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Density of alternative fuel stations and refueling availability
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 8 ( 2 0 1 3 ) 1 2 4 3 8 e1 2 4 4 5
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Density of alternative fuel stations and refueling availability Masashi Miyagawa* Department of Regional Social Management, University of Yamanashi, 4-4-37 Takeda, Kofu, Yamanashi 400-8510, Japan
article info
abstract
Article history:
This paper develops an analytical model for determining sufficient density of alternative
Received 15 March 2013
fuel stations required to achieve a certain level of service. The service level is represented
Received in revised form
as the probability that the vehicle can make the repeated round trip between randomly
16 July 2013
selected origin and destination. Distance is measured as the Euclidean distance on a
Accepted 19 July 2013
continuous plane. The probability is obtained for regular and random patterns of stations
Available online 12 August 2013
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 analytical expressions for the probability demonstrate how the density of stations, the vehicle range,
Keywords: Location Flow demand Vehicle range
and the trip length, as well as the refueling availability at origin and destination affect the service level. The result shows that the effect of the refueling availability at origin and destination is significant. Copyright ª 2013, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights
Trip length
reserved.
Round trip
1.
Introduction
Global warming, urban air pollution, and high oil prices have increased interests in alternative fuel vehicles powered by hydrogen, electricity, and biofuels. One of the most significant factors for the transition to alternative fuel vehicles is the adequate availability of refueling stations [1]. A sufficient number of alternative fuel stations has been calculated. Melaina [2] developed three estimation approaches based on the number of existing gasoline stations, metropolitan land areas, and lengths of major roads. Melaina and Bremson [3] proposed a sufficient level of station coverage that meets the refueling needs of the general population in urban areas. Nicholas et al. [4] developed a GIS model for siting hydrogen stations and examined the effect of the number of
stations on the average driving time to the nearest station. Nicholas and Ogden [5] studied the regional variation in the number of stations needed to achieve a travel time target. Honma and Kurita [6] obtained the optimal number of hydrogen stations that minimizes the sum of operation and transportation costs. A limitation of these studies is that they focus on the distance from customers to their nearest station. In other words, they assume that demand for service originates from nodes of a network or points on a plane. Refueling stations are, however, typical flow demand facilities in that demand for service can be expressed as flow. In fact, drivers usually refuel their vehicles on pre-planned paths from origin to destination. This type of refueling behavior should therefore be taken into account when discussing the sufficient number of refueling stations.
* Tel.: þ81 55 220 8338; fax: þ81 55 220 8666. E-mail address: [email protected]. 0360-3199/$ e see front matter Copyright ª 2013, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijhydene.2013.07.067
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 8 ( 2 0 1 3 ) 1 2 4 3 8 e1 2 4 4 5
Flow demand was first introduced into facility location models by Hodgson [7], who considered the location of facilities that minimizes the total deviation time from work journeys. Hodgson [8] and Berman et al. [9] formulated a location model, which Hodgson called the flow capturing model, for locating facilities on a network so as to maximize the total flow covered. Hodgson et al. [10] applied the model to the actual road network in Edmonton, Canada. The model was generalized by Berman et al. [11] to consider the possibility of deviations, by Averbakh and Berman [12] to take account of multi-counting, by Zeng et al. [13] to include consumers’ preferences, and by Tanaka and Furuta [14] to deal with facilities of different sizes and attractions. Berman [15] introduced four location problems that combine demand coverage problems with flow coverage problems. Zeng et al. [16] proposed a generalized flow-interception location-allocation model for effectively locating facilities on a network with flow-based demand. Miyagawa [17] derived the distribution of the deviation distance to visit a facility from pre-planned routes on a continuous plane. Flow demand location models have also been applied to alternative fuel stations. Kuby and Lim [18] extended the flow capturing model and developed the flow refueling location model (FRLM) for optimally locating refueling facilities. The FRLM locates p facilities to maximize the total flow volume that can be refueled. Kuby et al. [19] applied the FRLM to the location of hydrogen stations in Florida. Lim and Kuby [20] presented three heuristic algorithms for the FRLM. Capar and Kuby [21] proposed an efficient formulation of the FRLM that makes it possible to solve large problems. The FRLM was extended by Kuby and Lim [22] to add candidate sites along network arcs, by Upchurch et al. [23] to include the capacity of refueling facilities, and by Kim and Kuby [24] to allow drivers to deviate from their shortest paths. Upchurch and Kuby [25] compared the point-based p-median model and the flowbased FRLM. In this paper, we present an analytical model for determining sufficient density of refueling stations required to achieve a certain level of service. The major characteristic of the paper is that the service level is represented as the probability that the vehicle can make the repeated round trip between randomly selected origin and destination. The model explicitly incorporates flow demand, and will thus give a more appropriate estimate for the sufficient density of refueling stations. An important criterion in flow demand location models is the deviation distance, which is the additional travel distance to visit a facility from the shortest path. Note, however, that the deviation distance is insufficient for evaluating refueling availability, because even if the deviation distance is smaller than the vehicle range, the vehicle cannot always complete the trip. For example, if a station is on the shortest path (i.e., the deviation distance is zero) but the distance from origin to the station is greater than the vehicle range, the vehicle cannot reach the station. We therefore focus on whether the vehicle can make the round trip between origin and destination. The sufficient density of stations depends on the vehicle range, the trip length, and whether the vehicle can be refueled at origin and/or destination. In fact, plug-in electric vehicles can be charged at home. Since the driving range of alternative
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fuel vehicles is shorter than that of gasoline engine vehicles, more stations would be required. On the other hand, if the trip length is short and the vehicle can be refueled at home, even a small number of stations may satisfy refueling demand. We investigate these relationships by using an analytical approach. Analytical expressions provide fundamental understandings on the sufficient density of stations, thereby supplementing further location models of refueling stations. The remainder of this paper is organized as follows. The next section develops a model for determining the sufficient density of refueling stations. The following sections examine how the density of stations, the vehicle range, the trip length, and the refueling availability at origin and destination affect the service level of refueling stations. Three cases are considered: 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 penultimate section provides the density of stations required to achieve a specified level of service. The final section presents concluding remarks.
