6.1.
Conclusion
As presented in chapter 3, the research questions were the following:
1. How can a model be developed to provide online estimation of the traffic state of an urban network, taking into account measurements from sensors?
2. How accurately can this developed model estimate the fundamental diagram parameters of each link of the network?
To address the research questions, the developed solution uses Streamline as the process model. It is a higher order LWR model, which is more suitable than a first order model because of the more correlations between links that can be captured and lead to more opportunities for correction on links for which no measurements are available. In addition, Streamline incorporates a junction model (XSTREAM), which is a vital element of a model designed for urban networks, as the role of the junctions is decisive. The observation model was set up accordingly, using the appropriate equations (measurement functions) of the process model. Extended Kalman Filtering was selected as the data assimilation method, as it is suitable for working with non-linear equations and is computationally efficient. As it was required to develop a model suitable for online state estimation, the model does not require any pre-processing of the data and is capable of running faster than real-time, at least for relatively small networks. Finally, the state vector was set up to include the speeds and densities, which are needed to answer the first/main research question, as well as the fundamental diagram parameters, in order to cover the second/additional research question as well. To answer the research questions and assess the effectiveness of the developed approach, the model was validated through a series of tests using artificially created measurements.
Regarding the first research question, it can be concluded that the model could satisfactorily estimate the densities and speeds of all links of the network, as the average error values of all tests were at an acceptable level. Estimation improves over time, as the model “learns”
through the fusion of measurements. An increasing uncertainty level of the system leads to a reduced rate of improvement overall for the network. Estimation of the speeds appears in most tests to be more accurate than the estimation of the densities. This can be attributed to the fact that the densities are indirectly corrected through the fusion of flow measurements, while the speeds are directly corrected through the fusion of speed measurements. Another contributing factor is the method selected for fusing the flow measurements in the developed approach, due to technical reasons. By summing up flows from each direction instead of fusing each directional flow measurement, the correction effect of the flow measurements is slightly reduced.
Uncertainty and error on the measurements is the parameter with the highest impact on the results, followed by the inaccuracy of the fundamental diagram parameters and the
64 inaccuracy of the OD matrix. Combinations of these parameters in most cases led to an increased combined impact on the state estimation. Among the fundamental diagram parameters, the free flow speed has the most significant impact on the estimation of the speeds and consequently the densities.
The results of the last test, which combined all uncertainty parameters showed that, although estimation improved overall in that test as well, the results of certain links showed a deteriorating estimation, as the error values of these links kept increasing over time. This occurred in links on which the uncertainty parameters had the highest impact, e.g. links connected to centroids (and therefore affected by the OD matrix inaccuracy) while at the same time being situated away from any measured link, offering practically no opportunity to correct its state. This result especially underlined the importance of the availability of measurements. It showed, in addition, that very high uncertainty levels (perhaps even higher than those tested) may lead to a failure in estimation. Therefore, it is important to attempt to reduce uncertainty on parameters that can be pre-processed, such as OD matrix calibration or setting the initial fundamental diagram parameter values.
In cases where estimation of the fundamental diagram parameters is not of importance, the model can be modified to exclude them from the state vector. The immediate result would be a significant increase in simulation speed, as the dimensions of the Jacobian matrices would drastically decrease allowing for faster calculations. It can also be argued that estimation of speed and density will also slightly benefit from this change, as the model will have less parameters to calibrate when applying the correction. However, additional tests would be required to validate this hypothesis.
