A collaboration-strength-aware assessment method for vehicle-charger-grid coordination in distribution systems via multi-dimensional business coupling

1 Introduction

1.1 Research background

Against the background of the dual-carbon targets and transport electrification, electric vehicles are no longer merely transportation loads but are becoming key flexible resources linking transportation systems with distribution networks; large-scale EV integration may aggravate nodal voltage deviations, line loading, and local congestion, but can also support peak-valley regulation, flexible adjustment, and renewable energy accommodation (Deb, 2021; Mastoi et al., 2022). As aggregators, charging stations, and users become more deeply involved, vehicle-charger-grid interaction has shifted from simple charging scheduling to multi-agent coordination characterized by the coupling of information, energy, and value-price flows (Sadeghian et al., 2022; Srivastava et al., 2023). Most existing studies focus on EV integration impacts, coordinated charging, and infrastructure planning; however, a unified evaluation framework is still lacking for assessing inter-agent coordination strength and identifying critical links (Ray et al., 2023; Apata et al., 2023).

1.2 Literature review

1.2.1 Impact of electric vehicle integration on distribution network operation

Existing studies have developed a relatively systematic understanding of how electric vehicle integration affects distribution network operation. The literature widely suggests that large-scale EV integration can significantly influence distribution-network voltage levels, line loading, transformer loading, and power quality, with the impacts strongly dependent on access location, charging time, user behavior, and network topology (Gilleran et al., 2021; Stiasny et al., 2021; Almeida et al., 2023; Fescioglu-Unver and Yıldız Aktaş, 2023). Recent studies have moved beyond qualitative impact assessment toward refined analyses of hosting capacity, low-voltage sensitivity, and local congestion evolution, shifting the focus from the existence of impacts to their triggering conditions and mitigation measures (Nutkani et al., 2024; Chandra et al., 2024). Methodologically, these studies commonly evaluate system responses at different EV penetration levels using indices such as voltage violation, line congestion, and peak load growth, together with power flow simulation, probabilistic analysis, or scenario-based modeling (Li and Jenn, 2024).

1.2.2 Coordinated charging and charging infrastructure planning

Regarding coordinated charging, recent studies have explored the use of EV flexibility to relieve network constraints and reduce operating costs through orderly charging, aggregator-based scheduling, hierarchical control, and demand response (Ullah et al., 2024). Existing reviews have provided systematic summaries of typical models, objectives, and solution methods for coordinated charging, suggesting a future trend toward integrated “distribution-network-constraint, station-side-control, and user-response” frameworks (Aarabi et al., 2025; Guerrero-Silva et al., 2025). Meanwhile, charging infrastructure planning has extended beyond conventional siting and capacity configuration to include station-network coupling, service coverage, operation strategies, and multi-spatiotemporal coordinated planning, reflecting its transition from facility deployment to an integrated construction-operation-dispatch problem (Nasri et al., 2025; Alaraj et al., 2025).

Notably, despite extensive achievements in optimization strategies and engineering applications, most studies on coordinated charging and infrastructure planning still focus on peak-valley regulation, cost reduction, or service capability enhancement, with limited explicit characterization of multi-agent coordination structures. For instance, factors such as aggregator-station information exchange efficiency, distribution-network-station coordination, and user-aggregator incentive compatibility are usually not considered within a unified framework for coordination assessment (Jabr et al., 2025; Maghami et al., 2025).

1.2.3 Data-driven prediction, V2G, and integrated coordination

Driven by advances in data-driven methods, EV charging demand forecasting, user behavior identification, and interpretable spatiotemporal modeling have emerged as new research focuses in vehicle-charger-grid studies. Studies have shown significant heterogeneity among EV users in terms of arrival/departure time, SOC level, price elasticity, and response willingness, suggesting that uniform empirical parameters are insufficient to represent the actual flexibility available on the user side (Mansouri et al., 2025; Kreft et al., 2024). Building on this, more studies have linked load forecasting, price guidance, and coordinated charging to enable refined distribution-network regulation and improved station-side scheduling (Shang et al., 2025).

Meanwhile, V2G, PV-storage-charging integration, and renewable-energy coordination have shifted the vehicle-charger-grid problem beyond load coordination toward multi-energy coupling and value-based coordination. Recent evidence suggests that, in the absence of proper system-level coordination, combining distributed renewable generation with EV charging loads may not improve system operation and may instead introduce new voltage-vulnerable points and local constraint bottlenecks (Ahmad and Bilal, 2023). Therefore, future work needs to go beyond optimizing control results and develop a unified evaluation framework for characterizing inter-agent coordination, thereby explaining why a coordination mode performs better, where the critical links lie, and which agent interactions drive system performance improvement (Hanig et al., 2025).

1.2.4 Research gaps in existing studies

Existing studies have made considerable progress in integration impact analysis, coordinated charging, infrastructure planning, predictive modeling, and renewable-energy coordination. However, most existing evaluation methods remain centered on individual operational objectives, such as voltage security, charging cost, station planning, or prediction accuracy, while the internal coordination relationships among multiple participants have received comparatively limited attention. Consequently, the structural differences among coordination modes and their relationship with system-level operating performance have not yet been characterized within a unified assessment framework. These limitations motivate further research on interpretable and multi-level coordination assessment for vehicle-charger-grid systems.

