Revisiting Jiang s dynamic continuum model for urban cities. Abstract Jiang et al. (Transportation Research Part B, 2011, 45(2), ) proposed a

Transcription

1 *2. Manuscript Click here to view linked References Revisiting Jiang s dynamic continuum model for urban cities Jie Du 1, S.C. Wong 2, Chi-Wang Shu 3, Tao Xiong 4, Mengping Zhang, and Keechoo Choi 6 Abstract Jiang et al. (Transportation Research Part B, 11, 4(2), ) proposed a predictive continuum dynamic user-optimal (PDUO-C) model to investigate the dynamic characteristics of traffic flow and the corresponding route-choice behavior of travelers. The modeled region is a dense urban city, which is arbitrary in shape and has a single central business district (CBD). However, we argue that the model is not well posed due to an inconsistency in the route-choice strategy. To overcome this inconsistency, we revisit the PDUO-C problem, and construct an improved path-choice strategy. The improved model consists of a conservation law to govern the density, in which the flow direction is determined by the improved path-choice strategy, and a Hamilton-Jacobi equation to compute the total travel cost. The simultaneous satisfaction of both equations can be treated as a fixed-point problem. A self-adaptive method of successive averages (MSA) is proposed to solve this fixed-point problem. This method can automatically determine the optimal MSA step size using the least squares approach. Numerical examples are used to demonstrate the effectiveness of the model and the solution algorithm. Key Words: Continuum model, Dynamic traffic assignment, Predictive user equilibrium, Path-choice strategy, Conservation law, Hamilton-Jacobi equation, Fixed-point problem, Method of successive averages (MSA), Self-adaptive. 1 Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, P.R. China. dujie@mail.ustc.edu.cn 2 Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, P.R. China. hhecwsc@hkucc.hku.hk 3 Division of Applied Mathematics, Brown University, Providence, RI, USA. shu@dam.brown.edu 4 Department of Mathematics, University of Houston, Houston, TX, USA. txiong@math.uh.edu Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, P.R. China. mpzhang@ustc.edu.cn 6 Department of Transportation Engineering, TOD-based Sustainable Urban Transportation Center, Ajou University, Korea. keechoo@ajou.ac.kr 1

2 1 Introduction The traffic equilibrium problem has received much attention in recent decades. The large number of studies devoted to the problem can be generally classified into two categories: those that use the discrete modeling approach and those that use the continuum modeling approach. The discrete modeling approach is the conventional methodology used for detailed studies of travel patterns in road network systems, in which each road link within the network is modeled separately (Lee, 2). In contrast, the continuum modeling approach takes the modeling region as a continuum and focuses on the overall behavior of travelers at the macroscopic level. In the continuum modeling approach, the differences between adjacent areas within a network are assumed to be relatively small compared with the variation over the entire area, and hence the characteristics of the network, such as the flow intensity, density, and travel cost, can be represented by smooth mathematical functions (Vaughan, 1987). The majority of the studies that have used the continuum modeling approach for urban cities have been confined to static cases. Studies on the static continuum modeling of traffic equilibrium problems can be broadly divided into two categories: those that focus on specific city configurations and those that focus on arbitrary city configurations. In the studies on specific city configurations, the study area is approximated by an idealized domain for analysis. A circular region specification is most commonly adopted. For example, Blumenfeld and Weiss (197), Lam and Newell (1967), Williams and Ortuzar (1976), and Zitron (1978) assumed the modeling region to be circular. Other specifications, such as linear (Tong and Wong, 1997; Vaughan, 198; Zhang et al., 8), bilinear (Buckley, 1979), and rectangular (Dafermos, 198) specifications, have also been considered for various kinds of trip assignment and distribution problems. In contrast, a continuum model approach with an arbitrary city configuration can be used for regions of any arbitrary shape, and dose not require paths within the region to run in a predefined travel direction (Blumenfeld, 1977; Taguchi and Iri, 1982; Wong et al., 1998). 2

3 Extensions of the continuum modeling approach have also been proposed for multiple destinations and/or multiple user classes (Wong, 1998; Ho et al., 6, 13; Yin et al., 13). As the temporal variation of flow and cost are not considered in the static models, they cannot be used to study elements such as travelers departure/arrival time choices or dynamic traffic management and control. To address these problems, the dynamic traffic assignment (DTA) problem has received much attention in recent decades. An important component of DTA is the travel choice problem, which models travel behavior. Three major problems are considered in this component: the dynamic system-optimal (DSO) problem, which aims to minimize the total travel cost of the system (Chow, 9; Lo and Szeto, ), the reactive dynamic user-optimal (RDUO) problem, in which travelers choose their route using the instantaneous information and change their choice in a reactive manner (Boyce et al., 1993; Kuwahara and Akamatsu, 1), and the predictive dynamic user-optimal (PDUO) problem, in which travelers are assumed to have perfect information about the modeled domain with which to choose the route that minimizes the actual travel cost to the destination (Hoogendoorn and Bovy, 4; Lo and Szeto, 2; Szeto and Lo, 4; Tong and Wong, ). However, most of the continuum models for urban cities rely on the static assumption of the demand and supply sides of the transportation system. To improve the capacity of the continuum model to govern the dynamic changes and the practicality in largescale/dense networks, Jiang et al. (11) discussed the formulation of and proposed a solution algorithm for a predictive continuum dynamic user-optimal (PDUO-C) problem for a large, dense urban city with a single central business district (CBD). Travelers are continuously distributed across the city and travel to the CBD within the modeling region for a given time-dependent travel demand. The proposed model consists of a flow conservation equation to compute the density, and a time-dependent Hamilton- Jacobi equation and an Eikonal equation to compute the travel cost. In this paper, 3

