Optimizing Network Flows with Congestion-Based Flow Reductions
Douglas R. Bish, Edward P. Chamberlayne & Hesham A. Rakha
Networks and Spatial Economics A Journal of Infrastructure Modeling and Computation ISSN 1566-113X Netw Spat Econ DOI 10.1007/s11067-012-9181-3
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Author's personal copy Netw Spat Econ DOI 10.1007/s11067-012-9181-3
Optimizing Network Flows with Congestion-Based Flow Reductions Douglas R. Bish · Edward P. Chamberlayne · Hesham A. Rakha
© Springer Science+Business Media, LLC 2012
Abstract When optimizing traffic systems using time-expanded network flow models, traffic congestion is an important consideration because it can decrease both the discharge traffic flow rate and speed. One widely used modeling framework is the Cell Transmission Model (CTM) (see Daganzo, Transp Res-B 28(4):269–287, 1994, Transp Res-B 29(2):79–93, 1995), which is implemented in a linear program (LP) in Ziliaskopoulos (Transp Sci 34(1):37– 49, 2000). While the CTM models the reduction in speed associated with congestion and the backward propagation of congestion, it does not properly model the reduction in discharge flow from a bottleneck after the onset of congestion. This paper discusses this issue and proposes a generalization of the CTM that takes into account this important phenomena. Plainly, an optimization that does not consider this important negative result of congestion can be problematic, e.g., in an evacuation setting such an optimization would assume that congestion does not impact network clearance time, which can result in poor evacuation strategies. In generalizing the CTM, a fairly simple modification is made, yet it can have significant impacts on the results. For
D. R. Bish (B) Grado Department of Industrial and Systems Engineering (0118), Virginia Tech, Blacksburg, VA 24061, USA e-mail:
[email protected] E. P. Chamberlayne Charleston District, US Army Corps of Engineers, 69A Hagood Avenue, Charleston, SC 29403, USA H. A. Rakha Charles E. Via, Jr. Department of Civil and Environmental Engineering (0105), Virginia Tech, Blacksburg, VA 24061, USA
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instance, we show that for the generalized CTM the traffic holding (a result of the linearization of the CTM flow constraints) plays a more harmful role, which thus requires a scheme to eliminate traffic holding. In this paper, we propose a mixed binary program to eliminate traffic holding, along with methods to improve solvability. Keywords Cell Transmission Model (CTM) · Mixed binary programming · Time-expanded network flows · Congestion · Evacuation planning/modeling
1 Introduction Models that optimize network traffic flows over time are useful in many traffic planning applications; examples include dynamic traffic assignment and regional evacuation planning (our main area of interest). The Cell Transmission Model (CTM) introduced in Daganzo (1994, 1995) and incorporated into a linear program (LP) in Ziliaskopoulos (2000) is such a model. This framework has been used or studied in numerous papers, including those that study dynamic traffic assignment (Li et al. 2003; Ukkusuri and Waller 2008), ramp metering and traffic signal control (Lo 2001; Lin and Wang 2004; Gomes and Horowitz 2006), and evacuation modeling and planning (Tuydes and Ziliaskopoulos 2006; Sbayti and Mahmassani 2006; Liu et al. 2006; Chiu et al. 2007; Yazici and Ozbay 2008; Xie et al. 2009; Yao et al. 2009; Chung et al. 2011, 2012). Traffic flow is complex (see Kerner and Klenov 2006; Nagel et al. 2003; Schnhof and Helbing 2007, for instance) and optimization-based traffic flow models must include many simplifications in order to be tractable. The CTM is a first-order macroscopic traffic stream model that conforms to the hydrodynamic traffic flow and density relationships proposed by Lighthill and Whitham (1955), Richards (1956) (commonly referred to as the LWR model). The CTM provides a promising modeling framework for traffic flow optimization because it models the spillback propagation and speed reduction caused by congestion, and can be incorporated, in an approximate manner, within an LP (see Ziliaskopoulos 2000), which are generally computationally efficient. To gauge whether a particular modeling framework is appropriate, we can examine whether important higher level metrics are appropriately developed. In this case, we can do this using empirical data and higher fidelity microscopic traffic simulators. Examining the CTM from this perspective, we find a significant problem, which is that the CTM fails to capture the reduction in the f low discharge rate after the onset of congestion at a bottleneck. Traffic stream models typically have flows that initially increase with density until a critical density is reached, after which the flow decreases as density increases, eventually reaching a flow of zero at jam density (May 1990). The fact that congestion upstream of a bottleneck reduces the flow discharge rate from the bottleneck has been extensively shown in various empirical studies (Banks 1990; Hall and Agyemang-Duah 1991; Cassidy and Bertini 1999; Chung et al.
