Quick and Energy-Efficient Routes Computing Constrained Shortest Paths for Electric Vehicles

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1 Quick and Energy-Efficient Route Computing Contrained Shortet Path for Electric Vehicle Sabine Storandt FMI, Univerität Stuttgart Stuttgart, Germany ABSTRACT In thi paper we tudy multi-criteria routing problem related to Electric Vehicle (EV). Becaue EV till uffer from a rather mall cruiing range retricted by the battery capacity, and loading tation are pare and reloading i time intenive, previou work focued on computing the mot energy-efficient route efficiently. Unfortunately thee approache do not guarantee anything in term of ditance or travel time. But even a very eco-friendly driver might not be willing to accept a tour three time a long a the quicket one to ave ome energy. Therefore we preent new type of querie conidering energy-conumption and ditance or travel time and reloading effort, e.g. computing the hortet or quicket path on which the EV doe not run out of energy while requiring at mot k recharging event (with k being an input parameter). The repective optimization problem are intance of the contrained hortet path problem, which i NP-hard. Neverthele we will provide preproceing technique that allow for fat query anwering even in large treet graph. Categorie and Subject Decriptor G2.2 [DISCRETE MATHEMATICS]: Graph algorithm General Term Algorithm Keyword Route Planning, Contrained Shortet Path, E-Mobility 1. INTRODUCTION In recent year E-mobility ha been identified a important mean to reduce the conumption of foil fuel. Electric Vehicle (EV) are battery-powered and the neceary electricity can be produced from regenerative ource. Furthermore EV typically exhibit lower emiion to their immediate environment in term of combution gae or noie level. To accelerate the widepread tranition to E-Mobility everal Permiion to make digital or hard copie of all or part of thi work for peronal or claroom ue i granted without fee provided that copie are not made or ditributed for profit or commercial advantage and that copie bear thi notice and the full citation on the firt page. To copy otherwie, to republih, to pot on erver or to reditribute to lit, require prior pecific permiion and/or a fee. ACM SIGSPATIAL IWCTS 12, November 6, Redondo Beach, CA, USA Copyright c 2012 ACM ISBN /12/11...$15.00 Figure 1: Small example for the diverity of path: The energy-optimal path (green, bottom) i 9.203km long, the hortet path (purple, top) only 8.824km. The hortet path on which the EV doe not run out of energy (black, middle) i only 11m longer than the hortet one. government offer reduced taxe for EV compared to fuel driven car, and provide federal funding for the development of green technologie. Neverthele Electric Vehicle till wait for their great breakthrough. One reaon might be the retricted cruiing range of actual EV (up to 150km) reulting from the limited battery capacity. Indeed in ue, there i the poibility to recuperate energy during deceleration phae or when going downhill (a long a the battery capacity i not exceeded!), but till recharging will be neceary when driving longer tour. Here further inconvenience become noticeable: On the one hand loading tation (LS) are till pare in mot countrie certainly not a widepread a ga tation therefore one can not jut plan a trip and rely on the preence of nearby LS when the battery charge level drop. On the other hand reloading take everal hour hence it hould be avoided if poible. Therefore the energy-conumption along a path hould be a mall a poible to tay mobile. At the ame time people are certainly not willing to accept coniderably longer travel time jut to ave a few kwh of energy. Therefore a fair trade-off need to be achieved. The following quetion reflecting multi-criteria objective are natural from a uer perpective (ee Figure 1 for a bit of illutration): 1. What i the hortet or quicket path on which the EV doe not run out of energy? 2. What i the mot energy-efficient path which i at mot x time longer than the hortet one (e.g. x = 1.05)? A path obeying the battery contraint (i.e. the EV doe not run on energy while repecting the prohibition of overcharg-

