The Influence of Assignment Criteria for the Solution of ... - Core

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ScienceDirect Transportation Research Procedia 10 (2015) 850 – 859

18th Euro Working Group on Transportation, EWGT 2015, 14-16 July 2015, Delft, The Netherlands

The influence of assignment criteria for the solution of dynamic travel demand estimation Marialisa Nigroa,*, Ernesto Cipriania, Luca Di Pietrantonioa, Akmal Abdelfatahb a

Department of Engineering, Roma Tre University, Via Vito Volterra 62, Rome 00146, Italy b Department of Civil Engineering, American University of Sharjah, Sharjah, UAE

Abstract This study deals with an investigation of the influence of several assignment criteria, combined with a different set of information about traffic conditions, on the efficacy and stability of the solution of the off-line dynamic demand estimation problem. Motivations of such investigation derive from the need to better reproduce the real users’ behaviors during the demand estimation and benefit from the extensive information, which is provided by traffic monitoring systems that collect advanced traffic data ubiquitously distributed on the network. Several synthetic experiments have been conducted on a test network, obtaining interesting results about the common adoption of dynamic user equilibrium and about the possibility to introduce new types of information during the optimization. © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license © 2015 The Authors. Published by Elsevier B. V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and peer-review under responsibility ofTechnology Delft University of Technology. Peer-review under responsibility of Delft University of Keywords: Travel demand estimation; dynamic traffic assignment models; route choice models; SPSA.

1. Introduction This work deals with the off-line estimation of OD matrices for a within day dynamic framework, and specifically it investigates the impact of assignment criteria, combined with several information on traffic regime,

* Corresponding author. Tel: +39 06-573-33-441 E-mail address: [email protected]

2352-1465 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of Delft University of Technology doi:10.1016/j.trpro.2015.09.038

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during the estimation. Different approaches exist in the literature to solve the dynamic OD matrices estimation problem. For a review of early contributions, the reader is referred to Balakrishna et al. (2005), whilst in Toledo et al. (2015) more recent research attempts can be found. The bi-level approaches are the main ones considered in this paper. In such cases, the upper-level problem consists of the adjustment of a starting demand using traffic measurements. These traffic measurements are in turn linked to the dynamic demand and this link is generated through the Dynamic Traffic Assignment (DTA) problem at the lower level. As reported in Antoniou et al. (2015), relevant topics faced in literature about dynamic demand estimation deal with 1) non-linearity of the relation between ODs and traffic measurements; 2) high indeterminateness of the problem due to the dimension of the unknowns; 3) selection of traffic flow measurements to be adopted during the optimization. Respect to the latter point, Balakrishna (2006) and Cipriani et al. (2011; 2014) accurately investigated the role of link counts, speeds and occupancy; several authors adopted probe data from vehicle equipped by AVI tags (Dixon and Rilett, 2002; Eisenman and List, 2004; Antoniou et al., 2004; Zhou and Mahmassani, 2006; Caceres et al., 2007, Barcelò et al., 2012; Mitsakis et al., 2013; Cipriani et al., 2014). Also the influence of aggregated demand information derived by commonly adopted demand models, such as traffic emissions and attractions by zones, or the reliability of the starting demand matrix have played an important role in some literature (Iannò and Postorino, 2002; Cipriani et al., 2014; Cantelmo et al. 2014a-b, 2015). Less often the impact of the assignment criteria (i.e. the type of DTA), combined with the traffic measurements adopted in the optimization, has been investigated: this is the aim of the paper. The common approach to solve the DTA, required at the lower level of the demand estimation, is to follow a Dynamic User Equilibrium (DUE) principle in case of congested networks, while Dynamic Network Loading (DNL) is usually adopted for not congested networks or for networks without path’s options. However, despite the DUE has the advantage of smoothing out the inherent stochasticity of route choice through the iterative process needed to search for the equilibrium, the concept behind is a modelling construct that makes sense under specific hypotheses: 1) travelers are assumed to choose O-D routes that require minimum disutility; 2) travelers are assumed to know, and accurately perceive, travel times throughout the network; 3) O-D flows and roadway characteristics are assumed to be fixed and known (Chiu et al., 2001). If the modeling time horizon is not long enough or if advanced traveler information and management systems suggest a route option to inexperienced road users, the equilibrium principle can hardly be reached. Several synthetic experiments have been conducted on a test network consisting of 22 nodes (14 are signalized intersections), 68 links, 6 traffic zones, a whole planning horizon of 35 minutes discretized into 5 minutes intervals. Specifically, these experiments consider: x The Dynasmart model (Jayakrishnan et al., 1994) to solve the DTA; x Different assignment criteria for the DTA, specifically 1) Dynamic Network Loading (DNL), 2) Dynamic User Equilibrium (DUE) approach, 3) System Optimum (SO) approach, 4) different percentages of users that follow DUE or SO, 5) different percentages of users that follow the previous approaches with users following always the same path (“unresponsive users”). These different DTA options allow mimicking possible behaviors of users, even in presence of information and control systems on the network; x Different combinations of traffic measurements inside the objective function of the dynamic travel demand estimation, also proposing the adoption of new traffic measurements with respect to those commonly reported in literature; x The Simultaneous Perturbation Stochastic Approximation, Asymmetric Design, Polynomial Interpolation (SPSA AD-PI, Cipriani et al. 2010, 2011) method to solve the off-line estimation of dynamic OD matrices. This paper contains four sections including this introduction: Section 2 deals with the formulation of the off-line dynamic demand estimation and the method adopted to solve it. Section 3 presents the experimental design and the results obtained on the adopted test network. Conclusions follow in Section 4.

