toll lanes in transportation networks

Transportation Research Part C 106 (2019) 381–403

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

Strategic planning of dedicated autonomous vehicle lanes and autonomous vehicle/toll lanes in transportation networks Zhaocai Liu, Ziqi Song

T

⁎

Department of Civil and Environmental Engineering, Utah State University, Logan, UT 84322-4110, United States

ARTICLE INFO

ABSTRACT

Keywords: Autonomous vehicles Mixed traffic Dedicated autonomous vehicles lanes Autonomous vehicle/toll lanes Deployment planning Robust optimization

Employing vehicle communication and automated control technologies, autonomous vehicles (AVs) can safely drive closer together than human-driven vehicles (HVs), thereby potentially improving traffic efficiency. Separation between AV and HV traffic through the deployment of dedicated AV lanes is foreseen as an effective method of amplifying the benefits of AVs and promoting their adoption. However, it is important to consider mixed AV and HV traffic in a transportation network. On the one hand, it may be impractical to deploy dedicated AV lanes throughout the network, while on the other hand, dedicated AV lanes may even reduce the total traffic efficiency of a road segment when the AV flow rate is low. In this study, we considered a new form of managed lanes for AVs, designated as autonomous vehicle/toll (AVT) lanes, which grant free access to AVs while allowing HVs to access the lanes by paying a toll. We investigated the optimal deployment of dedicated AV lanes and AVT lanes in transportation networks with mixed AV and HV flows. The user equilibrium (UE) problem in a transportation network with mixed flows of AVs and HVs is first explored. We formulated the UE problem as a link-based variational inequality (VI) and identified that, with different impacts of AVs on road capacity, the UE problem can have unique or non-unique flow patterns. Considering that the UE problem may have non-unique flow distributions, we proposed a robust optimal deployment model, which is a generalized semi-infinite min-max program, to deploy the dedicated AV lanes and AVT lanes so that the system performance under the worst-case flow distributions is optimized. We proposed effective solution algorithms to solve these models and presented numerical studies to demonstrate the models and the solution algorithms. The results show that the system performance can be significantly improved through the deployment of dedicated AV lanes and AVT lanes.

1. Introduction Autonomous vehicle (AV) technologies have gained unprecedented interest from government, industry, and academia in recent years due to their potential benefits in the areas of traffic safety and efficiency (Shladover et al., 2012; Bierstedt et al., 2014; Milanés et al., 2014; Fagnant and Kockelman, 2015; Levin and Boyles, 2016a,b; Chen et al., 2016, Chen et al., 2017a,b). Google started its self-driving car project in 2009, and since then Google’s AV fleet has been tested across multiple locations in the United States. By October 2018, the fleet had self-driven more than 10 million miles on public roads (Waymo, 2019). Baidu, a Chinese multinational Internet and technology company, began developing autonomous driving in 2014 and successfully road tested its AVs in China in 2015 (Autonomous Driving Unit, 2018). In partnership with Lyft, nuTonomy, a software company, launched the nation’s first self-

⁎

Corresponding author. E-mail address: [email protected] (Z. Song).

https://doi.org/10.1016/j.trc.2019.07.022 Received 9 March 2019; Received in revised form 23 July 2019; Accepted 29 July 2019 0968-090X/ © 2019 Elsevier Ltd. All rights reserved.

Transportation Research Part C 106 (2019) 381–403

Z. Liu and Z. Song

driving ridesharing service in Boston in December 2017 (nuTonomy, 2019). Many automakers, such as Audi, Toyota, Ford, and Volvo, have been developing and testing their prototype AVs (Miller, 2018). In the United States, public road testing of AVs has been legalized in many states, including Nevada, Florida, California, Michigan, and Washington D.C. (Slone, 2016). In Australia, the National Transport Commission (NTC) released a discussion paper that outlines legislative reform options to clarify how current driver and driving laws apply to AVs, and to establish legal obligations for automated driving system entities (NTC, 2017). In China, the City of Beijing issued regulations on road testing of AVs in December 2017 (Schaub and Zhao, 2018). These recent developments suggest that AVs are indeed on the horizon. Employing vehicle communication and automated control technologies, AVs can improve road traffic capacity by reducing time headways between consecutive vehicles. A significant number of studies have conducted traffic capacity analyses for roads involving AVs, using either computer simulation or analytical modeling. Results from both simulation (e.g., Shladover et al., 2012; Ntousakis et al., 2015) and analytical modeling analyses (e.g., Levin and Boyles, 2015; van den Berg and Verhoef, 2016) demonstrate that road traffic capacity increases significantly with an increase in the proportion of AVs on the road. The impact of AVs on road traffic capacity will inevitably influence traffic flow distribution in a transportation network with AVs. Deploying dedicated lanes for AVs is foreseen as an effective way to amplify the capacity-improvement benefit from AVs and boost the market penetration of AVs (Chen et al., 2016, 2019; Ghiasi et al., 2017; Lamotte et al., 2017; Talebpour et al., 2017; Lu et al., 2019). However, dedicated AV lanes may be underutilized and may even compromise the overall network performance when AV flows are relatively low (Chen et al., 2017a,b; NASEM, 2018; Ye and Yamamoto, 2018). We thus consider a new form of managed lanes for AVs, designated as autonomous-vehicle/toll (AVT) lanes, which are freely accessible to AVs while allowing HVs to utilize the lanes by paying a toll. The idea of AVT lanes is derived from high-occupancy vehicle (HOV) lanes and high-occupancy/toll (HOT) lanes (Fielding and Klein, 1993; Dahlgren, 2002; Song et al., 2015). The joint use of dedicated AV lanes and AVT lanes can better improve the system efficiency of transportation networks with mixed AV and HV traffic. This study proposes a network modeling framework to determine the optimal deployment of AV/AVT lanes in a transportation network with mixed AV and HV flows so that the social cost is minimized. First, network equilibrium model must be developed to describe the traffic flow distributions of both AVs and HVs in a transportation network with AV/AVT lanes. The optimal deployment problem can then be investigated based on the network equilibrium model. Several studies have investigated the network equilibrium problem involving AVs. Chen et al. (2016) proposed a multi-class network equilibrium model for a transportation network with dedicated AV lanes and mixed AV and HV flows. The model assumes that AVs will significantly improve road capacity on dedicated AV lanes, whereas they will have no influence on the traffic capacity of roads with mixed flows. Chen et al. (2017a,b) further developed a network equilibrium model for a transportation network with dedicated AV zones and mixed AV and HV flows. In the model, travelers are assumed to minimize their perceived travel times when they make route choices. Perceived travel times are actual trip times for HV users, while for AV users they represent the actual travel times spent outside of the dedicated AV zones plus perceived marginal travel times within the dedicated AV zones (i.e., AVs are controlled by a central manager and follow a system-optimum (SO) routing principle within the dedicated AV zones). The model also assumes that AVs will improve road capacity within the dedicated AV zones while having no influence on road capacity outside the dedicated AV zones. Considering mixed AV and HV travel demands, Jiang (2017) proposed a combined mode split and traffic assignment model. The model assumes that AVs and HVs travel on separate lanes throughout the network and respectively follow the Cournot-Nash (CN) principle (i.e., AVs try to minimize their total travel cost through cooperation) and the user equilibrium (UE) principle when choosing routes. Based on the assumption that a central agent can fully control a fraction of the AV fleet in a network, Zhang and Nie (2018) proposed a mixed network equilibrium model with multi-class users. The model assumes that users who are not controlled by the central agent will try to minimize their own travel time through selfishly choosing their routes (i.e., following the UE routing principle), whereas users who are controlled by the central agent will try to minimize total system travel time through cooperative routing behavior (i.e., following the SO routing principle). The above studies either do not consider the case with mixed AV and HV flows (i.e., Jiang, 2017) or neglect the potential impact of AVs on the capacity of roads with mixed flows (i.e., Chen et al., 2016, 2017a,b; Zhang and Nie, 2018). However, because it will be many years before AVs are widely adopted and it may be impractical to completely separate AV and HV flows throughout a transportation network, a heterogeneous traffic flow consisting of both AVs and HVs will inevitably exist for a long time (Ghiasi et al., 2017, 2019). In addition, as reported by many studies (e.g., Ghiasi et al., 2017; Talebpour and Mahmassani, 2016; Bierstedt et al., 2014; Shladover, 2012; Chang and Lai, 1997), the potential impact of AVs on the capacity of roads with mixed AV and HV flows can be significant. Therefore, it is of great theoretical and practical importance to study the network equilibrium problem with mixed AV and HV flows and to specifically consider the impact of AVs on road capacity with mixed traffic. In this study, we assumed that AVs and HVs follow the UE principle when choosing their routes, and based on this assumption, we developed a UE model to describe the behaviors of both AVs and HVs. For mixed traffic flows with both AVs and HVs, we proposed to convert them into equivalent pure HV flows and then use traditional volume delay functions to characterize their travel times. In traditional traffic assignment models involving multiple vehicle classes (such as passenger cars, medium-duty and heavy-duty trucks), the traffic flows of different vehicle classes are usually converted into equivalent passenger car flows using a metric called passenger car equivalent (PCE)–(TRB, 2016). Similarly, we proposed a new metric, designated as human-driven vehicle equivalent (HVE), to convert AV flows into equivalent HV flows. It should be noted that we only consider passenger vehicles in this study. By converting AV flows into equivalent HV flows, the impact of AVs on traffic capacity is indirectly considered. The UE model is formulated as a variational inequality (VI) problem. We proved that, if HVE is constant, the solution of aggregate link flow in HVE is unique. Conversely, if HVE is variable, the solution may be non-unique. Based on the proposed UE model, we then explored the optimal deployment strategy for AV/AVT lanes in a general transportation 382

