The proliferation of electric cars presents a significant challenge to the safe and economic operation of power grids due to the fluctuating nature of their charging demand. This fluctuation stems largely from user behavior, which is often characterized by bounded rationality. Unlike perfectly rational agents assumed in classical models, electric car users face travel uncertainty and exhibit varying degrees of risk appetite, leading to unpredictable decision-making regarding travel routes and, consequently, charging needs. This inherent uncertainty complicates grid planning, load management, and market participation for entities like electric car aggregators. Therefore, a nuanced understanding and modeling of these bounded rational behaviors are crucial for accurate electric car charging demand analysis.
Prospect Theory (PT), a cornerstone of behavioral economics, provides a powerful framework for modeling decision-making under risk and uncertainty, aligning well with the concept of bounded rationality. It posits that people evaluate potential outcomes relative to a reference point, perceive gains and losses differently (loss aversion), and weigh probabilities in a non-linear fashion. In the context of electric car travel, users are not simply minimizing travel time or cost; they make route choices based on a subjective assessment of multiple uncertain attributes, filtered through their individual psychological traits. This study leverages a risk-based multi-attribute decision-making (MADM) method grounded in Prospect Theory to analyze the route selection behavior of electric car users under bounded rationality. We consider multiple uncertain factors influencing electric car travel and develop a variable-coefficient model that dynamically links user risk preferences to attribute reference points, ultimately constructing an electric car charging demand model based on comprehensive prospect values.
Foundations: Prospect Theory and Bounded Rationality in Electric Car Travel
Traditional rational choice models often fail to capture the complexities of real-world electric car user behavior. Bounded rationality acknowledges the cognitive limitations and subjective biases of decision-makers. For an electric car user, this manifests in two primary ways: travel uncertainty and risk preference heterogeneity.
Travel uncertainty arises because multiple factors affecting a journey are stochastic or imperfectly known. These include fluctuating travel times due to congestion, varying energy consumption rates, the availability of charging stations, and even qualitative factors like driving comfort. Users must make decisions with incomplete information. Risk preference heterogeneity refers to the fact that different electric car users have different tolerances for risk. Some may be “risk-seeking,” preferring routes with a chance of very short travel time despite a risk of long delays, while others are “risk-averse,” opting for more predictable, albeit potentially longer, journeys.
Prospect Theory formalizes these ideas. The value function in PT is defined asymmetrically for gains and losses relative to a reference point:
$$
V(x) = \begin{cases}
(x)^{\alpha} & \text{if } x \geq 0 \text{ (gain)} \\
-\lambda(-x)^{\alpha} & \text{if } x < 0 \text{ (loss)}
\end{cases}
$$
where \(x\) is the deviation from the reference point, \(\alpha\) (\(0 < \alpha \leq 1\)) is the risk preference coefficient, and \(\lambda\) (\(\lambda \geq 1\)) is the loss aversion coefficient. A smaller \(\alpha\) indicates lower sensitivity to changes and a more risk-averse attitude for gains, while a larger \(\lambda\) indicates stronger aversion to losses. The decision weight function \(\pi(p)\) transforms objective probabilities \(p\) into subjective decision weights, typically overweighting small probabilities and underweighting large ones.
In this study, we extend this core model to a multi-attribute context relevant to electric car routing. The choice of a travel path for an electric car is a multi-attribute decision problem. We categorize the influencing factors into three types based on how their outcomes are represented:
| Attribute Type | Description | Examples for Electric Car Travel | Representation |
|---|---|---|---|
| Type C₁ (Cost) | Attributes where smaller values are better. Subject to significant uncertainty. | Travel Time, Energy Cost | Interval Number \([x^{l}, x^{u}]\) (Normally distributed within the interval) |
| Type C₂ (Cost) | Attributes with clear, quantifiable values. | Traffic Congestion Rate, Toll Cost | Crisp Number |
| Type C₃ (Benefit) | Attributes where larger values are better, often qualitative. | Driving Comfort, Scenic Value, Charging Station Convenience | Triangular Fuzzy Number \((a, b, c)\) |
Each attribute \(j\) for a given path \(i\) under a specific state \(h\) (e.g., good, medium, bad traffic conditions) has an outcome \(x^h_{ij}\). The user has an expectation or reference point \(r^h_j\) for each attribute. The core of the modeling involves calculating the perceived gain \(G^h_{ij}\) or loss \(L^h_{ij}\) of the outcome relative to this reference point, using methods appropriate to the data type (interval, crisp, or fuzzy).