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2. Off-line dynamic demand estimation: formulation and solution method Given the period of analysis T, divided into nh intervals, a subset of links S ={1..ns}  A and nodes P ={1..np}  N equipped with sensors, a subset of monitored routes U ^1..nr ` R, the problem of off-line simultaneous dynamic demand adjustment can be generalized as: d1*...d*n arg min (x ...x ) ª« f1 x1...x n , d1 ...d n  ¦ f 2l y1...y n , yˆ 1...yˆ n ¦ f 3t z1...z n , zˆ 1...zˆ n ¦ f 4p w1...w n , wˆ 1...wˆ n º» (1) l t p ¬ ¼ h

1

nh

h

h

h

h

h

h

h

h

where: xi = estimated matrix for departing time interval i, i=1..nh di = seed matrix for departing time interval i, i=1..nh yi = simulated information on links  S in the time interval i, i=1..nh yˆ i = collected measurements on links  S in the time interval i, i=1..nh l = type of collected measurements on links zi = simulated information on nodes  P in the time interval i, i=1..nh zˆ i = collected measurements on nodes  P in the time interval i, i=1..nh t = type of collected measurements on nodes wi = simulated information on routes r for departing time interval i, i=1..nh wˆ i = collected measurements on routes r for departing time interval i, i=1..nh p = type of collected measurements on routes. The link between the measurements and the demand is captured by simulation: y1…ynh = Γ(x1…xnh) for each l z1…znh = Γ(x1…xnh) for each t w1…wnh = Γ(x1…xnh) for each p

(2)

with Γ = DTA Generally Γ is reported in previous works of the authors as a Dynamic User Equilibrium (DUE) assignment; however similar approaches in literature adopt also Dynamic Network Loading (DNL) when there is no need for route choice (Frederix et al., 2013, Cantelmo et al., 2014) and non-equilibrium approaches. The solution approach adopted, in this paper, is the SPSA AD-PI (Cipriani et al., 2010, 2011), where an average approximated gradient is computed based on a simultaneous asymmetric perturbation of each Origin-Destination (OD) variable. Then the solution is updated by a minimization of a third degree polynomial interpolation of the Objective Function (O.F.) (1) along the descendent direction.

3. Numerical experiments 3.1. Experimental design Synthetic experiments have been conducted on the test network reported in Figure 1, considering a planning horizon of 35 minutes discretized into 5 minutes intervals. The 6 traffic zones of the network generate a total value of 252 dynamic O-D variables to be estimated. Scenarios in terms of assignment models and traffic measurements adopted during the dynamic demand estimation are reported in Table 1: for each DTA, thus for each scenario, eight combinations of link measurements inside the O.F. have been investigated. Each term of the different O.F.s is expressed by the Normalized Root Mean Square Error statistic (NRMSE) as goodness of fit between simulated values and traffic measurements. All the required simulations have been conducted using Dynasmart (DYnamic Network Assignment Simulation Model for Advanced Road Telematics, Mahmassani et al. 2004).

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The experiments start from a traffic demand supposed as the “true” demand (18,900 vehicles), which is assigned on the network in order to procure the measurements to be included in the O.F. (ground truth conditions). The assignment criterion to generate the measurements changes depending on the considered scenario: For example, the measurements adopted in Scenario 1 derive from a DUE of the real demand, while for Scenario 2 they derive from a SO. Then, the “true” demand has been perturbed in order to obtain a “seed” demand (12,638 vehicles), which is the starting point for the optimization.

