Coordinating V2V Energy Sharing for Electric Fleets via Multi-Granularity Modeling and Dynamic Spatiotemporal Matching

Abstract

The increasing adoption of electric delivery fleets introduces significant challenges related to uneven energy utilization and suboptimal scheduling efficiency. Vehicle-to-Vehicle (V2V) energy sharing presents a promising solution, but its effectiveness critically depends on precise matching and co-optimization within dynamic urban traffic environments. This paper proposes a hierarchical optimization framework to minimize total fleet operational costs, incorporating a comprehensive analysis that includes battery degradation. The core innovation of the framework lies in coupling high-level path planning with low-level real-time speed control. First, a high-fidelity energy consumption surrogate model is constructed through model predictive control simulations, incorporating vehicle dynamics and signal phase and timing information. Second, the spatiotemporal longest common subsequence algorithm is employed to match the spatio-temporal trajectories of energy-provider and energy-consumer vehicles. A battery aging model is integrated to quantify the long-term costs associated with different operational strategies. Finally, a multi-objective particle swarm optimization algorithm, integrated with MPC, co-optimizes the rendezvous paths and speed profiles. In a case study based on a logistics network, simulation results demonstrate that, compared to the conventional station-based charging mode, the proposed V2V framework reduces total fleet operational costs by a net 12.5% and total energy consumption by 17.4% while increasing the energy utilization efficiency of EV-Ps by 21.4%. This net saving is achieved even though the V2V strategy incurs a marginal increase in battery aging costs, which is overwhelmingly offset by substantial savings in logistical efficiency. This study provides an efficient and economical solution for the dynamic energy management of electric fleets under realistic traffic conditions, contributing to a more sustainable and resilient urban logistics ecosystem.

1. Introduction and Literature Review

1.1. Introduction

Driven by global ‘dual-carbon’ goals, the electrification of the transportation sector is advancing at an unprecedented pace. In urban logistics, electric delivery fleets have become a cornerstone of green, low-carbon transport. However, these fleets face unique operational challenges. Due to factors such as task assignments, travel routes, dynamic traffic conditions, and varying initial states of charge, significant energy utilization imbalances often arise within a fleet. Some vehicles, termed electric vehicle-providers (EV-Ps), complete their tasks with substantial surplus energy, while others, electric vehicle-consumers (EV-Cs), risk premature energy depletion and face ‘range anxiety’.

The traditional solution involves directing energy-deficient vehicles to fixed charging stations (CSs). This approach, however, has numerous drawbacks, including suboptimal station distribution, limited charger availability, and long queuing times during peak hours. When a vehicle’s energy level becomes critical far from a suitable station, the resulting detour not only consumes extra energy due to complex road conditions but also causes severe task delays, increasing time-window penalty costs and degrading overall fleet efficiency. Ultimately, the challenges facing electric fleets are deeply rooted in the three pillars of sustainability. Environmentally, the goal is to minimize energy consumption and reduce the carbon footprint per delivery. Economically, fleet operators must reduce operational costs—including energy, time penalties, and long-term asset depreciation like battery aging—to ensure financial viability. Socially and operationally, enhancing the efficiency of logistics reduces traffic congestion and lessens the strain on public charging infrastructure, contributing to a more resilient and functional urban system. To address these challenges, Vehicle-to-Vehicle (V2V) energy sharing has emerged as a novel, distributed energy interaction paradigm [1]. However, achieving efficient V2V energy sharing is non-trivial. It demands a solution to a complex, multi-faceted optimization problem that traditional research has not fully addressed [2].

1.2. Literature Review

In recent years, V2V energy sharing technology has become a research hotspot in intelligent transportation and energy management [3]. Existing studies primarily focus on supply–demand matching mechanisms, path planning and scheduling optimization [4].

The primary step in achieving efficient energy sharing lies in accurately identifying and matching potential supply-and-demand pairs. Hosseini et al. [5] proposed a V2V supply and demand matching strategy based on the Hungarian algorithm, which realizes the dynamic matching of the minimum cost under the distributed fog computing architecture by constructing a bipartite graph model of energy supply vehicles and energy demand vehicles. For the first time, this strategy integrates the cost of electricity, driving, and time waiting, and it provides a low-cost distributed energy trading solution for the Internet of Vehicles. Shurrab et al. [1] proposed a two-layer stable matching framework, the first layer of which uses Gale–Shapley game to minimize the cost of generation and maximize the profit of stable pairing, to solve the problem of individual conflict of interest in traditional matching. The second layer innovatively introduces the user satisfaction model, which screens the pairs with high performance willingness through satisfaction ranking, and increases the transaction success rate by 25–40%, overcoming the industry pain point of high default rate caused by low satisfaction. Zhang et al. [6] proposed a V2V dynamic matching protocol based on bipartite graphs, which optimizes social welfare through the maximum weight algorithm and introduces stable matching to solve individual rational problems. Wang et al. [7] proposed a Spatio-temporal coordinated V2V exchange strategy, which realizes supply and demand matching through Mixed-Integer Nonlinear Programming (MINLP) and Branch Cutting External Approximation Algorithm (BCBOA), and incorporates the waiting time of traffic lights into the energy consumption model for the first time, which solves the problem of charging overload in suburban scenarios. Liu et al. [8] proposed a dynamic V2V pairing algorithm based on path proximity, combined with the Dynamic Time Regularization (DTW) algorithm to quantify the Spatiotemporal similarity of the two vehicle paths. The path fragments are extracted through the sliding window, and the minimum cumulative distance between the EV-C and EV-P path nodes is calculated to ensure the Spatiotemporal alignment of the matching vehicle path.

Once matching pairs are identified, the next critical challenge is to plan optimal rendezvous and subsequent travel paths to minimize the additional costs incurred during the interaction. Yang et al. [9] proposed a time-oriented improved Dijkstra algorithm to minimize the travel time of electric vehicles through the dynamic weight mechanism and the queuing theory model. Sari et al. [10] proposed a hybrid ant colony algorithm, the core feature of which is multi-strategy fusion: by combining the genetic algorithm, particle swarm optimization or artificial potential field method, multi-objective collaborative optimization of path length, number of turns and energy consumption can be realized. Liu et al. [11] developed a JPS-enhanced A* algorithm to significantly improve the computational efficiency. Li et al. [12] developed a Q-learning dynamic path planning framework that combined LSTM traffic prediction with a Kuhn–Munkres (KM) matching algorithm. The innovation lies in the establishment of a longitudinal dynamic energy consumption model to generate the optimal charging path through real-time road condition prediction. In the measured road network in Nanjing, the scheme reduces the charging waiting time by 15% and realizes cooperative navigation with low communication overhead. Liu et al. [8] designed a Spatio-temporal collaborative path planning algorithm based on the dynamic V2V pairing algorithm of path proximity, which solves the problem of additional travel time and energy waste caused by the traditional parking charging mode, significantly shortens the charging time and improves the service efficiency. Yang et al. [13] proposed a crowd perception-driven path planning framework that reduced travel time by 21.7% and charging cost by 14.5% in the Shenzhen case. Wu et al. [14] developed a PHEV path–velocity joint optimization model, which solves the defect of traditional TSP ignoring velocity elasticity through sub-gradient cutting and effective inequality, achieves a maximum of 59% reduction in energy consumption, and promotes the refined scheduling of electric logistics.

Closely related to path planning is scheduling optimization, which involves coordinating the entire fleet or multiple energy-sharing events at a macroscopic level to ensure overall system efficiency and stability. Merkuryeva et al. [15] developed a simulation-based vehicle scheduling algorithm to solve the problem of minimizing the idle time of trucks with time windows, which is characterized by dealing with multiple constraints (such as capacity and time windows) through OptQuest and evolutionary algorithms, and using simulation models to verify the feasibility. Rashmi et al. [16] proposed a local optimal and fair allocation scheduling algorithm to solve the problem of charging cost optimization of electric vehicles (EVs), which is characterized by the use of the PSO algorithm to deal with dynamic EV arrival and the sliding window mechanism to achieve efficient grid load balancing. Gupta et al. [17] designed a hierarchical scheduling algorithm (VAA/VSA) for wind–solar charging stations, which prioritizes the use of renewable energy in synergy with V2V, and only uses the grid as a backup. The contribution is to verify the feasibility of collaborative diSPATch in reducing grid dependence (energy consumption by 94%) and cost (by 68%), and providing practical solutions for green charging infrastructure. Fan et al. [2] proposed a fairness-aware MADDPG scheduling algorithm and introduced Noisy Net to enhance the exploration ability. By designing a reward function with a fairness factor, the scheduling robustness problem when some chargers fail. Abdolmaleki et al. [18] proposed a V2V charging scheduling framework based on dynamic programming, which realizes the real-time scheduling of vehicle formation charging through Spatiotemporal-energy network modeling. It solves the problems of insufficient charging facilities, high battery cost and range anxiety, and reduces the peak pressure of the power grid by 15%. Kabir et al. [19] proposed a mobile charging vehicle scheduling framework, first established an ILP model, but proved to be an NP difficult problem, and then proposed three heuristic algorithms: SWSPF (the strictest time window first), SDSPF (minimum power demand first), EASPF (early arrival first), and in the Montreal case, the SDSPF algorithm achieved a service efficiency of 5.33 EVs/truck (close to the ILP optimal solution of 6.66), which solved the problem of insufficient coverage of fixed charging piles

Despite these significant advancements, a critical review of the literature as shown in

Table 1. Summary of this study in comparison with related works.

Ref.	Scenario	Time Window (TW)	Optimization Algorithm	Multi-Objective	Energy Model	Signal Info (SI)	Spatio-Temporal Matching	Limitations
[1]	Areas without charging stations	×	Gale-Shapley Stable Match	Cost, profit, user satisfaction	Economic model	×	√	TW and lDM are not taken into account
[2]	Charging station energy exchange	×	MADDPG	Cost/equity	×	×	×	TW and lDM are not taken into account
[5]	Highway/city charging stations	×	Hungarian Algorithm	Cost	Linear model	×	×	TW and lDM are not taken into account
[6]	EV systems in IoE environment	×	Max-Weight Matching, Stable Matching	Social welfare, grid load	Linear model	×	×	TW and lDM are not taken into account
[7]	Suburban aggregation station	√	MINP, BCBOA	Cost/load balancing	×	√	√	lDM are not taken into account
[8]	Wireless V2V charging in traffic	×	DTW, Spatio-temporal path planning	Charging efficiency	Linear model	×	√	TW and lDM are not taken into account
[13]	Urban dynamic road network	×	Branch and Cut	Travel time, cost	Linear model	×	×	TW and lDM are not taken into account
[14]	Urban logistics distribution	√	Branch-and-Cut	Energy cost	Longitudinal dynamics model (LDM)	×	×	SI are not taken into account
[15]	Truck distribution scheduling in the logistics center	√	OptQuest	The total idle time of the truck	×	×	×	lDM are not taken into account
[17]	Wind–solar charging stations	√	Vehicle Admission/Scheduling Algorithm	Grid dependency	×	×	×	lDM are not taken into account
This paper	Urban logistics energy mutual assistance	√	MOPSO, STLCSS	Time window, energy consumption	Longitudinal dynamics model	√	√	Consider TW and SI and LMD with spatiotemporal matching

reveals two persistent gaps. First, the majority of existing V2V routing and scheduling models rely on oversimplified, macroscopic energy consumption models. As shown in our review, these models typically neglect the nonlinear dynamics of the vehicle powertrain and, more importantly, fail to incorporate the influence of real-time traffic signal information (SPAT), which is a key determinant of energy efficiency in urban driving. This leads to inaccurate energy and travel time predictions, undermining the reliability of the entire optimization process. Furthermore, a fundamental limitation of prior work lies in its decoupled approach, which often neglects the nuanced spatiotemporal dynamics of vehicle matching while treating path planning and speed control as separate problems. This fragmentation inherently leads to suboptimal system-wide performance. In contrast, our framework resolves this by holistically co-optimizing these interdependent variables, ensuring that the vehicle pairing, their rendezvous path, and their speed profiles are mutually optimal.

To address these limitations, this paper proposes a hierarchical framework that bridges the gap between microscopic vehicle control and macroscopic fleet coordination. Our primary contributions are threefold: (1) Develop a high-fidelity energy surrogate model using MPC and SPAT data to provide accurate cost estimations. (2) Employ a sophisticated spatiotemporal matching algorithm (STLCSS) to identify the most promising V2V pairs. (3) Introduce an integrated optimization approach that co-optimizes rendezvous paths and speed profiles simultaneously, leading to a more efficient and economical V2V energy-sharing strategy.

