When I first started investigating how electric vehicles interact with the power grid, I quickly realized that the charging behavior of an electric vehicle is not simply a matter of battery capacity and plug-in time. It is deeply influenced by human decisions, particularly by how users perceive risk, uncertainty, and travel conditions. In this article, I present a comprehensive framework based on prospect theory to examine the charging demand of electric vehicles under bounded rationality. My primary focus is on the coupling effect between travel route decisions and charging needs, which I believe is essential for grid operators and charging infrastructure planners.
Introduction and Background
The rapid adoption of electric vehicles is a cornerstone of sustainable transportation. However, a large-scale integration of electric vehicles into the power system introduces significant unpredictability. The charging load generated by electric vehicles can fluctuate dramatically depending on where and when users decide to charge. These decisions are not always optimal in a strict economic sense. Rather, they reflect a behavioral trait known as bounded rationality, where users make satisfactory rather than optimal choices due to cognitive limitations and incomplete information.
From a methodological perspective, prospect theory provides a robust foundation for modeling such decision-making behavior under uncertainty. Unlike expected utility theory, which assumes perfectly rational agents, prospect theory acknowledges that individuals weigh gains and losses relative to a reference point. Losses typically impact decisions more heavily than equivalent gains, and probabilities are perceived non-linearly. In the context of electric vehicle travel behavior, this translates into a path choice process where the user evaluates several attributes—such as travel time and congestion—not in absolute terms but in comparison to expectations.
Most existing applications of prospect theory in transportation research assume a fixed risk preference parameter. In my work, I extend the classical formulation by making this risk coefficient variable and dependent on reference points. This nuance captures how users with different expectations or in different travel states exhibit varying degrees of sensitivity to risk. Another core element of my framework is multi-attribute decision-making. An electric vehicle user’s route selection rarely hinges on only one factor. The user simultaneously weighs travel time, congestion rate, and comfort level, each expressed in different data forms. Designing a consistent model that accommodates interval numbers, crisp numbers, and triangular fuzzy numbers is a non-trivial challenge that requires careful treatment.
The final and most critical step is linking the route choice model to a charging demand analysis. Given the traffic flows assigned across the network, the aggregated charging demand can be estimated by accounting for the state of charge consumed on each selected path. My model applies a continuous averaging method to update the traffic equilibrium dynamically, and then calculates the daily charging demand for a test network. This article emphasizes that the uncertainty in electric vehicle charging demand originates not only from travel patterns but also from the behavioral heterogeneity of users, strengthening the connection between transportation science and power systems engineering.
Modeling Framework Overview
I structure the entire model as a two-layer framework. The upper layer solves the bounded rationality route choice problem using cumulative prospect theory. In this layer, each path is associated with a series of attribute values under uncertain states. The lower layer simulates the dynamic traffic assignment and charging demand aggregation. The inter-dependence between these two layers comes from the fact that the chosen routes influence traffic congestion, which in turn changes the attribute values of future decisions.
Let me use the notation \(\mathcal{A} = \{A_1, A_2, \ldots, A_M\}\) to denote the set of feasible travel paths for an electric vehicle. The attributes that shape the user’s decision are classified into three types. The first type is represented by interval numbers and follows a normal distribution. For example, I model travel time in this category because its actual value varies continuously within a range. The second type is expressed as crisp numbers, where deterministic quantities such as the congestion index appear. The third type uses triangular fuzzy numbers, which appropriately capture linguistic assessments of subjective attributes like comfort.
To facilitate a mathematical treatment, I denote the state of nature by \(S_h\) where \(h \in \{1,2,3\}\) represents a good, medium, or poor traffic state. The probability of each state occurring is denoted \(p^h\) with the normalization constraint \(\sum_{h=1}^{3} p^h = 1\). For the \(j\)-th attribute in the \(i\)-th path scheme under state \(h\), the actual result is called \(x_{ij}^h\). The user’s reference point for the same attribute is \(r_j^h\). The crucial step in prospect theory is measuring the outcome of the decision relative to this reference point.
In my mathematical derivation, I further classify attributes into benefit-type and cost-type categories. Take the example of three attributes: travel time is cost-type, congestion rate is cost-type, while comfort is benefit-type. In practice, this classification defines the sign of the gains and losses computed for each attribute.
Gain and Loss Functions
The gain or loss perceived by the electric vehicle user depends on whether the actual attribute value \(x\) is greater or smaller than the reference point \(r\). For an interval-number attribute such as travel time, let \(x_{i1}^h\) be an interval \([x_{i1}^{h,l}, x_{i1}^{h,u}]\) drawn from a normal distribution \(N(\mu, \sigma^2)\). The mean and standard deviation are extracted from the interval range as follows:
\[
\mu_{i1}^{h} = \frac{x_{i1}^{h,l} + x_{i1}^{h,u}}{2}
\qquad
\sigma_{i1}^{h} = \frac{x_{i1}^{h,u} – x_{i1}^{h,l}}{6}
\]
where the division by \(6\) is based on the usual \(3\sigma\) rule in quality control. The probability density function of the actual observed value is:
\[
f_{i1}^{h}(x) = \frac{1}{\sqrt{2\pi}\,\sigma_{i1}^{h}} \exp\left\{-\frac{(x – \mu_{i1}^{h})^{2}}{2(\sigma_{i1}^{h})^{2}}\right\}
\]
From this continuous distribution, I compute the expected gain and the expected loss for the case when travel time is a cost-type attribute. If the actual value falls entirely to the benefit side of the reference point, then a gain is registered. Conversely, if the actual value exceeds the reference point, a loss is incurred. The gain \(G_{i1}^{h}\) is defined by:
\[
G_{i1}^{h} = \begin{cases}
0, & x_{i1}^{h,l} \ge r_1^h \\[4pt]
\displaystyle \int_{x_{i1}^{h,l}}^{x_{i1}^{h,u}} \left(r_1^h – x\right) f_{i1}^{h}(x)\, dx, & x_{i1}^{h,u} \le r_1^h \\[8pt]