4 we show that one of the equations used to compute the travel cost is mathematically unnecessary. However, the user-optimal conditions in Jiang et al. (11) require both of these equations to be satisfied. The main problem with this inconsistency comes from the requirements in the path-choice strategy, as we illustrate below. To overcome this inconsistency, we revisit the PDUO-C problem and propose an improved path-choice strategy. The improved model consists of two PDEs: a conservation law, which governs the density in the city using the improved path-choice strategy, and a time-dependent Hamilton-Jacobi equation to compute the total travel cost, which differs from the equation in Jiang et al. (11). To solve our model numerically, we adopt the Lax-Friedrichs scheme, which is a conservative monotone scheme with properties such as the maximum principle and total variation diminishing (TVD). Unlike other problems in which the initial conditions of both equations are set at the beginning of the modeling period, the initial conditions of these two equations in our model are set differently, with one at the beginning and the other at the end of the modeling period. Therefore, we cannot solve them together as usual. In fact, this model can be treated as a fixed-point problem, which can be solved by iteration methods such as the method of successive averages (MSA) (Bar-Gera and Boyce, 6; Liu et al., 9; Nagurney and Zhang, 1996; Polyak, 199; Robbins and Monro, 191). Among the various types of MSA, the MSA with a constant step size (Bar-Gera and Boyce, 6) is very efficient. However, this method needs to estimate the optimal step size according to past experience and decrease it as necessary. Based on a formula proposed by Bar-Gera and Boyce (6), we construct a self-adaptive MSA that can automatically determine the optimal step size using the least squares method without prior information. The numerical results show that this new constructed self-adaptive MSA is much faster than the conventional methods. The remainder of this paper is organized as follows. The description of the PDUO- C problem is given in the next section. In Section 3, we review the formulation of the 4

5 original PDUO-C model in Jiang et al. (11) and point out the inconsistency. In Section 4 we discuss the formulation of an improved model and describe the solution algorithm in Section. In Section 6 we present the numerical results and compare the results with the MSAs for other step sizes. Our conclusions are presented in Section 7. 2 Problem description Figure 1: The modeling domain As Figure 1 shows, the modeled region is an urban city denoted by Ω. Let Γ o be the outer boundary of the city, let Γ c be the boundary of the compact CBD, and let Γ i be the boundary of an obstruction such as a lake, park, or undeveloped area, where traffic is not allowed to enter. Thus, the boundary of Ω is Γ = Γ o Γc Γi. The travelers homes are continuously located along (x, y) Ω. They travel to the CBD within the modeling region for a given time-dependent demand. The variables are denoted as follows. ρ(x, y, t) (in veh/km 2 ) is the density of travelers at location (x, y) at time t. We assume that no traveler is allowed to leave the city by crossing the boundary Γ o or to enter the obstruction though Γ i, so we have ρ(x, y, t) =, (x, y) Γ o Γi, t T, (1)

6 where T = [, t end ] (in h) is the modeling period. v = (u(x, y, t), v(x, y, t)) is the velocity vector at location (x, y) at time t. U(x, y, t) (in km/h) is the speed, which is the norm of the velocity vector, i.e. U = v, and is determined by the density as U(x, y, t) = U f e βρ2, (x, y) Ω, t T, (2) where U f (x, y) (in km/h) is the free-flow speed, and β (in km 4 /veh 2 ) is a positive scalar related to the road condition. F = (f 1 (x, y, t), f 2 (x, y, t)) is the flow vector at location (x, y) at time t, which is defined as F = ρv, (x, y) Ω, t T. (3) F is the flow intensity, which is the norm of the flow vector F, and is defined as F = ρu, (x, y) Ω, t T. (4) As U is a function of ρ, F can also be determined by the density, as shown in Figure 2. Let us define a critical density ρ c. When ρ is between and ρ c, it is called the non-congested condition. When the density is greater than ρ c, it is called the congested condition. F increases as ρ increases from to ρ c, and decreases as ρ increases beyond ρ c. The maximum flow intensity is achieved at a density of ρ c, which is denoted by F max. The flow-density relationship shown in Figure 2 is called the macroscopic fundamental diagram (Daganzo and Geroliminis, 8; Geroliminis and Daganzo, 8). q(x, y, t) (in veh/km 2 /h) is the travel demand at location (x, y) at time t, which is a non-negative and time-varying function. 6

7 Figure 2: The macroscopic fundamental diagram c(x, y, t) (in $/km) is the local travel cost per unit distance of travel at location (x, y) at time t, and is defined as c(x, y, t) = κ( 1 U + π(ρ)), (x, y) Ω, t T, () where κ is the value of time, 1 U represents the cost associated with the travel time, and π(ρ) represents other costs that are dependent of the density. ϕ(x, y, t) is the total travel cost incurred by a traveler who departs from location (x, y) at time t to travel to the CBD using the constructed path-choice strategy. Next, we need to construct a path-choice strategy and a model of this problem to investigate the relationship between different variables. Here, we assume that the road network is very dense and travelers can move freely around it. We also assume that travelers have perfect information about traffic conditions over time, so they will choose the path that minimizes the actual travel cost (not the instantaneous travel cost), thus resulting in a predictive user-equilibrium for a dynamic system. 3 Inconsistency of the original PDUO-C model 3.1 Original model Jiang et al. (11) constructed a PDUO-C model to solve the problem mentioned 7

8 above. In their path-choice strategy, the following two conditions are required: (u, v, 1) (ϕ x, ϕ y, ϕ t ) (6) and ϕ = cu U 2 + 1, (7) where = (,, ). It is proven that under these two conditions, travelers will choose x y t their path to the CBD in a dynamic predictive user-optimal manner. The model proposed in Jiang et al. (11) can be solved by ρ t + F = q, (x, y) Ω, t T, F = ρu ϕ ϕ, (x, y) Ω, t T, ϕ = cu (8) 2, (x, y) Ω, t T, U 2 +1 ϕ t + 1 ϕ =, (x, y) Ω, t T, U subject to the initial boundary conditions ρ(x, y, ) = ρ (x, y), (x, y) Ω, ρ(x, y, t) =, (x, y) Γ o Γi, t T, ϕ = ϕ CBD, (x, y) Γ c, t T, where = ( x, y ). 3.2 The inconsistency (9) Consider the model described in the equations in (8). The first three equations already represent a well posed problem in mathematics if proper initial boundary conditions are given, so the last equation is unnecessary. We now verify that a solution (ρ, ϕ) exists that satisfies the first three equations, but not the last. Take our domain to be a region of arbitrary configurations with a CBD in the center. The boundary of the CBD (i.e. Γ c ) is a circle located at (, ) with a diameter of 1. Consider a steady state problem, in which the density ρ(x, y, t) and the free-flow speed U f (x, y) are positive constants. Hence, the speed U(x, y, t) = U f e βρ2 and the local cost c(x, y, t) = κ( 1 U + π(ρ)) are also positive constants. Define the right side of the third equation in (8) as a = cu 2 U >, () 8