A Variable-Coefficient Risk-Based Multi-Attribute Decision Model for Electric Car Route Choice
This section details the construction of the decision model. Let \(A = \{A_1, A_2, …, A_M\}\) be the set of available paths for an electric car traveling from an origin to a destination.
Step 1: Define Attribute States and Reference Points
We consider three states \(S_h\) (\(h=1,2,3\)) for the travel environment (e.g., light, moderate, heavy traffic), each occurring with probability \(p_h\), where \(\sum_{h=1}^3 p_h = 1\). For each attribute \(C_j\) and state \(S_h\), the user’s reference point is \(r^h_j\). This could be based on past experience, stated preferences, or common expectations. The actual outcome of path \(A_i\) for attribute \(C_j\) in state \(S_h\) is \(x^h_{ij}\).
Step 2: Calculate Gains and Losses for Each Attribute Type
For Type C₁ (Interval Number, e.g., Travel Time): The outcome \(x^h_{i1} = [x^{hl}_{i1}, x^{hu}_{i1}]\). We assume the actual value \(x\) is normally distributed within this interval: \(x \sim N(\mu^h_{i1}, (\sigma^h_{i1})^2)\), with \(\mu^h_{i1} = (x^{hl}_{i1} + x^{hu}_{i1})/2\) and \(\sigma^h_{i1} = (x^{hu}_{i1} – x^{hl}_{i1})/6\). The gain or loss is the expected deviation from the reference point \(r^h_1\). For example, the gain (when outcomes are less than the reference for this cost attribute) is calculated as:
$$
G^h_{i1} = \int_{x^{hl}_{i1}}^{min(r^h_1, x^{hu}_{i1})} (r^h_1 – x) f^h_{i1}(x) \, dx
$$
where \(f^h_{i1}(x)\) is the probability density function of the normal distribution. The loss is calculated similarly for the portion where \(x > r^h_1\).
For Type C₂ (Crisp Number, e.g., Congestion Rate): The calculation is straightforward. For a cost attribute:
$$
G^h_{i2} = \max(r^h_2 – x^h_{i2}, 0); \quad L^h_{i2} = \min(r^h_2 – x^h_{i2}, 0)
$$
For Type C₃ (Triangular Fuzzy Number, e.g., Comfort): The outcome is a fuzzy number \(x^h_{i3} = (a^h_{i3}, b^h_{i3}, c^h_{i3})\) with a membership function \(\phi^h_{i3}(x)\). The gain (for this benefit attribute) is the expected positive deviation:
$$
G^h_{i3} = \int_{max(r^h_3, a^h_{i3})}^{c^h_{i3}} (x – r^h_3) \phi^h_{i3}(x) \, dx
$$
The loss is calculated over the range where \(x < r^h_3\).
These calculations yield a risk gain matrix \(G^h = [G^h_{ij}]_{M \times 3}\) and a risk loss matrix \(L^h = [L^h_{ij}]_{M \times 3}\) for each state \(h\).
Step 3: Incorporate Variable Risk Preference Coefficient
A key innovation is making the risk preference coefficient \(\alpha\) variable and dependent on the reference point. In standard PT, \(\alpha\) is often fixed (e.g., 0.88). However, an electric car user’s risk sensitivity is context-dependent. If the reference point for an attribute is high (e.g., expecting a very short travel time), the same absolute gain or loss may be perceived as less significant, implying a lower risk sensitivity. We model this as:
$$
\alpha^h_j = \left(1 – \frac{r^h_j}{\sum_{h=1}^3 r^h_j}\right)^\theta
$$
where \(\theta\) (\(0 \leq \theta \leq 1\)) is a scaling parameter that can reflect the size of the user population or other contextual factors. As \(\theta\) increases, \(\alpha^h_j\) decreases, moving the user’s behavior from risk-seeking towards risk-neutral.
Step 4: Compute the Comprehensive Prospect Value for Each Path
First, we calculate the prospect value for path \(A_i\) and attribute \(C_j\) using the variable \(\alpha^h_j\) and standard PT parameters \(\lambda\), \(\zeta\), and \(\delta\) for the weighting function.
