Optimal lane expansion model for electric vehicle transportation networks with bounded rational travelers

Xu Xin School of Economics and Management, Tongji University https://orcid.org/0000-0002-0780-156X Xiaoli Wang School of Economics and Management, Tongji University Kang Chen School of Maritime Economics and Management, Dalian Maritime University Cui Li School of Mechanical Engineering, Tongji University Yanran Liu (  yanranliu@tongji.edu.cn ) School of Economics and Management, Tongji University


Introduction
As reported by the World Meteorological Organization (WMO), in the past 50 years, more than 11,000 disasters caused by weather, climate, and water have caused 2 million deaths and 3.6 trillion US dollars in economic losses, and nearly 22 million people have become climate refugees (World Meteorological 2021). Carbon emissions are becoming a major practical issue affecting human survival and development (Chu et al. 2021;Zhao et al. 2020). Social and economic development processes have simultaneously put considerable pressure on the natural environment. It should be noted that global carbon dioxide emissions will continue to grow with the future growth of the economy, population and resource demand. Therefore, mitigating greenhouse gas emissions is among the most urgent and far-reaching issues in the international political economy, and the sustainable development of high-energy-consumption and high-carbon-emission industries, such as the transportation industry, must be promoted.
In recent years, with the strict management and control of carbon emissions by the governments of various countries, the battery electric vehicle (BEV) manufacturing industry has attracted considerable research attention (Xu et al. 2021). Substituting BEVs for traditional gasoline vehicles (GVs) is an effective way to reduce greenhouse gas (GHG) emissions and the dependence of the automobile industry on petroleum (Hu et al. 2019;Waraich et al. 2013). Therefore, various countries have announced plans to achieve carbon neutrality by the middle of the 21 st century, and their governments have introduced policies to promote the BEV industry; consequently, the production of BEVs has increased rapidly. According to data released by Statistics Canada, the number of newly registered BEVs nationwide increased from 25,163 in 2014 to 69,010 in 2018, a growth rate of 174% (Lin et al. 2021). In China, the number of BEV and plug-in hybrid vehicle (HEV) registrations increased from 1,430 in 2010 to 579,000 in 2017 (Cheng et al. 2020). Thus, BEVs will be a major part of future urban transportation networks. Not surprisingly, BEV transportation network design has become the focus of both the government and academia because it promotes green transportation (Montoya et al. 2016;Schiffer and Walther 2017).
In a BEV transportation network, the behavior of travelers is often different from that in a traditional GV transportation network. On the one hand, range anxiety makes the feasible travel path of BEV drivers significantly different from that of GV drivers. Range anxiety reflects travelers' fear of running out of battery energy before reaching their destination (or charging station) (Xu et al. 2020;Yuan et al. 2018). Due to the range anxiety phenomenon, travelers are concerned about their battery state-ofcharge (SOC), and they will make prudent charging decisions (Cheng et al. 2020;Miwa et al. 2017). It is manifested in the fact that travelers often do not allow the SOC of BEVs to be lower than a certain level during travel (Neubauer and Wood 2014). Under the influence of range anxiety, BEV drivers will choose the most secure travel route that ensures an available power supply (Xu et al. 2017;Yang et al. 2016). This constraint allows us to establish a different definition of feasible paths in the design of BEV transportation networks.
On the other hand, traveler behaviors are often associated with bounded rationality. In the traditional transportation network design problem, it is assumed that travelers choose transportation routes based on decision rules for utility maximization; i.e., the traffic flow distribution in the transportation network obeys the perfect rational user equilibrium (PRUE) principle. However, this assumption has been challenged by numerous scholars (Ilin and Rogova 2017;KAI-INEMAN and Tversky 1979).
Researchers have found that when choosing a departure time (Mahmassani and Chang 1986;Schwanen and Ettema 2009) or a transportation route (Zhou et al. 2014), travelers' choices are often associated with bounded rationality. Under the influence of the above phenomenon, the traffic flow distribution in a transportation network then forms a bounded rational user equilibrium (BRUE) state: an equilibrium state in which travelers' adjustments to travel routes will not cause their travel time to exceed an established limit (Eikenbroek et al. 2018;Mahmassani and Chang 1987). From an economic perspective, bounded rationality is associated with people's habits, inertia (Lotan 1997;Samuelson and Zeckhauser 1988), and myopia (Bogers et al. 2005). With bounded rationality, the equilibrium state of a BEV transportation network can highly vary from that of a traditional GV transportation network.
Motivated by the two abovementioned behavioral factors, in-depth research on the design of BEV transportation networks is conducted in this paper. Our paper aims to investigate a BEV transportation Network Design Problem considering two human choice behaviors (BNDP) to help governments better develop lane expansion schemes for specific regions. Specifically, the equilibrium conditions of the transportation network are first developed to describe the BRUE state of the BEV network. Then, a network design model is constructed to optimize the government's optimal lane expansion scheme and to minimize the system travel time. Finally, to solve the abovementioned model, a heuristic algorithm is designed to identify the local optimal solution based on the framework of the active set algorithm (ASA). Moreover, a column generation algorithm (CGA) is embedded in the abovementioned algorithm to avoid the enumeration of paths.
The contributions of this paper are threefold. First, we define a new BEV transportation network design problem. To the best of our knowledge, our paper is the first to design a BEV transportation network considering both range anxiety and BRUE factors. Second, we establish the relevant equilibrium conditions for the BRUE-based transportation network and extend the theory of bounded rationality in economics to the field of BEV transportation network design. Third, we design a heuristic algorithm that explores the local optimal solution of the problem based on an ASA framework. The abovementioned algorithm can provide an efficient solution for problems at practical scales.
The remainder of this paper is organized as follows. Section 2 reviews the literature related to BEV transportation network equilibrium and network design to clarify the current research gaps. Section 3 presents the assumptions, problem descriptions, and mathematical expressions associated with the BRUE state of the BEV transportation network. Section 4 models the BEV transportation network design problem considering range anxiety and BRUE factors. The algorithm for solving the model is presented in Section 5. In Section 6, numerical experiments are performed to verify the effectiveness of the model and algorithm. Finally, Section 7 concludes the paper and proposes several potential research directions.

