the time variation from the dynamic assignment to calibrate the static travel times creates a ...... When evaluating models, the likelihood ratio test against a model with param- ...... The two midpoints that define the intermediate segments differ ..... Table 5.2 Estimation Results for Models B1, B2 and B3, in which an additional.
Estimation of a Demand Model Based on Data from a Dynamic Tra c Assignment Model
NikitasNikolaos Papakatsikas Internal Supervisor: Marcus Sundberg External Supervisor: Svante Berglund
Master of Science Project in Systems Analysis and Economics
School of Architecture and the Built Environment Royal Institute of Technology
June 2017
Abstract Increasing congestion phenomena in large cities necessitate the introduction of more reliable data in the demand modelling process. Recent progress in dynamic tra c assignment has enabled the insertion of within-day variation in the network levelof-service data. This thesis employs such a procedure to acquire a dynamic travel time matrix and use it as input in the re-estimation of an existing destination and mode choice model which is based on static data. A departure time choice model is rst estimated to produce a dynamic, 15-minute sliced demand matrix for work trips during the morning commute in Stockholm. A mesoscopic network of Greater Stockholm is then used in two assignment simulations with reduced volumes to avoid extreme gridlock conditions. The primary destination-mode choice model estimation shows that the dynamic travel times of this work do not result in an improved model. Deeper analysis reveals patterns that eventually lead to the conclusion that using the time variation from the dynamic assignment to calibrate the static travel times creates a combined input that produces a better model overall.
3
Sammanfattning Ökande trängsel i storstäder gör det nödvändigt att introducera mer pålitliga data i efterfrågemodellers skattning. De senaste framstegen inom dynamisk utläggning möjliggör att använda timvariation i nätverksdata.
Examensarbetet använder sig
av en sådan process för att skapa en dynamisk restidsmatris och nyttjar den som indata i den skattning av en destinations- och färdmedelsval modell som har tidigare skattats på statiska data. En avresetidpunktsmodell är skattad för att ta fram en dynamisk, 15-minutersindelad efterfrågematris av arbetsresor under morgontra k i Stockholm. Ett mesoskopisk nätverk över Stockholms län används sedan i två utläggningar med reducerade volymer för att undvika extrema trängseltillstånd. Den grundläggande destinations- och färdmedelsval modell skattningen visar att de dynamiska restiderna i det här verket inte resulterar i en förbättrad modell. En djupare analys avslöjar mönster som leder till slutsatsen att användningen av restidsvariation som uppkommer från den dynamiska utläggningen i en kalibrering av de statiska restiderna skapar kombinerade indata som producerar en bättre modell generellt sett.
5
Dedication To my always supportive family
7
Acknowledgements I would like to thank my supervisors Marcus Sundberg at at
WSP Analys & Strategi
KTH
and Svante Berglund
for their advice and assistance in both the theoretical
and practical part of this thesis.
Our meetings and discussions have always been
productive and educational. Much credit is due to my colleagues Patryk Larek and Olivier Canella, who had extensively worked on the network that I started with and for their guidance along this project.
9
Contents
1 Introduction
17
1.1
Background
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1.2
Purpose
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.3
Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.4
Project Planning
21
1.5
Limitations and Scope
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Literature Review 2.1
2.2
23
Demand and Supply Models . . . . . . . . . . . . . . . . . . . . . . .
23
2.1.1
Why do we model? . . . . . . . . . . . . . . . . . . . . . . . .
23
2.1.2
What is travel demand?
. . . . . . . . . . . . . . . . . . . . .
23
2.1.3
What is travel supply? . . . . . . . . . . . . . . . . . . . . . .
24
2.1.4
What is the demand-supply interaction?
. . . . . . . . . . . .
24
. . . . . . . . . . . . . . . . . . . . . . . .
25
4-Step Demand Modelling 2.2.1
2.3
21
. . . . . . . . . . . . . . . . . . . .
25
Discrete Choice Models . . . . . . . . . . . . . . . . . . . . . . . . . .
How is demand modelled?
27
2.3.1
Random Utility Theory . . . . . . . . . . . . . . . . . . . . . .
27
2.3.2
The Multinomial Logit Model . . . . . . . . . . . . . . . . . .
27
2.3.3
Nested Logit . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.4
Estimation of the Model
. . . . . . . . . . . . . . . . . . . . . . . . .
29
2.5
Model Speci cation and Evaluation . . . . . . . . . . . . . . . . . . .
29
2.6
An Interpretation Issue: Endogeneity . . . . . . . . . . . . . . . . . .
30
The Fourth Step: Assignment
2.7
. . . . . . . . . . . . . . . . . . . . . .
31
2.7.1
Static Assignment Issues . . . . . . . . . . . . . . . . . . . . .
31
2.7.2
Dynamic Tra c Assignment . . . . . . . . . . . . . . . . . . .
32
2.7.3
DTA and Demand Coupling . . . . . . . . . . . . . . . . . . .
36
2.7.4
Departure Time Model and Importance for DTA
37
. . . . . . .
3 Data
39
3.1
Travel Habit Survey
. . . . . . . . . . . . . . . . . . . . . . . . . . .
39
3.2
A Priori Demand Matrix . . . . . . . . . . . . . . . . . . . . . . . . .
39
3.3
Network
39
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Methodology
41
4.1
Model Formulation
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
4.2
Model Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4.3
Model Inputs
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
4.3.1
Dynamic Tra c Assignment . . . . . . . . . . . . . . . . . . .
44
4.3.2
Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
11
Estimation of a demand model based on data from a dynamic tra c assignment model 4.3.3
Time-Slicing . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Results
49
57
5.1
Travel Time Data Selection
. . . . . . . . . . . . . . . . . . . . . . .
57
5.2
Data Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
5.3
Model Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
5.4
Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
5.5
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
5.5.1
General Analysis
60
5.5.2
Travel Time Parameters
5.5.3
Cost Parameters and Value of Time . . . . . . . . . . . . . . .
61
5.5.4
Density and Congestion
61
5.5.5
Nest Signi cance
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
. . . . . . . . . . . . . . . . . . . . . . . . .
62
. . . . . . . . . . . . . . . . . . . . . . . . . . .
63
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
Discussion and Future Work . . . . . . . . . . . . . . . . . . . . . . .
68
5.6
Deeper Investigation
5.7
Conclusions
5.8
Bibliography
71
A Network
75
A.1
Road Classes
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
A.2
De nition of Central Zones . . . . . . . . . . . . . . . . . . . . . . . .
77
A.3
Convergence Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
B Time-Slicing
79
B.1
Time-Slicing of Other Purpose Trips
. . . . . . . . . . . . . . . . . .
79
B.2
Time-Slicing of Additional Trips . . . . . . . . . . . . . . . . . . . . .
79
C Model Estimation C.1
81
Complete Model Speci cations . . . . . . . . . . . . . . . . . . . . . .
12
81
List of Tables
4.1
Simulated Trips Number . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.2
Departure Time Model Estimation Results . . . . . . . . . . . . . . .
53
5.1
Estimation Results (1)
. . . . . . . . . . . . . . . . . . . . . . . . . .
59
5.2
Estimation Results (2)
. . . . . . . . . . . . . . . . . . . . . . . . . .
60
5.3
Value of Time Results
. . . . . . . . . . . . . . . . . . . . . . . . . .
62
5.4
Peak and O-Peak Analysis
. . . . . . . . . . . . . . . . . . . . . . .
64
5.5
Zonal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
5.6
Final Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
5.7
Final Model Comparison . . . . . . . . . . . . . . . . . . . . . . . . .
67
A.1
Urban Road Class De nitions
. . . . . . . . . . . . . . . . . . . . . .
76
A.2
Rural Road Class De nitions . . . . . . . . . . . . . . . . . . . . . . .
76
B.1
Factors for Time-Slicing of Other Purpose Trips . . . . . . . . . . . .
80
C.1
Full Model Speci cations (1) . . . . . . . . . . . . . . . . . . . . . . .
82
C.2
Full Model Speci cations (2) . . . . . . . . . . . . . . . . . . . . . . .
83
C.3
Full Model Speci cations (3) . . . . . . . . . . . . . . . . . . . . . . .
84
13
List of Figures
1.1
Work ow of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.1
Land Use-Transport Feedback Cycle
25
2.2
Traditional Four-Step Model Structure
. . . . . . . . . . . . . . . . .
26
4.1
Destination-Mode Model Structure - Initial . . . . . . . . . . . . . . .
42
4.2
The Network
4.3
Speed-Density Diagrams
4.4
Roundabouts
4.5
Initial Departure Pro le
. . . . . . . . . . . . . . . . . . . . . . . . .
55
4.6
Balanced Departure Pro le . . . . . . . . . . . . . . . . . . . . . . . .
55
5.1
Travel Time Data Comparison . . . . . . . . . . . . . . . . . . . . . .
58
5.2
Value of Time - Car . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
5.3
Value of Time - Public Transport
. . . . . . . . . . . . . . . . . . . .
63
A.1
Speed-Density General Diagram . . . . . . . . . . . . . . . . . . . . .
75
A.2
Central Zones De nition
. . . . . . . . . . . . . . . . . . . . . . . . .
77
A.3
Convergence Chart (1)
. . . . . . . . . . . . . . . . . . . . . . . . . .
78
A.4
Convergence Chart (2)
. . . . . . . . . . . . . . . . . . . . . . . . . .
78
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
. . . . . . . . . . . . . . . . . . . . . . . . .
46
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
15
Chapter 1 Introduction
1.1 Background The development of urban areas has become signi cantly faster in the last few decades. While in 1950 less than a third of the world's population lived in urban settlements, in 2014 more than 50% of the population was urban.
This trend is
projected to continue at the same degree in the coming decades (DESA, UN, 2014). Taking into account the similarly increasing population growth, the absolute number of urban residents displays an even more impressive hike. Intranational and international migration towards capitals and other important cities, especially in already densely populated and developed regions, such as Western Europe, has resulted in large metropolitan areas, often organised in multi-core patterns. Apart from other problems of social background that arise from these facts, metropolitan areas usually also have to deal with transport-related problems. Solutions to these problems can be either pacity network or
aggressive,
for example building a higher ca-
passive, such as the introduction of mode shifting policies.
When
considering these options, sustainability is an important factor in decision making. Preserving environmental quality for the less advantaged, for future generations or even for the environmental diversity itself, as well as investigating opportunities for economic progress is the burdensome balance that has to be achieved in the pursuit of sustainability goals (Moavenzadeh and Markow, 2010).
Speci cally in
transport, these goals can be speci ed in objectives such as reducing greenhouse gas emissions, improving transport safety and accessibility and reducing congestion, thus respecting all three pillars of sustainability, i.e. its environmental, social and economic aspect (May, 2013). Several transport policies aspire to meet these targets.
Strategic choices that
range from performance improvement to applying the correct trade-os among users as well as tactical decisions such as congestion charges and the subsidisation of more advanced,
green
technologies are often implemented today. The evaluation of these
possible solutions and even long-term strategies needs to be based on a precise model of the transport system. This procedure must be continuous as well as constantly updated in order to realistically represent the current and future conditions of the network and its users. Traditionally, the four-step model is applied in strategic analyses of transport systems. It includes a sequentially leveled structure that describes every trip with the levels being generation, distribution, modal choice and assignment. The rst three
17
Estimation of a demand model based on data from a dynamic tra c assignment model steps result in mode-speci c origin-destination matrices, where each cell represents the number of trips between two speci ed zones of the network, in other words the travel demand. These are fed into an assignment routine that decides which of the possible routes will be actually used in the trip, eventually providing information about the network conditions, what is often called the network supply. One of the most recent problems related to the rapid development of urban areas is congestion. The signi cance of this issue becomes obvious from the number and variety of its implications, all posing a threat to the sustainability pillars mentioned above. The rising importance of saving time in a business-focused society, the environmental repercussions resulting from noise and emissions as well as the personi ed aspect of mental health lift the confrontation of congestion issues very high in the priority list of the relevant city agencies. Combining this volition with the current state-of-the-art transport modelling techniques reveals some inherent problems of the latter. It appears that the currently used procedures are not adequately exible and detailed to predict the tra c situation in large areas with high degree of congestion, like most metropoleis are today. In an eort to re ect the temporally and spatially dynamic phenomenon of congestion, researchers have focused on improving their modelling methods and replacing parts of their usual work ow with more complex concepts. Signi cant progress has been made in using dynamic routines for assignment, the last step of the model described above.
In the city of Stockholm, for example, the IHOP project (Alm-
roth et al., 2014) has reached the conclusion that the technical requirements for the replacement of the currently used static procedure by a dynamic procedure are ful lled. The report, however, also mentions various issues that appear due to such a replacement, for example inconsistencies between the input and the output data or higher degrees of sensitivity to simpli ed network modelling. Treating these issues will allow for a full implementation of dynamic tra c assignment in the demand models of cities that have to deal with congestion. This will eventually lead to more reliable results when evaluating and comparing measures, thus hopefully to better decision-making.
1.2 Purpose This master thesis aims to observe the eects of using dynamic network data in the estimation of a demand model by performing a sensitivity analysis to nd the portion of uncertainty that can be explained by the presumed higher accuracy of a dynamic assignment procedure in a congested urban network.
Furthermore, by
focusing on the current Swedish transport model, it intends to provide a stepping stone towards the full inclusion of a consistent dynamic assignment model in the future and identify elements of the method that need additional consideration.
1.3 Objectives The objectives of this master thesis have been set to:
Compiling a network that covers Stockholm county using recent and ongoing work, with certain enhancements when deemed necessary
18
CHAPTER 1. INTRODUCTION
Using a historical (a-priori) demand matrix and time-slicing it in a systematic way according to a departure time model
Inserting all data to a dynamic assignment software to acquire dynamic link ows and travel times
Estimating the demand model using dynamic travel times and existing survey data
and can be visualised as in Figure 1.1.
19
INSPECTION
Estimation of a demand model based on data from a dynamic tra c assignment model
Network
”A priori” O-D Matrix
Travel Habit Survey Data
Modifications
Time Slicing
Departure Time Model
Modified Network
Time-Sliced O-D Matrix
Dynamic Assignment
Dynamic Travel Times
Travel Habit Survey Data
CALIBRATION
Demand Model Estimation
Figure 1.1 Work ow of this thesis
20
CHAPTER 1. INTRODUCTION
1.4 Project Planning The project assists in on-going work, both in terms of carrying out practical work that can be used directly as well as in investigating potential issues for similar future applications.
It has therefore been organised according to the features and
requirements of these applications and parts of it have been developed in collaborative work. Some of the input data have been used without further changes (e.g. the previous demand model speci cation and the speed-density functions used in assignment), while some have been modi ed through individual work by the author (e.g. existing network improvements and demand matrix time separation).
