Optimal Toll Locations and Levels in Congestion Pricing Schemes: a Case Study of Stockholm

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1 Optml Toll Loctons nd Levels n Congeston Prcng Schemes: Cse Study of Stockholm Jokm Ekström, Leond Engelson nd Cls Rydergren Lnköpng Unversty Post Prnt N.B.: When ctng ths work, cte the orgnl rtcle. Ths s n electronc verson of n rtcle publshed n: Jokm Ekström, Leond Engelson nd Cls Rydergren, Optml Toll Loctons nd Levels n Congeston Prcng Schemes: Cse Study of Stockholm, 214, Trnsportton plnnng nd technology (Prnt), (37), 4, Trnsportton plnnng nd technology (Prnt) s vlble onlne t nformworldtm: Copyrght: Tylor & Frncs (Routledge): SSH Ttles Postprnt vlble t: Lnköpng Unversty Electronc Press

2 Optml Toll Loctons nd Levels n Congeston Prcng Schemes: Cse Study of Stockholm Jokm Ekström *, Leond Engelson b nd Cls Rydergren Deprtment of Scence nd Technology, Lnköpng Unversty, Norrköpng, Sweden *Correspondng uthor. Eml: jokm.ekstrom@lu.se 1

3 Optml Toll Loctons nd Levels n Congeston Prcng Schemes: Cse Study of Stockholm As congeston prcng hs moved from theoretcl des n the lterture to rel world mplementtons, the need for decson support when desgnng the prcng schemes hs become evdent. Ths pper dels wth the problem of fndng optml toll levels nd loctons n rod trffc network, nd presents cse study of Stockholm. The optmzton problem of fndng optml toll levels, gven predetermned cordon, nd the problem of fndng both optml toll loctons nd levels re presented, nd prevously developed heurstcs re used for solvng these problems. For the Stockholm cse study, the possble welfre gns of optmzng the toll levels n the current cordon, nd optmzng both the toll loctons nd ther correspondng toll levels re evluted. It s shown tht by tunng the toll levels n the current congeston prcng cordon used n Stockholm, the welfre gn cn be sgnfcntly ncresed, nd furthermore mproved by llowng toll on the bypss hghwy Essngeleden. It s lso shown tht by optmzng both the toll loctons nd levels, congeston prcng scheme wth welfre gn close to wht cn be cheved by mrgnl socl cost prcng, cn be desgned wth tolls beng locted on only fourth of the tollble lnks. Keywords: congeston prcng; rod tolls; blevel optmzton; user equlbrum, network desgn, 1 Introducton Wthout ny nterventon from the rod uthorty, the users of trnsportton network re ssumed to mke trvel decsons tht mnmze ther ndvdul trvel cost. Ths my led to n neffcent usge of the trnsportton fcltes, whch s experenced s ncresed congeston n everydy trffc, n metropoltn res cross the world. A more effcent usge cn, however, be cheved f the users mke trvel choces wth respect to the effect tht these choces hve on the other users' trvel cost. One wy of chevng such decsons s to ntroduce congeston prcng scheme, whch hs recently, n Sweden, been done n Stockholm nd Gothenburg. 2

4 As congeston prcng hs moved from n de n trnsportton lterture to prctcl mplementtons, the need for decson support when desgnng the prcng schemes hs become evdent. In ths pper, prevously developed methods (Ekström 28; Ekström, Engelson nd Rydergren 29), bsed on mthemtcl models wth the objectve to mxmze the socl surplus, for desgnng lnk-bsed congeston prcng schemes, re used n cse study of desgnng congeston prcng schemes n Stockholm. The study ncludes tunng of the current toll levels t exstng toll loctons, extendng the exstng cordon wth ddtonl toll loctons, nd completely new desgn of the congeston prcng scheme. It s, however, mportnt to recognze tht the mthemtcl model used n ths pper s lmted to wht cn be mesured nd modelled. When ntroducng congeston prcng scheme n prctce there wll be prctcl nd poltcl consdertons tht lso hve to be tken nto ccount n the fnl desgn, consdertons whch cnnot be ncluded n mthemtcl model. Generlly, the most effcent usge of scrce resource s ccomplshed when the users' cost s equl to the socl cost. Therefore, the optml usge of rod network s cheved when the tolls re equvlent to the mrgnl mpct tht one ddtonl user hs on the other users' totl trvel cost. Ths prcng prncple s often referred to s mrgnl socl cost prcng (MSCP) (Beckmnn, McGure nd Wnsten 1956) nd ws frst dscussed by Pgou (192), followed by Knght (1924). The de of MSCP n trffc network s theoretclly ppelng, but n prctce not very lkely to be mplemented. MSCP requres the collecton of tolls on every rod segment wth congeston, nd does not tke the setup nd opertonl costs of the toll collecton system nto ccount. Therefore, there s need for lterntve nd more prctcl congeston prcng schemes. When restrctons re mposed on the congeston prcng scheme desgn, the optml desgn s usully sd to be second-best (Verhoef, Njkmp 3

5 nd Retveld 1996) s opposed to frst-best when there re no restrctons. The lnk flow nd trvel demnd pttern n the trffc network, resultng from MSCP tollng, s usully referred to s system optml (SO). The MSCP tollng scheme s, however, not the only prcng scheme whch cheve SO flows nd demnds, nd n Yldrm nd Hern (25) method for fndng other prcng schemes whch cheve SO flows nd demnds s presented. For the cse when the demnd s elstc Lrsson nd Ptrksson (1998) show tht the totl toll revenue s unque for ll prcng schemes ledng to SO lnk flow nd trvel demnd (lter lso shown n Yn nd Lwphongpnch [29]). Lrsson nd Ptrksson (1998) further conclude tht even though the collecton of MSCP tolls cn be trnsferred between dfferent lnks on route, t s only possble n prctce when severl lnks cn be replced by sngle one n the SO soluton. Thus the dscusson n Yldrm nd Hern (25), relted to fndng the congeston prcng scheme wth the mnmum number of locted tolls whch stll gves SO lnk flows nd trvel demnds, s of lmted prctcl nterest. To set up nd operte congeston prcng scheme cn, even wth the dvnced technologes vlble tody, be costly tsk. Thus there s n nterest n fndng system whch produces hgh beneft t low cost. The objectve s to mxmze the socl surplus (SS), whch s the user surplus (US) plus the opertor surplus (OS). The user surplus s the user benefts (UB) (gven by the Mrshlln mesure [Zerbe nd Dvely 1994]) mnus the totl user costs (UC) (trvel tme nd pd tolls), nd the opertor surplus s the totl toll revenues (R), sometme referred to s the opertor beneft (Verhoef 22), mnus the cost of collectng the tolls (C), whch ncludes setup nd opertonl costs of the toll collecton system. The socl surplus cn thus be expressed s SS=US+OS=UB-UC+R-C. 4

