| Abstract (english) | Most of the fundamental traffic flow models developed in the last 90 years, including the HCM models developed in USA, are based on deterministic functions obtained by regression analysis between traffic flow parameters, so they cannot describe the stochasticity of real traffic flows. In Croatia, the deterministic models from foreign literature are most often used as the foundation for determining the capacity and level of service of roads and intersections, although they are not adapted to domestic traffic system conditions. In this research, a probabilistic fundamental traffic flow models, specifically adapted to domestic traffic system conditions are developed which describe the variability of flow and speed in different traffic flow regimes on high-performance roads in Croatia. Proposed models, compared to the traditionally used foreign deterministic models, enable a more accurate and precise description of real traffic flow characteristics on high-performance roads in Croatia.
Since the very beginnings of Traffic Flow Theory as a scientific discipline, researchers have been attempting to find a way to more precisely describe the highly complex spatio-temporal characteristics of real traffic flow and the mathematical relationships between the empirical values of macroscopic and microscopic traffic flow parameters. In pursuit of this goal, numerous different traffic flow models have been proposed. Traffic flow models that are developed so far can be classified according to several criteria. According to the level of aggregation, traffic flow models can be divided into macroscopic, mesoscopic and microscopic. Depending on the type of input and output variables and mathematical equations used in a model, they can be classified into deterministic and stochastic, as well as into continuous and discrete traffic flow models. They can also be classified according to the number of traffic flow regimes into one-regime, two-regime and multi-regime traffic flow models.
Even though numerous researchers have proposed a wide variety of traffic flow models over the past decades in their attempt to provide a more precise description and graphical representation of the mathematical relationships between the three fundamental traffic flow parameters, an optimal generic formulation of "speed-density", "flow-density", and "speed-flow" fundamental diagrams has not yet been found. The main limitation of the deterministic traffic flow diagram models arises from the fact that they cannot represent the variability that, in realistic traffic flow conditions, is present in the values of all macroscopic and microscopic traffic flow parameters.
Traffic flow, by its inherent characteristics, exhibits a pronounced stochastic behavior. This stochasticity primarily arises from the complex and variable spatio-temporal interactions among vehicles in the traffic flow and secondly it is caused by the differences in the type, complexity, and the structure of the traffic flow, as well as by the differences in relevant road design and infrastructure elements and road environment characteristics. Furthermore, differences in imposed speed limits, prevailing traffic conditions, vehicle dynamic characteristics, driver behavioral characteristics, and prevailing weather conditions also contribute to the stochastic nature of traffic flow. Consequently, the behavior of real traffic flow cannot be accurately represented solely based on deterministic analytical expressions that are used to describe the mathematical relations between the aggregated average values of speed, flow and density.
In order to mathematically express the random fluctuations of traffic flow parameters values, it is necessary to extend deterministic traffic flow equations by introducing random variables that can precisely simulate the oscillations observed in the empirical values of vehicle speeds, flow and density. Due to this fact, over the past two decades an increasing number of researchers have opted to develop various forms of stochastic traffic flow models. However, many of these models rely on highly complex systems of partial differential equations that were developed based on methods adopted from statistical physics, which complicates their practical application.
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The problem of this research stems from the fact that in the Republic of Croatia, only few research has been conducted so far that specifically address the problematics of developing deterministic forms of fundamental traffic flow diagrams by which it would be possible to more precisely describe the mathematical relations between average traffic flow parameter values on different road categories. Additionally, no research has specifically examined the possibility of developing stochastic forms of fundamental traffic flow diagram, which could be used to describe random fluctuations of speed, flow and density in real traffic flow conditions on Croatian roads.
A large number of road traffic engineers, transportation planners, and other transportation and traffic experts and researchers in the Republic of Croatia still primarily rely on knowledge, methodological approaches, and deterministic traffic flow models taken from foreign literature, primarily from various editions of the Highway Capacity Manual (HCM), as well as from the Handbook for the Design of Road Traffic Systems (HBS – German: Handbuch für die Bemessung von Strassenverkehrsanlagen), which is also known as German Highway Capacity Manual. Due to this fact, traffic flow models developed in other countries are often either directly used for conducting various traffic engineering and spatial planning analyses, as well as for the design of roads and intersections, or alternatively attempts are made to calibrate these same models to adapt them to the prevailing road traffic conditions in the Republic of Croatia.
