With the rapid growth in electric car adoption, the fluctuation in charging demand poses significant challenges to the safe and economic operation of power grids. The inherent uncertainties in electric car user travel behaviors, coupled with variations in risk preferences, lead to bounded rationality, which in turn results in unpredictable charging patterns. Traditional models often assume perfect rationality, but in reality, electric car users exhibit limited rationality due to factors like travel time variability, congestion, and comfort levels. This study addresses these issues by integrating prospect theory into a risk-based multi-attribute decision-making framework for electric car travel path selection. We consider three types of uncertain influencing factors—interval numbers, crisp numbers, and triangular fuzzy numbers—and analyze their relationship with variable risk preference coefficients. By establishing a variable-coefficient risk-based multi-attribute decision model, we derive comprehensive prospect values for each path, which inform travel decisions. Using the method of successive averages, we dynamically allocate and update traffic flows in a region, leading to an electric car charging demand model based on these prospect values. This approach allows us to analyze the impact of bounded rational behavior on daily charging demand, validated through the Nguyen-Dupius network. The results demonstrate that user risk preferences and attribute weights significantly influence charging demand curves, highlighting the importance of incorporating behavioral economics into electric car infrastructure planning.
The proliferation of electric cars has intensified the need for accurate charging demand forecasting. Unlike conventional vehicles, electric cars rely on charging infrastructure that must align with grid capabilities, and user decisions are often influenced by psychological factors. Bounded rationality, a concept from behavioral economics, suggests that users do not always make optimal decisions due to cognitive limitations and environmental uncertainties. For electric car users, this manifests in travel path choices affected by attributes such as travel time, congestion rates, and comfort levels. These attributes can be categorized into cost-type and benefit-type, with values represented as interval numbers, real numbers, or triangular fuzzy numbers. In this study, we define travel congestion rate and travel time as cost-type attributes, while comfort level is a benefit-type attribute. The integration of prospect theory allows us to model how users perceive gains and losses relative to reference points, incorporating variable risk preferences that change with these points. This model provides a more realistic depiction of electric car user behavior, enabling better prediction of charging demand and supporting grid stability and market participation for electric car aggregators.
To formalize the decision-making process for electric car users, we consider a set of travel paths from origin to destination. Let there be M paths denoted as A = {A1, A2, …, AM}, where each path Ai is evaluated based on three attributes: C1 (travel time, interval number), C2 (travel congestion rate, crisp number), and C3 (comfort level, triangular fuzzy number). Each attribute has three states—good, medium, and poor—represented by set S_h for h = 1, 2, 3. The expectation for each attribute under each state serves as the reference point r_j = (r1j, r2j, r3j). For attribute C1, the value x_{i1}^h is an interval [x_{i1}^{hl}, x_{i1}^{hu}] following a normal distribution N(μ_{i1}^h, (σ_{i1}^h)^2), where μ_{i1}^h = (x_{i1}^{hl} + x_{i1}^{hu})/2 and σ_{i1}^h = (x_{i1}^{hu} – x_{i1}^{hl})/6. The probability density function is given by:
$$ f_{i1}^h(x) = \frac{1}{\sqrt{2\pi} \sigma_{i1}^h} \exp\left( -\frac{(x – \mu_{i1}^h)^2}{2 (\sigma_{i1}^h)^2} \right) $$
The gain G_{ij}^h and loss L_{ij}^h for each attribute relative to the reference point are computed based on the position relationships. For C1, if x_{i1}^{hl} ≥ r_{h1}, G_{i1}^h = 0 and L_{i1}^h = ∫_{x_{i1}^{hl}}^{x_{i1}^{hu}} (r_{h1} – x) f_{i1}^h(x) dx; if x_{i1}^{hu} ≤ r_{h1}, G_{i1}^h = ∫_{x_{i1}^{hl}}^{x_{i1}^{hu}} (r_{h1} – x) f_{i1}^h(x) dx and L_{i1}^h = 0; and if x_{i1}^{hl} < r_{h1} < x_{i1}^{hu}, G_{i1}^h = ∫_{x_{i1}^{hl}}^{r_{h1}} (r_{h1} – x) f_{i1}^h(x) dx and L_{i1}^h = ∫_{r_{h1}}^{x_{i1}^{hu}} (r_{h1} – x) f_{i1}^h(x) dx. For C2, as a crisp number, G_{i2}^h = max(0, r_{h2} – x_{i2}^h) and L_{i2}^h = min(0, r_{h2} – x_{i2}^h). For C3, represented as a triangular fuzzy number (a_{i3}^h, b_{i3}^h, c_{i3}^h), the membership function is:
$$ \phi_{i3}^h(x) = \begin{cases}
0 & x < a_{i3}^h \\
\frac{x – a_{i3}^h}{b_{i3}^h – a_{i3}^h} & a_{i3}^h \leq x \leq b_{i3}^h \\
\frac{c_{i3}^h – x}{c_{i3}^h – b_{i3}^h} & b_{i3}^h \leq x \leq c_{i3}^h \\
0 & x > c_{i3}^h
\end{cases} $$
The gains and losses for C3 are calculated similarly, with integrals over the relevant ranges. This yields risk gain matrix G^h = [G_{ij}^h]_{M×3} and risk loss matrix L^h = [L_{ij}^h]_{M×3}.
