The rapid proliferation of electric vehicles (EVs) represents a pivotal shift in the global energy and transportation landscape. While offering a path towards decarbonization, the large-scale integration of these electric vehicle fleets into the power grid introduces significant operational challenges due to the inherent randomness and concentration of user charging behaviors. Unmanaged charging can exacerbate peak loads, cause voltage deviations, increase network losses, and threaten overall grid stability. Conversely, the collective battery capacity of an electric vehicle cluster constitutes a vast, distributed energy resource with remarkable potential for providing grid services such as peak shaving, frequency regulation, and renewable energy integration.
Accurately assessing the dispatchable or schedulable potential of these electric car aggregations is therefore fundamental for effective grid management and market design. Traditional evaluation methods often rely on deterministic models or probabilistic forecasts based solely on objective factors like State of Charge (SOC) and time-of-use electricity tariffs. A critical oversight in such approaches is the treatment of users as perfectly rational actors who comply unconditionally with dispatch signals. This simplification fails to capture the complex, subjective decision-making processes of electric vehicle owners, which are influenced by psychological factors such as range anxiety, price sensitivity, and personal convenience. Ignoring these subjective elements can lead to significant discrepancies between projected and actual response, resulting in either over-estimation or under-utilization of the electric vehicle car resource.
This paper addresses these gaps by proposing a novel, integrated framework for evaluating the boundary domain of an electric vehicle cluster’s schedulable potential. The core innovation lies in dynamically coupling user subjectivity with external incentives. The methodology unfolds in three principal stages. First, we move beyond objective data analysis to formally quantify the elusive subjective state of “range anxiety” by examining its coupling with observable charging metrics using a Decision-Making Trial and Evaluation Laboratory (DEMATEL) analysis informed by the Weber-Fechner Law. Second, we construct a Multi-Dimensional Semi-Cloud Model (MDSCM) to map users’ uncertain response willingness, creating a nuanced classification of users into non-responding, potentially responding, and always-responding clusters. Finally, we employ an Evolutionary Game Theory (EGT) model to simulate how a population of potentially responding electric vehicle users dynamically adjusts its charging strategies in response to varying compensation price signals from the grid operator. The equilibrium state of this evolutionary process directly determines the final composition of the responsive electric car fleet, from which the aggregate power and energy boundaries of the cluster’s schedulable potential are derived using Minkowski addition.
1. Modeling Electric Vehicle Load Response Characteristics Considering Subjective-Objective Coupling
The charging behavior of an electric vehicle owner is not a purely mechanical function of battery state and electricity price; it is a decision deeply infused with personal perception and psychological state. A key subjective factor is “Range Anxiety” (Z), a driver’s worry that their battery will deplete before reaching a destination or charger. To integrate this into a quantitative model, we first analyze empirical charging data to understand behavioral patterns, then establish the causal links between objective factors and this subjective anxiety.
We begin with a dataset comprising one month of operational records from public charging stations of comparable scale located in different functional zones (e.g., residential, commercial, industrial) of a city. After standard data cleaning and preprocessing, we apply a k-means clustering algorithm to extract typical daily load profile clusters K for each zone. To identify the zone with the most pronounced load volatility—and thus the greatest need for and potential from demand response—we calculate the standard deviation ($\phi$) and the maximum peak-to-valley difference ($PV_{max}$) for each cluster’s representative curve:
$$ \phi = \sqrt{ \frac{1}{95} \sum_{t=1}^{96} (l_t – \bar{l})^2 } $$
$$ PV_{max} = \frac{l_{t_{max}} – l_{t_{min}}}{l_{t_{max}}} \times 100\% $$
where $l_t$ is the load at time interval $t$, and $\bar{l}$ is the daily average load. The residential zone typically exhibits the highest volatility and is selected for further study.
We identify five primary factors influencing a user’s willingness to respond to a grid signal: three objective—Initial SOC, Required Charging Energy, and Electricity Price Period—and two intertwined—Charging Power (objective) and Range Anxiety (subjective). The causal relationships among these factors are not uniform across users. To model this, we first define a fuzzy direct-influence matrix $\mathbf{M} = [m_{ij}]_{5 \times 5}$, where $m_{ij}$ represents the direct influence of factor $i$ on factor $j$, with $m_{ii}=0$. The elements of $\mathbf{M}$ are randomly sampled from predefined plausible ranges for each pair, simulating individual perceptual differences via Monte Carlo simulation.