1.5 Limitations and Scope This work focuses in the region of Greater Stockholm and uses data from the period of 2005-2006.
The time constraints imposed by the thesis project plan limit the
accuracy of the dynamic assignment procedure to an imperfect, yet acceptable, level. The focus of the work is on destination and modal choice with other aspects of travel demand, such as generation or departure time, being treated independently, either by assumptions or by isolated methods.
The estimation of the destination-mode
choice model is only made by the use of dierent input data in the form of dynamic travel times and not by adding or removing input variables.
21
Chapter 2 Literature Review
2.1 Demand and Supply Models 2.1.1
Why do we model?
A model is a representation of one speci c part of a system of interest. It aims to make that particular part more understandable through isolation from the rest of the world. It can be described as a box with well-de ned borders that separate what is taken into account in the modelling process from what is not, i.e. the environment. Deciding the extent of the borders is crucial from a scienti c point of view since it will play a major role in achieving high accuracy and precision, in other words minimizing the systematic and random errors respectively. In transport science, mathematical models are essential tools to examine the current situation of a transportation system and reach decisions for the future, often with the help of scenarios that change some or all of the considered variables. It is important to note that transport modelling does not have a direct impact on society on its own. Instead, it is when the models are used to support planning decisions, either operational, tactical or strategic, that they become indispensable to the eld. This role of modelling was not as clear before the 1980s, in a period when innovative developments remained focused on a strictly academic level (de Dios Ortuzar et al., 1994). A transportation system is of course a complicated network of multiple elements and clear or less clear relationships among them. Narrow limits are hard to draw without loss of generality and broad ones make the model di cult to estimate or use. Despite this complexity, in essence, a transport system can be described as the relationship between two elements, demand and supply.
2.1.2
What is travel demand?
Demand in transport is derived and is not an end in itself (de Dios Ortuzar et al., 1994). Trips are made to ful ll a speci c purpose, such as going to work or school, shopping and entertainment and they would not occur without the existence of that purpose.
The demand is therefore subject to all the elements that are connected
to that purpose, including the user themselves, spatial and temporal characteristics of the trip. The wide range of variation in these elements results in a largely dierentiated demand, making predictions over it even harder.
23
Estimation of a demand model based on data from a dynamic tra c assignment model 2.1.3
What is travel supply?
On the other side, supply is de ned as the service that is provided to serve potential demand.
It consists of the infrastructure that supports the network, for example
roads, transit lines, public transport eets and centroids (points that for certain reasons raise interest to visit). Again, there is a wide range of factors that aect supply.
Time and monetary cost are important variables; infrastructure requires
long construction times and su cient funds. Politics are also very much of interest, both in terms of planning by the authorities and from the viewpoint of gaining or losing stakeholders.
2.1.4
What is the demand-supply interaction?
Demand and supply can in theory exist independently of each other.
A traveler
may need to travel to an island to ful ll a purpose even though there is no bridge or ferry to take them there. A road will still be provided after its construction even if it is not used by anyone. In reality, this relationship hardly stands, at least not in the long term. A purpose that no-one is capable to reach will be substituted by a similar one with better accessibility, while the unused road will not receive adequate maintenance, eventually becoming unusable. In short, demand generates supply and vice versa. The role of a planner is to ensure that a certain level of demand is satis ed (de Dios Ortuzar et al., 1994). Adding to this statement, the role of a good planner is to ensure that this demand is only marginally satis ed.
One reason for this is
that it is desirable that the costs of the required supply measures remain acceptable. Another reason is that supply is rather a service than a good and its utility cannot be stored for future demand. In other words, it would be cost-ine cient to build high capacity networks for the sake of a future hike in demand, because operating costs would exceed pro ts from the existing demand for a long time.
Finally, demand
and supply are in constant interaction; an excessive supply will be followed by an increase in demand, leading to the eventual ineectiveness of the taken measures. This is a result of the Land Use-Transport Feedback Cycle shown in Figure 2.1, which depicts how higher accessibility of an area leads to higher amount of activities that subsequently derive more travel demand (Wegener and Fürst, 2004). Considering all this, it is obvious that the estimation of reliable and accurate demand models is of great importance for the transportation eld. They will provide the information that all planners need to make correct decisions, especially from a strategic point of view.
24
CHAPTER 2. LITERATURE REVIEW
Figure 2.1 The land use - transport feedback cycle. The gure is an adaptation of
the gure in (Wegener and Fürst, 2004).
2.2 4-Step Demand Modelling 2.2.1
How is demand modelled?
Years of experimentation have resulted in a widely used structure, already present from the 1960s and only slightly altered since then.
This structure is called the
Classic Transport Model and consists of a more or less sequential method to deduce the travel pattern of the system in question. Although the order and the modelling techniques may vary in applications, the traditional sequence can be described here. The initial data would be a network separated into zones with their population in dierent classes as well as the activities that can be pursued in these zones and to which extent these generate trips. This will provide the basis to model the number of trips that originate from and head towards each zone. The next step is to distribute these trips in origin-destination pairs, i.e. producing a trip matrix. Modal split then separates the trips into dierent transport modes and the resulting mode-separated trip matrices are then assigned into the possible routes of the network in the last step (de Dios Ortuzar et al., 1994). Figure 2.2 shows the traditional sequential procedure of this 4-step model, consisting of Trip Generation, Trip Distribution, Modal Split and Assignment. This modelling strategy is widely used but not unique in the eld of transport modelling, especially not within the academia. Approaches that treat these steps as part of the same model or that emphasise in activities or tours rather than trips are some ground-gaining alternatives that are likely to improve our knowledge. Other methods adopt the 4-step model logic but modify the sequence or treat distribution and modal split simultaneously.
They can also be advantageous in certain cases.
All in all, the 4-step model provides a signi cant basis for formulating the demand estimation issue in an intuitive way and is still state-of-the-practice in many parts of the world today.
25
Estimation of a demand model based on data from a dynamic tra c assignment model
Figure 2.2 The traditional structure of a 4-step model process. The order or modelling techniques may vary, for example by combining the trip distribution and modal split steps into one model.
Generation Trip generation is the stage in which the total number of trips originating in each zone and destined for each zone is predicted. Equivalent terms for these two concepts are generation/production and attraction respectively. Certain characteristics may cause a zone to generate or attract more trips; or both. Population, employment, car ownership, commercial development and accessibility are some of the most important variables. Trips themselves can be categorised in accordance to their purpose (work, shopping, recreational), the time of day they are made (peak or non-peak period) and the individual that made them (socioeconomic data).
Distribution The allocation of trips in origin-destination pairs follows the generation stage.
A
trip matrix called the Origin-Destination (O-D) Matrix is the most frequent way to represent this distribution. The initial input data is survey results complemented by modelling procedures that incorporate zone-speci c variables. For every trip originating in zone i, the generalised cost of travel towards every zone j is calculated. This includes travel time and costs, weighted by parameters that express each variable's signi cance to the traveler. Various methods can then be used to calibrate the O-D matrix such that the generalised costs are correctly represented and the zone totals are consistent with the result of generation.
Modal Split A very important element in transport planning and policy making is the correct estimation of mode choice. The bene ts of choosing public transport over private cars in the nancial, social and environmental aspect are well-known. The demand model should re ect this and the third stage of the 4-step process ensures that the in uential factors on mode choice are accounted for. Those factors can be traveler,
26
CHAPTER 2. LITERATURE REVIEW journey or supply characteristics. Depending on the modellers' purposes and state-of-practice in dierent periods, modal split model have been applied either before or after trip distribution, preserving characteristics of the individual in the rst case and of the trip in the second. Simultaneous modelling is also possible and often required to better accommodate the characteristics of the trip maker. This leads to complicated model structures that include multiple destinations and multi-modal alternatives, with various degrees of correlation between them (de Dios Ortuzar et al., 1994).
Assignment The process of determining the path of each individual trip is the last step in a 4-step model. Assignment is dicussed in detail in Section 2.7.
Complementary models Additional models can be incorporated in the 4-step modelling system and assist in the prediction of factors that in uence the main choices on a secondary degree. These models are thoroughly examined in the literature and are often included in many applications. Frequent examples are a car ownership model that predicts the change of the car eet size and types as well as a departure time model that allows for additional costs in trips that deviate from a traveler's desired arrival time point.
2.3 Discrete Choice Models 2.3.1
Random Utility Theory
The models that have been discussed so far involve choices of individuals from a nite set of alternatives
I
and are called discrete choice models. The most common
formulation in this context is random utility theory, in which certain assumptions are made, including: i. Rational and fully-informed individuals that act on a utility maximization agenda ii. De ned alternatives and their availability regarding each individual iii. Utility functions that account for both the measurable and deterministic part as well as the unobserved and stochastic part (de Dios Ortuzar et al., 1994). The attractiveness of the alternatives
i∈I
for each individual
by a utility function, which consists of a deterministic part, eller, and a random part,
εin ,
n are represented
known to the mod-
also called the error.
uin = Vin + εin
2.3.2
Vin
(2.1)
The Multinomial Logit Model
The simplest and most popular model used in discrete choice is called the Multinomial Logit Model and assumes that the distribution of the random errors across
27
Estimation of a demand model based on data from a dynamic tra c assignment model the population is independently and identically Gumbel. Each individual following choice probability for every alternative
i
in his/her choice set
n
has the
In :
exp(βVin ) Pin = P exp(βVjn )
(2.2)
j∈In with
β = π/σ
√
6 and
σ
being the standard deviation of the error distribution.
The utility functions may include both generic and alternative speci c variables with universal and separate parameters respectively.
An alternative speci c con-
stant, i.e. a variable with a value of 1 for all individuals, can be speci ed for all alternatives but one, representing the inherent preference of the average person to that alternative in comparison to the alternative without this parameter. MNL satis es the axiom of independence of irrelevant alternatives, i.e. when two alternatives have a non-zero probability of being chosen, their ratio is not aected by the inclusion or exclusion of another alternative in the choice set.
This may
prove problematic if correlation between alternatives is suspected and in such cases a dierent speci cation is usually required.
2.3.3
Nested Logit
Nested Logit models allow for a more complicated covariance matrix between alternatives than MNL and therefore are useful in the cases of similar groups of alternatives. Their utilities can be separated in two portions, one part that is speci c to that alternative and another part that is common between some or all the alternatives. Alternatives are therefore placed in a nested structure where each nest
i with some part of shared utility. The nest is then represented by a composite alternative, j that competes with the other alternatives (composite or not) in the choice set J with its utility being
includes alternatives
Vj = φj · ln
X
exp(Vi/φj )
(2.3)
i∈Ij
φj natives i where
are structural parameters to be estimated and in nest
Ij
is the choice set of alter-
j.
The probability to choose a nested alternative
i ∈ Ij
is then equal to the joint
probability of choosing the nest and of choosing that alternative given the choice of the nest.
exp(Vi|j ) exp(Vj ) ·P j∈J exp(Vj ) i∈I exp(Vi|j )
Pi,j = Pj · Pi|j = P where
Vi|j
(2.4)
is the utility of the nested alternative.
Although nested logit is very often implemented in practice and can incorporate parts of the 4-step process in the same model (usually destination and mode choice are modelled this way), it still does not solve other issues. For example, it assumes that error variance is equal and coe cients are the same across individuals, ignoring taste variation. These weaknesses can be dealt with by more advanced formulations, such as Mixed Logit.
28
CHAPTER 2. LITERATURE REVIEW
2.4 Estimation of the Model The model is estimated by maximizing the log-likelihood function, which takes the form
LL(β) =
N X
ln Pn (β)
(2.5)
n=1 where
Pn (β)
is the probability of the observed outcome for decision maker
the sample size and
β
n, N
is
is the model parameters vector (Train, 2003).
2.5 Model Speci cation and Evaluation Choosing the correct formulation of the ones mentioned above or other models proposed in the literature is dependent on the available data, the general characteristics of the choice in question and the modeller's own experience and, more often than not, opinion. A basic problem is to de ne the choice set in terms of size and formation, for example when organising the nests in a nested logit model. The modeler needs to make a trade-o between complexity and relevance, always with data availability or lack thereof minimizing the range of options. Methods of sub-setting and aggregation can be contrasted with a fully inclusive choice set (de Dios Ortuzar et al., 1994). Specifying the model relates to the utility function types.
Linear expressions
are the norm in many contexts, but other choices call for transformations.
The
decision is based on data, theory and sometimes intrinsic non-linearities, such as in the destination choice model. With regards to estimation, the maximum likelihood method is normally used, which is based on the idea that the sample is more probable to have been taken from a certain population than from others. The parameters that generate the observed sample most often are the ones with the highest likelihood. For each parameter separately, the t-test is used to determine their signi cance in the model.
In many cases non-signi cant parameters may still be included if
there has been evidence that they are relevant according to theory, e.g. parameter.
the cost
Other decisions may be based on whether the parameter sign is as
expected and policy-relevant variables in decision making. The t-score is always in reference to a con dence level. This is often 95% and this value is also used in this work. When evaluating models, the likelihood ratio test against a model with parameters equaling zero is often used, in other words comparing to a completely random 2 choice among the alternatives. The R goodness of t index that shows how well the model ts the observed data is also important, although attention must be raised to cases where the model is a good t but a bad predictor (the case of over tting). Its value range is
[0, 1] and higher values mean better t.
When putting similar models
in comparison, with the dierence being the absence of one or more parameters, the likelihood ratio test is the suggested evaluation method.
A work-around solution
when comparing models with dierent functional forms is to separately evaluate both against a composite model with hybrid functions. Another method is to study the relative eect of the parameters.
For example, the ratio of the time parame-
ter to the cost parameter, often called Value of Time, can be compared between
29
Estimation of a demand model based on data from a dynamic tra c assignment model two models. The results can be contrasted to a realistic value measured in another study. Finally, validation samples can be used to judge the prediction power of the estimated model.
2.6 An Interpretation Issue: Endogeneity The results of the model consist of a set of estimated parameter values which need to be interpreted by the modeller. It is important to distinguish between causality and association or correlation. the value of another variable
y
The fact that a variable
x
is a good predictor of
does not imply any form of causal relationship. All
consistent, i.e. they
the implemented designs focus on providing parameters that are
converge to the true population parameter as the sample size approaches in nity. Consistency is an integral part of a causal analysis (Antonakis et al., 2014). High statistical signi cance and correct direction (parameter sign) is not in itself a basis of supporting the argument that a parameter is consistent and can be interpreted. The issue of
endogeneity
should not be ignored. Holding all other things
equal (
a change in
x
ceteris paribus )
should aect
y.
