Traffic automation and lane management for communicant, autonomous, and human-driven vehicles

Transportation Research Part C 111 (2020) 477–495

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Transportation Research Part C journal homepage: www.elsevier.com/locate/trc

Traffic automation and lane management for communicant, autonomous, and human-driven vehicles

T

Mahyar Amirgholya, Mehrdad Shahabib, H. Oliver Gaoc,d,

⁎

a

Department of Civil and Construction Engineering, Kennesaw State University, Atlanta, GA, 30060, United States Department of Civil and Environemental Engineering, University of Michigan, Ann Arbor, MI 48109, United States School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853, United States d Sino-US Global Logistics Institute, Antai College of Economics and Management, Jiaotong University, Shanghai 200030, China b c

ARTICLE INFO

ABSTRACT

Keywords: Heterogeneous demand Cooperative traffic control Automated highway system Macroscopic fundamental diagram San Francisco

The advent of autonomous driving technologies has created a crucial need for upgrading conventional traffic control and lane management strategies in large cities. In this research, we design an optimal lane management strategy for corridors with a heterogeneous demand of human-driven, autonomous, and communicant autonomous vehicles (HVs, AVs, and CAVs). In a monocentric city setting, we dynamically control the inflow of the network by optimizing the size of CAV platoons in the corridors based on the instantaneous condition of the integrated system. These corridors can potentially have three types of lanes for vehicles with different levels of automation technology. We model the multiple lane type corridors as sets of parallel bottlenecks with general distributions of multiclass demand. The dynamics of the congestion in the network is also modeled using the macroscopic network fundamental diagram (MNFD). To study the impacts of the rise in the penetration rate of AVs and CAVs on the performance of the system, we derive a closed-form representation of the model. We show that the increase of the delay in the network with the rise in the penetration rate of AVs and CAVs can have a stable, an unstable, or a hybrid pattern. To optimize the system, we minimize a weighted summation of the experienced delay in the corridors and the total travel time in the urban network by optimizing the number of lanes of each type and the dynamic size of the CAV platoons. The results of the San Francisco case study show that implementing an optimal lane management strategy can reduce the experienced delay in the corridors up to 78% with a rise in the AV/CAV penetration rate. By dynamically controlling the size of the CAV platoons in the automated highway of the Bay Bridge, we limit the increase of the travel time in the downtown network as low as 5%.

1. Introduction Recent advances in communication, computation, and automation technologies have created an enormous potential for a profound transformation in transportation systems. The advent of self-driving vehicles that can communicate with each other and with the transportation infrastructure has promoted the idea of cooperative traffic control in automated highways (Van Arem et al., 2006; Fernandes and Nunes, 2015). However, we may be several decades away from a homogeneous (fully automated) traffic condition because of the high price of the new technology and the relatively low willingness of users to pay for it (Litman, 2014; Viereckl et al., 2015; Shabanpour et al., 2018). During the transition phase, traffic management and control strategies need to be designed for the ⁎

Corresponding author at: School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853, United States. E-mail addresses: [email protected] (M. Amirgholy), [email protected] (M. Shahabi), [email protected] (H. Oliver Gao).

https://doi.org/10.1016/j.trc.2019.12.009 Received 31 October 2018; Received in revised form 12 November 2019; Accepted 16 December 2019 0968-090X/ © 2019 Published by Elsevier Ltd.

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integrated system of corridors and the urban network with heterogeneous demand of human-driven, autonomous, and communicant autonomous vehicles (HVs, AVs, and CAVs). AVs use a radar-based autopilot technology to independently follow their predecessors at a safe distance (Zohdy et al., 2015; Kockelman et al., 2017). Improvement in perception-reaction time in AVs increases highway capacity by reducing headway, smoothing acceleration/deceleration, and raising the speed of wave propagation in traffic flow (Bose and Ioannou, 2003; Fernandes and Nunes, 2012; Mahmassani, 2016). In highways, the capacity can be further increased by enhancing automation with cooperative control technology. Dynamic control of communicant autonomous vehicles (CAVs) along routes (longitudinal control) and between lanes (lateral control) increases highway capacity by reducing the headway between vehicles moving in platoons (Horowitz and Varaiya, 2000; Rajamani and Shladover, 2001). In a mixed traffic condition, implementing lane dedication strategies can significantly reduce congestion in highways when the penetration rate of AVs and CAVs is high enough. Improvement in the throughputs of interregional corridors (connecting suburban areas to urban regions), however, worsens the traffic condition in the urban network over peaks due to the hypercongestion effect. In fact, the performance of the urban network can be improved by dynamically controlling its inflow at the cost of increasing the delay in the corridors (Geroliminis et al., 2013; Ramezani et al., 2015). The key strategy for optimizing the integrated system is to strike a balance between the delay that users experience in the corridors and delay in their trips in the urban network. In this research, we design an optimal lane-management strategy for corridors having a heterogeneous demand of HVs, AVs, and CAVs. To optimize the network inflow, we dynamically control the size of the CAV platoons in the corridors based on the instantaneous condition of the integrated system. To this end, we first develop a macroscopic traffic model for the integrated system of corridors and the urban network in a monocentric city. The corridors can potentially have three types of lanes: (1) a conventional lane (CVL) accessible to HVs, AVs, and CAVs, (2) a high performance lane (HPL) dedicated to AVs and CAVs, and (3) an automated highway system (AHS) exclusively provided to CAVs. We model the multiple lane corridors as sets of parallel bottlenecks with general distributions of multiclass demand. We also use a stochastic choice model to dynamically assign multiclass demand to the multiple lane types. The effect of hypercongestion on the performance of the network is also captured by a trip-based model for a general shape of the MNFD. To investigate the impacts of a rise in penetration rate of AVs and CAVs on the traffic condition of the integrated system, we derive a closed-form representation of the model for a uniform distribution of the multiclass demand and a triangular MNFD. We show that the increase in the delay in the network from the rise in the penetration rate of AVs and CAVs can have a stable, an unstable, or even a hybrid pattern, depending on the characteristics of the network and demand. To optimize the system, we minimize a weighted summation of the experienced delay in the corridors and the total travel time in the urban network by optimizing the number of lanes of each type and the dynamic size of the CAV platoons in the AHS. We solve this optimization problem for the integrated system comprised of the corridors and downtown network of San Francisco, CA, for different penetration rates of AVs and CAVs. The remainder of the paper is organized as follows: Section 2 presents the integrated system of corridors and the urban network. In Section 3, we develop a dynamic traffic model for the integrated system. Section 4 formulates the traffic optimization problem. Section 5 presents the case study of San Francisco, CA. Lastly, the conclusions of the paper are summarized in Section 6. 2. Integrated system of corridors and the urban network In large metropolitan areas, a considerable portion of the daytime urban population live in suburban areas and commute daily to the urban region through a set of interregional corridors, I , that connect these areas, as illustrated in Fig. 1a. When traffic flow is a mixture of HVs, AVs, and CAVs, these corridors can potentially have three types of lanes, (1) CVL, accessible to HVs, AVs, and CAVs, (2) HPL, dedicated to AVs and CAVs, and (3) AHS, exclusively provided to CAVs, as illustrated in Fig. 1b. In this research, we use a bottleneck model to study the dynamics of congestion in the different lane-types of the corridors i.e., CVL, HPL, and AHS. The effect of hypercongestion on the performance of the urban network is also taken into account by the MNFD model.

