With the rapid growth in the adoption of electric cars, the fluctuation in charging demand poses significant challenges to the safe and economic operation of power grids. The uncertainty in electric car users’ travel behavior, driven by factors such as route choices and risk preferences, leads to bounded rationality, which in turn affects charging demand patterns. In this study, I explore a risk-based multi-attribute decision-making approach grounded in prospect theory to model the travel path selection behavior of electric car users under bounded rationality. I consider three types of uncertain influencing factors—interval numbers, crisp numbers, and triangular fuzzy numbers—and analyze their relationship with variable risk preference coefficients. By developing a variable-coefficient risk-based multi-attribute decision model for electric car travel, I derive comprehensive prospect values for each path to inform travel decisions. Using the method of successive averages, I dynamically allocate and update traffic flows of electric cars within a region, establishing a charging demand model based on these comprehensive prospect values. This allows me to analyze the impact of bounded rational behavior on daily charging demand for electric cars, validated through a case study on the Nguyen-Dupius network. The integration of these elements provides a robust framework for understanding and predicting electric car charging patterns, which is crucial for grid planning and market participation by electric car aggregators.
The proliferation of electric cars has introduced new dynamics in energy demand, particularly due to the intermittent nature of charging activities. Unlike conventional vehicles, electric cars rely on charging infrastructure that interacts directly with the power grid, making it essential to account for user behavior in demand forecasts. Bounded rationality, a concept from behavioral economics, suggests that users do not always make perfectly rational decisions; instead, their choices are influenced by uncertainties and personal risk attitudes. For electric car users, this manifests in travel decisions—such as route selection—that are affected by multiple attributes like travel time, congestion rates, and comfort levels. These attributes often involve uncertainties, which I model using interval numbers (e.g., for travel time), crisp numbers (e.g., for congestion rates), and triangular fuzzy numbers (e.g., for comfort levels). By applying prospect theory, I capture how users perceive gains and losses relative to reference points, incorporating variable risk preferences that change with these references. This approach moves beyond traditional models that assume fixed risk coefficients, offering a more realistic depiction of electric car user behavior.
To formalize the decision-making process for electric car travel, I define a set of alternative paths between origins and destinations. Let \( A = \{A_1, A_2, \dots, A_M\} \) represent the decision options, where each \( A_i \) corresponds to a specific path. The attributes influencing these decisions are categorized into three types: \( C_1 \) (e.g., travel time, modeled as interval numbers), \( C_2 \) (e.g., congestion rate, modeled as crisp numbers), and \( C_3 \) (e.g., comfort level, modeled as triangular fuzzy numbers). Each attribute can be further classified as either a benefit-type or cost-type attribute, denoted as \( N_1 \) and \( N_2 \), respectively. For instance, travel time and congestion rate are cost-type attributes, as users prefer lower values, whereas comfort level is a benefit-type attribute. The states of these attributes—good, medium, and poor—are represented by the set \( S_h \) for \( h = 1, 2, 3 \). The expected value of each attribute, \( r_j = (r_{1j}, r_{2j}, r_{3j}) \), serves as the reference point for prospect theory calculations. For each path \( A_i \) and attribute \( C_j \), I compute the gains and losses relative to these reference points under different states.