1.3 Main contributions of this paper

In response to the above research gaps, the main contributions of this paper are summarized in the following three aspects. In this paper, a service-oriented collaborative network refers to a multi-agent network in which participants are represented according to the coordination services they provide or receive, rather than solely as physical devices or electrical loads. Its inter-node edges describe the corresponding information, energy, and value-price interactions among different participants.

• A service-oriented collaborative network modeling method for vehicle-charger-grid systems is proposed for distribution network scenarios. Unlike conventional treatments that regard EVs merely as adjustable loads, this paper uniformly maps distribution network nodes, aggregators, charging stations, and EV user groups into a multi-agent collaborative network, elevating vehicle-charger-grid analysis from operational optimization to service-oriented coordination involving structure-behavior-operation coupling.
• A coordination strength assessment framework is developed by integrating information, energy, and value-price coupling. By defining edge-level coordination strength and incorporating coordination-consistency correction, node importance, and network-level comprehensive indicators, the proposed framework enables multi-level coordination assessment from local links to the entire network and supports unified comparison among different scenarios.
• A joint validation framework of “coordination structure-operational performance” is established. Rather than limiting collaborative network strength to abstract evaluation, this paper links it with physical indicators, including the main-grid power supply profile, nodal voltage deviation, line loading, and operational risk, to verify whether coordination is strengthened and whether such enhancement effectively improves system operation.

To provide an intuitive overview of the proposed method, the overall framework of the vehicle-charger-grid collaboration strength evaluation is illustrated in Figure 1. Starting from multi-source inputs and operating scenarios, the method first constructs a service-oriented collaborative network involving the distribution network, aggregators, charging stations, and EV user groups. On this basis, three-dimensional collaboration modeling is carried out from the perspectives of information coupling, energy coupling, and value-price coupling, followed by coordination-strength evaluation with consistency correction. Finally, the proposed framework is linked with scenario comparison and operational-performance validation, thereby forming a complete analytical chain from coordination structure modeling to performance-oriented assessment.

The five modules in Figure 1 correspond to the subsequent sections of the manuscript as follows. Module 1 summarizes the multi-source inputs and operating scenarios detailed in Section 4.1. Module 2 corresponds to the construction of the vehicle-charger-grid service-oriented collaborative network in Section 2.1, while Module 3 represents the information, energy, and value-price coupling models developed in Section 2.2. Module 4 is formulated through the coordination-consistency correction and hierarchical indicator models in Sections 2.3 and 2.4, and is implemented through the indicator extraction, preprocessing, calculation, and temporal aggregation procedures presented in Section 3. Module 5 corresponds to the comprehensive scenario comparison and operational-performance validation reported in Section 4. In addition, the coordinated operation model in Section 2.5 provides one possible source of the operational data required for the subsequent assessment.

2 Model formulation

2.1 Vehicle-charger-grid service-oriented coordination system

In distribution networks, large-scale EV integration is shifting the traditional mode of “independent vehicle charging, independent facility operation, and passive grid accommodation” toward multi-agent coordination among the distribution network, aggregators, charging stations, and user groups. Different from general static loads, the vehicle-charger-grid system features dispersed terminals, pronounced spatiotemporal load shifting, stochastic travel behavior, and time-varying distribution network operational constraints. Therefore, the system is abstracted as a time-varying service-oriented collaborative network, based on which a coordination strength model and an operation optimization model are developed.

Let the discrete time set be T = 1 , 2 , … , T and the network node set be V = V g ∪ V s ∪ V a ∪ V e, where V g, V s, V a, and V e denote the distribution-network node set, charging-station node set, aggregator node set, and EV user-group node set, respectively; the vehicle-charger-grid collaborative network is defined in Equation 1 as follows:

G t ≜ V , ε t , W t , t ∈ T (1)

Here, ε t represents the directed edge set at time t, and each ordered pair i , j ∈ ε t denotes a coordination interaction initiated by node i and received or acted upon by node j. The matrix W t = w i j t is the associated weighted adjacency matrix, where w i j t, subsequently represented by the final edge-level coordination strength ϕ i j t, characterizes the directional coordination strength from node i to node j. Therefore, ϕ i j t and ϕ j i t are evaluated separately and are not required to be equal. For example, the aggregator-to-station direction represents command delivery and station-side execution, whereas the station-to-aggregator direction represents measurement, operating-state, and execution-result feedback. The two strengths coincide only when the reciprocal interaction data and response characteristics are identical. Accordingly, the weighted adjacency matrix may be asymmetric. Because inter-node connections involve physical power exchange, control information interaction, and economic interest association, the edge set is represented in Equation 2 as follows:

ε t = ε t I ∪ ε t E ∪ ε t V (2)

Here, ε t I, ε t E, and ε t V denote the information-coupling edges, energy-coupling edges, and value-price-coupling edges, respectively.