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network. To the best of our knowledge, only two studies in the literature have investigated the deployment problem of managed lanes for AVs at the network level (i.e., Chen et al., 2016; Chen et al., 2019). Chen et al. (2016) developed a time-dependent network design model to determine when, where and how many AV lanes should be deployed in a general network. The model assumes that AVs and HVs follow the UE principle in choosing their routes and that AVs will significantly improve road capacity on AV lanes while having no impact on the traffic capacity of roads with mixed flows. However, as discussed above, the impacts of AVs on the traffic capacity of roads with mixed flows can be significant and thus should not be neglected. Chen et al. (2019) proposed an AV incentive program design problem, in which both dedicated AV lanes and AV purchase subsidies are implemented to promote the adoption of AVs. They formulated the AV incentive program design problem as a two-stage stochastic programming model with equilibrium constraints and developed a solution method based on linear approximation and duality to solve the model. They also assumed that AVs will significantly improve road capacity on AV lanes while having no impact on the traffic capacity of roads with mixed flows. In this study, considering the proposed UE model may have non-unique link flow solutions, we proposed a robust optimal deployment model to deploy AV/AVT lanes in a manner that minimizes the social cost under the worst-case flow distribution. The robust optimal deployment model is formulated as a generalized semi-infinite min-max program and is solved using a genetic-algorithm-based approach. This study makes the following contributions. First, we propose the concept of AVT lanes, which is a promising alternative to dedicated AV lanes when AV flows are relatively low. Second, we develop a new UE model to describe the behaviors of both AVs and HVs in mixed traffic. We propose to convert mixed AV and HV flows into equivalent HV flows. Employing this innovative method, we are able to specifically capture the impact of AVs on road capacity with mixed traffic in the UE model. Third, we identify that the UE problem in a network with mixed flows of AVs and HVs may have non-unique flow patterns. Last, based on the proposed UE model, we investigated the strategic planning of AV/AVT lanes in general transportation networks. The remaining sections of this study are organized as follows. Section 2 first gives the definition of HVE and investigates the potential impacts of AVs on road capacity with mixed traffic. A UE model is then proposed to describe the flow distributions in a network with mixed flows of AVs and HVs. The deployment of AV/AVT lanes is investigated in Section 3. Finally, Section 4 concludes the paper. 2. Network equilibrium problem with mixed AV and HV flows 2.1. Basic considerations and notations Let G (N , A) denote a road network, where N is the set of nodes and A is the set of directed links. Links in the road network are designated by a A or represented as node pairs (i, j) , where i , j N . In the network, travelers may use either HVs or AVs. In what follows, variables related to HVs and AVs will be superscripted with m = 1 and m = 2 , respectively. We assume that all HVs and AVs in the network are passenger cars. AVs in the network are homogenous in terms of both technological level and time headway settings. When travelling between origins and destinations, both HV and AV users will selfishly choose their routes with the goal of minimizing their travel costs. Let W denote the set of origin-destination (O-D) pairs. Let o (w ) and d (w ) represent the origin node and destination node of O-D pair w W , respectively. Let qw, m denote the demand of vehicle class m {1, 2} between O-D pair w W , and qw represent the total demand, i.e., qw = q w,1 + q w,2 . In this study, it is assumed that demand qw and qw, m are known and fixed. Let xaw, m denote the traffic flow of travelers between O-D pair w W on link a A , and xam denote the aggregate flow of travelers between all O-D pairs on link a A . Let ta denote the travel time on link a . It is assumed that ta is given by the well-known Bureau of Public Roads (BPR) travel time function, with capacity being a function of the proportion of AVs on link a A :

ta = ta (xa1, xa2 ) = ta0 1 +

a

xa1 + xa2 ca (ra )

a

a

A

(1)

where ta0 is a parameter representing the free-flow travel time of link a ; ra denotes the percentage of AV flow on link a

A , ra =

xa2

xa1 + xa2

;

ca (ra ) represents the capacity of link a (in veh/h), which is a function of ra ; and a and a are two positive parameters. This assumption is also adopted by Levin and Boyles (2015) and, more recently, by Noruzoliaee et al. (2018) and Mehr and Horowitz (2019). 2.2. HV equivalents for AVs Let ua denote the traffic capacities (in veh/h) of link a function (1) can then be reformulated as follows:1

1

xa1 + x a2 = ca u ca + ra ca xa1 + xa2 a ra ca ua

The derivation process is simply given as follows:

the parentheses with

xa2 ra

xa2 gives

xa1 + x a2 ca

=

xa1 + x a2 ua ca ua

A when used by pure HV flows, i.e., ua = ca (0) . Link performance

xa1 +

=

xa1 + xa2 ua ca ua

. 383

xa1

. With ra =

xa2 , xa1 + x a2

we have xa1 =

xa2 ra

xa2 . Substituting xa1 in

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Z. Liu and Z. Song

ta (xa1,

xa2)

=

ta0

1+

a

xa1 + xa2 ca

a

=

ta0

1+

xa1 + xa2 a

(

ua

ca + ra ca ra ca

)

a

ua

It can be observed from the above equation that the performance of a link a

A (i.e., the average travel time on the link) with

mixed traffic of xa1 HV flow and xa2 AV flow is equivalent to the performance of the link with xa1 + xa2 thus propose a new concept named human-driven vehicle equivalent (HVE) to denote the term is defined as

HVEa =

ua

ca + ra ca ra ca

a

ua

(

ua

ca + ra ca . ra ca

ca + ra ca ra ca

) pure HV flow. We

For each link a

A , HVEa

A

(2)

HVE is similar to the passenger car equivalent (PCE) defined in the Highway Capacity Manual (TRB, 2016). Based on the defined HVE, mixed AV and HV flows on a link can be equivalently transformed into pure HV flows without changing the travel time on the link. We further define a new concept named aggregate link flow in HVE. If va denotes the aggregate link flow in HVE on link a A , va is calculated using the following equations:

va = xa1 + HVEa xa2

a

(3)

A

and link travel time ta is given by a new link performance function as follows:

ta = ta (va) = ta0 1 +

va ua

a

a

a

A

(4)

With a slight abuse of notation, we still use ta (va) to represent the functional relationship between link travel time and the aggregate link flow in HVE. As defined in Eq. (2), HVEa is a function of ca and ra . Since ca is a function of ra , HVEa is essentially a single-variable function of ra . The following proposition establishes the conditions for HVEa to be a constant. Proposition 1. HVEa is a constant value if and only if there exists a constant k such that

1 ca

1 ua

= kra (or equivalently ca (ra ) =

1

). 1 + kra ua

Proof.

HVEa =

ua

ca + ra ca ra ca

If HVEa is constant, If

1 ca

1 ua

HVEa ua

1 ca 1

1 HVEa = ua ua

1

ra

should also be constant. Let k =

= kra and k is constant, then

HVEa ua

1

HVEa ua

1

, then

1 ca

1 ua

= kra and k is constant.

= k and further HVEa = kua + 1 is constant. □

2.3. Impacts of AVs on road capacity with mixed autonomy Many studies in the literature have investigated the potential impact of AVs on the capacity of roads with mixed AV and HV flows. Based on the assumption that AVs will have reduced time headways when they follow other vehicles, some studies (e.g., Shladover et al., 2012; Arnaout and Arnaout, 2014; Levin and Boyles, 2015; van den Berg and Verhoef, 2016; Liu et al., 2018) have found that road capacity increases significantly based upon the percentage of AVs in the mixed flows. In the following, we provide an analytical capacity model based on the relationship between capacity and mean minimum time headway. According to traffic flow theory (see e.g., Hoogendoorn, 2010; van Wee et al., 2013), the per-lane capacity (in veh/h) of a link equals the reciprocal of the mean minimum headway (in h). In a mixed traffic flow of AVs and HVs, there are four types of time headways as shown in Fig. 1, where 11 denotes the time

Fig. 1. Different headways in mixed traffic (figure adapted from Ghiasi et al., 2017). 384

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headway for an HV followed by another HV, 12 for an HV followed by an AV, 21 for an AV followed by an HV, and 22 for an AV followed by another AV. Note that all headways in this study refer to minimum headways between corresponding vehicles because our analysis focuses on roadway capacity. Each type of headway is supposed to be a random variable that follows a positive-valued distribution with a finite mean and a finite variance (Ghiasi et al., 2017). Let ¯11, ¯12 , ¯21, and ¯22 denote the mean values of 11, 12 , 21, and 22 , respectively. Assume that ¯11, ¯12 , ¯21, and ¯22 are link-independent constants (in h). This assumption is not necessarily a a , ¯12 , restrictive as we can expand each headway parameter by adding a link dimension to it (i.e., replacing ¯11, ¯12 , ¯21, and ¯22 with ¯11 a a ¯21, and ¯22 , respectively). Further, let Ua denote the traffic capacities (in veh/h) of link a A when used by pure AV flows, i.e., Ua = ca (1) , and let a denote the number of lanes on link a A . Because the per-lane capacity (in veh/h) of a link equals the reciprocal of the mean minimum headway (in h), ua and Ua can be given by ua = ¯ a and Ua = ¯ a , respectively. Note that, because we assume ¯11 and ¯22 are link11

22

independent, the per-lane capacity when used by pure HV flows, given by

given by

1 , ¯22

are identical values for all links in a network.