Fig. 1. Monocentric urban region with multiple lane type interregional corridors.

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Fig. 2. Increase of the AHS capacity, µi AHS , with rise in the platoon size,

i.

2.1. Corridors The improvement in perception-reaction time in AVs and CAVs increases the capacity of the corridors by reducing the headway between vehicles. In a heterogeneous traffic condition, the vehicle headway varies along the route and also with time of day. The capacity of ig lanes of type g {CVL, HPL} with a free flow speed of vfg in corridor i I , µig , can be equivalently calculated for the weighted average headway as below (see remarks of Daganzo (1997)):

µig =

i

d v + vfg

g g vf

m Mg

Nig, m m g ,m m Mg Ni

, (1)

where d v is the average vehicle length, and m is the safe vehicle headway for class m . Ni is the portion of the demand of vehicle class m Mg in corridor i that chooses lane-type g , and Mg denotes the set of vehicle classes that have access to lane-type g : MCVL = {HV, AV, CAV} and MHPL = {AV, CAV} . In automated highways, the capacity can be further increased by decreasing the headway between vehicles moving in platoons. The flow of CAVs is homogenous in the AHS. So, the AHS capacity can be derived by calculating the average density of platoons of size i vehicles with inter-platoon spacing sp and intra-platoon spacing s v (see remarks of Varaiya (1993)): g,m

µi AHS =

i

dv

i

AHS AHS vf i

+(

i

1) s v + sp

.

(2)

Cooperative traffic control in the AHS enables perimeter control of the network inflow in large urban regions. The AHS capacity is a monotonic function of the platoon size as illustrated in Fig. 2. By adjusting the platoon size, we can dynamically control the AHS throughput and travel time based on the instantaneous condition of the network. Under the equilibrium condition, travel time in the CVL and the HPL also varies with the consequent change in lane choice behavior of the AV and CAV users. Dynamic platoon control enables us to optimize the assignment of the demand and the inflow of the network by adjusting the AHS throughput. In a monocentric city, the improvement in the throughputs of the corridors connecting the suburban areas to the urban network also increases the inflow of the urban network, which can have adverse impacts on the network traffic condition due to the hypercongestion effect. In the next section, we use the MNFD to model the dynamics of the congestion in the urban network of the monocentric city. 2.2. Urban network The relationship between network flow, qr (veh/sec.lane), and density, kr (veh/m.lane) in large urban regions can be represented by the macroscopic network fundamental diagram (MNFD), Qr (kr ) , (Daganzo and Geroliminis, 2008). Research on the observed traffic data from the city of Yokohama network (Geroliminis and Daganzo, 2008) and the results of microsimulations of the traffic flows in the San Francisco (Geroliminis and Daganzo, 2007) and Nairobi (Gonzales et al., 2011) networks show that, in the uncongested state of the network (off peak), a rise in vehicular density increases network flow from zero to its maximum feasible value, as shown in Fig. 3. However, in the congested (hypercongested) state of the network (during the peak), the flow dramatically deceases back to zero, with further rise in the density of the network as the system moves towards a complete gridlock. In large urban regions, the macroscopic relationship between the flow and density of the network can be accurately estimated using the methods proposed in the MNFD literature (Saberi et al., 2014; Leclercq et al., 2014; Loder et al., 2017; Ambühl et al, 2018). The effect of cruising-for-parking on the traffic condition of the network also has been widely studied in the literature (Geroliminis, 2015; Liu and Geroliminis, 2016; Zheng and Geroliminis, 2016). In this research, we assume that cruising-for-parking constitutes a negligible portion of the users’ trip length and adopt a trip-based approach to model the dynamics of the congestion in the urban network. We capture the hypercongestion effect on the performance of the system using the MNFD. In this model, we calculate the travel time of individual users based on their arrival times to the network; the instantaneous network speed is derived as the ratio of the flow to the density at each point, and the flow-density relationship is presented by the MNFD, as explained in Section 3.2. Research shows the performance of urban networks significantly improves when there is no inhomogeneity in the flow of AVs/ 479

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Fig. 3. Macroscopic network fundamental diagram (MNFD): Yokohama (Geroliminis and Daganzo, 2008); San Francisco (Geroliminis and Daganzo, 2007); Nairobi (Gonzales et al., 2011).

