Stochastic traffic assignment of mixed electric vehicle and gasoline vehicle flow with path distance constraints

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1 Avilble online t ScienceDirect Trnsporttion Reserch Procedi 21 (2017) Interntionl Symposium of Trnsport Simultion (ISTS 16 Conference), June 23~25, 2016 Stochstic trffic ssignment of mixed electric vehicle nd gsoline vehicle flow with pth distnce constrints Wento Jing, Inhi Kim *, Mohsen Rmezni, Zhiyun Liu b Institute of Trnsport Studies, Deprtment of Civil Engineering, Monsh Univeity, Clyton, Victori 3800, Austrli. b School of Trnsporttion, Southest Univeity, Nnjing, Chin. Abstrct This pper ddresses generl stochstic user equilibrium (SUE) trffic ssignment problem (TAP) for trnsport networks with electric vehicles (EV), where EV pths re restricted by the EV driving rnge limits. A minimiztion model for pth-constrined SUE is fit proposed s n extension of pth-constrined deterministic user equilibrium (DUE) TAP, which lso extends the existing generl SUE models with link-bsed constrints to pth-bsed constrints. The resulting SUE model nd solution lgorithm cn be used for other conditions with similr pth-bsed constrints. The equilibrium conditions revel tht ny pth cost in the network is the sum of corresponding link costs nd pth specific out-of-rnge penlty term, while pth out-of-rnge term should equl to zero to ensure fesible flows. We develop modified method of successive verges (MSA) with predetermined step size sequence where both multinomil logit nd multinomil probit bsed loding procedure re pplied to solve the TAP. The suggested methods incorporte K-shortest pths lgorithm to generte the pth set on need bsis. Finlly, two numericl exmples re presented to verify the proposed model nd solution lgorithms. Copyright 2017 The Autho. Published by Elsevier B.V. Selection nd Peer-review under responsibility of Dept. of Trnsporttion Engineering, Univeity of Seoul. Keywords: Trffic ssignment; stochstic user equilibrium; pth distnce constrints; K-shortest pth lgorithm; Multinomil Logit; Multinomil Probit 1. Introduction Crbon-bsed emissions nd greenhouse gses (GHG) re criticl globl issues s ddressed by the Kyoto Protocol in 1998 (U.S. Envirionmentl Protection Agency, 2006). The trnsport sector is significnt contributor to * Corresponding uthor. Tel.: ; fx: E-mil ddress: inhi.kim@monsh.edu Copyright 2017 The Autho. Published by Elsevier B.V. Selection nd Peer-review under responsibility of Dept. of Trnsporttion Engineering, Univeity of Seoul /j.trpro

2 66 Wento Jing et l. / Trnsporttion Reserch Procedi 21 (2017) GHG emissions in most countries, comprising 23% (worldwide) of CO2 emissions from fossil fuel combustion in 2005 while utomobile trnsport is the principl CO2 production source. From the energy sfety point of view, the trnsport sector s whole is 98% dependent on fossil oil which is lso exceedingly ffected by chnges in energy resources (OECD-ITF Joint Trnsport Reserch Centre, 2008). So chnges to the current energy structure in trnsport sector re in urgent need. Alterntive fuels re ddressed s new fuel choice to reduce GHG emissions nd electric vehicles (EV)re believed to be sustinble solution (OECD-ITF Joint Trnsport Reserch Centre, 2008). Governments nd utomotive compnies hve recognized the vlue of these vehicles in helping the environment nd re encourging the ownehip of EV through economic incentives (Hcker et l., 2009). It is mentioned tht one million plug-in hybrid nd electric vehicles will be on the rod by 2015 in United Sttes to reduce greenhouse gs emission nd dependence on oil (Sber nd Venygmoorthy, 2009). According to Electric Drive Trnsporttion Assocition, the plug-in electric vehicles (PEV) in US hs exceeded 190,000 between Jnury of 2011 nd Mch of 2014 (Ghmmi et l., 2014). Although