Spatio-temporal coordinated dispatch of electric vehicle clusters and assessment of dispatchable potential, taking into account dynamic congestion at charging stations

1 Introduction

With the rapid growth in the number of electric vehicles (EVs) and the high proportion of distributed photovoltaic (PV) systems connected to distribution grids, the coupling between the transport and energy sectors is deepening. However, there is a significant temporal mismatch between the temporal characteristics of distributed PV—which generates high output at midday and low output in the evening—and the disorderly charging behavior of EVs, which typically charge late at night and stop charging early in the morning. Furthermore, influenced by user travel habits and geographical accessibility, a large volume of charging demand often concentrates in local hotspots, further inducing issues such as distribution network node overloads, voltage exceedances, and regional load imbalances. Consequently, how to achieve coordinated regulation of EV charging demand across temporal and spatial dimensions whilst meeting user travel needs, and to quantify its dispatchable potential, has become a key issue in the flexible operation of new-generation distribution networks.

Extensive research has been conducted on the modelling of EV charging loads and the assessment of their regulation potential. Regarding load forecasting, Jokinen and Lehtonen (2023) analysed data from the US Department of Transportation’s National Household Vehicle Survey. Assuming that EV user behavior is consistent with that of internal combustion engine (ICE) vehicles, the study fitted the probability distributions of key random variables using ICE vehicle behavior data and generated charging load curves through Monte Carlo simulation. To further improve accuracy, Tang et al. (2025) introduced the concept of trip chains, constructing a spatio-temporal distribution model that accounts for multi-purpose trips, thereby providing a more detailed characterization of user behavior. To further explore the deep interactions between networks, Wei et al. (2018) developed a system equilibrium model that couples the transport network with the distribution grid. This study combines the user equilibrium model of the transport network with the optimal power flow of the distribution network, providing an in-depth simulation of road network congestion characteristics, node electricity prices, and the route-selection behavior of individual electric vehicles seeking the lowest travel and charging costs. It offers an analytical framework for dissecting the spatio-temporal coupling mechanisms between traffic flow and the distribution network. Jawad and Liu (2023) further evaluated the mobility of charging loads within urban transport networks using spatio-temporal analysis methods, revealing the significant influence of dynamic traffic conditions and network topology on users’ choice of charging stations and the spatial redistribution of loads. Having clarified the evolution of the spatio-temporal distribution of loads, the key to further optimization lies in accurately quantifying and harnessing the dispatchable potential of vast numbers of dispersed vehicles. Barthel et al. (2021), through an empirical analysis of real-world electric vehicle charging data, indicates that the charging flexibility of different types of vehicle fleets is highly dependent on the actual parking time windows and inelastic energy demands of the vehicles. This suggests that the refined classification and identification of vehicles’ flexibility potential is a crucial prerequisite for subsequent aggregation modelling. On this basis, Al Taha et al. (2025) proposed an EV cluster modelling method based on Minkowski summation theory, aggregating EV clusters as generalised energy storage devices. It established dispatchability models for both individual EVs and clusters to quantify the dispatchable potential of the clusters. Research in Arnold et al. (2024) demonstrated that the synergy between ordered charging and energy storage systems can significantly reduce the peak-to-valley load difference in the power grid and enhance system operational stability.

However, existing research still suffers from two major shortcomings. Firstly, most existing transport–grid coupling studies focus on charging load forecasting; their output typically remains at the level of load distribution and has not yet established an effective link with the assessment of EV dispatchable potential, lacking a unified analytical framework that integrates ‘user travel behavior—spatial charging demand—cluster dispatchability’. Secondly, existing studies on electric vehicle cluster aggregation are predominantly based on modelling charging and discharging flexibility in the temporal dimension, typically assuming that vehicles can access charging facilities without constraints, whilst overlooking spatial constraints within the transport network. In fact, the dispatchability of electric vehicles depends not only on parking duration and state-of-charge limits, but is also jointly influenced by real-time traffic conditions, charging station accessibility, and local node capacity constraints. When large numbers of vehicles congregate at a few hotspot nodes due to geographical inertia, even if charging and discharging flexibility exists in the temporal dimension, the scheduling potential may be difficult to realise in practice due to local distribution network capacity bottlenecks, thereby overestimating the system’s true flexibility limits (Wang et al., 2025). Therefore, it is necessary to construct a spatio-temporal collaborative analysis framework that integrates both traffic spatial behavior and grid temporal scheduling, to achieve the rational allocation of electric vehicle charging demand and the quantitative assessment of true scheduling potential.

To systematically address the aforementioned issues, this paper proposes a framework for the allocation of electric vehicle energy demand and the assessment of dispatchable potential, tailored to a ‘vehicle-road-station-grid’ coupled system. Firstly, a spatiotemporal demand allocation model for electric vehicles is developed, taking into account the dynamic traffic network. By integrating trip-chain theory, origin-destination (OD) behavioral characteristics, and the Greenshields speed–density model, a driving behavior model for electric vehicles under traffic constraints is established. A charging station scheduling radius mechanism is proposed to achieve the rational allocation of user charging demand in the spatial dimension.

Secondly, a method for assessing the dispatchable potential of electric vehicle clusters under spatial constraints is proposed. Based on Minkowski summation theory, flexible and semi-flexible electric vehicles are modelled collectively; through rigid load shedding and boundary extraction, the dynamic dispatchability of the EVA is accurately quantified. Finally, a framework for the coordinated operation of photovoltaic systems, energy storage and EVs is constructed, and the boundary effects of spatial scheduling degrees of freedom are revealed. Through a coupled traffic network and distribution grid case study, the impact of spatio-temporal coordination strategies on system economics, load smoothing capability and spatial equilibrium is analysed, and it is demonstrated that the spatial scheduling radius has an optimal physical boundary for the specific Sioux Falls test system considered in this paper.

To clearly illustrate the overall physical logical boundaries of the strategy proposed in this paper, Figure 1 presents the overall logical relationships of the proposed spatio-temporal coordination framework across three levels: the transport domain, the power grid domain, and the dispatch domain. The transport domain is responsible for generating the spatio-temporal demand for electric vehicles, the power grid domain performs the aggregation and optimised dispatch of solar-storage-vehicle resources, whilst the dispatch domain carries out the quantitative assessment of dispatchable potential.

2 Modelling the spatio-temporal distribution of preliminary EV dispatch loads

EV charging demand exhibits significant spatiotemporal heterogeneity, and its formation is closely related to user travel patterns, parking behavior, and the state of the transport network. As different vehicle types exhibit distinct differences in travel purposes, operating modes, and parking characteristics, this paper constructs separate travel behavior models for private EVs and electric taxis to generate vehicle spatiotemporal trajectories that conform to real-world traffic patterns, thereby providing foundational inputs for subsequent charging demand allocation (Jenkins and Kockar, 2022).

2.1 Analysis of electric vehicle travel behavior characteristics

2.1.1 Travel characteristics of electric private car users based on the travel chain theory

To characterise the typical daily travel patterns of electric private car users, this paper establishes a residential vehicle travel model based on the travel chain theory. Given that residential travel exhibits strong temporal regularity and activity continuity, residential areas, office districts and commercial zones are treated as primary functional zones. As illustrated in Figure 2, four typical travel chain patterns are constructed to describe vehicle movement behavior throughout the day.

Based on statistical patterns of residential travel, the start times of electric vehicles’ first trips follow a normal distribution. Parking duration is determined by the nature of the destination; for private vehicles, parking duration in residential areas follows a two-parameter Weibull distribution. Parking duration in office and commercial districts is characterised using a generalised extreme value distribution to reflect differences in user parking behavior across different scenarios (Zhang et al., 2022), and Monte Carlo sampling is employed to generate individual vehicle trip plans, thereby ensuring that aggregate travel demand conforms to real-world traffic patterns at the macro level.