The purpose of this research primarily lies in the development of a fundamental probabilistic traffic flow model that, compared to the traffic flow models defined in foreign literature, will enable more accurate and more precise description of real traffic flow stochastic characteristics on high-performance roads in the Republic of Croatia. This model is intended to be used by road traffic engineers, transportation planners and other transportation and traffic professionals and researchers in the Republic of Croatia during the preparation of various traffic studies, projects, traffic technical reports and road safety revisions. Additionally, the purpose of this research stems from the need to establish a conceptual framework for the further development of both macroscopic and microscopic stochastic traffic flow models, specifically adapted for describing stochastic characteristics of real traffic flows on other road categories (state, regional, local and urban roads) in the Republic of Croatia.
In order to stop negative trend and reduce large research gap related to the development of fundamental traffic flow models that currently exists in Croatian science and practice, this research has examined the possibility of developing new probabilistic forms of fundamental
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traffic flow diagram which could, on the one hand, more accurately and more precisely describe the analytical relations between average values of basic traffic flow parameters on high-performance roads in the Republic of Croatia and on the other hand, also provide an additional capability to describe the occurrence of random oscillations in traffic flow parameter values around their average values.
In line with the above, the goal of this research was to develop a fundamental probabilistic traffic flow model suitable for describing the stochastic characteristics of traffic flow on high-performance roads in the Republic of Croatia.
The main research hypothesis is defined as follows:
• Based on the samples of empirical traffic flow parameter values collected on representative motorway and expressway sections it is possible to develop probabilistic "speed-density" and "flow-density" models, suitable for describing the behaviour of real traffic flow on high-performance roads in the Republic of Croatia.
In addition to the main research hypothesis, the following two auxiliary research hypotheses have been formulated:
• An adjustment factor, dependent on road category and traffic flow density, can be applied to determine the most appropriate forms of statistical probability density distributions for describing the variability of vehicle flow and speed in different traffic flow regimes.
• Probabilistic "speed-density" and "flow-density" models will provide a more accurate and precise description of the mathematical relations between traffic flow parameter values on the observed roads compared to models defined in foreign literature.
The expected original scientific contributions of this research stem from:
• Development and validation of probabilistic "speed-density" and "flow-density" models for describing stochastic traffic flow characteristics on high-performance roads in the Republic of Croatia.
• Selection of statistical probability density distributions suitable for describing the variability of vehicle speeds and flow, depending on the road category and traffic flow regime.
• The possibility of more accurate and more precise description of mathematical relations between traffic flow parameters values on high-performance roads in the Republic of Croatia.
In accordance with the defined problem, subject, purpose and goal of the research, as well as formulated set of hypotheses, the research was conducted through the following eight phases:
• In the first phase of the research, a review and analysis of the results and conclusions drawn from previous studies related to the problematic of development and validation of new fundamental traffic flow models, as well as to the calibration and comparison of existing fundamental traffic flow models was conducted. In order to collect data necessary for research, a detailed review of all available data sources was performed. Reviewed data sources primarily include scientific and other publicly available databases, as well as webGIS portals of relevant state institutions and organizations.
• In the second phase of the research, the parameters, characteristics, and features of traffic flow relevant for the development of a fundamental probabilistic traffic flow model for high-performance roads in the Republic of Croatia were defined. Representative road segments of high-performance roads on which it was necessary to conduct aerial recording of traffic flows were selected based on the K-means algorithm. For each selected road segment, characteristic peak hours and off-peak hours, representative for collecting the samples of empirical traffic flow parameter values were also defined. Furthermore, for each selected road segment, the minimum representative statistical sample size required for the development of fundamental probabilistic traffic flow model was determined.
• During the third phase of the research, aerial recording of traffic flows was conducted by Unmanned Aerial Vehicle (UAV) on selected representative segments of high-performance roads during defined peak hour and off-peak hour periods, as defined in prepared field research plan. Empirical traffic flow parameter values, extracted from recorded aerial video files were stored into output database.
• In the fourth phase of the research, a descriptive statistical analysis of the empirical data contained in the output database, created in the previous research phase, was conducted in order to gain insights into fundamental characteristics of real traffic flow on high-performance roads.
