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Integrated equilibrium model for electrified logistics and power systems

Rui Yao, Xuhang Liu, Anna Scaglione, Shlomo Bekhor, Kenan Zhang

TL;DR

The paper develops an integrated equilibrium model that couples electrified logistics with power systems by modeling e-truck routing/charging through a $PU$-$MDP$ for the $ELO$ and price setting through a $DC$-$OPF$ for the $PSO$. The two players interact via charging demand and Locational Marginal Prices, resulting in a fixed-point equilibrium whose existence is proven and which is computed for the Hawaii network using Anderson acceleration. A central contribution is the reward-design framework that aligns individual e-truck behavior with system-wide objectives, including a fixed-point characterization of optimal rewards and continuity results that ensure well-posedness of the coupled problem. The numerical study reveals that e-truck charging can meaningfully raise $LMP$ during congestion and that prices strongly influence charging decisions and spatial charging patterns, offering practical insights and guidelines for scheduling, pricing, and differential pricing strategies to enable sustainable electrified logistics. The framework advances understanding of bidirectional interactions between logistics and power systems and provides a tractable, implementable approach for planning and operation in urban environments.

Abstract

This paper proposes an integrated equilibrium model to characterize the complex interactions between electrified logistics systems and electric power delivery systems. The model consists of two major players: an electrified logistics operator (ELO) and a power system operator (PSO). The ELO aims to maximize its profit by strategically scheduling and routing its electric delivery vehicles (e-trucks) for deliveries and charging, in response to the locational marginal price (LMP) set by the PSO. The routing, delivery, and charging behaviors of e-trucks are modeled by a perturbed utility Markov decision process (PU-MDP) while their collective operations are optimized to achieve the ELO's objective by designing rewards in the PU-MDP. On the other hand, PSO optimizes the energy price by considering both the spatiotemporal e-truck charging demand and the base electricity load. The equilibrium of the integrated system is formulated as a fixed point, proved to exist under mild assumptions, and solved for a case study on the Hawaii network via Anderson's fixed-point acceleration algorithm. Along with these numerical results, this paper provides both theoretical insights and practical guidelines to achieve sustainable and efficient operations in modern electrified logistics and power systems.

Integrated equilibrium model for electrified logistics and power systems

TL;DR

The paper develops an integrated equilibrium model that couples electrified logistics with power systems by modeling e-truck routing/charging through a - for the and price setting through a - for the . The two players interact via charging demand and Locational Marginal Prices, resulting in a fixed-point equilibrium whose existence is proven and which is computed for the Hawaii network using Anderson acceleration. A central contribution is the reward-design framework that aligns individual e-truck behavior with system-wide objectives, including a fixed-point characterization of optimal rewards and continuity results that ensure well-posedness of the coupled problem. The numerical study reveals that e-truck charging can meaningfully raise during congestion and that prices strongly influence charging decisions and spatial charging patterns, offering practical insights and guidelines for scheduling, pricing, and differential pricing strategies to enable sustainable electrified logistics. The framework advances understanding of bidirectional interactions between logistics and power systems and provides a tractable, implementable approach for planning and operation in urban environments.

Abstract

This paper proposes an integrated equilibrium model to characterize the complex interactions between electrified logistics systems and electric power delivery systems. The model consists of two major players: an electrified logistics operator (ELO) and a power system operator (PSO). The ELO aims to maximize its profit by strategically scheduling and routing its electric delivery vehicles (e-trucks) for deliveries and charging, in response to the locational marginal price (LMP) set by the PSO. The routing, delivery, and charging behaviors of e-trucks are modeled by a perturbed utility Markov decision process (PU-MDP) while their collective operations are optimized to achieve the ELO's objective by designing rewards in the PU-MDP. On the other hand, PSO optimizes the energy price by considering both the spatiotemporal e-truck charging demand and the base electricity load. The equilibrium of the integrated system is formulated as a fixed point, proved to exist under mild assumptions, and solved for a case study on the Hawaii network via Anderson's fixed-point acceleration algorithm. Along with these numerical results, this paper provides both theoretical insights and practical guidelines to achieve sustainable and efficient operations in modern electrified logistics and power systems.
Paper Structure (16 sections, 5 theorems, 27 equations, 4 figures, 1 table, 2 algorithms)

This paper contains 16 sections, 5 theorems, 27 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. MODEL
  3. E-truck's routing, delivery, and charging problem
  4. ELO's reward design problem
  5. PSO's DC optimal power flow problem
  6. Equilibrium of integrated system
  7. NUMERICAL ANALYSIS
  8. Setting and solution algorithm
  9. Temporal impacts of charging demand on LMP
  10. Spatiotemporal analysis of LMP with e-truck operations
  11. Impact of LMP on e-truck operations
  12. CONCLUSIONS
  13. Proof of Proposition \ref{['prop:reward_fixed_point']}
  14. Proof of Proposition \ref{['prop:continuity_optimal_charging_action_flows']}
  15. Proof of Proposition \ref{['prop:LMP_continuity']}
  16. ...and 1 more sections

Key Result

Proposition 1

Given the PU-MDP specified in Sec. sec:PU-MDP, the optimal value $V^*$ is a continuously differentiable convex function of rewards $u$. In addition, the optimal action flow $x^*$ is a continuously differentiable function of rewards $u$, and further satisfies

Figures (4)

  • Figure 1: Oahu network (Transportation network: 36 zones, with depot marked as star; Power system: 31 substations, 37 buses, and 45 generators)
  • Figure 2: Impact of charging demand on LMP
  • Figure 3: Spatiotemporal distribution of LMPs
  • Figure 4: Impact of LMP on demands

Theorems & Definitions (8)

  • Proposition 1: Adapted from Prop. 2, Lemma 2 in yao2024PUME
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • proof
  • proof
  • proof