2. Research Object and Problem Statement

2.1. Research Object

Figure 1 presents an integrated framework designed to optimize vehicle operations, with a particular emphasis on enabling energy sharing through vehicle-to-vehicle (V2V) interactions. The core of this framework is V2V communication technology, which serves as the fundamental mechanism for inter-vehicle coordination and data exchange. Surrounding this core are three interconnected modules, each providing distinct functionalities that collectively support the system’s operation.

Located in the top-left quadrant, the SUMO Vehicle Modeling module establishes a high-fidelity simulation environment that accurately replicates vehicle dynamics and traffic scenarios. This module’s capability to capture the complexities of real-world traffic is pivotal, as it generates reliable input data essential for subsequent optimization stages. The simulation integrity ensures that vehicle interactions, road network behaviors, and traffic flow patterns are authentically represented, providing a robust foundation for system-wide operational analysis.

Positioned in the top-right quadrant, the Scheduled Speed and Path Optimization module performs dynamic adjustments to predefined routes and velocities. By employing the Frank–Wolfe algorithm for route computation combined with PSO (Particle Swarm Optimization) for multi-objective enhancement, this module ensures vehicles adhere to highly efficient trajectories according to operational schedules. Energy efficiency serves as the primary performance metric, driving the selection of routes and speeds that minimize energy consumption while satisfying all scheduling constraints and operational requirements.

The Supply–Demand-Based Vehicle Matching for Energy Sharing module (bottom), which utilizes the specialized STLCSS algorithm, facilitates energy-sharing partnerships by matching vehicles with surplus energy to those in need within the energy network. This process supports V2V energy transfer, improves overall energy utilization, and offers operational flexibility for vehicles with diverse energy requirements.

These three components constitute the core architecture of the V2V energy reciprocity system, enabling seamless interaction between urban vehicles and cloud-based central control through bidirectional energy information and routing data exchange. This integrated mechanism ensures optimal task execution with enhanced energy efficiency and minimized operational consumption, while facilitating real-time system-level coordination and adaptive decision-making capabilities.

2.2. Technical Route for V2V Energy Sharing

In this paper, a three-level topology (node-section-path arc) of urban road network based on graph theory is constructed. The node layer integrates real-time phase information (SPAT) of traffic lights, and dynamically analyzes timing parameters such as traffic light cycle and green light window through V2I communication. The road section layer integrates the road speed limit and the road length; The path arc layer characterizes the energy characteristics of the complete travel path. The digital map is used to extract the pre-planned path, and the geometric attributes (node information, road length) and signal light time series are modeled by the open-source traffic simulation tool Netedit, so as to form a dynamic road network basic framework with temporal and SPAT constraints.

A longitudinal vehicle dynamics model quantifies the balance between traction, aerodynamic drag, and rolling resistance, capturing discrepancies between commanded and actual acceleration. SPAT-driven target speed range calculations ensure signal-passing within green windows, reducing idling losses. A MPC framework formulates a multi-objective optimization problem: the cost function combines energy consumption (piecewise traction/regenerative braking), traffic light synchronization penalties, and acceleration smoothness, with constraints including speed limits and jerk thresholds. The CasADi-IPOPT solver enables efficient nonlinear optimization. A cubic polynomial surrogate model linking average speed to energy consumption supports path-level evaluation.

Based on the Floyd–Warshall algorithm, the global shortest path is calculated, and a logistics distribution model with time windows and dynamic constraints is established. Multi-objective Particle Swarm Optimization (MOPSO) is used to optimize the path velocity sequence: the particle position encodes the road section speed, the objective function joint minimizes the energy consumption proxy model output and the time window default penalty, and the inertia weight linear attenuation balances the global exploration and local development. At the same time, the SPATiotemporal Longest Common Subsequence Algorithm (STLCSS) is used to achieve supply and demand matching: the SPAT distance threshold (ε, δ) is set, the trajectory point alignment is recursively calculated, and the normalized similarity score is used to screen the optimal EV-P/EV-C pairing, so as to solve the SPAT dislocation problem of traditional matching.

Vehicles are classified as Energy-Providers (EV-Ps, surplus SOC) or Energy-Consumers (EV-Cs, deficit SOC) via non-charging simulation. For matched pairs, candidate rendezvous points (nodes/neighboring stations) along the EV-C’s remaining path are generated. Multi-objective optimization selects rendezvous locations by maximizing STLCSS similarity, minimizing arrival time differences, and satisfying energy conservation (EV-P post-transfer SOC suffices for return). Energy transfer compensates the EV-C’s deficit, with dynamic SOC updates. Post-transfer speed re-optimization and receding horizon control ensure time-window compliance and mission completion.

The framework achieves a breakthrough through three layers of innovation: (1) coupled modeling of road network–signal lights–dynamics improves the accuracy of energy consumption prediction; (2) the MOPSO-STLCSS synergy mechanism reduces the suboptimal risk of path planning and matching; (3) the dynamic mutual aid strategy activates the redundant energy circulation of the fleet. Compared with the traditional charging station mode, it significantly reduces the additional energy consumption and time window default costs caused by bypasses, effectively improves the utilization rate of redundant energy, and achieves system-level optimization of both operating costs and total energy consumption.

The technology roadmap is shown in Figure 2:

3. Modeling of Road Section Efficiency and Dynamic Cost Quantification Under Free Traffic Flow

The energy efficiency optimization of urban transportation systems is a key technological pathway to achieving the ‘dual carbon’ goals. Traditional energy consumption models are mostly based on static road network assumptions, neglecting the dynamic nature of traffic signals, vehicle power characteristics, and the coupling effects of road network topology. As a result, these models fail to meet the needs of real-time energy consumption prediction and optimization in intelligent and connected environments. To address this, this chapter proposes a three-stage modeling framework of ‘multi-granularity road network modeling–energy efficiency agent training–dynamic cost quantification,’ aiming to overcome the limitations of traditional static models and achieve refined energy consumption prediction in complex traffic scenarios. The main notations and variables used throughout this paper are summarized in the Nomenclature table in Appendix A for easy reference.

3.1. Multi-Granularity Road Network Energy Consumption Modeling

3.1.1. Energy Consumption Model Based on Vehicle Dynamics

In a connected environment, various vehicle information such as position and speed can be obtained through V2I technology and GPS systems. Since the speed optimization is performed solely based on V2V and V2I information, lateral and vertical control of the vehicle are neglected, and a vehicle dynamics model is established. The longitudinal dynamics model constructed in this study is as follows:

$$x={\left[\begin{array}{cc}s& v\end{array}\right]}^T$$

(1)

where s denotes the vehicle position and v represents the vehicle speed. The control variable u represents the acceleration a, and the vehicle dynamics are described by the following system of ordinary differential equations:

$$\left\{\begin{array}{l}{\displaystyle \frac{ds}{dt}}=v(t)\hfill \\ {\displaystyle \frac{dv}{dt}}=a_{\mathrm{real}}(t)\hfill \end{array}\right.$$

(2)

In the equation, the actual acceleration a_real is determined jointly by the commanded acceleration a_cmd and the resistance acceleration.

$$a_{\mathrm{real},k}=a_{\mathrm{cmd},k}-{\displaystyle \frac{F_{\mathrm{resist},k}}{m}}$$

(3)

Here, F_resist,k represents the total resistance force, and m denotes the vehicle mass. The total resistance comprises aerodynamic resistance F_aero and rolling resistance F_roll (road gradient is neglected for urban roads). The calculation formulas are as follows:

$$F_{\mathrm{resist},k}={\displaystyle \frac{1}{2}}{C}_d{A}_f{\rho }_{\mathrm{air}}{v}_k^2+C_{r}mg$$

(4)

Here, C_d is the aerodynamic drag coefficient, A is the frontal area of the vehicle, ρ is the air density, C_r is the rolling resistance coefficient, and g is the gravitational acceleration.

Since the system is already linear, linearization is not required. The continuous-state equations for the vehicle’s longitudinal motion are given by

$$\dot{x}=Ax+Bu$$

(5)

$$A=\left[\begin{array}{cc}0& 1\\ 0& -{\displaystyle \frac{1}{{\tau }_d}}\end{array}\right],\hspace{1em}B=\left[\begin{array}{c}0\\ {\displaystyle \frac{T_s}{{\tau }_d}}\end{array}\right]$$

(6)

Here, $\dot{x}$ denotes the system state variables, and u represents the system control variables. The kinematic equations are discretized using an explicit discretization method based on the analytical solution of uniformly accelerated motion. Under the assumption of approximately constant acceleration, this method offers high accuracy and fast computation, making it suitable for real-time Model Predictive Control (MPC).

By integrating the position s, the velocity

$$v(t)={\displaystyle \frac{ds}{dt}}$$

(7)

is obtained. Assuming the actual acceleration a_real remains constant, the displacement update equation is given by

$$s_{k+1}=s_k+{\displaystyle {\int }_{t_k}^{t_k+\Delta t}\left(v_k+a_{\mathrm{real},k}(t-t_k)\right)}dt=s_k+v_k\Delta t+{\displaystyle \frac{1}{2}}{a}_{\mathrm{real},k}\Delta t^2$$

(8)

Here, s_k denotes the vehicle position at step k, v_k is the vehicle speed (m/s) at step k, a_k represents the actual acceleration at step k, and Δt is the time step duration (seconds).

These constraints ensure the safety and feasibility of the vehicle’s operation in real-world conditions. Next, we consider the velocity update equation. By integrating the velocity v(t) and assuming that the actual acceleration a_real remains constant, the velocity update equation can be derived as follows:

$$v_{k+1}=v_k+a_{\mathrm{real},k}\Delta t$$

(9)

Here, v_k+1 represents the velocity at the next time step, v_k is the velocity at the current time step, a_real,k denotes the actual acceleration at the current time step, and Δt is the time step duration.

Based on the dynamic model, the energy consumption and recovery characteristics are further quantified. The relationship between the traction force F_traction and the instantaneous power P_k is expressed as

$$F_{\mathrm{traction},k}=m\cdot a_{\mathrm{actual},k}+F_{\mathrm{aero},k}+F_{\mathrm{roll},k}$$

(10)

$$P_k=F_{\mathrm{traction},k}\cdot v_k$$

(11)

Energy consumption calculations distinguish between driving and braking conditions. When P > 0, the vehicle consumes energy during acceleration or cruising; when P < 0, the vehicle decelerates and recovers energy through regenerative braking.

$$\left\{\begin{array}{ll}\Delta E_{\mathrm{consumed}}=P\cdot \Delta t\hfill & (P>0)\hfill \\ \Delta E_{\mathrm{recovered}}=|P\cdot \Delta t|\cdot {\eta }_{\mathrm{regen}}\hfill & (P<0)\hfill \end{array}\right.$$

(12)

Here, Δt denotes the simulation time step, and η_regen represents the regenerative braking efficiency.

During braking, kinetic energy is converted into heat or electrical energy:

$$\Delta KE={\displaystyle \frac{1}{2}}m\left(v_k^2-v_{k+1}^2\right)$$

(13)

$$E_{\mathrm{brake}}=(1-\eta )\cdot \Delta KE$$

(14)

The total energy consumption E_total is calculated by the integral formula:

$$E_{\mathrm{total}}={\displaystyle \sum _{k=0}^N\left[\max (P_k,0)\Delta t-{\eta }_{\mathrm{regen}}\min (P_k,0)\Delta t\right]}$$

(15)

To simplify the model, it is assumed that energy conversion occurs without additional losses (e.g., the motor efficiency is assumed to be constant at 1), focusing primarily on capturing the macroscopic energy consumption trends.

3.1.2. Construction of the Planned Road Network Model

To achieve dynamic optimization of urban road network energy efficiency, this study constructs a three-level topological model of ‘node–edge–arc’ based on graph theory, as shown in Figure 3. i and j: These are “customer points,” marked with blue stars. They represent key locations in the network, such as the starting point and destination of a journey, or specific pickup/delivery locations.

a and b: These are “intermediate nodes,” marked with red circles. They represent road intersections or waypoints along a route that are neither the origin nor the final destination. They are points that must be passed through when traveling between other locations.