6 In ths pper the rod users re ssumed to be dstrbuted ccordng to user equlbrum (Wrdrop 1952). To mxmze the socl surplus cn then be formulted s blevel optmzton problem (for n overvew of blevel optmzton problems see Luo, Png nd Rlph [1996]). On the upper level the socl surplus s optmzed by djustng the decson vrbles, whch re the toll loctons nd levels. On the lower level, the user equlbrum lnk flows nd trvel demnds re computed for fxed vlues on the decson vrbles, by solvng convex progrm (Sheff 1985), wth lnk flows nd trvel demnds beng the trffc stte vrbles. Ths mplct reltonshp between decson nd trffc stte vrbles mkes the blevel optmzton problem both nonconvex nd non-smooth, nd thus dffcult to solve for globl optml soluton (for n overvew of blevel optmzton problems n trffc plnnng context see e.g. Mgdls [1995]). To solve the blevel problem, whch wll be denoted s the toll desgn problem (TDP), dfferent (met-) heurstcs hve prevously been suggested. For nstnce Shepherd nd Sumlee (24) developed genetc lgorthm whch s extended n Sumlee (24) to fnd closed cordon schemes. In Zhng nd Yng (24) the problem of fndng closed cordon structures whch mxmze the socl surplus s ddressed by combnton of genetc lgorthms nd smulted nnelng. A closely relted problem s the problem of fndng toll levels when the toll loctons re predetermned. Ths s stll non-convex nd non-smooth optmzton problem, nd wll be referred to s the toll level problem (TLP). Note tht for the TLP, the opertor cost wll be constnt nd s therefore sometmes excluded from the optmzton problem. In Verhoef, Njkmp nd Retveld (1996) nd Lu nd McDonld (1999) propertes of the TLP re dscussed, nd n Verhoef (22), Lwphongpnch nd Hern (24) nd Shepherd nd Sumlee (24) methods for fndng optml toll levels re 5

7 developed. The congeston prcng trl n Stockholm 26, nd the permnent congeston prcng scheme, strted n 27, hs been studed n severl ppers. An overvew of the effects durng the trl perod s presented n Elsson et l. (29) nd the behvourl djustments nd equty effects re studed n Krlström nd Frnkln (29). A costbeneft nlyss of the current congeston chrgng system s gven n Elsson (29). These studes re bsed on mesured trffc dt nd trvel surveys, nd not on model estmtons. Before the trl n Stockholm, severl studes of the effects of dfferent desgns of the congeston prcng scheme were crred out (Trnsek, 23, 24) usng the Swedsh ntonl trvel demnd forecstng tool Smpers (descrbed n e.g. Beser nd Algers [22]), whch nclude detled network model of the rods n the Stockholm regon. Besdes Smpers, the sme network model of Stockholm hs been used together wth the regonl demnd forecstng models T/RIM (presented n e.g. Engelson nd Svlgård [1995]) nd SIMS (descrbed n e.g. Algers et l. [1996]), for estmtng the effects of dfferent desgns of the Stockholm congeston prcng scheme (for comprson of the dfferent demnd forecstng models see Trnsek [23]). In our cse study, more ggregted rod network s used, ncludng only the mn rods n Stockholm, nd the demnd model (bsed on dt from T/RIM) merely ncludes the choce between cr nd publc trnsport, gven fxed totl demnd. The purpose of the cse study presented n ths pper s to gve some nsght on optml desgn of congeston prcng schemes n Stockholm, but lso to show the pplcblty of the senstvty nlyss bsed method presented n Ekström (28) nd of the smoothenng technque presented n Ekström, Engelson nd Rydergren (29). The senstvty nlyss bsed heurstc s used for tunng the toll levels n the current congeston prcng scheme n Stockholm, n order to mxmze the socl surplus, nd the 6

8 smoothenng technque s used for mxmzng the socl surplus, by djustng both the toll loctons nd ther correspondng toll levels. The remnng prt of ths pper s outlned s follows. In Secton 2 the modellng frmework nd soluton lgorthms re presented. The Stockholm cse study s presented n Secton 3, nd n Secton 4 we conclude the results. 2 Methodology 2.1 The rod trffc model The trffc network s modelled by set of lnks A nd set of orgn destnton (OD) prs I. For ech lnk A there s trvel cost functon c of cr flow v nd lnk toll : c (, v ) t( v ), (1) where t ( v ) s the lnk trvel tme functon, the men vlue of tme (VOT) whch trnsforms tme nto monetry cost, nd the cr occupncy whch gve the men number of persons n ech cr. Note tht c ( v ) s the cost perceved by one trveller, not one cr, nd snce the toll,, s chrged for ech cr t s dvded by n (1), n contrst to t ( v ) whch s the trvel tme perceved by ech trveller n the sme cr. For the Stockholm cse study presented n ths pper the lnk trvel tme functons tkes on the form p 4, v v t( v ) L p 2, 1, p1, n p3, n (2) 7

9 where L s the lnk length, n the number of lnes, nd p1, p4, lnk type specfc prmeters. The rod cpcty for lnk s gven by p3,n. For ech OD pr I there s set of routes, wth f p trvellers on ech route p. The cr flow, v, on lnk s gven by v 1 f, p p I p where p tkes the vlue of 1 f route p trverses lnk, nd otherwse. The rod users re ssumed to be dstrbuted ccordng to Wrdrop's user equlbrum (UE) (Wrdrop, 1952), whch ssumes tht the rod users hve perfect nformton of trvel costs n the network nd mke decsons whch mxmze ther ndvdul utlty. The demnd s ssumed to be elstc, nd gven for OD pr I by the decresng demnd functon q D( ), wth ts nverse formulton D 1 ( q), where s the mnmum trvel cost (n monetry terms) nd q the trvel demnd (n the unt of trvellers per hour) n OD pr I. For the Stockholm cse study the elstc demnd functon wll be gven by the pvot pont verson of the bnoml logt model (Kumr, 198). For ech OD pr I, the cr trvel demnd s clculted s q A D ( ) T, A K e ( ) (3) where s the mnmum trvel cost by cr n the tolled (non-tolled) ( ) scenro respectvely, nd the prmeters A nd K re the number of trvellers by cr nd by publc trnsportton n the non-tolled scenro (correspondng to the cr trvel cost ). T s the totl demnd n OD pr, wth T A K, nd s the dsperson prmeter. In the pvot pont verson of the bnoml logt model the trvel tme for publc trnsport s ssumed to be fxed, nd not dependng on congeston on the 8

