TREBALL FINAL DE GRAU VALIDATION OF THE CL-CTM FREEWAY MERGE MODEL Treball realitzat per: Carles Cañero Martínez Dirigit per: Francesc Soriguera Martí Grau en: Enginyeria Civil Barcelona, 18 juny 213 Departament d Infraestructura del Transport i del Territori (ITT)
ABSTRACT This paper presents the empirical validation of the Capacity Lagged Cell Transmission Model (CL-CTM) which consists on an improvement of the well know Newell-Daganzo merge model called Cell Transmission Model (CTM). The model introduces the Lagged property as well as the capacity drop phenomenon in order to improve the accuracy of the method and bring in an observed phenomenon in empirical investigations. For this main objective the data used comes from a database of the B23 freeway in the vicinity of a merge point with congestion and free flowing states to encompass any possible situation. The essential parameters to run the model have been obtained from this database. Finally, the model is run for the wide range of data available and the results obtained are compared to the data measured.
1. INTRODUCTION In 1955 Michael J. Lighthill and Gerald B. Whitham firstly associate traffic flow with the movement of flood in rivers; with the introduction of shock-waves on the highway by Paul I. Richards in 1956 the LWR was completed. This macroscopic model consists on a Partial Differential Equation (PDE) that relates the main traffic flow variables: flow, density, speed, etc. Since then, lots of studies have found solutions to the LWR model, a remarkable one is the proposed by Gordon Newell and Carlos F. Daganzo named Cell Transmission Model (CTM) (1995) because of its low computational cost and highly reliable results. However, it has been proved afterwards that this model lacks of reality accuracy as it does not take into account the capacity drop as well as has problems when the shock wave propagation does not coincide with the time step chosen. Soriguera and Torné (214) propose some additions to the Newell-Daganzo merge model (CTM) incorporating the solution for both problems before mentioned. This additions lead to a more accurate modelling of freeway active management strategies, specifically in case of the presence of Dynamic Speed Limits (DSL). This new model is named Capacity Lagged Cell Transmission Model (CL-CTM) and has been proved to be theoretically correct but needs an empirical validation to be accepted completely. A validation of a traffic model is obviously more accurate as the results predicted resemble the actual behaviour of the traffic flow but in spite of its accuracy, if the data used is really short the model has not an enough credibility; it is necessary a great amount of data to encompass any given possibility. For this validation, the B23 freeway accessing to Barcelona is used, especially the vicinity of a merge point. Soriguera and Sala (213) constructed a database of the last 13 Km of the B23 freeway during 9 days between May and June 213. Some of the data available in this database is used for the validation, the one that corresponds to detectors 13 to 19 for 4 different days. Some traffic variables are analysed while pursuing the objective of obtaining the main parameters of the fundamental diagram to run the model and compare the real data measured to the parameters estimated. The rest of the paper is organized as it follows. Next, Section 2 describes CTM and CL-CTM models. Section 3 present briefly the test site used for the validation. Then, the parameters estimation is presented in Section 4. Model validation and the results are presented in Section 5 while Section 6 consists on the conclusions. Finally Section 7 and 8 present the acknowledgements and references.
2. MODEL DESCRIPTION The Cell Transmission Model (CTM) is a first order model that presents the Godunov s discrete approximation to the LWR s PDE; this model is particularly relevant for its low computational cost and reliable results. It consists on the discretization of space and time for the studied zone into cells (constructing a regular grid) and some simple rules for the flows received and sent for every cell at each time lapse. For this case, the studied zone is: For the validation of the model in this zone the next values have been considered: The CTM predicts the state of every cell at any time step starting from the initial conditions i.e. number of vehicles in the cell at a given time and the upstream and downstream boundary conditions. The CL-CTM adopts the CTM main structure but introduces the Lagged property and the capacity drop phenomenon.
