The rapid deployment of electric cars has changed the traditional coupling between transportation systems and power systems. In my research, I pay special attention to the fact that electric-car users are not perfectly rational decision-makers. Their travel behaviour is affected by experience, habit, perceived risk, and heterogeneous attitudes towards uncertainty. These behavioural features are usually described by the concept of bounded rationality. When the scale of electric cars becomes large, the fluctuation of their charging demand caused by bounded-rational travel choices may threaten the secure and economical operation of power grids. Therefore, my work aims at establishing a complete analytical chain from behavioural route choice to spatio-temporal charging demand, so that the resulting charging-demand information can be better used by distribution system operators, charging facility planners, and electric-car aggregators.
Prospect theory, which was developed from psychological experiments, is one of the most powerful frameworks for modelling bounded rationality. In my model, I adopt the risk-based multi-attribute decision-making method under prospect theory and combine it with the route-choice behaviour of electric-car users. The novelty of my study is not only the application of prospect theory to electric-car charging demand analysis, but also the treatment of the risk-preference coefficient as a variable rather than a fixed constant. Most previous studies set the risk-preference coefficient to the canonical value of \(0.88\). I argue that this setting cannot represent the diversity of electric-car users. In reality, a user whose reference point is high may react less sensitively to a certain gain or loss, and therefore the risk-preference coefficient should be related to the selected reference point. On the basis of this idea, I construct a variable-coefficient risk-based multi-attribute electric-car travel decision model. The model considers three types of uncertain influencing factors; their attribute values are expressed by interval numbers, crisp numbers, and triangular fuzzy numbers. Then, I calculate the prospect value of each alternative route and obtain a route-choice scheme based on the comprehensive prospect value.
After the travel decision model is established, I further analyse the impact of bounded-rational behaviour on the daily charging demand of electric cars. A traffic-flow updating process based on the method of successive averages is embedded into the framework. Each iterative step allocates the electric-car flow among the candidate routes according to the current comprehensive prospect values. The travel time of each road segment is updated by the classical Bureau of Public Roads function, which reflects the congestion effect of traffic flow. When the iterative traffic-flow differences become sufficiently small, the network reaches a behavioural equilibrium. Then, the obtained route-flow distribution is converted into electric-car charging demand with a simple battery model. The final charging-demand model therefore incorporates both travel uncertainty and users’ risk attitudes in a unified analytical framework. I validate the proposed model in a benchmark transport network composed of directed links and multiple origin-destination pairs. The simulation results show that the bounded-rational behaviour of electric-car users has a clear influence on the shape, peak level, and fluctuation of the daily charging-demand curve.
Uncertainty Factors in Electric-Car Travel Decisions
An electric-car trip is a multi-stage process. The user must choose not only the destination and departure time, but also the route from the origin to the destination. In an uncertain traffic environment, many subjective and objective factors affect the route-choice result, such as congestion, weather, travel time, travel cost, comfort, and the availability of charging facilities. These factors conflict with each other; it is usually impossible to optimize all of them simultaneously. Therefore, the electric-car user must evaluate each possible route in a multi-attribute manner and choose the route with the highest perceived utility. My model considers three different types of attributes that appear in practical travel choices:
- the first group of attributes is described by interval-valued quantities, such as the congestion level observed from a navigation system;
- the second group is represented by crisp numbers, such as the travel time of a route;
- the third group is evaluated by linguistic variables, such as the convenience and comfort perceived by the user, and is converted into triangular fuzzy numbers.
For each route, I suppose that the actual value of the \(j\)-th attribute can be one of three states. These states represent good, medium, and poor conditions of the traffic environment. The user has an expectation, or reference point, for each attribute in each state. The difference between the actual attribute value and the reference point determines whether the user perceives a gain or a loss. In my decision model, the first two attributes are cost-type attributes, meaning that smaller values are better. The third attribute is benefit-type, meaning that larger values are better. Table 1 and Table 2 summarize the attribute types and information representations used in my model.
| Category | Example | Attribute type | Representation |
|---|---|---|---|
| C1 | Congestion level | Cost type | Interval number |
| C2 | Travel time | Cost type | Crisp number |
| C3 | Comfort and convenience | Benefit type | Triangular fuzzy number |
| Linguistic variable | Triangular fuzzy number |
|---|---|
| Very low | (1, 2, 3) |
| Low | (2, 3, 4) |
| Moderately low | (3, 4, 5) |
| Medium | (4, 5, 6) |
| Moderately high | (5, 6, 7) |
| High | (6, 7, 8) |
| Very high | (7, 8, 9) |
Gain and Loss Functions under Risk
Let the decision scheme set of an electric-car user contain \(M\) routes. The alternative set is written as \(A = \{A_1, A_2, \ldots, A_M\}\). For the \(j\)-th attribute, the reference point vector is \(r_j = (r_{1j}, r_{2j}, r_{3j})\), where \(r_{hj}\) is the user expectation in state \(S_h\). The result of route \(A_i\) with respect to attribute \(C_j\) in state \(S_h\) is denoted by \(x^h_{ij}\). In the following paragraphs, I provide the calculation formulas for the gains and losses of the three attribute types.