2.
Model
Consider trips using alternative fuel vehicles of range r. Origins and destinations are selected at random within a study region. Distance is measured as the Euclidean distance on a continuous plane. Let t be the trip length and P(t) be the probability that the vehicle can make the repeated round trip between randomly selected origin and destination. Drivers are assumed to deviate from their shortest paths to refuel their vehicles and refueling is allowed only once for each one-way trip. Although only one refueling may not be enough for longer trips, the result will supply building blocks for further analyses with multiple refueling. For example, multiple refueling can be incorporated by adding transit points between origin and destination. Refueling stations are represented as points of regular and random patterns on a continuous plane, as shown in Fig. 1. Since actual patterns of stations can be regarded as the intermediate between regular and random, the theoretical results of these extremes will give an insight into empirical studies on actual patterns. In fact, the regular and random patterns have been used in location analysis [26e28]. The regular and random patterns are assumed to continue infinitely. This assumption enables us to examine the availability of stations without taking into account the edge effect.
3. Fuel is available at both origin and destination 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. If t r, the vehicle can reach D without refueling, fill up at D, and return to O, i.e., P(t) ¼ 1. If t > 2r, the vehicle cannot reach D, because more than one refueling is needed, i.e., P(t) ¼ 0. Hence, we focus on the case where r < t 2r. If r < t 2r, the vehicle can make the round trip if both O and D are within the distance r of a station. In fact, the vehicle can
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Fig. 1 e Regular and random point patterns.
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, P(t) is the probability that both origin and destination are within the distance r of a station.
3.1.
S ¼ 2r2 arccos
Fig. 2 e Calculation of the probability for the grid pattern.
(1)
P(t) is obtained as PðtÞ ¼
Grid pattern
Suppose that stations are regularly distributed on a square grid with spacing a(2r). The density of stations r, that is, the number of stations per unit area is then expressed as r ¼ 1/a2. The probability P(t) can be calculated by considering only one station, because we assume that the grid pattern continues infinitely. The study region is then confined to the region where a station is the nearest, which is given by the square centered at the station with side length a, as depicted in Fig. 2. For making the round trip, both O and D must be in the circle C centered at the station with radius r. This means that the midpoint of the OeD path must be in the shaded region in Fig. 2. This region is the intersection of the two circles which are obtained by moving the circle C by t/2 in the direction parallel to the OeD path and in the opposite direction. P(t) is then the probability that the midpoint of the OeD path lies inside the intersection of the two circles. P(t) is given by the ratio between the area of the intersection of the two circles and that of the outer square. Since the area of the intersection of the two circles is
ffi t t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4r2 t2 ; 2r 2
S a2
(2)
PðtÞ ¼ 2rr2 arccos
ffi t rt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4r2 t2 : 2r 2
(3)
P(t) for the grid pattern is shown in Fig. 4(a). Note that P(t) ¼ 1 for t r and P(t) ¼ 0 for t > 2r. As expected, P(t) decreases with the trip length t and increases with the density of stations r and the vehicle range r.
3.2.
Random pattern
Suppose that stations are uniformly and randomly distributed on a plane. For making the round trip, a station must be in the intersection of the two circles centered at O and D with radius r, as depicted in Fig. 3. P(t) is then the probability that the intersection of the two circles contains at least one station. The probability that a region of area S contains exactly x stations, denoted by P(x,S ), is given by the Poisson distribution as Pðx; SÞ ¼
ðrSÞx expðrSÞ x!
Fig. 3 e Calculation of the probability for the random pattern.
(4)
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Fig. 4 e Probability that the vehicle can make the repeated round trip.
where r is the density of stations [29]. From Eq. (1), P(t) is obtained as
that origin is within the distance r of a station and destination is within the distance r/2 of the station.
