Electric Vehicle Charging Demand Analysis Considering Bounded Rational User Behavior

Introduction
The large-scale adoption of electric vehicles (EVs) introduces significant volatility in charging demand, challenging power grid safety and economic operation. Traditional models often assume perfect user rationality, overlooking critical behavioral factors. In reality, electric vehicle users exhibit bounded rationality due to travel uncertainties and heterogeneous risk preferences. This directly translates to stochastic charging patterns that impact grid stability and market participation efficiency for electric vehicle aggregators. This work integrates Prospect Theory (PT) and multi-attribute decision-making (MADM) to model how user psychology shapes electric vehicle routing and charging demand, providing crucial insights for grid planning and market operations.

1. Bounded Rationality in EV Routing: A Risk-Based MADM Framework
Electric vehicle users face complex route choices influenced by multiple uncertain factors. We categorize these into three attribute types with distinct mathematical representations:

Table 1: EV Travel Uncertainty Attributes & RepresentationsAttribute TypeExamplesRepresentationProperty TypeC₁ (Interval)Travel Time, Costx₁ᵢʰ = [x₁ᵢʰ⁻, x₁ᵢʰ⁺] ~ N(μ₁ᵢʰ, (σ₁ᵢʰ)²)CostC₂ (Crisp)Congestion Rate, Charger DensityReal Number x₂ᵢʰCostC₃ (Triangular Fuzzy)Weather, Comfortx₃ᵢʰ = (aᵢʰ, bᵢʰ, cᵢʰ)BenefitWhere:
- i: Path index (i = 1, 2, ..., M)
- h: State index (h = 1 (Good), 2 (Medium), 3 (Poor))
- μ₁ᵢʰ = (x₁ᵢʰ⁻ + x₁ᵢʰ⁺)/2, σ₁ᵢʰ = (x₁ᵢʰ⁺ - x₁ᵢʰ⁻)/6 for C₁.
- C₁ and C₂ are Cost attributes (lower values preferred). C₃ is a Benefit attribute (higher values preferred).

1.1. Gain/Loss Calculation Relative to Reference Points
The user’s perceived “gain” or “loss” for an attribute value xⱼᵢʰ is computed relative to a reference point rⱼʰ (the expected value under state h). The calculation differs by attribute type:

C₁ (Interval Attributes):
Gain G₁ᵢʰ and Loss L₁ᵢʰ are computed via integration over the probability density function f₁(x) (Eq. 1), considering the position of r₁ʰ relative to the interval [x₁ᵢʰ⁻, x₁ᵢʰ⁺]:math复制下载f_1(x) = \frac{1}{\sqrt{2\pi}\sigma_{1i}^h} \exp\left(-\frac{(x-\mu_{1i}^h)^2}{2(\sigma_{1i}^h)^2}\right) \quad (1)math复制下载G_{1i}^h = \begin{cases} 0 & x_{1i}^{h+} > r_1^h \\ \int_{r_1^h}^{x_{1i}^{h+}} (x – r_1^h) f_1(x) dx & x_{1i}^{h-} \leq r_1^h \\ \int_{r_1^h}^{x_{1i}^{h+}} (x – r_1^h) f_1(x) dx & x_{1i}^{h-} < r_1^h < x_{1i}^{h+} \end{cases} \quad (2)math复制下载L_{1i}^h = \begin{cases} \int_{x_{1i}^{h-}}^{r_1^h} (r_1^h – x) f_1(x) dx & x_{1i}^{h-} > r_1^h \\ 0 & x_{1i}^{h+} \leq r_1^h \\ \int_{x_{1i}^{h-}}^{r_1^h} (r_1^h – x) f_1(x) dx & x_{1i}^{h-} < r_1^h < x_{1i}^{h+} \end{cases} \quad (3)
C₂ (Crisp Attributes):
Calculations are direct differences:math复制下载G_{2i}^h = \begin{cases} 0 & x_{2i}^h > r_2^h \\ r_2^h – x_{2i}^h & x_{2i}^h \leq r_2^h \end{cases} \quad (4)math复制下载L_{2i}^h = \begin{cases} r_2^h – x_{2i}^h & x_{2i}^h > r_2^h \\ 0 & x_{2i}^h \leq r_2^h \end{cases} \quad (5)
C₃ (Triangular Fuzzy Attributes):
Gain G₃ᵢʰ and Loss L₃ᵢʰ are computed via integration using the membership function φ₃ᵢʰ(x) (Eq. 6), considering the position of r₃ʰ relative to the support (aᵢʰ, cᵢʰ):math复制下载\varphi_{3i}^h(x) = \begin{cases} 0 & x < a_{3i}^h \\ (x – a_{3i}^h)/(b_{3i}^h – a_{3i}^h) & a_{3i}^h \leq x \leq b_{3i}^h \\ (c_{3i}^h – x)/(c_{3i}^h – b_{3i}^h) & b_{3i}^h \leq x \leq c_{3i}^h \\ 0 & x > c_{3i}^h \end{cases} \quad (6)math复制下载G_{3i}^h = \begin{cases} \int_{r_3^h}^{c_{3i}^h} (x – r_3^h) \varphi_{3i}^h(x) dx & a_{3i}^h > r_3^h \\ 0 & c_{3i}^h \leq r_3^h \\ \int_{r_3^h}^{c_{3i}^h} (x – r_3^h) \varphi_{3i}^h(x) dx & a_{3i}^h < r_3^h < c_{3i}^h \end{cases} \quad (7)math复制下载L_{3i}^h = \begin{cases} 0 & a_{3i}^h > r_3^h \\ \int_{a_{3i}^h}^{r_3^h} (r_3^h – x) \varphi_{3i}^h(x) dx & c_{3i}^h \leq r_3^h \\ \int_{a_{3i}^h}^{r_3^h} (r_3^h – x) \varphi_{3i}^h(x) dx & a_{3i}^h < r_3^h < c_{3i}^h \end{cases} \quad (8)This yields Risk Gain (G_h) and Loss (L_h) matrices for all paths and states.

1.2. Variable Risk Preference Integrated Prospect Value
Standard PT uses fixed parameters for the value function (α, λ) and probability weighting function (ξ, δ):

V^{(+)h}_{ij} = (G_{ji}^h)^\alpha, \quad V^{(-)h}_{ij} = -\lambda (-L_{ji}^h)^\alpha \quad (9)

\pi^{(+)h}_{j} = \frac{(p_h)^\xi}{[(p_h)^\xi + (1-p_h)^\xi]^{1/\xi}}, \quad \pi^{(-)h}_{j} = \frac{(p_h)^\delta}{[(p_h)^\delta + (1-p_h)^\delta]^{1/\delta}} \quad (10)

Where p_h is the probability of state h, α (risk preference), λ (loss aversion), ξ, δ (probability weighting curvature). Conventionally, α=0.88, λ=2.25, ξ=0.61, δ=0.69.

Critically, user risk preference (α) is not static but depends on the chosen reference points. A higher reference point generally lowers sensitivity to gains/losses (i.e., lower risk preference α). We model α as a variable dependent on attribute reference points and a scale parameter θ:

\alpha_j^h = \left(1 - \frac{r_j^h}{\sum_{k=1}^3 r_j^k}\right)^\theta \quad (11)

Where 0 ≤ θ ≤ 1 (θ=0: Strong Risk Seeking; θ=1: Risk Neutral). Substituting Eq. 11 into Eq. 9 gives the adapted value functions:

V^{(+)h}_{ij} = (G_{ji}^h)^{\alpha_j^h}, \quad V^{(-)h}_{ij} = -\lambda (-L_{ji}^h)^{\alpha_j^h} \quad (12)

The prospect value Vᵢⱼ for path i under attribute j aggregates gains and losses across states h:

V_{ij} = \sum_{h=1}^3 V^{(+)h}_{ij} \pi^{(+)h}_{j} + \sum_{h=1}^3 V^{(-)h}_{ij} \pi^{(-)h}_{j} \quad (13)

Normalizing Vᵢⱼ (Eq. 14-15) and weighting by attribute importance ωⱼ yields the Comprehensive Prospect Value (CPV) Uᵢ for path i:

V_j^{max} = \max_{i} |V_{ij}|, \quad V_{ij}^* = \frac{V_{ij}}{V_j^{max}} \quad (14, 15)

U_i = \sum_{j=1}^3 \omega_j V_{ij}^* \quad \text{where} \quad \omega_j \geq 0,  \sum_{j=1}^3 \omega_j = 1 \quad (16)

Paths are ranked by Uᵢ; higher Uᵢ indicates higher selection likelihood by the bounded rational electric vehicle user.

2. EV Charging Demand Model Based on CPV
2.1. Traffic Flow Assignment & CPV Update
The routing model (Section 1) determines initial path choices. As users converge on high-CPV paths, congestion increases, altering travel times and subsequently CPVs. We model this feedback loop using the Method of Successive Averages (MSA) and the Bureau of Public Roads (BPR) function:

Traffic Flow Update (MSA):math复制下载x_a^s = \left(1 – \frac{1}{s}\right) x_a^{s-1} + \frac{1}{s} F_a^s \quad (17)Where xₐˢ is the flow on link a at iteration s, Fₐˢ is the auxiliary flow assigned to link a in iteration s.
Link Travel Time Update (BPR):math复制下载T_a^s = t_a^0 \left[ 1 + 0.15 \left( \frac{x_a^s}{C_a} \right)^4 \right] \quad (18)Where Tₐˢ is the updated travel time (min) on link a, tₐ⁰ is free-flow time (min), Cₐ is link capacity (veh/h).
Link Speed (Derived):math复制下载v_a = \frac{1000 \cdot l_a}{60 \cdot t_a^0} \quad (\text{km/h}) \quad (19)Where lₐ is link length (km). Updated Tₐˢ feeds back into the path attribute C₂ (travel time), triggering recalculation of CPV Uᵢ for all paths until flow equilibrium (Wardrop’s Principle) is reached.

2.2. Charging Demand Calculation
The final CPV Uᵢ determines path choice probabilities uₖᵢ for each Origin-Destination (OD) pair k. Charging demand arises from energy consumed during trips exceeding the user’s range anxiety threshold. For an electric vehicle v on path i of OD pair k at time t:

Required Charging Time:math复制下载T_{k,i,v,t}^{chg} = \frac{(SOC_{k,i,v,t}^{end} – SOC_{k,i,v,t}^{res}) \cdot C_v}{P_v} \quad (20)Where:
- SOCᴿᴱˢ: Residual State of Charge (kWh) at trip start.
- SOCᴱᴺᴰ: Required SOC at trip end (must satisfy range anxiety).
- Cᵥ: Electric vehicle battery capacity (kWh).
- Pᵥ: Electric vehicle charging power (kW).
Range Anxiety Constraint:math复制下载SOC_{k,i,v,t}^{end} \geq SOC_{k,i,v,t}^{res} + d_k \cdot q_v \quad (21)Where dₖ is the distance (km) of path i in OD pair k, qᵥ is the electric vehicle energy consumption rate (kWh/km), dᵥ is the user’s range anxiety threshold (km).
Total Regional Charging Demand:math复制下载Q_t = \sum_{k=1}^{N_{OD}} \left[ p_{k,t} \cdot D_t \cdot \sum_{i=1}^{M_k} (u_{k,i,t} \cdot \Delta SOC_{k,i,v,t}) \right] \quad (22)math复制下载\Delta SOC_{k,i,v,t} = SOC_{k,i,v,t}^{end} – SOC_{k,i,v,t}^{res} \quad (23)Where:
- pₖ,ₜ: Proportion of regional electric vehicle trips using OD pair k at time t.
- Dₜ: Total regional electric vehicle travel demand (veh) at time t.
- uₖᵢ,ₜ: Probability of choosing path i for OD pair k at time t (driven by CPV Uᵢ). Paths with Uᵢ ≤ 0 are excluded (uₖᵢ,ₜ = 0).
- ΔSOCₖᵢᵥ,ₜ: Net energy charged for electric vehicle on path i (kWh).

