ASYMMETRIC TRAFFIC ASSIGNMENT WITH FLOW RESPONSIVE SIGNAL CONTROL IN AN URBAN NETWORK

Transcription

1 AYMMETRIC TRAFFIC AIGNMENT WITH FLOW REPONIVE IGNAL CONTROL IN AN URBAN NETWORK Ken'etsu UCHIDA *, e'ch KAGAYA **, Tohru HAGIWARA *** Dept. of Engneerng - Hoado Unversty * E-al: uchda@eng.houda.ac.p ** E-al: agayas@eng.houda.ac.p *** E-al: hagwara@eng.houda.ac.p 1 INTRODUCTION Intersecton delay s a sgnfcant factor of travel cost n urban networs. At the ntersecton, the opposng traffc affects the rght-turn vehcles but not vce versa. In expressng ths aneuver n equlbru assgnent probles, any techncal dffcultes arse. However, these probles are observed n networ probles, for exaple, probles nvolvng prorty uncton, ult-odal lane, flow responsve sgnal control and dynac choce odel (Waltng 1996). These probles generally becoe ones of asyetrc equlbru. Ths study develops a lane-based ntersecton delay odel consderng the opposng traffc. nce the ntersecton delay s affected by uncertanty factors (e.g., the varance of saturaton flow rate, the arrval dstrbuton of traffc etc.) and the drvers perceve t as a fuzzy nuber rather than a stochastc nuber, the ntersecton delay s therefore represented by a fuzzy nuber. Next, an equlbru assgnent odel wth flow responsve sgnal control s develops. th s P sgnal control polcy (th 198, 1993) that axzes networ capacty s appled to the odel. We carred out both the analytcal and the coputatonal tests of unqueness of equlbru. nce our odel s a non-separable proble, both tests should be conducted to obtan the property of the odel. 2 MODEL DECRIPTION Our odel supposes deternstc user equlbru assgnent wth fxed deand. The travel cost s coposed of arteral travel cost and ntersecton delay. The BPR (Bureau of Publc Roads) type functon s used for the arteral travel cost. 2.1 Intersecton delay odel The ntersecton delay s calculated as follows.

2 Opposng traffc q The ext of ln p(q ) 1. Lane-based capacty C C,C L,C R The ext of ln Inflow traffc q, q L, q R 1. C Capacty Fg. 1. Relaton of PTI 1 2 λ ax Inflow traffc tl tc tr t te Fg. 2. The bass of the delay estaton Passng Through Index (PTI) expresses the ease of passng through the ntersecton. The PTI of the straght-through, the left-turn and the rght-turn traffcs at the ext of ln are represented by µ, µ L and µ R, respectvely. The µ and µ L are defned by the functon p of q or q L whch expresses the stuaton of traffc (Fg.1). Where, q, and C ndcate the traffc flow, the saturaton flow rate and the capacty at the ext of ln wth oveent ax (= for straght through, =L for left-turn and =R for rght-turn), respectvely. Theλ ndcates the axu green te proporton assgned to the ext of ln. The C s calculated by Eq. 1, where λ ndcates the effectve green te proporton assgned to the ext of ln. The µ and µ L depend only on traffc n ther lanes. However, µ R depends on both traffcs on ts lane and opposng lane (Eq. 2). Where, q ndcates the straght-through traffc at the ext of opposng ln. C = λ (1) µ =p(q ) for = or L, µ =p(q ) p(q ) for =R (2) Next, the fuzzy nuber, whch s the bass of the estaton, s expressed by a trangular fuzzy nuber T(tl, tc, tr ) (Fg. 2), n whch tl, tc and tr are left-spread, center value and rght-spread of the fuzzy nuber, respectvely. The center value tc s shown by the average ntersecton delay (Eq. 3). 2 c(1 λ ) tc = (3) 2(1 λ ( q / C )) Where, c s the sgnal s cycle te. The tl and tr are expressed by usng paraeters β (for : β ; for L: β L ; for R: β R ) and γ (for : γ ; for L: γ L ; for R: γ R ), respectvely (Eq. 4).

3 tr = β tc tl =γ tc β 1. (4-a) <γ <1. (4-b) To ae the fuzzy ntersecton delay syetrcal when PTI equals 1., and to ae the fuzzy ntersecton delay have larger left and rght spreads when PTI decreases (Fg. 3), the followng translatons are ade. 1. q = q =q 1 q =q 2 <q 1 < q 2 Fg. 3. Intersecton delays n our odel t te PTI γ tc tc (2-γ ) t (β -γ +1) t te tl * tc d tr * Fg. 4. Estaton of ntersecton delays t te The lne segent 1, whch s on the left-hand sde of tc, s nverted 1, and the lne segent 2, whch s on the rght-hand sde of tc, s shfted rghtward by the dstance between tl and tr 2. The ntersecton ponts of 1 and 2 wth PTI are the left-spread (tl *) and the rght-spread (tr *) of the estated fuzzy ntersecton delay, respectvely. The representatve value for fuzzy ntersecton delay s the center of gravty d (Eq. 5). Ths value wll be used n sgnal control, whch wll be descrbed n secton 2.2. Consderng real traffc stuatons, t s natural that the set of paraeters ( β,γ ) tae dfferent values n undersaturaton ( q C ) and n oversaturaton (C <q < λ ax ). Thus, two sets of paraeters should be deterned. These paraeters are deterned by easurng the ntersecton delays. tc d = { µ ( β γ ) + β γ + 3} (5) 3