PðtÞ ¼ 1 Pð0; SÞ
(5)
4.1.
ffi t rt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4r2 t2 : PðtÞ ¼ 1 exp 2rr2 arccos þ 2r 2
(6)
For making the round trip, O must be in the circle C1 centered at the station with radius r and D must be in the circle C2 centered at the station with radius r/2, as depicted in Fig. 5. This means that the midpoint of the OeD path must be in the shaded region in Fig. 5. This region is the intersection of the two circles which are obtained by moving the circle C1 by t/2 in the direction parallel to the OeD path and moving the circle C2 by t/2 in the opposite direction. P(t) is then the probability that the midpoint of the OeD path lies inside the intersection of the two circles. P(t) is given by the ratio between the area of the intersection of the two circles and that of the outer square. The area of the intersection of the two circles is
P(t) for the random pattern is shown in Fig. 4(b). Note that P(t) for the random pattern is smaller than that for the grid pattern.
4. Fuel is available at either origin or destination 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. To complete the round trip, the vehicle is required to reach D with at least half a tank remaining. If t r/ 2, the vehicle can start at O with full tank of fuel, reach D, and return to O without running out of fuel, i.e., P(t) ¼ 1. If t > 3r/2, the vehicle cannot make the round trip without refueling more than once, i.e., P(t) ¼ 0. Hence, we focus on the case where r/2 < t 3r/2. If r/2 < t 3r/2, the vehicle can make the round trip if O is within the distance r of a station and D is 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, P(t) is the probability
Fig. 5 e Calculation of the probability for the grid pattern.
Grid pattern
4t2 þ 3r2 r2 4t2 3r2 þ arccos S ¼r2 arccos 8rt 4 4rt sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi 1 3 4r2 t2 r2 þ t2 : 2 4
Fig. 6 e Calculation of the probability for the random pattern.
(7)
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Fig. 7 e Probability that the vehicle can make the repeated round trip.
From Eq. (2), P(t) is obtained as 4t þ 3r rr 4t 3r þ arccos PðtÞ ¼rr2 arccos 8rt 4 4rt sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi r 3 4r2 t2 r2 þ t2 : 2 4 2
2
2
2
5. Fuel is available at neither origin nor destination
2
(8)
P(t) for the grid pattern is shown in Fig. 7(a). Note that P(t) ¼ 1 for t r/2 and P(t) ¼ 0 for t > 3r/2. It can be seen that P(t) is smaller than that of the case where fuel is available at both origin and destination (see Fig. 4(a)).
4.2.
Random pattern
For making the round trip, a station must be in the intersection of the circle centered at O with radius r and the circle centered at D with radius r/2, as depicted in Fig. 6. P(t) is then the probability that the intersection of the two circles contains at least one station. From Eqs. (5) and (7), P(t) is obtained as
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 [18]. If t > r, the vehicle cannot make the round trip, i.e., P(t) ¼ 0. Hence, we focus on the case where t r. If t r, the vehicle can complete the round trip with at least half a tank remaining if both O and D are within the distance r/2 of a 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, P(t) is the probability that both origin and destination are within the distance r/2 of a station.
5.1.
Grid pattern
P(t) for the random pattern is shown in Fig. 7(b).
For making the round trip, both O and D must be in the circle C centered at the station with radius r/2, as depicted in Fig. 8. This means that the midpoint of the OeD path must be in the shaded region in Fig. 8. This region is the intersection of the two circles which are obtained by moving the circle C by t/2 in the direction parallel to the OeD path and in the opposite direction. P(t) is then the probability that the midpoint of the OeD path lies inside the intersection of the two circles. P(t) is given by the ratio between the area of the intersection of the two circles and that of the outer square. The area of the intersection of the two circles is
Fig. 8 e Calculation of the probability for the grid pattern.
Fig. 9 e Calculation of the probability for the random pattern.
(
4t2 þ 3r2 rr2 4t2 3r2 arccos 8rt 4 4rt sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi) r 3 4r2 t2 r2 þ t2 þ : 2 4
PðtÞ ¼1 exp
rr2 arccos
(9)
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Fig. 10 e Probability that the vehicle can make the repeated round trip.
S¼
r2 t t pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 t2 : arccos r 2 2
(10)
From Eq. (2), P(t) is obtained as PðtÞ ¼
rr2 t rt pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 t2 : arccos r 2 2
(11)
P(t) for the grid pattern is shown in Fig. 10(a). Note that P(t) ¼ 0 for t > r. In this case, even short distance trips are not always possible.
5.2.
Random pattern
For making the round trip, a station must be in the intersection of the two circles centered at O and D with radius r/2, as depicted in Fig. 9. P(t) is then the probability that the intersection of the two circles contains at least one station. From Eqs. (5) and (10), P(t) is obtained as rr2 t rt pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 t2 : arccos þ PðtÞ ¼ 1 exp r 2 2
(12)
P(t) for the random pattern is shown in Fig. 10(b).
6.