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2007) and in traffic simulations (Chamberlayne 2011). In Papageorgiou (1998), the author states that LWR-based models like the CTM will seek to create congestion in order to maximize the network outflow. In this paper we present a generalization of the CTM that captures bottleneck flow discharge rate reductions after the onset of congestion. When incorporating the CTM into an LP (see Ziliaskopoulos 2000) the resulting solution often exhibits traf f ic holding, which is an artifact of linearizing the CTM. This linearization results in an under-constrained problem, which in turn allows the model to have an unrealistic level of control and restrict flows (i.e., traf f ic holding) at locations where it is not realistic or reasonable. This problem will be discussed in more detail in Section 3.1, and is especially relevant to our topic because traffic holding is significantly influenced by flow reductions at bottlenecks. To use optimization-based models to develop traffic management strategies, it is important that they model the effects of congestion sufficiently and do not allow traffic holding to produce unrealistic solutions. For instance, when considering network flows during an emergency evacuation, congestion can become severe, lasting for hours, and causing large reductions in the network flow capacity. For obvious reasons, this type of congestion should be avoided since it is not only an inconvenience but can threaten lives during an emergency evacuation. Furthermore, if a solution depends on traffic holding, yet there is no way to enforce this behavior, it will not be effective. This paper advances the understanding of this difficult problem of network flow over time with congestion by making the following specific contributions: 1. The development of a generalized CTM that models the reduction in flow discharge rates at a bottleneck after the onset of congestion. 2. The implementation of the generalized CTM within a mathematical program that eliminates unwarranted traffic holding, along with a discussion of traffic holding and congestion-based flow reductions. 3. Improvements to the computational performance of the mathematical program incorporating the generalized CTM. 4. Numerically demonstrates the generalized CTM and the mathematical program incorporating the generalized CTM, using two common freeway bottleneck examples. This paper seeks to capture the net effect of congestion within a mathematical program for the purposes of network flow optimization; it is not an attempt to advance traffic flow theory but is intended for use in transportation network planning and to better understand optimal traffic flows. The remainder of this paper is structured as follows. Section 2 reviews the CTM and proposes a generalized CTM that models the impacts of congestion more realistically, along with an illustrative example. Section 3 reviews a linear programming implementation of the generalized CTM, similar to the one introduced in Ziliaskopoulos (2000), discusses the problem of traf f ic holding, and proposes a mixed binary programming formulation to eliminate traffic
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holding. Also included is a freeway merge example and methods for reducing the computation effort required to solve this problem. Section 4 presents the research conclusions.