CAVs (Ambühl et al, 2016). In a heterogeneous traffic condition, however, enhancement in performance of networks remains relatively minor without upgrading the conventional control system (Mahmassani, 2016). In this case, the effect of heterogeneity of driving technology on the shape of the MNFD is negligible. Hence, the improvement in throughput of the interregional corridors can adversely affect the performance of the urban network by increasing the vehicular density inside the region. 3. Analytical model of the dynamic congestion In this section we develop a macroscopic traffic model for an integrated system of corridors and urban network in a monocentric city. We model the multiple lane type corridor as a system of parallel bottlenecks with a general distribution of multiclass demand. AV and CAV users can choose their lane-type at their arrivals to the corridor. We use a stochastic choice model to dynamically assign multiclass demand to the multiple lane types. The effect of hypercongestion on the performance of the network is also captured by a trip-based model for a general shape of the MNFD. To study the impacts of the rise in the penetration rate of AVs and CAVs on the traffic condition of the integrated system, we derive a closed-form representation of the traffic model for a uniform distribution of multiclass demand and a triangular MNFD, without losing the generality of the analytical results. Dedicating exclusive lanes to AVs/ CAVs can make the traffic condition worse in the corridors when the penetration rate of AVs and CAVs is relatively low. However, the experienced delay in the corridors sharply decreases with the rise in the penetration rate. We also show that the increase of the delay in the network with the rise in the penetration rate of AVs and CAVs can have a stable, an unstable, or a hybrid pattern. 3.1. Dynamics of congestion in the corridors Travel time in lane-type g G , G = {CVL, HPL, AHS} , of corridor i I , ig (t ) , can be expressed as the summation of its free flow travel time, ig, f , and the delay, ig (t ) , that users experience in their commutes depending on their arrival times to the corridor, t : g i (t )

=

g i, f

+

g

i

(t ),

(3)

where the free flow travel time can be expressed as the length of corridor i , li , divided by the free flow speed in lane-type g , g i, f

=

li . vfg

vfg : (4)

Fig. 4. Queueing diagram of the multiple lane types of the corridors. 480

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We also derive the delay in the multiple lane types of the corridors by modeling the dynamics of the congestion in a set of parallel bottlenecks. In the queueing diagram of Fig. 4a, the temporal distribution of demand for lane-type g in corridor i is generally represented by the cumulative counts of arrivals by time, t , Aig (t ) . The throughput of lane-type g in corridor i is also represented by the cumulative counts of departures by time t , Dig (t ) , where the departure rate is bounded from above by the bottleneck capacity, g Di (t ) µig . In this formulation of the problem, the commute delay that the user arriving to the bottleneck at time t experiences can be graphically illustrated as the horizontal distance between the cumulative arrival and departure curves (See Amirgholy and Gonzales, 2017): g

i

(t ) = Dig 1 (Aig (t ))

(5)

t,

where Dig 1 (·) denotes the inverse departure function. Following from the above, the total experienced corridor i , Tig , can be represented as the area enclosed by the arrival and departure curves (shaded area in

Tig =

tig, E g, S

(Aig (t )

ti

Dig (t )) dt .

delay in lane-type g of Fig. 4a): (6)

To split the multiclass demand between the multiple lane types, we consider a stochastic lane choice behavior for the users. The CAV users have the choice of traveling through CVL, HPL, or AHS at their arrivals to the corridors while this choice is restricted to CVL or HPL for the AV users and CVL for the HV users. By calculating the choice probabilities using a multinomial logit model, we assign the demand of each class to its accessible lane-types under a stochastic equilibrium condition. The cumulative arrival of vehicle class m to lane-type g by time t in corridor i , Aig, m (t ) , is then derived by splitting the temporal distribution of the demand of class m in corridor i , aim (·) , between the accessible lane-types based on the instantaneous travel time in these lanes, ig (·) : t

A ig , m ( t ) =

m g i ( )

e

aim ( )

tig , S

g Gm

e

m g i ( )

d ,

(7)

where Gm denotes the set of accessible lane-types to vehicle class m (GHV = {CVL}, GAV = {CVL, HPL}, GCAV = {CVL, HPL, AHS} ), and m is the constant parameter of the model that reflects the lane-type choice preferences of the users in class m , which can be estimated through calibration to reflect the aggregated preferences of the users in each class (Amirgholy et al., 2015, 2017). 3.1.1. Closed-form representation: lane type choice under equilibrium To study of the effects of the rise in the penetration rate of AVs and CAVs on the traffic condition in the corridors, we formulate the delay the in the multiple late types under dynamic user equilibrium. We assume that users choose their lanes at their arrivals to the corridors to minimize the travel time of their commutes to the urban region. The cumulative result of the individual lane-type decisions of the multiclass users over time leads to a dynamic user equilibrium condition in which no user can reduce his/her travel time by changing his/her choice of lane-type. The equilibrium condition can be generally presented under two cases in corridor i I , as described below. Case (I): User Equilibrium with Equal Travel Times User equilibrium holds when the multiclass demand splits between two or more of the lane-types such that the travel time through these lane-types remains exactly equal over time. In this case, user equilibrium corresponds to a split of the multiclass demand proportional to the capacity of these lane-types where the free flow speed is identical for g GiET (See Appendix A):

µi g

A ig ( t ) =

g GiET

µig

m

¯ g GiET Mg

qim (t ),

GiET ,

g

(8)

where qim (t )

is the cumulative distribution of the demand of vehicle class m and Ai (t ) is the cumulative arrivals by time t to lane-type ¯ g Mg is the set of classes that certainly g of corridor i . GiET denotes the set of lane-types with an equal travel time in corridor i , and M ¯ CVL = {HV} , M ¯ HPL = {AV} , and M ¯ AHS = {CAV} . use lane-type g if g GiET : M In this case, the total delay and the delay that individual users experience in their commutes through different lane-types can be derived by substituting Aig (t ) from (8) into (6) and (5), respectively. When the multiclass demand has a uniform distribution over time, Eqs. (5) and (6) can be simplified as below (See Fig. 4b): g

i

(t ) =

Tig = where

g

tiS

µig

+

m

2

m i denotes

m

¯ g GiET Mg g GiET

¯ g GiET Mg g GiET

µig

m i

m i (t

tiS ) t,

µig m

¯ g GiET Mg g GiET

µig

GiET ,

g m i

1 (tiE

(9)

tiS )2 ,

g

GiET ,

tiS

tiE :

(10)

the fixed demand rate of class m in corridor i from

Case (II): User Equilibrium with Unequal Travel Times 481

to

qim (t )

=

m i (t

tiS ) .