For attribute \( C_1 \) (interval numbers), the attribute value \( x_{i1}^h \) is represented as an interval \( [x_{i1}^{hl}, x_{i1}^{hu}] \), where \( x_{i1}^{hl} < x_{i1}^{hu} \), and it follows a normal distribution \( N(\mu_{i1}^h, (\sigma_{i1}^h)^2) \) with \( \mu_{i1}^h = (x_{i1}^{hl} + x_{i1}^{hu})/2 \) 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 \( L_{i1}^h \) for attribute \( C_1 \) are calculated based on the position of \( x_{i1}^h \) relative to \( r_{1}^h \):
$$ G_{i1}^h =
\begin{cases}
0, & x_{i1}^{hl} \geq r_{1}^h \\
\int_{x_{i1}^{hl}}^{x_{i1}^{hu}} (r_{1}^h – x) f_{i1}^h(x) \, dx, & x_{i1}^{hu} \leq r_{1}^h \\
\int_{x_{i1}^{hl}}^{r_{1}^h} (r_{1}^h – x) f_{i1}^h(x) \, dx, & x_{i1}^{hl} < r_{1}^h < x_{i1}^{hu}
\end{cases} $$
$$ L_{i1}^h =
\begin{cases}
\int_{x_{i1}^{hl}}^{x_{i1}^{hu}} (r_{1}^h – x) f_{i1}^h(x) \, dx, & x_{i1}^{hl} \geq r_{1}^h \\
0, & x_{i1}^{hu} \leq r_{1}^h \\
\int_{r_{1}^h}^{x_{i1}^{hu}} (r_{1}^h – x) f_{i1}^h(x) \, dx, & x_{i1}^{hl} < r_{1}^h < x_{i1}^{hu}
\end{cases} $$
For attribute \( C_2 \) (crisp numbers), the attribute value \( x_{i2}^h \) is a real number, and the gain and loss are simpler:
$$ G_{i2}^h =
\begin{cases}
0, & x_{i2}^h \geq r_{2}^h \\
r_{2}^h – x_{i2}^h, & x_{i2}^h < r_{2}^h
\end{cases} $$
$$ L_{i2}^h =
\begin{cases}
r_{2}^h – x_{i2}^h, & x_{i2}^h \geq r_{2}^h \\
0, & x_{i2}^h < r_{2}^h
\end{cases} $$
For attribute \( C_3 \) (triangular fuzzy numbers), the attribute value \( x_{i3}^h \) is represented as a triple \( (a_{i3}^h, b_{i3}^h, c_{i3}^h) \) with \( a_{i3}^h \leq b_{i3}^h \leq c_{i3}^h \), and its membership function is:
$$ \phi_{i3}^h(x) =
\begin{cases}
0, & x < a_{i3}^h \\
\frac{x – a_{i3}^h}{b_{i3}^h – a_{i3}^h}, & a_{i3}^h \leq x \leq b_{i3}^h \\
\frac{c_{i3}^h – x}{c_{i3}^h – b_{i3}^h}, & b_{i3}^h \leq x \leq c_{i3}^h \\
0, & x > c_{i3}^h
\end{cases} $$
The gain and loss for \( C_3 \) are computed as:
$$ G_{i3}^h =
\begin{cases}
\int_{a_{i3}^h}^{c_{i3}^h} (x – r_{3}^h) \phi_{i3}^h(x) \, dx, & a_{i3}^h \geq r_{3}^h \\
0, & c_{i3}^h \leq r_{3}^h \\
\int_{r_{3}^h}^{c_{i3}^h} (x – r_{3}^h) \phi_{i3}^h(x) \, dx, & a_{i3}^h < r_{3}^h < c_{i3}^h
\end{cases} $$
$$ L_{i3}^h =
\begin{cases}
0, & a_{i3}^h \geq r_{3}^h \\
\int_{a_{i3}^h}^{c_{i3}^h} (x – r_{3}^h) \phi_{i3}^h(x) \, dx, & c_{i3}^h \leq r_{3}^h \\
\int_{a_{i3}^h}^{r_{3}^h} (x – r_{3}^h) \phi_{i3}^h(x) \, dx, & a_{i3}^h < r_{3}^h < c_{i3}^h
\end{cases} $$
Using these calculations, I construct risk gain matrices \( G^h = [G_{ij}^h]_{M \times 3} \) and risk loss matrices \( L^h = [L_{ij}^h]_{M \times 3} \) for each state \( h \). Next, I apply prospect theory to derive value functions and probability weighting functions. The value functions for gains and losses are:
$$ V^{(+)}_{hij} = (G_{ij}^h)^{\alpha} $$
