Modeling Combined Travel Choices of Electric Vehicle Drivers with Variational Inequality Formulation

Plug-in electric vehicles become more attractive with government incentives and policies. Tractable modeling techniques that can incorporate electric vehicles are critical for developing realistic and effective travel behaviors models. It is shown in this paper that the special features of electric vehicles and the limited charging infrastructures lead to different travel behavior of electric vehicle drivers from gasoline vehicle drivers. This paper investigates both the temporal and spatial travel choices behavior of electric vehicles drivers. A multi-class quasi-dynamic model formulation is presented in the form of variational inequality. An optimization-based heuristic method is adopted to solve the model. Finally, numerical experiments are conducted to show that the model generates a network equilibrium solution of the combined choices. INTRODUCTION Battery electric vehicle (BEV) is a typical type of plug-in electric vehicle (PEV), which uses a battery to store the electrical energy that powers the motor. Unlike traditional gasoline vehicles (GVs) refilled by gas, BEVs are recharged by plugging a cord into an electric power source. The usage of BEVs will diversify the U.S. national energy usage, increase the energy independence from petroleum, and reduce harmful emissions from transportation fuels to the environment. Because of these opportunities, BEVs have received tremendous attention in recent energy policy discussions. While BEVs have many benefits on improving environments and reducing gas usage, they are facing some concerns as well. First, there are a limited number of public charging infrastructures for BEV drivers to recharge their vehicle, and the range limits are usually lower than that of GVs. Therefore, most BEV drivers will fully charge their vehicle at home before traveling and will tend to finish their whole trip chain before their car running out of electricity. If the whole trip chain cannot be finished with one full charged battery, the BEV drivers will have to recharging their vehicles at one of the intermediate destinations. These facts lead to the range anxiety and the park-and-charge concept. Second, with provided public charging infrastructures at work, shopping centers, or parking garages, the charging time for electric vehicles is usually up to a number of hours under the current technologies. For example, a midsize BEV with a 20 kWh battery pack may require 6 to 8 hours for a full charge with a level II charger (providing the 240 volt AC charging), and up to 20 hours with a level I charger (providing the 120 volt AC charging) [1] . The concerns on BEVs’ range limitation and on the availability of public charging infrastructures will affect the drivers’ travel behaviour, such as route choice and parking choice (as a result of the park-and-ride charging concept). In addition, the long-time recharging requirement for electric vehicles at an intermediate destination will affect the driver’s time spending at that destination. With the above facts, the electric vehicle drivers’ behaviour will be different from the GV driver’s behaviour. However, at this early stage with few electric vehicles on the road, the electric vehicle drivers’ behaviour has not been incorporated in the network based travel behaviour models. With the increasing adoption of BEVs, the existing network models for travel choices without considering the special recharging requirement and limitations of BEVs will lead to inaccurate predictions for transportation system conditions. Therefore, this paper focuses on building a network based travel behaviour model taking into account the recharging activities of individual BEV driver. The impacts of recharging activates on BEV drivers’ behaviour is both temporal and spatial. Temporally, BEV drivers’ responses to the recharging requirements may include travel choices such as departure time choice and duration of stay choice at destination. Spatially, the responses may include route choice and charging/parking location choice, etc. Knowing electric vehicle drivers’ responses and behaviours will help decision makers and agencies to obtain new traffic pattern information and set transportation policies accordingly. The overarching theme of this work is the need to build a model for quantifying the electric vehicle drivers’ above travel choices behavior. As the travel choices include several aspects, a combined model for joint choices modeling is required [2]. In this paper, a time-dependent joint choices model is presented to identify the traveling behavior of electric vehicle users on a network level with mix flows (including BEVs and GVs) on the road. Multiple user classes are considered in this problem and the user class is defined by the duration of stay, purpose of trips and the types of vehicles used together. The problem examined in this paper is not only timely and important on its own, but also can be mingled within the consideration of a network design problem for public charging infrastructure. The rest of this paper is structured as follows. The next section reviews the literature of related modelling techniques and solution methods for the research tasks we mentioned above. Section 3 discusses the problem settings and the network model for joint travel choices of electric vehicle drivers. Section 4 presents a solution algorithm for the model. Section 5 discusses the experimental results from applying the model and solution methods. Finally, Section 6 concludes the paper. LITERATURE REVIEW