the electric vehicle shortest-walk problem with battery ... · the electric vehicle shortest-walk...

The Electric Vehicle Shortest-Walk ProblemWith Battery Exchanges

Jonathan D. Adler & Pitu B. Mirchandani &Guoliang Xue & Minjun Xia

Published online: 24 January 2014# Springer Science+Business Media New York 2014

Abstract Electric vehicles (EV) have received much attention in the last few years.Still, they have neither been widely accepted by commuters nor by organizations withservice fleets. It is predominately the lack of recharging infrastructure that is inhibitinga wide-scale adoption of EVs. The problem of using EVs is especially apparent in longtrips, or inter-city trips. Range anxiety, when the driver is concerned that the vehiclewill run out of charge before reaching the destination, is a major hindrance for themarket penetration of EVs. To develop a recharging infrastructure it is important toroute vehicles from origins to destinations with minimum detouring when batteryrecharging/exchange facilities are few and far between. This paper defines the EVshortest-walk problem to determine the route from a starting point to a destination withminimum detouring; this route may include cycles for detouring to recharge batteries.Two problem scenarios are studied: one is the problem of traveling from an origin to adestination to minimize the travel distance when any number of battery recharge/exchange stops may be made. The other is to travel from origin to destination whena maximum number of stops is specified. It is shown that both of these problems arepolynomially solvable and solution algorithms are provided. This paper also presentsanother new problem of finding the route that minimizes the maximum anxiety inducedby the route.

Keywords Constrained vehicle routing .Constrained shortest paths . Shortest pathswithrefueling . Electric vehicles . Alternative-fuel vehicles

Netw Spat Econ (2016) 16: –DOI 10.1007/s11067-013-9221-7

J. D. Adler :G. Xue :M. XiaSchool of Computing, Informatics and Decision Systems Engineering, Arizona State University,Tempe, AZ 85281, USA

P. B. Mirchandani (*)School of Computing, Informatics and Decision Systems Engineering, Arizona State University,P.O. Box 878809, Tempe, AZ 85287-8809, USAe-mail: [email protected]

155 173

1 Introduction

The environmental, geopolitical, and financial implications of the world’s dependenceon gasoline-powered vehicles are well known and documented, and much has beendone to lessen our dependence on gasoline. One thrust on this issue has been theembracing of the electric vehicles (EV) as an alternative to gasoline powered automo-biles. These vehicles have an electric motor rather than a gasoline engine, and a batteryto store the energy required to move the vehicle. Governments and automotivecompanies have recognized the value of these vehicles in helping the environment(Hacker et al. 2009), and are encouraging the ownership of EVs through economicincentives. States and cities are assisting owners of electric vehicles by creatingcharging stations for EVs in busy areas (Senart et al. 2010). For many electric vehicles,such as the Nissan LEAF or Chevrolet VOLT, the current method of recharging thevehicle battery is to plug the battery into the power grid at places like the home or office(Bakker 2011; Kurani et al. 2008). Because the battery has a limited capacity before itrequires a recharge, this method has the implicit assumption that vehicle will be usedonly for driving short distances. EV companies are trying to overcome this limitedrange requirement with fast charging stations: locations where a vehicle can be chargedin only a few minutes to near full capacity. Besides being much more costly to operaterapid recharge stations, the vehicles still take a more time to recharge than a standardgasoline vehicle would take to refuel (Botsford and Szczepanek 2009). These inherentproblems, combined with a lack of refueling infrastructure, are inhibiting a wide-scaleadoption of electric vehicles. These problems are especially apparent in longer trips, orinter-city trips. Range anxiety, when the driver is concerned that the vehicle will run outof charge before reaching the destination, is a major hindrance for the market penetra-tion of EVs (Jeeninga et al. 2002; Sovacool and Hirsh 2009; Yu et al. 2011). Hybridvehicles, which have both an electric motor and a gasoline engine, have been successfulsince they overcome the range anxiety of their owners by running on gasoline whenneeded. Since hybrids still require gasoline these vehicles do not fully mitigate theenvironmental consequences (Bradley and Frank 2009; Shiau et al. 2010).

Another refueling infrastructure design is to have quick battery exchange stations(BEs). These stations will remove a pallet of batteries that are nearly depleted from avehicle and replace the battery pallet with one that has already been charged (Shemer2012). This method of refueling has the advantage that it is reasonably quick. Theunfortunate downside is that all of the vehicles serviced by the battery exchange stationare required to use identical pallets and batteries. It is assumed here that the developersof these battery pallets will coalesce around a single common standard, as has been thecase for other car parts such as tires and wipers. In conjunction to the battery exchangeconcept, it is required for there to be a viable business model that provides a reasonableprofit for companies that establish battery exchange facilities for the public. Batteryexchange stations have been tried out by taxi vehicles in Tokyo in 2010 (Schultz 2010).In fact the country of Denmark is investigating the possibility of having sufficientbattery exchange locations so that the country relies on none, or very few, gasoline-powered vehicles (Mahony 2011).

Although the research team is working on several issues related to the design andoperations of the infrastructure for battery exchange or battery charging stations, thispaper focuses on only one aspect of the problem: finding the shortest walk from an

156 J. D. Adler et al.

origination point s to a destination t with minimum cost. This is route referred to as a“walk” as opposed to a “path” since a detour to exchange or charge a battery mayinclude repeat arcs which are normally not included in shortest paths. In fact, thisproblem applies to any scenario where there only a limited number of places for“refueling” which is true not only for EVs but also other alternative fuel vehicleswhere a refueling infrastructure still needs to be developed, for example hydrogenpowered vehicles (Ogden et al. 1999) where empty tanks or canisters are exchanged forfull ones at special stations.