9 then we can solve the third equation in (8) with a proper boundary condition ϕ =, (x, y) Γ c, t T, (11) to obtain ϕ(x, y, t) = a( x 2 + y 2 1), (12) which is in fact a distance function. Now, the first two equations in (8) could be easily satisfied by adjusting the source term q(x, y, t). So far, we have constructed a solution (ρ, ϕ) that satisfies the first three equations. However, we can easily show that ϕ t + 1 U ϕ = a U >, (13) which contradicts the last equation in (8). The main reason of this inconsistency comes from the requirements in the path-choice strategy defined by Equations (6) and (7). Jiang et al. (11) required both of these equations to be satisfied before proving the predictive user-equilibrium principle for the dynamic system. However, these two equations may be contradictory, which we illustrate in detail as follows. Note that Equation (7) is an Eikonal equation, which can be solved by many numerical methods, such as the fast sweeping method. It is sufficient to obtain ϕ(x, y, t) by solving Equation (7), and Equation (6) is unnecessary. However, the ϕ(x, y, t) solved by Equation (7) may not satisfy Equation (6). In fact, the vector (u, v, 1) in Equation (6) is the velocity vector in the time-space domain, and is subject to u 2 + v 2 = U 2, () as mentioned in the problem description section. Now, define a vector (ũ, ṽ, w) which is parallel to (ϕ x, ϕ y, ϕ t ), where ϕ(x, y, t) is solved by Equation (7). If (ũ, ṽ, w) is the velocity vector according to the meaning of Equation (6), then with the condition ũ 2 + ṽ 2 = U 2, (1) 9

10 we can obtain but we cannot obtain Uϕt Φ w = Uϕ t ϕ, (16) = 1 from Equation (7). However, according to the meaning of Equation (6), w is the speed of the time, which must be 1h/h. Conversely, if Equation (6) is satisfied, we can assume u = aϕ x, v = aϕ y, 1 = aϕ t, (17) where a is a positive scalar to be determined. As u 2 + v 2 = U 2, () we can obtain a = U. (19) ϕ 2 x + ϕ 2 y Using the third equation in (17) and Equation (19), we have ϕ t + 1 ϕ =. () U Note that Equation () is a Hamilton-Jacobi equation that can be solved with proper initial boundary conditions, so Equation (7) is not necessary. Again, the ϕ(x, y, t) solved by Equation (6) may not satisfy Equation (7). In fact, ϕ(x, y, t) is the travel cost from location (x, y) at time t to the CBD along the speed vector (u, v), and it can be determined if (u(x, y, t), v(x, y, t)) is given. When the given (u(x, y, t), v(x, y, t)) is changed, the resulting ϕ(x, y, t) also changes. In the PDUO- C problem, the path-choice strategy should find only one suitable relationship between the travel direction (u, v) and the resulting ϕ(x, y, t), such that a traveler can reach the CBD by the route that minimizes the actual travel cost. Hence, the two requirements in Equation (6) and (7) are over-specified.

11 4 An improved model As there is an inconsistency in the original PDUO-C model, an improved model is needed to solve the PDUO-C problem. In Section 4.1, we construct a path-choice strategy for the case in which the only cost is the travel time. A general path-choice strategy for a general local cost function is presented in Section 4.2, and Section 4.3 describes the relationship between them. The initial boundary conditions and the complete system of our new model are discussed in Section The derivation of the path-choice strategy in the special case that the cost is the time Figure 3: The 3D trajectory of a vehicle First, let us consider the case in which the cost is the travel time in the domain. The path-choice strategy in this paper is discussed in the time-space domain, which is shown in Figure 3. We use the x-y plane to represent the space domain in which the vehicles drive. The vertical axis represents the time. Let the coordinate of point A in the x-y plane be (x,y). The curve that passes point (x,y,t) represents the 3D trajectory of a vehicle passing point A at time t and heading toward the CBD in the time-space domain. 11

12 Definition 1 (The departure time) Suppose the speed vector in the time-space domain is known and denoted by ( u(x, y, t), v(x, y, t) ). We denote z(x, y, t D ) as the departure time at point (x, y), such that a vehicle can arrive at the CBD at time t D along the speed trajectory. According to this definition, if the speed vector in the time-space domain is known, the departure time can be determined by tracing back from the CBD. Now, let us consider the derivation of the equation to compute z(x, y, t D ). Given a time, t D, we can view z(x, y, t D ) as a surface in the time-space domain. On this surface, a vehicle can reach the CBD at time t D when moving along the given speed trajectory. Let us define a path on this surface as (x(t), y(t), t) with the parameter t, and the position of the CBD is (x(t D ), y(t D )). Clearly, we have z(x(t), y(t), t D ) = t. (21) So, on this path, we have dz dt = z dx x dt + z dy y dt = 1. () According to the definition of this path, we know that dx dt represents the speed along the x axis, i.e., Similarly, dx dt = u(x, y, t) = u(x, y, z(x, y, t D)). (23) dy dt = v(x, y, t) = v(x, y, z(x, y, t D)). (24) We can then obtain the equation to compute z(x, y, t D ): { z(x,y,td ) x u(x, y, z(x, y, t D )) + z(x,y,t D) y v(x, y, z(x, y, t D )) = 1, z(cbd, t D ) = t D. (2) For each t D, the initial condition is set at time t D, and we can solve this equation outwards from the CBD, by backtracking in time. In the above, for each given time t D and the speed vector v = ( u(x, y, t), v(x, y, t) ), we can obtain the corresponding function z(x, y, t D ) by solving (2), and thus z can 12

13 be determined. Note that z = (z x, z y ) and v = ( u, v ) at (x, y, z(x, y, t D )) need not be parallel. In fact, v can be arbitrary. However, in the following, we illustrate our path-choice strategy. Theorem 1: If we choose the speed vector v such that the resulting z is always parallel to v and has the same direction, then the dynamic predictive user equilibrium in terms of total travel time to the CBD is satisfied. Proof: If z v, then we have z v = z v = z U, (26) where U = u 2 + v 2 is the given isotropic speed in the time-space domain. Using Equation (2), we can obtain z U = 1, (27) and hence z(x, y, t D ) = 1/U(x, y, z(x, y, t D )). (28) Next, consider Figure 4, in which the larger circle has a radius tu, which is the boundary of distance movement in t time, as the isotropic speed is U. Here, we assume that t is small enough such that z(x, y, t D ) can be treated as a linear function of x and y in such a small area. The dotted curve represents the used path from (x, y) to the CBD based on the parallel condition z v. The total travel time along this used path becomes the difference between the departure time z(x, y, t D ) from (x, y, z(x, y, t D )) and the arrival time t D at the CBD, and thus ϕ(x, y, z(x, y, t D )) = t D z(x, y, t D ). (29) We need to show that for any path from (x, y, z(x, y, t D )) that deviates from this used path (3D trajectory), the arrival time will be no earlier than t D ; that is, the total travel time will be no less than ϕ(x, y, z(x, y, t D )). We can prove this with the following two steps. 13