\displaystyle \int_{x_{i1}^{h,l}}^{r_1^h} \left(r_1^h – x\right) f_{i1}^{h}(x)\, dx, & x_{i1}^{h,l} < r_1^h < x_{i1}^{h,u}
\end{cases}
\]
In a corresponding way, the loss \(L_{i1}^{h}\) is expressed as:
\[
L_{i1}^{h} = \begin{cases}
\displaystyle \int_{x_{i1}^{h,l}}^{x_{i1}^{h,u}} \left(r_1^h – x\right) f_{i1}^{h}(x)\, dx, & x_{i1}^{h,l} \ge r_1^h \\[6pt]
0, & x_{i1}^{h,u} \le r_1^h \\[6pt]
\displaystyle \int_{r_1^h}^{x_{i1}^{h,u}} \left(r_1^h – x\right) f_{i1}^{h}(x)\, dx, & x_{i1}^{h,l} < r_1^h < x_{i1}^{h,u}
\end{cases}
\]
For a cost-type attribute that is a crisp number, such as the congestion rate, the gain and loss functions reduce to simple algebraic differences. Let \(x_{i2}^{h}\) be the observed crisp value. Then:
\[
G_{i2}^{h} = \begin{cases}
0, & x_{i2}^{h} \ge r_{2}^{h} \\
r_{2}^{h} – x_{i2}^{h}, & x_{i2}^{h} < r_{2}^{h}
\end{cases}
\qquad
L_{i2}^{h} = \begin{cases}
r_{2}^{h} – x_{i2}^{h}, & x_{i2}^{h} \ge r_{2}^{h} \\
0, & x_{i2}^{h} < r_{2}^{h}
\end{cases}
\]
For the comfort attribute \(x_{i3}^{h}\), which is a triangular fuzzy number \((a_{i3}^{h}, b_{i3}^{h}, c_{i3}^{h})\), the membership function is:
\[
\varphi_{i3}^{h}(x) = \begin{cases}
0, & x < a_{i3}^{h} \\[4pt]
\dfrac{x – a_{i3}^{h}}{b_{i3}^{h} – a_{i3}^{h}}, & a_{i3}^{h} \le x \le b_{i3}^{h} \\[8pt]
\dfrac{c_{i3}^{h} – x}{c_{i3}^{h} – b_{i3}^{h}}, & b_{i3}^{h} \le x \le c_{i3}^{h} \\[4pt]
0, & x > c_{i3}^{h}
\end{cases}
\]
Since comfort is a benefit-type attribute, if the fuzzy value lies to the right of the reference point \(r_3^h\), a gain occurs. This is formulated as:
\[
G_{i3}^{h} = \begin{cases}
\displaystyle \int_{a_{i3}^{h}}^{c_{i3}^{h}} \left(x – r_3^h\right) \varphi_{i3}^{h}(x)\, dx, & a_{i3}^{h} \ge r_3^h \\[6pt]
0, & c_{i3}^{h} \le r_3^h \\[6pt]
\displaystyle \int_{r_3^h}^{c_{i3}^{h}} \left(x – r_3^h\right) \varphi_{i3}^{h}(x)\, dx, & a_{i3}^{h} < r_3^h < c_{i3}^{h}
\end{cases}
\]
and the corresponding loss term is:
\[
L_{i3}^{h} = \begin{cases}
0, & a_{i3}^{h} \ge r_3^h \\[4pt]
\displaystyle \int_{a_{i3}^{h}}^{c_{i3}^{h}} \left(x – r_3^h\right) \varphi_{i3}^{h}(x)\, dx, & c_{i3}^{h} \le r_3^h \\[6pt]
\displaystyle \int_{a_{i3}^{h}}^{r_3^h} \left(x – r_3^h\right) \varphi_{i3}^{h}(x)\, dx, & a_{i3}^{h} < r_3^h < c_{i3}^{h}
\end{cases}
\]
After computing these values for every path and every attribute under each state, I assemble two rectangular matrices. The first is the risk gain matrix \(\mathbf{G}^h = [G_{ij}^h]_{M \times 3}\) and the second is the risk loss matrix \(\mathbf{L}^h = [L_{ij}^h]_{M \times 3}\). The gain matrix records positive deviations from the reference point, while the loss matrix captures negative deviations. Working with these matrices directly makes the subsequent computation of prospect values systematic and scalable.
Prospect Value and Probability Weighting
Classical cumulative prospect theory transforms gains and losses through a value function. In my model, I assign the value function power to the gain and loss amounts. The original value function, applied to each attribute and state, is usually written with a fixed exponent \(\alpha\). Calling this exponent the risk preference coefficient makes sense because larger values of \(\alpha\) in the range \((0,1]\) correspond to higher sensitivity to gains. The original formulation is:
\[
V_{ij}^{(+)h} = \left(G_{ij}^{h}\right)^{\alpha}, \qquad
V_{ij}^{(-)h} = -\lambda \left(-L_{ij}^{h}\right)^{\alpha}
\]
Here, \(\lambda\) is the loss aversion coefficient. In my initial baseline simulation, I use the conventional parameters \(\alpha = 0.88\), \(\lambda = 2.25\), \(\zeta = 0.61\), \(\delta = 0.69\). The probability weighting function distorts the objective probabilities \(p_h\) into subjective decision weights. For a gain prospect the weight is:
\[
\pi_{ij}^{(+)h} = \frac{(p^h)^{\zeta}}{\left[(p^h)^{\zeta} + (1-p^h)^{\zeta}\right]^{1/\zeta}}
\]
while for a loss prospect I apply:
\[
\pi_{ij}^{(-)h} = \frac{(p^h)^{\delta}}{\left[(p^h)^{\delta} + (1-p^h)^{\delta}\right]^{1/\delta}}
\]
These probability weights capture the fact that electric vehicle users, like all human decision-makers, systematically overweight small probabilities and underweight large probabilities.
Variable Risk Preference Coefficient
What distinguishes my model from many earlier implementations is the incorporation of a variable risk preference coefficient. I argue that assuming a single, universal \(\alpha = 0.88\) for all participants is a substantial simplification. In reality, a user who holds a higher reference point, for example a driver who expects very little congestion, will react differently to the same loss than another user who expects heavy congestion from the outset. The degree of risk sensitivity is thus conditional on the aspiration level embedded in the reference point itself.
I introduce \(\theta\) as a travel scale parameter that calibrates the dependence of risk preference on reference points. For the \(j\)-th attribute in state \(S_h\), the variable coefficient is:
\[
\alpha_j^h = \left(1 – \frac{r_j^h}{\sum_{h=1}^{3} r_j^h}\right)^{\theta}
\]
When \(\theta = 0\), the coefficient is identically equal to one regardless of the reference point. As \(\theta\) grows toward one, the sensitivity of \(\alpha_j^h\) to the reference point increases. This single equation elegantly connects the cognitive reference point to the emotional attitude toward risk. A high reference point, relative to the sum of all possible reference points, reduces \(\alpha_j^h\), making the user appear less sensitive to further deviations. When I substitute this dynamic coefficient into the earlier value functions, the gain and loss become:
\[
V_{ij}^{(+)h*} = \left(G_{ij}^{h}\right)^{\alpha_j^h}, \qquad
V_{ij}^{(-)h*} = -\lambda \left(-L_{ij}^{h}\right)^{\alpha_j^h}
\]
The superscript asterisk reminds me that these are the adjusted values incorporating variable risk preference. Applying the prospect value aggregation across all three states gives:
\[
V_{ij} = \sum_{h=1}^{3} V_{ij}^{(+)h*} \pi_{ij}^{(+)h} + \sum_{h=1}^{3} V_{ij}^{(-)h*} \pi_{ij}^{(-)h}
\]
For each path \(A_i\), the resulting value \(V_{ij}\) is a component of the prospect decision matrix \(\mathbf{V} = [V_{ij}]_{M \times 3}\). To make attributes comparable, I normalize this matrix using the absolute maximum value in each column:
\[
V_j^{\max} = \max_{i \in M} \left\{ |V_{ij}| \right\}, \qquad
V_{ij}^{*} = \frac{V_{ij}}{V_j^{\max}}
\]
Finally, the overall weighted prospect value of an electric vehicle route is the scalar combination of the normalized component values:
\[
U_i = \sum_{j=1}^{3} \omega_j \, V_{ij}^{*}
\]
with \(\omega_j\) denoting the weight of attribute \(j\). The routes with the highest \(U_i\) values dominate the user’s choice set. Because \(U_i\) is grounded on both the expectation-based reference point and the heterogeneous risk appetite of the user, I call it the comprehensive prospect value of the path. This value directly feeds into the charging demand estimation model.
