Dynamics and Optimization in Spatially Distributed Electrical Vehicle Charging

TL;DR

This work develops a fluid-dynamics model for dynamic, decentralized allocation of EV charging demands over a spatial network of charging stations. Arriving vehicles observe travel times and queueing delays, and routing decisions are based on a smooth, delay-driven criterion that couples with queue dynamics through a mean sojourn time $T$ and capacity constraints $c_j$, yielding a globally Lipschitz system with a unique equilibrium. The equilibrium is characterized as the optimum of a convex transport-type problem with entropy regularization, and duality provides a convergent Lyapunov-based proof that the system globally converges to this equilibrium; a Price of Anarchy analysis compares selfish routing to a social optimum. The authors extend the model to elastic demand via a thinning mechanism driven by utilities $U_i(r_i)$ and a saddle-point formulation $W(r,\mu)$, with convergence guarantees. Stochastic simulations corroborate the fluid predictions in a realistic spatial setting and illuminate robustness beyond the fluid limit, highlighting implications for scalable, distributed EV charging infrastructure design.

Abstract

We consider a spatially distributed demand for electrical vehicle recharging, that must be covered by a fixed set of charging stations. Arriving EVs receive feedback on transport times to each station, and waiting times at congested ones, based on which they make a selfish selection. This selection determines total arrival rates in station queues, which are represented by a fluid state; departure rates are modeled under the assumption that clients have a given sojourn time in the system. The resulting differential equation system is analyzed with tools of optimization. We characterize the equilibrium as the solution to a specific convex program, which has connections to optimal transport problems, and also with road traffic theory. In particular a price of anarchy appears with respect to a social planner's allocation. From a dynamical perspective, global convergence to equilibrium is established, with tools of Lagrange duality and Lyapunov theory. An extension of the model that makes customer demand elastic to observed delays is also presented, and analyzed with extensions of the optimization machinery. Simulations to illustrate the global behavior are presented, which also help validate the model beyond the fluid approximation.

Figures (5)

• Figure 1: Delay model \ref{['eq.mu of q']} and penalty barrier function \ref{['eq.barrier function']}.
• Figure 2: Rate splitting for the socially optimal (left) and selfish (right) routing policies as a function of input rate, for Example \ref{['example:poa']}.
• Figure 3: Cost comparison for the socially optimal and selfish routing policies.
• Figure 4: Charging station positions, minimum distance cells and attraction regions in equilibrium.
• Figure 5: Time evolution of station occupations for the stochastic system and fluid approximation (thick lines). The predicted equilibrium of Theorem \ref{['teo.sojourn-eq']} is shown in dotted lines.

Theorems & Definitions (19)

• Lemma 1
• Remark 1
• Remark 2
• Proposition 2
• Theorem 3
• Proposition 4
• Remark 3
• Example 1
• Lemma 5
• Proposition 6