flexible loads, such as electric vehicles, air conditioning systems, and household appliances (e.g., washing machines, dryers, dishwashers). Thermal flexible loads encompass heating for air conditioning and domestic hot water systems. The maximum energy demand shift is capped at 20% (Alipour et al., 2017). Emission factors for externally sourced electricity, heat, and natural gas are 0.95 kg/kWh, 0.43 kg/kWh, and 0.18 kg/kWh, respectively (Hou et al., 2021; Li and Yu, 2020). To evaluate the environmental impact of energy sharing, a carbon tax of $15.27/t is applied (Sun et al., 2020).

Multiple scenarios are devised to investigate the influence of building types and sharing strategies on the economic and environmental performance of the systems, as detailed in Table 3.

Case 1 serves as a baseline scenario where all prosumers engage in electricity and heat sharing. The load profile, depicted in Figure 3, comprises two residential clusters (R), one office cluster (O), and one commercial cluster (C), each spanning 50,000 square meters, totaling 200,000 square meters. Case 2, the no-sharing scenario, simulates a real-world situation where each prosumer operates autonomously. A comparative analysis of these two cases in Section 4.2 elucidates the economic and environmental advantages of energy sharing. To further explore the impact of diverse building types on energy sharing performance, Cases 3.1 and 3.2 are established, maintaining a constant total floor area but varying the composition of residential, office, and commercial clusters. This analysis is elaborated in Section 4.3. These scenarios facilitate a comprehensive examination of how building types and sharing strategies influence both economic and environmental outcomes, offering valuable insights into the benefits of multi-energy sharing models in urban contexts.

Figure 4 illustrates the financial outcomes for participants in Case 1 and Case 2. Case one represents a scenario where electricity and heat are shared among prosumers, while Case 2 simulates a real-world situation where all participants operate independently. A comparative analysis reveals substantial economic disparities between the two scenarios. For each entity (SO, Pro1, Pro2, and Pro3), expenses in Case 1 are consistently lower than in Case 2. In the following two subsections, the Shapley value is used to distribute the cooperative surplus. The other allocation methods will be compared in Section 4.4. The multi-energy sharing model in Case 1 results in a total cost of $8,276.45 for prosumers, which is 13.19% lower than the $9,533.97 in Case 2. Notably, Pro1 achieves the highest cost reduction (12%) through energy sharing, while Pro2 and Pro3 realize savings of 8% and 4%, respectively. The System Operator (SO) also derives financial benefits from the sharing arrangement. This cost reduction primarily stems from increased interactions among internal prosumers and decreased reliance on external energy entities, which typically impose higher time-of-use prices and offer lower feed-in tariff rates. These findings demonstrate that the multi-energy sharing framework effectively integrates local resources, minimizes overall system costs, and enhances benefits for all stakeholders, particularly Pro1.

Figure 5 presents a comparison of carbon emissions between Case 1 and Case 2 for different prosumers (Pro1, Pro2, and Pro3), including the percentage change in emissions for Case 2 relative to Case 1. Overall, total emissions in Case 1 are 16% lower than in Case 2. For Prosumer 1, emissions in Case 1 decreased dramatically by 82%, from 27.55 tons to 4.85 tons, indicating that individual operation was less efficient for Pro1, or that increased generation and grid transactions in Case 2 led to higher emissions. Conversely, emissions for Prosumer two and Prosumer three increased by 61% and 10%, respectively. This increase is attributed to Pro2 and Pro3 functioning as energy sellers during most periods. Although emissions for Pro2 and Pro3 rise, their costs decrease, and the total emissions of the entire system are reduced.

Figure 6 illustrates the energy sharing strategies of prosumers in Case 1, detailing both electricity and heat sharing patterns. Positive values indicate energy purchasing, while negative values represent energy selling in the sharing market. The electricity sharing strategies in Figure 6A reveal distinct patterns among the three prosumers. Pro1 predominantly acts as a net consumer of electricity, showing positive values for most of the day. In contrast, Pro2 and Pro3 alternate between being net consumers and suppliers at different times.