2 A generalized cell transmission model In this section, we provide an overview of a generalized CTM that is based on the work presented in Daganzo (1994, 1995). This framework better models congestion, specifically considering the the effects of congestion on the discharge traffic flow and speed through a bottleneck. The results are demonstrated using a freeway bottleneck example and compared to a microscopic traffic simulation that captures these effects. The CTM utilizes a discrete time-expanded network of cells and links (C, L) to represent the roadway system of interest. The set of cells, C, is composed of the following subsets: source cells (So ), roadway cells (R), and sink cells (Se ). Links represent allowable movements between cells. The planning horizon is divided into T time intervals of equal length. Roadway cells represent sections of roadway of a length such that vehicles traveling at free-flow speed can traverse them in one time interval. These roadway cells are further classified as ordinary, merging (when multiple links enter the cell), or diverging (when multiple links exit the cell), see Fig. 1. Additional parameters and variables are defined as follows: Parameters: Qi Ni i δi dit
: maximum flow into or out of roadway cell i in a single time interval, ∀i ∈ R : maximum number of vehicles that roadway cell i can hold (i.e., the jam density), ∀i ∈ R : potential flow out of roadway cell i at the maximum traffic density (xit = Ni ), i ∈ (0, Qi ] : ratio of wave speed for the propagation of congestion to free-flow speed for roadway cell i, ∀i ∈ R : number of vehicles that flow out of the source i in time interval t, ∀i ∈ So , t = 1, · · · , T
Decision Variables: xit yijt
: number of vehicles in cell i at the beginning of time interval t (i.e., the cell density), ∀i ∈ C, t = 1, · · · , T : number of vehicles that flow over link (i, j) in time interval t, ∀(i, j) ∈ L, t = 1, · · · , T
Note that while the Qi , Ni and δi -parameters do not vary with respect to time in this paper, it is easy to include time-varying parameters.
Author's personal copy Optimizing Network Flows with Congestion-Based... Fig. 1 A small network example with merging cell h, ordinary cells i and j, and diverging cell k
The generalized CTM determines the flow over link (i, j) using Eq. (1).1 When i = Qi , the second term of Eq. (1) simplifies to Qi , which then makes Eq. (1) equivalent to the flow equation in Daganzo (1994). Because of this, we will denote the original CTM as CTM=Q , and the more generalized form as CTM≤Q . yijt = min xit , Qi − xit − Qi (Qi − i )/(Ni − Qi ), Q j, δ j N j − xtj . (1) The CTM parameters have the following relationship for any roadway cell i: 0 < i ≤ Qi < Ni , while 0 < δi ≤ 1. Furthermore, Qi represents the maximum flow into or out of cell i, based on the parameters and density of cell i. This flow might not be achievable, for instance, if the adjacent downstream cell j had a lower Q-value. It does imply that the parameters be set such that δi [Ni − xit ] > Qi when xit < Qi , else the definition of Qi would be contradicted, that is, cell i itself would prohibit a flow of Qi from ever entering cell i. The minimum of the first two terms of Eq. (1) represent the potential f low out of cell i, based on its parameters and traffic density, which Fig. 2(a) and (b) illustrates for the CTM=Q and CTM≤Q when i < Qi , respectively. Likewise, the minimum of the last two terms of Eq. (1) represent the potential f low into cell j, i.e., the flow that cell j can accept based on its parameters and traffic density, which Fig. 2(c) illustrates. The actual flow is the minimum of these two potentials. Essentially, Fig. 2 shows how the components of the trapezoidal CTM flowdensity relationship are assigned to cells when determining flows, which allows us visualize the relationship better. To reduce the flow due to high traffic densities, the CTM relies on the downstream cell and the δ j[N j − xtj]-term of Eq. (1), but this does not work at a bottleneck, as we show below. When xit ≤ Qi , cell i is in the free-flow state, that is, cell i has the potential to send all xit vehicles to the downstream adjacent cell. When xit > Qi , cell i is in the congested state. For CTM=Q the congested state is defined by a horizontal line segment, see Fig. 2(a), which implies a reduction in traffic speed as density increases, but not a reduction in discharge flow. Thus, for a cell in isolation, CTM=Q has an unusual flow-density relationship, one where flow does not decrease, even at jam density. For CTM≤Q , when i < Qi , the congested state is defined by a downward sloping line segment, see Fig. 2(b), which implies a reduction
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slightly different equation is used for links that enter a merge cell or leave a diverge cell (see Daganzo 1994, for details). Also if i is a source, or j a sink, we can eliminate their respective terms.