2 ing the battery) i alo called feaible. If without reloading there exit no feaible path from ource to target, the deciion how often and where to recharge have to be made reaonably. So now we have to balance ditance or time to travel, energy conumption and reloading effort. Thi lead to lightly more complex optimality criteria, which are reflected in the following query type: 3. Find the hortet/quicket feaible path with at mot k recharging event. 4. Find a feaible path with a minimal number of recharging event and bounded ditance/travel time. Note that the problem demand the input of a parameter which allow the uer to modulate between eco-friendly and fat or hort route. Thee kind of querie are called cutomizable becaue the parameter i revealed at runtime only. Thee four optimization problem are intance of or involve the olution of the contrained hortet path problem (CSP), which i NP-hard in general and hence the exitence of polytime algorithm i doubtful. Neverthele we will preent technique which guarantee optimal query anwering in a reaonable time pan for real-world network. Thi enable uer to plan their EV trip according to their preference and permit navigation of EV in large treet network. 1.1 Related Work Becaue of long recharging time and the limited cruiing range of EV, the firt approach for enible route planning wa finding the path with minimal total energy-conumption. Computing uch an energy optimal path for an EV require a lot more effort than conventional route planning. On the one hand the recuperation capability introduce negative edge weight to the graph, a there might be a urplu of energy on a certain road egment. Thi inhibit the direct appliance of Dijktra algorithm, which i the crucial ingredient in mot route planning application. On the other hand the EV mut not run out of energy on the uggeted path and i alo not allowed to overcharge the battery (when recuperating). The problem of finding a path that meet thee requirement (the battery contraint) while minimizing the total energy conumption from ource to target wa introduced in [3]. Here, the author provided an extenion of the Bellman-Ford algorithm that olve the problem in O(nm) (n being the number of node and m the number of edge). In [4] proper cot model and preproceing technique were introduced that reduce both the theoretical and practical runtime of a query: Firt the contant edge cot repreenting energy conumption (e.g. in kwh) were tranformed into edge cot function which fulfil the FIFO-property, hence allowing traightforward ue of Bellman-Ford. Then the partly negative edge cot function were tranformed into non-negative one by uing an adaption of Johnon hifting technique. Upon that bai Dijktra algorithm can be applied, leading to ubequent query time of O(n log(n)+m). Finally the preproceing technique contraction hierarchie [5] wa adapted to the cenario, leading to a total peed-up of two order of magnitude. So very efficient retrieval of energy-optimal path i poible now. But the energy-conumption minimizing path might not alway be the deired route. Many owner of EV can recharge their EV in their garage; therefore driving home after work, the goal i rather to get there a fat a poible than to maximize the battery load at the detination. Accordingly the quicket or hortet path that i feaible i more appropriate here. The technique preented o far do not guarantee anything in term of travel time or ditance. Of coure they alo dealt with contraint, namely the battery contraint. But note that only one metric energy conumption wa involved up to now. So the contraint and the objective were both concerned with the ame metric. In contrat, incorporating travel time or ditance, a econd metric get introduced to the problem and putting a contraint on the one metric while minimizing the other, we end up with intance of CSP. Therefore we can no longer hope for a polytime algorithm like for the previouly tudied problem. Neverthele in [6] it wa hown that peed-up technique originally developed for the conventional hortet path problem like contraction hierarchie or arc-flag can be adapted to olve CSP, enabling practical query time for graph with up to a million node and edge. Note that for conventional CSP intance both involved metric are repreented a contant cot along the edge. But uing the energy conumption edge cot function from [4] a one of the metric, the etting get more complicated and the technique preented in [6] can not be applied traightforward. Alo reloading deciion have to be taken into account. Thi topic wa already tackled in [7]. Here the author aimed for a path with a minimum number of neceary recharging event. Thi problem can be olved in polytime becaue beide feaibility nothing ele i guaranteed for the reulting path. The approach i reaonable under the precondition of very time intenive reloading. But currently more and more o-called battery witch tation get etablihed. Here the complete battery ytem i wapped for a fully loaded one, taking coniderably le time than recharging. Still having to witch the battery i an inconvenience, but the uer might not accept arbitrary long detour anymore jut to avoid thi proce. 