Specifically, the road network graph G is decomposed into the following layers:

Node: Represents road intersections or key points (e.g., customer points i, j and intermediate nodes a, b). Traffic signal nodes are defined as special control points, and their states (red/green light phases and remaining time) are updated in real time via V2I (Vehicle-to-Infrastructure) communication.

Edge: Physical road segments that connect adjacent nodes (e.g., ia, ab, jb), integrating road segment speed limits and traffic flow density data.

Arc: A complete path composed of multiple consecutive edges (e.g., the feasible path from customer point i to j), whose energy efficiency characteristics are influenced by the coordination of all traffic signals along the path.

The model dynamically integrates SPAT information to enable online parsing of parameters such as the signal cycle t_c, green light window t_green, and red light window t_red. Specifically, the traffic signal uses the red light phase as the cycle start point, and its state switching logic satisfies the following conditions:

$$t_{\mathrm{current}}=\mathrm{mod}(t,t_c) \mathrm{and} K_w=\left[{\displaystyle \frac{t}{t_c}}\right]$$

(16)

In the equation, K_w is the cycle counter used to synchronize the signal timings of multiple intersections. This design provides the dynamic temporal constraints necessary for subsequent speed planning.

Traditional cruise control (CCC) and car-following models (such as the Gipps model) have the issue of stopping at red lights. To address this, this study proposes a speed planning method based on SPAT information. This method analyzes the status of the upcoming traffic signals (including the remaining green light time t_{green_remain} and the red light waiting time t_{red_wait}) to calculate the target speed range [v_ilb,v_ihb]. When the speed of the extended-range vehicle falls within this range, it ensures the vehicle can pass through the traffic signal within the current green light time window, thus reducing stop or idling time and alleviating traffic congestion. The calculation of the target speed range is shown in Figure 4.

The formulas for calculating the target speed range based on SPAT information are given by Equations (17)–(21). It should be noted that these formulas must satisfy the constraints of urban road speed limits and the longitudinal dynamics of the vehicle. Additionally, the formulas also provide the calculation method for the traffic signal states during the modeling process. In order to increase the number of extended-range vehicles passing through the traffic signal within the same green light time window, thereby reducing the time for a specific trip, this section sets the target speed of extended-range vehicles as the upper limit of the target speed range. The specific formulas are as follows:

$$v_{i\mathrm{t}\mathrm{h}\mathrm{b}}(t)=\left\{\begin{array}{ll}{\displaystyle \frac{d_{ia}(t)}{K_w{t}_c-t_g-t}}\hfill & \mathrm{Green} \mathrm{light}\hfill \\ v_{i\max }\hfill & \mathrm{Green} \mathrm{light} \mathrm{and} {\displaystyle \frac{d_{ia}(t)}{K_w{t}_c-t}}\le v_{i\max }\hfill \\ {\displaystyle \frac{d_{ia}(t)}{K_w{t}_c+t_r-t}}\hfill & \mathrm{Other} \mathrm{cases} \mathrm{when} \mathrm{green} \mathrm{light}\hfill \end{array}\right.$$

(17)

$$v_{ilb}(t)=\left\{\begin{array}{ll}{\displaystyle \frac{d_{ia}(t)}{K_w{t}_c-t}}\hfill & \mathrm{Red} \mathrm{light}\hfill \\ {\displaystyle \frac{d_{ia}(t)}{K_w{t}_c-t}}\hfill & \mathrm{Green} \mathrm{light} \mathrm{and} {\displaystyle \frac{d_{ia}(t)}{K_w{t}_c-t}}\le v_{i\max }\hfill \\ {\displaystyle \frac{d_{ia}(t)}{(K_w+1)t_c-t}}\hfill & \mathrm{Other} \mathrm{cases} \mathrm{when} \mathrm{green} \mathrm{light}\hfill \end{array}\right.$$

(18)

$$\mathrm{Traffic} \mathrm{light} \mathrm{status}=\left\{\begin{array}{ll}\mathrm{Red} \mathrm{light}\hfill & \mathrm{if} 0\le \mathrm{mod}\left({\displaystyle \frac{t}{t_c}}\right)<t_r\hfill \\ \mathrm{Green} \mathrm{light}\hfill & \mathrm{if} t_r\le \mathrm{mod}\left({\displaystyle \frac{t}{t_c}}\right)\le t_c\hfill \end{array}\right.$$

(19)

$$t\_c=t\_g+t\_r$$

(20)

$$K_w>{\displaystyle \frac{k}{t_c}}$$

(21)

In the formula, v_i^target denotes the target speed of the i-th vehicle; v_i^lb and v_i^hb represent the lower and upper bounds of the target speed range, respectively; t is the total travel time of the vehicle; d_ia is the distance from the i-th vehicle to the upstream traffic signal a; K_w is an integer describing the cycle count of the traffic signal; t_g and t_r denote the durations of the green and red lights, respectively; t_c represents the duration of one full signal cycle; and mod is the modulo function producing the remainder of t divided by t_c.

Within the dynamic programming framework, traffic signal coordination is quantified through a time-window penalty term. The estimated time of arrival (ETA) of a vehicle at traffic signal a is defined as

$$t_{\mathrm{ETA}}={\displaystyle \frac{d_{ia}}{v_k}}$$

(22)

If $t_{\mathrm{ETA}}\notin [K_w{t}_c,K_w{t}_c+t_{\mathrm{green}}]$, the departure penalty function is defined as

$$P_{\mathrm{traffic}\_\mathrm{light}}=\mu \cdot \left(\max (0,K_w{t}_c-t_{\mathrm{ETA}})+\max (0,t_{\mathrm{ETA}}-(K_w{t}_c+t_{\mathrm{green}}))\right)$$

(23)

In the equation, μ represents the penalty weight used to dynamically adjust the vehicle speed in order to synchronize with the traffic signal timing. This mechanism effectively reduces energy waste caused by stopping at red lights, including idling fuel consumption and braking energy losses.

Following the proposed methodology, a set of pre-planned candidate driving routes is first established based on node coordinates and real-world road network data, as visualized in Figure 5. Subsequently, focusing on the longitudinal vehicle dynamics (while temporarily neglecting lateral dynamic effects), each extracted route is meticulously modeled and post-processed using the open-source traffic network simulation tool Netedit. Specifically, for every pair of directly connected nodes in the route topology, a precise road segment model is constructed that reflects actual road geometric features such as the number of lanes, gradient, and curvature. Additionally, timing models of key traffic control devices, such as traffic signals, are integrated. Through this series of modeling steps, the original route planning tasks are ultimately abstracted and transformed into directed arcs within the road network, each possessing specific energy consumption attributes—referred to as ‘energy arcs’—thus laying a solid foundation for subsequent energy consumption optimization or route evaluation.

3.2. Optimal Control Problem Solution for Vehicle Speed Planning Based on MPC

Addressing the vehicle passage problem at continuously signal-controlled urban intersections, the objective function is constructed as a weighted sum of electric energy consumption, travel time, speed fluctuations, and overspeed penalties. Considering constraints such as vehicle longitudinal dynamics, road speed limits, and target speed range, an optimal control problem for economically efficient vehicle speed planning through continuous urban arterial intersections is formulated based on the state transitions of the vehicle speed state variables. The following sections detail the design of the state transition matrix, objective function, and constraint conditions within this optimal control problem.

3.2.1. State Transition Matrix

To accommodate real-time MPC, an explicit discretization method based on the uniform acceleration assumption is employed. We introduce the state vector x_k and the control vector u_k. The state vector x_k consists of position s_k and velocity v_k, while the control vector u_k includes the commanded acceleration a_k. The state update equation can be expressed as

$$x_{k+1}=A{x}_k+B{\mathrm{u}}_k+D(v_k)$$

(24)

Here, matrices A, B, and D(v_k) represent the system dynamics matrix, control input matrix, and resistance influence matrix, respectively. Their specific forms are as follows:

$$A=\left[\begin{array}{cc}1& \Delta t\\ 0& 1\end{array}\right]$$

(25)

$$B=\left[\begin{array}{c}{\displaystyle \frac{{(\Delta t)}^2}{2}}\\ \Delta t\end{array}\right]$$

(26)

$$D(v_k)=\left[\begin{array}{c}-{\displaystyle \frac{{(\Delta t)}^2}{2m}}{F}_{\mathrm{resist}}(v_k)\\ -{\displaystyle \frac{\Delta t}{m}}{F}_{\mathrm{resist}}(v_k)\end{array}\right]$$

(27)

Finally, based on the analytical solution of uniformly accelerated motion, the discretized state update equation can be expressed as

$$x_{k+1}=f(x_k,u_k)=\left[\begin{array}{c}{s}_k+v_k\Delta t+{\displaystyle \frac{1}{2}}\left(u_k-{\displaystyle \frac{F_{\mathrm{aero},k}+F_{\mathrm{roll}}}{m}}\right)\Delta t^2\\ v_k+\left(u_k-{\displaystyle \frac{F_{\mathrm{aero},k}+F_{\mathrm{roll}}}{m}}\right)\Delta t\end{array}\right]$$

(28)

Here, F_aero and F_roll represent the aerodynamic resistance and rolling resistance, respectively, and u_k denotes the commanded acceleration.

3.2.2. Constraints

The constraints include speed limits, acceleration bounds, and jerk (rate of change of acceleration) limits:

$$\left\{\begin{array}{l}{a}_{\min }\le a_k\le a_{\max }\hfill \\ 0\le v_k\le v_{\max }\hfill \\ v[k]\le v_{\mathrm{legal}}(s[k])\hfill \\ s_k\ge s_{k-1}\hfill \\ j_k\le j_{\max }={\displaystyle \frac{a_{\max }-a_{\min }}{\Delta t}}\hfill \\ d_{\mathrm{safe}}\ge d_{\min }=5 \mathrm{m}\hfill \\ s_{\mathrm{ego}}[k]\le s_{\mathrm{lead}}[k]-(d_{\mathrm{safe}}+v_{\mathrm{ego}}[k]\cdot t_{\mathrm{reaction}})\hfill \\ \mathrm{if} s[k]\in [s_{\mathrm{stop}\_\mathrm{line}}-ϵ,s_{\mathrm{stop}\_\mathrm{line}}+ϵ], \mathrm{then} v[k]=0\hfill \\ v[k]\le v_{\max }+{ϵ}_v,\hspace{0.33em}{ϵ}_v\ge 0,\hspace{1em}{w}_e\cdot {ϵ}_v\to \min \hfill \end{array}\right.$$

(29)

v_legal(s) is the speed limit function for the road segment; d_safe is the static safety distance; s_ego(k), v_ego(k) represent the position and speed of the ego vehicle at step k, respectively; s_lead(k) is the position of the lead vehicle at step k; t_reactiont is the driver’s reaction time; and ϵ is the relaxation variable, which converts the hard constraint into a soft constraint.

3.2.3. Objective Function

For the research subject, the range-extended vehicle, the primary performance metric for evaluating the vehicle’s driving economy is the consumed electric energy, while also considering the requirement to arrive at the destination on time. Therefore, the optimization objective of the model focuses on two core indicators: energy consumption and travel time, ensuring that the vehicle minimizes energy consumption while reaching the target location within the specified time. To improve driving smoothness, a mechanism is designed to mitigate situations with large speed fluctuations. This is achieved through acceleration smoothing and jerk (rate of change of acceleration) control, reducing abrupt acceleration and deceleration, thereby avoiding unnecessary energy loss and negative impacts on passenger comfort. Additionally, the model incorporates a penalty term for speed points exceeding the target speed range, ensuring the vehicle maintains a reasonable speed range during operation, thus enhancing driving efficiency while balancing economy and safety.

(1) Speed Tracking Cost Term

This cost term aims to optimize driving efficiency and reduce travel time by minimizing the deviation between the vehicle’s speed and the preset maximum allowable speed. Its mathematical expression is the weighted sum of the squared differences in speed at each time step within the prediction horizon:

$$C_{\mathrm{speed}}=w_v\cdot {\displaystyle \sum _{k=0}^N{\left(v_{\max }-v[k]\right)}^2}$$

(30)

In this control model, v_max represents the speed limit of the road or the maximum allowable speed of the vehicle, while v(k) is the vehicle’s speed at the k-th time step within the prediction horizon.

(2) Acceleration Smoothing Cost Term

To improve driving smoothness and reduce the mechanical system load, this cost term penalizes abrupt changes in acceleration between adjacent time steps, thus constraining the continuity of the control inputs.

$$C_{\mathrm{accel}\_\mathrm{smooth}}=w_a\cdot {\displaystyle \sum _{k=1}^N{\left(a[k+1]-a[k]\right)}^2}$$

(31)

Here, a(k) represents the acceleration control input at the k-th time step.