mny cities re plnning construction nd expnsion of chrging infrstructures for EV, it is likely tht in the foreseeble future EV commute will need to chrge their vehicles t home most of the time (Morrow et l., 2008). It is obvious tht the driving rnge limit inevitbly dds certin level of restrictions to EV drive trvel behvio, t lest in long future period prior to the coverge of rechrging infrstructures reching sufficient level (Jing et l., 2013). EV compnies re trying to overcome this limited rnge requirement with fst chrging sttions, where vehicle cn be chrged in only few minutes to ner full cpcity. Besides being much more costly to operte rpid rechrge sttions, the vehicles still tke more time to rechrge thn stndrd gsoline vehicle would tke to refuel (Botsford nd Szczepnek, 2009). However, the widespred doption of PEV clls for fundmentl chnges to the existing network flow modelling tools for properly cpturing chnged behvio nd induced constrints in forecsting trvel demnds nd evluting trnsporttion development plns (Jing et l., 2013). In order to tke into considertion of driving rnge limit nd insufficient chrging fcility sttus in trffic ssignment, Jing et l. (2012) proposed n pproch to restrict flow of pth to zero if the pth distnce is greter thn the driving rnge limit of EV. They employed pth trvel time function tht is the sum of the corresponding link cost such s the Bureu Public Rod (BPR) function nd showed the Lgrngin multiplier of its optiml solution stnds for the unit out-of-rnge trvel distnce cost. Clssic Frnk-Wolfe lgorithm with constrined shortest pth lgorithm s its subroutine cn be pplied to solve this problem. The deterministic user equilibrium (DUE) condition chrcterizes route choice behvior where use hve perfect trffic network informtion nd lwys choose the shortest pth ccurtely. A convex minimiztion model for DUE conditions cn be built by dding pth distnce constrints into the Beckmnn s conventionl DUE model. A more relistic nd generl sitution is tht trvel times re rndom vribles or trvel times re perceived by trvelle in imperfect, stochstic mnner. Although the stochstic user equilibrium (SUE) principle plys more relistic role thn DUE principle in ddressing rod user s route choice behvior, the SUE trffic ssignment problem with pth- distnce constrints hs received little ttention. To be consistent with the generlized DUE with pth distnce constrints, the SUE trffic ssignment model with generlized pth trvel times re referred to s generlized SUE trffic ssignment with pth distnce constrints. A milestone in formulting SUE conditions is Dgnzo s unconstrined minimiztion model (Dgnzo, 1982) of conventionl SUE conditions, which cn led to convergent lgorithm for solving the generl SUE trffic ssignment problem. However, dding side constrints (e.g. link cpcity constrints) into Dgnzo s model cnnot yield solution fulfilling generlized SUE conditions Literture Review It is well known tht the stndrd TAP under DUE cn be solved efficiently with Frnk-Wolfe type lgorithm

3 Wento Jing et l. / Trnsporttion Reserch Procedi 21 (2017) whose linerized sub-problem finds shortest pths for ech OD pir t ech itertion. The problem of finding the shortest pth for n EV ws originlly discussed by Ichimori et l. (1981),where vehicle hs limited bttery nd is llowed to stop nd rechrge t certin loctions. Lwler (2001) developed polynomil lgorithm for its solution. Adler et l. (2014) proposed n EV shortest-wlk problem to determine the shortest trvel distnce route which my include cycles for detouring to rechrging btteries from origins to destintions with minimum detouring. Kobyshi et l. (2011) nd Siddiqi et l. (2011) included bttery rechrging sttions in their shortest weight-constrined pth problem models, which is known to be NP-Complete (Desrosie et l., 1984; Desroche nd Soumis, 