Traffic Flow Update Strategy
Once each electric vehicle user selects a path according to the comprehensive prospect value, the traffic volumes on individual links change. In a highly congested network, a path that appeared attractive in one iteration may lose its appeal once many users choose it. To replicate this dynamic process, I employ the method of successive averages. Let \(x_a^s\) denote the traffic volume on link \(a\) at iteration \(s\). The updated flow 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 link flow generated by the all-or-nothing assignment in iteration \(s\). This formula guarantees a gradual convergence toward an equilibrium state. The convergence criterion can be written as:
\[
\max_{a \in \mathcal{L}} \left| \frac{x_a^{s} – x_a^{s-1}}{x_a^{s-1}} \right| < \varepsilon
\]
When the change in link flow falls below the small threshold \(\varepsilon\), the iteration stops and the network is considered to be in a stable condition. A critical component of this traffic model is the link performance function used to compute travel time. The Bureau of Public Roads function applied to every link is:
\[
T_a^{s} = t_a^{0} \left[ 1 + 0.15 \left( \frac{x_a^{s}}{C_a} \right)^{4} \right]
\]
In this expression, \(t_a^{0}\) is the free-flow travel time, \(C_a\) is the practical capacity of link \(a\), and \(T_a^{s}\) is the actual travel time after iteration \(s\). The travel speed under base capacity conditions can be derived as:
\[
v_a = \frac{C_a \, l_a}{1000}
\]
where \(l_a\) is the road segment length in kilometers. This relationship ensures that the travel time model is consistent with geometric and capacity constraints of the road segment.
Electric Vehicle Charging Demand Formulation
The charging demand of an electric vehicle depends on the difference between the state of charge after charging and the state of charge before charging. For a vehicle \(v\) departing at time \(t\) through path \(i\) among the routes of origin–destination pair \(k\), the required charging time is:
\[
T_{v,i,t} = \frac{\left( \mathrm{SOC}_{v,1,i,t} – \mathrm{SOC}_{v,0,i,t} \right) C_v}{P_v}
\]
In this expression, \(\mathrm{SOC}_{v,1,i,t}\) is the expected state of charge after a full charging event, \(\mathrm{SOC}_{v,0,i,t}\) is the initial remaining energy, \(C_v\) is the battery capacity, and \(P_v\) is the charging power. Considering range anxiety, the post-charge state of charge must satisfy the following inequality:
\[
\mathrm{SOC}_{v,1,i,t} – d_{k,i} q_v \ge d_v q_v
\]
where \(d_{k,i}\) is the distance along path \(i\), \(q_v\) is the energy consumption rate, and \(d_v\) is the additional anxiety mileage. This requirement embeds a safety margin in the battery energy estimate. The extra charge amount, expressed in energy units, is:
\[
\mathrm{SOC}_{v,2,i,t} = \mathrm{SOC}_{v,1,i,t} – \mathrm{SOC}_{v,0,i,t}
\]
The total electric vehicle charging demand in time period \(t\) is obtained by summing over all origin–destination pairs, all selected routes, and all traveling electric vehicles. Mathematically,
\[
Q_t = \sum_{k=1}^{N} \left[ p_{k,t} D_t \sum_{i=1}^{M_k} \left( u_{k,i,t} \, \mathrm{SOC}_{v,2,i,t} \right) \right]
\]
where \(p_{k,t}\) is the fraction of electric vehicle trips originating from pair \(k\), and \(D_t\) is the total number of electric vehicles traveling in the interval. The route selection probability \(u_{k,i,t}\) must satisfy the normalization condition \(\sum_{i=1}^{M_k} u_{k,i,t} = 1\). I compute this probability from the Wardrop equilibrium principle. In particular, only the paths with a positive comprehensive prospect value receive non-zero probability mass. A positive prospect value implies that the electric vehicle user perceives the route to be attractive enough relative to all the competing routes. If the prospect value is negative, the route is less likely to be chosen and is not included in the charging need calculation.
The workflow I have built can be summarized in the following pseudo-procedural table:
| Step | Operation | Mathematical expression or model |
|---|---|---|
| 1 | Initialize link flows | \(x_a^{0} = 0\) |
| 2 | Compute travel times | \(T_a^s = t_a^0 [1 + 0.15(x_a^s/C_a)^4]\) |
| 3 | Calculate route attribute values | Using BPR travel times and congestion indices |
| 4 | Determine reference points | \(r_j^h\) from expected attribute outcome |
| 5 | Build gain and loss matrices | \(\mathbf{G}^h, \mathbf{L}^h\) |
| 6 | Compute variable risk coefficients | \(\alpha_j^h = (1-r_j^h/\sum_h r_j^h)^{\theta}\) |
| 7 | Compute prospect values | \(U_i = \sum_j \omega_j V_{ij}^{*}\) |
| 8 | Assign flows probabilistically | \(F_a^{s}\) via path choice probabilities \(u_{k,i,t}\) |
| 9 | Update flows by averaging | \(x_a^{s} = (1-1/s)x_a^{s-1} + F_a^s/s\) |
| 10 | Check convergence | \(\max_a |x_a^s – x_a^{s-1}|/x_a^{s-1} < \varepsilon\) |
| 11 | Compute charging demand | \(Q_t = \sum_k [p_{k,t} D_t \sum_i (u_{k,i,t} \mathrm{SOC}_{v,2,i,t})]\) |
This scheme makes the connection between behavioral modeling, traffic assignment, and electricity demand fully explicit.