2.1.2 Travel patterns of electric taxi users based on origin-destination (OD) matrices

Compared with private cars, electric taxis are characterised by continuous operation, high-frequency movement and a high degree of randomness in travel purposes. This paper employs a dynamic travel modelling method based on the origin-destination (OD) matrix to generate the spatio-temporal trajectories of taxis by calculating the transfer probabilities between vehicle origins and destinations across different time periods, thereby characterising the dynamic migration patterns of the taxi fleet throughout the day (Li et al., 2019). OD transfer matrices for different time periods reflect the tidal migration patterns of taxis between residential areas, commercial districts and transport hubs, thereby ensuring that the model can effectively capture the spatial clustering characteristics during peak hours.

2.2 Dynamic traffic network modelling

The essential load generated by each EV user to meet their travel needs constitutes the initial dispatch load, which serves as the objective basis for assessing their dispatch potential. A spatio-temporal forecasting model for electric vehicles is constructed by taking into account user travel behavior, the dynamic characteristics of the urban transport network, and users’ charging behavior, in order to simulate the spatio-temporal distribution of the initial dispatch load for large-scale clusters of electric vehicles (Liu and Liu, 2023).

2.2.1 Dynamic traffic network model

In this paper, the urban traffic network is abstracted as a graph model, in which nodes represent traffic zones or charging station locations, and road connections are represented by edges. Given that the Sioux Falls road network focuses primarily on regional-level traffic accessibility analysis rather than micro-level traffic control issues, the traffic network is simplified to an undirected graph in order to reduce model complexity and improve computational efficiency. This is expressed in Equation 1.

G T = V , E , K , W V = v i | i = 1 , 2 , 3 , ⋯ , p E = v i j | v i ∈ V , v j ∈ V , i ≠ j K = k ∣ k = 1 , 2 , 3 , ⋯ , q W = w i j k | v i j ∈ E , k ∈ K (1)

In the formula: G T represents the transport network; V denotes the set of all nodes in graph G T; E denotes the set of all road sections in graph G T; K denotes the set of time slots, dividing the entire day into q time slots; W denotes the set of road section weights, representing the travel cost for that section, quantified using journey costs.

2.2.2 A practical speed-flow model based on real-time traffic volume statistics from the road network

As the road network forms the basis for vehicle travel and charging, graph theory methods are employed to establish a road network model in order to provide a quantitative description of the road structure. Vehicle speed is a key factor in calculating journey time and energy consumption. This study employs the Greenshields speed-density model to dynamically calculate vehicle speeds on road sections. Compared to complex traffic flow models, this model offers advantages such as fewer parameters, clear physical significance, and ease of coupling with optimization models, making it suitable for analysing traffic conditions at a regional scale. The average segment speed and travel time for electric vehicles between any two adjacent points i and j can be calculated by solving the Greenshields speed-density equation, the expression for which is shown in Equation 2.

q i j = c i j V i j − V i j 2 V i j , f (2)

In the equation, V i j , f represents free-flow speed (km/h), q i j represents traffic flow (veh/h), and c i j represents congestion density (veh/km). As road traffic density increases, the average vehicle speed gradually decreases, thereby affecting the time cost and spatial accessibility of vehicles reaching charging stations. Consequently, dynamic traffic conditions will further influence the spatial distribution of vehicle charging demand.

2.3 Charging decisions based on the optimal total charging cost

Let the transport network be represented as an undirected graph, assuming that the charging stations within the area are located at fixed nodes in the transport network, with the charging station numbered Ω CS = 1 , 2 , … , N. At time t, an EV with number Ω EV = 1 , 2 , … , M generates a charging demand. The travel time for the EV between any two adjacent points i and j is given by Equation 3.

T i j = D i j V i j (3)

In the equation, T i j represents driving time (h), D i j represents road length (km), and V i j represents the average section speed between points i and j (km/h), which can be calculated using the Greenshields speed-density equation.

When a vehicle triggers a charging requirement, its choice of charging station essentially constitutes a bounded-rationality decision-making problem that balances travel costs against charging benefits. Given that users are relatively sensitive to the costs of additional detours, this paper incorporates both time and distance costs into a generalised travel cost function and uses Dijkstra’s algorithm to find the optimal solution for candidate routes. When EV m m ∈ Ω EV, with origin o and destination d, triggers a charging demand, the vehicle selects charging station n.Based on the Dijkstra algorithm, the path o d and the minimum travel time T o d , n for this path are obtained. The travel cost of the EV comprises time cost and distance cost; thus, the travel cost C o d , m , n drive of path o d taken by EV m to reach charging station n is shown in Equation 4.

C o d , m , n drive = α R o d , n + β D o d , n T o d , n = T o C S n + T C S n d D o d , n = D o C S n + D C S n d (4)

In the formula: T o d , n represents the estimated time taken for EV m to complete the o d route; D o d , n represents the distance travelled by EV m along the o d route; T o C S n and D o C S n represent the travel time and distance, respectively, for EV m to travel from the origin o to charging station n; T C S n d and D C S n d represent the travel time and distance, respectively, for EV m to travel from charging station n to the destination d; α represents the time value coefficient (Yuan/h); β represents the cost per unit distance coefficient (Yuan/km).

To alleviate the concentration of vehicles in a few hotspots, this paper introduces a dynamic congestion weighting mechanism. Essentially, this is a traffic–grid coordinated guidance signal which, by increasing the generalised charging cost at local high-load nodes, directs some flexible vehicles to move to areas with lower load, thereby achieving active spatial balancing of charging demand.

Let the safe grid connection capacity of the transformer at charging station n be N c a p , n. At time t, the number of vehicles exceeding the safe capacity at the station is N o v e r f l o w , n t in Equation 5:

N o v e r f l o w , n t = max 0 , N E V , n t − 1 − N c a p , n (5)

Based on the real-time overflow status of the site, define the dynamic congestion weighting coefficient ω c , n t for charging station n as follows in Equation 6:

ω c , n t = min 1 + N o v e r f l o w , n t N c a p , n , ω max (6)

In the formula, ω max represents the upper weight threshold.

Based on the above mechanism, EV m will take into account both the cost of driving and the congestion status of the grid at the charging stations to assess the generalised station selection cost C s e l e c t , m , n t for travelling to candidate stations in Equation 7:

C s e l e c t , m , n t = C o d , m , n drive + ω c , n t · w pri t · Δ E (7)

In the equation, w pri t represents the real-time time-of-use tariff published by the grid, and Δ E represents the vehicle’s expected charge volume. It should be noted that the dynamic congestion weight ω c , n t acts solely as a ‘virtual decision-making resistance’ during the station selection phase, serving to generate spatial momentum and disperse concentrated traffic flows; however, once the vehicle is actually connected to a charging point, the settlement tariff applied to its interaction with the grid strictly adheres to the actual time-of-use tariff ω pri t.