To characterize the operating state of a node, the state vector of node i at time t is defined in Equation 3 as follows:

X i t = P i t Q i t θ i t v i t π i t τ i t ⊤ (3)

Here, P i t and Q i t represent the active and reactive power states, respectively, while θ i t, v i t, π i t, and τ i t denote the node service state, information observability, incentive state, and response delay. In practical implementation, θ i t is obtained from device-availability and service-mode records, v i t from measurement and communication-availability records, π i t from tariff or incentive signals and the corresponding response records, and τ i t from the timestamps of command issuance, reception, and acknowledgment; the first three quantities are normalized before being used in the assessment model. In addition, s o c e t is introduced as the state of charge for EV user-group nodes, P s c h t as the station-level total charging power for charging-station nodes, and u a t as the aggregation control command for aggregator nodes. Based on the above definitions, the system state evolution can be expressed in Equation 4 as follows:

X t + 1 = f X t , U t , ξ t (4)

Here, X t represents the system-wide state vector, U t denotes the control decision vector, and ξ t is the external disturbance term induced by traffic fluctuations, stochastic arrival behavior, price disturbances, and variations in distribution-network operating conditions.

2.2 Three-dimensional coupled coordination relationship modeling

Based on the three core coupling relationships-information coupling, energy coupling, and value-price coupling-in vehicle-charger-grid systems, this paper decomposes the basic coordination strength between any two nodes i and j into information, energy, and value-price coupling strengths, as expressed in Equation 5:

ϕ i j 0 t = α t ϕ i j I t + β t ϕ i j E t + γ t ϕ i j V t (5)

Here, ϕ i j I t, ϕ i j E t, and ϕ i j V t represent the information, energy, and value-price coupling degrees between nodes i and j, respectively, while α t, β t, and γ t are time-varying weights satisfying α t + β t + γ t = 1 , α t , β t , γ t ≥ 0. To capture the dynamic migration of coordination priorities across operating conditions, the three weights are determined using the state-driven softmax formulation in Equation 6:

α t , β t , γ t = softmax Λ r t (6)

Here, r t is a normalized state-feature vector composed of system security pressure, economic operation pressure, and service quality pressure. The security component reflects operating stress associated with voltage deviation and line loading; the economic component reflects changes in electricity price, main-grid power purchase, and operating cost; and the service-quality component reflects response delay, execution performance, and user-demand satisfaction. In the present study, Λ is preset according to the intended sensitivity of the information, energy, and value-price coupling dimensions to these three state components and is kept unchanged across all scenarios. The softmax mapping converts the resulting state-dependent scores into nonnegative weights whose sum equals one. Accordingly, security-related weights increase under heavy-load or abnormal conditions, whereas energy and value-price coupling receive greater emphasis during normal economic operation.

The three coupling models are expressed in a unified form in Equation 7 as follows:

ϕ i j k t = θ k ∑ m = 1 M k ω m k z i j , m k t + 1 − θ k s i j k t , k ∈ I , E , V (7)

Here, ∑ m = 1 M k ω m k = 1, with θ k ∈ 0 , 1 denoting the balancing coefficient; z i j , m k t represents the normalized evaluation indicator for the k-th coupling type, ω m k denotes its corresponding weight, and s i j k t is the similarity or complementarity correction term. For the information-coupling dimension, z i j , m I t consists mainly of the state sharing rate, data update frequency, control-command arrival rate, and prediction consistency. For the energy-coupling dimension, z i j , m E t is characterized by the interactive power intensity, local energy matching rate, peak-valley complementarity index, and aggregation execution rate. For the value-price-coupling dimension, z i j , m V t mainly covers price response consistency, benefit-sharing matching, and fulfillment consistency. Correspondingly, s i j I t, s i j E t, and s i j V t are used to characterize information-state similarity, power complementarity, and price-elasticity consistency, respectively.

More specifically, a larger s i j I t indicates that nodes i and j exhibit closer information states, such as similar observability, update frequency, command-arrival performance, and predicted-state variation. By contrast, a larger s i j E t indicates stronger complementarity between their power variations; for example, an increase in power injection or a reduction in demand at one node may offset an increase in charging demand at another node. Similarly, s i j V t becomes larger when the two nodes exhibit more consistent response tendencies to the same price or incentive signal. Therefore, the information and value-price terms mainly reward state or response consistency, whereas the energy term rewards compensating power variations.

2.3 Collaborative consistency model

Three-dimensional coupling alone cannot fully characterize the actual coordination quality; even with high information exchange and power interaction between two nodes, the coordination effect may degrade significantly if their control actions show clear time mismatch, delay, or execution deviation. Therefore, a coordination consistency coefficient η i j t is introduced to dynamically modify the basic coordination strength, and the consistency coefficient of node pair i , j at time t is defined as:

η i j t = exp − λ τ Δ τ i j t − λ e Δ e i j t · ρ i j t (8)

Here, Δ τ i j t represents the response-delay deviation, Δ e i j t the execution deviation, and ρ i j t ∈ 0 , 1 the action-direction consistency rate, while λ τ and λ e denote the penalty coefficients for delay and execution deviation, respectively. According to Equation 8, larger response-delay and execution deviations exponentially reduce η i j t and thereby attenuate the final edge-level coordination strength, whereas a higher action-direction consistency rate increases η i j t and preserves a larger proportion of the basic coordination strength. Therefore, the final edge-level coordination strength is expressed in Equation 9 as follows:

ϕ i j t = η i j t ϕ i j 0 t (9)

Based on this, the system-level weighted adjacency matrix is constructed as:

A t = ϕ i j t N × N (10)

This matrix serves as the basis for subsequent analyses of node importance, network density, and overall coordination level.

