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Optimal Pricing of Electric Vehicle Charging on Coupled Power-Transportation Network based on Generalized Sensitivity Analysis

Lyuzhu Pan, Hongcai Zhang

TL;DR

This paper tackles pricing EV charging in a coupled power-transportation network, where driver routing and charging decisions are captured by a user-equilibrium model, making the pricing problem non-convex and computationally challenging at scale. It introduces a generalized sensitivity-analysis approach to compute the gradient $\nabla_{\boldsymbol{\lambda}} \tilde{\boldsymbol{x}}^{\text{fcs}}$ of charging flows with respect to prices, and develops a gradient-descent framework (GDGSA) that solves two convex subproblems per iteration without embedding KKT conditions. A hyper-arc transformation extends traditional sensitivity analysis to charging behavior, with a linear-algebraic framework (ELI/ELD) ensuring Jacobian invertibility and enabling polynomial-time scaling in path sets and OD pairs. The method is validated across Nguyen-Dupuis, Eastern Massachusetts, and Winnipeg networks, achieving substantial speedups over conventional MIP-based approaches while incurring modest profit gaps, and revealing how optimal pricing can improve CSP profits at the cost of small increases in network losses and travel time. Overall, GDGSA offers a scalable, practical tool for evaluating and implementing price-based coordination in large-scale CPTNs, with potential extensions to dynamic pricing and vehicle-to-grid interactions.

Abstract

In the last decade, charging service providers are emerging along with the prevalence of electric vehicles. These providers need to strategically optimize their charging prices to improve the profits considering operation conditions of the coupled power-transportation network. However, the optimal pricing problem generally involves the user equilibrium model, which leads to a mathematical program with equilibrium constraints. As a result, the pricing problem is non-convex and computationally intractable especially for large-scale network. To address this challenge, we propose a generalized sensitivity analysis approach for optimal pricing of electric vehicle charging on coupled power-transportation network. Specifically, we adopt a sensitivity analysis to capture the best response of charging demand to charging price in the gradient form. Consequently, charging service providers can make pricing decisions based on the gradient information instead of the conventional KKT conditions of the user equilibrium model. We then propose a tailored gradient descent algorithm to solve the whole pricing problem. The mathematical proof of validity is given and the time complexity of the proposed algorithm is theoretically polynomial. Numerical experiments on different scales of networks verify the computational efficiency of the proposed algorithm, indicating its potential in evaluating the impact of the optimal pricing on the operational performance of large-scale coupled power-transportation network.

Optimal Pricing of Electric Vehicle Charging on Coupled Power-Transportation Network based on Generalized Sensitivity Analysis

TL;DR

This paper tackles pricing EV charging in a coupled power-transportation network, where driver routing and charging decisions are captured by a user-equilibrium model, making the pricing problem non-convex and computationally challenging at scale. It introduces a generalized sensitivity-analysis approach to compute the gradient of charging flows with respect to prices, and develops a gradient-descent framework (GDGSA) that solves two convex subproblems per iteration without embedding KKT conditions. A hyper-arc transformation extends traditional sensitivity analysis to charging behavior, with a linear-algebraic framework (ELI/ELD) ensuring Jacobian invertibility and enabling polynomial-time scaling in path sets and OD pairs. The method is validated across Nguyen-Dupuis, Eastern Massachusetts, and Winnipeg networks, achieving substantial speedups over conventional MIP-based approaches while incurring modest profit gaps, and revealing how optimal pricing can improve CSP profits at the cost of small increases in network losses and travel time. Overall, GDGSA offers a scalable, practical tool for evaluating and implementing price-based coordination in large-scale CPTNs, with potential extensions to dynamic pricing and vehicle-to-grid interactions.

Abstract

In the last decade, charging service providers are emerging along with the prevalence of electric vehicles. These providers need to strategically optimize their charging prices to improve the profits considering operation conditions of the coupled power-transportation network. However, the optimal pricing problem generally involves the user equilibrium model, which leads to a mathematical program with equilibrium constraints. As a result, the pricing problem is non-convex and computationally intractable especially for large-scale network. To address this challenge, we propose a generalized sensitivity analysis approach for optimal pricing of electric vehicle charging on coupled power-transportation network. Specifically, we adopt a sensitivity analysis to capture the best response of charging demand to charging price in the gradient form. Consequently, charging service providers can make pricing decisions based on the gradient information instead of the conventional KKT conditions of the user equilibrium model. We then propose a tailored gradient descent algorithm to solve the whole pricing problem. The mathematical proof of validity is given and the time complexity of the proposed algorithm is theoretically polynomial. Numerical experiments on different scales of networks verify the computational efficiency of the proposed algorithm, indicating its potential in evaluating the impact of the optimal pricing on the operational performance of large-scale coupled power-transportation network.

Paper Structure

This paper contains 19 sections, 2 theorems, 34 equations, 4 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. User Equilibrium Model
  4. Charging Service Pricing Model
  5. Complexities Analysis
  6. Gradient Descent Method based on Generalized Sensitivity Analysis
  7. Feasible Direction Method
  8. Generalized Sensitivity Analysis
  9. Optimal Stepsize Determination
  10. Solution Algorithm Summary
  11. Overall Solution Process
  12. An Illustrative Case
  13. Theoretical Analysis of the Algorithm
  14. Experiments
  15. Experiments
  16. ...and 4 more sections

Key Result

Proposition 1

Suppose the generalized arc cost is strictly monotone and twice differentiable in $\boldsymbol{x}$, then $\nabla_{\boldsymbol{x}}\boldsymbol{c}^\text{garc}$ is positive-definite and the user equilibrium model has the unique solution $\boldsymbol{x}$.

Figures (4)

  • Figure 1: Structure of the optimal pricing problem.
  • Figure 2: Illustrative transportation network network. The Roman number in the circle represents the index of the node. The number on the arrow is the arc index. Assume $\mathcal{N}^\text{fcs}=\{II,III\}$, $\mathcal{W}=\{(I,III,1.5),(I,V,2.0)\}$.
  • Figure 3: Hyper-arc transformation.
  • Figure 4: Charging service provider's profit function visualization.

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2