(3) Jerk Penalty Term

Jerk (the time derivative of acceleration) directly affects passenger comfort. To suppress the shocks caused by rapid acceleration or braking, this cost term applies a quadratic penalty to jerk:

$$C_{\mathrm{jerk}}=w_j\cdot {\displaystyle \sum _{k=1}^N{\left(\frac{a[k+1]-a[k]}{\Delta t}\right)}^2}$$

(32)

In the equation, Δt represents the discrete time step. According to the ISO 2631 standard [20] for human vibration comfort, the jerk threshold is generally set to 0.8 m/s³.

(4) Energy Consumption Cost Term

For energy efficiency optimization in electric vehicles, this cost term differentiates between traction and regenerative braking conditions and constructs a piecewise energy consumption penalty function:

$$C_{\mathrm{energy}}=w_e\cdot {\displaystyle \sum _{k=0}^N\left\{\begin{array}{ll}a{[k]}^2\hfill & (a>0,\mathrm{driving})\hfill \\ {({\eta }_{\mathrm{regen}}\cdot a[k])}^2\hfill & (a<0,\mathrm{regenerative} \mathrm{braking})\hfill \end{array}\right.}$$

(33)

Here, η_regen represents the regenerative braking efficiency coefficient.

(5) Traffic Light Time Window Penalty Term

To reduce intersection delays and improve traffic flow, this cost term guides the vehicle to pass through the traffic light within the predicted green light time window. Its mathematical form is:

$$C_{\mathrm{tl}}={\displaystyle \sum _{i,j}\mathrm{II}}(Constraints)\cdot {\left(T_{\mathrm{arrive},i,j}-{\mathrm{proj}}_{green light}(T_{\mathrm{arrive},i,j})\right)}^2$$

(34)

$${\eta }_{i,k}={\displaystyle \frac{\mathrm{tl}\_{\mathrm{pos}}_{\mathrm{i}}{-\mathrm{s}}_{\mathrm{k}}}{\max (v_k,0.1)}}$$

(35)

$T_{arrive,i,j}$ represents the arrival time of the vehicle; $tl\_po{s}_i$ is the position of the iii-th traffic light; $v_k$, $s_k$ are the speed and position of the vehicle at the kkk-th time step, respectively; $green\_star{t}_i$ and $green\_en{d}_i$ are the start and end times of the green light, respectively; ${\eta }_{i,k}$ is the predicted arrival time of the vehicle; and L_lookahead is the lookahead distance.

(6) Moderate Speed Cost Function

Moderate speed plays a multifaceted positive role in vehicle control strategies, helping to achieve energy savings, safety, efficiency, and comfort in travel. Its mathematical form is:

$$J_{\mathrm{mid}}=w_{\mathrm{mid}}{\displaystyle \sum _{k=0}^{N-1}{\left(v_k-v_{\mathrm{mid}}\right)}^2}$$

(36)

$w_{mid}$ is the weight corresponding to the moderate speed, and $v_{mid}$ is the preset moderate speed value.

(7) Multi-Objective Coordination and Weight Design

$$C_{\mathrm{total}}={\displaystyle \sum _{k=0}^N\left\{\begin{array}{l}{w}_v\cdot {(v_{\max }-v[k])}^2\hfill \\ +w_a\cdot {\left(a[k+1]-a[k]\right)}^2\cdot \mathbb{I}(k>0)\hfill \\ +w_j\cdot {\left(\frac{a[k+1]-a[k]}{\Delta t}\right)}^2\cdot \mathbb{I}(k>0)\hfill \\ +w_{\epsilon }\cdot \left\{\begin{array}{ll}a{[k]}^2\hfill & \mathbb{I}(a[k]>0)\hfill \\ {({\eta }_{\mathrm{regen}}\cdot a[k])}^2\hfill & \mathbb{I}(a[k]<0)\hfill \end{array}\right.\hfill \\ +C_{\mathrm{tl}}\hfill \\ +w_{\mathrm{mid}}{\displaystyle \sum _{k=0}^{N-1}{(v_k-v_{\mathrm{mid}})}^2}\hfill \end{array}\right\}}$$

(37)

where ${\delta }_{\mathrm{activate}}[k]$ is the activation function, which takes the value of 1 when approaching an intersection. The total cost function is achieved by the weighted fusion of multiple objectives, enabling comprehensive optimization of the vehicle’s longitudinal motion control. Its core consists of five cost terms: speed tracking, acceleration smoothing, jerk suppression, energy consumption efficiency, and traffic light coordination, with their respective priorities adjusted through weight coefficients w_v, w_a, w_j, w_e, w_t.

In Model Predictive Control (MPC), the weight coefficients of each cost term are determined through multi-objective optimization theory. The core idea is to unify heterogeneous objectives into a dimensionless weighted sum.

$$w_i={\displaystyle \frac{{\alpha }_i}{{\sigma }_i^2}}$$

(38)

where α_i is the subjective adjustment factor for the priority of the objective (determined by the scenario requirements), and σ_i represents the standard deviation of the historical data for the corresponding cost term (used to eliminate dimensional differences).

$$\left\{\begin{array}{l}{w}_v={\displaystyle \frac{{\alpha }_v}{{(v_{\max })}^2}}\hfill \\ w_a={\displaystyle \frac{{\alpha }_a}{{(a_{\max }-a_{\min })}^2}}\hfill \\ w_j={\displaystyle \frac{{\alpha }_j}{{(j_{\max })}^2}}\hfill \\ w_e={\displaystyle \frac{{\alpha }_e}{P_{\mathrm{base}}^2}}\hfill \\ w_t={\displaystyle \frac{{\alpha }_t}{{(\Delta t_{\mathrm{window}})}^2}}\hfill \\ w_{\mathrm{mid}}=\left\{\begin{array}{ll}1.2w_{\mathrm{mid}}^{\mathrm{base}},\hfill & v<0.3v_{\mathrm{mid}}\hfill \\ 0.8w_{\mathrm{mid}}^{\mathrm{base}},\hfill & v>0.7v_{\mathrm{mid}}\hfill \\ w_{\mathrm{mid}}^{\mathrm{base}},\hfill & \mathrm{otherwise}\hfill \end{array}\right.\hfill \end{array}\right.$$

(39)

(8) Dynamic Weight Adjustment and Weight Varying with Scenario Changes:

$$w=w_{\mathrm{base}}\cdot \alpha (d_{\mathrm{it}},\Delta v)$$

(40)

Traffic Light Distance Adaptation:

$$\alpha (d_{it})=\left\{\begin{array}{cc}3.75\hfill & d_{it}<100 \mathrm{m}\hfill \\ 2.0\hfill & 100 \mathrm{m}\le d_{it}<300 \mathrm{m}\hfill \\ 1.5\hfill & \mathrm{There} \mathrm{exists} \mathrm{a} \mathrm{secondary} \mathrm{traffic} \mathrm{light} (d_{it2}<500 \mathrm{m})\hfill \\ 1.0\hfill & \mathrm{Other} \mathrm{cases}\hfill \end{array}\right.$$

(41)

Speed Error Compensation:

$$\alpha (\Delta v)=\left\{\begin{array}{cc}1.25\hfill & \mathrm{if} |v-v_{\mathrm{des}}|<2 \mathrm{m}/\mathrm{s}\hfill \\ 1.0\hfill & \mathrm{Other} \mathrm{cases}\hfill \end{array}\right.$$

(42)

Dynamic weight adjustment involves real-time regulation of the priorities of different control objectives, enabling the vehicle to make optimal decisions in complex and dynamic traffic scenarios. Its core significance lies in achieving dynamic balance among multiple objectives: when the vehicle approaches a traffic light, the system automatically increases the weight of traffic flow efficiency, prioritizing the calculation of the optimal passing speed or stopping strategy; during cruising, the focus shifts to energy consumption optimization, minimizing unnecessary acceleration fluctuations to improve energy efficiency; when encountering deceleration by the leading vehicle or sudden obstacles, the weight of the safety distance becomes dominant to ensure driving safety.

3.2.4. Solution Method

This study utilizes the combination of CasADi symbolic modeling and the IPOPT solver to address nonlinear optimization problems. CasADi, through symbolic expressions and automatic differentiation techniques, enables intuitive construction of mathematical models for complex systems (such as vehicle dynamics) and efficiently computes gradients and Hessian matrices. Additionally, CasADi supports the generation of efficient code, significantly improving computation speed, making it particularly suitable for systems with high real-time computation demands.

As a nonlinear optimization solver, IPOPT uses the Interior Point Method to handle large-scale, non-convex, and highly constrained optimization problems. It features sparse matrix optimization and adaptive parameter adjustment, ensuring numerical stability and efficient convergence during the solving process. In practice, CasADi symbolizes the optimization problem and passes it to IPOPT, leveraging its powerful solving capabilities and optimization strategies to ensure the efficient solution of complex optimization problems. Additionally, through warm-starting and dynamic parameter updates (such as time-varying traffic light constraints), this combination significantly enhances the real-time performance and dynamic adaptability of the optimization problem solving.

The combination of CasADi and IPOPT is particularly suitable for the rolling optimization problems in Model Predictive Control (MPC), allowing real-time adjustment of control strategies based on the latest system states and constraints. This combination not only benefits from being open-source and free, reducing deployment costs, but also offers excellent cross-platform compatibility, supporting the entire process from prototype development to embedded deployment. Therefore, CasADi and IPOPT have become the preferred tools for multi-objective nonlinear optimization problems in fields such as autonomous driving and robotics, balancing modeling flexibility, solving efficiency, and engineering practicality.

3.2.5. Generation of the Energy Consumption Surrogate Model

Initially, pre-planned candidate driving routes were extracted from the digital map. These routes were then input into the open-source traffic network simulation tool Netedit for detailed modeling, with a primary focus on the vehicle’s longitudinal dynamics (temporarily ignoring the effects of lateral dynamics). Specifically, for each pair of directly connected nodes in the road network, a precise road segment model was constructed to realistically reflect the geometric characteristics of the road, including parameters such as the number of lanes, slope, and curvature.

Building upon this, the model further integrated the timing and scheduling model for traffic signals and other key traffic control facilities to accurately capture the impact of traffic signals on the driving process during the simulation. Through these modeling steps, the initial route planning task was gradually transformed into a physically meaningful directed arc model in the road network. Each directed arc represents a specific road segment in the network and is assigned a particular energy consumption attribute, referred to as the ‘energy arc’. The energy consumption value considers the road’s geometric characteristics, traffic signal timing, and the influence of other traffic control facilities.

This series of modeling and post-processing steps transforms the initial route planning task into a surrogate model that can be used for energy consumption assessment and optimization. To further accurately assess the energy consumption of each ‘energy arc’, a curve fitting of speed versus unit motor shaft energy consumption was performed, which was then used as the surrogate model for energy consumption.

A cubic polynomial least squares fitting method is used to establish a quantitative relationship model between the average speed (v) and unit motor shaft energy consumption (E):

Mathematical model:

$$E(v)=a{v}^3+b{v}^2+cv+d$$

(43)

Fitting Principle:

$$\min {\displaystyle \sum _{i=1}^n{\left[E_i-\left(a{v}_i^3+b{v}_i^2+c{v}_i+d\right)\right]}^2}$$

(44)

where (v_i,E_i) represents the i-th experimental data point, and nnn is the total number of data points. This optimization process is implemented using the polyfit function from the NumPy library. Input the 500 data points from Figure 6 into the function to obtain the coefficients [a, b, c, d]. By minimizing the sum of squared vertical distances between the experimental data points and the fitted curve, the optimal polynomial coefficients are obtained using NumPy’s polyfit function, leading to the final energy-speed equation:

$$E=-7.61\times {10}^{-5}{v}^3+5.05\times {10}^{-3}{v}^2-4.06\times {10}^{-2}v+2.35\times {10}^{-1}$$

(45)

This fitting method demonstrates high accuracy within the experimental speed range, with a coefficient of determination R² of 0.7 and an average relative error of less than 3%. The figure also shows a ±5% fitting error band, providing a visual representation of the model’s prediction confidence interval. The curve clearly reveals the nonlinear relationship between energy consumption and speed: in the low-speed region (<10 m/s), energy consumption increases super-linearly; in the medium-speed region (10–20 m/s), it approaches a linear increase; while in the high-speed region (>20 m/s), a significant cubic growth characteristic is observed, with a notable minimum energy consumption point around 8.5 m/s.

