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Sharing Storage in a Smart Grid: A Coalitional Game Approach
Pratyush Chakraborty, Enrique Baeyens, Kameshwar Poolla, Pramod P. Khargonekar, Pravin Varaiya
TL;DR
Consumers may share electricity storage either by operating already-owned devices together or by jointly acquiring capacity. The paper models both settings with cooperative game theory and shows that both games have nonempty cores with analytically computable cost allocations. It concludes that cooperative storage sharing can amortize storage costs and increase utilization.
Problem
The paper examines whether consumers can beneficially share electricity storage under time-of-use pricing in already-owned and jointly acquired storage settings.
Method
The paper models both storage-sharing scenarios as cooperative cost-sharing games, including storage operation, capacity decisions, and coalition allocations.
Results
Both cooperative games are balanced and therefore have nonempty cores, with core cost allocations that have analytical or closed-form, easy-to-compute expressions.
Takeaways & Limitations
Cooperative storage sharing can reduce costs, amortize storage expenses, increase utilization, and support renewable integration within the modeled settings.
Abstract
from arXiv · showhide
Sharing economy is a transformative socio-economic phenomenon built around the idea of sharing underused resources and services, e.g. transportation and housing, thereby reducing costs and extracting value. Anticipating continued reduction in the cost of electricity storage, we look into the potential opportunity in electrical power system where consumers share storage with each other. We consider two different scenarios. In the first scenario, consumers are assumed to already have individual storage devices and they explore cooperation to minimize the realized electricity consumption cost. In the second scenario, a group of consumers is interested to invest in joint storage capacity and operate it cooperatively. The resulting system problems are modeled using cooperative game theory. In both cases, the cooperative games are shown to have non-empty cores and we develop efficient cost allocations in the core with analytical expressions. Thus, sharing of storage in cooperative manner is shown to be very effective for the electric power system.
I. INTRODUCTION
The paper studies cooperative sharing of electricity storage among consumers under TOU pricing, covering both already-owned devices and jointly acquired capacity. It models these scenarios with coalition-based costs and storage operation assumptions.
- I. INTRODUCTION: Sharing underused electricity-grid resources, including storage capacity, is motivated by anticipated reductions in battery-storage prices.The paper positions storage sharing alongside sharing excess solar generation and flexible demand.
- I. INTRODUCTION: The study considers consumers who either share already-installed storage or jointly invest in a common storage system.The scenarios assume electrical connectivity for sharing and abstract from network capacity constraints, topology, and losses.
- I. INTRODUCTION: Daily storage costs combine amortized capacity investment, peak-period electricity purchases, and off-peak purchases used to cover peak consumption.Peak consumption is modeled as random, while peak and off-peak prices are fixed and known.
- I. INTRODUCTION: The second scenario models joint capacity acquisition as minimizing expected daily storage cost before cooperatively sharing that expected cost.The coalition first chooses how much capacity to acquire, then allocates the resulting expected cost among participants.
C. Quantifying the Benefit of Cooperation Benefit
The paper quantifies cooperation benefits in two storage-sharing settings: operating storage already owned by consumers and jointly deciding on new shared capacity.
- C. Quantifying the Benefit of Cooperation Benefit: Consumers with previously installed storage aggregate their capacities and operate them cooperatively to reduce realized daily storage costs.Their devices may differ in technology, acquisition time, and amortized daily capital cost.
- C. Quantifying the Benefit of Cooperation Benefit: The first scenario develops an efficient cost-allocation rule intended to be satisfactory for every participating consumer.The allocation distributes the daily storage cost after cooperative operation.
- C. Quantifying the Benefit of Cooperation Benefit: Consumers jointly acquiring storage first optimize the amount of capacity and then share the resulting expected daily cost cooperatively.The capacity decision minimizes expected daily storage cost, and the paper quantifies the resulting reduction.
III. BACKGROUND: COALITIONAL GAME THEORY FOR COST SHARING
The paper introduces cooperative games for cost sharing, where players form coalitions, assign costs to them, and seek allocations that no coalition would prefer to abandon. These concepts support the storage-sharing analysis.
