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.
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 [
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. [
The research effort was directed mainly to improve the efficiency and the effectiveness of the solution methods, by following different research lines [
As far as the first line, Yang [
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 [
As far as the third research line, Frederix et al. [
The research illustrated in this paper finds its starting motivations just in the results obtained by the contribution of Cipriani et al. [
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 [
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 [
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 [
The paper is organized in four sections including this introduction. Section
This formulation considers a network consisting of a set of arcs
Functions
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 [
Dynasmart modeling framework [
Drivers’ behaviour in response to en route real-time information is simulated according to the bounded rationality approach, which assumes that drivers will change their path according to travel time information received only if the new path yields travel time savings greater than given thresholds.
About the possibility of approximating the DTA, the QDTA approach by Fusco et al. [
The dynamics of traffic is introduced in the network loading process, which simulates the progression of all packets of flow in each time interval by moving them, according to the value of speed corresponding to the link flow, up to the position on the network reached in the time interval. Because of the link-cost functions used, the model does not simulate the processes of queue progression and clearance but computes anyway the traffic congestion in terms of link travel time.
It is assumed that users do not modify their route choice during the travel; thus, at each iteration, routes are computed only for the new users that enter the network. However, because of the time-dependent interaction of flows on the network, origin-destination (O-D) flows starting at a generic time interval may be affected by flows starting at a successive time interval and overlapping their route (consequently changing their travel time).
Since Dynasmart is assumed as term of reference in this laboratory application, the QDTA model has been calibrated by applying a Particle Swarm Optimization algorithm to determine the parameters of volume-delay functions that better approximate the results provided by Dynasmart on the test network used in this experiment.
In the following, the main principles of SPSA AD-PI, adopted in this paper as the solution method for the offline simultaneous dynamic demand estimation, are summarized. SPSA AD-PI was firstly proposed by Cipriani et al. in 2010 and it is based on the path search optimization method of Spall, [
Considering the generic iteration
The average approximated gradient
Each gradient approximation
The linear optimization in (
One-dimension Polynomial Interpolation (PI).
About the computational times of SPSA AD-PI, these are function of the number of traffic assignments. Each time the SPSA computes a gradient approximation (
Several experiments have been conducted on a test network consisting of 22 nodes (14 are signalized intersections), 68 links (composed by highway links, freeway links, onramp, and offramp links), 6 traffic zones, and a whole planning horizon of 35 minutes discretized into 5 minutes intervals (Figure
Test network.
In total, 12 links are monitored and monitoring sections are located between the following pairs of nodes:
All the experiments start from a traffic demand supposed as the “true” demand, which is assigned on the network in order to procure the measures to be included in the OF (ground truth conditions). Then, the “true” demand has been perturbed in order to obtain a “seed” demand, which is the starting point for the optimization. The OF in (
Specifically, only collected measures on links have been adopted inside the OF (link volume, speed, density, and queue length). Various combinations of types of information are assumed in the different tests, as reported in Table
Experiment design.
Experiments | Type of DTA | Measures inside the OF for each type of DTA |
---|---|---|
Set I | DUE | OF1: link volume |
Set II | SO | |
Set III | DNL | |
Set IV | 50% DUE – 50% SO | |
Set V | DNL + fixed DUE paths | |
Set VI | DNL + fixed DUE paths (highest O–D flows) | |
Set VII | DUE + fixed DUE paths (highest O–D flows) | |
| ||
Set VIII | QDTA | O.F.1: link volume |
It is worth noting that no term containing information on prior O–D matrix has been included in any OF in order to exploit network measurements in the estimation procedure as much as possible, avoiding anchoring it to the initial estimate. Thus, the goal of the experiments is to investigate the influence of different components of information and different assignment models on the O-D matrix estimation. In fact, strong differences are imposed, both in the total value of the trips (12,638 seed versus 18,900 true) and in the O–D distribution between true and seed.
Different
As a whole, eight sets of experiments have been conducted, each set following a specific DTA approach and containing a total number of eight types of analyzed OF for the first seven types of DTA criteria and four types of OF for the last QDTA criteria.