If instead the value of
x
de-
pends on other, non-modeled variables, which also change other modelled variables than
x,
then
x
is an endogenous variable and the results of the estimation are unin-
terpretable. Unless the endogeneity in the model is treated, estimated relationships may be spurious and do not provide any basis for a causal structure. In the eld of transport supply and demand interaction, endogeneity occurs as a result of both processes being modelled stochastically and in dependence to each other. The stochastic demand model is estimated based on observed route choices, which are a result of both modelled and non-modelled (
error )
parts. The demand
model travel times are then again used as supply data in a stochastic assignment process, in which these error parts are once again involved. The correlation between those stochastic elements means that endogeneity is an issue.
More speci cally,
the dierence between the demand and supply travel times that emanates from the combination of stochastic processes can be expressed by an error component that is correlated to the random part of the demand side
ε
η
because it accounts for
day-to-day uctuations which arise from this random part (Nagel and Flötteröd, 2012). In the case of the current work a destination-mode choice model will be estimated and compared to an existing model.
The travel times in both models originate
from the same source, a travel habit survey, however in the current work they are used as the initial data for a stochastic dynamic assignment process.
Therefore,
the route choices are based on the initial travel times and are used to produce new, dynamic travel times, which are then inserted as estimation variables.
The
apparent endogeneity issue means that the parameters cannot be interpreted in a detached manner, but only in the context of comparison, as is the case of a sensitivity analysis. Furthermore, the endogeneity issue is historically not taken into account in the estimation of the National Swedish Transport Model. An eort to treat it would be outside the scope of the thesis.
30
CHAPTER 2. LITERATURE REVIEW
2.7 The Fourth Step: Assignment Assignment is the process of distributing trips of known mode, origin and destination over the network, by setting the exact path of the trip through the network links and nodes.
The problem consists of a route choice model and the loading of the
total trip matrix to the routes identi ed by the model (de Dios Ortuzar et al., 1994). If it is assumed that travelers choose their route based on the same attributes and do not have any particular dierences in how they perceive them, they will inevitably select from the same choice set of shortest paths (usually equalling
cheapest routes ),
if their O-D pair and mode are identical. This is a result of the rational traveler assumption, which states that every individual tries to maximize their utility, in this case minimize their costs. This equilibrium seeking process leads to results that neglect to consider preference variation and capacity constraints. A stochastic approach will accommodate for variability in cost perception, allowing for the choice of alternative routes to the one with the objectively minimum cost to be made. A simulation-based method uses link-speci c cost perception distributions over individuals, so that for a number of individuals the least costly route (as a sum of link costs) is dierent than others, even if they travel between the same origin and destination as well as by the same mode (Burrell, 1968). A proportional stochastic method considers the links feeding trips into a node and uses logit-type formulas to decide the choice among reasonable alternative routes. Similarity of routes is not modelled, re ecting the weaknesses of MNL. Probit route choice, crossnested logit and hybrid logit models have also been proposed in this direction (Hall, 2012). In order to incorporate congestion into route choice, trips need to be assigned with some relation to this factor.
Volume-delay functions on a link level may be
used for a deterministic equilibrium approach, often called Wardrop's assignment (Wardrop and Whitehead, 1952). Equilibrium is reached when no driver can reduce their costs if they change their route (user optimum).
Another formulation de-
mands that under equilibrium conditions, the total travel cost is minimized (system optimum), although this is not as behaviorally sound. Combining the strive to an equilibrium as well as the stochastic methods that represent individual variability will lead to a realistic approach that models both capacity restraints and perceptual dierences. This model will result in a stochastic user equilibrium. The issues regarding this eort can be summarised in three components: the de nition of the choice set of reasonable routes, model estimation and integration in an equilibrium assignment framework (de Dios Ortuzar et al., 1994). Various solutions are proposed, such as the inclusion of a commonality factor to nest similar routes (Cascetta et al., 1996), a Path-Size Logit model that weighs the uniqueness of paths and the use of Mixed Logit to account for heteroscedasticity. A nested recursive logit model that formulates path choice as a series of link choices has also been proposed more recently (Mai et al., 2015).
2.7.1
Static Assignment Issues
Assignment has traditionally been dealt with as a static problem. In static models, the analysis period is generally long and typically de ned as the peak or o-peak time-of-day. In this framework, the assignment equilibrium that is reached can at
31
Estimation of a demand model based on data from a dynamic tra c assignment model best describe a solution to an averaging of the problem and is very inadequate when large deviations occur within the analysis period. Furthermore, a volume-delay function describes the performance of each link, allowing for an increase in travel time as the in ow increases. This increase is limitless and when applied in congested networks, the model produces results with links exceeding their physical capacity. The assigned volume should be more accurately considered as demand, rather than as actual ow.
Congestion is modelled by a
volume-capacity ratio that is higher than 1, without displaying connections to any physical measure (e.g. speed, density or queue), since these values of the ratio are obviously not possible (Chiu et al., 2011). In fact, congestion is speci cally related to the link with the exceeded capacity rather than eventually spreading to other links upstream, as is the case in reality, what is widely called the phenomenon of queue spillback.
This weakness of the
static model is explained by the absence of a detailed spatial representation of the link connections. Delay is predicted to occur only inside the link where the queue is formed and not upstream (Flügel et al., 2014). In complex networks with short links, such as in an urban environment, where many links are expected to demonstrate V /C > 1, this can lead to a very faulty analysis. Dierences in travel times between dierent congestion level scenarios will be underestimated, due to the failure of capturing the positive and non-linear relationship to neighborly links and time periods (Almroth et al., 2014). Static models are also simplistic representations, which leads to various other consequences. One of them is the enforcement of the rst in, rst out (FIFO) rule. As a result, the same travel time is experienced by all vehicles on a link, disallowing any overtaking movements, which are quite frequent in reality. A similar problem derives from the absence of any separation between the dierent lanes in a segment. A low-speed lane in a freeway due to an oversaturated o-ramp means that the other lanes are also moving at that speed, a representation far from the truth (Chiu et al., 2011). As mentioned above, static processes attempt to capture average tra c conditions which do not include any uncertainty or variability either in the network or the travel behavior. There is no way to model an infrequent event that will suddenly cause large parts of the network to collapse, such as the breakdown of a vehicle or a technical fault (Flügel et al., 2014). Static routines also only allow for only limited heterogeneity when it comes to individual travelers.
They make use of OD matrices to separate between a few
traveler or vehicle classes, thus missing out on important information that could be vital to an appraisal process that considers equity aspects. (Larek, 2015).
2.7.2 Dynamic Tra c Assignment De nition Assuming that individuals attempt to minimize their travel disutility and thus their travel times and costs, an equilibrium state is searched for, in which no traveler can further reduce their travel time by modifying their route choice. The concept of the equilibrium, however, requires that travelers have information on conditions through numerous trials and that demand as well as supply data are xed and known (Chiu et al., 2011).
32
CHAPTER 2. LITERATURE REVIEW The general assumption behind dynamic assignment is that route choice depends not only on previous information about travel times but also in what the traveler learns about the current situation in the network, either in the form of supplied information (delay announcements, congestion updates, noti cation ITS etc.)
or
based on what the traveler is experiencing during his trip. Also, travel times themselves, as a result of this process, are also temporally distinct, rather than stable during the whole analysis period.
Dynamic models generally do not accept that
tra c networks oer the same supply within the day. A dynamic user equilibrium therefore needs two extensions over the previous de nition. The rst of them is that travelers are aware not only of network conditions before their departure but of future conditions as well. This is more clearly de ned as a contrast between Instantaneous and Experienced Travel Times and forms the basis of the development of time-dependent shortest routes. The second extension is that travel times in a route are equal only for travelers with the same departure time, thus leading to a dynamic user equilibrium (DUE), that is dierent for each departure time (Chiu et al., 2011).
Framework and past representation eorts Various researchers have focused on creating a framework and solving the dynamic user equilibrium. Methodology varies and can generally be separated into two large groups, namely analytical and simulation methods. The initially examined and analytical approach is to use common optimization techniques. At rst, a discrete time setting for a one-destination network was considered and the problem was treated as a non-linear and non-convex mathematical programming problem. The resulting model was a generalization of the then conventional static system optimum assignment formulation and it was later referred to as the M-N model (Merchant and Nemhauser, 1978). It essentially showed that a system optimum can be reached as an optimal solution of a certain linear program, which was later proved to be obtainable by optimizing N+1 objective functions, where N is the number of discrete time periods (Ho, 1980). The next step was to develop the formulation for a multiple destination network (Carey, 1987). A shift towards the user equilibrium was made a few years later by including experienced travel times instead of instantaneous (Janson, 1991).
However, even later eorts
could not overcome the inherent limitation of applying the solutions in general networks in a realistic way (Peeta and Ziliaskopoulos, 2001). Optimal control theory assumes that origin-destination matrices are known continuous functions of time and are used to estimate link ows as continuous functions of time, thus eliminating the discretization in optimization techniques. Both the system and user optima were discussed for the one-destination case, with route choices being able to change continuously according to prevailing network conditions (Friesz et al., 1989). Multiple origins and destinations were considered in later work (Ran and Shimazaki, 1989). Limitations of this type of formulation in considering congestion and general networks have led to its abandonment. Variational Inequality is the last analytical approach in the DTA context. Earlier introduced in the static model, the method is more general and allows for greater exibility and realism in tra c scenarios, solving many aspects of the problem (Nagurney, 2013).
Despite this, the fact that it still is a solution in analytical
context and the higher computational burden compared to previous methods pose
33
Estimation of a demand model based on data from a dynamic tra c assignment model both theoretical and practical problems to the researchers (Peeta and Ziliaskopoulos, 2001). An alternative approach focuses on nding an approximation of the equilibrium and ranks the availability of a realistic and operational solution over the potential inadequacy in theoretical insights derived from it. This decision is mainly based on the vague meaning of convergence and uniqueness when discussing dynamic assignment solutions (Peeta and Ziliaskopoulos, 2001). Numerous models have been proposed in the literature, often with the use of dierent tra c simulators. Starting from fully known O-D matrices and xed de-
b
parture times (Peeta and Mahmassani, 1995 ) and soon extending to multiple user classes the method was quickly facing computational burden problems. This called for practical solutions to apply the rolling horizon approach, i.e. to use current and
a
near-term forecast information (Peeta and Mahmassani, 1995 ). Another simulator was proposed, that included the interaction of a demand and a supply simulator. It considered historical information as well as driver response to it in order to estimate future O-D demand and then used this as input to a simulator that determined ow patterns (Ben-Akiva et al., 1997). Obviously, as higher computational power became available, more complex networks and models were developed.
The modelling of a complex urban network
with thousands of O-D pairs and hundreds of thousands of vehicles is now possible (Ben-Akiva et al., 2012). Furthermore, complexity may also be increased from the viewpoint of multiple modes and traveler classes, as well as the development of a departure-time model to estimate the time-dependent demand (Meng et al., 2014). Complexity and analysis depth is also determined by the type of simulator. A microscopic transport model will represent every physical object and all spatial relationships and interactions among them (Flügel et al., 2014). Inputs are highly detailed, including even individual characteristics for every object and outputs will be fully disaggregate and will consist of each vehicle's route. Though these models are suggested for small networks of few intersections, the computational cost increase makes them unsuitable for larger systems (Almroth et al., 2014). On the other end of the spectrum, a macroscopic model will use aggregate measures and deterministically ensure that route shares equal the probability of choosing that route, even if they are stochastically de ned (Watling and Hazelton, 2003). Their solution is a result of a system of equations (Flügel et al., 2014). Considering the issues of these two types it is clear that the disaggregate representation of the micro-simulators is desirable and makes better use of the fact that the assignment is now dynamic.
The computational burden, however, leads to a
compromise of a few entities or interactions being modelled in a less detailed way. For example, vehicle behaviours can be aggregated to a link performance function, which in turn will describe the vehicle group's movement (Flügel et al., 2014). Mesoscopic tra c simulation models have been developed as a hybrid solution that combines elements from the two dierent kinds of simulators. The supply side is described in a macroscopic way with link performance functions, while demand is described in a microscopic way, with each driver's route choice being modelled individually. Link performance functions are most often in the form of a speed-density relationship, separate for each link, based on the fundamental diagram (Balakrishna et al., 2008). The number of elements from macroscopic or microscopic models and the degree to which they are implemented in a mesoscopic model is not xed and is
34
CHAPTER 2. LITERATURE REVIEW decided on a case to case basis. For example, the network size can be either regional or smaller, and the detail level may vary even within the network.
Advantages over a static model Dynamic tra c assignment models attempt to represent the temporal relationships between choices, ows and levels of service, including the interactivity of demand and the time-variation of the network.
Through iterative algorithmic procedures,
the route choices are adjusted so that minimum experienced travel time paths are chosen by all travelers, regardless of departure time (Chiu et al., 2011). In a dynamic tra c assignment model, travel time and congestion are closely linked. If the link out ow is lower than the in ow, density will increase and speed will decrease, thus travel time will increase.
The longer the in ow is larger than
the out ow, the bigger the accumulation of vehicles, which will eventually reach the entrance node, when the in ow will start reducing again. The upstream extension of congestion through the link and its eventual spread to neighbour upstream links is described by the link's own fundamental diagram (Chiu et al., 2011). The positive and convex relationships between adjacent links and time periods are su ciently represented and travel times in congested situations are much higher than with a static model. This was clearly demonstrated in a tolled road study (Boyles et al., 2006). The FIFO rule is generally not enforced in a dynamic model, unless it follows uid mechanics rules.
Vehicles can move on discrete lanes of the roadway, thus
overtaking can be modelled.
Finally, a typical simulation environment will allow
for at least some level of variation in the characteristics of the dierent lanes of a segment (Chiu et al., 2011). In general, it can be argued that the dynamic models will perform better than the static models, as they do not make as many assumptions and represent the system in a much more realistic way. It is not so much their superiority that is in question as is their applicability in planning procedures, especially their implementation in the currently used framework (Peeta and Ziliaskopoulos, 2001). Their theoretically better results require a very detailed description of the network and the demand pattern, as their sensitivity to error and potential misrepresentation is high. This is mainly the reason why dynamic assignment is not used in every model.
Recent and future developments With more computational power and higher interest by researchers, much progress has been made in examining a variety of solutions and concepts. One of the most important developments are the models that are based on agents, which try to incorporate the changes of network conditions into higher-level decisions, such as mode and departure time choice. Agent-based modelling consists of a microscopic model of behavioral decisions made by single travelers (agents). The decisions lead to individual travel plans that are evaluated in every iteration. An equilibrium is reached when no traveler can increase the utility of their travel plan by changing routes.