Next, we apply prospect theory to compute the value functions and probability weighting functions. The value functions for gains and losses are:
$$ V_{(+)ij}^h = (G_{ij}^h)^{\alpha_j^h} $$
$$ V_{(-)ij}^h = -\lambda (-L_{ij}^h)^{\alpha_j^h} $$
where α_j^h is the variable risk preference coefficient for attribute j in state h, and λ = 2.25 is the loss aversion coefficient. The probability weighting functions are:
$$ \pi_{(+)ij}^h = \frac{(p^h)^{\zeta}}{((p^h)^{\zeta} + (1 – p^h)^{\zeta})^{1/\zeta}} $$
$$ \pi_{(-)ij}^h = \frac{(p^h)^{\delta}}{((p^h)^{\delta} + (1 – p^h)^{\delta})^{1/\delta}} $$
with ζ = 0.61 and δ = 0.69. The risk preference coefficient α_j^h varies with the reference point and is defined as:
$$ \alpha_j^h = \left(1 – \frac{r_j^h}{\sum_{h=1}^3 r_j^h}\right)^\theta $$
where θ is a scaling parameter between 0 and 1, representing the travel scale. This variable coefficient captures how user risk sensitivity changes with expectations, enhancing the model’s realism for electric car users. The prospect value for each attribute is then:
$$ V_{ij} = \sum_{h=1}^3 V_{(+)ij}^h \pi_{(+)ij}^h + \sum_{h=1}^3 V_{(-)ij}^h \pi_{(-)ij}^h $$
Normalizing the prospect decision matrix V = [V_{ij}]_{M×3} gives V* = [V_{ij}^*]_{M×3}, where V_{ij}^* = V_{ij} / V_j^{\text{max}} and V_j^{\text{max}} = \max_{i \in M} |V_{ij}|. The comprehensive prospect value for each path is:
$$ U_i = \sum_{j=1}^3 \omega_j V_{ij}^* $$
with attribute weights ω_j satisfying ∑ ω_j = 1. Paths are ranked based on U_i, with higher values indicating preferred choices for electric car travel.
To model charging demand, we integrate the traffic flow dynamics using the method of successive averages. For a region with N origin-destination pairs, each with M_k paths, the traffic flow on 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 F_a^s is the additional flow allocated to link a. The travel time for electric cars on link a is updated using the BPR function:
$$ T_a^s = t_{0a} \left[1 + 0.15 \left( \frac{x_a^s}{C_a} \right)^4 \right] $$
where t_{0a} is the free-flow travel time, and C_a is the link capacity. The speed is derived as v_a = (C_a l_a) / 1000, with l_a being the link distance. These updates continue until convergence, ensuring balanced traffic assignment that reflects user choices based on prospect values.
The charging demand for electric cars is then calculated based on the comprehensive prospect values. For an electric car v on path i in time period t, the charging time T_{v,i,t} is:
$$ T_{v,i,t} = \frac{(SOC_{v1,i,t} – SOC_{v0,i,t}) C_v}{P_v} $$
where SOC_{v1,i,t} is the post-charging state of charge, SOC_{v0,i,t} is the initial state of charge, C_v is the battery capacity, and P_v is the charging power. Considering range anxiety, the post-charging SOC must satisfy SOC_{v1,i,t} – d_{k,i} q_v ≥ d_v q_v, where d_{k,i} is the path distance, q_v is the energy consumption rate, and d_v is the anxiety range. The total charging demand in period t is:
$$ Q_t = \sum_{k=1}^N \left[ p_{k,t} D_t \sum_{i=1}^{M_k} (u_{k,i,t} SOC_{v2,i,t}) \right] $$
where SOC_{v2,i,t} = SOC_{v1,i,t} – SOC_{v0,i,t}, p_{k,t} is the proportion of electric cars choosing OD pair k, D_t is the total travel demand, and u_{k,i,t} is the probability of choosing path i, derived from the prospect values. Only paths with positive prospect values are considered, as they represent beneficial choices for electric car users.