All the parameters to be defined for the application of SPSA AD-PI have been chosen according to the directions given in Cantelmo et al. [
First comments on the results are related to the efficacy of the SPSA AD-PI in terms of the average OF reduction achieved in each set of tests and the related standard deviation of OF reduction obtained by applying different OFs specifications (Table
Efficacy of the O–D estimation method in terms of OF reduction.
Experiments | OF reduction [%] | Standard deviation [%] |
---|---|---|
Set I | | 0.19 |
Set II | | 21 |
Set III | - | - |
Set IV | | 8 |
Set V | | 30 |
Set VI | | 28 |
Set VII | | 24 |
Set VIII | | 14 |
Figure
OF1 scan for DTA-DNL between the seed matrix (alpha = 0) and the true matrix (alpha = 1).
A similar picture is shown in Figure
OF1 scan for DTA-DUE.
It is clear that the DNL (Set III) prevents the algorithm from finding a descent direction for the dynamic O–D estimation problem, thus making the SPSA ineffective. This is due to the strong variations of the route choice generated by the dynamic shortest path algorithm when changing the demand matrix. However, scanning the OF1 when the DTA-DUE criterion is applied clearly individuates a descent direction, although with some noise. Similar OF trends can be recorded also for DTA-SO or mixing DUE and SO (resp., Set II and IV).
The reductions of the OFs have a direct impact on the reproduction of the traffic measures: Table
Efficacy of the O–D estimation method in terms of reduction of the errors in measures reproductions.
Experiments | Link volume | Speed | Density | Outflow | Left-turn movements | Queue length |
---|---|---|---|---|---|---|
Set I | ||||||
Average reduction [%] | | | | | | |
St.dev. [%] | 0.52 | 6.69 | 0.50 | 16.16 | 81.63 | 3.23 |
Set II | ||||||
Average reduction [%] | | | | | | |
st.dev. [%] | 20.97 | 23.74 | 20.50 | 22.71 | 66.86 | 23.50 |
Set IV | ||||||
Average reduction [%] | | | | | | |
St.dev. [%] | 7.02 | 12.41 | 8.50 | 37.34 | 34.77 | 13.23 |
Set V | ||||||
Average reduction | | | | | | |
st.dev. | 29.76 | 29.72 | 20.50 | 35.37 | 119.26 | 30.05 |
Set VI | ||||||
Average reduction [%] | | | | | | |
St.dev. [%] | 27.55 | 27.96 | 25.40 | 29.16 | 130.96 | 28.16 |
Set VII | ||||||
Average reduction [%] | | | | | | |
St.dev. [%] | 24.27 | 23.14 | 22.45 | 15.99 | 446.03 | 21.72 |
Set VIII | ||||||
Average reduction [%] | | | | - | - | - |
St.dev. [%] | 14.81 | 12.48 | 15.30 | - | - | - |
Left-turn error index scan for DTA-DUE.
The experiments show good results in terms of traffic measure reproduction, both directly (when measures are inside the OF) and indirectly (when measures are not considered in the OF), except for the left-turn movements. For this measure, improvements with respect to the starting point are quite limited (at most about 47%, Set IV, Table
Scatter plots between traffic measures and simulated measures obtained from the seed matrix (a) and from the final estimated matrix (b).
Summarizing, the solution of the dynamic O-D estimation in combination with different assignment criteria seems to underline the efficacy of the adopted algorithm. Also when the QDTA is applied, as an approximation of the DTA, the solutions founded confirm the trend of the other assignment criteria and the results obtained in previous studies: the reproduction of traffic measures is good and when link volumes are considered together with speeds and densities, the improvements are comparable with those obtained in the Dynasmart case by adopting an iterative dynamic assignment approach.
Some problems can be detected in case of instability of route choice to change in the demand, that is to say, when a DNL criteria is followed. This instability can be reduced working on the selected paths, as synthetically made in the experiments with fixed paths, assuming these paths as those selected by road users in real world.
