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This paper derives the distribution of the deviation distance to visit an alternative fuel station. Distance is measured as the Euclidean distance on a continuous plane. The distribution explicitly considers the vehicle range and whether the round trip between origin and destination can be made. Three cases are examined: fuel is available at both origin and destination, fuel is available at either origin or destination, and fuel is available at neither origin nor destination. The analytical expressions for the distribution demonstrate how the vehicle range, the shortest distance, and the refueling availability at origin and destination affect the deviation distance. The distribution will thus be useful to estimate the number of vehicles refueled at a station.

Alternative fuel vehicles powered by electricity, hydrogen, and biofuels have been promoted because of environmental, geopolitical, and financial concerns. The transition from gasoline engine vehicles to alternative fuel vehicles, however, would be difficult. Major barriers to the transition are the scarcity of refueling stations and the limited range of vehicles.

A sufficient number of alternative fuel stations has been calculated. Melaina [

An efficient location of alternative fuel stations has also been studied. Kuby and Lim [

Refueling stations are typical flow demand facilities in that demand for service can be expressed as flow rather than point. In fact, drivers usually refuel their vehicles on pre-planned paths from origin to destination. Other examples of flow demand facilities include convenience stores, automated teller machines, and nursery schools. Flow demand was first introduced into facility location models by Hodgson [

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In this paper, we derive the distribution of the deviation distance to visit an alternative fuel station. The distribution that shows how the deviation distance is distributed will supply building blocks for location models of refueling stations. For example, the distribution can be used to estimate the number of vehicles refueled at a station, because refueling demand generally decreases with the deviation distance. Given the location of stations and the set of origin-destination pairs, the distribution of the deviation distance can be numerically calculated. The result, however, depends on the specific data and cannot be applied to other situations. For examining fundamental characteristics of the deviation distance, analytical expressions are necessary. Analytical expressions are also useful to interpret and comprehend numerical results.

Although Miyagawa [

The remainder of this paper is organized as follows. The next section defines the deviation distance and the distribution of the deviation distance. The following sections derive the distribution of the deviation distance for three cases: fuel is available at both origin and destination, fuel is available at either origin or destination, and fuel is available at neither origin nor destination. The final section presents concluding remarks.

Consider trips using alternative fuel vehicles of range r. Distance is measured as the Euclidean distance on a continuous plane. Origins and destinations are assumed to be uniformly and independently distributed over the study region. This assumption facilitates the analytical treatment of the distribution of the deviation distance. In fact, the uniformity assumption has frequently been used in continuous transportation models [24-26]. In addition, the uniform demand serves as a basis for further analyses with non-uniform demand. For example, more generalized travel demand can be incorporated by using spatial interaction models [

Drivers are assumed to deviate from their shortest paths to refuel their vehicles. Let t and U be the shortest distance between origin and destination and the deviation distance to visit a station, respectively. The deviation distance is defined as the sum of the distances from origin O to the station and from the station to destination D. Let be the volume of flow such that. We call the distribution of the deviation distance. The region that a driver can cover within the deviation distance forms an ellipse, the foci of which are at O and D, as shown in

First, we assume that fuel is available at both origin O and destination D. Then the vehicle can start at O with full tank of fuel. Note that plug-in electric vehicles can be charged at home or work place. Note also that this assumption also applies to long distance trips where O and D represent other stations.

To refuel at a station and complete the round trip, both O and D must be within the distance r of the station. In fact, the vehicle can reach the station, fill up, go to D, fill up again, turn round, fill up again at that same station, and return to O. Thus, both O and D must be in the circle C centered at the station with radius r, as depicted in

For, the midpoint of the O-D path must be in the ellipse centered at the station, as discussed in the previous section. Set the origin of the coordinate system at the station, as shown in

where. is obtained by calculating the area of the ellipse in the intersection of the two circles as

where

The distribution of the deviation distance is shown in

Next, we assume that fuel is available at either origin O or destination D. Without loss of generality, we assume that fuel is available at only O. If the round trip is considered, the vehicle is required to reach D with at least half a tank remaining.

To refuel at a station and complete the round trip, O must be within the distance r of the station and D must be within the distance r/2 of the station. In fact, the vehicle can reach the station, fill up, go to D, turn round, fill up again at that same station, and return to O. Thus, O must be in the circle centered at the station with radius r and D must be in the circle centered at the station with radius r/2, as depicted in

is obtained by calculating the area of the ellipse (1) in the intersection of the two circles, as shown in

and if,

where

The distribution of the deviation distance is shown in

Finally, we assume that fuel is available at neither origin O nor destination D. We also assume that the vehicle starts at O with half a tank of fuel and reaches D with at least half a tank remaining, as suggested by Kuby and Lim [

To refuel at a station and complete the round trip, both and D must be within the distance r/2 of the station. In fact, the vehicle can reach the station, fill up, go to D, turn round, fill up again at that same station, and return to O. Thus, both O and D must be in the circle C centered at the station with radius r/2, as depicted in

is obtained by calculating the area of the ellipse (1) in the intersection of the two circles, as shown in

where

The distribution of the deviation distance is shown in

This paper has derived the distribution of the deviation distance to visit an alternative fuel station. The major characteristic of the paper is that the focus is on whether the round trip between origin and destination can be made. The distribution that explicitly considers the vehicle range will give a more appropriate framework for the deviation distance of alternative fuel vehicles.

The analytical expressions for the distribution demonstrate how the vehicle range, the shortest distance, and the refueling availability at origin and destination affect the deviation distance. Note that finding these relationships by using discrete network models requires computation of the deviation distance for each combination of the parameters. The relationships are useful to estimate the number of vehicles refueled at a station and sufficient capacity of the station. The sufficient capacity can be used as an input in location models of refueling stations.

In this paper, we implicitly assume the early stages of an alternative fuel industry where stations are sparse. If many stations are available, drivers will choose the best station that minimizes the deviation distance. Incorporating the competition among stations would be a topic for future research.