With the development of information technologies such as the Internet of Vehicles (IoV), electric vehicles have acquired schedulable characteristics. These characteristics exist in both spatial and temporal dimensions. At the spatial level, this paper proposes a charging station scheduling radius model. The definition of Ω R. Ω R is shown in Equation 8, which characterises the range within which EV users select charging stations given a finite tolerance. In a scenario involving multiple charging stations, once an EV generates a charging demand between a starting point and a destination, it is assumed that charging at station n * will result in the lowest time cost. However, the selection of n * considers only the driver’s interests; the charging load added to this distribution network node may increase the operational burden on the grid. Therefore, Ω R encompasses all charging stations that the driver is willing to make available for grid optimization scheduling within a finite tolerance. It can be seen that the set is jointly constrained by thresholds for both distance cost and time cost. An EV selects exactly one charging station within the dispatch radius for charging, as expressed by Equation 9. For any EV generating a charging demand, the total charging cost of travelling to each charging station within the charging radius is a deterministic quantity. The EV user’s selection of a charging station is illustrated in Figure 3:

Ω R = n D o d , n − D o d , n * ≤ D max ∩ n T o d , n − T o d , n * ≤ T max (8)

In the equation: D max represents the maximum distance radius (km), and T max represents the maximum time radius (h).

∑ n ∈ Ω R z m , n EV = 1 ∑ n ∉ Ω R z m , n EV = 0 (9)

Let Z be a Boolean variable representing the EV’s choice of charging station, where 1 indicates that the charging station is selected and 0 indicates that it is not selected.

In particular, the charging and discharging power of EV m at charging station n can be expressed as P m , n EV, as shown in Equation 10.

− P n dis , ⁡ max z m , n EV ≤ P m , n EV ≤ P n ch , ⁡ max z m , n EV (10)

In the formula: P n ch , ⁡ max represents the maximum charging power of EV charging station n, and P n dis , ⁡ max represents the maximum discharging power of EV charging station n. All power variables are expressed in kW.

3 Aggregated storage-like and generator-like resource models for PV-storage-charging systems

To enhance physical interpretability, we classify resources into two categories: storage-like resources (including flexible/semi-flexible EV clusters aggregated as EVA, and stationary BESS) and generator-like resources (distributed PV). Storage-like resources have bidirectional power flow capability and energy constraints, while generator-like resources have unidirectional power output with optional curtailment.

Given that large-scale electric vehicle fleets are characterised by their sheer numbers, dynamically changing connectivity statuses and significant individual heterogeneity, attempting to schedule individual vehicles directly would not only entail extremely high computational complexity but would also make it difficult to meet the real-time optimization requirements of large-scale scenarios. It is therefore necessary to employ aggregate modelling methods to transform large-scale electric vehicle fleets into equivalent resources that can be scheduled uniformly.

The Minkowski sum effectively describes the geometric superposition characteristics of the feasible regions of multiple heterogeneous energy storage units, significantly reducing the optimization dimension whilst ensuring the interpretability of physical constraints. Consequently, this paper constructs an Electric Vehicle Aggregator (EVA) model based on the Minkowski sum theory and further couples it with station-side photovoltaic and energy storage systems to form a framework for the coordinated operation of photovoltaic, energy storage and electric vehicles (Al Taha et al., 2025).

3.1 Charging and discharging model for a single EV

By determining the initial state of charge (SOC) of a single EV, as well as the start and end times of parking, a corresponding load model can be established, as shown in Equations 11, 12.

0 ⩽ P m EV , ch t ⩽ P m , ⁡ max EV , ch x m EV , ch t 0 ⩽ P m EV , dis t ⩽ P m , ⁡ max EV , dis x m EV , dis t t ∈ t m EV , arrive , t m EV , leave 0 ⩽ x m EV , ch t + x m EV , dis t ⩽ x m EV t (11)

s m , ⁡ min EV ⩽ s m EV t ⩽ s m , ⁡ max EV s m EV t = s m EV t − 1 + η ch · P m EV , ch t − P m EV , dis t / η dis Δ t t ∈ t m EV , arrive , t m EV , leave s m EV t m EV , leave ≥ s min required (12)

In the equation: P m EV , ch t and P m EV , dis t represent the charging and discharging power of EV m at time t (kW),respectively; s m EV t represents the battery charge of EV m at time t (kWh.); η ch and η dis represent the charging and discharging efficiencies of the EV; P m , ⁡ max EV , ch and P m , ⁡ max EV , dis represent the maximum charging and discharging power of EV m (kW), respectively; s m , ⁡ max EV and s m , ⁡ min EV represent the upper and lower limits of EV m’s battery charge, respectively (kWh.); t m EV , arrive and t m EV , leave represent the start and end times of EV m’s parking, respectively; s min required represents the amount of battery charge required for an EV to meet basic travel needs (kWh.), the electric vehicle charges and discharges between the start and end times of parking to meet the user’s energy requirements. Let x m EV t denote the stopped state of EV m at time t, t takes the value 1 when the vehicle is stopped and 0 when it is not stopped; Let x m EV , ch t and x m EV , dis t be Boolean variables representing the charging and discharging states of EV m at time t, respectively; when in the charging state, x m EV , ch t = 1 and x m EV , ch t = 0 hold; when in the discharging state, x m EV , dis t = 1 and x m EV , dis t = 0 hold; x m EV , ch t + x m EV , dis t ≤ 1 indicates that charging and discharging cannot occur simultaneously at time t.

In a vehicle-road-station-network system containing a large number of EVs, if the charging data reported by all EV users were to be incorporated into system regulation, this would generate a vast amount of high-dimensional variable data during the dispatching process, which would hinder the optimization of the dispatching plan. Therefore, EV clusters should first be aggregated by an EV aggregator before being incorporated into the power system dispatching process.

3.2 Storage-like resource model: EVA and BESS

Before constructing a generalised energy storage model for an EV cluster, vehicles must be classified according to their charging patterns. Rigid EVs refer to vehicles that, due to their low remaining state of charge, require immediate charging at rated power upon arrival to meet basic mobility needs. The charging behavior of this category of vehicles lacks flexibility; their power profiles are treated as uncontrollable rigid loads during the dispatch process. Semi-flexible EVs refer to vehicles that possess limited time-shifting capability, provided that the user’s required State of Charge (SOC) upon departure is guaranteed. Such vehicles allow for adjustments to charging timing within the parking window, but the range of power variation is constrained by the departure time and minimum energy requirements. Flexible EVs refer to vehicles with ample parking time and a high degree of adjustment flexibility in their state of charge. They are capable of participating in bidirectional charging and discharging (Vehicle-to-Grid, V2G) in response to electricity price signals, photovoltaic output and grid dispatch requirements, thereby providing flexible regulation capabilities to the system.

Given the significant differences in control flexibility among various types of electric vehicles, aggregating all vehicles uniformly would introduce interference from non-adjustable loads into the assessment of cluster flexibility, thereby overestimating the system’s actual dispatch capacity. Therefore, this paper adopts a rigid decoupling strategy, defining the charging demand of rigid vehicles as an exogenous variable—representing the total power that charging station s must rigidly meet during time slot t—and forcing it to participate in power balancing as a fixed parameter. Subsequent Minkowski summation and EVA parameter extraction are performed exclusively for flexible and semi-flexible vehicles, ensuring that the resulting aggregation boundary accurately reflects the cluster’s dispatchable potential. Concurrently, during the Minkowski aggregation process, the system deducts the energy deviation introduced into the cluster by the initial SOC of rigid vehicles upon arrival. This ensures that the extracted EVA aggregation parameters accurately reflect the cluster’s dynamic dispatchable potential over time.

Minkowski summation is a summation theory applicable to Euclidean spaces, which can be used to determine the dilated sets of variable spaces sharing the same domain. For variable spaces M and N with the same domain, the Minkowski summation procedure is as follows in Equation 13:

M ⊕ N = m ′ + n ′ ∣ m ′ ∈ M , n ′ ∈ N (13)

In the formula: M⊕N denotes the Minkowski sum of two variable spaces, forming a larger space; m ′ and n ′ are elements of the variable spaces M and N, respectively.