Attribute represented by an interval number
For the first attribute, I use an interval number \(x^h_{i1} = [x^{hl}_{i1}, x^{hu}_{i1}]\), where \(x^{hl}_{i1} < x^{hu}_{i1}\). The actual attribute value \(x\) is assumed to follow a normal distribution in the interval, with expectation \(\mu^h_{i1} = (x^{hl}_{i1}+x^{hu}_{i1})/2\) and variance parameter \(\sigma^h_{i1} = (x^{hu}_{i1}-x^{hl}_{i1})/6\). The probability density function is:
$$
f^h_{i1}(x) = \frac{1}{\sqrt{2\pi}\sigma^h_{i1}} \exp\Bigl(-\frac{(x-\mu^h_{i1})^2}{2(\sigma^h_{i1})^2}\Bigr).
$$
The gain of attribute \(C_1\), represented by interval values relative to the reference point \(r_{h1}\), is:
$$
G^h_{i1} =
\begin{cases}
0, & x^{hl}_{i1} \ge r_{h1}, \\[4pt]
\displaystyle\int_{x^{hl}_{i1}}^{x^{hu}_{i1}} (r_{h1}-x) f^h_{i1}(x)\, dx, & x^{hu}_{i1} \le r_{h1}, \\[6pt]
\displaystyle\int_{x^{hl}_{i1}}^{r_{h1}} (r_{h1}-x) f^h_{i1}(x)\, dx, & x^{hl}_{i1} < r_{h1} < x^{hu}_{i1}.
\end{cases}
$$
The corresponding loss is:
$$
L^h_{i1} =
\begin{cases}
\displaystyle\int_{x^{hl}_{i1}}^{x^{hu}_{i1}} (r_{h1}-x) f^h_{i1}(x)\, dx, & x^{hl}_{i1} \ge r_{h1}, \\[6pt]
0, & x^{hu}_{i1} \le r_{h1}, \\[4pt]
\displaystyle\int_{r_{h1}}^{x^{hu}_{i1}} (r_{h1}-x) f^h_{i1}(x)\, dx, & x^{hl}_{i1} < r_{h1} < x^{hu}_{i1}.
\end{cases}
$$
Because the first attribute is cost-type, an actual value larger than the reference point implies a worse outcome. The above formulas naturally generate positive gains when the actual outcome is better than the expectation, and negative losses when the outcome is worse.
Attribute represented by a crisp number
For the second attribute, the attribute value \(x^h_{i2}\) is a crisp number. Two positional relations with the reference point are possible: \(x^h_{i2} \ge r_{h2}\) or \(x^h_{i2} < r_{h2}\). The gain is:
$$
G^h_{i2} =
\begin{cases}
0, & x^h_{i2} \ge r_{h2}, \\
r_{h2} – x^h_{i2}, & x^h_{i2} < r_{h2}.
\end{cases}
$$
The loss is:
$$
L^h_{i2} =
\begin{cases}
r_{h2} – x^h_{i2}, & x^h_{i2} \ge r_{h2}, \\
0, & x^h_{i2} < r_{h2}.
\end{cases}
$$
For the second attribute, which is also cost-type, a gain is obtained only if the actual travel time or congestion measure is strictly smaller than the reference value. If the actual value is larger than the reference, the user suffers a loss.
Attribute represented by a triangular fuzzy number
For the third attribute, the linguistic evaluation is converted into a triangular fuzzy number \(x^h_{i3} = (a^h_{i3}, b^h_{i3}, c^h_{i3})\), with \(a^h_{i3} \le b^h_{i3} \le c^h_{i3}\). Its membership function is:
$$
\varphi^h_{i3}(x) =
\begin{cases}
0, & x < a^h_{i3}, \\[4pt]
(x-a^h_{i3})/(b^h_{i3}-a^h_{i3}), & a^h_{i3} \le x \le b^h_{i3}, \\[4pt]
(c^h_{i3}-x)/(c^h_{i3}-b^h_{i3}), & b^h_{i3} \le x \le c^h_{i3}, \\[4pt]
0, & x > c^h_{i3}.