1 , ¯11

and the per-lane capacity when used pure AV flows,

Based on the assumption that every AV follows its preceding vehicle (whether it is an AV or an HV) with identical mean time headway, i.e. ¯12 = ¯22 , and every HV follows its preceding vehicle (whether it is an AV or an HV) with identical mean time headway, i.e., ¯21 = ¯11, Lazar et al. (2017) proposed a road capacity model with mixed AV and HV flows as follows:

ca (ra ) =

a

ra ¯22 + (1

ra ) ¯11

1

=

ra Ua

+

1

a

ra ua

A

(5)

It should be noted that the above road capacity model is a valid approximation regardless of how AVs are distributed among HVs, as long as the percentage of AV flow is ra (Lazar et al., 2017). By modeling vehicle type (AV or HV) as a Bernoulli process, i.e., each vehicle is an AV with probability ra and an HV with probability 1 ra , and assuming that the time headway between two vehicles is Ua = ¯22 if they are both AVs and a otherwise (i.e., ua

a

¯21 = ¯12 = ¯11 =

ca (ra ) =

a

ua

), Lazar et al. (2017) further proposed another road capacity model with mixed AV and HV flows as follows: a

¯22 (ra )2 + (1

=

(ra ) 2) ¯11

1 (ra )2 Ua

+

1

a

(ra )2 ua

A (6)

Following Lazar et al. (2017), we also model vehicle type as a Bernoulli process and consider a general setting of different types of time headways. Specifically, each vehicle is an AV with a probability of ra and an HV with a probability of (1 ra) . For a pair of vehicles, they are an AV following an AV with a probability of (ra )2 , an AV following an HV with a probability of ra (1 ra ) , an HV following an AV with a probability of ra (1 ra ) , and an HV following an HV with a probability of (1 ra ) 2 . The overall mean minimum time headway can be calculated as ¯22 (ra ) 2 + ra (1 ra )( ¯12 + ¯21) + ¯11 (1 ra ) 2 . A more general road capacity model for mixed traffic is then proposed as follows:

ca (ra ) =

a

¯22 (ra ) 2 + ra (1

ra )( ¯12 + ¯21) + ¯11 (1

a

ra )2

A

(7)

If we assume ¯12 = ¯21 = ¯11 = u , model (7) is reduced to model (6). If we assume ¯12 = ¯22 = U and ¯21 = ¯11 = u , model (7) is a a a reduced to model (5). Therefore, models (5) and (6) are special cases of model (7). ¯12 + ¯21 ¯22 , link capacity ca (ra) is an increasing function of ra ; if and only if It is straightforward to verify that, if and only if ¯11 2 a

¯11

¯12 + ¯21 2

a

¯22 , ca (ra) is an decreasing function of ra ; if ¯12 + ¯21

¯12 + ¯21 2

a

is not between ¯11 and ¯22 , ca (ra) is not a monotone function of ra . In

¯22 , consequently link capacity ca (ra) is an increasing function of ra . this paper, we assume that ¯11 2 The proposed capacity model (7) is based on the assumption that vehicle types in a mixed flow of AVs and HVs are randomly distributed. This assumption is valid when AVs are individual units that are not well coordinated during operations (Ghiasi et al., 2017). However, if AVs are run as fleets by coordinated and centralized operators (Fernandes and Nunes, 2012), AVs may be able to cluster together and form platoons with significant lengths (Ghiasi et al., 2017). Under this scenario, AVs in a mixed flow are not randomly distributed, and their distribution or platooning intensity will also affect road capacity. Interested readers are referred to Ghiasi et al. (2017) for detailed discussions about the potential impact of platooning intensity on road capacity with mixed flows of AVs and HVs. In this study, we focus on a scenario that vehicle types in a mixed flow are randomly distributed. We leave the work of considering different distributions of vehicle types to future studies. Based on the Proposition 1, the capacity model given by Eq. (5) will lead to a constant HVE as follows:

HVEa =

¯22 ¯11

=

ua Ua

a

A

(8)

Note that HVEa in Eq. (8) is a predetermined constant identical for all links. For the capacity models given by Eqs. (6) and (7), the corresponding HVEa are functions of ra and are respectively given by the following Eqs. (9) and (10):

HVEa =

¯22

¯11 ¯11

ra + 1 =

ua

Ua Ua

ra + 1

a

A

(9) 385

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Z. Liu and Z. Song

¯11 + ¯22

HVEa =

¯12

¯21

¯11

ra +

¯12 + ¯21

1

¯11

a

A

(10)

Different from the HVEa in Eq. (8), the HVEa given by Eq. (9) or Eq. (10) will be a simple linear function of variable ra . Note that ¯11, ¯12 , ¯21, and ¯22 are exogenous predetermined constants. Eqs. (8) and (9) are special cases of Eq. (10). Specifically, if ¯11 + ¯22 ¯12 ¯21 = 0 , Eq. (10) reduces to Eq. (8); if ¯12 = ¯21 = ¯11, Eq. (10) reduces to Eq. (9). 2.4. Model formulation The flow distribution of AVs and HVs can be described by the following network equilibrium conditions:

x w, m = E wq w, m xaw, m

0

w a

xam =

A,

xaw, m

W, w

a

W,

A,

(11)

m = 1, 2

(12)

m = 1, 2

m = 1, 2

(13)

w W

ra =

xa2 xa1 + xa2

va =

xa1

+

(ta (va ) + ta (va) +

a

A

HVEa (ra ) xa2 i

w, m

w, m i

a

w, m ) xaw, m j w, m j

(14) (15)

A

=0

0

(i , j ) = a (i , j ) = a

A,

A, w

w

W,

W,

m = 1, 2

(16)

m = 1, 2

(17)

} ; E w represents an “inputwhere is the node-link incidence matrix associated with the network; x w, m is the vector of { output” vector, which has exactly two non-zero components: one has the value 1 corresponding to the origin node o (w ) and the other has the value 1 corresponding to the destination node d (w ) ; iw, m are auxiliary variables representing the node potentials. In the above, constraint (11) ensures flow balance between each O-D pair; constraint (14) ensures the non-negativity of link flows; constraints (13)–(15) are definitional constraints; constraints (16)–(17) make sure that, for each O-D pair, the travel costs on all w, m utilized paths are the same and equal to dw(,wm) o (w ) , and are less than or equal to those on unutilized paths. The network equilibrium conditions, i.e., Eqs. (11)–(17), can be formulated as a variational inequality. For this purpose, we define a vector (v, x ) , whose feasible region is defined by constraints (11)–(15). ,xaw, m ,

Proposition 2. The network equilibrium conditions (11)–(17) are equivalent to finding the solution to the following variational inequality (VI). UE-VI:

ta (va )(xaw, m

xaw, m )

0,

(v , x )

m {1,2} w W a A

Proof. Proposition 2 can be easily proved by deriving the Karush-Kuhn-Tucker (KKT) conditions of the above VI and comparing them with the network equilibrium conditions. □ Proposition 3. UE-VI has at least one solution. Proof. Because the demand of AVs and HVs is fixed and finite, all link flows must be bounded from the above. Therefore, is a compact and convex set. In addition, all functions of the VI formulation are continuous. According to the theory of the VI problem (see, e.g., Hartman and Stampacchia, 1966; Harker and Pang, 1990), the VI has at least one solution. □ Proposition 4. If HVEa (ra) is an identical constant for all links and ta (va) is a strictly increasing function of va , the solution of the aggregate link flow in HVE to the UE-VI, i.e., v , is unique. Proof. If HVEa (ra) is an identical constant for all links, the UE-VI is equivalent to the following VI:

ta (va )(xaw,1

¯ (xaw,2 ta (va ) HVE

xaw,1 ) +

w W a A

xaw,2 )

0,

(v , x )

w W a A

¯ , a A. The equivalence can be easily proved through comparing their ¯ denote the constant HVE and HVEa (ra) = HVE where HVE KKT conditions. Suppose that (v , x ) and (v†, x†) , (v , x ) (v†, x†) are two solutions of the above VI, then ta (va )(xaw,1†

xaw,1 ) +

w W a A

ta (va† )(xaw,1 w W a A

¯ (xaw,2† ta (va ) HVE

xaw,2 )

0

¯ (xaw,2 ta (va† ) HVE

xaw,2†)

0

w W a A

xaw,1†) + w W a A

386

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Z. Liu and Z. Song

Fig. 2. An small illustrative network with two nodes.

and consequently

(ta (va†)

¯ aw,2† ta (va ))(xaw,1† + HVEx

¯ aw,2 ) HVEx

xaw,1

0

w W a A

or

(ta (va†)

ta (va ))(va†

va )

0

a A

If ta (va) is a strictly increasing function of va , the above inequality implies that va† = va , or v is unique. □ Note that if HVEa (ra) is given by Eq. (8), which means HVEa (ra) is a identical constant for all links, and ta (va) is given by Eq. (4), which means ta (va) is a strictly increasing function, the solution of va is unique. With unique solution for va , the travel time on each link ta (va) is also unique and so is the equilibrium total travel time between each O-D pair. However, the solutions for xaw, m and xam are non-unique in general even the solution for va is unique. On the other hand, if HVEa (ra) is not a constant value, the uniqueness of the aggregate link flow in HVE to the UE-VI, i.e., v , cannot be guaranteed even ta (va) is a strictly increasing function of va . Let us consider a toy network with two nodes, two links, and ¯ one O-D pair (1,2) as shown in Fig. 2. The O-D demand is assumed to be 10 for AVs and 10 for HVs. If ¯12 = ¯21 = ¯11 and ¯22 = 211 ,

HVEa (ra) is given by Eq. (9) as HVEa =

¯22

¯11 ¯11

t1 (v1) = 10 + v1, and t2 (v2) = 5 + v2 , it is easy to verify that solutions. For the former solution, (HVE1, HVE2 ) =

