SYSTEM DESIGN AND ITS IMPACT ON TRAFFIC FLOW

Transcription

1 CENTER FOR T R A N S P O R T A T I O N STUDIES I T S I N S T I T U T E ADAPTIVE CRUISE CONTROL SYSTE DESIGN AND ITS IPACT ON TRAFFIC FLOW Rajesh Rajaman Davd Levnson Panos chalopoulos J. Wang Kumaragovndhan Santhanakrshnan X Zou Department of Cvl Engneerng Unversty of nnesota CTS 5- HUAN-CENTERED TECHNOLOGY TO ENHANCE SAFETY AND OBILITY

2 Techncal Report Documentaton Page. Report No Recpents Accesson No. CTS 5-4. Ttle and Subttle 5. Report Date Adaptve Cruse Control System Desgn And Its Impact on Traffc Flow ay Author(s 8. Performng Organzaton Report No. Rajesh Rajaman, Davd Levnson, Panos chalopoulos, J. Wang, Kumaragovndhan Santhanakrshnan, X Zou 9. Performng Organzaton Name and Address. Project/Task/Work Unt No. Unversty of nnesota Department of Cvl Engneerng 5 Pllsbury Drve SE nneapols, N CTS Project Number: 24. Contract (C or Grant (G No. 2. Sponsorng Organzaton Name and Address 3. Type of Report and Perod Covered Intellgent Transportaton Systems Insttute Center for Transportaton Studes Unversty of nnesota 5 Washngton Avenue, SE Sute 2 nneapols, N Supplementary Notes 6. Abstract (Lmt: 2 words Fnal Report 4. Sponsorng Agency Code Ths study resolves the controversy over the stablty of constant tme-gap polcy for hghway traffc flow. Prevous studes left doubt as to the effectveness of constant tme-gap polces and whether they mantan stablty n all traffc condtons. The results of ths study prove that the constant tmegap polcy s n fact stable to a lmt. At ths lmt, dependng on the boundary condtons, condtons lose ther stablty. Ths study develops alternatve ways to mantan the balance between safety and traffc flow for ACC vehcles that does not rely on constant tme-gap polces. New spacng polces wll create more stablty, and therefore safer condtons, and allow for greater traffc capacty. 7. Document Analyss/Descrptors 8.Avalablty Statement Traffc Flow Cruse control ACC System Spacng Safety No restrctons. Document avalable from: Natonal Techncal Informaton Servces, Sprngfeld, Vrgna Securty Class (ths report 2. Securty Class (ths page 2. No. of Pages 22. Prce Unclassfed Unclassfed 67

3 Adaptve Cruse Control System Desgn And Its Impact on Traffc Flow Fnal Report Prepared by: Rajesh Rajaman Davd Levnson Panos chalopoulos J. Wang Kumaragovndhan Santhanakrshnan X Zou Department of Cvl Engneerng Unversty of nnesota Aprl 25 Intellgent Transportaton Systems Insttute Unversty of nnesota CTS 5-

4 Acknowledgements The authors wsh to recognze those who made ths research possble. Ths study was funded by the Intellgent Transportaton Systems (ITS Insttute at the Unversty of nnesota. The ITS Insttute s a federally funded program admnstered through the US Department of Transportaton s Research & Innovatve Technology Admnstraton.

5 Table of Contents Chapter Development of an Evaluaton Framework for ACC Vehcles... Chapter 2 Should ACC Systems be Desgned to antan a Constant Tme-Gap Between Vehcles?...6 Chapter 3 Overall Evaluaton of the Constant Tme-Gap Spacng Polcy...3 Chapter 4 Addressng the Trade-Off Between Safety and Traffc Flow...32 Chapter 5 Development of an Ideal Spacng Polcy...43 Chapter 6 Smulaton and Analyss of xed ACC/ anual Traffc...52 References...7 Appendx A...A- Appendx B...B- Appendx C...C-

6 Notaton A summary of major symbols used n the report x l ε δ nertal longtudnal poston of th vehcle length of th vehcle spacng error for th vehcle spacng error for th vehcle L des desred nter-vehcle spacng at zero speed τ h λ Q tme constant for st order lag model of acceleraton trackng by vehcle tme-gap control gan used n CTG, VTG and other control laws traffc flow volume rate traffc densty n vehcles/ meter v f speed parameter used n VTG spacng polcy m densty parameter used n VTG spacng polcy g ( x& desred spacng as a functon of vehcle velocty

7 Executve Summary. SHOULD ACC SYSTES BE DESIGNED TO AINTAIN A CONSTANT TIE-GAP BETWEEN VEHICLES? The desred spacng that an ACC vehcle attempts to mantan wth respect to the precedng vehcle s called the spacng polcy. In Fg., the desred spacng s the desred value of x x l. The desred dstance s typcally a functon of the ACC vehcle velocty [] but could also be a constant or a functon of other varables such as the relatve velocty. l x x x + Fg. Strng of adaptve cruse control vehcles The spacng polcy s mportant because t determnes vehcle safety, traffc flow as well as user-acceptance of the ACC system. The most common spacng polcy used n ACC systems by researchers as well as automotve manufacturers s the constant tme-gap spacng polcy. The constant tme-gap spacng polcy s gven by δ = ε + hx& + where the nter-vehcle spacng s ε = x x L ( (2

8 Is t a good dea for an ACC system to be desgned so as to mantan a constant tme-gap? There has sgnfcant controversy about the mplcaton of the constant tme-gap polcy for traffc flow. [], [2]. A recent result by Swaroop, et. al. [] stated that the traffc flow obtaned on a hghway s unstable when all vehcles on the hghway use the constant tme-gap polcy. Traffc flow nstablty here refers to the unattenuated upstream propagaton of dsturbances that occurs when a densty perturbaton s ntroduced nto the traffc flow []. In the proof of ths result, Swaroop consders an open stretch of hghway where all vehcles use the constant tme-gap polcy and there s a (small constant nflow of vehcles from a ramp. The result by Perry, et. al. [2] appears to contradct that of Swaroop []. L s paper consders a crcular hghway wth no nlets or exts for vehcles to enter or leave the hghway. It shows that the consequent traffc flow obtaned wth the constant tme-gap spacng polcy s stable. The results contaned n the present report resolve the above controversy. Further, the report objectvely evaluates the performance of the constant tme-gap spacng polcy n terms of safety and traffc flow. The questons we seek to answer n the report are Is the traffc flow obtaned wth the constant tme-gap polcy stable? 2 What does traffc flow nstablty mply from a practcal pont of vew? 3 How does stablty depend on operatng condtons/ boundary condtons? Can any spacng polcy be stable for all operatng condtons? 4 If we choose an alternate spacng polcy (other than constant tme-gap, what traffc flow and safety benefts can we obtan? The major results obtaned wth respect to the constant tme-gap polcy are The results of both Swaroop [] and L [2] are found to be mathematcally accurate wth no contradcton.

9 a Stablty of the traffc flow, n the case of the constant tme-gap polcy, s found to depend on the boundary condtons. The flow s stable for some boundary condtons, unstable for others. b If a spacng polcy could be desgned whch resulted n a steady-state flow-densty Q curve wth a postve slope >, then the traffc flow would be stable for all boundary condtons. 2 The practcal mplcatons of the mathematcal stablty results are a In the case of the constant tme-gap polcy, nflow from a ramp can be accommodated only when there s slack n the hghway,.e. when the manlne flow decreases to a level where the vehcles swtch from spacng control to speed control. b In the absence of a slack n the hghway, nflow from an nlet ramp wll eventually cause traffc to come to a stop. c The use of a ramp meter to allow vehcles to enter from a ramp only when there s slack on the manlne would be valuable. The answer to the queston Should ACC systems be desgned to mantan a constant tme gap between vehcles? s NO. It s easy to fnd alternate spacng polces wth better stablty propertes. 2. ALTERNATE SPACING POLICIES Q Havng shown that the CTG polcy does not satsfy the stablty condton >, we look at alternate spacng polces. Alternate spacng polces can be developed n whch Q > can be ensured over a range of operatng denstes. In such alternate spacng polces, the nter-vehcle spacng would be a nonlnear functon of the ACC vehcle velocty. One such alternate spacng polcy s:

10 S( x& = x& ( m v f where m s a densty parameter and v f s a speed parameter. In Fg.2, the nter-vehcle spacng of the CTG and VTG polces are compared as a functon of velocty. Values of m =, L =5 and v L f = 75mph are used. One can see that spacng ncreases wth velocty n the case of the VTG polcy, but not proportonally. Fg. 2 Desred spacng as a functon of ACC vehcle velocty The resultng traffc flow from the above spacng polcy s shown to be stable for a wde range of operatng denstes. Sgnfcantly hgher traffc capacty can also be obtaned. Whle new spacng polces can be desgned wth superor stablty propertes (such as the one above, t has been found that there are some fundamental constrants one wll encounter : a No matter what spacng polcy s chosen, there wll be a certan crtcal densty beyond whch the traffc flow wll be unstable. b The crtcal parameters that can be determned by desgn of the spacng polcy are the value of the crtcal densty and the value of the traffc flow that can be acheved at the crtcal densty. c In case of the constant tme-gap polcy, the crtcal densty turns out to be the same as the densty at whch spacng control s ntated. Hence the ACC control

11 system s ntated only after the capacty of the system has already been exceeded. d There are safety Vs traffc flow trade-offs n choosng the values of crtcal densty and maxmum traffc flow. Detaled smulaton results are presented n the report comparng the safety and traffc flow performance of the new spacng polcy wth that of the CTG polcy. 3. ADDRESSING THE TRADE-OFF BETWEEN SAFETY AND TRAFFIC FLOW New spacng polces whch are nonlnear functons of the ACC vehcle velocty can mprove traffc capacty and ensure traffc flow stablty. However, as can be deduced from Fg. 2, they nherently have a trade-off n safety. Is t at all possble to acheve traffc flow mprovements wthout any deteroraton n safety? Results show that f the spacng polcy s explctly made a functon of relatve velocty, then sgnfcant mprovements n safety can be obtaned wthout any change n steady state traffc flow characterstcs. The nonlnear spacng polcy developed n secton 2 can be modfed to take relatve velocty nto account. The steady state traffc flow characterstcs then reman the same. Safety s analyzed analytcally and through a number of smulaton scenaros, ncludng A vehcle mergng at short range nto the path of the ACC vehcle. The ACC vehcle closng-n on a sgnfcantly slower movng vehcle. The leadng vehcle n a strng of ACC vehcle decelerates suddenly to a lower speed or to a complete stop.

12 4. ANALYSIS AND SIULATION OF IXED TRAFFIC (ACC AND ANUAL VEHICLES ACC vehcles wll coexst wth manually drven vehcles on the exstng roadway system long before they become unversal. Smulaton results of varous mxed fleet scenaros are presented n the report. The analyss of the effect of mxng on capacty and stablty of traffc are based on these smulaton results. It has been found that throughput ncreases wth the proporton of ACC vehcles under below capacty condtons. But above capacty, speed varablty ncreases and speed drops wth the CTG system compared to human drvers.

13 . Development of an Evaluaton Framework for ACC Vehcles. INTRODUCTION TO ADAPTIVE CRUISE CONTROL Adaptve cruse control (ACC systems are currently beng developed by automotve manufacturers for hghway vehcle automaton [9],[]. An ACC system enhances regular cruse control by usng an on-board radar to mantan a desred spacng from a precedng vehcle that has been detected n the same lane on the hghway. Frst-generaton ACC systems are beng developed prmarly from the pont of vew of ncreased drvng comfort wth some addtonal potental for an ncrease n safety. The long-term mpact of ACC systems on hghway traffc has been nadequately studed [4]. Under a futurstc scenaro where a large number of hghway vehcles operate under ACC, the mpact of the ACC control algorthm on hghway traffc flow dynamcs and hghway safety needs to be carefully analyzed. Ths report focuses on the desgn and analyss of the nter-vehcle spacng polces and control laws used by ACC systems. The spacng polcy refers to the desred dstance x x l (see Fg. that the ACC system attempts to mantan from the precedng vehcle. The desred dstance s typcally a functon of the ACC vehcle velocty [] but could also be a constant or a functon of other varables such as the relatve velocty. A varety of dfferent spacng polces have been developed by researchers [5], [6], [8].

14 l x + x x Fg. Strng of adaptve cruse control vehcles Ths chapter develops a framework for the desgn and evaluaton of spacng polces for adaptve cruse control. The followng framework s proposed to evaluate the spacng polcy and the assocated control law a The spacng polcy and assocated control law should guarantee stablty of the ndvdual vehcle. b The spacng polcy and assocated control law should guarantee strng stablty n a strng of vehcles that possess the ACC system [3]. c The spacng polcy should yeld stable traffc flow [4]. d The control effort requred by the control law should be wthn practcal vehcle lmtatons. e Spacng polces that satsfy (a, (b and (c should be compared based on the traffc capacty they yeld at hghway speeds. The followng paragraphs defne and descrbe the terms ndvdual vehcle stablty, strng stablty and traffc flow stablty. The spacng error for the th vehcle (the ACC vehcle under consderaton s defned as x x + L δ = des (please see Fg.. Here des L s the desred spacng and ncludes the precedng vehcle length l. The desred spacng L des could be chosen as a functon of 2

15 varables such as the vehcle speed x&. The ACC control law s sad to provde ndvdual vehcle stablty f the followng condton s satsfed & x δ ( In other words, the spacng error of the ACC vehcle should converge to zero f the precedng vehcle s operatng at constant velocty. If the precedng vehcle s acceleratng or deceleratng, then the spacng error s expected to be non-zero. Snce the spacng error s expected to be non-zero durng acceleraton/ deceleraton of the precedng vehcle, t s mportant to descrbe how the spacng error would propagate from vehcle to vehcle n a strng of ACC vehcles that use the same spacng polcy and control law. The strng stablty of a strng of ACC vehcles refers to a property n whch spacng errors are guaranteed not to amplfy as they propagate towards the tal of the strng ([3], [8]. For example, strng stablty ensures that any errors n spacng between the 2 nd and 3 rd cars does not amplfy nto an extremely large spacng error between cars 7 and 8 further down n the strng of vehcles. In ths paper, the followng condton s used to determne f the system s strng stable : ˆ s Hs $ ( (2a where H ( s the transfer functon relatng the spacng errors of consecutve vehcles ˆ δ H ( s = δ. (2b In addton to (2a, a condton that the mpulse response functon ht ( correspondng to H ˆ ( s does not change sgn s sometmes consdered desrable ([4], [5]. The reader s referred to [4] for detals. In desgnng the controller to acheve ndvdual vehcle stablty and strng stablty, the followng plant model s utlzed & x = u (3 Thus, the acceleraton of the car s assumed to be the control nput. However, due to the fnte bandwdth assocated wth the engne, engne controller, brake controller, etc., each car s actually expected to track ts desred acceleraton mperfectly. The performance specfcaton s 3

16 therefore re-stated as that of meetng vehcle stablty and strng stablty robustly n the presence of a frst-order lag n trackng the desred acceleraton: & x = && x _ des = u (4 τs + τs + Equaton (3 s thus assumed to be the nomnal plant model whle the performance specfcatons have to be met even f the actual plant model were gven by equaton (4. Ths report assumes a lag of τ =.5 sec for analyss and smulaton. The maxmum acceleraton and deceleraton possble are assumed to be.5g and.5g respectvely. The traffc flow stablty of a spacng polcy refers to a macroscopc property assocated wth the traffc flow that would be obtaned on a hghway f all the vehcles on the hghway adopted ths partcular spacng polcy. For purposes of ths paper, we wll consder a one-lane hghway wth all the vehcles on the hghway beng ACC vehcles that follow the same spacng polcy. We wll then defne traffc flow to be stable f the gradent of the traffc flow volume wth respect to hghway vehcle densty s postve.e. Q / > (5 Here Q s the traffc flow volume on the hghway descrbed n unts such as vehcles/hour and s the traffc densty descrbed n unts such as vehcles/km. Once the spacng polcy has been defned, the steady state relaton between Q and for the hghway can be determned, as wll be shown n secton 3. The traffc flow stablty of the spacng polcy can then be evaluated. The sgnfcance of the above traffc flow stablty condton can be understood from the characterstc shown n Fg. 2. Ths s a typcal Q Q characterstc that s obtaned on today s hghways wth manually drven vehcles. The traffc flow frst ncreases wth ncreasng vehcle densty.e. wth the entry of more vehcles nto the hghway. However, after a crtcal densty, the gradent Q / becomes negatve and the traffc flow s sad to be unstable n ths regon. In the unstable regon, as more vehcles enter the hghway (as densty ncreases, the traffc flow actually decreases. 4