\end{cases}
$$
Three positional relations with the reference point \(r_{h3}\) are possible. The gain of the benefit-type attribute \(C_3\) is:
$$
G^h_{i3} =
\begin{cases}
\displaystyle\int_{a^h_{i3}}^{c^h_{i3}} (x-r_{h3}) \varphi^h_{i3}(x)\, dx, & a^h_{i3} \ge r_{h3}, \\[6pt]
0, & c^h_{i3} \le r_{h3}, \\[6pt]
\displaystyle\int_{r_{h3}}^{c^h_{i3}} (x-r_{h3}) \varphi^h_{i3}(x)\, dx, & a^h_{i3} < r_{h3} < c^h_{i3}.
\end{cases}
$$
The loss is:
$$
L^h_{i3} =
\begin{cases}
0, & a^h_{i3} \ge r_{h3}, \\[4pt]
\displaystyle\int_{a^h_{i3}}^{c^h_{i3}} (x-r_{h3}) \varphi^h_{i3}(x)\, dx, & c^h_{i3} \le r_{h3}, \\[6pt]
\displaystyle\int_{a^h_{i3}}^{r_{h3}} (x-r_{h3}) \varphi^h_{i3}(x)\, dx, & a^h_{i3} < r_{h3} < c^h_{i3}.
\end{cases}
$$
In this way, for every route and every state, I can obtain a risky gain matrix \(G^h = [G^h_{ij}]_{M \times 3}\) and a risky loss matrix \(L^h = [L^h_{ij}]_{M \times 3}\). The gain terms are non-negative, while the loss terms are non-positive. These matrices form the basis for computing the value function in the prospect-theory framework.
Variable-Coefficient Risk Preference and Prospect Values
Prospect theory distinguishes between the value function and the probability weighting function. The value function captures the asymmetric perception of gains and losses. The probability weighting function captures the tendency of humans to overweight small probabilities and underweight moderate or large probabilities. In my model, the value and probability weighting functions are:
$$
V(+)^h_{ij} = \bigl(G^h_{ij}\bigr)^{\alpha},
\qquad
V(-)^h_{ij} = -\lambda \bigl(-L^h_{ij}\bigr)^{\alpha},
$$
and
$$
\pi(+)^h_{ij} =
\frac{(p^h)^{\zeta}}{\Bigl((p^h)^{\zeta} + (1-p^h)^{\zeta}\Bigr)^{1/\zeta}},
\qquad
\pi(-)^h_{ij} =
\frac{(p^h)^{\delta}}{\Bigl((p^h)^{\delta} + (1-p^h)^{\delta}\Bigr)^{1/\delta}}.
$$
Here \(p^h\) is the probability of state \(S_h\). The parameter \(\alpha \in (0,1]\) describes the risk sensitivity of the user. A larger \(\alpha\) means that the user is more sensitive to risk. The parameter \(\lambda \ge 1\) is the loss-aversion coefficient. The parameters \(\zeta, \delta \in (0,1)\) determine the curvature of the probability weighting functions. In the conventional implementation, the parameters are usually fixed as \(\alpha=0.88\), \(\lambda=2.25\), \(\zeta=0.61\), and \(\delta=0.69\). I first adopt these conventional values as a baseline case. However, I also emphasize that a fixed coefficient cannot capture the heterogeneity of electric-car users.
In practice, the user’s risk preference is related to the selected reference point. If the reference point is high, then a user with the same gain or loss is less sensitive to the change of outcome. The user becomes more conservative. I therefore extend the risk-preference coefficient as an endogenous variable. The novel variable-risk-preference formulation is:
$$
\alpha^h_j = \Bigl(1 – \frac{r^h_j}{\sum_{h=1}^{3} r^h_j}\Bigr)^{\theta}.
$$
Here \(\theta\) is an investigation-scale parameter satisfying \(0 \le \theta \le 1\). With the increase of \(\theta\), the user gradually moves from risk-seeking behaviour toward risk-neutral behaviour. The value functions become:
$$
V(+)^h_{ij*} = \bigl(G^h_{ij}\bigr)^{\alpha^h_j},
\qquad
V(-)^h_{ij*} = -\lambda \bigl(-L^h_{ij}\bigr)^{\alpha^h_j}.
