An improved car-following model considering variable safety headway distance. Yu-han Jia a,*, Jian-ping Wu a

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1 A improved ar-followig model osiderig variable safety headway distae Yu-ha Jia a,*, Jia-pig Wu a ( a Departmet of Civil Egieerig, Tsighua Uiversity, Beijig 00084, Chia) Abstrat: Cosiderig high speed followig o expressway or highway, a improved ar-followig model is develed i this paper by itroduig variable safety headway distae. Stability aalysis of the ew model is arried out usig the otrol theory method. Fially, umerial simulatios are implemeted ad the results show good osistey with theoretial study. Re śume : A l égard du phéomèe de ar-followig, est-à-dire la suite de voiture, àgrade vitesse sur l autoroute, et artile présete u modèle ar-followig basésur la distae de séuritémiimum de variable. Esuite, l aalyse de la stabilité du ouveau modèle e utilisat la théorie de la ommade est effetué, suivie par u grad ombre d expérimetatios simulées mises e oeuvre. Efi, les résultats motret que l étude théorique du ouveau modèle est orrete. Key words: Car-followig model; Variable safety headway distae; Stability aalysis. * Correspodig author: Yu-ha Jia address: jiajiayuyuhaha@63.om

2 . Itrodutio Car-Followig theory is a method to desribe how vehiles follow oe aother o a roadway. I the model ars are regarded as disrete ad iteratig partiles without overtakig to aalyze traffi flow harateristis. So far, a variety of models have bee develed, iludig safety distae model, stimulus-respose model ad fuzzy logi based model -8. ew ar-followig model osiderig variable safety headway distae (VSHD). I setio the ew model is develed ad stability aalysis is arried out i setio 3. The umerial simulatio experimets are performed to verify the theoretial study i setio 4. The summary is give i setio 5.. Improved model The typial OVM is preseted as Amog those models, the timal veloity model d x t () [ V ( x( t)) v( t)] dt () (OVM) is well kow for auray ad ratioality 3,4,9. Afterwards the OVM has bee exteded by itroduig geeralized fore, full veloity differee ad lateral effets 0-4. However, the models metioed aot be used to preisely reflet the ar-followig pheomeo i expressway or highway with higher speed ad larger headway (mostly loger tha 0m), i whih ase the hyperboli taget part ( tah( ) ) i where x ad v are the positio ad veloity of the th vehile; x is the headway distae betwee the th ad its leadig vehile; is the sesitivity parameter of the driver; V () is the OV futio desribed as where roadway; vmax V ( x ( t)) [tah( x ( t) h ) tah( h )] () v max is the maximum veloity o a partiular h meas the safety headway distae. OVM remais as a ostat. Therefore the safety However, as otied i the study o expressway headway distae parameter i OVM eeds to be modified. where vehiles have higher speed ad larger headway, the x h part is larger tha o regular road Based o previous work, this paper ivestigates a whih makes the tah( ) remai as. Hee the fol-

3 lowig vehile gets little or o ifluee despite x () t is hagig, whih otradits the real traffi situatio. dv [ Vew ( x ( t), v ( t)) v ( t)] dt v dy v ( t) v( t) dt, (4) To avoid the metioed problem, the fixed parameter h i OV futio is replaed by VSHD as where y ( t) x ( t) x ( t ). We assume the first vehile is ot iflueed by others ad rus ostatly at hf bv ts h, where t s is the time step uit. The speed v 0, the the steady state is give by physial meaig of h f is that the aeptable safety headway distae of a driver is dyami hagig T ( ), ( ), ( 0) T ew v t y t v V v. (5) based o v level. The liearized system of (9) a be alulated Referrig to previous study 4,5, the veloity aroud steady state (0) as differee betwee the th ad its leadig vehile v () t is itrodued, makig the ew futio as d x [ V ( ( ), ( )) ( )] ew x t v t v t dt v v V x t v t x t h tah( hf )] hf bv ts h max ew ( ( ), ( )) [tah( ( ) f ) 3. Stability aalysis. (3) Aordig to stability aalysis method 5-7, the stable oditio of the modified ar-followig model a be show as d v y ()Λ t v tλ v t dt, (6) ( v t vt ) d y v t vt dt where v t v t v, () partial derivatives ad Vew ( x ( t), v ( t)) Λ v y t y t V v, Vew ( x ( t), v ( t)) Λ x ( t) y tv ew ( v0 ) After Laplae trasformatio, we have s Λ. y tv ew ( v0 ) V s V s Y s ps () s Λ, (7) where V s L( v t ), Y s L( y t ), L () is, the Laplae trasformatio ad the harateristi polyomial is p s s s s Λ s Λ. The 3

