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Resource Allocation in Electricity Markets with Budget Constrained Customers

Lila Perkins, Baosen Zhang

Abstract

In electricity markets, customers are increasingly constrained by their budgets. A budget constraint for a user is an upper bound on the price multiplied by the quantity. However, since prices are determined by the market equilibrium, the budget constrained welfare maximization problem is difficult to define rigorously and to work with. In this letter, we show that a natural dual-ascent algorithm converges to a unique competitive equilibrium under budget constraints. Further, this budget-constrained equilibrium is exactly the solution of a convex welfare maximization problem in which each user's utility is replaced by a modified utility that splices the original utility with a logarithmic function where the budget binds. We also provide an explicit piecewise construction of this modified utility and demonstrate the results on examples with quadratic and square root utility functions.

Resource Allocation in Electricity Markets with Budget Constrained Customers

Abstract

In electricity markets, customers are increasingly constrained by their budgets. A budget constraint for a user is an upper bound on the price multiplied by the quantity. However, since prices are determined by the market equilibrium, the budget constrained welfare maximization problem is difficult to define rigorously and to work with. In this letter, we show that a natural dual-ascent algorithm converges to a unique competitive equilibrium under budget constraints. Further, this budget-constrained equilibrium is exactly the solution of a convex welfare maximization problem in which each user's utility is replaced by a modified utility that splices the original utility with a logarithmic function where the budget binds. We also provide an explicit piecewise construction of this modified utility and demonstrate the results on examples with quadratic and square root utility functions.
Paper Structure (18 sections, 4 theorems, 17 equations, 5 figures, 2 tables, 1 algorithm)

This paper contains 18 sections, 4 theorems, 17 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Setup
  4. Price Equilibrium in the Unconstrained Case
  5. Budget-Constrained Competitive Equilibrium
  6. Algorithm
  7. Existence and Uniqueness of Equilibrium Price
  8. Primal Interpretation via Modified Utility
  9. Construction of Modified Utility
  10. Primal Equivalence Theorem
  11. Examples
  12. Quadratic Utilities (two crossover points)
  13. Square Root Utilities (one crossover point)
  14. Simulations
  15. Two users with quadratic utilities
  16. ...and 3 more sections

Key Result

Theorem 1

Under the assumptions on $u_i$ and $C$ in Sec. sec:setup, there exists a unique price $\lambda^* \in (0, \infty)$ at which supply equals demand and the market clears. Furthermore, Algorithm alg:constrained-price-update converge to $\lambda^*$ from any initial condition $\lambda^0 > 0$ where $x_i^*(\

Figures (5)

  • Figure 1: Original utilities $u_i$ (dashed) and modified utilities $\hat{u}_i$ (solid) for the two user quadratic examples. The modified utility replaces the original utility with $b \log x + \textrm{const.}$ in regions where the budget binds (shaded gray), resulting in a slower-growing function which captures budget-constrained utility behavior.
  • Figure 2: Original utilities $u_i$ (dashed) and modified utilities $\hat{u}_i$ (solid) for the two user square root examples. The modified utility replaces the original utility with $b \log x + \textrm{const.}$ in regions where the budget binds (shaded gray), resulting in a slower-growing function which captures budget-constrained utility behavior.
  • Figure 3: Convergence of the price iterate $\lambda^k$ in Algorithm \ref{['alg:constrained-price-update']} to the equilibrium price $\lambda^* = 2.743$ for the two-user quadratic utility example.
  • Figure 4: Supply and demand curves for the two-user quadratic utilities example. The competitive equilibrium price $\lambda^*$ is where aggregate demand (gray) intersects supply (black).
  • Figure 5: Convergence of the price iterate $\lambda^k$ in Algorithm \ref{['alg:constrained-price-update']} to the equilibrium price $\lambda^* = 4.845$ for the five-user mixed utility example.

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Lemma 1
  • Theorem 2: Primal Equivalence
  • proof
  • Lemma 2