The employed method is the same and described next. In both models, there is two different kind of cells, the ones that receive flow from one unique cell, which are called ordinary cells; and those which receive from two cells converging to into it, they are called merging cells. The formulation for these two kinds of cells is different. Ordinary cells: Two equations describe the two main parameters in CTM, the number of vehicles in the cell and the flow received. The first equation is a mass balance and it is presented below: Where: The second equation describes the flow between two consecutive cells: [ ( ) ] The first component is the capacity of the sending cell while the second one is the maximum flow available of this same sending cell. The third condition is the capacity of the cell studied; the one which receives the flow while the fourth is the maximum flow that can be allocated in the cell. In this equation the Lagged property is already introduced with f and l parameters that appear in the second and fourth components of the equation. Those parameters improves the model accuracy as the CTM do not describe reality correctly when the shock wave propagation is not proportional to the time lapse Δt chosen for the validation; this is due to the combination of the use of a discrete model for a continuous variable as time. The expressions of these parameters are:
As it was mentioned before, the second equation is divided in two components from the sending cell and two from the receiving one, so it can also be written: [ ] [ ( )] Using the second equation or this pair separating the sending and receiving cell, the is determined by the binding constraint of all 4 components. If the fourth component (from now on: ) is binding means that the queue has arrived to the cell and the capacity drop must be taken into account because the capacity is reduced. It can be rewritten as: { Where. If the binding constraint is, it is a bottleneck. Merging cells: In this case a new sending cell must be considered so the first equation is now formulated: Where is the flow per time lapse that sends the complemented cell. In addition, the model considers that when the available capacity at the merge is not enough to serve the demand from both cells (i.e. ) the merging flows and must satisfy: [ ] [ ] Where and are the fractions of the total merging flow coming from the mainstream and the onramp when both are congested.
In case the capacity at the merge is enough to serve de demand: Exactly as in the ordinary cells case, the capacity drop phenomenon is introduced to achieve a more realistic approximation. In order to include the loss of flow by the convergence of two cells into one, a new parameter is introduced, Beta (β). Then, the free flowing capacity at the merge cell is: Where is the maximum free flowing capacity if there was no merge and is the maximum queue discharge rate. The capacity drop α and is determined by the relation between and while the Beta coefficient is defined by the relation between flows in both cells.
3. TEST SITE DESCRIPTION It is important to remember the main objective of this paper, which is the empirical validation of a theoretical model. An empirical validation needs a great amount of values to be reliable so the location chosen must be according to this purpose. The vicinity of the B23-A2 merge point is the adequate location as it has plenty of detectors as well as a DSL strategy management is used and CL-CTM is especially suitable for these cases. The B23 freeway is in the metropolitan area of Barcelona and the direction chosen is towards the city. Here is shown the geographical location of the freeway with beginning of the freeway in red: In order to be more specific, the next scheme shows the structure of the B23 zone that has been used, from Km. 7,28 to Km. 4,23 which corresponds to detectors 19 to 13, respectively.
The detectors in the scheme are all named ETD but in the next chapter the difference between them will be explained. In addition, from now on, the section where the detectors are placed will be called with the same number.
4. PARAMETERS ESTIMATION Soriguera and Sala (213) constructed a database of this last 13 Km which includes the junction with A2 freeway that is going to be really helpful while trying to calibrate de model. This database was registered during May and June 213 in 9 different days only between 7: AM and 1: AM. Those days and its nomenclature from now on are: Date Nomenclature Thu. 3 th May 213 Day #1F Tue. 4 th June 213 Day #1 Wed. 5 th June 213 Day #4F Thu. 6 th June 213 Day #5 Tue. 11 th June 213 Day #3 Wed. 12 th June 213 Day #4 Thu. 13 th June 213 Day #6 Tue. 18 th June 213 Day #7 Wed. 19 th June 213 Day #2 For this project, only days #1F, #1, #6 and #2 are considered. For each day, and each one of the 26 detectors, the database has per minute: Number of vehicles Average speed Lane occupancy Those 18 values for each one of the three variables are registered, for each section, in two ways: by lane and aggregated. While the number of vehicles and occupancy can be added directly from all the lanes to obtain the sectional value, the speed aggregation is a little bit more complex. The arithmetic average would not have taken into account the different weight that has each lane depending on how many vehicles pass through, so a weighted average has been used. Where is the number of lanes.
