Ex-post Stable and Fair Payoff Allocation for Renewable Energy Aggregation

Aggregating statistically diverse renewable power producers (RPPs) is an effective way to reduce the uncertainty of the RPPs. The key question in aggregation of RPPs is how to allocate payoffs among the RPPs. In this paper, a payoff allocation mechanism (PAM) with a simple closed-form expression is proposed: It achieves stability (in the core) and fairness both in the “ex-post” sense, i.e., for all possible realizations of renewable power generation. Furthermore, this PAM can in fact be derived from the competitive equilibrium in a market. The proposed PAM is evaluated in a simulation study with ten wind power producers in the PJM interconnection.

💡 Research Summary

The paper addresses the problem of how to allocate the total payoff of an aggregation of renewable power producers (RPPs) in a two‑settlement electricity market (day‑ahead (DA) and real‑time (RT)) so that the allocation is stable, fair, and budget‑balanced for every possible realization of renewable generation (“ex‑post”). Each RPP i independently chooses a DA contract quantity c_i and later realizes generation x_i. If x_i < c_i it must purchase the shortfall in the RT market at price p_r,b; if x_i > c_i it can sell the excess at price p_r,s (with p_r,s < p_r,b to avoid arbitrage). The individual payoff when acting alone is given by (1): P_sep,i = p_f c_i – p_r,b·max{c_i–x_i,0} + p_r,s·max{x_i–c_i,0}.

When the RPPs form an aggregation, the aggregator sells the sum of the contracts c_N = Σ_i c_i in the DA market, collects the realized total generation x_N = Σ_i x_i, and settles the net deviation in the RT market using the same price rules. The aggregator’s realized payoff is P_A = p_f c_N – p_r,b·max{c_N–x_N,0} + p_r,s·max{x_N–c_N,0}. The central design question is how to split P_A among the participants.

The authors list five desirable ex‑post properties:

1. Budget balance (Σ_i P_i = P_A);
2. Individual rationality (P_i ≥ P_sep,i for all i);
3. Fairness (if a producer’s net surplus c_i–x_i is larger, its allocated payoff should be larger);
4. No‑exploitation (if a producer has no shortfall, its payoff equals p_f c_i);
5. Core stability (for any subset T, Σ_{i∈T} P_i ≥ v(T), where v(T) is the value the subset could obtain by forming its own aggregation).

Lemma 1 proves that to satisfy individual rationality and budget balance for all realizations, the aggregator must commit exactly the sum of the individual contracts (c_N = Σ_i c_i). This ensures that the aggregation always yields a non‑negative excess profit relative to the sum of separate payoffs (Corollary 1).

The paper then defines the sets of surplus producers S⁺ = {i | x_i – c_i ≥ 0} and deficit producers S⁻ = {i | x_i – c_i < 0}. Lemma 2 expresses the excess profit of the aggregation as Excess = (p_r,b – p_r,s)·min{ Σ_{i∈S⁺} (x_i – c_i), Σ_{i∈S⁻} (c_i – x_i) }. Thus the excess profit is limited by the smaller of total surplus and total deficit and is shared at the spread between the RT buying and selling prices.

Proposed Payoff Allocation Mechanism (PAM)
The authors propose a simple closed‑form allocation (Equation 10):

• If the total aggregation has a deficit (x_N < c_N):
   P_i = p_f c_i – p_r,b·(c_i – x_i) for deficit producers (i ∈ S⁻),
   P_i = p_f c_i + (p_r,b – p_r,s)·(x_i – c_i) for surplus producers (i ∈ S⁺).
• If the total aggregation has a surplus (x_N > c_N):
   P_i = p_f c_i – p_r,s·(c_i – x_i) for surplus producers,
   P_i = p_f c_i + (p_r,b – p_r,s)·(c_i – x_i) for deficit producers.
• If the total exactly matches the contract (x_N = c_N): any price p_∘ satisfying p_r,s < p_∘ < p_r,b can be used in the same formula.

Intuitively, when the coalition as a whole is short, the deficit RPPs receive exactly what they would have earned alone, while the surplus RPPs share the excess profit at the spread (p_r,b – p_r,s). The opposite occurs when the coalition is over‑producing. When there is no net deviation, any price between the RT buying and selling prices yields a core allocation, reflecting the non‑uniqueness of the core in that special case.

Theoretical Results
Theorem 1 states that this PAM satisfies all five ex‑post properties. Budget balance follows directly from the construction; fairness and no‑exploitation are evident from the piecewise definitions. Core stability is proved by considering any subset T and the three possible aggregate deviation cases (deficit, surplus, or balanced). In each case, the sum of allocated payoffs to T is shown to be at least v(T) using the price ordering p_r,b ≥ p_r,s. Consequently, individual rationality is also guaranteed.

Derivation from Competitive Equilibrium
Section V shows that the same allocation can be obtained as the competitive equilibrium of a market with transferable payoff. Each RPP i is modeled with a concave “production” function f_i(x) = P_sep,i (the same payoff they would obtain acting alone). The market’s core is defined as the set of allocations that maximize Σ_i f_i(z_i) subject to Σ_i z_i = Σ_i x_i (redistributing total realized generation). Lemma 3 proves that this market core coincides with the cooperative game defined earlier (v(T)). Solving the competitive equilibrium yields exactly the piecewise allocation (10), confirming that the PAM is not ad‑hoc but grounded in economic equilibrium theory.

Numerical Study
A case study uses real wind generation data from ten producers in the PJM interconnection. The authors compute the PAM for each realized hour, compare it with alternative mechanisms from the literature (e.g., proportional cost sharing, ex‑ante optimal contracts), and evaluate the resulting individual payoffs and total surplus. Results demonstrate that the proposed PAM always lies in the core (no subset can improve by deviating) and distributes the excess profit in a way that respects the fairness criterion. Moreover, the mechanism is computationally trivial (closed‑form) and requires only the realized generation and contract quantities, making it attractive for practical implementation.

Conclusions
The paper contributes a practically implementable, analytically simple payoff allocation rule that guarantees ex‑post core stability, fairness, budget balance, individual rationality, and no‑exploitation for renewable aggregations in a two‑settlement market. By linking the rule to a competitive equilibrium, the authors provide a solid economic justification. The work offers a valuable tool for aggregators and policymakers aiming to promote renewable integration while ensuring that all participants are treated equitably under every possible generation outcome.