1. Introduction
The large-scale integration of electric vehicles (EVs) poses significant challenges to power grid stability due to fluctuating charging demands. Traditional models often assume perfect user rationality, overlooking critical behavioral factors like travel uncertainty and heterogeneous risk preferences. These omissions lead to inaccurate demand forecasts, causing financial penalties for EV aggregators in electricity markets and suboptimal grid planning. Prospect Theory (PT) offers a psychological framework to model bounded rationality, where users evaluate gains/losses relative to reference points rather than absolute outcomes. This work integrates PT with risk-based multi-attribute decision-making to:
- Quantify how EV users’ risk attitudes and travel uncertainties jointly influence path choices.
- Develop a dynamic charging demand model sensitive to behavioral heterogeneity.
- Validate outcomes using traffic-flow equilibrium principles.
2. Risk-Based Multi-Attribute Decision Model for EV Route Choice
2.1 Uncertainty Attributes in EV Travel
EV route selection involves three stochastic attributes, each modeled distinctly:
- Attribute C₁ (Cost-Type, Interval Number): Travel time, represented as xijh=[xL,ijh,xU,ijh]xijh=[xL,ijh,xU,ijh], following a normal distribution N(μijh,(σijh)2)N(μijh,(σijh)2), where μijh=(xL,ijh+xU,ijh)/2μijh=(xL,ijh+xU,ijh)/2, σijh=(xU,ijh−xL,ijh)/6σijh=(xU,ijh−xL,ijh)/6.
- Attribute C₂ (Cost-Type, Crisp Number): Congestion rate (ratio of EVs to road capacity), denoted as a real number xijhxijh.
- Attribute C₃ (Benefit-Type, Triangular Fuzzy Number): Comfort/convenience, expressed linguistically and mapped to fuzzy sets (Table 1).
Table 1: Linguistic Variables and Triangular Fuzzy Numbers
| Linguistic Variable | Triangular Fuzzy Number |
|---|---|
| Very Low | (1, 2, 3) |
| Low | (2, 3, 4) |
| Medium | (4, 5, 6) |
| High | (6, 7, 8) |
| Very High | (7, 8, 9) |
2.2 Gain/Loss Calculation Relative to Reference Points
For a path AiAi under state ShSh, gains (GG) and losses (LL) are computed relative to user-specific reference points rjhrjh:
- C₁ (Interval Number):Gijh={0,xL,ijh>rjh∫xL,ijhrjh(rjh−x)fij(x)dx,xU,ijh≤rjh∫xL,ijhrjh(rjh−x)fij(x)dx,xL,ijh<rjh<xU,ijhGijh=⎩⎨⎧0,∫xL,ijhrjh(rjh−x)fij(x)dx,∫xL,ijhrjh(rjh−x)fij(x)dx,xL,ijh>rjhxU,ijh≤rjhxL,ijh<rjh<xU,ijhLijh={∫rjhxU,ijh(x−rjh)fij(x)dx,xL,ijh≥rjh0,xU,ijh≤rjh∫rjhxU,ijh(x−rjh)fij(x)dx,xL,ijh<rjh<xU,ijhLijh=⎩⎨⎧∫rjhxU,ijh(x−rjh)fij(x)dx,0,∫rjhxU,ijh(x−rjh)fij(x)dx,xL,ijh≥rjhxU,ijh≤rjhxL,ijh<rjh<xU,ijhwhere fij(x)fij(x) is the PDF of N(μijh,(σijh)2)N(μijh,(σijh)2).
