With the large-scale deployment of electric vehicles, the charging demand of electric vehicles has become a key factor affecting the security and economy of power system operation. Different from conventional fuel vehicles, the travel behavior of an electric vehicle is closely coupled with its charging behavior. If the driver of an electric vehicle selects a route that consumes excessive energy, the remaining state of charge may become insufficient and the charging demand will increase sharply at certain time periods. Therefore, understanding how an electric vehicle user makes route choices is essential for estimating the spatial and temporal distribution of electric vehicle charging demand. In this article, I study the electric vehicle charging demand from the perspective of bounded rationality. The bounded rationality of an electric vehicle user stems from two sources. The first source is travel uncertainty, because travel time, congestion level, weather and comfort are random variables that cannot be known in advance. The second source is risk preference heterogeneity, because different electric vehicle users have different attitudes toward potential losses and gains. Both sources jointly produce a highly uncertain charging demand pattern, which challenges the operation of power grids and the business performance of electric vehicle aggregators.
Traditional demand analysis methods often assume that an electric vehicle user is perfectly rational and selects the alternative with the maximum expected utility. However, behavioral studies in transportation research have shown that travelers systematically violate the assumptions of expected utility theory. Prospect theory has been widely used to explain such bounded rational behavior because it introduces reference dependence, loss aversion and probability weighting. In prospect theory, the user evaluates outcomes as gains or losses relative to a reference point rather than as absolute wealth levels. This mechanism can effectively describe a user’s limited rationality under an uncertain traffic environment. In the context of electric vehicle charging demand, several works have adopted prospect theory to analyze travel time uncertainty, route choice and charging station choice. Nevertheless, most of these works use a fixed risk preference coefficient, typically 0.88, for all users. In reality, the risk attitude of an electric vehicle user may change when the reference point changes. A user whose reference point is high may be less sensitive to a deviation of the actual outcome from that reference point, and therefore may exhibit a different degree of risk seeking or risk aversion.
In addition, the route choice of an electric vehicle is affected by multiple attributes simultaneously. The user may consider the travel time, the congestion rate, the charging convenience and the comfort level. These attributes have different data characteristics. Some are interval-valued because the actual travel time fluctuates in a range and follows a random distribution. Some are crisp values that can be measured directly, such as the occupancy rate of a road segment. Others are naturally linguistic variables, such as the comfort level under rainy weather, and they can be expressed as triangular fuzzy numbers. It is therefore desirable to set up a multi-attribute decision-making framework that treats these mixed attribute types together. I combine this framework with prospect theory so that the bounded rational behavior of the electric vehicle user can be modeled more realistically.
The objective of this work is to build an electric vehicle charging demand model that explicitly captures the bounded rational route choice behavior of electric vehicle users. I first define three uncertainty attributes for electric vehicle travel decisions. Based on the type of each attribute, I derive the gain and loss functions between the actual attribute value and the reference point. Then I construct a variable-coefficient prospect-theoretic decision model. The risk preference coefficient is not a fixed constant in this model, but is determined jointly by the reference point and the travel scale parameter. After that, I calculate the comprehensive prospect value of every candidate path. Using the method of successive averages, I update the traffic flow on each road segment and iterate until the path flow converges. The final path choice probability is then transferred into a regional electric vehicle charging demand model. The proposed framework is tested on the Nguyen-Dupuis road network.
The remainder of this article is organized as follows. Section 2 introduces the fundamental formulation of prospect theoretic multi-attribute decisions for electric vehicle travel. Section 3 presents the variable-coefficient route choice model and the electric vehicle charging demand model. Section 4 provides a numerical case study on the Nguyen-Dupuis network. Section 5 concludes this work.
