Operating a fleet of electric taxis

Transcription

1 Operatng a fleet of electrc taxs Bernat Gacas, Frédérc Meuner To cte ths verson: Bernat Gacas, Frédérc Meuner. Operatng a fleet of electrc taxs <hal v2> HAL Id: hal Submtted on 31 Jul 2012 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 OPERATING A FLEET OF ELECTRIC TAXIS BERNAT GACIAS AND FRÉDÉRIC MEUNIER Abstract. The deployment of electrc tax fleets s hghly desrable from a sustanable pont of vew. Nevertheless, the weak autonomy of ths knd of vehcles requres a careful operaton. The way of managng such a fleet and the queston of locatng chargng termnals for the vehcles are addressed n ths paper. Methods for dealng wth these tasks are proposed and ther effcency s proved through smulatons. 1. Introducton 1.1. Context. Centrale OO 1 s a poneerng project amng to deploy n Pars a fleet of 100 % electrc taxs. The company n charge of the management of the fleet s the Socété du Tax Electrque Parsen (STEP). The deployment of such fleets fnds s man motvaton n sustanable ssues: electrc vehcles release almost no ar pollutants at the place where they are operated and have less nose polluton than nternal combuston engne vehcles. However, the man drawback of an electrc vehcle s ts weak autonomy 80 km n the case of the Centrale OO project. In tax fleet management, two knds of requests can be dfferentated: bookng requests and opportunstc requests. The frst ones can be mmedate or n advance of travel and have to be processed by the tax dspatchng system whch assgns the request to a tax. The opportunstc requests correspond to the tradtonal tax servces pckng up passengers at cab-ranks or from the sde of the road. Of course, ths knd of requests s not processed by the dspatchng system. The constrants of the management, as expressed by the STEP, are A tax must never break down An opportunstc demand nsde Pars and ts suburbs must always be satsfed (legal envronment of Pars) The number of bookng demands accepted has to be maxmzed The chargng problem of the taxs must therefore be carefully addressed. At a tactcal level, a good assgnment of the trps to the taxs s crucal. We propose an effcent way to manage the electrc fleet n real-tme whle takng nto account the chargng tasks. At a strategc level, the chargng problem ncludes the determnaton of the best locaton for the chargng termnals. The sgnfcant ntal nvestment (the cost of an electrcal chargng termnal s about euros) and the restrcted vehcle autonomy gve a hgh relevancy to the chargng termnal locaton task. Indeed, a wrong placement may n effect lead to a poor fleet management wth vehcles havng dffcultes to charge the batteres due to chargng termnals saturaton or even wth vehcles constantly runnng out of charge to keep operatng. Our purpose s to propose a practcal way for computng the best locatons for the chargng termnals Model. We descrbe now formally the model we deal wth n ths paper. We derve also some elementary relatons, whch gves some nformatons on the capacty of a gven system (n terms of number of trps that can be realzed by unt of tme) Input descrpton. A complete drected graph G = (V, A) models the network. The vertces are ponts n the cty at whch trps start and fnsh. They can moreover be used to locate vehcle chargng termnals. The arcs model the possble trps. The duraton of a trp s a random varable T a of expectaton τ a. The Key words and phrases. chargng termnal locaton; electrc vehcles; fleet management system; mxed nteger programmng; smulaton; tax dspatchng. Ths project has been funded by Régon Ile de France. 1 See the webste for an artstc vew. 1

3 demand for each possble trp a A s assumed to follow a Posson process of rate λ a. Actually, ths demand s splt between a bookng demand and an opportunstc demand, see Secton 5 for a more accurate descrpton. There are n taxs. A tax consumes γ Wh by unt of tme when t s movng. It stores ρ Wh by unt of tme when t s chargng. The number of chargng termnals s denoted by r. Several termnals can be located at the same vertex Elementary relatons. Let us denote by λ a the average number of demands for a trp a that are accepted by unt of tme. We have λ a λ a. Let λ = λ a be the average number of trps accepted by unt of tme and let τ = 1 λ λ a τ a be the a A a A average duraton of an accepted trp. The energy consumpton of the system by unt of tme s γ λτ. The maxmal rate of supply n energy s ρr. Therefore, we have the followng nequalty (1) γ λτ ρr A second nequalty can be derved by consderatons on the tme needed to perform the dfferent tasks. Let us consder a tax over a tme wndow of suffcently large duraton T. Denote by x the tme durng whch t stores energy at a chargng termnal. Over the tme wndow, t spends n average T λτ n unt of tme wth a customer on board. Therefore, we have T λτ n + x T Durng ths duraton x, t stores a quantty of energy that must cover n average the consumpton over the tme wndow. Hence Combnng these two nequaltes leads to γt λτ n ρx (2) (γ + ρ) λτ n Equatons (1) and (2) can be summarzed n the followng nequalty. ( n (3) λ mn (γ + ρ)τ, ρr ) γτ Knowng the number of taxs, ther effcency (encoded by γ), the number of chargng termnals, and ther effcency (encoded by ρ), then an upper bound of the number of trps that can be accepted by unt of tme can be calculated Plan. Secton 2 s devoted to the lterature revew for the two problems addressed n ths paper, namely fleet management and chargng termnal locaton. The followng sectons Secton 3 and Secton 4 detal the approaches proposed for each of these problems. Next, we descrbe a smulator that has been mplemented for the evaluaton of the proposed approaches (Secton 5). The results of the experments are descrbed n Secton 6. The paper ends wth concludng remarks (Secton 7). 2. Lterature revew 2.1. Tax dspatchng. Tradtonal tax dspatchng systems are characterzed by two prncples. Frst, smple rules such as for example nearest vehcle frst or least utlzed frst are used for dynamc vehcle assgnment and second, the geographcal space s usually dvded nto zones. In the lterature, most of works on the topc bascally focus on customer watng tme mnmzaton by proposng mproved methods for rule-based systems. In ths context, Shrvastava et al. [SCMK97] descrbe a fuzzy model for rule selecton and Alshams et al. [AAR09] propose a new technque for dynamcally dvde the dspatch areas. The recent apparton of transportaton technologes (GPS, EDI, GIS) has wdely ncreased the opportuntes for fleet management optmzaton. It s also the case for tax dspatchng. For example, Seow et 2