1989), nd proposed heuristic techniques s solution methodologies. There hs been some recent considertion of the effect of EV on trffic ssignment nd DUE. (Jing et l., 2012) studied the effect of restricted the EV pth distnces nd ssumes chrging events only occur t OD nodes, which corresponds to the rel circumstnce of insufficient chrging fcilities. As rtionl extension to the DUE, the stochstic user equilibrium (SUE) principle cn be dopted to formulte the TAP. Meng et l. (2007) found tht dding link cpcity constrints into Dgnzo s model would led to linerly constrined minimiztion problem. Nevertheless, ny optiml solution of the induced minimiztion model did not fulfil the generlized SUE conditions. This indictes tht the typicl technique used in modelling the generlized DUE conditions ws not vilble for the generlized SUE conditions except the logit-bsed generlized SUE conditions formulted by Bell (1995b). Meng et l. (2007) proposed generl stochstic user equilibrium (SUE) trffic ssignment problem with link cpcity constrints, inspired by Mher et l. (2005), who proposed formultion for stochstic socil optimum (SSO) with the objective of minimizing the totl perceived trvel time nd found tht the solution to SSO cn be chieved by solving SUE problem using the mrginl cost function. Meng et l. (2007) found tht SUE flow pttern cn be generted by solving SSO problem pplying modified link trvel time function. Previous studies on generl SUE trffic ssignment problem minly tckled link-bsed constrints [see, e.g., (Meng et l., 2007; Meng nd Liu, 2011; Meng et l., 2014)]. However, for EV use, the route choice is restricted by their driving rnge limit, which imposes pth distnce constrints to the generl SUE model. Erly lgorithms developed to solve the unconstrined logit-bsed SUE problem were link-bsed [e.g. Mher (1998)]. These link-bsed lgorithms do not require pth storge nd often use Dil s STOCH lgorithm or Bell s lterntive s the stochstic loding step (Dil, 1971; Bell, 1995). Pth-bsed lgorithms require explicit pth storge to directly compute the logit route choice probbilities. Olof et l. (1996) developed pth-bsed lgorithm bsed on the disggregted simplicil decomposition lgorithm to solve the multinomil logit (MNL) SUE problem. Bekhor nd Toledo (2005) compred pth-bsed lgorithms for the MNL SUE problem, nd showed tht the disggregted simplicil decomposition lgorithm is superior to the pth-bsed method of successive verges (MSA) lgorithm. Among the pth-bsed lgorithms for the trffic equilibrium problem with dditive pth costs, much of the recent ttention hs been focused on the disggregte simplicil decomposition (DSD) lgorithm, which ws proposed by Lson nd Ptriksson (1992), nd the grdient projection (GP) lgorithm. A comprison work between these two pth-bsed lgorithm could be found in Chen nd Lee (1999). Xu et l. (2012) investigted different strtegies for determintion of step size of the pth-bsed lgorithms developed to solve the C-logit SUE models bsed on n dpttion of the GP method. Three strtegies were investigted: () predetermined step size (Ngurney nd Zhng, 1996), (b) Armijo line serch (Lrry, 1966; Bertseks, 1976), nd (c) self-dptive line serch (He et l., 2002; Chen et l., 2012). To hve fir comprison of different step size strtegies, Bekhor et l. (2006). Used working route set, obtined from route choice set genertion lgorithm, such s lbelling, link penlty, link elimintion nd simultion, in pth-bsed problem. Behviorlly, it hd the dvntge of explicitly identifying those routes tht were most likely to be used nd lso llowed greter flexibility to include route-specific ttributes tht might not be obtinble directly from the link ttributes (Cscett et l., 1997; Bekhor et l., 2006). A column genertion procedure could lso be redily embedded in the GP lgorithm (Chen nd Jykrishnn, 1998).