- C₂ (Crisp Number):Gijh=max(rjh−xijh,0),Lijh=max(xijh−rjh,0)Gijh=max(rjh−xijh,0),Lijh=max(xijh−rjh,0)
- C₃ (Fuzzy Number):Gijh={∫rjhcijh(x−rjh)ϕij(x)dx,aijh>rjh0,cijh≤rjh∫rjhcijh(x−rjh)ϕij(x)dx,aijh<rjh<cijhGijh=⎩⎨⎧∫rjhcijh(x−rjh)ϕij(x)dx,0,∫rjhcijh(x−rjh)ϕij(x)dx,aijh>rjhcijh≤rjhaijh<rjh<cijhLijh={0,aijh>rjh∫aijhrjh(rjh−x)ϕij(x)dx,cijh≤rjh∫aijhrjh(rjh−x)ϕij(x)dx,aijh<rjh<cijhLijh=⎩⎨⎧0,∫aijhrjh(rjh−x)ϕij(x)dx,∫aijhrjh(rjh−x)ϕij(x)dx,aijh>rjhcijh≤rjhaijh<rjh<cijhwhere ϕij(x)ϕij(x) is the membership function of triangular fuzzy number (aijh,bijh,cijh)(aijh,bijh,cijh).
2.3 Variable Risk Preference Coefficient
Traditional PT fixes risk preference coefficients (e.g., α=0.88α=0.88), ignoring individual heterogeneity. We propose a variable coefficient αjhαjh tied to reference points:αjh=(1−rjh∑h=13rjh)θ,θ∈[0,1]αjh=(1−∑h=13rjhrjh)θ,θ∈[0,1]
where θθ scales with trip size, modulating risk sensitivity. This yields adaptive value functions:Vij(+)h=(Gijh)αjh,Vij(−)h=−λ(−Lijh)αjhVij(+)h=(Gijh)αjh,Vij(−)h=−λ(−Lijh)αjh
with λ=2.25λ=2.25 (loss aversion), and probability weighting functions:πj(+)h=(ph)γ(ph)γ+(1−ph)γ,πj(−)h=(ph)δ(ph)δ+(1−ph)δπj(+)h=(ph)γ+(1−ph)γ(ph)γ,πj(−)h=(ph)δ+(1−ph)δ(ph)δ
where γ=0.61γ=0.61, δ=0.69δ=0.69, and phph is state probability.
2.4 Comprehensive Prospect Value
The prospect value VijVij for path AiAi and attribute jj is:Vij=∑h=13[Vij(+)hπj(+)h+Vij(−)hπj(−)h]Vij=h=1∑3[Vij(+)hπj(+)h+Vij(−)hπj(−)h]
Normalized by Vj∗=maxi∣Vij∣Vj∗=maxi∣Vij∣:Vij∗=VijVj∗Vij∗=Vj∗Vij
The comprehensive prospect value UiUi determines path rankings:Ui=∑j=13ωjVij∗,∑ωj=1Ui=j=1∑3ωjVij∗,∑ωj=1
Higher UiUi indicates better paths.
3. EV Charging Demand Model with Bounded Rationality
3.1 Traffic Flow Allocation
Using the Method of Successive Averages (MSA), road flows evolve iteratively to equilibrium:xas=(1−1s)xas−1+1sFasxas=(1−s1)xas−1+s1Fas
where xasxas is flow on link aa at iteration ss, and FasFas is auxiliary flow. Travel time TaTa updates via Bureau of Public Roads (BPR) function:Ta=ta0[1+0.15(xaCa)4]Ta=ta0[1+0.15(Caxa)4]
with CaCa = link capacity, ta0ta0 = free-flow time.
3.2 Charging Demand Calculation
For an electric vehicle vv on path ii during period tt, charging time is:TT
where = residual charge, CvCv = battery capacity, PvPv = charging power. must satisfy range anxiety:SOCSOC
where didi = path distance, qvqv = energy consumption rate, dada = anxiety range (e.g., 20 km).
Total regional charging demand QtQt aggregates over KK OD pairs:Qt=∑k=1K[pk,tDt∑i=1Mk(uk,i,t⋅ΔSOCv,i,t)]Qt=k=1∑K[pk,tDti=1∑Mk(uk,i,t⋅ΔSOCv,i,t)]ΔSOCv,i,t=SOCΔSOCv,i,t=SOC
Here, pk,tpk,t = proportion of electric vehicles in OD pair kk, DtDt = total EV demand, and uk,i,tuk,i,t = probability of selecting path ii (derived from Ui>0Ui>0).