4 the trasfer futio a be obtaied as The suffiiet oditio a be obtaied as s Λ Gs 0 ps s Λ. (8) s Λ s Λ p s s s s s Λ Λ Based o stability theory, the traffi jam will Λ Λ Λ Λ 0 0, whih a be rewritte as, (9) ever happe i this system if G s. ps is stable ad Λ Λ Λ Λ 0. (0) The eutral stability surfae i the spae of As d x () t dt is i positive orrelatio with (Λ,Λ, ) with differet b are show i Fig.. [ V ( x ( t), v ( t)) v ( t)], we have 0. Beause of ew the OV futio harateristi, we have Λ >0, so the oditio for p s to be stable is Λ. The we osider expressed as G( j) G( j) G( j) G s whih a be Λ. Λ ( Λ ) Whe b=0, the model equals the full veloity differee model (FVDM) 4, ad the figure shows a eutral stability lie i Fig.(a). Aordig to Fig.(b) - (d), with higher b, the peak of stability surfae moves to larger x whe v ireases. As a result, to avoid traffi jam at higher speed, driver eeds to be more arefully to keep above the stability surfae. Fig. Neutral stability surfae with differet b: (a) b=0 (FVDM); (b) b=0.; () b=0.3; (d) b=0.5. 4

5 4. Numerial simulatios Let us osider a 00-vehile system ruig o a expressway without overtakig. I the simulatios, the iitial values are set as v (0) =5m/s, v max h =7m, = =0.5. =0m/s, aeleratio or deeleratio to keep the traffi system i stable situatio. 5. Summary I this paper, we prosed a ew ar-followig Firstly, we assume i the simulatio proess the leadig vehile has a radom flutuatio i aeleratio or deeleratio. Fig.(a) shows the spae-time plot whe b =0 (FVDM) ad we a observe the osillatig headway distaes i the lower vehiles. Fig.(b) represets the ew model with b =0.3, idiatig traffi jam ever happes. model osiderig the variable safety headway distae based o the typial OVM. The stability oditio of ew model is aalyzed by applyig the otrol theory. Fially, umerial simulatios are give, ad the results are osistet with the theoretial study. I olusio, the ew model suppresses the traffi jam i typial OVM with high speed ad larger headway Seodly, we osider the situatio with disturbae where the leadig vehile sts suddely distae. from t=0s to t=5s. The spae-time plot ad veloity behavior of three vehiles are desribed i Fig.3-5. We a see heavy traffi jams whe b =0, ad as b ireases the traffi jams are suppressed. Simulatios illustrate that with a larger headway distae but higher speed, vehiles i ew model are Akowledgemet The researh i this paper was oduted as part of the projet The ourree ad evolutio of traffi gridlok i mega-ity uder storm rai oditios, whih is fuded by Beijig Natural Siee Foudatio (Projet No. 9300). more sesitive to the variatio of leadig vehile s 5

6 (a) (b) Fig. Spae-time plot of: (a) Typial model (b=0); (b) New model (b=0.3). (a) (b) Fig.3 (a) Spae-time plot; (b) Veloity of the first, 5 th ad 50 th vehiles (b=0). (a) (b) Fig.4 (a) Spae-time plot; (b) Veloity of the first, 5 th ad 50 th vehiles (b=0.05). 6

7 (a) (b) Fig.5 (a) Spae-time plot; (b) Veloity of the first, 5 th ad 50 th vehiles (b=0.3). 7

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9 List of Figures Captios Fig. Neutral stability surfae with differet b: (a) b=0 (FVDM); (b) b=0.; () b=0.3; (d) b=0.5. Fig. Spae-time plot of (a) Typial model (b=0); (b) New model (b=0.3) Fig.3 (a) Spae-time plot (b=0); (b) Veloity of the first, 5 th ad 50 th vehiles (b=0) Fig.4 (a) Spae-time plot (b=0.05); (b) Veloity of the first, 5 th ad 50 th vehiles (b=0.05) Fig.5 (a) Spae-time plot (b=0.3); (b) Veloity of the first, 5 th ad 50 th vehiles (b=0.3) 9