Density of stations
Using the probability P(t) for the grid and random patterns, we can calculate the density of stations required to achieve a specified level of service. Table 1 shows the density of stations required to achieve a probability P(t) a (a ¼ 0.2, 0.4, 0.6, 0.8) for the random pattern. Note that, as the target probability increases and the trip length becomes longer, more stations are required. Note also that the effect of the refueling availability at origin and destination is significant. If fuel is available at origin and destination, no station is needed for short distance trips, because the vehicle can make the round trip without refueling. If fuel is available at neither origin nor destination, more stations are required than the other two cases to achieve the same level of service. The sufficient service level varies among regions according to the traffic condition. If fuel is available at many homes and workplaces, we can assume that fuel is available at both origin and destination. In the region where long distance trips are dominant, we should use a large value for the trip length t. Once the density of stations required to achieve the service
Table 1 e Density of stations required to achieve a probability P(t) ‡ a: (a) fuel is available at both origin and destination; (b) fuel is available at either origin or destination; (c) fuel is available at neither origin nor destination. a
(a)
(b)
(c)
0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8
Range of 1
Range of 2
Trip length t
Trip length t
0.25
0.50
0.75
1.00
1.25
1.50
0.50
1.00
1.50
2.00
2.50
3.00
0 0 0 0 0 0 0 0 0.41 0.95 1.70 2.99
0 0 0 0 0 0 0 0 0.73 1.66 2.98 5.24
0 0 0 0 0.37 0.85 1.53 2.69 1.97 4.51 8.09 14.20
0 0 0 0 0.64 1.46 2.61 4.59 e e e e
0.27 0.63 1.12 1.97 1.71 3.91 7.02 12.32 e e e e
0.49 1.13 2.02 3.55 e e e e e e e e
0 0 0 0 0 0 0 0 0.10 0.24 0.43 0.75
0 0 0 0 0 0 0 0 0.18 0.42 0.75 1.31
0 0 0 0 0.09 0.21 0.38 0.67 0.49 1.13 2.02 3.55
0 0 0 0 0.16 0.36 0.65 1.15 e e e e
0.07 0.16 0.28 0.49 0.43 0.98 1.75 3.08 e e e e
0.12 0.28 0.51 0.89 e e e e e e e e
“e” means the round trip is impossible.
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level is calculated, the sufficient number of stations can be obtained by multiplying the density by the area of the region.
7.
Conclusion
This paper has developed an analytical model for determining sufficient density of alternative fuel stations. The probability that the vehicle can make the repeated round trip between randomly selected origin and destination has been obtained for regular and random patterns of stations. The model that explicitly incorporates flow demand will give a more appropriate estimate for the sufficient density of refueling stations. The analytical expressions for the probability demonstrate how the density of stations, the vehicle range, and the trip length, as well as the refueling availability at origin and destination affect the service level of refueling stations. Note that finding these relationships by using discrete network models requires computation of the number of originedestination pairs that can make the round trip for each combination of the parameters. The relationships help planners to estimate the sufficient number of alternative fuel stations required to achieve a certain level of service. The estimated number of stations can be used as an input in location models of refueling stations. Comparing the effects on the service level will also be useful to prioritize investments for the transition to alternative fuel vehicles (e.g. subsidy for refueling stations). The present model assumes that drivers refuel their vehicles at most once for each one-way trip. If multiple refueling is allowed, the sufficient number of stations decreases but the inconvenience of drivers increases. Thus, there exists a tradeoff between the construction cost of stations and the travel cost of drivers. Incorporating the tradeoff into the estimation would be a topic for future research.
Acknowledgments This research was supported by Grant-in-Aid for Scientific Research (C) (24510187). I wish to thank anonymous reviewers for helpful comments and suggestions.
references
[1] Greene D. Survey evidence on the importance of fuel availability to the choice of alternative fuels and vehicles. Energy Studies Review 1996;8:215e31. [2] Melaina M. Initiating hydrogen infrastructures: preliminary analysis of a sufficient number of initial hydrogen stations in the US. International Journal of Hydrogen Energy 2003;28:743e55. [3] Melaina M, Bremson J. Refueling availability for alternative fuel vehicle markets: sufficient urban station coverage. Energy Policy 2008;36:3233e41. [4] Nicholas M, Handy S, Sperling D. Using geographic information systems to evaluate siting and networks of hydrogen stations. Transportation Research Record 2004;1880:126e34.