$$
Then the prospect value of route \(A_i\) with respect to attribute \(C_j\) is obtained by combining gains and losses with their probability weights:
$$
V_{ij} =
\sum_{h=1}^{3} V(+)^h_{ij*} \, \pi(+)^h_{ij}
+
\sum_{h=1}^{3} V(-)^h_{ij*} \, \pi(-)^h_{ij}.
$$
Because the three attributes have different scales, the prospect values must be normalized before aggregation. The normalization is:
$$
V^{\max}_j = \max_{i \in M} \bigl\{ |V_{ij}| \bigr\},
\qquad
V^*_{ij} = \frac{V_{ij}}{V^{\max}_j}.
$$
The comprehensive prospect value of route \(A_i\) is then:
$$
U_i = \sum_{j=1}^{3} \omega_j V^*_{ij},
$$
where \(\omega_j\) is the weight of attribute \(C_j\), satisfying \(\omega_j \ge 0\) and \(\sum_{j=1}^{3} \omega_j = 1\). I regard \(U_i\) as the final utility index for the route-choice decision. Routes with larger positive comprehensive prospect values are more attractive to bounded-rational electric-car users. Table 3 lists the key notations of the model for easy reference.
| Symbol | Meaning |
|---|---|
| \(M\) | Number of candidate routes |
| \(C_j\) | The \(j\)-th attribute |
| \(S_h\) | The \(h\)-th traffic state |
| \(r_{hj}\) | Reference point of attribute \(C_j\) in state \(S_h\) |
| \(x^h_{ij}\) | Actual attribute value of route \(A_i\) |
| \(G^h_{ij}\), \(L^h_{ij}\) | Gain and loss matrices |
| \(\alpha^h_j\) | Variable risk-preference coefficient |
| \(\omega_j\) | Weight of attribute \(C_j\) |
| \(U_i\) | Comprehensive prospect value |
Traffic-Flow Assignment and Charging-Demand Evaluation
The route-choice model presented above determines the route attractiveness from the perspective of an individual user. In a real traffic network, however, the route attractiveness depends on traffic flow. If too many electric-car users choose the same route, congestion becomes severe and the travel time increases. As a result, the comprehensive prospect value of that route decreases. A stable flow pattern cannot be obtained by a single assignment process. I therefore use the method of successive averages to update the route flows iteratively.
Suppose that the study region contains \(N\) origin-destination (OD) pairs. For OD pair \(k\), there are \(M_k\) candidate paths. Let \(x^s_a\) be the flow on link \(a\) at iteration \(s\). The traffic-flow updating rule is:
$$
x^s_a = \Bigl(1 – \frac{1}{s}\Bigr) x^{s-1}_a + \frac{1}{s} F^s_a,
$$
where \(F^s_a\) is the auxiliary flow assigned to link \(a\) in iteration \(s\). The auxiliary flow is obtained by assigning all electric-car users to the route with the largest comprehensive prospect value. The procedure stops when the difference between two consecutive flow patterns becomes smaller than a pre-specified tolerance.
The travel time of each link is updated according to the Bureau of Public Roads function. In my model, the travel time of link \(a\) in iteration \(s\) is:
$$
T^s_a = t_{0a} \Bigl[1 + 0.15 \Bigl(\frac{x^s_a}{C_a}\Bigr)^4\Bigr],
$$
where \(C_a\) is the capacity of link \(a\), and \(t_{0a}\) is the free-flow travel time. The corresponding travel speed in the prediction update is:
$$
v_a = \frac{C_a l_a}{1000},
$$
where \(l_a\) is the length of link \(a\) in kilometers. These speed and time values are then transformed into the attribute values that feed into the risky multi-attribute decision model. In this way, the decision model and the traffic-flow assessment are fully coupled. Table 4 gives the iterative calculation procedure.
| Step | Action |
|---|---|
| 1 | Initialize all link flows and compute free-flow travel times; |
| 2 | For each OD pair, calculate the three attribute values of each route; |
| 3 | Build gain and loss matrices with respect to each reference point; |
| 4 | Compute the variable risk-preference coefficients and prospect values; |
| 5 | Obtain the comprehensive prospect value \(U_i\) of each route; |
| 6 | Assign the OD traffic to the route with the highest positive prospect value; |
| 7 | Update link flows by the method of successive averages; |
| 8 | Check convergence; if not satisfied, return to Step 2; |
| 9 | Convert the equilibrium flow into the time-varying electric-car charging demand. |
After the traffic-flow assignment converges, the charging demand model can be evaluated. For electric-car \(v\) travelling on route \(i\) of OD pair \(k\) in time interval \(t\), the required charging time is:
$$
T_{v,i,t} = \frac{\bigl(SOC_{v1,i,t} – SOC_{v0,i,t}\bigr) C_v}{P_v},
$$
where \(SOC_{v0,i,t}\) is the remaining battery energy before charging, \(SOC_{v1,i,t}\) is the battery energy after charging, \(C_v\) is the battery capacity, and \(P_v\) is the charging power.