2.4 Node-level and network-level coordination indicator modeling

Based on the edge-level coordination strength, node-level and network-level indicators are further developed to assess the global coordination state of the vehicle-charger-grid system. The comprehensive coordination importance of node i is defined in Equation 11 as follows:

H i t = μ 1 C i s t + μ 2 C i b t + μ 3 C i c t (11)

Here, C i s t, C i b t, and C i c t denote the weighted strength centrality, betweenness centrality, and closeness centrality of the node, respectively, while μ 1 , μ 2 , μ 3 represent the corresponding weighting coefficients that sum to one.

At the network level, the network-level comprehensive coordination index is defined in Equation 12 by jointly considering the average edge-level coordination strength, network density, weighted clustering coefficient, network efficiency, average coordination response delay, and operational risk:

Ψ ¯ net t = κ 1 ϕ ¯ t + κ 2 D t + κ 3 C t + κ 4 E t − κ 5 τ ¯ t − κ 6 R ¯ t (12)

Here, ϕ ¯ t represents the average edge-level coordination strength, D t denotes the network density, C t denotes the weighted clustering coefficient, E t denotes the network efficiency, τ ¯ t denotes the average coordination response delay, R ¯ t denotes the operational risk index, and κ 1 , ⋯ , κ 6 denote the evaluation weights.

Given the significant time-varying nature of the vehicle-charger-grid system, a temporal aggregation method is further used to formulate the periodic comprehensive coordination score in Equation 13 as follows:

Φ sys = λ 1 Φ ¯ edge + λ 2 Φ ¯ node + λ 3 Φ ¯ net (13)

In this expression, Φ ¯ edge, Φ ¯ node, and Φ ¯ net represent the periodic aggregation values of the edge layer, node layer, and network layer, respectively; λ 1 , λ 2 , λ 3 denote the corresponding evaluation weights, whose sum equals one. The larger Φ sys is, the higher the overall coordination level of the system over the evaluation period.

2.5 Coordinated operation optimization model for vehicle-charger-grid systems

Building on the collaborative network model, coordination assessment is further coupled with operation optimization to establish an integrated “coordination assessment-scheduling decision-making” model. Since vehicle-charger-grid systems need to enhance coordination while considering power purchase cost, voltage quality, network losses, and battery degradation, the optimization objective is constructed in Equation 14 as follows:

min ⁡ F = − ω 1 Φ sys + ω 2 C op t + ω 3 C volt t + ω 4 C loss t + ω 5 C deg t (14)

Here, the first term maximizes the overall coordination effectiveness of the system, while the second to fifth terms represent the operating cost, voltage-deviation penalty, network-loss penalty, and battery degradation cost, respectively; all weighting coefficients are nonnegative.

The detailed expressions of these terms are given in Equation 15 as follows:

C op t = λ buy t P grid + t Δ t − λ sell t P grid − t Δ t + ∑ e ∈ V e c e inc t r e t − ∑ s ∈ V s c s m y s t C volt t = ∑ n ∈ V g V n t − V n ref C loss t = ∑ m , n ∈ L r m n P m n 2 t + Q m n 2 t V m 2 t + ε C deg t = ∑ e ∈ V e c e deg P e ch t η e ch + η e dis P e dis t Δ t (15)

Here, P grid + t and P grid − t represent the power purchased from the main grid and the power injected back into the grid, respectively; λ buy and λ sell are the electricity purchase price and selling price, respectively; r e t denotes the user response incentive, y s t the station operating or on/off status, r m n the resistance of line m , n, and c e deg the EV battery degradation cost coefficient.

2.6 Constraint formulation for vehicle-charger-grid coordinated operation

To guarantee both physical feasibility and practical executability, the joint optimization model is subject to the following constraints:

EV nodes are subject to charging/discharging power constraints, SOC dynamic constraints, and minimum departure SOC constraints; charging stations should satisfy station-level aggregated power and access capacity constraints; the distribution network should meet nodal power balance, voltage upper/lower limit, and line capacity constraints; moreover, aggregation control and coordinated execution constraints are introduced to ensure that the aggregation coordination strategy can be implemented at both the station and user sides. These constraints can be compactly formulated in Equation 16 as follows:

0 ≤ P i ch t ≤ y i t P i , ⁡ max ch , 0 ≤ P i dis t ≤ 1 − y i t P i , ⁡ max dis , y i t ∈ 0 , 1 SOC i min ≤ SOC i t ≤ SOC i max , SOC i t i dep ≥ SOC i req 0 ≤ ∑ i ∈ Ω s P i ch t ≤ P ¯ s cs , 0 ≤ ∑ i ∈ Ω s P i dis t ≤ P ¯ s dis g P t , Q t , V t = 0 , V i min ≤ V i t ≤ V i max , S l t ≤ S ¯ l P s tar t − P s act t ≤ ε s , u s t − u agg t ≤ δ s , τ s t ≤ τ ¯ P i ch t = P i dis t = 0 , t ∉ t i arr , t i dep (16)