17 Fg. 2 Typcal Q curve Traffc engneers have known for many years that shock waves occur n the regon where Q / < [5]. Swaroop [4] has also presented results that show that when Q / <, densty and velocty dsturbances that occur n the steady state flow propagate wthout attenuaton upstream from the source. However, there s also some controversy related to ths result. The result by Perry, et. al. [2] appears to contradct that of Swaroop [4]. L s paper consders a crcular hghway wth no nlets or exts for vehcles to enter or leave the hghway. It shows that for ths crcular hghway, the consequent traffc flow obtaned wth the constant tme-gap spacng polcy s stable. Chapter 3 of ths report resolves the above controversy and shows that the condton Q / < guarantees uncondtonal traffc flow stablty n whch traffc flow s stable ndependent of the boundary condtons at the nlets and outlets of the hghway. Thus the condton Q / < s a very desrable condton to satsfy. A better understandng of traffc flow stablty/ nstablty can be obtaned from the descrpton and results n Chapter 3. 5

18 2. Should Adaptve Cruse Control (ACC Systems Be Desgned to antan a Constant Tme-Gap Between Vehcles? Ths chapter addresses the stablty of traffc flow on a hghway when the vehcles operate under an adaptve cruse control (ACC system. ACC systems are commonly desgned to mantan a constant tme-gap between vehcles durng vehcle followng. Prevous researchers n lterature have produced contradctory results on whether the traffc flow s stable when the constant tme gap spacng polcy s used. Ths chapter resolves the contradcton and shows that the boundary condtons used at the nlets and exts nfluence traffc flow stablty n the case of the constant tme-gap polcy. Further, the chapter shows that t s possble to desgn an uncondtonally stable spacng polcy,.e. a spacng polcy, whch guarantees traffc stablty under all boundary condtons. The practcal mplcatons of nstablty are shown through traffc smulaton results. The advantages of an uncondtonally stable spacng polcy over the constant tme-gap polcy are demonstrated. The answer to the queston Should ACC systems be desgned to mantan a constant tme gap between vehcles? s NO. It s qute easy to develop alternate spacng polces wth superor stablty propertes.. INTRODUCTION As descrbed earler, the desred spacng that an ACC vehcle attempts to mantan wth respect to the precedng vehcle s called the spacng polcy. In Fg., the desred spacng s the desred value of x x l. The desred dstance s typcally a functon of the ACC vehcle velocty [6] but could also be a constant or a functon of other varables such as the relatve velocty. 6

19 w x + x x Fg. Strng of adaptve cruse control vehcles The spacng polcy s mportant because t determnes: Vehcle safety: The nter-vehcle dstance determnes the tme avalable to brake and the tme avalable for the drver to take manual control of the vehcle. 2 Traffc flow on the hghway: Smaller nter-vehcle dstances (wthout a correspondng decrease n speed can lead to hgher traffc flow utlzaton of the hghway. 3 User-acceptance: A spacng that s too large or too small s lkely to make the drver uncomfortable. A large spacng can lead to cut-ns from other vehcles makng the drver queston the value of the ACC system. A small spacng can make the drver feel unsafe. Ths chapter deals wth the frst two ssues of vehcle safety and hghway traffc flow. The most common spacng polcy used n ACC systems by researchers as well as automotve manufacturers s the constant tme-gap (CTG spacng polcy. In the CTG spacng polcy, the desred spacng of the -th vehcle s hx& + L where L s a constant that ncludes the vehcle length w of the precedng vehcle. The spacng error under the CTG spacng polcy s gven by δ = ε + x& L ( h + where the nter-vehcle spacng s ε x x (2 = A control law that ensures that the spacng error δ converges to zero s gven by [6] 7

20 & x des = ( x& x& + λδ = ( & ε + λδ (3 h h Is t a good dea for an ACC system to be desgned so as to mantan a constant tme-gap? There s sgnfcant controversy about the mplcaton of the constant tme-gap polcy for traffc flow. ([3], [6]. A recent result by Swaroop, et. al. [6] states that the traffc flow obtaned on a hghway s unstable when all vehcles on the hghway use the constant tme-gap polcy. Traffc flow nstablty here refers to the unattenuated upstream propagaton of dsturbances that occurs when a densty perturbaton s ntroduced nto the traffc flow [6]. In the proof of the above result, Swaroop, et al consder an open stretch of hghway where all vehcles use the constant tme-gap polcy. The open stretch hghway has nlets and exts for vehcle to enter and leave the hghway. For any gven nlet flow condtons, there are correspondng equlbrum condtons on the hghway that are acheved at steady state. The stablty about these equlbrum condtons s analyzed. Swaroop, et al [6] show mathematcally that the entry of addtonal vehcles from a ramp results n perturbatons due to whch the traffc flow condtons are dsturbed from the equlbrum condtons and never come back to equlbrum. The result by L, et. al. [3] appears to contradct that of Swaroop, et al. [6]. L s paper consders a crcular hghway wth no nlets or exts for vehcles to enter or leave the hghway. It shows that the consequent traffc flow obtaned wth the constant tme-gap spacng polcy s stable.e. densty perturbatons attenuate wth tme and the traffc flow returns to equlbrum. The present paper ams to resolve the above mathematcal controversy. The paper shows that dfferent boundary condtons can lead to dfferent conclusons on traffc flow stablty. Ths explans the strkngly dfferent conclusons about the CTG polcy n the results [3] and [6]. Further, the paper shows that t s possble to desgn a spacng polcy such that t leads to 8

21 stable traffc flow for all boundary condtons. Based on ths result, a new spacng polcy that leads to stable traffc flow s developed. The paper also dscusses the practcal mplcatons of nstablty. It presents smulaton results to explan how an unstable polcy can cause traffc flow to come to a stop n the absence of a slack (or relef n demand. The new spacng polcy developed n the paper allows traffc to contnue flowng smoothly even when there s no slack n demand. 2. BOUNDARY CONDITIONS INFLUENCE TRAFFIC FLOW STABILITY Consder the spatally dscrete model of a ppelne hghway shown n Fg. 2. The hghway s parttoned nto N sectons of equal length. Let l be the length of each secton. The densty n the -th secton s denoted by dvded by the length of the secton. and equals the number of vehcles n the secton, v q q - q - + q N Fg. 2: Traffc flow n a sectoned ppelne hghway The dynamcs of each secton can be descrbed by the followng equaton. & = [ q q ] =,, N (4 l where q s the flow rate out of secton. Each q n equaton (4 s defned to be a convex combnaton of the two deal upstream and downstream flow condtons [3,6]: q = v + ( + v+ (5 In equaton (5,, =,,N are the mxng coeffcents that nclude the nfluence of the upstream and downstream flow condtons. Assumng all the vehcles on the hghway use the constant tme-gap polcy 9

22 v = [ L] h where h s headway tme and L ncludes the length of the precedng vehcle. (6 From equaton (4 and (5 we get (7 & = [ v ( v + ( + + v ( l Substtutng from (6 nto (7, we can obtan dynamc equatons for. The stablty of the system can then be analyzed. Consder three dfferent boundary condtons as dscussed n Cases, 2 and 3 below. The 3 dfferent cases lead to dfferent conclusons on stablty. ] Case : Traffc flow s stable for a crcular hghway In the crcular hghway case of nvestgaton of [3], the effect of boundary condtons was elmnated by assumng a crcular hghway wth no nlets and exts. The traffc dynamcs for the crcular hghway can be wrtten as & = A (8 where = [,..., and T N ] A= h L l ( N ( N ( ( + L L L L K N L N ( + N N It turns out that the above system s stable when <. 5 for =,,N [3], and n steady state we obtan n ss = ( Nl where n s the total number of vehcles on the whole hghway and N s the number of sectons n the hghway. (9

23 Case 2: Traffc flow s stable on an open-stretch hghway for these boundary condtons The dummy N + th secton s assumed to be empty,.e.. In the nvestgaton n [3], the followng boundary condtons were suggested N + = q = ( v + r ( q N = N v N N where r s some exogenous nput sgnal representng the traffc demand[3]. Then the traffc flow dynamcs for the N -sectoned hghway can be wrtten as: where A= h L l u =, lh ( + (2 & = A + Bu (3 ( ( + L L L L N ( N L N + N and B = N When <. 5 for =,,N, the real parts of all the egenvalues of matrx A n equaton (4 are negatve and the system s stable. The secton denstes converge to equlbrum values [3]. (4 One should note that the above boundary condtons result n an outflow from the N th secton of qn = N N vn and an nflow nto the N th secton of q v v. The fact that q N < q N for N, N <. 5 N = ( N N N + N N N seems to ndcate that these boundary condtons are unreasonable. There wll be an accumulaton of vehcles n the N-th secton of the hghway because the outlet flow of that secton s always less than ts nlet flow.

24 Case 3: Traffc flow s unstable on an open stretch hghway for these boundary condtons In the nvestgaton n [6], the boundary condtons were selected as q = constant, q N = v (.e.. Then we obtan & = A + Bu where N N A= h L l = N = ( ( + L L L L N L N N u = lh and B = (5 The system s unstable because some real parts of the egenvalues of matrx A n equaton (5 are postve [6]. Concluson From the results n ths secton, we see that stablty of the traffc flow depends on the boundary condtons. Even for the same spacng polcy (CTG, we get dfferent conclusons about stablty dependng on how the boundary condtons are chosen. 3. UNCONDITIONALLY STABLE TRAFFIC FLOW From the prevous secton, we concluded that stablty results can vary wth boundary condtons. However, as we shall now show, f the followng condton s satsfed at equlbrum : Q > then the traffc flow s stable for all of the boundary condtons consdered. Ths wll be called uncondtonal traffc flow stablty. 2

25 Theorem : Consder an open stretch of hghway, dvded nto N sectons of equal length, as n Fg. 2. For smplcty, we wll consder no ramp nflows or outflows from any secton. The nflow and outflow occur only nto the st secton and from the Nth secton respectvely. (These results are, however, extendable to the case where there are ntermedate ramp nflows and outflows. Let all the vehcles on the hghway follow a spacng polcy v = and let the g( equlbrum condtons for the traffc flow be, v, q for the th secton. The equlbrum pont (, and (3 f q s locally stable for all the boundary condtons consdered n Cases (, (2 ( g( = >. Proof : For each secton q & q = (6 l where l s the length of each secton. As before, the flow through secton boundary wll be assumed to be q = v + ( + v+ (7 Let the equlbrum condtons be denoted by, v, q for the th secton. For the boundary condtons descrbed n Case 3 of the prevous secton, we fnd that the equlbrum condtons wll satsfy the followng equatons For each secton q = q v ( 2 v ( v (8 2 At the nlet = v + ( v = q (9 3 At the outlet 2 2 v + ( v v (2 N N N N N N = 3

26 4 Let the spacng polcy be gven by g( v = (2 and let ( ( ( g v d = =.e. ( ( g g d + = (22 For equlbrum condtons ( ( g g d + = > (23 Consder a lnearzaton of the system about the equlbrum condtons. Let ~ = The nonlnearty ( g can be lnearzed usng the Taylor seres expanson as ( ] ( [ ( ( g g g + = g g g g ~ ( ( ( ( + + or d g g ~ ( ( + (24 Then l & & ( ( ~ = = v v v v = l ( 2 ( v v v or l & ~ ( ( ( ~ 2 ( ( 2 ( ~ ( ~ = u d h d h d h = l ~ ( ~ 2 ( ~ d d d

27 5 ( ( ( 2 ( ( [ h h h l From equaton (8, the second set of terms of the above equaton equals zero. Hence, we obtan the lnearzed equatons as l & ~ ( ~ 2 ( ~ ~ = d d d (25 At the nlet ~ = & & = l q q = ] ( [ 2 2 v v q + l l = ] ~ ( ( ~ [ d v d v q l l = ~ ~ ] ( [ d d v v q l l l The frst set of terms s zero from equaton (9. Hence 2 2 ~ ( ~ ~ d d l l & = (26 At the outlet ~ = N N & & = l N q N q = l ( ( N N N N N N N N v v v v ( N + =

28 6 = l N N N N v v 2 ( + = l N N N N N N N N d v d v ~ 2 ( 2 ( ~ = l 2 ( N N N N v v + ] ~ 2 ( ~ [ N N N N d d + + l The frst set of terms are agan zero at the equlbrum pont. Hence we fnd that the equatons about the equlbrum pont are gven by ] ~ ( ~ [ ~ 2 2 d d = l & (27a ] ~ ( ~ 2 ( ~ [ ~ = d d d l & (27b ] ~ ( ~ [ ~ N N N N N d d + = l & (27c In matrx form, t turns out that the stablty of the followng matrx determnes stablty of the overall system + 2 ( ( 2 ( ( ( N N d d d d d d d L L L L L L Ths matrx can be wrtten as L L L L L L 2 2 N N d d d d O (28 and the stablty of the frst matrx n the product then determnes stablty of the overall system.

29 7 Smlarly, equlbrum condtons for the boundary condtons n Case 2 can be shown to satsfy ( 2 ( 2 2 = v v (29 2 ( = + N N N N v v (3 and ( 2 ( = v v v (3 Equatons about the above equlbrum ponts for Case 2 turn out to be ] ~ ( ~ 2 [( ~ 2 2 d d = l & ] ~ ( ~ 2 ( ~ [ ~ = d d d l & ] ~ 2 ( ~ [ ~ N N N N N d d + = l & In matrx form, the stablty of the followng matrx ( ( 2 ( ( 2 ( N N d d d d d d d L L L L L L = L L L L L L 2 N N d d d d O (32

30 8 determnes stablty of the overall system. Agan the stablty of the frst matrx n the product determnes overall stablty. In the case of the crcular hghway (Case, the equatons about the equlbrum pont can be represented by the matrx equaton ~ ~ A = & where the matrx A = ( ( ( 2 ( ( 2 ( N N N d d d d d d d d d L L L L L L = L L L L L L 2 N N d d d d O (33 It can be shown that the matrces n equatons (28 and (32 are asymptotcally stable as long as 5 >.. Smlarly, for >.5, the matrx n equaton (33 has one egen value at the orgn wth all other egen values beng n the open left-half plane. The egenvector correspondng to the egen value at the orgn s [ ] T L. Hence, n ths case the system converges to equal denstes n all sectons. Values of greater than.5 make good physcal sense and are reasonable to use when traffc flow s not congested. Prevous researchers who developed and expermentally valdated traffc smulaton models have shown that ranges from.5 to ([4], [5], [9] and [] when traffc flow s un-congested (.e. > Q

31 End of proof Stablty of the CTG polcy For the CTG spacng polcy, the steady-state spacng s gven by aggregate hghway velocty. Ths leads to a steady-state densty of = L + hv Solvng for v n terms of, one can get δ = L + hv,where v s v = ( L h and the traffc flow s Q = ( L (34 h The gradent Q s thus always negatve whch means the CTG polcy s not uncondtonally stable. 4. THE PRACTICAL IPLICATIONS OF TRAFFIC FLOW INSTABILITY The theoretcal results of the prevous secton showed that when all the vehcles on a hghway use the CTG polcy, a densty perturbaton would cause the traffc flow to move away from equlbrum under realstc boundary condtons. In ths secton, we verfy the same result through mcroscopc smulatons. The smulatons wll also llustrate the mpact of ths nstablty from a practcal pont of vew. Frst, note that n a practcal system, spacng control s ntated by an ACC system only n the presence of a precedng vehcle. In the absence of a precedng vehcle, the ACC vehcle s expected to mantan a constant speed equal to the speed lmt or a user-set value. In terms of steady state traffc flow, the swtchng between speed and spacng control can be smplfed nto the followng representaton. A crtcal densty can be obtaned above whch spacng control s ntated. 9