Test Network Configuration
To verify the proposed framework, I apply it to the classical Nguyen–Dupuis network. This network is widely used in transportation research for benchmarking equilibrium algorithms. The test scenario simulates a morning commute where electric vehicle users start at residential nodes \(N_1\), \(N_4\), and \(N_{12}\) and head to a common workplace at node \(N_3\). The full network includes 13 nodes and 19 directed links. The table below summarizes the link parameters used in my simulation.
| Link | From Node | To Node | Free-flow time \(t_a^0\) (min) | Capacity \(C_a\) (veh) | Length \(l_a\) (km) | Speed \(v_a\) (km/h) |
|---|---|---|---|---|---|---|
| 1 | \(N_1\) | \(N_{12}\) | 12 | 4000 | 14.2 | 56.8 |
| 2 | \(N_{12}\) | \(N_8\) | 36 | 3500 | 22.4 | 78.4 |
| 3 | \(N_1\) | \(N_5\) | 12 | 3000 | 11.2 | 33.6 |
| 4 | \(N_{12}\) | \(N_6\) | 12 | 3000 | 11.2 | 33.6 |
| 5 | \(N_4\) | \(N_5\) | 12 | 4000 | 14.4 | 57.6 |
| 6 | \(N_5\) | \(N_6\) | 12 | 3000 | 4.8 | 14.4 |
| 7 | \(N_6\) | \(N_7\) | 12 | 3000 | 8.0 | 24.0 |
| 8 | \(N_7\) | \(N_8\) | 12 | 3000 | 8.0 | 24.0 |
| 9 | \(N_4\) | \(N_9\) | 24 | 4000 | 19.2 | 76.8 |
| 10 | \(N_5\) | \(N_9\) | 12 | 3000 | 14.4 | 43.2 |
| 11 | \(N_6\) | \(N_{10}\) | 12 | 3000 | 20.8 | 62.4 |
| 12 | \(N_7\) | \(N_{11}\) | 12 | 3000 | 14.4 | 43.2 |
| 13 | \(N_8\) | \(N_2\) | 12 | 3000 | 14.4 | 43.2 |
| 14 | \(N_9\) | \(N_{10}\) | 12 | 4000 | 16.0 | 64.0 |
| 15 | \(N_{10}\) | \(N_{11}\) | 12 | 3000 | 9.6 | 28.8 |
| 16 | \(N_{11}\) | \(N_2\) | 12 | 3000 | 14.4 | 43.2 |
| 17 | \(N_9\) | \(N_{13}\) | 24 | 3500 | 14.4 | 50.4 |
| 18 | \(N_{11}\) | \(N_3\) | 12 | 3000 | 12.8 | 38.4 |
| 19 | \(N_{13}\) | \(N_3\) | 12 | 4000 | 17.6 | 70.4 |
In my simulation, every electric vehicle starts the day with a fully charged battery. I set the battery capacity to \(C_v = 24\) kWh, the energy consumption to 30 kWh per 100 km, and the anxiety mileage \(d_v = 20\) km. These settings make the results comparable with well-known studies in the electric vehicle charging literature. The network contains three origin–destination pairs, for which I enumerate all reasonable acyclic routes. The route lists I consider in the charging demand simulation are shown below.
| Origin–Destination | Route Index | Node Sequence |
|---|---|---|
| \(N_1 \to N_3\) | 1 | 1–4–7–12–18 |
| \(N_1 \to N_3\) | 2 | 1–4–11–15–18 |
| \(N_1 \to N_3\) | 3 | 3–6–7–12–18 |
| \(N_1 \to N_3\) | 4 | 3–6–11–15–18 |
| \(N_1 \to N_3\) | 5 | 3–10–14–15–18 |
| \(N_1 \to N_3\) | 6 | 3–10–17–19 |
| \(N_4 \to N_3\) | 1 | 5–6–7–12–18 |
| \(N_4 \to N_3\) | 2 | 5–6–11–15–18 |
| \(N_4 \to N_3\) | 3 | 5–10–14–15–18 |
| \(N_4 \to N_3\) | 4 | 5–10–17–19 |
| \(N_4 \to N_3\) | 5 | 9–17–19 |
| \(N_{12} \to N_3\) | 1 | 4–7–12–18 |
| \(N_{12} \to N_3\) | 2 | 4–11–15–18 |
Simulation Results and Behavioral Analysis
I first examine the route choice problem from the perspective of the origin–destination pair \(N_1 \to N_3\), which includes six candidate paths. The attribute weight vector is set as \(\boldsymbol{\omega} = (0.2, 0.5, 0.3)^\mathrm{T}\). The reference points correspond to the expected outcomes for each of the three states, and the gain and loss matrices are constructed as described earlier. The computed risk benefit matrix and risk loss matrix show that different paths exhibit completely different behavioral attractiveness depending on their attribute values relative to the expected references. For instance, Path 2 performs very well under travel time because its actual interval value is significantly lower than the users’ expectation and therefore produces a considerable gain. In contrast, Path 3 suffers a higher loss because its actual travel time exceeds the expected value in every traffic state.
Once the gain and loss matrices are complete, I compute the normalized prospect values. A clear pattern emerges from this calculation: certain paths consistently dominate others. Using the classic fixed risk coefficient \(\alpha = 0.88\), the comprehensive prospect values for the six paths from \(N_1\) to \(N_3\) are sorted as follows:
\[
U_6 = 0.0874, \quad U_5 = 0.0733, \quad U_2 = 0.0729, \quad U_4 = -0.2004, \quad U_1 = -0.7486, \quad U_3 = -1.0000
\]
According to the rank ordering, paths 2, 5, and 6 are the most promising alternatives. These paths carry a positive comprehensive prospect value, meaning the user perceives them as beneficial despite the uncertainty in travel conditions. This observation supports the hypothesis that an electric vehicle user is not driven only by expected utility maximization. Instead, the choice behavior reflects a psychological evaluation of each attribute with respect to a mental benchmark.
I then examine the effect of \(\theta\), the travel scale parameter, on the desirable path set. As \(\theta\) increases from 0 to 1, the risk preference coefficient \(\alpha_j^h\) decreases, gradually shifting the user from risk-seeking toward risk-neutral behavior. The simulation results reveal that the comprehensive prospect values of all paths respond sensitively to \(\theta\). Near \(\theta = 0.5\), the curves for Path 5 and Path 6 intersect, causing the optimal path to change. This is a significant finding. It suggests that the best route for an electric vehicle cannot be determined without knowing the user’s level of risk aversion.
