Journal of Advanced Transportation

Volume 2018, Article ID 2085625, 18 pages

https://doi.org/10.1155/2018/2085625

## Dynamic O-D Demand Estimation: Application of SPSA AD-PI Method in Conjunction with Different Assignment Strategies

^{1}Department of Engineering, Roma Tre University, Rome, Italy^{2}American University of Sharjah, Sharjah, UAE^{3}Department of Civil, Constructional and Environmental Engineering, Sapienza University of Rome, Rome, Italy

Correspondence should be addressed to Marialisa Nigro; ti.3amorinu@orgin.asilairam

Received 31 October 2017; Revised 26 February 2018; Accepted 8 April 2018; Published 23 May 2018

Academic Editor: Luca D’Acierno

Copyright © 2018 Marialisa Nigro et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper examines the impact of applying dynamic traffic assignment (DTA) and quasi-dynamic traffic assignment (QDTA) models, which apply different route choice approaches (shortest paths based on current travel times, User Equilibrium: UE, and system optimum: SO), on the accuracy of the solution of the offline dynamic demand estimation problem. The evaluation scheme is based on the adoption of a bilevel approach, where the upper level consists of the adjustment of a starting demand using traffic measures and the lower level of the solution of the traffic network assignment problem. The SPSA AD-PI (Simultaneous Perturbation Stochastic Approximation Asymmetric Design Polynomial Interpolation) is adopted as a solution algorithm. A comparative analysis is conducted on a test network and the results highlight the importance of route choice model and information for the stability and the quality of the offline dynamic demand estimations.

#### 1. Introduction

Dynamic Traffic Assignment (DTA) models are among the most effective tools for analysis and prediction of traffic conditions, especially in congested road networks. To provide accurate and reliable estimates, DTA models need information on the distribution of the trips in space and time (dynamic demand matrices) that are assigned to the network. It is straightforward that a better estimation of the dynamic demand matrices leads to a better estimation and prediction of traffic conditions.

This paper considers the offline estimation of the dynamic origin-destination (O-D) demand matrices as a starting point that can be upgraded to deal with real-time information for online demand estimation. The offline estimate of the dynamic demand matrices assumes a starting demand value to be known based on the available information on traffic conditions on the network. This is a highly undetermined, nonlinear, nonconvex problem, which was the object of a relevant research effort in the last years [1].

The offline dynamic estimation problem is usually approached as a bilevel problem. The upper-level problem consists of the adjustment of a starting demand using traffic measures, which are in turn linked to the dynamic demand. This link is generated from the dynamic traffic network assignment problem at the lower level, solved by using a dynamic traffic assignment (DTA) model.

Cascetta et al. [2] approached the dynamic O–D matrix estimation problem by introducing two different estimators: simultaneous and sequential. The former estimates all the matrices for all time slices using the whole set of traffic counts; the latter estimates at each step the matrix for a given time slice expressed as a function of the traffic counts within the same time slice and the already estimated previous demand matrices. The simultaneous estimator is more robust and provides better estimates than the sequential one. However, it requires knowledge of the dynamic assignment matrix, which has huge dimensions in real networks, and is computationally very expensive. On the other hand, Marzano et al. [3] pointed out that the highly indeterminacy of the problem is often cause of poor performances of the solution.

The research effort was directed mainly to improve the efficiency and the effectiveness of the solution methods, by following different research lines [4]: (i) introducing some approximation to the optimization method to reduce the computational effort; (ii) including more variables exploiting all available sources of information on traffic performances; (iii) assuming some simplification into the traffic assignment model.

As far as the first line, Yang [5] provided two heuristic solution approaches: iterative estimation-assignment and sensitivity-analysis based algorithms. The iterative approach does not guarantee the convergence to the solution. The iterative approach is theoretically not fully satisfactory, because the upper-level problem neglects the dependence of link flows on O–D matrix. On the other hand, sensitivity-analysis algorithm needs to approximate the derivatives through simulation for each O–D pair and each time interval at every iteration. Balakrishna and Koutsopoulos [6] introduced gradient approximation methods within a simultaneous perturbation stochastic approximation (SPSA) framework in order to reduce the number of simulation runs when calculating numerical derivatives or gradients. Cipriani et al. [7] proposed some modifications of the basic SPSA, introducing the Asymmetric Design (AD) for gradient computation and the Polynomial Interpolation (PI) of the objective function along the gradient direction. One of the recent contributions, related to the bilevel approach, is to try to jointly solve the offline demand estimation with the user equilibrium (UE) DTA problem [8]. In the same year, Djukic et al. [9] applied the principal component analysis method to reduce the O–D demand variables. Then, Cantelmo et al. [11] extended the SPSA AD-PI method to a second-order approximation by applying a Quasi-Newton method. In addition, they proposed an adaptive approach that computes, at each iteration, the weights in the gradient computation according to the relevance of any O–D pair as computed. Lu et al. [12] introduced an enhanced SPSA algorithm, which incorporates spatial and temporal correlation between parameters and measurements to minimize the noise generated by uncorrelated measurements and reduce the gradient approximation error. While in the original SPSA the objective function is a single scalar, in the weighted SPSA it is a vector, whose gradient components are weighed by a matrix that expresses the correlations between parameters and measurements. Numerical tests to a small size and a real-size road network highlighted that the enhanced method improved the efficiency and the accuracy of the original SPSA estimation method.