Figure 1. Test network Table 1. Experiment design Experiments Scenario I Scenario II Scenario III Scenario IV Scenario V Scenario VI Scenario VII

Type of DTA DUE SO DNL 50% DUE – 50% SO DNL + fixed DUE paths DNL + fixed DUE (highest O–D flows) DUE + fixed DUE (highest O–D flows)

paths paths

Measurements inside the O.F. for each type of DTA O.F.1: Link Volume O.F.2: Link Volume + Speed O.F.3: Link Volume + Density O.F.4: Link Volume + Queue length O.F.5: Link Volume + Speed + Density O.F.6: Link Volume + Speed + Queue length O.F.7: Link Volume + Density + Queue length O.F.8: Link Volume + Density + Speed + Queue length

Respect to the commonly adopted DTA-DNL (Dynamic Network Loading), DUE (Dynamic User Equilibrium) and SO (System Optimum) criteria, a set of experiments has been conducted considering one-half of the users following the DUE criterion and the remaining following an SO criterion (Set IV: 50% DUE + 50% SO). From Scenario V to Scenario VII, DNL or DUE approaches have been mixed with fixed route choices for a subset of O–D pairs: in Scenario V some fixed paths (specifically, 28 paths) have been randomly selected from the paths used under DUE conditions. In Set VI and VII, the fixed paths still belong to the DUE paths, but they are selected among the set of O–D pairs with the highest flows in order to cover at least the 33% of the total demand. About the measurements adopted inside the O.F., it is interesting to notice the queue length: this new measure could add more accurate information on congestion phenomena, as compared to the commonly adopted speed and density, thus requiring a thorough investigation. In addition, other link measurements were available as output of simulations by Dynasmart. Specifically, these are left turn movements and outflows. These types of measurements, although potentially significant for the estimation procedure, cannot be adopted during the experiments due to the high noise generated in the O.F. for different demand values and distribution, as illustrated in Figure 2. Before proceeding with the demand estimation procedure, a sensitivity analysis has been conducted in order to fix the number of iterations for each type of assignment. This has been set as a function of the “violations”, i.e. the statistic adopted by Dynasmart to measure the convergence (Figure 3), computed as the number of times the minimal difference (in vehicles) of assignment levels between two consecutive iterations, for all O-D pairs, for all departure time intervals, is exceeded.

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Figure 2. Lefturn and Outflow error trend for different demand matrices

Figure 3. Number of violations as a function of assignment (DUE and SO) iterations 3.2. Results Table 2 shows that the different traffic measurements can influence the efficacy and stability of the estimation, regardless of the assignment method adopted. Link volumes alone are not satisfactory and also the adoption of speed cannot be sufficient. Table 2. Influence of different measurements on efficacy and stability of the estimation. Measurements adopted inside O.F.

Average O.F. reduction [%]

St.dev.

Volume Volume+speed Volume+density Volume+Queue Volume+speed+density Volume+speed+queue Volume+density+queue Volume+density+speed+queue

-62 -77 -92 -92 -85 -90 -100 -98

41 29 12 18 23 14 1 2

It is required to add other measurements that are able to represent congestion phenomena as densities and queue length. The queue length is absolutely information to be taken into account: it is demonstrated not only by the improvements in terms of the O.F. reduction, but also by the stability of the estimation. In fact the stability can be influenced both by the type of measurements and the type of DTA: however, it is always high (low standard deviation) when the queue length is considered. Link volumes and speeds are instead very sensitive to the change of

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assignment model. Table 3 shows again the efficacy and the stability of the estimation, but as a function of the type of DTA, regardless of the traffic measurements adopted inside the O.F. Table 3. Influence of DTA approaches on efficacy and stability of the estimation.

Figure 4.

Figure 5.

DTA

Average O.F. reduction [%]

St.dev.

DUE SO 50%DUE+50%SO DNL + fixed DUE paths DNL + fixed DUE paths (highest O–D flows) DUE + fixed DUE paths (highest O–D flows)

-99 -92 -96 -68 -76

0.2 21 8 30 28

-91

24

Link volume error trend for different demand matrices (with DNL alone)

Link volume error trend for different demand matrices (DNL with fixed paths)

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Figure 6.

