PaperlayerView exact revision on arXiv

Source-linked AI summary

Sharing the Gains of Aggregation: Cooperative Imbalance Cost Allocation

Asmus Winther Eriksen, Jalal Kazempour

arXiv:2609.39398v1eess.SYcs.SY

TL;DR

Facilitators manage consumers’ residual imbalance costs, but allocating the savings from pooling raises fairness and stability questions. The paper models these costs as a cooperative game, compares six allocation mechanisms, and analyzes their properties. In the Danish case study, aggregation reduces cost by 10%, while the allocation rule strongly affects how consumers share the savings.

  • Problem

    Allocating residual imbalance costs across a facilitator’s consumer portfolio raises questions of fairness and stability.

    Source: §I · pp. 1–2

  • Method

    The paper models coalition imbalance costs as a cooperative game and compares six allocation mechanisms across computational tractability, budget balance, group rationality, and additivity.

    Source: §I · p. 2

  • Results

    10% lower grand-coalition imbalance cost than the sum of standalone costs; allocation outcomes depend strongly on the rule, and flat-rate allocation charges some consumers more than twice their standalone cost.

    Source: B. Results: Cost Savings · p. 8, §VI · p. 10

  • Takeaways & Limitations

    All budget-balanced game-theoretic mechanisms examined belong to the core for the Danish dataset, whereas the flat-rate allocation violates individual rationality.

    Source: §VI · p. 10

Abstract

from arXiv · show

A facilitator is an intermediary that offers renewable producers and consumers fixed-price contracts and, acting as their balance responsible party, manages the residual imbalances in the market. Pooling imperfectly correlated residuals nets consumers' imbalances and reduces the portfolio's total imbalance cost, but raises an allocation question: how should these savings be divided among heterogeneous consumers? We formulate this as a cooperative game, the imbalance netting game, and compare six allocation mechanisms under a two-price imbalance settlement in terms of computational requirements, budget balance, group rationality, and additivity. We establish three analytical results: a necessary and sufficient condition for budget balance of the marginal cost contribution mechanism, group rationality of the marginal cost contribution and Vickrey-Clarke-Groves mechanisms, and a necessary and sufficient condition for when the Gately point is well-defined. On Danish 2024 data for 19 consumers, aggregation reduces imbalance cost by 10%, but how these savings are shared depends strongly on the allocation rule. A flat-rate allocation proportional to consumption, used as a benchmark for socialized imbalance pricing, charges some consumers more than twice their standalone cost, whereas all the game-theoretic mechanisms studied produce stable allocations in practice.

I. INTRODUCTION

Facilitators manage consumers’ residual balancing risk, and pooling imperfectly correlated residuals can reduce imbalance costs; the paper asks how to allocate that saving fairly and stably. It models coalition costs under two-price settlement and compares allocation rules, including how coalition bids and costs are computed.

I. INTRODUCTION

  • The study fixes PPA contracts and focuses on allocating residual imbalance costs for inelastic consumers under asymmetric two-price settlement.Helping deviations settle at the day-ahead price, while deviations that aggravate the system imbalance face a less favorable imbalance price.Source: §I · pp. 1–2
  • Pooling imperfectly correlated residuals can reduce portfolio imbalance costs, but consumers differ in forecastability, size, and contribution to imbalance, making allocation consequential.The paper asks how to allocate aggregation savings among consumers in a way that is fair, stable, budget-balanced, and implementable.Source: §I · pp. 1–2
  • The paper compares six allocation mechanisms across computational tractability, budget balance, group rationality, and additivity, and establishes game-specific conditions for selected mechanisms.The reported analytical results include budget-balance conditions for marginal cost contribution, group rationality for marginal cost contribution and VCG, and a well-definedness condition for the Gately point.Source: §I · p. 2

II. THE IMBALANCE NETTING GAME

  • The paper formulates coalition imbalance costs as a cooperative game, with each coalition’s characteristic value equal to its imbalance cost under two-price settlement.The imbalance netting game’s coalition values are computed through bidding and ex-post settlement before an allocation mechanism divides the grand-coalition cost.Source: §II · p. 2
  • The facilitator centrally forecasts renewable production and each consumer’s consumption, so consumers cannot influence the game by misreporting their forecasts.Source: §II · p. 3

A. Bidding Mechanism

  • Coalition bids sum consumers’ individually optimized newsvendor-quantile bids; netting occurs at ex-post settlement when offsetting deviations reduce the coalition’s aggregate imbalance cost.The singleton bids preserve additive consumer imbalances; coalition-level quantile bids could lower coalition costs further but would make individual imbalances coalition-dependent.Source: A. Bidding Mechanism · p. 3

B. Ex-post Imbalance Cost Calculation

Coalition imbalance is the sum of member imbalances, and its cost is calculated from the coalition’s realized position and the settlement spread. Under two-price settlement, helping deviations incur no imbalance cost, while harming deviations incur a nonnegative cost.