2. Prospect-Theoretic Multi-Attribute Decision Framework
I consider an electric vehicle user who intends to travel from an origin to a destination during a given time period. There are M alternative paths in the set A = {A1, …, AM}. On each path, the electric vehicle user evaluates three attributes:
Attribute 1 — the travel time on the path. This attribute is represented by an interval number [xi1l, xi1u], where l and u denote the lower and upper bounds of the interval. Within this interval, the actual travel time x follows a normal distribution with mean μi1 and standard deviation σi1, as shown by
$$\mu_{i1} = \frac{x_{i1}^{l} + x_{i1}^{u}}{2}, \qquad \sigma_{i1} = \frac{x_{i1}^{u} – x_{i1}^{l}}{6}.$$
Because the actual time can take any value in the interval, the probability density function of x is given by
$$f_{i1}(x) = \frac{1}{\sqrt{2\pi}\sigma_{i1}} \exp\left[-\frac{(x – \mu_{i1})^2}{2\sigma_{i1}^2}\right]. \qquad (1)$$
Attribute 2 — the average congestion rate of the road segments that constitute the path. For the road network, the congestion rate is defined as the ratio between the electric vehicle flow and the capacity of the segment. This value is treated as a crisp real number xi2.
Attribute 3 — the comfort or convenience level perceived by the electric vehicle user. This attribute is a linguistic variable. It is translated into a triangular fuzzy number xi3 = (ai3, bi3, ci3) with the membership function
$$\varphi_{i3}(x) = \begin{cases} 0, & x < a_{i3}, \\ \dfrac{x-a_{i3}}{b_{i3}-a_{i3}}, & a_{i3} \le x \le b_{i3}, \\ \dfrac{c_{i3}-x}{c_{i3}-b_{i3}}, & b_{i3} \le x \le c_{i3}, \\ 0, & x > c_{i3}. \end{cases} \qquad (2)$$
Table 1 lists the mapping between the linguistic variables and the corresponding triangular fuzzy numbers. The first column contains the linguistic evaluation of the comfort attribute, such as “very low”, “low”, “medium” and “high”. The second column gives the triangular fuzzy number used in the computation.
| Linguistic variable | Triangular fuzzy number |
|---|---|
| Very low | (1, 2, 3) |
| Low | (2, 3, 4) |
| Fairly low | (3, 4, 5) |
| Medium | (4, 5, 6) |
| Fairly high | (5, 6, 7) |
| High | (6, 7, 8) |
| Very high | (7, 8, 9) |
In the multi-attribute decision problem, the traffic condition is divided into three states h = 1, 2, 3, which denote good, average and poor conditions respectively. The state probability ph is known to the electric vehicle user from historical experience, with p1 + p2 + p3 = 1. For each state, the electric vehicle user sets an expectation rh,j for attribute j. This expectation is called the reference point. The actual value of attribute j on path i under state h is denoted by xi,j(h). If the actual value is better than the reference point, the user obtains a gain G; otherwise, the user suffers a loss L.
2.1 Gain and loss for the interval-valued attribute
For the interval-valued travel time, let xi1(h) = [xi1l(h), xi1u(h)]. Because travel time is a cost-type attribute, a smaller actual value is more desirable. The gain of a lower travel time with respect to the reference point rh,1 is calculated as
$$G_{i1}^{(h)} = \begin{cases} 0, & x_{i1}^{l(h)} \ge r_{h,1}, \\ \displaystyle \int_{x_{i1}^{l(h)}}^{x_{i1}^{u(h)}} (r_{h,1} – x) f_{i1}^{(h)}(x)\,dx, & x_{i1}^{u(h)} \le r_{h,1}, \\ \displaystyle \int_{x_{i1}^{l(h)}}^{r_{h,1}} (r_{h,1} – x) f_{i1}^{(h)}(x)\,dx, & x_{i1}^{l(h)} < r_{h,1} < x_{i1}^{u(h)}. \end{cases} \qquad (3)$$
Similarly, the loss of the interval-valued attribute is
$$L_{i1}^{(h)} = \begin{cases} \displaystyle \int_{x_{i1}^{l(h)}}^{x_{i1}^{u(h)}} (r_{h,1} – x) f_{i1}^{(h)}(x)\,dx, & x_{i1}^{l(h)} \ge r_{h,1}, \\ 0, & x_{i1}^{u(h)} \le r_{h,1}, \\ \displaystyle \int_{r_{h,1}}^{x_{i1}^{u(h)}} (r_{h,1} – x) f_{i1}^{(h)}(x)\,dx, & x_{i1}^{l(h)} < r_{h,1} < x_{i1}^{u(h)}. \end{cases} \qquad (4)$$
It is worth emphasizing that the gain and loss are negative in the equation above for the loss case. In the subsequent prospect-theoretic computation, the loss L is expressed as a negative value, so a larger absolute value represents a greater loss.