As far as the second research line, considerable attention has been given to the role of different traffic measures adopted inside the O–D estimation procedure, for offline and online applications in addition to the usually adopted link counts, specifically speed and link occupancy [7, 8, 13], probe data from vehicle equipped by AVI tags [14–21]; aggregate demand data, such as traffic emissions and attractions by zones [7, 22, 23]; measure link speeds and path travel times [24].

As far as the third research line, Frederix et al. [25] provided a linear approximation of the relationship between O–D flows and link flows, based on a marginal computation method that performs a perturbation analysis using the kinematic wave theory. Lu et al. [26] presented a single-level nonlinear path flow-based optimization model, which does not require explicit dynamic link-path incidences and applies a Dynamic Network Loading model based on Newell’s simplified kinematic wave theory in the DUE assignment process. Cascetta et al. [27] proposed a “quasi-dynamic” framework for estimation of O–D flows, in which O–D shares are assumed constant across a reference period, while total flows leaving each origin are assumed varying for each subperiod within the reference period. Cipriani et al. [28] applied a quasi-dynamic traffic assignment model (QDTA [29]) that approximates the dynamic traffic model by steady-state intervals and applies approximate performance functions in order to reduce the computational burden to solve the estimation problem. Cantelmo et al. (2016) proposed a utility-based formulation for the demand estimation, which is able to incorporate activity duration and to consider different activity patterns. The building block of this methodology is to adopt a utility-based departure time choice model in the DTA and to exploit this model to derive the demand temporal distribution.

The research illustrated in this paper finds its starting motivations just in the results obtained by the contribution of Cipriani et al. [28], where the approximation of the DTA with the QDTA model to solve the O-D estimation problem was able to generate interesting results in terms of traffic measures reproduction. Specifically, the SPSA AD-PI algorithm was adopted in conjunction with the QDTA on a subnetwork of the city of Rome, consisting of 113 traffic zones, 757 links, and 335 nodes, where the traffic measures, in order to adjust the starting value of the demand, were true data of flows and speeds on 41 links of the network. These true data have been derived by Floating Car Data (FCD) collected during a national project (Pegasus Project) for the whole metropolitan area of Rome: a fleet of 103,000 floating vehicles, travelling 9 million trips, provided 104 million records containing positions and speeds during one month (May 2010). About 80,000 vehicles of these 103,000 floating vehicles crossed the study area, thus generating a huge amount of data with high disaggregation. Simulated link data, at the end of the O-D estimation, resulted in a high correspondence with the true data, obtaining improvement of approximately 65% in terms of relative mean error with respect to the starting conditions.

Thus, on one hand these results seem to show that there is the challenge to approximate DTA models, reducing both the calibration efforts and the computation time and laying the foundation for applications to time-dependent O-D estimation problems on real-size networks; on the other hand, there is a question worth of investigation, that is, the degree of approximation that can be introduced by using the QDTA to simulate the user’s behaviours.

Such considerations have to be integrated into a recent analysis of a large dataset of FCD collected in Rome that highlights that users moving from the same origin at the same time interval (or with quite close departure times) choose multiple routes with different observed travel times to reach the same destination [30, 31]. The finding of different actual route travel times experienced by drivers within the same time interval raises some issues related to the concept of equilibrium in the dynamic traffic assignment phase.

The first issue concerns the realism of the behavioural assumptions underlying the dynamic assignment models: dynamic equilibrium traffic models provide detailed information regarding the temporal profile of performance metrics (travel times, speeds, and densities), extend the equilibrium concept introduced in the static model, and then assume the same assumptions that drivers are rational and have perfect information on network conditions they will face approaching their destinations. Whether the real route choice mechanism should be based on instantaneous rather than on experienced travel times (or a combination of them) is still an open research topic [32]. Even if dynamic equilibrium principles are often referred to experienced travel times, loading vehicles on the network according to instantaneous travel times reflects the fact that either drivers choose their shortest routes on the basis of an instantaneous picture of the network (conditions) taken at their departure time, as more and more frequently now happens, through websites visited at home (pretrip information), or they adjust their route while travelling, on the basis of information received on real-time updated mobile devices while on the route (en route information).

