With the rapid growth of electric vehicle adoption, particularly in China’s EV market, the fluctuation in charging demand poses significant challenges to the safe and economic operation of power grids. The uncertainty in electric vehicle users’ travel behavior, driven by factors such as route choices and risk preferences, leads to bounded rationality, which in turn affects charging demand patterns. This paper explores a risk-based multi-attribute decision-making approach grounded in prospect theory to model electric vehicle users’ path selection under bounded rationality. We consider three types of uncertain influencing factors—interval numbers, clear numbers, and triangular fuzzy numbers—and analyze their relationship with risk preference coefficients. A variable-coefficient risk-based multi-attribute decision model for electric vehicle travel is established, forming a travel decision scheme based on comprehensive prospect values. Using the method of successive averages, we dynamically allocate and update traffic flows of electric vehicles within a region, developing a charging demand model that incorporates comprehensive prospect values. This model analyzes the impact of bounded rational behavior on daily charging demand, validated through the Nguyen-Dupius network. The integration of these elements provides a robust framework for understanding and predicting electric vehicle charging patterns, essential for grid stability and planning in the context of China’s expanding EV infrastructure.
The proliferation of electric vehicles, especially in China, has intensified the need for accurate charging demand forecasts. Traditional models often assume perfect rationality, overlooking the psychological factors that influence user decisions. In reality, electric vehicle users exhibit bounded rationality due to uncertainties in travel time, congestion, and comfort levels, as well as variations in risk appetite. This paper addresses these gaps by incorporating prospect theory, which accounts for how people perceive gains and losses relative to a reference point. For electric vehicle users, this means that path choices are not solely based on objective metrics but are influenced by subjective evaluations of attributes like travel time (interval numbers), congestion rates (clear numbers), and comfort levels (triangular fuzzy numbers). By modeling these attributes and their interplay with dynamic risk preferences, we can better capture the complexities of electric vehicle charging behavior. The subsequent sections detail the mathematical formulation, model implementation, and case study analysis, emphasizing the role of China’s EV policies and market trends in shaping these dynamics.
To formalize the electric vehicle travel path selection, we define a set of decision alternatives \( A = \{A_1, A_2, \ldots, A_M\} \), where each \( A_i \) represents a path from origin to destination. The attributes influencing decisions are categorized into three types: \( C_1 \) for travel time (interval numbers), \( C_2 \) for congestion rate (clear numbers), and \( C_3 \) for comfort level (triangular fuzzy numbers). Each attribute can be further classified as cost-type (e.g., \( C_1 \) and \( C_2 \), where lower values are preferred) or benefit-type (e.g., \( C_3 \), where higher values are preferred). The states of nature—good, medium, and poor—are denoted by \( S_h \) for \( h = 1, 2, 3 \), with corresponding probabilities \( p_h \) such that \( \sum_{h=1}^3 p_h = 1 \). The expected value of each attribute \( r_j = (r_{1j}, r_{2j}, r_{3j}) \) serves as the reference point for prospect theory calculations. For attribute \( C_1 \), the value \( x_{i1}^h \) is an interval number \( [x_{i1}^{hl}, x_{i1}^{hu}] \), assumed to follow a normal distribution \( N(\mu_{i1}^h, (\sigma_{i1}^h)^2) \), where \( \mu_{i1}^h = (x_{i1}^{hl} + x_{i1}^{hu})/2 \) and \( \sigma_{i1}^h = (x_{i1}^{hu} – x_{i1}^{hl})/6 \). The probability density function is given by:
$$f_{i1}^h(x) = \frac{1}{\sqrt{2\pi}\sigma_{i1}^h} \exp\left(-\frac{(x – \mu_{i1}^h)^2}{2(\sigma_{i1}^h)^2}\right)$$
The gain \( G_{i1}^h \) and loss \( L_{i1}^h \) relative to the reference point \( r_{1}^h \) are computed based on the position of \( x_{i1}^h \) relative to \( r_{1}^h \). For instance, if \( x_{i1}^{hl} < r_{1}^h < x_{i1}^{hu} \), the gain is:
$$G_{i1}^h = \int_{x_{i1}^{hl}}^{r_{1}^h} (r_{1}^h – x) f_{i1}^h(x) \, dx$$
And the loss is:
$$L_{i1}^h = \int_{r_{1}^h}^{x_{i1}^{hu}} (r_{1}^h – x) f_{i1}^h(x) \, dx$$
For attribute \( C_2 \), the value \( x_{i2}^h \) is a clear number, and the gain and loss are straightforward:
$$G_{i2}^h = \begin{cases} 0 & \text{if } x_{i2}^h \geq r_{2}^h \\ r_{2}^h – x_{i2}^h & \text{if } x_{i2}^h < r_{2}^h \end{cases}$$
$$L_{i2}^h = \begin{cases} r_{2}^h – x_{i2}^h & \text{if } x_{i2}^h \geq r_{2}^h \\ 0 & \text{if } x_{i2}^h < r_{2}^h \end{cases}$$
For attribute \( C_3 \), represented as a triangular fuzzy number \( (a_{i3}^h, b_{i3}^h, c_{i3}^h) \), the membership function is:
$$\phi_{i3}^h(x) = \begin{cases} 0 & \text{if } x < a_{i3}^h \\ \frac{x – a_{i3}^h}{b_{i3}^h – a_{i3}^h} & \text{if } a_{i3}^h \leq x \leq b_{i3}^h \\ \frac{c_{i3}^h – x}{c_{i3}^h – b_{i3}^h} & \text{if } b_{i3}^h \leq x \leq c_{i3}^h \\ 0 & \text{if } x > c_{i3}^h \end{cases}$$
The gain \( G_{i3}^h \) and loss \( L_{i3}^h \) are calculated similarly, considering the integral of the difference from the reference point weighted by the membership function. These computations yield risk gain matrices \( G^h = [G_{ij}^h]_{M \times 3} \) and risk loss matrices \( L^h = [L_{ij}^h]_{M \times 3} \). The value functions under prospect theory are defined as:
$$V_{(+)ij}^h = (G_{ij}^h)^{\alpha_j^h}, \quad V_{(-)ij}^h = -\lambda (-L_{ij}^h)^{\alpha_j^h}$$
where \( \alpha_j^h \) is the risk preference coefficient for attribute \( j \) in state \( h \), and \( \lambda = 2.25 \) is the loss aversion coefficient. The probability weighting functions are:
$$\pi_{(+)ij}^h = \frac{(p_h)^\zeta}{((p_h)^\zeta + (1-p_h)^\zeta)^{1/\zeta}}, \quad \pi_{(-)ij}^h = \frac{(p_h)^\delta}{((p_h)^\delta + (1-p_h)^\delta)^{1/\delta}}$$
with \( \zeta = 0.61 \) and \( \delta = 0.69 \). The risk preference coefficient \( \alpha_j^h \) is modeled as a variable to reflect its dependence on the reference point:
$$\alpha_j^h = \left(1 – \frac{r_j^h}{\sum_{h=1}^3 r_j^h}\right)^\theta$$
where \( \theta \in [0,1] \) is a scale parameter representing the sample size effect. This variable coefficient allows the model to adapt to different user risk attitudes, enhancing the realism of electric vehicle path choices. The prospect value for each path and attribute is then:
$$V_{ij} = \sum_{h=1}^3 V_{(+)ij}^h \pi_{(+)ij}^h + \sum_{h=1}^3 V_{(-)ij}^h \pi_{(-)ij}^h$$
Normalizing these values gives \( V_{ij}^* = V_{ij} / V_j^{\max} \), where \( V_j^{\max} = \max_{i \in M} |V_{ij}| \). The comprehensive prospect value for each path is:
$$U_i = \sum_{j=1}^3 \omega_j V_{ij}^*$$
with attribute weights \( \omega_j \) satisfying \( \sum_{j=1}^3 \omega_j = 1 \). Paths are ranked based on \( U_i \), with higher values indicating better choices under bounded rationality.