[5] Nicholas M, Ogden J. Detailed analysis of urban station siting for California hydrogen highway network. Transportation Research Record 2006;1983:121e8. [6] Honma Y, Kurita O. A mathematical model on the optimal number of hydrogen stations with respect to the diffusion of fuel cell vehicles. Journal of the Operations Research Society of Japan 2008;51:166e90. [7] Hodgson M. The location of public facilities intermediate to the journey to work. European Journal of Operational Research 1981;6:199e204. [8] Hodgson M. A flow-capturing location-allocation model. Geographical Analysis 1990;22:270e9. [9] Berman O, Larson R, Fouska N. Optimal location of discretionary service facilities. Transportation Science 1992;26:201e11. [10] Hodgson M, Rosing K, Storrier A. Applying the flow-capturing location-allocation model to an authentic network: Edmonton, Canada. European Journal of Operational Research 1996;90:427e43. [11] Berman O, Bertsimas D, Larson R. Locating discretionary service facilities, II: maximizing market size, minimizing inconvenience. Operations Research 1995;43:623e32. [12] Averbakh I, Berman O. Locating flow-capturing units on a network with multi-counting and diminishing returns to scale. European Journal of Operational Research 1996;91:495e506. [13] Zeng W, Hodgson M, Castillo I. The pickup problem: consumers’ locational preferences in flow interception. Geographical Analysis 2009;41:107e26. [14] Tanaka K, Furuta T. A hierarchical flow capturing location problem with demand attraction based on facility size, and its Lagrangian relaxation solution method. Geographical Analysis 2012;44:15e28. [15] Berman O. Deterministic flow-demand location problems. Journal of the Operational Research Society 1997;48:75e81. [16] Zeng W, Castillo I, Hodgson M. A generalized model for locating facilities on a network with flow-based demand. Networks and Spatial Economics 2010;10:579e611. [17] Miyagawa M. Distributions of rectilinear deviation distance to visit a facility. European Journal of Operational Research 2010;205:106e12. [18] Kuby M, Lim S. The flow-refueling location problem for alternative-fuel vehicles. Socio-Economic Planning Sciences 2005;39:125e45. [19] Kuby M, Lines L, Schultz R, Xie Z, Kim J, Lim S. Optimization of hydrogen stations in Florida using the flow-refueling location model. International Journal of Hydrogen Energy 2009;34:6045e64. [20] Lim S, Kuby M. Heuristic algorithms for siting alternative-fuel stations using the flow-refueling location model. European Journal of Operational Research 2010;204:51e61. [21] Capar I, Kuby M. An efficient formulation of the flow refueling location model for alternative-fuel stations. IIE Transactions 2012;44:622e36. [22] Kuby M, Lim S. Location of alternative-fuel stations using the flow-refueling location model and dispersion of candidate sites on arcs. Networks and Spatial Economics 2007;7:129e52. [23] Upchurch C, Kuby M, Lim S. A model for location of capacitated alternative-fuel stations. Geographical Analysis 2009;41:85e106. [24] Kim J, Kuby M. The deviation-flow refueling location model for optimizing a network of refueling stations. International Journal of Hydrogen Energy 2012;37:5406e20. [25] Upchurch C, Kuby M. Comparing the p-median and flowrefueling models for locating alternative-fuel stations. Journal of Transport Geography 2010;18:750e8.
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[28] Miyagawa M. Joint distribution of distances to the first and the second nearest facilities. Journal of Geographical Systems 2012;14:209e22. [29] Clark P, Evans F. Distance to nearest neighbor as a measure of spatial relationships in populations. Ecology 1954;35:85e90.
Available online at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/he
Density of alternative fuel stations and refueling availability Masashi Miyagawa* Department of Regional Social Management, University of Yamanashi, 4-4-37 Takeda, Kofu, Yamanashi 400-8510, Japan
article info
abstract
Article history:
This paper develops an analytical model for determining sufficient density of alternative
Received 15 March 2013
fuel stations required to achieve a certain level of service. The service level is represented
Received in revised form
as the probability that the vehicle can make the repeated round trip between randomly
16 July 2013
selected origin and destination. Distance is measured as the Euclidean distance on a
Accepted 19 July 2013
continuous plane. The probability is obtained for regular and random patterns of stations
Available online 12 August 2013
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 analytical expressions for the probability demonstrate how the density of stations, the vehicle range,
Keywords: Location Flow demand Vehicle range
and the trip length, as well as the refueling availability at origin and destination affect the service level. The result shows that the effect of the refueling availability at origin and destination is significant. Copyright ª 2013, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights
Trip length
reserved.
Round trip
1.
Introduction
Global warming, urban air pollution, and high oil prices have increased interests in alternative fuel vehicles powered by hydrogen, electricity, and biofuels. One of the most significant factors for the transition to alternative fuel vehicles is the adequate availability of refueling stations [1]. A sufficient number of alternative fuel stations has been calculated. Melaina [2] developed three estimation approaches based on the number of existing gasoline stations, metropolitan land areas, and lengths of major roads. Melaina and Bremson [3] proposed a sufficient level of station coverage that meets the refueling needs of the general population in urban areas. Nicholas et al. [4] developed a GIS model for siting hydrogen stations and examined the effect of the number of
stations on the average driving time to the nearest station. Nicholas and Ogden [5] studied the regional variation in the number of stations needed to achieve a travel time target. Honma and Kurita [6] obtained the optimal number of hydrogen stations that minimizes the sum of operation and transportation costs. A limitation of these studies is that they focus on the distance from customers to their nearest station. In other words, they assume that demand for service originates from nodes of a network or points on a plane. Refueling stations are, however, typical flow demand facilities in that demand for service can be expressed as flow. In fact, drivers usually refuel their vehicles on pre-planned paths from origin to destination. This type of refueling behavior should therefore be taken into account when discussing the sufficient number of refueling stations.