32 Fg. 3 CTG Q curve Below the crtcal densty, vehcles wll be assumed to operate under speed control. If the speed lmt s denoted by v m, then for the CTG polcy, the crtcal densty equals For operaton below crtcal densty, speed control results n a flow rate of results n a postve slope for the ntated after the crtcal densty and n ths regon the L + hv m Q = vm. Ths Q as shown n Fg. 3. The CTG spacng control s Q curve has a negatve slope. Ths means the CTG polcy always works n the unstable regon of traffc flow!. In the followng smulatons the traffc behavor of the CTG ACC system operatng n the unstable regme (at a densty beyond the crtcal densty s llustrated. A sngle lane freeway wth an nlet ramp s consdered, as shown n Fg. 4. All the vehcles n the smulaton operate under the CTG ACC algorthm. The ramp wll be used to ntroduce densty perturbatons nto the manlne flow. A tme gap h of sec and a control gan λ =.4 are selected for ths smulaton. A lag of τ =. sec for vehcles to track the desred acceleraton s assumed. The vehcles maxmum acceleraton and deceleraton lmts are assumed to be.3g and -.5g respectvely. The length of each vehcle ( L s unformly set to be 5m n ths smulaton. A vehcle mergng nto the automated traffc from the on-ramp s placed halfway between the vehcles closest to the ramp and has an ntal speed equal to the average velocty of the closest vehcles. Ths smulaton s set up such that the lead vehcle n the freeway always attempts to mantan the speed lmt (65 mph. 2

33 Vehcles enterng from the man lane enter at the speed lmt of 65 mph. We assume the traffc has reached steady state before enterng the smulaton ppelne. Ths means the nflow rate s calculated from the ACC spacng law at equlbrum.e. or where = L + hv Q = v = v /( L + hv (35 nflow Q nflow s the nflow rate at the man nlet, s steady state or equlbrum densty, v s ntal velocty equal to the speed lmt, h s tme gap and L s vehcle length. Several scenaros wth dfferent nflow rates on the on-ramp were run n the smulatons. The smulaton results shown n Fg. 5 ndcate that a small vehcle nflow rate from the ramp (equal to.8 vehcles / sec results n all the vehcles on the hghway eventually comng to a stop upstream. The 3-D plot shows the vehcles speed values n the space (whole ppelne tme (whole smulaton perod doman. Fg. 5a Traffc nflow from man lane and ramp (wthout nflow relef Fg. 5b Smulaton result (wthout relef The mnmum amount of ramp nflow that trggers traffc to come to a stop depends on the downstream length. As the downstream length s ncreased, the mnmum ramp nflow 2

34 requred to trgger a stop decreases. In the lmt, as the downstream length s made nfnte, even an nfntesmally small ramp nflow s enough to trgger the traffc to come to a stop. Ths s n agreement wth the theory the traffc flow condtons are not uncondtonally stable under these condtons and there exst boundary condtons that wll cause traffc to come to a stop. The exstence of a fnte upstream length provdes relef, snce vehcles can ext at the speed lmt. Smulatons showed that the traffc flow s stable under the followng condtons a The downstream length s fnte, provdng a relef boundary condton b The ramp nflow s below a threshold, the value of the threshold beng determned by the downstream length. Thus the smulaton results verfy the theoretcal result of nstablty under certan boundary condtons when the uncondtonal stablty condton s not satsfed. To avod traffc comng to a stop n the man lane, the nflow from the ramp can be metered so that nflow s only allowed when there s a slack n the hghway,.e. when the manlne flow decreases to a level where the vehcles swtch from spacng control to speed control. Ths s shown n Fg. 6a. In Fg. 6b, vehcles do not come to a stop n spte of an nflow from the ramp that exceeds the crtcal threshold. Hence, the ntroducton of a ramp meter to control the vehcles enter from a ramp s an extremely good dea f vehcles were to operate under the CTG ACC polcy. Fg. 6a Traffc nflow from man lane and ramp (wth nflow relef Fg.6b Smulaton result (wth relef 22

35 5. CAN WE FIND A SPACING POLICY BETTER THAN THE CONSTANT TIE- GAP POLICY? Is t possble to fnd a spacng polcy that leads to uncondtonally stable traffc flow? We propose the followng ACC spacng polcy (adapted from an dea suggested n [6]. We shall term the new spacng polcy a varable tme-gap (VTG polcy. The VTG polcy s a nonlnear functon of vehcle velocty. The desred spacng under the proposed new polcy s gven by spacng error s gven by x& ( m v f and the δ = ε + (36 x& ( m v f where m s a densty parameter and v f s a speed parameter. In Fg.7, the nter-vehcle spacng of the CTG and VTG polces are compared as a functon of velocty. Values of m =, L =5 and v L f = 65mph were used. One can see that spacng ncreases wth velocty n the case of the VTG polcy, but not proportonally. At low speeds, the desred spacng of the VTG polcy s less than that of the CTG s. And at hgh speeds, the spacng of the VTG polcy s larger than that of CTG s. Settng & δ = λδ by dfferentatng equaton (36, the desred acceleraton wth the VTG polcy can be obtaned as x& & x ( ( des = m v f x& ( & ε + λδ (37 v f Fg. 7 Inter-vehcle spacng as a functon of velocty for the CTG and VTG polces 23

36 For the VTG polcy, the densty at steady state s gven by = m ( v. One can then v see that the aggregate velocty n terms of densty would be gven by v = v f ( and the traffc flow would be gven by Q = v f (. Fg. 8 llustrates the traffc flow and densty characterstc curves of both the CTG and VTG systems. We see that the VTG ACC system s stable ( Q > when the spacng control s ntated at a densty of.3 vehcles per meter. It only becomes unstable at a densty beyond / 2 (=., a sgnfcantly hgher densty. The CTG polcy on the other hand has Q < rght from the ntaton densty of.3 vehcles per meter. m f m m Fg. 8 Q- curve of CTG and VTG From the comparson of the CTG and VTG smulaton results n Fg. 9 and Fg. respectvely, t s easy to see that the VTG gves better traffc behavor than the CTG does. Not only do no vehcles stop n the man lane n the case of the VTG system, but also the man lane traffc remans at a level equal to (manlne + ramp nflow even as the ramp flow persstently remans at.2 vehcle/sec for a long tme. 24

37 Fg. 9a Traffc nflow from man lane and ramp (CTG Fg. 9b Smulaton result (CTG Fg. a Traffc nflow from man lane and ramp (VTG Fg. b Smulaton result (VTG In Fg. a and Fg. b, t can also be seen that the densty dsturbance ntroduced by the ramp at the mddle of the ppelne gets attenuated as t propagates upstream. The total travel (TT and total travel tme (TTT of the two ACC systems are lsted n Table. It s apparent that the VTG ACC system provdes far better traffc moblty than the CTG system. Table oblty comparson of two ACC system Constant Tme Gap polcy (.s Varant Tme Gap polcy Total Travel (kmvehcle Total Travel Tme (hvehcle System Speed (km/h

38 An addtonal mportant consderaton n an ACC system s safety. It s mportant to ensure that the traffc flow mprovement s not obtaned at the expense of safety. Whle safety s not rgorously analyzed here, the followng smulaton verfes that the VTG system s able to handle hard-brakng by the lead car. A strng of 9 cars operatng under the new control algorthm was smulated. In the smulaton, all the cars are ntally movng at steady state at the speed lmt (65mph. The lead car performs a hard brake maneuver wth a deceleraton of (-.5g to come to a stop. Then, after all the cars stop, the leadng car slowly accelerates back to the speed lmt. Fg. shows that there s no car collson durng the hard deceleraton and acceleraton of the lead car. Fg. Inter-vehcle spacng wth the VTG system n the safety smulaton study 6. STRING STABILITY In ACC systems, t s mportant to descrbe how the spacng error would propagate from vehcle to vehcle f a strng of ACC vehcles used the same spacng polcy and control law. The strng stablty of a strng of ACC vehcles refers to a property n whch spacng errors are guaranteed not to amplfy as they propagate towards the tal of the strng ([4], [9]. For example, strng stablty ensures that any errors n spacng between the 2 nd and 3 rd cars does not amplfy nto an extremely large spacng error between cars 7 and 8 further down n the strng of vehcles. In ths paper, the followng condton s used to determne f the system s strng stable : 26

39 Hs $ ( (38a ˆ s where H ( s the transfer functon relatng the spacng errors of consecutve vehcles ˆ δ H ( s = δ. (38b In addton to (38a, a condton that the mpulse response functon ht ( correspondng to H ˆ ( s does not change sgn s sometmes consdered desrable ([9]. The reader s referred to [9] for detals. In desgnng the controller to acheve ndvdual vehcle stablty and strng stablty, the followng plant model s utlzed & x = u (39 Thus, the acceleraton of the car s assumed to be the control nput. However, due to the fnte bandwdth assocated wth the engne, engne controller, brake controller, etc., each car s actually expected to track ts desred acceleraton mperfectly. The performance specfcaton s therefore re-stated as that of meetng strng stablty robustly n the presence of a frst-order lag n trackng the desred acceleraton: & x = && x _ des = u (4 τs + τs + Equaton (39 s thus assumed to be the nomnal plant model whle the performance specfcatons have to be met even f the actual plant model were gven by equaton (4. Ths paper assumes a lag of τ =. sec for analyss and smulaton. The maxmum acceleraton and deceleraton possble are assumed to be.5g and.5g respectvely. It has also been shown [9] that the above controller guarantees strng stablty for h > 2τ, where τ s the lumped lag assocated wth the actuators, engne and drve-lne and descrbed n equaton (4. Hence, ths spacng polcy (and controller satsfes the strng stablty crteron as long as a suffcently large headway tme h s mantaned. 27

40 The lnearzed representaton of the VTG spacng polcy error s gven by: S δ = ε + S ( x& + ( x& x& (4 x& where x& =equlbrum velocty. The varable S & x s thus equvalent to the tme-gap h n S the CTG spacng polcy. Hence from a lnearzaton pont of vew f > 2τ, then the & spacng polcy wll have strng stablty [9]. For VTG spacng polcy, S x& v = ( v m f f x& 2 > 2τ s requred to guarantee strng stablty. x In our case m =.2, v f = 75mph and τ =.s, so strng stablty s guaranteed for all speeds above 4.47m/s or.mph. 7. CONCLUSIONS The followng ponts summarze the conclusons obtaned from the analyss n the paper. a Should ACC systems be desgned to mantan a constant tme-gap between vehcles? Based on the results presented n ths paper, the answer s clearly NO. Whle the CTG system can be stable under certan boundary condtons, t s not uncondtonally stable. It requres relef at ether the upstream nlet or at a downstream outlet n order to be able to accommodate nflow from a ramp. b It s possble to develop alternate spacng polces whch are uncondtonally stable for a range of operatng denstes. Such polces do not requre relef at ether the upstream or downstream ends n order to accommodate ramp nflow. c No matter what spacng polcy s chosen, however, there wll be a certan crtcal densty beyond whch the traffc flow wll be unstable. Ths concluson s derved from the assumpton that the traffc flow has to fall to zero at maxmum densty (jam densty. For 28

41 ths reason, there wll be some part of the (, Q( curve where Q <. Hence, t s not possble to acheve traffc flow stablty for all. d The crtcal parameters that can be determned by desgn of the spacng polcy are the value of the crtcal densty and the value of the traffc flow that can be acheved at the crtcal densty. There are safety Vs traffc flow trade-offs n choosng the values of crtcal densty and maxmum traffc flow. However, a new spacng polcy can be desgned n whch hgher traffc capacty can be obtaned whle stll mantanng safety by use of relatve velocty n the spacng polcy. 29

42 3. Overall Evaluaton of the Constant Tme-Gap Spacng Polcy As dscussed prevously, the most common spacng polcy used n ACC systems by researchers as well as automotve manufacturers s the constant tme-gap spacng polcy. The constant tmegap spacng polcy s gven by δ = ε + x& L ( h + where the nter-vehcle spacng s ε (2 = x x The correspondng longtudnal controller assocated wth the above spacng polcy s descrbed by u _ des. = ε& + λδ (3 h The present chapter evaluates the constant tme gap polcy from the pont of vew of the evaluaton framework proposed n Chapter 2. It can be easly shown [] that the above control provdes ndvdual vehcle stablty.e. f & x&, then δ. It has also been shown [2] that the above controller guarantees strng stablty for h > 2τ, where τ s the lumped lag assocated wth the actuators, engne and drvelne and descrbed n equaton (4 of Chapter 2. Hence, ths spacng polcy (and controller satsfes the requred crteron (b as long as a suffcently large headway tme h s mantaned. The traffc flow stablty of the constant tme-gap controller can be analyzed by evaluatng the sgn of the correspondng Q /. Snce the steady-state spacng s gven by δ = L + hv, where v = & s aggregate hghway velocty, ths leads to a steady state densty of x = L + hv (4 3

43 By nvertng equaton (4, one can then see that the aggregate velocty n terms of densty would be gven by v = ( L (5 h and the traffc flow would be gven by q = ( L (6 h The varable Q / s thus always negatve wth the adopton of the constant tme-gap spacng polcy and hence the traffc flow s always unstable for all traffc denstes. The constant tme-gap spacng polcy thus volates crteron (c of the evaluaton framework, although t satsfes crtera (a and (b n the same evaluaton. 3

44 4. Addressng the Trade-Off Between Safety and Traffc Flow Ths chapter addresses the trade-off between safety and traffc flow and deals wth the desgn of new adaptve cruse control (ACC systems that can mprove traffc flow whle at the same tme ensurng safe operaton on today s hghways. The new spacng polcy, referred to as a varable tme-gap (VTG polcy developed n chapter 3, s shown analytcally to lead to better traffc flow and a hgher hghway capacty. Practcal advantages of usng the new spacng polcy have been demonstrated through traffc smulatons. However, a detaled analyss of safety shows that the tradtonal CTG polcy s superor n several scenaros. The VTG polcy s therefore modfed by explctly takng nter-vehcle relatve velocty nto account n the defnton of desred spacng. The resultng new spacng polcy s shown to retan the advantages of stable traffc flow and a hgher capacty whle provdng the same level of safety as the CTG polcy.. HIGHER CAPACITY AT THE COST OF SAFETY? The traffc flow advantages of the new VTG spacng polcy from Chapter 3 have been clearly demonstrated. The VTG polcy provdes a hgher traffc capacty and the ablty to accommodate ramp nflows at hgh operatng denstes. However, the nter-vehcle spacng as a functon of speed s smaller than that of the CTG polcy for a wde range of speeds, as seen from Fg.. Fg. Inter-vehcle spacng as a functon of velocty for the CTG and VTG polces 32

45 Ths leads to the queston of the safety of operaton of ACC systems usng the VTG polcy. In ths secton, the safety propertes of the CTG and VTG polces are compared based on the followng operatonal scenaros: The precedng vehcle decelerates suddenly to a lower speed or brakes to a stop. The ACC vehcle encounters a slower movng precedng vehcle. A vehcle cuts nto the path of the ACC vehcle at short range. The above set of tests does not cover all the drvng scenaros an ACC system s lkely to encounter. Ths secton s not ntended to be an exhaustve study of safety but a qualtatve study of the relatve performances of the CTG and VTG systems. In our smulatons we wll consder not just one ACC vehcle but a strng of ACC vehcles wth the lead ACC vehcle encounterng the scenaros descrbed above.. The Lead Vehcle n a Strng of ACC Vehcles Suddenly Brakes to a Stop A common safety-crtcal scenaro encountered on the hghway s one where the lead vehcle n a strng of vehcles decelerates suddenly to a sgnfcantly lower speed or to a complete stop. The ACC system must guarantee that there wll not be an upstream collson n such a scenaro. Consder the case where there s a strng of eght ACC vehcles all usng the same spacng polcy. Intally, all the vehcles are n steady state.e. the ntal speed of every vehcle s equal to the speed lmt (65 mph and the nter-vehcle spacng s equal to the desred steady state spacng. Durng the frst 3 seconds of the smulaton, the lead vehcle n the strng performs a hard brake maneuver wth a deceleraton of.5 g to come to a complete stop. After all the vehcles n the platoon stop, the leadng car accelerates back up agan to the speed lmt. Fgs. 2 and Fg. 3 show how the vehcle headways (nter-vehcle spacng change wth tme for the CTG and VTG systems respectvely. Note the transent behavor durng the brakng of the lead vehcle and then durng the subsequent gentle acceleraton. The lengths of the vehcles are assumed unformly to be 5 m. 33