• During the fifth phase of the research, the strength and the direction of correlations between the observed traffic flow parameters were determined. Regression functions suitable for describing “speed-density” and “flow-density” relations in realistic conditions of traffic flow on high-performance roads were identified based on selected linear and nonlinear regression methods, whereby traffic flow density was considered as the independent variable and vehicle speed and flow as dependent variables of regression model.
• In the sixth phase of the research, an analysis of residual probability density distributions of the proposed regression models has been conducted. Based on a results obtained by performing a series of distribution fitting statistical tests (Xo2 test, Kolmogorov-Smirnov test, Anderson-Darling test) and graphical methods for comparing the quantiles and cumulative probabilities of considered empirical and theoretical probability distributions (Q-Q plot and P-P plot), an optimal types and shapes of probability density functions, suitable for describing the empirical distributions of vehicle speeds and flow in different traffic flow regimes on high-performance roads in the Republic of Croatia have been identified. The final mathematical formulation of the fundamental probabilistic traffic flow model has been defined by combining the mathematical formulations of proposed regression models with mathematical formulations of optimal forms of probability density functions, selected based on the conducted distribution fitting tests.
• In the seventh phase of the research, validation of proposed fundamental probabilistic traffic flow model for high-performance roads in the Republic of Croatia has been performed. In order to validate the proposed model, the collected statistical sample of traffic flow parameters empirical values was first divided into five subsamples of equal size. Each of the obtained subsamples was then used in iterative procedure of cross-validation, whereby during each iteration, one of the five subsamples, which includes 20% of observations randomly selected from the original sample, was used for the validation of the model developed (learned) based on the data contained in remaining four subsamples (remaining 80% of observations). Based on the conducted cross-validation procedure, the values of relevant performance indicators for the proposed fundamental probabilistic traffic flow model have been determined, including the values of Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), Normalized Root Mean Squared Error (NRMSE), Mean Percentage Error (MPE), Mean Absolute Percentage Error (MAPE), Mean Absolute Deviation (MAD), Root Mean Squared Percentage Error (RMSPE), coefficient of determination (r2) and confidence interval (CI). Obtained values of relevant performance indicators were then analyzed in order to confirm the possibility of using the proposed fundamental probabilistic "speed-density" and "flow-density" diagrams for accurate and precise description of the relations between traffic flow parameters on high-performance roads in the Republic of Croatia.
• In the final phase of the research, a concise overview of the obtained results has been provided, together with brief reflection on the advantages and disadvantages of the proposed fundamental probabilistic traffic flow model for the high-performance roads in the Republic of Croatia. Particular emphasis is given to the conclusions that are related to research goal, main research hypothesis, and the two auxiliary hypotheses of the research. Lastly, suggestions and guidelines for future research are also provided.
• The correlation and regression analysis between empirical speed and density values, and vehicle flow and density, was conducted in the OriginLab OriginPro software environment for statistical data processing. This analysis was based on software modules for linear function fitting and nonlinear curve fitting, examining the suitability of adapting more than 200 different linear and nonlinear functions to the collected empirical sample. From this analysis, seven regression functions were identified as potentially suitable for describing the relationships between average values of speed and density, and vehicle flow and traffic flow density. These functions include linear, logarithmic, negative exponential, cubic, power, logistic, and sigmoid regression functions. To identify the most suitable form of regression function, eight quantitative and two qualitative performance indicators were determined for each of the seven selected regression functions, including Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), F-statistic, correlation coefficient, coefficient of determination, mean squared error, root mean squared error; mean absolute error, the model complexity (number of model parameters), and the model interpretability. Moreover, a detailed analysis of standardized residual deviations was conducted for each observed regression model. Based on the conducted correlation and regression analysis, the negative exponential model was chosen as the most suitable for describing the deterministic component of the fundamental probabilistic "speed-density" and "flow-density" traffic flow models for high-performance roads in the Republic of Croatia.
• Based on the results of the regression fitting and conducted goodness-of-fit tests for more than 60 different theoretical probability distributions, it was concluded that variations in traffic flow speeds at different densities can be simulated by 11 theoretical probability distributions. The Generalized Extreme Value (GEV) distribution was identified as the best. On the other hand, to describe the variability of vehicle flow, 12 theoretical probability distributions can be applied according to the obtained results, among which the five-parameter Wakeby distribution proved to be the best.
• The final mathematical formulations of the fundamental probabilistic "speed-density" and "flow-density" models were obtained based on the sum of the deterministic component of the model represented by negative exponential “speed-density” regression function and the derived asymmetric second order "flow-density" curve, and the stochastic component of the model represented by the adapted forms of the Generalized Extreme Value distribution.