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Fig. 5. Variations of the total delay in a corridor, Ti , with the AV/CAV penetration rate,

m i .

User equilibrium may also emerge as unequal travel times in some or all the lane-types. In this case, the lane-type with the lowest travel time gets the demand of the classes that have access to it while the lane-types with higher travel times remain unused by those classes:

A ig ( t ) =

qim (t ),

¯ i, g m M

GiUT ,

g

(11)

denotes the set of used lane-types with unequal travel times in corridor i , and M¯ i, g

Mg is the set of classes that use lanewhere ¯ i, g in type g in corridor i , i.e., lane-type g has the lowest travel time among the lane-types accessible to each and every class m M ¯ Mg . lane-type i : Mi, g = Mg ET UT

GiUT

g

(Gi

Gi

) M

g

Mg

The individual user delay and the total delay in different lane-types can be then derived by plugging Aig (t ) from (11) into (5) and (6), respectively. For the case of uniformly distributed demand, Eqs. (5) and (6) can be simplified as below (See Fig. 4b): g

i

(t ) = tiS +

Tig =

1 2

¯ i, g m M

m i (t

tiS )

t,

µig m i

¯ i, g m M

¯ i, g m M

µi

g

m i

g

1 (tiE

GiUT ,

tiS ) 2,

(12)

g

GiUT .

(13)

3.1.2. Corridor system performance Dedicating exclusive lanes to AVs/CAVs can make the traffic condition worse in the corridors when the penetration rate of AVs and CAVs is relatively low. However, the performance of the corridor significantly improves as the AV/CAV penetration rate increases. Fig. 5a illustrates the decrease of the total delay in corridor i , Ti , with the rise in the penetration rate of class m {AV, CAV} , m i ,

{AV, CAV} and im = 0 . When im = 0 , the delay is at its upper bound, TiU , since the entire demand of the where m m corridor commutes through the CVL. The delay starts decreasing with the rise in im since the users of class m switch to the lane-type that has a lower travel time under the equilibrium in Case I. With further rise in im beyond im ,12 , travel time becomes equal in different lane-types in some periods of time during the peak where Case II starts emerging (if the demand is high enough), while Case I still holds in the rest of the peak period in the transition phase from Case I to Case II.1 The decrease of the delay continues with the L rise in the penetration rate beyond im ,1 as Case I expands over the entire peak. The total delay finally reaches its lower bound (Ti ) m when i = 1. In the case that the demand has a uniform distribution over time, the transition from Case I to Case II occurs inm m stantaneously with the rise in the penetration rate beyond im ,1 = i,12 , as shown in Fig. 5b. In this case, i,1 becomes equal to the ratio of ¯ g , to the total capacity of corridor i : the capacity of lane-type g , m M m i,1

=

µi g g GiET

µig

.

(14)

The upper and lower bounds of the delay then can be derived for the extreme cases of 0 and 1 penetration rates using relations (13) and (10), respectively, as presented in Fig. 5b. 1

The transition from Case I to Case II largely depends on the distribution of the multiclass demand and the capacity of different lane types. 482

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Fig. 6. Trip-based MNFD model.

3.2. Dynamics of congestion in the urban network The travel time that users experience in their trips inside the urban region can be dynamically approximated based on the macroscopic traffic condition of the network over time. In this research, we use the MNFD model to calculate the travel time of individual users based on their arrival times to the network. In this model, the instantaneous speed of the network is derived as the ratio of the flow to the density at each point in time:

qr (t )

vr (t ) =

k r (t )

,

(15)

where the network flow is a function of the vehicular density as illustrated in Fig. 6a: qr (t ) = Qr (k (t )) . The density is also derived as the ratio of the vehicular accumulation (number of circulating vehicles inside the region) to the total lane-length of the urban network, Lr . By expressing the instantaneous accumulation of the system as the difference between the cumulative arrivals (counts of vehicles that enter or start their trips inside the region), Ar (t ) and departures (counts of vehicles that exit or park inside the region), Dr (t ), at each point in time, t , the variations in the density of the network over the peak can be captured as below:

k r (t ) =

Ar (t )

Dr (t ) Lr

.

(16)

In the queueing diagram of Fig. 6b, the accumulation of the network at each point in time can be represented graphically as the vertical distance between the arrival and departure curves. The horizontal distance between theses curves at each point also illustrates the duration of the individual trips inside the region, which is the solution of the following integral equation for r (t ) as explained in Fosgerau (2015): t + r (t )

l r (t ) =

t

vr ( ) d .

(17)

Here lr (t) is the trip length and r (t ) denotes the inside-the-region trip duration for the user who arrives to the network at time t . In the integrated system of corridors and the urban network, the cumulative arrivals to the network, Ar (t ) , is equal to the summation of the cumulative departures from different lane types of the corridors by time t , Dig (t ) , and the cumulative counts of the trips inside the network by time t , Air (t ) : Ar (t ) = i I g G Dig (t ) + Air (t ) . The cumulative departures from the network, Dr (t ) , is also determined by trip-based MNFD model (17). The total duration of the trips in the network, r , is then presented as an integral of

the difference between the cumulative arrival and departure curves over the study period, t S to t E , illustrated as the shaded area surrounded by the arrival and departure curves in the queueing diagram of Fig. 6b: r

=

tE tS

(

i I

g G

Dig (t ) + Air (t )

)

Dr (t ) dt .

(18)

The trip-based MNFD model accounts for variations in network density within the life of individual trips of heterogeneous length; the departure time of users can be dynamically approximated based on their arrival times to the network using the “numerical resolution” method presented in Mariotte et al. (2017). Eq. (17), however, does not have a straightforward closed-form solution. To derive a closed-form representation for the traffic model, we adopt the concept of the macroscopic network exit function (MNEF). Under the assumption that network inflow varies slowly over time in comparison with the system relaxation time2, the network outflow can be expressed as a function of the instantaneous accumulation over time using the MNEF (accumulation-based MNEF model). A comprehensive analysis of the accuracy of the trip- and accumulation-based models is presented in Mariotte et al. (2017). 2

The time it takes the system to reach a steady state after a change occurs in the density. 483

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Fig. 7. Accumulation-based MNEF model.