From the analysis of the most problematic links in each test, it was observed that an inaccurate OD matrix proved to have a higher impact on the estimation of the densities of links connected to a centroid (destination/origin). In addition, in tests with higher uncertainty it was observed that state estimation of links situated on parts of the network where no measurements were available did not improve over time, underlining the importance of the number of measurement points on the network, as well as their location. Regarding the second research question, the model succeeds in improving the estimate of the free flow speed over time but fails to estimate the speed at capacity and capacity per lane. The improvement in the estimation of the free flow speed proved to be independent of the initial values set. The standard deviations which control the variation allowed per time step for each fundamental diagram parameter could be increased in order to achieve faster convergence of the free flow speed. However, a high standard deviation value would lead to more nervous behavior of the estimated value of the free flow speed, possibly leading to a higher average error over time. Therefore, a series of preliminary tests could be run beforehand, in order to decide on their values. Finally, the inclusion of a warm up phase, to allow the parameters to approach their actual values, before actual state estimation begins, is recommended whenever possible.
65 The most probable cause of the observed failure of the estimation of the speed at capacity and the capacity per lane appears to be the fact that in the tests performed the network was mostly in free flow conditions. As there were almost no measurements near capacity, the system cannot estimate the relevant fundamental diagram parameters. However, this hypothesis needs further validating through relevant tests on networks on near-capacity conditions.
A recommendation would be to use sets of common fundamental diagram parameter values
for consecutive/”similar” links in the network, instead of separate values per link. By following this strategy, it is expected that the system will converge to the correct fundamental diagram parameter values and respond to changes in their values faster even in cases with lower availability of measurements, as the EKF will have more sources of information for the same parameters. However, this hypothesis has not been tested yet. Finally, regarding implementation, it was observed that changing the order of fusing the measurements had a negligible impact to the state estimation. In addition, the importance of setting reasonable thresholds for the state variables needs to be stressed, as it can prevent the estimation from diverging, especially in cases with high uncertainty.
6.2.
Suggestions for future research
Future research could focus on the alleviation of the inefficiencies of the developed model, its expansion, as well as topics that were derived from the discussion of the results of this research.
An important element of the developed model that could be improved is the definition selected for the outflow limit factors. Defining and using outflow limit factors per turn direction, as well, would lead to an improvement in the fusion of flow measurements and consequently an improvement in state estimation, especially the estimation of the densities. Introducing sets of common fundamental diagram parameters is another suggestion that could improve estimation of the fundamental diagram parameters, as well as the speeds and densities.
Applying the model on a more congested network would shed light on its ability to estimate the speed at capacity and capacity per lane, too, instead of only the free flow speed. By fusing more measurements near capacity, it is expected that the estimation of those fundamental diagram parameters would improve, as the model would have more information on conditions related to these parameters.
A possibly determining factor that has not been examined at all in this research is latency. It is assumed that the measurements for each minute are immediately available at the end of the 60th second. However, in a real network, this assumption is not realistic, as
66 factors, such as the data size, transmission speed and the required processing of sensor data. A suggestion offered as a starting point when researching how to handle latency in a real network is the following: The traffic state of each time step (e.g. minute) would be predicted from the process model, based on the traffic state of the previous time step. Kalman Filtering would be applied for a time step several minutes in the past, as soon as measurement data for that time step is made available, resulting to a corrected traffic state for that time step. Using this corrected traffic state, the predictions of the traffic states of all subsequent time steps would then be recalculated. For example, assuming a latency of five minutes, the process would begin with the traffic state being predicted solely by the process model for the first six minutes. At the end of minute 6, the measurements for the first minute would be made available. The traffic state of minute 1 would then be corrected accordingly, by fusing the measurements for minute 1. This updated traffic state of minute 1 would be used to predict again the traffic state of minutes 2-7 using the process model. Similarly, at the end of minute 7, measurements for minute 2 would be available, leading to an updated traffic state for minute 2 and a new prediction for all subsequent minutes etc. Finally, the use of the Kalman filter provides the opportunity to estimate other parameters as well. For example, routing could be affected through the use of the turning fractions, which could be included in the state vector and changed by the Kalman filter with the information from the measurements. The turning fractions would then replace the relevant
junction modelling factor (“inflow reduce factor”) in the equations, as the factor practically
imitates the turning fractions. OD matrix calibration could also be included as an additional process to be followed at the end of each time step.
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