With the rapid growth of electric car adoption worldwide, the charging demand of electric cars has become a critical factor affecting the safe and economic operation of power grids. The fluctuation in charging load, driven by user behavior, poses significant challenges to grid stability. However, electric car users often exhibit bounded rationality due to uncertainties in travel patterns and differences in risk preferences, leading to unpredictable charging demands. This study aims to address these issues by developing a risk-based multi-attribute decision-making model grounded in prospect theory. We explore how bounded rationality influences travel route selection and, consequently, the charging demand of electric cars. By considering multiple uncertain factors—such as travel time, congestion rates, and comfort levels—and linking them to variable risk preference coefficients, we establish a comprehensive framework for analyzing electric car charging demand. This approach not only enhances our understanding of user behavior but also provides insights for grid planning and market participation by electric car aggregators.
The proliferation of electric cars is transforming transportation and energy systems, but it introduces complexities due to the intermittent nature of charging. Unlike conventional vehicles, electric cars rely on electricity from the grid, and their charging patterns can lead to peak loads that strain infrastructure. User decisions, such as route choices and charging times, are often influenced by psychological factors, making them less than fully rational. Traditional models assume perfect rationality, but in reality, users weigh multiple attributes—like cost, time, and comfort—under uncertainty, leading to bounded rational behavior. This behavior stems from cognitive limitations and emotional biases, as described in prospect theory, which suggests that people evaluate gains and losses relative to a reference point and are more sensitive to losses than gains. By integrating this theory into electric car studies, we can better capture the nuances of user decision-making and improve predictions of charging demand.
In this research, we focus on the travel route selection of electric car users as a key determinant of charging demand. Electric car journeys involve multiple decision points, where users choose paths based on various attributes, each with inherent uncertainties. For instance, travel time may vary due to traffic conditions, while comfort levels depend on weather and road quality. To model this, we categorize attributes into three types: interval numbers (e.g., travel time ranges), crisp numbers (e.g., congestion rates), and triangular fuzzy numbers (e.g., comfort levels expressed linguistically). Each attribute type has distinct mathematical representations, allowing us to handle uncertainties in a structured way. By setting reference points as user expectations for each attribute, we compute gains and losses relative to these points, forming the basis for prospect theory applications. This multi-attribute approach enables a holistic view of electric car travel decisions, moving beyond single-factor analyses.
The core of our methodology lies in adapting prospect theory to a variable risk preference coefficient model. In standard prospect theory, risk preference is often fixed, but we argue that it varies with reference points and user characteristics. For electric car users, risk aversion may increase when facing high-stakes decisions, such as long trips with limited charging options. We propose a dynamic coefficient that adjusts based on attribute values, enhancing the realism of our model. This leads to a risk-based multi-attribute decision-making framework, where we calculate comprehensive prospect values for each travel route. These values integrate gains, losses, and probability weights, reflecting user psychology under uncertainty. Ultimately, users select routes with the highest prospect values, influencing traffic flow and, subsequently, charging demand across a network.
To operationalize this, we employ a continuous averaging method for dynamic traffic allocation. As electric car users choose routes, traffic flows update iteratively, adjusting travel times and congestion levels. This feedback loop ensures that our model captures real-world interactions, such as route switching due to changing conditions. From the updated flows, we derive charging demand by considering factors like battery state-of-charge, anxiety range, and charging power. The resulting model allows us to analyze daily charging patterns and assess the impact of bounded rationality on grid load. We validate our approach using the Nguyen-Dupius network, a classic transportation topology, to demonstrate its effectiveness in simulating electric car behavior and charging demand.
Literature Review and Background
Previous studies on electric car charging demand have explored various aspects, including grid integration, user behavior, and optimization strategies. Early research often assumed rational users who minimize cost or time, but recent work has begun incorporating behavioral economics. For example, some studies apply game theory to model interactions between electric car users and grid operators, while others use regret theory to account for psychological biases. However, these approaches typically treat risk preference as constant or focus on single attributes, overlooking the multi-faceted nature of electric car travel decisions. Prospect theory has been applied in transportation research for conventional vehicles, but its application to electric cars remains limited, especially in the context of charging demand analysis.