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Flow demand was first introduced into facility location models by Hodgson [7], who considered the location of facilities that minimizes the total deviation time from work journeys. Hodgson [8] and Berman et al. [9] formulated a location model, which Hodgson called the flow capturing model, for locating facilities on a network so as to maximize the total flow covered. Hodgson et al. [10] applied the model to the actual road network in Edmonton, Canada. The model was generalized by Berman et al. [11] to consider the possibility of deviations, by Averbakh and Berman [12] to take account of multi-counting, by Zeng et al. [13] to include consumers’ preferences, and by Tanaka and Furuta [14] to deal with facilities of different sizes and attractions. Berman [15] introduced four location problems that combine demand coverage problems with flow coverage problems. Zeng et al. [16] proposed a generalized flow-interception location-allocation model for effectively locating facilities on a network with flow-based demand. Miyagawa [17] derived the distribution of the deviation distance to visit a facility from pre-planned routes on a continuous plane. Flow demand location models have also been applied to alternative fuel stations. Kuby and Lim [18] extended the flow capturing model and developed the flow refueling location model (FRLM) for optimally locating refueling facilities. The FRLM locates p facilities to maximize the total flow volume that can be refueled. Kuby et al. [19] applied the FRLM to the location of hydrogen stations in Florida. Lim and Kuby [20] presented three heuristic algorithms for the FRLM. Capar and Kuby [21] proposed an efficient formulation of the FRLM that makes it possible to solve large problems. The FRLM was extended by Kuby and Lim [22] to add candidate sites along network arcs, by Upchurch et al. [23] to include the capacity of refueling facilities, and by Kim and Kuby [24] to allow drivers to deviate from their shortest paths. Upchurch and Kuby [25] compared the point-based p-median model and the flowbased FRLM. In this paper, we present an analytical model for determining sufficient density of refueling stations required to achieve a certain level of service. The major characteristic of the paper is that the service level is represented as the probability that the vehicle can make the repeated round trip between randomly selected origin and destination. The model explicitly incorporates flow demand, and will thus give a more appropriate estimate for the sufficient density of refueling stations. An important criterion in flow demand location models is the deviation distance, which is the additional travel distance to visit a facility from the shortest path. Note, however, that the deviation distance is insufficient for evaluating refueling availability, because even if the deviation distance is smaller than the vehicle range, the vehicle cannot always complete the trip. For example, if a station is on the shortest path (i.e., the deviation distance is zero) but the distance from origin to the station is greater than the vehicle range, the vehicle cannot reach the station. We therefore focus on whether the vehicle can make the round trip between origin and destination. The sufficient density of stations depends on the vehicle range, the trip length, and whether the vehicle can be refueled at origin and/or destination. In fact, plug-in electric vehicles can be charged at home. Since the driving range of alternative
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fuel vehicles is shorter than that of gasoline engine vehicles, more stations would be required. On the other hand, if the trip length is short and the vehicle can be refueled at home, even a small number of stations may satisfy refueling demand. We investigate these relationships by using an analytical approach. Analytical expressions provide fundamental understandings on the sufficient density of stations, thereby supplementing further location models of refueling stations. The remainder of this paper is organized as follows. The next section develops a model for determining the sufficient density of refueling stations. The following sections examine how the density of stations, the vehicle range, the trip length, and the refueling availability at origin and destination affect the service level of refueling stations. Three cases are considered: 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 penultimate section provides the density of stations required to achieve a specified level of service. The final section presents concluding remarks.
2.
Model
Consider trips using alternative fuel vehicles of range r. Origins and destinations are selected at random within a study region. Distance is measured as the Euclidean distance on a continuous plane. Let t be the trip length and P(t) be the probability that the vehicle can make the repeated round trip between randomly selected origin and destination. Drivers are assumed to deviate from their shortest paths to refuel their vehicles and refueling is allowed only once for each one-way trip. Although only one refueling may not be enough for longer trips, the result will supply building blocks for further analyses with multiple refueling. For example, multiple refueling can be incorporated by adding transit points between origin and destination. Refueling stations are represented as points of regular and random patterns on a continuous plane, as shown in Fig. 1. Since actual patterns of stations can be regarded as the intermediate between regular and random, the theoretical results of these extremes will give an insight into empirical studies on actual patterns. In fact, the regular and random patterns have been used in location analysis [26e28]. The regular and random patterns are assumed to continue infinitely. This assumption enables us to examine the availability of stations without taking into account the edge effect.
3. Fuel is available at both origin and destination 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. If t r, the vehicle can reach D without refueling, fill up at D, and return to O, i.e., P(t) ¼ 1. If t > 2r, the vehicle cannot reach D, because more than one refueling is needed, i.e., P(t) ¼ 0. Hence, we focus on the case where r < t 2r. If r < t 2r, the vehicle can make the round trip if both O and D are within the distance r of a station. In fact, the vehicle can
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Fig. 1 e Regular and random point patterns.
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, P(t) is the probability that both origin and destination are within the distance r of a station.
3.1.
S ¼ 2r2 arccos
Fig. 2 e Calculation of the probability for the grid pattern.