1.2 Contribution and Outline For the firt two problem computing the hortet feaible path and the mot energy-efficient one with bounded ditance we preent in Section 3 an adaption of contraction hierarchie for CSP which can alo deal with edge cot function intead of contant value only. Moreover we take loading tation into account and decribe on that bai the contruction of an auxiliary graph, which help to find the optimal path efficiently. Upon that, in Section 4 we propoe preproceing technique and algorithm, that olve the more complex optimization problem 3. and 4. where recharging i limited or minimized. Finally we evaluated the propoed approache on real-world data. The repective experimental reult can be found in Section PRELIMINARIES Baically we are given the capacity of the EV battery M R +, a treet (di)graph G(V, E), a cot function repreenting ditance or travel time d : E R and a function f e on each edge e E repreenting energy conumption. Thee function are modelled in conitency with [4] a follow: Let c : E R be the contant energy cot (in kwh), which might be partly negative (due to recuperation). To atify the battery contraint, it mut be aured that no

3 overcharging and no running out of energy i poible. So let e = (v, w) be ome edge with contant cot c(e), and b(v) the battery load of the EV at node v. If c(e) > 0 the battery load b(v) might be inufficient to ue the edge. To make ure that no optimal path will contain e in uch a cae, we et the repective edge cot to if c(e) > b(v). So for poitive contant cot the function look a follow: { b(v) < c(e) f e(b(v)) = c(e) otherwie If c(e) < 0 the edge can alway be ued, but one mut be careful not to overcharge the battery, i.e. if b(v) c(e) > M the final load at w will till be M only. Thi can be achieved uing the following edge cot function: { c(e) b(v) < M + c(e) f e(b(v)) = b(v) M otherwie So a feaible path p =, v 1,, v k, t from ource = v 0 V to target t = v k+1 V i characterized by b(v i) [0, M] for all i = 0,, k 1. In congruency with [7] we call a node w reachable from v if there exit a feaible path p = v,, w. The et of all node reachable from v i denoted by R(v), the et of node from which v can be reached by R 1 (v). Node in the latter et are alo called invere reachable from v. Both et can be computed efficiently a decribed in [7]. Label Setting: In the general CSP etting, there are contant cot c : E R + and contant reource conumption r : E R +. Moreover on query time, we are given a ource/ target, t V a well a a bound B R + on the reource conumption. The goal i to find the path from to t which minimize the cot while not exceeding the reource bound B. One claical approach to olve intance of CSP i the label etting computation (LSC) [1]. A LSC aign to each node v the lit of all pareto-optimal tuple (c(p), r(p)) for an -v-path p. Pareto-optimal mean that there i no dominating path p with c(p ) c(p) and r(p ) r(p) (and inequality hold at leat once). Thee label are created by an approach very imilar to Dijktra algorithm for computing plain hortet path. Here alo label are tored in a priority queue (PQ). A label i a triple coniting of a node ID, cot and reource conumption. The PQ ort the label in the increaing order of cot. The initial PQ contain only the label (, 0, 0). In every round the label (v, c, r) with minimal cot c i extracted and for the repective node v all outgoing edge e = (v, w) become relaxed, i.e. if (c = c + c(e), r = r + r(e)) i pareto-optimal for w the new label (w, c, r ) get puhed into the PQ. If (c, r ) dominate any olution that wa already aigned to w the dominated olution get pruned. Simple Pruning: The number of label that can be aigned to a ingle node i exponential and o i the runtime of LSC. Therefore pruning the earch pace i of utmot importance to achieve practical runtime in larger graph. A imple attempt to reduce the graph ize i baed on minimal reource label [2]. Here the minimal reource label r min of a node v i the minimal reource conumption of an -tpath that viit v. Obviouly, all node with r min(v) > B can never be on a feaible path, and hence thee node a well a their adjacent edge can be excluded a priori. Thi condition can be checked efficiently for all node by running two conventional Dijktra computation on the reource conumption tarting in, and t (on the revered graph). Afterward the two reulting label r (v) and r t(v) are ummed for each node to receive r min(v). Contraction Hierarchie: The peed-up technique contraction hierarchie (CH) wa introduced in [5]. The baic idea i augmenting the graph with hortcut that allow to kip a lot of edge relaxation