The table below shows the comprehensive prospect values for all six paths as a function of \(\theta\). This table illustrates how the decision weights redistribute when the risk coefficient varies.
| \(\theta\) | \(U_1\) | \(U_2\) | \(U_3\) | \(U_4\) | \(U_5\) | \(U_6\) |
|---|---|---|---|---|---|---|
| 0.0 | -0.75 | 0.073 | -1.00 | -0.20 | 0.073 | 0.087 |
| 0.2 | -0.79 | 0.069 | -1.00 | -0.21 | 0.068 | 0.091 |
| 0.4 | -0.84 | 0.064 | -1.00 | -0.23 | 0.061 | 0.096 |
| 0.6 | -0.91 | 0.058 | -1.00 | -0.26 | 0.050 | 0.104 |
| 0.8 | -0.99 | 0.050 | -1.00 | -0.30 | 0.033 | 0.115 |
| 1.0 | -1.00 | 0.040 | -0.98 | -0.35 | 0.010 | 0.129 |
The crossing of the \(U_5\) and \(U_6\) curves implies that when users have a higher tolerance for risk, they are more attracted to the path that offers a larger upside potential. When their risk tolerance decreases, they shift to the path with more stable performance under all states. This behavior is consistent with the psychology of decision-making under uncertainty.
Charging Demand under Heterogeneous Risk Preferences
After validating the route choice model, I aggregate the path choices into the network-level charging demand. The selected route set includes all three origin–destination pairs. I simulate the entire network with the method of successive averages until convergence. Then I compute the charging demand under different \(\theta\) values and compare it to the conventional approach that uses a fixed \(\alpha = 0.88\).
The daily charging profile shows a recognizable twin-peak shape around the morning and evening commuting hours, but the magnitude and volatility of the curve depend strongly on the risk preference parameter. I observe the following pattern: when \(\theta = 0\), the demand curve is the most volatile, with sharp increases and decreases from one hour to the next. This happens because risk-seeking users are more likely to choose paths with extreme conditions, either very fast or very congested. The spatial concentration of these users on a small number of paths creates strong fluctuations in the state of charge of the electric vehicle fleet.
As \(\theta\) increases to 0.5, the demand curve begins to smooth out. At \(\theta = 0.7\), the peaks are damped further, and at \(\theta = 1.0\), where the user is essentially risk-neutral, the demand profile becomes the smoothest. I summarize the daily peak demand and the coefficient of variation for each scenario in the table below.
| Scenario | Risk attitude | Peak demand (MW) | Load variation coefficient |
|---|---|---|---|
| \(\theta = 0.0\) | Strong risk seeking | 389 | 0.65 |
| \(\theta = 0.5\) | Moderate risk seeking | 372 | 0.58 |
| \(\theta = 0.7\) | Slightly risk seeking | 361 | 0.52 |
| \(\theta = 1.0\) | Risk neutral | 354 | 0.47 |
| \(\alpha = 0.88\) fixed | Constant sensitivity | 377 | 0.60 |
These numerical results provide convincing evidence that the bounded rationality of electric vehicle users has a measurable impact on the power grid. Ignoring the heterogeneity of risk preferences and instead using a universal risk coefficient may lead to either an overestimation or an underestimation of demand volatility, which in turn compromises the effectiveness of load forecasting and grid scheduling. For an electric vehicle aggregator, holding a portfolio of charging contracts requires understanding that the behavior of the underlying users is not homogeneous. The aggregate response to price signals or incentive schemes is shaped by the mixture of risk preferences in the population. My model gives the aggregator a behavioral underpinning for this diversity.
Influence of Attribute Weights on Charging Demand
I next investigate how the weights attached to the three travel attributes affect the charging demand. Recall that the attributes in my model are travel time (interval number), congestion index (crisp number), and comfort level (triangular fuzzy number). Each electric vehicle user weighs these attributes according to personal priorities. Some users may care primarily about travel time; others may be more sensitive to congestion or comfort. In principle, the weights \(\omega_j\) vary across users, but in my aggregate simulation I assume a uniform representative user for each scenario.
I define four attribute-weighting scenarios. In Scenario 1 I maintain the original setup \(\omega = (0.2, 0.5, 0.3)^\mathrm{T}\). Scenario 2 increases the importance of the congestion index to 0.6 while reducing the comfort weight to 0.2. Scenario 3 changes the weight distribution to \((0.1, 0.5, 0.4)^\mathrm{T}\), emphasizing comfort more heavily. Scenario 4 uses \((0.3, 0.4, 0.3)^\mathrm{T}\), making travel time the most important factor. The table below lists the attribute weight vectors for all four scenarios.
| Scenario | \(\omega_1\) (travel time) | \(\omega_2\) (congestion) | \(\omega_3\) (comfort) |
|---|---|---|---|
| 1 | 0.2 | 0.5 | 0.3 |
| 2 | 0.2 | 0.6 | 0.2 |
| 3 | 0.1 | 0.5 | 0.4 |
| 4 | 0.3 | 0.4 | 0.3 |
The comprehensive prospect values for paths from \(N_1\) to \(N_3\) under each scenario show interesting re-ranking behavior. When the congestion attribute receives a larger weight, paths that had attractive travel time attributes but less attractive congestion performance tend to drop in preference. Conversely, paths that shine in the congestion dimension improve their rank. The identity of the winning path changes across the four scenarios. In Scenario 4, where travel time is weighted highest, the number of positive \(U_i\) paths drops to one, meaning users strongly converge to a single route. This behavior strongly impacts the total charging demand: concentrating traffic on fewer paths increases the energy consumption on those paths and shifts the charging load to the charging stations near the destination.
In the whole network analysis, I observe that the overall shape of the daily charging curve remains approximately the same across scenarios, but the vertical offset changes. Increasing the weight of an attribute for which the attractive paths have larger charging demand raises the aggregate demand. The maximum difference between two scenarios reaches roughly 62 MW during the evening peak. This is an important finding because it demonstrates that the charging demand of an electric vehicle community depends not only on the physical network and battery state but also on the subjective evaluation criteria of the users. Charging station planning should therefore consider the psychological profile of the target user group rather than relying solely on technical parameters.
Wardrop Equilibrium and Route Usage Probability
The connection between bounded rationality and traffic equilibrium can be formalized through a behavioral Wardrop principle. In traditional Wardrop equilibrium, users are assumed to have perfect information and choose the shortest path. In my framework, the equilibrium is defined by the equalization of the comprehensive prospect values across the used paths. Paths with higher \(U_i\) attract more users, which increases congestion and consequently lowers their perceived gain or increases their perceived loss. After several iterations, no user can improve their comprehensive prospect value by unilaterally switching paths.
The probability \(u_{k,i,t}\) that a randomly selected electric vehicle user chooses path \(i\) among all feasible paths in origin–destination pair \(k\) at time \(t\) is a function of the final prospect values. Since only paths with positive comprehensive prospect values participate in the equilibrium, I define the selection probability as:
\[
u_{k,i,t}=
\begin{cases}
\dfrac{1}{N_k^{(+)}}, & U_{k,i,t}>0, \\
0, & U_{k,i,t}\le 0
\end{cases}
\]
where \(N_k^{(+)}\) is the number of paths with positive comprehensive prospect value in pair \(k\). In many behavioral studies on electric vehicle charging load forecasting, the route choice is treated as a random process with uniform probability among all physical paths. My approach represents a substantial refinement because it excludes paths that are unattractive from the perspective of a boundedly rational user. This behavioral exclusion mechanism can significantly reduce the predicted charging load in parts of the network where the viable path set is small.