By extending the grid connection times of individual EVs with varying characteristics to a common feasible time domain, the model for a single EV becomes Minkowski-additive. Consequently, the aggregated model of the EVA model for the EV cluster can be expressed as in Equations 14–16:

∑ m ∈ Ω EV s m , ⁡ min EV x m EV t ⩽ ∑ m ∈ Ω EV s m EV t ⩽ ∑ m ∈ Ω EV s m , ⁡ max EV x m EV t t ∈ t m EV , arrive , t m EV , leave (14)

0 ⩽ ∑ m ∈ Ω EV P m EV , dis t ⩽ ∑ m ∈ Ω EV P m , ⁡ max EV , dis x m EV , ch t 0 ⩽ ∑ m ∈ Ω EV P m EV , ch t ⩽ ∑ m ∈ Ω EV P m , ⁡ max EV , ch x m EV , ch t t ∈ t m EV , arrive , t m EV , leave (15)

∑ m ∈ Ω EV s m EV t = ∑ m ∈ Ω EV t − 1 + ∑ m ∈ Ω EV P m EV , ch t − P m EV , dis t ∆ t + ∑ m ∈ Ω EV s m EV , arrive x m EV t x m EV t − x m EV t − 1 − t ∈ t m EV , arrive , t m EV , leave ∑ m ∈ Ω EV S m EV , leave x m EV t − 1 x m EV t − 1 − x m EV t (16)

In the equation, s m EV , arrive and s m EV , leave represent the battery charge levels at the start and end of the EV parking session (kWh.), respectively. The aggregation of the EV cluster is achieved through a Minkowski summation process, thereby extracting the EVA variable and parameters, as shown in Equation 17.

P EVA , dis t = ∑ m ∈ Ω EV P m EV , dis t , P EVA , ch t = ∑ m ∈ Ω EV P m EV , ch t P max EVA , dis t = ∑ m ∈ Ω EV P m , ⁡ max EV , dis x m EV , dis t P max EVA , ch t = ∑ m ∈ Ω EV P m , ⁡ max EV , ch x m EV , ch t S EVA t = ∑ m ∈ Ω EV s m EV t S max EVA t = ∑ m ∈ Ω EV s m , ⁡ max EV x m EV t , S min EVA t = ∑ m ∈ Ω EV s m , ⁡ min EV x m EV t Δ S EVA t = ∑ m ∈ Ω EV s m EV , arrive x m EV t x m EV t − x m EV t − 1 − ∑ m ∈ Ω EV s m EV , leave x m EV t − 1 x m EV t − 1 − x m EV t (17)

In the equation: P EVA , ch t and P EVA , dis t represent the charging and discharging power of the EVA at time t (kW), respectively; S EVA t represents EVA’s battery charge at time t (kWh.); P max EVA , ch t and P max EVA , dis t represent the maximum charging and discharging power of the EVA at time t (kW), respectively; S max EVA t and S min EVA t represent the upper and lower limits of the EVA’s battery charge at time t (kWh.), respectively; Δ S EVA t represents the change in the EVA’s battery charge at time t due to the EV being out of service (kWh.). The final set of extracted EVA parameters comprises P ch EVA t , P dis EVA t , S max EVA t , S min EVA t , Δ S EVA t elements. We have obtained the macro-level behavioral boundaries of the EV cluster, which quantitatively reflect the cluster’s dispatchable potential. In this section, through the aggregation and modelling of large-scale electric vehicle clusters participating in grid dispatch, we introduce the Minkowski sum to aggregate the preliminary dispatch potential of each EV into a unified cluster EVA model.

The energy storage system model is constructed as follows in Equation 18:

E t BESS = E t − 1 BESS 1 − β BESS + η BESS , ch P t BESS , ch Δ t − P t BESS , dis Δ t η BESS , dis (18)

Where: E t BESS is the real-time energy stored at time t (kWh); β BESS is the self-discharge rate (0.001/h); η BESS , ch is the charging efficiency; P t BESS , ch is the charging power at time t; η BESS , dis is the discharge efficiency (kW); P t BESS , dis is the discharge power at time t (kW); Δt is the time step(h).

In particular, the charging and discharging power constraints are as follows in Equation 19:

0 ⩽ P BESS , ch ⩽ P max BESS , ch 0 ⩽ P BESS , dis ⩽ P max BESS , dis (19)

The energy storage battery’s charge constraints are as follows in Equation 20:

E min BESS ⩽ E BESS ⩽ E max BESS (20)

3.3 Generator-like resource model: PV

The solar-storage charging station is powered jointly by a photovoltaic power generation system and the grid, thereby enabling power exchange with the grid. The model of the photovoltaic system in the solar-storage charging station is constructed as follows (Shi et al., 2023) in Equation 21.

P PV = N PV P max PV g t 1000 g 0 1 + θ T T 0 − T r (21)

In the equation: P PV represents the actual output power of the photovoltaic system (kW); N PV represents the number of photovoltaic modules installed; P max PV represents the rated output power of a photovoltaic module (kW); g t represents the surface solar irradiance at time t (W/m²); g 0 represents the standard irradiance (1000 W/m²); θ T represents the power temperature coefficient; T 0 represents the actual temperature of the photovoltaic panel (°C); T r represents the reference temperature (°C).

3.4 Distribution network model

The safe operation of the distribution network is a prerequisite for optimised dispatch. This paper establishes the DistFlow model to implement node power balance constraints (Cui et al., 2024), which ensures that active and reactive power are balanced within the system and that node voltages and line currents remain within safe limits.

P t , u v = ∑ z ∈ δ ζ P t , u r + P t , u g r i d + r u v I t , u v 2 Q t , u v = ∑ z ∈ δ ζ Q t , u r + Q t , u b a s e + x u v I t , u v 2 U t , u 2 = U t , v 2 − 2 r u v P t , u v + x u v Q t , u v + r u v 2 + x u v 2 I t , u v 2 I t , u v 2 = P t , u v 2 + Q t , u v 2 U t , v 2 (22)

In the Equation 22: U t , v and I t , u v denote the voltage amplitude at node v and the branch current in branch u v, respectively; δ ζ denotes the set of terminal nodes in the downstream branches of node v; P t , u v and Q t , u v denote the active and reactive power flowing from branch u v to node v, respectively; r u v and x u v denote the resistance and reactance values of line uv; P t , v g r i d denotes the net active power exchanged between the charging station and the distribution network at node v; Q t , v base denotes the base reactive load at node v.

To accurately characterise the true flexibility boundaries of electric vehicle clusters, this paper explicitly decouples rigid loads from adjustable resources within the station-level power balance constraints. In this context, the rigid charging demand generated by vehicles requiring emergency recharging is treated as an exogenous load and directly incorporated into the system power balance, while flexible and semi-flexible vehicles are modelled as adjustable virtual energy storage resources using a Minkowski aggregation model. The net active power exchange between charging station node u and the grid during time slot t is explicitly reconstructed as follows:

P t , v s t o , c h = P t , v E V A , c h + P t , v B E S S , c h P t , v s t o , d i s = P t , v E V A , d i s + P t , v B E S S , d i s (23)

P t , v g r i d = P t , v s t o , c h − P t , v s t o , d i s + P t , v , R i g i d − P t , v P V − P t , v c u r t a i l (24)

In the Equations 23, 24: P t , v s t o , c h and P t , v s t o , d i s storage-like resources (EVA and BESS) are aggregated into net charging/discharging power, reflecting the total flexibility available at the station (kW).