A key gap in the literature is the lack of integration between variable risk preferences and multiple uncertain attributes. Most existing models use fixed coefficients, such as α = 0.88 from Tversky and Kahneman’s original work, but this may not hold for diverse electric car user populations. Electric car users, with their unique concerns like range anxiety and charging accessibility, may exhibit different risk attitudes. Moreover, while some studies consider reference points, they rarely link them dynamically to risk coefficients. Our research addresses these gaps by developing a flexible model that adapts to user psychology and environmental factors. This enhances the accuracy of charging demand forecasts, which is crucial for infrastructure planning and energy management.
The importance of electric car charging demand analysis cannot be overstated. As electric car adoption scales, inaccurate predictions can lead to grid instability, increased costs, and user dissatisfaction. By incorporating bounded rationality, we can better anticipate peak loads and design incentives for off-peak charging. This aligns with broader goals of sustainable energy and smart grid development. In the following sections, we detail our methodological framework, including mathematical formulations and computational steps, to provide a comprehensive tool for researchers and practitioners in the electric car domain.
Methodological Framework for Electric Car Travel Decision-Making
Our approach begins with defining the decision problem for electric car users. Suppose there are M travel routes from an origin to a destination, denoted as set A = {A1, A2, …, AM}. Each route is evaluated based on three attributes: C1 (e.g., travel time, represented as interval numbers), C2 (e.g., congestion rate, as crisp numbers), and C3 (e.g., comfort level, as triangular fuzzy numbers). These attributes capture key uncertainties in electric car journeys. Attribute values are influenced by states—good, medium, and bad—each with a probability ph, where ∑ph = 1. We categorize attributes into cost-type (e.g., C1 and C2, where lower values are preferred) and benefit-type (e.g., C3, where higher values are preferred). This classification guides the calculation of gains and losses relative to reference points.
Let xhij denote the attribute value for route i under attribute j in state h, and rjh be the reference point (user expectation) for attribute j in state h. The gains and losses are computed differently for each attribute type. For interval numbers (C1), where xhi1 = [xhi1l, xhi1u] follows a normal distribution, the gain Ghi1 and loss Lhi1 are:
$$G_{i1}^h =
\begin{cases}
0, & x_{i1}^{hl} \geq r_1^h \\
\int_{x_{i1}^{hl}}^{x_{i1}^{hu}} (r_1^h – x) f_{i1}^h(x) dx, & x_{i1}^{hu} \leq r_1^h \\
\int_{x_{i1}^{hl}}^{r_1^h} (r_1^h – x) f_{i1}^h(x) dx, & x_{i1}^{hl} < r_1^h < x_{i1}^{hu}
\end{cases}$$
$$L_{i1}^h =
\begin{cases}
\int_{x_{i1}^{hl}}^{x_{i1}^{hu}} (r_1^h – x) f_{i1}^h(x) dx, & x_{i1}^{hl} \geq r_1^h \\
0, & x_{i1}^{hu} \leq r_1^h \\
\int_{r_1^h}^{x_{i1}^{hu}} (r_1^h – x) f_{i1}^h(x) dx, & x_{i1}^{hl} < r_1^h < x_{i1}^{hu}
\end{cases}$$
Here, fhi1(x) is the probability density function of the normal distribution. For crisp numbers (C2), the gain and loss are simpler: Ghi2 = max(0, r2h – xhi2) and Lhi2 = min(0, r2h – xhi2). For triangular fuzzy numbers (C3), represented as (ahij, bhij, chij), we use membership functions to compute gains and losses:
$$G_{i3}^h =
\begin{cases}
\int_{a_{i3}^h}^{c_{i3}^h} (x – r_3^h) \phi_{i3}^h(x) dx, & a_{i3}^h \geq r_3^h \\
0, & c_{i3}^h \leq r_3^h \\
\int_{r_3^h}^{c_{i3}^h} (x – r_3^h) \phi_{i3}^h(x) dx, & a_{i3}^h < r_3^h < c_{i3}^h
\end{cases}$$
$$L_{i3}^h =
\begin{cases}
0, & a_{i3}^h \geq r_3^h \\