10 rod or n the publc trnsportton system tself. Note tht the number of cr users n OD pr s lwys less thn T nd when ncreses the number of trvellers by cr wll decrese nd the number of trvellers by publc trnsport ncrese. The user equlbrum wth elstc demnd, gven toll vector, cn be formulted s the complementrty problem f p c (, v ) p, p, I A f p c (, v ) p, p, I A q D q I 1 ( ), q D q I, 1 ( ), whch sttes tht ny used route n OD pr must hve trvel cost equl to the mnmum trvel cost n OD pr. An equvlent convex progrm cn be formulted s (see e.g. Sheff [1985] nd Ptrksson [1994]) (,, ) v 1 mn (, ) q ( ), qv A I G q v c x dx D w dw subject to fp q, I p f, p, I p (4) q, I v 1 f, p p I p A. 9

11 When the reltonshp between trvel cost nd trvel demnd s gven by the pvot pont verson of the bnoml logt model, we cn rewrte (4) s (Ekström, 28) v q A ( T q ) T T mn q, v G(, q, v) c (, ) ln ln x dx q A I Kq T q subject to fp q, I p f, p, I p (5) q, I v 1 f, p p I p A. whch cn be solved by the prtl lnerzton method presented n Evns (1976), n whch fxed demnd UE problem s solved repetedly nd the demnd s updted n between. For solvng the fxed demnd UE problem, the Dsggregted Smplcl Decomposton (DSD) lgorthm (Lrsson nd Ptrksson, 1992) wll be used. The DSD lgorthm cn be restrted from prevous soluton, wth the demnd dt updted, nd re-optmze the prevous soluton effcently. Ths s useful feture when solvng the fxed demnd UE problems n Evns prtl lnerzton method. 2.2 The toll level problem In the toll level problem (TLP) the objectve s to mxmze the socl surplus nd the decson vrbles re the toll levels for ech toll locton. The toll loctons re gven exogenously nd the set of tolled lnks (T ) s gven by where M s lrge postve number, nd 1 f lnk s tollble nd otherwse. T :{ g M}, (6) g s prmeter whch tkes on the vlue of The socl surplus mesure (SS) conssts of the user surplus (US), 1

12 q ( ) 1 US(, q( ), v( )) D ( ) ( ( )) ( ), w dw t v v (7) I A n whch the frst sum s the user benefts (gven by the Mrshlln mesure [Zerbe nd Dvely, 1994]) nd the second sum the user costs, plus the opertor surplus (OS) (8) OS(, q( ), v( )) v ( ) C g, A whch s the collected tolls mnus the cost of collectng the tolls. q() nd v() re gven by the soluton to the user equlbrum problem, ether n the form of (4) or (5), nd C equls the cost of settng-up nd opertng toll on lnk. Let SS be the socl surplus ssocted wth the zero toll vector, nd let SS(, q( ), v( )) be the socl surplus ssocted wth toll vector. The objectve to mxmze cn then be formulted s mx F ( ) SS(, q( ), v( )) SS. T TLP Note tht SS s constnt, nd wth (7) nd (8) nserted, the objectve cn be expressed s TLP mx F (, q( ), v( )) US, q( ), v( ) OS, q( ), v( ) T q ( ) 1 D ( ) ( ( )) ( ) ( ) w dw t v v v Cg SS I A A q ( ) 1 ( ) ( ( )) I A A D w dw t v v ( ) C g SS. (9) Insertng the demnd functon (3) n (9), the optmzton progrm cn be expressed s (for detls see Ekström [28]) ( ( )) 1 Ae K mx FTLP ( v( ), ( ), ) T ln v ( ) C g, T T (1) I where s the mnmum OD trvel cost n OD pr n I, nd cn esly be extrcted from the soluton to (5). The TLP, ether n the form of (9) or (1) s blevel progrm A 11

13 whch n the generl cse s non-convex. Even though t s possble to trnsform the problem nto sngle level mthemtcl progrm wth equlbrum constrnts (MPEC) (Lwphongpnch nd Hern 24), the non-convex nture s not chnged (Luo, Png nd Rlph 1996). Also, F TLP s non-dfferentble t solutons for whch the set of used routes (n the user equlbrum problem) wth mnml trvel cost s chnged. To solve the TLP dfferent methods hve prevously been suggested. Dervtve bsed methods re presented n Yng nd Lm (1996), Verhoef (22), (Lwphongpnch nd Hern 24) nd Ekström (28), nd n Shepherd nd Sumlee (24) nd Yng nd Zhng (23) stochstc serch methods re ppled. In Ptrksson nd Rockfellr (23) frmework for performng senstvty nlyss of trffc equlbr wth elstc demnd s presented. The results from Ptrksson nd Rockfellr hve been ppled to obtn grdents (or subgrdents) n vrous blevel progrms, n whch the lower level progrm represents user equlbrum problem; n Josefsson nd Ptrksson (27) for network desgn problem, n Ekström (28) nd Ekström, Engelson nd Rydergren (29) for toll level nd locton problems, nd n Lundgren nd Peterson (28) nd Ledermn nd Wynter (211) for OD estmton problems. In ths pper the scent method presented n Ekström (28) nd Ekström, Engelson nd Rydergren (29) wll be used to solve the TLP. By performng senstvty nlyss on the current equlbrum lnk flows, OD flows, nd OD trvel costs * v, * q nd *, t the toll level, the subgrdents v ( ),, A nd q ( ), I, cn be computed (Josefsson nd Ptrksson 27) by solvng one senstvty nlyss problem for ech tollble lnk. Also, the subgrdent ( ), I cn be computed for the cse when the demnd functon s gven by (3) (for detls see Ekström [28]). The senstvty nlyss problem s essentlly user 12

