3.4.2.1 Case study description
A comparison is made between the use of flow and speed data versus the use of density data in dynamic OD estimation. Both approaches are tested by performing a dynamic OD estimation on the highway network around the city of Antwerp in Belgium
Figure 3.12. The choice of this network is motivated by various factors. First of all, since we are focusing on the effect of congestion dynamics, we need a network where there is little influence of route choice. Secondly, the amount of congestion is substantial in this network, as can be seen in the spatio-temporal plots of mean speeds along the inner ring of Antwerp (R1) in Figure 3.13. For this network, both flow and speed measurements are available every 5 minutes. Density 'measurements' are approximated by dividing flow by speed measurements. The supply parameters of the links on this network hav
previous study for the Flemish Traffic Centre.
Figure 3.12: Ringway around Antwerp and highway E313 from Liege
Ring of Antwerp R1
of magnitude, less importance is given to the data type of lesser magnitude. This should be corrected for (e.g. by considering normalized values). Also, there might be
data measured by the same detector. When this is not accounted for, some information in the network might get a higher weight than it should get. To address this problem, it is common practice to weigh with an inverse covariance matrix. However, iance matrix assumes the relationship between two data sets to be linear, which does not hold in many cases, for example in the case of the relationship between flows and speeds, or in the case of the relationship between flows and densities. In the follow
decide to first assess the proposed methodology without this correction. As will become clear later on, we will decide to continue our research in a different direction. However, should the proposed methodology be further developed , this problem will need to be addressed.
In the next section we test the different approaches described above in a real
, and analyze whether they result in an estimated OD matrix that produces flows and
y description
A comparison is made between the use of flow and speed data versus the use of density data in dynamic OD estimation. Both approaches are tested by performing a dynamic OD estimation on the highway network around the city of Antwerp in Belgium, schematized in . The choice of this network is motivated by various factors. First of all, since we are focusing on the effect of congestion dynamics, we need a network where there is little te choice. Secondly, the amount of congestion is substantial in this network, temporal plots of mean speeds along the inner ring of Antwerp this network, both flow and speed measurements are available every 5 minutes. Density 'measurements' are approximated by dividing flow by speed measurements. The supply parameters of the links on this network have been calibrated in a previous study for the Flemish Traffic Centre.
: Ringway around Antwerp and highway E313 from Liege
Ring of Antwerp
E313 Liege-Antwerp
of magnitude, less importance is given to the data type of lesser magnitude. This should be corrected for (e.g. by considering normalized values). Also, there might be a correlation data measured by the same detector. When this is not accounted for, some information in the network might get a higher weight than it should get. To address this problem, it is common practice to weigh with an inverse covariance matrix. However, iance matrix assumes the relationship between two data sets to be linear, which does not hold in many cases, for example in the case of the relationship between flows and In the following, we decide to first assess the proposed methodology without this correction. As will become clear later on, we will decide to continue our research in a different direction. However, should the
ill need to be addressed.
n a real-world case , and analyze whether they result in an estimated OD matrix that produces flows and
A comparison is made between the use of flow and speed data versus the use of density data in dynamic OD estimation. Both approaches are tested by performing a dynamic OD , schematized in . The choice of this network is motivated by various factors. First of all, since we are focusing on the effect of congestion dynamics, we need a network where there is little te choice. Secondly, the amount of congestion is substantial in this network, temporal plots of mean speeds along the inner ring of Antwerp this network, both flow and speed measurements are available every 5 minutes. Density 'measurements' are approximated by dividing flow by speed e been calibrated in a
: Ringway around Antwerp and highway E313 from Liege
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We estimate the OD flows every 15 minutes. Four different OD estimation methods, as described in the previous subsection, are tested. In each method a different goal function is used. The goal function penalizes deviations from:
I. both flow and speed measurements (goal function (3.12)) II. density measurements only (goal function (3.13))
III. density measurements in a first stage (goal function (3.13)) flow measurements in a second stage (goal function (3.14)) IV. density measurements in a first stage
both flow and density measurements in a second stage (goal function (3.15))
In all four cases quadratic deviations are used as a distance function. Because we are interested in the influence of different data sources on the estimation process, without any other influence, the term z1( , )x x that accounts for the distance between the target matrix and the estimated matrix is ignored, i.e. it is given a weight of zero. To correct for the different magnitude of the different data types, all terms that account for flow deviations are given a weight of one, while all terms that account for speed deviations are given a weight equal to the flow measurement divided by the speed measurement. Goal function (3.12) thus becomes:
(
)
2(
)
2 1 1 ( ) ( ) ( ) k K I k k i k k i i k i i k i i y g y y v v v = = = − + − ∑∑
x x x (3.16)Similarly goal function (3.15) becomes:
(
)
2 2 1 1 ( ) ( ) ( ) k K I k k k k i i i i i k k i i y g y y v v ρ = = = − + − ∑∑
x x x (3.17)The goal function is again optimized using the SPSA algorithm (see section 3.3.1). The parameter c was set to 1, A was set to 1000 to reduce rapid decrease of the step size with the number of iterations, and a was chosen such that the average step size in the first iteration of the SPSA was 10 veh/h. LTM is used to assign the OD matrix and derive the simulated flows, speeds, and densities. The initial OD matrix in all four cases is an OD matrix that produces free flow all over the network. Therefore we are in an analogous situation as depicted in Figure 3.11. Convergence of the estimation process is assumed when the difference in RMSN after 100 iterations is smaller than 0.001. The performance is evaluated by the RMSE and RMSN of the link flows and speeds, and by qualitative comparison of the congestion patterns.