- III. BACKGROUND: COALITIONAL GAME THEORY FOR COST SHARING: Cooperative cost-sharing games model players who minimize joint cost and distribute the resulting cost among coalition members.The game is represented by players and a value function assigning total cost to every coalition.
- III. BACKGROUND: COALITIONAL GAME THEORY FOR COST SHARING: A coalition is any subset of players, while the grand coalition contains all players.The possible coalitions form the power set of the player set.
- III. BACKGROUND: COALITIONAL GAME THEORY FOR COST SHARING: A subadditive cost game has combined coalition costs no greater than the sum of costs when disjoint coalitions act separately.This property formalizes an advantage of cooperation in cost sharing.
- III. BACKGROUND: COALITIONAL GAME THEORY FOR COST SHARING: A cost allocation assigns each player a cost, and an imputation is efficient while remaining individually rational.Efficiency requires total allocation to equal grand-coalition cost; individual rationality bounds each player’s assigned cost by its standalone value.
- III. BACKGROUND: COALITIONAL GAME THEORY FOR COST SHARING: The core contains allocations under which no coalition can obtain a lower cost than the sum of its members’ assigned costs.The core is the central solution concept for cooperative games.
- III. BACKGROUND: COALITIONAL GAME THEORY FOR COST SHARING: The Bondareva-Shapley theorem states that a coalitional game has a nonempty core if and only if it is balanced.The paper applies balancedness to its storage cost-sharing games.
1) Coalitional Game and Its Properties:
The first scenario models consumers with existing storage as a cost-sharing cooperative game. The game is subadditive and balanced, yielding a nonempty core and an analytically computable stabilizing allocation.
- The game is subadditive, so joint daily investment cost is no greater than the sum of individual daily investment costs.
- The game is balanced, which guarantees a nonempty core and cost allocations that stabilize the grand coalition.
- Nucleolus and minimum-worst-case-excess allocations stabilize the coalition but require linear programs with exponentially many constraints.
- Allocation 1 defines a cost allocation for consumers with existing storage capacities and different daily capital costs.
- Theorem 4 establishes that Allocation 1 belongs to the game's core.
- The analytical allocation enables consumers with independently acquired storage to reduce costs through sharing under two-period TOU pricing.
1) Coalitional Game and Its Properties:
The second scenario studies consumers jointly investing in storage under uncertain peak consumption and expected-cost minimization. Its game is subadditive and balanced, with an analytically defined allocation in the core.
- The risk-neutral consumer chooses storage capacity that minimizes the expected daily cost.
- The common-storage scenario assumes equal daily capital costs because consumers buy the same storage technology at the same time.
- Joint storage investment is chosen using random daily peak consumption and its marginal cumulative distribution function.
- The cooperative investment cost-sharing game is subadditive, so cooperation provides a reduction in cost.
- The game is balanced and therefore has a stabilizing allocation.
- The proposed cost allocation has an analytical formula, can be efficiently computed, and belongs to the core.
2) Stable Sharing of Expected Cost:
The paper defines an analytical expected-cost allocation for jointly acquired storage and proves that it lies in the cooperative game's core. It also develops realized daily allocations that are budget balanced and converge to the fixed allocation.
- Stable Sharing of Expected Cost: Allocation 2 defines an analytical rule for sharing the expected daily storage cost of a coalition.The rule is given by ζi := πℓE[xi] + πSE[xi | xN ≥ C∗N].
- Stable Sharing of Expected Cost: Theorem 8 proves that the Allocation 2 cost vector belongs to the core of the cost-sharing cooperative game.
- Sharing of Realized Cost: The expected allocation supports long-term savings, while realized allocations vary with daily consumption randomness.
- Sharing of Realized Cost: The realized daily allocation is budget balanced and becomes strongly consistent with the fixed allocation ζi as the number of days increases.
- Stable Sharing of Expected Cost: Joint storage operation reduces total cost, with individual reductions determined by the coalition's aggregate consumption and storage condition.