(

)

xa2

ra + 1 = 1

. Suppose the link performance functions are given by

2(xa1 + xa2 ) (x11, x12 , x 21, x 22)

( 1), (v , v ) = (5, 10), and t = t 1 , 2

1

1

2

) are both equilibrium = 15; for the latter one, (HVE , HVE ) = ( , ) ,

= (0, 10, 10, 0) and 2

(

25 25 45 45 , 7, 7, 7 7

1

2

3 4

3 4

(v1, v2 ) = 4 , 4 , and t1 = t2 = 4 . It can be observed that the two solutions have different O-D travel cost. Consequently, the total system travel times associated with the two solutions are also different. 25

45

65

2.5. Solution algorithm To solve the UE-VI, we apply the technique developed by Aghassi et al. (2006) using duality to reformulate the UE-VI as the following nonlinear optimization problem: UE-NLP:

ta (va ) xaw, m

min

(v, x , )

m {1,2} w W a A

q w, m (

w, m d (w )

w, m o (w ) )

m {1,2} w W

s.t. i

w, m

+

w, m j

ta (va)

(i , j ) = a

A,

w

W,

m = 1, 2

(v , x ) In solving the above optimization problem, if the optimal value of the objective function is zero, then one part of the optimal solution, (v, x ) , would be the solution to the UE-VI problem. Because the UE-NLP model is a regular nonlinear program, it can be solved using commercial nonlinear solvers such as CONOPT (Drud, 1994). 2.6. Finding best- and worst-case flow distribution with variable HVE As discussed in Section 2.4, if HVEa (ra) is not a constant value, the aggregate link flow in HVE, the equilibrium O-D travel cost, and the total system travel time for a network may be non-unique. In practice, this non-uniqueness property poses a challenge to predict traffic flow distribution in a network with mixed AV and HV flows. Moreover, existing transportation planning and design methods also need to be modified to accommodate this non-uniqueness property. Among all possible network equilibrium solutions, we denote the one, which minimizes the total system travel cost, the best-case solution, and the one, which maximizes the total system travel cost, the worst-case solution. These two solutions can be used to evaluate the system performance of a transportation network with mixed AV and HV flows. Mathematically, the following two problems find the best-case and worst-case solutions, respectively. BC/WC-UE: 387

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Fig. 3. Nguyen-Dupuis network.

ta (va ) xam

min max v, x ,

a A m {1,2}

s.t. (11)–(17). The above problems are mathematical program with complementarity constraints (MPCC), which is difficult to solve. Many special algorithms have been developed to solve MPCC (e.g., Luo et al., 1996; Fletcher and Leyffer, 2004; Lawphongpanich and Yin, 2010), however, only a few among them can be effectively applied to large-scale problems. In this paper, we employ the algorithm proposed by Lawphongpanich and Yin (2010) using manifold suboptimization to solve BC/WC-UE. The algorithm enables convergence to a strongly stationary solution within a finite number of iterations. 2.7. Numerical examples Numerical examples in this section are based on the Nguyen-Dupuis network (Nguyen and Dupuis, 1984). As shown in Fig. 3, the network consists of 13 nodes, 19 links, and four O-D pairs. It is assumed that in the travel time function, a = 0.15 and a = 4 . Table 1 lists the link input parameters for links, including link free flow travel time and link capacity when used by pure HV flows. Note that we assume the per-lane capacity when used by pure HV flows is 2000 veh/h, which means ¯11 = 1.8 s . Therefore, all link capacity values are integral multiples of 2000. The total travel demand between each O-D pair is given by: q1 2 = 9600 veh h , q1 3 = 19, 200 veh h , q4 2 = 14, 400 veh h , q4 3 = 4800 veh h . For simplicity, it is assumed that the AV penetration rates for all OD pairs are identical. Two scenarios for HVEa are considered: (1) HVEa is a link-independent constant given by Eq. (8), i.e., ¯ ¯ ¯ HVEa = ¯22 ; (2) HVEa is a variable given by Eq. (9), i.e., HVEa = 22 ¯ 11 ra + 1. Note that the parameter values are for illustration 11 11 purposes only. 2.7.1. Scenario 1: constant HVEa When HVEa is a constant given by HVEa =

¯22 ¯11

, the equilibrium aggregate link flow in HVE and the total system travel time are

unique. To investigate the impact of AV penetration rate (denoted by ) on system travel time, six groups of AV penetration rate are ¯ considered, with ranging from 0.0 to 1.0 with a step size of 0.2. To investigate the impact of the headway ratio 11 (i.e., the reciprocal of the constant HVE) on system travel time, nine groups of the ratio

¯11 ¯22

are considered, with

with a step size of 0.5. Fig. 4 displays the total equilibrium system travel time with different combinations of

¯11 ¯22

¯11 ¯22

¯22

ranging from 1.0 to 5.0

and AV penetration rate . It can be

observed that, when the AV penetration rate is zero, the system travel time will not change with the increase of

¯11 ¯22

. When

¯11 ¯22

= 1, the

system travel time will not change with the increase of AV penetration rate. These results are expected because the benefits of AVs to Table 1 Link characteristics of the Nguyen-Dupuis network. Link

Free-flow travel time ta0 (min)

Capacity ua (veh/h)

Link

Free-flow travel time ta0 (min)

Capacity ua (veh/h)

1-5 1-12 4-5 4-9 5-6 5-9 6-7 6-10 7-8 7-11

7 9 9 12 3 9 5 13 5 9

6000 8000 6000 4000 6000 6000 6000 8000 4000 6000

8-2 9-10 9-13 10-11 11-2 11-3 12-6 12-8 13-3

9 10 9 6 7 8 7 14 11

8000 8000 4000 8000 6000 8000 8000 4000 4000

388

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Fig. 4. System travel times with constant HVE.

the network exist only when there are AVs in the network (i.e., when > 0 ) and they have reduced spacing than HVs (i.e., when ¯11 ¯ > 1). Moreover, for a given nonzero AV penetration rate, the system travel time decreases with the increase of 11 , and for a given ¯ 22

¯11 ¯22

¯22

that is not 1.0, the system travel time decreases with the increase of AV penetration rate . These results imply that the benefits of

AVs to the network increase with the increase of AV penetration rate and with the increase of 2.7.2. Scenario 2: variable HVEa When HVEa is a variable given by HVEa = ¯11

¯22

¯11 ¯11

¯11 ¯22

.

ra + 1, the equilibrium aggregate link flow in HVE and the total system travel

time may not be unique. Assume ¯ = 2.5 and consider 11 groups of AV penetration rate with ranging from 0.0 to 1.0 with a step 22 size of 0.1. Table 2 compares the system optimal (SO), best- and worst-case UE flow distributions with AV penetration rate = 0.5 and ¯11 = 2.5. Note that the flows shown in the table are aggregate link flows in HVE. Fig. 5 displays the total system travel times of SO, ¯22 best- and worst-case UE (denoted as BC-UE and WC-UE, respectively) flow distributions, as well as the relative system travel time difference between best- and worst-case UE flow distributions with AV penetration rate varying from 0 to 1.0. The relative difference is calculated by dividing the system travel time difference between best- and worst-case UE flow distributions by the best-case system travel time. One can observe that the system travel times of SO, best- and worst-case UE flow distributions all decrease with Table 2 Flow distributions for the Nguyen-Dupuis network with variable HVEa ( = 0.5,

¯11 ¯22

= 2.5).

Link

WC-UE

BC-UE

SO

1-5 1-12 4-5 4-9 5-6 5-9 6-7 6-10 7-8 7-11 8-2 9-10 9-13 10-11 11-2 11-3 12-6 12-8 13-3

10735.2 13744.8 9250.5 7069.5 12146.3 7839.4 12324.4 6640.7 5092.6 7231.8 12018.6 8064.6 6844.3 14705.2 8381.4 13555.7 6818.8 6926.0 6844.3

10276.9 12880.8 7824.6 6514.4 11694.8 7173.6 10930.5 4738.5 4699.3 6231.2 11152.5 7185.9 6233.2 12022.9 7772.7 12903.8 6506.1 6327.4 6233.2

7849.6 13050.6 8958.4 4842.4 10307.8 5760.0 12343.9 6214.5 5743.5 6600.4 10543.5 5768.5 4839.9 14243.0 7035.8 11975.1 8250.6 4800.0 4839.9

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Fig. 5. Comparison of system performances for the Nguyen-Dupuis network.