However, the reduction of the OF and the correct measures reproduction could not be sufficient to appreciate the goodness of the solution. In fact, the aim of the O-D estimation is to find a really reliable demand matrix in order to use it for traffic management and planning applications. For this reason, in the following paragraph, the results have been analyzed as for the capability of O–D estimation method to reproduce the true O–D matrix. This usually cannot be done, unless for laboratory experiments (this is the case), in which the true demand is known.
Results reported in Table
Evaluation of the estimated demand as the Euclidean distance from the true O–D matrix and the total number of trips.
Reference values | Euclidean distance | Total trips | Total trips |
---|---|---|---|
Experiments | Euclidean distance | Euclidean distance | Total trips |
Set I | |||
Average value | 826.17 | 820.07 | 17,230 |
St.dev. | 144.04 | 125.74 | 1,714 |
Set II | |||
Average value | 817.15 | 786.86 | 17,786 |
St.dev. | 92.47 | 83.58 | 2.648 |
Set IV | |||
Average value | 782.41 | 770.05 | 17,379 |
St.dev. | 153.42 | 91.24 | 2,641 |
Set V | |||
Average value | 437.04 | 548.82 | 15,256 |
St.dev. | 218.20 | 102.33 | 2,115 |
Set VI | |||
Average value | 414.52 | 527.81 | 15,183 |
St.dev. | 199.09 | 80.78 | 1,496 |
Set VII | |||
Average value | 775.20 | 760.26 | 17,403 |
St.dev. | 212.36 | 152.68 | 2,062 |
Set VIII | |||
Average value | 192.48 | 465.05 | 13,307 |
St.dev. | 63.46 | 12.11 | 466 |
The distribution in space and time of each O-D has been evaluated measuring the Euclidean Distance (ED) between the estimated matrix and both the true and the seed. Then, these statistics are compared with the reference values of Table
Correlation founded when using DUE, SO, and their combination.
In the cases of Sets V and VI, as shown in Table
OF1 scan for DNL + fixed UE paths (Set V).
This noise reduction becomes stronger when the highest O–D flows are intercepted (see Set VI, Table
Evaluation of the OF noise (average absolute difference of two following OF values for a mono-dimensional scan between seed and true demand).
Experiments | OF noise evaluation |
---|---|
Set I | 32.19 |
Set II | 29.18 |
Set III | 29.62 |
Set IV | 55.30 |
Set V | 12.20 |
Set VI | 10.94 |
Set VII | 32.33 |
In case of adoption of the QDTA model (Set VIII), the optimization algorithm does not find the need to move so far from the starting matrix as it does in Dynasmart simulations in order to reproduce the traffic measures. In consequence, the total value of the estimated demand is quite far from the true one (about 13,000 trips with respect to 18,900). However, the O–D demand pattern is much closer to the true one than that estimated by applying Dynasmart; in all tests the Euclidean Distance from the true demand is comparable with the corresponding value between starting and true matrix (approximately 459).
In Set VII, the DUE route choice approach, which exhibited the highest convergence capability, has been mixed with the approach that fixes some UE paths with the highest O–D flows, which showed the best capacity of mitigating the OF noise. However, this experiment resulted in noise comparable with the basic DUE case (Table
The reason for these results can be explained considering that when adding such fixed DUE paths to the DNL, the route choice variation generated by the dynamic shortest path according to a change in travel demand is strongly limited.
Previous results showed in general a high difficulty in finding the right spatial and temporal distribution of the true demand, even though the total right level can be reached; however there is a need to understand if the problem is in the spatial or in the temporal correlation between O-Ds.
It is worth noting that the test designed was very challenging, since the seed matrix was very different from the true one (ED = 459, average difference between O-Ds = 25, maximum difference between O-Ds = 60, and standard deviation between O-Ds = 39) and the time resolution for O–D estimation (5 minutes) was much smaller than that usually adopted in dynamic traffic assignment problems (15–20 minutes).
Table
Evaluation of the estimated demand in terms of O–D distribution in space.