Consider the underlying scenario. Taking a trip, especially one through lowlypopulated areas, requires the driver to plan when the vehicle will need to be refueled.Given the abundance of gasoline stations for standard vehicle, drivers of these vehiclesusually consider refueling only when their fuel tank is low. The search for a goodrefueling point can be further aided by navigation systems and smart phone apps, suchas Google Maps, that provide motorists the location of gasoline stations in the vicinity.In the case of EVs, planning refueling is more important than for gasoline vehicles,since there are few places to recharge. Thus, the vehicles will likely have to detour fromthe most direct route. Likewise, understanding routing would be even more critical forbattery exchange facilities, since the infrastructure would gradually involve so, at leastinitially, the density of battery exchange stations would be very low. Hence, one needsto develop models which look for the shortest routes from origins to destinations thatinclude detouring when necessary.

2 Literature Review

The problem of finding the shortest path for an electric vehicle was originally discussed byIchimori et al. (1981), where a vehicle has a limited battery charge and is allowed to stopand recharge at certain locations. This paper improves on that work by adding a limit tothe number of times the vehicle can stop, as well as running empirical tests to validate thealgorithm and describing several special case modifications (such as minimizing themaximum anxiety). We also provide an illustrated example of the algorithm. The problempresented by Ichimori et al. was also discussed by Lawler (2001) who sketched apolynomial algorithm for its solution. If each arc requires an amount of fuel that doesnot depend on the length of the arc, and the goal is to find the shortest path constrained onthe amount of fuel used (and the vehicle cannot stop to refuel), then the problem is exactlythe shortest weight-constrained path problem (Garey and Johnson 1979). This problem isNP-hard and has been discussed extensively in the literature (Handler and Zang 1980;Beasley and Christofides 1989; Desrochers and Soumis 1988; Xiao et al. 2005). Addingrefueling stations to the shortest weight-constrained path problem has been discussed byLaporte and Pascoal (2011) and Smith et al. (2012), where, since the fuel and lengthcomponents of the arcs are not related, the problem is still NP-hard. The problemdescribed in this paper can be solved in polynomial time since the fuel and lengthcomponents of each arc are equal; this allows for more efficient solutions. Sachenbacher(2010) has studied route planning for electric vehicles on a single battery use where partsof the trip can return energy to the battery.

Of course, there is a complementary location problem (not addressed here) where wewish to locate “refueling” stations (battery recharging, battery exchanging and, other

The Electric Vehicle Shortest-Walk Problem With Battery Exchanges 157

alternative refueling options can all addressed similarly) in a region where there arecurrently none. The problem of optimally locating such refueling stations has beeninvestigated by several researchers (Kuby 2005; Kuby and Lim 2007; Upchurchet al. 2009; Capar et al. 2012). Typically, they use modifications of flow capturingor flow interception models (Hodgson 1990; Berman et al. 1992; Rebello et al.1995), to cover as many origin-destination routes as possible with a given numberof stations. To compare proposed models, standard p -median and p -centerproblems (e.g. Mirchandani and Francis 1990) have been used as proxies formaximizing proximity to stations and coverage by stations, respectively, forlocating the stations. The developed models have been compared empiricallyfor specific scenarios in order to choose one location model over another (Limand Kuby 2010). However, these models do not take into consideration thelikely possibility of vehicles making detours to refuel; therefore the directconsideration of locating facilities to minimize detouring distances and orminimizing detouring stops have not been included in the model developments.Extensive research has been done into when drivers will detour off of the direct route(Ramaekers et al. 2013).

Another location issue relates to the effect of locating battery exchange-stationson origin-destination traffic patterns of the driving population. If travelers from originsto destinations do not choose the shortest routes but instead chose routes thatminimize their detouring costs due to refueling, then the addition of BE stationswould change traffic patterns to a new equilibrium. There has been some recentconsideration of the effect of EVs on traffic assignment and traffic equilibrium, butthe research is only on restricting the distances EV can travel and assumes norefueling. (Jiang et al. 2012). There have also been investigations into how the powerload caused by charging electric vehicles can be managed by altering the prices at publicEV charging stations (He et al. 2013).

3 The Shortest EV-Walk Problem

We will assume a network model. The prototypical problem is to find the “best”walk from a given origin to a given destination, possibly making battery exchanges(we will refer this generically to “refueling” when the context is clear). Anysolution walk must have no segment without refueling with a length greater thanc, where c is the distance the vehicle can travel on a full battery. The vehicle mustmake no more than p battery exchanges, where p>0 (when p=0 this is the trivialshortest path problem). We use the general term route to refer to the movement ofa EV on the network that includes a set of refueling stops along the EV walk.While there are many different criteria that could be used, initially consider theobjective being optimized is the minimization of travel distance of the route, i.e.the length of the walk the EV route takes.

Let G=(V,E) be an undirected network with node set V and edge set E, and let n=|V|and m=|E| be the number of vertices and edges respectively. Let vertices s, t∈Vrepresent the starting and ending points of a trip by the EV. Let dij denote the lengtheach edge (i,j)∈E and let R⊆V be the given set of BE locations. We define the EVshortest-walk problem (EV-SWP) as the problem of finding the shortest walk in G


starting at s and ending at t such that any walk contained in the path starting and endingat nodes in {s,t}∪R has length at most c and the vehicle stops to recharge at mostp times. In the unconstrained-stops version of the problem as many refueling stopsmake be taken as necessary (p=∞).