at query time. To that end, in a preproceing phae an importance value i aigned to each node, and node are orted in increaing order of importance. Afterward the node get removed/ contracted one by one in that order, while preerving all hortet path ditance in the remaining graph by inerting additional edge (o called hortcut). More preciely, when contracting a node v for every path u, v, w the ditance from u to w mut tay unchanged. Therefore the edge e = (u, w) ha to be added if the only hortet path from u to w i u, v, w. The cot of e equal the chained cot of the edge (u, v) and (v, w). If there exit a path with cot le than the one of u, v, w, a o called witne path i found (normally via a Dijktra run from u to w) and the hortcut can be omitted. After all node have been removed, a new graph G i created coniting of all node and edge of the original graph and all hortcut. An edge e = (v, w) (original or hortcut) i called upward if the importance of v i maller than that of w and downward otherwie. In G -t-querie can be anwered bidirectionally, with the forward run (tarting at ) conidering only upward edge and the backward run (tarting at t) conidering excluively downward edge. Therefore thi trategy prune the repective earch pace dramatically. If unidirectional query anwering i neceary, all edge on downward path ending in t can be marked via a run of breadth-firt-earch and the forward run from conidering edge on upward path and the marked edge will provide the optimal olution. In [4] the technique wa adapted to the energy conumption function. Here one mut aure that all energy-optimal path are maintained during the contraction proce. In particular thi mean, that for every poible battery load b(u) [0, M] the optimal path to w i not allowed to viit v in order to omit the hortcut. A CH-variant for CSP wa introduced in [6]. In thi cenario a witne path mut dominate the reference path u, v, w in order to make the hortcut (u, w) uperfluou. In the following we will decribe how thee two approache can be plugged together to olve our CSP intance with energy conumption function a cot or reource conumption. 3. SHORT & ENERGY-EFFICIENT PATHS Energy-optimal Path with Bounded Ditance: Even very eco-friendly EV owner might not accept driving a route three time a long a the repective hortet path in order to ave energy. But mot likely they would be willing to drive a detour of e.g. 10% of the hortet path ditance. Hence in a realitic etting we earch for a route which obey the battery contraint, but i not longer than ome given ditance bound B. So we are confronted with an intance of CSP, where the cot are given by a function repreenting the energy con-

4 umption, the reource uage i equivalent to the edge length, and the reource bound i given by B = x π(, t) with x 1 being choen by the uer and π(, t) being the hortet path ditance. Hence a LSC will aign tuple (c(p), r(p) = d(p)) to the node coniting of ummed cot and ditance. Note that the ummed cot equal the difference between initial and actual battery charge tatu, therefore we could alo aign tuple (b, d(p)), where the ummed cot are replaced with the actual battery load. Here of coure the tuple dominate (b, d (p)), if b b and d(p) d (p). Simple pruning can be performed conventionally in thi cae by running two Dijktra computation tarting at and t repectively and conidering only the ditance value. To peed up the LSC via CH, we firt have to clarify how a witne i characterized in thi cenario. Of coure we can only omit a hortcut (u, w) for a path p = u, v, w if for any initial battery load b(u) = I [0, M] there exit an alternative path which i not longer than p and lead to at leat the ame final battery load in w a p. Oberve that equivalent to the CH contruction for uncontrained energyoptimal path, there i no neceity that thi i the ame path for every I. Therefore a witne can be decribed a et of path q 1,, q t with min(f q1,, f qk ) f p (f being the chained energy conumption function of all path edge) and d(q i) d(p) i = 1,, k. To perform the witne earch efficiently we chooe a et of initial battery load I 1, like in [4] but tart a LSC with reource bound d(p) intead of a imple Dijktra computation for every b(u) = I. Shortet/Quicket Feaible Path: If the objective i reaching the detination a fat a poible without running out of energy, no additional parameter i required to determine the reource bound. Intead the bound i alway given implicitly by the battery capacity M. In thi cenario the cot are contant value (ditance or travel time) while the reource conumption i decribed by the energy conumption function. Accordingly the imple pruning technique ha to be adapted. Here