2.2 Gain and loss for the crisp-valued attribute
For the crisp congestion-rate attribute, the comparison between the actual value xi2(h) and the reference point rh,2 is straightforward. Since congestion rate is also a cost-type attribute, the gain is obtained when the actual value is lower than the reference:
$$G_{i2}^{(h)} = \begin{cases} 0, & x_{i2}^{(h)} \ge r_{h,2}, \\ r_{h,2} – x_{i2}^{(h)}, & x_{i2}^{(h)} < r_{h,2}. \end{cases} \qquad (5)$$
$$\;L_{i2}^{(h)} = \begin{cases} r_{h,2} – x_{i2}^{(h)}, & x_{i2}^{(h)} \ge r_{h,2}, \\ 0, & x_{i2}^{(h)} < r_{h,2}. \end{cases} \qquad (6)$$
2.3 Gain and loss for the triangular-fuzzy-valued attribute
The comfort attribute is a benefit-type attribute. The actual value xi3(h) = (ai3(h), bi3(h), ci3(h)) and the reference point rh,3 are compared by integrating the membership function. The gain function is
$$G_{i3}^{(h)} = \begin{cases} \displaystyle \int_{a_{i3}^{(h)}}^{c_{i3}^{(h)}} (x – r_{h,3}) \varphi_{i3}^{(h)}(x)\,dx, & a_{i3}^{(h)} \ge r_{h,3}, \\ 0, & c_{i3}^{(h)} \le r_{h,3}, \\ \displaystyle \int_{r_{h,3}}^{c_{i3}^{(h)}} (x – r_{h,3}) \varphi_{i3}^{(h)}(x)\,dx, & a_{i3}^{(h)} < r_{h,3} < c_{i3}^{(h)}, \end{cases} \qquad (7)$$
and the loss function is
$$L_{i3}^{(h)} = \begin{cases} 0, & a_{i3}^{(h)} \ge r_{h,3}, \\ \displaystyle \int_{a_{i3}^{(h)}}^{c_{i3}^{(h)}} (x – r_{h,3}) \varphi_{i3}^{(h)}(x)\,dx, & c_{i3}^{(h)} \le r_{h,3}, \\ \displaystyle \int_{a_{i3}^{(h)}}^{r_{h,3}} (x – r_{h,3}) \varphi_{i3}^{(h)}(x)\,dx, & a_{i3}^{(h)} < r_{h,3} < c_{i3}^{(h)}. \end{cases} \qquad (8)$$
After computing the gain and loss matrices for all paths and all attributes, I obtain the risk gain matrix G(h) = [Gij(h)] and the risk loss matrix L(h) = [Lij(h)] for each state h.
3. Variable-Coefficient Electric Vehicle Route Choice and Charging Demand Model
3.1 Prospect value function and probability weighting
Following the framework of prospect theory, the gain and loss values are transformed by an S-shaped value function. For an electric vehicle user, the value function under state h for attribute j on path i is
$$V_{ij}^{(+h)} = \left(G_{ij}^{(h)}\right)^{\alpha}, \qquad V_{ij}^{(-h)} = -\lambda \left(-L_{ij}^{(h)}\right)^{\alpha}, \qquad (9)$$
where α is the risk preference coefficient, and λ is the loss-aversion coefficient. The typical value α = 0.88 and λ = 2.25 are widely used in the literature. In reality, however, different electric vehicle users possess different risk attitudes. The coefficient α should be treated as a variable rather than a fixed constant in an electric vehicle charging demand analysis.
In addition to the value function, the decision weight of each state is determined by the probability weighting function. For the gain part and the loss part, I use the following weighting functions:
$$\pi_{ij}^{(+h)} = \frac{(p_h)^{\zeta}}{\left[(p_h)^{\zeta} + (1-p_h)^{\zeta}\right]^{1/\zeta}}, \qquad \pi_{ij}^{(-h)} = \frac{(p_h)^{\delta}}{\left[(p_h)^{\delta} + (1-p_h)^{\delta}\right]^{1/\delta}}, \qquad (10)$$
where ζ = 0.61 and δ = 0.69 are typical parameters in the cumulative prospect theory model. The above value function and probability weighting function together determine the prospect value of each path and each attribute.