14 Figure 4: The used and unused paths Step1. Let us first consider an unused path that deviates from the used path only in the first t time and then reverts back to the used path. As shown in Figure 4, we consider the path that moves away from the starting point with a vector tṽ = t(ũ, ṽ), where ṽ = U is the isotropic speed, followed by the dashed curve that represents the 3D used trajectory from ( x, ỹ) to the CBD, where ( x, ỹ) is the position at which the vehicle arrives at the dotted circle boundary in t time with the vector tṽ. As ṽ = (ũ, ṽ) is not parallel with z = (z x, z y ), by defining the angle between v and ṽ as θ, we can show that the change in z at point (x, y, z(x, y, t D ) along the movement vector tṽ is z( x, ỹ, t D ) z(x, y, t D ) = z tṽ = z ṽ cos θ t = z U cos θ t z U t. (3) Here, the first equal sign comes from the assumption that z(x, y, t D ) is a linear function of x and y in this small area. Even without this assumption, as t is small enough, the nonlinear item in the Taylor expansion is the high order infinitesimal, which will make no difference to our discussion and can be neglected. Using Equation (28), we have z( x, ỹ, t D ) z(x, y, t D ) t U = t, (31) U

15 that is z(x, y, t D ) + t z( x, ỹ, t D ). (32) This means that the arrival time at ( x, ỹ) along this unused path, i.e. z(x, y, t D ) + t, will not be earlier than the departure time from ( x, ỹ) that is required to reach the CBD at time t D, i.e. z( x, ỹ, t D ). The central question is whether a vehicle that departs from ( x, ỹ) at a later time could arrive at the CBD earlier than t D, even if the vehicle follows the used path for the reminder of the journey. We can show that this is not possible by contradiction. Figure : Two used paths from ( x, ỹ) to the CBD To illustrate this, consider Figure. Suppose that it is possible to arrive at the CBD at an earlier time t D, as shown by the dotted-dashed line, even if the vehicle departs later. As z(x, y, t D ) is a closed surface emanating from (CBD, t D ), the dotted-dashed curve that leaves ( x, ỹ) at time z(x, y, t D ) + t z( x, ỹ, t D ) and arrives at the CBD at t D < t D must cut through the closed surface z(x, y, t D ) somewhere at (ˆx, ŷ, z(ˆx, ŷ, t D )). However, by definition, if a vehicle departs from this time-space point, it can reach the CBD at both t D and t D, which is impossible. Therefore, if the vehicle moves in any direction other than v = (u, v) in the first t 1

16 time, it will not arrive at the CBD earlier than t D, thus giving a total travel time no less than ϕ(x, y, z(x, y, t D )) along the used path. Step2. Now, let us consider the more general case. In step 1, if a vehicle moves along ṽ = (ũ, ṽ) in the first t time, it will arrive at the CBD at time t D t D. However, if it does not revert back to the used path in the next step and continues to move in a direction other than the speed vector, then by the same token, it will arrive even later at t D t D t D. Therefore, for any unused path, the total travel time must be no less than that of the used path. 4.2 The derivation of the path-choice formula with a more general cost So far, we have used the total travel time to represent the cost. Now, we need to construct a more general path-choice strategy for a general local cost function in the time-space domain. If the speed vector ( u(x, y, t), v(x, y, t) ) in the time-space domain is known, then the cost potential function ϕ(x, y, t) can be determined (discussed in more detail below), so we can obtain (ϕ x, ϕ y ). In the above, (u, v) can be arbitrary, and (u, v) and (ϕ x, ϕ y ) need not be parallel. However, in the following, we show that if we choose the vector (u, v) such that the resulting (ϕ x, ϕ y ) is parallel to (u, v), then the dynamic user equilibrium is satisfied. Theorem 2: If (u, v) ( ϕ x, ϕ y ), then the predictive dynamic user equilibrium principle is satisfied. Proof: Recall from the definition in the problem description section that ϕ(x, y, t) is the total travel cost incurred by a traveler who departs from location (x, y) at time t to travel to the CBD using the constructed path-choice strategy. This means traveling along the path that satisfies (u, v) ( ϕ x, ϕ y ), i.e, the used path. Now, we need to show that for any unused path from O(x, y, t ) that deviates from the used path, 16

17 the total cost must be no less than ϕ(x, y, t ). We can prove this using the following two steps. Step1. We first consider the unused path that deviates from the used path only at the initial t time and reverts back to the used path thereafter. Again, we assume that t is small enough such that we can neglect the high order infinitesimal in the Taylor expansion. Consider Figure 6, in which the circle has a radius tu; that is, Figure 6: The used and unused paths the boundary of moving t time from location O(x, y, t ). A(x, y) is the position at which the vehicle arrives along the used path with a vector t(u, v) based on the parallel condition (u, v) ( ϕ x, ϕ y ), where U = u 2 + v 2 is the given isotropic speed in the time-space domain. Ã( x, ỹ) is the position at which the vehicle arrives along the unused path with a vector t(ũ, ṽ). Obviously, the arrival times at A and à are both t + t. We can show that the change in ϕ along the used path is ϕ(x, y, t + t) ϕ(x, y, t ) = (ϕ x, ϕ y, ϕ t ) (u, v, 1) t = t[(ϕ x, ϕ y ) (u, v) + ϕ t ] = t( ϕ U + ϕ t ). (33) 17