The number of lanes varies between 2 and 4 depending on the section. For this project, only detectors 13 to 19 are considered. There are two sections that must be studied deeply: Detector 17 is placed in the mainstream and an on-ramp, this means that the first and second lanes are placed in the B23 freeway while the third and fourth on the on-ramp coming from the A2 freeway. Consequently, the first two lanes must be taken into account separated from the third and fourth. That is why detector 17 is divided in s17_12 and s17_34. In addition, trucks are forced to leave the freeway just at the section where the detector 19 is located. In this section there are 4 lanes but two of them lead to the off-ramp where trucks leave so they are not considered in the study of detector 19. So, finally our 6 detectors considered are: s13, s16, s17_12, s17_34, s18 and s19. It is also relevant to discuss the different kind of detectors that has been used. There are three types of detectors that can be treated as two groups: DT detectors: Triple technology, non-intrusive traffic detectors. They use Doppler radar, ultrasound and passive infrared detection in order to obtain the vehicle count, average speed and detector occupancy per lane and per minute. S detectors: Simple loop detectors. Installed under the pavement they measure vehicle count and detector occupancy per lane and per minute but not speed. ETD detectors: Traditional double loop detectors. Also installed under the pavement, they are able to measure what simple loops can and average speed measurements. As it is explained later, S and ETD detectors are going to be treated equally while DT detectors are going to be evaluated as another group. With the data proportioned from these 6 detectors, we are able to calculate the density, an important value in traffic analysis. There are two options to find it but both of them have its inherent errors as they contain hypotheses that might not adjust to reality. First option: Every day, there are some detectors that measure individual data like vehicle speed and the time (in milliseconds) that the vehicle has stood over the detector. With these two values and the length of the detector, is possible to calculate the length of every vehicle that passes by that detector between 7: AM and 1: AM. Those detectors are 13 and 19. Individual vehicle length is calculated with:
Where: However, not for all days the same detector with individual data is available. For days #1F, #1 and #6 detector 13 is used while for day #2 the detector with individual data is 19. In addition, those two detectors are not the same kind because 13 is a DT detector when it measures individual data while detector 19 is an ETD. This difference is important because DT detectors have not length; they are not under the pavement because they use radar, infrared and ultrasound technology. However, ETD detectors have a 2 meters length and it must be considered in order to calculate the individual vehicle length. This means that the individual vehicle length ( ) for days #1F, #1 and #6 is only correct for detectors DT while for the S and ETD detectors 2 meters must be added. The opposite must be done for day #6, the individual vehicle length is correct only for detectors S and ETD while for DT detectors 2 meters must be subtracted. The detector, its kind and number of lanes is presented below: Detector Kind # Lanes s13 S 3 s16 DT 3 s17_12 S 2 s17_34 DT 2 s18 S 2 s19 ETD 2 Once finally the correct individual vehicle length is achieved we are able to calculate the density (k). Starting from the definition of occupancy we get to the following equation: [ ] Where: This first option uses individual data measured in some detectors to obtain density in other detectors assuming that the vehicles that pass through a detector also gets to the others but there are on-ramps
and off-ramps in the zone studied. We assume that, in spite of this hypothesis, the density is accurate because trucks are not allowed in this section and cars length does not change significantly. Second option: In this case, the hypothesis is to assume that the speed measured by the detectors is the space mean speed which is not absolutely correct because is the temporary mean speed. With this assumption we can calculate the density for every minute, using the fundamental traffic relation: Those two densities are both calculated with a certain degree of error but they should not be much different from each other so, to prove it a relation is done, section 13 is presented below while the other sections are on Annex 1. This case is particularly in section 13 but it is just an example of how clearly observable is that the dispersion is over the unity, mostly for low densities while it is more dispersed as the densities grow. For the rest of the sections the same phenomenon is observed; this higher dispersion is due to the hypothesis done for the second option density; it is less accurate as speeds are lower. The lower dispersion and bias of the first option is the main reason to choose it and, from now on, work only with it. With the density chosen and the occupancy, it is possible to calculate the average length for the vehicles for every minute. However, for the use of different detectors explained in the first density option, this length must be, in average, 2 meters different for days #1F, #1 and #6 from day #2. An average for every 5 minutes (36 values) is done to observe clearer the distribution of average lengths.