14 equlbrum problem wth lner lnk cost nd nverse demnd functons. The lnk cost functon for lnk s replced by * c ( v, ) (, ) v v c v nd the nverse demnd functon for OD pr s replced by 1 * 1 D ( q) ( ) q D q q, where nd cn be nterpreted s vectors of drectonl dervtves n the drectonl chnge of lnk toll levels, gven by the vector. Settng 1 for lnk lnks b nd for ny other b wll gve the lnk flow nd demnd perturbtons, nd, whch correspond to postve unt chnge of the toll level on lnk b. By solvng the senstvty nlyss problem for ech lnk b A, wth 1 nd for every lnk b, the subgrdents v ( ) nd q ( ) cn be constructed. b If TLP F ( v ( ), ( ), ) s dfferentble wth respect to n the pont, v v, q q nd q, f F TLP ( v ( ), ( ), ) s non-dfferentble n then the subgrdents wll be used to pproxmte the grdents. By the rule of chn F TLP cn then be computed, nd used n n scend method. Snce there cn exst cses when F TLP ( v ( ), ( ), ) s non-dfferentble t s not gurnteed tht FTLP wll lwys gve n scent drecton, nd the lne serch (n the scent method) s therefore used wth mnmum step length n Ekström (28). Snce the senstvty nlyss problem essentlly s user equlbrum problem Josefsson nd Ptrksson (27) dscuss how to extend the DSD lgorthm for solvng the senstvty nlyss problem, for the cse when the demnd s fxed. For more detls on the senstvty nlyss problem we refer to Ptrksson nd Rockfellr (23) nd Josefsson nd Ptrksson (27), nd for detls on the senstvty nlyss bsed scent 13

15 method used for solvng the TLP (ether on the form (9) or (1) we refer to Ekström (28). 2.3 The toll desgn problem In the toll desgn problem (TDP) the objectve s to fnd both the toll loctons nd ther correspondng levels, whch mxmze the socl surplus. The set of tollble lnks s stll gven by (6), but the cost ssocted wth ech toll locton s now gven by C sgn( ), where the functon sgn of rel rgument x s defned s 1, f x sgn( x), f x. By usng the sgn-functon, there wll only be cost ssocted wth the toll locton f the toll level s postve, nd the toll loctons re gven mplctly by the postve toll levels. Wth the demnd gven by (3), the toll desgn problem s formulted s ( ( )) 1 Ae K mx FTDP ( v( ), ( ), ) T ln v ( ) C sgn( ). T T (11) I To solve ths problem heurstc procedure (referred to s the pproxmton pproch n Ekström, Engelson nd Rydergren (29) s ppled, whch combnes smoothng technque of the objectve functon wth the senstvty nlyss bsed lgorthm descrbed n the prevous secton. The generl de of the smoothng technque s to replce the sgn functon n (11), wth contnuous functon. Note tht due to the blevel nture of the problem, the objectve functon my stll be non-smooth wth respect to the toll levels. Comprng the TLP (1) wth the TDP (11), the sgn functon n (11) gves the problem combntorl structure whch mkes dervtve bsed soluton methods A 14

16 unsutble. In the heurstc pproch we wll smoothen the dscrete prt of the objectve functon by pproxmtng the sgn functon wth the contnuous functon 2 (, k) 1, k 1 e where k determnes the ccurcy of the pproxmton. Note tht (, k ) t k, nd for ny, (, k ) tends to sgn( ) when k tends to nfnty. Replcng sgn( ) n problem (11) by (, k ) gves ( ( )) 1 Ae K mx FTDP-S( v( ), ( ), ) T ln v ( ) C (, k), T T (12) I whch s solved by the sme senstvty nlyss bsed scent method s the TLP. Strtng wth low vlue of k, problem (12) s solved to locl optmum. The prmeter k s then ncresed nd (12) s re-solved wth the prevous soluton s the new strtng soluton. The tertons re repeted untl the dfference between the A objectve functon vlues n (11) nd (12) s smll, nd s k s ncresed F TDP-S symptotclly converges to F TDP. Note tht the nformton trnsferred between the tertons s the toll levels, nd the toll loctons re gven mplctly by the toll levels. When the ccurcy of the pproxmton s grdully ncresed, t s, however, less lkely tht the toll loctons re chnged, nd n the fnl terton t s n prctce TLP tht s solved. Note tht settng k n (12) results n TLP wth the fxed set of tollble lnks gven by T, nd f ll lnks re tollble the lnk flow nd demnd wll correspond to the SO soluton. Solvng problem (1), wth the senstvty nlyss bsed scent method, wll n ech terton requre t lest one user equlbrum problem to be solved nd lso one senstvty nlyss problem to be solved for ech tollble lnk. Note tht solvng problem (11), wth the smoothenng technque requre seres of pproxmton problems (12) to be solved. Thus the computtonl burden of solvng 15

17 (1) nd (11) wll depend on the network sze (number of lnks nd OD prs), the level of congeston, nd the number of tollble lnks. For more detled descrpton of the method, nd the complete lgorthm, see Ekström (28). 3 Cse study 3.1 The Stockholm model The Stockholm network hs 392 lnks (312 f the connectors to the orgn/destnton zones re not counted) nd 4 zones whch gve 1 56 OD prs, nd wll model the cr trffc n the Stockholm regon durng the mornng rush hour. To reduce the computtonl tme, one cn dentfy nodes n the network whch only connect two lnks (only countng one drecton), nd locte toll to ether one of these, but not both. For the Stockholm network, ths reduces the number of tollble lnks from 312 to 291. In ths study, n ggregted trffc network of the Stockholm regon hs been used (Fgure 1), wth trvel tme functons on the form (2) nd trvel demnd functons on the form (3). The totl demnd of cr nd publc trnsport (for trvellers wth ccess to cr) durng the mornng rush hour, for ech OD-pr, s bsed on dt from the demnd forecst model T/RIM. The T/RIM dt s, however, gven for full Stockholm network wth bout 11 lnks nd 125 zones. The zones n the full network hve thus been ggregted to the zones n the smller ggregted network used n ths study. The dsperson prmeter.7 nd the cr occupncy 1.13 trvellers per cr re lso provded from the T/RIM model. It s possble tht the ggregton of the trvel demnd model nd ts usge wth the ggregted trffc network, wthout further clbrtons, wll result n hgher lnk flows compred wth results from other models. In Tble 1, lnk flows produced by the ggregted model for the no toll soluton s compred to results presented n Trnsek (23) from usng Smpers, SIMS nd T/RIM 16