A more realistic version requires a stochastic user equilibrium search, in
which travelers can select from a number of travel plans based on discrete choice theory. (Nagel and Flötteröd, 2012)
35
Estimation of a demand model based on data from a dynamic tra c assignment model Methods to achieve faster convergence have also been developed. Convergence is often de ned as the relative gap between the iteration results and the equilibrium solution. This solution is a situation in which users have updated their paths and cannot further improve their travel time by changing them anymore. Although there is no mathematical proof that such an equilibrium can actually be reached using simulations, and therefore the travel times that correspond to it are not known beforehand, it is still reasonable to measure convergence using this gap, since the results are generally expected to be more realistic than those of other routines. The widely used in assignment procedures Method of Successive Averages is a common recursive averaging method, with which travel times over the last two iterations are averaged to be used in the following iteration. It has been found that another method, Polyak Iterate Averaging, can achieve faster convergence, using a step size that does not decrease as fast as that of the MSA. A dynamically adjusted step size was shown to be similarly e cient, however these results need to be examined in larger and more complicated networks. (Taale and Pel, 2015). Another comparative study in larger networks found that gradient-based heuristics did not produce notably dierent results, except that the rst few iterations that might lead to a merely acceptable solution converged at a higher pace than MSA (Levin et al., 2015).
2.7.3
DTA and Demand Coupling
The transport model consists of two parts. Firstly, the demand model, which de nes the number of trips, their origins and destinations and other characteristics (modal split, departure time) of each trip. Most often, the output is aggregated into mode-speci c OD matrices. Secondly, the supply model, which assigns all trips into dierent routes and provides the network performance levels, in the form of link ows and travel times. With the dynamic tra c assignment process, these levels of service are time-dependent and form an impedance matrix that is de ned for every time interval in the model. Supply and demand models are estimated separately and sequentially, holding one set of inputs and parameters constant while estimating the remaining set. The outputs of the two models are inconsistent with each other. The O-D matrix fed into the DTA model will be dierent from the O-D matrix produced by the supply data that derived from that DTA process. Similarly, the impedance matrix used to estimate the initial O-D matrix is not identical to the one produced by the DTA model. In order to treat this inconsistency, there has to be an iterative process between the two models, until some level of convergence is reached. Thus, the dynamic travel times and ows from the DTA need to be used as input for the estimation of a new O-D matrix which will be then assumed to be the demand for a new DTA procedure. Only when the outputs of the two models are consistent with each other can they be safely used for further analysis, for example in a cost-bene t analysis. The problem has been described in the literature as DTA-based demand calibration, under the assumption of holding driving behavior parameters constant and adjusting only the O-D matrix and the assignment output. Dierent categories of studies are DTA-based supply calibration, in which driving behavior parameters such as speed-density models are calibrated and eorts of joint demand and supply calibration, either iterative or simultaneous (Omrani and Kattan, 2012).
36
CHAPTER 2. LITERATURE REVIEW Returning to the eld of focus, the inconsistencies between the estimated O-D matrix and the ows from the assignment process can be addressed in a number of ways. In essence, it can be formulated as an algorithm of bi-level optimization. Traditionally, the upper level concerns the O-D matrix estimation under known ows and the lower level concerns the assignment step, given the obtained O-D matrix from the upper level.
Finally, due to the increasing need to implement dynamic
processes in complicated urban networks, challenges such as the size issue and the multiple vehicle classes need to be addressed, leading to studies that propose various methods to overcome computational requirements (Verbas et al., 2011). Although these eorts have been extremely forward-thinking and progressed quickly, despite the limited amount of literature in the subject, they all mostly treat the inconsistency issues from a short time-horizon perspective, i.e. an operational planning mindset. This is why most studies use sensor data or other real-time tra c measurements to obtain ow counts. It is therefore also useful to see if a long-term, strategic planning can also deal with the coupling issues that arise from dynamic procedures.
In other words, it would be interesting to look at the problem from
a transport planner's view in contrast to the data scientist's view that has been adopted so far.
2.7.4
Departure Time Model and Importance for DTA
The O-D matrix that is used as input in a dynamic tra c assignment model is time-sliced, i.e. there is a dierent O-D matrix per time interval. These intervals are typically set to 15 minutes (Camus et al., 1997). The partial demand in each period can be obtained in a number of ways. A simple method is to weight the matrix according to the shares of dierent purpose trips occurring in each period.
The weights should actually vary among
O-D pairs, with respect to the variation of departure times in dierent areas. The inability of this approach to cover possible future changes in these shares because of imposed measures makes the inclusion of some kind of departure choice model almost necessary (Almroth et al., 2014). Adding a level in the demand model nested logit in which broad time intervals will be chosen upon can be adequate in some cases.
For example, if the choices
are the peak and o-peak periods and the target is to estimate the departure times of non-binding trips (social, shopping etc), the changes will be re ected to some extent. Problems occur when the peak spreading eect of the congestion is relevant (Almroth et al., 2014). The only remaining solution is to implement a departure choice model with short time intervals. There have been formulation proposals of a continuous choice model as well (Bellei et al., 2006). The correlation between consecutive time intervals deems the multinomial logit model generally not ideal for the departure time model. Two alternative model definitions are the error component (mixed) logit and the Order Generalized Extreme Value models, which can accommodate correlation and heteroscedasticity between alternatives (Börjesson, 2006). The model variables often include the preferred arrival or departure times, usually with a penalty for schedule deviation. These can be set as mixed variables, in order to capture heterogeneity among travelers.
A
simpli ed variation is to separate them in population segments, for example based
37
Estimation of a demand model based on data from a dynamic tra c assignment model on their value of time and exibility (Kristoersson, 2007).
A very recent piece
of work developed the departure choice model as part of the trip distribution step using arrival time preferences as input. The advantage of this method is that the data is easier to collect and the demand pro les vary per O-D pair, although not per traveler class (Levin et al., 2016).
38
Chapter 3 Data
3.1 Travel Habit Survey An existing survey is used for the estimation of the demand model as well as the departure time model.
The survey includes data from the period 2005-2006 and
concerns trips made within Stockholm county (both origin and destination). Tripspeci c and traveler-speci c data are available for a total of 30,054 such trips. Selections from this initial database, related to the trip purpose or individual characteristics are made in various stages of the thesis. The same survey has been used in the past to estimate the demand model that is to be compared to the estimated model of this work.
3.2 A Priori Demand Matrix The trip matrices that are used originate from the travel habit survey and SamPers. Three groups of matrices are inserted, one for the work trips that consists of X trips, one for other trips that consists of Y trips and nally an additional matrix with Z trips that accounts for tra c from delivery trucks, taxis etc. These are later split into time-dependent matrices in dierent ways. A departure time model is applied for the work trips matrix, hourly factors are multiplied with the other trips matrix and combinations of high and low tra c volumes are used for the additional tra c.
3.3 Network A network covering the region of Stockholm county is used in this thesis.
The
creation of the network is part of another project that includes a mesoscopic coding of the region.
18,321 links and 15,040 nodes constitute the network, in varying
degrees of detail, mostly depending on their centrality.
The area de ned as city
centre (having Nacka, Farsta, Täby and Jakobsberg as borders) is coded in higher detail.
39
Chapter 4 Methodology
4.1 Model Formulation A demand model aims to be a good predictor of the number and characteristics of the trips made in the network. It consists of a group of models which deal with a dierent aspect of the trips. Structures can vary, however the 4-step demand model is usually the basis of most applications, often enhanced with additional side-models. Destination and mode choice are modelled simultaneously in a nested logit model with two levels in this work. This is common practice, as mentioned in the literature study chapter, since this step is assumed to be a bi-dimensional choice. The combined alternatives are correlated, since the choices may include dierent destinations by the same mode or the same destination by dierent modes.
However,
taste variation and heteroscedasticity may not necessarily pose signi cant problems, which is the reason a nested logit model is suitable. The correlation is captured by the unobserved part of the utility in nested logit models, with the distribution of the error terms re ecting the correlation patterns. Alternatives within the nest have correlated errors, whereas alternatives in dierent nests do not. The joint probability of choosing a mode
m
and destination
j
exp(Vm|j ) exp(Vj ) ·P Pm,j = Pj · Pm|j = P k exp(Vk ) m exp(Vm|j )
is:
(4.1)
and consists of the probability of choosing a destination and the probability of choosing a mode given the destination choice. The sum of the mode utilities in a nest expresses the expected maximum utility of all the alternatives of the nest and can be formulated as:
Lj = φj · ln
X
exp(Vm|j )
(4.2)
m∈Mj with
φj
being the log-sum parameter of nest j.
Each lower nest is represented by a composite alternative that is competitive to the other nests and inserted in the upper choice (destination) through its composite utility. The deterministic part of the utility of each destination nest is expressed in the following way:
Vj = V˜j + Lj = V˜j + φj · ln
X m∈Mj
41
exp(Vm|j )
(4.3)
Estimation of a demand model based on data from a dynamic tra c assignment model where
V˜j
is the destination-speci c part of the utility added to the expected max-
imum utility of the destination nest.
Note that the logsum parameter is only a
multiplier of the logsum and does not divide the utility of each mode as the following expression, often used in the literature, would suggest:
Vj = V˜j + φj · ln
X
exp(Vm/φj )
(4.4)
m As a result, the units between nest levels are not consistent, as the scaling does not cancel out. The advantages of this formulation are mostly simplicity in programming but also exibility in scale variability, if that is required in some applications. The conditional probability of choosing a mode alternative speci c destination
j
m
to travel to a
is:
exp(Vm|j ) Pm|j = P m exp(Vm|j ) The mode-speci c part of the utility,
Vm|j ,
(4.5)
consists of the same variables in every
nest and only diers in value in the application of this work.
4.2 Model Structure The structure of the model is initially assumed to be identical to a pre-existing model, which is part of on-going work from other researchers and will be used later for results comparison. The structure of the model is as follows and is summed up in Figure 4.1.
An upper nest of 1240 destination alternatives, each representing a zone in the greater Stockholm network
1240 lower nests, each comprising of ve mode alternatives (car driver, car passenger, public transport, walk and bike)
A total of 6200 alternatives of combined destination and mode
Figure 4.1 The structure of the pre-existing destination-mode model which is used
as the basis of the estimated model 42
CHAPTER 4. METHODOLOGY The utility functions of the mode choice level for individual tination
j
n
travelling to des-
are:
Vcar_dr|jn = − 0.02863 ∗ carcost − 0.2048 ∗ ln(0.01 + carcost) − 0.03965 ∗ cartime + 3.471 ∗ hh_car − 1.029 ∗ carcomp − 1.167 ∗ woman − 0.009 ∗ density Vcar_pass|jn = − 0.4465 − 0.02863 ∗ carcost − 0.2048 ∗ ln(0.01 + carcost) − 0.06281 ∗ cartime VP T |jn = + 4.507 − 0.02863 ∗ P T cost − 0.2048 ∗ ln(0.01 + P T cost) − 0.04361 ∗ P T Invtime − 0.04637 ∗ P T Auxtime − 0.3062 ∗ P T N oBoard − 0.03585 ∗ P T F stW ait Vwalk|jn = + 3.096 − 0.4105 ∗ Dist − 0.4348 ∗ ln(0.01 + Dist) − 0.06126 ∗ Dist_Diag − 0.9311 ∗ V illa Vcycle|jn = + 0.3829 − 0.2333 ∗ Dist − 0.2397 ∗ ln(0.01 + Dist) − 1.570 ∗ W inter − 0.4593 ∗ woman − 0.0024 ∗ density where hh_car
whether a car exists in the household (0 if not, 1 otherwise)
carcomp
car competition in the household (0 if not, car licenses/cars otherwise)
PTInvtime
in-vehicle time in public transport
PTAuxtime
time between origin / destination and public transport
PTNoBoard number of boardings PTFstWait
waiting time for the rst boarding
Dist
distance between origin and destination
Dist_Diag
distance that is used for trips within the same zone (matrix diagonal)
Villa
dummy variable (1 if household is detached, 0 otherwise)
Winter
dummy variable (1 if the trip occured in weeks 1-12 or 49-52, 0 otherwise)
density
zone-related variable that represents the density of points of interest in the destination zone
The utility functions of the destination choice level are:
V˜j|n = log (ZoneEmpl − 50 ∗ T est(zone)) where ZoneEmpl number of employment points in the destination zone Test(zone) variable used for destination zones without employment points
43
Estimation of a demand model based on data from a dynamic tra c assignment model The logsum parameter
φ
was estimated equal to 0.5876 and common for all the
nests. The estimated model in this work may eventually adopt a dierent structure, depending on the value of the estimated log-sum parameter, otherwise called the structural or tree coe cient. Consistency to utility theory demands that this parameter lies in the interval
(0, 1], with φ = 1 collapsing the nested logit to a multinomial
logit model. The structure of the tree is responsible for values outside this range. In general, alternatives with greater cross-elasticity should place the respective choice in the lower nest. Failing to account for this will normally result in
φ ≤ 0.
If this is
the case in the estimation carried out in this thesis, the structure will be reversed, with mode lying in the upper nest instead.
0 < φ ≤ 1, it may with φ < 0, an increase
To better understand the theoretical need of the condition be useful to refer to equation (4.2). It is easy to see that
in the utility of an alternative within a nest would decrease the overall probability of choosing the nest, while with
φ = 0,
the nest probability would be independent
of the utilities of the alternatives in the nest.
On the other hand, if
φ > 1,
the
respective increase of the nest probability would also increase the probabilities of the other alternatives in the nest, which are directly competitive to the alternative of the initially increased utility. For some values of the explanatory variables, the model is not consistent with utility maximization behaviour in this case (Train, 2003).
4.3 Model Inputs 4.3.1
Dynamic Tra c Assignment
The dierence of the model in this work compared to the model estimated in previous work is that the network levels of service, i.e. the travel times between origindestination pairs, will be provided by a dynamic assignment routine and therefore will vary per time interval. Assignment will be performed with the software TransModeler, which allows for a hybrid (mesoscopic/macroscopic) dynamic simulation. As mentioned by most researchers in the eld of Dynamic Tra c Assignment, the two important inputs for this routine are the underlying network on which the assignment is performed as well as a time-varying demand matrix which accurately represents the temporal distribution of the trips in the modelled area. The following sections describe the steps to ensure that these inputs are of high quality and will produce realistic and accurate results.
4.3.2
Network Stockholms län ),
The network covers the whole region of Stockholm County ( 2 area of more than 6,500 km and where more than 2 million people live.
an
In the
model, the road network is represented by 18,321 links and 15,040 nodes (Figure 4.2). The detail is higher in the centre of the region, where most of the population and centroids are located in reality, however it is only minor roads that have been excluded in the outer regions. The simulator can be given instructions on which level of delity (microscopic, mesoscopic or macroscopic) to use. Only mesoscopic delity is used in the links of
44
CHAPTER 4. METHODOLOGY
Figure 4.2 An image of the complete network. The area within the red frame has
been modelled in higher detail than the rest of the map.