The EV-SWP can be formulated as a binary integer program. In our solution, the EVwill likely make several trips between different refueling stations. Let these walksbetween {s,t}∪R be called subtrips. The vehicle will make at most p+1 of thesesubtrips, since the vehicle can only stop at p refueling stations. Define the decisionvariable xijk for (i,j)∈E and k∈{1,…,p+1} as whether or not the EV will travel from ito j during its k th subtrip. Let yik for i∈R∪{t} and k∈{1,…,p} represent the decisionfor whether or not the vehicle stops at station i after subtrip k. The index i can also takethe value of t to represent the vehicle reaching the destination without having toexchange batteries the full p times and so it artificially exchanges at the end point.The integer program is now the following:

MinimizeX

k¼1

pþ1Xi; j: i; jð Þ∈E dijxijk ð1Þ

s:t:X

j: j;ið Þ∈E xjik−X

j: i; jð Þ∈E xijk ¼ 0 ∀i∈V∖ R∪ s; tf gð Þ; k ¼ 1;…; pþ 1 ð2Þ

Xj: j;ið Þ∈E xjik þ yik−1−

Xj: i; jð Þ∈E xijk−yik ¼ 0 ∀i∈R∪ tf g; k ¼ 2;…; p ð3Þ

Xj: j;ið Þ∈E xji1−

Xj: ijð Þ∈E xij1−yi1 ¼ 0 ∀i∈R∪ tf g ð4Þ

Xj: i; jð Þ∈E xijpþ1 þ yip−

Xj: j;ið Þ∈E xjipþ1 ¼ 0 ∀i∈R ð5Þ

Xj: s; jð Þ∈E xsjk−

Xj: j;sð Þ∈E xjsk ¼ 0 k ¼ 2;…; pþ 1 ð6Þ

Xj: s; jð Þ∈E xsj1−

Xj: j;sð Þ∈E xjs1 ¼ 1 ð7Þ

Xj: j;tð Þ∈E xjtpþ1 þ ytp−

Xj: t; jð Þ∈E xtjpþ1 ¼ 1 ð8Þ

Xi∈R∪ tf g yik ¼ 1 k ¼ 1;…; p ð9Þ

Xi∈V

Xj: i; jð Þ∈E dijxijk ≤c k ¼ 1;…; pþ 1 ð10Þ


xijk∈ 0; 1f g ∀i; j∈V : i; jð Þ∈E; k ¼ 1;…; pþ 1 ð11Þ

yik∈ 0; 1f g ∀i∈R∪ tf g; k ¼ 1;…; p ð12ÞConstraints (2) ensure the conservation laws hold for each vertex not in R∪{s,t}.

Constraints (3) through (6) ensure that conservation laws hold for the vertices inR∪{s,t}, since special care is need to ensure that when a battery is exchanged the nextsubtrip starts. Constraints (7) and (8) ensure that the vehicle starts at origin s and ends atdestination t. Constraint (9) ensures that between each subtrip exactly one batteryexchange station (or t) is visited. Constraint (10) ensures that the vehicle can indeedtraverse each subtrip without running out of battery power. Because the problem isformulated as a binary integer program here, using an off-the-shelf optimizationprogram to solve it will likely take non-polynomial time.

If there is no limit on the number of stops (i.e. the vehicle can make up to |R| stops)and the only concern is to minimize total distance, then it can be shown that theproblem can be easily solved in polynomial time using the standard shortest pathlabeling algorithm (Ahuja et al. 1993), albeit up to |R| times. We will first describethe algorithm with an illustration before we analyze it. Suppose we wish to travel fromvertex s to vertex t in the network of Fig. 1. The colored nodes in the figure arerefueling stations.

When the range c is large, say greater than 50, then the shortest path from s to t canbe found using a shortest path algorithm such as Dijkstra’s (Ahuja et al. 1993); the boldpath shown in Fig. 1 is the shortest path of length 39. Note that this path does not passany refueling points.

If the range was 20 then the vehicle will have to refill at least once to reach vertex t.The shortest path tree from s to all reachable nodes within distance c is shown in Fig. 2.As shown in the figure, two stations are reachable from s, the green station at vertex 5and the mauve station at vertex 9.

We then can do the same with these two stations as the starting points. Figure 3shows the range limited shortest path tree starting at the green station. Now from thegreen station at vertex 5 we can reach three stations, mauve (vertex 9), tan (vertex 11)and orange (vertex 13) at distances 4, 20, and 18 as shown in the boxed labels.

Fig. 1 The shortest unconstrained path from origin s to destination d


We can repeat this for each of the reachable stations and will come up with thefollowing network of range-limited shortest paths between stations, origin and desti-nation, where each of the edges correspond to a path in a range-limited shortest pathtree. We refer to this undirected network as the refueling shortest path network (RSPN)denoted by G'=(V',E'), and let n0=|V'| and m0=|E'|. Observe that RSPN can be obtainedin, at most, (|R|+1) iterations of the shortest path algorithm: one iteration for the startingnode and one for each of the stations. Figure 4 shows the RSPN for the example.

Now the shortest path in this network is s – 5 (green station) – 13 (orange station) – twith a distance of 43. This corresponds to EV walk in the original network s – 1 – 4 – 5(refuel) – 4 – 7 – 13 (refuel) – 14 – t. The paths s – 5, 5 – 13, and 13 – t are each subtripsin the solution. Note that the walk includes a cycle 4 – 5 – 4 and a detour 7 – 13 – 14 ascompared to the shortest segment 7 – 12 – 14 in the shortest path from s to t (see Fig. 5).

Theorem 1 The EV-shortest walk problem can be solved inO Rj j nlog2nþ mð Þð Þ time.