This matrix is normalized to obtain $\mathbf{F}$, and then the total influence matrix $\mathbf{Q}$ is calculated, capturing both direct and indirect effects:
$$ \mathbf{F} = \frac{\mathbf{M}}{\max_{1 \le i \le 5}(\sum_{j=1}^{5} m_{ij})} $$
$$ \mathbf{Q} = \mathbf{F} (\mathbf{I} – \mathbf{F})^{-1} = [q_{ij}]_{5 \times 5} $$
From $\mathbf{Q}$, we compute the influence degree ($d_i$), the influenced degree ($c_i$), the central degree ($m_i$), and the cause degree ($g_i$) for each factor $i$:
$$ d_i = \sum_{j=1}^{5} q_{ij}, \quad c_i = \sum_{j=1}^{5} q_{ji} $$
$$ m_i = d_i + c_i, \quad g_i = d_i – c_i $$
The comprehensive impact degree $T_i$ of factor $i$ on the overall user response willingness is then derived by combining its centrality and causality:
$$ T_i = \frac{\sqrt{m_i^2 + g_i^2}}{\sum_{i=1}^{5} \sqrt{m_i^2 + g_i^2}} $$
The vector $\mathbf{T} = (T_1, T_2, …, T_5)$ ranks the factors. Our analysis consistently shows that Range Anxiety (Z) holds the highest comprehensive impact degree among all factors.
To quantify the subjective range anxiety $Z$, we use the Weber-Fechner Law, which posits a logarithmic relationship between perceived intensity and physical stimulus. The “stimulus” here is a weighted sum of the objective factors that influence anxiety. The weight $y_i$ for objective factor $i$ is derived from its total influence on anxiety, $q_{i5}$, from matrix $\mathbf{Q}$:
$$ y_i = \frac{q_{i5}}{\sum_{i=1}^{4} q_{i5}} $$
The quantified Range Anxiety is then calculated as:
$$ Z = c_y + k_1 \log(Y_1) + k_2 \log(Y_2) + k_3 \log(Y_3) + k_4 \log(Y_4) $$
where $Y_i$ are the actual values of the four objective factors (Initial SOC, Charging Energy, etc.), $k_i$ are scaling coefficients, and $c_y$ is a constant. This provides a novel, psychologically-grounded metric for a previously qualitative user state.
2. Characterizing Electric Vehicle User Response via a Multi-Dimensional Semi-Cloud Model
With the key influencing factors identified and quantified, the next step is to model the user’s final decision: their willingness to respond (B). This willingness is inherently uncertain and varies continuously. We model it using a Multi-Dimensional Semi-Cloud Model (MDSCM), which excels at translating qualitative concepts (like “high willingness”) into quantitative, probabilistic distributions.
We first define three key indicator margins that serve as inputs to the cloud model, derived from the factors analyzed in Section 1:
- Charging Idle Time Margin ($\eta_1$): The flexibility in a user’s schedule.
$$ \eta_{1,i} = \frac{(T_{i,leave} – T_{i,in}) – T_{i,min}}{T_{i,leave} – T_{i,in}} $$
where $T_{i,in}$, $T_{i,leave}$, and $T_{i,min}$ are the connection time, departure time, and minimum charging time for electric vehicle $i$. - Low-Price Time Margin ($\eta_2$): The economic incentive within the available window.
$$ \eta_{2,i} = \frac{T_{i,price\_low}}{T_{i,leave} – T_{i,in}} $$
where $T_{i,price\_low}$ is the duration of low electricity price periods during the connection. - Range Anxiety Relief Margin ($\eta_3$): The psychological comfort based on battery state.
$$ \eta_{3,i} = \frac{Z_{i,exp}}{Z_{i,in}} $$
where $Z_{i,exp}$ is the driver’s expected (target) range anxiety level upon departure, and $Z_{i,in}$ is the initial anxiety level at connection, calculated from Eq.