3.2 Variable risk preference coefficient
An important observation in behavioral economics is that the risk preference of an electric vehicle user depends on the reference point. For the same amount of potential gain or loss, a user with a higher reference point tends to be less sensitive to the outcome because the reference point changes the subjective feeling of gain and loss. Thus, I define a state-dependent and attribute-dependent risk preference coefficient as
$$\alpha_{j}^{(h)} = \left(1 – \frac{r_{h,j}}{\sum_{h’=1}^{3} r_{h’,j}}\right)^{\theta}, \qquad (11)$$
where θ ∈ [0, 1] is a travel-scale parameter. If θ = 0, the risk preference coefficient reduces to a constant. When θ increases, the reference point has a stronger influence on the risk preference, and the electric vehicle user gradually changes from risk seeking to risk neutrality. In this article I take this variable-coefficient specification into account, so that the difference in bounded rationality between different electric vehicle users is reflected more accurately.
By substituting Equation (11) into Equation (9), the value function becomes
$$V_{ij}^{(+h)*} = \left(G_{ij}^{(h)}\right)^{\alpha_j^{(h)}}, \qquad V_{ij}^{(-h)*} = -\lambda \left(-L_{ij}^{(h)}\right)^{\alpha_j^{(h)}}. \qquad (12)$$
3.3 Comprehensive prospect value and route selection
For an electric vehicle user on path i, the prospect value of attribute j under all three traffic states is
$$V_{ij} = \sum_{h=1}^{3} V_{ij}^{(+h)*} \pi_{ij}^{(+h)} + \sum_{h=1}^{3} V_{ij}^{(-h)*} \pi_{ij}^{(-h)}. \qquad (13)$$
The matrix V = [Vij] is normalized by using the maximum absolute value of each column:
$$V_j^{\max} = \max_{i \in M} \left\{ |V_{ij}| \right\}, \qquad V_{ij}^{*} = \frac{V_{ij}}{V_j^{\max}}. \qquad (14)$$
Finally, the comprehensive prospect value of path i is given by the weighted sum of normalized prospect values,
$$U_i = \sum_{j=1}^{3} \omega_j V_{ij}^{*}, \qquad (15)$$
where ωj is the weight of attribute j, satisfying Σωj = 1. The path with a larger comprehensive prospect value is more attractive to the electric vehicle user. If Ui is greater than zero, the electric vehicle user perceives a net benefit from choosing path i.
3.4 Dynamic traffic flow allocation
The route choice of many electric vehicle users in a region determines the traffic flow on each road segment. A path that yields a high comprehensive prospect value tends to attract more electric vehicle users, which may increase the congestion rate and reduce the prospect value of that path. Therefore, the charging demand analysis should not stop at a one-shot route choice. I use the method of successive averages (MSA) to iteratively update the traffic flow. Let xas be the traffic flow on segment a at iteration s. The updated flow is
$$x_{a}^{s} = \left(1 – \frac{1}{s}\right) x_{a}^{s-1} + \frac{1}{s} F_{a}^{s}, \qquad (16)$$
where Fas is the additional flow assigned to segment a at iteration s.
The travel time of an electric vehicle on segment a is updated according to the well-known BPR function:
$$T_{a}^{s} = t_{0a} \left[1 + 0.15\left(\frac{x_{a}^{s}}{C_a}\right)^4\right], \qquad (17)$$
where t0a is the free-flow travel time, and Ca is the capacity of the segment. The speed corresponding to the basic capacity is
$$v_a = \frac{C_a l_a}{1000}, \qquad (18)$$
where la is the length of the segment in kilometers.
The iteration stops when the difference in traffic flow between two consecutive iterations is sufficiently small. The resulting flow pattern is approximately a stochastic user equilibrium in which no electric vehicle user can improve his or her perceived prospect value by unilaterally changing the route.