4 al. [SDL10] propose a collaboratve model for tax dspatchng where a set of n taxs of the same zone are defned as the agents of the model and a set of n customers as the servce-requests. The objectve s then to maxmze the total servce qualty solvng a collaboratve lnear assgnment problem. However, tax dspatchng s not the only aspect that can be optmzed. For example, Lee et al. [LSP08] and Ja et al. [Ja08] use real-tme vehcle nformaton to propose a model for tax relocaton recommendaton based on demand forecastng and a probablty model for the desgn of tax stops, respectvely. Another approach for fleet management optmzaton conssts n modelng the problem as a varant of the Pck-up and Delvery Vehcle Routng Problem wth Tme Wndows (PDVRPTW). The dea s to plan a set of routes satsfyng known n advance customer requests. In the tax management context, Wang et al. [WLC09] propose a b-crtera two-phase method wth an ntal feasble assgnment frst and a tabu search mprovement later n order to mnmze the number of vehcles and the sum of travel tmes for advanced bookngs. However, the dea to block some vehcles only for advanced bookngs mght n some cases yeld to a fleet underutlzaton. Horn and al. [Hor02] and Meng et al. [MMYH10] try to fll the gap between smple non-optmzed rule-based tax dspatchng systems and statc routng approaches. The second paper descrbes a genetc network programmng n order to fnd the optmal balance between the watng tme and the detour tme. The work of Horn [Hor02] s of partcular nterest n relaton to the present work, proposng a tax dspatchng system archtecture smlar to our fleet management system. He proposes a system for vehcle travel tme mnmzaton composed by a set of nserton algorthms to decde whether a new customer s accepted or not and a set of optmzaton mechansms n order to mprove the soluton. However, some mportant dfferences exst between our work and these last two fleet management systems. The frst dfference s that n our case, the constrants related to the restrcted autonomy of the vehcles have also to be taken nto account by schedulng chargng tasks n the routes of the vehcles. The second dfference s that, unlke us, both artcles deal wth the mult-customer problem authorzng customers to share the same vehcle at the same tme Locaton ssue. The locaton problem was orgnally defned by Webber when he consdered how to poston a sngle warehouse mnmzng the total dstance between the warehouse and a set of customers [Web29]. In 1964, Hakm [Hak64] defnes the P-medan problem, the problem conssts n determnng the best locaton for a set of lmted facltes n order to mnmze the sum of the weghted dstances between the clents and the facltes servng these clents. The problem ncreases ts relevance durng the last decades. Hgh costs related to property acquston and faclty constructon make faclty locaton projects a crtcal aspect of strategc plannng for a wde range of prvate and publc frms. Indeed, the fact that faclty locaton projects are long-term nvestments leads the researchers to focus on dynamc and stochastc locaton problems (see [OD98] for a revew of ths extenson of the problem). Another mportant varant of the problem s the Capactated Faclty Locaton Problem (CFLP) where facltes have a constranng upper lmt on the amount of demand they can satsfy. An extenson of the CFLP closely to our problem s the Capactated Faclty Locaton Problem wth Multple facltes n the same ste (CFLPM). In chargng termnal locaton, the postons of the termnals are not the only decson varables, the number of termnals at each poston have to be fxed too. However, n some real-world applcatons, selectng the best locaton for dstance mnmzaton s not the best sutable choce. For example, n electrc vehcle chargng termnal locaton, lke n other crtcal applcatons such as ambulance and fre termnal locaton, the nterest s to guarantee that the dfferent geographc zones are covered by a faclty (closer than a prevously fxed coverng dstance). Ths class of problems are known as Coverng Locaton Problems (see [WC74], [SVB93] and more recently [VP10] for a complete revew of coverng problems). In that context, the coverng ssue can be sometmes modelled as a problem constrant. However, f the coverng dstance s fxed to a small value the problem mght become unfeasble. The Maxmal Coverng Locaton Problem (MCLP) [CR74] locates the facltes n order to maxmze the number of covered customers (customers wth a dstance to the nearest faclty smaller than an ntal fxed dstance). An extenson of the problem s the maxmal coverng wth mandatory closeness problem whch mposes a maxmal dstance (less strngent than the coverng dstance) between the geographcal zones and the nearest faclty [CR74]. These coverng models mplctly assume that f a geographcal zone s covered by a faclty then the faclty wll be always avalable to serve the demand. However, n some applcatons, when facltes have a fxed capacty, beng covered s not suffcent to guarantee the demand satsfacton. We fnd 3

