Optimal Sharing and Fair Cost Allocation of Community Energy Storage

I. INTRODUCTION

• Customer-level storage deployment is impeded by high capital costs and long payback periods, despite its potential to help consumers reduce bills through renewable synchronization and grid-price response.The passage describes this as a barrier to customer-level deployment.Source: §I · p. 1
• Sharing storage can increase utilization through complementary building profiles, avoid duplicate installation costs, capture economies of scale, and relieve space concerns.These benefits motivate a comprehensive business model for cooperative storage sharing.Source: §I · p. 1
• Sizing and operation are interdependent, while rule-based cost allocation does not provide a notion of fairness.The paper identifies fair allocation as important because users are self-interested and independent stakeholders.Source: §I · pp. 1–2

A. Main contributions

• The coalition-game formulation integrates optimal energy-storage sizing, operation, and cost allocation for multiple buildings.Buildings seek economic benefits from local renewable integration and grid-price arbitrage.Source: A. Main contributions · p. 2
• Less than 1% of the characteristic function is required for cost allocation with 20 buildings, while case studies report fairness and computational efficiency over Shapley and proportional methods.The shared-storage model also improves building-wise and community-wise economic benefits over individual storage, with overall value about 1.83 times higher.Source: A. Main contributions · p. 2

II. RELATED WORKS

• Earlier studies commonly addressed cooperative storage sizing and operation separately, despite their interdependence, and often used rule-based cost allocation without a notion of fairness.Related work also includes a two-period grid-arbitrage setting and third-party-led sharing models with different objectives.Source: §II · p. 2

A. The Configuration of ES Sharing

• The model has buildings with on-site wind and solar that cooperatively invest in and share community storage for renewable utilization and grid-price arbitrage.Buildings may charge from local renewable generation or grid purchases, discharge to meet demand, and sell energy back to the grid.Source: A. The Configuration of ES Sharing · pp. 2–3
• The analysis projects long-term sizing onto a daily basis and uses discretized time.The authors cite computation burden as the reason for this projection.Source: A. The Configuration of ES Sharing · p. 3

B. Main Assumptions

• The assumptions exclude energy trading between buildings through storage and restrict demand to non-elastic demand, with demand response not discussed.The authors state that the model could be extended to incorporate energy sharing.Source: B. Main Assumptions · p. 3

C. Coalition Game Formation

• A coalition is any subset of buildings that cooperatively shares storage, and its value is the total cost including electricity bills and storage capital cost.The characteristic function assigns a value to each coalition, and the number of possible coalitions grows as O(2^N).Source: C. Coalition Game Formation · p. 3
• The two-stage stochastic optimization couples first-stage storage sizing with second-stage operating cost across representative scenarios.The first stage includes amortized storage capital cost and weighted operating cost; the second stage optimizes operation subject to capacity and scenario realization.Source: C. Coalition Game Formation · p. 3
• The operating formulation enforces charging and discharging limits, stored-energy tracking, supply-demand balance, peak demand, and non-simultaneous charging and discharging.With complementary constraints relaxed under the stated efficiency and grid-price conditions, the two-stage min-min problem can be converted to a single-stage convex problem solvable by commercial solvers.Source: C. Coalition Game Formation · p. 4

D. ES Coalition Game Properties

• The energy-storage coalition game is subadditive, so the value of a union of disjoint coalitions is no greater than the sum of their separate values.The authors infer that merging disjoint groups will not make them worse off and that buildings are therefore inclined to form a grand coalition.Source: D. ES Coalition Game Properties · p. 4

IV. COST ALLOCATION BASED ON NUCLEOLUS

• Fair ex-post allocation matters because building-wise gains depend on participants being satisfied and having no motivation to leave or disrupt cooperation.Source: §IV · p. 4
• Core and Shapley approaches also face characteristic-function computation challenges; core additionally raises existence and uniqueness concerns.Source: §IV · p. 4
• The nucleolus pursues fairness by minimizing player dissatisfaction and is nonempty and unique, although its computation remains challenging.Source: §IV · p. 4

A. Definitions

• A cost allocation assigns each player a share of the coalition game's cost; an imputation is efficient and individually rational for the grand coalition.Source: A. Definitions · p. 4
• The core consists of imputations for which no subset of players has an incentive to leave the grand coalition.Source: A. Definitions · p. 5
• For cost-minimization games, coalition excess represents dissatisfaction, and the nucleolus is the imputation with the lexicographically minimal ordered excess vector.Source: A. Definitions · p. 5

B. An algorithm to find the nucleolus

• For 20 buildings, computing the full characteristic function would require solving the stochastic optimization problem more than 10^7 times.Source: B. An algorithm to find the nucleolus · p. 5
• Constraint generation exploits the sparse structure of the lexicographic LPs: at most N constraints bind, so identifying them can limit the coalition values needed.The problem has N decision variables and 2^N − 1 linear constraints.Source: B. An algorithm to find the nucleolus · p. 5
• The algorithm first computes the least core, then solves successive lexicographic minimization problems until reaching the unique nucleolus.Source: B. An algorithm to find the nucleolus · p. 5