Value function for gains/losses:
$$
V^{(+)h*}_{ij} = (G^h_{ij})^{\alpha^h_j}; \quad V^{(-)h*}_{ij} = -\lambda(-L^h_{ij})^{\alpha^h_j}
$$
Decision weight functions:
$$
\pi^{(+)h}_{ij} = \frac{(p_h)^\zeta}{((p_h)^\zeta + (1-p_h)^\zeta)^{1/\zeta}}; \quad \pi^{(-)h}_{ij} = \frac{(p_h)^\delta}{((p_h)^\delta + (1-p_h)^\delta)^{1/\delta}}
$$
The overall prospect value for attribute \(C_j\) on path \(A_i\) is the weighted sum across all states:
$$
V_{ij} = \sum_{h=1}^3 \left( V^{(+)h*}_{ij} \pi^{(+)h}_{ij} + V^{(-)h*}_{ij} \pi^{(-)h}_{ij} \right)
$$
This forms a prospect decision matrix \(V = [V_{ij}]_{M \times 3}\). We then normalize this matrix to get \(V^* = [V^*_{ij}]_{M \times 3}\), where \(V^*_{ij} = V_{ij} / V^{max}_j\) and \(V^{max}_j = \max_{i \in M} \{ |V_{ij}| \}\).
Finally, the comprehensive prospect value \(U_i\) for each electric car path \(A_i\) is computed by aggregating the normalized attribute values, weighted by the importance of each attribute \(\omega_j\) (where \(\sum_{j=1}^3 \omega_j = 1\)):
$$
U_i = \sum_{j=1}^3 \omega_j V^*_{ij}
$$
The path with the highest \(U_i\) is considered the most attractive to a bounded rational electric car user under the modeled conditions. This value integrates the effects of multiple uncertain travel attributes filtered through the user’s dynamic risk preferences.
Dynamic Traffic Flow Update and Electric Car Charging Demand Modeling
The choice model above determines the probability \(u_{k,i,t}\) that a user chooses path \(i\) between origin-destination (OD) pair \(k\) at time \(t\). However, as more electric car users select the path with the highest prospect value, its traffic flow increases, which in turn degrades its attributes (like travel time and congestion), altering its prospect value. This feedback loop must be captured. We employ the Method of Successive Averages (MSA) for dynamic traffic assignment.
Let \(x^s_a\) be the flow on road segment \(a\) in iteration \(s\). It is updated as:
$$
x^s_a = \left(1 – \frac{1}{s}\right)x^{s-1}_a + \frac{1}{s} F^s_a
$$
where \(F^s_a\) is the auxiliary flow assigned to segment \(a\) in iteration \(s\) based on the current shortest (highest prospect value) paths. The travel time on segment \(a\) is then updated using the Bureau of Public Roads (BPR) function:
$$
T^s_a = t^0_a \left[ 1 + 0.15 \left( \frac{x^s_a}{C_a} \right)^4 \right]
$$
where \(t^0_a\) is the free-flow travel time and \(C_a\) is the capacity of segment \(a\). This updated travel time feeds back into the attribute \(x^h_{i1}\) for the paths, leading to a recalculation of comprehensive prospect values \(U_i\). The process iterates until the flows converge to a stable state, representing a behavioral equilibrium where no electric car user can improve their perceived prospect value by unilaterally changing routes.
Electric Car Charging Demand Model Based on Comprehensive Prospect Value
Once the equilibrium flows and the final set of comprehensive prospect values \(U_i\) for all paths are obtained, we can model the aggregate charging demand. The charging demand is derived from the energy consumed on the chosen paths. Only paths with a positive \(U_i\) are considered for charging demand calculation, as these are the ones perceived as beneficial by users.
For an electric car \(v\) on path \(i\) at time \(t\), the required charging time is:
$$
T_{v,i,t} = \frac{(SOC_{v1,i,t} – SOC_{v0,i,t}) C_v}{P_v}
$$
where \(SOC_{v0,i,t}\) and \(SOC_{v1,i,t}\) are the initial and target state-of-charge, \(C_v\) is the battery capacity, and \(P_v\) is the charging power. The target SOC must satisfy the user’s range anxiety:
$$
SOC_{v1,i,t} \geq SOC_{v0,i,t} + (d_{v} + d_{k,i}) q_v
$$
where \(d_{k,i}\) is the distance of path \(i\) for OD pair \(k\), \(q_v\) is the electric car’s energy consumption rate, and \(d_v\) is the anxiety range (the extra range a user desires as a buffer).