3.2.1. Closed-form representation: accumulation-based MNEF model To shed light on the adverse impacts of the rise in the penetration rate of AVs and CAVs on the performance of the network, we derive a closed-form representation of the traffic model for a linear MNEF and a uniform distribution of endogenous demand ( Air (t ) = ir for t S = trS t trE = t E ; zero otherwise). We use the analytical results to show that the delay in the urban network of a monocentric city is an increasing function of the AV/CAV penetration rate when the network conventional control system is not upgraded. Note that the generalized version of the problem does not have a straightforward closed-form solution. The change in the distribution of the demand and the shape of the MNFD affects the variation of the delay with a rise in the AV/CAV penetration rate; however, the variation pattern remains generally increasing. The MNEF is a rescale of the macroscopic relationship between the flow and density of the network, presented by the MNFD, to express the network outflow as a function the vehicular accumulation (Daganzo, 2007). Hence, the dynamics of the congestion over the peak can be modeled just using the declining part of the MNEF. The network outflow, µr (t ) = (Lr lr ) qr (t ), can be then formulated as a function of the adjusted accumulation, nr (t ) = Lr (kr (t ) kr ) , by horizontally shifting the vertical axis to the turning point of the MNEF (See Amirgholy and Gao, 2017). In the case of the linear MNEF of Fig. 7a, the network outflow-accumulation relationship can be expressed as below:

µr (t ) = µ r 1

n r (t ) , nr , j

(19)

where µr and nr , j denote the capacity (maximum feasible outflow) and the adjusted jam accumulation, respectively, of the network. The queueing diagram of Fig. 7b illustrates the dynamics of the congestion in a network with a linear MNEF. Here, the arrival curve is a piecewise linear function with slope r = i I g G µi g + ir > µr in t S t t E and zero elsewhere, while the slope of the departure curve at each point in time equals the network instantaneous outflow, determined by the MNEF model (19). By excluding the constant free flow travel time in the network from the calculations, the delay that individual users experience can be graphically represented as the horizontal distance between the arrival and departure curves at each point, while the vertical distance between the arrival and departure curves represents the instantaneous (adjusted) accumulation of the system, nr (t ) = Ar (t ) Dr (t ) . We now describe the performance of the network by splitting the peak into two congestion regimes – deteriorating regime and recovering regime. (1) Deteriorating Regime Performance of the urban network declines with a rise in the accumulation of the system (hypercongestion effect) from t S to t E in which µr (t ) < r , as shown in Fig. 7b. In this case, the dynamics of the congestion in the network can be described by the ordinary differential equation (ODE) resulting from setting µr (t ) from (19) equal to the network outflow at time t , Dr (t ) , and substituting nr (t ) = r t Dr (t ) :

Dr (t )

mr Dr (t ) + mr

rt

(20)

µr = 0,

where mr = µr nr , j . The cumulative departures from the network, Dr (t ) , is derived by solving ODE (20) under initial condition Dr (0) = 0 :

Dr (t ) =

(

r

µ r )(1 mr

e mr t )

+

r t.

(21)

The total delay in Deteriorating Regime, is then calculated as the integral of the difference between the cumulative arrival and departure curves from time t S to t E , graphically presented as the area enclosed by the arrival and departure curves in the queueing diagram of Fig. 7b:

TrSE ,

484

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TrSE =

(

µ r )(e mr t

r

E

mr t E

1)

mr2

.

(22)

(2) Recovering Regime The congestion built up in time span t S to t E gradually diminishes as the performance of the network improves with the decline in

accumulation of the system from t E to t E in which µr (t ) > Ar (t ) = 0 . To simplify the presentation of the formulation, we make use of an auxiliary coordinate system in the queuing diagram of Fig. 7b with the origin at E , t E , No , and both axes in opposite directions

~ t = tE t and N = No N ).3 Then, we formulate the dynamic congestion model as an ODE by setting from the original system (~ ~ ~ ~ t ): µr (t ) from (19) equal to Dr ( t ) , and substituting nr (t ) = Dr ( ~ ~ ~ Dr (~ t ) + mr Dr (~ t ) µr = 0, (23) ~ ~ t in the auxiliary system, and is derived by solving ODE where Dr ( t ) denotes the cumulative departures from the network by time ~ ~ (23) under initial condition Dr (0) = 0 :

µ (1 ~ Dr (~ t) = r

mr ~ t)

e mr

.

(24)

The total delay in the recovering regime, TrEE , equals the integral of the difference between the cumulative arrival and departure ~ curves from time t E to t E . Equivalently, the total delay can be calculated as the shaded area under Dr (t ) from time zero to ~ E E E E t =t t in the auxiliary coordinate system. Here, t is determined by solving the equation that implies differentiability of the

~ departure curve at the transition point between the congestion regimes, Dr (t E ) = Dr t E : µr

ln

r + (µ r

tE = tE +

m tE r )e r

mr

.

(25)

t E in the Having t E , the total delay in the recovering regime,TrEE , is calculated as the integral of the departure curve from zero to ~ auxiliary coordinate system:

TrEE =

(

r

ln

E

emr t )

µ r )(1

mr2

+

µr r + (µ r

m tE r )e r

mr2

µr .

