Study of Distribution of Free Flow Speeds on Urban Road Sections Depending on Their Functional Purpose and One-Way Traffic—Evidence from Kharkiv (Ukraine)

Abstract

Data on the distribution of the free flow speed (FFS) of cars are used to solve a wide range of tasks in the field of road transport, starting from road design and ending with the development of traffic modeling and simulation programs. The purpose of this study was to obtain the distribution of vehicle speeds on typical sections of the city road network, characterized by the presence of one-way traffic. The data were obtained by field observations using a portable radar. As a result, statistical characteristics and speed distribution laws for four sections of streets in the city of Kharkiv were analyzed. It was shown that the characteristics of FFS distributions differ depending on the functional class of the streets. Average FFS values on main street segments were on average 19 km/h higher. The one-way traffic has less impact on the FFS distribution, especially for arterial streets. The characteristics of FFS distributions differ depending on the type and functional class of streets; they can be described with sufficient accuracy by typical distribution laws, such as Normal, Log-normal, Gamma, and Chi-square. The results of this study can be useful for traffic modeling problems.

1. Introduction

One of the main areas of solving the problem of ensuring the efficiency and safety of road traffic in cities is the creation of an effective system for controlling the speed of vehicles. A necessary condition for the development and maintenance of the functioning of such a system is the monitoring and research of the flow speed and the individual vehicle speed in real conditions.

In accordance with generally accepted judgments, a vehicle can be considered free when its speed is not influenced by the speed of the vehicle traveling ahead. In [1], the author shows that the optimal characteristic of a car that moves under free conditions is an interval to the vehicle ahead of it that is at least 6 s.

It is known that the speed of a single car is a complex function of a large set of factors and parameters of the “driver—car—road—environment” system. Therefore, the study of distributions of speeds, which are obtained as a result of statistical processing of a sufficient amount of data, is of practical value.

Many classic traffic flow models such as Greenshields, Gipps’, Newell’s, Lighthill–Whitham–Richards, and BPR Function contain FFS or its derivatives as model parameters. These models are critical for studying traffic dynamics, optimizing signal timing, and improving road capacity. The importance of the FFS parameter in traffic modeling was noted in [1,2,3]. Although our study focused on the potential usefulness of the results for traffic modeling, the FFS parameter has wider applications, for example, in the design of road infrastructure and speed limit decisions, and it can also be considered a key factor in road safety [4].

The main area where the results of the FFS study can be used is traffic modeling, including the construction of computer simulation models using specialized software products, such as PTV VisSim. The need for road traffic modeling may be particularly high in Ukraine in the future, as post-war urban reconstruction will require infrastructure solutions, including reconstruction of the road network.

Since some simulation software products use a specific driver behavior model, more reliable results will be shown by models built on FFS data collected in that particular region. Such data can take into account the specific national mentality of the driver when driving and include the choice of speed. Therefore, it is relevant to investigate the modes of automobile traffic in real conditions of different cities and regions of the world.

Conducting research in Ukraine has another obvious advantage, which is the very small use of automatic speed control on the roads. It is obvious that in conditions of speed control the driver will not be free to choose the speed and, therefore, it is impossible to study the influence of road conditions on FFS.

Of particular interest for traffic modeling is the study of the statistical characteristics of FFS distributions, as well as finding the parameters of theoretical distribution laws. In this study, the impact of one-way traffic was analyzed in analyzing the FFS distributions for different sections of urban streets, as this factor has not received much attention in the scientific literature.

2. Statement of the Problem and Analysis of Publications

2.1. Review of Factors Affecting the Free Flow Speed

Since FFS is a parameter of many mathematical models of traffic flow, a large number of publications have been devoted to studies of this parameter since the middle of the last century.

It is known that the speed of a single car on a road section is a random value that is chosen directly by the driver and is realized as a result of a psychophysiological process, which is based on the reflection of goal setting, motivation, awareness, and perception of the totality of road conditions and circumstances environment. The driver constantly strives to choose the most appropriate speed mode based on the criteria of minimizing travel time and ensuring comfortable and safe driving conditions. At the same time, the driver’s choice of speed is also influenced by his qualification, psychophysiological state, the purpose of the trip, and other factors [5,6,7,8,9]. Experience shows that the driver drives the car at maximum speed only in exceptional cases and for a short time, as it is associated with excessive mental stress. Studies conducted in the same road conditions on the same type of car showed that the average speed of the car in different highly qualified drivers can vary within ±10% of the average value. In inexperienced drivers, this difference is greater [7].

As observations show, the actual range of instantaneous FFS on horizontal sections of main streets and roads is from 50 to 120 km/h, regardless of the established speed limits [6,10].