Here, P i ch and P i dis represent the EV charging and discharging power at time t, respectively; y i t denotes the charging/discharging mutual-exclusion state variable of EV i, where y i t = 1 indicates the charging state, while y i t = 0 indicates the discharging-or-idle state. P i , ⁡ max ch and P i , ⁡ max dis represent the maximum charging and discharging power of EV i, respectively; SOC i t denotes the SOC of EV i at time t, and SOC i min, SOC i max, and SOC i req correspond to the lower limit, upper limit, and minimum departure requirement of SOC, respectively. t i arr and t i dep represent the arrival and departure times of EV i, respectively; Ω s denotes the set of EVs connected to charging station s. P ¯ s cs and P ¯ s dis represent the upper charging and discharging capacity limits of charging station s, respectively; g · = 0 denotes the power flow balance equations of the distribution network. V i t denotes the voltage at node i at time t, with V i min and V i max representing the lower and upper voltage limits; S l t and S ¯ l denote the apparent power and capacity limit of line l at time t, respectively. P s tar t and P s act t represent the target and actual executed power of charging station s, respectively; u s t and u agg t denote the station-side control command and aggregator coordination command, respectively. ε s and δ s represent the station-level execution error tolerance and aggregation coordination deviation tolerance, respectively; τ s t and τ ¯ denote the coordination delay of charging station s and its maximum allowable value. The final constraint captures the direct impact of travel behavior on the charging/discharging feasible region, indicating that an EV can engage in energy exchange only within its arrival-departure interval.

3 Assessment method for service-oriented collaborative network strength in vehicle-charger-grid scenarios

3.1 Assessment procedure for collaborative network strength

To quantitatively characterize the coordination relationships in the vehicle-charger-grid system, a mapping from operational data to coordination strength indicators is further developed based on the coordination modeling framework established in Section 2. Using the state variables, control variables, and distribution network operation results over the evaluation period, this paper develops a service-oriented collaborative network strength assessment method for vehicle-charger-grid scenarios, including raw indicator extraction, dimensionless preprocessing, hierarchical indicator calculation, temporal aggregation, and grade classification, to support unified quantification of coordination structures and operational characteristics under different scenarios.

If the system operation data over the evaluation period are obtained from the optimization model or scenario simulation in Section 2, the collaborative network strength assessment problem can be formulated in Equation 17 as follows:

R = F X 1 : T , U 1 : T , Y 1 : T = A t , H i t , Ψ net t , Φ sys , L (17)

Here, X 1 : T represents the state-variable sequence over the evaluation period, U 1 : T the control-variable sequence, and Y 1 : T the distribution-network operation-result sequence; A t denotes the weighted coordination matrix at time t, H i t the coordination importance of node i, Ψ net the network-level comprehensive coordination index, Φ sys the periodic comprehensive coordination score, and L the final coordination grade.

3.2 Raw indicator system construction and preprocessing

To convert the abstract three-dimensional coupling model in Section 2 into computable quantities, an edge-oriented raw evaluation indicator system is first established. For an arbitrary node pair i , j, the raw indicator vector for information coupling at time t is defined in Equation 18 as follows:

Z i j I , raw t = r i j shr t , r i j upd t , r i j cmd t , r i j pre t (18)

Here, r i j shr t, r i j upd t, r i j cmd t, and r i j pre t represent the state sharing rate, information update-frequency consistency, control-command arrival rate, and prediction consistency, respectively. The corresponding information-coupling indicators are calculated using Equation 19 as follows:

r i j shr t = O i t ∩ O j t O i t ∪ O j t + ε r i j upd t = 1 − f i upd t − f j upd t f i upd t + f j upd t + ε r i j cmd t = n i j arr t n i j send t + ε r i j pre t = 1 − P ^ i t − P ^ j t P ^ i t + P ^ j t + ε (19)

Here, O i t and O j t represent the observable information sets of nodes i and j at time t, respectively; f i upd t and f j upd t are their information update frequencies; n i j send t denotes the number of control commands sent from node i to node j, and n i j arr t denotes the number of commands successfully received; P ^ i t and P ^ j t are the predicted values of the relevant power or state variables of nodes i and j, respectively; ε is a small positive constant introduced to prevent division by zero. Similarly, the raw indicator vector for energy coupling is defined in Equation 20 as follows:

Z i j E , raw t = q i j int t , q i j mat t , q i j cmp t , q i j exe t (20)

Here, q i j int t, q i j mat t, q i j cmp t, and q i j exe t represent the interactive power intensity, local energy matching rate, peak-valley complementarity index, and aggregation execution rate, respectively. The corresponding expressions are formulated in Equation 21 as follows:

q i j int t = P i j t max τ ∈ T P i j τ + ε q i j mat t = 1 − P i ava t − P j dem t P i ava t + P j dem t + ε q i j cmp t = 1 − ρ i j P t 2 q i j exe t = 1 − P i act t − P i ref t + P j act t − P j ref t P i ref t + P j ref t + ε (21)

Here, P i j t represents the interactive power between nodes i and j at time t; P i ava t is the adjustable power of node i at time t, and P j dem t is the regulation demand of node j; ρ i j P t denotes the power correlation coefficient between nodes i and j within a sliding event window; P i act t and P i ref t are the actual executed power and reference power of node i, respectively, and P j act t and P j ref t are defined analogously. The raw indicator vector for value-price coupling is defined in Equation 22 as follows:

Z i j V , raw t = v i j pri t , v i j ben t , v i j con t ⊤ (22)

Here, v i j pri t, v i j ben t, and v i j con t represent price response consistency, benefit-sharing matching, and fulfillment consistency, respectively, with the corresponding expressions given in Equation 23 as follows:

v i j pri t = 1 − Δ P i t − Δ P j t Δ P i t + Δ P j t + ε v i j ben t = 1 − B i t − B j t B i t + B j t + ε v i j con t = n i j ful t n i j tot t + ε (23)

Here, Δ P i t and Δ P j t represent the power response changes of nodes i and j under price incentives, respectively; B i t and B j t denote the coordination benefits of the two nodes at time t; n i j tot t is the total number of contract executions for node pair i , j, and n i j ful t is the number of fulfilled executions. Furthermore, to align with the coordination consistency model developed in Section 2.3, the raw indicator vector for consistency correction is defined in Equation 24 as follows:

Z i j C , raw t = Δ τ i j t , Δ e i j t , ρ i j t ⊤ (24)

Here, Δ τ i j t represents the response-delay deviation between nodes i and j at time t, Δ e i j t represents the execution deviation, and ρ i j t represents the action-direction consistency rate.

Because the indicators differ in dimension and value range, direct weighting is inappropriate, and dimensionless preprocessing is therefore conducted first. For an arbitrary raw indicator z, the normalized value is denoted by z ∼ and is calculated using the unified form in Equation 25:

z ∼ = z − z min z max − z min + ε , z ∈ Z + z max − z z max − z min + ε , z ∈ Z − (25)

Here, Z + and Z − represent the positive-indicator set and negative-indicator set, respectively; z max and z min denote the maximum and minimum values of the indicator over the evaluation period, and ε is a small positive constant introduced to prevent division by zero.

To improve the stability of the assessment results under random fluctuations and short-term noise in vehicle-charger-grid operation, exponential smoothing is further used to temporally filter the normalized sequence as shown in Equation 26:

z ^ t = α z ∼ t + 1 − α z ^ t − 1 (26)

Here, z ^ t represents the smoothed indicator value, while α ∈ ( 0 , 1 is the smoothing coefficient. After normalization and smoothing, standardized input sequences are obtained for three-dimensional coupling calculation and consistency correction.

3.3 Edge-level coordination strength calculation method

Based on the preprocessed indicator sequences, the input vectors for information, energy, and value-price coupling are constructed in Equation 27 as follows:

Z ^ i j k t = z ^ i j , 1 k t , z ^ i j , 2 k t , ⋯ , z ^ i j , M k k t ⊤ , k ∈ I , E , V (27)

Substituting the above expressions and the consistency-correction input vector Z ^ i j C t into the three-dimensional coupling and coordination consistency models developed in Section 2 yields the edge-level coordination strength ϕ i j t at time t. For subsequent graph-structure analysis, all edge-level coordination strengths are arranged into the weighted adjacency matrix in Equation 10.

Because inter-node coordination propagation is strongly associated with the path structure, the edge-weight-based distance is further defined in Equation 28 as follows:

d i j t = 1 ϕ i j t + ε (28)

Based on this, the shortest-path distance matrix D * t at time t is constructed for later computation of node centrality and network efficiency.

3.4 Node-level and network-level indicator calculation method

At the node level, the weighted strength centrality of each node is first defined in Equation 29 as follows:

C i s t = ∑ j ≠ i ϕ i j t + ϕ j i t (29)

This indicator characterizes the direct coordination connectivity of a node within the collaborative network. Furthermore, the weighted betweenness centrality and weighted closeness centrality of each node are respectively given in Equation 30:

C i b t = ∑ p ≠ q ≠ i σ p q i t σ p q t , C i c t = N − 1 ∑ j ≠ i D i j * t (30)

Here, σ p q t represents the number of shortest paths from node p to node q, σ p q i t represents the number of such paths passing through node i, and D i j * t denotes the shortest weighted distance. Substituting the above expressions into the node comprehensive coordination importance model defined in Section 2 yields H i t.

At the network level, network density, weighted clustering coefficient, network efficiency, and average coordination response delay are constructed to characterize the overall coordination state of the system and are expressed in a unified form in Equation 31 as follows:

D t = 1 N N − 1 ∑ i ≠ j 1 ϕ i j t ≥ ζ C t = 1 N ∑ i = 1 N C i tri t E t = 1 N N − 1 ∑ i ≠ j 1 D i j * t τ ¯ t = ∑ i ≠ j τ i j t 1 ϕ i j t ≥ ζ ∑ i ≠ j 1 ϕ i j t ≥ ζ + ε (31)

Here, ζ is the effective coordination threshold, and c i tri t denotes the local weighted clustering coefficient of node i. Since system operational risk may significantly affect coordination performance, the network-level risk indicator is further defined in Equation 32 as follows:

R t = ω v R v t + ω l R l t + ω u R u t (32)

Here, R v t, R l t, and R u t represent voltage risk, line overload risk, and unmet user demand risk, respectively; ω v , ω l , ω u denote the corresponding risk weights, whose sum equals one. Substituting the above two expressions into the network-level comprehensive coordination index model defined in Section 2 yields Ψ net t.