B. Ex-post Imbalance Cost Calculation

  • Coalition imbalance is additive across consumers, with a positive value denoting surplus and a negative value denoting deficit.The coalition’s realized position combines its bid and contracted renewable share with realized production and consumption.Source: B. Ex-post Imbalance Cost Calculation · p. 3
  • Under two-price settlement, a coalition’s spread is zero when its imbalance helps the power system and positive when it harms it.The coalition’s direction determines which settlement spread applies; under the price-taking assumption, the spread changes with direction rather than imbalance volume.Source: B. Ex-post Imbalance Cost Calculation · p. 3
  • The characteristic cost sums each coalition’s imbalance priced at its spread over the study horizon, so helping deviations incur no cost and harming deviations incur nonnegative cost.Source: B. Ex-post Imbalance Cost Calculation · pp. 3–4

C. Game Properties and Definitions

Sub-additivity means pooling does not increase total cost, but the imbalance netting game is not concave, so concavity cannot be relied on to ensure stable allocations. The paper defines stability through coalition incentives to break away from the grand-coalition allocation.

C. Game Properties and Definitions

  • Sub-additivity ensures that merging coalitions costs no more than operating separately because opposing imbalances can cancel while aligned imbalances sum.This result holds under the price-taking assumption.Source: C. Game Properties and Definitions · p. 4
  • Concavity requires a consumer’s marginal cost contribution not to increase as the coalition it joins grows, a condition that guarantees a stable allocation exists.Source: C. Game Properties and Definitions · p. 4
  • The imbalance netting game is not concave: a consumer can reduce one coalition’s cost by offsetting its shortfall yet leave a larger coalition’s cost unchanged.Thus stability does not follow from concavity and must be checked for each allocation mechanism.Source: C. Game Properties and Definitions · p. 4
  • An imputation distributes the grand-coalition cost and leaves each consumer no worse off alone, but it may still let a subgroup benefit by breaking away.Source: C. Game Properties and Definitions · p. 4
  • The core contains imputations under which no coalition can reduce its total charge by leaving, making core allocations stable.The core can be empty, and finding an allocation inside a nonempty core is non-trivial.Source: C. Game Properties and Definitions · pp. 4–5

III. IMBALANCE COST ALLOCATION MECHANISMS

The section presents three allocation mechanisms with different cost-evaluation requirements and stability implications. The Shapley value averages marginal contributions across coalitions, MCC charges grand-coalition marginal contributions, and VCG allocates participation externalities.

A. Shapley value

  • The Shapley value averages a consumer’s marginal cost contribution over coalitions, but requires 2^|N| cost evaluations and is not guaranteed to lie in the core for this non-concave game.It is the unique allocation satisfying the standard cooperative-game fairness axioms.Source: A. Shapley value · p. 5

B. Marginal Cost Contribution (MCC)

  • MCC charges each consumer its marginal contribution to grand-coalition cost, requiring only |N| + 1 coalition-cost evaluations.These are the |N| leave-one-out coalitions and the grand coalition.Source: B. Marginal Cost Contribution (MCC) · p. 5

C. VCG

  • VCG assigns each consumer the externality it imposes on the remaining consumers, and only pivotal consumers receive non-zero allocations.A consumer is pivotal when its participation changes the imbalance price spread faced by the others; most consumers therefore receive zero, while those paid leave the facilitator with a budget deficit.Source: C. VCG · p. 5

D. Nucleolus

This section describes the nucleolus, marginal price allocation, and Gately point, then compares mechanisms on computational effort and game properties. The comparison highlights coalition-scale costs, conditional guarantees, and budget deficits for this game.

D. Nucleolus

  • The nucleolus lexicographically maximizes coalition excesses, prioritizing the coalition with the strongest incentive to deviate, but requires constraints for all coalitions.Its computation uses a sequence of linear programs, making it challenging for large portfolios.Source: D. Nucleolus · p. 5

E. Marginal Price

  • Marginal price allocation settles each consumer’s imbalance at the grand-coalition spread regardless of direction, using an internal one-price scheme.It requires only the grand-coalition imbalance cost and is the most computationally efficient mechanism considered.Source: E. Marginal Price · pp. 5–6

F. Gately point

  • The Gately point minimizes the largest propensity to disrupt, balancing departure incentives, and requires 2|N| + 1 coalition costs.Those costs are the grand coalition, each leave-one-out coalition, and each singleton coalition.Source: F. Gately point · p. 6