The dynamic allocation of electric vehicle traffic flows is crucial for updating path choices and charging demand. Using the method of successive averages, the traffic flow on link \( a \) at iteration \( s \) is updated as:
$$x_a^s = \left(1 – \frac{1}{s}\right) x_a^{s-1} + \frac{1}{s} F_a^s$$
where \( F_a^s \) is the auxiliary flow assigned to link \( a \) in iteration \( s \). The travel time on link \( a \) is updated via the BPR function:
$$T_a^s = t_{0a} \left[1 + 0.15 \left(\frac{x_a^s}{C_a}\right)^4\right]$$
Here, \( t_{0a} \) is the free-flow travel time, and \( C_a \) is the link capacity. The speed under basic capacity conditions is derived as \( v_a = C_a l_a / 1000 \), where \( l_a \) is the link distance. These updates ensure that the model reflects real-time traffic conditions, influencing electric vehicle users’ path selections and subsequent charging behavior.
The charging demand model integrates the comprehensive prospect values to estimate daily charging needs. For an electric vehicle \( v \) on path \( i \) in time period \( t \), the charging time is:
$$T_{v,i,t} = (SOC_{v1,i,t} – SOC_{v0,i,t}) C_v / P_v$$
where \( SOC_{v1,i,t} \) is the post-charging state of charge, \( SOC_{v0,i,t} \) is the initial state of charge, \( C_v \) is the battery capacity, and \( P_v \) is the charging power. Considering range anxiety, the post-charging state must satisfy:
$$SOC_{v1,i,t} – d_{k,i} q_v \geq d_v q_v$$
with \( d_{k,i} \) being the path distance, \( q_v \) the energy consumption rate, and \( d_v \) the anxiety range. The total charging demand in period \( t \) is:
$$Q_t = \sum_{k=1}^N \left[ p_{k,t} D_t \sum_{i=1}^{M_k} (u_{k,i,t} SOC_{v2,i,t}) \right]$$
where \( p_{k,t} \) is the proportion of electric vehicles choosing origin-destination pair \( k \), \( D_t \) is the total travel demand, \( SOC_{v2,i,t} = SOC_{v1,i,t} – SOC_{v0,i,t} \) is the charging requirement, and \( u_{k,i,t} \) is the probability of selecting path \( i \), derived from the comprehensive prospect values. Only paths with \( U_i > 0 \) are considered, as they represent beneficial choices under bounded rationality. This approach aligns with Wardrop’s equilibrium principle, where traffic flows stabilize over iterations.
To validate the model, we apply it to the Nguyen-Dupius network, a standard test case in transportation studies. The network comprises multiple nodes and links, with electric vehicles originating from nodes like N1, N4, and N12, and destined for N3. Key parameters include link free-flow travel times, capacities, and distances, as summarized in the table below. For instance, electric vehicle battery capacity is set to 24 kWh, with an energy consumption rate of 30 kWh per 100 km, and an anxiety range of 20 km, reflecting typical values for China’s EV market.
| Link | Node Sequence | Free-Flow Time (min) | Capacity (veh) | Distance (km) | Speed (km/h) |
|---|---|---|---|---|---|
| 1 | N1↔N12 | 12 | 4000 | 14.2 | 56.8 |
| 2 | N12↔N8 | 36 | 3500 | 22.4 | 78.4 |
| 3 | N1↔N5 | 12 | 3000 | 11.2 | 33.6 |
| 4 | N12↔N6 | 12 | 3000 | 11.2 | 33.6 |
| 5 | N4→N5 | 12 | 4000 | 14.4 | 57.6 |
| 6 | N5→N6 | 12 | 3000 | 4.8 | 14.4 |
| 7 | N6↔N7 | 12 | 3000 | 8.0 | 24.0 |
| 8 | N7→N8 | 12 | 3000 | 8.0 | 24.0 |
| 9 | N4→N9 | 24 | 4000 | 19.2 | 76.8 |
| 10 | N5↔N9 | 12 | 3000 | 14.4 | 43.2 |
| 11 | N6→N10 | 12 | 3000 | 20.8 | 62.4 |
| 12 | N7↔N11 | 12 | 3000 | 14.4 | 43.2 |
| 13 | N8→N2 | 12 | 3000 | 14.4 | 43.2 |
| 14 | N9→N10 | 12 | 4000 | 16.0 | 64.0 |
| 15 | N10↔N11 | 12 | 3000 | 9.6 | 28.8 |
| 16 | N11→N2 | 12 | 3000 | 14.4 | 43.2 |
| 17 | N9↔N13 | 24 | 3500 | 14.4 | 50.4 |
| 18 | N11↔N3 | 12 | 3000 | 12.8 | 38.4 |
| 19 | N13↔N3 | 12 | 4000 | 17.6 | 70.4 |