The heat sharing strategies in Figure 6B exhibit more pronounced fluctuations and larger magnitudes compared to electricity sharing. This suggests that thermal energy sharing plays a crucial role in the community’s energy balance. Pro1 consistently acts as a major heat consumer during both nighttime and daytime hours. Pro2 and Pro3 display complementary behaviors in heat sharing: when one is a net supplier, the other tends to be a net consumer. This complementarity indicates effective load balancing and resource utilization within the community.

The results demonstrate that energy sharing significantly enhances interactions among internal prosumers. By enabling prosumers to exchange energy based on their individual surpluses and deficits, the community improves both economic and environmental performance, as evidenced in Figures 4, 5. Energy sharing also achieves better overall energy balance, potentially reducing dependence on external grids and improving self-sufficiency. The complementary roles of different prosumers in both electricity and heat sharing contribute to a more efficient and balanced local energy system, underscoring the potential benefits of collaborative energy management strategies.

This section maintains a constant total floor area of 200,000 square meters while analyzing cost-saving and emission reduction ratios by adjusting the proportions of different building types. This approach allows us to examine how various building compositions influence energy sharing. As outlined in Section 4.1 and Table 3, we establish Case 3.1 and Case 3.2 based on Case 1 by reducing residential floor area and increasing office and commercial building cluster areas, respectively.

Figure 7 illustrates the cost-saving and emission reduction ratios when comparing individual operation scenarios under different load profiles in Cases 1, 3.1, and 3.2. Figure 7A depicts cost-saving ratios for different prosumers across cases, clearly showing that adjusting building type proportions significantly impacts each prosumer’s cost-saving ratio. Figure 7B presents overall cost-saving and emission reduction ratios for each case. Case 3.2 achieves the highest cost-saving ratio (19.44%), followed by Case 1 (13.19%) and Case 3.1 (7.26%). The increasing cost-saving ratio from Case 3.1 to Case 1 and Case 3.2 indicates that modifying the proportions of residential, office, and commercial buildings can substantially enhance cost efficiency. Similarly, emission reduction ratios are highest in Case 3.2 (23.35%), followed by Case 1 (16.36%) and Case 3.1 (10.83%). The parallel trends in emission reduction and cost-saving ratios compared to individual operation scenarios suggest that optimizing building type proportions can significantly improve both economic and environmental performance. Among the three cases, Case 3.2 demonstrates superior performance in both cost savings and emission reduction, indicating a synergistic effect of optimizing building type proportions for both economic and environmental benefits. These findings support the strategic adjustment of building type proportions as an effective means to optimize energy sharing, reduce costs, and minimize carbon emissions.

Figure 8 comprises four sub-figures (a, b, c, and d) illustrating various aspects of energy sharing among prosumers in Cases 1, 3.1, and 3.2. The electricity sharing patterns, heat sharing patterns, total energy sharing quantities, and total prosumer costs are analyzed to provide insights into how building type composition affects energy sharing strategies. Electricity sharing patterns in Figure 8A vary significantly across the three cases and throughout the day. Case 3.2 exhibits the highest peaks in electricity sharing, particularly during hours 9–21. Cases 3.1 and one display lower overall sharing quantities, with Case 3.1 showing the least variation. Heat sharing demonstrates distinct patterns compared to electricity sharing, as shown in Figure 8B. Case 1 exhibits the highest heat sharing quantities, with significant peaks during early morning hours (1–7) and evening hours (21–24). Cases 3.1 and 3.2 display lower heat sharing quantities, with Case 3.2 showing the least variation. These differences likely stem from the varying building type compositions in each case.

As illustrated in Figure 8C, total energy sharing quantities are substantially higher for heat sharing than for electricity sharing across all cases. Moreover, Case 1 shows the highest total heat sharing, while Case 3.2 shows the highest electricity sharing. The predominance of heat sharing over electricity sharing suggests that multi-energy management offers greater potential for prosumers. The variation across cases further emphasizes how building type composition significantly influences energy sharing opportunities. The cost comparison in Figure 8D reveals that energy sharing leads to substantial cost reductions compared to individual operation. By comparing the contribution of cost reduction from electricity sharing and heat sharing, it is found that electricity sharing alone accounts for a 9% cost reduction, indicating a more substantial impact on cost savings than heat sharing. This is the primary reason for the high cost-saving ratio in Case 3.2 shown in Figure 7B.

