Link flow destination distribution estimation based on observed travel times for traffic prediction during incidents

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(1)LiU-ITN-TEK-A--20/025--SE. Link flow destination distribution estimation based on observed travel times for traffic prediction during incidents Anna Danielsson Gabriella Gustafsson 2020-06-12. Department of Science and Technology Linköping University SE-601 74 Norrköping , Sw eden. Institutionen för teknik och naturvetenskap Linköpings universitet 601 74 Norrköping.

(2) LiU-ITN-TEK-A--20/025--SE. Link flow destination distribution estimation based on observed travel times for traffic prediction during incidents The thesis work carried out in Transportsystem at Tekniska högskolan at Linköpings universitet. Anna Danielsson Gabriella Gustafsson Norrköping 2020-06-12. Department of Science and Technology Linköping University SE-601 74 Norrköping , Sw eden. Institutionen för teknik och naturvetenskap Linköpings universitet 601 74 Norrköping.

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(4) Abstract In a lot of big cities, the traffic network is overloaded, with congestion and unnecessary emissions as consequence. Therefore, different traffic control methods are useful, especially in case of an incident. One key problem for traffic control is traffic prediction and the aim of this thesis is to develop, calibrate and evaluate a route flow model using only observed travel times and travel demand as input. The route flow model was used to calculate the metric link flow destination distribution, that presents to which destinations the travelers on a link are going in percentage. A route set was generated using a shortest path algorithm, and the implemented route flow model creates route probabilities for a pre-generated route set. The route flow model was applied on a network in the northeast part of Los Angeles and a model over the network in the simulation tool Aimsun already exists. The route flow model was calibrated against link flows and route probabilities from the existing simulation model based on a statistical comparison. Thereafter, the link flow destination distribution was calculated and compared against the link flow destination distribution from a microsimulation as evaluation, both during normal and incident state. A microsimulation with an incident was considered the ground truth and the route flow model was evaluated on if the link flow destination distribution from that simulation could be recreated with the route flow model. The result showed that even if the link flows had a good correspondence to the existing simulation model, the route flows were not recreated. Since the route flows were not recreated, the estimation of link flow destination distribution was challenging. The deviation to the result from the existing model was not acceptable according to the statistical comparison. Even if an incident affected the result, it did not affect the overall estimation of link flow destination distribution. Factors that could have affected the result negatively is for example the similarity of the route sets used in the models, using the same parameters for all OD-pairs and time periods, and the loading method. Also using travel times as the only input to predict travel behavior could be inadequate and an investigation of the most important route choice aspects for predicting route flows is left for further work. To use the implemented route flow model for operational traffic control, it could be simplified to only cover a smaller part of the network around the incident.. i.

(5) Acknowledgement First of all, we want to sincerely thank our supervisor David Gundlegård and examiner Clas Rydergren at Linköping University for all the support and interesting discussions during the thesis work. We also want to express out very great appreciation for being given the opportunity to write this thesis in collaboration with University of California, Berkeley. Our special thanks to Anthony Patire and Sakib Mahmud Khan and their colleagues at UCB for providing us with valuable knowledge and support during our stay in Berkeley. Our visit in Berkeley was a truly great experience even though it was suddenly interrupted due to the world pandemic caused by the Corona virus. Furthermore, we want to thank Nikolaos Tsanakas and Rasmus Ringdahl at Linköping University for technical support, guidance and help during our thesis work. Finally, we want to thank our families, classmates and everybody involved, for the support and motivation during our thesis work as the grand finale of the Master of Science program in Communications, Transport and Infrastructure at Linköping University. Anna Danielsson and Gabriella Gustafsson Norrköping, June 2020. ii.

(6) Table of content 1.. 2.. 3.. 4.. 5.. 6.. INTRODUCTION ............................................................................................................................................ 1 1.1. BACKGROUND .....................................................................................................................................................1. 1.2. AIM ...................................................................................................................................................................3. 1.3. METHOD ............................................................................................................................................................3. 1.4. LIMITATIONS .......................................................................................................................................................4. 1.5. OUTLINE .............................................................................................................................................................5. LITERATURE REVIEW ..................................................................................................................................... 6 2.1. INCIDENT MANAGEMENT AND TRAFFIC PREDICTION .....................................................................................................6. 2.2. TRAFFIC FLOW MODELS .........................................................................................................................................8. 2.3. DYNAMIC TRAFFIC ASSIGNMENT ............................................................................................................................11. METHODOLOGY .......................................................................................................................................... 22 3.1. PROBLEM FORMULATION .....................................................................................................................................22. 3.2. OVERVIEW OF METHODOLOGY ..............................................................................................................................25. 3.3. SIMULATION......................................................................................................................................................26. 3.4. ROUTE SET GENERATION ALGORITHM .....................................................................................................................27. 3.5. ROUTE FLOW MODELING......................................................................................................................................29. 3.6. LINK FLOW DESTINATION DISTRIBUTION ..................................................................................................................32. 3.7. COMPUTER TOOLS ..............................................................................................................................................34. SIMULATION STUDY ................................................................................................................................... 37 4.1. EXISTING SIMULATION MODEL...............................................................................................................................37. 4.2. NORMAL STATE SIMULATION SETUP .......................................................................................................................41. 4.3. INCIDENT STATE SIMULATION SETUP .......................................................................................................................43. RESULTS AND ANALYSIS.............................................................................................................................. 44 5.1. PARAMETERS FOR ROUTE SET GENERATION ..............................................................................................................44. 5.2. CALIBRATION OF ROUTE FLOW MODEL ....................................................................................................................49. 5.3. EVALUATION OF ROUTE FLOW MODEL.....................................................................................................................54. DISCUSSION ................................................................................................................................................ 66 6.1. PARAMETERS FOR ROUTE SET GENERATION ..............................................................................................................66. 6.2. CALIBRATION OF ROUTE FLOW MODEL ....................................................................................................................67. 6.3. EVALUATION OF ROUTE FLOW MODEL.....................................................................................................................68. iii.

(7) 6.4 7.. FUTURE WORK ...................................................................................................................................................70. CONCLUSIONS ............................................................................................................................................ 71. REFERENCES......................................................................................................................................................... 72. iv.