$$ V^{(-)}_{hij} = -\lambda (-L_{ij}^h)^{\alpha} $$
where \( \alpha \) is the risk preference coefficient, \( \lambda \) is the loss aversion coefficient (typically set to 2.25), and the probability weighting functions are:
$$ \pi^{(+)}_{hij} = \frac{(p^h)^\zeta}{((p^h)^\zeta + (1 – p^h)^\zeta)^{1/\zeta}} $$
$$ \pi^{(-)}_{hij} = \frac{(p^h)^\delta}{((p^h)^\delta + (1 – p^h)^\delta)^{1/\delta}} $$
Here, \( p^h \) is the probability of state \( h \), with \( \sum_{h=1}^3 p^h = 1 \), and parameters \( \zeta = 0.61 \) and \( \delta = 0.69 \) are used based on empirical studies. Traditionally, \( \alpha \) is fixed at 0.88, but I introduce a variable risk preference coefficient that depends on the reference points:
$$ \alpha_j^h = \left(1 – \frac{r_j^h}{\sum_{h=1}^3 r_j^h}\right)^\theta $$
where \( \theta \) (0 ≤ θ ≤ 1) is a scaling parameter that reflects the sample size or travel scale. Substituting this into the value functions gives:
$$ V^{(+)*}_{hij} = (G_{ij}^h)^{\alpha_j^h} $$
$$ V^{(-)*}_{hij} = -\lambda (-L_{ij}^h)^{\alpha_j^h} $$
The prospect value for each path and attribute is then:
$$ V_{ij} = \sum_{h=1}^3 V^{(+)*}_{hij} \pi^{(+)}_{hij} + \sum_{h=1}^3 V^{(-)*}_{hij} \pi^{(-)}_{hij} $$
I normalize the prospect decision matrix \( V = [V_{ij}]_{M \times 3} \) to obtain \( V^* = [V_{ij}^*]_{M \times 3} \), where \( V_j^{\text{max}} = \max_{i \in M} \{ |V_{ij}| \} \) and \( V_{ij}^* = V_{ij} / V_j^{\text{max}} \). The comprehensive prospect value for each path \( A_i \) is:
$$ U_i = \sum_{j=1}^3 \omega_j V_{ij}^* $$
where \( \omega_j \) is the weight of attribute \( C_j \), with \( \sum_{j=1}^3 \omega_j = 1 \). Paths are ranked based on \( U_i \), with higher values indicating better options. This model accounts for the bounded rationality of electric car users by incorporating variable risk preferences and multiple uncertainties.
To translate these travel decisions into charging demand, I model the traffic flow dynamics using the method of successive averages. Assume there are \( N \) origin-destination pairs, with \( M_k \) paths for the \( k \)-th pair. The traffic flow on each path is updated iteratively to achieve equilibrium. Let \( x_a^s \) be the flow on link \( a \) at iteration \( s \), 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 using the Bureau of Public Roads (BPR) function:
$$ T_a^s = t_{0a} \left[1 + 0.15 \left(\frac{x_a^s}{C_a}\right)^4\right] $$
where \( T_a^s \) is the travel time, \( t_{0a} \) is the free-flow travel time, and \( C_a \) is the capacity of link \( a \). The speed on link \( a \) is derived from the basic capacity formula:
$$ v_a = \frac{C_a l_a}{1000} $$
where \( l_a \) is the length of link \( a \). These updates continue until convergence, ensuring that flows reflect user choices based on comprehensive prospect values.