I also consider the range anxiety of electric-car users. The charged energy must guarantee that the user can complete the remaining planned driving distance and keep an anxiety margin \(d_v\). Therefore, the post-charging energy level should satisfy:
$$
SOC_{v1,i,t} – d_{k,i} q_v \ge d_v q_v,
$$
where \(d_{k,i}\) is the distance of path \(i\), and \(q_v\) is the energy consumption rate per kilometer of the electric car. The energy demand for charging is:
$$
SOC_{v2,i,t} = SOC_{v1,i,t} – SOC_{v0,i,t}.
$$
Finally, the total charging demand in the study region at time interval \(t\) is:
$$
Q_t = \sum_{k=1}^{N} \Bigl[p_{k,t} D_t \sum_{i=1}^{M_k} \bigl(u_{k,i,t} SOC_{v2,i,t}\bigr)\Bigr].
$$
In the above expression, \(p_{k,t}\) is the fraction of electric cars travelling from OD pair \(k\) at time \(t\), and \(D_t\) is the total number of electric cars travelling in the region at time \(t\). The variable \(u_{k,i,t}\) is the selection probability of route \(i\) in OD pair \(k\). In my model, the selection probability is determined by the comprehensive prospect value. Routes with negative comprehensive prospect values are usually not selected by users; only routes whose values are greater than zero are considered as attractive choices.
Simulation Set-Up of the Benchmark Network
To illustrate the effectiveness of my model, I simulate the morning commuting behaviour in a benchmark transport network. The network consists of 13 nodes and several directed road segments. I select two types of nodes: residential origin nodes and workplace destination nodes. The user population is divided into three OD pairs. In the baseline case, all electric cars depart from home with a fully charged battery. The battery capacity of an electric car is set to 24 kWh, the energy consumption is 30 kWh per 100 kilometers, and the user’s anxiety distance is 20 kilometers. The multi-attribute weights are initially chosen as \(\omega = (0.2, 0.5, 0.3)^T\), meaning that more attention is paid to travel time than to congestion and comfort. The travel routes are summarized in Table 5.
| Route index | Origin 1 to destination 3 | Origin 4 to destination 3 | Origin 12 to destination 3 |
|---|---|---|---|
| 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 | — | — |
I first validate the route-choice model on the six routes from origin 1 to destination 3. For each route, I calculate the reference values of the three attributes. Then I obtain the risky gain and loss matrices. The gain matrix shows that Route 2 obtains relatively large gains because its actual congestion-related attribute is lower than the reference value in many states. Route 6 also has a high benefit from comfort and convenience. The loss matrix shows that Route 3 has the largest loss because it performs worse for both congestion and comfort. These behavioural tendencies are consistent with the expectation of prospect theory: users evaluate an alternative not only by its absolute performance but also by whether the performance is better or worse than the reference point.
After computing the normalized prospect values and the comprehensive prospect values, I obtain the baseline results shown in Table 6. The conventional fixed risk coefficient \(\alpha=0.88\) is used in this baseline validation.
| Route | Comprehensive prospect value | Ranking |
|---|---|---|
| Route 1 | -0.7486 | 5 |
| Route 2 | 0.0729 | 3 |
| Route 3 | -1.0000 | 6 |
| Route 4 | -0.2004 | 4 |
| Route 5 | 0.0733 | 2 |
| Route 6 | 0.0874 | 1 |
In the baseline case, the route-choice ranking is Route 6, Route 5, Route 2, Route 4, Route 1, and Route 3. Route 6 is the most attractive route because it offers the best balance between the uncertain congestion attribute, the travel-time attribute, and the comfort attribute. Route 2 and Route 5 are close to Route 6 in terms of prospective utility. The positive comprehensive prospect values confirm that these routes are acceptable choices for bounded-rational electric-car users. Negative values indicate that the corresponding routes are dominated by other alternatives under the given reference points and attribute weights.