Link volume error trend for different demand matrices (DUE with fixed paths)

It can be noticed that scenario III (DNL alone) is not reported: this is due to the strong instability of the O.F., where also little changes in the travel demand can result in very different route choices. This is reported in Figure 4, where a typical monodimensional scan of the O.F. has been performed on the test network of Figure 1, using a convex combination between the seed matrix (alpha = 0) and the real matrix (alpha = 1). It is obvious that in such a case, a demand estimation method based on a gradient approach (as the SPSA) cannot be applied. Instead, when the DNL is mixed with some fixed paths (Scenario IV and V) the O.F. trend shows low noise levels and a clear descendent direction (Figure 5). Regardless of the type of information adopted in the O.F., using the DUE criterion resulted in greater stability and efficacy to the estimation procedure when compared with the other assignment models/route choice options. When DUE is mixed with fixed paths, results are not as good as one might expect: in such conditions, the available capacity of the network is lower for users who can choose their path, thus creating problems in achieving the equilibrium. The monodimensional scan of the O.F between the seed and the real matrix for this type of assignment criterion seems to confirm this hypothesis (Figure 6). Table 4 illustrates the analyses of the average error reduction that can be achieved on several link variables as a function of the type and number of measurements adopted inside the O.F. Link variables analyzed are the same ones adopted in the O.F. (i.e. link volumes, speeds, densities and link queues) and other measurements as the outflows and the left turn movements, which for the reasons previously explained, have not been added in the O.F. In this later case, it is interesting to investigate the impact of the estimation on reproducing measurements that are not directly adopted during the optimization. Table 5 presents the analyses of the standard deviation of the error reduction for the different O.F.s adopted. This variability is strictly correlated to the DTA criterion adopted. Results show again the need for measurements as density and queue length to obtain a solution that is able to represent the traffic state correctly: however, there is a strong difficulty in reproducing measurements that are not directly adopted during the optimization. This is true for the outflows, but it is much more evident for the left turn movements, where high deterioration is detected. It must be emphasized that the left turn movement could be a measure able to represent the real route choices of users, while the other link measurements adopted, in the experiments, for the estimation lose this kind of information. Thus, despite the excellent O.F. and link error reductions obtained, it is clear how the estimation still fails in reproducing the real user’s choices and how information that is more focused to capture these choices are required during the procedure. Surely, the ability of the algorithm itself to work on such a complex problem cannot be denied. Figure 7 reports some examples of the O.F. scans between the solutions obtained during the optimization procedure. Where there are horizontal lines, it means that the algorithm does not find better solutions between iterations. Instead, the peaks are

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what the algorithm is able to overtake: it is clear that the algorithm has been able to optimize even though it is facing scratchy and noisy functions. Finally about the stability, we have not only underlined the capacity of the DUE approach, but also the number and the “quality” of the information contained in the O.F., which can significantly influence the stability of reproducing the link variables (Table 5). Table 4. Influence of the different measurements inside O.F. on reproducing link variables. Average error reduction on link variables [%] Measurements adopted inside O.F.

volume

speed

density

outflow

Volume Volume+speed Volume+density Volume+Queue Volume+speed+density Volume+speed+queue Volume+density+queue Volume+density+speed+queue

-62 -77 -93 -93 -85 -90 -99 -98

-58 -73 -88 -87 -83 -89 -97 -96

-62 -77 -93 -93 -85 -90 -99 -98

-43 -49 -75 -71 -66 -73 -78 -66

Left turn +106 +16 +110 +37 +22 +309 +61 +150

queue -59 -72 -89 -92 -81 -90 -99 -98

Table 5. Influence of the different measurements and DTA approaches on the stability of link variables reproduction. Standard deviation on the error reduction on link variables Measurements adopted inside O.F.

volume

speed

density

outflow

Volume Volume+speed Volume+density Volume+Queue Volume+speed+density Volume+speed+queue Volume+density+queue Volume+density+speed+queue

41 29 12 18 23 14 0.4 3

39 30 12 17 27 15 3 2

41 29 12 18 23 14 0.5 3

26 43 30 25 25 34 15 32

Figure 7.

Left turn 264 227 358 137 114 517 210 345

queue 37 30 12 18 25 15 1 2

Examples of the O.F. scans between solutions obtained during iterations

4. Conclusions This paper considers the problem of off-line dynamic demand estimation, formulated as a bi-level optimization problem and solved by using a simulation approach based on a gradient approximation computation (SPSA AD-PI method). Specifically, the combined effects of several traffic measurements and assignment criteria adopted during the

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optimization are investigated, based on synthetic experiments conducted on a test network. These experiments consider different combinations of traffic measurements inside the objective function of the problem and different DTA, simulated by the Dynasmart tool. Results showed the importance of a new measure as the queue length for the efficacy of the procedure in terms of both the O.F. reduction and traffic measurements reproduction. Stability of the results are influenced by the type of DTA adopted (in this case, the DUE approach seems to be the most stable approach) and by the number and quality of information adopted inside the O.F. Finally, the importance of capturing the actual users’ route choices, within the process, has been demonstrated. In fact, link measurements are not able alone to define the real traffic pattern and this is reflected in solutions that do not represent the real users’ behaviors. 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