(8) List of figures FIGURE 1 - EXPLANATION OF LINK FLOW DESTINATION DISTRIBUTION ............................................................................................2 FIGURE 2 - GENERAL IDEA OF THE METHOD..............................................................................................................................4 FIGURE 3 - FUNDAMENTAL RELATIONS IN DIFFERENT PLANES .......................................................................................................8 FIGURE 4 - SCHEMATIC SKETCH OVER AN EXAMPLE NETWORK ....................................................................................................22 FIGURE 5 - OVERVIEW OF METHODOLOGY .............................................................................................................................25 FIGURE 6 - FLOW CHART OF ROUTE SET GENERATION ALGORITHM ..............................................................................................28 FIGURE 7 - FLOW CHART OF CALIBRATION OF THE ROUTE FLOW MODEL .......................................................................................32 FIGURE 8 - FLOW CHART OF LINK FLOW DESTINATION DISTRIBUTION CALCULATION ........................................................................33 FIGURE 9 - THE NETWORK IN LOS ANGELES WITH ALL LINKS ......................................................................................................37 FIGURE 10 - EXPLANATION OF NODES, CENTROIDS, LINKS AND CENTROID CONNECTORS IN A SMALL PART OF THE NETWORK...................38 FIGURE 11 - EXPLANATION OF NODES, CENTROIDS, NEW NODES, LINKS AND CENTROID CONNECTORS IN A SMALL NETWORK ..................40 FIGURE 12 - LOCATION OF THE SIMULATED INCIDENT AND LINKS WITH LONGER TRAVEL TIMES DUE TO THE INCIDENT ............................43 FIGURE 13 - VISUALIZATION OF THE TWO CHOSEN OD-PAIRS ....................................................................................................44 FIGURE 14 - NUMBER OF ROUTES VARYING WITH DIFFERENT LEVELS OF THRESHOLD, TH .................................................................45 FIGURE 15 - DEGREE OF SIMILARITY VARYING WITH DIFFERENT LEVELS OF THRESHOLD, TH ..............................................................45 FIGURE 16 - NUMBER OF ROUTES VARYING WITH DIFFERENT LEVELS OF PENALIZING PARAMETER 𝜇 ...................................................46 FIGURE 17 - DEGREE OF SIMILARITY VARYING WITH DIFFERENT LEVELS OF PENALIZING PARAMETER 𝜇 ................................................47 FIGURE 18 - ROUTES FOR THE LONG OD-PAIR WITH TRAVEL TIMES IN MINUTES ............................................................................51 FIGURE 19 - EVALUATION OF ROUTE PROBABILITIES WITH DIFFERENT VALUES ON 𝜃 FOR THE LONG OD-PAIR ......................................51 FIGURE 20 - ROUTES FOR THE SHORT OD-PAIR WITH TRAVEL TIMES IN MINUTES ...........................................................................52 FIGURE 21 - EVALUATION OF ROUTE PROBABILITIES WITH DIFFERENT VALUES ON 𝜃 FOR THE SHORT OD-PAIR .....................................52 FIGURE 22 - COMPARISON OF LINK FLOW (LEFT) AND ROUTE PROBABILITIES (RIGHT)......................................................................54 FIGURE 23 - COMPARISON OF LINK FLOW BETWEEN MICROSIMULATION AND ROUTE FLOW MODEL WITH ANOTHER DATA SET ................55 FIGURE 24 - COMPARISON OF LINK FLOWS WHEN USING ROUTE SETS WITH MORE OPTIONS .............................................................57 FIGURE 25 - COMPARISON OF LINK FLOW DESTINATION DISTRIBUTION (LFDD) .............................................................................58 FIGURE 26 - VISUALIZATION OVER LINK FLOW DESTINATION DISTRIBUTION (LFDD) DIFFERENCE AGGREGATED OVER LINKS ....................60 FIGURE 27 - DIFFERENCE IN LINK FLOW DESTINATION DISTRIBUTION (LFDD) AGGREGATED OVER DESTINATIONS .................................61 FIGURE 28 - DESTINATIONS WITH LARGE LINK FLOW DESTINATION DISTRIBUTION DIFFERENCE AGGREGATED OVER DESTINATIONS ...........61 FIGURE 29 - LINK FLOW DESTINATION DISTRIBUTION (LFDD) DURING NORMAL STATE FOR THE EXAMPLE LINK ....................................62 FIGURE 30 - LINK FLOW DESTINATION DISTRIBUTION (LFDD) DURING INCIDENT STATE FOR THE EXAMPLE LINK ...................................63 FIGURE 31 - COMPARISON OF LINK FLOW DESTINATION DISTRIBUTION (LFDD) TO CURRENT APPROACH AND ROUTE FLOW MODEL .........64 FIGURE 32 - COMPARISON OF LINK FLOW DESTINATION DISTRIBUTION (LFDD) DURING INCIDENT STATE............................................65. v.

(9) List of tables TABLE 1 - LIST OF NOTATIONS .............................................................................................................................................23 TABLE 2 - LIST OF NOTATIONS USED IN ROUTE FLOW MODEL .....................................................................................................30 TABLE 3 - ATTRIBUTES FOR LINKS AND CENTROID CONNECTORS..................................................................................................39 TABLE 4 - ATTRIBUTES FOR NODES AND CENTROIDS .................................................................................................................39 TABLE 5 - RESULT OF HOW NUMBER OF GENERATED ROUTES AND DEGREE OF SIMILARITY ................................................................48 TABLE 6 - CHOSEN PARAMETERS FOR ROUTE SET GENERATION ALGORITHM ..................................................................................48 TABLE 7 - CALIBRATION OF ROUTE FLOW MODEL WITH DIFFERENT VALUES OF 𝜃 AND 𝛽 ..................................................................50 TABLE 8 - CALIBRATION OF ROUTE FLOW MODEL WITH WIDER RANGE OF SCALE FACTOR 𝜃 ..............................................................53 TABLE 9 - VALIDATION OF LINK FLOW DEPENDING ON DATA SET .................................................................................................55 TABLE 10 - EVALUATION OF LINK FLOW DEPENDING ON ROUTE SET.............................................................................................56. vi.

(10) Abbreviations GEH. Geoffrey E. Havers’ statistic. HOV. High-occupancy vehicle. I-210. Interstate 210 freeway. LFDD. Link flow destination distribution. MNL. Multinomial logit model. OD. Origin-destination. PA. Path assignment. PPM. Prediction process manager. PRMS Percentage of root-mean-square error RSGA Route set generation algorithm VDF. Volume delay function. vii.

(11) 1. Introduction In a lot of big cities, the traffic network is overloaded with congestion and unnecessary emissions as consequence. According to the United States Environmental Protections Agency, 29% of the greenhouse gas emissions in the United States are caused by traffic (EPA 2019). Using traffic control, such as controlling green time at traffic lights, ramp metering and spreading information on variable message signs, the time each traveler spend in the network can be reduced. Decreasing travel times includes clearing the road after an incident, redirecting traffic and avoiding secondary incidents and congestion. These actions are important, both from a social, economical and environmental perspective. From a social perspective, a faster redirection of traffic after an incident leads to less irritation and fewer delays caused by the congestion and could also get the emergency service faster to the location of the incident. The economical and environmental factors go hand in hand, since less congestion leads to less unnecessary use of gas, which in terms leads to less emissions. To evaluate different traffic control actions, traffic models are useful tools. With models, a network can be analyzed both for normal conditions and when the network is changed, for example in case of an incident. The current traffic situation can be determined using sensors in the network and from that, different scenarios and actions can be evaluated and compared.. 1.1 Background In Los Angeles, the traffic volumes are high, which makes the network sensitive for unexpected events. Therefore, it is of high interest to analyze what happens in case of a change in the existing network, for example closed links or lanes due to an incident. A change affects the amount of traffic on the roads in the network due to modified route choices. A model, made in the traffic simulation tool Aimsun, over the northeast part of the city already exists, with which travel times and link flows can be estimated. This model is part of a system aiming for increased understanding of traffic prediction for traffic control, called I-210 Connected Corridors Integrated Corridor Management System by California PATH (Partners for Advanced Transportation Technology). The system receives data from several stakeholders to a data hub. It could be information about travel time, speed, density and measured flow affected by for example road works or incidents. A 1.

(12) decision support system creates an estimation of the current traffic state (called initial state) based on the data from the data hub. Based on this initial state, a response plan to manage traffic control is generated using a prediction process manager (PPM) containing the simulation model. Several response plans are evaluated and ranked, and it is then up to the stakeholders in charge to decide which actions to take. Link flow destination distribution is a metric for estimation of the initial state. Link flow destination distributions present to which destinations the travelers on all links in the network are going. It is defined as the proportion of link flow that is heading for a certain destination. An example is presented in Figure 1, where the link flow destination distribution for link 1 is 50% to destination A and 50% to destination B.. Figure 1 - Explanation of link flow destination distribution, that is 50% to destination A and 50% to destination B for link 1. Adding link flow destination distribution as input to the simulation model, a more accurate traffic prediction in case of an incident can be made. Therefore, it is a part of the updating process of the prediction in the PPM. Thus, more relevant actions of traffic control can be developed for evaluation to mitigate the consequences of an incident. In order to generate the link flow destination distributions after an incident with the existing model, the simulation needs to be run again but with an incident in the network. To avoid this, a faster and simpler model that calculates the link flow destination distributions for a specific incident would be useful. Then the initial state could be obtained faster without a simulation and thus the response plans to prevent congestion after an incident are generated faster.. 2.