45
Estimation of a demand model based on data from a dynamic tra c assignment model 120
0, 32
30
9, 32
25
80
Speed (km/h)
Speed (km/h)
100
35
0, 108 8, 108
25, 69 60 40
20
46, 17 15 10
20
5
130, 0
130, 0 0
0 0
50
100
150
0
Density (veh/km)
(a)
Motorway with speed limit of 110
50
100
150
Density (veh/km)
km/h
and 3 lanes
(b)
Urban road with speed limit of 30
km/h
and 1 lane
Figure 4.3 Speed-Density diagrams used in two dierent road classes
the network. In links with mesoscopic simulation, individual vehicles are grouped into platoons, also called tra c streams. The behaviour of each platoon is derived from the link performance function of the link the platoon is moving on. The simulator uses speed-density functions to de ne link performance. It is the speed and density of the platoon that determines the vehicle individual behaviour and not acceleration or lane changing models. Two important variables that are associated with the speed-density relationship are jam density and minimum speed.
Jam density is the density at which tra c
ceases to move and both speed and ow equal zero. Minimum speed is used as the minimum of the curve and is dierent from zero for practical reasons; for example so that travel times are not calculated as in nite for the period when tra c at a link is stopped. Various pre-loaded functions are available to use or manual functions can be constructed. Functions can be categorised per road class to account for class-speci c eects. All the functions are piecewise linear and consist of three linear segments of dierent gradient. The segments correspond to free ow, transition and congested conditions, with negative slopes, as speed decreases with higher road density. The end points of the function represent the maximum speed and jam density conditions respectively. The two midpoints that de ne the intermediate segments dier according to the road type, speed limit and number of lanes. They determine the density values at which conditions change. The functions were inferred by the speed- ow relationships that are recommended by the Swedish Tra c Agency (Tra kverket, 2017).
They are separated
per road type (motorway, urban and rural roads), speed limit and number of lanes. Some examples can be seen in Figure 4.3 and a full table can be found in the Appendix. Although the pre-existing work of a mesoscopic network had been extensive, various modi cations were made upon it to further improve the network reliability and to suit the purpose of this master thesis. The road classes were thoroughly checked and applied to the whole network, including motorways, as they are currently de-
Tra kverket -
ned in the Swedish Tra c Agency online national road database (
Nationell vägdatabas,
n.d.).
The same source provided speed limit data for those
links that lacked this attribute.
Roundabouts were also checked outside the cen-
46
CHAPTER 4. METHODOLOGY
Figure 4.4 A screenshot of a roundabout (Ekelundsvägen, Solna). Entrance nodes
use a Capacity discharge model (red) whereas exit nodes do not use a discharge model (green).
tral area and corrected or re-done where required. Links in roundabouts were also speci ed as a separate road class, with their own speed-density function. A signi cant attribute that needed special consideration was the discharge model to be used for the network nodes, which determines the delay calculation at intersections. The software allows for three basic options, namely
None.
Capacity, Delay
and
If the capacity model is used, then the delay would be regulated in a way that
vehicle departure headways are not less than headways at capacity which would be unrealistic. The delay model calculates delay at all intersections according to suggested formulas (HCM, 2010).The third option does not specify any particular rules for the discharge model. According to the software manual, urban congested networks should not be combined with a delay calculation model, as this would lead to excessive delays and an overrepresentation of congestion. Following this instruction, the option used as default in the network.
None
was
An exception was made in roundabouts, where
the software recommends the use of the
Capacity
functions for the entrance nodes
(Figure 4.4). The simulator was set to stop when the sum of the relative gaps in all intervals (Equation 4.6) between the current solution and the equilibrium is lower than 0.001 or when 100 iterations have been run without reaching the threshold. Relative gaps can be also observed during the simulation after every iteration, both in table form and graphically. The equilibrium is de ned using the shortest paths for every OD pair.
Since reaching the equilibrium has not been proven possible and due to
the complexity of the problem, it was not expected to surpass the threshold and the procedure would be stopped when the relative gaps appeared stable for a few iterations. Paths were re-assigned for every trip that lasted at least 5% longer than the expected travel time.
Initial route choice calculations are based on free ow
travel times and the output travel times are averaged between iterations according to the Method of Successive Averages. The updated travel times are then used in
47
Estimation of a demand model based on data from a dynamic tra c assignment model the next iteration until convergence.
Gapτ =
T otal Experienced T ravel T imeτ − T otal Expected T ravel T imeτ T otal Expected T ravel T imeτ (4.6)
with
T otal Experienced T ravel T imeτ =
XX
fkτ tτk
i∈I k∈K τ
T otal Expected T ravel T ime =
X
dτi tτmin,i
i∈I where
Gapτ
Relative gap in interval
I
Set of all O-D pairs
Ki
Set of used paths for O-D pair
fkτ
Path ow for path
tτk
Travel time in path
dτi
Demand departing in time interval
tτmin,i
Travel time on shortest path for O-D pair
τ
i
k
i
in time interval
k
τ
in time interval
τ
τ i
in time interval
τ
(Caliper, 2014) The simulations proved to be very time-demanding, not as much because of the running times, but mainly due to the gridlock or near-gridlock conditions that arose in every iteration. Vehicles that could not make their intended turning movement blocked the movements of other vehicles, leading to large parts of tra c staying completely still. Central parts of the network were very sensitive to small changes in volume and due to the realistic representation of queue spillback, the oversaturated situation exteded quickly to neighbour links in both directions. The reasons for this eect can be numerous: small coding mistakes in critical locations, a broad selection of routing and other settings the result of which was not always clear and the large volumes and long analysis period which both make convergence di cult to reach. All these problems were treated in a long series of tests as suggested by the literature, for example with a
at
matrix, i.e. with an
equal split of demand over the period, lower volumes, isolation of possible problem locations and extremely long cool-down periods with zero incoming vehicles to allow the gridlock to solve itself (Chiu et al., 2011). Despite the heavy work in testing and improving the network, it was impossible to reach a situation without gridlock in the central part of the network and queues in far too many links, that would not dissolve even after 6 hours of cooldown periods. According to literature regarding dynamic tra c assignment, it is not advisable to use results of a process that includes gridlock, because there are many links with no ow and in nite travel times (Sloboden et al., 2012). Furthermore, the gridlocked
48
CHAPTER 4. METHODOLOGY links disallowed all trips that would originate from them, resulting in a large number of queued trips and a low ratio of completed trips. This aects the validity of the results and further delays any eorts for convergence.
The primary target of the
qualitative analysis of the results is to discover a stable solution, free of gridlock and with familiar congestion patterns (Chiu et al., 2011). The solution that is deemed appropriate for the current purpose calibrates the demand matrix so that the trips starting from and/or ending in the city centre are reduced by 50%. The departure time distribution of work trips was also adjusted in the high tra c periods, as will be shown in the next section. A cooldown period of 4 hours with no additional tra c was used. This combination of volume and departure pro le avoids long periods of gridlock, yet causes heavy congestion in critical locations and therefore can generally be considered realistic. Another assignment was also run with a further 20% reduction in the trip matrices overall as well as a further 50% reduction in trips originating from the regions of Nacka and Värmdö which were deemed critical for creating gridlock. The number of simulated trips in each case is shown in Table 4.1.
Table 4.1 Number of trips included in the two assignment simulations. Purpose
Assignment 1
Assignment 2
Additional
197 188 73 283 125 852
153 597 57 328 98 639
Total
396 323
309 564
Work Other
4.3.3
Time-Slicing
Generating the set of time-varying demand matrices from a matrix including the total trips in the analysis period is known by the term time-slicing.
Methods to
perform this procedure have been described in the literature review. As mentioned there, it is vital to include some kind of departure time choice model when dealing with a dynamic assignment process, instead of relying on common weight factors for each period. Short time intervals or continuous choice approaches can be tested and various types of models are deemed more or less suitable. In this thesis, the departure time choice model will be mainly used for acquiring a reliable time-sliced matrix and to later implement in a demand estimation process as an additional level of choice. The model is applied to split the work trips matrix, that constitutes the largest percentage of the total trips. For the purposes of this work it is considered suitable to use a Multinomial Logit model, despite the weakness mentioned earlier regarding the inherent correlation between the alternatives. This has been partially treated with calibration methods, as discussed later. A continuous approach would lead to unnecessarily high computational burden and instead, the common practice of 15-minute intervals is applied. The departure model that is implemented in this work is based on unpublished work which was used after the IHOP project and is available on request. Data from travel habit surveys were used to estimate a departure time model for the morning
49
Estimation of a demand model based on data from a dynamic tra c assignment model work trips (4:00 to 11:00) in the greater Stockholm area, the administrative area. A Multinomial Logit model with 28 alternatives was used, one for each quarter in the analysis period. High income, gender and distance to workplace were identi ed as signi cant parameters for the for the departure time choice and were inserted in suitable intervals. An additional dummy parameter was inserted for the departure intervals that started at a whole or half hour, in order to account for unintentional rounding by the responders and avoid a pattern of waves in the network loading phase. Using this reasoning as a basis, a new departure time model has been estimated in this thesis. Data from the 2005-2006 travel habit survey have been used. They include responses from around 30,000 individuals. First, an initial selection has been made from the data. Only work trips starting from home in the area of Stockholm have been used for the estimation.
A second work trip departing from home per
individual has been a frequent occurrence, as some respondents return home for their lunch break and then travel back to work. Those trips have been cleared o the database, leaving only one trip per individual.
Finally, trips made with any
other mode than car, either as driver or passenger, have not been considered. Trip origins are available at the level of tra c analysis zones, therefore the model can be estimated with a higher precision. Distances of each zone to the center was extracted from a shape le and matched with the individual trips. To account for a socio-economic characteristic as well, the share of high incomes per zone was also calculated. In zones with no available data regarding this question, the overall share was assumed to stand.
Since, as described later, income was
incorporated as a dummy variable (higher than 300,000 SEK/annum or not), this simpli cation is not so important, which becomes even clearer from the number of no-data zones. The classes in the data included a class for incomes 280,000-320,000. Higher income share for a zone is calculated as the share of respondents that belong in a higher class added to half the share of that class. Average income was avoided as a variable, because the highest class was open-ended, leaving room for possible misrepresentation of the real average income in the survey. These two variables are the ones that make the result origin-related, i.e. departure time distribution pro le is dependent on the origin zone. This is an improvement from the previous method of constant factors used in IHOP. The model is multinomial logit with 28 discrete alternatives representing the consecutive 15-minute intervals in the period 04:00-11:00. Each interval has a span of 15 minutes, for example the rst interval includes all trips starting at any point in time between 04:00 and 04:14. The last interval also includes trips starting exactly at 11:00. The model consists of the following variables:
20 alternative speci c constants for the alternatives that represent departure times from 5:00 to 9:59. Normalisation is made towards the rst alternative and the alternatives without ASC did not exhibit large dierences in total departures from the rst alternative.
A common variable for alternatives 2-4 (i.e. departure time between 04:1504:59).
A dummy variable for the alternatives that represent trips starting in intervals of whole or half hours, starting from 06:30 (alternative 11) and ending at 10:44
50
CHAPTER 4. METHODOLOGY (alternative 27). Respondents tend to round up their departure time in whole or half hours, which is obvious from the histogram of the data in Figure 4.5.
A dummy variable for gender in alternatives 13-24 (i.e. 07:00-9:59). This is based on the assumption that women tend to depart later for their work trips for reasons such as the caretaking and preparation of children to school and an observed tendency to work closer to home.
A dummy variable for high income, de ned as higher than 300,000 SEK per annum, in alternatives 12-19 (i.e. 06:45-08:44). Higher income individuals are associated with professional roles that demand commuting during peak hour to reach the workplace at a suitable time.
A variable for distance between origin and the city centre in alternatives 211 (i.e. 04:15-06:44).
Longer distance trips, which are assumed to be well
represented by this variable, are expected to start earlier in the day. Various speci cations with dierent positioning and transformations of the variables and data were tested. Eventually, the model that proved to be the best t for the data used a logarithm transformation for the distance variable (as opposed to a square root or no transformation).
51
Estimation of a demand model based on data from a dynamic tra c assignment model The utility functions for the time intervals are as follows:
V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12 V13 V14 V15 V16 V17 V18 V19 V20 V21 V22 V23 V24 V25 V26 V27 V28
=0 = − 1.5740 + 0.7997 ∗ ln dist = − 1.5740 + 0.7997 ∗ ln dist = − 1.5740 + 0.7997 ∗ ln dist = − 0.6575 + 0.7997 ∗ ln dist = − 0.6575 + 0.7997 ∗ ln dist = − 0.1875 + 0.7997 ∗ ln dist = − 0.1875 + 0.7997 ∗ ln dist = − 0.3721 + 0.7997 ∗ ln dist = + 0.4072 + 0.7997 ∗ ln dist = + 0.2226 + 0.7997 ∗ ln dist + k00 = + 2.3700 + 0.8753 ∗ high_inc = + 1.5470 + 0.8753 ∗ high_inc + 1.314 ∗ wom + k00 = + 2.0690 + 0.8753 ∗ high_inc + 1.314 ∗ wom = + 2.1550 + 0.8753 ∗ high_inc + 1.314 ∗ wom + k00 = + 2.3700 + 0.8753 ∗ high_inc + 1.314 ∗ wom = + 1.6100 + 0.8753 ∗ high_inc + 1.314 ∗ wom + k00 = + 1.3250 + 0.8753 ∗ high_inc + 1.314 ∗ wom = + 0.9914 + 0.8753 ∗ high_inc + 1.314 ∗ wom + k00 = + 1.8130 + 1.314 ∗ wom = + 0.7731 + 1.314 ∗ wom + k00 = − 0.1796 + 1.314 ∗ wom = + 0.4164 + 1.314 ∗ wom + k00 = + 0.3313 + 1.314 ∗ wom = + k00 =0 = + k00 =0
where dist
the origin-to-city centre distance
high_inc dummy variable for average zone income
≥ 300, 000
SEK/annum
wom
dummy variable for gender (1 for female, 0 for male)
k00
parameter for whole/half hours (=
0.3622
after calibration)
As shown in Table 4.2, all parameters bar some alternative speci c constants are signi cant at the 95% con dence level, with the t-values of income, gender and distance parameters exceeding the signi cance limit comfortably. The signs of these variables are sensible and in accordance with the assumptions described above. Note
52
CHAPTER 4. METHODOLOGY Table 4.2 Results of the estimation of the departure time model. Alternative
speci c constants are referenced to the respective quarter. ASC 2 was used in quarters 2-4. ASC 14, ASC 15, ASC 16 and ASC 17 were slightly adjusted from these values to produce a more even distribution in the peak hour. Parameter
Value
t-Test Value
High Income
+0.8753
+5.0
Woman
+1.3140
+6.7
Distance
+0.7997
+6.1
Half/Whole Hour
+0.9445
ASC 2
−2.7 −2.7 −1.1 −1.1 −0.3 −0.3
ASC 9
−1.5740 −0.6575 −0.6575 −0.1875 −0.1875 −0.3721
ASC 10
+0.4072
+0.7
ASC 11
+0.2226
+0.5
ASC 12
+2.3700
+5.7
ASC 13
+1.5470
+4.7
ASC 14
+2.0690
+4.9
ASC 15
+2.1550
+6.8
ASC 16
+2.3700
+5.7
ASC 17
+1.6100
+4.9
ASC 18
+1.3250
+2.9
ASC 19
+0.9914
+2.8
ASC 20
+1.8130
+4.1
ASC 5 ASC 6 ASC 7 ASC 8
+0.7
ASC 21
+0.7731
+2.1
ASC 22
−0.1796
−0.3
ASC 23
+0.4164
+1.0
ASC 24
+0.3313
+0.6
that the distance variable was logarithmically transformed, hence the parameter means that for every 10% increase in the origin-centre distance, the utility of that
dist · ln(110/100) ' dist · 0.095 ' 0.076. above, the dummy variable k00 was inserted to
alternative increases by As mentioned
account for the
inherent tendency of the respondents to round their departure time to the nearest half/whole hour, which leads to misrepresentation of the even-numbered alternatives in the model that include departures in intervals XX:15-XX:29 and XX:45-XX:59. The parameter is signi cant and therefore remains in the model to t the data better, however this observed wave pattern is not desirable when actually performing a dynamic vehicle loading of the network, as the departure time pro le needs to be as realistic as possible. To handle this issue, a more balanced distribution is formed in the following steps:
The probabilities of each alternative are obtained from the model and for each interval the sum of its probability and the probabilities of its immediately
53
Estimation of a demand model based on data from a dynamic tra c assignment model preceding and immediately following interval is calculated. For the rst and the last interval only two addends are used.