Literature review
Research on the design of BEV transportation networks is inseparable from the discussion of the equilibrium state of the networks. This is because the calculation of the equilibrium state of a transportation network is the basis for evaluating the design scheme of the network. The concept of transportation network equilibrium was first defined by Wardrop in 1952(Wardrop 1952. Wardrop (1952) assumed that all drivers choose their travel route based on the principle of the shortest travel time and proposed the famous Wardrop's first principle. In recent years, with the growth of BEV ownership, research on the equilibrium state of BEV transportation networks has expanded. In the early stage, He et al. (2014) first proposed a BEV transportation network equilibrium model considering range anxiety. The authors noted that the difference between the equilibrium state of a BEV transportation network and that of a GV network is associated with the phenomenon of range anxiety, and range anxiety seems unrealistic to eliminate. Based on the consideration of the above phenomenon, the authors constructed two network equilibrium models, one of which further considered flow-dependent energy consumption.
Considering the characteristics of wireless charging lanes, Chen et al. (2016) established a comparatively complicated BEV transportation network equilibrium model. The authors assumed that BEV drivers could choose to charge wirelessly in a charging lane. To ensure that a BEV has sufficient battery power, the driver can adjust the charge received by changing the travel speed of the vehicle. Xu et al. (2017) suggested that the conditions proposed by He et al. (2014) were incomplete, and they constructed a set of network equilibrium conditions. These conditions allowed both BEVs and GVs to exist in the studied transportation network. To reflect real-world conditions in the model, the authors further introduced the road grade effect. When a BEV is driving on roads of different grades, the dwell time and swapping cost will vary. However, the authors assumed that only swapping stations existed in the studied transportation network. This assumption limited the impact of dwell time on network equilibrium during the modeling process. Liu and Song (2018)  Based on the abovementioned studies, two research gaps are identified. First, the existing research has not broadly focused on network equilibrium or BEV transportation network design considering the bounded rationality of travelers; thus, studies of BRUE in BEV transportation networks have been inadequate. Second, algorithm development for network design models with bounded rationality has been limited, especially for BEV transportation networks. Notably, algorithms that can obtain local (or global) optimal solutions to problems have not been studied in detail. Therefore, to fill these research gaps, we first define the bounded rationality of travelers in BEV transportation networks and then give a mathematical expression of the BRUE state in these networks.
Next, a mathematical model is constructed to solve the transportation network design problem. Finally, considering the characteristics of the model, a heuristic algorithm is designed to obtain a solution. Overall, the above model and algorithm can provide necessary decision support for the government to rationally construct transportation infrastructure.