VI. CASE STUDY
The case study uses Pecan St. household data and a two-period time-of-use tariff to illustrate both storage-investment and realized-cost sharing. The experiments examine five households, their peak-consumption distributions, and cost allocations over 2016.
- VI. CASE STUDY: The case study uses Pecan St. 2016 data with a two-period tariff and assumes a storage cost of πS = 15¢/KWh.The tariff parameters are πh = 55¢/KWh and πℓ = 20¢/KWh; the assumed storage price reflects a projected reduction in storage cost.
- VI. CASE STUDY: Five households are evaluated using estimated individual and joint daily peak-consumption CDFs for Texas peak hours.The study considers non-holidays and non-weekends from 7h to 23h, and reports similar CDF shapes across households.
- VI. CASE STUDY: Table II reports optimal storage investments, minimal expected storage costs, and grand-coalition expected cost allocations.
- VI. CASE STUDY: Scenario I allocates realized aggregate costs for the first ten days and tracks average household allocations throughout 2016.
- VI. CASE STUDY: The average realized allocations are compared with optimal expected costs and converge to limiting values under stationary peak-consumption assumptions.
APPENDIX
The appendix establishes balancedness and core membership for the cost-sharing game by proving subadditivity and verifying budget balance and individual rationality of the allocation.
- APPENDIX: The function J is shown to be subadditive: J(xS + xT, CS + CT) ≤ J(xS, CS) + J(xT, CT).
- APPENDIX: Because the storage cost function is u(S) = J(xS, CS), subadditivity transfers to the cost-sharing cooperative game.
- APPENDIX: Positive homogeneity together with subadditivity proves that the cost-sharing game is balanced.
- APPENDIX: The allocation ξ is budget balanced and individually rational, so it is an imputation.
- APPENDIX: Coalitional inequalities then establish that ξ belongs to the core of the cooperative game.
D. Proof of Theorem 6
The proof establishes subadditivity of the expected coalition cost by applying the corresponding inequality for J and then taking expectations. This yields subadditivity of the cooperative game's value function.
- D. Proof of Theorem 6: The proof considers two arbitrary disjoint nonempty coalitions and aims to show that Φ(xS) is subadditive.
- D. Proof of Theorem 6: Starting from the definition of J, the argument derives the needed inequality for coalition costs.
- D. Proof of Theorem 6: Taking expectations preserves the inequality and proves subadditivity of Φ.
- D. Proof of Theorem 6: Since v(S) = Φ(xS), subadditivity of Φ implies subadditivity of the cost-sharing cooperative game.
E. Proof of Theorem 7
The proof establishes positive homogeneity of Φ by analyzing how scaling affects distribution functions and daily storage cost. This property then yields balancedness of the cost-sharing cooperative game.
- Scaling a random variable by α > 0 transforms its CDF according to Fα(θ) = F(θ/α).
- For α ≥ 0, the quantile relation γ = F(C) is equivalent to γ = Fα(αC).
- Taking expectations after scaling storage decisions and costs gives Φ(αxS) = αΦ(xS), proving positive homogeneity and implying game balancedness.
F. Proof of Theorem 8
The proof verifies that the allocation in (9) is budget balanced and satisfies the coalition inequalities required for core membership. It concludes that the allocation is an imputation in the core.
- The proof begins by showing that the cost allocation given by (9) satisfies budget balance.
- Core membership requires proving v(S) ≥ ∑_{i∈S} ζi for every coalition S ⊂ N.
- The coalition inequality also includes individual rationality as part of the same condition.
- The proof partitions feasible aggregate storage decisions into A+ = {xN ∈ R+ | xN ≥ CN} and A− = R+\A+, then defines an auxiliary function ψ(xN).
- Using the joint distribution function F(xS, xN) of peak consumptions, the proof shows that the allocation satisfies ∑_{i∈S} ζi ≤ v(S).
- Consequently, the cost allocation {ζi : i ∈ N} is an imputation in the core.