(26)

Remark (gridlock prevention condition). The hypercongestion diminishes in the recovering regime, i.e., t E < and TrEE < , only if the network jam accumulation, nr , j , is large enough such that the deteriorating regime does not end to a complete gridlock, Dr (t E ) > 0 . The resulting condition should hold to prevent a complete gridlock over the peak:

nr , j >

µr No r ln

(

r

µr

r

)

. (27)

3.2.2. Network performance In a monocentric city, the network inflow increases as the corridor throughputs improve with the rise in the penetration rate of AVs and CAVs. The enhancement in performance of the network, however, remains relatively minor without upgrading the conventional control system (Mahmassani, 2016). In this section, we demonstrate the impacts of the rise in the AV/CAV penetration rate on the traffic condition of the urban network. In a network with a fixed demand of No , the total delay, Tr = TrSE + TrEE , is an increasing function of the network inflow, Ar (t ), since the first order (partial) derivative of the delay with respect to the network inflow always has a positive value, as shown for the closed-form representation of the model:

Tr r

µr

=

mr2

(

1+ r

No mr r

µ r )e

No mr r

e

No mr r

> 0,

(28)

r

since the numerator and denominator of (28) are both negative under the gridlock prevention condition (27), 3

~ The definition of the slope does not change in the auxiliary coordinate system due to inversion of both axes: Dr (t ) = Dr ( t ) . 485

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Fig. 8. Variations of the network delay, Tr , with the AV/CAV penetration rate,

m i .

µr )) No . The network inflow, Ar (t ) = r , is also an increasing function of the AV/CAV penetration rate: r i I , m {AV, CAV} . So, the total delay in the network must also be an increasing function of the AV/CAV penetration rate according to the differentiation chain rule, Tr im 0 . The resulting decline in performance of the network with the rise in the accumulation of the system may or may not eventually lead to complete gridlock. In fact, the increase of the delay in the network with the rise in the AV/CAV penetration rate can have an unstable, a stable, or even a hybrid pattern, depending on the characteristics of the network and demand.

mr < r

r ln( r m 0 i

(

(a) Unstable Pattern A rise in the penetration rate of the AVs and CAVs can move the system towards a complete gridlock when the network inflow is already high enough. In this case, the total network delay is an increasing convex function of the AV/CAV penetration rate as illustrated in Fig. 8a:

Tr

>0

m i

and

2T r ( im ) 2

0,

0

m i

<

m i, crt ,

(29)

m in corridor i I at which a complete gridlock occurs in the network. im where im , crt denotes the critical penetration rate of class , crt can be derived by solving the equation resulting from setting the slope of the departure curve at the end of the deteriorating regime E equal to zero: Dr (t ) = 0 . (b) Stable Pattern In this pattern, the network traffic condition continues worsening, but slows down with the rise in the penetration rate of the AVs and CAVs when network inflow is relatively low. In this case, the total network delay is still an increasing, but concave function of the AV/CAV penetration rate, as illustrated in Fig. 8b:

Tr

>0

m i

and

(

2T r m 2 i )

0,

0

m i

1.

(30)

(c) Hybrid Pattern Variation of the total experienced delay in the urban network with a rise in the penetration rate of the AVs and CAVs can have a hybrid pattern starting with a stable (unstable) pattern in the first part, but continuing as unstable (stable) in the second part with further rise in the penetration rate of the AVs and CAVs, as illustrated in Fig. 8c. In this case, the total network delay is an increasing quasilinear4 function of the AV/CAV penetration rate, one with an inflection point at the transition between the stable and unstable patterns: Tr

m i

Tr

>0

m i

where, 4

m i, inf

>0

and and

2T r ( im )2

< 0,

2T r ( im )2

> 0,

m i

0 0

<

m i

<

m i, inf m i, inf

and

2T r ( im )2 2T r ( im )2

and

denotes the penetration rate of class m in corridor i

A function that is both quasiconvex and quasiconcave is quasilinear. 486

0, 0,

m i, inf m i, inf

m i

<

m i

m i, crt

<1

I at the inflection point.

Stable

Unstable

Unstable

Stable

(31)

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4. Lane management and cooperative traffic control Lane dedication and highway automation strategies can reduce congestion in corridors when demand is a mixture of HVs, AVs, and CAVs. This improvement in the corridors, however, worsens the traffic condition in the urban network. In this section, we + , in each optimize the number of lanes of each type g G , ig and the dynamic size of the CAV platoons in AHS, i (t ) corridor i I to control the inflow of the network based on the instantaneous condition of the integrated system. The design problem is formulated as a mathematical program that minimizes a weighted summation of the experienced delay in the corridors and the total travel time in the urban network, and the total number of non-conventional lanes:

(

min z= g i , i (t )

i I

i

g G

g

Tig +

r

r

)+

i I

g {HPL, AHS }

i

g

,

(32)

subject to CVL i

i

AHS

i (t )

+

i

HPL

i

=

AHS, U

U,

i,

,

i i

i

(33)

I,

(34)

I,

(35)

I,

where i denotes the weight of the total experienced delay in lane-type g of corridor i , and r is the weight of the total travel time in the network in the objective function. is also a nominal penalty weight considered for lane dedication and highway automation in the corridors to keep the total number of non-conventional lanes minimized. Here, the total number of lanes to split between CVL and HPL is fixed, and the optimal number of AHS lanes is limited to an upper bound, i AHS, U , in each corridor, as expressed by constraints (33) and (34), respectively. The size of the CAV platoons also has an upper bound, U , imposed by safety criteria, as formulated in constraint (35). The total travel time in the network ( r ) is calculated for a general distribution of the inflow and a general shape of MNFD using Eq. (18). The experienced delay in different lane-types of the corridor (Tig ) is also derived using a bottleneck model (6) for a general distribution of the arrivals where the capacity of CVLs, HPLs, and AHSs are calculated using (1) and (2). The multiclass demand is split between the multiple lane types based on the instantaneous travel time in these lanes using a dynamic logit model (7). In this optimization problem, the objective function (32) is nonlinear and nonconvex. In Section 5, we discretize time and dynamically solve the design problem for the integrated system of corridors and the downtown network of San Francisco, CA, using GAMS/Baron, a global solver for handling general optimization problems (see remarks of Tawarmalani and Sahinidis (2004)). g

5. Case Study: San Francisco The city of San Francisco, CA, is located on the tip of a peninsula bounded by the Pacific Ocean to the west and San Francisco Bay to the East and North, as depicted in Fig. 9a. Access to the city is provided through four interregional corridors: (1) Golden Gate Bridge from the northeast, (2) San Francisco – Oakland Bay Bridge from the east, (3) US-101 from the southeast, and (4) I-280 from the south. A significant portion of the commuters to the city through these corridors have a destination in downtown San Francisco (Financial District and South of Market Area), which has an area of 5 (km2) and a grid arterial network with a total lane-length of Lr = 76.2 (lane.km). The dynamics of the congestion in downtown network is modeled using the MNFD we fit to the results of the traffic simulation adopted from Geroliminis and Daganzo (2007), as plotted in Fig. 9b.