Effect of Variable Risk Preference on Route Choice
I next analyse the effect of the variable risk-preference coefficient on the route-choice result. The parameter \(\theta\) is sampled from 0 to 1. When \(\theta=0\), the coefficient \(\alpha^h_j\) is identically equal to 1; the user is fully sensitive to all gains and losses. As \(\theta\) increases, \(\alpha^h_j\) decreases and the user becomes less risk-seeking. I calculate the new comprehensive prospect values for the six routes from origin 1 to destination 3. The result shows that the ordering of routes changes with \(\theta\). In the low-\(\theta\) region, the optimal route is Route 5. When \(\theta\) exceeds about 0.5, Route 6 overtakes Route 5 and becomes the optimal route. This occurs because Route 5 has a relatively large dispersion of possible outcomes, whereas Route 6 provides a more stable performance across the three attributes. A risk-seeking user is willing to accept a volatile outcome in exchange for a possibly high gain. A less risk-seeking user prefers the stable route. Table 7 provides a qualitative overview of the effect of \(\theta\) on the most preferred route.
| Parameter \(\theta\) | Risk attitude | Most preferred route |
|---|---|---|
| \(\theta \le 0.4\) | Risk-seeking | Route 5 |
| \(\theta\) near 0.5 | Transitional | Route 5 and Route 6 are comparable |
| \(\theta \ge 0.7\) | Near risk-neutral | Route 6 |
This observation is important for electric-car charging-demand analysis. If the planner uses the fixed risk coefficient \(\alpha=0.88\), the flow pattern may always be directed to one particular route. In reality, however, the distribution of risk preferences among electric-car users varies. The charging demand at different charging stations will therefore be different from the result obtained by a single fixed coefficient. My variable-coefficient model provides a more conservative and realistic prediction of route-flow diversity.
Impact of Risk Appetite on Daily Charging Demand
After the route-choice analysis, I simulate the multiple-OD network until the traffic-flow assignment reaches convergence. I choose four values of \(\theta\), namely \(\theta=0\), \(\theta=0.5\), \(\theta=0.7\), and \(\theta=1.0\), and compare them with the conventional case \(\alpha=0.88\). The daily charging-demand curves under different risk appetites show a clear peak in the morning commuting period. In the low-\(\theta\) case, many electric-car users choose a single optimal route with a high prospect value. This concentrated behaviour produces a sharp charging-demand peak. With the increase of \(\theta\), the risk-seeing tendency decreases, and users become more willing to choose neutral and stable routes. The route-flow distribution becomes more balanced. Consequently, the fluctuation amplitude of the daily charging-demand curve is reduced. Table 8 lists the relative change in the charging-demand peak characteristics.
| Parameter setting | Peak-demand level | Fluctuation tendency |
|---|---|---|
| \(\theta = 0\) | Highest | Strong fluctuation |
| \(\theta = 0.5\) | Moderately lower | Moderate fluctuation |
| \(\theta = 0.7\) | Lower | Weaker fluctuation |
| \(\theta = 1.0\) | Lowest | Weakest fluctuation |
| Fixed \(\alpha = 0.88\) | Intermediate | Intermediate fluctuation |
The comparison reveals that the risk appetite of electric-car users not only affects their route-choice behaviour but also affects the amplitude of charging demand. If risk-seeking behaviour dominates, the charging peaks become steeper and more concentrated. If risk-neutral behaviour dominates, the charging demand is smoother. This result has direct implications for grid operation. A charging-demand curve with strong fluctuation requires more reserve capacity and may increase the operation risk of the distribution system. Through the behavioural analysis of electric-car users, my model enables system operators to anticipate whether the charging peak will be sharp or smooth under different user compositions.
Effect of Attribute Weights on Charging Demand
In practice, different electric-car users may assign different importance to congestion, travel time, and comfort. To study the influence of attribute weights, I define four simulation scenarios. Scenario 1 uses the baseline weights \(\omega=(0.2,0.5,0.3)^T\). Scenario 2 increases the weight of travel time and decreases the weight of comfort while keeping the congestion weight unchanged: \(\omega=(0.2,0.6,0.2)^T\). Scenario 3 increases the comfort weight and reduces the congestion weight: \(\omega=(0.1,0.5,0.4)^T\). Scenario 4 uses a more balanced setting: \(\omega=(0.3,0.4,0.3)^T\).