4 68 Wento Jing et l. / Trnsporttion Reserch Procedi 21 (2017) To the best of our knowledge, it is still n open question to find n exct solution method for solving the generl SUE problem with pth distnce constrints on trnsport network with EV Objectives nd Contributions Since SUE relxes the perfect informtion ssumption of the DUE model by incorporting rndom error term in the pth cost function to model trvele imperfect perceptions of trvel times, it would be more relistic to pply generl SUE model to ssign EV flows to the trnsport network. It is interesting to compre the ssignment results between stochstic models nd deterministic one. Hence, the results from proposed model re compred with tht from DUE with pth distnce constrints. The pth-bsed pproch requires pth genertion/enumertion, where the size of pth set between single O-D pir cn be extremely lrge. Thus predetermined pth set is pplied in this pth-bsed problem. Meng et l. (2007) proposed solution frmework for generl SUE problem with link-bsed constrints. For the generl SUE trffic ssignment problem ddressed in this study, we investigte the properties of the pth distnce constrints nd the solution method frmework. Furthermore, K-shortest pths lgorithm is pplied to void pth enumertion. To sum up, the contributions of this study re twofold. Fit, holistic methodology is proposed for generl SUE trffic ssignment model with pth distnce constrints on EV scheme, in which the clssic unconstrined SUE model cn be used to incorporte pth distnce constrints by modifying MSA lgorithm nd finding the distnceconstrined K-shortest pths in stochstic network loding process. It is ssumed tht the EV route choices re restricted by the distnce EV cn trvel with single chrge. Second, comprison results between SUE nd DUE re provided. The mjor prt of this pper is discussion of the modelling nd solution methods for the SUE trffic ssignment problem with pth distnce constrints. The reminder of this pper is orgnized in the following order. In Sections 2 & 3, we elborte the problem formultion, nd nlyse its solution properties. Section 4 presents the solution lgorithms for both Logit-bsed nd Probit-bsed stochstic network loding models, while Section 5 presents the numericl results from pplying the lgorithm procedure for smll network nd Sioux Flls network s well s the comprison work between SUE TAP with pth distnce constrints nd DUE TAP with pth distnce constrints. In the end, Section 6 provides few concluding remrks. 2. Nottion, ssumptions nd problem description Consider strongly connected network, denoted by G N, A, where N nd A re sets of nodes nd links, respectively. (r, s) stnds for certin ordered pi of nodes, r R nd s S,where node r is n origin nd node s is destintion. R N nd S N re sets of origins nd destintions, respectively. There re non-negtive trvel demnd q of n-th vehicle type between r, s. q q,n T, r, s is column vector for ll the n n trvel demnds. Let K be the set of pths connecting O-D pir r, s, pth k K, f T f be trffic flow of n-th vehicle type on f kn, n,k K be column vector of ll these pth flows between OD pir r, s, nd kn f f T, r, s be column vector of ll the pth flows over the entire network. Let v denote trffic flow on T link A nd v v, A is column vector of ll the link flows. The pth flows nd link flows should comply with fundmentl flow conservtion equtions:

5 Wento Jing et l. / Trnsporttion Reserch Procedi 21 (2017) v f kn,k, A (1) n (r,s) k f kn q n, (r, s), n k (2) f 0, (r, s), n, k K (3) kn where =1 if pth k K between O-D pir (r, s) trvees link A, nd 0 otherwise.,k Let t v denote the seprble trvel time function of link A, which is ssumed to be positive, strictly incresing, convex nd continuously differentible function of the trffic flow on the link. All the link trvel time functions re grouped into column vector t v t v T, A. Trvel time on pth k K between O-D pir (r, s) cn be considered s function of ll the pth flows, denoted by c (f ) with the expression c k (f ) t (v ),k k (4) Given ny positive fesible pth flow pttern f, f stisfies the conventionl SUE conditions ssocited with the pth trvel time functions, c ( f ) c T,, (r, s), n, k K k is column vector of ll these pth trvel time between OD pir r, s,nmely kn = ( ( f )) (5) where P is the probbility tht vehicle type n choose pth k between O-D pir (r, s) Pth distnce constrints nd insufficient chrging fcility Bsed on EV s mrket potentil, it is expected tht in the future gsoline vehicles (GV) nd EV will coexist in the utomobile mrket. For this reson, the proposed model includes multiple clsses of vehicles, nmely GV nd EV, which distinguish from ech other in terms of driving distnce rnge nd trvel cost composition. To derive the theoreticl properties of the problem, we consider set of ssumptions regrding demnd heterogeneity nd trvel behvior. Fit without loss of generlity, it is ssumed tht the demnd popultion is only comprised of GV nd EV. Plugin hybrid electric vehicle (PHEV) re not explicitly considered since they cn be simply treted s n in-between clss of GV nd EV in terms of the technologicl nd economic fetures (i.e., driving rnge limit nd trvel cost composition), or specil type of GV with lower operting costs. Redily multiple types of EV with different driving rnge limits nd operting costs cn be incorported into the model. Second, we ssume the totl trvel demnd between ech O-D pir for every vehicle type is deterministiclly known -priori. SUE concept is devised for route choice procedure, in which ech trveler chooses route tht minimizes his/her perceived trvel cost while no one cn reduce his/her perceived trvel cost by unilterlly switching to n lterntive route. For n individul GV trveler, stochstic user equilibrium simply implies conventionl stochstic trffic ssignment problem of serching for perceived minimum cost (trvel time); wheres for n EV trveler, it poses pth distnce-constrined perceived minimum cost problem. In this pper, we scrutinize the integrted effect of different vehicle types with vrious pth constrints.