3.5 Time-domain aggregation evaluation method

Because the vehicle-charger-grid system is highly time-varying, coordination indicators at a single time instant cannot fully characterize the overall coordination level over the evaluation period. Therefore, temporal aggregation is further performed. First, time weights are constructed through exponential decay weighting in Equation 33 as follows:

ξ t = exp − δ T − t ∑ τ = 1 T ⁡ exp − δ T − τ , t = 1 , 2 , … , T (33)

Here, δ ≥ 0 denotes the time decay coefficient. The larger δ is, the higher the weight assigned to coordination behaviors near the end of the evaluation period. Based on this, the edge-level, node-level, and network-level results are temporally aggregated in a unified form in Equation 34 as follows:

A ¯ = ∑ t = 1 T ξ t A t , H ¯ i = ∑ t = 1 T ξ t H i t , Ψ ¯ net = ∑ t = 1 T ξ t Ψ net t (34)

Here, A ¯ represents the time-weighted coordination matrix, H ¯ i the time-weighted node importance, and Ψ ¯ net the time-weighted network-level comprehensive coordination index. The periodic aggregation values of the edge, node, and network layers are further calculated in Equation 35 as follows:

Φ ¯ edge = 1 N N − 1 ∑ i ≠ j a ¯ i j , Φ ¯ node = 1 N ∑ i = 1 N H ∼ i , Φ ¯ net = Ψ ¯ net (35)

Here, a ¯ i j represents the corresponding element of A ¯, while H ∼ i denotes the normalized node importance. Substituting the above expressions into the periodic comprehensive coordination score model defined in Section 2 yields Φ sys for different scenarios.

3.6 Coordination level classification method

To enable cross-scenario comparison, the periodic comprehensive coordination score Φ sys is further classified into different coordination levels. Since fixed thresholds are less adaptable to different case scales and parameter settings, a quantile-driven adaptive threshold method is adopted. Let the comprehensive coordination scores of all scenarios form the set S, and the adaptive classification thresholds are defined in Equation 36 as follows:

q 33 = Q 0.33 S , q 67 = Q 0.67 S (36)

Here, Q 0.33 · and Q 0.67 · represent the 33\% and 67\% quantiles, respectively. Based on these thresholds, the system coordination level is classified in Equation 37 as follows:

L = Weak , Φ sys < q 33 Medium , q 33 ≤ Φ sys < q 67 Strong , Φ sys ≥ q 67 (37)

Without relying on empirical thresholds set manually, this method adaptively classifies different coordination scenarios and is consistent with the weak, medium, and strong coordination levels identified in the subsequent case study.

4 Case analysis

4.1 Case configuration and scenario description

To validate the proposed assessment method for vehicle-charger-grid service-oriented collaborative network strength, the IEEE 33-bus distribution system is used as the test system, and a multi-agent coordination case is constructed with the distribution network, aggregators, charging stations, and EV user groups. Considering the spatial distribution and response-behavior differences of EV charging loads, several charging stations are integrated at representative feeder nodes, and the corresponding user-side resources are aggregated into EV groups to represent practical vehicle-charger-grid interaction scenarios. The configuration of the IEEE 33-bus test system and the interaction relationships among the distribution network, aggregators, charging stations, and EV user groups are illustrated in Figure 2.

Based on this setup, three operating scenarios are considered, namely, the non-collaborative scenario (Scenario A), basic collaborative scenario (Scenario B), and enhanced collaborative scenario (Scenario C). Specifically, charging stations operate independently based on local demand in the non-collaborative scenario; a basic coordination mechanism between aggregators and stations is introduced in the basic collaborative scenario; and the enhanced collaborative scenario further incorporates the joint effects of information, energy, and value-price coupling. The effects of the three scenarios on vehicle-charger-grid operating performance are analyzed by comparing main-grid power supply, network-level coordination index, voltage quality, line loading, and other indicators.

The evaluation indicators considered in this section include the main-grid power supply curve, network-level comprehensive coordination index, system comprehensive coordination score, peak-valley difference, voltage deviation, risk index, time-weighted coordination strength matrix, node importance, and voltage profiles of representative feeder nodes.

4.2 Analysis of main-grid power supply characteristics and network-level coordination response

Figure 3 shows the 24 h profiles of power drawn from the main grid under the three scenarios. The non-collaborative scenario reaches its maximum value of 0.4223 p. u. during 20–21 h, and its peak-valley difference is 0.2132. Both the basic and enhanced collaborative scenarios substantially reduce evening peak grid power consumption and smooth the daily power profile. In particular, the peak-valley difference decreases to 0.1866 and 0.1862, respectively. These results show that both collaborative scenarios effectively mitigate the overlap between EV charging demand and the evening base-load peak, while the enhanced collaborative scenario achieves the best overall smoothing performance.