In spite of having two averages lengths between 4 and 4,5 meters in day #2, the main part of the 36 values are between 6 and 6,5 meters while for the other days nearly all values are between 4 and 4,5; the 2 meters difference that was expected. However, adding or subtracting the 2 meters when needed, the sample is homogenised are the results obtained are nearly identic for all days with an average length a little higher of 6 meters:
occ [%] occ [%] Length 35 3 25 2 y =,5751x 15 Day #2 1 5, 1, 2, 3, 4, 5, k [veh/km] Length 2 175 y =,6557x y =,648x 15 125 y =,6283x 1 75 Day #1F Day #1 Day #6 5 25, 5, 1, 15, 2, 25, 3, k [veh/km]
q [veh/h] So the average lengths for every day are the followings: In addition, flow-density relations are done to observe the appearance of the fundamental traffic diagram for every section. Particularly, section 13 is presented below with a 5 minutes aggregation; the others Fundamental Diagrams as well as those calculated with are presented in Annex 2. 7 Fundamental Diagram s13 (5 min) 6 5 4 3 2 Day #1F Day #1 Day #6 Day #2 1, 5, 1, 15, 2, 25, k [veh/km] An essential value in traffic modelling can be obtained from the analysis of the flow-density relation which is speed. There are two main phases in this diagram, the free flow and the congested part. The first one has a positive slope, it means that as well as density increases, flow also does and this slope is called free flow speed. Then a maximum flow is reached (called the section capacity), from then, while density increases the flow is decreasing, the slope in this second part is negative and called shock wave speed (w). This second speed will be presented afterwards while the first flow speed will be commented briefly. The free flow speed is approximately the percentile 9 for all speeds registered and this comparative is done and presented below:
q [%] Section Speed [km/h] Day #1F Day #1 Day #6 Day #2 s13 Speed.9 7,2 75,59 64,16 74,7 Free Speed 69,4 75,2 6,1 67,8 s16 Speed.9 89, 89,6 89,8 89,1 Free Speed 73,5 67,7 67,2 69,3 s17_12 Speed.9 87,3 86,1 84,1 85, Free Speed 116, 95,4 98,5 9,1 s17_34 Speed.9 85,6 83,5 83,7 85,3 Free Speed 13, 92,6 88,8 86,2 s18 Speed.9 14,5 13,3 1,8 99,3 Free Speed 11,5 19,3 12,1 1,8 s19 Speed.9 12,1 98,3 96,5 99,2 Free Speed 13,3 13,1 13, 99,2 Also, flow-occupancy is presented in a 5 minutes aggregation for section 13, the other sections are presented in Annex 3: 7 q-occ s13 (5min) 6 5 4 3 2 Day #1F Day #1 Day #6 Day #2 1 2 4 6 8 1 12 14 k [veh/km]
NT graphics A cumulative arrival curve is a graphical method that plots the number of vehicles that passes through a section to time, and identifies changes in traffic state; in addition, the slope of the representation is the flow. However, in these representations the slope is always positive and therefore, hard to identify subtle changes of traffic flow. In order to facilitate the analysis of this method, Micheal J. Cassidy and John R. Windover described in 1995 in their article Methodology for Assessing Dynamics of Freeway Traffic Flow a new concept called background flow ( ). The background flow is a coefficient, approximately the mean arrivals per elapsed time, and it reduces cumulative counts uniformly. There is a background associated to each section and day. Then, the curve plotted is: From now on, the curve is called oblique N curve and it is no longer positive anyway, only when the flow is higher than the background flow. This simple change allows an easier analysis of traffic states as it is clearer when a flow drop occurs. This same method is also used with the occupancy and it has its own background coefficient, the background occupancy ( ). It allows seeing the fluctuations along the three hours of study for all sections each day. So the oblique occupancy curve is: For all days, those curves have been plotted for sections 13 to 19. The oblique N and T curves for Day #6 are presented below.