46 Fg. 2 Inter-vehcle spacng wth CTG n scenaro Fg. 3 Inter-vehcle spacng wth VTG n scenaro Under the CTG polcy, the vehcles come to a stop wth the nter-vehcle spacng equal to 5 meters. The transent behavor of the VTG system s poorer than that of the CTG system. However, the nter-vehcle spacng when the vehcles come to a stop s only margnally less than that of the CTG system. For ths partcular smulaton scenaro, the CTG system s only margnally better than the VTG system. Ths s n part expected because the vehcles n the strng all start wth the same speed and the same safe nter-vehcle spacng. The desred nter-vehcle spacng at a complete stop s equal for both the CTG and VTG systems, though t s dfferent at the ntal speed. Snce both systems are able to track the desred spacng, they acheve the same fnal nter-vehcle spacng. The transent performance of the VTG system, however, s poorer..2 The Strng of ACC Vehcles Encounters a Slower ovng Vehcle Ths test s amed at checkng how the ACC vehcles respond when a sgnfcantly slower vehcle s encountered n the same lane, a lkely scenaro on the hghway. The same strng of 8 vehcles s used n the smulatons. All the ACC vehcles are ntally n steady state wth desred nter-vehcle spacng. At a tme t=3 s, the leadng ACC vehcle detects a precedng vehcle that s 5 m/s slower. Fgs. 4 and Fg. 5 show the tme hstores of the nter-vehcle spacng. 34

47 Fg. 4 Spacng of CTG n safety smulaton Fg. 5 Spacng of VTG n safety smulaton One can see that both the CTG and VTG systems make the ACC vehcles smoothly and safely reach a new steady state n whch they have the same velocty as the slower movng precedng vehcle. The transent performance of the VTG system s slower (poorer compared to that of the CTG system. Ths s because the desred nter-vehcle spacng changes more rapdly at the operatng speed of 65 mph n the case of the VTG system compared to that of the CTG system..3 A Vehcle Cuts nto the Path of the ACC Vehcle at Short Range An mportant scenaro n whch safety should be analyzed s one where a slower movng vehcle cuts nto the path of an ACC vehcle at short range. Ths scenaro occurs, for example, very commonly n the mergng area of an nlet-ramp. A strng of 8 ACC vehcles s consdered n the smulatons. At the begnnng, the vehcles run at steady state, all the 8 vehcle at the speed lmt (65 mph. At a tme t=3 s, a vehcle wth a speed slower by 5m/s suddenly merges nto the path of the second ACC vehcle of the strng at a dstance of 5 meters. The lengths of the vehcles are assumed unformly to be 5m. Fgs. 6 and Fg. 7 show the tme hstores of the nter-vehcle spacng for the CTG and VTG spacng polces respectvely. Only the spacng plots of the vehcles behnd the mergng vehcle are shown. 35

48 Fg. 6 ergng smulaton of CTG Fg. 7 ergng smulaton of VTG Comparng Fg. 6 and Fg. 7, one can see that the CTG system n general ensures larger spacng between vehcles durng the transent perod. For nstance, vehcle 2 n the strng has a mnmum dstance of 27 meters n the case of the CTG polcy and 24 meters n the case of the VTG polcy. Further, the transent performance s agan poorer for the VTG system wth a sgnfcantly longer tme beng taken to reach steady state. 2. ODIFIED VARIABLE TIE-GAP (VTG POLICY The VTG system developed n chapter 3 acheves a sgnfcantly hgher traffc capacty than the CTG system and ensures traffc flow stablty over a wde range of operatng denstes. However, ths mprovement n traffc flow does come at the cost of decrease n safety, as descrbed through the smulatons n secton. The smaller desred nter-vehcle spacng of the VTG system over most operatng speeds, as shown n Fg., s of course responsble. From a safety pont of vew, a larger nter-vehcle spacng s much more mportant durng transents where there s a speed dfference between successve vehcles than n the steady state where all the vehcles have the same speed. The larger the spacng that the followng vehcle can keep from the precedng one durng transents, the more the room the followng vehcles have to deal wth cut-ns as well as other unexpected events. To mantan the same steady state traffc flow characterstcs as the VTG system and yet sacrfce no safety, the VTG spacng polcy s modfed to take nto account the dfference between veloctes of the 36

49 current vehcle and ts precedng one. The proposed desred spacng of the modfed VTG polcy s as follows: S = x& ( m v f + r & ε ( wth m and v f beng the same varables as those n Chapter3, &ε = x & x& beng the relatve velocty between th vehcle and -th vehcle and r a (postve coeffcent whch determnes the weght of the relatve velocty varable n the desred spacng. Under the modfed VTG (VTG polcy, the desred nter-vehcle spacng s the same as that of the VTG polcy when there s no speed dfference wth respect to the target vehcle (as shown n Fg.. The desred spacng, however, wll be larger when the ACC vehcle s speed s hgher than that of the precedng vehcle. In the case where the ACC vehcle s speed s lower than that of the precedng vehcle (no safety needs to be consdered, the desred spacng becomes smaller than that of the VTG system. The steady state traffc flow characterstcs of the TVG system are the same as that of the VTG system. The spacng error of the modfed VTG ACC spacng polcy s gven by: δ = ε + x& ( m v f + r & ε Settng & δ = λδ, dfferentatng equaton (2, the desred acceleraton s gven by: (2 x& & x ( ( des = m v f x& ( & ε + r& ε + λδ (3 v f Ths s the control law for the VTG system and wll ensure that the desred VTG spacng s accurately tracked by the ACC vehcle. To calculate the desred acceleraton & x& des addtonal measurement (or estmate, relatve acceleraton ε& &, s needed., an 37

50 Implcatons of Traffc Flow Stablty The same mcroscopc smulaton setup as earler s used to llustrate the traffc flow behavor of the VTG system. Smulaton results wth relatve velocty weght coeffcents r chosen as. and 5. are shown n Fg. 8a and Fg. 8b respectvely. It can be seen that although the densty dsturbances at the mddle of the ppelne propagate a longer dstance upstream compared to the VTG system,, the dsturbances are stll strctly attenuated as they propagate upstream. Thus the VTG system yelds stable traffc flow. Ramp nflows can be accomodated over a wde range of operatng denstes durng spacng control. Fg. 8a Traffc nflow from man lane and ramp (relatve Fg. 8b Smulaton result (r =. Fg. 9a Traffc nflow from man lane and ramp (relatve Fg. 9b Smulaton result (r =5. The total travel (TT and total travel tme (TTT of each of the three ACC systems usng the CTG, VTG and VTG algorthms are lsted n Table. It s readly seen that the VTG and 38

51 the VTG systems have sgnfcantly better traffc moblty and productvty values than the CTG system. Table oblty and Productvty Comparson of Three ACC System Constant Tme Gap Polcy (.s Varable Tme Gap Polcy odfed Varable Tme Gap Polcy ( r =. / 5. Total Travel / 2.43 (kmvehcle Total Travel /.388 Tme (hvehcle System Speed (km/h / SAFETY RECOVERED USING THE VTG POLICY The safety performance of the control system usng the VTG polcy s descrbed for the same three smulaton scenaros n ths secton. 3. The Lead Vehcle Suddenly Brakes to a Stop Agan a strng of 8 ACC vehcles are used n ths smulaton. The vehcles are ntally n steady state, at a speed of 65 mph and at desred nter-vehcle spacng. Durng the frst 3 seconds of the smulaton, the leadng vehcle performs a hard brake maneuver (-.5 g to come to a complete stop. After all the vehcles n the platoon stop, the leadng car accelerates back agan to the 65mph speed. Fg. Fg. show how the vehcle space headways change wth tme for the VTG polcy wth r = and 5 respectvely. 39

52 Fg. VTG Spacng n smulaton (r=. Fg. VTG Spacng n smulaton (r=5. One can see that safety s mproved by havng ntroduced relatve velocty nto the VTG polcy n the sense that the vehcles now mantan more nter-vehcle spacng as they stop compared wth both the CTG and VTG polces. The bgger the value of r, the larger the spacng that s mantaned between the vehcles as they stop, hence the safer system s n ths scenaro. The spacng at a complete stop s also larger. The transent performance appears to be poorer. Ths s only because the desred spacng s smaller for most ntermedate operatng speeds and then suddenly ncreases sharply as we approach the speed lmt (see Fg The Strng of ACC Vehcles Encounters a Slower ovng Vehcle In ths smulaton scenaro, the same strng of 8 ACC cars s used. Intally all the ACC vehcles are at steady state. At a tme t=3 s, the leadng ACC vehcle detects a precedng vehcle that s 5 m/s slower than the ACC platoon speed n the same lane on the hghway. Followng Fg. 2 Fg. 3 show how the vehcle space headways change wth tme for the modfed VTG spacng polces wth r = and 5 respectvely. Fg. 2 Spacng of relatve smulaton (r=. Fg. 3 Spacng of relatve n safety (r=5. 4

53 One can see that the VTG polcy has a performance smlar to the other two systems. All the three spacng polces ensure that the ACC platoon reaches steady state wth the same velocty as the precedng vehcle smoothly. 3.3 A Vehcle Cuts nto the Path of the ACC Vehcle at Short Range Agan a strng of 8 ACC vehcles s consdered. At the begnnng, the vehcles are at steady state, all the 8 vehcles wth the same speed as the speed lmt (65 mph. At t=3 s, a vehcle wth a speed 5m/s less than the lead vehcle speed suddenly merges nto the path n front of the (now second ACC vehcle of the platoon at a dstance of 5 meters. The lengths of the vehcles are assumed unformly to be 5m. The followng fgures Fg. 4 Fg. 5 show how the vehcle space headways (spacng change wth tme n the transent perod after the cut-n for the modfed VTG polcy wth r = and 5 respectvely. Only the space headways of the vehcles behnd the mergng vehcle are shown. (Every lne n the plot ndcates the spacng of a vehcle Fg. 4 ergng smulaton of relatve (r=. Fg. 5 ergng smulaton of relatve (r=5. In Fg. 4, we see that the relatve velocty weght coeffcent r = allows the new polcy to keep larger space headways compared to the VTG system. Furthermore, n Fg. 5, wth the relatve velocty weght coeffcent r ncreased to 5, vehcles usng ths spacng polcy can mantan much larger space headways than that of the VTG system and even larger than that of the CTG system. The transent performance seems poorer but s actually due to the fact 4

54 that the desred spacng s smaller for most of the operatng speeds and ncreases sharply only towards the speed lmt regon. 4. CONCLUSIONS Ths chapter analyzed the desgn of new adaptve cruse control (ACC systems from the pont of vew of mprovng traffc flow whle at the same tme ensurng safe operaton on today s hghways. A new nter-vehcle spacng polcy called a varable tme-gap (VTG polcy that s a nonlnear functon of vehcle speed was developed n Chapter 3. The new spacng polcy led to stable traffc flow and a hgher capacty. Practcal advantages of usng the new spacng polcy were demonstrated through traffc smulatons. However, an analyss of safety showed that the tradtonal CTG polcy was superor n several scenaros. The VTG polcy was then modfed by takng nter-vehcle relatve velocty nto account. The resultng new spacng polcy was shown to provde stable traffc flow, a hgher capacty and the same level of safety as the CTG polcy. 42

55 5. Development of an Ideal Spacng Polcy Ths chapter proposes an deal spacng polcy that evolves naturally from the evaluaton framework proposed n Chapter 2. The proposed spacng polcy s a nonlnear functon of speed. It provdes strng stablty and traffc flow stablty as well as a hgher traffc flow capacty compared to the standard tme-gap controller. An assocated result proved n the chapter s that traffc flow stablty mples strng stablty for flow volumes up to a descrbed maxmum value.. STRUCTURE OF THE PROPOSED IDEAL SPACING POLICY As a startng pont, the structure of the proposed deal spacng polcy s assumed to be δ = ε + g( x. ( where g x& s a nonlnear functon of vehcle speed that wll be defned later so as to satsfy ( crtera (a, (b and (c of the evaluaton framework. The assocated controller s obtaned usng feedback lnearzaton. Dfferentatng equaton (.. = ε + x g x... δ (2 Settng δ. = λδ the desred acceleraton s gven by x& =. _ des ε + λδ. g / x & (3 Ths control law automatcally ensures ndvdual vehcle stablty, so that crteron (a s satsfed. 2. STRING STABILITY The proposed controller has the same structure as the constant tme-gap controller but wth the headway tme constant replaced by a velocty dependent tme headway. The correspondng lnearzed spacng polcy s gven by 43

56 . g.. δ = ε + g( x o + ( x. x o (4 x where ẋ o = equlbrum velocty. From a lnearzaton pont of vew f g. x > 2τ (5.. for all ponts on the ( x, g( x curve, then the spacng polcy wll guarantee strng stablty. We assume that satsfyng equaton (5 s adequate to satsfy crteron (b of the evaluaton framework. Smulaton results also ndcate that satsfacton of equaton (6 s adequate to ensure strng stablty. Snce traffc flow capacty and traffc flow stablty are better studed on the (, Q( curve, the crtera (5 on g (v curve needs to be transformed nto a condton on the Q( curve. Let Q( be the steady state traffc flow obtaned when the steady state densty s. Let v be the correspondng steady state velocty. Then Q v Q = v = v + (6 The steady state densty s gven by = g( v (7 The dervatve g. x can then be evaluated n terms of and Q as follows : 2 v g( v = g / v 2 Q g( v = v = v g / v g / v (8 (9 g v = ( v Q / ( Imposng the condton (5 on g / v, we obtan 44

57 2τ ( v Q / ( or Q v 2τ Q Q 2τ (2 (3 It can be readly seen that the constant tme headway polcy satsfes ths nequalty as an equalty. If C s the maxmum traffc flow acheved, then (3 leads to Q C 2τ (4 If t s assumed that the traffc flow capacty drops to zero (velocty becomes zero at maxmum densty then ths condton mples that at very hgh denstes the Q( curve for the gven spacng polcy must le below the Q( curve of the constant tme-gap polcy. 3. TRAFFIC FLOW STABILITY Traffc flow stablty ensures that densty dsturbances do not propagate downstream wthout attenuaton. Usng conservaton of mass prncple and consderng traffc velocty dynamcs, Swaroop [4] has shown that small perturbatons to the steady state densty and velocty can magnfy downstream wthout attenuaton n spte of the spacng polcy achevng strng stablty. The conservaton of mass equaton s gven by Q + = t x and the traffc velocty dynamcs s modeled by where v v + v = t x τ ( h( v v = h( gven by the spacng polcy µ > s a functon of alone. µ ( x (5 (6 45

58 Usng densty and velocty perturbatons of the form ~ ~ kx+ wt kx+ wt p = e and v p = v e (7 Swaroop [4 ] has shown that, for densty perturbatons about the steady state, v ( o o Q( o / w = (8 2v τ and w < for traffc flow stablty. o The traffc flow s stable as long as Q / >. As descrbed earler, n the case of the constant tme-gap spacng polcy, Q / < for all and hence the traffc flow s always unstable. Snce t s assumed that the traffc flow has to fall to zero at maxmum densty, there wll be some part of the (, Q( curve when Q / <. Hence t s not possble to acheve traffc flow stablty for all. However, traffc flow stablty can be acheved for all denstes that are below a value close to the maxmum densty. If the maxmum traffc flow acheved, C, s (the maxmum traffc flow acheved by 2τ constant tme headway polcy, then t can be seen that traffc flow stablty ( > mples strng stablty.e., the correspondng nequalty s satsfed. Q 4. DESIGN OF THE IDEAL SPACING POLICY Assumng τ =.5 sec n (3, we get Q Q (9 46