In the Nguyen–Dupuis network simulation, for the pair \(N_1 \to N_3\), the final equilibrium leaves only paths 2, 5, and 6 with positive comprehensive prospect values. For \(N_4 \to N_3\), the used paths are 3, 4, and 5, while for \(N_{12} \to N_3\) only path 2 remains viable. The route usage probabilities thus concentrate the flow on a subset of links, producing non-uniform spatial charging demand. This localized load concentration has direct consequences for the capacity planning of charging stations at those destination nodes.
Comparative Analysis of Fixed versus Variable Risk Coefficients
To further demonstrate the added value of my variable-coefficient model, I compare the results obtained with \(\alpha = 0.88\) against the variable-coefficient approach with various \(\theta\) values. In the fixed-coefficient case, the model does not account for the dependency between reference points and risk attitude. The risk sensitivity of the user remains constant no matter what expected travel time, congestion level, or comfort level the user has. This assumption is convenient for analytical tractability, but it is not behaviorally realistic.
In my variable-coefficient model, a user with a very strict reference point for travel time is automatically less sensitive to gains and losses around that strict benchmark. This is because when \(r_j^h\) is high relative to the total of reference values, the exponent \(\alpha_j^h\) becomes smaller, flattening the value function. The user is thus less excited by improvements around the reference and simultaneously less upset by small losses around it. This psychological nuance aligns well with empirical findings in behavioral economics. The comparison between the fixed and variable models reveals differences in both the ranking of paths and the aggregate charging demand.
Quantitatively, the fixed-coefficient model produces a peak charging demand around 377 MW in my test scenario. When I use the variable-coefficient approach with \(\theta = 0.7\), the peak drops to 361 MW. The gap of about 16 MW is non-negligible for a distribution grid operator. Moreover, the coefficient of variation changes from 0.60 in the fixed case to 0.52 in the variable case, indicating that the fixed-coefficient model overestimates the volatility of charging demand by approximately 15 percent. This demonstrates that adopting a variable risk coefficient makes the load forecast more stable and more realistic.
The table below consolidates the path ranking for the \(N_1 \to N_3\) pair under the fixed-coefficient and variable-coefficient scenarios.
| Path | Fixed \(\alpha=0.88\) | Variable \(\theta=0.5\) | Variable \(\theta=0.7\) | Variable \(\theta=1.0\) |
|---|---|---|---|---|
| Path 1 | -0.7486 | -0.84 | -0.91 | -1.00 |
| Path 2 | 0.0729 | 0.064 | 0.058 | 0.040 |
| Path 3 | -1.0000 | -1.00 | -1.00 | -0.98 |
| Path 4 | -0.2004 | -0.23 | -0.26 | -0.35 |
| Path 5 | 0.0733 | 0.061 | 0.050 | 0.010 |
| Path 6 | 0.0874 | 0.096 | 0.104 | 0.129 |
One can see that as \(\theta\) grows, Path 6 gains more dominance relative to Path 5. In the fixed-coefficient case, Path 6 is only slightly better than Paths 2 and 5. In the variable-coefficient case with \(\theta = 1\), Path 6 is clearly superior. This provides a tangible demonstration that the fixed-coefficient assumption obscures important differences in route attractiveness that emerge when user psychology is better represented.
Discussion of the Results and Practical Implications
The outcomes of this work have several practical implications for the management of the power distribution network and for the design of charging tariffs. First, the variable risk preference framework supplies a systematic way to explain the fluctuation in the daily charging curve. When the user population is more risk-seeking, peaks in charging demand are higher and more concentrated. This suggests that dynamic pricing schemes could be used to nudge risk-seeking users toward off-peak charging by altering the reference point through information campaigns or personalized recommendations.
Second, by modeling the travel behavior as a bounded rational process rather than a purely deterministic utility maximization, I can better predict how an electric vehicle aggregator should bid in the electricity market. The aggregator’s portfolio depends on the aggregate charging behavior of many individual users. Knowing that this behavior is a function of psychological parameters and reference points allows the aggregator to estimate the uncertainty band around its expected charging demand more accurately. This uncertainty band is a key input for setting reserve requirements and for calculating the penalty risk associated with deviation from contracted quantities.
Third, the finding that attribute weights significantly alter charging demand demonstrates the importance of user segmentation. A fleet of electric vehicles used for commuting may exhibit strong emphasis on travel time. A fleet used for leisure trips may be more sensitive to comfort. The charging demand in a commercial district is therefore not directly transferable to a residential area, even if the electric vehicle penetration rate and battery characteristics are identical. Location-specific behavioral parameters are necessary for accurate load modeling.
Fourth, the integration of uncertainty attributes expressed as interval numbers, crisp numbers, and triangular fuzzy numbers into one unified decision framework broadens the scope of the analysis. In many engineering studies, it is convenient to treat all uncertain quantities as random variables with known distributions. However, subjective attributes such as comfort are not naturally captured by empirical distributions. The fuzzy set representation is more natural and yields a mathematically consistent integration with the prospect-theoretic weighting scheme. This hybrid representation is one of the contributions of my model.
Methodological Limitations and Future Opportunities
I recognize that the work presented here is not without limitations. The calibration of \(\theta\), the travel scale parameter, requires field survey data or stated-preference experiments. Since such data are not publicly available for my test network, I selected a reasonable range of \(\theta\) values and performed a sensitivity analysis. Future research should collect real travel behavior data for electric vehicle users through GPS tracking and charging logs. Such data would make it possible to estimate the distribution of risk preferences across the population and calibrate the model to a specific urban context.
Another limitation is the static nature of the traffic assignment. Even though I apply the method of successive averages to reach an equilibrium, this equilibrium is deterministic and not stochastic. In reality, traffic conditions vary randomly from day to day. A stochastic version of the equilibrium model, where link travel times fluctuate according to a probability distribution, would add an extra layer of realism to the charging demand analysis. In that case, the state probabilities \(p_h\) would be derived from the stochastic traffic simulation rather than fixed a priori.
My model also treats the charging location as the destination of the trip. In reality, an electric vehicle user may decide to charge at an intermediate charging station rather than at the final destination. Routing decisions and charging decisions are therefore coupled in a more complex way when the battery state of charge is low. Designing a joint route-choice and charging-station-choice model under bounded rationality remains an important research direction. The infrastructure layout of charging stations would substantially influence the available strategies, and the user’s range anxiety would play a stronger role in the behavioral model.
On the methodological side, I plan to extend the variable risk coefficient to the loss aversion parameter \(\lambda\). The empirical literature suggests that loss aversion is not a stable individual trait either. Users who set high reference points or who have experienced recent losses may exhibit stronger loss aversion. Making both \(\alpha\) and \(\lambda\) reference-dependent would further improve the predictive capacity of the model.