It should be noted that most of the papers analyzing the effect of road conditions on FFS consider suburban highways. Among the road factors that influence a driver’s choice of speed on urban roads in different regions of the world, the researchers noted the following: Speed Limits, Land-use type (commercial, residential, industrial); Road Geometry; Surface Type and Quality; Weather Conditions; Roadside Environment (the presence of buildings, trees, or other barriers); Access Points (like driveways or intersections); Lane Configuration (the number of lanes, their arrangement); and roadside frictional (parking, pedestrian and bicycle activity, etc.) [11,12,13,14,15]. It is noted in [11,12] that gradients, curvilinear sections, and road irregularities lead to a decrease in FFS. In urban environments, FFS is further affected by speed limits, the presence of pedestrian traffic, parking, and other factors [14,15]. Studies [1,16] show that factors such as road class and area type have a stronger effect on the FFS of passenger cars, for which the average values for different streets range from 58 to 80 km/h. The average FFS values for urban highways turned out to be on average 22–25 km/h higher than those for local traffic streets [1,8,12,17].

Since one-way traffic is very common in cities in different regions of the world, traffic conditions on such streets are often the object of modeling [18,19,20]. Therefore, studying the differences in FFS for one-way traffic conditions can improve the accuracy of traffic flow description. In addition, the use of models of traffic characteristics and parameters once obtained for some conditions, such as in cities in developed countries, may not work well when modeling traffic in other regions of the planet [21]. Therefore, expanding the geographic scope of the study of traffic parameters, particularly FFS, is also relevant.

For the one-way traffic, although in some works there are the results of FFS analysis for city streets with one-way traffic, we have not come across any work where the influence of this factor would be studied separately from others. Our work, it seems to us, can fill this gap.

2.2. Review of Experience in Studying Velocity Distributions

Since the actual speed of a car is affected by many factors of both a technical and social nature, FFS is a random value, including for the flow of cars of the same type in a given road section. When studying this problem, the methods of analysis of speed distributions are widely used [6,11,12,16,22,23,24].

Usually, the speed distribution is characterized by a normal distribution law or close to it. Although, as some studies show, even for a flow of homogeneous cars, the distribution of free speeds may deviate from the normal law [25]. In particular, violations of the normal speed distribution can be observed for road sections in the presence of specific road factors or the speed control zone. In this case, other analytical distributions (Log-normal, Gamma distribution, Weibull, and others) can be used to approximate the speed distribution, while the use of distributions with three or more parameters is impractical since such parameters of the distributions as the coefficient of asymmetry and kurtosis are not revealed to have a stable correlation with road conditions.

Graphical form is often used to present the results of velocity distribution analysis. Differential distribution graphs or histograms show how many cars are moving in specified speed intervals. Accumulation graphs (cumulative curves) make it possible to determine the percentage of cars moving at a speed less than any given value. The value of the 15th percentile speed describes the speed of the slowest part of the flow of cars, which leads to the complication of traffic conditions and an increase in the probability of road accidents [26]. When restrictions on the movement of slow-moving vehicles are introduced on the road, the value of this speed is often taken as the minimum permissible [27]. The 50th percentile speed values characterize the average speed of the flow of cars. A very important characteristic is the 85th percentile speed, which characterizes the maximum speed of the main part of the flow of cars. This value is usually used when introducing a maximum speed limit [28]. The 95th percentile speed values usually correspond to the calculated speed of single cars in the given road conditions.

Thus, the study of the law of the distribution of FFS on street sections remains an actual direction of scientific research. In addition, the time period in which the studies were conducted is of some importance when examining the FFS distribution. In general, modern vehicles, especially electric vehicles, have better dynamics, are equipped with more advanced safety systems, and may have ADAS and unmanned driving systems, which can influence the driver’s speed selection behavior. Therefore, other things being equal, it is advisable to favor more recent studies of the FFS distribution.

3. Description of the Research Methodology

3.1. Analysis of Methods for Obtaining Data on Free-Flowing Vehicle Speeds

Since, as mentioned above, FFS plays an important role in the planning, operational analysis, and modeling of traffic, there are requirements for the use of reliable methods for determining this parameter. However, the determination of FFS in real traffic conditions is time consuming because it requires obtaining a sufficient sample size of vehicle speeds at appropriate times when flow intensity is minimal and at appropriate locations with uniform geometric, traffic, and regulation characteristics.

Published scientific papers use different methods of measuring vehicle speeds, which differ in both speed recording devices and data processing technology. Their comparative analysis is given in [29]. Data from traffic monitoring or control systems can be used to obtain data on car speeds, but this requires the use of additional equipment and software for data processing, such as in most cases in which scientists are dealing with one-time measurements on specially selected sections of streets and roads carried out by a small team of researchers.

Therefore, the use of portable radar-type measuring instruments may be a suitable method for measuring velocities. They allow you to measure the speed of a single vehicle and a vehicle moving in a group. The error in measuring speed with modern radars usually does not exceed 1 km/h. One of the disadvantages of using such devices is that they cannot reliably record the speed of several vehicles moving in different lanes; however, when measuring the speed of a single vehicle in free conditions, this limitation does not matter.