18 The change in ϕ along the unused path is ϕ( x, ỹ, t + t) ϕ(x, y, t ) = (ϕ x, ϕ y, ϕ t ) (ũ, ṽ, 1) t = t[(ϕ x, ϕ y ) (ũ, ṽ) + ϕ t ] = t( ϕ U cos θ + ϕ t ) t( ϕ U + ϕ t ). (34) Hence, we have ϕ( x, ỹ, t + t) ϕ(x, y, t ) ϕ(x, y, t + t) ϕ(x, y, t ), (3) ϕ( x, ỹ, t + t) ϕ(x, y, t + t). (36) Note that the travel costs from point O(x, y, t ) to A and à are both tuc(x, y, t). Clearly, we have ϕ(x, y, t ) = tuc(x, y, t ) + ϕ(x, y, t + t), (37) and the cost along the unused path that deviates from the used path only in the first t period is ϕ = tuc(x, y, t ) + ϕ( x, ỹ, t + t). (38) As ϕ( x, ỹ, t + t) ϕ(x, y, t + t), the total cost along the unused path must be no less than ϕ(x, y, t ). Step2. In the above case, if a vehicle moves along (ũ, ṽ) in the first t period, it will arrive at the CBD with the cost ϕ ϕ(x, y, t ). However, if it does not revert back to the used path in the next step and continues to move in a direction other than that of the speed vector, then by the same token it will arrive at the CBD with a cost ϕ ϕ ϕ(x, y, t ). Therefore, for any unused path, the total cost must be no less than that of the used path.

19 Figure 7: δt > Figure 8: δt < Now, we compute ϕ(x, y, t) when the speed vector in the time-space domain is given. In fact, we only need to choose the vector (u, v) such that the resulting (ϕ x, ϕ y ) is parallel to (u, v). As shown in Figures 7 and 8, let (δx, δy, δt) be a vector at location (x, y, t) along the used path, that is { (δx, δy, δt) (u, v, 1) if δt >, (δx, δy, δt) (u, v, 1) if δt <. (39) As (ϕ x, ϕ y ) is parallel to (u, v), we have { (δx, δy) (ϕx, ϕ y ) if δt >, (δx, δy) (ϕ x, ϕ y ) if δt <. (4) 19

20 As u 2 + v 2 = U 2, we have { δt = 1 U (δx)2 + (δy) 2 if δt >, δt = 1 U (δx)2 + (δy) 2 if δt <. (41) Assuming the length of the vector (δx, δy, δt) is small enough, we can obtain ϕ(x + δx, y + δy, t + δt) ϕ(x, y, t) = ϕ x δx + ϕ y δy + ϕ t δt { (4) ϕ (δx)2 + (δy) = 2 + ϕ t δt if δt > ϕ (δx) 2 + (δy) 2 + ϕ t δt if δt < { (41) (δx)2 + (δy) = 2( ϕ t /U ϕ ) if δt > (δx)2 + (δy) 2( ϕ ϕ t /U ) if δt <. (42) Hence, we have ϕ(x,y,t) ϕ(x+δx,y+δy,t+δt) (δx) = ϕ ϕ t /U if δt >, 2 +(δy) 2 ϕ(x+δx,y+δy,t+δt) ϕ(x,y,t) (δx) = ϕ ϕ t /U if δt <. 2 +(δy) 2 (43) If we take the limiting function δx and δy along the direction of the speed vector, clearly we also have δt according to Equation (41). Note that the left side of (43) includes both the local costs c(x, y, t) in this limiting function, so we can obtain the equation to compute ϕ(x, y, t) which is a Hamilton-Jacobi equation. 1 U ϕ t ϕ = c, (44) 4.3 The relationship between the special case and the general case In Section 4.1, we considered the case in which the cost is the travel time in the domain. An important question is whether this is a special case of the general strategy in Section 4.2. If the answer is yes, we will have more confidence in our new model. In Section 4.1, we denoted z(x, y, t D ) to be the departure time from point (x, y) such that the vehicle can arrive at the CBD at time t D. Thus, in the special case in which

21 the cost ϕ(x, y, t) is the travel time from point (x, y, t) to the CBD, we have ϕ ( x, y, z(x, y, t D ) ) = t D z(x, y, t D ). (4) Then, we can obtain i.e., { ϕx + ϕ t z x = z x, ϕ y + ϕ t z y = z y, { ϕx = (ϕ t + 1)z x, ϕ y = (ϕ t + 1)z y. Before we continue, let us consider the following theorem: (46) (47) Theorem 3: In the case that the cost ϕ(x, y, t) is the travel time in the domain, we have ϕ t 1. Figure 9: Two used paths Proof: The proof is similar to the proof of the dynamic user equilibrium principle in Section 4.1. As shown in Figure 9, suppose a vehicle can arrive at the CBD at time t D along the used path from point (x, y, t ), and can arrive at the CBD at time ˆt D along the used path from (x, y, t + t). From Section 4.1, we know that if a vehicle departs from (x, y) at a later time, it will arrive at the CBD later, so we have { ˆt D t D, if t, ˆt D t D, if t <. (48) 21

22 Now we can estimate the value of ϕ t as follows: ϕ(x, y, t + t) ϕ(x, y, t ) ϕ t = lim t t [ˆt D (t + t)] [t D t ] = lim t = lim t ˆt D t D t Using Equation (48), we get ϕ t 1. t 1. (49) From (47), we get (ϕ x, ϕ y ) (z x, z y ). As ϕ t 1, we also know that they point in the same direction. Hence, the requirement of (u, v) (ϕ x, ϕ y ) in Section 4.2 and the requirement of (u, v) (z x, z y ) in Section 4.1 are the same. In Section 4.2, the formula for ϕ is 1 U ϕ t ϕ = c. () In the case that the travel cost is the travel time in the domain, the local cost is in fact the travel time per unit distance of movement, c(x, y, t) = 1/U(x, y, t). Thus, we can get 1 U ϕ t ϕ = 1 U (47) = 1 U ϕ t (ϕ t + 1) 2 (z 2 x + z 2 y) = 1 U T heorem3 = 1 U ϕ t (ϕ t + 1) z = 1 U = z = 1 U. (1) We then obtain a formula to compute z, which is the same as the formula in Section 4.1. Using (47) and Theorem 3, we have proved that in the special case in which the only cost is the travel time, the path-choice strategies and the equations to compute the travel cost ϕ in Section 4.1 and Section 4.2 are actually the same.