59 Fg. Set of Q curves satsfyng equaton (9 Fg. shows the set of (, Q( curves that would satsfy (3 as an equalty. The (, Q( curve correspondng to the constant tme headway polcy s part of the set. A maxmum densty of.429 Vehcles/m (correspondng to a mnmum spacng of 7m was assumed. It can be seen that any spacng polcy wth the controller gven by (3 and achevng greater traffc flow than the constant tme headway polcy wll have non-zero traffc flow at maxmum densty. So n order to acheve more traffc flow and have zero traffc flow at maxmum densty (jam densty, the spacng polcy wll have to be strng unstable at some denstes. The followng propertes were assumed whle desgnng the nonlnear spacng polcy Assumng a speed lmt of 65 PH and 2-65 PH to be the range of vehcle speeds observed on today s hghways, the spacng polcy should guarantee hgher traffc flow than constant tme headway polcy. The spacng polcy should guarantee strng stablty n the range of speeds 2-65 PH. The spacng polcy should guarantee traffc flow stablty n ths range. 47

60 A (, ( Q curve s desgned to satsfy the constrants (8 and (9 n the speed range 2-65 PH. A lookup table of v Vs g(v can be formed from v = Q / g( v = / (2 The functon Q( s gven by where = traffc densty Q ( v = a = Q ( v = a 2 = Q ( v = a Q = Q ( + Q ( + Q ( (2 + a + a + a ( a elsewhere 22 + a elsewhere a a a + a for v 3 for 65PH v 2PH 3 for > 65PH v < 2PH The above structure was adopted to reduce the complexty of Q ( n satsfyng all the stated requrements. Further more to ensure smooth transtons n the spacng functon, the followng propertes are satsfed by proper choce of a j. Q Q2 = Q2 Q3 = at v = 65PH at v = 2PH (22 Fg. 2 shows the desgned (, Q( curve and the correspondng desred spacng as a functon of speed s shown n Fg

61 Fg. 2 Desgned Traffc Flow Capacty Curve Fg. 3 Desgned Desred Spacng Curve The desgned curve guarantees strng stablty and traffc flow stablty n the speed range 2-65 PH whle provdng hgher traffc flow capacty than the standard constant tme headway 49

62 polcy. Beyond 8 PH (Fg. 5 g. x (changes rapdly and so the control effort for any spacng error. Thus the strng stablty results wll not hold n that regon. 5. SIULATION RESULTS For smulaton, the lead car performed the followng maneuver. Startng from 2 PH and correct spacng, the car accelerates to 65 PH and then undergoes a hard brake maneuver (-.5 g deceleraton to drop back to a speed of 2 PH. λ =.4 was used for the smulaton. Fg. 4 shows the control effort for the cars n a strng of cars followng each other. The control effort reduces downstream. Gven that the lead car can brake at the maxmum deceleraton, the followng cars requre lesser acceleraton hence assurng that they wll be capable of achevng the desred control effort durng a hard brake scenaro. Fg. 4 Control Efforts of Cars n a Platoon Fg. 5 shows the spacng errors durng the acceleraton and deceleraton maneuvers. The errors n spacng decrease downstream. Ths ensures that f the frst car doesn t collde wth the lead car, then there wll not be a collson downstream between the th th and the ( + car ( >. 5

63 Fg. 5 Spacng Error n a Platoon 6. CONCLUSIONS A general framework for the desgn and evaluaton of ACC spacng polces was developed n chapter 2 of ths report. A specfc nonlnear spacng polcy that could be consdered deal evolved naturally from the framework. The spacng polcy used autonomously avalable nformaton. Analytcal calculatons showed that the spacng polcy would guarantee strng stablty and traffc flow stablty whle mantanng smaller steady state spacng and hence provdng larger traffc flow capactes n the speed range 2-65 PH. The smulaton results n the paper confrmed the theoretcal results. As opposed to nonautonomous spacng polces, ths spacng polcy can be readly mplemented on ACC vehcles on today s hghways. 5

64 6. Smulaton and Analyss of xed Adaptve Cruse Control / anual Traffc Adaptve (Intellgent Cruse Control (ACC Vehcles wll coexst wth manually drven vehcles on the exstng roadway system long before they become unversal. Ths mxed fleet scenaro creates new capacty and safety ssues. In ths chapter, measurement of traffc qualty s dscussed. A defnton of traffc flow stablty s proposed. Smulaton results of varous mxed fleet scenaros are presented. The analyss of the effect of mxng on capacty and stablty of traffc system s based on these results. It s found that throughput ncreases wth the proporton of ACC vehcles when flow s below capacty condtons. But above capacty, speed varablty ncreases and speed drops wth Constant Tme Headway (CTH control ACC compared wth human drvers. Ths chapter also addresses the mpact of Adaptve Cruse Control laws on the traffc flow. Smulaton results of varous mxed fleet scenaros under dfferent ACC laws are presented. Explct comparson of two ACC laws, Constant Tme Headway and Varable Tme Headway (VTH, are based on these results. It s found that VTH has better performance n terms of capacty and stablty of traffc. Throughput ncreases wth the proporton of CTH vehcles when flow s below capacty condtons. But above capacty, speed varablty ncreases and speed drops wth the CTH traffc compared wth manual traffc, whle the VTH traffc always performs better than CTH.. LITERATURE REVIEW. DESIGN CONSIDERATIONS OF ADAPTIVE CRUISE CONTROL Whle vehcle manufacturers hope that ACC systems wll mprove the drver s comfort and safety, research on the propertes of fully automated vehcle platoons has shown the potental benefts for capacty and safety (Van Arem et al. 996; Broqua et al. 99; nderhoud and Bovy 998. It seems an appealng scenaro that all of the vehcles on a hghway are automated. But, t s more reasonable to magne that at the ntal stage of deployng automated or semautomated vehcles, they wll coexst wth conventonal manually drven vehcles. For nstance, 52

65 Ter One forecasts that by 2, 2 percent of cars wll be equpped wth ether ACC-Collson Warnng or other headway control systems (Ter One, 2b. Ths mxed control scenaro rases complex capacty and safety ssues on traffc flow that we must probe before ACC becomes realty. The mpacts of the deployment of ACC on the traffc flow pattern and ts control must be taken nto consderaton n the very early stage of ACC desgn. Up to now, mportant desgn consderatons of ACC systems largely nclude: ( antanng a safe dstance between vehcles: the system may fully control the vehcle, and t must guarantee that the vehcle wll not enter an unsafe state as a result of the control; (2 Characterstcs of real-tme response: the response tme of the vehcle to control nputs must be short; (3 All-weather capablty: Performs well n poor vsblty condtons and should not be adversely affected by poor weather condtons; (4 Performs well durng road turns, bumps and slopes; (5 Smplcty of use: A drver wth no pror experence can use t correctly. Patterson (998 studed ACC s mpacts on capacty and safety. Hs thess compared ACC wth conventonal cruse control and manual drvng at both the macroscopc and mcroscopc level. At the macroscopc level, t s found that ACC was used more n smlar trps and the number of brake nterventons n ACC vehcles s larger than that n CCC vehcles. At the mcroscopc level, t s found that manual drvng results n larger headway. But ACC and CCC have smlar speed-headway profles. Because of some advantages to fuzzy logc models, Wu et al. (998 gave a complete descrpton of the fuzzy sets for both car followng and lane changng n FLOWSI whch offers a user defned update rate and apples acceleratons. The fuzzy nference model for car followng s based on the dvergence of the rato of vehcle dstance to desred vehcle dstance and the relatve speed of two vehcles. Holve et al. (995a suggested that the ACC system has to meet the expectatons of the human drver to a certan degree. They proposed an adaptve fuzzy logc controller that s flexble n dfferent drvng stuatons and comprehensble for the drver. Holve et al. (995b also proposed a scheme to generate fuzzy rules for the ACC controller, n whch the drver s a component of the ACC control loop. Ther Fuzzy-ACC has been tested n normal road traffc. Smlar work can be found n Chakroborty et al. (999, n whch relatve speed, 53

66 dstance headway and acceleraton/deceleraton rate of leadng vehcle are the nputs to a fuzzy logc model. arsden et al. (2 employed Wu s car followng model n smulaton. An ACC algorthm based on a manufacturer prototype was also employed n whch the acceleraton rate of the ACC vehcle s related to the vehcle mass, the gap headway, the rate of change of gap headway and the velocty of the equpped vehcle. Ther results showed that ACC could reduce the varaton of acceleraton compared to manual drvng (arsden et al., 2. It should be noted that the authors also ponted out the lmtatons of mcroscopc smulaton n modelng the mpacts of ACC because of the lack of knowledge of the behavor and nteracton between the drver and the ACC system n dfferent traffc condtons (arsden et al., 2. Ioannou et al. (994 proposed ther ACC scheme for constant tme vehcle followng. Ther results showed that the scheme could mantan a steady state of nter-vehcle spacng wthout reducng the drver comfort..2 Study In the Aspect of Traffc Flow The desgners of ACC algorthms often consder the strng stablty as the prmary crteron (Darbha and Rajagopal 999, Fancher and Bareket 995, L and Shrvastava 2. Lang and Peng (999 presented a two-level ACC synthess method whch calculates desred acceleraton rate and controls vehcles to acheve the desred rate accurately. They suggested that ther method can guarantee strng stablty and yeld mnmum mpact on vehcles nearby. Furthermore, they (Lang and Peng, 2 suggested a framework to analyze strng stablty and defned a margnal ndex to gve a quanttatve measurement of ACC desgns. any smulaton studes have evaluated the mpacts of ACC systems (Van Arem et al. 996; Broqua et al. 99; nderhoud and Bovy 998. However, the traffc flow characterstcs that ACC wll brng are dffcult to quantfy. And t s not possble to make drect comparson among these documents, because these studes have employed dfferent ACC algorthms, dfferent drver behavor models, and dfferent drvng envronments. What we can do s to fnd some common trend and make some qualtatve explanatons. In ther work, Broqua et al. (99 estmated that gans n throughput are 3% wth 4% of vehcles equpped wth constant-spaceheadway ACC when the target tme-gap of the system was second. Van Arem et al. (996 and 54

67 nderhoud and Bovy (998 have found a decrease n average speed caused by a collapse of speed n the fast lane for ACC target tme-gaps of.4 s and above. Bose and Ioannou (2 reported ther studes on the mxed traffc of ACC vehcles and manual drven vehcles. Ther results showed that % presence of ACC vehcles smoothed traffc flow n the case of rapd acceleraton of the leadng vehcle, whch results n less fuel consumpton and polluton levels than pure manual drven traffc. Sponsored by the Natonal Hghway Traffc Safety Admnstraton (NHTSA, an ACC system evaluaton project (Kozol et al. 999 was mplemented by the Unversty of chgan Transportaton Research Insttute from July 996 to September 997. Based on data obtaned from the experments, the Volpe Natonal Transportaton Systems Center (Volpe Center nvestgated ACC s mpacts. They concluded that deployment of ACC results n safer drvng (Kozol et al Fancher et al. (998 reported the ICC Feld Operatonal Test of NHTSA and UTRI. They concentrated on the safety and comfort ssues of ICC (ACC system. They found that ACC s very attractve to most drvers and s used n many traffc condtons (Fancher et al., 998. On the other hand, they found some ssues ndcate potental mpacts on safety and traffc; the mportance of human-centered desgn s also hghlghted (Fancher et al., 998. In the aspect of assessng mpacts on traffc flow, they suggested that the current results are not enough for analyss (Fancher et al., 998. Swaroop and Huandra (999 studed the desgn problem of the ACC algorthm. Based on analyss and numercal smulaton, they demonstrated that a good ACC spacng polcy must satsfy the condton that the slope of the correspondng fundamental traffc characterstcs s always postve. VanderWerf et al. (2 studed the mpacts of autonomous ACC (AACC and cooperatve ACC (CACC on traffc based on ther mcroscopc smulaton. They found that AACC have very small mpact on hghway capacty. The capacty gan from % to 2% AAC penetraton s greater than that from 2% to 4%. And there s no capacty ncrease wth more AACC penetraton (VanderWerf et al., 2. Cooperatve ACC, on the other hand, can potentally ncrease capacty quadratcally along wth CACC penetraton (VanderWerf et al., 2. All of ths research provded the estmates of mpacts of ACC n some specfed stuatons. Ther results are meanngful for the traffc operators to outlne the potental mpacts of the ACC 55

68 system. Our research wll begn wth some smplfed scenaros. Then more complex stuatons are smulated n our mcroscopc traffc smulaton program. We wll try to summarze the mpacts of ACC from a large number of smulatons n whch some stochastc mechansms make the results more realstc..3 PURPOSE OF THIS RESEARCH The man purpose of our research ncludes: ( To develop a framework to evaluate adaptve cruse control (ACC algorthms; and (2 To develop new autonomous vehcle followng algorthms that overcome the shortcomngs of exstng ACC algorthms. To acheve these goals, both theoretcal analyss and smulaton work are needed. The conventonal methods n control theory, such as Laplace transform and frequency doman analyss can be used to nvestgate the propertes of the vehcle platoons. However, these methods cannot be used n the analyss of mxed traffc n whch the relatonshp between vehcles can t be descrbed by a unform dfferental equaton or ts derved forms. The only way to provde an explct observaton of the mxed traffc s behavor s mcroscopc smulaton n whch each vehcle s controlled by ts own control algorthm, ether a carfollowng algorthm whch emulates the human drver s behavor statstcally, or an adaptve cruse control law. ost of our conclusons n ths report are obtaned from the smulaton results. The steps of our work nclude: ( To develop mcroscopc traffc smulaton tools; (2 To evaluate the mpacts of ACC algorthms on traffc flow; (3 To compare varous ACC algorthms and fnd the optmal algorthm and the mode of operaton that benefts the effectveness of traffc flow. In ths report, we wll frst dscuss methodology and crtera that can be used to evaluate the mpacts of ACC algorthms on traffc flow. A number of cases wth mxed ACC and manually controlled traffc are smulated and analyzed usng a mcroscopc traffc smulaton program. To smplfy the analyss, a one-lane hghway s studed. The sem-automated vehcles are equpped wth an ACC algorthm that allows them to keep a constant tme headway or varable tme headway whle followng. The newly-developed varable tme headway control algorthm s mplemented n smulaton and compared wth the constant tme headway algorthm. Gpps model (Gpps, 98 s used to smulate manually drven vehcles. The densty and speed profles as a functon of the proporton of ACC vehcles are nvestgated, whch show the potental 56

69 benefts of the sem-automated vehcles. Dfferent vehcle followng scenaros wth sudden deceleratons and acceleratons are analyzed n order to study the effect of the response of ACC vehcles n mxed traffc. In chapter 2, the evaluaton ssues of traffc flow, such as stablty, robustness and safety, are dscussed. Chapter 2 also descrbes the mxed traffc scenaro that s nvestgated and the smulaton program. Chapter 3 to Chapter 5 summarze the smulatons on the dfferent level of hghway traffc as a functon of the proporton of ACC vehcles. The stablty and transent response of traffc flow n dfferent mxed traffc stuatons are llustrated n the results. Chapter 3 presents the smulaton results of traffc flow mxed by constant tme headway ACC and manually drven cars. Chapter 4 compares the performance of varable tme headway and constant tme headway ACC. Chapter 5 shows the smulaton result n whch headways of ACC and manually drven vehcles are randomly dstrbuted. Comparsons are made to show the mpacts of random factors. Chapter 6 summarzes the smulaton of mxed traffc based on AISUN and GETRA Extenson. Some concludng remarks n Chapter 7 complete the report. 2 EVALUATION OF ADAPTIVE CRUISE CONTROL 2. INTRODUTION The evaluaton of the mpacts of ACC on traffc flow s a based on measurements of traffc qualty. Traffc operators want ( a hgh capacty of traffc flow n the current nfrastructure; (2 stable traffc flow n the cases of hgh demand and ncdents and (3 guaranteed safety for each drver. Hgh capacty can only be obtaned when the average tme headway s reduced whle keepng the same speed. That means vehcles are closer to each other n the same speeds. ACC provdes a very short response and may take the place of the drver n longtudnal control. However, ths mprovement wll rase a safety problem f the response tme of the drver-vehcle system cannot be reduced n the same scale. The tradeoff pont n terms of capacty and safety can be chosen based on the safety requrements, whch are determned by the mechancs of vehcle and 57