(13) 1.2 Aim The aim of this master thesis is to develop, calibrate and evaluate a simple route flow model for estimating link flow destination distribution using observed travel times and travel demand as input. The aim includes an investigation of if the route flow model can be used to estimate the link flow destination distribution after an incident in an acceptable way. The aim can be concretized in the following research questions: -. How can a route flow model, based on observed travel times, be developed to recreate observed route flows?. -. How can a route flow model, based on observed travel times, be used to estimate link flow destination distribution?. -. How can a route flow model, based on observed travel times, be used to estimate link flow destination distribution in case of an incident?. 1.3 Method In this thesis a route flow model is implemented and calibrated for the traffic network in the northeastern part of Los Angeles. In order to generate route choices, a route set must already exist. Therefore, a route set between every defined start and end point in the network (origin and destination) is generated based on a shortest path algorithm. The route flow model is a C-logit model based on only observed link travel times, that in this case are simulated in a microsimulation of the existing model. Combined with the travel demand (origin-destination (OD) flows), the route flows are estimated. Using the concept “direct loading”, the route flows are converted to link flows, that will be compared to the link flows from the existing simulation model to calibrate the route flow model. Thus, the goal of the calibration is to get the link flows from the route flow model and the simulated link flows as similar as possible. When the route flow model is calibrated, the destination specific link flows obtained is used to calculate link flow destination distributions of the links in the network. The link flow destination distributions are evaluated against the simulated link flow destination distributions from the existing model. Link flow destination distribution estimation makes a good indication of how to update the initial state in a traffic prediction simulation after an incident. With a better estimation 3.

(14) of the initial state, the PPM can improve traffic predictions and create more effective response plans. The general idea of the method is summarized in Figure 2.. Figure 2 - General idea of the method. 1.4 Limitations To regulate the magnitude of the thesis, some limitations to the work were defined. One limitation is that the analysis only covers the network in Los Angeles. Therefore, it cannot be established that the results are valid also for other networks. Another limitation is that the route flow model assumes that the entire route flow is on all links in the route at the same time, called “direct loading”. This means that a vehicle is assumed to be on all links of the route at the same time, which of course is not possible in reality. To avoid this limitation, some other loading method could be used. In addition, there is no capacity limit in the route flow model. Thus, the entire travel demand will be assigned to a route, regardless of what the flow on the links are. This simplification is expected to affect the result with different extent during different time periods. If congestion in the network occurs, the capacity limit in the simulated model will be reached causing a lower flow than in the route flow model. This is partly regulated with the observed travel times since the travel time increases when there is congestion. In the analysis in the thesis, the observed travel time data is simulated and not observed in reality. This is not expected to affect the results significantly. Although, the data might not fully reflect the reality. Also, the data only consider cars and no other type of vehicles. In Los Angeles, Highoccupancy vehicle lanes (HOV) are common to increase the number of travelers sharing a vehicle (defined as a specific vehicle type in the model), but those lanes are not considered in this thesis, since the data only includes the vehicle type car.. 4.

(15) 1.5 Outline The thesis is structured as follows. In chapter 2, a literature review is presented that includes theory about incident management and traffic prediction, different types of traffic models, route set generation and route flow modeling. A description of the methodology used for the route set generation, the route flow modeling and the link flow destination distribution calculation is presented in chapter 3. In the next chapter, chapter 4, the network and simulation setups are described. The following chapters, chapter 5 and 6, contain results, analysis, discussion and future work. Lastly, in chapter 7, the conclusion of the thesis is declared.. 5.

(16) 2. Literature review In this chapter, relevant theory for the thesis is presented. The chapter includes a review of incident management and traffic prediction, traffic flow models and dynamic traffic assignment. The dynamic traffic assignment includes theory regarding route set generation, route flow modeling, how dynamic traffic simulations are made in the simulation software Aimsun and network loading.. 2.1 Incident management and traffic prediction Incidents negatively impact the capacity on a lane or the entire road, resulting in decreased throughput. Incident management mainly aims to mitigate the consequences on the network after an incident (reactive methods), but also to prevent major incidents to occur (proactive methods). The reactive incident management, that aims to easing the consequences on the network after an incident has occurred, consist of procedures to speed up incident detection, verification and response. Wilmink et al. (1998) mention two categories of management measures that aims at reducing the clearance time of the incident: information spreading and measures to increase the safety and efficiency of emergency services. Information can be distributed through for example variable message signs (VMS) or radio messages, which should result in a reduction of the number of secondary incidents and less rubbernecking causing congestion in the opposite travel direction according to Wilmink et al. (1998). The VMS can also be used to warn road users about the emergency services. The proactive incident management assumes that dangerous situations in the traffic can be prevented. According to Wilmink, Immers, Barcelo, Zwaneveld, Montero & Barcelo (1998) the road geometry characteristics combined with weather data can be used to estimate incident probability. The probability can be alleviated by applying proactive traffic management, for example lowering the speed limit through variable speed signs, ramp metering or traveler information (Wilmink et al. 1998). Today, the trends for traffic management is to work more proactive and avoid congestion according to Allström (2016). When deciding which information to distribute to travelers and how to manage traffic control, predictions of the traffic state are useful. Chen and Chen (2019) agrees and states that traffic flow prediction is one of the main problems in traffic control. For a traveler 6.

(17) the predicted experienced travel time, the time it takes between two points with a specific start time, is more relevant than a travel time based on an estimated current traffic state. However, a prediction of the future traffic state is in general based on the latest traffic state estimate. The traffic state prediction is often a more complex problem than estimating the traffic state, due to the unknown future (Allström 2016). Accurate travel time predictions are also useful information to travelers when deciding departure time, route and transport mode. Allström (2016) states that it is a general assumption that accurate travel time information can be supplied to users in order to make an adequate route choice and choice of departure time. Such an ability would result in a better experience for the individual and more reliable travel times (Allström 2016). In order to predict the behavior of the travelers in case of an incident, a simulation model is a good tool. Burgout, Koutsopoulos & Andreasson (2010) suggest one such tool, a mesoscopic simulation model called MEZZO used to provide decision support for incident management. Burgout et al. (2010) state that one challenge developing such tools is prediction of the drivers’ behavior in terms of route choice and the effect on the network. For this, they use logit models with the following approach: 1. Find all affected upstream links from all routes passing the incident link. 2. For each affected link and destination, check for alternative routes that already exists in the route set for all origins that passes through the link and leads to the destination. 3. If there is no existing route, calculate a new shortest path with a large penalty on the incident link and the affected links. The components of the MEZZO model by Burgout et al. (2010) were tested in a small case study that investigates the effects of providing information to the road users after an incident. The result showed that even a small delay in providing incident information can have a large effect on the resulting traffic performance, which means that timely traffic control actions are important.. 7.