23 4.4 The complete model and the discussion of initial boundary conditions So far, we have constructed a general path-choice strategy. In this section, we present some proper equations and their initial boundary conditions to construct a complete model using this new path-choice strategy The conservation law and its initial boundary conditions Similar to flow conservation in fluid dynamics, the density ρ(x, y, t) is governed by the following flow conservation law: ρ t + F = q, (x, y) Ω, t T, (2) where ρ t = ρ and = (, ). Using the path-choice strategy in Section 4.2, we know t x y that if we choose the vector v = (u, v) such that the resulting (ϕ x, ϕ y ) is parallel to (u, v), then the dynamic user equilibrium is satisfied, so we have F = ρv = ρu ϕ ϕ. (3) Assuming that no vehicle is allowed to enter the obstruction through the boundary Γ i or leave the city through Γ o, we have the boundary condition ρ(x, y, t) =, (x, y) Γ o Γi, t T. (4) In the physical sense, the density in the current time results from the density and its variation in the past. Hence, we set the initial time as t = and solve the conservation law along the positive time direction. Here, let ρ (x, y) be the density of traffic at location (x, y) at the beginning of the modeling period, and set the initial condition as ρ(x, y, ) = ρ (x, y), (x, y) Ω. () The Hamilton-Jacobi equation and its initial boundary conditions To solve the above conservation law, we still need to determine the travel cost ϕ to compute the flow vector in Equation (3). The travel cost can be solved by the 23

24 Hamilton-Jacobi equation, which was presented in Section 4.2: Let 1 U ϕ t ϕ = c, (x, y) Ω, t T. (6) ϕ(x, y, t) = ϕ CBD, (x, y) Γ c, t T, (7) which represents the boundary value of ϕ on Γ c and can be interpreted as the cost to the traveler of entering the CBD. There is still the problem of how to set the initial condition. Note that the travel cost to the CBD only depends on the events that will occur in the future, and has nothing to do with events that happened in the past. Hence, it seems reasonable to solve the Hamilton-Jacobi equation along the negative time direction, thus we set the initial time at t = t end. Here, we assume that all travelers have entered the CBD and there is no traffic in the city at t = t end. This can be considered as a static state and the travel cost to the CBD is the instantaneous cost. Following Huang et al. (9), we use a 2D Eikonal equation to solve the initial values ϕ (x, y): { ϕ (x, y) = c(x, y, t end ) (x, y) Ω, ϕ (x, y) = ϕ CBD, (x, y) Γ c. (8) The complete model We can write our model in two parts. The CL part is ρ t + F = q, (x, y) Ω, t T, F = ρu ϕ, (x, y) Ω, t T, ϕ ρ(x, y, t) =, (x, y) Γ o Γi, t T, ρ(x, y, ) = ρ (x, y), (x, y) Ω. The HJ part is 1 ϕ U t ϕ = c, (x, y) Ω, t T, ϕ(x, y, t) = ϕ CBD, (x, y) Γ c, t T, ϕ(x, y, t end ) = ϕ (x, y), (x, y) Ω, (9) (6) 24

25 where the initial value ϕ (x, y) is computed by a 2D Eikonal equation: { ϕ (x, y) = c(x, y, t end ), (x, y) Ω, ϕ (x, y) = ϕ CBD, (x, y) Γ c. (61) Here, ρ(x, y, t) is governed by the conservation law, and ϕ(x, y, t) is computed using the Hamilton-Jacobi equation. Note that the initial time in the conservation law is set as t =, whereas the initial time in the Hamilton-Jacobi equation is t = t end. Solution algorithm In this section, we describe the solution algorithms used to solve the improved model, including the Lax-Friedrichs scheme to solve the conservation law Equation (9), the Lax- Friedrichs scheme to solve the Hamilton-Jacobi Equation (6), the fast sweeping method to solve the Eikonal Equation (61), and the self-adaptive MSA to solve the fixed-point problem..1 Lax-Friedrichs scheme to solve the conservation law In this subsection, we assume that the cost potential function ϕ(x, y, t) is known for all (x, y) Ω and t T, and focus on the numerical method to solve the conservation law: ρ t + F = q, (x, y) Ω, t T, F = ρu ϕ, (x, y) Ω, t T, ϕ ρ(x, y, t) =, (x, y) Γ o Γi, t T, ρ(x, y, ) = ρ (x, y), (x, y) Ω. We use the conservative difference scheme to approximate the point values ρ n i,j ρ(x i, y j, t n ): ρ n+1 i,j = ρ n i,j t x ( ˆf i+ 1 2,j ˆf i 1 2,j) t y (ĝ i,j+ 1 2 (62) ĝ i,j 1 ) + q i,j t, (63) 2 where q i,j = q(x i, y j, t n ) is the given demand at location (x i, y j ) at time t n, x and y are the mesh sizes in the x and y directions, respectively, and ˆf i+ 1 2,j and ĝ i,j+ 1 2 numerical fluxes in the x and y directions, respectively. Here, we use the Lax-Friedrichs are 2

26 flux, which is a monotone flux: ˆf i+ 1 2,j = 1 2 [f(ρn i,j) + f(ρ n i+1,j) α f (ρ n i+1,j ρ n i,j)] (64) ĝ i,j+ 1 2 where α f = max f (ρ) and α g = max g (ρ). = 1 2 [g(ρn i,j) + g(ρ n i,j+1) α g (ρ n i,j+1 ρ n i,j)], (6).2 Lax-Friedrichs scheme to solve the time-dependent Hamilton- Jacobi equation In this subsection, we suppose that the density ρ(x, y, t) is known for all (x, y) Ω and t T, and focus on the numerical method to solve the Hamilton-Jacobi equation: 1 ϕ U t ϕ = c, (x, y) Ω, t T, ϕ(x, y, t) = ϕ CBD, (x, y) Γ c, t T, (66) ϕ(x, y, t end ) = ϕ (x, y), (x, y) Ω. Note that the initial time is t = t end and the initial value ϕ (x, y) is computed by the Eikonal Equation (61). We give the numerical method for solving the Eikonal equation in the Appendix, and in this subsection we assume that ϕ (x, y) is known. As the initial time is t = t end, we define τ = t end t, Φ(x, y, τ) = ϕ(x, y, t end τ), (67) thus we rewrite the time-dependent HJ equation into the usual form: 1 Φ U τ + Φ = c, (x, y) Ω, τ T, Φ(x, y, τ) = ϕ CBD, (x, y) Γ c, τ T, Φ(x, y, ) = ϕ (x, y), (x, y) Ω. (68) Define then the scheme to solve Φ τ + H(Φ x, Φ y ) = is H(Φ x, Φ y ) = U( Φ c), (69) where Φ n+1 i,j = Φ n i,j tĥ((φ x) i,j, (Φ x) + i,j, (Φ y) i,j, (Φ y) + i,j ), (7) (Φ x ) i,j = Φ i,j Φ i 1,j x, (Φ x ) + i,j = Φ i+1,j Φ i,j, (71) x 26