18 on the full sze Stockholm network. The totl flows re compred for the bypss hghwy Essngeleden nd for the crossng of the current toll cordon. For Essngeleden the dfference between ll four models re smll, but for the crossng of the current toll cordon the totl flow rnge between nd vehcles per hour, nd the totl flow produced by the ggregted model s very close to the flow from the T/RIM model. It should be noted tht ll four models ssgn flow close to the model cpcty on Essngeleden. For Smpers, SIMS nd T/RIM the sme network lyout (the full network model) nd trvel cost functons re used, nd for the ggregted model the trvel cost functons for Essngeleden re smlr to wht s used n the full network model. Thus the vrton n lnk flows cn be expected to be smll for Essngeleden, between the dfferent models. The ggregted model nd the Smpers model re further compred n Tble 2, both for the flows wth the no toll soluton, nd for the predcted effect of the current congeston prcng cordon n Stockholm. Even though the totl flow on Essngeleden s smlr for the two models, the ggregted model gves hgher flow for the northbound trffc nd lower flow for the southbound trffc. The northbound flow on Essngeleden from the ggregted model s close to the model cpcty n the non-tolled cse, nd whle Smpers gve n ncrese of 8% wth the current congeston prcng scheme n Stockholm, the flow s only ncresed by 1% wth the ggregted model. Also, the reducton n outbound flow cross the current congeston prcng cordon seems to be hgh compred wth the Smpers model. 17

19 Fgure 1: The Stockholm network. Tolled lnks n the currently mplemented cordon n Stockholm re mrked by grey brs. The unt of ccount for costs nd benefts n the objectve functon cn ether be consumer prces or producer prces. The users' benefts nd costs of trvellng re usully gven n consumer prces to reflect tht the users relte ther costs nd benefts of trvellng to consumpton of lterntve goods wth prces ncludng vlue dded tx. The opertors nvestment costs re, however, usully gven n producer prces to reflect tht the opertor s not chrged vlue dded tx on nvestment costs. To convert from producer prces to consumer prces fctor equl to 1.21, whch represents the verge ndrect tx on consumer goods n Sweden, s therefore used (SIKA 29). In ths cse study ll costs nd benefts re gven n Swedsh currency (SEK) nd n consumer prces, wth the vlue of tme set to 1.2 SEK per mnute. The collecton cost (n 18

20 consumer prces) s gven per pek hour, nd s for ech lnk computed s C 5 5n, where n s the number of lnes for lnk. The men collecton cost s SEK per pek hour nd toll locton (1 189 SEK n producer prces). Tble 1: Trffc flow predctons, n vehcles per hour, computed wth dfferent models for the mornng rush hour. Source for Smpers, SIMS nd T/RIM: Trnsek (23). Model Essngeleden ll drectons Across the current cordon ll drectons Smpers SIMS T/RIM Aggregted model Tble 2: Modelled trffc flows for Stockholm, wth nd wthout the current congeston Essngeleden Both drectons Essngeleden Northbound Essngeleden Soutbound Across cordon Both drectons Across cordon Inbound Across cordon Outbound prcng cordon n Stockholm. Source for Smpers: Trnsek (24). Smpers Wthout (vec/h) Wth (chnge of flow n %) Aggregted model Wthout (vec/h) Wth (chnge of flow n %) Optmzng toll loctons nd levels n the Stockholm network A totl of sx scenros wll be evluted wth respect to the socl surplus. The frst nd second re the MSCP scheme nd the current congeston prcng cordon n Stockholm (wth the toll levels set to 2 SEK). In the thrd scenro, the toll loctons re gven by the current cordon but wth the toll levels optmzed. The fourth nd ffth scenros re bsed on the current cordon but wth tolls (one n ech drecton) dded on the bypss 19

21 hghwy Essngeleden (mrked n Fgure 1); the fourth scenro wth ll toll levels beng set to 2 SEK (denoted extended current cordon) nd no optmzton performed, nd the ffth wth ll the toll levels optmzed (denoted extended current cordon optmzed). Fnlly, n the sxth scenro, (11) s solved to optmze both toll loctons nd toll levels (denoted TDP soluton), usng the smoothenng technque descrbed n Secton 2.3. When solvng the TLP, the senstvty nlyss bsed scent method hs been ntlzed wth toll levels equl to 2 SEK. For the smoothenng technque, the ntl vlue of k s set to 1, nd the ntl soluton ( ) s set to zero tolls for ll lnks. Note tht the hgher k s n the frst terton, the fewer toll loctons wll be consdered when ll lnks re ntlly set to zero. On the other hnd, hgher ntl vlue on k wll reduce both the totl number of tertons, nd wthn ech terton fewer senstvty computtons hve to be crred out, reducng the computtonl burden. Our experence s tht k 1 nd.25, gves resonble blnce between soluton qulty nd computtonl tme. The rnge of the toll levels, the number of toll fcltes, the number of lnks wth flow bout cpcty nd the totl cr demnd, re for ech scenro gven n Tble 3. In Tble 4, the resultng consumer surplus, opertor surplus nd socl surplus ssocted wth ech scenro re presented. wll mply comprson wth the no toll soluton, nd SS wll denote the chnge n socl surplus f the set-up nd opertonl costs re excluded. The mxmum vlue on SS cn be obtned by mxmzng (11) wth C for every lnk n A, whch s one prtculr cse for whch (11) cn be solved to globl optmlty by usng MSCP tolls. In Fgure 1, the current cordon, the bypss hghwy Essngeleden nd the centrl busness dstrct (CBD) re dsplyed. 2

22 Tble 3: Toll levels ( ), number of toll collectng fcltes ( N ), number of lnks wth flow exceedng cpcty ( E ) nd totl cr demnd (q) n number of cr trps per hour. Scheme (SEK) N E q. No toll MSCP Current cordon Current cordon optmzed Extd. current cordon Extd. current cordon optmzed TDP soluton Tble 4: Consumer surplus ( CS ), opertor surplus ( OS ), nd socl surplus wth ( SS nd SS ), n SEK for ech scenro. Scheme 1. MSCP Current cordon Current cordon optmzed Extd. current cordon Extd. current cordon optmzed TDP soluton In the TDP soluton the toll levels rnge between 1.1 nd 34.8 SEK, nd the demnd s reduced by cr journeys per hour from wth the no toll soluton. The number of lnks wth flow bove model cpcty s reduced from 8 to 24. Wth no tolls n the trffc network 35 lnks hve flows whch exceed cpcty by more thn 15%, nd wth the TDP soluton the flow on only three lnks exceeds the cpcty wth more thn 125%. Ths cn be compred wth the current closed cordon congeston scheme whch hs 6 lnks wth flow bove cpcty, nd 23 lnks wth flow tht exceeds 125% of the cpcty. In Tble 4 t s cler tht the mxmum vlue of SS s obtned wth MSCP tolls. MSCP tollng (Scenro 1), however, requre lrge number of tolls to be locted whch reduce the vlue of SS the current closed cordon schemes come close to the. It cn lso be noted tht none of the solutons bsed on 21 SS vlue of the TDP soluton.