Fig. 9. San Francisco Network.

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Fig. 10. Queuing diagrams of the interregional corridors and the downtown network.

The queueing diagrams of Fig. 10a–e illustrate the dynamics of the traffic in the corridors and the downtown network5, in the absence of AVs and CAVs, over the morning period on an average Wednesday in mid Fall, 2016.6 To investigate the impacts of the proposed lane dedication and traffic automation strategies on the performance of the integrated system, we plot the variations of the experienced delay in the corridors, with one dedicated HPL and one additional AHS lane, and the total travel time in the downtown network with the rise in the penetration rate of AVs and CAVs, in Fig. 11a–e. In this example, the demand of different classes of vehicles has an identical distribution over time in each corridor. Characteristics of the different vehicle classes, lane-types, and corridors are also described by the parameters listed in Tables 1–4. Dedicating exclusive lanes to AVs/CAVs makes the traffic condition worse in the corridors when the penetration rate of AVs and CAVs is low. However, the experienced delay in the corridors sharply decreases with the rise in the penetration rate as plotted in 5 The Downtown network does not become hypercongested in the base case scenario for an average trip length lr = 1.74 (km) and with no new technologies on the roads yet, as can be inferred by comparing the queueing diagrams of Figs. 10e and 6b. 6 The traffic count data in the nine consecutive Wednesdays of October and November 2016 is provided by the San Francisco County Transportation Authority (SFCTA). The outflow of the corridors to the downtown network and the through-the-network inflow are estimated based on the origin-destination demand matrices of the Bay Area.

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Fig. 11. Variation of the delay in the corridors, Ti , and the travel time in the downtown network, m , m {AV, CAV} . i

r,

with the penetration rate of AVs and CAVs,

Table 1 Number of available lanes and potential AHS lanes. Corridor i i

AHS, U

Golden Gate

Bay Bridge

US-101

I-280

3 2

5 4

4 3

3 2

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Table 2 Free flow speed in different lane-types. Lane-type

CVL

HPL

AHS

v fg (km/hr)

80

100

120

Table 3 Reaction time and lane-type choice model parameters. Vehicle Class m (sec) m

HV

AV

CAV

2.0 —

1.5 0.8

1.5 0.8

Table 4 Parameters of the platoon control system in the AHS. Parameters

dv (m)

sp (m)

sv (m)

AHS

5

150

5

U

7

Table 5 Optimal lane management strategies for different penetration rates of AVs and CAVs. Design (

i

AV

,

CAV ) i

Corridor

CVL i

Delay (1000×min) HPL i

i

AHS

TiCVL

TiHPL

TiAHS

(0,0)

Golden Gate Bay Bridge US-101 I-280

3 5 4 3

0 0 0 0

0 0 0 0

648.00 3699.72 1467.19 788.00

N/A N/A N/A N/A

N/A N/A N/A N/A

(20,15)

Golden Gate Bay Bridge US-101 I-280

3 5 4 3

0 0 0 0

0 1 0 0

498.88 2634.08 1173.99 663.78

N/A N/A N/A N/A

N/A 0.00 N/A N/A

(40,30)

Golden Gate Bay Bridge US-101 I-280

2 4 3 2

1 1 1 1

0 1 0 0

336.43 710.67 879.63 433.42

7.08 38.15 33.72 9.51

N/A 17.35 N/A NA

(60,30)

Golden Gate Bay Bridge US-101 I-280

2 4 3 2

1 1 1 1

0 1 0 0

269.04 655.79 798.65 375.62

20.80 63.26 53.77 22.81

N/A 29.85 N/A N/A

Fig. 11a–d.7 The numerical results presented in Fig. 11e also show that the increase of the travel time in the network of downtown San Francisco with the rise in the penetration rate of AVs and CAVs, m = im i I , follows a hybrid pattern in the case study presented in this paper. To optimize the system, we solve design problem (32)–(35) for different penetration rates of the new technologies, where ig = 1 i, g , = 0.01. The design problem falls within the category of nonlinear optimization problems that can be solved numerically r = 2 , and using the GAMS/BARON platform. Here, we solve the numerical problem by setting the optimality gap for Baron to 1% and the time limit for the solver to 3600 sec . With this presetting, the average and maximum computational times for solving the design problem are 2.8 sec and 105.19 sec , respectively. Table. 5 compares the optimal design of the HPL and AHS facilities in the corridors for different penetration rates of AVs and CAVs. The results indicate that when the penetration rate of the AVs and CAVs is relatively low, dedicating exclusive lanes to AVs/CAVs may worsen the traffic condition in the corridors. So, in the first scenario, with the AV penetration rate of 20% and CAV penetration rate of 15%, it is not optimal to have HPLs in any of the corridors, while it is just as efficient to provide one AHS lane for the CAVs on the Bay Bridge, which has the worst traffic condition among the corridors. However, with the increasing use of the new technologies, it becomes beneficial to dedicate one lane in each corridor to AVs/CAVs, and keep the AHS lane on the Bay Bridge.

7 Note that substituting the user equilibrium with the dynamic stochastics model has no substantial effect on variation pattern of the delay in the corridors and network.

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Fig. 12. Dynamic control of the CAV platoon size in the AHS of Bay Bridge,

2 (t ) .