we firt compute R() and R 1 (t) and tore the repective battery load b and b min for all contained node. All node v V with b(v) < b min(v) are not part of any feaible path from to t and hence can be ignored completely (thi i epecially true for v / R() R 1 (t)). Oberve that the CH-graph can be obtained jut like decribed in the lat ection (or reued), becaue the definition for a witne propoed there i till valid in thi cenario. Regarding Loading Station: The preence of loading tation might affect the hortet feaible route or might be the bai for the exitence of a feaible path at all. A LS i a node v V at which the EV completely recharge whenever it viit. The et of LS i denoted by L V. To incorporate LS efficiently, we make the following obervation: Firtly an optimal path viit at mot L (all) LS once, becaue any cycle tarting and ending at a LS would increae the ditance, but could not lead to a higher battery charge tatu at that vertex. Secondly the pairwie hortet feaible path between LS are not influenced by the choice of and t; therefore thoe path can be preproceed. Thee two obervation give rie to the contruction of an auxiliary graph imilar to the one propoed in [7]. Here we create a node for every loading tation l L and edge (l, l ), if there exit a feaible path from l to l (without the uage of any further LS). In contrat to the approach decribed in [7], we now weigh the edge with the hortet feaible path ditance (we refer to thi graph alo a reach graph). At query time we have to add the node and t and proper edge. For thi purpoe we firt compute R() and R 1 (t) and interect thee et with L to derive the directly (invere) reachable LS. For every l R() L we have to compute the hortet feaible path from to l. Of coure thi can be achieved imultaneouly for all of the concerned LS via a ingle LSC. The ame yield for t reverely. After adding the edge (, l), l R() L and (l, t), l R 1 (t) to the auxiliary graph, a ingle Dijktra run from v to v t return the deired equence of LS on the hortet feaible path from to t. To benefit from applying CH when computing R() and R 1 (t) we leave all LS node uncontracted. Therefore they all lay on upward path from and downward path to t, i.e. exploring the repective graph i ufficient to determine the et of (invere) reachable LS. Moreover thi peed up the computation of the auxiliary graph, becaue the hortet feaible ditance from one LS to all the other can be computed in the remaining (much maller) graph with a plain LSC. Naturally uch a route might contain a large number of recharging event, becaue a fully loaded battery might allow for driving a much horter/quicker tour than a heavily dicharged one. In cae of minimizing travel time, we can find a more reaonable path by conidering alo the time needed for the recharging event. Thee penalty time have to be added to all edge in the auxiliary graph which end in a LS. Then again a ingle run of Dijktra algorithm lead to the optimal equence of LS, ee Figure 2 for a mall example (+3) 8(+3) v 1 v (+3) 2(+3) Figure 2: Computing the quicket feaible path in the preence of loading tation. If no penalty time for viiting a LS are aigned, the optimal path (orange, c = 11) include two recharging event. If penalty time are added (value in bracket, here 3 unit for each LS), the new optimal path (green, c = ) require recharging only once. 4. CONTROLLING THE NUMBER OF RECHARGING EVENTS In the lat ection we howed how one can incorporate penalty time for recharging event in order to keep their number mall on quicket feaible path. But it i difficult to etimate the penalty time reaonably, a they actually include poibly waiting time, time to intall the new battery (if it i completely witched, otherwie time to recharge), time to pay and o on. Moreover if ditance i minimized intead of travel time, the optimal path till might include a lot of recharging event. Therefore we will now preent two way to control the number of recharging event without accepting long detour. t