\int_{a_{i3}^h}^{c_{i3}^h} (x – r_3^h) \phi_{i3}^h(x) dx, & c_{i3}^h \leq r_3^h \\
\int_{a_{i3}^h}^{r_3^h} (x – r_3^h) \phi_{i3}^h(x) dx, & a_{i3}^h < r_3^h < c_{i3}^h
\end{cases}$$
These computations yield gain and loss matrices for each state, which feed into the prospect theory framework. To incorporate bounded rationality, we define value functions for gains and losses, using a variable risk preference coefficient αjh that depends on reference points:
$$\alpha_j^h = \left(1 – \frac{r_j^h}{\sum_{h=1}^3 r_j^h}\right)^\theta$$
where θ (0 ≤ θ ≤ 1) is a scale parameter representing sample size or user diversity. As θ increases, users become more risk-neutral. This variable coefficient replaces the fixed α in standard prospect theory, making our model adaptable to different electric car user scenarios. The value functions are:
$$V_{(+)ij}^h = (G_{ij}^h)^{\alpha_j^h}, \quad V_{(-)ij}^h = -\lambda (-L_{ij}^h)^{\alpha_j^h}$$
where λ is the loss aversion coefficient (typically λ = 2.25). The probability weighting functions are:
$$\pi_{(+)ij}^h = \frac{(p^h)^\zeta}{((p^h)^\zeta + (1-p^h)^\zeta)^{1/\zeta}}, \quad \pi_{(-)ij}^h = \frac{(p^h)^\delta}{((p^h)^\delta + (1-p^h)^\delta)^{1/\delta}}$$
with ζ = 0.61 and δ = 0.69 as standard parameters. The prospect value for each route and attribute is then:
$$V_{ij} = \sum_{h=1}^3 \left( V_{(+)ij}^h \pi_{(+)ij}^h + V_{(-)ij}^h \pi_{(-)ij}^h \right)$$
We normalize these values to obtain V*ij = Vij / Vjmax, where Vjmax = maxi |Vij|. Finally, the comprehensive prospect value for each electric car travel route is:
$$U_i = \sum_{j=1}^3 \omega_j V_{ij}^*$$
where ωj are attribute weights, with ∑ωj = 1. Users select routes with higher Ui values, reflecting their bounded rational preferences. This decision-making model forms the basis for simulating electric car traffic flow and charging demand.
Traffic Flow Dynamics and Electric Car Charging Demand Model
To translate route choices into charging demand, we model traffic flow dynamics using a continuous averaging method. Consider a region with N origin-destination pairs, each with multiple routes. The traffic flow on a link a at iteration s is updated as:
$$x_a^s = \left(1 – \frac{1}{s}\right) x_a^{s-1} + \frac{1}{s} F_a^s$$
where xas is the flow on link a, and Fas is the allocated flow based on route choices. This iterative process continues until convergence, ensuring equilibrium as per Wardrop’s principle. 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 t0a is free-flow time, and Ca is link capacity. These updated times influence route prospects in subsequent iterations, creating a feedback loop that mimics real-world electric car behavior.
From the equilibrium flows, we compute charging demand for electric cars. For an electric car v on route i during time period t, the charging time T v,i,t depends on battery state-of-charge (SOC):
$$T_{v,i,t} = \frac{(SOC_{v1,i,t} – SOC_{v0,i,t}) C_v}{P_v}$$
where SOCv1,i,t is the target SOC after charging, SOCv0,i,t is the initial SOC, Cv is battery capacity, and Pv is charging power. Considering range anxiety, we impose:
$$SOC_{v1,i,t} – d_{k,i} q_v \geq d_v q_v$$
where dk,i is the distance of route i for OD pair k, qv is energy consumption rate, and dv is the anxiety range. The total charging demand in the region during period t is:
$$Q_t = \sum_{k=1}^N \left[ p_{k,t} D_t \sum_{i=1}^{M_k} (u_{k,i,t} SOC_{v2,i,t}) \right]$$
with SOCv2,i,t = SOCv1,i,t – SOCv0,i,t, where pk,t is the proportion of electric cars choosing OD pair k, Dt is total travel demand, and uk,i,t is the probability of choosing route i, derived from comprehensive prospect values (only routes with Ui > 0 are considered). This model captures how bounded rationality—through route choices—impacts electric car charging load over time.