The results presented in Table. 5 also show that the rise in the penetration rate of the AVs and CAVs can reduce the delay in the corridors when the system is optimally designed, whereas the drop in the experienced delay in the conventional lanes, i.e., CVL, is always larger than the summation of the rise of delay in the non-conventional lanes, i.e., in HPLs and AHSs, in this case study. Note the delay can be further reduced in the corridors by dedicating/providing multiple non-conventional lanes to the vehicles with new technologies. However, further improvement of the mobility in the corridors upsets the optimal balance between the experienced delay in the corridors and the experienced delay in the network that minimizes the total cost of the integrated system. To optimize the inflow of the network over the peak, we dynamically optimize the size of the CAV platoons along the corridor based on the instantaneous traffic condition of the integrated system, as illustrated in Fig. 12. It is worth pointing out here that, in the last two scenarios with higher-penetration-rate AVs and CAVs, we deliberately allow an optimal amount of delay in the AHS by limiting the average size of the CAV platoons to 4 vehicles at most in order to optimize the assignment of AVs and CAVS to the CVL, HPL, and AHS, and consequently the total inflow of the corridor under a stochastic user equilibrium condition. In other words, we improve the overall traffic condition of the system by creating optimal amounts of congestion in the corridors. Fig. 13a-c plot the variations of the components of the objective function (32) with their relative weights, r ig for different penetration rates of AVs and CAVs. Note that these are conflicting objectives that cannot be improved simultaneously; the optimal

Fig. 13. Variations of the components of the objective function with their relative weights,

491

r

g i .

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Fig. 14. Variations of the optimal delay in the corridors, Ti , with the AV/CAV penetration rate,

m , i

m

{AV, CAV} .

travel time in the downtown network decreases as its relative weight increases in the objective function, while the optimal delay in the corridors increases with the corresponding decline in its relative weight. To study the effects of the growth in market share of AVs and CAVs on the optimal performance of the system, we plot the variations of experienced delay in the corridors and the downtown network with the rise in the penetration rate of AVs and CAVs in Fig. 14a–d and Fig. 15a. The results show that low penetration rates of AVs and CAVs have no significant effect on the performance of

Fig. 15. Variations of the optimal travel time in the downtown network, penetration rate, m = im , m {AV, CAV} .

r

492

, and the total delay cost of the integrated system, T , with the AV/CAV

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the system. However, implementing optimal lane management strategies significantly reduces the experienced delay in the corridors as the penetration rate increases: by 55% for the Golden Gate Bridge, by 84% for the Bay Bridge, by 92% for US-101, and by 49% for I-280. By dynamically controlling the size of the CAV platoons in the Bay Bridge AHS, we limit the increase of travel time in the downtown network to 5%. As graphically illustrated in Fig. 15b, the proposed lane management and traffic control strategies can reduce the total delay cost of the integrated system, T , by 66% as the penetration rate of AVs and CAVs increases. Further improvement in the traffic condition of the integrated system, however, requires enhancing the performance of the urban network by upgrading conventional control systems. 6. Conclusion Dedicating exclusive lanes to AVs and CAVs can significantly reduce congestion by increasing the throughput of corridors. The resulting increase in the network inflow, however, worsens the traffic condition in the urban network over the peaks. In this research, we show that the key to optimizing the integrated system is to strike a balance between the delay that users experience in the corridors and the delay in their trips in the network. We then design an optimal lane management strategy for the corridors with a heterogeneous demand of HVs, AVs, and CAVs. To optimize the network inflow, we dynamically control the size of the CAV platoons in the AHS based on the instantaneous condition of the integrated system. We solve the traffic optimization problem for the integrated system of the corridors and downtown network of San Francisco, CA. The results show that low penetration rates of AVs and CAVs have no significant effect on the performance of the system, and dedicating exclusive lanes to AVs/CAVs in this case can just make the traffic condition worse in the corridors. However, as the penetration rate increases, implementing optimal lane management strategies significantly reduces the experienced delay in the corridors: by 55% for the Golden Gate Bridge, by 84% for the Bay Bridge, by 92% for US-101, and by 49% for I-280. Dynamic control of the CAV platoons in the AHS of Bay Bridge also limits the increase in travel time in the downtown network to 5%. To further improve the traffic condition of the integrated system, we need to improve the performance of the urban network by upgrading conventional control systems, as we explain in Amirgholy et al. (2019). Acknowledgements This work was supported in part by the United States Department of Transportation (USDOT) Center for Transportation, Environment, and Community Health (CTECH), the National Science Foundation [project CMMI-1462289], and the Lloyd’s Register Foundation, UK. The authors are grateful to the two anonymous reviewers for their valuable comments and also to Dan Tischler and the San Francisco County Transportation Authority, Eric J. Gonzales, and Mohsen Ramezani, for providing essential data. Appendix A. . User equilibrium with equal travel times

g

The equilibrium assignment of the multiclass demand to the multiple lane types of corridor i GiET : g i (t )

=

i (t ),

g

I results in equal travel times in (A.1)

GiET .

The travel time through lane-type g in corridor i at time t , ig (t ) , can be expressed as the summation of the free flow travel time and the delay that users experience in their commutes as formulated in Eq. (3). For the sake of simplicity of formulation, we consider that g,S the peak starting time is identical in g GiET , tiS = ti . By setting the origin of the coordinate system of the queueing diagram at point S, as shown in Fig. A1, Eq. (3) can be further simplified:

Fig. A1. Queueing diagram of lane-type g of corridor i over the peak. 493

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g i (t )

=

g i, f

+

Aig (t ) µig

t.

(A.2)

By substituting from (A.2) in (A.1), the equilibrium condition for the lane-types with equal travel times in corridor i can be generally formulated as follows: g i (t )

g i, f

µig + Aig (t ) µig

=

i (t ),

g

GiET .

(A.3)

In the case that the free flow speed is identical in the multiple lane types, the share taken by lane-type g of corridor i can be derived by simplifying Eq. (A.3):

A ig ( t ) =

µi g g GiET

µig

m

¯ g GiET Mg

qim (t ),

g

GiET

from the demand

GiET ,

(A.4)

where qim (t )

¯g denotes the cumulative distribution of the demand of vehicle class m by time t in lane-type g of corridor i , and M ¯ CVL = {HV} , M ¯ HPL = {AV} , and M ¯ AHS = {CAV} . the set of classes that certainly use lane-type g if g GiET : M

Mg is

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