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Vehicle-to-Vehicle Charging: Model, Complexity, and Heuristics

Cláudio Gomes, João Paulo Fernandes, Gabriel Falcao, Soummya Kar, Sridhar Tayur

Abstract

The rapid adoption of Electric Vehicles (EVs) poses challenges for electricity grids to accommodate or mitigate peak demand. Vehicle-to-Vehicle Charging (V2VC) has been recently adopted by popular EVs, posing new opportunities and challenges to the management and operation of EVs. We present a novel V2VC model that allows decision-makers to take V2VC into account when optimizing their EV operations. We show that optimizing V2VC is NP-Complete and find that even small problem instances are computationally challenging. We propose R-V2VC, a heuristic that takes advantage of the resulting totally unimodular constraint matrix to efficiently solve problems of realistic sizes. Our results demonstrate that R-V2VC presents a linear growth in the solution time as the problem size increases, while achieving solutions of optimal or near-optimal quality. R-V2VC can be used for real-world operations and to study what-if scenarios when evaluating the costs and benefits of V2VC.

Vehicle-to-Vehicle Charging: Model, Complexity, and Heuristics

Abstract

The rapid adoption of Electric Vehicles (EVs) poses challenges for electricity grids to accommodate or mitigate peak demand. Vehicle-to-Vehicle Charging (V2VC) has been recently adopted by popular EVs, posing new opportunities and challenges to the management and operation of EVs. We present a novel V2VC model that allows decision-makers to take V2VC into account when optimizing their EV operations. We show that optimizing V2VC is NP-Complete and find that even small problem instances are computationally challenging. We propose R-V2VC, a heuristic that takes advantage of the resulting totally unimodular constraint matrix to efficiently solve problems of realistic sizes. Our results demonstrate that R-V2VC presents a linear growth in the solution time as the problem size increases, while achieving solutions of optimal or near-optimal quality. R-V2VC can be used for real-world operations and to study what-if scenarios when evaluating the costs and benefits of V2VC.
Paper Structure (29 sections, 1 theorem, 9 equations, 6 figures, 3 tables, 1 algorithm)

This paper contains 29 sections, 1 theorem, 9 equations, 6 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. V2VC Model Definition
  4. Assumptions
  5. Parameters
  6. Time-Space Network Graph
  7. Binary Decision Variables
  8. Linear Constraints
  9. Path Constraints
  10. Battery Constraints
  11. G2VC Constraints
  12. V2VC Constraints
  13. Dimensions
  14. Objective function
  15. NP-Completeness Proof
  16. ...and 14 more sections

Key Result

Theorem 1

Under the assumption that $\mathcal{F}^\mathrm{R}$ is a separable convex objective function, R-V2VC can be solved in time polynomial to the size of the problem.

Figures (6)

  • Figure 1: Illustration of a reduction from a 3SAT instance to a V2VC instance.
  • Figure 2: Number of variables used in scenarios B1--B11. Note that the units are different: the blue axis ranges in the thousands ($10^3$), while the red axis ranges in the billions ($10^9$), as shown by the SI prefixes $k$ and $G$, respectively.
  • Figure 3: Least Squares Logarithmic Regression of the solution time (ms) against the number of variables in randomly generated scenarios with R-V2VC. The regression is fit with a degree of 4, after which the three first decimal digits of the Mean Squared Error (MSE) do not change. The Mean Absolute Error (MAE) of the fit is $0.969$. A total of 900 data points were computed (36 scenarios times 5 random seeds times 5 runs).
  • Figure 4: Plot of the objective function value of each of the R-V2VC solutions for the scenarios Q1--Q6 against the minimum and maximum objective function values obtained with the original formulation.
  • Figure 5: Illustration of a solution for a scenario using the original formulation, in its equivalent $G_{\mathrm{TS}}$. The needy EV B is charged by the helper EV A at $t=0$ for 3 time steps and then charges the other needy EV C at $t=4$.
  • ...and 1 more figures

Theorems & Definitions (3)

  • Definition 1
  • Theorem 1
  • proof