6 70 Wento Jing et l. / Trnsporttion Reserch Procedi 21 (2017) Third, without loss of generlity, we ssume tht both GV nd EV trvele use common form of systemtic trvel cost function for determining their trvel choices. The link trvel time functions re ssumed to be seprble between different network links nd identicl for different vehicle clsses, implying the trvel time on prticulr link only depends on its own trffic flow. These functions re ssumed to be positive, monotoniclly incresing, nd strictly convex. In our network equilibrium nlysis, it is implicitly ssumed tht ll EV re fully chrged t their origins. The possible vilbility of commercil bttery-chrging or bttery-swpping sttions emerging in urbn res cn be considered in the future when chrging infrstructures chieve certin level of coverge. EV use would choose pth whose distnce l is less thn or equl to the driving rnge limit of the vehicle type, denoted by D. Hence, k ny fesible pth flow pttern should stisfy the pth distnce constrints: f (D l ) 0, (r, s), n, k K (6) kn n k n which mens tht if the flow of tht clss of EV use going through this pth is positive, the pth distnce is smller thn or equl to the driving rnge of given clss of EV; otherwise, the trip flow should equl to zero. 3. Mthemticl model Due to the complexity of probit-bsed SUE problem, directly dding side constrints into tht generl SUE model does not give us n equivlent minimiztion model to the probit-bsed SUE trffic ssignment with side constrints (Meng nd Liu, 2011). However, we cn still dd the pth distnce constrint into this minimiztion model developed by Sheffi (1985) s follows, only if predetermining pth set to ensure distnces of ll the used pths re less thn the rnge limit for ech O-D pir. min Z (v) q S [c (v)] v t (v ) t ( )d (7) n v v n 0 s.t.: (1)(2)(3)(6) Compred to Sheffi s model which cn be solved s n unconstrined minimiztion problem nd still yield solution tht stisfies the flow conservtion constrints (1)(2)(3), the extr pth distnce constrints re the constrints tht needs creful considertion. To prove the equivlence between the solution of the problem given in Eq.(7) nd the SUE equtions, the fit-order derivtive of this problem hve to coincide with the SUE conditions. L(x,μ) q S [c (x)] x t (x ) t ( )d f (D l ) (8) n x 0 kn kn n k n n k The fit-order derivtive require tht L(x,μ) 0 (9) The grdient is tken with respect to link flow vector x, nd the derivtives of fit three summtion terms of Eq. (7) cn be clculted s Note tht the extr pth distnce constrints could be infesible. For some OD pi, if none of those pths connecting them stisfies the rnge limit of certin clss of vehicles, the trvel demnd between them of this clss of vehicles cnnot be ssigned to the network nd the problem hs no fesible solution.