For the charging demand forecast, I also intend to integrate the model with a realistic power distribution system simulator. The Nguyen–Dupuis network serves well as a proof of concept, but practical applications require the actual topology of the urban road network and the distribution grid. In that context, the spatial resolution of charging demand would allow distribution grid operators to identify the specific substations and feeders that are prone to overload during the evening peak. This information could drive investment decisions for network reinforcements and battery storage placement.
Sensitivity Analysis of Model Parameters
To better understand the robustness of my conclusions, I perform additional sensitivity analyses. A key parameter in the routing model is the attribute weight vector \(\boldsymbol{\omega}\). I compare how the daily charging demand responds to changes in the weight vector. Four weighting scenarios are analyzed and compared to a baseline. The results show that daily energy consumption varies by about 6 percent across scenarios, while the peak power demand varies by about 14 percent. This makes clear that the shape of the charging curve is more sensitive to behavioral assumptions than the total energy amount is. This observation is reassuring from the grid perspective because the total energy is determined mainly by the fleet’s mileage and efficiency, whereas the peak demand is determined by the temporal concentration of charging events.
In my sensitivity analysis, I also examine the role of the anxiety mileage \(d_v\). Increasing the anxiety mileage from 10 km to 30 km raises the total charging energy by 3 to 5 percent because users charge to a higher state of charge to maintain an extra safety buffer. More importantly, the spatial distribution of charging demand shifts slightly toward the charging stations that are closer to the origin of the trip, because users prefer to reduce range anxiety by charging as early as possible. My framework captures this by increasing the target state of charge in the charging demand equation. This effect can be quantitatively represented by modifying \(\mathrm{SOC}_{v,1,i,t}\).
The following table summarizes the sensitivity outcomes for the peak demand under different anxiety mileages and day types.
| Anxiety mileage \(d_v\) (km) | Total daily energy (MWh) | Morning peak (MW) | Evening peak (MW) |
|---|---|---|---|
| 10 | 2672 | 288 | 346 |
| 20 | 2751 | 297 | 361 |
| 30 | 2834 | 306 | 377 |
The increase in peak demand with higher anxiety mileage is intuitive, but the model makes the magnitude precise. Including range anxiety in the charging demand analysis is therefore not just an optional refinement, it is a necessary step for realistic charging infrastructure planning.
I also test the impact of the route choice state probability vector \(\mathbf{p} = (p^1,p^2,p^3)\). In my baseline, I assume \((0.3,0.5,0.2)\) corresponding to a higher chance of medium traffic conditions. I test scenarios with more optimistic and more pessimistic state probabilities. The table below shows how the peak load shifts with these assumptions.
| State probabilities \(\mathbf{p}\) | Interpretation | Evening peak load (MW) |
|---|---|---|
| (0.2, 0.3, 0.5) | Pessimistic traffic outlook | 356 |
| (0.3, 0.5, 0.2) | Baseline | 361 |
| (0.5, 0.3, 0.2) | Optimistic traffic outlook | 375 |
These variations reinforce the conclusion that the electric vehicle charging demand model must account for the users’ subjective assessments of the future traffic conditions. A purely deterministic view of the transportation network would hide the behavioral roots of the charging load volatility.
Algorithmic Implementation and Computational Complexity
The implementation of my model in MATLAB R2018b follows a modular structure. In the first module, the network topology and the route set are defined. In the second module, given current link travel times, the gain and loss matrices are computed for all routes and all attribute types. This module includes integration over interval-valued probability distributions and over triangular fuzzy membership functions. The integration is performed analytically where possible, and numerically via Gaussian quadrature where closed-form expressions are not available.
In the third module, the variable risk preference coefficients are computed based on the reference points of the attributes. This module allows the user to specify \(\theta\) and the state probabilities. The fourth module computes the comprehensive prospect values, determines the viable path set for each origin–destination pair, and returns the route choice probability distribution. The fifth module is the traffic assignment loop. It runs the method of successive averages until the convergence criterion is satisfied, typically requiring several hundred iterations depending on the network congestion level. In the final module, the charging demand is aggregated from the converged traffic flows.
The computational cost of the entire framework is modest. For the Nguyen–Dupuis network with 13 nodes and 19 links, the entire simulation including a 24-hour demand profile takes under five minutes on a standard desktop computer. This makes the framework scalable to larger urban networks with several hundred nodes. In such larger networks, the principal computational bottleneck would be the enumeration of the feasible route set for each origin–destination pair. A k-shortest-path algorithm could be applied to generate the candidate route set without enumerating all paths.
The route enumeration problem becomes more complex when electric vehicle range limitations are taken into account. If the battery capacity is insufficient to cover certain paths without charging, those paths may be infeasible for users with a low initial state of charge. In my current model, the initial state of charge is assumed to be full for all trips, which simplifies the feasibility analysis. Future work could integrate the battery state dynamics more deeply with the route choice model.
Interdisciplinary Impact and Grid Integration Challenges
The modeling framework presented here sits at the intersection of transportation engineering, behavioral economics, and power grid analysis. This interdisciplinary viewpoint is necessary because the electric vehicle is fundamentally a mobile load whose position and charging behavior are shaped by a vast array of social and technical factors. The power grid sees the electric vehicle only as a load at a specific node and time, but the path that leads the vehicle to that node is the result of complex decision-making processes.
In a deregulated electricity market, the electric vehicle aggregator plays the role of intermediary between the grid and the vehicle users. To bid accurately into the day-ahead market, the aggregator needs an accurate forecast of its expected charging load for each settlement period. This forecast must reflect both the physical driving patterns and the behavioral response of users to price signals. My model provides a transparent way to compute the aggregated demand from the bottom up, starting from individual route decisions. The behavioral parameters of the model can be updated using historical charging data and traffic data, creating a data-driven yet behaviorally grounded forecasting pipeline.
Grid operators are also interested in the flexibility potential of electric vehicle charging. If the charging load can be shifted from peak hours to valley hours, the need for additional peaking capacity is reduced. My analysis of risk preferences shows that risk-averse electric vehicle users tend to charge less aggressively during peak periods because their route choices already smooth the arrival pattern at charging stations. A grid operator that understands the risk profile of the local electric vehicle fleet could better estimate the achievable flexibility from smart charging programs. For example, a fleet dominated by risk-seeking drivers offers more peak-shifting potential because a larger share of their charging events occurs during narrow time windows that could be moved by appropriate price incentives.
Detailed Steps of the Iterative Charging Demand Procedure
I now outline the complete iterative procedure for simulating the charging demand. This step-by-step description helps practitioners implement the model for a real-world network.
Step 1. Define all origin–destination (O-D) pairs and prepend the route set for each pair. Let \(M_k\) be the number of feasible routes for O-D pair \(k\). Initialize link traffic flow \(x_a^0 = 0\) for all links \(a\).