Fig. 2 The road network where the bold lines indicate the range-limited shortest path tree from s, and theboxed labels are the distances to reachable nodes from s

s

3

6

14

5 7 10

911

4

9

8

7

4

4

13

8

6

12 74

510

7

911

7

10

8

7

5

4

15

9

8

6

t

9

12

10

12

6

4

0

1

10

20

15

8

18

2

1

1

3

2

1

4

5

6

7

8

9

10

11

12

13

14

16

15

Fig. 3 The road network where the bold lines indicate the range-limited shortest path tree from vertex 5, andthe boxed labels are the distances to nodes reachable from vertex 5


Proof First note that we solved |R|+1 shortest path problems for starting node s and one foreach station in set R. The best known bound for a shortest path algorithm isO(nlog2n+m)where n is the number of vertices and m the number of edges in the originalnetwork. Finally, there is the step that solves the problem on the refueling shortest pathnetwork (see, e.g., Fig. 4) which has complexityO Rj j log2 Rj j þ m0ð Þð Þ if solved by thebest known algorithm; this is asymptotically dominated by O Rj j nlog2nþ mð Þð Þ. Thetheorem follows.

3.1 The Restricted Shortest EV-walk Problem

Up to now the number of stops to reach the destination has not been restricted. In therestricted case p<|R| stops, we need to find the solution to a stop-limited walk in RSPN.That is, we need the solution to the shortest walk between vertices s and t that has atmost p+1 edges in RSPN. It is not immediately clear if this problem still polynomialtime solvable. Still the structure of the problem allows us to develop a preprocessingpolynomial network transformation to the classical shortest path problem which ofcourse is polynomial solvable.

Fig. 4 The Refueling Shortest Path Network, RSPN, for the example, here the boxed numbers are the lengthsof the shortest feasible paths

s

3

6

14

5 7 10

911

4

9

7

4

4

13

8

6

12 74

510

7

911

7

10

8

7

5

4

15

9

8

6

t

2

1

1

3

2

1

4

5

6

7

8

9

10

11

12

13

14

16

15

Fig. 5 The solution EV shortest walk for the example, with different styled bolded lines indicating differentedges in the RSPN


Notice that for a given graph G and r1,r2∈R, there is a single shortest path to getbetween the two refueling stations r1 and r2 that does not depend on s and t. If we knowthat the vehicle is going to refuel at r1 then refuel next at r2, we do not need to knowany other information to find the path the vehicle will take between these two refuelingstations. If the length of the shortest path between r1 and r2 is greater than c, then nofeasible solution can have the vehicle refuel at r1 then refuel next at r2. So now we willcreate a directed graph where the vertices represent the refueling stations as well as sand t, and directed edges represent paths from stations and the start/end nodes to otherstations and start/end nodes that are reachable with fuel c. If we are restricted to prefueling stops, then p+2 copies of these directed arcs are needed. One may think ofeach network copy signifies the reachable nodes with a fully-charged battery (or with afull fuel tank). The RSPN for the example gives the multi-level network shown inFig. 6 when we are restricted to a maximum of 2 refueling stops.

We create a directed graph G''=(V'',A'') that is a transformation of RSPN, G'=(V',A').The vertex set V ''={x[i] :x∈{s,t}∪R,i∈{0,1,…,p+1}} has p+2 copies of each vertexin {s,t}∪R. We define the arc set as A''=A1

''∪A2'' where A1

''={(a[i],b[i+1]):(a',b')∈E',i∈{0,1,…,p}} and A2

''={(t[i],t[i+1]):i∈{0,1,…p}}. The distance mapping is definedfor arcs in A1

'' as d''(a[i],b[i+1])=d'(a',b') and for arcs in A2'' as d''(t[i],t[i+1])=0. Thus,

the distances in this new graph between levels are the same as those betweenvertices in the RSPN, except for the addition of zero distance edges which allowthe vehicle to go from any of the t[i] nodes to the t[p+1] node penalty free. Thisgraph has the property that any path from s[0] to t[p+1] contains exactly p+1 arcs,and corresponds to a path in G ' that travels from s to t in at most p+1 edges. Thus,to find the stop-limited shortest walk, we need to find the shortest path in G ' ' froms[0] to t[p+1]. The shortest path will contain exactly p intermediate nodes, and thenumber of refueling stations the vehicle will stop at corresponds to the number ofintermediate nodes in the path until reaching the first termination node t[i]. The bold

112

111

11

92

9

161313

16165

9134 169

15

15

9

2051

1820

52132

131

s2

s1

ts

t1

t2

91

49

18

161313

16164 169

15

15

9

2018

49

18

113

112

93

9

52

20

53

133

s3t3

9

161313

16164 169

15

15

9

2018

20

49

18

Level 1

Level 2

Level 3

0

0

0

Fig. 6 The multilevel directed network when the number of refueling stops is restricted to a maximum of two


edges in Fig. 6 show this shortest path for the example: s0 to 51 (stop at green station 5),51 to 132 (stop at orange station 13) and then to destination t3, with the travel distance 43as discovered earlier. For illustrative purpose we have indicated another path with adashed line in Fig. 6 from s0 to t3: s0 to 91 (stop at mauve station 9), 91 to 112 (stop at tanstation 11) and then to destination t3, with the travel distance 44 which does notcorrespond to a shortest walk.

Theorem 2 The p-stops limited EV-shortest walk problem can be solved inO p Rj j nlog2nþ mð Þð Þ time.