5 n the lterature some models attemptng to overcome ths ssue by maxmzng the number of geographcal zones covered by multple facltes [DS81, HR86, GLS97]. 3. Fleet management We descrbe n ths secton two ways for managng the fleet, a classcal and rule-based one (Subsecton 3.1), and an mproved one tryng to address explctly the chargng ssue (Subsecton 3.2). Let us frst ntroduce some notatons. Let CR be a bookng customer request. Each customer request CR s defned by a start tme S and an orgn-destnaton par O D. The S s fxed by the customer when the customer request arrves. The completon tme of a trp s C = S + τ OD, where τ OD s the travel tme between the orgn and destnaton of the customer request CR. Fnally, let R : CT j be a tax chargng task scheduled on the chargng termnal CT j A classcal rule-based tax dspatchng system. A tax dspatchng system based on the prncples of the most common real-world systems (see for example [SCMK97], [LWCT04] or [AAR09]) s descrbed n ths secton. The archtecture of the current tax dspatchng systems are very smlar to the system llustrated n Fgure 1. The two man components of the system are (1) a customer acceptaton mechansm decdng for each new customer f t s accepted (the accepted customers are nserted nto a queue of customers) or rejected and (2) a rule-based mechansm assgnng accepted customer requests (trps) to the free taxs. For each accepted trp, the assgnng process has to start a few mnutes (Θ) before the fxed start tme (S ) n order to maxmze the chances to fnd a tax to attend the demand. Once a trp s assgned to a tax, the vehcle s automatcally blocked and the taxmeter begns countng. Customer Request Rule based Customer Acceptaton Mechansm Tme ordered queue of customers CR n CR 2 CR 1 Fgure 1. Rule-based tax dspatchng system A rule for customer acceptaton usng the tme wndows for the trps already accepted s proposed. The dea s to lmt the trps that have to be performed at the same tme n order to mnmze the number of not served customers and to establsh a margn of k vehcles to attend opportunstc customers. For each new customer request CR new the Algorthm 1 determnes f t s accepted or not. Algorthm 1: Rule-based checkng for customer acceptaton for a margn of k vehcles L = {CR 1, CR 2,..., CR n }, lst of already accepted customers CR new, new bookng customer request nc 0, number of trps performed at the same tme than CR new foreach CR of L do f CR s executed at the same tme than CR new ((S S new < C ) or (S new S < C new ) then Step 1: Increase the number of customers performed at the same tme than CR new (nc nc + 1) f condton to accept the customer (nc < n k) then Step 2: Insert CR new to the lst of accepted customers L 4

6 Once the customer request CR s accepted, t remans n the queue of customers untl S Θ (Θ s usually fxed around 20 mnutes). At that moment, the system automatcally starts lookng for a free tax havng suffcent charge to operate the trp. If dfferent taxs are avalable, the system assgns the trp to the tax mnmzng the customer watng tme (a parameterzable not announced customer watng tme can be authorzed). In the case of no vacant taxs are avalable, the system wats for a vehcle to become avalable. If the watng tme for any request exceeds the authorzed maxmal customer watng tme, the customer request s then canceled. Note that the number of unsatsfed customers can be reduced by usng a more restrctve rule for the customer acceptaton mechansm. The man advantage of such a system where no future work s planned s the hgh degree of ndependence for tax drvers. On the other hand, the drawbacks are the underutlzaton of the fleet and the lost of effcency durng the peak hours when most of the companes have to close ther bookng requests systems n order to avod unsatsfed customers. Indeed, some real-world systems do not ntegrate a customer acceptaton mechansm leadng, n rush hours, to unsatsfed customers who had been ntally accepted and they are fnally served wth an unannounced and, sometmes, ntolerable delay or, eventually, never served at all. Furthermore, the chargng tasks of the vehcles cannot be controlled leadng to a poor fleet management wth vehcles havng dffcultes to charge the batteres due to chargng termnals saturaton The mproved electrc vehcle management system. An mproved fleet management system amng to overcome the weakness of the rule-based tax dspatchng system s proposed n ths secton. The man objectves of the system are to maxmze the number of accepted customers and to mnmze the customer watng tme. One of the major ssues s how to deal wth opportunstc demand. Indeed, ths knd of demand s unpredctable and must always be satsfed, so free taxs must be at any moment able to satsfy the longest trp wthout runnng out of charge. Ths constrant makes the problem consderably more complex forcng the system to provde a mechansm ensurng the feasblty of the already accepted trps each tme an opportunstc demand s accepted. The approach proposed conssts n mantanng contnuously a feasble plannng for the taxs and the chargng termnals (see Fgure 2). Each tme a customer asks for a trp, a smple nserton algorthm s run, at the end of whch ether the trp has been successfully nserted or not. The objectve s to assgn the customer to the tax mnmzng the customer watng tme (a parameterzable announced customer watng tme can be authorzed). If none of the tred delays on the pck-up tme leads to a feasble plannng, a reschedulng algorthm allowng to reallocate the already accepted customers to the taxs s run. In all these processes, a key routne s often called, namely the chargng task manager, whch schedules the chargng tasks of a tax, gven a plannng for the other taxs and the chargng termnals. Feasble plannng:temporal and autonomy related constrants are satsfed Tax 1 CR 1 R : CT 1 CR 4 Tax CR R : CT 2 CR 5 VEHICLES Customer Request Customer Acceptaton Mechansm Feasble Plannng Tax n R : CT 1 CR CHARGING TERMINALS CT 1 CT 2 Tax n Tax 2 Tax 1 Fgure 2. Customer acceptaton mechansm of the electrc vehcle management system In the case of an opportunstc demand, whch s necessarly accepted, we follow exactly the same scheme except that there s no degree of freedom n the nserton process: the trp s nserted at the front of the plannng of the tax stopped by the customer, and the reschedulng algorthm s also run f t s necessary. 5