Figure 4 shows the time-series variation of the network-level comprehensive coordination index Ψ net. The non-collaborative scenario exhibits the lowest overall level, and its maximum value reaches only 0.2634. The basic collaborative scenario is significantly improved, with values maintained above 0.24 for most of the time. The enhanced collaborative scenario achieves the best performance during 8–10 h and 22–24 h, and attains the overall peak value of 0.2870 at 23 h. The results show that stronger coordination continuously enhances cross-layer interactions among the distribution network, aggregators, charging stations, and users, and that the enhanced collaborative scenario delivers a higher level of overall coordination capability during key operating periods.

4.3 Comparison of comprehensive effectiveness indicators

Table 1 summarizes the comprehensive indicator results for the three scenarios. The non-collaborative scenario shows the lowest system comprehensive coordination score of 0.4355, with voltage deviation and operational risk reaching 0.0405 and 0.0045, respectively, indicating the weakest overall performance. Under the basic collaborative scenario, the system comprehensive coordination score is improved to 0.4955, while the peak-valley difference is reduced to 0.1866. The enhanced collaborative scenario further improves the system comprehensive coordination score to 0.5031, while reducing the voltage deviation, operational risk, and maximum line loading to 0.0203, 5.55 × 10⁻⁵, and 0.4090, respectively. These results demonstrate that the enhanced collaborative scenario achieves the best performance in overall coordination, operational security, and voltage quality.

TABLE 1. Comprehensive indicator comparison under different scenarios.

4.4 Analysis of coordination structure characteristics

Figure 5 shows the time-weighted coordination strength matrices for the three scenarios. In the non-collaborative scenario, stronger coordination is mainly confined to interactions among charging stations and local user-side clusters. Under the basic collaborative scenario, the links between the aggregator, charging stations, and user groups become stronger. In the enhanced collaborative scenario, cross-layer coupling is further enhanced. For instance, AGG-CS1 increases from 0.3146 to 0.3770, AGG-EV1 from 0.1996 to 0.2654, and DN-EV1 from 0.1731 to 0.2037. These results show that the enhanced collaborative scenario not only reinforces local links, but also significantly enhances the overall interaction capability among the distribution network, aggregator, charging stations, and users.

4.5 Analysis of typical node voltage response and line-loading characteristics

Figure 6 shows the voltage profiles of Bus6, Bus13, Bus25, and Bus33. Under the non-collaborative scenario, Bus13 and Bus33 reach minimum voltages of 0.9070 p. u. and 0.9144 p. u., respectively, showing clear voltage-weak characteristics. Under the basic collaborative scenario, the minimum voltages of the two nodes rise to 0.9322 and 0.9364 p. u., respectively, and further increase to 0.9509 and 0.9555 p. u. under the enhanced collaborative scenario. In addition, Bus6 and Bus25 generally maintain higher voltage levels under the enhanced collaborative scenario. These results show that enhanced coordination provides stronger voltage support for weak nodes and effectively improves voltage quality in terminal and heavily loaded areas of the distribution network.

Figure 7 shows the time-series variation of the maximum feeder line loading. The maximum line loading in all three scenarios remains well below the 1.0 p. u. limit, indicating that the system has no overload risk. The non-collaborative scenario reaches the highest line stress during 20–21 h, with a maximum value of 0.4413 p. u. Under the basic collaborative scenario and the enhanced collaborative scenario, the maximum line loading is reduced to 0.4103 p. u. and 0.4090 p. u., respectively, both markedly lower than that in the non-collaborative scenario. These results show that both collaborative scenarios effectively relieve evening-peak line loading, while the enhanced collaborative scenario achieves the lowest maximum loading and maintains a better line security margin.

5 Conclusion

To address the lack of a unified quantitative representation of multi-agent coordination in vehicle-charger-grid systems, this paper proposes a service-oriented collaborative network strength assessment method for distribution systems. The distribution network, aggregators, charging stations, and EV user groups are represented within a unified multi-agent network, and their information, energy, and value-price interactions are integrated with coordination-consistency correction. On this basis, the proposed framework characterizes direct interaction strength at the edge level, identifies structurally important participants at the node level, and evaluates the global and time-varying coordination state at the network level. It thereby establishes a structured assessment path from local coordination relationships to the overall system state.

The IEEE 33-bus case studies show that the proposed method clearly distinguishes the non-collaborative, basic collaborative, and enhanced collaborative scenarios. The enhanced collaborative scenario achieves the highest comprehensive coordination score of 0.5031, while its voltage deviation and operational risk decrease to 0.0203 and 5.55 × 10⁻⁵, respectively. More importantly, the edge-level coordination matrix reveals the strengthening of representative cross-layer links, while the network-level comprehensive index captures the overall improvement in multi-agent coordination. The agreement between these assessment results and the observed changes in nodal voltage and line loading demonstrates that the proposed framework can relate coordination-structure evolution to physical operating performance rather than merely rank different scenarios.

Overall, the proposed method provides an interpretable multi-level tool for identifying local coordination links, locating critical participants, and evaluating the global coordination state of vehicle-charger-grid systems. It can therefore support the comparison of alternative coordination modes and provide a quantitative basis for subsequent coordination optimization, active distribution network regulation, and multi-agent interaction mechanism design.

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Footnotes

Edited by: Cong Zhang, Hunan University, China

Reviewed by: Yucui Wang, North China Electric Power University (Baoding), China

Rui Cheng, North China Electric Power University, China