Proof Let T1 represent the time required to transformG' intoG'', and let T2 represent thetime required to find the shortest path in G'' from s[0] to t[p+1]. Note thatT1 ¼ O p n0 þ m0ð Þð Þ since it creates p+1 copies of V' and p copies of E'. Sincen 0=|R|+2, this means T 1 ¼ O p Rj j þ m0ð Þð Þ. Also, by the construction of G'' notethat |V ''(p+2)(|R|+2) and |E '' |≤(p+1)m 0. Thus finding the shortest path in G'' willtake T 2 ¼ O p Rj jlog2p Rj j þ m0ð Þð Þ time.

The overall run time T of the p-stops limited EV walk algorithm thereforeis from first finding G', the RSPN, whose complexity is O(|R|(n log2n+m)),plus T1 ¼ O p Rj j þ m0ð Þð Þ and plus T2 ¼ O p Rj jlog2p Rj j þ m0ð Þð Þ time. All terms aredominated byO p Rj j nlog2nþ mð Þð Þ. The theorem follows. The p -stops limited case hasa higher complexity due to having to make p+2 copies of the RSPN graph, which is notneeded if the number of stops is unconstrained.

4 Experimental Results

We tested the algorithm on randomly generated data to determine effectiveness of thealgorithm. When generating random networks, we wanted to ensure that the randomnetworks reasonably resembled potential real world road networks. This required thenetwork to be planar, and for vertices in the network to be positioned in ℝ2. Further theedges between vertices had to relate to the distances between the vertices in ℝ2. Thus,the random network data was generated in the following way. On input n, the nodes Vsuch that |V|=n were randomly selected from {1,2,…,100}2, representing points in adiscretized plane. The edges E were defined as the Delaunay triangulation of the pointsV, and for each edge (vi,vj)∈E, the length of the edge was set to be the L1 (Manhattan)distance between vi and vj. This method of generating the edges was used since it wouldallow for the edges to be both planar and reasonably sized in relation to the nodelocations. The vehicle was set to have a distance capacity c=35.

To determine which vertices should be the refueling stations, the following algo-rithm was implemented:

1. Initialize R as a random vertex v∈V.2. For each v∈V, find the minimum distance δ(v) between v and any element ofR∖{v}.3. Let �v be the vertex with the maximum distance. If δ �vð Þ≤c , then stop. Otherwise,

let ��v ∈V∖R such that ��v has the maximum distance of all vertices that are notalready refueling stations. Add ��v to R and repeat step 2.

n' m'

n' m'

m'

m'

m' m'


Picking station locations in this manner has the advantage that all of the vertices inthe network will be within battery distance of a refueling station. After R is determined,s and t are selected randomly from the vertices V∖R.

OnceG, R, s and t are generated, the graph is checked to ensure there exists a feasibleway for the vehicle to get from s to t using the refueling stations of R. If there is not, therandomly generated road network is thrown out. While this method generated data thathad many of the properties of real-world road networks, it did not allow for directcontrol of the number of refueling stations and the number of edges. Figure 7 shows anexample random network.

We generated 1,000 random networks with n=100, and compared three differentshortest routes

1. The shortest route if there was no distance constraint, i.e. the shortest path betweens and t.

2. The shortest walk if the vehicle had a fuel-distance constraint, but had no limit onthe amount of times it could stop to refuel.

3. The shortest walk if the vehicle was allowed to make one less stop thanthat in walk of (2).

In the vast majority of cases (985) there was no feasible stop-restricted walk, i.e. theshortest walk with distance constraints already also had the least number of stops. Thisis not unexpected, since detouring stops increase walk distances and, therefore, shorterfeasible routes tend to have fewer stops. However, as the number of battery-exchangestations increases so too should the chance that there exists a feasible path with fewerstops than the shortest walk. Figure 8 shows the runtime of the shortest walk algorithmwithout stop limit as a function of the number of refueling stations in the network. Eachtrial had a different number of edges, which explains some of the variance in theruntime. Overall, the runtime increases linearly as the number of charging stationsincreases, which is expected.

For the stop-restricted algorithm, Fig. 9 shows the number of allowed stops versus theruntime of the algorithm. This graph only includes generated instances where thenumber of allowed stops was at least 1. There does not appear to be a strong relationshipbetween the number of stops versus the runtime. This is due to the fact that finding theRSPN takes substantially more time than finding the shortest path in the multi-levelnetwork. Figure 9 includes a 3rd degree polynomial trend line fitted to the data.

Intersection

BE station

Start/end vertex

Fig. 7 An example randomly generated network and its corresponding RSPN


Figure 10 shows the length of the fuel constrained shortest walk compared to theunconstrained shortest path between the starting and ending nodes. In most cases theincrease in route length due to detouring is less than 10 %, however it can be over twiceas long due to unfortunate refueling station placement.

Fig. 8 The number of charging stations in the network versus the runtime of the algorithm

Fig. 9 The number stops allowed in the restricted algorithm versus the runtime of the algorithm


We also compared the solutions from our algorithm to the solutions generated byusing CPLEX to solve the integer programming formulation in equations (1)-(12). Wetook each of the 1,000 randomly generated networks used earlier and solved them usingthe IBM ILOG CPLEX IDE. We found that the CPLEX solver gave the same solutionfor each of the problems, typically in a matter of seconds, provided a solution existed.In the event that there was no feasible solution the algorithm did not seem to halt, evenafter twenty minutes. Thus to ensure a solution existed, for each of the 1,000 problemswe set p to be either the number of stops used in the unconstrained-stops version of theproblem, or 1 if the solution to the unconstrained-stops problem was a route withoutany battery-exchanges. A comparison of the runtimes of the integer program on the1,000 problems is shown in Fig. 11. While the vast majority of the problems ran inunder 5 seconds, several of the sample problems took well over 20 seconds and fourhad to be cut off after running for 10 minutes. This is compared to our algorithm inMATLAB which consistently ran under a tenth of a second. Table 1 shows the runtimesand route lengths for the EV-SWP algorithm in MATLAB compared to the CPLEX

