Balancing Fairness and Efficiency in Energy Resource Allocations

TL;DR

This paper formalizes the problem of fair energy resource allocation and introduces the framework for aggregators, which reveals optimized allocation schemes that lie on the Pareto front, balancing fairness and efficiency in resource allocation strategies.

Abstract

Bringing fairness to energy resource allocation remains a challenge, due to the complexity of system structures and economic interdependencies among users and system operators' decision-making. The rise of distributed energy resources has introduced more diverse heterogeneous user groups, surpassing the capabilities of traditional efficiency-oriented allocation schemes. Without explicitly bringing fairness to user-system interaction, this disparity often leads to disproportionate payments for certain user groups due to their utility formats or group sizes. Our paper addresses this challenge by formalizing the problem of fair energy resource allocation and introducing the framework for aggregators. This framework enables optimal fairness-efficiency trade-offs by selecting appropriate objectives in a principled way. By jointly optimizing over the total resources to allocate and individual allocations, our approach reveals optimized allocation schemes that lie on the Pareto front, balancing fairness and efficiency in resource allocation strategies.

Figures (5)

• Figure 1: 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 2: This graph plots the PoF and PoE for various $\alpha$-fairness criteria as a function of the number of users. The shaded areas represent the 90% confidence interval (from the 5th to the 95th percentile) for each parameter setting. Fairness parameters closer to the socially optimal ($\alpha=0.0$) tend to have a lower PoF and higher PoE. On the other hand, fairness parameters closer to the max-min solution ($\alpha=\infty$) tend to have higher PoF and lower PoE
• Figure 3: This graph plots the PoF and PoE for various $\alpha$-fairness criteria and number of users. Each bar represents the mean value of PoF/PoE for a specific number of users, with error bars indicating the standard deviation. Fairness parameters closer to the socially optimal ($\alpha=0.0$) tend to have a lower PoF and higher PoE. On the other hand, fairness parameters closer to the max-min solution ($\alpha=\infty$) tend to have higher PoF and lower PoE.
• Figure 4: These plots compare the distribution of allocations (top) and surpluses (bottom) under the social welfare solution (SW) and the proportional fairness solution (PF) for Class 1 in (blue) and Class 2 (orange). The probability densities are shown in the shaded regions and the black lines indicate the minimum, median, and values.
• Figure 5: These plots illustrate the gains in allocation (top) and surplus (bottom) when switching from the social welfare solution to the proportional fairness solutions for Class 1 in (blue) and Class 2 (orange). Positive values indicate that the proportional fairness solution provides higher allocations or surpluses compared to the social welfare solution, while negative values show the opposite. The probability densities are shown in the shaded regions and the black lines indicate the minimum, median, and values.

Theorems & Definitions (9)

• Lemma 1
• proof
• Definition 1: Pareto Optimality
• Definition 2: Pareto Front
• Conjecture 2
• Theorem 3
• proof
• Lemma 4
• proof