4. Space-Time Coordination and Path Planning for Vehicles

4.1. Coordinated Speed-Route Optimization for Individual Vehicles

4.1.1. Problem Definition and Modeling

In urban logistics distribution systems, route planning must comprehensively consider multiple factors including road network topology, vehicle dynamics constraints, and energy supply–demand balance. The urban road network is defined as a weighted graph G = (V,E), where the node set V contains n delivery points (e.g., universities, warehouses), and the edge set E represents roadway connectivity. Each vehicle departs from the origin and sequentially visits multiple delivery points while satisfying the following constraints:

Time Window: Each delivery point has a strict time window [t_earliest,t_latest]; vehicles must arrive within this interval.

Dynamics: v∈[v_min,v_max].

Energy Balance: Vehicle energy consumption must align with charging/supply facility SPATiotemporal distribution.

Objective Function:

$$\mathrm{Total} \mathrm{Cost}=\min {\displaystyle \sum _{i=1}^n\left(E_i+P_e\cdot \Delta t_{early,i}+P_l\cdot \Delta t_{late,i}+C_C\right)}$$

(46)

where n is the fleet size, P_e and P_l are penalties for early/late arrivals, and C_C is the charging cost.

4.1.2. Inter-Node Distance Calculation (Haversine Formula)

For geographical coordinates (latitude/longitude) in urban road networks, we calculate spherical distances using the Haversine formula:

$$a={\sin }^2\left({\displaystyle \frac{\Delta \varphi }{2}}\right)+\cos \left({\varphi }_1\right)\cos \left({\varphi }_2\right){\sin }^2\left({\displaystyle \frac{\Delta \lambda }{2}}\right)$$

(47)

$$c=2\cdot \mathrm{atan}2\left(\sqrt{a},\sqrt{1-a}\right)$$

(48)

$$d=R\cdot cd$$

(49)

where R is the Earth’s radius, ϕ₁, ϕ₂ are start/end latitudes (radians), and λ_1,λ₂ are start/end longitudes (radians).

$$\Delta \phi ={\phi }_2-{\phi }_1$$

(50)

$$\Delta \lambda ={\lambda }_2-{\lambda }_1$$

(51)

4.1.3. Global Shortest Path Calculation (Floyd–Warshall Algorithm)

To obtain the shortest path matrix between all node pairs:

(i) Initialization: Adjacency matrix D⁽⁰⁾(i,j) stores initial edge weights (direct distances)

(ii) Iterative update: For each intermediate node k∈[1,n], update all node pairs (i,j):

$$D_k(i,j)=\min \left(D_{k-1}(i,j),D_{k-1}(i,k)+D_{k-1}(k,j)\right)$$

(52)

(iii) Termination: When k = n, D_(n)(i,j) contains global shortest paths.

4.1.4. Battery Aging Cost Model

To provide a holistic assessment of the V2V energy sharing strategy and address its long-term economic sustainability by accounting for the impact on fleet assets, we incorporated a battery aging (BA) cost model [21]. The total BA cost for the entire fleet, denoted as C_aging, is the summation of costs from every operational segment (e.g., driving, charging, V2V transfer) for every vehicle in the fleet. The aging cost for a single vehicle v during a specific operational segment ss, denoted as C_s(v,s), is modeled as follows:

$$C_s(v,s)=E_{cycled}(v,s)\cdot \beta \cdot {\alpha }_{soc}(v,s)$$

(53)

The total fleet aging cost is therefore

$$C_{\mathrm{aging}}={\displaystyle \sum _{v\in \mathrm{Fleet}}{\displaystyle \sum _{s\in \mathrm{Segments}(v)}{C}_s}}(v,s)$$

(54)

E_cycled(v,s) is the absolute amount of energy (in kWh) that is either charged into or discharged from the battery of vehicle v during segment s.

Β is the base degradation cost factor, a constant representing the baseline monetary cost per kWh of energy cycled under normal conditions (e.g., β = 0.1 ¥/kWh).

α_soc(v,s) is the dimensionless SOC Stress Factor, which applies a penalty for operating at non-ideal states of charge. It is defined as a piecewise function of the average SOC during the segment, SOC_avg(v,s):

$${\alpha }_{\mathrm{soc}}(v,s)=\left\{\begin{array}{cc}{\alpha }_{\mathrm{high}}\hfill & \mathrm{if} SO{C}_{\mathrm{avg}}(v,s)<20\% \mathrm{or} SO{C}_{\mathrm{avg}}(v,s)>80\%\hfill \\ {\alpha }_{\mathrm{low}}\hfill & \mathrm{if} 20\%\le SO{C}_{\mathrm{avg}}(v,s)\le 80\%\hfill \end{array}\right.$$

(55)

In our simulation, we set high = 1.5 and αlow = 1.0 to reflect the accelerated aging that occurs at the extremes of the SOC range. This comprehensive model ensures that every process affecting battery life—driving, conventional charging, and V2V energy sharing—is fairly quantified and contributes to the total aging cost.

4.1.5. Cost Calculation

This section calculates the total operating costs under both charging station and mutual energy supply modes through weighted summation of multi-dimensional cost components to achieve comprehensive cost evaluation.

Energy Costs:

Charging Station Mode:

$$C={\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{num}\_\mathrm{vehicles}}(E_{\mathrm{I},\mathrm{k}}+E_{\mathrm{C},k}\cdot \gamma -E_{\mathrm{F},k})}$$

(56)

Here, E_I,k is the initial energy of the k-th vehicle (kW·h), $E_{C,k}$ is the energy replenished at charging stations (kW·h), $E_{F,k}$ is the final remaining energy (kW·h), and γ is the electricity price (¥/kW·h).

Mutual Supply Mode:

$$\begin{array}{l}C={\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{num}\_\mathrm{vehicles}}(E_{I,\mathrm{k}}+E_{C,k}\cdot i-E_{F,k})}\\ i=\pm 1\end{array}$$

(57)

In this mode, i is a weighting coefficient: i = −1 for energy-supplying vehicles and i = 1 for energy-demanding vehicles.

Time Window Penalty Costs:

$$C_{\mathrm{P}}={\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{num}\_\mathrm{vehicles}}{C}_{\mathrm{P},k}}$$

(58)

where C_P represents the time window penalty cost (¥) for all vehicle, arising from early or late arrivals. C_P,k represents the time window penalty cost (¥) for the k-th vehicle, arising from early or late arrivals.

Time Consumes Costs:

$$C_t={\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{num}\_\mathrm{vehicles}}\rho T_k}$$

(59)

where C_t represents the cost of time (¥) incurred by all vehicles to drive, recharge, and perform delivery tasks. T_k denotes the total time (¥) consumed by the kth car.

4.1.6. Coordinated Route-Speed Optimization

This section uses MOPSO method to optimize vehicle speed along the shortest path derived by the Floyd–Warshall algorithm. The particle swarm optimization algorithm treats the solution of the optimization problem as a ‘particle’ in a multidimensional search space. Each particle has a position (representing the candidate solution) and velocity (representing the search direction and step size). For urban logistics scenarios, we focus on minimizing energy consumption and time window violations. Given the path node sequence R = [r₁, r₂, …, r_m] (m nodes), the adjacent node spacing di = distance (r_i, r_i 1). The velocity vi∈[v_min, v_max] needs to be assigned to each segment of the path i∈[1, m − 1], and the objective function consists of two parts:

Energy Consumption Items:

$$E(v_i,d_i)=[a{({\displaystyle \frac{v_i}{3.6}})}^3+b{({\displaystyle \frac{v_i}{3.6}})}^2+c({\displaystyle \frac{v_i}{3.6}})+d]\cdot d_i/100$$

(60)

The coefficients a = −7.61 × 10⁻⁵, b = 0.00505, c = −0.0406, d = 0.235 were obtained by fitting the measured data.

Time Window Penalties:

$$P_i=\left\{\begin{array}{l}{P}_e\times \max (t_{\mathrm{e}}-t_a,0)\hfill \\ P_l\times \max (t_a-t_l,0)\hfill \end{array}\right.$$

(61)

where [ej,lj] is the time window for node rj and tiarr is the arrival time.

ej represents the earliest allowed arrival time at task node j;

lj represents the latest allowed arrival time at task node j.

Objective function:

$$f(v)={\displaystyle \sum _{i=1}^{m-1}[E(v_i,d_i)+P_i]}$$

(62)

Constraints:

$${\mathrm{SOC}}_k={\mathrm{SOC}}_0-{\displaystyle \sum _{j=1}^{k}E(v_j,d_j)}\ge {\mathrm{SOC}}_{\mathrm{threshold}}\begin{array}{c}\hspace{1em}\forall k\in [1,m-1]\end{array}$$

(63)

Solutions that violate constraints are marked as infeasible (fitness is set to ∞).

The particle position represents the velocity sequence:

$$x=\left[v_1,v_2,\dots ,v_{m-1}\right]\in {ℝ}^{m-1}$$

(64)

Speed Update Equation:

$$v^{\left(t+1\right)} = {\omega }^{(t)}{v}^{(t)}+ c_1{r}_1\left(p_{best,t} -x^{(t)}\right) +c_2{r}_2\left(g_{best} -x^{(t)}\right)$$

(65)

Position Update Equation:

$$x^{\left(t+1\right)}=x^{\left(t\right)}+v^{\left(t+1\right)}$$

(66)

where i is the number of the particle, v^(t) and x^(t) represent the velocity and position of the particle at time t, p_best,t represent the historical optimal position of particle i, and g_best represents the historical optimal position of the population. ω denotes the inertia weight; c₁ and c₂ denote the learning factor, adjusting the weight of individual and group experiences, respectively; usually c₁ = c₂∈[1,2]; r₁, r₂ denotes the random number of [0,1]; and the search randomness is introduced to avoid local optimality.

Inertia weights decrease linearly:

$${\omega }^{(t)}={\displaystyle \frac{{\omega }_{max}-\left( \left({\omega }_{max}- {\omega }_{\min }\right)\cdot t \right)}{ T_{max}}}$$

(67)

where ω^(t) (the current inertia weight) represents the dynamic weight value at iteration step t, which directly controls the strength of the inertia of the particle motion; ω_max (maximum inertia weight) is the weight benchmark value at the initial iteration (t = 0), usually in the range of [0.8, 0.9]. ω_min (minimum inertia weight) is the lower limit of the weight when the algorithm is terminated (t = T_max), and the typical value range is [0.4, 0.6]. t (the current number of iterations) represents the execution progress of the algorithm, and its value ranges from 0 to T_max. T_max (maximum number of iterations) defines the overall threshold for the number of steps executed by the algorithm, and the optimization process is terminated when t reaches T_max. The linear decreasing design smoothly attenuates the inertia weight from the initial ω_max to ω_min, maintaining a large inertia in the early stage of optimization to enhance the global search ability, and reducing the inertia weight in the later stage to improve the efficiency of local fine development.

4.2. SPATIotemporal Energy Matching and Dynamic Route Planning

4.2.1. D Energy Situation Matrix

The energy situation matrix M(x,y,t,q) is defined as a four-dimensional representation, where x,y is geographic coordinates (SPATial dimensions), t is timestamp (temporal dimension), q is energy supply/demand quantity (positive values indicate demand, negative values indicate supply). Heuristic algorithms populate this matrix to quantify energy interaction requirements across SPATiotemporal units, providing a foundation for path merging decisions.

4.2.2. SPATiotemporal Longest Common Subsequence (STLCSS)

In multi-vehicle route optimization, evaluating similarity between vehicle trajectories is critical for optimizing scheduling and resource allocation. The SPATiotemporal Longest Common Subsequence (STLCSS) method, an enhanced version of the classic LCSS algorithm, integrates SPATial and temporal information to measure trajectory similarity. By defining SPATial and temporal thresholds, STLCSS identifies the longest subsequence where corresponding points in two trajectories meet both SPATial and temporal alignment criteria.

Given two trajectories:

$$\begin{array}{l}{T}_1=\left\{\left(x_{1i},y_{1i},t_{1i}\right)\right\}\\ T_2=\left\{\left(x_{2j},y_{2j},t_{2j}\right)\right\}\end{array}$$

(68)

Matching Conditions:

SPATial Match: The Euclidean distance between two points must satisfy:

$$d_s\left(i,j\right)=sqrt{(x_{1i}-x_{2j})}^2+{(y_{1i}-y_{2j})}^2\le ϵ$$

(69)

where ϵ is the SPATial distance threshold.