(18) 2.2 Traffic flow models Traffic flow models are used to understand, describe and predict traffic flows. Many models, of different types, have been developed since the beginning of the twentieth century and they are all based on the fundamental diagram (Wageningen-Kessels, Lint, Vuik & Hoogendoorn 2015). The original fundamental diagram, by Greenshield, relates the distance between vehicles to the velocity, (d) in Figure 3. However, now there are other shapes as well, for example expressed in density (that is the average number of vehicles per length unit of a road) and flow (that is the average number of vehicles per time unit).. Figure 3 - Fundamental relations in different planes (source: Wageningen-Kessels, Lint, Vuik and Hoogendoorn, 2015). The density-flow diagram, (a), shows that flow increases with increasing density until a limit, the critical density, is reached. (b) in Figure 3 shows that the velocity decreases with increasing density, which is a result of congestion. Part (c) in the figure illustrates that the same flow is obtained in two levels of velocity, few vehicles with a high velocity or with a lot of vehicles with a low velocity. The diagram that relates the spacing to the velocity, (d), shows that with increased spacing between vehicles, the velocity increases. A traffic flow model can be of three different types depending on the investigation: macroscopic, microscopic and mesoscopic models. The macroscopic models consider traffic streams and groups of drivers, the microscopic models are in terms of individual drivers and the mesoscopic model is a mix of these two. Burghout, Koutsopoulos and Andreasson (2006) states that a mesoscopic model maintains individual vehicles representation but with a more aggregated representation of traffic dynamics. 2.2.1 Macroscopic traffic models Macroscopic traffic flow models are commonly used for large areas and long analysis time periods (Tsanakas 2019). The driving behavior is described for an aggregated group of vehicles and is assumed to be homogeneous for the group. The behavior is often described in terms of flow, 8.

(19) density and velocity for the group. Wageningen-Kessels et al. (2015) states that macroscopic models describe the traffic flow as if it was a one-dimensional fluid. That leads to one important assumption in macroscopic models, the conservation of flow. That means that no vehicles appear or disappear on a road segment other than those entering or exiting the segment (Tsanakas 2019). The major branches of macroscopic models are the kinematic wave models and high-order models (Wageningen-Kessels et al. 2015). The first macroscopic traffic models were a prototype of the kinematic wave model, also known as LWR model named after the inventors Lighthill, Whitham and Richards. Wageningen-Kessels et al. (2015) claim that the model is simple, mainly because it assumes that vehicles reach the new equilibrium velocity immediately after a change in the traffic state, which implies infinite acceleration. The model does not consider any capacity drop in case of congestion, but instead the transition from free flow to congestion always occurs at the same density. Wageningen-Kessels et al. (2015) also mention multi-class kinematic wave models, that assume there are two types of drivers: slugs and rabbits. The slugs are slow drivers who have little incentive to overtake while the rabbits attempt to drive fast and overtake more often. The high-order models, on the other hand, includes an equation describing the acceleration towards the equilibrium velocity (Wageningen-Kessels et al. 2015). They are based on the fundamental relation and two-coupled partial differential equations describing the conservation of vehicles and velocity dynamics. 2.2.2 Microscopic traffic models Microscopic traffic models aim to simulate movement of individual vehicles rather than flow and density. According to Wageningen-Kessels et al. (2015), they are based on the assumption that drivers adjust their behavior after the leading vehicle, which includes theories regarding carfollowing, lane-changing and gap-acceptance. The vehicle considered for analysis in a microscopic model is denoted as 𝑛, while the leader is ordered as 𝑛 − 1 and the follower as 𝑛 + 1. The behavior of the vehicles is described in terms of the position, velocity and acceleration, and the basic output is a trajectory, which is combined information of the three components (Tsanakas 2019). Microscopic traffic models are useful for analysis of geometric design configurations of individual intersections and interactions of different transportations modes (Chiu, Bottom, Mahut, Paz, Balakrishna, Waller & Hicks 2011). Microscopic models are built by using many stochastic choice mechanisms and are therefore required to run several replications. The relatively high computation 9.

(20) time make them more useful for smaller networks and since the models often involve many parameters, the calibration is of high importance (Florian, Mahut & Tremblay 2008). Wageningen-Kessels et al. (2015), presents a description of three branches of microscopic models, which all are car-following models: safe-distance models, stimulus-response models and actionpoint models. The safe distance models are the earliest developed car-following models and are based on safe following distance, where the position of the leader is a function of the position of the follower. The safe distance is, according to Wageningen-Kessels et al. (2015), assumed to be large enough for the follower to be able to avoid a collision even if the leader would act unpredictable. The stimulus-response models on the other hand, assume that the drivers accelerate or decelerate as a reaction of three stimuli: their own current velocity, spacing with respect to the leader or relative velocity with respect to leader. The first stimuli assumes that the drivers accelerate or decelerate to their optimal velocity (Wageningen-Kessels et al. 2015). The third branch of microscopic models described by Wageningen-Kessels et al. (2015) is the action point models. They are based on the assumption that drivers only react if they perceive that they approach a vehicle. It is assumed that there is a threshold that the headway must reach before the driver reacts to accelerate or decelerate. The models take into consideration that the driving behavior is only influenced by the leader if the headway is small enough and if the change in velocity is large enough to be perceived. 2.2.3 Mesoscopic traffic models When defining a mesoscopic traffic model, it could be placed somewhere between macroscopic and microscopic traffic models. The traffic flow in a mesoscopic model is modelled at an aggregate level, for instance by homogeneous groups of traffic entities. The reason for placing a mesoscopic traffic model between macroscopic and microscopic traffic models is because each one of the groups is following an individual behavior (Tsanakas 2019). According to Florian et al. (2008) the need for handling larger networks with reasonable computation times led to the development of mesoscopic simulation models. They are less precise in the representation of traffic behavior and simpler computationally. Although, the mesoscopic models still have a traffic representation that capture the basic temporal congestion phenomena.. 10.

(21) According to Burghout et al. (2006) a mesoscopic traffic model provides a middle ground with their ability to model large networks with limited network coding and calibration effort, while providing a better representation of the traffic dynamics and individual travel behavior than their macroscopic counterparts. The calibration and validation process for microscopic and mesoscopic requires more effort than for a macroscopic model (Hoogendoorn 2001). Mesoscopic models are used for both planning and real time operations. They are more flexible than macroscopic models for modeling for example route choices. However, a mesoscopic model is still limited in its ability to represent detailed traffic operations. According to Hoogendoorn (2001) the traffic flow operation for a mesoscopic model is when analyzing the behavior of drivers without explicitly distinguishing their time-space behavior. Therefore, a microscopic traffic model would be suitable for investigating a lane-change maneuver for example because it represents individual vehicles as an instantaneous event, where the decision to perform a lane-change is based on for example relative lane densities. The most popular branch of mesoscopic models, according to Wageningen-Kessels et al. (2015), is Gas-kinetic models. They were developed from models describing the motion of large numbers of small particles in gas. They describe the dynamics of velocity distribution functions of vehicles, including a term describing the acceleration towards equilibrium velocity.. 2.3 Dynamic traffic assignment A traffic model can be both dynamic and static, and in which state it is depends on the aim of the analysis and what the purpose of the model is. A static traffic model is usually considered suitable for long-range planning and can be the basis of large capacity expansion projects. A static traffic model is not appropriate when analyzing traffic congestion since such a model cannot reflect either conditions and changes in characteristics of the system or variations in traffic flow over a certain time (Chiu et al. 2011). A dynamic traffic model on the other hand, includes the effects of the interaction between travel choice, traffic flow, time and cost in the model. The aim of a dynamic traffic model is to describe a time-varying network and demand interaction using a behavior approach. Areas where dynamic traffic assignment (DTA) models usually are applied are areas of real-time operational control of vehicular traffic systems and decision-making considerations. 11.