As it has to be, all the curves starts from at 7: AM, which corresponds to time and finish also at because the background coefficients has been measured taking the last value of the cumulative curves, divided by the 18 minutes the data is measured.
In both kinds of curves the fluctuations are clearly seen and they are more evident when they are displayed in their own scale, not sharing the axes with the maximum curve that is given for section 16, especially in T curves. From now on, section 19 won t be analysed more as it is not relevant for the model. One of the main important conclusions that can be obtained from the oblique NT curves is the congestion periods which are essential to detect bottlenecks. A congestion causes a decrease of flow while an increase of occupancy at the same time, which is quite intuitive to understand. The opposite phenomenon is a free flow state, where the flow increases and so does the occupancy. Plotting the oblique NT curves together for a chosen day and section, the congestion states are clearly detected. As an example, section 16 for day #6 and section 18 for Day #1F are presented below while the other sections and days are on Annex 4.
The zones in green are the time lapses where the section is congestioned, so in Day #6 section 16 is congestioned twice while in Day #1F section 18 is not anytime. As it is shown in the Annex 4, these congestion zones vary from day to day in number, time length and starting time. In addition, the maximum flow for the day and section is also presented. In spite of these possible changes, there is a tendency observed for each section. Section 13 usually shows a few minutes congestion before another larger; the first one is due a little fluctuation of few minutes and it is called a spillback. The second one starts around 8:3 AM and usually finishes near 9: AM. Section 16 has an earlier congestion, commonly of 3 minutes and starts at 7:3 AM while a second one is also done for some days. In addition, section 17 presents, at both pairs of lanes, short congestion states approximately at the same time section 16 does. Finally, section 18 seldom shows any. Another important parameter that can be observed in oblique NT curves is the derivatives, the flow as the derivative of N curve and occupancy for T curves. So, every congestion state has its own congestion flow and occupancy associated Q d and occ d respectively. In addition, as it was presented before, there is a relation between occupancy and density ruled by the average length so, it can be obtained also the density associated to those congestion states, k d. Moreover, the maximum slope in N curves i.e. the maximum flow is also easily calculated and an essential parameter in order to construct the fundamental diagram and it will be also related to a maximum occupancy and density, Q f, occ f and k f.
q [veh/h] q [veh/h] Plotting for every section the pairs of values Q f -k f and Q d -k d for every congestion for all days, a scheme of the fundamental diagram is obtained, from which it is possible to obtain two essential parameters to calibrate the model, the slope in congestion state (w) and Jam Density (K). As it is said, not all sections have an equal number of congestions so the graphic is more accurate for those sections where more values are available. Those values that had been obtained from intervals shorter than 5 minutes were depreciated and, with the other congested values, a least squares approximation has been used. The results for sections 16 and 18 are presented below while the others are on section 5: 7 6 s 16 5 4 3 2 1 5 1 15 2 25 3 k [veh/km] Day #1F Day #1 Day #6 Day #2 3 25 2 15 1 5 s 18 2 4 6 8 k [veh/km] Day #1F Day #1 Day #6 Day #2 Once the approximation to w is done, the maximum density K is easily found. In addition, it was shortly commented that section 18 has only one day with a congestion, as it is seen in the graphic above, is Day #6. With that unique value and coming from an only 5 minutes interval, it is not possible to do the least squares approximation so parameters w and K are not available for this section.