(1)
P(t) is obtained as PðtÞ ¼
Grid pattern
Suppose that stations are regularly distributed on a square grid with spacing a(2r). The density of stations r, that is, the number of stations per unit area is then expressed as r ¼ 1/a2. The probability P(t) can be calculated by considering only one station, because we assume that the grid pattern continues infinitely. The study region is then confined to the region where a station is the nearest, which is given by the square centered at the station with side length a, as depicted in Fig. 2. For making the round trip, both O and D must be in the circle C centered at the station with radius r. This means that the midpoint of the OeD path must be in the shaded region in Fig. 2. This region is the intersection of the two circles which are obtained by moving the circle C by t/2 in the direction parallel to the OeD path and in the opposite direction. P(t) is then the probability that the midpoint of the OeD path lies inside the intersection of the two circles. P(t) is given by the ratio between the area of the intersection of the two circles and that of the outer square. Since the area of the intersection of the two circles is
ffi t t pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4r2 t2 ; 2r 2
S a2
(2)
PðtÞ ¼ 2rr2 arccos
ffi t rt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4r2 t2 : 2r 2
(3)
P(t) for the grid pattern is shown in Fig. 4(a). Note that P(t) ¼ 1 for t r and P(t) ¼ 0 for t > 2r. As expected, P(t) decreases with the trip length t and increases with the density of stations r and the vehicle range r.
3.2.
Random pattern
Suppose that stations are uniformly and randomly distributed on a plane. For making the round trip, a station must be in the intersection of the two circles centered at O and D with radius r, as depicted in Fig. 3. P(t) is then the probability that the intersection of the two circles contains at least one station. The probability that a region of area S contains exactly x stations, denoted by P(x,S ), is given by the Poisson distribution as Pðx; SÞ ¼
ðrSÞx expðrSÞ x!
Fig. 3 e Calculation of the probability for the random pattern.
(4)
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Fig. 4 e Probability that the vehicle can make the repeated round trip.
where r is the density of stations [29]. From Eq. (1), P(t) is obtained as
that origin is within the distance r of a station and destination is within the distance r/2 of the station.
PðtÞ ¼ 1 Pð0; SÞ
(5)
4.1.
ffi t rt pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4r2 t2 : PðtÞ ¼ 1 exp 2rr2 arccos þ 2r 2
(6)
For making the round trip, O must be in the circle C1 centered at the station with radius r and D must be in the circle C2 centered at the station with radius r/2, as depicted in Fig. 5. This means that the midpoint of the OeD path must be in the shaded region in Fig. 5. This region is the intersection of the two circles which are obtained by moving the circle C1 by t/2 in the direction parallel to the OeD path and moving the circle C2 by t/2 in the opposite direction. P(t) is then the probability that the midpoint of the OeD path lies inside the intersection of the two circles. P(t) is given by the ratio between the area of the intersection of the two circles and that of the outer square. The area of the intersection of the two circles is
P(t) for the random pattern is shown in Fig. 4(b). Note that P(t) for the random pattern is smaller than that for the grid pattern.
4. Fuel is available at either origin or destination 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. To complete the round trip, the vehicle is required to reach D with at least half a tank remaining. If t r/ 2, the vehicle can start at O with full tank of fuel, reach D, and return to O without running out of fuel, i.e., P(t) ¼ 1. If t > 3r/2, the vehicle cannot make the round trip without refueling more than once, i.e., P(t) ¼ 0. Hence, we focus on the case where r/2 < t 3r/2. If r/2 < t 3r/2, the vehicle can make the round trip if O is within the distance r of a station and D is 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, P(t) is the probability
Fig. 5 e Calculation of the probability for the grid pattern.
Grid pattern
4t2 þ 3r2 r2 4t2 3r2 þ arccos S ¼r2 arccos 8rt 4 4rt sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi 1 3 4r2 t2 r2 þ t2 : 2 4
Fig. 6 e Calculation of the probability for the random pattern.
(7)
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Fig. 7 e Probability that the vehicle can make the repeated round trip.
From Eq. (2), P(t) is obtained as 4t þ 3r rr 4t 3r þ arccos PðtÞ ¼rr2 arccos 8rt 4 4rt sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi r 3 4r2 t2 r2 þ t2 : 2 4 2
2
2
2
5. Fuel is available at neither origin nor destination
2
(8)
P(t) for the grid pattern is shown in Fig. 7(a). Note that P(t) ¼ 1 for t r/2 and P(t) ¼ 0 for t > 3r/2. It can be seen that P(t) is smaller than that of the case where fuel is available at both origin and destination (see Fig. 4(a)).
4.2.
Random pattern
For making the round trip, a station must be in the intersection of the circle centered at O with radius r and the circle centered at D with radius r/2, as depicted in Fig. 6. P(t) is then the probability that the intersection of the two circles contains at least one station. From Eqs. (5) and (7), P(t) is obtained as
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 [18]. If t > r, the vehicle cannot make the round trip, i.e., P(t) ¼ 0. Hence, we focus on the case where t r. If t r, the vehicle can complete the round trip with at least half a tank remaining if both O and D are within the distance r/2 of a 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, P(t) is the probability that both origin and destination are within the distance r/2 of a station.
5.1.
Grid pattern
P(t) for the random pattern is shown in Fig. 7(b).
For making the round trip, both O and D must be in the circle C centered at the station with radius r/2, as depicted in Fig. 8. This means that the midpoint of the OeD path must be in the shaded region in Fig. 8. This region is the intersection of the two circles which are obtained by moving the circle C by t/2 in the direction parallel to the OeD path and in the opposite direction. P(t) is then the probability that the midpoint of the OeD path lies inside the intersection of the two circles. P(t) is given by the ratio between the area of the intersection of the two circles and that of the outer square. The area of the intersection of the two circles is
Fig. 8 e Calculation of the probability for the grid pattern.