(22) (Priya 2016). The reason for using DTA is that static traffic assignment cannot capture traffic dynamics and if time varying flows and queues needs to be taken into consideration a DTA is required (Priya 2016). The linkage between travel time and congestion that a traveler experiences in reality is affecting dynamic traffic models. If the outflow is lower than the inflow for a link, link density will increase and then according to the fundamental speed-density relationship the link travel time will increase. Comparatively, the linkage between travel time and congestion does not affect vehicles travelling on the same link in a static traffic model, all vehicles experience the same travel time and also, the inflow is equal to the outflow. The reason for this is that a static traffic model does not variate during a time period, while a dynamic model does (Chiu et al. 2011). The goal of a traffic assignment in a modelling process is to determine the network flows as a result of the interactions among the route choices travelers make on their way from their origin to their destination (Chiu et al. 2011). A dynamic traffic assignment is a modeling framework where the vehicles in the model act with a dynamic state. 2.3.1 Route set generation Route set generation is a process to generate possible alternative routes form a choice set. To generate a route set is the first process of a two-stage route flow modeling process, where the second stage is to calculate the probability that a given route is chosen from the specified route set. The resulting routes from a route set generation could be numerous alternative routes for an origindestination pair (OD-pair). Most of the possible routes may be overly circuitous, but the goal is to identify all the routes that any traveler might consider (Bekhor, Moshe & Ramming 2006). One method for generating a route set is the K-shortest Path algorithm (Nassir, Ziebarth, Sall & Zorn 2014). The idea of this method is to generate the first “k” shortest paths for an OD-pair. Depending on the network´s design, this method can generate much overlap between the routes, since a link can be used by several routes and the geographical divergence can be close to the shortest path. It is on the other hand possible to find totally unique routes with no overlapping by applying link elimination. The method generates the shortest path between an origin and destination and then the used links in the first path is removed from the link set. When applying a method for finding 12.

(23) the shortest path again, the second shortest path in the new network will be found and that route´s links will be removed and so on. This link elimination can according to Nassir et al. (2014) eliminate essential links or bridges when they have been used once and can therefore, depending on the network, generate an unreasonable route set due to no overlapping links. Because, when traveling from A to B most likely some links will be used by serval routes and it is this overlapping that makes the route set generation difficult. According to Nassir et al. (2014) another method for route set generation is link penalty, where the impedance of all links in the shortest path is gradually increased, in order to find a reasonably overlapping between the links. According to Bovy and Fiorenzo-Catalano (2007) this algorithm adds a penalty term 𝑝 on the links in the obtained shortest path, defined in equation (1). 𝑝(𝑎) =. 𝑙𝑎 𝑙𝑛 ∑ 𝛿𝑎𝑗 𝜇𝐿. (1). 𝑗∈𝐶𝑡. where 𝑎 = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑙𝑖𝑛𝑘 𝑙𝑎 = 𝑔𝑒𝑛𝑒𝑟𝑎𝑙𝑖𝑧𝑒𝑑 𝑐𝑜𝑠𝑡 𝑜𝑓 𝑢𝑠𝑖𝑛𝑔 𝑙𝑖𝑛𝑘 𝑎 𝜇 = 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝐿 = 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑐𝑜𝑠𝑡 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡ℎ𝑒 𝑂𝐷 − 𝑝𝑎𝑖𝑟 𝐶𝑡 = 𝑠𝑒𝑡 𝑜𝑓 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑 𝑟𝑜𝑢𝑡𝑒𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑜𝑟𝑖𝑔𝑖𝑛 𝑎𝑛𝑑 𝑑𝑒𝑠𝑡𝑖𝑛𝑎𝑡𝑖𝑜𝑛 𝑎𝑡 𝑒𝑛𝑑 𝑜𝑓 𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑡 𝛿𝑎𝑗 = 𝑒𝑞𝑢𝑎𝑙𝑠 1 𝑖𝑓 𝑙𝑖𝑛𝑘 𝑎 𝑖𝑠 𝑖𝑛 𝑟𝑜𝑢𝑡𝑒 𝑗, 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 When using this penalty term, the cost for using a link is increased every time it is generated in the shortest path. The pseudo code for generating T paths according to Nassir et al. (2014) is described below. Initialization: 𝑡 ∶= 1; 𝐶𝑡 ∶= ∅ (𝑒𝑚𝑝𝑡𝑦 𝑠𝑒𝑡) Set the link cost to the generalized cost and the penalty term for link 𝑎 to 0; Set 𝐿 to the minimum cost between origin and destination; Step 1: Find the shortest path (based on link cost and the penalty term added) between the origin and destination pair and assign it to path i; 13.

(24) Step 2: 𝐶𝑡 ∶= 𝐶𝑡 ∪ {𝑖}; Step 3: Calculate the penalty term 𝑝(𝑎) as in equation (1) and update the cost for all links 𝑎 in path 𝑖; Step 4: 𝑡 ∶= 𝑡 + 1; if 𝑡 < 𝑇, go to Step 1, otherwise stop. When the route set is generated, the demand of each OD-pair is distributed among the routes in order to complete the route flow modeling. Dijkstra´s algorithm Deciding the shortest path from a node 𝑛𝑟 to a node 𝑛𝑠 in a network is one of the most fundamental problems in network optimization. If the network consists of nonnegative link costs, one algorithm that can obtain the shortest path is Dijkstra´s algorithm (Lundgren, Rönnqvist & Värbrand 2010). When solving the shortest path problem, the network is assumed to have the following characteristic: •. Directed arcs.. •. Node 𝑛𝑠 is achievable from node 𝑛𝑟 .. •. No cycles with negative cost.. Dijkstra´s algorithm finds the shortest path between node 𝑛𝑟 to node 𝑛𝑠 in a network with the set of nodes 𝑁 and the set of links B. The following four steps explain the algorithm (Lundgren et al. 2010). Step 0: Divide the set of nodes into the set 𝐴 = {𝑠𝑒𝑎𝑟𝑐ℎ𝑒𝑑 𝑛𝑜𝑑𝑒𝑠} = ∅ and 𝐷 = {𝑢𝑛𝑠𝑒𝑎𝑟𝑐ℎ𝑒𝑑 𝑛𝑜𝑑𝑒𝑠} = 𝑁.. Mark. the. start. node. 𝑛𝑟. with. (𝑝𝑟 , 𝑦𝑟 ) =. (𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑛𝑜𝑑𝑒, 𝑛𝑜𝑑𝑒 𝑝𝑟𝑖𝑐𝑒) = (−,0). This means that the node has no previous node and the current node price 𝑦𝑟 is zero. All other nodes in the network are initialized with (−, ∞), which means that the node has no previous node and the current node price, 𝑦𝑟 , is infinite. Step 1: Identify the node 𝑖 ∈ 𝐷 with the lowest node price: 𝑦𝑖 = min 𝑦𝑘 . 𝑘∈𝐷. Step 2: Search node 𝑛𝑖 , meaning investigate all outgoing links (𝑖, 𝑗) ∈ 𝐵 from node 𝑛𝑖 . If (𝑦𝑖 + 𝑐𝑖𝑗 ) < 𝑦𝑗 a cheaper way between 𝑛𝑟 and 𝑛𝑗 through 𝑛𝑖 has been found. Mark node 𝑛𝑗 with (𝑝𝑗 , 𝑦𝑗 ) = (𝑖, 𝑦𝑖 + 𝑐𝑖𝑗 ). 𝑐𝑖𝑗 is the link-cost between node 𝑖 and 𝑗. 14.