Finally, all the parameters presented until there are presented in the following table, it means: the maximum and congested occupancy, flow and density with its time lapse and capacity drops associated, the maximum density (K), w and the average length: #1F #1 #6 #2 s13 s16 Q f cong 1 cong 2 cong 3 Hores 7:12-7:52 7:16-7:57 7:16-7:45 7:8-8: Q f [veh/h] 593 586 5796 567 k f [veh/km] 18,4642367 115,1685393 12,165526 118,177529 occ f [%] 71,12 73,8 75,5 67,96 Hores 7:52-7:55 8:8-8:1 7:49-7:51 8:9-8:11 Q d [veh/h] 4698 3264 468 2784 k d [veh/km] 125,5147171 2,9987516 176,8263568 225,8737611 occ d [%] 82,3 128,8 111,1 129,9 Capacity drop α[%] 26,2239257 79,53431373 42,47787611 13,6637931 Hores 8:23-8:26 8:25-9: 8:24-9: 8:31-9:9 Q d [veh/h] 342 445 435 498 k d [veh/km] 151,2886991 155,2746567 141,7316569 177,126934 occ d [%] 99,2 99,5 89,5 11,8 Capacity drop α[%] 73,39181287 44,872114 33,24137931 38,361757 Hores 8:36-8:51 - - - Q d [veh/h] 4388 - - - k d [veh/km] 116,5166997 - - - occ d [%] 76,4 - - - Capacity drop α[%] 35,14129444 - - - w [km/h] 21,8 K [veh/km] 35 Hores 7:1-7:28 7:9-7:3 7:7-7:26 7:13-7:27 Q f cong 1 cong 2 Q f [veh/h] 5768 5574 5484 5532 k f [veh/km] 82,2632314 62,17228464 92,15342989 68,56198922 occ f [%] 53,94 39,84 57,9 39,43 Hores 7:28-8:2 7:3-7:44 7:43-8:9 7:27-7:57 Q d [veh/h] 4881 4826 4368 546 k d [veh/km] 189,4158914 158,551812 194,8113958 151,278386 occ d [%] 124,2 11,6 122,4 87 Capacity drop α[%] 18,1725563 15,49937837 25,5494555 9,63139121 Hores - 8:42-8:49 8:3-8:53 8:14-8:16 Q d [veh/h] - 336 368 1641 k d [veh/km] - 57,1161487 113,95832 256,4771344 occ d [%] - 36,6 71,6 147,5
s17_12 s17_34 s18 Capacity drop α[%] - 68,625483 49,2173913 237,1115174 w [km/h] 2,6 K [veh/km] 376 Hores 7:11-7:29 7:18-7:33 7:17-7:26 7:21-7:45 Q f cong 1 cong 2 Q f [veh/h] 261 2658 2694 2532 k f [veh/km] 35,24477657 14,91763 18,3335827 18,6545992 occ f [%] 23,11 9,3 11,5 1,7 Hores 7:29-7:44 7:33-7:38 7:26-7:28 8:15-8:18 Q d [veh/h] 2598 236 138 96 k d [veh/km] 61,61354278 22,31585518 68,53413974 51,29542688 occ d [%] 4,4 14,3 43,6 29,5 Capacity drop α[%],461893764 15,26452732 95,2173913 163,75 Hores - - 7:55-7:57 Q d [veh/h] - - 834 - k d [veh/km] - - 113,48 - occ d [%] - - 71,3 - Capacity drop α[%] - - 223,215827 - w [km/h] 17,2 K [veh/km] 193 Hores 7:1-7:29 7.12-7:24 7:6-7:26 7:12-7:24 Q f cong 1 cong 2 Q f [veh/h] 324 3111 2754 3264 k f [veh/km] 13,619339 12,68726592 14,7381824 14,25838985 occ f [%] 8,93 8,13 9,26 8,2 Hores 7:29-8:11 7:34-7:53 7:26-7:48 7:26-7:32 Q d [veh/h] 258 2538 2238 2928 k d [veh/km] 33,5671811 11,9382225 41,1193427 18,43157712 occ d [%] 22,1 7,65 25,83 1,6 Capacity drop α[%] 29,1866287 22,57683215 23,56327 11,4754984 Hores - 7:54-7:58 8:14-8:21 Q d [veh/h] - - 1357,8 2394 k d [veh/km] - - 53,87553716 28,6966249 occ d [%] - - 33,85 16,5 Capacity drop α[%] - - 12,828143 36,3485213 w [km/h] 16,6 K [veh/km] 18 Hores 7:15-7:45 7:18-7:35 7:12-7:39 7:14-7:43 Q f Q f [veh/h] 2634 2574 2522 2448 k f [veh/km] 35,9926954 24,18851436 29,28537323 26,46496262 Occ f [%] 23,6 15,5 18,4 15,22 cong 1 Hores - - 7:56-8:1 -