P t , v E V A , c h and P t , v E V A , d i s represent the virtual charging and discharging powers of flexible and semi-flexible electric vehicle clusters, respectively, subject to Minkowski aggregation boundaries (kW); P t , v B E S S , c h and P t , v B E S S , d i s represent the charging and discharging powers of the fixed energy storage system at the station, respectively (kW); P t , v , R i g i d represents the non-adjustable, rigid charging load identified through Monte Carlo micro-simulation of the transport network, which is not treated as a decision variable in top-level grid optimization (kW); P t , v P V represents the theoretical output of the distributed photovoltaic system (kW); P t , v c u r t a i l represents the active curtailed power at the photovoltaic-storage charging station (kW).

The boundary constraints for the safe operation of the distribution network are expressed as follows in Equation 25:

U min 2 ≤ U t , v 2 ≤ U max 2 − P v , Max ≤ P t , v g r i d ≤ P v , Max 0 ≤ P t , v c u r t a i l ≤ P t , v P V (25)

In the formula: U min and U max represent the permissible lower and upper limits of the node voltage respectively; P v , Max represents the upper limit of the physical reverse-feed/grid-connection capacity of the substation distribution transformer.

4 Calculation of dispatchable potential

In the method proposed in this paper, the EVA quantifies its dispatchable potential through two interrelated stages. Firstly, prior to optimization in the temporal dimension, the system has already completed the spatial matching of EVs and charging stations through a pre-established dynamic comprehensive charging cost guidance mechanism. Subsequently, based on the determined spatial allocation results and the Minkowski aggregation model, charging and discharging optimization is carried out centrally along the time axis, thereby generating reference charging and discharging powers for EVA across various time slots. Based on the optimization results from the preliminary scheduling stage, the second stage calculates the dispatchable potential, quantifying the adjustable ranges for both power and capacity. Based on the reference power obtained from the preliminary dispatch, the dispatch power used to reduce the system’s net load is termed upward regulation power, whilst the corresponding dispatch power utilising EVAs to increase the system load is termed downward regulation power. Energy storage devices, as components of the photovoltaic-storage-charging system, participate in grid regulation in conjunction with EVAs. By comprehensively considering both EVAs and energy storage, the total dispatch potential can be determined. Taking the dispatchable power of EVAs as an example, a schematic diagram is shown in Figure 4.

4.1 Preliminary scheduling optimization methods

To eliminate differences in units of measurement between the multiple optimization objectives, the model constructs a multi-objective normalised fitness function. The initial weights of each objective were determined following normalisation and sensitivity analysis. Subsequently, a Pareto frontier analysis was performed to validate the chosen weight combination, demonstrating that it achieves a balanced trade-off rather than representing an arbitrary selection. The model takes into account economic arbitrage, load smoothing, depreciation of EV users, and depreciation of BESS. The overall objective function is defined as in Equations 26–30:

min ⁡ F = ω 1 F 1 + ω 2 F 2 + ω 3 F 3 + ω 4 F 4 (26)

C n EV , ch = ∑ m ∈ Ω EV n ∑ t = t m EV , arrive t m EV , leave w pri t P m EV , ch t − P m EV , dis t Δ t F 1 = ∑ n = 1 N E V A C n EV , ch F 1 , ref (27)

F 2 = η 1 ∑ t = 1 T P t o t a l t − P m e a n 2 F 2 , r e f v a r + η 2 ∑ t = 1 T − 1 P t o t a l t + 1 − P t o t a l t 2 F 2 , r e f s m o o t h (28)

C loss , n EV = ∑ m ∈ Ω EV n ∫ t m EV , arrive t m EV , leave w EV t P m EV , dis t d t F 3 = ∑ n = 1 N E V A C loss , n EV F 3 , ref (29)

C loss , n BESS = w BESS ∑ i = 1 ψ ε i F 4 = ∑ n = 1 N E V A C loss , n BESS F 4 , ref (30)

In the equation: F represents the objective function value, comprising economic arbitrage for EV users, load smoothing, depreciation of EV users, and depreciation of BESS; ω 1 to ω 4 are the weighting coefficients for each indicator. F 1 is the economic arbitrage cost; C n EV , ch represents the global net electricity purchase cost for EVA in response to time-of-use pricing; w pri t is the charging cost for all EV users; F 2 is the time-of-use tariff; P t o t a l t represents the grid load smoothing metric, designed to limit the sharp power fluctuations caused by large-scale vehicle traffic; this is composed of the system’s overall peak-to-valley difference (variance penalty) and the smoothness of hard ramping in adjacent time slots (difference penalty); P m e a n represents the total active load of the distribution network system at time t, and represents the system’s average active load over the entire dispatch cycle; F 3 represents the depreciation cost of the EV battery fleet; C loss , n EV represents the depreciation cost per discharge cycle for all users in EVA n; w EV t represents the unit energy cost of EV discharge at time t; P m EV , dis t represents the power discharged by EV_m to the grid at time t (Zhang and Sun, 2017); F 4 represents the depreciation cost of station-side stationary energy storage (BESS) participating in charge-discharge cycles; C loss , n BESS represents the daily depreciation cost of the energy storage batteries at photovoltaic-storage charging station n; w BESS denotes the loss cost coefficient of the energy storage batteries; ε i represents the depth of discharge for the ith cycle on that day; ψ ε i denotes the damage to battery life caused by a single cycle (Gama et al., 2025). F 1 , ref represents the baseline cost of random charging; F 2 , r e f v a r and F 2 , r e f s m o o t h are the variance and differential reference values corresponding to the original base load, respectively, used to eliminate differences in the units of measurement between different physical quantities; η 1 and η 2 are the weighting coefficients for the sub-items. F 3 , ref represents the marginal depreciation cost resulting from the EV cluster charging at the theoretical maximum power during all available time slots; F 4 , ref represents the marginal depreciation cost of the station-side BESS operating at full power for charging and discharging throughout all time slots, used for normalisation.

To further justify the selected weights, we conducted a Pareto frontier analysis by varying the relative weights of economic cost ω 1 and load smoothing ω 2, while keeping ω 3 = 0.3 and ω 4 = 0.2 fixed. Figure 5 shows the trade-off between total electricity procurement cost and peak-to-off-peak difference for five representative weight combinations: (0.50, 0.00) (0.375, 0.125), (0.25, 0.25) (0.125, 0.375) and (0.00, 0.50). The weight set used in this paper ω 1 = ω 2 = 0.25 yields a solution near the “knee” of the Pareto front, providing a balanced compromise between cost efficiency and load smoothing. This evidence-based approach confirms that the selected weights are not arbitrary but reflect a rational design preference. We also note that the proposed optimization model is a convex quadratic programming (QP) problem, which is solved to global optimality using the commercial solver Gurobi 11.0. The convex QP formulation guarantees the same rigor as a mixed-integer linear programming (MILP) approach, ensuring that each Pareto point is globally optimal for its given weight combination.

4.2 Calculation of EVA’s dispatchable potential

The dispatchable potential of the EVA is calculated based on the preliminary scheduling results and the aggregated feasible region obtained via Minkowski summation (Section 3.2). The maximum charging/discharging powers P max EVA , ch t and P max EVA , dis t (kW), as well as the energy boundaries S min EVA t and S max EVA t (kWh), are directly retrieved from the EVA model parameters in Equation 17. The preliminary scheduling power P pre − sche EVA , ch t and P pre − sche EVA , dis t , P pre − sche EVA t, are the optimal solution of the preliminary scheduling stage Section 4.1 (kW). All indices and bounds are therefore perfectly consistent with the Minkowski aggregation process.