The charging demand for electric cars is then calculated based on the selected paths. For an electric car \( v \) on path \( i \) during time period \( t \), the charging time \( T_{v,i,t} \) is:
$$ T_{v,i,t} = (\text{SOC}_{v1,i,t} – \text{SOC}_{v0,i,t}) C_v / P_v $$
where \( \text{SOC}_{v1,i,t} \) is the post-charging state of charge, \( \text{SOC}_{v0,i,t} \) is the initial state of charge, \( C_v \) is the battery capacity, and \( P_v \) is the charging power. To account for range anxiety, the post-charging state of charge must satisfy:
$$ \text{SOC}_{v1,i,t} – d_{k,i} q_v \geq d_v q_v $$
where \( d_{k,i} \) is the distance of path \( i \) for O-D pair \( k \), \( q_v \) is the energy consumption rate, and \( d_v \) is the anxiety range. The total charging demand in the region during period \( t \) is:
$$ Q_t = \sum_{k=1}^N \left[ p_{k,t} D_t \sum_{i=1}^{M_k} (u_{k,i,t} \text{SOC}_{v2,i,t}) \right] $$
where \( \text{SOC}_{v2,i,t} = \text{SOC}_{v1,i,t} – \text{SOC}_{v0,i,t} \) is the charging demand per electric car, \( p_{k,t} \) is the proportion of electric cars choosing O-D pair \( k \), \( D_t \) is the total travel demand, and \( u_{k,i,t} \) is the probability of choosing path \( i \), derived from the comprehensive prospect values \( U_i \). Only paths with \( U_i > 0 \) are considered, as they represent beneficial choices under bounded rationality.
In the case study on the Nguyen-Dupius network, I apply this model to analyze electric car charging demand. The network includes multiple origins and destinations, with paths consisting of various links. For example, paths from node N1 to N3 include options like 1-4-7-12-18 and 3-10-17-19. I assume attributes such as travel time (interval number), congestion rate (crisp number), and comfort level (triangular fuzzy number), with weights \( \omega = (0.2, 0.5, 0.3)^T \). The comprehensive prospect values are computed for each path, and traffic flows are updated iteratively. The results show that paths with higher \( U_i \) values are preferred, influencing the spatial and temporal distribution of charging demand. For instance, as the risk preference parameter \( \theta \) varies, the charging demand curve shifts—higher \( \theta \) reduces risk sensitivity, leading to smoother demand patterns. This highlights how bounded rationality affects electric car charging behavior, emphasizing the need for adaptive grid management strategies.
The following table summarizes an example of attribute values and prospect values for paths from one origin to destination, illustrating how different risk preferences impact decisions:
| Path | Travel Time (min) | Congestion Rate | Comfort Level | Comprehensive Prospect Value (\( U_i \)) |
|---|---|---|---|---|
| A1 | [10, 15] | 0.6 | (5, 6, 7) | -0.75 |
| A2 | [12, 18] | 0.4 | (6, 7, 8) | 0.07 |
| A3 | [14, 20] | 0.7 | (4, 5, 6) | -1.00 |
| A4 | [11, 16] | 0.5 | (5, 6, 7) | -0.20 |
| A5 | [9, 14] | 0.3 | (7, 8, 9) | 0.07 |
| A6 | [13, 17] | 0.4 | (6, 7, 8) | 0.09 |
Another table demonstrates how changing attribute weights affects the comprehensive prospect values and, consequently, path choices:
| Scenario | Attribute Weights (\( \omega \)) | Paths with \( U_i > 0 \) | Impact on Charging Demand |
|---|---|---|---|
| 1 | (0.2, 0.5, 0.3) | A2, A5, A6 | Moderate demand |
| 2 | (0.2, 0.6, 0.2) | A2, A5 | Lower demand |
| 3 | (0.1, 0.5, 0.4) | A5 | Higher demand |
| 4 | (0.3, 0.4, 0.3) | A2 | Variable demand |
These findings underscore the importance of incorporating bounded rationality into electric car charging demand analysis. The variable risk preference model provides a more accurate representation of user behavior compared to fixed-coefficient approaches. For grid operators and electric car aggregators, this means that charging demand forecasts should account for psychological factors and uncertainties in travel decisions. Future research could explore additional influences on risk preferences, such as user demographics or trip purposes, to further refine the model. Moreover, integrating real-time data from electric car networks could enhance the dynamic allocation of traffic flows and charging resources. In conclusion, understanding the bounded rationality of electric car users is essential for managing the growing impact of electric cars on power systems and ensuring sustainable energy integration.