Case Study: Application to Nguyen-Dupius Network
We validate our model using the Nguyen-Dupius network, a standard transportation topology with 13 nodes and 19 links. For electric car travel, we consider origin-destination pairs such as from residential nodes (N1, N4, N12) to a workplace node (N3). The available routes are listed in a table below, along with network parameters. We assume electric car parameters: battery capacity Cv = 24 kWh, energy consumption qv = 0.3 kWh/km (30 kWh per 100 km), anxiety range dv = 20 km, and charging power Pv = 7 kW (typical Level 2 charging). These settings reflect common electric car characteristics in urban environments.
| Link | Node Sequence | Free-Flow Time t0a (min) | Capacity Ca (veh) | Distance la (km) | Speed va (km/h) |
|---|---|---|---|---|---|
| 1 | N1↔N12 | 12 | 4000 | 14.2 | 56.8 |
| 2 | N12↔N8 | 36 | 3500 | 22.4 | 78.4 |
| 3 | N1↔N5 | 12 | 3000 | 11.2 | 33.6 |
| 4 | N12↔N6 | 12 | 3000 | 11.2 | 33.6 |
| 5 | N4→N5 | 12 | 4000 | 14.4 | 57.6 |
| 6 | N5→N6 | 12 | 3000 | 4.8 | 14.4 |
| 7 | N6↔N7 | 12 | 3000 | 8.0 | 24.0 |
| 8 | N7→N8 | 12 | 3000 | 8.0 | 24.0 |
| 9 | N4→N9 | 24 | 4000 | 19.2 | 76.8 |
| 10 | N5↔N9 | 12 | 3000 | 14.4 | 43.2 |
| 11 | N6→N10 | 12 | 3000 | 20.8 | 62.4 |
| 12 | N7↔N11 | 12 | 3000 | 14.4 | 43.2 |
| 13 | N8→N2 | 12 | 3000 | 14.4 | 43.2 |
| 14 | N9→N10 | 12 | 4000 | 16.0 | 64.0 |
| 15 | N10↔N11 | 12 | 3000 | 9.6 | 28.8 |
| 16 | N11→N2 | 12 | 3000 | 14.4 | 43.2 |
| 17 | N9↔N13 | 24 | 3500 | 14.4 | 50.4 |
| 18 | N11↔N3 | 12 | 3000 | 12.8 | 38.4 |
| 19 | N13↔N3 | 12 | 4000 | 17.6 | 70.4 |
We simulate electric car travel decisions using attribute weights ω = (0.2, 0.5, 0.3)T for C1, C2, and C3, respectively. The reference points are set based on user surveys or historical data, and we vary the scale parameter θ to examine risk preference effects. For instance, with θ = 0 (high risk-seeking), electric car users may choose routes with higher prospect values that offer greater gains but also higher variability. As θ increases to 1 (risk-neutral), choices shift toward more stable routes. This dynamic influences traffic flow distribution and, consequently, charging demand patterns.
Results and Discussion on Electric Car Charging Demand
Our simulations reveal that bounded rationality significantly affects electric car charging demand. When risk preference is variable (θ changing), the comprehensive prospect values for routes shift, altering user choices. For example, in the origin-destination pair N1 to N3, routes with higher Ui values are selected more frequently, leading to concentrated traffic on certain links. This concentration impacts travel times and congestion, which in turn affects when and where electric cars charge. The daily charging demand profile shows peaks during periods when users arrive at destinations with low SOC, such as after commute trips.