23 Equl toll levels (Scenro 2 nd 4) my stll be ppelng from prctcl perspectve. The MSCP scheme yelds the second hghest ncrese n socl surplus, nd comprng SS for the dfferent scenros, optmzng toll locton nd levels results n 96% of the mxmum SS, but wth only 69 tolled lnks (24% of the tollble lnks) compred wth the 291 tolled lnks whch re requred to rech MSCP. Exmnng the schemes bsed on the current cordon t s obvous tht by tollng Essngeleden, the current scheme cn be mproved, nd wth toll of 2 SEK on every tolled lnk n the extended scheme, SS s ncresed by 42%, compred wth the current cordon. We lso notce tht by re-optmzng the toll levels, nsted of usng the sngle one of 2 SEK, SS cn be mproved between 35 nd 39%. The proportons between the consumer surplus (CS) nd opertor surplus (OS) for the two schemes bsed on the current Stockholm cordon (wth nd wthout toll on Essngeleden ), but wth dfferentted toll levels, s dfferent from the other prcng schemes. The reoptmzed schemes yeld reduced dsutlty for the rod users, but the opertor surplus s lso reduced. It s, however, necessry to remember the non-convex nture of the problem, nd the re-optmztons hve been ntlzed wth n equl toll level of 2 SEK for ll tollble lnks. If zero toll levels would nsted be used s ntl toll levels n senstvty bsed scent method, the re-optmzton of the current cordon (no toll on Essngeleden ) would result n SS SEK, but wth CS SEK nd OS SEK. Thus, both the dsutlty for the rod users nd the opertors surplus hs ncresed compred wth the current cordon. It s, however, no surprse tht the fnl soluton wll depend on the strtng pont when usng steepest scent method to solve non-convex optmzton problem. In ths cse t s nterestng to see tht even though the two solutons lmost hve the sme objectve functon vlue, there re lrge dfferences n the dsutlty experenced by the rod users, nd the totl sum of the 22

24 collected tolls. If the opertor s more nterested of low experenced dsutlty for the rod users, the prcng scheme wth CS SEK s preferble, but f the opertor prefers hgh proft CS SEK s the one to choose. The chnges n cr trvel tmes for four dfferent OD prs hve been studed more crefully, comprng the TDP soluton, the current cordon (equl tolls, 2 SEK) nd the no tolled soluton. One OD pr wth orgn wthn the CBD, defned by the current closed cordon n Stockholm (see Fgure 1) nd destnton n the southern suburbs (), the second one orgntng n the southern suburbs nd endng wthn the CBD (b), the thrd one wth both orgn nd destnton wthn the CBD (c), nd the fourth wth orgn n the northern nd destnton n the southern suburbs (d). Tble 5 dsplys the mxml nd the mnml trvel tmes, the mxml nd the mnml pd tolls, nd the totl trvel costs for the four OD prs nd the three scenros. For trvellers n OD pr () nd (c) the reductons n trvel tme s smll, nd for OD pr (c) the reducton do not dffer very much between the TDP soluton nd the current cordon, even though the users trvelng wthn the current cordon tody do not need to py ny tolls. Also, for OD pr () nd (c) the reducton n trvel tme s lrger wth the current cordon, compred to the TDP soluton. In the TDP soluton OD pr (c) gets reducton n trvel tme tht s lrger thn the tme equvlent to the pd tolls. For the current closed cordon scheme we notce tht no tolls hve to be pd for journeys wthn OD pr (c) nd (d), whch s not surprsng snce (c) s journey wthn the current cordon nd rod users n OD pr (d) do not need to pss the current cordon to rech ther destnton. Focusng on the TDP soluton the lrge reductons n trvel tme cn be found for rod users n OD pr (b) nd (d), nd these re lso the journeys wth the hghest pd tolls. 23

25 Tble 5: Trvel tme, pd tolls nd totl cost for cr journey n four dfferent OD prs. OD pr b c d Scheme TDP soluton Current cordon No toll TDP soluton Current cordon No toll TDP soluton Current cordon No toll TDP soluton Current cordon No toll Trvel tme (mnutes/trveller) Pd tolls (SEK/cr) Totl cost (SEK/trveller)

26 Fgure 2: Lnk flow chnges wth the TDP soluton (upper) nd wth the current closed cordon (lower) compred to the no toll soluton. Reduced flow s mrked wth green nd ncresed wth red. The lnk flows n the TDP nd n the current cordon solutons hve been compred to the lnk flows n the no toll soluton. Fgure 2 shows lnk flow comprson for the mjor prt of the Stockholm network. Green brs ndcte reducton n lnk flow nd red brs n ncrese. The TDP soluton gves reducton of the trffc flow, not only 25

27 wthn the CBD but lso n the surroundng network, whle the current cordon only hs lrger mpct on the trffc flow n nd out of the CBD. Tble 6 gves the chnge n trffc flow cross the current congeston prcng cordon n Stockholm. The TDP nd MSCP solutons hve smlr effect on the flow crossng the cordon n both drectons nd the chnge n the nbound flow from these solutons re lso close to wht s cheved wth the current cordon. For the outbound flow, both the TDP nd MSCP solutons gve less reducton. The optmzed versons of the current cordon (both wth nd wthout toll on Essngeleden ) shows lrger chnge n the nbound trffc flow, nd smller chnge n the outbound trffc flow, compred wth the current cordon. As the current cordon n Stockholm s constructed, t s lkely tht the trffc flow on Essngeleden wll ncrese nd crete severe congeston. In Tble 7 the volume/cpcty (V/C) rto, from ech scenro, s gven for Essngeleden. Both the current cordon nd ts re-optmzed verson hve smll effects on the northbound trffc on Essngeleden, whch s the drecton wth the most sever congeston. For the southbound drecton there s n ncrese by 14-17%. The extended cordon decreses the V/C rto between 12 nd 19%, nd wth the toll levels optmzed, ths cordon hs mjor mpct wth reducton round 4% n both drectons. The TDP nd MSCP schemes show smlr V/C rtos, wth reductons between 23 nd 26% n the southbound drecton nd round 32% n the northbound drecton. Over ll the TDP soluton ncreses the flow on very few lnks nd none of the lnks wth ncresed flow exceeds the lnk cpcty. 26