Assumptions and notations
To facilitate modeling, we introduce the basic assumptions (or considerations) for the BNDP.
(1) Behavior of BEV drivers: When travelers choose transportation routes, they will follow the principle of bounded rationality. Under the influence of the abovementioned principle, when the difference between the travel times along two paths is within a threshold value, these paths are considered alternative paths for travelers. The network equilibrium state based on this travel choice behavior is called the BRUE state.
(2) Charging time of BEVs: Unlike traditional GVs, BEVs require charging times that generally take several hours. Therefore, if a BEV needs to be charged during travel, the charging time needs to be added to the travel time. We assume that the BEV driver can charge their BEV at a charging station in the transportation network used for travel and that the charging time is proportional to the amount of battery that must be charged. In a BEV transportation network, each charging station has only one type of charging pile, and the number of charging piles at a charging station is assumed to be sufficient. In other words, we do not consider the case of BEV queuing at charging stations.
(3) BEV energy consumption and the range anxiety of travelers: Each BEV has a fixed battery storage capacity; thus, we assume that each BEV has the same initial state of charge (i.e., the SOC of the battery before the trip). Note that this assumption can be relaxed by introducing different types of BEVs. The battery power will decrease linearly as the transportation distance increases. Due to the existence of range anxiety, BEV drivers will not allow the SOC of the battery to fall below a predetermined value at any time.
(4) Link performance function: The function between the traffic flow and the transportation time along each link is called the performance function. We assume that in the BEV transportation network, the performance function for each link follows the form of the Bureau of Public Roads (BPR) function.
Moreover, the travel times on different links are additive.
(5) The government's decision: The government aims to establish a reasonable lane expansion scheme with the goal of minimizing the system travel time under a limited budget. We assume that the government will not construct additional links and will only expand lanes based on existing links. In addition, for modeling convenience, we assume that the government expands each link by 3 lanes at most. The above assumptions can also be relaxed by introducing additional binary variables.
For convenience, the notations frequently used in this paper are given as follows.

Formulation of the BRUE state
Now, we discuss the BRUE state in the BEV transportation network. In this paper, we assume that all vehicles in the transportation network are BEVs. Notably, with the strict environmental policies being established by various governments, BEVs will come to dominate transportation networks in the foreseeable future. According to the principle of bounded rationality, each traveler can choose their travel path freely when the travel times along several paths do not exceed a predetermined threshold. However, due to battery power restrictions and range anxiety, some paths will be inappropriate. To describe the above situation, we introduce the following definitions.

Definition 1 (Useable path).
A path is useable if and only if the traveler can complete the journey by selecting this path and the SOC of the battery will not cause the traveler to feel range anxiety at any time during the trip (He et al. 2014).
Definition 1 indicates that in a BEV transportation network, the SOC at any point along a usable path must be greater than or equal to the minimum battery power that causes range anxiety. Figure 1 further illustrates Definition 1.

Fig. 1. Schematic diagram of a BEV transportation network
A BEV network with 4 nodes and 5 links is shown in Figure 1, and the length of each link is marked above it. There are charging stations at Node 2 and Node 3. It is assumed that the traveler's trip goes from Node 1 to Node 4. Therefore, there are three path options: Path 1: 1-2-4, Path 2: 1-4, and Path 3: 1-3-4. We set the initial SOC of the traveler's BEV to 15 kWh and the electricity consumption of the BEV to 1 kWh per kilometer. Without considering range anxiety, Path B becomes unusable because the battery power consumed by the BEV on link ( ) 1, 4 is 20 kWh. When we define the range anxiety of travelers as 1 kWh, Path A further becomes unusable. This is because (1),  (1) to (6) can be transformed as follows.
(1), (5), where ( ) Remark. We can easily verify that when w  equals 0, the variable w  represents the equilibrium travel time for O-D pair w in the PRUE state. When 0 w   , the traffic flow distribution that satisfies (1), (5), and (7)

