With the rapid growth of electric vehicle adoption, particularly in China EV markets, the fluctuating charging demand poses significant challenges to the safe and economic operation of power grids. Electric vehicle users exhibit bounded rationality due to travel uncertainties and varying risk preferences, leading to uncertainties in charging behavior. This study employs a risk-based multi-attribute decision-making approach grounded in prospect theory to analyze electric vehicle users’ travel path selection under bounded rationality. We consider three types of uncertain influencing factors—interval numbers, crisp numbers, and triangular fuzzy numbers—and establish a variable-coefficient risk-based multi-attribute decision model for electric vehicle travel. By dynamically allocating traffic flow using the method of successive averages, we develop a charging demand model based on comprehensive prospect values. The impact of bounded rationality on daily charging demand is analyzed and validated using the Nguyen-Dupuis network.
The proliferation of electric vehicles, especially in China EV sectors, has intensified the need for accurate charging demand forecasts. Traditional models often assume full rationality, ignoring psychological factors that influence user behavior. Prospect theory provides a framework to capture how users perceive gains and losses relative to reference points, incorporating risk attitudes. In this study, we extend prospect theory to multi-attribute decision-making, where attributes like travel time, congestion rate, and comfort level are evaluated under uncertainty. The integration of variable risk preference coefficients allows for a more realistic representation of user behavior, enhancing the prediction of electric vehicle charging patterns.
Electric vehicle travel decisions involve multiple attributes, each with inherent uncertainties. We categorize these attributes into three types: interval numbers (e.g., travel time), crisp numbers (e.g., congestion rate), and triangular fuzzy numbers (e.g., comfort level). For each attribute, we define reference points as user expectations and compute gains and losses relative to these points. The value function in prospect theory is modified to include variable risk preference coefficients, which depend on the reference points. This approach captures the essence of bounded rationality, where users’ risk sensitivity changes with their expectations. The comprehensive prospect value for each path is calculated, guiding the selection of optimal travel routes.
The charging demand for electric vehicles is derived from the traffic flow distribution obtained through the path selection model. We use the method of successive averages to update link flows iteratively, ensuring convergence to a stable state. The charging time for an electric vehicle depends on the state of charge before and after charging, battery capacity, and charging power. Additionally, we account for range anxiety by ensuring that the post-charging state of charge suffices for the remaining trip plus a safety margin. The total charging demand in a region is aggregated over all origin-destination pairs and paths, weighted by the probability of path selection based on comprehensive prospect values.
To formalize the multi-attribute decision model, let \( A = \{A_1, A_2, \ldots, A_M\} \) represent the set of available paths for an electric vehicle trip. Each path is evaluated based on attributes \( C_1, C_2, C_3 \), corresponding to travel time, congestion rate, and comfort level, respectively. Attribute values under different states (good, medium, poor) are denoted as \( x_{ij}^h \), where \( i \) is the path index, \( j \) is the attribute index, and \( h \) is the state index. The reference point for each attribute is the expected value \( r_j^h \). Gains and losses are computed as follows:
For interval-number attributes (e.g., \( C_1 \)), the attribute value \( x_{i1}^h = [x_{i1}^{hl}, x_{i1}^{hu}] \) follows a normal distribution. The gain \( G_{i1}^h \) and loss \( L_{i1}^h \) are calculated using integrals over the distribution. For example, the gain is:
$$ G_{i1}^h = \begin{cases} 0, & \text{if } x_{i1}^{hl} \geq r_1^h \\ \int_{x_{i1}^{hl}}^{x_{i1}^{hu}} (r_1^h – x) f_{i1}^h(x) \, dx, & \text{if } x_{i1}^{hu} \leq r_1^h \\ \int_{x_{i1}^{hl}}^{r_1^h} (r_1^h – x) f_{i1}^h(x) \, dx, & \text{if } x_{i1}^{hl} < r_1^h < x_{i1}^{hu} \end{cases} $$
where \( f_{i1}^h(x) \) is the probability density function. Similarly, for crisp-number attributes (e.g., \( C_2 \)), gains and losses are straightforward differences. For triangular fuzzy number attributes (e.g., \( C_3 \)), we use the membership function to compute integrals for gains and losses.
The value function in prospect theory is 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 variable risk preference coefficient for attribute \( j \) in state \( h \), and \( \lambda \) is the loss aversion coefficient. The probability weighting functions are:
$$ \pi_{ij}^{h+} = \frac{(p^h)^\zeta}{((p^h)^\zeta + (1-p^h)^\zeta)^{1/\zeta}}, \quad \pi_{ij}^{h-} = \frac{(p^h)^\delta}{((p^h)^\delta + (1-p^h)^\delta)^{1/\delta}} $$
where \( p^h \) is the probability of state \( h \). The prospect value for each path and attribute is:
$$ V_{ij} = \sum_{h=1}^3 V_{ij}^{h+} \pi_{ij}^{h+} + \sum_{h=1}^3 V_{ij}^{h-} \pi_{ij}^{h-} $$
After normalizing the prospect decision matrix, the comprehensive prospect value for path \( A_i \) is:
$$ U_i = \sum_{j=1}^3 \omega_j V_{ij}^* $$
where \( \omega_j \) is the weight of attribute \( j \). Paths with higher \( U_i \) are preferred.