4 2.2 gnal Control Polcy Consderng a four-way ntersecton there are four lns. If one of these lns s ln, there are three exts classfed by the oveent of vehcle ={, L, R}. It s supposed that the four lns are classfed nto two stages, stage 1 and stage 2. The stage ndcates a set of lns that allows pass through at the ntersecton sultaneously. Effectve green te proportons λ 1, λ 2 are assgned to stage 1 and stage 2, respectvely. The ntersecton delay at the ext of ln wth oveent (d ) s expressed by Eq.6. d =d (q,, λ ) (6) gnal control polcy (expanson of th s odel 198, 1993),.e. Eq. 7, s consstent wth the results at each ntersecton. Choose λ = ( λ 1, λ 2 ) so that λ 1 + λ 2 = 1., q λ ax and d ( q,, λ ) = d ( q,, λ ) (7) ln stage1 oveent ln stage 2 Mn f c L R q q q L R v) dv + d ( w) dw + d ( x) dx + d ( y, q ) dy oveent ' ( (8-a) rs rs rs s.t. f = f, f, f = q + q L + q R, q < λ ax and Eq. 7 (8-b) ' ' ' ' Our odel s forulated by Eq. 8. Where, f rs, f rs and c ndcate OD flow between OD par rs, -th route flow between OD par rs and arteral cost functon for ln, respectvely. 3 TET OF UNIQUE EQUILIBRIUM In our odel, the postve defnte condton s volated as to the ntersecton delay. That s, n the ntersecton delay odel of rght-turn traffc, the donant explanatory factor s not the traffc on that lane but the opposng traffc when q R < C R and q > C (Eq.9). Where, q s the straght-through traffc of opposng ln. The Eq.9 ndcates that the odel has no or ultple stable equlbra (th 1979, 1982). d R / q R < d R / q (9) We conducted the coputatonal test wth respect to convergence and unqueness of equlbru. Test networ s a real networ,.e. apporo dtown networ, whch conssts of 44 lns (Fg. 5). The traffc deand s assued to be hourly traffc volue. We set fve ntal flow patterns. Table 1 shows the convergence result of base soluton. The base soluton ndcates the soluton startng fro one ntal flow pattern, whch s

5 arbtrarly selected fro the solutons. The errors ndcate the nuber of lns that do not satsfy the convergence condton. The nu travel costs ndcate the nu costs fro orgn A to destnaton B and fro orgn C to destnaton D, n each teraton (Fg. 5). Table 2 shows the travel costs between each OD par when each soluton s obtaned. ol ol. 4. ndcate the solutons startng fro four dfferent ntal flow patterns, excludng the base soluton. The unqueness of the soluton s evaluated based on both the Proporton of lar Intal Ln Flows (PILF) and the Proporton of lar Equlbru Ln Flows (PELF) (Meneguzzer 1995). PILF (or PELF) ndcates the proporton of ntal (or equlbru) ln flows that satsfy the condton that relatve error n ters of the ntal (or equlbru) base soluton s less than 1%. Ideally, a successful test of unqueness denotes a sall PILF and a large PELF. Table 3 shows these proportons. These tables show that applyng our odel to the urban networ wll gve reasonably close equlbru flow patterns. 4 CONCLUION Ths paper develops the ntersecton delay estaton odel consderng lane-based aneuver of vehcles. Next, the user equlbru assgnent odel, whch ntroduces the flow responsve sgnal control polcy, s developed. Ths equlbru assgnent odel volates the analytcal condton as to the unqueness of equlbru. Therefore, we carred out the coputatonal tests. As a result, t s clarfed that applyng our odel to the urban networ wll gve reasonably close equlbru flow patterns. C A apporo staton B Node Centrod D Under path Fgure 5. Test networ

6 Table 1. Convergence test Iteraton Errors Mnu travel costs A --> B C --> D Table 2. Travel costs of each soluton Table 3. Test of unqueness olutons Travel cost (sec) Case PILF PELF A --> B C --> D ol. 1 vs. Base soluton Base ol. 2 vs. Base soluton ol ol. 3 vs. Base soluton.6.84 ol ol. 4 vs. Base soluton ol ol REFERENCE C. Meneguzzer (1995). An Equllbru Route Choce Model Wth Explct Treatent of the Effect of Intersectons. Transportaton Research 29B, pp M. J. th (1979). The Exstence, Unqueness and tablty of Traffc Equlbru. Transportaton Research 13B, pp M. J. th (198). A Local Traffc Control Polcy Whch Autoatcally Maxzes the Overall Travel Capacty of an urban networ. Traffc Engneerng & Control. June 198, pp M. J. th (1982). Juncton Intersectons and Monotoncty n Traffc Assgnent. Transportaton Research 16B, pp.1-3. M. J. th (1993). Traffc Equlbru wth Responsve Traffc Control. Transportaton cence. Vol.27, No.2, pp D. Waltng (1996). Asyetrc Probles and tochastc Process Models of Traffc Assgnent. Transportaton Research 3B, pp