Finding best-and worst-case scenarios for a network
where ( )

Robust BEV network design model
For (1), (5), (7), (8), (9), (10), (12), (13), and (14), The BNDPM is established as a min-max programming model. Equation (15) is the objective function, and the lane expansion scheme u must minimize the worst-case scenario (i.e., the flow distribution associated with longest system travel time) in the BEV transportation network. Constraint (16) restricts the government's investment from exceeding the budget ceiling. Constraints (17) and (18)  infinite min-max problem (Polak and Royset 2005). Solving such problems has always been a challenge. To date, only a few studies (e.g., Royset et al. (2003) and Polak and Royset (2005)) have discussed the solution to this kind of problem. Moreover, the inner problem is an MPCC because of the existence of Constraints (12)  design a heuristic method, as described in Section 5.

Solving the best-and worst-case
In this section, we solve for the BRUE state of the BEV transportation network based on the modified ASA proposed by Lou et al. (2010). The core concept of the ASA is to rewrite the two parts of one equilibrium condition into two different constraints. For example, in BC/WC-BRUE, the mathematical expression of the equilibrium condition (12) can be rewritten as shown in Equation (19).
Step 1: Let 1  = , and solve for the PRUE state of the transportation network with = u0 by applying the technique proposed by Xin et al. (2021). To avoid the enumeration of paths, one can learn from the CGA proposed in Section 5.3. To obtain good strongly stationary solutions, one may also apply this step with multiple initial solutions (Lou et al. 2010). Based on the model solution, set 1 P  and 1 P  , and go to Step 2.

f y ε λ solve the R-BC-BRUE model, and go to
Step 3.

Column generation to avoid path enumeration
According to our definition, not all paths are usable for an O-D pair in the BEV transportation network. This requires us to calculate the usable paths before solving for the BRUE state of the BEV transportation network to avoid enumerating the paths for each O-D pair. In this context, we apply the CGA proposed by Cheng et al. (2020).
Based on the above model, we propose the following method to explore the usable paths for each O-D pair by applying the CGA.
Step 1 Step 2: Solve for the BRUE state of the network based on set w P , and obtain the optimal solution ( ) ; then, go to Step 3.
Step 3 Note that CGA is an ordinary mixed-integer linear programming model that can be easily solved with commercial solvers such as CPLEX (CPLEX 2008). In addition, since the number of usable paths in a BEV transportation network is finite, the abovementioned procedure can be terminated after a finite number of iterations (Cheng et al. 2020;He et al. 2014).

Solution procedure for the robust optimization model
In this section, a heuristic algorithm is designed to solve the robust optimization model, i.e., the BDNPM. Since the inner problem and outer problem of the original model affect each other, we first consider separating these problems. Therefore, we convert

BNDPM-IN into P-BNDPM-IN by adding a penalty function. With this approach, we
can obtain an ordinary semi-infinite optimization problem that can be solved by the cutting-plane method proposed by Lawphongpanich and Hearn (2004).   . Based on the above descriptions, we give the solution procedure for the BNDPM as follows.
Step 1: Initialization. Set = u0 , and solve the WC-BRUE model to obtain an initial solution: 1  . Set 1  = and   1   =  , and go to Step 2.
Go to Step 4.
Step 4: If , stop; in this case,  u is an optimal robust lane expansion scheme. Otherwise, go to Step 5.