1) Computing least core:

• The least core minimizes the maximum coalition excess; constraint generation repeatedly adds the most violated coalition until none remains.Source: 1) Computing least core · p. 5
• The least core may contain multiple allocations, so further episodes minimize subsequent entries of the ordered excess vector within the previously fixed optima.The example distinguishes allocations that tie on maximum excess by comparing their next excess values.Source: 1) Computing least core · p. 5

2) Lexicographically optimization:

• At episode k, the method minimizes the k-th ordered excess while fixing the optima from earlier episodes.Source: 2) Lexicographically optimization · p. 6

3) Constraint generation:

• The method searches for the most dissatisfied coalition not yet included in the current constraint set, representing player membership with binary variables.Source: 3) Constraint generation · p. 6
• The coalition-search formulation excludes empty, grand, and already included coalitions, and accounts for the stochastic optimization defining coalition value.Source: 3) Constraint generation · p. 6
• For buildings in the identified coalition, grid procurement must satisfy demand; for buildings outside it, the load-balance constraints are relaxed.Source: 3) Constraint generation · p. 6
• The search and allocation update alternate until no violated coalition remains, signaling that the current least-core or lexicographic optimization is approached.Source: 3) Constraint generation · p. 6
• The algorithm has nested loops: an inner constraint-generation loop and an outer loop over lexicographic episodes, with binding constraints carried forward.Source: 3) Constraint generation · pp. 6–7

V. CASE STUDY

• The case study evaluates the nucleolus-based cost allocation for fairness and computational efficiency.Source: §V · p. 7
• It then examines the economic benefits of energy-storage sharing relative to individual energy storage.Source: §V · p. 7
• The reported numerical analysis covers both cost allocation and the shared-versus-individual storage comparison.Source: §V · p. 7

A. Simulation setup

• The setup uses 10 representative scenarios from one year of real building demand and wind and solar generation profiles.It models a daily cycle in 24 one-hour slots, with charging and discharging efficiencies of 0.9, a 6% annual interest rate, and a 10-year storage lifetime.Source: A. Simulation setup · p. 7

B. Fairness and computational efficiency

• The comparison tests five coalition sizes, N={3, 5, 8, 10, 20}, against proportional and Shapley allocation methods.The proportional method is empirical and easy to compute; Shapley is more sophisticated but computationally intensive.Source: B. Fairness and computational efficiency · p. 7
• Fairness is assessed through minimum coalition excess: allocations are fair when DSAT ≤0, with more negative DSAT preferred.The study assumes building players are profit-oriented and have no other preferences.Source: B. Fairness and computational efficiency · pp. 7–8
• At N=8 and N=10, only nucleolus and Shapley ensure fairness; proportional allocation may fail.At N=3 and N=5, all three methods ensure fairness.Source: B. Fairness and computational efficiency · p. 8
• At N=20, nucleolus requires less than 1% of characteristic-function values, versus the full characteristic function for Shapley.At N=10, nucleolus requires 2.54% (26/1023); its computation cost is slightly higher than proportional but significantly lower than Shapley.Source: B. Fairness and computational efficiency · p. 8

C. Economic benefits of ES sharing

• For N=5, overall cost falls 9.0% with CES versus 2.5% with IES; CES + Share cuts B3's cost by 59.5%.For B3, CES reduces cost by 16.7%, compared with 4.9% for IES.Source: C. Economic benefits of ES sharing · p. 9, Figure 2 · p. 9
• Larger sharing coalitions increase cost reductions for buildings common to N=5 and N=10, though they may require a more powerful central coordinator.B1–B5 all gain higher cost reduction at N=10, except for a marginal decrease for B1 attributed to computation accuracy.Source: C. Economic benefits of ES sharing · p. 9
• Community VoS with CES is about 1.8 times that with IES, and CES + Share further increases community VoS over CES.Individual-building VoS is higher with CES than IES; CES + Share community VoS is reported because individual capital costs cannot be separated from the allocation.Source: C. Economic benefits of ES sharing · p. 9, Figure 3 · p. 9

VI. CONCLUSION

• The study integrates storage sizing, operation, and fair ex-post cost allocation through a coalition game using the nucleolus.Constraint generation approaches the unique nucleolus using the sparse problem structure.Source: §VI · pp. 9–10
• The model does not account for storage-capacity degradation from charging and discharging cycles because quantifying it is complex.The conclusion identifies incorporating a convex rainflow cycle-based model as future work.Source: §VI · p. 10

• The proof constructs alternative feasible operations that lower cost, contradicting the assumption of simultaneous charging and discharging.The argument uses charging and discharging efficiencies below 1 while preserving the energy state.Source: Appendix A · p. 10

• Linearity of the objective and constraints makes the sum of optimal solutions for disjoint coalitions feasible for their union.Source: Appendix B · p. 10