The dispute between instantaneous and experienced travel times is strongly related to the second issue that raises from FCD observations: the convergence of assignment procedures. As it is well known, the procedure is considered to have converged, approaching equilibrium conditions, when no simulated driver can improve his travel time by shifting to an alternative route; thus, no change in experienced travel times can be detected and no change in traffic pattern occurs on the network, even if running additional iterations. This implies that the network has reached a condition of stability, which is the third issue, one of the major interests for this study and has motivated the present paper. Stability of the equilibrium condition is reflected by the algorithm progression as the condition of detecting no change (or negligible change) in network conditions when running additional iterations after the equilibrium has been reached. This condition can be alternatively seen from the supply side by producing a minor change to the network features, for instance, by changing the speed limit on a link and detecting only local changes of traffic conditions or, from the demand side, by slightly changing the demand, for instance, adding one vehicle to an O–D pair, and detecting only negligible changes of traffic flow patterns.

The latter example is of great relevance in the present paper because it affects the shape of the objective function being minimized in the demand estimation problem: if the traffic assignment is not stable then the objective function is very noisy and, moreover, may exhibit no descent direction towards the real demand matrix.

Such observations have motivated the investigation of the paper, which is to solve the demand estimation problem under different traffic assignment conditions and criteria, also adopting approximation of DTA models, thus investigating the impact of different route choice modeling on the convergence and the accuracy of the estimation. The offline dynamic demand estimation problem has been solved on a test network with the adoption of a bilevel approach based on the SPSA AD-PI algorithm. Traffic assignments required at the lower level have been performed by using the QDTA model [29] and different route choice options given by the Dynasmart model [33–37].

The paper is organized in four sections including this introduction. Section 2 deals with the offline dynamic O-D matrices estimation (DODME) problem: it describes the main concepts behind the DTA models adopted in the study and the solution approach based on the SPSA AD-PI method. Section 3 presents the results of numerical tests carried out by applying the solution method to a test network in combination with different kinds of traffic measurements and different assignment models. Conclusions follow, in Section 4.

#### 2. Problem Formulation

This formulation considers a network consisting of a set of arcs , a set of nodes , and a set of routes . Given the period of analysis , divided into intervals, a subset of links and nodes equipped with sensors, a subset of monitored routes , and a set of centroids , where the demand is assumed to have origin and destination, the problem of offline simultaneous dynamic demand estimation with multiple sources of information can be formulated as follows:where = estimated O–D matrix for departing time interval , ; = seed O–D matrix for departing time interval , ; = simulated values on subset of links** L **in the time interval , ; = collected measures on subset of links** L **in the time interval , ; *l *= type of collected measures on links** L;** = simulated values on subset of nodes** Q **in the time interval , ; = collected measures on subset of nodes** Q **in the time interval , ; *q *= type of collected measures on nodes** Q;** = simulated values on subset of routes** P **for departing time interval , ; = collected measures on subset of routes** P **for departing time interval , ; = type of collected measures on routes** P**.

Functions , , , and are goodness of fit functions related to different kinds of information about demand and traffic patterns that in general may be available. If a reliable prior O–D matrix (called “seed matrix”) is available, the objective function to be minimized can suitably include function as the distance with respect to the seed matrix. This specification of the estimation problem is usually referred to as demand adjustment in literature. Moreover, the formulation reported in (1) permits the introduction of different types of data collected on the network in order to better reproduce the demand in space and time. Examples of such types of data are as follows: (i) measures on links as flow, speed, occupancy, queue length, and density; (ii) measures on nodes, for example, turning movements; (iii) measures on routes as travel time, distance travelled, or path flow fractions. In a dynamic framework, the relationship between the values of the measured variables and the demand is captured by simulation is presented asOperator is a generic traffic assignment model able to represent, given a dynamic demand as an input, the time variation of the measures along the different elements of the network. Generally is reported in previous works as a Dynamic User Equilibrium Traffic Assignment (DTA-DUE); however similar approaches in literature adopt also Dynamic Network Loading (DTA-DNL), usually when there is no need for route choice [10, 38]. Nonequilibrium approaches, such as system optimum approach (DTA-SO) or quasi-dynamic traffic assignment (QDTA), can be also adopted. It is easy to assume that the choice of Γ can influence the results of the demand estimation, which is the focus of this paper.

To apply different assignment strategies and simulate different behavioural assumptions on route choice, the state-of-the-art DTA software Dynasmart (DYnamic Network Assignment Simulation Model for Advanced Road Telematics) is used [39]. This is a simulation assignment model that integrates traffic flow models, path processing methodologies, behavioural rules, and information supply strategies. The traffic model is a mesoscopic simulation model, which applies a macroscopic concept for moving the vehicles on the links. For the movement of vehicles at intersections, Dynasmart adopts a microscopic simulation concept. The modeling framework for Dynasmart is presented in Figure 1. The path selection process may apply any of the following rules: pretrip information, en route real-time information, Dynamic User Equilibrium (DUE), and system optimum (SO).