Reference values [ref.] | Euclidean distance | |||
---|---|---|---|---|
Experiments | Euclidean distance | Difference with respect to ref. [%] | Euclidean distance | Difference with respect to ref. [%] |
Set I | 750 | | 1,107 | |
Set II | 825 | | 1,122 | |
Set IV | 1,075 | | 1,095 | |
Set V | 518 | | 1,015 | |
Set VI | 512 | | 957 | |
Set VII | 936 | | 924 | |
Set VIII | 192 | | 1,124 | |
In such a case, an improvement with the estimated demand is always obtained and this improvement is higher when fixed paths are adopted: fixed paths are able to simplify the route choice, thus reaching improvement of about 20% on O-D spatial distribution. In some tests, specifically when they are combined with DNL also improvement of about 30% has been recorded. The improvement decreases to 10% if fixed paths are not adopted.
In the QDTA case (Set VIII), the new computed ED values emphasize the difficulty of moving away from the seed matrix and this results in an insufficient ability to reproduce the space distribution of O-Ds (only 7% of improvement respect to the starting conditions).
To complete the analysis of the results in terms of estimated demand, the ED values are computed aggregating four-time intervals at time by adopting a rolling horizon average method (Table
Evaluation of the estimated demand in terms of O–D aggregated distribution in time.
Reference values [ref.] | Euclidean distance seed-true | Time interval (minutes) | |||
---|---|---|---|---|---|
0–20 | 5–25 | 10–30 | 15–35 | ||
700 | 700 | 693 | 682 | ||
Set I | Euclidean distance seed-estimated | | | | |
[% diff. respect to Ref] | | | | | |
Euclidean distance true-estimated | | | | | |
[% diff. with respect to ref] | | | | | |
| |||||
Set II | Euclidean distance seed-estimated | | | | |
[% diff. with respect to ref.] | | | | | |
Euclidean distance true-estimated | | | | | |
[% diff. with respect to ref.] | | | | | |
| |||||
Set IV | Euclidean distance seed-estimated | | | | |
[% diff. with respect to ref.] | | | | | |
Euclidean distance true-estimated | | | | | |
[% diff. with respect to ref.] | | | | | |
| |||||
Set V | Euclidean distance seed-estimated | | | | |
[% diff. with respect to ref.] | | | | | |
Euclidean distance true-estimated | | | | | |
[% diff. with respect to ref.] | | | | | |
| |||||
Set VI | Euclidean distance seed-estimated | | | | |
[% diff. with respect to ref.] | | | | | |
Euclidean distance true-estimated | | | | | |
[% diff. with respect to ref.] | | | | | |
| |||||
Set VII | Euclidean distance seed-estimated | | | | |
[% diff. with respect to ref.] | | | | | |
Euclidean distance true-estimated | | | | | |
[% diff. with respect to ref.] | | | | | |
| |||||
Set VIII | Euclidean distance seed-estimated | | | | |
[% diff. with respect to ref.] | | | | | |
Euclidean distance true-estimated | | | | | |
[% diff. with respect to ref.] | | | | |
This last analysis highlights a difficulty in the estimation as time advances, that is, as more demands come through the network and as congestion increases. This is also due to the number of measurements used for the estimation of the demand in any time interval: the demand of the initial time interval which estimated using measurements from the 1st to the 7th interval, the demand of the second-time interval using measurements from the 2nd to the 7th time interval, and so on up to using only one time interval measurements for the estimation of the demand of the last time interval. This difficulty results in worsening the ED for the last two intervals. Again, when fixed paths are introduced, a slight improvement is reported also for these two intervals, where the best result is obtained for Set VI.
At this point, it can be stated that the introduction of information on the followed paths can be a fundamental element to substantially improve the O–D estimation in terms of reproduction of the true demand value and of its distribution in space and time: this is surely suggested by the results obtained for Sets V and VI but it is also underlined by the difficulties of the O–D estimation procedure in reproducing indirectly some measurements as the left-turn movements. As already mentioned, the left-turn movements are strongly related to the route choices. Since many matrices can reproduce quite perfectly link measures as volumes and speeds and also link queues and outflows, without being the true demand, an effective way to reproduce the true conditions on the traffic networks, even for short-term planning and management operations, is to be able to reproduce the correct path choices.