The network equilibrium models are useful tools for long term transportation planning. Many researchers proposed network equilibrium models which combine different choices together. Such combined choices include the mode and routes choices, destination choices and route choices, departure time and route choices, etc. [2-5]. The combined choices have been proposed for dozens of years on different levels of choice combinations. Sheffi (1985) presented a hypernetwork approach to accommodate the joint travel choices[6]. The hypernetwork consists of hyperlinks for trip generation, mode choice, destination choice and the route choice. There are also works for combining the trip distribution and traffic assignment into one model, where the two steps in planning are solve simultaneously to obtain more consistent results [7, 8]. Later, the convex optimization formulation is extended by introducing origin and/or destination constraints [8-10] and is also used for combining trip distribution, mode choice and traffic assignment models with a set of hierarchical choices[11, 12]. Lam and Huang (1992) and Boyce and Bar-Gera (2001) Some work formulated multimode, multiclass network equilibrium models to incorporate trip distribution with traffic assignment [13, 14]. In addition, the logit model has been adopted in many research for investigating many choices such as mode choices, destination choices, route choices, departure time choices and the joint/combined choices [11, 15-19]. Variational Inequality (VI) is widely used for the combined choices [20-23], especially for the time-dependent combined travel choices [24-26].Yang et al. (1998) proposed a space-time expended network (STEN) for the departure time and route choice in a queuing network with elastic demand to determine the optimal variable congestion tolls [27]. Friesz et. al. (1993) first formulated an infinitedimensional VI model for the combined choices of departure time and route choice without providing the solution approaches[28]. Later, Wei et.al.(1995) developed a discretized VI formulation for simultaneous route and departure time choice equilibrium problem[29]. In addition, they presented a heuristic algorithm but with no convergence established. Zhou et. al. (2007) developed a VI model and a heuristic procedure to describe and to solve the combined mode, departure time and route choices in multimodal urban transportation network[30]. Zhang (2007) considered simultaneous departure time and route choices using VI formulation [31]. Florian et.al. (2002) formulated a VI formulation for a multi-class multi-mode variable demand equilibrium, in which the joint choices model was a hierarchical Logit function [19]. Wu and Lam (2003) proposed a network equilibrium model that predicts the mode choice and route choice simultaneously in the VI form [32]. Lam et al. (2006) proposed a time-dependent network equilibrium VI formulation for simultaneously departure time, route, parking location and parking duration choice in deterministic user equilibrium [33]. It should be noted that the time-dependent model in this paper is actually a quasi-dynamic model for long-term planning purposes. Quasi-dynamic model for strategic planning purposes have been proposed in some previous research as well [33]. It means that in the model developed, the travel demand in each time interval is in steady-state equilibrium. The connection between successive time intervals is represented by the charging stations occupancy that is carried over to the next interval, which is similar to the problem setting in [34]. PROBLEM STATEMENT AND NOTATION It is assumed that drivers have a preferred time window for their arrivals at destinations and they will encounter a schedule delay cost if arriving at the destination out of the time window. During different time of the day, congestion levels are different. Therefore, different departure times will result in different travel time on a certain route. The departure time and the network conditions together will affect the drivers' arrival time at destinations. Moreover, the electricity-charging time at destinations plays an important role in the duration of stay for the BEVs drivers. Before traveling, BEV drivers will have an expected duration of stay at their destinations. However, this expected duration of stay might be different from their actual duration of stay at destinations in the end. For example, if a BEV driver needs to recharge his/her vehicle at the destination for 2 hours to get enough electricity for his/her following trip, while his/her expected duration of stay at destination is 1 hours, his/her actual duration of stay at destination will be longer than the expected duration of stay. We define a cost for this extra duration of stay time (which is 1 hour in the above example) at the destination for BEV drivers, named extra-charging time cost. A combined problem of departure time choice, duration of stay choice and route choice for electric vehicle drivers is considered in this paper. The problem is modeled using a nested-logit (NL) structure and formulated as an equivalent variational inequality formulation in this paper. The NL structure is shown in Figure 1: In the upper-level, the drivers make choices on the departure time and the expected duration of stay at destination. The alternative of departure time and duration of stay combination is a generic alternative in this level. In the lower-level, the drivers choose the perceived cheapest route directing to the destination. This hierarchical choice structure has been adopted by many related studies [25, 33, 35].