G. Properties and Comparison

  • The Shapley value and nucleolus require costs for all 2^|N| coalitions; computing the nucleolus may require up to 2^|N| − 1 linear programs, rendering both mechanisms computationally intractable as the portfolio grows.Because the game is non-concave, the Shapley value is not guaranteed to lie in the core.Source: G. Properties and Comparison · p. 6
  • For this game, VCG is not budget-balanced except when c(N) = 0, and its deficit exceeds MCC’s by exactly c(N).The table notes MCC is budget-balanced if and only if its stated condition holds; otherwise the facilitator is in deficit.Source: G. Properties and Comparison · p. 6

IV. ANALYTICAL RESULTS

The analytical results characterize when MCC balances its budget and establish how its charges relate to portfolio cost. For the imbalance netting game, failure of the budget-balance condition always leaves the facilitator in deficit.

A. Budget Balance of MCC

  • A sufficient condition for MCC budget balance is that removing any one consumer leaves the grand-coalition price spread unchanged in every hour.The condition can be checked hour by hour because the game decomposes additively over hourly subgames.Source: A. Budget Balance of MCC · p. 7
  • When the budget-balance condition fails, MCC always leaves the facilitator in deficit rather than producing a surplus.The charges collected fall short of the imbalance cost.Source: A. Budget Balance of MCC · p. 7
  • The hourly decomposition relies on temporal separability; intertemporal coupling such as storage state of charge prevents that decomposition and the associated additivity.The game’s cost is a sum of independent per-hour terms in the separable setting.Source: A. Budget Balance of MCC · p. 7
  • MCC is budget-balanced if and only if marginal cost contributions sum to the grand-coalition cost, a property called additivity of marginal contributions at the top level (AMC).When this condition fails, consumer charges do not cover the imbalance cost borne by the facilitator.Source: A. Budget Balance of MCC · p. 7

B. Group Rationality of MCC and VCG

MCC and VCG are group rational in the imbalance netting game. The Gately-point results characterize when it exists and relate its allocation to MCC charges and standalone costs.

B. Group Rationality of MCC and VCG

  • Both MCC and VCG are group rational for the imbalance netting game.The supplied result states that MCC allocates a lower cost than a group-rational mechanism, while under VCG no consumer makes a positive net payment.Source: B. Group Rationality of MCC and VCG · p. 7

C. Existence and Uniqueness of the Gately point

  • The Gately point is defined only when cooperation yields cost savings relative to consumers acting individually; the paper characterizes when it is well-defined for this game.When well-defined, it blends MCC charges with standalone costs to restore budget balance and coincides with MCC when condition (16) holds.Source: C. Existence and Uniqueness of the… · pp. 7–8
  • When consumers’ imbalances are perfectly aligned in every hour, the game is inessential and its unique core allocation charges each consumer its standalone imbalance cost..Source: C. Existence and Uniqueness of the… · p. 8

V. NUMERICAL RESULTS

The 19-consumer Danish case study finds that aggregation reduces imbalance costs, while coalition composition and allocation rules shape who benefits. Flat-rate billing can charge some consumers more than twice their standalone cost, unlike the similar ratios produced by budget-balanced mechanisms.

A. Input Data

  • The case study uses 2024 hourly data for 19 Danish consumers, whose demand and PV coverage differ substantially; consumer A's demand exceeds all others combined.The portfolio shares a PV plant, so production uncertainty is common and cannot be reduced through netting.Source: A. Input Data · p. 8

B. Results: Cost Savings

  • The average coalition allocation ratio falls to 10% for the grand coalition, while coalitions of equal size vary and some achieve savings exceeding 20%.The ratio is coalition cost divided by the sum of its members’ standalone costs.Source: B. Results: Cost Savings · p. 8, Figure 4 · p. 8
  • Consumer allocation ratios measure each consumer's grand-coalition cost relative to its standalone imbalance cost.A ratio at or below 100% means the consumer would prefer the grand coalition to operating alone.Source: B. Results: Cost Savings · pp. 8–9
  • Flat-rate allocation charges some consumers more than 200% of standalone cost, while budget-balanced mechanisms yield similar ratios markedly different from the flat rate.The flat rate has the largest consumer, A, at a lower ratio than under stable mechanisms, so smaller consumers subsidize A.Source: B. Results: Cost Savings · p. 9, Figure 5 · p. 9

C. Results: Stability of Mechanisms

For this dataset, all budget-balanced game-theoretic mechanisms are stable, while the flat-rate allocation is not. Empirical checks also confirm when MCC is budget-balanced, when the Gately point is well-defined, and how VCG transfers concentrate on pivotal consumers.