70 nfrastructure, such as vehcle dynamcs, road surface frcton, effcency of Ant-lock Brake System and even weather. Smooth traffc s also desred by users and operators because t means less travel tme and less potental ncdents. The problem s how to reduce congeston and make traffc more stable. A clear defnton of traffc flow stablty s stll unavalable n prevous studes because of the nonlnear, dynamc and stochastc nature of traffc and the fuzzy am of term defnng. Stablty, whch s tradtonally used to descrbe the nternal characterstc of a system to mantan a bounded movement, s dfferent from robustness, whch s used to evaluate the external characterstc of system: the nsenstvty of a system to nput varaton. In traffc studes, these two terms are often blurred and treated as the same quantty. But dstngushng these two quanttes s sgnfcant when one consders the upstream traffc as the nput of downstream traffc n traffc flow operaton. Unfortunately, up to now, there are no explct defntons of stablty or robustness. A promsng approach to nvestgate these characterstcs s the study of traffc congeston n whch many reasons cause oscllatons n speed, densty and flow rate. Traffc congeston s such a complex phenomenon that no up-to-date theores, such as carfollowng model (Rothery 2, queung theory (Newell. 982, knetc theory (Prgogne And Herman 97, cellular automata (Nagel and Shreckenberg 992, hgher-order model (Kuhne and chalopoulos 2 and knetc wave theory (Lghthll and Whtham 955, can soundly descrbe t, though everyone experences t daly. In ths research, the prmary focus s to characterze the performance of ACC systems, drver behavor under normal drvng condtons, and drver nteracton wth partally automated vehcles. Although some progress has been made n these areas, much work s stll requred to better understand the relatonshps between benefts, user acceptance, ACC system requrements, drver behavor, and the drvng envronment. oreover, the mpact of drvers or automatc control dynamcs on system capablty and benefts s not well understood. In a word, the crtera now beng used are just sutable for an dealzed traffc system. The crtera that quantfy drvers and system benefts need further research. 2.2 EASUREENTS OF TRAFFIC QUALITY 2.2. TRAFFIC FLOW STABILITY 58

71 any nvestgatons have been conducted to study the stablty ssues of traffc flow. But most of them are concentrated on strng stablty. Strng stablty s stablty wth respect to ntervehcular spacng. It ensures the poston and speed of each vehcle n a strng change wthn small boundares of error, and dsturbances n vehcle speeds do not be amplfed when they are propagated upstream n traffc. Normally, no vehcle enters or leaves the strng n the study of strng stablty. It should be noted that guaranteeng strng stablty doesn t mean guaranteeng traffc flow stablty. Strng stablty mght ensure that the space between vehcles n the strng reman the same constant, but can not keep the speed of vehcles n the strng from decreasng to a level that blocks a motorway secton and thus causes an nstable state of traffc flow. At a macroscopc scale, traffc flow s the aggregaton of strngs and sngle vehcles n many sectons of motorway. Traffc flow stablty deals wth the evoluton of aggregate velocty and densty n response to change n the flow rate. So vehcles enter and leave specfed traffc n the study of flow stablty. Darbha and Rajagopal (999 proposed that, Traffc flow stablty can be guaranteed only f the velocty and densty solutons of the coupled set of equatons s stable,.e., only f stablty wth respect to automatc vehcle followng and stablty wth respect to densty evoluton s guaranteed. Ther defnton s n the sense of Lyapunov Stablty. A formal defnton of Lyapunov stablty s (urray et al. 994: The equlbrum pont x= of x & = f ( x, t s stable at t=t f for any ε > there exsts a δ(t, ε> such that: x ( t δ t > t ( < x( t < ε, Furthermore, the asymptotc stablty s defned as (urray et al. 994: The equlbrum pont x= of x & = f ( x, t s asymptotc stable at t = t f. x= s stable, and 2. x= s locally attractve;.e. there exsts δ(t such that x( t < δ lm x( t = (2 t Here a pont x s an equlbrum pont of x & = f ( x, t where f ( x, t. Darbha and Rajagopal defne Traffc Flow Stablty for automated vehcle traffc as: 59

72 Let v ( x,, k ( x, denote the nomnal state of traffc. Let v p ( x, t, k p ( x, t be the speed and t t densty perturbatons to the traffc, consstent wth the boundary condtons and are such that v p ( x, t, ( x, t k p x xu. The traffc flow s stable, f. gven ε >, there exsts a δ > such that sup v x x u p ( x, k p ( x, < δ supsup v t x x u p ( x, t k p ( x, t < ε (3 and 2. lm sup v ( x, t k ( x, t = t x x u p p (4 Where sup(. s the abbrevaton for supremum or least upper bound, meanng, for a gven set of numbers S, the smallest element of a set U s the upper bound of S. Ths defnton of stablty can be descrbed n Fgure. v, s a steady state of traffc. Here the traffc flow s defned to ( k be stable when the dsturbance v, k does not exceed a boundary f ts ntal value s wthn a ( p p lmt, and, n the end, the dsturbance becomes zero. As shown n the enlarged fgure n Fgure, f the ntal state s wthn the boundary (the trangle pont, t goes to the equlbrum pont ( k v, n the end; but f the ntal state s beyond the boundary (the rectangle pont, t does not converge to the equlbrum pont but goes to other ndefnte states. Ths defnton s very strong n that the dsturbance must be elmnated over tme by ts own movement. In real-world traffc, dsturbances or unstable traffc usually are elmnated because of lght demand nflows whch happen from tme to tme. On the other hand, t s also a loose defnton because t doesn t present the traffc state change from free flow to congeston n whch the change of the nomnal state of traffc tself s the source of traffc flow nstablty. Thus, a Lyapunov stablty defnton n terms of speed-densty relaton may not work well n descrbng traffc stablty. 6

73 v ( v, k p p δ ( v, k p p ( v, k ( v, k k Fgure. Stablty n Term of Speed-Densty Relaton In Zou and Levnson (2, a new crteron functon of the relaton between densty and flow rate s proposed to detect potental traffc breakdown. ( The convoluton s defned as: C ( f, h = f ( t h( u t du (5 The movng-average of densty s defned as: k low = C( k( t, P( t t2 < t < t P( t = t < t2, t > t (2 The fltered hgh frequency components k ( t are restored by subtractng low frequency components from the orgnal sgnal, whch s defned as Densty Dynamcs: k ( t = k( t k ( t (7 low (3 Cross-Correlaton s always used to detect the dversty of measurements: f, h = f ( t h( u + t du (8 We conduct a template cross-correlaton calculaton of densty dynamcs and flow rate: corr( k, q = max k q ( (6 6

74 where k and q are templates of k and q, respectvely, move smultaneously. Template means a consecutve porton of data n a seres. The new crteron functon s defned as: d z ( t = ( corr( k, q ( dt The computaton results based on real world traffc data justfed the effectveness of ths d functon. It s shown that there s a traffc breakdown only when z ( t = ( corr( k, q > z, dt where z could be a threshold obtaned from experence. If the changng rate of the crosscorrelaton exceeds the threshold, the transton s unreturnable. Another mportant property s that the crteron functon has a sngular peak n the onset of the phase transton. These results ndcate that the crteron functon mght be a mathematcal descrpton of phase transton n traffc flow whch presents the traffc flow movng from stable to unstable states. In dervng the crteron functon, we fnd concentratons of traffc states n both free flow and congested traffc by means of movng average computaton, as shown n Fgure 2. Ths ntutvely provdes support for prevous studes that suggest dstnct phases n traffc flow. Also, the transtons that happened between relatvely stable phases represent cases when the system loses or regans stablty. Thus the transent condton of traffc flow we obtaned sheds lght on the conceptons of traffc stablty and robustness. If the traffc transferrng from free flow to congeston s consdered as an unstable state, we can provde stablty and nstablty crtera as: Stablty Crteron The traffc flow wll reman stable f the changng rate of the cross-correlaton between flow rate d and densty dynamcs s always wthn a boundary,.e., z( t = ( corr( k, q z. dt Instablty Crteron The traffc flow wll be unstable f the changng rate of the cross-correlaton between flow rate d and densty dynamcs exceed a boundary,.e., z ( t = ( corr( k, q > z. dt Fgure 3 llustrates the basc movement of traffc flow n losng and reganng stablty. Our study shows that: f the ntal state s wthn the boundary of stable states cluster, and f the state transton satsfes the stablty crteron, the traffc wll reman stable. Otherwse, traffc wll loss 62

75 stablty and goes to congeston states. Ths s a knd of Lyapunov stablty. By Lyapunov stablty, we mean that the state dsturbances, that satsfy the boundary condtons, reman bounded. 63

76 Fgure 2 (~k s k n these graphs; A means movng-average 64

77 q Free Flow Speed (Lose Stablty Stable States (Regan Stablty Congeston k Fgure 3. Stablty Change of Traffc Flow Robustness of Traffc Flow can also be defned. A robust system s a system that can restore ts normal condton after beng dsturbed by nternal or external nose or dsturbances. Formally, a robust system s defned as a system that behaves n a controlled and expected manner when expected varatons arse n ts domnant parameters, but also n the face of unexpected varatons (EAS GmbH, 2. In traffc systems, typcal varatons nclude the acceleraton nose of vehcles, nternal dsturbances such as the sudden brakng of a vehcle n the strng and external dsturbances such as the change of demand at the entry of the road. We can qualtatvely judge the robustness of the traffc system by observng the profles of flow, speed, and densty. Our results show that f the dsturbances are wthn the boundary, the phase of traffc flow wll not change. So the measurement of robustness of traffc flow s the boundary of the changng rate of the cross-correlaton between flow rate and densty dynamcs. As one can see n Fgure 4, the traffc experenced a dsturbance at some tme and run away from the free flow curve, but t ddn t result n a congeston but went back to stable traffc (the crcled porton. At another tme, the boundary was exceeded and jam presented. 65

78 Fgure 4. Though ths crteron functon cannot be easly appled to the evaluaton of ACC, t at least ndcates that the densty dynamcs should not be too large n the case of hgh flow rate. Otherwse, traffc wll be unstable. From the mcroscopc pont of vew, t means that the response tme of vehcles should not be too small. It s a reasonable nducton n that drvers responses to the densty change ahead. If the change s drastc, drvers wll respond more serously, whch causes bg dsturbances that affect upstream traffc adversely. The ACC desgners should consder ths nfluence of ACC s behavor n mxed traffc SAFETY Traffc safety s a problem related wth headway, response tme of drver/automated system, vehcle mechancal delay and road surface condton. Normally safety can be guaranteed f the headway s greater than the summaton of response tme and delays. Normally, the desgn of ACC algorthms s headway control desgn. Whle the headway control can be acheved effectvely, there are stll some practcal problems to be solved. ACC headway desgn must consder all these factors. Such as: ( Who should be responsble for emergency brakng? The maxmum of deceleraton of current ACC system s not enough for t. If t s the drver s responsblty, when should the drver be alerted? That s a lablty problem. An deal soluton mght be an ACC system combned wth collson warnng/avodng system. In the latter one, ACC system s connected to ABS to apply brakng commands. 66

79 (2 Should ACC change headway n the cases of specal weather condton such as cy, sleetng and rany days? The preset headway n slppery surface should be longer than those n dry surface n the same speeds. Should the drver be responsble for dentfyng the envronment changes and changng the preset headway? How much should the drver change? The ACC desgner must provde gudance for these stuatons. (3 Should the ACC system provde headway optons accordng to drver s response capablty? Should t determne the lower headway lmt for drver wth longer response tme? How s the lablty f the drver uses the least headway? ACC desgners may take conservatve values of preset headway to avod rsks, but the effcency of traffc mght deterorate. On the other hand, the system mght test the response tme of drver by recordng ther behavor to mmnent vehcles approachng. 2.3 ICROSCOPIC SIULATION SYSTE CONFIGURATION The above dscusson on traffc flow stablty and safety s meanngful for the evaluaton of ACC algorthms n that t provdes some qualtatve and quanttatve measurements. But the evaluaton of the mpacts of ACC cannot be easly solved analytcally, because the mxed traffc flow s the aggregaton of vehcles wth dfferent control behavor and ts propertes cannot be obtaned from the unform mathematcal model or dfferental equatons. So we use smulaton tools to observe traffc behavor and summarze ACC s mpacts on the capacty, stablty and safety of traffc before we get a breakthrough n the theoretcal analyss SYSTE CONFIGURATION: IXED TRAFFIC SCENARIO When traffc s comprsed of vehcles controlled by dfferent knds of controllers, adaptve cruse control or/and human drvers, we consder t to be mxed. For ths smulaton, Constant Tme Headway (CTH control and Varable Tme Headway (VTH control ACC algorthms were selected. Although a number of drvng smulator studes have been undertaken, these have focused on crtcal safety aspects of ACC use, such as Nlsson (Nlsson, 995. Very few studes exst on how drvers ncorporate the functonalty of ACC nto ther drvng cycle. For the purpose of ths research, t has been assumed that f there s an ACC system equpped n the vehcle, t s used. However, 67

80 accordng to the US Feld Operatonal Test trals, ACC was used for just over 5% of all mles drven at speeds of above 35 mph. In addton, usage rates for ndvduals vared between 2% and % (Fancher et al Nevertheless, the purpose of the smulaton s to defne the range of traffc effects that could be found. So we don t stochastcally change modes of control n our smplfed smulatons. A smple scenaro of a one-lane hghway secton, 3.2 km long, wth one entry and one ext was establshed. No lane changng s consdered n ths smulaton work. Sgnfcant nter-vehcle nteracton s present throughout the smulaton. The scenaros were desgned to test whether or not ACC could generate a hgher capacty whle guaranteeng stable drvng. ACC vehcles are allowed onto the current hghway system used by manually drven vehcles. eterng s done at the entrance to guarantee enough ntal headway on the hghway, and whle watng to enter the hghway, ACC vehcles are treated just lke manual vehcles. The role of the drver of the ACC vehcle s the same n these scenaros. On reachng the target lane, the drver engages the automated control system of the vehcle that takes over the longtudnal control of the vehcle. The drver s responsble for all drvng functons as n a manually drven vehcle except for the longtudnal control. The drver dsengages the headway control of the ACC vehcle and accelerates to a maxmum speed f the hghway s clear before hm and at last exts the lane. The maxmum deceleraton of the ACC equpped vehcle when under tme headway control mode s lmted to 2 m/s 2 whle the maxmum acceleraton under ACC s.5 m/s 2. Three typcal scenaros are of most nterest, these nclude: (a No-ACC traffc: All vehcles on the road are controlled by Gpps car-followng model. Ths s the scenaro to smulate the current manually controlled traffc. (b xed traffc: ACC vehcles mx wth Gpps vehcles wth certan penetraton. We wll hghlght ths scenaro as the ntermedary stage of ACC deployment. The safety and stablty ssues n ths scenaro are expected to be more complcated than others. (c Pure ACC traffc: All vehcles are controlled by ACC. It can be called semautomated n whch each vehcle ndvdually assgns desred headway and desred speed. 68

81 2.3.2 DYNAIC ODELS OF THE COPONENTS OF IXED TRAFFIC The man dynamc models used n the smulaton are the vehcle dynamcs, the ACC algorthm and the car-followng model. Vehcle Dynamcs The vehcle dynamcs s smplfed to a thrd-order dfferental equaton: x = ( xdes x τ (2 where: x s the jerk of vehcle ; x s the acceleraton of vehcle ; x des s the desred acceleraton of vehcle whch s generated by the car-followng model or ACC algorthm. Adaptve Cruse Control Polcy The most conventonal ACC algorthm s Constant Space Headway control, whch s n form of: xdes = kε k2 ε ε = x x + L (3 Though t takes advantages of the relatve poston and relatve speed as the control nput, t has been proven that ths control law cannot guarantee strng stablty (Darbha and Rajagopal 999. So we don t pursue ths control law. Constant Tme Headway (CTH control can be represented as xdes = ε + λδ h δ = x x + L + h x CTH takes advantage of the relatve speed and contans an extra term to fulfll tme headway control. It has been proven that ths control law can guarantee strng stablty (Darbha and Rajagopal 999 and thus becomes a promsng alternatve to the constant space gap law. In ths report, we use CTH control law as the prmary one to test the mpacts of ACC on the traffc flow. Varable Tme Headway (VTH control (Wang and Rajaman, 2 takes the relatve velocty nto account n the desred spacng, whch s gven by as follows: (4 69