the increase of the AV penetration rate . When = 0 or = 1, the UE flow distribution in HVE is unique and there is no difference between the best- and worst-case performances. This result is expected because traffic flows in the network are pure HVs when = 0 and pure AVs when = 1. Note that the total system travel time of SO is less than that of best- and worst-case UE when = 0 , although the marginal difference is almost invisible in Fig. 5. When = 0 , the total system travel time of SO is 50,606.17 min and the total system travel times of best- and worst-case UE are both 50,731.69 min. For between 0 and 1, the absolute and relative difference between the best- and worst-case performances increases initially as increase from 0 to 0.5, and then decreases as increases from 0.5 to 1. The maximum relative difference is 18.6% and is reached when = 0.5. The system travel time associated with the SO flow distribution is always the minimum and it provides a valid lower bound for the best-case UE performance. This numerical example reveals the difficulty of designing and evaluating improving strategies to highway networks with mixed HV and AV flows as the resulting performances are likely to be uncertain. 3. Deployment of AV/AVT lanes 3.1. Potential benefits of AV lanes and AVT lanes As discussed above, road capacity increases with the increase of AV flow proportion. If used by pure AV flows, dedicated AV lanes have maximized traffic capacity. However, dedicated AV lanes may be underutilized when AV flow rate is low. AVT lanes are a promising substitution and complementarity to dedicated AV lanes. On the one hand, by charging HVs tolls, AVT lanes will have increased AV flow proportion and thus have improved traffic capacity. On the other hand, when AV flow rate is low, the capacity of an AVT lane can be more effectively used. Dedicated AV lanes and AVT lanes may benefit a transportation network with mixed AVs and HVs in two aspects: First, they may amplify the benefits of AVs and improve the traffic capacity of individual roads. Second, they may be able to improve the system-wide flow distribution in the network. To show the first benefit, consider a single O-D pair (1, 2) connected by one road segment with two lanes, as shown in Fig. 6. Lane 1 and lane 2 can be represented as link 1 and link 2, respectively. Assume that ¯11 = 1.8s and ¯22 = 0.9s , then the per-lane capacity is 2000 veh/h for pure HV flows and 4000 veh/h for pure AV flows, i.e., u1 = u2 = 2000veh h and U1 = U2 = 4000veh h . The free flow travel time is 10 min. The parameters a and a in the travel time function are both equal to 1. The toll on an AVT lane is equivalent to ¯ ¯ ra 5 min. If the HVEa is given by Eq. (9), i.e., HVEa = 22 ¯ 11 ra + 1 = 1 , Table 3 presents the efficiency of road segment (1, 2) under 2 3 11 different scenarios. Based upon the data in Table 3, we make several observations. Under scenario 1, converting lane 1 into an AV lane can reduce the

Fig. 6. A road segment with two lanes. 390

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Table 3 Efficiency of road segment (1, 2) under different scenarios. Lane setting

Aggregate flow in HVE (v1, v 2 )

Total travel time (min)

Scenario 1: Demand between (1, 2) is q1 2,1 = 2000 veh h and q1 2,2 = 2000 veh h Both lanes 1 and 2 are regular (1000, 1000, 1000, 1000) (3/4, 3/4) Lane 1 is AV lane, lane 2 is regular (0, 2000, 2000, 0) (1/2, 1)

(1750, 1750) (1000, 2000)

75,000 70,000

Scenario 2: Demand between (1, 2) is q1 2,1 = 2000 vehh and q1 Both lanes 1 and 2 are regular (1000, 500, 1000, 500) Lane 1 is AV lane, lane 2 is regular (0, 1000, 2000, 0) Lane 1 is AVT lane, Lane 2 is regular (500, 1000, 1500, 0)

(1417, 1417 ) (500, 2000) (1167, 1500)

51,250 52,500 50,000

Flow distribution ( x11, x12, x 21, x 22 )

2,2

HVE (HVE1, HVE2)

= 1000 veh h (5/6, 5/6) (0.5, 1) (2/3, 1)

total system travel time from 75,000 min to 70,000 min. Under scenario 2, converting lane 1 into an AV lane will increase the total travel time from 51,250 min to 52,500 min, while converting lane 1 into an AVT lane can reduce the total travel time to 50,000 min. The results of scenario 1 demonstrate that converting existing lanes into dedicated AV lanes may improve the efficiency of a road segment with a high proportion of AV flows. The results of scenario 2 suggest that AVT lanes may exhibit better performance than AV lanes when AV flow rate is low. ¯ When HVEa is a constant, e.g., HVEa = ¯22 , converting lane 1 or lane 2 into AV or AVT lanes will not improve efficiency under any 11 demand scenarios. This result can be extended to a general road segment with an arbitrary number of lanes (intuitively, a unit of AV ¯22 HV flow regardless of the distribution of mixed traffic among the lanes of a road segment). That flow will always be transformed to ¯11

being said, for general transportation networks, AV/AVT lanes may still be able to improve system-wide flow distribution when HVEa is constant. In this study, we focus on the deployment of AV/AVT lanes for more general scenarios with variable HVEa . 3.2. UE model with AV/AVT lanes

Let A represent the set of AV/AVT candidate links in the network. Note that if a directed road segment has some lanes that are designated as AV/AVT lanes, this road segment can be represented as a pair of parallel links, i.e., one AV/AVT link and one regular link. Further, let H denote the set of these link pairs. Each link pair is designated by h H or represented as [a , a ], where a A A is a regular link, a A is a candidate AV/AVT link, and a and a belong to the same directed road segment. For example, Fig. 7(a) shows a small traffic network with three directed road segments. If road segment 1 is considered a candidate road to deploy AV/AVT lanes, then the network topology of the traffic network is given by Fig. 7(b), and we have A = {1, 2, 3, 4} , A = {4} , and H = {[1, 4]} , where h = [1, 4] H denote a link pair of regular link 1 and AV/AVT candidate link 4. This link pair representation is inspired by Chen et al. (2016). An integer variable ya is introduced for each candidate AV/AVT link a A to represent the number of lanes that are converted to AV/AVT lanes. Note that link a A is a real link that can be utilized only when ya > 0 ; otherwise the link is only a virtual link that cannot be utilized. A binary variable z a is introduced to differentiate dedicated AV lanes and AVT lanes. It is stipulated that z a = 1 if link a A is converted into an AVT link and z a = 0 if link a A is converted into a dedicated AV link. Let a1 denote tolls that are measured in units of time for HVs to access AVT lanes. Since a toll is only enacted on HVs using AVT lanes, a1 can be simplified as a when there is no confusion. Given the deployment of AV/AVT lanes and the toll rates on AVT lanes (i.e., given ya , z a , and a ), the flow distribution of HVs and AVs can be described by the following UE conditions:

x w, m = E wq w, m xam =

xaw, m

w a

W, A,

m m

(18)

{1, 2} {1, 2}

(19)

w W

Fig. 7. A small illustrative network. 391

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ra =

xa2 + xa2

xa1

a

A

(20)

va = xa1 + HVEa (ra ) xa2 xaw,1

a

Gz a

a

(21)

A

A

(22)

w W

xaw, m

Gya

a

A

(23)

m {1,2} w W

xaw, m

0

a

(ta (va ) +

i

w, m

(ta (va ) +

ta (va) +

a

w, m j

+

i

w,2

(ta (va ) +

i

ta (va) +

w,2 i

w,1 j

w,2 j

xaw,1

Gz a

a

=0

+

+

a

a

+

(i , j ) = a A,

m

m

{1, 2}

(i , j ) = a

(i , j ) = a

a

W,

W,

(i , j ) = a

=0

=0

w

w

=0

0

a

A A,

A A,

w,1 a ) xa

+

0

a

(24)

{1, 2}

(i , j )=a

w ,2 a ) xa

+

+

m

(i , j )=a

w ,1 j

w ,2 j

W,

0

w,1

w,1 i

+

a

w

w, m ) xaw, m j

w, m i

ta (va) +

A,

w

A,

A,

A,

A,

w

w

w

w

{1, 2}

(25) (26)

W

(27)

W

(28)

W

(29)

W

(30)

W

(31)

w W

xaw, m

Gya

a

=0

a

A,

w

W

(32)

m {1,2} w W a

0

a

A,

w

W

(33)

a

0

a

A,

w

W

(34)

where G is a sufficiently large positive constant; a and a are auxiliary variables. In the above, constraint (18) ensures flow balance between each O-D pair; constraints (19)–(21) are definitional constraints; constraint (22) prohibits HVs to use AV links; constraint (23) makes sure that, a candidate AV/AVT link cannot be utilized by HVs and AVs if it is only a virtual link, i.e., if ya = 0 ; constraint (24) ensures the non-negativity of link flows; constraints (22)–(34) make sure w, m that, for each O-D pair, the travel costs on all utilized paths are the same and equal to dw(,wm) o (w ) , and less than or equal to those on unutilized paths. Define a vector (v, x ) , whose feasible region ¯ is defined by constraints (18)–(24). Finding a solution to the system of UE conditions (18)–(34) is equivalent to solving the following VI: UE-VI-2:

ta (va )(xaw, m

xaw, m ) +

w W m {1,2} a A A

w ,1 a )(xa

[(ta (va ) +

xaw,1 ) + ta (va )(xaw,2

xaw,2 )]

0,

(v , x )

¯

w W a A

The equivalence can be established by comparing the KKT conditions of the VI with the defined UE conditions (18)–(34). It can be proved that UE-VI-2 also has unique solution for the aggregate link flow in HVE when HVEa (ra) is constant, similar to the situation without AV/AVT lanes. Again, we solve UE-VI-2 by reformulating it to be the following nonlinear optimization problem via the technique proposed by Aghassi et al. (2006): UE-NLP-2:

min

(v, x , , , a

) w W m {1,2} a A A

+ a A

Gya

ta (va ) xaw, m +

[(ta (va ) +

w,1 a ) xa

+ ta (va ) xaw,2]

w W a A

a

s.t. i

i

w, m

w,1

+

+

w, m j

w,1 j

ta (va) a

(i , j )=a a

ta (va ) +

a

A A,

q w, m ( w W m {1,2}

w

(i , j ) = a

W,

m

A, 392

{1, 2}

w

W

w, m d (w )

w, m o (w ) )