C. Results: Stability of Mechanisms

  • All budget-balanced game-theoretic mechanisms are stable on this dataset, while flat-rate allocation is not; MCC and VCG satisfy coalition rationality but under-recover the grand-coalition cost.Among stable, budget-balanced mechanisms, marginal price allocation has the smallest stability margin; the nucleolus is the lexicographically most stable benchmark, and Gately has the largest minimum excess among computationally tractable, budget-balanced mechanisms.Source: Table II · p. 9, C. Results: Stability of Mechanisms · p. 9, §VI · p. 10

D. Empirical Verification of Analytical Results

  • 76.65% of hours satisfy the MCC budget-balance condition; its entire €10,995 shortfall occurs in the remaining 23.35%, when removing a consumer flips the portfolio imbalance sign.VCG incurs a deficit even in hours when MCC is budget-balanced.Source: D. Empirical Verification of… · p. 9
  • 86.77% of the 2024 hours have mixed consumer imbalance signs, making the Gately point well-defined and unique for the dataset and any subset containing at least one of those 7,622 hours.The required condition is at least one hour with mixed consumer imbalance signs.Source: D. Empirical Verification of… · p. 9
  • VCG transfers concentrate on pivotal consumers: A receives €9,548 and is pivotal in 1,372 hours (15.6%), while S’s transfer is negligible and S is pivotal twice (0.02%).The paper links non-zero VCG transfers to consumers whose participation changes the portfolio imbalance direction.Source: D. Empirical Verification of… · pp. 9–10

VI. CONCLUSION

  • The paper establishes a necessary and sufficient MCC budget-balance condition, group rationality for MCC and VCG, and a necessary and sufficient condition for Gately existence and uniqueness.Further work is proposed on symmetric one-price settlement, flexible demand and decentralized forecasting, and coalition-level bidding or price-making portfolios.Source: §VI · p. 10

APPENDIX

MCC is additive across settlement periods. It is budget-balanced when the price spread remains unchanged upon any one consumer’s exit, but sign-flipping hours cause under-recovery.

A. Additivity of MCC

  • MCC is additive over the horizon: each consumer’s charge is the sum of its hourly marginal cost contributions.The game’s cost decomposes into hourly subgames, and the marginal contribution in each hour is the hourly MCC charge.Source: A. Additivity of MCC · p. 11

B. Conditions for Budget Balance of MCC

  • MCC is budget-balanced when a consumer’s departure leaves the hourly price spread unchanged; additivity then extends hourly balance to the full horizon.Under that condition, the marginal contributions sum to the hourly grand-coalition cost.Source: B. Conditions for Budget Balance of MCC · p. 11

C. Budget Deficit of MCC

  • Whenever an hour has a consumer whose removal flips the portfolio imbalance sign, MCC charges sum to less than that hour’s cost, leaving a facilitator deficit that favors consumers.The result extends to any game containing such an hour.Source: C. Budget Deficit of MCC · p. 11

D. Group Rationality of MCC and VCG

MCC and VCG satisfy group rationality, although VCG’s non-positive consumer payments do not recover the grand-coalition cost. The Gately point is unique under the stated strict-surplus condition, with an inessential-game exception.

D. Group Rationality of MCC and VCG

  • MCC is group rational because it is componentwise no greater than marginal-price allocation, which is group rational for this class of games.This comparison yields nonnegative coalition excess for every coalition.Source: D. Group Rationality of MCC and VCG · pp. 11–12
  • VCG is group rational because no consumer has a positive net payment and every coalition’s cost is nonnegative.Its allocation is also additive over the per-period decomposition.Source: D. Group Rationality of MCC and VCG · p. 12
  • The Gately point is uniquely defined when the grand-coalition cost is strictly below the sum of singleton costs and each consumer’s singleton cost is at least its marginal contribution.When every hour has same-sign consumer imbalances, the game is inessential and the unique core allocation equals each consumer’s singleton cost.Source: D. Group Rationality of MCC and VCG · p. 12

F. Full Allocation Results

The flat-rate allocation generally benefits larger consumers and disadvantages smaller ones. In Gately-based billing, group rationality requires an individualization grade above 0.70, so stability constrains how much imbalance cost can be socialized.

F. Full Allocation Results

G. Practical Billing and Cost Individualization

  • The billing rule interpolates between expected socialized charges and realized individualized allocations; individualization reallocates balancing risk between consumers and the facilitator rather than eliminating it.At full individualization, consumers bear the realized allocation; at full socialization, the facilitator bears the difference between collected charges and realized portfolio cost.Source: G. Practical Billing and Cost… · p. 13
  • Group rationality under the Gately-based hybrid allocation holds only above individualization grade 0.70; the facilitator therefore cannot socialize more than 30% of the imbalance charge in this case study.Below the threshold, at least one coalition could reduce its total charge by leaving the grand coalition and choosing full individualization.Source: G. Practical Billing and Cost… · p. 13, Figure 7 · p. 13
Loading 2609.39398v1…