(25) Step 3: Move node 𝑛𝑖 from subset 𝐷 with unsearched nodes to subset 𝐴 with search nodes. Step 4: If all nodes has been searched, 𝐴 = 𝐷, break. Otherwise, go to step 1. When all nodes have been searched the shortest path can be obtained by the node labeling. According to Lundgren et al. (2010) the shortest paths from one node to all other nodes are generated since all nodes are searched in the algorithm. Overlap between routes For a generated route set, it is possible to investigate the similarity of the routes for an OD-pair. The similarity of routes can be interpreted as the degree of overlap of the links used in the routes. Hu & Chiu (2015) have defined the degree of similarity between routes as follows. Equation (2) calculates the total degree of similarity for each route, 𝑟 𝑘 . 𝑘 ∑𝑛𝑎=1 𝑟𝑎𝑘 𝑟 = 𝑚−1. 𝑘. (2). The degree of similarity for all routes within an OD-pair, 𝑟, can be calculated by equation (3). 𝑘 ∑𝑚 𝑘=1 𝑟 𝑟= 𝑚 ∑𝑘=1 𝑛𝑘. (3). The notations for equation (2) and (3) are: 𝑟𝑎𝑘 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑙𝑖𝑛𝑘 𝑎 𝑖𝑛 𝑟𝑜𝑢𝑡𝑒 𝑘 𝑠ℎ𝑜𝑤 𝑢𝑝 𝑖𝑛 𝑎𝑙𝑙 𝑜𝑡ℎ𝑒𝑟 𝑚 − 1 𝑟𝑜𝑢𝑡𝑒𝑠 𝑛𝑘 = 𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑙𝑖𝑛𝑘𝑠 𝑖𝑛 𝑟𝑜𝑢𝑡𝑒 𝑘 𝑟 𝑘 = 𝐷𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 𝑜𝑓 𝑟𝑜𝑢𝑡𝑒 𝑘 𝑟 = 𝐷𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑙𝑖𝑛𝑘 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑜𝑡𝑎𝑙𝑖𝑡𝑦 𝑜𝑓 𝑚 𝑟𝑜𝑢𝑡𝑒𝑠 The value of 𝑟 will be in the interval [0 1], where 0 means that the routes are completely unique, thus, there is no overlap between the routes and 1 means that the routes are identical (Hu & Chiu 2015). The higher the value of 𝑟 is, the higher degree of overlap will be observed. The desirable degree of similarity is different depending on the application of the route set. It may not be desirable to have almost identical routes in the route set, but it is not desirable to have completely unique routes either. Hu & Chiu (2015) uses a route set with 25-35% degree of similarity.. 15.

(26) 2.3.2 Route flow estimation To see how the flow for an OD-pair allocate among a given route set and calculate the flow for a network, different methods for estimation can be used. The route flow becomes an estimation since travelers make their own decision and it is impossible to know exactly how people will behave and make their decisions. One method of route flow estimation is to find the user equilibrium with Wardrop´s principle or using a logit model. These two methods are described in the following sections. Wardrop’s principle Wardrop’s principle is believed to be a reasonable approximation of traveler route choice. All travelers in a network are in practice assumed to choose the best route between their origin and destination. Best as in the route having the lowest cost or travel time. Although, the travel time and the cost depend on the route choices of the other travelers in the network, whom themselves are trying to choose the best route for them, due to congestion. The average travel time can therefore be seen as a function of the flow, called volume delay functions (VDF). According to Kucharski and Drabicki (2017), one purpose of the VDF is to reproduce congestion effects in a macroscopic model. Chiu et al. (2011) states that when every traveler succeeds to find their best route, every used route in an OD-pair has the same travel time, which is the shortest possible. This state is known as user equilibrium. Traffic assignment algorithms aim at finding the link volumes and travel times that satisfy the user equilibrium condition iteratively. Logit model A multinomial logit model (MNL) is a discrete choice model that can be used to analyze and predict a decision maker’s choice of one alternative from a set of different alternatives (Koppelman & Bhat 2006). The decision maker is assumed to choose the alternative with the greatest utility, which is defined as a sum of the utility observed by the analyst and an error term. The error term is the difference between the unknown utility used by the individual and the utility estimated by the analyst and is represented by a random variable. The observed utility is a function of attributes related to the decision maker and the alternative. The utility is, according to Koppelman and Bhat (2006), represented by equation (4). 𝑈𝑖𝑡 = 𝑉𝑖𝑡 + 𝜀𝑖𝑡. 16. (4).

(27) where 𝑈𝑖𝑡 is the true utility of alternative 𝑖 for decision maker 𝑡, 𝑉𝑖𝑡 is the estimated observable portion of the utility and 𝜀𝑖𝑡 is the error or the unknown portion of the utility. An example of a utility function is 𝑉𝑖𝑡 = −𝜃 ∗ 𝑡𝑟𝑎𝑣𝑒𝑙 𝑡𝑖𝑚𝑒, where 𝜃 is a parameter, and the goal with the function is that routes with a longer travel time should get a smaller utility and are therefore not as likely to be chosen. Assuming that the error term is Gumbel distributed, identically and independent across all alternatives and across all decision makers, the error term does not need to be considered in the probability of selecting alternative 𝑖 (Koppelman & Bhat 2006). The general expression for the probability of choosing alternative 𝑖 from a set of available alternatives, 𝐽, by the logit model is shown in equation (5). 𝑃𝑟 (𝑖) =. 𝑒 𝑉𝑖 ∑𝑗∈𝐽 𝑒 𝑉𝑗. (5). where 𝑃𝑟 (𝑖) is the probability of selecting alternative 𝑖 and 𝑉𝑗 is the estimated observable portion of the utility component of the utility of alternative 𝑗. One application of the logit model is to assign demand on routes in an OD-pair depending on different parameters. In such cases, often a C-logit modification is used. In the C-logit model the general formulation is used, added by a factor called commonality factor, 𝐶𝐹𝑖 . The purpose of the commonality factor is that heavily overlapping routes should have a larger communality factor and thus a smaller utility (Russo & Vietta 2003). The observable portion of the utility is then modified as in equation (6). 𝑉∗𝑖 = 𝑉𝑖 − 𝐶𝐹𝑖. (6). where 𝑉∗𝑖 is the new estimated observable portion of utility and 𝐶𝐹𝑖 is the commonality factor. The commonality factor is based on the length of the links in common in proportion to the lengths of route 𝑖 and 𝑗 and is by Zhengfeng, Zhaodong, Pengjun and Wenjun (2018) defined as equation (7).. 17.

(28) 𝐶𝐹𝑖 = 𝛽 ∗ 𝑙𝑛 (∑. 𝐿𝑖𝑗. √𝐿𝑖 ∗ 𝐿𝑗 𝑗∈𝐽. ). (7). where 𝛽 is a parameter, 𝐿𝑖𝑗 is the length of the links that route 𝑖 and route 𝑗 have in common, and 𝐿𝑖 is the length of route 𝑖. 𝛽 regulates the importance of the overlap and with 𝛽 = 1 the choice probability tends to be 1/N of the choice probability calculated with MNL logit model where there are N coincident paths. Zhengfeng et. al. (2018) assumes 𝛽 = 1 for all OD-pairs. Calibration metrics A vital part to be able to use a route flow model is calibration, to see that the result of the model corresponds to the reality in an acceptable way. In traffic models, this is often done by comparing link flows and one way to do that is by looking at the GEH statistic described in Balakrishna, Antoniou, Ben-Akiva & Koutsopoulos (2007), see equation (8).. 𝐺𝐸𝐻 = √. 2 ∗ (𝑌𝑛𝑠 − 𝑌𝑛0 )2 (𝑌𝑛𝑠 + 𝑌𝑛0 ). (8). where 𝑌𝑛𝑠 is the simulated traffic volume and 𝑌𝑛0 is the observed traffic count. According to Balakrishna et al. (2007) a GEH statistic below 5 is considered a good match between the estimated volume and the observed counts and an acceptable result is to have at least 85% of the links with a GEH-value below 5. The GEH statistic is named after its inventor Geoffrey E. Havers. Another performance metric is the percentage root mean square error, PRMS. Which is the root mean square error (RMS) divided by the mean of the observed data. It is more intuitive than the RMS, described by equation (9) (Brockfeld & Wagner 2006).. 𝑁. 1 𝑅𝑀𝑆(𝜑) = √ ∑(𝑞𝑖 (𝜑) − 𝑞̂𝑖 (𝜑))2 𝑁 𝑖=1. 18. (9).