For each margin $e$ ($e=1,2,3$), we process historical or survey data for $N$ users to derive the digital characteristics $(E_e, S_e, H_e)$ of a semi-cloud model using backward cloud transformation. Here, $E_e$ is the expected value, $S_e$ is the entropy (width of the distribution), and $H_e$ is the hyper-entropy (dispersion of the entropy).
For a specific electric vehicle car $i$ with a given margin value $\eta_{i,e}$, a conditional forward cloud generator creates a stochastic response willingness $b_{i,e}$ for that dimension:
$$ b_{i,e} = \begin{cases}
\exp\left( -\frac{(\eta_{i,e} – E_e)^2}{2 A^2} \right), & \text{if } \eta_{i,e} < E_e \\
1, & \text{if } \eta_{i,e} \ge E_e
\end{cases} $$
where $A$ is a normally distributed random number with mean $S_e$ and variance $H_e^2$. This captures the idea that willingness saturates at 1 once the margin is sufficiently high, but is uncertain and follows a half-normal distribution below that threshold.
The final, comprehensive response willingness $B_i$ for user $i$ is a weighted sum of the willingness from all three margins:
$$ B_i = \omega_1 b_{i,1} + \omega_2 b_{i,2} + \omega_3 b_{i,3} $$
The weights $\omega_e$ are determined objectively using a combined CRITIC-Entropy method, which accounts for both the contrast intensity (CRITIC) and the information uncertainty (Entropy) of each margin’s data. The combined weight for margin $e$ is:
$$ \omega_e = \frac{\alpha W_e + \beta w_e}{\sum_{e=1}^{3} (\alpha W_e + \beta w_e)} $$
where $W_e$ and $w_e$ are the weights from the CRITIC and Entropy methods, respectively, and $\alpha$, $\beta$ are coefficients balancing the two methods. The output is a cloud drop $(\eta_{i,1}, \eta_{i,2}, \eta_{i,3}, B_i)$ for each electric vehicle in the population.
By aggregating all users’ $B_i$ values and analyzing their distribution, we can classify the electric car fleet into three behavioral clusters with distinct scheduling implications:
- Always-Responding Cluster: Users with $B_i$ close to 1. They will consistently follow orderly charging schedules regardless of minor incentive changes.
- Potentially-Responding Cluster: Users with $B_i$ in a middle range. Their behavior is malleable and highly sensitive to external incentives like compensation prices.
- Non-Responding Cluster: Users with $B_i$ close to 0. They are unlikely to participate in demand response programs under normal circumstances.
The identification of the Potentially-Responding Cluster is crucial, as it represents the dynamic, schedulable portion of the electric vehicle fleet whose behavior can be shaped by market mechanisms.
3. Evaluating Electric Vehicle Cluster Dispatchable Potential via Evolutionary Game Theory
The Potentially-Responding Cluster of electric vehicles does not make decisions in isolation. Each user observes the environment and may imitate the seemingly successful strategies of others. This population dynamics problem is perfectly suited for Evolutionary Game Theory (EGT). We model the strategic interaction where each electric vehicle user (player) can choose one of two strategies: Strategy 1 (Orderly Charging, responding to the grid signal) or Strategy 2 (Disorderly Charging, ignoring the signal).
A user’s payoff or utility for choosing a strategy depends on two dimensions: price satisfaction and convenience. For electric vehicle $i$, the utilities are:
$$ U_{i,1} = f_1 \cdot R_i \cdot (C_{i,in} – p_c) + f_2 \cdot J_i $$
$$ U_{i,2} = f_1 \cdot R_i \cdot C_{i,in} + f_2 \cdot J_i $$
where:
– $C_{i,in}$ is the time-of-use electricity price at connection.
– $p_c$ is the compensation price offered for orderly charging.
– $R_i$ is the user’s price satisfaction preference (higher $R_i$ means greater sensitivity to cost savings).
– $J_i$ is the convenience factor for the chosen strategy (e.g., Disorderly Charging might have a higher $J$ as it’s simpler).
– $f_1, f_2$ are the weights for price and convenience, with $f_1 + f_2 = 1$ and their values differ slightly between strategies (e.g., orderly charging emphasizes price, disorderly emphasizes convenience).
Note that the utility for orderly charging, $U_{i,1}$, includes the benefit of the compensation $p_c$.