3.4.2.2 Results
Table 3.2 shows the results for the whole motorway network depicted in Figure 3.12 in terms of RMSE and RMSN for the four different methods. The first method performs best in terms of flows. However it does not perform equally well in terms of speeds. Also, as one can see in Figure 3.14(a), this method does not reproduce a correct congestion pattern. If one would
use this OD matrix for travel time prediction or for some form of decision support system one could suggest erroneous management strategies, like sending more flows where congestion is already serious.
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(a)
(c)
(d)
Figure 3.14: Estimated speed contour plot using (a) method I, (b) method II, (c) method III, and (d) method IV
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The second method performs better in terms of speed. Also, the congestion pattern resembles the actual congestion pattern better than method 1, although the results are not yet satisfactory. In terms of flow this method does not perform well. This can be explained by the high sensitivity of the flows to the density in the free flow regime: a larger error in the flow will be perceived as a small error in the density.
The third and fourth method start from the estimated OD flows of the second method. In the third method the correspondence with the flows becomes better, but both the speeds and the congestion pattern become worse. This problem was anticipated, as in the fourth method also the densities were included to keep the congestion pattern correct. This fourth method shows the best fit in terms of speeds and, looking at Figure 3.14(d), it seems to reproduce quite satisfactorily the congested area. The correspondence with the flows is similar as for the first method.
I II III IV
RMSE link flows (veh/h) 782 1102 866 845
RMSN link flows (%) 29.6 41.8 32.8 32.0
RMSE link speeds (km/h) 24.8 23.6 26.9 23.3
RMSN link speeds (%) 34.05 32.4 37.0 32.0
Table 3.2: Comparison of RMSE and RMSN for four different OD estimation approaches on the Antwerp network
It should be pointed out that the RMSE and RMSN values are quite high. Furthermore, the difference in performance in Table 3.2 seems rather small. Further analysis learns that this is caused by a number of factors:
• In the case study, there are stop-and-go patterns present on the E313. These stop-and- go patterns cannot be reproduced by a simplified first-order traffic model such as LTM. As a result, there is a high residual error both in terms of speed and flow. This high residual error is dominant compared to the other errors, and as a result the difference in RMSE between the four methods is small. Furthermore, these stop-and-go patterns and the resulting residual error also have an impact on the optimization algorithm. SPSA uses a coarse approximation of the gradient; at a certain point it does not find a proper descent direction anymore, while there is still room for improvement. The reason is that the residual errors are dominant over the other errors when determining the approximate gradient. This causes very slow convergence.
• A triangular fundamental diagram is used, which assumes that in free flow regime the speed is indifferent to the flow rate. In the measurements there is some variability of the speed in free flow regime. This is one of the causes for the high RMSE of the speeds. Also, since the congestion area over the entire network is relatively small compared to the free flow area, the gain of estimating the congestion area well is small. This is another reason for the small differences in RMSE of the speeds.
• The 'measured' densities that are derived by dividing the measured flows by the measured speeds are not all located on the fundamental diagram. When pursuing this density, the resulting flow will be erroneous. This can explain why for method 4 the
congestion pattern has a decent resemblance with the actual congestion pattern, but the RMSE values for speed and flow do not score equally well.