5 k L R() L L L L R 1 (t) L t Figure 3: Layered graph to find the hortet feaible path from to t viiting with at mot k loading tation. Each node ha an edge to every node in the next layer (black) apart from edge that indicate the viit of the ame LS conecutively. Node in R 1 (t) L have an additional edge to t (blue). Edge are weighted according to the hortet feaible path ditance in G neglecting LS ( if no feaible path exit). Limiting Recharging Event: While the natural limit of recharging event i L, a driver i normally only willing to accept a few recharging event, e.g. one for mall tour and three for longer tour. Therefore we would like to compute the hortet feaible path that require at mot k recharging event. For that purpoe we turn the auxiliary reach-graph propoed in the lat ection into a layered graph a depicted in Figure 3. The main and invariant part of the graph conit of L layer each containing a vertex v l for every loading tation l L. We call i(v l) {1,, L } the layer index of v l. Between equent layer j and j + 1, there exit all edge (v l, v l ) l l with i(v l ) = j and i(v l ) = j +1. Edge are weighted according to the hortet feaible path ditance that can be achieved from l to l in G without uing any other LS. To anwer an -t-query we have to augment the layered graph a well. We inert v into the layered graph at layer 0 and connect it to the firt layer by adding edge (v, v l ) l R() L, i(v l ) = 1 with the precomputed ditance. Then we inert v t at layer L + 1. To enable path with an arbitrary number of LS, we have to connect every layer to t. Hence we add edge (v l, v t) to the layered graph with l R 1 (t) and i(v l ) = 1,, L. For a given bound k on the number of recharging event, we only connect the layer 1,, k to t or ignore all layer with an index of k + 1 or higher. In the reulting graph a plain Dijktra computation reveal the optimal path. Minimizing the Number of Recharging Event under a Ditance Contraint: A uer could alo be intereted to input a limit on the path length and ak for the path with a minimum number of recharging event and the length not exceeding the given limit. For thi purpoe we could ue the ame layered graph a for finding the hortet feaible path regarding LS, but now let the ditance be the reource conumption and the actual cot are 1 for all edge not adjacent to t and 0 otherwie, ee Figure 4 (left) for a mall example. A LSC from to t in the layered graph with the given ditance limit anwer the query. But a the layered graph i a DAG (directed and acyclic) and the reource conumption for all node except t i equivalent to the node layer index, the LSC will aign only one pareto-optimal tuple to every node except t, namely the index combined with the hortet path ditance from to the node in the layered graph. Accordingly there will be at mot L + 1 tuple aigned to t a there are only L + 1 different cot value poible. The label for v V \ {t} could alo be obtained by computing the hortet path ditance for all node in the original layered graph without conidering edge adjacent to t. A thi graph i alo a DAG the ditance label are computed layerwie. For the firt layer thi require time O( L ) a there are only L incoming edge, for every ubequent layer we need time O( L 2 ). A oon a the ditance label of a node v i ettled, we relax the edge (v, t) (if it exit). If we aign a ditance label to t that doe not exceed the limit, we are done and the minimal number of recharging event k min equal the layer index i(v). Accordingly the runtime of thi approach i in O(k min L 2 ) (excluding the runtime for creating the graph with edge from/to /t). Of coure, we could alo retrieve the et of all pareto-optimal olution for t by toring the tuple (i(v), d(v) + c(v, t)) in a lit and pruning dominated one (ee Figure 4, right). Becaue the node get pared in layer order we only have to compare a new tuple to the tail of the lit to decide whether one of them need to be pruned. Hence we need L comparion and the overall runtime i O( L 3 ). Oberve that if for all node v in layer j 1 the tuple (j, d(v)+c(v, t)) i not pareto-optimal for t, then for all node w in a layer with an index j j the tuple (j, d(w)+c(w, t)) i not pareto-optimal for t a well. Accordingly we can abort the earch for pareto-optimal label for t a oon a we derive no pareto-optimal olution from a certain layer. While thi doe of coure not change the theoretical runtime, it might ave a lot of time in practice. 5. EXPERIMENTAL RESULTS We implemented the decribed approache in C++ and evaluated their performance on three tet graph, conidering varying cruiing range (CR) and number of loading tation (where taken into account). The tet graph are cutout of the treet network of Germany and baed on OpenStreetMap data: Winnenden (W) with node and edge, Baden-Württemberg (B) with node and edge, and Southern Germany (S) with node and edge. All timing (t) were taken on a ingle core of an Intel i3-2310m proceor with 2.1 GHz and 8 GB RAM. Firt we computed quicket feaible path. Without any preproceing the LSC take everal econd and about 5M priority queue poll in Southern German (ee the third and fouth column of Table 1).The CH only add about CR LSC CH CH-LSC (km) poll t(m) t() edge poll t(m) W B S Table 1: Experimental reult for CH-contruction and computation of quicket feaible path (averaged over 1000 random querie).