Temporal Match: The time difference between two points must satisfy:

$$d_t\left(i,j\right)=\left|t_{1i}-t_{2j}\right|\le \delta$$

(70)

where δ is the temporal difference threshold.

Recurrence Relation:

The STLCSS length L(i,j) is computed as

$$L\left(i,j\right)=\left\{\begin{array}{ll}0,\hfill & i=0 or j=0\hfill \\ L\left(i-1,j-1\right)+1,\hfill & d_s\left(i,j\right) \le ϵ and d_t\left(i,j\right)\le \delta \hfill \\ d_s\left(i,j\right)\le ϵ and d_t\left(i,j\right) \le \delta ,\hfill & otherwise\hfill \end{array}\right.$$

(71)

L(i − 1, j − 1) + 1: Increments the subsequence length when both SPATial and temporal matches occur.

max(L(i − 1, j), L(i, j − 1)): Skips a point to continue searching for the longest subsequence.

Similarity Calculation:

The normalized similarity score is

$$Similarity\left(T1,T2\right)=L\left(n,m\right)/min\left(n,m\right)$$

(72)

where L(n,m) is the final STLCSS length, and n,m are the number of points in T1 and T2, respectively.

4.2.3. Dynamic Route Planning

Paths are merged and re-optimized based on similarity thresholds. A mixed-integer program minimizes total energy consumption:

$$\min {\displaystyle \sum _{i=1}^{\mathrm{num}\_\mathrm{vehicles}}{\displaystyle \sum _{k=1}^K{E}_{i,k}}}{x}_{i,k}$$

(73)

Subject to vehicle capacity, time windows, path continuity, and energy balance.

Here, x_i,k∈{0,1} is a Binary decision variable (vehicle I selects path k or not), and E_i,k is an Energy consumption coefficient for vehicle i on path k, (kW·h).

4.3. Energy Mutual Assistance

4.3.1. SPaTiotemporal Matching Based on Energy State Classification

To facilitate V2V energy sharing, vehicles are first classified into energy-providing vehicles (capable of providing energy) and energy-demanding vehicles (requiring energy) based on their final State of Charge (SOC) after completing their assigned delivery routes under a non-charging simulation scenario. This classification process is as follows: Non-charging Simulation:

Energy State Classification: The first step is to determine each vehicle’s energy status by simulating its entire delivery mission without any charging events. This involves using the multi-objective particle swarm optimization to determine the optimal speed profile for each route segment, thereby minimizing a combined cost of energy consumption and time window penalties.

Final SOC Calculation: To classify each vehicle’s energy state, we first calculate its final State of Charge (SOCF) after completing the simulated delivery route without any charging events. This calculation is based on the vehicle’s initial State of Charge (SOCI), its total battery capacity, and the total energy consumed on its assigned path.

First, the total energy consumed, E_consumed (in kW·h), is calculated by summing the energy consumption over all m segments of the vehicle’s route, using our high-fidelity energy surrogate model (as detailed in Equation (57)):

$$E_{consumed}={\displaystyle \sum _{k=1}^m\left[\left(-7.61\times {10}^{-5}{v}_k^3+5.05\times {10}^{-3}{v}_k^2-4.06\times {10}^{-2}{v}_k+2.35\times {10}^{-1}\right)\cdot d_k\right]}$$

(74)

$$SO{C}_{\mathrm{F}}=SO{C}_{\mathrm{I}}-{\displaystyle \frac{E_{\mathrm{consumed}}}{C_{\mathrm{total}}}}$$

(75)

• SOC_I: The initial State of Charge of the vehicle (dimensionless, e.g., 0.9 for 90%).
• SOC_F: The final State of Charge of the vehicle after completing the tasks (dimensionless).
• E_consumed: The total energy consumed throughout the journey, as calculated by Equation (71) (in kW·h).
• C_total: The total rated capacity of the vehicle’s battery (in kW·h).
• v_k: The optimized speed on the k-th route segment (in m/s).
• d_k: The distance of the k-th segment (in meters).

$$\begin{array}{l}{S}_P=\left\{v_i\in v\hspace{0.33em}|\hspace{0.33em}SO{C}_F(v_i)>0\right\}\\ S_C=\left\{v_i\in v\hspace{0.33em}|\hspace{0.33em}SO{C}_F(v_i)\le 0\right\}\hspace{0.33em}\end{array}$$

(76)

$V=\{v_1,v_2,\dots ,v_N\}$ is the set of all vehicles in the fleet.

Vehicles in $S_p$ are classified as EV-P. The condition ${SOC}_F\left(v_i\right)>0$ indicates that the vehicle possesses remaining charge that can be utilized for sharing.

Vehicles in $S_c$ are classified as EV-C. The condition ${SOC}_F\left(v_i\right)\le 0$ signifies a depleted or critically low energy state, requiring external energy replenishment.

Following the classification of vehicles into Energy-Providers (EV-Ps) and Energy-Consumers (EV-Cs), the subsequent challenge is to identify the most compatible pairs for energy exchange. The classification serves as a critical prerequisite, transforming the fleet-wide coordination problem into a targeted bipartite matching problem between a ‘supply pool’ (EV-Ps) and a ‘demand pool’ (EV-Cs). The STLCSS algorithm is then employed as the core matching mechanism to quantify the spatiotemporal compatibility between potential provider-consumer pairs, ensuring that only feasible and efficient matches are considered.

Given two trajectories:

T1 = {(x1i,y1i,t1i)} (e.g., a EV-P’s trajectory)

T2 = {(x2j,y2j,t2j)} (e.g., a EV-C’s trajectory)

Matching Conditions:

(1)

PATial Match:

$$d_s\left(i,j\right)=sqrt{(x_{1i}-x_{2j})}^2+{(y_{1i}-y_{2j})}^2\le ϵ$$

(77)

The Euclidean distance between corresponding points must not exceed a SPATial threshold ϵ.

(2)

Temporal Match:

$$d_t\left(i,j\right)=\left|t_{1i}-t_{2j}\right|\le \delta$$

(78)

The time difference between corresponding points must not exceed a temporal threshold δ.

(3)

Recurrence Formula:

$$L\left(i,j\right)=\left\{\begin{array}{ll}0,\hfill & i=0 or j=0\hfill \\ L\left(i-1,j-1\right)+1,\hfill & d_s\left(i,j\right) \le ϵ and d_t\left(i,j\right)\le \delta \hfill \\ d_s\left(i,j\right)\le ϵ and d_t\left(i,j\right) \le \delta ,\hfill & otherwise\hfill \end{array}\right.$$

(79)

The algorithm recursively calculates the length of the longest SPATiotemporally aligned subsequence. If a pair of points meets both SPATial and temporal thresholds, the subsequence length increments; otherwise, it skips mismatched points to continue searching.

Similarity Calculation:

$$Similarity\left(T1,T2\right)=L\left(n,m\right)/min\left(n,m\right)$$

(80)

The similarity score is normalized by dividing the STLCSS length by the minimum trajectory length, ensuring a value between 0 (no similarity) and 1 (perfect alignment)

$$0\le L\left(n,m\right)\le 1$$

(81)

Optimal Supplier Assignment:

For each EV-C C_j∈D:

Compute STLCSS similarity scores with all EV-Ps P_i∈S, forming a score matrixM_|S|×|D|.

Assign the EV-P with the highest score to C_j:

$$s_i^{*}=\arg {\max }_{s_i\in S}M\left(i,j\right)$$

(82)

where S = {s₁, s₂, …, s_|S|} is a set of EV-Ps, and M(i,j) is the STLCSS score between EV-P P_i and EV-C C_j. This process ensures energy-efficient pairing based on SPATiotemporal trajectory alignment.

To provide a clear and reproducible implementation of this logic, the STLCSS algorithm is detailed in Algorithm 1.

Algorithm 1: Spatio-Temporal Longest Common Subsequence (STLCSS)

1: FUNCTION STLCSS(TrajA, TrajB, delta_d, delta_t): 2: // 1. Initialization 3: m = length(TrajA) 4: n = length(TrajB) 5: // Create an (m + 1) × (n + 1) Dynamic Programming (DP) table. 6: CREATE a 2D array dp of size (m + 1) × (n + 1) 7: INITIALIZE all elements of dp to 0 8: // 2. Fill the DP table 9: FOR i FROM 1 TO m: 10: FOR j FROM 1 TO n: 11: // 2.1 Check for a spatio-temporal match 12: spatial_distance = HaversineDistance(TrajA[i].coordinates, TrajB[j].coordinates) 13: temporal_difference = abs(TrajA[i].timestamp − TrajB[j].timestamp) 14: // 2.2 Apply the recurrence relation 15: IF spatial_distance ≤ delta_d AND temporal_difference ≤ delta_t THEN 16: dp[i][j] = 1 + dp[i − 1][j − 1] 17: ELSE 18: dp[i][j] = max(dp[I − 1][j], dp[i][j − 1]) 19: END IF 20: END FOR 21: END FOR 22: // 3. Return Result 23: RETURN dp[m][n] 24: END FUNCTION

Explanation of the Algorithm:

The algorithm utilizes a dynamic programming (DP) approach to efficiently compute the length of the STLCSS between two trajectories, TrajA and TrajB.

• Initialization: The algorithm begins by initializing an (m + 1) × (n + 1) DP table, named dp, where m and n are the number of points in each trajectory. Each cell dp[i][j] is designed to store the STLCSS length for the subtrajectories TrajA [1..i] and TrajB[1..j]. All values are initialized to zero.
• Dynamic Programming Recurrence: It then iterates through each pair of points from the two trajectories. For each pair (TrajA[i], TrajB[j]), it checks if they constitute a spatio-temporal match by comparing their spatial distance and temporal difference against predefined thresholds (delta_d and delta_t).

If the points match: They form a common point in the subsequence. The length of the STLCSS is thus incremented by one based on the length of the STLCSS of the preceding sub-trajectories (1 + dp[i − 1][j − 1]).

If the points do not match: No new common point is found. The length of the STLCSS is carried over from the longest subsequence found in the preceding comparisons, which is the maximum of dp[i − 1][j] and dp[i][j − 1].

3.

Final Result: After the table is fully populated, the value in the final cell, dp[m][n], represents the length of the STLCSS for the complete trajectories.

4.3.2. Mutual Assistance Rendezvous Point Optimization

The objective of rendezvous point optimization is to identify an optimal meeting point s* for supply and EV-Cs, enabling efficient SPATiotemporal coordination for energy mutual assistance. This process must satisfy the following criteria:

SPATiotemporal Synchronicity: Minimize the arrival time difference between vehicles at s*.

Energy Transfer Efficiency: Minimize energy loss during transfer while fulfilling the EV-C’s energy deficit.

Route Feasibility: Ensure paths to s* comply with topological constraints (e.g., road network connectivity).

Optimization Objectives:

$$s^{*}=\arg {\max }_{s\in C}[\alpha M(s)-\beta |t_P(s)-t_C(s)|-\gamma E_{\mathrm{t}}(s)]$$

(83)

$$M(s)={\displaystyle \frac{\mathrm{STLCSS}({\mathrm{Traj}}_P,{\mathrm{Traj}}_C)}{\max (L_P,L_C)}}$$

(84)

$$E_{\mathrm{t}}(s)=\max \left(0,E_{\mathrm{n}}(C)-E_{\mathrm{a}}(P)\right)$$

(85)

Maximize the SPATiotemporal matching score (M(s)), while minimizing both arrival time difference (|t_s(s) − t_d(s)|) and energy transfer cost (E_transfer), weighted by coefficients α,β,γ.

Here, L_P and L_C are the trajectory lengths of EV-Ps and EV-Cs, E_n is the Energy deficit of the EV-C (kW·h), and E_a is the transferable energy from the EV-P at s* (kW·h).

Optimization Process:

(1)

Candidate Set Construction:

Generate candidate points from Nodes along the EV-C’s remaining route.

$$\mathcal{C}=\{s\hspace{0.33em}|\hspace{0.33em}s\in {\mathrm{Path}}_C\}\cup \{s\hspace{0.33em}|\hspace{0.33em}\mathrm{dist}(s,{\mathrm{Path}}_C)\le {\delta }_s\}$$

(86)

Topology-based neighboring nodes (e.g., charging stations, intersections) within a SPATial radius δ_s (e.g., 1 km).