This sum is divided by three (by two in the cases of the rst and last interval) and the result is the Balanced Probability.
c ∈ [0, 1] in the survey, a target probability is T arget probability = c · M odel probability + (1 − c) · Balanced probability . A con dence of 0.25 was selected. Using a con dence degree
calculated for each interval, with
Value of
k00
is calibrated and the model probabilities are recalculated until the
sum of squared errors between them and the target probabilities is minimized. The derived value for
k00 after this algorithm is 0.3622 and gives a more balanced
distribution. Finally, the distribution in the peak hour (07:15-08:15) is further adjusted, by lowering the alternative speci c constants of the intervals 15 and 16 by 0.1 and simultaneously increasing those of the intervals 14 and 17 by 0.1. The algorithm for calibrating
k00
is repeated and its nal value is 0.2389. The nal distribution is
shown in Figure 4.6. The time-slicing of the matrices of the other trips is based on hourly factors and a combination of low and high tra c volumes. given in Appendix A.
54
The details of this procedure are
CHAPTER 4. METHODOLOGY
18.00% 16.00%
Percentage of Trips
14.00% 12.00% 10.00% 8.00%
6.00% 4.00% 2.00%
10:00
10:15
10:30
10:45
10:15
10:30
10:45
09:45
10:00
09:30
09:15
09:00
08:45
08:30
08:15
08:00
07:45
07:30
07:15
07:00
06:45
06:30
06:15
06:00
05:45
05:30
05:15
05:00
04:45
04:30
04:15
04:00
0.00%
Departure Time (Start of Interval)
Figure 4.5 Departure pro le according to the 2005-2006 survey
18.00% 16.00%
12.00% 10.00% 8.00% 6.00%
4.00% 2.00%
Departure Time (Start of Interval)
Figure 4.6 Departure pro le after balancing calibration
55
09:45
09:30
09:15
09:00
08:45
08:30
08:15
08:00
07:45
07:30
07:15
07:00
06:45
06:30
06:15
06:00
05:45
05:30
05:15
05:00
04:45
04:30
04:15
0.00% 04:00
Percentage of Trips
14.00%
Chapter 5 Results
5.1 Travel Time Data Selection The demand model estimations were made for various sets of travel time matrices, based on dierent assignment settings and result combinations. The use of the 15minute matrix results was found to be less realistic, due to the very high travel times observed in certain parts of the network. A comparative analysis with static results revealed that travel times after a certain quarter deviated signi cantly and unrealistically much from static results in a large number of zone-to-centre O-D pairs. That quarter depended on the tra c volume that was loaded in each assignment. Therefore, results from early quarters within the high tra c period were used for the later unrealistic quarters, whereas results from quarters before the high tra c period were also used for quarters after the high tra c period. The models that are presented in the next sections are based on travel time matrices originating from: A. An assignment with 50% fewer trips from, to and within the central zones 1. with travel times in 06:30-06:44 used for the whole period 2. with travel times in 06:00-06:14 used for o-peak (04:00-06:59 and 09:00-11:00) and travel times in 06:30-06:44 used for peak hour (07:00-08:59) B. An assignment with 50% fewer trips from, to and within the central zones, an additional reduction of 20% in the whole network and a 50% reduction of trips originating in the Nacka-Värmdö region 1. with travel times in 07:00-07:14 used for o-peak (04:00-06:59 and 09:00-11:00) and travel times in 08:00-08:14 used for peak hour (07:00-08:59) 2. with travel times in 07:15-07:29 used for o-peak (04:00-06:59 and 09:00-11:00) and travel times in 08:15-08:29 used for peak hour (07:00-08:59) 3. with travel times in 07:15-07:29 used for o-peak (04:00-06:59 and 09:00-11:00) and travel times in 08:00-08:14 used for peak hour (07:00-08:59) These models are to be referred to as models
A1, A2, B1, B2
and
B3
respectively,
while the initial model that was estimated on static travel times will be referred to as
Model 0.
57
Estimation of a demand model based on data from a dynamic tra c assignment model 70
60
60
50
Travel Time (min)
Travel Time (min)
50 Google-low 40
Google-high Dynamic1
30
Dynamic2 IHOP
20
40
Google-low Google-high
30
Dynamic1 Dynamic2
20
IHOP
Static
Static 10
Free flow
10
0
Free flow
0
Time (hh:mm)
(b)
Vällingby - Årstadal
60
60
50
50
40
Travel Time (min)
Travel Time (min)
(a)
Time (hh:mm)
Google-low Google-high
30
Dynamic1
Dynamic2 20
IHOP
Brommaplan - Centralen
40
Google-low
Google-high 30
Dynamic1 Dynamic2
20
IHOP Static
Static 10
10
Free flow
Free flow
0
0
Time (hh:mm)
Time (hh:mm)
(c)
(d)
Skarpnäck - Medborgarplatsen
Kungsholmen - Nybroplan
Figure 5.1 Comparison of dynamic assignments output with other sources.
5.2 Data Validation The decision on which quarters to use as representative of o-peak and peak times was based on a validation comparison with the travel time patterns that are available by the Google Maps Navigation function for some O-D pairs. The tool provides a low and high limit of travel time for every minute. Travel times at the start of the 15-minute intervals were collected. Free- ow times are also available from the same tool. Static data that were used in the initial model as well as data taken from a previous assignment in the IHOP project are also included for reference. Figure 5.1 shows that the quarters that were selected generally depict a tra c situation that is an average of the low and high limits oered by Google.
The
static model data generally follows the pattern of the low limit, while the IHOP travel times are near free- ow.
There is also a clear dierence between the two
assignments, however both seem to produce very high travel times after a certain quarter, due to probable gridlocks at some point in the path between the origin and destination. In the rst few quarters, travel times seem to be below free ow. These periods should be treated as warm-up periods.
58
CHAPTER 5. RESULTS
5.3 Model Structure The structure of the model, as mentioned in the Methodology section, would be bound to change from the comparison model if the logsum parameter was not in the interval
(0, 1].
This was not the case, therefore the structure remained identical.
The upper nest was the destination choice level (1240 alternatives) and the lower nest was the mode choice level (5 alternatives).
5.4 Model Parameters The same set of parameters in all utility functions as in Model 0 has been used. None of their values was constrained during the estimation. The most important parameter values for all models can be viewed in Tables 5.1 and 5.2.
Full model
speci cations are available in the Appendix.
Table 5.1 Estimation Results for Models A1 and A2, in which 50% of the trips
from, to and within the centre have been removed. Model 0
Parameter
Model A1
Model A2
Value
t-Score
Value
t-Score
Value
t-Score
Density (Dr.)
−0.02863 −0.2048 −0.03965 −0.06281 −0.04361 −0.009343
−12.4 −4.9 −10.0 −7.5 −14.6 −4.1
−0.03375 −0.2691 −0.05382 −0.08393 −0.04027 −0.01098
−15.3 −6.8 −8.0 −5.3 −14.2 −4.9
−0.03313 −0.2619 −0.06199 −0.1056 −0.04057 −0.01137
−14.9 −6.6 −8.3 −5.7 −14.3 −5.0
Logsum Parameter
+0.5876
+33.7
+0.6123
+36.0
+0.6099
+35.8
Cost
log(Cost) Tr. Time (Dr.) Tr. Time (Pass.) INV Time
1825 observations Log-Likelihood
R2
−12078.2
−12101.9
−12097.9
0.1968
0.1952
0.1955
59
Estimation of a demand model based on data from a dynamic tra c assignment model Table 5.2 Estimation Results for Models B1, B2 and B3, in which an additional 20% of the trips have been removed in the whole network and an additional 50% of trips from Nacka and Värmdö have been removed. Model B1 Parameter
Model B2
Model B3
Value
t-Score
Value
t-Score
Value
t-Score
Density (Dr.)
−0.03414 −0.2588 −0.0497 −0.09394 −0.03999 −0.01129
−15.9 −6.5 −8.7 −5.8 −14.2 −5.0
−0.0374 −0.26 −0.04275 −0.08317 −0.03985 −0.01102
−16.4 −6.6 −8.7 −5.8 −14.2 −4.9
−0.03364 −0.2582 −0.05053 −0.08965 −0.0403 −0.01081
−15.6 −6.5 −8.9 −5.9 −14.3 −4.8
Logsum Parameter
+0.6146
+36.2
+0.6159
+36.3
+0.6132
+36.1
Cost
log(Cost) Tr. Time (Dr.) Tr. Time (Pass.) INV Time
1825 observations Log-Likelihood 2
−12092.6
−12091.3
−12091.0
0.1958
0.1959
0.1959
R
5.5 Analysis 5.5.1
General Analysis
The models as described above did not perform better than the older model. In all of them, it appears that the travel time parameter signi cance has dropped, which leads to the conclusion that the used data was less reliable than the static travel times. Further tests are described later to identify and isolate the weaknesses of the new data and make eorts to achieve a better model. It is clear that the reduction of tra c volumes, which led to shorter gridlock periods in the dynamic assignment and less extreme results in the peak periods, produced travel time data that resulted in better models. Between
Model A2
Model A1
and
it appears that the inclusion of variation in travel times improves the
model. An important decision is to select a quarter that represents the situation of a longer period in the most accurate way.
This was not always an intuitive or
straightforward choice, since travel times were realistic for some O-D pairs in a certain quarter, but unrealistic for other ones. In the end, an exhaustive analysis was made among the most
likely
quarter combinations, given the overall patterns
in the travel time matrices. Not surprisingly, for the less-loaded network of the
B
models, later quarters proved to t the data better, even when peak hour quarters were used for o-peak periods. The opposite was true for
5.5.2
A models.
Travel Time Parameters
The travel time parameter was expected to be higher (lower in absolute terms) than the initial model's parameter, since the dynamic travel times should in general be higher during the congestion periods. This was indeed the case when peak quarters were used in estimations, however their inclusion resulted in far worse tting models and very low signi cance values for the time parameters. In the presented models,
60
CHAPTER 5. RESULTS the parameter is in fact lower than in
Model 0, however it behaves in accordance to
expectations, i.e. when later quarters from higher volumes are used, the parameter is even lower (for example if models
A2
and
B1
are compared). This does not hold
only between B1 and B3, where the earlier quarter in the o-peak period results in a higher parameter. The minimal dierence in value and the slight dierence in signi cance do not make the analysis of this deviation from the overall pattern very meaningful. The time parameter in the car passenger alternative followed its equivalent in the car driver alternative, both in values and signi cance. alternative was also not left intact.
The public transport
Small variations, generally greater with later
quarters included, were observed accompanied by a slight drop in signi cance as well.
5.5.3
Cost Parameters and Value of Time
All models demonstrated lower cost parameters than the model estimated on static travel times, combined with higher levels of signi cance. This can be explained by the value of time concept, i.e. that individuals will make a choice based on their valuation of time in monetary terms, which is intrinsic and should not be bound to change. Therefore, value of time should remain relatively constant per individual. In this case, with two cost parameters, where one is logarithmically transformed, the value of time is given by a function of cost.
V oT =
∂Vcar/∂β
time
∂Vcar/∂βcost
βtime
=
βcost +
(5.1)
βlog(cost) Cost
Indeed, for the estimated models with the best t, the value of time curve for car drivers appears the most similar to the initial model's (Figure 5.2). This can be another measure for the model's reliability, i.e. whether it is based on realistic values of time. There is a noticeable dierence between models
B1-B3
and
B2.
The
decrease in parameter values occured with dierent proportions for cost and time, which produced this result. Considering the value of time for public transport travelers, an important nding was made. The old model demonstrates a similar and slightly higher value of time for this alternative, which is not in accordance to other ndings (WSP, 2010). Instead, the new models consistently estimate a lower value of time for these commuters, largely at around the same values (Figure 5.3). In Table 5.3 all results are summarised and compared with the referenced study values. A possible explanation can be the non-inclusion of car parking costs in neither the demand model nor the assignment. Especially in the central zones, parking costs can be a signi cant portion of the overall trip costs and this leads to the car costs being underestimated, the cost parameter having a lower (higher absolute) value than it should and the value of time for car drivers being lower than in reality. Another reason may be the non-dynamic nature of public transport travel times, which assumes that the transit system works without issues even at peak hour.