7 Wento Jing et l. / Trnsporttion Reserch Procedi 21 (2017) However, those infesible OD pi cn be found esily by checking distnce of the shortest pth from the strt, nd if the distnce of shortest pth is physiclly longer thn rnge limit of certin clss of vehicles, then there would be no fesible pth for the SUE TAP between this OD pir for this clss of vehicles. If it is fesible, which mens there exists t lest one pth between ech OD pir tht is within the rnge limit of certin clss of vehicles. Then the term kn Dn l ls k, which is pth out-of-rnge cost incurred when the pth length exceeds the distnce limit of tht clss of vehicles should equl to zero. kn is proxy of equivlent trvel time vlue of the out-of-rnge cost per unit distnce. Therefore, by ensuring the distnce of ech chosen pth less thn rnge limit, the derivtive of the SUE objective function with respect to link-flow vrible becomes Assume tht link performnce functions re strictly incresing, the grdient becomes zero if nd only if The bove eqution expresses the SUE link flows when it hs ny fesible solution, nmely whenever we ssign trvel demnd to pths between ech OD pir, pth distnce should be less thn the rnge limit of tht clss of vehicles. Following the sme procedure of demonstrting uniqueness in pge 319, Chp. 12, Sheffi (1985), it is obvious tht the Hessin mtrix of the SUE objective function is positive definite, becuse the second derivtive of propositions s follows. with resperc to pth flow equls to zero. Therefor the model possesses two vitl Proposition 1: Any locl fesible minimum x * of this model stisfies the generlized SUE conditions, nd the Lgrngin multiplie ssocited with pth distnce constrint (6) re pth out-of-rnge costs. Proposition 2: The SUE link flow pttern induced by ny locl minimum solution of the liner constrined minimiztion model is unique. 4. Solution method 4.1. Modified MSA lgorithm It ws shown tht the MSA lgorithm cn still be pplied to the pth distnce constrined stochstic trffic ssignment problem (STAP) with direction-finding step different from tht of clssic STAP nd fesibility check step. For multinomil logit model (MNL), the steps re s follows: Step 0: Fesibility check. For ech OD pir, find the shortest pth ccording to physicl distnce. If the distnce of this pth is longer thn the rnge limit of certin type of vehicle nd the corresponding trvel demnd is positive, then there is no fesible pth for this type of vehicle between this OD pir. Record this OD pir nd infesible vehicle type to Set A. If Set A is empty, go to the next step; if not, stop nd disply Set A. Step 1: Initiliztion. Set x (0) 0, t t [x (0)]. For ech OD pir, find K shortest pth for ech clss of vehicles in terms of free flow trvel time.

8 72 Wento Jing et l. / Trnsporttion Reserch Procedi 21 (2017) If the pth distnce is greter thn the rnge limit of this clss of vehicles, set the pth trvel time to infinite. Clculte the probbility of choose ech pth, record them s initil pth set nd perform stochstic network loding to ssign ll the demnd of ech clss of vehicles between this OD pir to the corresponding K shortest pths. This yields x (1). Set itertion counter n 1. Step 2: Updte. Clculte new link cost in terms of t t [x (1)],. Step 3: Direction finding. Follow the sme procedure described in step 1 to find K shortest pth for ech clss of vehicles bsed on the current set of link trvel times,{t n }. If ll the K pths between n OD pir exceed rnge limit of this type of vehicle, use initil pth set in step 1 nd perform stochstic network loding. This yields n uxiliry link flow pttern {y n }. Step 4: Step size. { n } is predetermined step size sequence stisfying the three conditions: 0 n 1 nd lim n 0 n n n 1 2 n n 1 There re few step size sequences fulfilling the bove conditions; for exmple n, n 1, 2,..., n where prmeter 0 1 Step 5: Move. Find the new flow pttern by setting x n+1 x n (1/ n)(y n x n ). Step 6: Convergence test. Let n x 1 ( m xn x n 1 x n m 1 ) If the convergence criterion n 1 n n / x is met, stop nd {x n+1 } is the set of equilibrium link flows; otherwise, set n n 1 nd go to step Modified probit-bsed loding lgorithm For multinomil probit model (MNP), the steps re Step 0: Fesibility check. This step is the sme s tht of logit model. Step 1: Smpling. Set itertion counter n 1. Smple T n from T n ~ N(t, t ) for ech link. Step 2: All-or-nothing ssignment. For ech OD pir, find the distnce-constrined shortest pth bsed on