28 Tble 6: Chnge n trffc flow cross the current cordon n Stockholm compred wth the no toll soluton. Scheme Totl Inbound Outbound 1. MSCP -23% -28% -16% 2. Current cordon -37% -32% -47% 3. Current cordon optmzed -47% -49% -42% 4. Extd. current cordon -36% -31% -46% 5. Extd. current cordon optmzed -42% -45% -37% 6. TDP soluton -23% -3% -12% Tble 7: Volume/cpcty (V/C) rto for Essngeleden, chnge (n %) compred wth the no toll soluton. Scheme Southbound Northbound. No toll MSCP.5 (-23%).7 (-31%) 2. Current cordon.74 (+14%) 1.3 (+1%) 3. Current cordon optmzed.76 (+17%) 1.3 (+1%) 4. Extd. current cordon.57 (-12%).83 (-19%) 5. Extd. current cordon optmzed.39 (-4%).56 (-45%) 6. TDP soluton.48 (-26%).68 (-33%) In prctce, frctonl toll levels re not lkely to be ntroduced. Gven toll level soluton to ether (1) or (11), the contnuous toll levels re therefore replced by the closest multple of 5 SEK. Ths s done for one dscrete toll level t tme, strtng wth the lnks wth tolls closest to the lowest dscrete toll level, nd n between the remnng non-fxed toll levels re re-optmzed. Fgure 3 shows the toll loctons nd the dscretzsed toll levels bsed on the TDP soluton. For the dscrete toll levels, SS s SEK whch s 97% of tht wth the orgnl TDP soluton. The toll levels re 5, 1, 15, 2, 25 nd 35 SEK, snce no toll cme close to 3 SEK. In the process one toll s set to zero (wth n ntl toll level below 2.5 SEK). As comprson we round ll toll levels gven by the TDP soluton to the closest dscrete level (multples of 5 SEK), wthout ny re-optmzton n between, whch only reduce SS by nother SEK. Thus the beneft from re-optmzng the toll levels durng the dscretzton process, for ths cse, s neglectble. 27

29 Fgure 3: The complete set of toll loctons nd dscrete levels gven by the dscretzsed soluton. The upper network shows the mjor prt of the Stockholm network, nd the lower one s detled vew of the centrl prt. 3.3 Optml expnson of the current congeston prcng scheme n Stockholm How to extend n exstng congeston prcng scheme s n nterestng queston. Snce the toll collectng fcltes re lredy n plce, t s both prctcl nd economcl to mke use of the exstng fcltes when plnnng n extenson of the current congeston 28

30 prcng scheme. Ths s resonble lso from the trvellers perspectve, s they re lredy used to the current system. The smoothenng technque cn esly be dopted to solve ths problem, by fxng the toll loctons of the current toll collectng fcltes. When consderng n optml expnson of the current congeston prcng cordon n Stockholm, the current loctons (Fgure 1) nd toll levels (2 SEK) re gven s fxed nd the queston s where to locte ddtonl tolls. It s further ssumed tht t s only prctcl to toll the lrger rterl rods. Ths leves totl of 174 possble toll loctons. Applyng the smoothenng technque result n 4 new toll loctons n ddton to the 2 current loctons. For the ddtonl toll loctons the toll levels rnge from 3. to 25.4SEK, wth SS , only ncludng the setup nd opertonl costs of the ddtonl toll loctons, wth SS whch mkes up 86% of the mxmum SS. The sme procedure for fndng dscrete toll levels bsed on the contnuous ones, s dscussed n the prevous secton, hs lso been ppled to the expnson. The ddtonl toll loctons nd dscrete toll levels re presented n Fgure 4, nd the dscrete toll levels yelds 98.4% of SS wth contnuous ones. 29

31 Fgure 4: Addtonl toll loctons nd dscrete toll levels n multples of 5 SEK for the expnded prcng scheme. Conclusons nd further reserch The cse study presented n ths pper shows tht the smoothenng technque, used for solvng the TDP, s pplcble for networks of nturl sze. It s further shown tht t s possble to cheve 96% of the mxmum socl surplus (dsregrdng ny setup or opertonl cost) by only loctng toll on 24% of the tollble lnks, whch sgnfcntly reduce the sze nd cost of the toll collecton system, compred wth the MSCP soluton. The results lso provde nsght nto benefts tht cn be cheved by optmzton of the toll levels n the current cordon n Stockholm, nd/or ddng tolls on Essngeleden. Allowng dfferentted toll levels opens for ndvdul optmzton of 3

32 the level t ech toll pont, nd just by optmzng the toll levels n the current cordon the socl surplus cn be ncresed by 35%, wthout ny new toll loctons beng dded. It hs lso been shown tht even though two dfferent locl optml solutons to the TLP (found wth dfferent ntl solutons) hve smlr objectve functon vlues, they my dffer sgnfcntly when t comes to the level of consumer nd opertor surpluses. The Wrdrop equlbrum ssumes tht the rod users hve perfect nformton bout the trvel cost n the rod network. Allowng contnuous toll levels my, however, mke t more dffcult for the rod users to overvew the cost of mkng trp n the network. From ths perspectve dscrete toll levels re more ppelng n prctce. The ntroducton of dscrete toll levels would, however, mke ny knd of dervtve bsed lgorthm useless n the optmzton process. For the cse presented n ths pper t hs been shown how contnuous toll levels esly cn be replced by dscrete ones, wth only smll reducton of the socl surplus. These results of course depend on the number of dscrete toll levels, but for the Stockholm network the use of sx dscrete toll levels only reduce the mprovement n socl surplus by 3%. There cn be severl resons to keep the exstng toll collecton fcltes, nd the heurstc pproch for loctng the tolls cn be ppled to fnd ddtonl toll loctons. For the ggregted Stockholm network t s possble to rech 86% of the mxmum socl surplus, wth 4 ddtonl toll loctons. In ths study, only the mornng rush hour s studed. Evluton nd optmzton of the toll loctons gven the dfferent demnd pttern durng dfferent tme perods s n nterestng re for further reserch. Alterntve functonl form for pproxmtng the dscrete vrbles my further mprove the results. When t comes to lrge networks the computtonl burden wll be lrge, nd to fnd ncrementl schemes of the k 31