5.5.4
Density and Congestion
Because of the fact that static travel times underestimated the real travel times in congested areas,
density
was used in
Model 0 61
to capture the eect of congestion. It
Estimation of a demand model based on data from a dynamic tra c assignment model 120 Model A2
Value of Time (SEK/h)
100 Model A1 Model B3 Model B1 Model 0
80
Model B2 60
40
Model 0 Model A1 20
Model A2 Model B1 Model B2
0
0
20
40
60
80
100
120
140
160
180
200
Model B3
Cost (SEK)
Figure 5.2 Value of Time curves for car drivers. Models with the best t produce
a curve similar to the initial model.
Table 5.3 Detailed Value of Time results for all models and a survey for
comparison. The values refer to an individual with cost of travelling equal to 100 Swedish crowns. The study concerns train trips. For bus trips the value is 52 SEK/h. (in SEK/h) VoT Car Driver VoT Car Pass. VoT PT PT/Driver Ratio
0
A1
A2
B1
B2
B3
Study 136.0
77.5
88.6
104.0
81.2
68.7
83.7
122.8
138.2
177.2
153.5
133.6
148.5
85.3
66.3
68.1
65.3
64.0
66.8
1.09
0.74
0.65
0.80
0.93
0.80
68.0 0.50
was calculated as the sum of population and employment points in a zone divided by its area. This eect should be partially or wholly covered by the use of dynamic assignment and the better representation of congestion in this procedure. The estimated models show that
density remained a signi cant parameter with similar value,
therefore congestion was not captured. Even quarters with higher travel times did not manage to achieve an important decrease in the
5.5.5
density
value or signi cance.
Nest Signi cance
The results showed a slight increase in the logsum parameter value. This increase generally follows the pattern of change of the travel parameter within the newly estimated models, i.e. the higher the travel time parameter the higher the logsum parameter. There is no clear indication that this is necessarily meaningful, since the dierences are rather small. If any meaning is to be assigned to it, it seems that the nested structure slightly weakens with the increase in the travel time parameter. This would hint that an individual would be inclined to change destination as opposed to only mode more easily than the previous model suggests.
62
CHAPTER 5. RESULTS 100 90
Model 0
Value of Time (SEK/h)
80
Model A2 Model B3
70
Model A1
60
Model B1 Model B2
50 40
30
Model 0 Model A1
20
Model A2 10
Model B1 Model B2
0 0
20
40
60
80
100
120
140
160
180
200
Model B3
Cost (SEK)
Figure 5.3 Value of Time curves for public transport users. The new models
showcase lower values than the static data model.
5.6 Deeper Investigation In the following section, the best model was analysed further with regards to partial tests that include only part of the data and in an attempt to identify the reasons that the dynamic data did not help in estimating a better model. A successful eort to combine dynamic and static data was also made.
Peak / O-Peak analysis The speci cations of the initial model and of the one that performed best according to the log-likelihood index among the ones with dynamic data were used to reestimate them in separate peak and o-peak samples. The results (Table 5.4) showed 2 that both models are better in predicting choices in peak hour, as the R index suggests.
B3
There is not much dierence in which model is used, i.e. both
0
and
are roughly equally better in peak hour than in o-peak. Concerning the car
travel time parameters, however, it is obvious that while in o-peak,
B3
0
shows a small increase
has a notable (and desirable) decrease. The travel time variation
between peak and o-peak in
B3
led to a correct behaviour of the parameter, i.e.
an inverse uctuation to the change in trip duration. The car passenger alternative also appears to be explained by travel time in a higher degree for the o-peak trips with model
0
(larger signi cance), whereas the dierence for
B3
is not that large.
Zonal analysis The models were also estimated separately for samples including only the central zones and the non-central ones. From the results in Table 5.5 dierences between the results in the two cases can be observed.
Travel time parameters for the car
alternatives are dissimilar in value and insigni cant when central zones are involved.
63
Estimation of a demand model based on data from a dynamic tra c assignment model Table 5.4 Estimation Results for Models 0 and B3 on samples containing only
individuals travelling in peak or o-peak times respectively. Peak (07:00-09:00)
O-Peak
1072 individuals Model 0 Parameter
753 individuals
Model B3
Model 0
Model B3
Value
t-Score
Value
t-Score
Value
t-Score
Value
t-Score
Density (Dr.)
−0.033 −0.143 −0.042 −0.046 −0.042 −0.007
−10 −2.4 −7.9 −3.7 −10 −2.4
−0.038 −0.201 −0.049 −0.074 −0.040 −0.008
−12 −3.6 −7.1 −3.4 −10 −3.0
−0.024 −0.266 −0.039 −0.074 −0.046 −0.012
−7.1 −4.3 −6.4 −6.4 −10 −3.3
−0.029 −0.327 −0.058 −0.104 −0.042 −0.014
−9.1 −5.6 −5.3 −4.7 −9.8 −3.7
Logsum
+0.607
+26
+0.631
+28
+0.557
+21
+0.581
+22
Cost
log(Cost) Tr.Time (Dr.) Tr.Time (Pass.) INV Time
LL
−7027.2
−7033.0
−5016.7
−5028.3
R2
0.2057
0.2051
0.1896
0.1877
This eect is even more intense with
B3, which in fact proved to be a slightly better
tting model in this sample. It is assumed that the high parking costs in the centre, which were not in fact modelled, may have accounted for this result. Furthermore, the passenger alternative travel time parameter passed the t-test for to do so for
0.
The positive sign and low value of
density
B3
but failed
is also worth noting; it
appears that the variable is less signi cant and the intuitively wrong sign might lead to the suggestion that it doesn't function as intended. When all trips that involve a central zone are taken away, the comparison between the two models gives similar results to the comparison with the full samples. The nesting parameter is of higher value, a sign that mode and destination choice are less connected for trips in or between suburbs. Additionally, the goodness of t is substantially better. It appears that central zones, supposedly with more congestion and larger within-day travel time variation are equally well (or rather, badly) modelled by both models.
Factors from dynamic data into static data In this last test, the following assumption was made. The dierence between peak and non-peak hours is better captured by the dynamic model, however the actual travel times are less accurate due to practical issues during assignment and result in a worse model.
Two sub-tests were performed.
In the rst one, for all the
individuals that travelled o-peak, their static travel times were multiplied by a factor, calculated according to Formula 5.2.
Factor
=
Travel time at o-peak (07:15-07:29) Travel time at peak hour (08:15-08:29)
(5.2)
This is interpreted as follows: The travel times used from the static procedures are assumed to be a good representation of peak hour. Therefore, travel times for the o-peak periods should be reduced by being multiplied to the above factor< Factors are OD-pair dependent.
64
1.
CHAPTER 5. RESULTS Table 5.5 Estimation Results for Models 0 and B3 on samples containing only
individuals travelling from and/or to a central zone and all others respectively. Centre
Non-Centre
799 individuals Model 0 Parameter
1026 individuals
Model B3
Model 0
Model B3
Value
t-Score
Value
t-Score
Value
t-Score
Value
t-Score
INV Time
−0.025 −0.253 −0.005 −0.025 −0.029
−6.4 −2.2 −0.9 −1.9 −7.4
−0.026 −0.266 −0.001 −0.058 −0.028
−6.8 −2.4 −0.1 −2.2 −7.5
Density (Dr.)
+0.012
+4.7
+0.012
+4.7
−0.024 −0.113 −0.047 −0.084 −0.046 −0.035
−7.9 −2.6 −9.6 −7.2 −11 −7.6
−0.031 −0.183 −0.061 −0.109 −0.043 −0.035
−12 −4.6 −8.8 −5.5 −11 −7.8
Logsum
+0.629
+22
+0.635
+23
+0.665
+27
+0.686
+28
Cost
log(Cost) Tr.Time (Dr.) Tr.Time (Pass.)
LL
−5425.8
−5424.8
−6480.5
−6490.8
R2
0.1783
0.1784
0.2317
0.2305
In the second test, the opposite (and more probable) assumption was made: The static travel times are a good representation of the o-peak periods. Therefore, high tra c times should be increased, by being multiplied to the inverse of the above factor. Models
C1
and
C2
can be reviewed in Table 5.6. Even though the assumptions
were polar, they both ended up with a better model than the ones estimated so far. This is explained by the more realistic values of the static travel times, a large part of which are used untouched in the tests (58.7% of the sample in the rst test and 41.3% in the second).
The second assumption, though, i.e. that the static travel
times are too low to represent peak hour, seems to be con rmed by the fact that a better model than model
0
has been estimated. The realism of the static travel
times was complemented by a (to a certain extent) realistic within-day travel time variation of a dynamic process leading to this positive result.
Table 5.6 Estimation Results for two models with an adjustment of the static
travel times in the o-peak and the peak period respectively. Model C1
Parameter
Model C2
Value
t-Score
Value
t-Score
Density (Dr.)
−0.03199 −0.2199 −0.03511 −0.06203 −0.04152 −0.01111
−14.4 −5.3 −9.1 −6.8 −14.4 −4.9
−0.03159 −0.2172 −0.02726 −0.04982 −0.04147 −0.009095
−14.5 −5.3 −10.2 −7.1 −14.4 −4.0
Logsum Parameter
+0.5981
+34.7
+0.6028
+35.2
Cost
log(Cost) Tr. Time (Dr.) Tr. Time (Pass.) INV Time
Log-Likelihood 2
R
−12088.4
−12072.965
0.1961
0.1971
The stability of the result is enhanced by the increased signi cance of both the
65
Estimation of a demand model based on data from a dynamic tra c assignment model cost and the time parameters, as well as by the behaviour of the time parameter, which decreases in absolute value, as was expected. The passenger and public transport alternatives demonstrate a slightly lower, yet still strong signi cance in their time parameters. Even
density
appears with a marginally lower value and signi -
cance. A complete parameter comparison between
0
and
C2
can be inspected in
Table 5.7. All parameters have been explained in detail in Section 4.2.
66
CHAPTER 5. RESULTS
Table 5.7 Comparison between the initial fully static data model 0 and the
best- tting estimated model of this work, C2. Model 0
Alternative
Parameter
Car
Linear Cost
Driver
Transport
Travel Time
−14.5 −5.3 −10.2
Car in household
+3.471
+12.0
+3.560
+12.1
Car competition
−1.029 −1.167 −0.009343
−7.3 −8.5 −4.1
−1.040 −1.150 −0.009095
−7.3 −8.3 −4.0
Travel Time
−0.4465 −0.02863 −0.2048 −0.06281
−1.1 −12.4 −4.9 −7.5
−0.2940 −0.03159 −0.2172 −0.04982
−0.7 −14.5 −5.3 −7.1
ASC
+4.507
+12.6
+4.630
+13.3
Linear Cost
−0.02863 −0.2048 −0.04361 −0.04637 −0.3062 −0.03585
−12.4 −4.9 −14.6 −7.3 −6.9 −4.3
−0.03159 −0.2172 −0.04147 −0.04490 −0.3054 −0.03150
−14.5 −5.3 −14.4 −7.1 −6.9 −3.8
log(Cost)
ASC Linear Cost
log(Cost) Inv Time Aux Time Boardings First Wait Walk
ASC
+3.096
+7.9
+3.214
+8.2
Linear Distance
Villa
−0.4105 −0.4348 −0.06126 −0.9311
−6.1 −4.7 −3.0 −3.3
−0.4078 −0.4260 −0.05879 −0.9031
−6.1 −4.7 −3.0 −3.2
ASC
+0.3829
+1.0
+0.5638
+1.4
log(Dist) Diag. Dist
Bike
t-Score
−0.03159 −0.2172 −0.02726
log(Cost) Public
Model C2 Value
−12.4 −4.9 −10.0
Density
Passenger
t-Score
−0.02863 −0.2048 −0.03965
Woman
Car
Value
Linear Distance
−0.2333
−9.6
−0.2282
−9.4
log(Dist)
+0.4037
+4.0
+0.3751
+3.8
Winter Woman
−1.57 −0.4593
−4.6 −1.9
−1.556 −0.4490
−4.6 −1.9
Density
+0.002397
+0.6
+0.002028
+0.5
Logsum
+0.5876
Log-Likelihood 2
+33.7
+0.6028
+35.2
−12078.181
−12072.965
0.1968
0.1971
R
67
Estimation of a demand model based on data from a dynamic tra c assignment model
5.7 Conclusions A demand model has been estimated based on travel time data derived from a dynamic assignment process with time-varying demand in a large urban/sub-urban network. The model was compared to an identically speci ed model that used static data and achieved similar likelihood and goodness of t. The conclusions that can be drawn from the complete span of this work are numerous. The demand model proved more sensitive to the new data than expected. Travel time parameters were mostly aected, by a degree that varied heavily according to dierent data combinations.
They dragged the cost time parameters towards the
same direction and the fact that those were not alternative-speci c aected the rest of the variables as well. A deeper analysis of the patterns in the model t revealed interesting observations.
The insertion of some degree of travel time variation between peak and
o-peak periods always had some positive result to the model.
A better t was
found for the non-central zones, no matter the travel time source. A combination of static travel times that were increased by OD-pair dependent factors in peak periods resulted in a better model than the initial one. The signi cance of a correct method to time-slice the demand matrix has been underlined by the results. The parameters and likelihood were aected by the choice of the implemented travel time intervals. After consideration, travel times in quarters that resembled similarities to the true times were used, even if they were not nominally suitable, for example a quarter in the peak hour for o-peak trips estimation. The departure time model that was estimated as part of the thesis and its calibrated results ensured that at least some early intervals of the high tra c period can be used with a certain degree of realism and produce a reasonable demand model. The dynamic assignment procedure was more sensitive to modelling details of the network and speed-density relationships than to changes in overall demand. It was only after a large overall demand reduction that the model managed to produce results that can be utilised and congestion patterns that did not end up in longlasting gridlock situations. Instead, excessive network modi cations and corrections as well as a focused demand reduction (for example in those zones where the gridlock used to start forming) proved to be much more meaningful and e cient techniques. Although it was not the main purpose of this thesis, it would have been a positive result to have estimated a better predictor than the existing model using only dynamic travel times. It would have not only con rmed that dynamic assignment provides more realistic network data, but it would have also provided grounds for deeper analysis and safer conclusions. However, the fact that the variation pattern arising from a far from perfect dynamic assignment procedure was applied to static travel times and achieved a better overall t is in itself an important development.
5.8 Discussion and Future Work With the growing urbanisation in most parts of the world and the ever-increasing need for e cient and reliable transportation networks, modelling densely populated areas with high congestion and within-day volume variation has become an imperative challenge for transport planners. The constantly changing network conditions
68
CHAPTER 5. RESULTS and users' dierent ways of responding to them call for dynamic processes in meeting this goal. This work has shown the issues that must be overcome when attempting to acquire dynamic network data through a dynamic tra c assignment. An accurate O-D matrix, both in terms of demand level as well as in its distribution over the day, a detailed network model and careful assignment parameter selection are all vital aspects of a reliable result. The requirements in time, computational power and judgment may exceed initial expectations.