The population state is described by $x_{i,n}(\theta)$, the proportion of user $i$’s strategy $n$ at iteration $\theta$. Users are boundedly rational; they periodically review their strategy based on a comparison of payoffs. The dynamic evolution of the strategy proportion follows the replicator dynamics with a logit-based revision protocol, which gives the probability of switching to a strategy proportional to the exponential of its payoff:
$$ \rho_{n \rightarrow m}[U] = \frac{\exp(U_m)}{\sum_{k=1}^{2} \exp(U_k)} $$
The discrete-time evolutionary dynamic for user $i$ is then:
$$ x_{i,n}(\theta+1) = x_{i,n}(\theta) + \lambda \left[ \rho_{m \rightarrow n}(U_i(\theta)) \cdot x_{i,m}(\theta) – \rho_{n \rightarrow m}(U_i(\theta)) \cdot x_{i,n}(\theta) \right] $$
where $\lambda$ is the learning/adaptation step size. This process iterates until an Evolutionary Stable Strategy (ESS) is reached, where the population proportions no longer change. The final $x_{i,1}$ for all potentially responding electric vehicles indicates whether they have evolved to choose orderly charging (1) or not (0).
3.1. Aggregate Schedulable Potential Boundary Domain
Once the evolutionary game reaches equilibrium, we know the final composition of the responsive electric car fleet: the Always-Responding Cluster plus those from the Potentially-Responding Cluster who have adopted Strategy 1. For this aggregated cluster of $\Psi$ vehicles, we calculate its aggregate schedulable potential in terms of maximum power and available energy boundaries using Minkowski addition.
The maximum aggregate charging power $P_t^{max}$ at time $t$ is the sum of the individual maximum powers of all connected and responsive EVs:
$$ P_t^{max} = \sum_{i=1}^{\Psi} P_{i}^{max} \cdot X_{i,t} $$
The aggregate battery energy state $G_t$ has lower and upper bounds defined by the sum of individual SOC limits:
$$ G_t^{min} = \sum_{i=1}^{\Psi} SOC_{i}^{min} \cdot X_{i,t}, \quad G_t^{max} = \sum_{i=1}^{\Psi} SOC_{i}^{max} \cdot X_{i,t} $$
The dynamics of the aggregate energy are governed by:
$$ G_t = G_{t-1} + \vartheta_c \cdot P_t \cdot \Delta t + \Delta G_t $$
where $\vartheta_c$ is the average charging efficiency, $P_t$ is the actual dispatched charging power ($0 \le P_t \le P_t^{max}$), and $\Delta G_t$ accounts for the net energy change due to electric vehicles arriving or departing. The dispatchable potential boundary domain is thus the time-varying envelope defined by the constraints:
$$ 0 \le P_t \le P_t^{max} $$
$$ G_t^{min} \le G_t \le G_t^{max} $$
This domain visually and quantitatively represents the total “flexibility space” that the grid operator can utilize, shaped fundamentally by the compensation price $p_c$ through its influence on the evolutionary game.
4. Case Study and Analysis
To validate the proposed framework, we conduct a simulation based on real-world data from public charging stations in a residential area. The dataset contains one month of charging records for 1500 private electric vehicles. A time-of-use tariff is applied.
4.1. Analysis of Subjective-Objective Coupling
Initial clustering of load profiles from different city zones confirmed the residential area had the most volatile electric vehicle charging load, making it the prime candidate for demand response. Applying the DEMATEL analysis yielded the following comprehensive impact degrees for the five key factors on user response willingness:
| Influencing Factor | Comprehensive Impact Degree T |
|---|---|
| Range Anxiety (Z) | 37.7% |
| Initial State of Charge | 25.5% |
| Electricity Price Period | 13.7% |
| Required Charging Energy | 13.0% |
| Charging Power | 10.2% |
This result underscores the dominant role of the subjective psychological factor—Range Anxiety—in electric vehicle user decision-making, justifying its explicit inclusion in our model. The coupling weights from objective factors to range anxiety were also calculated, enabling the quantification of Z via the Weber-Fechner Law.