Table 4 presents a comparative fairness evaluation of three distribution schemes—Shapley value, Nucleolus, and N-H solution—within a multi-energy sharing framework in Case 1. The participants, denoted as SO, Pro1, Pro2, and Pro3, represent various agents involved in the energy sharing process. The fairness evaluation employs three indices: Shapley-Shubik power based Fairness Index (FI), Jain’s Fairness Index (JFI), and F Fairness Index (FFI).

The Shapley value method yields a balanced distribution among participants, as evidenced by its moderate fairness indices: a Shapley-Shubik FI of 0.495 and a Jain’s FI of 0.803. The F FI of 0 suggests that this scheme most closely approximates an ideal fair distribution according to the Shapley value criterion. The Nucleolus scheme exhibits the highest Shapley-Shubik FI at 0.854, indicating a lower degree of fairness, while its Jain’s FI is lower at 0.371. The F FI of 0.127 implies that this scheme deviates somewhat from the ideal Shapley distribution. The N-H solution demonstrates the lowest Shapley-Shubik FI at 0.301, signifying a fairer distribution compared to the other methods. It also boasts the highest Jain’s FI at 0.917, reflecting a more equitable distribution of resources among participants. Its F FI of 0.039 indicates that this distribution closely aligns with the Shapley value distribution.

The Shapley-Shubik power based FI offers insights into the fairness of distribution based on the concept of power in coalitions, with the N-H solution performing best in this regard. However, Jain’s FI, which measures the equality of distribution, ranks the N-H solution highest. This suggests that while the Nucleolus may prioritize fairness in coalition excess reduction, it may not necessarily ensure equal distribution among participants.

The selection of a distribution scheme depends on the specific objectives. If minimizing excess and ensuring coalition fairness are paramount, the Nucleolus scheme is most appropriate. Conversely, if equal distribution among participants is the primary concern, the N-H solution offers a more balanced approach. The Shapley value strikes a middle ground, providing a moderate balance between fairness and equality. These findings underscore the trade-offs between different fairness objectives and highlight the significant impact of the distribution mechanism on participants’ perceived fairness.

The results of our study highlight several important potential impacts. First, the demonstrated cost savings for participants could serve as a strong incentive for the broader adoption of distributed energy resources and energy-sharing schemes. Additionally, the observed reductions in carbon dioxide emissions indicate that this multi-energy sharing framework could play a valuable role in supporting decarbonization efforts in the energy sector. Furthermore, the integration of thermal energy management into the sharing framework has the potential to optimize energy efficiency and cost-effectiveness, especially in diverse building environments that require both electricity and heat management.

To achieve multi-energy sharing, several challenges must be addressed, along with potential solutions. (1) Infrastructure upgrades. Implementing the framework requires advanced metering, communication, and control systems to enable real-time energy sharing and coordination. Upgrading the existing infrastructure, especially in older buildings and communities, may pose significant financial and logistical hurdles. A phased approach could be considered, starting with pilot projects in newer buildings or communities that already have compatible infrastructure, and then gradually expanding the framework as technologies become more accessible and cost-effective. (2) Prosumers’ engagement. Another challenge is that prosumers may be reluctant to participate due to the perceived complexity of the system or concerns about losing control over their energy usage. To address this, we propose the development of user-friendly interfaces and automated systems that simplify participation. In addition, educating users about the long-term benefits of energy sharing, including potential cost savings and environmental impacts, could help increase participation. Providing clear, transparent information on how the system operates will also build trust and encourage engagement.

However, we also acknowledge certain limitations. First, this study does not account for external factors such as energy policy and market competition, both of which can have significant impacts on the actual implementation of energy-sharing models. In future research, we plan to explore how these external factors affect the economic and environmental outcomes of energy sharing, and how the framework can be adapted to address these influences. Second, while this study focuses on economic and environmental benefits, we recognize the importance of incorporating risk-sharing mechanisms in future iterations of the framework. As energy systems are increasingly exposed to extreme weather events and other hazards, resilience will become a critical factor. Future research will aim to extend our current framework by integrating risk-sharing strategies among prosumers to ensure equitable distribution of both benefits and risks. This will help develop a more robust and resilient energy-sharing model suitable for a variety of future applications.