For this reason, some final experiments have been conducted adding path choices information during the estimation in terms of O-D travel times inside the OF.
Recent ICT enhancements provide increasing deployment of identification sensors. In addition to traditional monitoring devices that collect information on the whole traffic stream at fixed locations, identification sensors allow tracking individual vehicles both at fixed locations on the road and on the network ubiquitously. Examples of fixed identification sensors are image recognition cameras and radio sensors that can capture public information from bluetooth and Wi-Fi devices on board of vehicles. Ubiquitous monitoring consists of sampling vehicles equipped with GPS location devices and mobile cellular data transmitters along their paths. The latter form of monitoring is particularly efficient, because it uses the same devices to provide information to drivers and get information by them, is very effective, and provides information on origin-destination of individual trips as well as on the routes travelled by the drivers.
This information is fundamental to get an accurate knowledge of travel demand and achieve reliable, well-calibrated road traffic models [
Today, Floating Car Data (FCD) provide a huge amount of true-time updated datasets on vehicle positions and speeds, which allow estimating both travel times and routes followed by vehicles tracked. They should be exploited to improve demand estimation and modeling assumptions.
In this last section, O-D travel times by FCD have been added inside the OF (
These travel times have been practically obtained as a result of the assignment of the true demand (with different assignment criteria), as already done for the collected link measurements in previous experiments.
The following sets have been analyzed through adding the O-D travel times to (i) Set I, since it is based on DUE criterion, which theoretically represents the real user behaviour in case of congested conditions; (ii) Set VI, where promising results have been obtained in the previous experiments including fixed paths; (iii) Set VIII, in order to continue the investigation of approximating the DTA with the QDTA.
Tables
Evaluation of the estimated demand as the Euclidean distance from the true O–D matrix and the total number of trips (adding O-D travel times).
Reference values | Euclidean distance | Total trips | Total trips |
---|---|---|---|
Experiments | Euclidean distance | Euclidean distance | Total trips |
Set I | |||
Average value | 654.62 | 578.86 | 18,635 |
St.dev. | 158.46 | 70.58 | 2,662 |
Set VI | |||
Average value | 566.22 | 539.25 | 17,335 |
St.dev. | 16.68 | 17.68 | 155 |
Set VIII | |||
Average value | 211.00 | 508.69 | 12,845 |
St.dev. | 75.87 | 28.37 | 147 |
Evaluation of the estimated demand in terms of O–D distribution in space (adding O-D travel times).
Reference values [ref.] | Euclidean distance | |||
---|---|---|---|---|
Experiments | Euclidean distance | Difference with respect to ref. [%] | Euclidean distance | Difference with respect to ref. [%] |
Set I | 893 | | 838 | |
Set VI | 1,003 | | 891 | |
Set VIII | 313 | | 1,184 | |
Evaluation of the estimated demand in terms of O–D aggregated distribution in time (adding O-D travel times).