4.2. User Classification via Multi-Dimensional Semi-Cloud Model
Using survey data to calibrate the semi-cloud models, the digital characteristics for the three margins were obtained. The combined CRITIC-Entropy method determined the final weights for aggregation: $\omega_1=0.360$ (Idle Time), $\omega_2=0.327$ (Price Time), $\omega_3=0.313$ (Anxiety Relief).
The comprehensive response willingness $B_i$ was calculated for all 1500 electric car users. Plotting the distribution of users against $B_i$ values revealed clear inflection points, leading to the following classification:
| User Behavior Cluster | Willingness Range (B_i) | Number of EVs |
|---|---|---|
| Always-Responding | [0.8, 1.0] | 344 |
| Potentially-Responding | [0.7, 0.8) | 539 |
| Non-Responding | [0.0, 0.7) | 617 |
This classification provides the critical input for the evolutionary game: the 539 electric vehicles in the Potentially-Responding Cluster are the agents whose behavior will evolve based on compensation.
4.3. Evolutionary Dynamics and Schedulable Potential Boundaries
We simulate the evolutionary game for the 539 potentially responding electric vehicles under different compensation prices $p_c$. The evolution clearly shows that as $p_c$ increases, the proportion of users evolving to choose orderly charging (Strategy 1) monotonically increases until it converges to 1 (full participation). The speed of convergence depends on the user’s initial electricity price context. Electric vehicles connecting during low-price (off-peak) periods converge to orderly charging faster with a given $p_c$ increase, as the compensation represents a larger relative saving on their total charging cost. Importantly, the final evolutionary equilibrium is independent of the initial strategy distribution, confirming the robustness of the model.
The final cluster composition after evolution directly determines the aggregate schedulable potential boundary domain. We analyze two key grid service periods: the midday peak (12:00-16:00) and the overnight valley (20:00-06:00).
For a compensation price $p_c = 0.6$ CNY/kWh, the boundary domains are significantly expanded compared to the scenario with no compensation ($p_c=0$). The results demonstrate a strong time-of-day effect:
- Overnight Valley Period: The upper energy boundary $G_t^{max}$ increased by over 100%, and the lower boundary $G_t^{min}$ by approximately 55%. This indicates that the electric vehicle car fleet can provide substantial energy shifting (e.g., absorbing excess wind power) during this period.
- Midday Peak Period: The increases were more modest but still significant (approx. 52% for $G_t^{max}$ and 29% for $G_t^{min}$), highlighting potential for peak shaving, though constrained by shorter parking durations and higher baseline electricity prices that dilute the incentive effect.
Further analysis reveals a law of diminishing marginal returns for the compensation price. While increasing $p_c$ from 0.4 to 0.8 CNY/kWh causes a large jump in schedulable capacity, the incremental gain from 0.8 to 1.0 CNY/kWh is smaller. This indicates an optimal range for compensation pricing to maximize cost-effectiveness for the grid operator.
5. Conclusion
This paper has developed a comprehensive, user-centric framework for evaluating the dynamic schedulable potential boundary domain of electric vehicle aggregations. By integrating a psychologically-grounded quantification of subjective user states, a fuzzy-probabilistic model of response willingness, and a dynamic evolutionary game model of population behavior under incentives, we bridge a critical gap in existing methodologies that treat electric car users as passive or perfectly rational assets.
The key conclusions are as follows. First, subjective factors, particularly range anxiety, play a dominant role in shaping electric vehicle user willingness to participate in demand response. Grid operators and aggregators must design programs that acknowledge and mitigate these psychological barriers to unlock the full potential of the electric vehicle car fleet. Second, the Multi-Dimensional Semi-Cloud Model provides an effective tool for classifying users based on their malleable response characteristics, identifying the crucial “swing” population whose behavior can be influenced. Third, compensation price is a powerful but nuanced tool for shaping the electric vehicle cluster’s flexibility. Its effect is highly time-dependent, with greater leverage during off-peak and valley periods, and exhibits diminishing marginal returns, implying the existence of an economically optimal incentive level.
The proposed framework offers a more realistic and actionable assessment of electric vehicle cluster potential for grid operators, aiding in the design of effective market mechanisms, day-ahead scheduling, and real-time control strategies. Future work will extend this model by incorporating spatial constraints from the transportation network and charging infrastructure layout, moving towards a fully integrated traffic-user-grid co-simulation platform.