The variable risk preference coefficient \( \alpha_j^h \) is modeled as:
$$ \alpha_j^h = \left(1 – \frac{r_j^h}{\sum_{h=1}^3 r_j^h}\right)^\theta $$
where \( \theta \) is a parameter representing the scale of travel. This formulation links risk preferences to reference points, reflecting how users become less risk-sensitive as references increase.
For charging demand analysis, the traffic flow on each link is updated iteratively. Let \( x_a^s \) be the flow on link \( a \) at iteration \( s \). The update rule is:
$$ x_a^s = \left(1 – \frac{1}{s}\right) x_a^{s-1} + \frac{1}{s} F_a^s $$
where \( F_a^s \) is the auxiliary flow. The travel time on link \( a \) is updated using the BPR function:
$$ T_a^s = t_{0a} \left[1 + 0.15 \left(\frac{x_a^s}{C_a}\right)^4\right] $$
where \( t_{0a} \) is free-flow travel time and \( C_a \) is link capacity. The speed on link \( a \) is derived as \( v_a = \frac{C_a l_a}{1000} \), where \( l_a \) is link length.
The charging time for an electric vehicle \( v \) on path \( i \) at time \( t \) is:
$$ T_{v,i,t} = \frac{(SOC_{v1,i,t} – SOC_{v0,i,t}) C_v}{P_v} $$
where \( SOC_{v1,i,t} \) and \( SOC_{v0,i,t} \) are the post-charging and initial states of charge, \( C_v \) is battery capacity, and \( P_v \) is charging power. To address range anxiety, we enforce:
$$ SOC_{v1,i,t} – d_{k,i} q_v \geq d_v q_v $$
where \( d_{k,i} \) is path distance, \( q_v \) is energy consumption rate, and \( d_v \) is anxiety range. The total charging demand at time \( t \) 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 O-D pair \( k \), \( D_t \) is total travel demand, \( u_{k,i,t} \) is the probability of choosing path \( i \), and \( SOC_{v2,i,t} \) is the charging demand.
In the case study using the Nguyen-Dupuis network, we analyze paths from multiple origins to a common destination. The network parameters include link free-flow times, capacities, and lengths. We assume electric vehicles have a battery capacity of 24 kWh and an energy consumption rate of 30 kWh per 100 km. The anxiety range is set to 20 km. The impact of risk preference variations and attribute weights on charging demand is examined.
The results show that as the risk preference parameter \( \theta \) increases, users become less risk-sensitive, leading to reduced fluctuations in charging demand. For instance, when \( \theta = 0 \), users are highly risk-seeking, and charging demand curves exhibit sharp peaks. As \( \theta \) approaches 1, demand smooths out. Different attribute weights also influence charging demand levels. For example, higher weights on travel time increase demand during peak hours, while comfort-focused weights shift demand to less congested periods. The comprehensive prospect value effectively captures these behaviors, providing a robust basis for charging infrastructure planning.
The following table summarizes the key parameters used in the model for electric vehicle charging demand analysis:
| Parameter | Description | Value/Range |
|---|---|---|
| \( C_v \) | Battery capacity | 24 kWh |
| \( q_v \) | Energy consumption rate | 30 kWh/100 km |
| \( d_v \) | Anxiety range | 20 km |
| \( \lambda \) | Loss aversion coefficient | 2.25 |
| \( \zeta \) | Probability weighting parameter | 0.61 |
| \( \delta \) | Probability weighting parameter | 0.69 |
| \( \theta \) | Risk preference scale parameter | 0 to 1 |
Another table illustrates the attribute types and their characteristics in the multi-attribute decision model for electric vehicle travel:
| Attribute Type | Description | Example | Value Representation |
|---|---|---|---|
| Interval Number | Uncertain values in a range | Travel time | \( [x^{hl}, x^{hu}] \) |
| Crisp Number | Exact numerical values | Congestion rate | Real number |
| Triangular Fuzzy Number | Linguistic variables with fuzzy sets | Comfort level | \( (a, b, c) \) |
The integration of prospect theory with multi-attribute decision-making offers a nuanced approach to modeling electric vehicle user behavior. By incorporating variable risk preferences, the model adapts to diverse user psychologies, which is crucial for accurate charging demand forecasting in China EV markets. The charging demand model, coupled with dynamic traffic assignment, provides insights into how bounded rationality influences grid load patterns. This can inform strategies for charging station deployment, time-of-use pricing, and grid management.
In conclusion, the bounded rationality of electric vehicle users significantly affects charging demand through path selection behaviors. The variable risk preference coefficient captures variations in user risk attitudes, while multi-attribute decision-making accounts for multiple travel uncertainties. The proposed framework enhances the realism of charging demand predictions, supporting the sustainable integration of electric vehicles into power systems. Future work could explore the impact of other factors, such as user types and trip purposes, on risk preferences and charging behavior.
The mathematical formulation of the prospect-based model ensures consistency with behavioral economics principles. For example, the value function’s curvature reflects diminishing sensitivity to gains and losses, while the probability weighting function overweights small probabilities and underweights large ones. These features are essential for simulating real-world decision-making in electric vehicle contexts. The use of the Nguyen-Dupuis network for validation demonstrates the model’s applicability to realistic transport networks, highlighting its potential for urban planning and energy management.
Overall, this study underscores the importance of psychological factors in electric vehicle charging demand analysis. As the adoption of electric vehicles, including China EV models, continues to grow, accounting for bounded rationality will be key to developing effective policies and infrastructures. The methods presented here offer a foundation for further research into user-centric models of electric vehicle integration.