Once the capacity and power limits of the EVA have been predicted and the preliminary load calculations for cluster scheduling have been completed, it is then possible to determine the dispatchable potential in terms of power and capacity, thereby enabling a quantitative assessment of the EVA’s dispatchable potential.

The dispatch power for EVA comprises upward dispatch power and downward dispatch power, as shown in Equation 31:

P up EVA t = P max EVA , dis t − P pre − sche EVA , dis t + P pre − sche EVA , ch t P down EVA t = P max EVA , ch t − P pre − sche EVA , ch t + P pre − sche EVA , dis t (31)

In the equation: P up EVA t and P down EVA t represent the upward and downward control powers of the EVA at time t (kW), respectively; P pre − sche EVA , ch t and P pre − sche EVA , dis t represent the preliminary dispatch charging and discharging powers of the EVA at time t (kW), respectively.

Dispatch capacity comprises upward dispatch capacity and downward dispatch capacity, as shown in Equation 32:

S up EVA t = S pre − sche EVA t − S min EVA t S down EVA t = S max EVA t − S pre − sche EVA t (32)

In the equation: S up EVA t and S down EVA t represent the upper and lower regulation capacities of the EVA at time t (kWh), respectively; S pre − sche EVA t represents the preliminary dispatch volume of the EVA at time t (kWh). The term Δ S EVA t accounts for the energy variation caused by EV arrivals and departures during time step t, and is directly retrieved from the EVA parameters in Equation 17.

The upward regulation power is the amount by which the EVA can reduce its net power consumption from the grid (or increase power injection to the grid) compared to the preliminary dispatch, schedule. It consists of the ability to curtail ongoing charging P pre − sche EVA , ch t and the ability to increase discharging beyond the scheduled level P max EVA , dis t − P pre − sche EVA , dis t. Conversely, downward regulation power reflects the ability to increase grid consumption. The regulation capacities represent the energy that can be further released or absorbed, computed relative to the current energy state S pre − sche EVA t. These definitions are consistent with the feasible region extracted by Minkowski summation: the power bounds are directly taken from Equation 17, and the energy bounds define the maximum and minimum possible energy storage levels of the aggregated EVA.

4.3 Calculation of the dispatchable potential of BESS

The method for calculating the dispatchable potential of energy storage is similar to that for EVA. Upward dispatchable power potential, downward dispatchable power potential, as shown in Equation 33:

P u p BESS t = P r a t e d BESS , dis − P t BESS , dis + P t BESS , ch P d o w n BESS t = P r a t e d BESS , ch − P t BESS , ch + P t BESS , dis (33)

In the equation: P u p BESS t and P d o w n BESS t represent the upper and lower control powers of the fixed energy storage system at time t (kW), respectively; P r a t e d BESS , ch and P r a t e d BESS , dis represent the rated charging and discharging powers of the fixed energy storage system (kW), respectively; P r a t e d BESS , dis and P t BESS , dis represent the charging and discharging powers of the fixed energy storage system (kW).

The upper adjustment capacity and lower adjustment capacity are shown in Equation 34:

S u p BESS t = E t BESS − E min BESS S d o w n BESS t = E max BESS − E t BESS (34)

In the equation: S u p BESS t and S d o w n BESS t represent the upper and lower regulation capacities of the fixed energy storage system at time t (kWh), respectively; E t BESS represents the real-time energy of the fixed energy storage system at time t (kWh); E min BESS and E max BESS represent the maximum and minimum capacities of the fixed energy storage system (kWh), respectively.

4.4 Total dispatchable potential of storage-like resources

The total dispatchable potential of the storage-like resources at a given station is the sum of the contributions from the two sub-units: the EVA (aggregated flexible/semi-flexible EVs) and the station-side BESS. Although both belong to the same “storage-like” category, they are physically independent devices with separate power and energy limits; therefore, their potentials can be summed directly. The total dispatchable power P u p s t o t and P d o w n s t o t (kW), and the total dispatchable capacity S u p s t o t and S d o w n s t o t (kWh), are given by Equations 35, 36:

Its total dispatchable power:

P u p sto t = P u p BESS t + P up EVA t P d o w n sto t = P d o w n BESS t + P down EVA t (35)

Total available capacity:

S u p s t o t = S u p BESS t + S up EVA t S d o w n s t o t = S d o w n BESS t + S down EVA t (36)

The total dispatchable potential of storage-like resources quantifies the maximum regulation capacity of the PV-storage-charging system relative to the base load, providing a basis for operators to participate in the electricity market.

It is important to note that the total dispatchable potential calculated by Equations 35, 36 represents the theoretical maximum flexibility of the storage-like resources, assuming that each resource can be dispatched independently without considering shared infrastructure constraints. In practice, the achievable dispatchable potential is further limited by the distribution network constraints, particularly the transformer capacity P v , Max at each charging station node. The DistFlow model in Section 3.4 ensures that the net power exchange P t , v g r i d satisfies the constraint − P v , Max ≤ P t , v g r i d ≤ P v , Max​ (Equation 25), which limits the simultaneous charging/discharging of EVA and BESS at the same node. Therefore, the flexibility actually utilized in the top-level optimization is always a subset of the theoretical potential. This decoupling between theoretical and achievable flexibility is a key contribution of our framework, preventing overestimation of the system’s regulation capability.

5 Case study analysis

5.1 Case setup and basic data

The underlying spatial physical architecture of the case study in this paper is formed by the deep coupling of the Sioux Falls urban transport network with a modified IEEE 33-node power distribution network. Figure 6 shows a map of the urban transport network in Sioux Falls. A spatial scaling factor (0.2) is introduced to scale down the model, transforming it into a medium-scale functional zone network to align with users’ tolerance for spatial detours. In terms of spatial distribution, the Sioux Falls transport network is divided into three typical functional zones: the office district, the commercial district and the residential district. A total of 2,000 electric vehicles are deployed across these zones, comprising 1,500 private cars with tidal travel patterns and 500 electric taxis with high-frequency random travel patterns. The battery capacity of all vehicles is uniformly set at 30 kWh, with a rated charging and discharging power of 15 kW. To facilitate physical interaction between ‘vehicles, roads, stations and the grid’, 13 integrated photovoltaic-storage-charging fast-charging stations have been strategically deployed within the system. The traffic network nodes at which these 13 charging stations are located are strictly anchored, via a spatial mapping matrix, to key load nodes or end-of-line high-load nodes within an IEEE 33-node distribution network. In terms of the physical parameter configuration of station-end equipment, each fast-charging station integrates an independent distributed photovoltaic system and a stationary energy storage system. Specifically, the installed capacity of the distributed PV system at each station is 200 kW; the station-side battery energy storage system (BESS) has a rated capacity of 400 kWh and a maximum charge/discharge power of 80 kW. Furthermore, taking into account the physical safety boundaries of the distribution network nodes and line thermal stability constraints, the maximum grid-connected/reverse power limit for the transformers at each PV-storage-charging station is set at 350 kW. The results presented are the average of 10 simulation runs. The complete set of simulation parameters—including EV classification thresholds, objective-function weights, traffic and battery data, station-to-grid mapping, and solver configuration—is provided in Table 1 and the Station-to-Grid Mapping Table.

TABLE 1. Simulation parameters.

It should be noted that the spatial scaling factor of 0.2 is introduced to adapt the Sioux Falls network (originally a macroscopic city-level topology) to a medium-scale functional zone that aligns with users’ realistic detour tolerance. While this simplification is common in co-simulation studies for computational tractability, it may alter actual topological characteristics such as inter-node distances and detour ratios. Consequently, the exact quantitative findings—e.g., the optimal dispatch radius of 1.0 km—are case-specific and may vary when applied to a real urban network with different scale and layout. Nevertheless, the qualitative advantages of the proposed spatio-temporal coordination framework are expected to hold across different scales because the underlying mechanisms (dynamic congestion pricing and Minkowski aggregation) are topology-agnostic.

