In order to investigate the accuracy of the nominal model (with previously identified parameters) under different conditions a new data set has been chosen and used for exciting. The data collected on 31/01/2006 and 02/02/2006 used for validating the nominal model and illustrated in Figure D.4 in terms of density values.
Accordingly, the non-linear model with the nominal parameter set collected in Table D.1 has been excited with the new data sets and compared with the measuredvm4 and qm4 values of the middle detector station. The comparison of the model responses are illustrated in Figure D.5. One can conclude that the tracking capability of the non-linear model is preserved, although a performance degradation can be seen, especially under rapidly varying traffic conditions. In order to characterize the accuracy change due to different training data, the optimization procedure has been repeated with the new data set, and the individually optimized parameters are collected in Table D.2. A new simulation has been carried out and evaluated in terms of (D.9), and a relative difference between the nominal and individually optimized results was found to be6%, in line with the findings of [CP81].
Finally, the sensitivity analysis has been carried out, in order to investigate the effect of parameter changes. For this purpose the parameter values were perturbed by
±5% from their nominal values (in Table D.1). Simulations have been then performed by using the perturbed parameters respectively, and the results of these individual simulations were compared by introducing a sensitivity index as in [CP81]:
σ(δΘ) = J(Θn+δΘ)−J(Θn)
J(Θn) 100%, (D.11)
where Θn denotes the nominal parameter vector, while δΘ is the perturbation. The results of the sensitivity analysis are summarized in Table D.3. As one can depict, the model is not very sensitive to small parameter changes, which underlines the transfer-ability and flexibility of the second-order model [CP81].
(a) 31/01/2006
Time [h]
Space[km]
20 40 60
6.00 AM 7.00 AM 8.00 AM 9.00 AM 10.00 AM
41.7 42.025 42.35 42.725 43.1 43.45 44.106 44.706 45.306
(b) 02/02/2006
Time [h]
Space[km]
20 40 60 80
6.00 AM 7.00 AM 8.00 AM 9.00 AM 10.00 AM
41.7 42.025 42.35 42.725 43.1 43.45 44.106 44.706 45.306
Figure D.4: Visualization of traffic data sets used for model validation
(a)
Time [ h]
Space-meanspeed km h
Detector measurements Non-linear model response
6.00 AM0 7.00 AM 8.00 AM 9.00 AM 10.00 AM
20 40 60 80 100 120
(b)
Time [ h]
Trafficflow veh h
Detector measurements Non-linear model response
6.00 AM0 7.00 AM 8.00 AM 9.00 AM 10.00 AM
2000 4000 6000 8000
Figure D.5: Space-mean speed D.4(a) and flow D.4(b) responses of the non-linear model with identified parameters compared with the testing data set
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