6 (0,25) (0,4) v 1 v 3 (1,6) (0,4) (1,10) (0,12) (1,3) (1,5) v 2 v 4 (0,12) t 10 3 v 1 v v 3 v 4 v l(v) d(v) c(v, t) label(t) (0,25) v (1,14) v (1,15) v (2,12) v (2,28) Figure 4: Example for finding the path with a minimum number of recharging event under a ditance contraint. On the left the augmented layered graph i given on which a label etting computation will output the optimal path. On the right an alternative approach i preented: Firt the hortet path ditance of all node v {, v 1, v 2, v 3, v 4} are computed in the diminihed layered graph. Then all poible pareto-optimal label for t are derived by paring through the node in layer order and combine their layer index with the minimal ditance from to t over v. Dominated tuple get pruned (croed out in the table). # CR reach-graph query reach avg/max LS (km) t() edge t() able # reload W / /0 B / / /4 S / / / / / /5 Table 2: Experimental reult for computing quicket feaible path regarding loading tation (averaged over 10 randomly choen et of LS and 100 ubequent random querie each). reachable denote the fraction of target to which a feaible path from the ource (via LS) exited. time the number of original edge a hortcut to the graph (ee the total number of original + hortcut edge in column 6) which i comparable to the conventional cae for hortet path. Note that a mall cruiing range lead to more hortcut inertion, becaue a greater diverity of path i optimal for different battery charge level. Therefore and becaue the runtime for pure LSC correlate with the CR the peed-up for the 25km range i only about 4-55, while for 250km we get up to a factor of 354. Incorporating LS increae the runtime, becaue more of the cruiing range ha to be explored to find all (invere) reachable LS from (from t) and the Dijktra run on the auxiliary (reach-)graph take additional time, ee Table 2. Neverthele we till end up with practical runtime about a econd on maximum. Moreover we oberved that the average number of reloading event eem to be acceptable for the conidered cenario, while the maximum number indicate that our approach alo output ome unreaonable route (ee the lat column of Table 2). Adding reloading penaltie reduce both value of coure, but chooing them in a realitic fahion there a till optimal route no uer would be willing to take. Therefore we performed experiment with a limit of k = 2 and k = 3 on the number of recharging event in Southern Germany. Unurpriingly we got comparable query time, becaue the layered graph i only about k time a large a the original reach-graph. Comparing the pure hortet feaible route to the one with the recharging limit, we aw that the ratio of reachable target remain almot unchanged (reduction by only 4% for k = 2) and the average travel time (without reloading time) increae lightly by about 3 10%. So taking reloading effort into account lead to more practical route in ome cae. Similar reult were obtained for minimizing the number of recharging event under a ditance contraint: Setting the bound to 1.05 time the hortet feaible path ditance we could already ave 1-2 recharging event on average, reulting in much more ueful EV route. 6. CONCLUDING REMARKS In thi paper we howed that the peed-up technique contraction hierarchie work for contrained hortet path even if one of the metric i repreented by a function intead of a contant value. Thi allow for anwering EV route planning querie, like computing the hortet path on which the EV doe not run out of energy, efficiently. Moreover we propoed preproceing approache for incorporating reloading deciion, allowing for minimizing recharging under a ditant contraint or limiting the number of uch event. Partly our algorithm are cutomizable, i.e. the uer can trade between eco-friendly and hort route with an input parameter. All our approache guarantee optimality and were proven to work well on real-world intance. Future work include conideration of incomplete reloading and dynamic change in the energy conumption function, e.g. due to traffic volume, trunk load or weather condition. 7. REFERENCES [1] V. Aggarwal, Y. Aneja, and K. Nair. Minimal panning tree ubject to a ide contraint. In 32nd ACM Sympoium on Theory of Computing (STOC), page , [2] Y. P. Aneja, V. Aggarwal, and K. P. K. Nair. Shortet chain ubject to ide contraint. Network, 13(2): , [3] Andrea Artmeier, Julian Haelmayr, Martin Leucker, and Martin Sachenbacher. The hortet path problem reviited: Optimal routing for electric vehicle. 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