I first compare the comprehensive prospect values of the six routes from origin 1 to destination 3. In Scenario 1, Routes 2, 5, and 6 have positive comprehensive prospect values. Increasing the weight of travel time, as in Scenario 2, is favourable to Route 5 but unfavourable to Routes 2 and 6. This is because Route 5 has the best travel-time attribute and Route 6 has a relatively weaker travel-time attribute compared with its comfort attribute. When the weight of comfort is increased in Scenario 3, Route 2 gains more advantage, while Route 5 loses some attractiveness. In Scenario 4, only Route 2 remains positive for this OD pair. For origin 4 to destination 3, Scenario 1 selects Routes 3, 4, and 5. Scenario 2 and Scenario 4 select Routes 3 and 5. Scenario 3 selects only Route 5. For origin 12 to destination 3, Route 2 is the only positive route in all four scenarios. Table 9 summarizes the positive-prospect routes under the four attribute-weight scenarios.
| Scenario | Attribute weight vector | Positive routes for OD 1-3 | Positive routes for OD 4-3 | Positive routes for OD 12-3 |
|---|---|---|---|---|
| 1 | (0.2, 0.5, 0.3) | Routes 2, 5, 6 | Routes 3, 4, 5 | Route 2 |
| 2 | (0.2, 0.6, 0.2) | Routes 2, 4, 5 | Routes 3, 5 | Route 2 |
| 3 | (0.1, 0.5, 0.4) | Routes 2, 4, 6 | Route 5 | Route 2 |
| 4 | (0.3, 0.4, 0.3) | Route 2 | Routes 3, 5 | Route 2 |
The daily charging-demand curves under the four scenarios show that the overall height of the curve varies with the change in attribute weights. The reason is intuitive: if users place a greater importance on an attribute whose favorable routes are spatially concentrated, then more electric cars are concentrated in the same set of charging stations and the charging demand becomes higher. Conversely, if users care more about an attribute that leads to dispersed route choices, the charging demand is spread across the network. In my simulation, different attribute weights change the magnitude of the demand peak, while the timing of the peak remains similar. This indicates that travel-uncertainty factors affect the location and intensity of charging demand, while the time pattern is largely governed by the trip-purposed departure distribution.
Discussion on the Charging-Demand Mechanism
The simulation results consistently show that the bounded rationality of electric-car users has two important consequences. The first consequence is heterogeneity in route choice under the same traffic condition. Because users have different risk preferences and reference points, even identical electric cars may choose completely different routes. The second consequence is the dynamic reallocation of traffic flow. As route flows change, travel times and congestion levels change accordingly, which feeds back into the prospect values of all routes. This feedback mechanism is crucial for charging-demand analysis. A static trip assignment would fail to capture such strategic interactions among electric-car users.
My model includes three types of uncertainty in the multi-attribute decision process: interval-valued congestion, crisp travel time, and fuzzy comfort. The interval-valued attribute captures the uncertainty caused by stochastic traffic conditions. The crisp attribute supplies an objective measure of time cost. The fuzzy attribute describes subjective perceptions that are difficult to quantify precisely. Such a hybrid attribute representation is closer to the real decision process of electric-car users than the single-attribute travel-time-based assignment used in many conventional studies.
Another contribution of my model is the coupling between reference points and risk preferences. In the conventional prospect-theory model, the value-function exponent \(\alpha\) is often fixed at 0.88. However, the same absolute gain has different psychological significance for different users. If the user has a high reference point, the gain appears small and the user may be less willing to take risk. If the reference point is low, the same gain appears large, and the user may behave more aggressively. My variable-coefficient formula explicitly relates \(\alpha^h_j\) to the reference point of attribute \(C_j\). The parameter \(\theta\) controls the overall degree of this effect. This design makes the model more flexible and more realistic, especially when an electric-car aggregator or grid operator needs to classify users according to their behavioural characteristics.
Comparative Observations from the Case Study
To provide a clearer overview of my simulation observations, Table 10 summarizes the key effects of the model inputs on the outputs. The rows of the table represent three types of model parameters: risk-preference modulation, attribute weights, and traffic-state probabilities. The columns represent the route-choice result, the charging-demand peak, and the fluctuation of the demand curve.
| Model parameter | Effect on optimal route | Effect on peak demand | Effect on fluctuation |
|---|---|---|---|
| Risk-preference scale \(\theta\) | Optimal route may switch | Higher \(\theta\) reduces the peak | Higher \(\theta\) smooths the curve |
| Attribute weights \(\omega_j\) | Route ranking changes | Changes the demand level | Limited influence on fluctuation |
| Reference points \(r_{hj}\) | Determine gain or loss | Indirect influence | Strong influence via risk appetite |
The detailed numerical results indicate that the bounded-rational travel behaviour of electric-car users should not be ignored in the planning of public charging infrastructure. In a network where most electric-car users are risk-seeking, the charging-demand in a few popular charging stations may reach high values, causing transformer overload and serious voltage deviation. If the users become more risk-neutral, the same amount of electric-car energy demand is dispersed over a larger number of routes and stations. Therefore, demand forecasts based on perfect rationality may systematically overestimate the required capacity of some charging stations and underestimate the need in others.