82 7 + + = + + = f m f f m des b v x b v x x v x ε ε δ λδ ε ε (5 Where, m s the maxmum densty of the hghway, at whch pont traffc wll stop (we assume L m =, L s the unform vehcle length; f v s the free flow speed; = x x ε s the relatve velocty between th vehcle and -th vehcle; b s a postve coeffcent whch determne the how much the relatve velocty contrbutes to the desred spacng. Car-followng odel any models are developed to emulate the human drver s drvng behavor, such as the G model, Greensheld s model, Drew s model and Gpps model (Gpps 98. In our smulaton, we use Gpps odel to represent the acceleraton and deceleraton of manually controlled vehcles. Ths model states that, the maxmum speed to whch a vehcle (n can accelerate durng a tme perod (t, t+t s gven by: (, (.25 (, ( ( 2.5, (, ( n x t n x n x t n x T n x t n x T t n xa + + = + (6 where: x (n,t s the speed of vehcle n at tme t; x (n s the desred speed of the vehcle (n for the current secton; x (n s the maxmum acceleraton for vehcle n; T s the reacton tme = updatng nterval = smulaton step. On the other hand, the maxmum speed that the same vehcle (n can reach durng the same tme nterval (t, t+t, accordng to ts own characterstcs and the lmtatons mposed by the presence of the leader vehcle s:

83 xb ( n, t + T = d( n T + 2 d( n T 2 d( n 2 x( n, t s( n, t x( n, t 2 x( n, t x( n, t T d' ( n (7 where: d(n (< s the maxmum deceleraton desred by vehcle n; x(n,t s poston of vehcle n at tme t; x(n-,t s poston of precedng vehcle (n- at tme t; s(n- s the effectve length of vehcle (n-; d'(n- s an estmaton of vehcle (n- desred deceleraton. In any case, the defntve speed for vehcle n durng tme nterval (t, t+t s the mnmum of those prevously defned speeds: x( n, t + T = mn xa ( n, t + T, xb( n, t + T (8 Then, the poston of vehcle n nsde the current lane s updated takng ths speed nto the movement equaton: x ( n, t + T = x( n, t + x( n, t + T T ( GENERATION OF TRAFFIC As we have mentoned before, traffc generaton comples wth gven traffc demand profles. Normally, we use a constant nflow rate or pulse nflow rate to test the system. Each vehcle enterng the road s controlled by a mechansm that changes the ntal headway accordng to the specfc nflow rate at that tme. The demand profle employed for the study was chosen to overload road capacty durng the mddle 5 seconds (from 2 to 35 second of the smulaton. The other ssue s to control the proporton of ACC vehcles. In our smulaton, ACC vehcles are n traffc flow followng a unform dstrbuton. 7

84 Fgure 5. Typcal Traffc Demand Profle 2.4 TRAFFIC SIULATION PROGRA A mcroscopc smulaton program s developed n the programmng language C++. The flowchart of the program s shown n Fgure 6. There s a man cycle of calculaton n whch the states of vehcles and the traffc flow are updated n a sngle samplng tme duraton. The samplng tme s. second n our smulaton, whch s the response tme of the ACC equpped vehcle. The man cycle ncludes: (a Vehcle entry procedure that determnes whether a new vehcle should enter the road. If so, t generates a new vehcle wth a randomly selected control law, ether Gpps model or ACC algorthm; (b Vehcle ext procedure that determnes whether the leadng vehcle should ext from the road. If so, t deletes the leadng vehcle and modfes the second vehcle to be the leadng vehcle. In our smulaton, the leadng vehcle wll be free to accelerate untl t reaches the maxmum speed; (c Vehcle state calculaton calls the functons to update the states of each vehcle n current samplng duraton. The car dynamcs functon wll call the Runge Kutta algorthm (Press et al. 992 that solves the dfferental equatons. Ether Gpps car-followng model or ACC algorthm wll generate the desred acceleraton for each ndvdual vehcle; (d Road state calculaton procedure gets the nstantaneous mean densty, space mean speed, nflow rate etc. n the current samplng tme. 72

85 The mportant parameters used n the smulaton are summarzed n Table and Table 2. Table. System Confguraton Road length 322 meters axmum sze of vehcles 4 meters Intal speed of vehcles 7.79 m/s (4 mph axmum Speed 28.9 m/s (65 mph Parameters of operaton Sample tme (calculaton cycle. second Smulaton tme duraton 6 ~9 seconds Table 2. Parameters of ACC Algorthms Parameters of Gpps odel Constant Tme Headway Desred Speed 28.9 m/s (65 mph λ.2 axmum Tme Headway.~.2 seconds Acceleraton.7 m/s 2 Varable Tme Headway axmum 3.78 m/s Free Flow Speed Deceleraton (7.5 mph -3.4 m/s 2 λ.2 Tme Headway 2 seconds 73

86 Program Intalzaton Platoon Intalzaton Vehcle Entry Procedure Car Dynamcs Vehcle Ext Procedure Car Followng Runge Kutta Alg. Vehcle State Calculaton Car Calculaton Veh. ID/ Tme Veh. Poston Vehcle State Storage Veh. Speed... Densty Road States Calculaton Space ean Speed Inflow Rate... Road States Storage Update Tme & Counter Fgure 6 Flowchart of smulaton Program 74

87 3 SIULATION OF CONSTANT TIE HEADWAY AND ANUAL CARS 3. SINGLE VEHICLE FOLLOWING BEHAVIOR The sngle vehcle followng behavor ncludes the behavors of vehcles wth varous controls under dfferent settngs. Some typcal results are shown n Fgure 7. In these smulatons, the preset tme headway of ACC vehcle s second, whle that of Gpps vehcle s 2 seconds. The two vehcles n the par start up wth the same ntal speed and wth a 2-meter dstance. It s shown that ACC vehcles wll have a qucker response and thus smaller transent tme than manual drven vehcles. So n these scenaros, Gpps vehcles cannot catch the leadng vehcles because the leadng vehcles are free to accelerate. In contradcton, ACC vehcles can always mantan the constant tme-gap. Ths result hghlghts an mportant advantage of ACC compared to manual vehcles: small tme headway s more easly acheved by ACC vehcles; thus ACC vehcles generate capacty. 75

88 Gpps Car Followng Gpps Car Gpps Car Followng ACC Car 35 Speed of Car Speed of Car2 Tme Headway Poston of Car Poston of Car Speed of Car Speed of Car2 Tme Headway Poston of Car Poston of Car2 6 Speed (m/s Headway (s Poston (m Speed (m/s Headway (s Poston (m Tme (s (a Tme (s (b ACC Car Followng Gpps Car ACC Car Followng ACC Car 35 Speed of Car Speed of Car2 Tme Headway Poston of Car Poston of Car Speed of Car Speed of Car2 Tme Headway Poston of Car Poston of Car2 6 Speed (m/s Headway (s Poston (m Speed (m/s Headway (s Poston (m Tme (s (c Fgure 7. Sngle Vehcle s Followng Behavors Tme (s (d 76

89 3.2 HEADWAY RESPONSE OF VEHICLES The headway response s the basc behavor that mpacts the safety and capacty of the road. We expermented wth the response of ACC vehcles and Gpps vehcles to the preset headway under car-followng scenaros. The basc condtons nclude: (a Two vehcles have the same ntal speed (7.7 m/s; (b The leadng vehcle s ntal poston s 2 meters from the entry, whle the followng vehcle s located n the entry pont; (c The maxmum speed for both vehcles s 28.9 m/s. The orgnal verson of Gpps model doesn t have a mechansm for achevng certan tme headway. In the smulaton, we added a tme headway term that can affect the speed of vehcle to realze the headway control,.e. f (space headway/(speed of followng vehcle < (desred tme headway, then (the defntve speed <= (current speed. Ths modfed Gpps model s more realstc wth respect to the real condton that most drvers adjust ther speeds accordng to estmated tme headway (Koppa, 998. Fgure 8 shows the tme headway response of ACC vehcles and Gpps vehcles. The settng (desred tme headway changes from. second to 3 seconds, whch s represented by the sold curve n each graph. In fact, headways under.5 seconds are rarely used because most drvers wll change to space control under such condtons. But these smulatons are meanngful to show the responses of these models n emergency stuatons or n the case of congeston. The real headway response s shown by the curve wth a marker n each graph. It s shown that ACC vehcles can always acheve the settng headway wth a small error. Changng the parameters of the algorthm cannot elmnate ths error, whch s largely from lack of an ntegral component n the controller. From (c we can see, the Gpps vehcle cannot catch the ACC vehcle. That s because the ACC vehcles can get to the maxmum speed more quckly. On the other hand, as shown n (d, the headway errors for Gpps vehcles are qute large compared to ACC vehcles. Though t s an nnate dsadvantage of Gpps model, we expect the same amount of errors for human drvers. 77

90 Real Headway (s ACC followng ACC Actual Headway Desred Headway Real Headway (s ACC followng Gpps Actual Headway Desred Headway Desred Headway (s 2 3 Desred Headway (s (a (b Real Headway (s Gpps followng ACC Actual Headway Desred Headway 2 3 Desred Headway (s Real Headway (s Gpps followng Gpps Actual Headway Desred Headway 2 3 Desred Headway (s (c (d Fgure 8. Headway Response Of Sngle Vehcle By comparng these results, we conclude that the headway response of ACC vehcles can fulfll the requrement that wll brng nto potental of hgh capacty. Gpps model can 78

91 emulate the human drvers behavor to some extent. The traffc flow comprsed by these two knds of vehcles s a mxed flow wth heterogeneous headway behavors. 3.3 SYSTE RESPONSE TO INTERNAL PULSE The system response to an nternal dsturbance s shown n Fgure 9. The nternal dsturbance s generated by a sudden brakng of a vehcle n the strng. After a whle, the speed of that vehcle s restored to normal condtons. The vehcles behnd the brakng vehcle wll be affected. As shown n Fgure 9, whch s the case of % ACC penetraton, the speeds of some vehcle are reduced to mantan safe dstance. After the speed of the leadng vehcle s restored, the affected vehcles can return to normal speeds. densty veh/km speed m/s Internal pulse response SPEED DENSITY tme s Fgure 9. Densty and Speed Profles n an Internal Pulse Furthermore, the restored platoon s runnng under a one-second tme headway, whch s smaller than that of normal condton. So there s a capacty gan that can compensate the loss caused by brakng. It should be noted that ths gan could only be obtaned when the normal runnng of traffc s below the capacty of the system. Ths result shows that ACC has the potental to stablze the traffc under small dsturbances. 3.4 SYSTE RESPONSE TO EXTERNAL PULSE What we are most nterested n s the response of the mxed traffc to the external dsturbance that s generated by the pulse demand as shown n the Fgure 5. Ths s because ths knd of dsturbance s a typcal case n real traffc. The scenaros wth 79

92 dfferent penetraton of ACC are smulated and the results are shown from Fgure to Fgure 3. As we can see: (a The denstes and space mean speeds of the system n the dsturbance are always wthn a boundary and can return to normal after the pulse. (b The densty-speed curves of these scenaros largely comply wth nverse proportonal relaton, whle hgh penetraton of ACC can ncrease the system speed n the pulse. (c The densty-flow rate curves show a lnear relatonshp n the un-congested regon. (d Hgh penetraton of ACC vehcles can reduce the system densty and the speed drop durng the pulse. Ths means there are potental capacty gans under hgh penetraton of ACC vehcles. (e The mxed traffc has larger speed oscllatons than the cases of pure ACC traffc or pure manual traffc. The oscllatons may come from the dfferent acceleraton behavors among ACC vehcles and manual vehcles. k (veh/km v (m/s Densty & Speed Profles: % ACC DENSITY SPEED tme (s 8

93 k-v : % ACC k-q: %ACC v (m/s k (veh/km q veh/s k veh/km Fgure. Response of %ACC system to External Impulse 8

94 Densty & Speed Profles: % ACC 5 45 k (veh/km v (m/s DENSITY SPEED tme (s k-v : % ACC k-q: %ACC v (m/s k (veh/km q veh/s k veh/km Fgure. Response of %ACC system to External Impulse 82

95 4 Densty & Speed Profles: 9% ACC k (veh/km v (m/s DENSITY SPEED tme (s k-v : 9% ACC k-q: 9%ACC v (m/s k (veh/km q veh/s k veh/km Fgure 2. Response of 9%ACC system to External Impulse 83

96 4 Densty & Speed Profles: % ACC 35 k (veh/km v (m/s SPEED DENSITY tme (s k-v : % ACC k-q: %ACC v (m/s k (veh/km q veh/s k veh/km Fgure 3. Response of %ACC system to External Impulse 84

97 3.5 SPEED PROFILES OF TRAFFIC FLOW WITH DIFFERENT ACC PENETRATION After mposng the same dsturbances n the system wth dfferent ACC vehcle penetratons we can compare the result speed profles and get the mpacts of ACC on the mxed traffc, as shown n Fgure 4. Speed (m/s ~%ACC response to external mpulse:.78 veh/s; headway :2s % 95% 9% 8% 6% 99% 4% 2% % tme (s Fgure 4. Speed Profles under Dfferent ACC Penetratons As we can see, the penetraton of ACC wll sgnfcantly affect the speed profles: (a The system uses less tme to get to the normal runnng state wth hgher ACC penetraton; (b The system wth hgher ACC penetraton uses less tme to restore to normal state after a dsturbance by the external pulse; (c The reductons of speed drop n the pulse are not lnear wth ACC penetraton. The most remarkable change s happened between 9% and % penetraton. Ths means that hgh penetraton of ACC reduce speed loss. (d A questonable result n ths graph s the speed profle wth % ACC penetraton. The Space mean speed ncreases nstead of decreasng n the pulse. A tentatve explanaton of ths phenomenon s that because of the hgh nflow rate of the demand, 85

98 more vehcles on the road accelerate to the maxmum speed than that under the normal case. In other words, a smaller porton of vehcles on the road are n the process of acceleraton. As we can see n Fgure 5, the proporton of low speed vehcles s zero n the peak of the speed profle. In the calculaton, a vehcle wth the speed lower than 2 m/s s called a low speed vehcle. Thus a hgher mean speed s obtaned n the peak where most vehcles are hgh speed. However, at the peak of the pulse, some vehcles cannot enter the system. They are queued at the bottleneck watng to enter the system and are not counted. 3 Speed Profle n the Pulse: % ACC. Speed (m/s Speed Proporton of Low Speed Vehcles Tme (s Fgure 5. Speed Profle n Pulse 3.6 K-V AND K-Q RELATION IN IXED TRAFFIC The typcal relatonshps among densty, flow rate and space mean speeds are meanngful n analyzng the mpacts of ACC on the traffc system. In our work, two types of these relatons result from the smulaton. The frst k-v and k-q relatons are obtaned from the dynamc process that the system encounters a saddle demand and s restored to normal state. Fgure 6 shows the k-q and k-v curves for a % ACC system that encounters an over-capacty demand. It s shown that k-q curve s lnear below capacty, and descends and ascends n the saddle demand part. In contrast, the k-v curve s nearly constant n under-capacty part. That means a pure ACC system can keep the free-flow speed before enterng the congested regon. Fgure 7 compares the cases that % ACC system and % ACC system encounter at 86

99 near-capacty saddle demand. It s obvous that % ACC system has a hgher speed and lower densty than a % ACC system. Fgure 6. k-q & k-v Relaton of Dynamc Process (Beyond Capacty 87

100 Fgure 7. k-q & k-v Relaton of Dynamc Process (Under Capacty 3.7 INFLUENCE OF HEADWAY OF VEHICLES Because we use constant tme gap ACC n our smulaton, the preset headway wll determne the throughput of the system. ean tme headway can be computed as: ha = haccp+ hman( p (8 where: h a : average headway h acc : headway of ACC vehcles h man : headway of manual vahcles p: proporton of ACC vehcles Throughput for sem-automated vehcles wth ACC can be obtaned from: q= 36/ ha (9 So we can ncrease the throughput by reducng the preset headway of ACC vehcles. 88