33 prmeter, possbly ndvdul for ech lnk, could both mprove the results nd cut the computtonl tme, by reducng the number of overll tertons. Acknowledgements The uthors would lke to thnk Torbjörn Lrsson for comments on n erler drft of ths pper. References Algers, S., A. Dly, P. Kjellmn, nd S. Wdlert ``Stockholm model system SIMS: Applcton. In World Trnsport Reserch. Proceedngs of the 7th World Conference on Trnsport Reserch, edted by Hensher, D. A., J. Kng, nd T. H. Oum: Oxford: Elsever. Beckmnn, M., C. McGure, nd C. B. Wnsten Studes n the Economcs of Trnsportton. New Hven: Yle Unversty Press. Beser, M., nd S. Algers. 22. ``SAMPERS, The New Swedsh Ntonl Trvel Demnd Forecstng Tool. In Ntonl Trnsport Models: Recent Developments nd Prospects, Advnces n Sptl Scence, edted by Lundkvst, L., nd L. G. Mttsson: Berln: Sprnger Verlg. Ekström, Jokm. (28).``Desgnng Urbn Rod Congeston Chrgng Systems: Models nd Heurstc Soluton Approches Lcentte Thess, Lnköpng Unversty. Ekström, J., L. Engelson, nd C. Rydergren. 29. ``Heurstc lgorthms for secondbest congeston prcng problem. NETNOMICS: Economc Reserch nd Electronc Networkng 1 (1): Elsson, J. 29. ``A cost-beneft nlyss of the Stockholm congeston chrgng system. Trnsportton Reserch Prt A: Polcy nd Prctce 43 (4):

34 Elsson, J., L. Hultkrntz, L. Nerhgen, nd L. S. Rosqvst. 29. ``The Stockholm congeston - chrgng trl 26: Overvew of effects. Trnsportton Reserch Prt A: Polcy nd Prctce 43 (3): Engelson, L., nd S. Svlgård ``The trvel demnd nd lnd-use model for Stockholm regon. Techncl report, Inreg AB. Evns, S. P ``Dervton nd nlyss of some models for combnng trp dstrbuton nd ssgnment. Trnsportton Reserch 1 (1): Josefsson, M. nd M. Ptrksson. 27. ``Senstvty nlyss of seprble trffc equlbrum equlbr wth pplcton to blevel optmzton n network desgn. Trnsportton Reserch Prt B: Methodologcl 41 (1): Krlström, A. nd J. P. Frnkln. 29. ``Behvorl djustments nd equty effects of congeston prcng: Anlyss of mornng commutes durng the Stockholm trl. Trnsportton Reserch Prt A: Polcy nd Prctce 43 (3): Knght, F ``Some fllces n the nterpretton of socl cost. Qurterly Journl of Economcs 38 (4): Kumr, A ``Pvot pont modelng procedures n demnd estmton. Trnsportton Engneerng Journl 16 (6): Lrsson, T. nd M. Ptrksson ``Smplcl decomposton wth dsggregted representton for the trffc ssgnment problem. Trnsportton Scence 26 (1): Lrsson, T. nd M. Ptrksson ``Sde constrned trffc equlbrum modelstrffc mngement through lnk tolls. In Equlbrum nd dvnced trnsportton modelng, edted by P. Mrcotte nd S. Nguyen, Norwell: Kluwer Acdemc. Lwphongpnch, S. nd D. W. Hern 24. ``An MPEC pproch to second-best toll prcng. Mthemtcl Progrmmng 11 (1):

35 Ledermn, R. nd L. Wynter (211). ``Rel-tme trffc estmton usng dt expnson. Trnsportton Reserch Prt B: Methodologcl 45 (7): Lu, L. N. nd J. F. McDonld ``Economc effcency of second-best congeston prcng schemes n urbn hghwy systems. Trnsportton Reserch Prt B: Methodologcl 33 (3): Lundgren, J. T. nd A. Peterson. 28. ``A heurstc for the blevel orgn-destntonmtrx estmton problem. Trnsportton Reserch Prt B: Methodologcl 42 (4): Luo, Z., J. Png, nd D. Rlph Mthemtcl progrms wth equlbrum constrnts. Cmbrdge: Cmbrdge Unversty Press. Mgdls, A ``Blevel progrmmng n trffc plnnng: Models, methods nd chllenge. Journl of Globl Optmzton 7 (4): Ptrksson, M The Trffc Assgnment Problem: Models nd Methods. Utrecht: VSP. Ptrksson, M. nd R. T. Rockfellr. 23. ``Senstvty nlyss of ggregted vrtonl nequlty problems, wth pplcton to trffc equlbr. Trnsportton Scence 37 (1): Pgou, A. C Welth nd Welfre. London: McMlln. Sheff, Y Urbn Trnsportton Networks: Equlbrum Anlyss wth Mthemtcl Progrmmng Methods. New Jersey: Prentce-Hll. Shepherd, S. nd A. Sumlee. 24. ``A genetc lgorthm bsed pproch to optml toll level nd locton problems. Networks nd Sptl Economcs 4 (2):

36 SIKA. 29. Värden och metoder för trnsport\-sektorns smhälls\-ekonomsk nlyser - ASEK 4. SIKA Rpport 29:3. SIKA - Sttens nsttut för kommunktonsnlys. Sumlee, A. 24. ``Optml rod user chrgng cordon desgn: A heurstc optmzton pproch. Computer-Aded Cvl nd Infrstructure Engneerng 19(5): Trnsek. 23. Mljövgfter Stockholm, nlys v nuvrnde förslg tll utformnng. Techncl report, Trnsek AB. Trnsek. 24. Mljövgfter Stockholm, effekter v kommunstyrelsens förslg tll utformnng december 23. Techncl report, Trnsek AB. Verhoef, E. T. 22. ``Second-best congeston prcng n generl sttc trnsportton networks wth elstc demnds. Regonl Scence nd Urbn Economcs 32 (3): Verhoef, E. T., P. Njkmp, nd P. Retveld ``Second-best congeston prcng: the cse of n untolled lterntve. Journl of Urbn Economcs 4 (3): Wrdrop, J `` Some theoretcl spects of rod trffc reserch. In Proceedngs of the Insttute of Cvl Engneers 1 (2): Yng, H. nd W. H. K. Lm ``Optml rod tolls under condtons of queueng nd congeston. Trnsportton Reserch Prt A: Polcy nd Prctce 3 (5): Yng, H. nd X. Zhng (23). ``Optml toll desgn n second-best lnk-bsed congeston prcng. Trnsportton Reserch Records 1857: Yldrm, B. nd D. W. Hern. 25. ``A frst best toll prcng frmework for vrble demnd trffc ssgnment problems. Trnsportton Reserch Prt B: Methodologcl 39 (8):

37 Yn, Y. nd S. Lwphongpnch 29. ``Alterntve mrgnl-cost prcng for rod networks. NETNOMICS: Economc Reserch nd Electronc Networkng 1 (1): Zerbe, R. O. J. nd D. D. Dvely Beneft-cost nlyss n theory nd prctce. New York: HrperCollns. Zhng, X. nd H. Yng (24). ``The optml cordon-bsed network congeston prcng problem. Trnsportton Reserch Prt B: Methodologcl 38 (6):