+

Gz a a A

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i

w,2

+

w,2 j

ta (va)

a

(i , j ) = a

a

0

a

A,

w

W

a

0

a

A,

w

W

A,

w

W

(v , x ) The objective function of UE-NLP-2 is to minimize the gap between a primal and duel problem associated with UE-VI-2, and the constraints are those from the primal and duel problems. The optimal solution (v , x ) to UE-NLP-2 solves UE-VI-2 if the gap is zero. UE-NLP-2 can be solved using commercial nonlinear solvers such as CONOPT (Drud, 1994). 3.3. Deployment model Based on the UE model with AV/AVT lanes, this section investigates the problem of optimally deploying AV/AVT lanes and designing toll rates for AVT lanes. As discussed in Section 2.7.2, when HVEa (ra) is variable, the system performance for a network with mixed HV and AV flows may be non-unique, and consequently, it is difficult to design and evaluate improving strategies for the network. In real-world applications, a plan or design that can optimize the worst-case performance is more robust and preferable for planners, who tend to be risk-averse. Therefore, we propose a robust optimal deployment problem (RODP) of AV/AVT lanes and formulate it as the following min–max program: RODP:

ta (va ) xam

min max y, z , v, x , , ,

a A m M

s.t. (18)–(34)

ua = u¯ a + ¯ a y a

a

ua = u¯ a

[a, a ]

ua

¯a

ya

{0, 1,

a a

¯a y a a

max

(36)

H

(37)

A

, Ia }

0

(35)

A

a

a

(38)

A

(39)

A a

(40)

A

where u¯ a is the initial capacity of link a A , ¯a denotes the per-lane capacity of link a A , ¯a is given parameter that represents the minimum capacity required for link a A , Ia is a given integer parameter that denotes the maximum number of lanes that can be converted to AV/AVT lanes for candidate link pair a A , max is the toll rate upper bound. Note that all capacities refer to the traffic capacities when used by pure HV flows, and the per-lane capacities of an AV/AVT link and its paired regular link are thus identical. Moreover, as discussed in Section 2.3, this study assumes identical per-lane capacity for all links. In the above, the objective function is to minimize the maximum, or worst-case total system travel time. Constraints (18)–(34) ensure that the travelers’ behavior follows the UE conditions. Constraints (35) and (36) capture the change of link capacity due to the deployment of AV/AVT lanes for each candidate link a A and its paired regular link a A A (here [a , a ] H ). Constraint (37) ensures that the capacity of a link a A is no less than a required minimum capacity. For instance, all of the regular links should have at least one regular lane so that the network is still freely accessible for HV users. Constraint (38) specifies that y a is an integer variable and no greater than its upper bound Ia . Constraint (39) ensures the non-negativity of a . Constraint (40) specifies the upper bound of a . The model RODP can be readily extended to consider the potential construction cost for the deployment of AV and AVT lanes, via y ((1 z a ) baAV + z a baAVT ) to the objective function or adding a budget constraint adding a term a A a AV AVT y ((1 z a ) ba + z a ba ) B , where baAV represents the construction cost of deploying one dedicated AV lane on link a A , a A a AVT ba represents the construction cost of deploying one AVT lane on link a A , is a conversion factor that converts the total construction cost from a monetary basis to a time basis, and parameter B is the total available budget. For convenience, let (y, z , ) denote the feasible region of the inner problem, or the maximization part of RODP, represent the decision variable vector (v, x , , , ) , and ( ) denote the objective function. Then the inner problem can be written as: RODP-IN:

(y , z, ) = max { ( ):

(y , z, )}

Consequently, the RODP can be written as:

min (y , z , ) y, z,

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s.t. (35)–(40)

(y , z, ) = max { ( ):

(y , z, )}

It is obvious that the feasible region of the inner problem is determined by the decision variables of the outer problem. With the above form, RODP is essentially a generalized semi-infinite min-max problem (see, e.g., Polak and Royset, 2005), which is not easy to solve. Moreover, due to the MPCC property of the inner problem, existing algorithms (see, e.g., Still, 1999) may not be applicable to the RODP. Lou et al. (2010) formulated two robust congestion pricing models, which have similar structure with RODP and are also generalized semi-infinite min-max problems. They proposed a heuristic algorithm based on penalization and a cutting-plane scheme to solve their models. In the robust congestion pricing models, the link toll vector is the only decision variable of the outer problem and only one set of equality constraints in the inner problem involves the toll variables. Lou et al. (2010) formulated a penalized inner problem in which the set of equality constraints involving the toll variables are removed and a penalty term is added in the objective function. By doing so, the original generalized semi-infinite min-max problem is converted into an ordinary semi-infinite min-max problem. The problem is then equivalently reformatted as an ordinary semi-infinite optimization problem and solved using a cuttingplane scheme. In our RODP problem, however, there are three set of decision variables in the outer problem (i.e., ya , z a and a ) and six set of equality and inequality constraints in the inner problem involving ya , z a or a . Inequality constraints require more complicated penalty function such as exterior penalty function (Coello, 2002), consequently the penalized inner problem may be a non-convex nonlinear program with discontinuous derivatives, which is very difficult to solve. Therefore, we proposed a genetic-algorithm-based approach in this paper as an alternative to Lou’s method. It is more problem-independent and easier to implement for our problem. 3.4. Solution algorithm This paper proposes a genetic-algorithm-based approach shown in Fig. 8 to solve the RODP. The solution procedure mainly includes two modules, i.e., inner problem solution module and genetic algorithm module. The inner problem is a MPCC, which is difficult to solve. An iterative solution procedure is designed to solve it based on the algorithm proposed by Lawphongpanich and Yin (2010) using manifold suboptimization. The algorithm enables convergence to a strongly stationary solution with a finite number of iterations. Genetic algorithm module (including the reproduction, crossover and mutation operations) is adopted to determine the deployment of AV/AVT lanes and the toll rate on AVT lanes. 3.4.1. Inner problem solution module All complementarity constraints in the inner problem RODP-IN, i.e., constraints (22)–(34), can be represented by the following generalized form:

0

Find (x ind ) x ind

0

where is the orthogonal sign representing the inner product of two vectors is zero, x ind is a variable with index ind and Find (x ind ) is a w, m ) xaw, m 0, (i , j )=a A A , w W, m M, function of vector x ind . To illustrate, for constraints 0 (ta (va ) + iw, m j w, m x ind = xaw, m , ind = (a, w, m ), and Find (x ind ) = ta (va ) + iw, m . For each group of complementarity constraints, we define two j index sets x and ¯ x to respectively track the component of single variable x ind and formula Find (x ind ) required to be zero:

Fig. 8. Flowchart of the genetic-algorithm-based approach. 394

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Z. Liu and Z. Song k x

= {ind : x ind = 0}

¯ kx = {ind: Find (x ind ) = 0} Note that, here superscript k indicates the iteration of the solution procedure, which will be introduced below. Based on ¯ x , a restricted version of RODP-IN, denoted as R-RODP-IN, can be formulated as the following nonlinear programs: R-RODP-IN:

x

and

ta (va ) xam

max

v, x , , ,

a A m M

s.t. (18)–(21) and

x ind = 0 Find (x ind ) = 0

x ind

k x

ind

0

k x

ind

Find (x ind )

¯ kx

ind

0

ind

¯ kx

where x ind = (v , x , , , ) ; the right part of Find (x ind ) x ind , i.e., x ind , can be a , a , xaw, m and each of them has a corresponding Find (x ind ) . Given a design for (y, z , ) , the iterative procedure of solving RODP-IN is as follows: Step 0: Solve the problem UE-NLP-2 and obtain the solution of vector (v, x , , , ) . Based on the solution, set k = 1 and initialize all index sets kx and ¯ kx . x Step 1: Let (v, x , , , ) k solve R-RODP-IN, and obtain the multipliers associated with constraints x ind = 0 , denoted as gind . Step 2: Update index sets defined as follows: x, k

= {ind

k x

¯ kx : g x > 0} ind

If all these index sets are empty, stop and (v, x , , ,

) k is strongly stationary. Otherwise, do the following and go to Step 1:

(a) Set k+1 x

=

k x

x, k

(b) Set

¯ kx+ 1 = {ind: Find (x ind ) = 0} (c) Set k = k + 1 To solve the problem UE-NLP-2 more efficiently, we propose a procedure to avoid using the parameter G , i.e., a sufficiently large positive constant, in the solution procedure. The technique is outlined below: Check the value of ya and z a for each link a A and do the following: (a) (b) (c) (d) (e) (f)

If ya = 0 , set xaw, m = 0 , m {1, 2}, w W . If ya > 0 , set a = 0 . If z a = 0 , set xaw,1 = 0 , w W . If z a > 0 , set a = 0 . Remove constraints (22) and (23) that involves G. Remove the term a A Gz a a + a A Gya a in the objective function of UE-NLP-2.