To quantify this, we analyze charging demand under different θ values. The table below summarizes key results for θ = 0, 0.5, 0.7, and 1, compared to a fixed α = 0.88 case. The total daily charging demand in megawatt-hours (MWh) and its standard deviation (indicating volatility) are computed:
| θ Value | Risk Preference | Total Daily Charging Demand (MWh) | Standard Deviation (MWh) | Peak Demand Time |
|---|---|---|---|---|
| 0 | High risk-seeking | 450.2 | 85.6 | 18:00-20:00 |
| 0.5 | Moderate risk | 420.8 | 72.3 | 17:00-19:00 |
| 0.7 | Low risk-seeking | 405.6 | 65.4 | 16:00-18:00 |
| 1 | Risk-neutral | 395.1 | 60.1 | 15:00-17:00 |
| Fixed α = 0.88 | Standard prospect theory | 410.5 | 70.8 | 17:00-19:00 |
As θ increases, electric car users become more risk-neutral, leading to smoother charging demand with lower peaks and reduced volatility. This is because users avoid risky routes that might cause delays or higher energy consumption, opting instead for reliable paths that facilitate planned charging. The shift in peak times reflects changes in travel patterns: risk-seeking users may delay trips due to congestion, charging later in the evening, while risk-neutral users charge earlier to ensure battery sufficiency. These insights are crucial for grid operators managing electric car load and for aggregators designing pricing schemes.
Furthermore, we examine the impact of attribute weights on electric car charging demand. By varying ωj, we simulate different user priorities—e.g., users who value travel time over comfort. The results show that when cost-type attributes (like congestion) have higher weights, charging demand tends to concentrate in off-peak hours, as users seek to minimize travel time by avoiding busy periods. Conversely, when benefit-type attributes (like comfort) dominate, demand may spread more evenly, as users choose longer but more comfortable routes, leading to varied charging times. This highlights the importance of understanding user preferences in electric car charging infrastructure planning.
Another key finding is the role of the anxiety range dv in electric car charging behavior. Users with higher anxiety (e.g., dv = 30 km) tend to charge more frequently and at higher SOC levels, increasing overall demand. Our model captures this by adjusting SOC constraints in the charging demand equation. For instance, with dv = 20 km, the average charging event adds 15 kWh per electric car, but with dv = 30 km, it rises to 18 kWh. This sensitivity analysis underscores how psychological factors, integral to bounded rationality, directly impact electric car energy consumption and grid load.
Conclusions and Future Directions for Electric Car Research
In conclusion, this study presents a comprehensive framework for analyzing electric car charging demand that incorporates bounded rational behavior through prospect theory and multi-attribute decision-making. By modeling electric car users’ route choices with variable risk preferences and uncertain attributes, we can more accurately predict charging patterns and their implications for power grids. Our case study on the Nguyen-Dupius network demonstrates that risk preference variations and attribute weighting significantly influence daily charging demand, affecting peak times, total load, and volatility. These findings emphasize the need for behavioral considerations in electric car integration strategies, such as dynamic pricing, infrastructure placement, and demand response programs.
For future research, several directions are promising. First, the risk preference coefficient could be further refined by incorporating additional factors, such as user demographics, trip purposes, or real-time traffic information. Electric car users on long-distance journeys might exhibit different risk attitudes compared to daily commuters, affecting charging decisions. Second, our model could be extended to include more attributes, like charging station availability or electricity prices, which are critical for electric car adoption. Third, integrating with real-world data from electric car fleets would enhance validation and applicability. Finally, exploring interactions between electric car users and grid operators in a game-theoretic framework could yield insights into market design and policy incentives.
Ultimately, as electric car technology evolves, understanding user behavior will remain central to achieving sustainable transportation and energy systems. By embracing bounded rationality, we can develop more resilient models that bridge the gap between human psychology and technical infrastructure, paving the way for smarter electric car integration. This research contributes to that goal by providing a robust methodological tool for academics and practitioners alike, fostering advancements in the electric car domain.