(2)

Trajectory Extraction & Similarity Calculation:

Extract trajectory segments Traj_P (EV-P) and Traj_C (EV-C) before reaching candidate point s. Compute similarity score M(s) using Equation (3.40).

(3)

Speed Coordination:

Optimize speeds vs (EV-P) and vd (EV-C) to minimize arrival time difference:

$${\min }_{v_P,v_C}\left|{\displaystyle \frac{d_P}{v_C}}-{\displaystyle \frac{d_P}{v_C}}\right|\hspace{1em}\mathrm{s}.\mathrm{t}.\hspace{1em}{v}_{\min }\le v_P,v_C\le v_{\max }$$

(87)

where v_P is the speed of the EC-P, v_C is the speed of the EV-C, d_P is the distance of the EC-P, d_C is the distance of the EC-C.

(4)

Energy Conservation Constraint:

Ensure the EV-P’s post-transfer energy suffices to return to the distribution center:

$$E_{\max }(s^{\ast })={\mathrm{SOC}}_P(s^{\ast })\cdot \mathrm{Capacity}-E_{\mathrm{r}}(s^{\ast })$$

(88)

where SOC_P (s*) is State of Charge (energy reserve) of the supply vehicle at meeting point s*, E_r(s*) is energy consumed during the return trip from s*, E_max(s*) is Maximum transferable energy from the EV-P (kW·h).

This framework balances energy efficiency, temporal coordination, and route feasibility for robust mutual assistance.

4.3.3. Energy Transfer

Energy Transfer Calculation Model

The energy transfer calculation aims to determine the optimal energy transfer amount Etransfer at the rendezvous point s*, balancing the EV-C’s energy deficit while preventing excessive discharge from the EV-P.

Core Formulas:

(1)

Energy Transfer Amount:

$$E_{\mathrm{t}}=\min \left(E_{\mathrm{n},\mathrm{c}},E_{\mathrm{a}}(s^{*})\right)$$

(89)

where E_n,c is Energy deficit of the EV-C (kW·h).

(2)

Energy Deficit Calculation:

$$E_{\mathrm{n},\mathrm{c}}=\mathrm{Capacity}\cdot \left(1-SO{C}_C(s^{*})\right)$$

(90)

where SOC_C(s*) is EV-C’s SOC at s*.

(3)

Available Energy Calculation:

$$E_{\mathrm{a}}(s^{*})=\mathrm{Capacity}\cdot \left(SO{C}_P(s^{*})-SO{C}_{\min }\right)$$

(91)

where Capacity is battery capacity (kW·h), SOC_C(s*) is remaining charge percentage of the EV-C upon reaching s*, SOC_P(s*) is Remaining charge percentage of the EV-P upon reaching s*, SOC_min is minimum safe discharge threshold for the EV-P.

Constraints:

(1)

Energy Conservation:

$$SO{C}_C^{\mathrm{new}}=SO{C}_C(s^{*})+{\displaystyle \frac{E_t}{\mathrm{Capacity}}}$$

(92)

The energy transfer must comply with energy conservation principles.

(2)

Post-Transfer SOC Requirement:

$$SO{C}_C^{\mathrm{new}}\ge SO{C}_{\mathrm{r}}$$

(93)

Transfer Time and Distance:

(1)

Transfer Time:

$$T_{\mathrm{t}}={\displaystyle \frac{E_{\mathrm{t}}}{P_e}}$$

(94)

where P_e is Mutual compensation power (kW).

Calculated based on energy transfer rate.

(2)

Target Distance:

$$D_{\mathrm{t}}=v_{\mathrm{t}}\cdot T_{\mathrm{t}}$$

(95)

Determined by transfer time and vehicle speed.

Energy Consumption Model for Mobility:

This model calculates energy consumption (Econsumed) for a vehicle traversing a route segment, linking speed (v), distance (d), and energy usage.

Core Formula:

$$E_{\mathrm{c}}={\displaystyle \frac{E}{100}}\cdot d$$

(96)

Energy consumption is proportional to distance traveled.

Post-Segment SOC Update:

$$SO{C}_F=SO{C}_I-{\displaystyle \frac{E_{\mathrm{c}}}{\mathrm{Capacity}}}\cdot 100\%$$

(97)

The remaining SOC after completing a route segment is dynamically updated based on distance and speed.

4.3.4. Post-Transfer Speed-Route Optimization

Remaining Route Planning

After energy transfer, the EV-C must complete its remaining delivery tasks. This requires:

Calculating the optimal route from the rendezvous point s* to the destination.

Ensuring the updated SOC (SOC_F) supports the remaining distance.

Satisfying time window constraints at all delivery points.

Optimization Objective:

$${\mathrm{Path}}_{\mathrm{remaining}}=\arg {\min }_{\mathrm{Path}}\left({\displaystyle \sum _{k=1}^K{E}_{\mathrm{c},k}}+{\displaystyle \sum _{k=1}^K{P}_{\mathrm{p},k}}\right)$$

(98)

Minimize total costs, including energy consumption (E_c,k) and time window penalties (P_p,k) for each route segment k.

Dynamic SOC Update Model

The SOC is dynamically updated for each segment of the remaining route based on distance, speed, and energy consumption:

Update Rule:

$${\mathrm{SOC}}_j={\mathrm{SOC}}_i-{\displaystyle \frac{E}{100}}\cdot {\displaystyle \frac{d_{ij}}{\mathrm{Capacity}}}\cdot 100\%$$

(99)

The SOC at the next node (SOC_j) depends on the current SOC (SOC_i), speed (v_ij), and distance (d_ij) between nodes i and j.

Time Window Constraints and Penalty Handling

SOC Lower Bound Constraint:

$${\mathrm{SOC}}_j\ge {\mathrm{SOC}}_{\mathrm{r},j}\hspace{1em}\forall j\in \mathrm{Path}$$

(100)

Ensure SOC ≥ SOC_required throughout the remaining route. If violated, adjust routes or speeds.

Time Window Penalty:

$${\mathrm{Penalty}}_j=P_e\cdot \max (0,t_{\mathrm{e},j}-t_{\mathrm{a},j})+P_l\cdot \max (0,t_{\mathrm{a},j}-t_{\mathrm{l},j})$$

(101)

Penalize deviations from delivery time windows using a multi-objective ant colony algorithm to optimize speeds.

5. Simulation Analysis

To validate the effectiveness and economic viability of the proposed energy-efficient V2V sharing framework for electric delivery vehicles, incorporating SPATio-temporal matching and path-speed co-optimization, a series of simulation experiments were designed. The experimental environment is based on an actual delivery network of a logistics company, employing light-duty electric vans as the subject vehicles. This section first details the experimental parameter settings, including network topology, node information, vehicle specifications, and cost components. Subsequently, the energy predicament under the original diSPATch plan is presented. Then, the superiority of multi-objective speed planning is analyzed. Finally, a comparative analysis is conducted, focusing on the operational costs, energy efficiency, and time-window management of the proposed V2V mutual-aid mode versus the conventional station-based charging mode.

5.1. Case Study of V2V Energy Sharing

The simulation scenario for this study is set within the delivery area of a logistics company. A total of 26 critical nodes are established, comprising 1 distribution center (Node 26, serving as the origin and destination for all vehicles), 20 customer nodes (Nodes 1–20; assuming task point 21 from original

Table 2. Task Point Coordinates and Related Information.

Node Type	Node Coordinates (°)	Time Window (h)
Task point	116.321, 39.959	8.0, 8.5
	116.365, 39.961	7.0, 7.5
	116.357, 39.981	8.0, 8.5
	116.328, 40.002	9.0, 9.5
	116.223, 39.991	9.0, 9.5
	116.461, 39.909	7.0, 8.0
	116.447, 39.937	7.5, 8.0
	116.403, 39.934	7.0, 7.5
	116.519, 39.924	8.0, 8.5
	116.548, 39.962	9.0, 9.5
	116.385, 39.907	7.5, 8.0
	116.331, 39.903	8.0, 8.5
	116.423, 39.916	7.0, 7.5
	116.422, 39.874	8.0, 8.5
	116.408, 39.904	7.0, 7.5
	116.495, 39.879	8.0, 9.5
	116.417, 39.892	8.5, 9.0
	116.354, 39.870	7.0, 7.5
	116.302, 39.895	9.0, 9.5
	116.374, 39.899	8.0, 8.5
	116.278, 39.904	9.0, 9.5

is re-indexed or the count is adjusted to 20 distinct task points for consistency), and 5 charging stations (CS, Nodes 22–25) incorporated to facilitate comparison between different replenishment strategies. Detailed coordinates and designated delivery time windows for task points are provided in Table 2, while coordinates and charging tariffs for stations and the depot are listed in

Table 3. Charging Station and Depot Coordinates and Related Information.

Node Type	Node Sequence Number	Node Coordinates (°)	Charging Price (¥)
CS	22	116.270, 39.960	1.5
	23	116.350, 39.920	1.5
	24	116.450, 39.920	1.5
	25	116.550, 39.900	1.5
Distribution center	26	116.472, 39.992

.

Acknowledging the complexity of urban road networks, a sparse network model was constructed where each node is connected to its four nearest neighbors.

The vehicle model is based on light-duty electric vans, with key parameters (e.g., gross vehicle mass, battery capacity, maximum/minimum speed, charging/discharging power) itemized in

Table 4. Vehicle-related Parameters and Cost Parameters.

Parameter Symbol	Parameter	Data	Unit
×	Fleet Size	9	Vehicles
m1	Gross vehicle weight	4495	kg
C	Rated capacity of the battery	200	Ah
Ubattery	Rated voltage	400	V
×	EV Battery Capacity	80	kWh
×	Initial SOC Distribution	[80.8, 19.6, 92.1, 91.0, 75.1, 14.4, 90.0, 18.3, 12.8]	%
Vmax	Maximum speed of electric vans	60	km/h
Vmin	Minimum speed for electric vans	10	km/h
Pcharge	Charging station charging power	50	kW
Pexchange	V2V transfer power	30	kW
×	Charging/Transfer Efficiency	97	%
×	SOC Threshold (for CS decision)	10.5	%
×	SOC Threshold (for V2V decision)	10.0	%
ξ	The unit time cost coefficient	5	¥/h
Pe	Penalties for early arrivals	5	kW·h/h
Pl	Penalties for late arrivals	10	kW·h/h

. Operational cost parameters, including daily vehicle usage cost, time consumption cost coefficient, and penalties for early/late arrivals at time windows, are summarized in Table 4. The critical vehicle model is based on light-duty electric vans, each with a battery capacity of 80 kWh. Key parameters, including gross vehicle mass, initial SOC distribution, operational thresholds, and cost components, are itemized in Table 4. For this study, our primary focus was on the logistical optimization of matching and routing. Therefore, to simplify the energy transfer model, the efficiency of both station-based charging and V2V energy transfer was assumed to be 97%. The critical parameters for the Multi-Objective Particle Swarm Optimization (MOPSO) algorithm employed in this study are detailed in

Table 5. Particle Swarm Optimization Parameters.

Parameter Symbol	Parameter	Data
ωmax	Maximum inertia weight	0.9
ωmin	Minimum inertia weight	0.4
Tmax	Maximum number of iterations	800
c1	The learning factor	1.5
c2	The learning factor	1.5
r1,r2	The random number	(0,1)
N	Number of particles	80

Detailed Parameter Description:

EV Battery Capacity: Each electric van in the fleet is modeled with a battery capacity of 80 kWh, which is representative of common light-duty electric logistics vehicles.

Delivery Demand Distribution: The demand is modeled as a fixed set of predefined mandatory customer nodes for each of the 9 vehicles, creating a deterministic and repeatable daily task assignment. The specific route assignments for each vehicle are detailed in

Table 6. Vehicle Route Assignments.

Vehicle ID	Assigned Route (Node Sequence, Depot = 26)
1	[26-13-21-26]
2	[26-11-12-4-26]
3	[26-15-3-26]
4	[26-9-19-26]
5	[26-2-18-7-26]
6	[26-5-22-26]
7	[26-16-1-20-26]
8	[26-8-14-26]
9	[26-10-6-17-26]

below. This setup ensures a realistic scenario where different vehicles have varying route lengths and energy requirements.