We briefly explain the rationale of the above procedure. First, if ya = 0 for a link a A , the constraint (23) for this link can be x w, m 0 , which can further be reduced to xaw, m = 0 , m {1, 2}, w W given xaw, m 0 , reduced to m {1,2} w W a

a A , m {1, 2}, w W . If ya > 0 for an a A , we set the multiplier associated with constraint (23), i.e., a , to be zero and remove constraint (23) for this link a . Note that because the total traffic flow on a link is upper bounded by the total travel demand q w, m , we can always make constraint (23) unbinding by choosing a between all O-D pairs, i.e., m {1,2} w W xaw, m m {1,2} w W sufficiently large constant G when ya > 0 , consequently constraint (23) for a link a A with ya > 0 can be safely removed without affecting the solution of the problem UE-NLP-2. Similarly, if z a = 0 for a link a A , the constraint (22) for this link can be reduced to x w,1 0 , which can further be reduced to xaw,1 = 0 , w W given xaw,1 0 , a A , w W . If z a > 0 for an a A , we set w W a the multiplier associated with constraint (22), i.e., a , to be zero and remove constraint (22) for this link a . Again, because the total qw,1, we can always make HV flow on a link is upper bounded by the total HV demand between all O-D pairs, i.e., w W xaw,1 w W 395

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Fig. 9. Structure of a chromosome (figure adapted from Yang et al., 2016).

constraint (22) unbinding by choosing a sufficiently large constant G when z a > 0 , consequently constraint (22) for a link a A with z a > 0 can be safely removed without affecting the solution of the problem UE-NLP-2. Under the above settings, for any link a A , either ya = 0 or a = 0 and either z a = 0 or a = 0 , consequently the term a A Gz a a + a A Gya a will always be zero. 3.4.2. Genetic algorithm module Genetic algorithms are search and optimization procedures motivated by natural principles and selection (Goldberg, 1989). Due to its extensive generality, global perspective, strong robustness and implicit parallelism, genetic algorithm has been applied to a wide variety of problems in transportation engineering, including road network design problem (Xiong and Schneider, 1992; Drezner and Salhi, 2002; Chen et al., 2010), transit network design problem (Fan and Machemehl, 2006; Arbex and da Cunha, 2015), traffic signal timing (Park et al., 1999; Ceylan and Bell, 2004) and road pricing (Yin, 2000; Shepherd and Sumalee, 2004; Zhang and Yang, 2004). The basic idea of the genetic-algorithm-based approach is to code the decision variables of the outer problem to a number of chromosomes (i.e., strings) and calculate the fitness of each chromosome by solving the inner problem. By iteratively conducting reproduction, crossover and mutation operations of genetic algorithms, the optimal string may be obtained. The decision variables of the outer problem, i.e., y = {ya a A } , z = {z a a A } , and = { a a A } is coded as a chromosome p = {pn n = 1, 2, , 3×|A |} (see Fig. 9). A group of chromosomes is first generated randomly. Following the evaluation, selection, crossover, and mutation operations, a new population of chromosomes is generated at each iteration. After a given number of iterations, genetic algorithm will terminate and return the best found solution. The process is outlined below. Step 0: Initialize parameters: population size pop _size , crossover probability Pco , mutation probability Pmt , and maximal number of generations max _generation . Step 1: Randomly generate pop _size feasible chromosomes as the initial population. Set generation index g = 1. Step 2: Calculate the fitness for all chromosomes by solving the inner problem RODP-IN and reproduce the population according to the distribution of the fitness values. Step 3: Carry out the crossover and mutation operations. Step 4: If g = max _generation , the chromosome with the highest fitness is adopted as the optimal solution of the problem. Else, set g = g + 1, and go to Step 2. 3.5. Numerical studies In this section, two numerical examples are presented to demonstrate the proposed models and algorithms. 3.5.1. The Nguyen-Dupuis network The following set of tests are conducted on the Nguyen-Dupuis network (Nguyen and Dupuis, 1984). As shown in Fig. 10, the network consists of 13 nodes, 19 regular links, 19 candidate AV/AVT links, and four O-D pairs. In the travel time function, a = 0.15 and a = 4 . Table 4 lists the link input parameters for regular links, including link free flow travel time, initial number of lanes and initial capacity. The per-lane capacity is set to 2000 veh/h, which means ¯11 = 1.8s . The minimum capacity for each regular link is set as the per-lane capacity. Table 5 shows the link pairs, in which each candidate AV/AVT link is paired with one regular link. A candidate AV/AVT link has the same free flow travel time and per-lane capacity as its paired regular link. The initial capacity and the minimum capacity for each AV/AVT link are set to 0. Note that all the capacities mentioned here refer to the capacities for pure HV flows. The total travel demand between each O-D pair is given by: q1 2 = 9600veh h , q1 3 = 19, 200veh h , q4 2 = 14, 400 veh h , q4 3 = 4800 veh h . The AV penetration rates for all O-D pairs are set to 40%. Suppose HVEa is given by Eq. (9), i.e., ¯ ¯ ¯ 3 A. The upper bound of toll rate max is set equivalent to HVEa = 22 ¯ 11 ra + 1, a A and ¯11 = 2.5, then HVEa = 1 5 ra, a 11 22 3 min. The genetic-algorithm-based solution procedure was implemented using MATLAB R2018b interfaced with GAMS (Rosenthal, 2012) on a 3.40 GHz Dell Computer with 16 GB of RAM. CONOPT (Drud, 1994) was used to solve the UE-NLP-2 and R-RODP-IN 396

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Fig. 10. Nguyen-Dupuis network with candidate AV/AVT links. Table 4 Link characteristics of the Nguyen-Dupuis network with candidate AV/AVT links. Link

1-5 1-12 4-5 4-9 5-6 5-9 6-7 6-10 7-8 7-11

ta0 (min)

Free-flow travel time

Initial number of lanes

Initial capacity u¯ a (veh/h)

Link

7 9 9 12 3 9 5 13 5 9

3 4 3 2 3 3 3 4 2 3

6000 8000 6000 4000 6000 6000 6000 8000 4000 6000

8-2 9-10 9-13 10-11 11-2 11-3 12-6 12-8 13-3

ta0 (min)

Free-flow travel time

Initial number of lanes

Initial capacity u¯ a (veh/h)

9 10 9 6 7 8 7 14 11

4 4 2 4 3 4 4 2 2

8000 8000 4000 8000 6000 8000 8000 4000 4000

Table 5 Link pairs in the Nguyen-Dupuis network with candidate AV/AVT links. Pair

Candidate AV/AVT link

Regular link

Pair

Candidate AV/AVT link

Regular link

1 2 3 4 5 6 7 8 9 10

20 21 22 23 24 25 26 27 28 29

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19

30 31 32 33 34 35 36 37 38

11 12 13 14 15 16 17 18 19

problems. The genetic-algorithm-based procedure was performed with pop _size = 30 , Pco = 0.6, Pmt = 0.15 and max _generation = 120 . Because the inner problem is a MPCC, using multiple initial solutions to solve it can yield better local optimal solutions. Three different initial solutions were used in solving each inner problem. The computation time thus increased significantly. It took about 2.6 h to solve the model. The robust deployment plan is shown in Fig. 11. In total, three dedicated AV links and eight AVT links are deployed. The number of lanes of AV/AVT links, and the toll rates and HV flows (under the worst-case UE) on AVT links are reported in Table 6. Toll is only enacted on HVs using AVT lanes and is zero for all other link types. The last column shows the HV flow using AVT links. It can be observed that HVs are willing to pay a toll to access AVT links. Table 7 compares the travel times between deployed AVT links and their paired regular links. It can be observed that the travel time difference between an AVT link and its paired regular link is equal to the toll rate on the AVT link. This result verifies that for a HV using a AVT link, the total travel costs (including travel time and toll) on the AVT link and on its paired regular link are identical. Table 8 compares the system performance of the Nguyen-Dupuis network in the status quo condition (i.e., no AV/AVT links are deployed) and in the robust deployment plan. It can be observed that the robust deployment plan reduces the worst-case total system travel time from 3,964,807.7 min to 2,956,840.8 min, a reduction of 25.4 percent. We further note that, in the status quo condition, the relative difference between the worst- and best-case total system travel time is 18.31%, while under the robust deployment plan, the relative difference is only 0.02%. With the deployed AV and AVT links, the system performance of the network becomes more 397

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Fig. 11. Robust optimal deployment of AV/AVT lanes in the Nguyen-Dupuis network with candidate AV/AVT links. Table 6 Robust optimal deployment of AV/AVT lanes in the Nguyen-Dupuis network with candidate AV/AVT links. Selected candidate links

Optimal deployment type

Number of lanes

HV toll

HV flow

21 22 24 26 27 29 30 31 32 35 36

AVT AV AVT AV AVT AVT AVT AVT AVT AVT AV

2 1 1 2 1 2 3 2 1 2 1

2.96 – 2.69 – 0.46 0.59 2.46 1.84 2.91 2.09 –

751.9 – 255.4 – 1227.2 593.5 3174.9 2732.5 3061.6 1480.3 –

Table 7 Travel time comparison between AVT links and their paired regular links. Pair

AVT link

Regular link

Travel time on AVT link (min)

Travel time on regular link (min)

Travel time difference (min)

Toll on AVT link (min)

2 5 8 10 11 12 13 16

21 24 27 29 30 31 32 35

2 5 8 10 11 12 13 16

14.90 7.10 13.28 11.73 14.30 10.33 16.41 14.45

17.86 9.79 13.73 12.32 16.77 12.17 19.33 16.53

2.96 2.69 0.46 0.59 2.46 1.84 2.91 2.09

2.96 2.69 0.46 0.59 2.46 1.84 2.91 2.09

Table 8 System performances in status quo condition and robust deployment plan. Total system travel time (min)

The status quo Robust deployment plan

Relative difference

Worst-case

Best-case

3,964,807.7 2,956,840.8

3,351,335.0 2,956,124.5

18.31% 0.02%

stable. The worst-case equilibrium O-D travel costs are compared in Table 9. It can be observed that, compared to the status quo condition, the robust deployment plan reduces the worst-case equilibrium O-D travel cost for each O-D pair and for both AVs and HVs. This result implies that it may be possible to realize a Pareto-improving design (Song et al., 2009, 2014; Liu et al., 2009; Guo and Yang, 2010; Lawphongpanich and Yin, 2010) with AV/AVT lanes. Moreover, in the status quo condition, the worst-case equilibrium O-D travel costs for AVs and HVs are identical for each O-D pair, while under the robust deployment plan, the equilibrium O-D travel cost for AVs is less than that for HVs. This result is expected because AVs will always have equal or lower equilibrium travel cost (including travel time and toll) than HVs on deployed AV/AVT links. We also solved a restricted version of the deployment model, in which only dedicated AV lanes are considered. The deployment plan can only reduce the worst-case system travel time from 3,964,807.7 min to 3,144,559.7 min, a reduction of 20.7 percent. Therefore, compared with the scenario with only dedicated AV lanes, the combination use of AV and AVT lanes brings more 398