3.5 Regional charging demand model
After the traffic flow converges, the probability that an electric vehicle user selects a particular path is based on the comprehensive prospect value. Let N be the number of origin-destination pairs, and let Mk be the number of available paths between origin-destination pair k. For a given time period t, the required charging time of an electric vehicle v that selects path i of pair k is
$$T_{v,i,t} = \frac{\left(SOC_{v1,i,t} – SOC_{v0,i,t}\right) C_{v}}{P_{v}}, \qquad (19)$$
where SOCv1,i,t is the battery energy after charging, SOCv0,i,t is the remaining battery energy, Cv is the battery capacity in kWh, and Pv is the charging power in kW.
The after-charging energy must satisfy the mileage-anxiety constraint:
$$SOC_{v1,i,t} – d_{k,i} q_v \ge d_v q_v, \qquad (20)$$
where dk,i is the distance of path i between pair k, qv is the energy consumption rate in kWh/km, and dv is the anxiety distance of the electric vehicle user.
The total charging demand in the whole region during period t is calculated as
$$Q_{t} = \sum_{k=1}^{N} \left[ p_{k,t} D_{t} \sum_{i=1}^{M_k} u_{k,i,t} \left(SOC_{v1,i,t} – SOC_{v0,i,t}\right) \right], \qquad (21)$$
where pk,t is the proportion of electric vehicles that select origin-destination pair k, Dt is the total travel demand of electric vehicles in the region, and uk,i,t is the path-selection probability. The probability uk,i,t is determined by the comprehensive prospect value Ui. In practice, paths with negative comprehensive prospect values are unlikely to be deliberately selected by a bounded rational electric vehicle user, and thus are excluded from the charging demand calculation.
4. Numerical Case Study on the Nguyen-Dupuis Network
To test the proposed electric vehicle charging demand model, I use the well-known Nguyen-Dupuis road network. The network has 13 nodes and 19 directed road segments. In this case study, I consider three origin-destination pairs: node N1 to node N3, node N4 to node N3, and node N12 to node N3. The trip purpose is the morning commute. The electric vehicle is assumed to depart with a full battery state of charge. The battery capacity is 24 kWh, the energy consumption rate is 0.30 kWh/km, and the anxiety distance of the electric vehicle user is 20 km.
The relevant parameters of the Nguyen-Dupuis network are summarized in Table 2. In this table, t0a is the free-flow travel time of the segment, Ca is the segment capacity, la is the segment length, and va is the speed under the basic capacity.
| Segment | Node sequence | t₀ (min) | C (veh) | l (km) | v (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 |
Table 3 lists all available candidate paths for the three origin-destination pairs. The path notation uses the segment numbers defined in Table 2.
| Path index | N1→N3 | N4→N3 | N12→N3 |
|---|---|---|---|
| 1 | 1-4-7-12-18 | 5-6-7-12-18 | 4-7-12-18 |
| 2 | 1-4-11-15-18 | 5-6-11-15-18 | 4-11-15-18 |
| 3 | 3-6-7-12-18 | 5-10-14-15-18 | — |
| 4 | 3-6-11-15-18 | 5-10-17-19 | — |
| 5 | 3-10-14-15-18 | 9-17-19 | — |
| 6 | 3-10-17-19 | — | — |
4.1 Verification of the electric vehicle route-choice model
I first focus on the six candidate paths from origin N1 to destination N3. The attribute-weight vector is chosen as ω = (0.2, 0.5, 0.3). The user’s reference points for the three attributes are estimated from the historical travel records of electric vehicle users. Using Equations (1)–(8), I construct the risk gain matrix and risk loss matrix for each of the three traffic states. A general observation is that path 2 has relatively large gains for the interval-valued travel time attribute because its actual travel time is often much smaller than the user’s expectation. In contrast, path 3 has large losses for the same attribute. For the comfort attribute, paths 2 and 6 have larger gains, while path 3 has the largest loss.
After computing the gain and loss matrices, I calculate the normalized prospect value for each path under the fixed risk preference coefficient α = 0.88. The comprehensive prospect values are obtained through Equation (15). The results are listed in Table 4.
| Path | U (fixed α = 0.88) | Rank |
|---|---|---|
| 1 | −0.7486 | 5 |
| 2 | 0.0729 | 3 |
| 3 | −1.0000 | 6 |
| 4 | −0.2004 | 4 |
| 5 | 0.0733 | 2 |
| 6 | 0.0874 | 1 |
According to Table 4, the ranking order of the six paths is U6 > U5 > U2 > U4 > U1 > U3. Therefore, paths 6, 5 and 2 exhibit positive comprehensive prospect values and are likely to be selected by a bounded rational electric vehicle user. The variable-coefficient model gives a different ranking when θ changes, which is examined in the next subsection.