0

100

200

300

400

500

600

700

1.0-

1.1

1.1-

1.2

1.2-

1.3

1.3-

1.4

1.4-

1.5

1.5-

1.6

1.6-

1.7

1.7-

1.8

1.8-

1.9

1.9-

2.0

2.0+

Nu

mb

er o

f tr

ials

Ratio of battery constrained length to unconstrained lengthFig. 10 A comparison of path lengths between the unconstrained shortest path and the EV shortest walk

0

100

200

300

400

500

600

700

0-1

1-2

2-3

3-4

4-5

5-6

6-7

7-8

8-9

9-1

0

10-

11

11-

12

12-

13

13-

14

14-

15

15-

16

16-

17

17-

18

18-

19

19-

20

20

+

Nu

mb

er o

f tr

ials

Run time (sec)Fig. 11 A histogram of the runtimes for the CPLEX formulation for the EV-SWP


implementation (when it successfully completed) and the unconstrained shortest pathalgorithm solved in MATLAB.

5 Special Cases

The algorithm can be adjusted to handle the case that the vehicle begins the trip withonly cL fuel, where cL<c. The only change required is to instead of using a givendistance function d, use a distance function dL defined as

dL eð Þ ¼ d eð Þ þ c−cLð Þ e adjacent to sd eð Þ otherwise

�

Using this distance function will cause the vehicle to artificially travel a less thandistance c−cL after the trip begins, which is equivalent to starting with only cL fuel.

The algorithm can also be altered for the case when the vehicle needs to go from s tot and back to s. Given G=(V,E), s, t, R, d, c, p, create a new undirected instanceG*=(V*,E*), s*, t*, R*, d*, c*, and p* where:

V � ¼ vi : v∈V ; i∈ 1; 2f gf g

E� ¼ a1; b1ð Þ : a; b∈V ; a; bð Þ∈Ef g∪ a2; b2ð Þ : a; b∈V ; a; bð Þ∈Ef g∪ t1; t2ð Þf g

d� ai; bið Þ ¼ d a; bð Þ ∀a; b∈V ; a; bð Þ∈E; i ¼ 1; 2f g

d� t1; t2ð Þ ¼ 0

s� ¼ s1; t� ¼ s2; R� ¼ ri : r∈R; i∈ 1; 2f gf g; c� ¼ c; p� ¼ p:

This new graph G* is two copies of the original graph G, where the vertices andedges are labelled by which copy they are in. The starting point is the vertex s in thefirst copy and the ending point is the vertex s in the second copy. The graph also has adistance zero edge between t1 and t2 (the vertex t in the two copies) which forces the

Table 1 A comparison of the runtimes for the EV-SWP in MATLAB and the formulation solved in CPLEX,along with the runtimes for unconstrained shortest path algorithm in MATLAB

n=1000 Runtime (seconds) Route Length

EV-SWP(MATLAB)

EV-SWP(CPLEX)

Shortest Path(MATLAB)

EV-SWP Unconstrained

Mean 0.0599 3.5755 0.0004 79.403 65.889

Median 0.0567 1.8570 0.0004 75.500 65.500

95th percentile 0.0641 5.7682 0.0005 162 118


vehicle to travel through the point t1, the original destination, on its way from s1 to s2.After the shortest walk is found, the {1,2} labels can be removed and the shortestroute corresponds to the optimal solution in the original graph. The algorithm forgoing to and from a destination is the same as the original problem on a graphthat is twice as large.

6 Routes that Minimize Anxiety

We now define a related problem shortest route problem that considers driver’s anxiety.Here we define the anxiety level of the driver as a monotonically increasing function ofthe charge (fuel) used from the battery from full; the lower the charge (fuel) the higherthe anxiety. We assume that like in previous sections the level of charge in the batterydecreases as the vehicle travels, until it reaches a refueling station. Hence, the level ofanxiety monotonically increases with distance traveled since fully charged. Every timethe vehicle recharges the driver’s anxiety drops to its minimum. Figure 12 gives atypically trajectory of anxiety as driver travels from s to t.

Notice that since anxiety is monotonic the driver’s anxiety is lowest when thevehicle is fully charged and reaches the highest point when the vehicle has traveledfurthest after refueling. Therefore the problem of minimizing the maximum anxiety issimply that of finding a walk from s to t with the minimum maximum edge in theRSPN. Because the RPSN is an undirected graph, finding a minimum spanning tree(MST) in the RSPN gives all the min-max paths in RSPN, in particular the min-maxpath from s to t (see Ahuja et al. 1993); the best complexity of finding MST is simplyO m0log2n

0ð Þ using Kruskal’s algorithm. As an example, the MST for the exampleRSPN is given in the bold lines in Fig. 13.

Now themin-max anxiety path is s - 1 - 4 - 5 (refuel) - 9 (refuel) - 8 - 11 (refuel) - 15 - twith the total travel distance of 44 whereas the walk that minimized distance haddistance length of 43. Notice that this route has three refueling stops with the maximumanxiety corresponding to the arc of length 16 from station 9 to station 11. The overallcomplexity for the unconstrained-stop case, including the time for building the RSPN, isO(|R|(n log2n+m)) since finding the minimum spanning tree is dominated bycomplexity in creating the RSPN.