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Fig. 2 The flow-density relationship for ordinary link (i, j) for the CTM (a) the potential flow out of i when i = Qi , (b) the potential flow out of i when i < Qi , and (c) the potential flow into j, based on Eq. (1)
in both traffic speed and the queue discharge flow rate as density increases. This generalization is a simple and logical change, which is more consistent with traffic flow theory and is similar to the common triangular flow-density relationship (e.g., see Rakha and Crowther 2002), but modified for use in a cellular framework (e.g., having a i = 0 would be problematic, and would not model flow at a bottleneck well). As we show later, this change only impacts congested bottleneck flows, but does not change the CTM performance in other situations or locations. The reduction in discharge flow after the onset of congestion accounts for the delay associated with vehicle accelerations from congested speeds to higher speeds (Chamberlayne et al. 2011). These losses increase as the speed difference between the congested and free-flow state increases, as was demonstrated by El-Metwally and Rakha (2011). This generalization of the CTM has some important modeling implications, as we discuss later. We make the following observation for roadway cell i in the free-flow state. Observation 1 If a roadway cell i is in the free-f low state (xit ≤ Qi , i ∈ R), then it will remain in the free-f low state unless the downstream adjacent cell j is a bottleneck (Qi > Q j) or in the congested state such that Qi > δ j[N j − xtj]. This observation implies that if the system is initially in the free-flow state (xi1 ≤ Qi , i ∈ R) then congestion can only start because of a bottleneck. While this is fairly straight forward, bottlenecks are extremely important in determining network performance. For the CTM=Q , bottlenecks are problematic. To demonstrate, consider the following example, which consists of a simple bottleneck caused by a lane drop. Example 1 Consider a section of freeway that contains a lane drop from three to two lanes, thus forming a bottleneck, see Fig. 3. Each roadway cell is 0.5455 miles in length, the free-flow speed is 65.45 miles/h, and the length of each time interval is 0.5 min. We assume a maximum flow rate of 1920 vehicles/lane/hour, which yields Q = 48 for the three lane segments (Cells 1–8) and Q = 32 for the two-lane segment (Cell 9). The N-parameters (based on a maximum or
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Fig. 3 Example 1: a bottleneck caused by a freeway lane drop
jam density of 198 vehicles/mile/lane) are 324 and 216 vehicles/cell for the three-lane and two-lane segments, respectively. All links in this network are ordinary. The flow out of the source (So ) is 48 vehicles/interval for 20 intervals (which will activate the bottleneck). We set δ = 1 for every roadway cell.2 One major implication of the CTM=Q is that the flow through a bottleneck is not reduced after the onset of congestion at the bottleneck. Figure 4 shows that the density in Cell 8 increases through time until an equilibrium density of N8 − Q9 /δ8 = 292 vehicles/cell is reached, which yields a traffic speed of Q9 /(N8 − Q9 /δ8 ) = 0.1096 cells/interval (the traffic speed when xt8 = N8 is Q9 /N8 = 0.0988 cells/interval). Despite the high level of congestion, the flow over Link (8, 9) is never reduced below Q9 = 32 vehicles/interval. This is because Eq. (1) has no mechanism for reducing the potential flow over a link based on high traffic densities in the upstream cell when = Q. The fourth term of Eq. (1) is never activated for Link (8, 9) as it is meant to limit flow into Cell 9 when Cell 9 has a high traffic density (in this example Cell 9 will remain in the free-flow state). It is well known that congestion can reduce throughput of a bottleneck (e.g., as mentioned in Papageorgiou et al. 2003), as shown empirically in many studies of traffic data (see, for instance Banks 1990; Hall and Agyemang-Duah 1991; Cassidy and Bertini 1999; Bertini and Leal 2005; Chung et al. 2007). To put this into context, for this example the equilibrium density of 292 vehicles/cell is equivalent to 178.4 vehicles/mile/lane. The Highway Capacity Manual places this firmly in the F-Level of Service range for a similar highway section, yet despite this, the CTM=Q produces flows through the bottleneck that are not negatively impacted by the congestion. To illustrate why this is a concern, consider using CTM=Q in an optimization-based evacuation planning setting (see, for instance, Liu et al. 2006; Tuydes and Ziliaskopoulos 2006; Chiu et al. 2007; Shen et al. 2007; Yao et al. 2009; Chung et al. 2011, 2012; Bish et al. 2011). For evacuation problems, an important metric is the network clearance time (e.g. Chalmet et al. 1982; Han et al. 2007; Bish and Sherali 2012), the elapsed time required for all demand to clear the system (i.e., reach a sink). When traffic flow is modeled using CTM=Q , congestion does not negatively impact network clearance time, and thus congestion is immaterial for this metric (or perhaps even desirable, as it
Kalafatas and Peeta 2006; Dixit et al. 2008) recommend setting the δ-parameter to 1 if Qi /(Ni − Qi ) is in the range of 1/6–1/4 for freeway modeling.