Practical Implications for Electric-Car Aggregators
Electric-car aggregators often purchase energy in the wholesale market and sell it to users through charging stations. The imbalance between their predicted demand and actual demand leads to penalty costs. My behavioural charging-demand model can provide a more realistic forecast of time-varying charging load. By simulating different behavioural compositions of electric-car users, the aggregator can estimate the probability range of peak demand. If the share of risk-seeking users is high, the aggregator should reserve more flexible capacity in the morning peak. If the share of risk-averse users is high, the aggregator can make a more economical plan because the load curve is smoother.
I also note the impact of user uncertainty on charging-station utilization. In the baseline scenario, Route 6 and Route 5 are selected by many bounded-rational users. This may create an unexpected burden on charging stations located near the destination area, while stations along the less attractive routes remain underutilized. My model can identify which route in a multiple-OD network is likely to attract more electric-car flow under bounded rationality. Such information is useful for both the location planning of new charging facilities and the operation of existing facilities.
Extensions and Future Research Directions
Although my model has produced promising results, several limitations remain. First, the risk-preference coefficient is modelled as a function of the attribute reference point and a single scaling parameter \(\theta\). In reality, the risk preference of a user is also affected by income, age, trip purpose, and the state of charge of the electric car. A more complete model should classify users into groups with different behavioural parameters. Second, the current charging-demand model uses a simplified battery model. The state of charge at the beginning of a trip is assumed to be full in the morning. For an all-day simulation, the charging history and trip chains of electric-car users should be included. Third, my model uses a benchmark network with deterministic origin-destination demands. The integration of stochastic trip-generation models would improve the realism.
Another possible extension is the joint optimization of electric-car charging and routing. In the present work, I only consider the charging demand triggered by route choice. If the electric-car user has to charge during the trip, the charging station location becomes an additional decision dimension. A future model can combine the charging-station choice with the path choice under bounded rationality. The prospect value of a path then depends not only on congestion and comfort, but also on the expected waiting time at the charging station and the probability of finding an available charger.
Finally, the integration of information technology should be addressed. In the era of connected vehicles and smartphone-based navigation, electric-car users receive real-time travel information. The information may change their perceived reference points and risk attitudes. For example, if a user learns from a navigation application that the traffic condition is bad, the user may raise the reference of acceptable travel time. A future dynamic model should update the reference points on a day-to-day basis and study the evolution of charging demand over multiple days.
Conclusion
In my study, I have established a systematic framework for analyzing the charging demand of electric cars from the perspective of bounded rationality. The framework contains three essential components. The first component is a risk-based multi-attribute decision model based on prospect theory, which considers the joint influence of interval-valued congestion, crisp travel time, and fuzzy comfort. The second component is a variable risk-preference coefficient that relates the risk attitude of an electric-car user to the attribute reference point. The third component is a traffic-flow updating and charging-demand aggregation model that converts the route-choice behavior into the time-varying charging demand in the entire region.
Through the simulation in the benchmark transport network, I have reached the following conclusions:
- Bounded-rational behavior significantly affects the route selection of electric-car users. The route with the best conventional performance is not necessarily the route with the largest comprehensive prospect value. The reference point and the risk attitude of the user play decisive roles in the final choice.
- The risk-preference coefficient in prospect theory should not be treated as a constant. Its variation affects both the ranking and the positive-negative sign of the comprehensive prospect value. A change from risk-seeking to risk-neutral behavior can move the selected route from a volatile but potentially attractive path to a more stable path.
- The charging-demand curve in a region is influenced by both the risk appetite and the attribute weights of electric-car users. Risk appetite mainly changes the fluctuation amplitude of the charging demand, while attribute weights mainly change the overall height and spatial concentration of the demand.
- The proposed model provides useful information for electric-car aggregators and distribution network operators. By considering the finite rationality of users, the model improves the robustness of charging-demand forecasting and thereby helps the grid to operate more safely and economically.
I believe that the combination of behavioral economics and power-system analysis will become increasingly important in future low-carbon transportation systems. The bounded rationality of electric-car users is not a negligible disturbance; it is an intrinsic feature of human decision-making. My work provides one step toward a more realistic integration of electric-cars into transportation and power networks. Future research should continue to refine the behavioral model and extend it to larger city-scale networks with real-world travel data, so that the interaction between electric-car mobility and power-grid operation can be better understood.