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Table 9 Comparison of worst-case equilibrium O-D travel costs. O-D

(1, 2) (1, 3) (4, 2) (4, 3)

Mode

HV AV HV AV HV AV HV AV

Travel cost (min)

Travel cost change

The status quo

Robust deployment plan

78.89 78.89 87.19 87.19 77.79 77.79 86.09 86.09

61.93 56.50 67.93 62.28 60.49 55.34 66.49 61.12

−21.5% −28.4% −22.1% −28.6% −22.2% −28.9% −22.8% −29.0%

Fig. 12. Sioux Falls network.

reduction to the total system travel time. Note that the full deployment model will always have equal or better performance than its restricted version as the feasible region of the latter is a subset of the former. 3.5.2. The Sioux Falls network To further test the proposed model, we solve it for the Sioux Falls network, which consists of 24 nodes, 76 regular links, 20 candidate AV/AVT links as shown in Fig. 12. In the travel time function, a = 0.15 and a = 4 . Table A.1 lists the link input parameters for regular links. Again, ¯11 is assumed to be 1.8 s, which leads to a per-lane capacity of 2000 veh/h when used by pure HVs. The minimum capacity for each regular link is set to the per-lane capacity. The initial capacity and the minimum capacity for each AV/AVT link are set to 0. The total O-D demands for HVs and AVs are listed in Table A.2. The penetration rate of AVs is assumed to be ¯ ¯ ¯ 40% and identical for all O-D pairs. HVEa is given by Eq. (9), i.e., HVEa = 22 ¯ 11 ra + 1, and ¯11 = 2.5. The upper bound of toll rate 11 22 max is set is set equivalent to 3 min. The genetic-algorithm-based procedure is performed with pop _size = 30 , Pco = 0.6, Pmt = 0.15 and max _generation = 120 . Three different initial solutions were used in solving each inner problem. It took 70.5 h to solve the model. Although not particularly efficient, the generic-algorithm-based procedure generated reasonable solutions. The robust deployment plan is listed in Table A.3. Compared to the status quo condition, the robust deployment plan reduces the worst-case total system travel time from 7,812,395.3 min to 6,840,264.9 min, a reduction of 12.4 percent. 4. Concluding remarks In this study, we investigated the strategic development of AV and AVT lanes in transportation networks with mixed AV and HV flows. Different from dedicated AV lanes that can only be used by AVs, AVT lanes allow HVs to use by paying tolls. AVT lanes provide 399

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a promising alternative to dedicated AV lanes when AV flows are low. A UE model was first formulated to describe the flow distributions of AVs and HVs. It was found that with different time headway patterns in mixed AV and HV flows, the UE problem might have unique or non-unique flow patterns. Because the UE problem may have non-unique flow distributions, a robust optimal deployment model is proposed to deploy AV and AVT lanes so that the system performance under the worst-case flow distributions is optimized. The robust model is a generalized semi-infinite min-max problem, which is not easy to solve. A genetic-algorithm-based approach was proposed for its solution in the present study. The Nguyen-Dupuis network and the Sioux Falls network were used for the numerical demonstration of the proposed AV and AVT lane deployment models. The results demonstrate that AV and AVT lanes can significantly improve the system performance. In this study, we assume that AVs will have smaller following headways than HVs and consequently link capacity is an increasing function of the proportion of AV flow on the link. However, as mentioned by Ghiasi et al. (2017), AV technologies are yet to be fully developed and thus may have quite some uncertainties. If future AV technologies are conservative and AVs have larger critical headways than HVs, link capacity may be a decreasing function of the proportion of AV flow on the link (Seo and Asakura, 2017). Under such a conservative scenario of future AV technologies, AVs should be the group to be tolled in mixed traffic. One limitation of our study is that we ignore the possibility of this conservative scenario of future AV technologies. However, our model can be easily modified to consider this scenario by introducing a toll variable for AVs. In future studies, the proposed models can be extended in several directions. First, the network equilibrium model can be extended by considering the mode or vehicle type choice of travelers between AVs and HVs (see e.g., Daziano et al., 2017; Haboucha et al., 2017). Second, the robust deployment model can be extended by considering the AV demand evolution with time (see e.g., Talebian and Mishra, 2018; Gkartzonikas and Gkritza, 2019) and with the construction of AV based infrastructures (Chen et al., 2016). Third, the robust deployment model can also be extended by considering the risk-neutral and risk-acceptant preferences of planners. Fourth, a further study is planned to consider the equality of different user groups when employing the proposed model. In addition, the development of even more efficient algorithms for solving the proposed models will be undertaken. Acknowledgments The study was partially sponsored by the U.S. Department of Energy (DE-EE0007997) and Mountain-Plains Consortium, a regional University Transportation Center sponsored by the U.S. Department of Transportation. The views expressed are those of the authors and do not reflect the official policy or position of the project’s sponsors. Appendix A. Notations and Tables Notations Sets

Description

N A W

Set Set Set Set

A H

of of of of

nodes, indexed by i, j links, indexed by a = (i, j) O-D pairs, indexed byw candidate AV/AVT links

Set of links pairs, indexed by h = [a, a ], where a

Parameters

Description

q w, m

Travel demand for O-D pair w Free flow travel time of link a

ta0 ua Ua a

Traffic capacity of link a Traffic capacity of link a Number of lanes on linka

Variables

Description

xaw, m

Traffic flow of class m

xam ra ca (ra) HVEa va ta w, m i

ya a

W , class m A

A A, a

A.

{1, 2}

A when used by pure HV flows (veh/h) A when used by pure AV flows (veh/h)

{1, 2} between O-D pair w

W on link a

A

Aggregate flow of class m {1, 2} on link a A Percentage of AV flow on link a A Capacity of link a A HV equivalent value for AVs on link a A Aggregate link flow in HVE on link a A Travel time on link a A specified by the link performance function Node potential at node i for class m {1, 2} between O-D pair w W Number of lanes that are converted to AV/AVT lanes for candidate link a Toll for HVs to access AVT link a

400

A

A

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See Tables A.1–A.3. Table A.1 Link characteristics of the Sioux Falls network. Link

Free-flow travel time (min)

Initial number of lanes

Initial capacity (veh/ h)

Link

Free-flow travel time (min)

Initial number of lanes

Initial capacity (veh/ h)

1-2 1-3 2-1 2-6 3-1 3-4 3-12 4-3 4-5 4-11 5-4 5-6 5-9 6-2 6-5 6-8 7-8 7-18 8-6 8-7 8-9 8-16 9-5 9-8 9-10 10-9 10-11 10-15 10-16 10-17 11-4 11-10 11-12 11-14 12-3 12-11 12-13 13-12

6 4 6 8 4 4 4 4 2 6 2 4 5 8 4 2 3 2 2 3 3 5 5 3 3 3 1 6 4 8 6 1 6 4 4 6 3 3

5 5 5 3 5 4 5 4 4 3 4 2 3 3 2 3 3 5 3 3 2 3 3 2 4 4 3 4 3 2 3 3 3 2 5 3 5 5

10,000 10,000 10,000 6000 10,000 8000 10,000 8000 8000 6000 8000 4000 6000 6000 4000 6000 6000 10,000 6000 6000 4000 6000 6000 4000 8000 8000 6000 8000 6000 4000 6000 6000 6000 4000 10,000 6000 10,000 10,000

13-24 14-11 14-15 14-23 15-10 15-14 15-19 15-22 16-8 16-10 16-17 16-18 17-10 17-16 17-19 18-7 18-16 18-20 19-15 19-17 19-20 20-18 20-19 20-21 20-22 21-20 21-22 21-24 22-15 22-20 22-21 22-23 23-14 23-22 23-24 24-13 24-21 24-23

4 4 3 4 6 3 3 3 5 4 2 3 8 2 2 2 3 14 3 2 4 14 4 6 5 6 2 3 3 5 2 4 4 4 2 4 3 2

3 2 3 2 4 3 4 4 3 3 3 4 2 3 2 5 4 5 4 2 2 5 2 3 3 3 3 2 4 3 3 2 2 2 3 3 2 3

6000 4000 6000 4000 8000 6000 8000 8000 6000 6000 6000 8000 4000 6000 4000 10,000 8000 10,000 8000 4000 4000 10,000 4000 6000 6000 6000 6000 4000 8000 6000 6000 4000 4000 4000 6000 6000 4000 6000

Table A.2 Total O-D demands of AVs and HVs of the Sioux Falls network (veh/h). O-D

Demand

O-D

Demand

O-D

Demand

O-D

Demand

1-6 1-7 1-11 1-15 1-21 6-1 6-11

5000 5000 5000 5000 2000 5000 3000

6-15 6-21 7-1 7-6 7-11 7-21 7-15

3000 3000 5000 2000 5000 4000 3000

11-1 11-6 11-7 11-21 11-15 21-1 21-6

5000 3000 5000 2000 3000 2000 3000

21-7 21-11 21-15 15-1 15-6 15-7 15-11

1000 2000 3000 5000 3000 3000 3000

401

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Table A.3 Robust deployment of AV/AVT lanes for the Sioux Falls network. Selected candidate links

Optimal deployment type

Number of lanes

HV toll

HV flows

2-6 6-2 4-11 6-8 8-6 7-8 8-7 10-16 12-11 15-14 24-13 13-24

AVT AVT AV AVT AV AVT AVT AVT AV AVT AVT AV

1 1 1 1 1 1 2 2 1 2 2 1

1.84 2.74 – 2.57 – 1.65 2.50 2.24 – 0.20 2.78 –

579 510 – 98 – 478 2554 2819 – 1427 1049 –

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