27 where (Φ y ) i,j = Φ i,j Φ i,j 1, (Φ y ) + i,j y = Φ i,j+1 Φ i,j, (72) y Ĥ is a Lipschitz continuous monotone flux consistent with H. Here we use the global Lax-Friedrichs flux: Ĥ(u, u +, v, v + ) = H( u + u + 2, v + v + ) αx (u + u ) 1 2 αy (v + v ), (73) where α x and α y are the viscosity constants and are defined as α x = max H 1(u, v), α y = max H 2(u, v). (74) A u B,C v D A u B,C v D Here, H 1 (H 2 ) is the partial derivative of H with respect to Φ x (Φ y ), [A, B] is the value range of u ± and [C, D] is the value range of v ±..3 Fixed-point problem and MSA Note that the two parts of the model are closely interconnected. When computing the density ρ by solving the CL (9), we must have known the total cost ϕ of getting to the CBD from every point, and thus we can decide on the direction of flow needed to compute the density at the next time level. When computing the cost ϕ by solving the HJ (6), we need the density information to obtain the local cost. However, neither ρ nor ϕ is known in advance, and these two equations cannot be solved together as they have different initial times. As mentioned in the introduction, this model is in fact a fixed-point problem that can be solved by the MSA. We illustrate this in detail in this subsection. Define the vector of the numerical solutions at each grid point and each time level as ρ = {ρ n i,j, i = 1,, N x, j = 1,, N y, n = 1,, N t } (7) ϕ = {ϕ n i,j, i = 1,, N x, j = 1,, N y, n = 1,, N t }, (76) where N x, N y and N t are the number of grid points in x, y and t, respectively. Now, let us give the definition for one iteration, which contains two steps, as shown in Figure. 27

28 Figure : Definition of one iteration Step 1. With a given vector ϕ old, we solve the CL (9) from t = to t = t end using the Lax-Friedrichs scheme described in Section.1, thus we obtain the vector ρ. We denote this step as ρ = g( ϕ old ). (77) Step 2. Solve the HJ Equation (6) from t = t end to t = using the Lax-Friedrichs scheme described in Section.2 to obtain an updated vector ϕ new. We denote this step as ϕ new = h( ρ). (78) We consider Step 1 and Step 2 as one iteration and denote it as ϕ new = h(g( ϕ old )) = f( ϕ old ). (79) With the definition of one iteration and the function f, the model translates to a fixedpoint problem ϕ = f( ϕ), (8) which can be solved by the MSA. The MSA was first introduced by Robbins and Monro (191) to solve the fixed-point problem. A general MSA is an iterative process. If we denote the solution before the kth 28

29 iteration as ϕ k, then computing the MSA to obtain ϕ k+1 involves the following steps: Step 1. Solve a temporary solution y k = f( ϕ k ) during the kth iteration. Step 2. Choose a step size λ k and use the following equation to get ϕ k+1 : ϕ k+1 = (1 λ k ) ϕ k + λ k y k, k = 1, 2, (81) Convergence is declared if ϕ k+1 ϕ k δ, (82) where δ is a given convergence threshold value. We use δ = 2 and L 2 as the norm in our computation. The selection of the step size sequence {λ k } is important to guarantee the convergence and efficiency of this method. In the conventional MSA, Robbins and Monro (191) suggested a predetermined step size sequence of {λ k = 1/k}. Performing the MSA with this step size sequence has two main weaknesses: i) the step sizes for the first few iterations are very large, which may amplify the error and more iterations will be required to reduce the error; and ii) the step sizes become too small after a large number of iterations, such that the weight of y k is too small in the computation of x k+1 and the convergence speed will be extremely slow. Hence, the conventional MSA often needs a large number of iterations to reach a convergent solution. To overcome these two weaknesses, some other methods consider the information generated from the iteration procedure and adjust the step size accordingly. For example, Liu et al. (9) constructed a self-regulated averaging method by slowing down the decrease in {λ k } when current iterations converged and speeding up the decrease in {λ k } when the solutions diverged. This method is much faster than the conventional MSA, but the convergence speed will still become extremely slow as long as the step sizes continue decreasing. Bar-Gera and Boyce (6) constructed a type of MSA with a constant step size sequence, which efficiently solved travel forecasting problems. They assumed that f is 29

30 smooth, and derived the following formula: ϕ k+1 f( ϕ k+1 ) 2 ϕ k f( ϕ k ) 2 r (λ), (83) where λ is the constant step size and r (λ) is a convex quadratic function which is pointing up. In addition, r () = 1 and r () <. Any constant step size such that λ r (λ) < 1 will result in convergence. Specially, there exits an optimal constant step size < λ 1 such that r (λ ) = min λ r (λ). However, the constant step size is related to the properties of f( ϕ), thus some prior information is needed to determine the step size. Bar-Gera and Boyce (6) did not offer a general method for determining the optimal constant step size for different problems. In the next subsection, we construct a self-adaptive MSA based on (83) derived by Bar-Gera and Boyce (6), and automatically determine the optimal step size for the MSA using the least squares method without prior information..4 A self-adaptive MSA Based on Formula (83), we can use the least squares method to fit the curve r (λ), and set the minimum point λ as the updated step size. The procedure for determining the optimal step size used in the self-adaptive MSA is as follows. 1. For the first several iterations, we use the predetermined step sizes λ 1 = 1., λ 2 =.4, λ 3 =.3, λ 4 =.2, λ =.1, λ 6 =.1, λ 7 =.. (84) 2. After the kth iteration (k > 2), we record the step size λ k 1 used before this iteration and the resulting ratio of the error r k 1 = ϕ k f( ϕ k ) 2 ϕ k 1 f( ϕ k 1 ) 2, (8) thus constituting a discrete point (λ k 1, r k 1 ) to fit the curve r (λ). 3