Q s17_34 Q d [veh/h] - - 2178 - k d [veh/km] - - 68,75696323 - occ d [%] - - 43,2 - Capacity drop α[%] - - 15,794367 - w [km/h] - K [veh/km] - Le [m] 6,557 6,48 6,283 5,751 Finally the last two parameters for the validation of the CLCTM are presented next: Beta Merge Ratio. and the The Merge Ratio is the existent relation between the flows in the first two lanes of section 17, the ones that are in the B23, and the two second ones, those that comes from the A2. So the MR is the relation between both flows in the merge point: Its graphical representation when both sections are congested is presented next: MR (γ) 4 35 3 25 2 15 y = 1,145x y = 1,1325x y = 1,2618x y =,9442x Day #1F Day #1 Day #6 1 Day #2 5 5 1 15 2 25 3 35 4 Q s17_12 Day #1F Day #1 Day #6 Day #2 MR (γ),9442 1,2618 1,145 1,1325 Using the values measured for the time intervals where both sections are congested, the average value for MR is:
Beta Estimated s17_12 [veh/h] 2497,5 s17_34 [veh/h] 262,5 Average MR ( ) 1,424242 Beta is a coefficient that determines how the general flow of B23 is affected by the relation between flows at the merge point. For the calculation of this value, an average of the Q d in section 16 is needed as well as an average of capacity drop in section 16, Q f in section 13 and the values of flow in section 16 while section 17 is congested in both pairs of lanes (Q fm ). Q d 16, Q f 13 and capacity drop α 16 are presented next while Q fm is different for each congestion once or twice: 16 [veh/h] 4792,94238 13 [veh/h] 5868,581818 capacity drop 16,224421543 In addition, two different Betas has been analysed; the measured and estimated. The first one is calculated with the values presented above, while the estimated has been calculated with the relation between the Gamma found in every interval of congestion in all section 17 that coincides with a uniform flow in section 16, and the average Gamma presented before. Moreover, for all congestion states, those which were just before the congestion has been treated separately than the others as shown below: 1,2 1,8,6 y =,9683x y = 1,1719x Just before congestion Others,4,2,2,4,6,8 1 Beta Measured Lineal (Just before congestion) Lineal (Others) Also, the relation between Q d (1+α βe) and Q fm in section 16 is presented as well as its relative error.
(Q d (1+α βe)-q fm 16)/Q fm 16 [Fraction] Q d (1+α βe) [veh/h] 7 6 5 4 y = 1,167x 3 2 1 1 2 3 4 5 6 7 Q fm 16 [veh/h],1,8,6,4,2 -,2 1 2 3 4 5 6 7 -,4 -,6 Q fm 16 [veh/h] Finally, it has been proved that when the Beta (β) coefficient is nearer to 1, so the flows are approximately the same, the flow in B23 is less affected. Probably this solution is given due to if there was no flow in one of the sections the other one would have quite more lane changes that surely affect to reach the maximum possible flow.