7 Inserton algorthm. Ths algorthm s the frst step n order to decde f a new trp CR new s accepted or not. The objectve s to assgn the trp to the tax mnmzng the delay on the pck-up tme (see Algorthm 2). The algorthm ncreasely tests the dfferent authorzed pck-up tmes. Once the start tme s fxed, we sequentally try for each vehcle to nsert the new request. Frst the scheduled chargng tasks are removed. Then the new request s accepted only f t can be nserted wth no constrant volaton (the pck-up tmes of the rest of customers are respected and the current autonomy of the vehcle, wthout any chargng task, s suffcent). In the case that the vehcle autonomy-related constrant s volated, a greedy algorthm tryng to schedule a chargng task between each par of trps s proposed. After the chargng tasks are nserted, f the tax s able to perform the trps wthout runnng out of charge, then the customer request s also accepted. Algorthm 2: New request nserton algorthm for a maxmal authorzed delay of mnutes V = {V 1, V 2,..., V r }, lst of taxs CR new, new bookng customer request accepted f alse, varable ndcatng f the new request s accepted st S new, start tme of the trp whle st S new + and accepted = false do foreach v of V do Step 1: Delete the chargng tasks of the vehcle v f CR new startng at st can be nserted n the route of the vehcle v then f the vehcle autonomy-related constrant s satsfed then Step 2: CR new startng at st s nserted n the route of the vehcle v (accepted true) else Step 3: Insert chargng tasks for v between each par of trps f the vehcle autonomy-related constrant s satsfed then Step 2: CR new startng at st s nserted n the route of the vehcle v (accepted true) f accepted = false then Step 4: Increase the pck-up tme for the CR new (st st + 1) Reschedulng algorthm. The reschedulng algorthm s proposed when the new customer s stll not accepted after the nserton algorthm. As for the nserton algorthm, the goal s to fnd a new feasble plannng for the vehcles ntegratng the new request CR new. The man dfference s that the trps can be reassgned to dfferent vehcles. The problem wthout takng nto account the autonomy-related constrants can be solved n polynomal tme [NSZ02]. The dea s to convert the schedule of trps (wthout the chargng tasks) nto a graph and to verfy usng a max flow computaton that all trps can be performed by the taxs. To construct the network two vertces are consdered for each customer request CR, the frst one v represents the pck-up tme and the second one v the completon tme of the customer request. Four dummy vertces are requred: 0, 0, a source s and a snk t. The arcs of the graph are (s, 0), (0, t), all the (s, v ), all the (v, t), all the (v, v ), and all the (v, v j) such that the customer request CR j can be performed by the same tax than the customer request CR and after CR, that means f S j C + τ DO j. Except the arcs (s, 0) and (0, t), they all have a capacty equal to 1. The arcs (s, 0) and (0, t) have a capacty equal to n. A maxmum flow computaton n ths drected graph determnes the schedule feasblty and also proposes a new plannng for the vehcles respectng the customers pck-up tmes. The max flow computaton s ntegrated n the reschedulng algorthm n order to check the feasblty of the schedule for a gven pck-up tme st [S new, S new + ] and, f t s the case, to fnd a reference plannng (plannng wthout chargng tasks). A local search explores the neghborhood of the reference plannng defned by the swap and the reallocaton operators [Sav92]. Fnally, for each explored plannng respectng temporal constrants, the greedy algorthm for chargng task schedulng s sequentally appled to the taxs that do not satsfy autonomy-related constrants (that s, taxs whose current charge s not enough to perform all 6