To comprehensively validate the effectiveness of the electric vehicle energy demand allocation strategy proposed in this paper under spatio-temporal coupling, this section sets out four progressive experimental scenarios for comparison: Scenario S0: Pure unordered charging without photovoltaics or energy storage, where all vehicles charge at maximum power immediately upon arrival. The distribution network bears the full brunt of the raw physical load impact. Scenario S1: Passive local PV-storage control. The charging station incorporates PV and fixed energy storage, with the storage system employing a local greedy control strategy (storing surplus electricity and discharging when power is insufficient), whilst electric vehicles continue to charge in a disorderly manner. Scenario S2: Time-only coordinated optimization. The spatial routing of electric vehicles remains disordered, following natural arrival patterns, but the charging power for semi-flexible and flexible vehicles is optimised through global reallocation along the time axis. Scenario S3: Spatio-temporal dual-coordination optimization. The strategy proposed in this paper, designed for spatio-temporal coordinated dispatch scenarios, further introduces a spatial dispatch mechanism on top of temporal optimization to achieve active migration of vehicle charging demand across different stations. Through comparative analysis of the scenarios, the study focuses on verifying the impact of spatial dispatch flexibility on the dispatchable boundary of the EVA and the system’s operational performance.

5.2 Comprehensive comparison of economic benefits and overall load smoothing

The total grid load for different scenarios is shown in the figure. As can be seen from Figure 7: In Scenario S0, the peak EV charging load coincides with the baseline electricity peak, further widening the grid’s peak-to-valley difference. In Scenario S1, the introduction of PV and stationary energy storage results in a deep valley formed by the reduction of the daytime peak; at the onset of the evening peak, a flattened slope is observed compared to Scenario S0. After the energy storage is depleted in the evening, the total load curve during the night—when PV output is zero—is identical to the original curve with unregulated charging. The total load curves for Scenarios S2 and S3 are broadly similar, achieving peak shaving and valley filling, with both peaks significantly reduced.

The system’s macro-level operational indicators directly reflect the overall benefits of different dispatch strategies at the distribution network level. Table 2 presents the key operational indicators for each scenario. It can be seen that, when comparing the unregulated S0 scenario with the S1 scenario, relying solely on passive absorption via on-site photovoltaic and storage systems can significantly reduce electricity procurement costs. However, the peak-to-off-peak difference in S1 is reduced by less than 300 kW, indicating that passive control cannot cope with the physical impact on the grid caused by a large number of unregulated EVs. Following the introduction of the S2 strategy, the global peak-to-valley difference decreased markedly by encouraging EVs to shift their charging times. Although the peak-to-valley difference in S3 is slightly higher than in S2, its electricity procurement cost is the lowest among the four scenarios. The S3 strategy directs a large number of flexible EVs to charge collectively during the midday period (when PV generation is high and electricity prices are at their lowest), essentially absorbing surplus green PV electricity into a vast mobile storage pool at extremely low cost; Subsequently, during the evening peak, V2G provides discharge support to the grid. Battery degradation under S3 is nearly double that of S1, indicating that S3 creates greater arbitrage opportunities for the fleet of vehicles. Ultimately, even after deducting the high battery depreciation costs, the S3 strategy still minimises the total system operating costs.

TABLE 2. Comparison of key performance indicators across different scenarios.

5.3 Evaluation of spatial load balancing under spatio-temporal decoupling

In the actual operation of distribution networks, spatial congestion at local nodes (transformers) often triggers safety crises earlier than overall system-wide overloads. This paper employs the Gini coefficient and spatial standard deviation to quantitatively analyse the spatial distribution equilibrium of the S2 and S3 strategies; the results are shown in Table 3.

TABLE 3. Evaluation of spatial load balancing under different scheduling strategies.

The core of evaluating spatio-temporal coordinated dispatching strategies lies in quantifying their ability to mitigate local spatial polarisation and load concentration effects in distribution networks. The spatial Gini coefficient has been introduced into transport networks as an internationally recognised statistical benchmark for measuring the geographical imbalance in EV charging demand and the spatial equity of road access rights (Li et al., 2022; Cai et al., 2024). Drawing on the aforementioned spatial evaluation framework and taking into account the highly time-varying characteristics of the ‘vehicle-road-station-grid’ coupled system, this paper extends the classical cross-sectional Gini coefficient to define three multidimensional evaluation indicators: the station-average Gini coefficient, the inter-station Gini coefficient at peak times, and the instantaneous average Gini coefficient. The station-average Gini coefficient reflects the degree of similarity in the average utilisation rates of each station from an overall, round-the-clock perspective. First, the time-averaged number of vehicles parked at each charging station is calculated over the entire simulation period to obtain a baseline value representing the station’s long-term utilisation level; subsequently, the Gini coefficient between the averages of each station is computed. The inter-station Gini coefficient at peak times measures whether charging demand during peak periods is overly concentrated at a few core stations. This indicator focuses on the critical moment when the system’s total load reaches its maximum value, by calculating the spatial Gini coefficient of the number of vehicles parked at each station during this most critical instant. The average Gini coefficient at each time step characterises the overall dynamic level of spatial distribution imbalance throughout the entire scheduling cycle. This indicator calculates the inter-station Gini coefficient for the distribution at every discrete time point within the simulation cycle, and subsequently computes the mathematical expectation of the Gini coefficients for all time points.

Based on the above metrics, a quantitative comparison of the spatial balance in scenarios S2 and S3 is carried out. During peak periods when traffic tidal effects are pronounced, Scenario S2, due to a lack of spatial dispatch capability, sees a large amount of load clustering in accordance with geographical inertia, resulting in a relatively high peak inter-station Gini coefficient. In contrast, the strategy in Scenario S3 reduces this Gini coefficient to 0.3178 through a dynamic station selection mechanism. Scenario S3 also exhibits lower per-station Gini coefficients and average Gini coefficients per time step compared to Scenario S2. The reduction in the per-station Gini coefficient indicates that the spatial imbalance in EV charging demand has been alleviated, meaning that load concentration has been mitigated. It should be noted that this study does not aim for absolute fairness in allocation, but rather seeks to reduce the risk of localised load aggregation through limited spatial dispatching, thereby enhancing the system’s overall dispatchability.

To further analyse the mechanism by which spatio-temporal coordination strategies enhance system performance, this paper conducts a comparative analysis of the spatio-temporal distribution characteristics of charging station node load factors under the time-only coordination scenario S2 and the spatio-temporal coordination scenario S3. The node load factor is defined as the ratio of the node’s actual operating power to its rated capacity; it reflects the degree of local load concentration and differences in operational pressure across different scenarios. Figure 8 illustrates the spatio-temporal distribution of charging station node load factors under different coordination strategies. It can be observed that in the time-only coordination scenario S2, some charging station nodes exhibit significant load concentration during peak hours (approximately 17:00–22:00), with persistent high-load operating conditions in local areas. This indicates that when optimization relies solely on the temporal dimension, vehicle charging demand still tends to concentrate at a few stations, resulting in high operational pressure at local nodes. In contrast, in the spatiotemporal coordination scenario S3, the number of areas with high node load factors is significantly reduced, and load hotspots exhibit a more dispersed spatial distribution. This indicates that under the influence of the spatial dispatch mechanism, some charging demand can be transferred across stations within the constraints of the dispatch radius, thereby reducing the concentration of load at local nodes and improving the spatial balance of charging load across different nodes. From the perspective of operational mechanisms, a more balanced distribution of node loads helps alleviate local node capacity pressures, providing a more stable operating environment for aggregated EV resources to participate in charging and discharging dispatch. Consequently, the spatio-temporal coordination strategy not only improves the local operational status of the distribution network but also provides favourable conditions for the subsequent release of dispatchable potential.