Reference values [ref.] | Euclidean distance seed-true | 700 | 700 | 693 | 682 |
---|---|---|---|---|---|
Set I | Euclidean distance seed-estimated | 607 | 593 | 481 | 516 |
[% diff. with respect to ref.] | | | | | |
Euclidean distance true-estimated | | | | | |
[% diff. with respect to ref.] | | | | | |
| |||||
Set VI | Euclidean distance seed-estimated | | | | |
[% diff. with respect to ref.] | | | | | |
Euclidean distance true-estimated | | | | | |
[% diff. with respect to ref.] | | | | | |
| |||||
Set VIII | Euclidean distance seed-estimated | | | | |
[% diff. with respect to ref.] | | | | | |
Euclidean distance true-estimated | | | | | |
[% diff. with respect to ref.] | | | | |
Previous experiments adopting the DUE (Set I), without adding information on travel times, demonstrated the ability to obtain the right level of total demand, but problems in terms of reproducing its temporal distribution were detected. The introduction of O-D travel times (Table
A detailed analysis of the results in Set I derives the fact that the adoption of the new information on travel times in the objective function enables the estimation procedure to reproduce the correct path choices. In fact, not only improvements on link measures are as high as those already obtained in previous tests (without O-D travel times) for volumes, speeds, and densities (higher than 95%), but, in addition, outflows (80% of improvement), queue length (99%) and, more importantly, left-turn movements (73%) are now properly estimated. Such occurrence indicates that with O-D travel time information the estimation procedure can simulate the correct traffic regime and the correct route choices. This is demonstrated also by the values reported in Tables
In case of Set VI, the O-D travel times allowed us to obtain the correct quantity of total demand with respect to the underestimation of the previous tests. Instead, the ED between real and estimated demand is almost the same (compare Set VI, Tables
Finally, adopting the QDTA (Set VIII), the O-D travel times do not seem to add further information during the estimation procedure. Results are quite similar to those obtained without travel times; indeed it seems to get worse as the solution moves away from the seed matrix.
The paper has dealt with the impact of introducing different sources of information and different route choice criteria in the O–D demand estimation problem, solved by the Simultaneous Perturbation Stochastic Approximation with Asymmetric Design and Polynomial Interpolation (SPSA AD-PI) method. A systematic analysis, performed on a test network, has showed that the estimation procedure is characterized by good or excellent convergence properties in all the cases examined other than when Dynamic Network Loading (DNL) method is adopted for performing assignment phase; in this case, the assignment criteria prevent the estimation algorithm from converging to a solution.
Nevertheless, the estimated O–D matrix in most cases is not satisfactory when properly compared to the true solution and may be even farther from it than the initial seed O-D matrix.
This is not surprising, provided that the problem is highly undetermined and highly nonconvex and that the distance from the initial matrix was not included into the objective function nor the was search limited by any constraint. This result highlights the importance of including a good initial estimate of the O–D matrix in the estimation process.
Among the different variables examined, the most important contribution to demand estimation has been, in most cases, achieved by using measurements of traffic counts, link speed, and queue length.
Apart from the various performance measures, the analysis has shown that the route choice is a main factor, since it affects the spatial correlation between link performances and O–D flows.
Indeed, when some routes are fixed, the performances of the method are improved. Specifically, better results are obtained when only some routes are fixed for some given O–D pairs and the other routes can change over time, as in the case in which DNL is applied. Worse results are obtained when the other routes can change over both the time and the space (as in the case in which DUE is applied).
The paper has dealt with the research line concerning the way to improve the effectiveness of the estimation method introducing information on route choices observed from Floating Car Data (FCD) collected in the field; this research line, still ongoing, seems to be a promising approach. Achieving direct information on drivers’ route choices and the related experienced route travel times opens promising perspectives also for a joint calibration of the O–D demand and the traffic assignment model and is expected to provide new insights into the real existence of dynamic equilibria [
A second research line, which will require further study with respect to the results here obtained, concerns the suitable degree of approximation for the dynamic O–D estimation problem. While the availability of many sources of information makes it possible to better appreciate the contribution of sophisticated models, it also enlarges the problem of dimensions and requires longer computation times. On the other hand, approximate traffic assignment models, such as quasi-dynamic assignment models, reduce both the calibration efforts and the computation time and might result in more advantageous for applications to time-dependent O–D estimation problems, where approximate algorithms are introduced to solve the problem on real-size networks. A question worth of investigation is the degree of approximation that can be introduced in the different steps of the estimation method, that is, the SPSA algorithm, which provides some numerical approximation of the gradient method; QDTA model, which approximates the dynamic model by steady-state intervals and applies approximate performance functions; a priori hypotheses on the time-space structure of the demand, assuming some correlations as fixed, as done by Cascetta et al. [
Finally, a third research line, which refers to the contents analyzed in this paper, concerns the structure of the objective function and specifically the investigation of the performances of SPSA AD-PI solution method when applied to the vector formulation recently introduced by Lu et al. [
The authors declare that they have no conflicts of interest.