Certain compromises might need
be made to achieve an applicable result, simultaneously ensuring that it represents reality to an acceptable extent. The output of this procedure has proved to be fruitful.
Despite several sim-
pli cations and only a slight insertion of within-day variation, equally good if not better than the pre-existing demand models can be estimated. Especially the last application that combined static and dynamic data appears promising for inspiring future work. The elements that additional attention should be drawn to are not few. First of all, a thorough examination of the complete network for modelling errors is important. The range of available settings oered by the software is large and each decision should be made with caution; the process has been very sensitive even to minor mishandlings. Dierent route choice methods that are more realistic than the stochastic shortest path can be experimented upon. A reliable method to time-slice the demand matrices is also required. Both realism, e.g. with a large sample survey, and applicability, e.g. with suitable adjustments should be taken into consideration. Results should be validated by being compared to known tra c conditions. Finally, during model estimation, it may be the case that a fully dynamic travel time matrix may result in a worse model, given the high sensitivity of the underlying assignment model. Choosing those intervals that are more representative may be a more successful technique. A conceptual point that should also be reviewed is the full integration of the departure time choice model in the destination-mode choice model.
It has been
assumed here that departure time is independent from the other two choices, but that is not necessarily the case.
Some of the proposed solutions in Section 2.7.4
can be implemented in the future. Another point of view may propose the dynamic modelling of public transport times, which would take into account the network weaknesses during peak hour. The subject of demand modelling with dynamic data is promising for the eld. Investigating ways to properly apply the results of a dynamic assignment may lead to signi cantly better demand models and consequently to further accomplishments in improving our transport networks.
69
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74
Appendix A Network
A.1 Road Classes Highly detailed speed-density relationships are vital for an accurate modelling of a mesoscopic network. The relationships that are used are those suggested by the Swedish National Road Database.
They are piecewise-linear functions consisting
of three sections that represent free ow, transition and congested conditions. The four de ning points for each of the 34 road classes is given in Tables A.1 and A.2. Slight variations of less important roads are not presented.
Speed (km/h)
0, Vmax Klow,Vhigh
Khigh, Vlow
Kmax, 0 Density (veh/km)
Figure A.1 General Diagram used in Speed-Density Relationships
75
Estimation of a demand model based on data from a dynamic tra c assignment model Table A.1 De nition of the critical points in the road class piecewise linear
functions for urban areas
Road
Speed Limit
Class
(km/h)
Motorway
70 80 90 100
Other
30
Lanes
Vmax
Vhigh
Klow
Vlow
Khigh
Kmax
(km/h)
(km/h)
(v/km)
(km/h)
(v/km)
(v/km)
2
77.0
77.0
11.7
52.4
35.1
130.0
3
77.0
77.0
11.7
52.4
32.5
130.0
2
85.0
85
10.4
57.0
35.1
130.0
3
85.0
85.0
10.8
57.0
31.5
130.0
2
91.5
91.5
9.9
60.0
33.7
130.0
3
91.5
91.5
10.0
60.0
30.5
130.0
2
100.5
100.5
9.1
65.3
31.7
130.0
3
100.5
100.5
9.1
65.3
28.5
130.0
1
27.0
27.0
9.1
6.8
91.0
130.0
2
31.0
31.0
14.3
14.0
65.0
130.0
40
1
37.0
37.0
9.1
29.0
31.5
130.0
2
41.0
41.0
14.2
32.0
33.1
130.0
50
1
43.0
43.0
7.8
33.0
27.7
130.0
2
48.0
48.0
13.0
17.8
66.3
130.0
1
56.5
56.5
6.0
36.0
29.4
130.0
2
58.0
58.0
11.4
40.0
32.5
130.0
1
67.0
67.0
6.5
13.4
79.3
130.0
2
67.0
67.0
10.4
25.5
55.9
130.0
60 70
Table A.2 De nition of the critical points in the road class piecewise linear
functions for rural areas
Road
Speed Limit
Class
(km/h)
Motorway
Vmax
Vhigh
Klow
Vlow
Khigh
Kmax
(km/h)
(km/h)
(v/km)
(km/h)
(v/km)
(v/km)
2
93.5
93.5
9.3
60.7
31.9
130.0
3
93.5
93.5
9.3
60.7
28.6
130.0
2
102.5
102.5
8.5
66.0
29.3
130.0
3
102.5
102.5
8.5
66.0
26.3
130.0
2
108.0
108.0
8.1
69.0
28.1
130.0
3
108.0
108.0
8.0
69.0
25.1
130.0
70
1
75.5
75.5
1.9
59.5
27.5
130.0
2
75.0
75.0
11.5
52.0
36.1
130.0
80
1
85.5
85.5
1.7
62.5
26.2
130.0
2
85.0
85.0
10.4
57.0
35.1
130.0
90
1
91.0
91.0
1.6
65.5
25.0
130.0
2
91.5
91.5
9.5
59.7
32.4
130.0
90 100 110
Other
100 110
Lanes
1
98.0
98.0
1.5
66.5
24.6
130.0
2
100.5
100.5
8.7
65.0
29.8
130.0
1
99.0
99.0
1.5
67.0
24.4
130.0
2
106.0
106.0
8.2
68.0
28.5
130.0
76
APPENDIX A. NETWORK
A.2 De nition of Central Zones
Figure A.2 Map of Stockholm with the zones that were treated as Central being
highlighted in red
77
Estimation of a demand model based on data from a dynamic tra c assignment model
A.3 Convergence Charts
Figure A.3 Convergence chart of the assignment process with a 50% decrease in
volume in the central zones.
Figure A.4 Convergence chart of the assignment process with a 50% decrease in
volume in the central zones, an additional 20% decrease in volume overall and an additional 50% of the trips originating in Nacka or Värmdö removed.
78
Appendix B Time-Slicing
B.1 Time-Slicing of Other Purpose Trips Hourly factors for the other purpose trips were generated according to the travel habit survey for a full 24-hour period (Table B.1).
Demand within the hour was
assumed to be equally split in the quarters, so one fourth of the demand of the respective full hour was used.
B.2 Time-Slicing of Additional Trips Two types of additional trip hourly matrices were generated from the Swedish national transport model, SamPers (a base prediction of April 2014), one for hightra c periods and one for low-tra c periods. The quarterly matrices were calculated as follows:
For quarters during 07:00-08:59, high tra c matrices were used, after being divided by 4.
For quarters during 06:00-06:59 and 09:00-09:59, the average of high and low tra c matrices was used, after being divided by 4.
For the rest of the period, low tra c matrices were used, after being divided by 4.
79
Estimation of a demand model based on data from a dynamic tra c assignment model
Time interval
Quarterly Factor
00:00-00:59
0.00000000
01:00-01:59
0.00000000
02:00-02:59
0.00039379
03:00-03:59
0.00000000
04:00-04:59
0.00000000
05:00-05:59
0.00109903
06:00-06:59
0.00431021
07:00-07:59
0.01089455
08:00-08:59
0.01845799
09:00-09:59
0.01986400
10:00-10:59
0.01871754
11:00-11:59
0.01934760
12:00-12:59
0.01199895
13:00-13:59
0.01282949
14:00-14:59
0.01904062
15:00-15:59
0.01540970
16:00-16:59
0.01925631
17:00-17:59
0.02510856
18:00-18:59
0.02248002
19:00-19:59
0.01677454
20:00-20:59
0.00877256
21:00-21:59
0.00275295
22:00-22:59
0.00146329
23:00-23:59
0.00102833
Table B.1 Quarterly factors used per hourly time interval to split the other
purpose trips into the 15-minute intervals
80
Appendix C Model Estimation
C.1 Complete Model Speci cations
81
Estimation of a demand model based on data from a dynamic tra c assignment model
Table C.1 Full speci cation of models A1 and A2 Model A1 Alternative
Parameter
Car
Linear Cost
Driver
log(Cost) Travel Time
−14.9 −6.6 −8.3
Car in household
+3.521
+12.0
+3.524
+12.0
Car competition
−1.029 −1.207 −0.01098
−7.2 −8.7 −4.9
−1.028 −1.192 −0.01137
−7.2 −8.6 −5.0
Travel Time
−0.824 −0.03375 −0.2691 −0.08393
−2.2 −15.3 −6.8 −5.3
−0.7465 −0.03313 −0.2619 −0.1059
−2.0 −14.9 −6.6 −5.7
ASC
+5.075
+14.5
+5.028
+14.3
Linear Cost
First Wait
−0.03375 −0.2691 −0.04027 −0.04428 −0.2912 −0.03149
−15.3 −6.8 −14.6 −7.1 −6.7 −3.8
−0.03313 −0.2619 −0.04057 −0.04442 −0.2932 −0.03201
−14.9 −6.6 −14.3 −7.1 −6.7 −3.9
ASC
+3.301
+8.5
+3.297
+8.4
Linear Distance
Villa
−0.4021 −0.4269 −0.05947 −0.8952
−6.0 −4.7 −2.9 −3.2
−0.4033 −0.4266 −0.05964 −0.8995
−6.0 −4.7 −3.0 −3.2
ASC
+0.6652
+1.7
+0.6501
+1.7
Linear Distance
−0.2238
−9.3
−0.2251
−9.3
ASC
Passenger
Linear Cost
log(Cost)
log(Cost) Inv Time Aux Time Boardings
log(Dist) Diag. Dist
Bike
t-Score
−0.03313 −0.2619 −0.06199
Car
Walk
Model A2 Value
−15.3 −6.8 −8.0
Density
Transport
t-Score
−0.03375 −0.2691 −0.05382
Woman
Public
Value
log(Dist)
+0.3636
+3.7
+0.3694
+3.7
Winter Woman
−1.573 −0.439
−4.6 −1.8
−1.577 −0.437
−4.7 −1.8
Density
+0.002204
+0.6
+0.002263
+0.6
Logsum
+0.6123
Log-Likelihood
+36.0
+0.6099
+35.8
−12101.9
−12097.9
0.1952
0.1955
R2
82
APPENDIX C. MODEL ESTIMATION
Table C.2 Full speci cations of models B1 and B2 Model B1 Alternative
Parameter
Travel Time
−16.4 −6.6 −8.7
Car in household
+3.563
+12.1
+3.568
+12.1
Car competition
−1.031 −1.169 −0.01129
−7.2 −8.4 −5.0
−1.030 −1.167 −0.01102
−7.2 −8.4 −4.9
Travel Time
−0.6241 −0.03414 −0.2588 −0.09394
−1.7 −15.9 −6.5 −5.8
−0.6085 −0.03474 −0.26 −0.08317
−1.7 −16.4 −6.6 −5.8
ASC
+5.099
+14.5
+5.117
+14.6
Linear Cost
First Wait
−0.03414 −0.2588 −0.03999 −0.04424 −0.2962 −0.03111
−15.9 −6.5 −14.2 −7.0 −6.8 −3.7
−0.03474 −0.26 −0.03985 −0.04414 −0.2950 −0.02981
−16.4 −6.6 −14.2 −7.0 −6.8 −3.6
ASC
+3.339
+8.5
+3.345
+8.6
Linear Distance
Villa
−0.4023 −0.4232 −0.05888 −0.8949
−6.0 −4.7 −2.9 −3.2
−0.4024 −0.4220 −0.05843 −0.8878
−6.0 −4.7 −2.9 −3.2
ASC
+0.7147
+1.8
+0.735
+1.9
Linear Distance
−0.224
−9.3
−0.2237
−9.3
Woman
Car
ASC
Passenger
Linear Cost
log(Cost)
log(Cost) Inv Time Aux Time Boardings
log(Dist) Diag. Dist
Bike
t-Score
−0.03474 −0.26 −0.04275
Density
Walk
Model B2 Value
−15.9 −6.5 −8.7
Linear Cost
log(Cost)
Transport
t-Score
−0.03414 −0.2588 −0.0497
Car Driver
Public
Value
log(Dist)
+0.3614
+3.7
+0.357
+3.6
Winter Woman
−1.567 −0.4386
−4.6 −1.8
−1.564 −0.4423
−4.6 −1.8
Density
+0.002239
+0.6
+0.002149
+0.6
Logsum
+0.6146
Log-Likelihood
+36.2
+0.6159
+36.3
−12092.6
−12091.3
0.1958
0.1959
R2
83
Estimation of a demand model based on data from a dynamic tra c assignment model
Table C.3 Full speci cations of models B3 and C1 Model B3 Alternative
Parameter
Car
Linear Cost
Driver
log(Cost) Travel Time
−14.4 −5.3 −9.1
Car in household
+3.558
+12.1
+3.519
+12.0
Car competition
−1.027 −1.176 −0.01081
−7.2 −8.5 −4.8
−1.037 −1.135 −0.01111
−7.2 −8.2 −4.9
Travel Time
−0.6675 −0.03364 −0.2582 −0.08965
−1.8 −15.6 −6.5 −5.9
−0.4220 −0.03199 −0.2199 −0.06203
−1.1 −14.4 −5.3 −6.8
ASC
+5.069
+14.5
+4.767
+13.3
Linear Cost
First Wait
−0.03364 −0.2582 −0.0403 −0.0444 −0.2946 −0.03093
−15.6 −6.5 −14.3 −7.1 −6.7 −3.7
−0.03199 −0.2199 −0.04152 −0.04508 −0.3116 −0.03507
−14.4 −5.3 −14.4 −7.2 −7.1 −4.2
ASC
+3.336
+8.5
+3.164
+8.0
Linear Distance
Villa
−0.4029 −0.424 −0.05929 −0.8954
−6.0 −4.7 −2.9 −3.2
−0.4051 −0.4323 −0.05941 −0.9223
−6.0 −4.7 −3.0 −3.3
ASC
+0.7083
+1.8
+0.4916
+1.3
Linear Distance
−0.2244
−9.3
−0.2292
−9.5
ASC
Passenger
Linear Cost
log(Cost)
log(Cost) Inv Time Aux Time Boardings
log(Dist) Diag. Dist
Bike
t-Score
−0.03199 −0.2199 −0.03511
Car
Walk
Model C1 Value
−15.6 −6.5 −8.9
Density
Transport
t-Score
−0.03364 −0.2582 −0.05053
Woman
Public
Value
log(Dist)
+0.3623
+3.7
+0.3908
+3.9
Winter Woman
−1.565 −0.441
−4.6 −1.8
−1.575 −0.4497
−4.7 −1.9
Density
+0.002244
+0.6
+0.002153
+0.6
Logsum
+0.6132
Log-Likelihood
+36.1
+0.5981
+34.7
−12091.0
−12088.4
0.1959
0.1961
R2
84