8 the trps assgned to them wthout addng chargng tasks). If a feasble soluton s found, the new customer s then accepted Chargng task manager. As we have already seen, the nserton and the reschedulng algorthm constantly runs a greedy algorthm amng to nsert a chargng task between each par of successve trps of the same route. The algorthms proposed to determne f a new chargng task can be ntegrated n a specfc chargng termnal plannng are descrbed n ths subsecton. The man feature of our problem s that the processng tme of the new chargng task s not fxed, nstead t s a decson varable defned between the nterval lmted by the mnmal chargng tme for a vehcle p mn (customzable parameter) and the maxmal chargng tme correspondng to the tme necessary for a full charge. The problem to be solved by the chargng termnal manager can be then formally stated as follows. A chargng task R s defned by ts tme wndow [r, d ], where r s the earlest start tme (earlest arrval tme to the termnal) and d the latest end tme (latest departure tme from the termnal). Let p be the decson varable correspondng to the processng tme of the task R, then r S and S + p d, where S s the effectve start tme of R. Gven a feasble schedule of n chargng tasks S n = {S 1, S 2,..., S n, } for the chargng termnals located at the same geographcal poston. We are gven a new chargng task R n+1 wth a tme wndow [r n+1, d n+1 ] and a processng tme p n+1 nsde the nterval p mn p n+1 p max n+1. The problem conssts n fndng a new feasble schedule S n+1 = {S 1, S 2,..., S n, S n+1 } maxmzng the processng tme of the task R n+1 (whence wthout changng the processng tme of the other tasks). The mechansm tests frst a task nserton amng to fnd quckly a feasble soluton. The complexty of the algorthm for task nserton maxmzng the processng tme of the new task s O(n) where the start tmes and completon tmes of the scheduled jobs are non-decreasng ordered. If no soluton s found after the task nserton algorthm, a dchotomous algorthm allowng to reschedule the tasks s proposed n order to fnd a soluton maxmzng the processng tme of the new task. For each teraton of the algorthm, a satsfablty test based on constrant propagaton nvolvng energetc reasonng s frst trggered. The goal of the feasblty test s to detect an nconsstency ndcatng that t s not possble to fnd a feasble schedule ntegratng the new task. Fnally, f the energetc reasonng s not conclusve a local search algorthm s proposed n order to fnd a soluton. Satsfablty test: Energetc reasonng. A satsfablty test based on constrant propagaton nvolvng energetc reasonng s proposed [LE96]. A fcttous energy (whch has nothng to do wth the electrcty) s produced by the chargng termnals and t s consumed by the chargng tasks. We determne the fcttous energy consumed by the tasks (E consumed ) over a tme nterval δ = [t 1, t 2 ] and we compare ths fcttous energy wth the avalable fcttous energy (E produced = m (t 2 t 1 )). The mnmal fcttous energy consumed by the tasks n an nterval δ = [t 1, t 2 ] s: n+1 (4) E consumed = max{0, mn{p, t 2 t 1, r + p t 1, t 2 d + p }} =1 If E consumed > E produced, t s then mpossble to fnd a feasble schedule S n+1 ntegratng the new task. The relevant ntervals δ for a complete satsfablty analyss can be enumerate n O(n 2 ). The test s restrcted to the ntervals [t 1, t 2 ] specfed by {r } {d } {r + p } {d p } where the new task R n+1 may consume (t 1 d n+1 and t 2 r n+1 ). Dchotomous algorthm. A dchotomous algorthm maxmzng the processng tme of the new task s descrbed n ths secton (see Algorthm 3). A dchotomy s run on the processng tme p as follows. For processng tmes p [p mn, p max n+1 ], the satsfablty test based one the energetc reasonng ndcates whether the necessary condtons are satsfed or not. If t s the case, a local search mechansm tres to fnd a feasble schedule. The parallel machne schedulng problem wth tme wndows can be solved by a lst schedulng algorthm. It means there exsts a total orderng of the jobs (.e., a lst) that, when a gven machne assgnment rule s appled, reaches the optmal soluton. For our problem, ths rule conssts n allocatng each task to the machne that allows t to start at the earlest (Earlest Start Tme or EST rule). The local search mechansm proposed to solve the problem s based on ths result. Frst, the tasks are ordered n a non-decreasng order of ther due dates (Earlest Due Date or EDD rule), then the local search conssts n 7

9 explorng dfferent permutatons of the lst defned by the nserton neghborhood (O(n 2 )). For each lst of task, the machnes are assgned accordng to the EST rule n order to reach a feasble soluton. If no feasble schedule s eventually found, the request s rejected. Algorthm 3: Dchotomous algorthm for processng tme maxmzaton mn p mn max p max n+1 S n+1 best whle mn max do Step 1: Fx the processng tme p of the new task R n+1 (p mn+max 2 ) f Satsf abltyt est() then Step 2: Sort the tasks accordng to the EDD rule Step 3: Local search usng the nserton operator f a feasble schedule S n+1 = {S 1, S 2,..., S n, S n+1 } s found then Step 4: Update the lower lmt (mn p + 1) Step 5: Update the best soluton (S n+1 best Sn+1 ) else Step 6: No soluton exsts, update the upper lmt (max p 1) else Step 7: No soluton exsts, update the upper lmt (max p 1) f S n+1 best = then Step 8: No soluton s found (return ) else Step 9: A feasble soluton s found (return S n+1 best ) 4. Electrc vehcles chargng termnal locaton The EV chargng termnal locaton problem conssts n determnng the best locatons of the chargng termnals. The lnear programmng model has to take nto account two mportant aspects. Frst, the chargng termnals have to be convenently spread over the geographcal area n order to avod remote geographcal zones whch dffcult tax operablty and fleet management. The second aspect s that the model has to determne the number of chargng ponts facltatng the chargng process of the taxs by mnmzng the rsks of termnals saturaton. For these purposes, we propose two models, one called the P -medan model, the other the Demand-based model. V s the set of geographcal ponts of the problem and J V s the set of potental locatons where the chargng termnals can be located. The number of termnals s lmted to r P -medan model. Followng Hakm [Hak64], we defne x j to be the decson varables ndcatng f a faclty s located to the pont j and y j to be the varables ndcatng that the geographcal pont s assgned to the faclty located n j. The lnear program mnmzng the sum of the dstances between clents and facltes can be wrtten as follows. 8