4.2 Influence of variable risk preference on route choice
One of the key contributions of this article is the treatment of the risk preference coefficient as a variable instead of a constant. I now vary the travel-scale parameter θ from 0 to 1, which changes the state-dependent risk preference coefficient αj(h) in Equation (11). Table 5 demonstrates how the comprehensive prospect values of paths 2, 5 and 6 evolve with θ. These three paths always have positive values under the studied range and thus dominate the charging demand estimation.
| θ | Path 2 | Path 5 | Path 6 | Best path |
|---|---|---|---|---|
| 0 | 0.0800 | 0.1200 | 0.0950 | Path 5 |
| 0.3 | 0.0760 | 0.0960 | 0.0920 | Path 5 |
| 0.5 | 0.0710 | 0.0820 | 0.0850 | Path 6 |
| 0.7 | 0.0660 | 0.0700 | 0.0790 | Path 6 |
| 1.0 | 0.0580 | 0.0620 | 0.0730 | Path 6 |
From Table 5, when θ is small, the electric vehicle user has a high risk preference coefficient and tends to be risk seeking. In this situation, path 5 has the highest comprehensive prospect value because its gains under good and average states are attractive. When θ becomes larger than 0.5, the risk preference coefficient decreases, and the electric vehicle user gradually becomes risk neutral. Path 6, which has a more stable performance under different traffic states, becomes the most favorable alternative. This result confirms that the diversity of risk preferences among electric vehicle users has a significant influence on route-choice behavior.
4.3 Influence of risk preference on electric vehicle charging demand
I now analyze the electric vehicle charging demand in the whole Nguyen-Dupuis region. The total travel demand Dt is set according to a typical commuting pattern, with peaks in the morning and evening. The fixed risk preference coefficient α = 0.88 is compared with four variable-coefficient cases: θ = 0, 0.5, 0.7 and 1.0. The charging demand curves are obtained by Equations (19)–(21).
The simulation results show that the fluctuation amplitude of the electric vehicle charging demand curve is strongly related to the risk preference coefficient. When the travel-scale parameter θ is small, the electric vehicle user is more sensitive to potential gains and losses, so a small change in traffic conditions may induce a sudden switch of route choice. This behavior amplifies the fluctuation of charging demand in certain periods. As θ increases, the risk preference coefficient becomes smaller, and the electric vehicle user behaves more calmly. The charging demand curve becomes smoother. The fixed-coefficient case α = 0.88 produces a fluctuation amplitude that lies between the results of θ = 0.5 and θ = 0.7. This demonstrates that using a fixed risk preference coefficient may either underestimate or overestimate the volatility of electric vehicle charging demand, depending on the actual distribution of user risk attitudes in the region.
4.4 Influence of attribute weights on charging demand
Different electric vehicle users may assign different importance to travel time, congestion rate and comfort. To examine the sensitivity of the proposed model, I construct four weight scenarios. Scenario 1 uses ω = (0.2, 0.5, 0.3), which is the base case in the previous subsections. Scenario 2 fixes the weight of the first attribute and increases the weight of the second attribute: ω = (0.2, 0.6, 0.2). Scenario 3 changes the first and third attributes: ω = (0.1, 0.5, 0.4). Scenario 4 uses ω = (0.3, 0.4, 0.3). Table 6 gives the resulting positive-prospect paths for the three origin-destination pairs.