• The main hypothesis of the research is confirmed by the fact that during the period from July 7, 2023, to August 8, 2023, field research and recording of traffic flows were conducted on 16 selected representative freeway and expressway segments in Croatia. Based on data obtained from the processing of recorded aerial videos, an empirical sample of traffic flow parameter values was obtained, from which the fundamental probability "speed-density" and "flow-density" models were successfully developed, validated and specifically adapted to describe conditions in the real traffic flows on highways and expressways in Croatia.
• Two auxiliary hypotheses of the research have also been confirmed. The final mathematical formulations of fundamental probabilistic "speed-density" and "flow-density" models include adjustment factors dependent on the category of high-performance roads, which allows the deterministic component of the model, namely its negative exponential function V(g) and asymmetric regression curve q(g), to be reduced to specific mathematical forms that can more precisely describe real traffic flow conditions on freeways and expressways in the Republic of Croatia. The developed fundamental probabilistic ''speed-density'' and ''flow-density'' equations also include the position, scale, and shape factors dependent on traffic flow density, based on which the most appropriate forms of statistical probability density distributions can be selected for the purpose of simulating the variability of vehicle speeds and flow in different traffic flow regimes.
• The results of the comparative analysis of the proposed fundamental probabilistic traffic flow model for high-performance road in Croatia with selected fundamental traffic flow models defined in foreign literature have also confirmed the value of the proposed probabilistic ''speed-density'' and ''flow-density'' mathematical formulations. Based on the conducted comparative analysis it was discovered that proposed fundamental probabilistic “speed-density’’ and “flow-density” models allow for a more accurate and precise description of the relationships between traffic flow parameter values on the observed roads than the selected fundamental traffic flow models defined in foreign literature.
When interpreting the results and conclusions of the research conducted within this doctoral dissertation, it is necessary to consider the following methodological and spatio-temporal limitations and research constraints:
• The influence of traffic flow structure on the shape of regression functions that are used to describe mathematical relations between average values of traffic flow density as independent variable and vehicle speeds and flow as dependent variables was not directly considered during the development of fundamental probabilistic "speed-density" and "flow-density" models. The impact of traffic flow heterogeneity on the shape of probability density functions which are used to simulate the occurrence of random oscillations in traffic flow parameter values was also not directly considered. In the scope of this research, these influences were considered indirectly, through converting real heterogeneous traffic flow into conditionally homogeneous traffic flow, by multiplying the different vehicle types that are present in traffic stream (passenger cars, motorcycles, light and heavy vehicles, buses and other types of vehicles) with corresponding Equivalent Passenger Car Units (PCU).
• For the purpose of developing the fundamental probabilistic traffic flow model, the empirical values of observed traffic flow parameters, including the traffic flow density, traffic flow speed, vehicle flow and vehicle time headway, obtained based on the automatic analysis of aerial video footage, were aggregated into 30-second and 1-minute intervals. The impacts of higher levels of time aggregation of observed traffic flow parameters empirical values on the shape of regression functions used to describe the relations between average traffic flow parameters values and on the shape of probability density functions used to describe the stochastic traffic flow characteristics were not considered in this research.
• Since the maximum flight duration of the unmanned aerial vehicle (UAV), used in this research for recording aerial video footage of traffic flow, was limited to 25 minutes, in order to collect minimal statistical samples required for developing the fundamental probabilistic traffic flow model for high-performance roads in the Republic of Croatia, it was necessary to record four aerial video files lasting between 15 and 25 minutes on each selected representative road segment, whereby two video files were recorded during representative peak hour flow period and remaining two video files during representative off-peak hour period. In that way, it was ensured that a minimum of 30-minutes of usable aerial video footage was recorded both during the representative peak hour and off-peak hour flow periods.
• This research focuses only on the basic freeway and expressway segments under uninterrupted traffic flow conditions. Segments of high-performance roads located in immediate proximity to road junctions were not considered during the model development.
• All freeway and expressway road segments covered by this research have two traffic lanes in each direction (on each carriageway). Therefore, in the scope of this research it was not possible to examine the influence of the number of traffic lanes on the shapes of obtained regression and probability density functions.