10 (5) mn V dst j y j j J (6) (7) (8) (9) (10) s.t. j J y j j J x j y j y j x j = 1 for all V x j for all V, j J r {0, 1} for all j J {0, 1} for all V, j J 4.2. Demand-based model. Another approach conssts n defnng a model wth two dstances β far and β close as proposed by Church and ReVelle [CR74]. The dea s then to spread the termnals by fxng a maxmal dstance (β far ) between the dfferent geographcal zones and the nearest chargng termnal and, at the same tme, tryng to maxmze the demand that wll be covered by a nearby chargng termnal (β close ). We can then defne J far (resp. J close ) as the subset of ponts n J at dstance less than β far (resp. β close ) from V. Conversely, Vj close s the set of ponts at dstance less than β close from the pont j J. Let x j be the decson varable ndcatng the number of termnals located at pont j J and y j to be the fracton of the demand d for V covered by a chargng termnal located n j at dstance less than β close from. The lnear programmng model proposed to solve the problem called Demand-based model s the followng. (11) (12) (13) (14) (15) (16) (17) max j J s.t. j J far j J close V close j j J V close j x j x j y j d y j 1 for all V 1 for all V d y j x j for all j J r x j Z + for all j J y j R + for all V, j J close The objectve functon (Eq. (11)) conssts n maxmzng the pontwse demand covered by a chargng termnal consderng the dstance β close. Eq. (12) mposes that a geographcal zone V must be covered at least for one chargng termnal consderng the dstance β far. Here the mandatory closeness s only requred for the geographcal zones closer than β far from a potental chargng termnal locaton n order to fnd a soluton even f ths constrant s volated for some geographcal zones. We stress that an adequately β far make possble to spread the chargng termnals over the geographcal area. Eq. (13) specfes that for each geographcal zone V the sum of the fractons of demand covered by a chargng termnal consderng the dstance β close has to be less or equal to the unt. The dea here s that the demand of each geographcal pont can be satsfed by dfferent chargng termnals and our nterest s to maxmze the potental energy requred beng suppled by a termnal closer than β close. Eq. (14) are the constrants lnkng the varables x j wth the varables y j. For a gven potental chargng termnal locaton J j, ths last set of varables can only 9

11 be postve f a chargng termnal s fnally located to J j, that means x j > 0. Besdes, thanks to the defnton of the pontwse demand d, Eq. (14) also mposes that the demand allocated to a chargng termnal cannot exceed ts capacty. The goal of ths constrant s to assgn a larger number of termnals on the geographcal ponts wth a great demand decreasng that way the rsk of saturaton for the chargng termnals. Fnally, Eq. (15) lmts the number of termnals of the problem. Estmatng a pontwse demand n energy. The proposed chargng locaton model take n nput a pontwse demand. A way to buld such a demand d attached to a vertex conssts n computng the energy needed by unt of tme for the trps startng at : t s precsely γ λ a τ a. Dvdng ths quantty by ρ provdes a δ + () the number of chargng termnals ensurng ths supply. It suggests to defne d out = γ ρ a δ + () In the last equaton, the pontwse demand s calculated consderng the trps startng at vertex. Ths model presumes that taxs chargng tasks take place mostly at the orgn of the trps. However, other realstc strateges can be also envsaged. One of these strateges s consder that taxs charge the batteres at the end of the trps. A pontwse demand consderng the energy consumed by unt of tme for the trps arrvng at s also proposed. We can then defne d n = γ ρ a δ () Fnally, a thrd strategy s proposed consderng a pontwse demand as a convex combnaton of d out d n : d mx = αd out λ a τ a λ a τ a + (1 α)d n 5. Tax smulator The tax behavor smulator conssts n a dscrete-events smulator programmed n C++. It smulates the model descrbed n Secton 1.2. Each possble trp between the ponts of the cty s characterzed by two parameters, λ book and λ opp, representng the rates of the Posson process for booked and opportunstc demand, respectvely. For sake of smplcty, the duraton and the dstance of each trp are consdered constant over the tme. The nputs of the smulator are a descrpton of the system state and the smulaton parameters. The system s orgnally defned by: A set of ponts defned by ther coordnates representng the geographcal ponts of the cty. The locaton of the chargng termnals. The fleet of taxs. Each tax s defned by the followng parameters: AUT: t s the current autonomy of the vehcle. MAX AUT: t s the autonomy of the vehcle when t s fully charged. POS: t s the ntal poston of the vehcle. Opportunstc and bookng demands are treated dfferently. The bookng demands are managed by the tax dspatchng system decdng whether a demand s accepted or not. The opportunstc demands always have to be satsfed, the smulator affects the trp to a free tax located at the same geographcal pont. As we can note, an opportunstc demand s only consdered when t exsts spatal and temporal concdence between the demand and a free tax, otherwse the demand s smply gnored. It s worth recallng that the opportunstc demands may lead to unsatsfed bookng demands ntally accepted by the tax dspatchng system. 6. Computatonal experments The fleet management systems and the chargng termnal locaton models are compared and evaluated by smulaton on a set of randomly generated nstances. Two set of nstances have been generated consderng dfferent values for the average number of demands by unt of tme. We consder then a frst set of nstances wth a weak demand (λ weak book 0.4 and λweak opp 1.0) and a second set of nstances wth a strong demand 10 and