(29) where 𝑞̂𝑖 is observed data and 𝑞𝑖 is simulated data, depending on a set of parameters, 𝜑, of the model. For the models tested by Brockfeld and Wagner (2006), the PRMS was around 15 − 20%. Another way of evaluating a calibration is with a R2-value. The R2-value gives a statistical measure of how well data correlates to a fitted regression line. The R2-value is always between 0 − 100%, where 0% indicates that the model explains none of the variability of the response data around its mean, while 100% indicates that the model explains all the variability of the response data around its mean (Blom 2017). 2.3.3 Aimsun routes In the simulation software Aimsun, routes are generated and chosen using either macro-, meso- or microsimulation through the steps described below: path definition and path selection. Path Definition In order to generate possible routes in the simulation, Aimsun can define available paths between an OD-pair with three different methods. The first one is called O/D routes and it corresponds to the idea of well-known paths or the most familiar paths for drivers according to the analyst’s knowledge of the modelled network. The second approach is called Paths from Assignment results. These paths are the result of applying traffic assignment using Aimsun Macro or a dynamic traffic assignment based on dynamic user equilibrium using Aimsun Micro or Meso. The third method is called Shortest paths. The calculations for this are based on the network’s links and nodes and the link cost function associated to each link in the network. Three types of link cost functions are used for the shortest path calculations, k-initial shortest path cost function, initial cost function and dynamic cost function. All three methods use a default cost function which is the link travel time in seconds. The Aimsun user can choose which of the functions to use but by default the initial cost function is used in the beginning of the simulation since no travel times has been calculated. Then, when travel time data is available, the dynamic cost function is used. The k-initial shortest path cost function can only be used when the k-initial shortest path calculations are available. It is also possible to use a user-defined link cost function. The shortest path algorithm is then calculated with a variation of Dijkstra’s algorithm based on the link cost functions and uses penalties associated with turning movements (TSS 2013).. 19.

(30) Path Selection Aimsun’s route choice model is called Path Selection and it estimates path flow rates based on discrete route choice models or on a user-defined assignment. From a given set of alternative paths, explained in the previous section about Path Definition, the Path Selection models the driver’s decision of which path to take between an origin and destination. This path selection process is used both when a vehicle is entering the system, called Initial assignment in Aimsun and during the trip when new alternatives could be available, called Enroute assignment. The main idea of the path selection process is to calculate the probability of each available path and then the driver’s decision is modelled by randomly selecting an alternative path according to the probabilities of each alternative (TSS 2013). The three methods of path selection in Aimsun is based on a hierarchy, where the O/D routes comes first, then the Paths from Assignment results and last is the Shortest paths method. When creating an Aimsun experiment, the user can choose which portion each of the methods should have. The first value that is considered when a vehicle is entering the network is the O/D routes percentage, that is the probability that a vehicle is assigned to a route from the O/D routes. If the vehicle is not assigned to a route from the O/D routes, the probability of being assigned to a path from the Paths from Assignment Result is checked. If a vehicle is not assigned to neither a route from the O/D routes nor the Paths from Assignment Result the route from the Shortest Paths is chosen (TSS 2013). This hierarchy can be further explained with an example. For example, the user defines 70% of the vehicles to follow O/D routes and 60% to follow Paths from Assignment result. Then 70% of the vehicles will follow O/D routes, 18% will follow Paths from Assignment result, which is calculated from 60 ∗ (100 − 70) = 18% and the last 12% (100 − 70 − 18) will take a route calculated from Aimsun’s Shortest paths model (TSS 2013). The default route choice models available in Aimsun are: Proportional, Multinomial Logit, C-logit and a user-defined function. The route choice models are formulated in terms of negative utility because the most common value associated to a trip is the travel time or travel cost. The parameters related to each route choice model in Aimsun are, Proportional: alpha factor, Logit: scale factor or C-logit: scale factor, beta and gamma (TSS 2013).. 20.

(31) 2.3.4 Network loading Network loading assigns vehicles to the links in the network. In a dynamic traffic assignment, the demand is not static and if a time-sliced OD-matrix is used, dynamic network loading can be used. This means that the arrival time at a link is different than the departure time because the flow is assigned to the link depending on the duration of the route until that time (Tsanakas 2019). By the network loading, it is possible to compute link flows from route flows (Dell´Orco 2006). Xu, Wu, Florian, Marcotte and Zhu (1999) describes that a dynamic network loading problem aims to finding, on a congested network, temporal link volumes, link travel times, and path travel times given time-dependent route flow rates for a given time period. Network loading tries to capture congestion and ensure vehicle conservation by propagating the assigned route volume to the route links. A dynamic network loading can be classified as microscopic, macroscopic and mesoscopic. Traditional network loading techniques commonly employ macroscopic traffic flow models or microscopic simulation (Yperman 2007).. 21.

(32) 3. Methodology In this chapter the methodology is presented. First, a small example to describe the problem of the thesis is shown and then an overview of the methodology is described. Furthermore, descriptions of how simulation was used and how the route set, route flow model and link flow destination distribution have been generated are presented. Lastly, the used computer tools are described.. 3.1 Problem formulation Link flow destination distribution is a useful metric for evaluation of traffic control actions. The problem is that in case of an incident link flow destination distribution is then changed compared to the normal state. Therefore, the simulation model needs to be ran again but with an incident in the initial state, which is time consuming. When there is no time to rerun the simulation, it would be useful to obtain the link flow destination distribution through a faster model. The problem can be clarified with the example in Figure 4, which shows a schematic sketch over a traffic network during two states of the network. The left part of the figure shows the normal condition of the network (normal state), while the right part of the figure shows the network with one unavailable link, link 3, due to an incident (incident state).. Figure 4 - Schematic sketch over an example network. The left part of the figure shows the network during normal conditions (normal state) and the link flow destination distribution (LFDD) for link 2 and 3. The right part of figure shows the network when link 3 is unavailable due to an incident (incident state) and the new link flow destination distribution (LFDD) for link 2 and 3.. A summary of the notations used in the problem formulation is presented in Table 1.. 22.

