Strategic and Fair Aggregator Interactions in Energy Markets: Mutli-agent Dynamics and Quasiconcave Games

Abstract

The introduction of aggregator structures has proven effective in bringing fairness to energy resource allocation by negotiating for more resources and economic surplus on behalf of users. This paper extends the fair energy resource allocation problem to a multi-agent setting, focusing on interactions among multiple aggregators in an electricity market. We prove that the strategic optimization problems faced by the aggregators form a quasiconcave game, ensuring the existence of a Nash equilibrium. This resolves complexities related to market price dependencies on total purchases and balancing fairness and efficiency in energy allocation. In addition, we design simulations to characterize the equilibrium points of the induced game, demonstrating how aggregators stabilize market outcomes, ensure fair resource distribution, and optimize user surplus. Our findings offer a robust framework for understanding strategic interactions among aggregators, contributing to more efficient and equitable energy markets.

Figures (13)

• Figure 1: Markets with aggregators can be thought as having a three-layer architecture. The aggregators interact with the market and compete strategically with each other. Within an aggregation, the resources (or benefits) are allocated to each of the users.
• Figure 2: The figure illustrates the feasible regions and Pareto fronts for two-user systems with quadratic utilities. The top panel ($U_1(x_1) = -x_1^2 + 3x_1$ and $U_2(x_2) = -x_2^2 + 6x_2$) shows a convex feasible region, while the bottom panel ($U_1(x_1) = -x_1^2 + 40x_1$ and $U_2(x_2) = -x_2^2 + 4x_2$) shows a non-convex feasible region. Optimal $\alpha$-fairness solutions lie on the Pareto front (the upper right boundary of the feasible region) and increasing $\alpha$ traces out a portion of the Pareto front starting with the least fair social welfare solution ($\alpha=0$) to the most fair max-min solution ($\alpha=\infty$).
• Figure 3: Evolution of small users' consumption and surplus under direct market participation: the top plot illustrates the convergence of consumption level over multiple iteration steps, while the bottom plot shows the convergence of surplus. Both plots display the results for 400 small users directly participating in the market. The figures demonstrate how the consumption and surplus dynamics quickly stabilize, converging to a market equilibrium.
• Figure 4: Convergence of strategic purchasing amounts for two aggregators using best-response dynamics: This figure illustrates the convergence of purchase amounts for two aggregators, each representing around 100 small users, as they interact with one large user in the market. The aggregators adjust their strategies iteratively using best-response dynamics, and the market dynamics converge smoothly to an equilibrium point within a few iteration steps. The rapid convergence demonstrates the efficiency of best-response updates in achieving equilibrium in this multi-aggregator setting.
• Figure 5: This figure illustrate how the average surplus of the $N$ small users change as the magnitude of the large user increases by changing the value $M$ from 0 to 400. The x-axis is the $K$ value that signifies and y-axis represents the average surplus amount. We observe that the average surplus decreases as $M$ increases.

Theorems & Definitions (4)

• Proposition 1: Existence of Nash Equilibrium for Quasiconcave Games
• Theorem 2
• Conjecture 1: Uniqueness of the Nash Equilibrium
• Definition 1