If we were restricted to at most p stops on the s - t route, then the same solutionapproach will apply the the problem as to that of the restricted case for the shortest walk

Distance

Anxiety

Start Refuel 1 Refuel 2 EndFig. 12 Typical anxiety trajectory for a trip; here the highest anxiety is just before first refuel


problem. Again create a (p+2)-level directed RSPN network using the sameprocedure as Section 3 and look for a min-max directed path from s to t in thisnetwork. Such a problem was referred to as the bottleneck shortest path by Kaibeland Peinhardt (2006), where the bottleneck corresponds to the largest arc in thepath. Such a path can be found by modifying Dijkstra’s algorithm so that thetemporary labeling updates to next node j from node i uses the followingoperation:

New temporary label on node j ¼ max permanent label on i; new arc distancedij� �

Hence, the overall complexity is the same as for the shortest walk caseO p Rj j nlog2nþ mð Þð Þ: Kaibal and Peinhardt propose an algorithm for the bottleneckshortest path problem which runs in O m0loglogm0ð Þ time, and if implemented wouldgive a complexity of O Rj j nlog2nþ mð Þ þ m0loglogm0ð Þ , which may or may not belower depending on the graph. Returning to the example, if we were restricted tomaximum of 2 stops, then the solution min-max path is s0 – 91 – 112 – t3 which also hasa max arc length of 16 and a distance of 44 units.

This approach to finding the min-max anxiety can also handle the special cases fromSection 5. In the case where the trip begins with partially charged battery, simply add adummy arc (so,s) of distance corresponding to charge c−cL and repeat aboveprocedure to find min-max anxiety path with or without stop-restrictions. To find theshortest route from s to t and back, generate a new graph which has two copies of theoriginal graph using the procedure in Section 5, then run the min-max anxiety pathalgorithm on it.

7 Concluding Remarks

Although electric vehicles have been around for a while, they have neither been widelyaccepted by commuters nor by organizations with service fleets. The benefits forowning EVs are many, which include little or no emissions, energy efficiency, lessdependency on non-renewable resources and the ability to recharge at home or theoffice. It is predominately the “range anxiety” that discourages people and organizationsfrom owning EV, which can be mitigated by proper network planning. Establishing andoperating a battery exchange (or recharging) infrastructure is essential for EVs to have a

Fig. 13 The MST of the example RSPN


larger market share. The design of such an infrastructure requires one to minimize thedetouring necessary for battery recharging through proper vehicle routing, and requiresminimize waiting times at the stations to pick up recharged batteries by locating andsizing battery exchange network. This paper addressed the first problem of routing tominimize detouring.

For minimizing detouring, the EV shortest-walk problemwas defined to determine theroute from a starting point to a destination; this route may include cycles for detouring torecharge batteries. Two such problem scenarios were studied: one was the problem oftraveling from an origin to a destination to minimize the travel distance when one ormore battery recharge/exchange stops may be made; the other was to travel from originto destination when a maximum of p stops can be made. It was shown that both of theseproblems are polynomially solvable. The first problem requires several runs of a shortestpath algorithm for determining shortest path trees totalingO Rj j nlog2nþ mð Þð Þ elemen-tary operations, where |R|, n, and m are the number of located stations, number of nodesin the network, and number of arcs, respectively. The second problem requires anadditional network transformation and also several shortest path trees, totalingO p Rj j nlog2nþ mð Þð Þ operations. The algorithms were tested on randomly gener-ated data, and it was shown that they are quick and efficient to run. Other cases ofrouting with battery-exchange stops were analyzed, specifically the case when thevehicle starts partially charged, the case when the vehicle needs to make a roundtrip, and the case where the goal is to minimize driver anxiety.

The work presented in this paper could be extended to handle routing the vehicle in anetwork with stochastic arc lengths so that the vehicle has a limited probability of runningout of battery power. This is similar to finding reliable shortest pathswhich has been studiedfor gasoline vehicles (Chen et al. 2012) only the vehicle now can increase reliability bystopping to swap batteries. Rather than routing a single car, the problem could be modifiedfor routing many vehicles through stations that have limited batteries. The problem couldalso be merged with the refueling station location problem to find locations for batteryexchange stations that assume the users may take detours to reach them.

Acknowledgments This material is based upon work supported by the National Science Foundation underGrant No. 1234584 and by the U.S. Department of Transportation Federal Highway Administration under theDwight David Eisenhower Transportation Fellowship Program. Any opinions, findings, and conclusions orrecommendations expressed in this material are those of the authors and do not necessarily reflect the views ofthe above organizations.

References

Ahuja RK, Magnanati TL, Orlin JB (1993) Network Flows: Theory, Algorithms, and Applications. PrenticeHall, Englewood Cliffs, NJ

Bakker J (2011) Contesting range anxiety: the role of electric vehicle charging infrastructure in the transpor-tation transition. Eindhoven University of Technology

Beasley JE, Christofides N (1989) An algorithm for the resource constrained shortest path problem. Networks19:379–394

Berman O, Larson RC, Nikoletta F (1992) Optimal location of discretionary service facilities. Transp Sci 26:201–211

Botsford C, Szczepanek A (2009) Fast charging vs. slow charging: pros and cons for the new age of electricvehicles, in EVS24 International Battery, Hybrid and Fuel Cell Electric Vehicle Symposium, 1–9


Bradley TH, Frank A (2009) Design, demonstrations and sustainability impact assessments for plug-in hybridelectric vehicles. Renew Sust Energ Rev 13:115–128

Capar I, Kuby M, Rao B (2012) A new formulation of the flow-refueling location model for alternative-fuelstations considering candidate locations on arcs. IIE Trans 44(8):622–636

Chen BY, Lam WHK, Sumalee A, Li Q, Shao H, Fang Z (2012) Finding reliable shortest paths in roadnetworks under uncertainty. Networks and Spatial Economics 13(2):123–148