2 (See
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ensures maximum throughput). This is also true of another evacuation related metric, the average time the demand is in the system (before reaching a sink). Thus, here CTM=Q fails to exhibit the desired high-level network behavior (we explore this further below). For Example 1 the network clearance time is 40 intervals when = Q. To estimate a network clearance time for Example 1 in a more realistic setting we use the microscopic traffic simulator INTEGRATION, see Van Aerde and Rakha (2007a, b) for more details on the simulator3 and Chamberlayne (2011) for more details on the study. While the traffic simulator is a high fidelity model which displays more realism (e.g., Chamberlayne et al. (2011) validated the post break-down flow discharge drop from INTEGRATION against empirical observation), we hope to be able to produce similar highlevel results, especially for important metrics such as the network clearance time. Figure 5 displays the network clearance time for Example 1 from INTEGRATION and from the generalized CTM for various -values. We observe that when = 0.2Q the network clearance time is 60 time intervals, which closely matches the simulation results and that CTM=Q underestimates the network clearance time by 33 % compared to the (more realistic) simulator. Much like the other CTM parameters, should be estimated based on empirical data, and can vary based on roadway characteristics and traffic composition. Figure 6 shows the traffic flows through the bottleneck (i.e., out of Cell 8) and the traffic densities for Cell 8 for Example 1 when = 0.2Q. An equiδ N 2 −N Q8 −δ8 N8 Q8 +Q8 8 librium density of 8 8δ8 (N88 −Q = 312.3 vehicles/cell (this equilibrium 8 )−Q8 +8 is for all cells that correspond to three lane roadway segment) is attained.
3 INTEGRATION
has been validated against standard traffic flow theory (Rakha and Crowther 2002, 2003; Dion et al. 2004) and been utilized for the evaluation of real-life applications (Rakha et al. 1998, 2005).
Author's personal copy Optimizing Network Flows with Congestion-Based... Fig. 5 The network clearance time for Example 1 for various -values and from INTEGRATION
The equilibrium density is the density where the second term of Eq. (1) for Link (8, 9) (i.e., Q8 − (xt8 − Q8 )(Q8 − 8 )/(N8 − Q8 )), and the fourth term of Eq. (1) for Link (7, 8) (i.e., δ8 [N8 − xt8 ]) are equal. In other words, the congested density where the flow into and out of Cell 8 are equalized. At this level of congestion the flow is 11.68 vehicles/interval and the traffic speed is 0.0374 cells/interval. Figure 6 shows that this equilibrium density lasts from time interval 24 until time interval 44. Papageorgiou (1998) used a similar example to illustrate this very problem for LWR models, which includes CTM=Q . Setting = Q is consistent with hydrodynamic models; the flow of water through a funnel is not reduced as the funnel fills with water. Traffic, which is a flow of heterogeneous particles, behaves in a much more complex manner, and we do not expect macroscopic models to exhibit this complexity (in fact it is not necessarily desirable), but they should exhibit the important high-level network effects of this complexity. The following important property of the CTM describes how congestion builds and dissipates.
Fig. 6 Flows out of the bottleneck and density for Example 1 using CTM