5.4 Preliminary scheduling and analysis of scheduling potential

As can be seen from Figure 9A, during the peak tariff period from 10:00 to 15:00, and taking into account charging costs and battery degradation, electric vehicles prioritise reducing their charging power due to the support provided by photovoltaic (PV) generation. As PV generation increases, electric vehicles increase their charging power to absorb the surplus PV output. During the evening peak period from 17:00 to 20:00, electric vehicles, having been recharged during the midday period using PV and off-peak electricity rates, discharge during the peak load period and peak-rate hours. In the time-only coordination scenario S2, due to the high concentration of load at a few nodes, this physically results in the flexible charging and discharging envelope of S2 being severely compressed. As the S3 strategy distributes the load evenly across 13 charging nodes, the power envelope of S3 is widened, enabling deeper charging power during price troughs and greater V2G discharge support during price peaks. In the energy boundary diagram shown in Figure 9B, the actual cluster energy storage state of S3 possesses a broader upper and lower buffer margin than S2, and its trajectory consistently remains strictly above the minimum exit threshold. Whilst S2 executes deep arbitrage and aggressive peak shaving and valley filling, it also unleashes the flexibility potential of vehicle-grid interaction.

As can be seen from Figure 10, between 00:00 and 06:00, EVs and energy storage systems utilise off-peak electricity tariffs to recharge, resulting in a decrease in downward regulation capacity and an increase in upward regulation capacity. Between 07:00 and 09:00, as vehicles consume electricity during their journeys, both upward regulation power and upward regulation capacity decrease. With the contribution of photovoltaic output, EVs can recharge gradually during peak hours. Between 14:00 and 16:00, when electricity prices are standard, EVs are recharged, introducing higher upward regulation power and capacity. This allows them to discharge during the evening rush hour between 17:00 and 20:00, when electricity prices are relatively high. Simultaneously, as discharge occurs during the evening rush hour, downward regulation power and capacity are introduced between 17:00 and 20:00, in preparation for charging during the early morning off-peak hours.

5.5 Sensitivity analysis of spatial scheduling distance radius and analysis of optimal boundaries

To investigate the non-linear impact of spatial dispatching flexibility on the system’s overall performance, this paper gradually increases the user’s maximum acceptable detour radius D max from 0.5 km to 5.0 km under the spatio-temporal coordination scenario S3. Figures 9, 10 illustrate the evolution of the system’s multi-dimensional performance under different dispatching radius values. As shown in Figure 11A, both the system’s total electricity procurement cost and the peak-to-off-peak difference reach a global optimum at a scheduling radius of D max = 1 K m as the radius increases. When D max is relaxed from 0.5 km to 1.0 km, the total cost drops to its lowest point, and the peak-to-off-peak difference also reaches a minimum. When D max is further increased, the system’s operating cost and peak-to-off-peak difference rise rather than fall. Combined with Figure 11B, it can be seen that the average detour distance for users is relatively short at D max = 1 K m, at which point the cluster’s total dispatchable capacity is fully utilised. However, as D max continues to expand, although the system attempts to seek more favourable tariff nodes through long-distance routing, users incur unnecessary detour losses within the urban road network. This not only increases energy consumption during travel but also causes the system’s actual available flexible capacity to fluctuate significantly or even decline, ultimately undermining overall economic efficiency.

Figure 12 shows that the spatial Gini coefficient generally decreases with increasing radius, indicating that greater dispatch flexibility objectively promotes spatial equilibrium of the overall load. However, in practical engineering applications, the pursuit of absolute spatial equilibrium comes at the expense of system economy and flexibility. In the spatial routing of V2G, a larger dispatch range is not necessarily better. The system has a physically optimal boundary within which the traffic network’s operational efficiency and the distribution network’s operational economy are optimally balanced. It should be emphasized that the dispatch radius of 1.0 km represents the best value among the tested cases for this specific Sioux Falls test system; the key insight is the existence of such an optimal boundary rather than the exact numerical value, which may differ for other network topologies and operating conditions.

5.6 Robustness analysis

To assess the robustness of the proposed spatio-temporal coordination strategy under varying operating conditions, we performed sensitivity analyses by varying the EV penetration (set to 2,500 vehicles, representing a high-penetration scenario) and PV generation (reduced to 60% of the baseline, representing a low-PV scenario). Table 4 presents a comparison of the key performance indicators for S2 (time-only coordination) and S3 (spatio-temporal coordination) under these two scenarios.

TABLE 4. Sensitivity analysis results under different scenarios.

The results show that S3 consistently outperforms S2: the total electricity procurement cost is reduced by 9.3% in the high-penetration scenario (17,096.35 vs. 18,843.42) and by 8.2% in the low-PV scenario (19,933.41 vs. 21,720.81), compared to a 12.7% reduction in the baseline. Furthermore, the peak-to-off-peak difference under S3 remains close to that of S2 in all tested scenarios, confirming that the spatial coordination mechanism does not compromise load smoothing. These findings demonstrate that the advantage of spatio-temporal coordination is robust to variations in EV fleet size and PV output. A more comprehensive stochastic or robust optimization framework will be pursued in future work to explicitly account for multi-source uncertainties.

6 Conclusion

Against the backdrop of the large-scale integration of electric vehicles, this paper addresses the issues of spatio-temporal imbalance in charging demand and the overestimation of dispatchable potential. It proposes a framework for the spatio-temporal coordinated allocation of EV demand and the assessment of dispatchable potential, taking into account the coupling between the transport and distribution grids. By constructing a transport network model, a spatial allocation mechanism for charging demand, and an EVA aggregation model based on Minkowski summation, the framework enables the quantitative assessment of the dynamic regulation capacity of EV clusters.

The results of the case studies demonstrate that the method proposed in this paper can effectively alleviate the concentration of load at local nodes, increase the system’s capacity margin, and expand the EVA scheduling boundary. Compared with scheduling strategies that consider only the temporal dimension, the spatio-temporal coordination mechanism exhibits superior performance in terms of load balancing, system operational economy, and flexibility. A Pareto frontier analysis further confirms that the selected objective weights provide a balanced trade-off between economic efficiency and load smoothing, and a sensitivity analysis on EV penetration and PV output demonstrates the robustness of the proposed approach. For the specific Sioux Falls test system, the best system performance was observed at a dispatch radius of 1.0 km among the tested values; this finding is case-specific and should not be interpreted as a universally applicable global optimum.

Future research could further examine the impact of herd behavior among electric vehicle users on the spatial concentration of charging demand, and develop a collaborative scheduling framework for electric vehicles that balances privacy protection with real-time responsiveness, taking into account real-time traffic conditions, dynamic pricing and distributed optimization methods. In addition, the current model treats travel demand, traffic flow, and PV output as deterministic inputs. Future work will extend the proposed framework to a stochastic or robust optimization formulation to explicitly account for multi-source uncertainties. The proposed framework will also be validated on a real-world, large-scale coupled transportation and distribution network to assess its scalability and practical applicability.

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Footnotes

Edited by: Hao Wang, Monash University, Australia

Reviewed by: Yongxi Zhang, Changsha University of Science and Technology, China

Zhiqi Xu, Southeast University, China