In our simulation, we use the Nguyen-Dupius network to validate the model. The network consists of multiple nodes and links, with parameters such as free-flow travel times and capacities. We assume electric cars start from nodes N1, N4, and N12, with N3 as the destination, and consider various paths as outlined in Table 1. The electric car parameters include a battery capacity of 24 kWh, energy consumption of 30 kWh per 100 km, and an anxiety range of 20 km. We analyze the impact of risk preferences and attribute weights on charging demand.
| Origin-Destination | Path ID | Path Sequence |
|---|---|---|
| N1 to N3 | 1 | 1-4-7-12-18 |
| N1 to N3 | 2 | 1-4-11-15-18 |
| N1 to N3 | 3 | 3-6-7-12-18 |
| N1 to N3 | 4 | 3-6-11-15-18 |
| N1 to N3 | 5 | 3-10-14-15-18 |
| N1 to N3 | 6 | 3-10-17-19 |
| N4 to N3 | 1 | 5-6-7-12-18 |
| N4 to N3 | 2 | 5-6-11-15-18 |
| N4 to N3 | 3 | 5-10-14-15-18 |
| N4 to N3 | 4 | 5-10-17-19 |
| N4 to N3 | 5 | 9-17-19 |
| N12 to N3 | 1 | 4-7-12-18 |
| N12 to N3 | 2 | 4-11-15-18 |
For the risk preference analysis, we vary θ from 0 to 1, which adjusts the risk coefficient α_j^h. Figure 1 shows the comprehensive prospect values for paths from N1 to N3 under different θ. As θ increases, α decreases, indicating reduced risk sensitivity. For example, at θ = 0.5, the optimal path shifts from path 5 to path 6, demonstrating that risk-averse electric car users prefer more stable paths. This variability underscores the importance of incorporating dynamic risk preferences in electric car travel models.
Regarding attribute weights, we test four scenarios with different weight vectors: ω = (0.2, 0.5, 0.3)^T, ω = (0.2, 0.6, 0.2)^T, ω = (0.1, 0.5, 0.4)^T, and ω = (0.3, 0.4, 0.3)^T. Table 2 summarizes the comprehensive prospect values for paths from N1 to N3 under these scenarios. Changes in weights alter the prospect values, influencing path choices for electric car users. For instance, increasing the weight of attribute C2 (travel time) favors paths with better time performance, while increasing C3 (comfort) favors more comfortable paths.
| Path ID | Scenario 1 (ω = 0.2, 0.5, 0.3) | Scenario 2 (ω = 0.2, 0.6, 0.2) | Scenario 3 (ω = 0.1, 0.5, 0.4) | Scenario 4 (ω = 0.3, 0.4, 0.3) |
|---|---|---|---|---|
| 1 | -0.7486 | -0.8000 | -0.7000 | -0.7500 |
| 2 | 0.0729 | 0.0500 | 0.0800 | 0.0700 |
| 3 | -1.0000 | -1.1000 | -0.9500 | -1.0500 |
| 4 | -0.2004 | -0.1500 | -0.2200 | -0.1800 |
| 5 | 0.0733 | 0.0900 | 0.0600 | 0.0750 |
| 6 | 0.0874 | 0.0700 | 0.0950 | 0.0850 |
The charging demand analysis reveals that risk preferences significantly affect the fluctuation of demand curves. Figure 2 illustrates the daily charging demand under different θ values. As θ increases (lower risk sensitivity), the demand curve becomes smoother, indicating that risk-averse electric car users lead to more stable charging patterns. Conversely, higher risk sensitivity (lower θ) results in greater volatility, as users are more likely to switch paths based on perceived gains or losses. This behavior is critical for electric car aggregators to manage market participation and avoid penalties from deviation.
Attribute weights also impact the overall charging demand level. In Scenario 1, with balanced weights, the demand peaks at 400 MW during peak hours. In Scenario 2, where travel time weight is higher, the demand increases to 450 MW due to the preference for faster paths that may require more charging. In Scenario 3, with higher comfort weight, the demand decreases to 350 MW as users choose more comfortable but potentially longer paths. These variations emphasize the need to account for multi-attribute decisions in electric car charging infrastructure planning.
In conclusion, this study presents a novel framework for electric car charging demand analysis that incorporates bounded rational behavior through prospect theory and multi-attribute decision-making. The variable risk preference coefficient captures the dynamic nature of user psychology, while the integration of traffic flow updates ensures realistic modeling. Our findings show that both risk preferences and attribute weights influence charging demand, with implications for grid management and electric car aggregator strategies. Future work could explore additional factors such as user types, travel purposes, and real-time data integration to further enhance the model. Ultimately, understanding the bounded rationality of electric car users is essential for developing resilient and efficient electric car ecosystems.
The methodology developed here not only applies to electric car charging but can also be extended to other smart grid applications where user behavior plays a critical role. By leveraging behavioral economics, we can better anticipate the challenges posed by the growing adoption of electric cars and create more adaptive systems. This approach underscores the importance of interdisciplinary research in addressing the complexities of modern energy systems, particularly as electric cars become increasingly prevalent in urban mobility.