5.2. The Advantages of Multi-Objective Speed Planning

Figure 7 illustrates the baseline operational route for a fleet of electric vans, which was establishd prior to the implementation of V2V (vehicle-to-vehicle) energy sharing. To demonstrate the enhanced performance of multi-objective speed planning, this section provides a case study focused on vehicle 1, which was assigned to the delivery tasks at points 13 and 21. A comparative analysis was carried out to compare the energy consumption, time window penalty cost, and time consumption cost generated by two different algorithm methods: (i) the single-objective particle swarm (PSO) algorithm, which specifically prioritizes time window compliance, and (ii) the proposed multi-objective PSO, which optimizes both time window compliance and energy saving. The optimized travel path and corresponding speed profile for Vehicle 1, derived from the multi-objective strategy, are depicted in Figure 8. This visualization demonstrates that the multi-objective planning, by strategically modulating the vehicle’s speed, effectively achieves a balance between minimizing time window associated costs and reducing energy expenditure.

Further extending this analysis to the fleet level, the application of multi-objective speed planning yielded remarkable results. As shown in Figure 9, the speed optimization of the entire fleet has little impact on the penalty cost of the time window, and the time consumption cost is increased by ¥30.9. However, this was largely offset by a significant reduction in energy costs, which cost ¥79.7 per trip. As a result, the integrated approach resulted in a net reduction of ¥37.3 in overall operating costs per trip per diSPATch cycle, translating into an overall optimization efficiency of 7.7%.

5.3. Comparative Analysis of Energy Replenishment Strategies

This section compares the performance of two energy replenishment modes (charging station-based replenishment and V2V energy sharing) through simulation experiments, focusing on three dimensions: cost efficiency, time window compliance rate, and redundant energy utilization.

5.3.1. SPATio-Temporal Matching for V2V

Following the energy state classification (as per Section 4.1), vehicles were designated as either energy-providing (EV-P) or energy-consuming (EV-C). Specifically, vehicles #2, #6, #8, and #9 were identified as EV-Cs, with the remainder classified as EV-Ps. The SPATio-Temporal Trajectory Local Similarity Search (STLCSS) algorithm was employed to identify potential V2V exchange opportunities by analyzing the SPATio-temporal trajectories of EV-Ps and EV-Cs. The resulting STLCSS matching score matrix is presented in

Table 7. STLCSS Matching Score.

	EV-C2	EV-C6	EV-C8	EV-C9
EV-P1	50.0000	25.0000	12.5000	33.3333
EV-P3	14.2857	28.5714	14.2857	16.6667
EV-P4	40.0000	11.1111	25.0000	33.3333
EV-P5	11.1111	22.2222	25.0000	33.3333
EV-P7	16.6667	33.3333	16.6667	16.6667

. The similarity scores predominantly ranged between 30 and 50, indicating the algorithm’s efficacy in capturing latent SPATio-temporal interaction opportunities between potential donor and recipient vehicles.

5.3.2. Scenario Illustration: V2CS vs. V2V Replenishment

In the V2CS mode, energy-deficient EVs are diSPATched to the nearest feasible charging station. For instance, consider the pair EV-P7 and EV-C6 (where EV-C6 requires charging). The operational trajectory for EV-C6, including a detour to charging station CS24 for replenishment, is depicted in Figure 10. Although EV-C6 completed its deliveries post-recharge, the time incurred for accessing and utilizing the charging station led to significant deviations from its designated service time windows.

Conversely, the dynamic V2V energy exchange mechanism, illustrated with the donor–recipient pair (EV-P7 → EV-C6) in Figure 11, facilitates direct energy transfer at optimized rendezvous points along their routes (e.g., between nodes S8 and S2). In this instance, EV-P1 transferred 19.76 kW·h to EV-C6, enabling EV-C6 to continue its mission without detouring to a charging station. The comprehensive V2V fleet-wide route plan, integrating such dynamic exchanges, is presented in Figure 12.

5.3.3. Performance Evaluation and Discussion

The V2V energy exchange strategy demonstrated marked superiority over the V2CS mode in terms of cost control and efficient energy utilization. To provide a comprehensive and fair comparison, our analysis was updated to include battery aging (BA) costs, as detailed in Section 4.1.4. Figure 13 illustrates the changes in individual vehicle operational costs, while the revised Figure 14 provides a comparative summary of total fleet operational costs, now including the BA component, under both replenishment paradigms.

Our analysis reveals a key trade-off. As shown in Figure 14, the V2V strategy incurs a slightly higher total battery aging cost (25.8 ¥) compared to the V2CS strategy (23.2 ¥). This result is expected and validates our model. The increase is primarily due to the “extra cycle” inherent in V2V sharing: energy is first discharged from the provider vehicle (EV-P) and then charged into the consumer vehicle (EV-C). This double-handling of energy naturally leads to slightly higher cumulative battery wear across the fleet.

However, and most importantly, this modest increase in battery aging cost (2.6 ¥) is overwhelmingly offset by the substantial savings in other operational areas. The V2V strategy achieves significant reductions in:

Time Window Penalty (TWP) costs: A reduction of 12.5% (from 142.9 ¥ to 125.0 ¥).

Energy Consumption (EC) costs: A reduction of 17.4% (from 141.7 ¥ to 117.0 ¥).

Time Costs (TC): A reduction of 11.6% (from 165.1 ¥ to 145.9 ¥).

When all cost components are aggregated, the net effect is a 12.5% reduction in overall operating costs compared to the V2CS model. These findings highlight the robust economic benefits of internalizing the remaining energy cycle within the fleet, demonstrating that the logistical efficiencies gained far outweigh the marginal increase in long-term battery degradation.

The V2V exchange model excels at utilizing the fleet’s remaining energy capacity. Figure 15 depicts the energy transfer relationship established in this mode. In stark contrast, the V2CS approach relies entirely on external grid power, leaving a large portion (typically 50–70%) of the remaining SOC that is not utilized in the vehicles that provide energy (e.g., EV-P1, EV-P4, EV-P5, EV-P7). The energy transmission network further shows that 66.2 kW·h of redundant energy is used. As shown in Figure 16, the V2V energy sharing model achieves a 24.7 kW·h reduction in total energy consumption per fleet mission (17.4 creases) and a 21.4% increase in energy efficiency (EU) for EV-PS, demonstrating that V2V not only improves the energy efficiency of the entire fleet, but also significantly reduces operational energy consumption.

5.4. Sensitivity Analysis

To address the potential impact of real-world uncertainties and enhance the confidence in our findings, we conducted a sensitivity analysis to evaluate the robustness of our proposed V2V framework. The original results were point estimates based on our high-fidelity energy consumption surrogate model. However, the accuracy of this model can be influenced by unmodeled factors such as payload variations, changing road conditions, or tire pressure. This analysis, therefore, assesses the stability of our framework’s performance against systematic perturbations in the core energy model.

Methodology:

We performed four sets of simulation runs, each comparing our proposed V2V sharing mode against the conventional charging station (V2CS) mode:

Baseline: Using the original energy consumption model.

Perturbation 1: Increasing the energy consumption of all vehicles by 1%.

Perturbation 2: Increasing the energy consumption of all vehicles by 2%.

Perturbation 3: Increasing the energy consumption of all vehicles by 3%.

For each run, we calculated the total operational costs for both modes and the resulting percentage cost reduction achieved by our V2V framework. The results of this analysis are summarized in

Table 8. Sensitivity Analysis of Total Operational Costs (¥) under Energy Model Perturbations.

Level	V2V Sharing Cost (¥)	Charging Station Cost (¥)	Cost Reduction (¥)	Cost Reduction (%)
0% (Baseline)	416.11	472.81	56.70	11.99%
+1%	417.02	473.16	56.14	11.86%
+2%	418.79	474.62	55.83	11.76%
+3%	420.56	477.04	56.48	11.84%
Mean	418.12	474.41	56.29	11.86%
Variance	3.72 (¥2)	3.35 (¥2)	0.13 (¥2)	0.009 (%)2

.

The analysis clearly demonstrates the robustness of our proposed framework. Despite introducing up to a 3% uncertainty in the energy consumption model, the performance advantage of the V2V sharing strategy remains highly consistent. The percentage reduction in total operational cost stayed within a very narrow range (11.76% to 11.99%), with a mean reduction of 11.86%. The extremely low variance across all scenarios (e.g., 0.13 (¥²) for cost reduction and 0.009 (%)² for the percentage) quantitatively confirms this stability. This indicates that the benefits of our framework—such as avoiding long detours, minimizing time window penalties, and utilizing surplus fleet energy—are fundamental structural advantages and are not overly sensitive to minor inaccuracies in model parameters. This analysis provides a stronger, statistically supported validation of our conclusions.

5.5. Practical Implications and Scalability

Our proposed framework has significant practical implications for various stakeholders.

For Logistics Providers: The primary benefit is direct operational cost reduction, as demonstrated by the 13.8% decrease in our case study. This stems from reduced energy expenses and lower time-window penalties. Furthermore, by internalizing energy replenishment within the fleet, companies can reduce their reliance on public charging infrastructure, which minimizes vehicle downtime spent queuing at stations. This increases asset utilization and enhances the fleet’s operational resilience, especially in areas with sparse charging coverage. Fleet managers can integrate this framework into their existing dispatch systems as an intelligent energy management module.

For City Planners and Grid Operators: The V2V energy sharing model promotes a decentralized energy ecosystem. By reducing the fleet’s collective demand on fixed charging stations during peak hours, it helps alleviate strain on the local power grid. This can defer the need for costly grid infrastructure upgrades. Moreover, the integration of SPAT information for speed optimization contributes to smoother traffic flow and reduced congestion at intersections, aligning with smart city initiatives for efficient and green urban mobility.

Scalability: The scalability of our hierarchical approach is a key consideration. The low-level MPC-based speed control is inherently scalable as it operates on individual vehicles. The high-level optimization, involving MOPSO and STLCSS, is more computationally intensive. For small-to-medium fleets (e.g., up to 50–100 vehicles), the proposed centralized optimization is feasible with modern computing power. For larger fleets, a decentralized or clustered approach would be necessary to manage computational complexity. Future work could explore dividing the fleet into regional clusters, with intra-cluster V2V optimization and inter-cluster coordination, or employing distributed machine learning algorithms like Multi-Agent Reinforcement Learning (MARL) for fully decentralized decision-making.

6. Conclusions

This paper successfully developed and validated a hierarchical optimization framework for V2V energy sharing that addresses critical sustainability challenges in electric delivery fleets. By tightly coupling macroscopic path planning with microscopic speed control, our approach delivers quantifiable benefits across the environmental, economic, and operational pillars of sustainability. The key innovation lies in the tight coupling of macroscopic path planning with microscopic speed control. This was achieved through a multi-stage process: constructing a high-fidelity energy surrogate model via MPC and SPAT data, employing the STLCSS algorithm for precise SPATiotemporal matching of donor and recipient vehicles, and utilizing a MOPSO algorithm for the integrated optimization of rendezvous routes and speed profiles.

Simulation results, based on a realistic logistics network, quantitatively confirmed the framework’s superiority. Compared to traditional station-based charging, our V2V approach demonstrated robust performance, reducing total operational costs by an average of 11.9% (with low variance against model perturbations) and energy consumption by 17.4%. Furthermore, a sensitivity analysis confirmed that these benefits are stable and not overly dependent on the precise accuracy of the energy model, highlighting the structural advantages of the proposed framework. Crucially, the framework unlocked the fleet’s latent energy potential, and the utilization of surplus energy from provider vehicles increased by 21.4%. This not only offers a direct economic benefit but also contributes to a more resilient and sustainable logistics ecosystem. By bridging the gap between detailed vehicle dynamics and fleet-level coordination, this work presents an effective and economical strategy for the advanced energy management of modern electric fleets.

For future work, several avenues warrant exploration to facilitate practical deployment. First, conducting a real-world pilot study with a logistics partner would be essential to validate the model’s performance under unpredictable real-world conditions, including communication delays and unexpected traffic events. Second, the framework relies heavily on reliable Vehicle-to-Infrastructure (V2I) and Vehicle-to-Vehicle (V2V) communication. Future work should focus on developing secure, low-latency communication protocols tailored for V2V energy-sharing transactions, ensuring data integrity and protecting against cyber threats. Finally, developing scalable, decentralized optimization algorithms, such as those based on multi-agent reinforcement learning, will be critical for applying this framework to very large-scale fleets, ensuring real-time performance without prohibitive computational overhead.