4. Case Study: Nguyen-Dupuis Network
4.1 Network Parameters and Paths
Simulations used the Nguyen-Dupuis network (Figure 2 in source) with parameters:
Table 2: Nguyen-Dupuis Network Parameters
| Link | Nodes | ta0ta0 (min) | CaCa (veh) | lala (km) | vava (km/h) |
|---|---|---|---|---|---|
| 1 | N1↔N5N1↔N5 | 12 | 4,000 | 14.2 | 56.8 |
| 2 | N1↔N9N1↔N9 | 36 | 3,500 | 22.4 | 78.4 |
| … | … | … | … | … | … |
Table 3: EV Paths for OD Pairs
| OD Pair | Paths (Link Sequences) |
|---|---|
| N1→N3N1→N3 | 1-4-7-12-18, 1-4-11-15-18, 3-6-7-12-18, … |
| N4→N3N4→N3 | 5-6-7-12-18, 5-6-11-15-18, 5-10-14-15-18, … |
Assumptions:
- Battery capacity Cv=24Cv=24 kWh, consumption qv=0.3qv=0.3 kWh/km.
- Attribute weights ω=(0.2,0.5,0.3)Tω=(0.2,0.5,0.3)T.
4.2 Key Findings
- Path Selection: At θ=0θ=0 (high risk-seeking), Path 5 (U5=0.0733U5=0.0733) was optimal; at θ=0.7θ=0.7 (risk-neutral), Path 6 (U6=0.0874U6=0.0874) dominated (Figure 5). Risk-sensitive electric vehicle users preferred paths with higher uncertainty (e.g., shorter but congested routes).
- Charging Demand vs. Risk Preference: Higher θθ (lower αα) reduced demand volatility (Figure 7). A 0.5 increase in θθ flattened peak demand by ~18%, demonstrating that risk-averse electric vehicle users stabilize grid load.
- Attribute Weight Sensitivity: Increasing ω2ω2 (congestion weight) from 0.5 to 0.6 raised charging demand by 12% (Figure 10), as users prioritized less congested but longer paths, increasing energy consumption.
5. Conclusions and Future Work
5.1 Conclusions
- Bounded Rationality Dominates EV Behavior: Users’ path choices are highly sensitive to dynamic risk preferences and multi-attribute uncertainties. The comprehensive prospect value UiUi effectively captures this synergy.
- Variable Risk Coefficients Are Crucial: Static risk models (e.g., fixed α=0.88α=0.88) misestimate electric vehicle routing. Our αjhαjh model shows risk-seeking users amplify demand volatility by ≤25%.
- Grid Implications: Risk aversion reduces charging demand fluctuations, while attribute weights (e.g., congestion sensitivity) shift demand magnitude. Aggregators must incorporate behavioral heterogeneity to minimize imbalance penalties.
5.2 Future Work
- Enhance αjhαjh by incorporating trip purpose, user demographics, and real-time data.
- Extend the charging demand model to include V2G interactions and renewable energy variability.
- Integrate game-theoretic frameworks for EV aggregators to optimize pricing under bounded rationality.
Mathematical Notation Summary
| Symbol | Definition |
|---|---|
| xijhxijh | Attribute value for path ii, attribute jj, state hh |
| rjhrjh | Reference point for attribute jj, state hh |
| αjh,θαjh,θ | Risk preference coefficient and scaling parameter |
| UiUi | Comprehensive prospect value of path ii |
| Tch,v,tTch,v,t | Charging time for EV vv at time tt |
| QtQt | Total regional charging demand at time tt |
This work bridges behavioral economics and power systems engineering, offering a robust framework for electric vehicle charging infrastructure planning and grid management amid growing EV adoption.