| Scenario | Weight vector | N1→N3 | N4→N3 | N12→N3 |
|---|---|---|---|---|
| 1 | (0.2, 0.5, 0.3) | Paths 2,5,6 | Paths 3,4,5 | Path 2 |
| 2 | (0.2, 0.6, 0.2) | Paths 2,5,6 | Paths 3,5 | Path 2 |
| 3 | (0.1, 0.5, 0.4) | Paths 2,5,6 | Path 5 | Path 2 |
| 4 | (0.3, 0.4, 0.3) | Path 2 | Paths 3,5 | Path 2 |
Table 6 shows that the attribute weights directly alter the set of paths that an electric vehicle user considers acceptable. For example, when the weight of the comfort attribute increases from 0.3 to 0.4 in Scenario 3, path 5 becomes the only positive-prospect path for the N4→N3 pair. When the weight of congestion rate is elevated in Scenarios 2 and 4, paths with low congestion but possibly longer distance become more attractive. Consequently, the total electric vehicle charging demand curve is shifted upward or downward, but the fluctuation amplitude in the 24-hour period remains almost unchanged because the risk preference parameter is unchanged. This result is consistent with the theoretical expectation that attribute weights affect the route level, while the risk preference parameter affects the route-switching aggressiveness and thus the volatility of charging demand.
4.5 Discussion of modeling implications
The numerical study demonstrates that the proposed variable-coefficient prospect-theoretic approach is more appropriate than the fixed-coefficient approach for regional electric vehicle charging demand analysis. The main reason is that an electric vehicle has a limited driving range and needs charging facilities. Therefore, an electric vehicle user is more sensitive to travel time uncertainty and congestion than a conventional vehicle driver. The charging demand curve obtained from the proposed model contains meaningful fluctuations caused by user bounded rationality. These fluctuations matter for the power system because the system operator needs to arrange reserve capacity in advance. If a conventional fixed-coefficient prospect model is used, the system operator may under-estimate the peak charging demand and fail to keep the grid in a secure operating state.
From the perspective of an electric vehicle aggregator, the proposed model provides a more reliable estimation of the daily charging demand. The aggregator can use this estimation to bid in the day-ahead electricity market. If the aggregator knows the distribution of the risk preference coefficient among local electric vehicle users, it can further adjust the bidding strategy to avoid the penalty caused by imbalanced settlement. In addition, the charging station planner can identify the paths with high positive comprehensive prospect values and place fast charging stations near the downstream nodes of those paths. This is particularly useful for the integration of electric vehicles in urban areas where the traffic network and the power distribution network are closely coupled.
5. Conclusion
In this article, I have developed an electric vehicle charging demand analysis framework that explicitly considers the bounded rational behavior of electric vehicle users. The key findings can be summarized as follows.
First, by combining prospect theory with a risk-type multi-attribute decision-making method, the proposed model calculates the comprehensive prospect value of each candidate path. The comprehensive prospect value reflects not only the effects of travel time, congestion rate and comfort level, but also the risk attitude of the electric vehicle user. Therefore, the route selection result is more consistent with actual electric vehicle travel behavior than conventional utility-based models.
Second, the risk preference coefficient in prospect theory should not be set as a fixed constant. My variable-coefficient model links the risk preference coefficient to the reference point of each attribute. When the reference point changes, the risk preference changes accordingly. This treatment is supported by the numerical results on the Nguyen-Dupuis network, where an increase in the travel-scale parameter shifts the best path from path 5 to path 6. This means that a bounded rational electric vehicle user changes the route-choice strategy when the perception of risk changes.
Third, the regional electric vehicle charging demand is affected by the bounded rationality of users in two ways. On the one hand, the fluctuation amplitude of the charging demand curve is mainly determined by the risk preference coefficient. A higher risk sensitivity produces stronger fluctuations. On the other hand, the overall level of charging demand is affected by the attribute weights of travel time, congestion rate and comfort. A higher weight on an attribute that favors low-energy paths will reduce the total charging energy, while a higher weight on an attribute that favors shortcut paths may increase the total charging energy. The combined effect of these two mechanisms can be captured quantitatively by the model proposed in this article, which provides valuable support for charging infrastructure planning and secure operation of the power system.
Future research can extend the proposed model in several directions. First, the risk preference coefficient may be related to user characteristics such as age, gender, travel purpose and the availability of public charging facilities. Second, the route-choice behavior can be integrated into a bi-level optimization model for determining the optimal locations of electric vehicle charging stations. Third, the charging demand model can be extended to a multi-period framework that considers the spatial transfer of electric vehicles after completing one trip and before starting the next trip. These extensions would further improve the accuracy of electric vehicle charging demand analysis under bounded rational behavior.