31 3. For the n + 1(n 7) step size, we use the least squares method to fit the discrete points (λ k, r k ), k = 1,, n and obtain the fitted quadratic curve r (λ). Define the new step size λ n+1 as the minimum point of r (λ), i.e. r (λ n+1 ) = min λ r (λ). 4. If the new step size is invalid, such that λ n+1 or λ n+1 1, then we abandon it and set λ n+1 =.λ n. There are several points to note about the self-adaptive MSA: The quadratic curve r (λ) must pass though the point (, 1), i.e. r () = 1, (86) so in fact only two parameters need to be determined to fit the curve. For the first several iterations, the accuracy of the discrete points (λ k, r k ) is low due to the nonlinearity of f, so we do not use the least squares method until the 8th step. We distribute the first several step sizes λ k, k = 2,, 7 on the interval [,.4], to avoid the step size distribution being too concentrated in a narrow range, which may lead to a poor least squares estimate. The descent speed of λ k, k = 1,, 7 is much faster than the conventional step size λ k = 1/k (Robbins and Monro, 191), hence avoiding too large an amplification of the error. The reason for resetting λ n+1 =.λ n when λ n+1 or λ n+1 1 is to avoid trying the same non-optimal step size repeatedly, which would not produce a better estimate of the quadratic curve. A small step size can also guarantee convergence. The self-adaptive MSA uses the information from the iterations and adjusts the step size accordingly. The step size may be inaccurate at the beginning, but be- 31

32 comes increasingly accurate as the number of iterations, k, increases. The numerical result shows that the step size is almost a constant after a number of iterations.. Solution procedure Figure 11: Solution procedure The question of how to set the initial value for the self-adaptive MSA remains. To start, let us compute the density along the positive time direction. However, we have no information about the actual total cost of traveling to the CBD. What we can do is compute the instantaneous total cost to the destination at each time level by solving a 2D Eikonal equation, and use the route strategy that satisfies the reactive dynamic user equilibrium principle (Huang et al., 9). Hence, we can compute the density ρ(x, y, t) by solving the following model: ρ t + F = q, (x, y) Ω, t T, F = ρu ϕ, (x, y) Ω, t T, ϕ ϕ = c(x, y, t), (x, y) Ω, t T, subject to the initial boundary conditions ρ(x, y, t) =, (x, y) Γ o Γi, t T, ρ(x, y, ) = ρ (x, y), (x, y) Ω, ϕ(x, y, t) = ϕ CBD, (x, y) Γ c, t T. (87) (88) 32

33 We then obtain the density vector based on the reactive dynamic user equilibrium principle and denote it as ρ 1. We can compute the actual total cost to the CBD by solving the HJ Equation (6) by backtracking in time to get y 1 = h( ρ 1 ), and set ϕ 2 = y 1. The solution procedure is as follows. 1. Compute the density ρ 1 from t = to t = t end using Equation (87), then compute the actual total cost y 1 = h( ρ 1 ) (89) from t = t end to t = by solving the HJ Equation (6), thus we get ϕ 2 = (1 λ 1 ) ϕ 1 + λ 1 y 1 = y Use the kth solution vector ϕ k to complete the kth iteration, k = 2,, i.e. { ρ k = g( ϕ k ), y k = h( ρ k ) = f( ϕ k ). (9) 3. Compute the step size λ k (k > 7) using the method described in Section.4. The step sizes λ k, k = 1,, 7 are predetermined. 4. Compute the (k + 1)th solution vector ϕ k+1 = (1 λ k ) ϕ k + λ k y k. (91). Stop the iteration process when (82) is satisfied. 6 Numerical example To demonstrate the effectiveness of the improved model and the proposed algorithm, we present a numerical example in this section and compare the result with the conventional MSA (Robbins and Monro, 191) and the self-regulated averaging method (Liu et al., 9). 33

34 Figure 12: The modeling domain 6.1 Problem setting As shown in Figure 12, we consider a rectangular domain that is 3km long and 2km wide in the numerical computation. The center of the compact CBD is located at (km, km). We assume that there is no traffic at the beginning or the end of the modeling period, (i.e., ρ (x, y) =, (x, y) Ω), and that no cost is incurred by entering the CBD (ϕ CBD =, (x, y) Γ c, t T ). We set t end = 6, so the modeling period is T = [h, 6h]. The traffic demand function q is defined as q(x, y, t) = q max [1 γ 1 d(x, y)]g(t), (92) where q max = 23veh/km 2 /h is the maximum demand,γ 1 =.1km 1 is a positive scalar, and d(x, y) = (x ) 2 + (y ) 2 is the distance from location (x, y) to the center of the CBD. The factor [1 γ 1 d(x, y)] is used to express the higher traffic demand generated in the domain closer to the CBD, where more of the population is located. g(t) is a non-negative and time-varying function defined by t, t [h, 1h], 1, t [1h, 2h], g(t) = 4(t 3) + 1, t [2h, 3h], 1, t [3h, h],, t [h, 6h]. (93) 34

35 The speed function is defined as U(x, y, t) = U f e βρ2, in which β = 2 6 km 4 /veh 2 and U f (x, y) is the free-flow speed given by U f (x, y) = U max [1 + γ 2 d(x, y)], (94) where U max = 6km/h is the maximum speed and γ 2 = 4 3 km 1. The factor [1 + γ 2 d(x, y)] is used to express the faster free-flow speed in the domain far from the CBD, where there are fewer junctions. With this definition, we can compute the critical density of ρ c as veh/km 2. The local travel cost per unit distance of travel is defined as c(x, y, t) = κ( 1 + π(ρ)), where κ = 9$/h and π(ρ) = U 8 ρ 2. We assume that the capacity of the CBD is large enough to accommodate all of the travelers in the city, so F should maintain the maximum flow intensity F max at the CBD boundary under the congested condition. We now use the composed algorithm described in the previous section to perform the numerical simulation. A uniform mesh with an N x N y grid is used. The mesh grids inside the CBD are not computed. The numerical boundary conditions are summarized as follows. 1. On the solid boundary Γ o, the outer boundary of the city, we set ρ = at the ghost points outside the city and thus the flow intensity F = ρu =. In the Hamilton-Jacobi equation, the boundary values of ϕ are obtained by extrapolation from inside the computational domain. In the Eikonal equation, we set ϕ = 12 at the ghost points outside the city. 2. On the boundary of the compact CBD, i.e. Γ c, we set ϕ = in both the HJ equation and the Eikonal equation. The boundary conditions for ρ inside the CBD are obtained by extrapolation from the grids outside the CBD. To maintain the maximum flow intensity F = F max on the boundary of the CBD under the congested condition, we set U(x, y, t) = U f inside the CBD. 3