(33) Table 1 - List of notations. 𝑨. Set of all links. 𝑹. Set of all origins. 𝑺. Set of all destinations. 𝑲𝒓𝒔. Set of all routes between every OD-pair 𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆. 𝒈𝒓𝒔. Demand of OD-pair 𝑟 ∈ 𝑅, 𝑠 ∈ 𝑆. 𝒑𝒉𝒓𝒔𝒌. ℎ Percentage of the demand 𝑔𝑟𝑠 travelling through route 𝑘 ∈ 𝐾𝑟𝑠 from origin. 𝑟 ∈ 𝑅 to destination 𝑠 ∈ 𝑆 during state ℎ. 𝒇𝒉𝒓𝒔𝒌 Flow on route 𝑘 ∈ 𝐾𝑟𝑠 from origin 𝑟 ∈ 𝑅 to destination 𝑠 ∈ 𝑆 during state ℎ 𝒕𝒂. Travel time on link 𝑎 ∈ 𝐴. 𝒗𝒉𝒂𝒔. Flow on link 𝑎 ∈ 𝐴 to destination 𝑠 ∈ 𝑆 during state ℎ. 𝒅𝒉𝒂𝒔. Link flow destination distribution to destination 𝑠 ∈ 𝑆 of link 𝑎 ∈ 𝐴 during state ℎ in percentage. The set of all links 𝐴 = {1, … , 6} in the example form a network with one defined origin, 𝑅 = {𝐴}, and two defined destinations, 𝑆 = {𝐵, 𝐶}. The set of routes from origin 𝐴 to the destinations 𝐵 and 𝐶, 𝐾𝐴𝐵 = {1,2} and 𝐾𝐴𝐶 = {3,4}, have been generated through a Shortest path calculation using Dijkstra´s algorithm. Route 1 consist of link 1 − 2 − 5, route 2 of link 1 − 3 − 4 − 6 − 5, route 3 of link 1 − 2 − 6 and route 4 of link 1 − 3 − 4. The total travel demand, stated in an OD-matrix, for OD-pair AB is 10, 𝑔𝐴𝐵 = 10. The demand is divided over the routes 𝑘 ∈ 𝐾𝐴𝐵 as 100% of 𝑔𝐴𝐵 1 on route 1 by the route flow model calculating route probabilities. This results in 𝑝𝐴𝐵1 = 1 and 1 𝑝𝐴𝐵2 = 0, which is the percentage of the demand 𝑔𝑟𝑠 traveling through route 𝑘 ∈ 𝐾𝑟𝑠 from origin. 𝑟 ∈ 𝑅 to destination 𝑠 ∈ 𝑆 during normal state (state index: ℎ = 1). Thus, the flows on the routes ℎ 1 1 during normal state, 𝑓𝑟𝑠𝑘 , are 𝑓𝐴𝐵1 = 10 and 𝑓𝐴𝐵2 = 0. The total demand for OD-pair AC, 𝑔𝐴𝐶 = 1 1 20, is divided as 50% each on the two routes, 𝑝𝐴𝐶3 = 0.5 and 𝑝𝐴𝐶4 = 0.5. Consequently, the flow ℎ 1 1 on the routes for OD-pair AC are 𝑓𝐴𝐶3 = 10 and 𝑓𝐴𝐶4 = 10. The route probabilities, 𝑝𝑟𝑠𝑘 , are based. on the travel time for each route, which is a sum of the travel time for all links in the route, ∑𝑎∈𝑙𝑖𝑛𝑘𝑠 𝑖𝑛 𝑘 𝑡𝑎 . A link can be used in several routes with different destinations. The flow on link 2 to the 1 1 destinations 𝑆 during in the normal state is 𝑣2𝐵 = 10 and 𝑣2𝐶 = 10. From this, the destination. 23.

(34) 10. 10. 1 1 distribution of flow on link 2 can be calculated as 𝑑2𝐵 = 10+10 = 0.5 and 𝑑2𝐶 = 10+10 = 0.5. ℎ Thus, the sum over all destinations for every link is 1, ∑𝑠∈𝑆 𝑑𝑎𝑠 = 1, ∀ 𝑎 ∈ 𝐴, ℎ ∈ 𝐻.. In the right part of the figure, in the incident state (state index: ℎ = 2), when an incident has occurred, the percentage distribution between the routes has changed for OD-pair AC. For OD2 2 pair AC the demand, 𝑔𝐴𝐶 = 20, is now distributed as 𝑝𝐴𝐶3 = 1 and 𝑝𝐴𝐶4 = 0. The distribution. between the routes for OD-pair AB in the incident state remains the same. Note that the routes for both OD-pairs, 𝐾𝑟𝑠 , remain the same before and after the incident. Also the link flow destination distribution has changed between the normal and incident state due to changed route probabilities. In the right part of Figure 4, when link 3 is unavailable, the flow to 2 destination B on link 2, 𝑣2𝐵 , has increased compared to the normal state. Vehicles that has planned. to take route 4 now change to route 3, which results in a modified destination distribution of the 10. 20. 2 2 flow on link 2 between destination B and C, 𝑑2𝐵 = 10+20 = 0.33 and 𝑑2𝐶 = 10+20 = 0.66.. 24.

(35) 3.2 Overview of methodology The method includes several components that contribute with different data and results. An overview of what is generated where and how the data is connected to each other in the thesis is clarified in Figure 5.. Figure 5 - Overview of methodology. Input to the route flow model (which is a C-logit model), regardless of state, is travel times on all links in the network, 𝑡𝑎 , and a route set of all the routes between the origins 𝑟 ∈ 𝑅, and destinations 𝑠 ∈ 𝑆, 𝐾𝑟𝑠 , stated in the top of Figure 5. The travel times were generated from a microsimulation using the existing Aimsun-model, but real travel times could also have been used. The route set 25.

(36) was generated through a shortest path algorithm in Python and stored in a PostgreSQL database. The route flow model was then implemented in MATLAB, and the output is probabilities that each ℎ route in the route set is chosen by a traveler, 𝑝𝑟𝑠𝑘 . Using the route probabilities and an OD-matrix ℎ extracted from the simulation model, the route flows, 𝑓𝑟𝑠𝑘 , were calculated. The route flows were. allocated on each link in the route, by direct loading, to get the total link flows in the network. To see how the route flow model deviate to the microsimulation, the calculated link flows and route probabilities were compared to simulated link flows and route probabilities. Depending on the comparison result, the route flow model was calibrated, which adjusts the route probabilities leading to changed route flows and link flows. Through this process, the route flow model was iteratively calibrated to match the simulation as good as possible. Once the route flow model was calibrated the link flows were used to calculate the destination distribution of the flow for each ℎ link, 𝑑𝑎𝑠 . The link flow destination distribution was compared to simulated link flow destination. distribution from the existing simulation model to see how well the route flow model performs link flow destination distribution. The comparison of both link flows, route probabilities and link flow destination distributions were made in MATLAB.. 3.3 Simulation Both micro- and macrosimulations have been made on the network in the existing model. A static macrosimulation has been used to generate a route set (called Paths from Assignment results in section 2.3.3) and route probabilities for those routes. The route probabilities were then used as input to the microsimulation, but also for comparison with the route probabilities calculated with the implemented route flow model as calibration. The microsimulations have been used to generate link travel times to use as input to the implemented route flow model and link flows to use for calibration of the route flow model. From the microsimulation, also link flow destination distribution for all combinations of links and destinations were obtained and considered ground truth in the evaluation of the route flow model. Thus, the goal was to recreate the results from the existing simulation model with a faster model.. 26.

(37) 3.4 Route set generation algorithm Before the route flow model can be run, a route set over the analyzed network must exist. The method for generating a route set was an iterative process of finding the shortest path using Dijkstra´s algorithm. The shortest path calculations were based on the travel time for each link. The initial travel time was the length divided by the speed limit for each link in the network. If a link was used during a route the travel time was increased with a penalty term. The reason for this was to obtain different routes during the iteration process. Between each iteration of finding a new route the travel time on all links in the previously created route was penalized with the penalty term showed in equation (10). 𝑝(𝑎) =. 𝑙𝑎 ln (1 + ∑ 𝛿𝑎𝑗 ) 𝜇𝐿 𝑗∈𝐶𝑡. (10). where 𝑎 is the current link, 𝑙𝑎 is the generalized cost of using link 𝑎, 𝜇 is the penalize parameter regulating the size of the penalty term, 𝐿 is the minimum cost between the OD-pair, 𝐶𝑡 is the set of generated routes between the OD-pair at the end of iteration 𝑡 and 𝛿𝑎𝑗 is a binary variable that is 1 if link 𝑎 is in route 𝑗, 0 otherwise. This penalty term is different to the one described in equation (1) in section 2.3.1 proposed by the literature. The difference is that the logarithm term has a value of one added. The reason for this is, when the first route for an OD-pair is generated the current generated route set 𝐶𝑡 will only consist of that route. In that case the links in the route will not be penalized since ln(1) = 0. To avoid creating the exact same route again this value of one is added. Only unique routes are added in the route set and they are only allowed to be a specified threshold longer than the shortest path, otherwise they are not included. Figure 6 shows a flow chart of the route set generation algorithm (RSGA).. 27.

(38) Figure 6 - Flow chart of route set generation algorithm. 28.

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