9 Wento Jing et l. / Trnsporttion Reserch Procedi 21 (2017) perceived link trvel time T n. Clculte the pth distnce of shortest pth for this clss of vehicles, if the pth distnce exceeds rnge limit, clculte second shortest one, etc. If ll the K pth distnces re greter thn rnge limit, use the initil shortest pth generted in Step 0. Assign the trvel demnd of ech clss of vehicles between this OD pir to the distnce-constrined shortest pth bsed on perceived link trvel time T n. This yields the set of link flow X n. Step 3: Flow verging. Let x n [(n 1)x n 1 x n ] / n. Step 4: Stopping test. Let n n 1nd go to step Numericl exmple n 1 m n 2 σ n [X x ], if mx σ n / x n,stop; otherwise, set n(n 1) m 1 Two numericl exmples re dopted in this section to ssess the proposed methodology Smll network exmple Origin Destintion Fig. 1. Smll network The fit exmple consists of 9 nodes, 18 links, nd 4 OD pi: (1,3), (1,4), (2,3), nd (2,4), s shown in Figure 1. The free-flow trvel time is used s proxy for the link length for ech link. Trvel time on ech link is defined by the following BPR (Bureu of Public Rod) type function where t 0 is the free flow trvel time, H is the link cpcity nd is prescribed prmeter. OD demnds re ssumed to be the sme for both GV nd EV (Given in Tble 1). Free-flow trvel time nd link cpcity re indicted in Tble 2. Tble 1. OD demnd of smll network exmple. Destintion Origin

10 74 Wento Jing et l. / Trnsporttion Reserch Procedi 21 (2017) We use this exmple to evlute performnce of proposed lgorithms for solving both logit-bsed nd probit- bsed SUE TAP with pth distnce constrints. The link flow ptterns under two different cses, MNL nd MNP with two clsses of use (EV nd GV), re estimted nd compred in Tble 2. EV nd GV rnge limits re set s 20 nd 100, respectively. K is set to be 8. Tble 2. The comprison of equilibrium EV nd GV flows on the smll network. Link # Link length Link Cpcity MNL flow MNP flow EV, D=20 GV, D=100 EV+GV EV, D=20 GV, D=100 EV+GV It is observed tht the SUE ssignment with different pth distnce limit resulted in different equilibrium link flows. For some links, e.g. link 1 & 2, both GV nd EV obtined similr ssignment results, while for the other links, it cn be seen tht the more EV use certin link, the less GV choose tht link, becuse the EV s pth choices re restricted by its driving rnge nd EV user prefer pths of short distnce. When EV use crowded into those links with short distnce, they becme over-sturted, thus incresing corresponding link trvel time, nd GV user would rther use those unsturted links to reduce their trvel time to obtin equilibrium. Compring MNL with MNP, it is cler tht there re 7 links close to zero flow in MNL while ll the links hve been ssigned some flow in MNP. This result might come from Independence of Irrelevnt Alterntives (IIA) property of MNL, which mkes MNP more relistic even if it suffe from its low efficiency.

11 Wento Jing et l. / Trnsporttion Reserch Procedi 21 (2017) Sioux flls network exmple Fig. 2. Sioux Flls network As shown in Figure 2, Sioux Flls network hs totl of 24 nodes nd 76 links. The trvel demnd tble used in the ppliction re from Suwnsirikul et l. (1987).The link flow ptterns under the sme scenrios re compred. Without loss of generlity, for the fit three cses, only one clss of vehicles re considered. Bsed on free-flow trvel time, the rnge limit for EV is set to where in this exmple mximum of ll shortest pth between ech O-D pir nd 1is prmeter to ensure there is t lest one fesible pth connecting ech OD pir. The K is set to 3 in K-shortest pth lgorithm. A similr OD trvel demnd is pplied to ll the four experiments, including UE, MNL, MNP & MNL with MCU (Multiclss use). Two clsses of use, i.e. EV nd GV use, re involved in MNL with MCU, where the trvel demnd of ech clss is hlf of originl trvel demnd, which mens, the EV mrket shre is 50%. For GV use, the rnge limit is set to 10. The results illustrte tht the equilibrium flows chnge significntly on number of links. A few exmple links were selected rndomly nd their flow vritions were observed in terms of different rnge limits (Figure 3). As cn be seen from Figure 2-6 tht when begins to increse, the network flows on these links behve in different mnner; UE flow ptterns chnge little s increses becuse the bse rnge limit ( 1 ) is the mximum distnce of shortest pths mong ll OD pi while UE usully requires shortest pth (All-or-nothing Assignment, for exmple). Compring with MNP, flow ptterns resulting from MNL model chnge drmticlly.