q17_34 [veh/h] 5. MODEL RESULTS & VALIDATION Finally, all the parameters found in the previous chapter allow running the Capacity Lagged Cell Transmission Model and compare the results predicted by the model to those measured. In next pages 7 different graphics will show some important variables from the model compared to the measured data. First of all, a comparative of the flows coming from mainstream and on-ramp in the merging point. The vertical and horizontal green lines shows the capacity that have those sections and, as they never arrived at capacity flow for the days measured, it has been calculated proportionally with the lanes of a section that did arrived i.e. 2/3 of in section 13. In addition, the diagonal green line represents this which can t be overpassed. As an example, Day #1 is presented below while the others are on Annex 6: 7 6 5 4 Day #1 3 2 model measured 1 1 2 3 4 5 6 7 q17_12 [veh/h] Few minutes measured overpass the second condition that the sum of the flows can t be higher than but with empirical data there are always data errors induced by the measure instruments. However, all the values proportioned by the model accomplish both conditions which support the validation of the model. Then, the relation between those flows in section 17, previously called Merge Ratio (γ) is presented. The approximation is better for merge ratios near 1 and the dispersion increases as the values get further. Day #1F is presented as an example; the other days are on Annex 7:
Absolute error MR measured 4 3,5 3 2,5 2 1,5 1,5 MR model-measured Day #1F y =,9448x y =,8737x,5 1 1,5 2 2,5 3 3,5 4 MR model Others Congestion Lineal (Others) Lineal (Congestion) For all the 18 values, those which correspond to a congestion state simultaneously for sections 17_12 and 17_34 has been signalised differently and as the graphic shows, have a more accurate approximation. To ensure the approximation, the absolute and relatives errors for the merge ratios have been plotted for the same day studied, Day #1F: 3 2 Day #1F 1-1 -2-3 -4 5 1 15 2 25 Occupancy [%] Others Congestion
q [veh/h] Relative error 3 2,5 2 1,5 1,5 -,5-1 -1,5 Day #1F 5 1 15 2 25 Occupancy [%] Others Congestion First of all, the occupation axe is the sum of the occupation for each lane, so it can be higher than 1% as it is for some values. Then, as it was seen in the Merge Ratio graphic, the values in congestion state are more accurate; their relative error does not overpass the,5 limit while the others have more dispersion. However, the main part of these other values is also between this limits so the model is being validated for any case. Also a flow-occupancy graphic for the section immediately after the merge point is presented as well as the error in this approximation; Day #1F is still the chosen but this time the data is presented with a 5 min aggregation to make the results clearer if 18 values were plotted for the model and as many for the measured, the other days are presented in Annex 8 without the 5 minutes aggregation. 7 6 5 4 Day #1F (5min) 3 2 1 25 5 75 1 125 15 Occupancy [%] model measured
N17_12 - N18 [veh/h] (q model-q measured)/ q measured [veh/h] Day #1F 4 3 2 1-1 -2 5 1 15 2 25 Occupancy [%] In spite of the data with a relative error of nearly 4, the values present a correct approximation for low occupancy values as for higher ones, which correspond to congestion states. Finally, an analysis of the queue between section 18 and 17 in the mainstream part is presented. The queue has been calculated as the difference of flow between section 17 in lanes 1 and 2 minus the flow in section 18. Day #1F is presented next while the other days are on Annex 9. 4 Queue Day #1F 2-2 -4 2 4 6 8 1 12 14 16 18 model measured -6 min The fluctuations of the difference between flows are quite symmetric so there is no queue generated in this section; in addition, the model predicts the real situation precisely, especially for short differences.
6. CONCLUSIONS The CL-CTM is not a revolutionary model which creates a new vision for active traffic management but a model that takes a consolidated one (CTM) and introduces some changes to make it a more accurate resolution with the Lagged property as well as the capacity drop phenomenon to capture an already observed phenomenon. A theoretical model is validated as it represents accurately the reality in any possible situation. This paper tries to include all those possible cases so the test site is in the vicinity of a merge point, to consider ordinary and merging cells. In addition, the interval of time when the data was measured was large enough to ensure congestion sates as well as free flowing. Some parameters as the merge ratio (γ) related to and mentioned in the Model Description- or Beta (β) that were introduced in the theoretical part are actually as relevant as it was thought so the capacity of the cell is precisely predicted by the model. Then, with all the parameters found the model is run and the results obtained are compared to the measured and registered values in the database. The approximation is correct for all days, and all cases (congestion or not, merge point or the mainstream part of the freeway) as it has been proved in the last chapter, so the Capacity Lagged Cell Transmission Model is correctly validated. 7. ACKNOWLEDGMENTS The author would like to thank the invaluable help of Josep Maria Torné and Francesc Soriguera for their advices and constant dedication in this paper. Also, mentioning the assistance provided by Marcel Sala for an important part of this validation project.
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