Desrochers M, Soumis FC (1988) A generalized permanent labelling algorithm for the shortest path problemwith time windows. INFOR 26(3):191–212

Garey MR, Johnson DS (1979) Computers and Intractability: A Guide to the Theory of NP-Completeness.W. F, Freeman, New York, NY

Hacker F, Harthan R, Matthes F (2009) Environmental impacts and impact on the electricity market of a largescale introduction of electric cars in Europe

Handler GY, Zang I (1980) A dual algorithm for the constrained shortest path problem. Networks 10:293–309He F, Yin Y, Wang J, Yang Y (2013) Optimal prices of electricity at public charging stations. Networks and

Spatial Ecnomics (Sustantability SI)Hodgson MJ (1990) A flow-capturing location-allocation model. Geogr Anal 22:270–279Ichimori T, Ishii H, Nishida T (1981) Routing a vehicle with the limitation of fuel. J Oper Res Soc Jpn 24(3):

277–281Jeeninga H, Arkel WGV, Volkers CH (2002) Performance and acceptance of electric and hybrid vehicles:

determination of attitude shifts and energy consumption of electric and hybrid vehicles used in theELCIDIS project. Muncipality of Rotterdam, Rotterdam, Netherlands

Jiang N, Xie C, Waller ST (2012) Path-constrained traffic assignment: model and algorithm. 91st AnnualMeeting of the Transportation Board

Kaibel V, Peinhardt MAF (2006) On the bottleneck shortest path problem. Konrad-Zuse-Zentrum fürInformationstechnik Berlin.

Kuby M (2005) The flow-refueling location problem for alternative-fuel vehicles. Socio Econ Plan Sci 39:125–145

Kuby M, Lim S (2007) Location of alternative-fuel stations using the flow-refueling location model anddispersion of candidate sites on arcs. Networks and Spatial Economics 7:129–152

Kurani KS, Heffner RR, Turrentine T (2008) Driving plug-in hybrid electric vehicles: reports from U.S.drivers of HEVs converted to PHEVs, circa 2006–07. Transportation Research Record: Journal of theTransportation Research Board 38–45

Laporte G, Pascoal MMB (2011) Minimum cost path problems with relays. Computers & OperationsResearch 38:165–173

Lawler EL (2001) Combinatorial Optimization: Networks and Matroids. 93Lim S, Kuby M (2010) Heuristic algorithms for siting alternative-fuel stations using the Flow-Refueling

Location Model. Eur J Oper Res 204:51–61Mahony H (2011) Denmark to be electric cars guinea pig. In: EUObserver.com. http://euobserver.com/

transport/32458Mirchandani PB, Francis RL (1990) Discrete Location Theory 1–576Ogden JM, Steinbugler MM, Kreutz TG (1999) A comparison of hydrogen, methanol and gasoline as fuels for

fuel cell vehicles: implications for vehicle design and infrastructure development. J Power Sources 79:143–168

Ramaekers K, Reumers S, Wets G, Cools S (2013) Modelling route choice decisions of car travelers usingcombined GPS and diary data. Networks and Spatial Economics 13(3):351–372

Rebello R, Agnetis A, Mirchandani PB (1995) The inspection station location problem in hazardous materialtransportation. INFOR 33:100

Sachenbacher M (2010) The shortest path problem revisited: optimal routing for electric vehicles. KI 2010:Advances in Artifical Intelligence, 33rd Annual German Conference on AI. pp 309–316

Schultz J (2010) Better place opens battery-swap station in tokyo for 90-Day taxi trial. In: The New YorkTimes. http://wheels.blogs.nytimes.com/2010/04/29/better-place-opens-battery-swap-station-in-tokyo-for-90-day-taxi-trial/. Accessed 2 Feb 2012

Senart A, Kurth S, Roux GL, et al. (2010) Assessment framework of plug-in electric vehicles strategies. SmartGrid Communications, 2010 First IEEE International Conference on 155–160

Shemer N (2012) “Better place unveils battery-swap network,” Jerusalem PostShiau C-SN,Kaushal N, HendricksonCTet al (2010) Optimal plug-in hybrid electric vehicle design and allocation

for minimum life cycle cost, petroleum consumption, and greenhouse gas emissions. J Mech Des 132:1–11Smith OJ, Boland N, Waterer H (2012) Solving shortest path problems with a weight constraint and

replenishment arcs. Computers and Operations Research 39:964–984


http://euobserver.com/transport/32458

http://euobserver.com/transport/32458

http://wheels.blogs.nytimes.com/2010/04/29/better-place-opens-battery-swap-station-in-tokyo-for-90-day-taxi-trial/

http://wheels.blogs.nytimes.com/2010/04/29/better-place-opens-battery-swap-station-in-tokyo-for-90-day-taxi-trial/

Sovacool BK, Hirsh RF (2009) Beyond batteries: An examination of the benefits and barriers to plug-in hybridelectric vehicles (PHEVs) and a vehicle-to-grid (V2G) transition. Energy Policy 37:1095–1103

Upchurch C, Kuby M, Lim S (2009) A model for location of capacitated alternative-fuel stations. Geogr Anal41:85–106

Xiao Y, Thulasiraman K, Xue G, Jüttner A (2005) The constrained shortest path problem: algorithmicapproaches and an algebraic study with generalization. AKCE International Journal of Graphs andCombinatorics 2(2):63–86

Yu ASO, Silva LLC, Chu CL, et al. (2011) Electric vehicles: struggles in creating a market. TechnologyManagement in the Energy Smart World (Portland, OR), 1–13


the electric vehicle shortest-walk problem with battery ... · the electric vehicle shortest-walk...

Documents