Step 2. At the beginning of the iteration loop with counter \(s\), calculate the link travel time \(T_a^s\) from the link congestion function described above. Sum link travel times to find the total travel time for each route. Similarly, compute the route congestion rate by dividing the expected link volumes by the link capacities and averaging over the route links. The convenience level is computed from fuzzy evaluations conditioned on road conditions.
Step 3. Determine the reference point of each attribute under each state \(S_h\). The reference point is the expectation of the route attribute outcomes. For the first attribute, which is interval-valued, the expectation references the mean value of the interval. The decision maker’ s personal expectations could also be used if survey data are available.
Step 4. Compute the outcome matrix with dimensions \(M_k \times 3 \times 3\), where the first dimension indexes routes, the second denotes the state \(S_h\), and the third denotes the attribute type. For each state–attribute pair, calculate gains and losses as described in the gain and loss equations, using interval probability integration for the travel time attribute, crisp subtraction for the congestion attribute, and fuzzy membership integration for the comfort attribute.
Step 5. Compute the variable risk coefficients \(\alpha_j^h\) using the reference point formula. Then transform gains and losses through the power value functions. Multiply the transformed gains by the probability weights \(\pi^{(+){h}}\) and transformed losses by \(\pi^{(-)h}\). Summing over states yields the component prospect value \(V_{ij}\). Then normalize each attribute column by its maximum absolute value.
Step 6. Assign an attribute weight vector \(\boldsymbol{\omega}\) that reflects the importance of travel time, congestion, and comfort in the user population. Compute \(U_i\) for all routes. Eliminate all routes with \(U_i < 0\) from further consideration. Among the remaining alternatives, assign equal choice probability. If no route has positive \(U_i\), continue with the route having the highest \(U_i\).
Step 7. Multiply the route choice probability by the total travel demand \(D_t\) for the time period to obtain the number of electric vehicles on each route. This is the auxiliary flow \(F_a^s\). The relationship between route flows and link flows is governed by the link-route incidence matrix.
Step 8. Use the method of successive averages to update link flows. Compute the maximum relative difference between current and previous link flows. If this difference is smaller than a specified tolerance \(\varepsilon\), the assignment has converged. Otherwise, increase the iteration counter by 1 and return to Step 2.
Step 9. At convergence, calculate the charging demand for each time period. The energy consumption of each electric vehicle depends on the selected route distance, the consumption rate, and the anxiety mileage. The charging demand is computed with the formula for \(Q_t\). Aggregate the result to produce a daily load profile.
Discussion of Attribute Forms in the Model
The use of three distinct attribute forms—interval numbers, crisp numbers, and triangular fuzzy numbers—enriches the model but also complicates the integration. In this paragraph I want to emphasize the modeling value of each attribute form. Interval numbers are appropriate for attributes whose variations follow a stochastic process. Travel time is perhaps the best example since it has well-documented statistical distributions that can be linked to traffic volume fluctuations. The normal distribution assumption is a pragmatic choice that fits empirical observations under moderate congestion levels.
Crisp numbers are appropriate when the attribute is accurately measurable at the time of the decision. The congestion index, defined as the ratio of link flow to capacity, is known with reasonable certainty from the traffic management system. The driver receives this information in real time through navigation apps. Therefore, modeling it as a deterministic crisp value is behaviorally adequate.
Triangular fuzzy numbers are an ideal tool for subjective attributes such as driving comfort. Since comfort is a perception rather than an objective measurement, different users will likely rate the same route differently. The triangular membership function offers a minimal yet expressive representation of this human vagueness. Since the linguistic scale is easily translated into triangular fuzzy numbers through the conversion table, the model is accessible to practitioners who collect subjective ratings.
Another crucial design choice in my model is the use of reference points as expected outcomes. I compute the reference point for each attribute by taking the average outcome across alternatives under the same state. This means that the reference point is endogenous to the choice set. It changes as traffic conditions change. This formulation aligns with the psychological literature: the same absolute outcome can feel like a gain if the expectation is low, or a loss if the expectation is high. The endogenous reference point makes my model capable of capturing the dynamic evolution of user expectations over time.
Summary of Contributions and Concluding Remarks
In this work, I have systematically developed a charging demand analysis framework that treats electric vehicle users as boundedly rational human beings rather than perfect optimizers. My contributions can be grouped into the following areas.
First, I constructed a behavioral travel route choice model that integrates the prospect theory with multi-attribute decision-making. Three heterogeneous attribute types were accommodated: interval stochastic numbers, crisp numbers, and triangular fuzzy numbers. The model yields a comprehensive prospect value for each route, combining cognitive expectations and subjective risk attitudes.
Second, I put forward a variable-coefficient formulation for the risk attitude parameter. Unlike a fixed \(\alpha = 0.88\), the variable coefficient depends on reference points, thereby capturing the interplay between expectation levels and risk sensitivity. Through simulation, I showed that the ranking of routes can change substantially when the risk parameter is allowed to vary.
Third, the route choice behavior was embedded into a dynamic traffic assignment procedure using the method of successive averages. The final equilibrium flow pattern reflects the aggregate behavior of boundedly rational electric vehicle users. Based on these flows, I developed a charging demand model that accounts for energy consumption along selected paths and the range anxiety buffer of the user.
Fourth, I provided a case study on the Nguyen–Dupuis network to validate the model and to quantify the sensitivity of charging demand to behavioral factors. The simulation results clearly show that both the users’ risk preference and their attribute weights have a distinct impact on the daily charging profile. These findings underline the need for behavioral models to be included in the planning and market-operation tools used for power systems with a substantial share of electric vehicles.
For the electric vehicle aggregator, my model directly supports the estimation of deviation risks in the charging energy volume. By simulating multiple behavioral scenarios for the user population, the aggregator can construct an uncertainty bound for the day-ahead charging forecast. This uncertainty bound is essential for deciding how much deviation penalty to tolerate and how much balancing capacity to reserve in the market.
In conclusion, the analysis of electric vehicle charging demand cannot be separated from the analysis of human behavior. Charging events are not independent random processes; they are downstream consequences of travel decisions that inherently involve bounded rationality under uncertainty. As electric vehicles continue to scale up, it is my conviction that behaviorally realistic charging demand models, such as the one introduced in this article, will play an increasingly prominent role in ensuring the safe and economical operation of the power grid. Let me close by restating that the comprehensive prospect value acting as the key behavioral variable is not a mere theoretical construct, but a powerful computational instrument. It integrates the many factors that define the daily reality of electric vehicle drivers and transforms those factors into a single quantitative measure of route attractiveness. By linking this measure to the charging demand, my framework establishes a solid foundation for further research and for practical engineering applications aimed at supporting the low-carbon energy transition.