Fig. 9 e Calculation of the probability for the random pattern.
(
4t2 þ 3r2 rr2 4t2 3r2 arccos 8rt 4 4rt sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi) r 3 4r2 t2 r2 þ t2 þ : 2 4
PðtÞ ¼1 exp
rr2 arccos
(9)
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Fig. 10 e Probability that the vehicle can make the repeated round trip.
S¼
r2 t t pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 t2 : arccos r 2 2
(10)
From Eq. (2), P(t) is obtained as PðtÞ ¼
rr2 t rt pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 t2 : arccos r 2 2
(11)
P(t) for the grid pattern is shown in Fig. 10(a). Note that P(t) ¼ 0 for t > r. In this case, even short distance trips are not always possible.
5.2.
Random pattern
For making the round trip, a station must be in the intersection of the two circles centered at O and D with radius r/2, as depicted in Fig. 9. P(t) is then the probability that the intersection of the two circles contains at least one station. From Eqs. (5) and (10), P(t) is obtained as rr2 t rt pffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 t2 : arccos þ PðtÞ ¼ 1 exp r 2 2
(12)
P(t) for the random pattern is shown in Fig. 10(b).
6.
Density of stations
Using the probability P(t) for the grid and random patterns, we can calculate the density of stations required to achieve a specified level of service. Table 1 shows the density of stations required to achieve a probability P(t) a (a ¼ 0.2, 0.4, 0.6, 0.8) for the random pattern. Note that, as the target probability increases and the trip length becomes longer, more stations are required. Note also that the effect of the refueling availability at origin and destination is significant. If fuel is available at origin and destination, no station is needed for short distance trips, because the vehicle can make the round trip without refueling. If fuel is available at neither origin nor destination, more stations are required than the other two cases to achieve the same level of service. The sufficient service level varies among regions according to the traffic condition. If fuel is available at many homes and workplaces, we can assume that fuel is available at both origin and destination. In the region where long distance trips are dominant, we should use a large value for the trip length t. Once the density of stations required to achieve the service
Table 1 e Density of stations required to achieve a probability P(t) ‡ a: (a) fuel is available at both origin and destination; (b) fuel is available at either origin or destination; (c) fuel is available at neither origin nor destination. a
(a)
(b)
(c)
0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8
Range of 1
Range of 2
Trip length t
Trip length t
0.25
0.50
0.75
1.00
1.25
1.50
0.50
1.00
1.50
2.00
2.50
3.00
0 0 0 0 0 0 0 0 0.41 0.95 1.70 2.99
0 0 0 0 0 0 0 0 0.73 1.66 2.98 5.24
0 0 0 0 0.37 0.85 1.53 2.69 1.97 4.51 8.09 14.20
0 0 0 0 0.64 1.46 2.61 4.59 e e e e
0.27 0.63 1.12 1.97 1.71 3.91 7.02 12.32 e e e e
0.49 1.13 2.02 3.55 e e e e e e e e
0 0 0 0 0 0 0 0 0.10 0.24 0.43 0.75
0 0 0 0 0 0 0 0 0.18 0.42 0.75 1.31
0 0 0 0 0.09 0.21 0.38 0.67 0.49 1.13 2.02 3.55
0 0 0 0 0.16 0.36 0.65 1.15 e e e e
0.07 0.16 0.28 0.49 0.43 0.98 1.75 3.08 e e e e
0.12 0.28 0.51 0.89 e e e e e e e e
“e” means the round trip is impossible.
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level is calculated, the sufficient number of stations can be obtained by multiplying the density by the area of the region.
7.
Conclusion
This paper has developed an analytical model for determining sufficient density of alternative fuel stations. The probability that the vehicle can make the repeated round trip between randomly selected origin and destination has been obtained for regular and random patterns of stations. The model that explicitly incorporates flow demand will give a more appropriate estimate for the sufficient density of refueling stations. The analytical expressions for the probability demonstrate how the density of stations, the vehicle range, and the trip length, as well as the refueling availability at origin and destination affect the service level of refueling stations. Note that finding these relationships by using discrete network models requires computation of the number of originedestination pairs that can make the round trip for each combination of the parameters. The relationships help planners to estimate the sufficient number of alternative fuel stations required to achieve a certain level of service. The estimated number of stations can be used as an input in location models of refueling stations. Comparing the effects on the service level will also be useful to prioritize investments for the transition to alternative fuel vehicles (e.g. subsidy for refueling stations). The present model assumes that drivers refuel their vehicles at most once for each one-way trip. If multiple refueling is allowed, the sufficient number of stations decreases but the inconvenience of drivers increases. Thus, there exists a tradeoff between the construction cost of stations and the travel cost of drivers. Incorporating the tradeoff into the estimation would be a topic for future research.
Acknowledgments This research was supported by Grant-in-Aid for Scientific Research (C) (24510187). I wish to thank anonymous reviewers for helpful comments and suggestions.
references
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