101 3 ~% ACC Pulse Response:.2 veh/s Pulse; 2 sec Preset Headway % 2% % 6% Speed (m/s 5 8% 9% 99% % tme (s Fgure 8. Speed Profles under Dfferent ACC Penetratons On the other hand, there s a serous dsadvantage of constant tme headway control. If the demand flow rate s hgher than the nverse of the preset headway of ACC vehcles, a rapd drop of speed wll happen, as we can see n Fgure 8. In ths case, the dfferences of the system n pulse wth varous proportons of CTH vehcles are rather small and the benefts of hgh penetraton of ACC are non-exstent. There s an adverse effect f penetraton of ACC s hgher than a lmt. Ths result shows CTH s not capable of reducng congeston n hgh demand condton. 3.8 THE VARIANCE OF SPEED IN THE EQUILIBRIU STATE The rpples n the pattern of the speed can be evaluated by the varance. The speed dscussed here s the space mean speed n the equlbrum state. It seems that CTH ACC vehcles wll generate more oscllatons n the patterns of the speed, as shown n Fgure 9. Ths effect s more serous f the proporton s very hgh (greater than 95%. In the stable range wth low proporton of CTH vehcles, the varance s always small. 89

102 .8 Speed Varance under dfferent nflow.7 speed varance % 2% 4% 6% 8% 9% 95% 99% Inflow veh/s Fgure 9. Speed Varance under Dfferent Inflow and Dfferent ACC Penetraton On the other hand, the speed varances dscussed here may not accurately represent the real value, because the road s rather short (3.2 km and the number of vehcles s small. We can expect smaller varance n a larger system. 3.9 QUEUE ON THE ROAD AND TRAVEL TIE Another method to evaluate mxed traffc s to measure the length of queue and the travel tme of vehcles. Fgure 2 compares the number of vehcles on the road under dfferent ACC penetratons. In ths case, the pulse of the nflow rate, as shown n Fgure 5, s just the upper lmt of ACC capacty. As we can see, the numbers of vehcles on the road durng hgh ACC penetraton scenaros are smaller than that of pure manual traffc n the pulse. It means less congeston n ths secton of the road. On the other hand, the duraton of congeston s shorter wth hgh ACC penetratons than wth pure manual traffc. 9

103 Number of vehcles Queue and Travel Tme:, 9 & % ACC; veh/s pulse # Enterng Vehcles Tme (s 9% % % # Extng Vehcles Fgure 2. As we dscussed before, t s a property of CTH ACC system that t cannot cope wth hgh demand flow rate beyond ts capacty, whch s determned by ts preset headway. In the case that the nflow rate s hgher than the capacty of ACC, we see a decrease of capacty. As shown n Fgure 2, the numbers of vehcles under hgh ACC penetratons are larger than the former cases. Number of vehcles Queue and Travel Tme:, 9 & % ACC;.2 veh/s pulse # Enterng Vehcles Tme (s 9% % % # Extng Vehcles Fgure 2. 9

104 4 COPARISON OF VARIABLE TIE HEADWAY AND CONSTANT TIE HEADWAY 4. SINGLE VEHICLE FOLLOWING BEHAVIOR The typcal sngle car followng behavors s shown n Fgure 22. In these smulatons, the preset tme headway of CTH vehcle s second. The two vehcles n the par start up wth the same ntal speed and wth a 2 meters dstance. It s shown that vehcle controlled by VTH has slower response and takes a relatvely long tme to get to steady state. On the other hand, all vehcles can ultmately attan enough dstance and mantan the constant tme-gap. For CTH vehcles, t happens shortly after the arrval of the followng vehcle. For VTH vehcles, t happens after two vehcles get to the maxmum speed. 92

105 3 Varable Tme Headway Speed m/s 24 Leadng Veh Followng Veh Tme s (a 3 Constant Tme Headway Speed m/s Leadng Veh Followng Veh Tme s Fgure 22. Sngle Vehcle s Followng Behavor (b 4.2 SPEED PROFILES OF TRAFFIC FLOW WITH DIFFERENT ACC PENETRATION 93

106 Under dfferent constant demands, the mxed traffc of VTH cars performs better than those of CTH cars. As shown n Fgure 23, under the same demand, the speed of VTH traffc s always hgher than CTH traffc except the % case, and they always have shorter response tme to reach the steady state. The mxed traffc has larger speed oscllatons than the cases of pure ACC traffc or pure manual traffc. The rpples n the pattern of the speed can be evaluated by the varance. The speed dscussed here s the space mean speed n the equlbrum state. It seems that CTH control ACC vehcles wll generate more oscllatons n speed, as shown n Fgure 24. Ths effect s more serous f the proporton s very hgh (greater than 95%. In the stable range wth low proporton of CTH vehcles, the varance s always small. On the other hand, a lttle hgher speed varance s found wth VTH vehcles, as shown n Fg. 26. But ths phenomenon s reversed n the cases of very hgh ACC penetraton, such as 99% and %. 94

107 Varable Tme Headway 28 % ACC 26 Space ean Speed m/s % ACC Tme s (a Constant Tme Headway 28 % ACC 26 Space ean Speed m/s % ACC Tme s Fgure 23. (b 95

108 28 Steady Speed under Dfferent ACC Penetraton Stead Speed m/s CTH VTH Penetraton of ACC % Fgure Speed Varance of Stead State under Dfferent ACC Penetraton.35 Speed Varance m/s CTH VTH ACC Penetraton % Fgure

109 4.3 SYSTE RESPONSE TO EXTERNAL PULSE After exertng the same dsturbances n the system wth dfferent ACC vehcle penetratons we can compare the speed profles and get the mpacts of ACC on the mxed traffc, as shown n Fgure 26. As we can see, the penetraton of ACC wll sgnfcantly affect the speed profles: (a The system s restored to the normal state more quckly wth hgher VTH penetraton than wth CTH. (b Hgh penetraton of VTH can reduce the system densty and the speed drop durng the pulse compared to a smlar penetraton of CTH cars. Under hgh demand, the drop of space mean speeds of the VTH traffc n the dsturbance are always smaller than manual traffc and can easly return to normal after the pulse, whle CTH traffc may experence serous speed drop that s even worse than that of pure manual traffc. 4.4 K-V AND K-Q RELATION IN IXED TRAFFIC The frst k-v and k-q relatons are obtaned from the dynamc process that the system encounters a saddle demand, whch s comprsed of a lnearly ncreasng part (5 seconds and a lnearly decreasng part (5 seconds. Fgures 27 and 28 show the k-q and k-v curves for a % ACC system that encounters an over-capacty demand. For CTH traffc, t s shown that k-q curve s lnear below capacty, and descends and ascends n the saddle demand part. In contrast, the k-q curve s nearly lnear for VTH traffc. That means a VTH system can keep the free-flow speed n a longer range. Fgure 28 compares the k-v curves of VTH and CTH traffc encounterng an over-capacty saddle demand. It s shown that VTH traffc has a hgher speed and lower densty than CTG traffc. 97

110 Space ean Speed m/s Varable Tme Headway % ACC % ACC Tme s 3 25 Constant Tme Headway % ACC Space ean Speed m/s % ACC Tme s Fgure 26. Speed Profles of Traffc Flow wth Dfferent ACC Penetraton 98

111 .2. Trangle Inflow Response VTH CTH Flowrate veh/s Densty veh/km Fgure k-v: Trangle Inflow Response Speed m/s VTH CTH Densty veh/km Fgure 28. Under the condton of very hgh demand nflow, VTH traffc decreases the speed and mantans the densty untl the demand s released, as shown n Fg. 3. The response process s shown n Fg. 3. As one can see, the system stops to accommodate more 99

112 vehcles after the speed gets to a low pont. In ths case, the nflow rate s not the ndcaton of the demand but the reflecton of system capacty. Flow Rate veh/s Densty vs. Flow Rate: Response to Over-Capacty Saddle Demand VTH Veh CTH Veh Densty veh/km Fgure % VTH Vehcles: Response to Saddle Demand 4 6 Densty Densty veh/km Speed m/s Speed Flow Rate 3 2 Flow Rate veh/s 5 Tme s Fgure 3.

113 5 SIULATION WITH SOE RANDO EFFECTS 5. INTRODUCTION As a smplfed analyss, the smulatons n Chapter 3 do not nclude the varaton of headways among ACC and manual drven vehcles. To make the smulaton more realstc, a randomly chosen headway s mplemented n smulaton. It s assumed that each drver keeps hs/her favorte headway all the tme. A new attrbute of vehcle class: vehcle.headway s preset n the vehcle generaton procedure, whch determnes the headway choce of each vehcle and cannot be changed durng smulaton. Though t s stll a smplfed stuaton, ts result s mportant n that t separates the mpacts of drvers personal headway choces. So we can compare them wth the former results we obtaned. The tme headway of Gpps vehcle s normally dstrbuted wth mean=2 sec and a gven standard devaton and a -second mnmum. Tme headway of CTH vehcle s normally dstrbuted wth mean= sec and a gven standard devaton and a.8-second mnmum. A normally dstrbuted random number s generated n functon: float gasdev(long dum. Expermental results are summarzed below: 5.2 IXED TRAFFIC OF CTH AND ANUAL DRIVEN VEHICLES Fve combnatons of CTH and manual drven vehcles wth dfferent headway dstrbutons are smulated, whch nclude: ( CTH= sec; Gpps= 2 sec, as shown n Fgure 3; (2 CTH= sec; Gpps= max(2+ N(,, sec, as shown n Fgure 32; (3 CTH= sec; Gpps= max(2+ N(,2, sec, as shown n Fgure 33; (4 CTH= max(.+ N(,/2,.8 sec; Gpps= max(2+ N(,2, sec, as shown n Fgure 34; (5 CTH= max(.+ N(,/4,.8 sec; Gpps= max(2+ N(,2, sec, as shown n Fgure 35. As one can see, the random headways of manual vehcles do not have much nfluence on the performance of traffc. In contrast, the random headways of CTH ACC vehcles greatly affect the traffc. Hgher headway devaton of CTH vehcles wll lead to hgher speed drop and oscllatons, especally when the ACC penetraton s very hgh. In the

114 case of % CTH ACC penetraton, hgher headway devaton results n serous speed drop and longer tme to recover. Fgure 36 presents the comparson of the average speed n the fve CTH experments; Fgure 37 presents the comparson of the speed varance. As one can see, hgh penetraton of CTH ACC ncreases the average speed n most of cases. But hgh headway devaton deterorates ths effect. On the other hand, the speed varance s not sgnfcant n most cases, except the case of % ACC penetraton under hgh headway devaton. It can be concluded that these results can not provde support of the clam that CTH ACC wll add traffc capacty. 3 CTH= sec; Gpps= 2 sec 25 Speed m/s % 6% 99% % tme s Fgure 3. 2

115 3 CTH= sec; Gpps= max(2+ N(,, sec 25 Speed m/s % 6% 99% % tme s Fgure CTH= sec; Gpps= max(2+ N(,2, sec 25 Speed m/s % 6% 99% % tme s Fgure 33. 3

116 3 CTH= max(.+ N(,/4,.8 sec; Gpps= max(2+ N(,2, sec 25 Speed m/s 2 5 % 6% 99% 5 % tme s Fgure CTH= max(.+ N(,/2,.8 sec; Gpps= max(2+ N(,2, sec 25 Speed m/s 2 5 % 6% 99% 5 % tme s Fgure 35. 4

117 Fgure 36. Fgure 37. 5

118 5.3 IXED TRAFFIC OF VTH AND ANUAL DRIVEN VEHICLES Three combnatons of VTH and manual vehcles wth dfferent headway dstrbutons are smulated, whch nclude: ( VTH and Gpps =2 sec, as shown n Fgure 38; (2 VTH and Gpps = max(2+ N(,, sec, as shown n Fgure 39; (3 VTH and Gpps = max(2+ N(,2, sec, as shown n Fgure 4. Because VTH does not have preset headway, ts headway s always n changng. The only random factor here s the random headway of human drvers n manual drven vehcles. It s shown that VTH ACC always performs well facng dfferent headway devaton of human drver. Fgure 4 presents the comparson of the average speed n the three VTH experments; Fgure 42 presents the comparson of the speed varance. There s no sgnfcant dfference n terms of speed and speed varance. These results further justfy the advantage of VTH ACC compared wth CTH ACC. 28 VTH; Gpps= 2 sec 26 Speed m/s % 99% 2 8 6% % tme s Fgure 38. 6

119 28 VTH; Gpps= (2 + N(. sec 26 Speed m/s % 99% 2 8 6% % tme s Fgure VTH; Gpps= (2 + N(.2 sec 26 Speed m/s % 99% 8 % 6% tme s Fgure 4. 7

120 Fgure 4. Fgure 42. 8

121 6. SIULATION OF IXED TRAFFIC IN AISUN Smulatons of one-lane traffc shown above are some deal scenaros. To evaluate mpacts of ACC on traffc flow n a more realstc way, we need the observaton of mxed traffc n mult-lane, on-ramp and off-ramp scenaros. These smulatons cannot be easly realzed based on the ppelne smulaton program. On the other hand, commercal traffc smulaton software such as AISUN provdes a good base for complex traffc smulaton (TSS 2. GETRA Extenson provdes nterfaces to obtan nformaton from AISUN and modfy some nformaton durng runnng, as shown n Fgure 43. In ths research, we take advantage of ths package to mplement ndvdual vehcle control. To realze ACC polces and control of traffc, some functons of GETRA Extenson are used. Fgure 43. Functon of GETRA Extenson In the frst smulaton of mxed traffc n AISUN, a ppelne scenaro s tested, wth whch we can compare our former smulaton results. A problem wth AISUN s that the smulaton step whch s also the reacton tme of all vehcles s lager than.5 seconds. But normally the reacton tme of ACC system s. second. And we must smulate the case n whch vehcles are wth dfferent reacton tmes. To solve ths problem, a modfed verson of AISUN was obtaned from the developer (TSS. In ths verson, the smulaton step ranges from. second to second. In each smulaton step, the program reads the nformaton of each vehcle on the road from AISUN, calculates ts new states and updates them. A problem s that the only functon can be used to update vehcle states s the functon to modfy vehcle speed. So 9

122 n ths smulaton, we cannot fully control vehcle behavors. The shapes of smulaton results are dentcal to those of former smulatons, as shown n Fgure 44 comparng to Fgure 6(d. k-q: % ACC;.2 veh/s.8 q veh/s k veh/km Fgure 44. Smulaton Results n AISUN: % CTH,.2 veh/s In the second smulaton, we bult up a smple scenaro, as shown n Fgure 45, n whch the characterstcs of mxed traffc can be observed. In ths scenaro:. All vehcles are tracked all the tme. Vehcles enter from the left sde of secton and secton 2 accordng to programmable demand curve; 2. If a vehcle exts from secton 2, a new vehcle s put n secton at the ntersecton; 3. The speed of the new vehcle s the mean of speeds of vehcles before and after the ntal poston; 4. The vehcle put n the ntersecton keeps the same followng mode (Gpps or ACC as the one exts from secton 2;

123 5. When a new vehcle s put n the ntersecton, ts poston s ether 35m (mddle pont or 343m (the left pont, dependng on whether there s a vehcle n the ntersecton; 6. If there s a vehcle n secton near the ntersecton, a vrtual vehcle s put at the end of secton 2; otherwse, the road ahead of the frst vehcle n secton 2 s clear. By ths we means that vehcles n secton 2 wll be affected by the traffc condton n the ntersecton; 7. All vehcles adjust speeds accordng to states of tself and the leader. If a new vehcle s nserted n, the follower treats t as a leader n the next cycle. Secton Secton Fgure Smulaton Scenaro n AISUN Fgure 46 shows the graphc nterface of AISUN smulaton. The nfluence of on-ramp traffc to the manstream traffc can be easly observed.

124 Fgure 46. Smulaton of xed Traffc AISUN (Screen Copy Fgure 47 shows some examples of the AISUN smulaton. Pulse demands are mposed n entrances both of sectons, n whch the normal demand s.3 veh/s, the demand n pulse s.6 veh/s. Fgure 47(a shows the average densty and speed of the man lane wth % ACC traffc whle Fgure 47(b wth % ACC traffc. These results present smlar patterns as those n former smulaton results. Wth % ACC traffc, the speed drop s not as serous as our former result. Ths s because the traffc enterng the ntersecton from the ramp s affected by the man lane traffc. The traffc wll be congested on the ramp so that the demand contrbuted by the ramp s lmted. 2