Numerical example
We perform numerical experiments based on the classic Sioux Falls network (Leblanc 1975;Long et al. 2010), as shown in Figure

Sensitivity analysis for the degree of bounded rationality
In this section, we adjust the parameter α to observe the impact of different degrees of rational behavior on the vehicle distribution in the transportation network. We set the range of α from 0 to 0.20 and performed 20 experiments with 0.01 as the step size. The best/worst-case total travel times in different situations are shown in Figure 3 and Figure 4. As parameter α gradually increased, the system total travel time in the best case gradually decreased and tended to stabilize. Moreover, the system travel time for the worst case gradually increased. The above situation is in line with our expectations.
As the parameter α increases, the choice behaviors of travelers in the system become increasingly irrational. Additionally, a traveler can make concessions to facilitate the travel of other individuals. Therefore, the system travel time in the best case will gradually tend to system optimal (SO) state. The opposite trend occurs in the worst case.
With the increase in irrationality, the behaviors of travelers may hinder the travel of other individuals in extreme cases, thus increasing the total travel time. Therefore, the difference between the best and worst-case BRUE states increases. Specifically, the difference between the two increased from 100.  Table 2 and Table 3, respectively.
Notably, as α increases, the flow distribution in the PRUE state will not change; however, the difference between the best and worst-case BRUE states becomes more obvious. For example, when 0.05  = , the flow difference for link (1, 2) in the best and worst cases is only 0.41. However, when  increases to 0.20, this flow difference increases to 6.67. The above phenomenon is consistent with the trend reflected in Figures 1 and 2. Another noteworthy phenomenon is that the flows along some links in Table 2 and Table 3 are 0. Specifically, electric vehicles have limited mileage capacities, and the range anxiety of travelers may limit vehicle use. Moreover, some links may not be included in a usable path due to the particularities of the corresponding geographical locations.

Sensitivity analysis for the government investment scale
In this section, we fix the parameter  and explore the changes in the lane expansion scheme for the BEV transportation network by adjusting the scale of government investment. We set 0.10

 =
, and the investment scale is increased from 0 to 100 in steps of 20. Therefore, six scenarios (C0 to C5) are explored. The optimal lane expansion schemes and system travel times in different scenarios are shown in Table 4.
The system travel times in different scenarios are also shown in Figure 5. In addition, Figure 5 shows that the system travel time decreases most significantly when the investment amount increases from 20 to 40. Subsequently, the effect of investment is not as obvious as in the previous scenarios. This result suggests that government investment is characterized by diminishing marginal returns. In other words, with increasing government investment, the corresponding rate of decrease in system travel time will first increase and then decrease. Based on the current parameters, the government's optimal investment scale should be 40. The above phenomenon indicates that for different investment goals, the government should scientifically set the investment scale to maximize social welfare.

Conclusions and recommendations
In this paper, a BEV transportation network design problem is investigated. To address issues related to the mileage limitations of BEVs, traveler range anxiety, and bounded rationality, the problem is formulated as an MPCC. The model aims to minimize the total travel time in the transportation system and establish an optimal lane expansion scheme based on the available investment budget. A heuristic algorithm is proposed based on the framework of the ASA. Through column generation, we avoid the enumeration of paths in the transportation network.
Numerical tests are performed to assess the impacts of different traveler behaviors on travel times. Moreover, a sensitivity analysis of the investment scale is performed to explore the optimal level of government investment. The experimental results show that different levels of rationality among travelers influence the best-and worst-case scenarios for the flow distribution to various degrees. The greater the degree of irrationality is, the closer the best case is to the SO state of the BEV transportation network. Government investment is characterized by the law of diminishing marginal returns. Most of the links included in the investment scheme are connected to nodes with charging stations. Therefore, the experimental results verify that there are some bottleneck links in the BEV transportation network. Mitigating these bottlenecks is the key to improving the efficiency of the network. In addition, the solution approach can be efficiently used to support the proposed the network design scheme and can potentially be applied to even larger networks.
Overall, the model and algorithms we propose can effectively provide decisionmaking support for the government in infrastructure construction in BEV transportation networks. We extend the bounded rationality model proposed by Lou et al. (2010) for BEV transportation network design. Through the ongoing adjustment of the model, the robust joint optimization of congestion tolls and network design could be achieved.
Future work could focus on the collaborative optimization of different transportation network design schemes based on BEV characteristics. For example, the collaborative optimization of construction schemes for charging facilities (e.g., wireless charging roads and charging stations) considering bounded rationality is an interesting research direction. In addition, the electric energy consumption of BEVs varies for different traffic flows. Determining how to incorporate these variations into the proposed model framework is also a potential research direction.