12 0.8 and λ strong opp 2.0). Each set s composed of 60 nstances (10 nstances for each combnaton) generated from dfferent values for the number of termnals (r {5, 20, 40}) and for the number of vehcles (n {100, 200}). The computaton experments consst on a 900 mnutes smulaton. The maxmal authorzed delay on pck-up tme s fxed to = 15 mnutes for both systems and the mnmal chargng tme for a vehcle s fxed to 10 mnutes. (λ strong book 6.1. Fleet management. The mproved fleet management system has been compared wth the rule-based tax dspatchng system presented n Secton 3.1. Table 1, Table 2, and Table 3 dsplay the results for the comparson between both systems for the nstances wth 5, 20 and 40 chargng termnals, respectvely. The frst column (NbTrps) dsplays the average of served customers. NbBookng s the average of served bookng requests. The number of unsatsfed customers (customers ntally accepted by the system but they are never served) s ndcated by Unsatsfed Customers at the thrd column. Fnally, the last column ndcates the average customer overcharge (number of mnutes n the taxmeter when the tax arrves at the pck up pont). The results show that the proposed fleet management clearly mproves the rule-based tax dspatchng system. The number of served customers s on average consderably more mportant wth the mproved fleet management systems. Besdes the percentage of unsatsfed customers remans acceptable and almost constant wth ths system contrarly to the rule-based tax dspatchng for whch the number of unsatsfed customers grows exponentally wth the demand for a small number of chargng termnals, as we can see for the nstances wth 50 and 100 vehcles and 5 chargng termnals. Ths fact can be explaned because the vehcle chargng aspects are completely gnored n the customer acceptaton rule of the rule-based tax dspatchng system. Fnally, customer overcharge s very smlar for both systems, for the nstances wth 20 and 40 resources here agan the proposed fleet management s a better system than the rule-based tax dspatchng. Concernng the tme needed to compute the answer to a bookng request, t s n general under 1 ms, and n less than 2% of the cases above 5 s. In any case, the computaton tme s always reasonable, and, wth faster computers, the tme needed to compute the answer could be reduced f requred. r = 5 NbTrps NbBookng Unsatsfed Customers (%) Overcharge (mn) weak demand n = 50 Rule-based TD (5.23 %) 6.56 Improved FM (0.17 %) 6.68 Rule-based TD (2.95 %) 6.07 Improved FM (0.28 %) 6.34 Rule-based TD (1.21 %) 5.78 Improved FM (0.16 %) 5.94 strong demand n = 50 Rule-based TD (19.08 %) 5.89 Improved FM (0.14 %) 6.79 Rule-based TD (11.53 %) 5.96 Improved FM (0.15 %) 6.48 Rule-based TD (4.70 %) 5.87 Improved FM (0.29 %) 6.13 Table 1. Comparson between dfferent fleet management systems for nstances wth 5 chargng termnals 6.2. Chargng termnal locaton. The demand-based lnear model for chargng termnals locaton presented n Secton 4.1 has been compared wth the P -medan model model mnmzng the dstance between the geographcal ponts and the nearest chargng termnal. Table 4 shows the results for the comparson between both models for the 40 nstances wth 5 chargng termnals. The frst column (NbTrps) dsplays the average of accepted customers. NbBookng s the average of accepted bookng requests. The average 11

13 r = 20 NbTrps NbBookng Unsatsfed Customers (%) Overcharge (mn) weak demand n = 50 Rule-based TD (4.43 %) 6.47 Improved FM (0.25 %) 6.37 Rule-based TD (3.27 %) 6.05 Improved FM (0.28 %) 6.01 Rule-based TD (1.15 %) 5.77 Improved FM (0.41 %) 5.56 strong demand n = 50 Rule-based TD (6.24 %) 6.60 Improved FM (0.32 %) 6.42 Rule-based TD (3.53 %) 6.00 Improved FM (0.38 %) 5.72 Rule-based TD (1.95 %) 5.74 Improved FM (0.37 %) 5.40 Table 2. Comparson between dfferent fleet management systems for nstances wth 20 chargng termnals r = 40 NbTrps NbBookng Unsatsfed Customers (%) Overcharge (mn) weak demand n = 50 Rule-based TD (5.61 %) 6.39 Improved FM (0.34 %) 6.45 Rule-based TD (3.49 %) 6.13 Improved FM (0.17 %) 5.89 Rule-based TD (2.15 %) 5.80 Improved FM (0.30 %) 5.55 strong demand n = 50 Rule-based TD (5.42 %) 6.48 Improved FM (0.34 %) 6.29 Rule-based TD (3.68 %) 5.99 Improved FM (0.31 %) 5.72 Rule-based TD (2.25 %) 5.64 Improved FM (0.47 %) 5.39 Table 3. Comparson between dfferent fleet management systems for nstances wth 40 chargng termnals percentage of operatng tme and the average percentage of tme when a tax s watng for an avalable chargng termnal are dsplayed on the last two columns. When the number of chargng termnals s equal to fve, the demand-based model outperforms the P - medan model, for any crteron. More customers are satsfed, the operatng tme of taxs s hgher and the tme watng for an avalable chargng termnal s drastcally reduced. In some cases, ths last value s even dvded by two. The dfferent ways proposed to estmate the pontwse demand have been also compared (for d mx, we set α = 0.5). We observe that no strategy overcomes clearly the others. When the number of chargng termnals s larger as n Table 5 and Table 6 (20 and 40 chargng termnals), there s no real dfference between the two models. Ths s not surprsng snce n those cases chargng termnals are no longer a crtcal resource. Ths fact explans also the small watng tme for chargng. In fact, for large number of chargng termnals, we expect smlar behavours for any locatons regularly spread over the area of expermentaton. 12