In the case study, we examine paths from N1 to N3, with attribute weights set to \( \omega = (0.2, 0.5, 0.3)^T \). The comprehensive prospect values for each path are computed, and paths with positive values are selected for charging demand calculation. For example, paths 2, 5, and 6 from N1 to N3 show higher prospect values, indicating they are preferred under bounded rationality. Similarly, for other origin-destination pairs, only paths with positive prospect values contribute to the charging demand. The impact of risk preference variation is analyzed by adjusting the parameter \( \theta \). As \( \theta \) increases from 0 to 1, the risk preference coefficient \( \alpha_j^h \) decreases, shifting user behavior from risk-seeking to risk-neutral. This change affects the fluctuation in charging demand curves; higher risk preferences lead to greater volatility, while lower preferences result in smoother patterns. This underscores the importance of incorporating dynamic risk attitudes in electric vehicle charging models, particularly for China’s EV market, where user behavior can vary widely.
Furthermore, we explore the effect of attribute weights on charging demand. By varying the weights \( \omega_j \), we observe shifts in path selections and subsequent charging patterns. For instance, increasing the weight of travel time (a cost-type attribute) amplifies the influence of paths with lower travel times on the comprehensive prospect value, thereby altering the charging demand distribution. The table below illustrates how different weight scenarios affect the comprehensive prospect values for paths from N1 to N3, highlighting that changes in attribute importance can significantly impact electric vehicle charging behavior. This flexibility allows the model to adapt to diverse user preferences and policy interventions, such as incentives for off-peak charging or congestion reduction in urban areas.
| Scenario | Attribute Weights \( \omega \) | Paths with \( U_i > 0 \) (N1 to N3) | Impact on Charging Demand |
|---|---|---|---|
| 1 | (0.2, 0.5, 0.3) | 2, 5, 6 | Moderate fluctuation |
| 2 | (0.2, 0.6, 0.2) | 2, 5 | Reduced peak demand |
| 3 | (0.1, 0.5, 0.4) | 5 | Stable demand curve |
| 4 | (0.3, 0.4, 0.3) | 2 | Increased baseline demand |
The integration of prospect theory with multi-attribute decision-making provides a powerful tool for analyzing electric vehicle charging demand under bounded rationality. By accounting for uncertainties in travel attributes and dynamic risk preferences, the model offers insights into how electric vehicle users make path choices and how these choices aggregate into regional charging patterns. The case study results demonstrate that risk preference variations and attribute weighting significantly influence charging demand curves, emphasizing the need for personalized approaches in managing electric vehicle integration into the grid. For China’s EV sector, this model can inform infrastructure planning, demand response programs, and policy design to mitigate grid impacts and promote sustainable electric vehicle adoption. Future work could extend this framework to include more attributes, such as environmental factors or real-time pricing, and incorporate machine learning techniques for enhanced prediction accuracy in evolving electric vehicle markets.
In conclusion, the bounded rationality of electric vehicle users introduces complexities in charging demand analysis that traditional models often overlook. Our approach, leveraging prospect theory and dynamic traffic assignment, captures these nuances by modeling path selections based on comprehensive prospect values. The variable risk preference coefficient and multi-attribute framework allow for a realistic representation of user behavior, essential for accurate forecasting in the context of China’s growing electric vehicle population. As electric vehicle adoption continues to rise, such models will be crucial for ensuring grid stability, optimizing resource allocation, and supporting the transition to sustainable transportation systems. The methodologies and findings presented here contribute to a deeper understanding of electric vehicle dynamics, paving the way for more resilient and adaptive energy management strategies.