3. Case Study: Nguyen-Dupuis Network
3.1. Network & EV Parameters

Table 2: Nguyen-Dupuis Network Link ParametersLinkNodest_a^0 (min)C_a (veh/h)l_a (km)v_a (km/h)1N1 ↔ N512400014.271.02N1 ↔ N236350022.437.33N4 ↔ N512300011.256.0………………18N9 ↔ N312300017.688.0
OD Pairs: (N1→N3), (N4→N3), (N12→N3) – 11 total paths.
EV Parameters: C_v = 24 kWh, q_v = 0.3 kWh/km, d_v = 20 km, P_v = 7 kW (assumed).
Initial State: Full battery (SOCᴿᴱˢ = 24 kWh) for morning commute.
Attribute Weights (Scenario 1): ω = (0.2, 0.5, 0.3)^T (Congestion Rate, Travel Time, Comfort).

3.2. Key Findings

Impact of Variable Risk Preference (θ): Path CPVs and consequently optimal path choices change significantly with θ (Fig 5b). Higher θ (lower risk preference α) shifts user preference towards less volatile paths, reducing charging demand fluctuations.
Impact of Attribute Weights (ω): Varying ω changes which paths have Uᵢ > 0 and their relative rankings (Fig 8, Fig 9). This alters the spatial distribution of electric vehicle flows and the magnitude of regional charging demand Qₜ (Fig 10), though demand volatility patterns persist if risk preference (θ) is fixed.
Charging Demand Characteristics: User risk preference (θ) primarily controls the volatility amplitude of the electric vehicle charging demand curve Qₜ. Attribute weights (ω) primarily control the baseline magnitude of Qₜ. Bounded rationality significantly shapes the spatio-temporal distribution of electric vehicle charging load.

4. Conclusions & Future Work
This work establishes a robust link between electric vehicle user psychology, travel behavior, and grid-impacting charging demand:

CPV for Realistic Routing: Modeling electric vehicle path choice via CPV under Prospect Theory and variable risk preference (α_j^h) captures bounded rationality more accurately than deterministic or fixed-risk models. Paths with Uᵢ > 0 reflect user-perceived utility considering uncertainty and risk attitude.
Dynamic Risk Preference is Crucial: Assuming a fixed α=0.88 oversimplifies user behavior. The variable coefficient model (Eq. 11) shows risk-seeking users (θ low, α high) prefer potentially optimal but risky paths, increasing charging volatility. Risk-averse users (θ high, α low) prefer stable paths, dampening volatility. Integrating risk preference with reference points is essential for realistic electric vehicle behavior modeling.
Drivers of Charging Demand: Bounded rationality directly shapes electric vehicle charging profiles. User risk preference (θ) governs charging demand fluctuation intensity. Attribute weights (ω) influence the overall demand level. Understanding these links is vital for grid operators and electric vehicle aggregators.
Validated Framework: The integrated risk-based MADM routing and MSA-BPR traffic flow model, coupled with the CPV-driven charging demand calculation, provides a validated framework for analyzing electric vehicle impacts under behavioral uncertainty.

Future research will focus on:

Enhancing the variable risk preference model (α) by incorporating user demographics, trip purpose, and real-time contextual factors beyond reference points.
Developing more granular electric vehicle charging demand models integrating this framework with activity-based travel chains and detailed grid topology.
Designing market mechanisms or incentives for electric vehicle aggregators explicitly accounting for bounded rational user behavior to optimize grid integration and profitability.

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