The total charging demand \(Q_t\) in the region at time \(t\) is the sum of the energy needs of all electric cars across all OD pairs and chosen paths:
$$
Q_t = \sum_{k=1}^{N} \left[ p_{k,t} D_t \sum_{i=1}^{M_k} \left( u_{k,i,t} \cdot (SOC_{v1,i,t} – SOC_{v0,i,t}) \right) \right]
$$
where:
- \(N\) is the number of OD pairs.
- \(p_{k,t}\) is the proportion of total regional electric car travel demand \(D_t\) attributed to OD pair \(k\).
- \(u_{k,i,t}\) is the probability of choosing path \(i\) for OD pair \(k\) at time \(t\), derived from the normalized comprehensive prospect values \(U_i\) of the positive-prospect paths (e.g., \(u_{k,i,t} \propto U_i\) for \(U_i > 0\), normalized per OD pair).
This model directly links the psychological decision-making process of individual electric car users (captured in \(U_i\)) to the macroscopic, time-varying charging load on the power grid.
Implications and Analysis of Key Modeling Factors
The proposed framework allows for the analysis of how bounded rationality shapes electric car charging demand. Two factors are particularly influential: the risk preference scaling parameter \(\theta\) and the attribute weight vector \(\omega = (\omega_1, \omega_2, \omega_3)^T\).
Impact of Variable Risk Preference (\(\theta\)): As \(\theta\) varies from 0 to 1, the effective risk coefficient \(\alpha\) changes. A lower \(\theta\) (higher \(\alpha\)) corresponds to users who are more sensitive to gains/losses and more risk-seeking. In simulations, this often leads to a charging demand profile with greater volatility. Users are more likely to gamble on routes that could be very fast (low energy use) but could also be slow (high energy use), leading to a wider spread in possible charging times and energy needs. Conversely, a higher \(\theta\) (lower \(\alpha\)) models more risk-averse or risk-neutral electric car users who prefer stable, predictable routes. This results in a smoother, less peaky aggregate charging demand curve, which is easier for the grid to accommodate. This highlights that the collective risk culture among electric car owners is a key determinant of grid load patterns.
Impact of Attribute Weights (\(\omega_j\)): The relative importance users place on travel time (\(\omega_1\)), cost/fcongestion (\(\omega_2\)), and comfort/convenience (\(\omega_3\)) fundamentally alters route choices and thus charging demand. If electric car users heavily weight travel time (\(\omega_1\) is high), they will select routes that minimize time, which may not be the most energy-efficient. This could concentrate charging demand in specific areas (near fast highways) and at specific times (post-commute). If charging convenience or comfort is prioritized (\(\omega_3\) is high), demand may shift towards locations with better amenities, even if travel time is longer, potentially spreading the charging load more evenly. The model can test different policy scenarios, such as the effect of congestion pricing (affecting \(C_2\)) or the deployment of premium charging stations (affecting \(C_3\)), on the resulting electric car charging demand by observing shifts in the chosen paths and their associated energy consumption.
Synthesis and Future Perspectives
In conclusion, the integration of a variable-coefficient, risk-based multi-attribute decision model within a Prospect Theory framework provides a powerful and psychologically realistic method for analyzing electric car charging demand. It moves beyond the assumption of perfect rationality by explicitly modeling travel uncertainty and heterogeneous risk preferences. The comprehensive prospect value \(U_i\) serves as a robust metric that encapsulates the complex trade-offs an electric car user makes. When integrated with dynamic traffic assignment, this approach can simulate how the collective bounded rational behavior of electric car users shapes spatial and temporal charging loads on the power grid.
Future research can extend this work in several meaningful directions. First, the model for the variable risk preference coefficient \(\alpha^h_j\) can be refined to incorporate other contextual factors such as trip purpose (commute vs. leisure), user socio-demographics, or real-time information provision. Second, while this study focuses on route choice, the framework can be expanded to a joint choice model that simultaneously considers departure time, route, and charging location/type, creating an even more holistic view of electric car user behavior. Third, the charging demand model output can be directly integrated into grid optimization algorithms, market clearing mechanisms for electric car aggregators, or infrastructure planning tools, providing a crucial bridge between behavioral transportation science and power systems engineering. Ultimately, acknowledging and accurately modeling the bounded rationality of electric car users is not just an academic exercise; it is a necessary step towards building a resilient, efficient, and user-centric integrated mobility and energy system.