12 76 Wento Jing et l. / Trnsporttion Reserch Procedi 21 (2017) As continuously increses, the flow rtes chnge mildly nd finlly converge to vlues without rnge limit constrints. However, these chnges my not be necessrily monotone Sensitivity nlysis Fig. 3. Equilibrium flow pttern () link 1 (b) link 6 (c) link 21 (d) link 51. We conduct sensitivity test with respect to the totl demnd in Sioux Flls network, multiplied its vlue by constnt fctor nd performed SUE ssignment to observe the effect of the congestion level on lgorithm performnce. Since the Sioux Flls mtrix is quite congested (Bekhor nd Toledo, 2005), the fctor rnges from 0.1 to 1.5, in intervls of 0.1. The number of itertions required by MNL nd MNP to rech given convergence rte is presented in Figure 5. For ll levels of demnd, the modified MSA lgorithm of MNL model requires less thn 50 itertions to rech 0.01% precision level especilly when demnd level is low. However, compring with MNL, the convergence rte of MNP model is quite slow, which requires more thn 200 itertions to chieve 0.1 precision. The performnce of MNL nd MNP for the Sioux Flls network is quite different in terms of convergence rte. For efficiency, the MNL model outperforms MNP. However, s is known to ll, MNL model suffe from IIA property, nd MNP might be better when distributed computing pproches re pplied.

13 Wento Jing et l. / Trnsporttion Reserch Procedi 21 (2017) Itertion number ,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2 1,3 1,4 1,5 Demnd fctor Itertion needed to rech precision for MNL Itertion needed to rech 0.1 precision for MNP Fig. 4. Sensitivity of the lgorithm performnce to the level of demnd. 6. Conclusions This pper worked on the trffic ssignment models with pth distnce constrints, where new SUE TAP is formulted, solved nd numericlly nlysed. SUE models, which include perception error of trvel time, re considered more rtionl thn UE model. Multiclss use in SUE model represents simplified cse of current trffic networks tht crry both EV nd GV. More clsses of use with vrious rnge limit cn lso be tken into considertion. The vehicles rnge limit is determined bsed on its trvel distnce only, while rtionlly the rnge limit should be relted to both trvel distnce nd trvel time. This pper shows tht t the equilibrium point the selected pths to ssign the trvel demnd re different from tht of bsic SUE TAP. The distnce of ech pth must be less thn the rnge limit of tht clss of vehicles. The wellknown nd widely used MSA procedure nd probit-bsed network loding method re dopted nd modified to solve this problem, following the ide of putting the pth distnce constrints into the pth selection rules of stochstic network loding procedure. The direction finding step for MNL, involves finding K fesible pths to lod the trvel demnd between ech OD pir, while it requires finding fesible shortest pth for MNP in ll-or-nothing ssignment step. The proposed lgorithm is esy to undetnd nd implement. The ppliction of the lgorithms in Sioux Flls network justifies the pplicbility of the solution procedures to generl network with pth-bsed constrints. The numericl nlysis results show the impct of rnge limit on network equilibrium flows. References Adler, J.D., Mirchndni, P.B., Xue, G.L., Xi, M.J., The electric vehicle shortest-wlk problem with bttery exchnges. Networks nd Sptil Economics, Bekhor, S., Ben-Akiv, M.E., Rmming, S., Evlution of choice set genertion lgorithms for route choice models. Annls of Opertions Reserch 144, Bekhor, S., Toledo, T., Investigting pth-bsed solution lgorithms to the stochstic user equilibrium problem. Trnsporttion Reserch Prt B: Methodologicl 39, Bell, M.G.H., Alterntives to Dil's logit ssignment lgorithm. Trnsporttion Reserch Prt B: Methodologicl 29, Bell, M.G.H., 1995b. Stochstic user equilibrium ssignment in networks with queues. Trnsporttion Reserch Prt B: Methodologicl 29, Bertseks, D.R., On the Goldstein-Levitin-Polyk grdient projection method. IEEE Trnsctionson Automtic Control 21, Botsford, C., Szczepnek, A., Fst chrging vs. slow chrging: pros nd cons for the new ge of electric vehicles. EVS24 Interntionl Bttery, Hybrid nd Fuel Cell Electric Vehicle Symposium 1-9.

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