• Considering the fact that the aerial videos of traffic flow were recorded on representative segments of high-performance roads in relatively short time intervals (from 15 to 25 minutes), based on the collected statistical samples collected in this research it was not possible to conduct a detailed analysis of the variations that are present in the empirical values of relevant traffic flow parameters during longer time periods (on daily, weakly, monthly and yearly level).
• The impact of road environment characteristics, relevant road design and road infrastructure elements and on the shapes of obtained regression and probability density functions was not considered in the scope of this research.
• Regardless of the fact that in the scope of this research the traffic flow density values have been determined based on the processing and reviewing the recorded aerial video files, these values still do not represent the exact empirical values, since they have been calculated based on the number of vehicles that were present in discrete time intervals on observed road segments, approximately 100 meters in length, depending on the flight altitude of the unmanned aerial vehicle.
• The field research during which aerial recording of traffic flows was performed by unmanned aerial vehicle equipped with high resolution video camera on selected representative segments of high-performance roads, has been conducted in the period between 7th of July and 8th of August 2023. This means that the statistical sample collected during this research does not contain the empirical traffic flow parameter values representative for the spring, autumn and winter periods and therefore it is not possible to gain full and detailed insight into traffic flow seasonal characteristics from the collected empirical data. Due to this fact, in order to maximize the accuracy and precision of the results obtained by proposed fundamental probabilistic traffic flow model, in future research it is necessary to further validate and calibrate proposed model based on the additional statistical samples that need to be collected during spring, autumn and winter periods of the year.
• In this research, the following assumptions were used to determine the minimum statistical sample size that needs to be collected on each selected representative segment of high-performance roads in the Republic of Croatia: (1) The statistical population size (N) was determined based on the data on the total number of registered road motor vehicles in the Republic of Croatia (without tractors and trailers); (2) The margin of error (e) (confidence interval) was set to ±5%; (3) The population proportion (d) was set to 50%. By using these assumptions, it was ensured to obtain the conservative estimate of the required minimum sample size. By varying the values of the population size, the margin of error and the population proportion parameters, different requirements can be set regarding the number of empirical values of traffic flow parameters that need to be collected for the purpose of model development and validation. Therefore, the results obtained in this research may vary to a certain extent if significant changes are made to the assumptions relevant for determining the minimum size of representative statistical sample.
• This research has not considered that the proportion of autonomous vehicles in traffic stream will increase in the future periods. Due to the fact that fundamental probabilistic traffic flow model for high-performance roads in the Republic of Croatia is developed based on the statistical sample which does not contain the empirical traffic flow parameter values for autonomous vehicles, obtained fundamental probabilistic "speed-density" and "flow-density" models cannot be directly applied for describing the stochastic characteristics of heterogeneous traffic flow composed of different percentages of autonomous and conventional vehicles. Changes in the proportions of conventional and autonomous vehicles in traffic stream can significantly alter the interactions between individual vehicles in the traffic flow, as well as the shapes of empirical distributions of all traffic flow parameters in different traffic flow regimes. It is expected that these changes will be especially pronounced during the adaptation period in which the drivers of conventional vehicles will get used to the presence of autonomous vehicles in traffic stream.
Based on the obtained results, derived conclusions and defined limitations of performed research, the following suggestions and guidelines for future research are proposed:
• The future research should examine the possibilities of using the unmanned aerial vehicles for simultaneous collection of data on the values of relevant traffic flow parameters on the larger number of observed road network segments, as well as the possibilities of implementing more advanced computer vision systems for automatic detection and tracking vehicle movements in traffic stream, which would enable the extraction of data related to individual vehicle trajectories and automatic analysis of empirical values of relevant traffic flow parameters.
• In the future research, it is also important to investigate the possibilities of utilizing the advanced data mining techniques for the purpose of clustering and calibrating collected data on traffic flow, as well as for the automatic identification of different operating conditions in traffic flow. Furthermore, the possibilities of combined use of data collected via mobile devices, GPS system and different types of detectors for the purpose of automatic construction of the optimal form of fundamental traffic flow diagram should be examined.
• To ensure that developed fundamental probabilistic traffic flow model will be applicable for describing the stochastic characteristics of mixed traffic flows composed of different proportions of conventional and autonomous vehicles, in future research it is necessary to calibrate the model based on additional samples of empirical traffic flow parameter values collected in traffic flows with different ratios of conventional and autonomous vehicles. |