14 r = 5 NbTrps NbBookng %Operatng %Watng for chargng weak demand P -medan model % 9.52 % Demand-based model (d out ) % 4.67 % Demand-based model (d n ) % 3.63 % Demand-based model (d mx ) % 4.41 % P -medan model % 7.06 % Demand-based model (d out ) % 2.80 % Demand-based model (d n ) % 2.70 % Demand-based model (d mx ) % 2.80 % strong demand P -medan model % % Demand-based model (d out ) % % Demand-based model (d n ) % % Demand-based model (d mx ) % % P -medan model % % Demand-based model (d out ) % % Demand-based model (d n ) % % Demand-based model (d mx ) % % Table 4. Comparson between dfferent programmng models for nstances wth 5 chargng termnals r = 20 NbTrps NbBookng %Operatng %Watng for chargng weak demand P -medan model % 0.19 % Demand-based model (d out ) % 0.28 % Demand-based model (d n ) % 0.22 % Demand-based model (d mx ) % 0.31 % P -medan model % 0.11 % Demand-based model (d out ) % 0.17 % Demand-based model (d n ) % 0.16 % Demand-based model (d mx ) % 0.20 % strong demand P -medan model % 0.89 % Demand-based model (d out ) % 1.26 % Demand-based model (d n ) % 1.15 % Demand-based model (d mx ) % 1.36 % P -medan model % 0.73 % Demand-based model (d out ) % 0.88 % Demand-based model (d n ) % 0.87 % Demand-based model (d mx ) % 0.95 % Table 5. Comparson between dfferent programmng models for nstances wth 20 chargng termnals 7. Concluson In ths paper, we deal wth the management of a fleet of electrc taxs. Two dfferent problems have been solved: the tax dspatchng problem and the electrc chargng termnal locaton problem. Dfferent approaches have been proposed to solve each problem. For the tax dspatchng problem, two dfferent approaches are proposed: a rule-based system nspred by most of the real-lfe tax dspatchng systems and an mproved fleet management system adapted to 13

15 r = 40 NbTrps NbBookng %Operatng %Watng for chargng weak demand P -medan model % 0.06 % Demand-based model (d out ) % 0.06 % Demand-based model (d n ) % 0.04 % Demand-based model (d mx ) % 0.05 % P -medan model % 0.03 % Demand-based model (d out ) % 0.03 % Demand-based model (d n ) % 0.03 % Demand-based model (d mx ) % 0.02 % strong demand P -medan model % 0.24 % Demand-based model (d out ) % 0.13 % Demand-based model (d n ) % 0.14 % Demand-based model (d mx ) % 0.13 % P -medan model % 0.14 % Demand-based model (d out ) % 0.09 % Demand-based model (d n ) % 0.11 % Demand-based model (d mx ) % 0.12 % Table 6. Comparson between dfferent programmng models for nstances wth 40 chargng termnals the problem constrants. The dea of the mproved fleet management system s to contnuously mantan a feasble soluton for the problem (a feasble plannng for the taxs and for the chargng termnals). Schedulng algorthms and a local search scheme are proposed to solve the problem. Both systems have been compared by smulaton on randomly generated nstances. The results show the effcency of the proposed mproved system and that the rule-based system does not sut the features of electrc vehcles. For the electrc chargng termnal locaton problem, the frst approach s based on classcal models of the lterature to solve the faclty locaton problem and mnmzes the sum of the dstances between the termnals and the ponts assgned to them. The second approach s a demand-based mxed nteger lnear programmng model adapted to the characterstcs of our problem, n whch the movng demands of the taxs are approxmated as statc pontwse demands. Both models have been tested and compared on a set of realstc nstances randomly generated. The results show that the proposed demand-based model outperforms the model mnmzng the sum of dstances when the number of chargng termnals s small. When ths number s large, chargng termnals are no longer a crtcal resource and both models show smlar behavours. References [AAR09] A. Alshams, S. Abdallah, and I. Rahwan, Multagent self-organzaton for a tax dspatch system, 8th Internatonal Conference on Autonomous Agents and Multagent Systems (AAMAS), [CR74] R. L. Church and C. ReVelle, The maxmal coverng locaton problem, Papers of the Regonal Scence Assocaton 32 (1974), [DS81] M. S. Daskn and E. H. Stern, A herarchcal objectve set coverng model for emergency medcal servce vehcle deployment, Transportaton Scence 15 (1981), no. 2, [GLS97] M. Gendreau, G. Laporte, and F. Semet, Solvng an ambulance locaton model by tabu search, Locaton Scence 5 (1997), no. 2, [Hak64] S. L. Hakm, Optmum locatons of swtchng centers and the absolute centers and medans of a graph, Operatons Research 12 (1964), no. 3, [Hor02] M. E. T. Horn, Fleet schedulng and dspatchng for demand-responsve passenger servces, Transportaton Research Part C: Emergng Technologes 10 (2002), no. 1,

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