Recording speed directly by a competent investigator using handheld radar has another advantage over other known methods such as the use of detectors, video surveillance, or GPS track data. The researcher is able to more accurately select cars moving in free conditions, and he can also exclude unique cases from measurements. These include cars whose drivers choose or are forced to choose speed due to special circumstances. For example, cars that are moving with technical faults, carrying oversized cargo, police, or emergency service vehicles.

After carrying out the required number of measurements, a table of the results is formed. Standard or specialized software, such as Excel, Matlab, and Statistica, is used for further data processing.

3.2. Characteristics of the Experimental Sections of the Streets and the Equipment Used

When measuring the cars’ speed by the method of a stationary measuring site, the first stage is the selection of a section of the road that meets the specified requirements. Since free traffic is subject to a number of mandatory conditions that exclude the influence of other road users on the driver’s speed decision, the range of suitable sections of the city road network for measuring FFS is quite limited.

When measuring FFS, outliers of values in the low-speed range can occur when the driver has just entered the road or has resumed driving in front of the measuring station or is about to make a turn or stop a short distance from it. Therefore, the location of the measuring post should be remote from road intersections and pedestrian crossings, parking lots, and places with traffic lights. Then, the driver will maintain the selected speed throughout the entire section of the street with stable road conditions.

Streets in the downtown area may not meet conditions due to high traffic volumes where free-flowing conditions may occur during limited periods of the day, primarily at night or early morning. Random factors such as weather conditions, obstacles, or road surface defects will also be able to contribute to the measurement results.

In order to check the effectiveness of the chosen methodology and obtain experimental distributions of FFS on the street and road network of the city of Kharkiv, four sections were selected, where measuring sites were located.

The street sections were divided into two groups according to their functional class. This division is contained in the state standard of Ukraine (БДH B.2.3-5) [30]. “Main streets” provide traffic in major directions and transportation connections between different parts of the city. “Local streets” provide transportation connections between adjacent areas of the city and between arterial streets. Street sections were also divided according to the traffic management feature (one-way or two-way traffic).

Two sites were located on sections of city-wide main streets (Ave. Heroiv Kharkova and Ave. Traktorobudivnykiv), and two sites were located on sections of local streets (St. 12-ho Kvitnia and St. Myru). Sections on Ave. Heroiv Kharkova and St. 12-ho Kvitnia had one-way traffic, sections along Ave. Traktorobodivnykiv and St. Myru had two-way traffic.

The selected sections did not have longitudinal slopes and curves, and the measuring sites were at least 300 m away from the nearest intersection. The sections had an asphalt concrete surface without defects that would affect the speed of vehicles. On all streets, the roadway had a raised curb. Brief characteristics of road conditions at the sites are given in

Table 1. Characteristics of road conditions at speed measurement areas.

No.	Name of the Street	Description	Number of Lanes, Total Width of the Roadway
Station 1	Ave. Heroiv Kharkova	Main street; one-way traffic; there is no pedestrian traffic; there is no development in the vicinity of the roadway; green spaces on the right side of the road	3 lanes, 9.0 m.
Station 2	Ave. Traktorobudivnykiv	Main street; two-way traffic; there is no pedestrian traffic; there is no development in the vicinity of the roadway; green spaces on the right side of the road	4 lanes (2 in each direction), 12.3 m.
Station 3	St. 12-ho Kvitnia	Local street; one-way traffic; sidewalks on both sides of the street; there is no construction in the vicinity of the roadway; green spaces on both sides of the street	2 lanes, 8.6 m.
Station 4	St. Myru	Local street; two-way traffic; the sidewalk of one street; there is no construction in the vicinity of the roadway; green spaces on both sides of the street	2 lanes (1 in each direction), 6.9 m.

. Figure 1 shows the location of the measuring sites on the city map.

Bushnell II Speed Radar Gun 101911 (Overland Park, KS, USA) was used for measurement. The radar allows you to easily measure the speed of the car with a measurement accuracy of ±1 km/h. The radar of this brand uses DSP (Digital Signal Processing) digital technology to ensure accurate measurement in real time.

During the measurement, the radar was directed at a single-passenger car moving along the road section at a distance of 80 to 100 m. At the same time, the operator tried to stand as close as possible to the roadway in order to reduce the measurement error due to the sine effect. In the process of working with the radar, it was recorded that some drivers began to reduce their speed when an operator with the device pointed at them came into their field of vision, so it was decided to record the speed of cars when they moved away from the operator.

The speeds of passenger cars were measured, the number of which is more than 90% of the traffic flow. The sample size was 50 cars for each section. In order to match the driving mode of the car whose speed is measured to the conditions of free movement, single vehicles or those that are the leaders of a pack of cars are selected.

4. Results of Experimental Studies

4.1. Statistical Characteristics and Graphs of Distributions

Standard methods of mathematical statistics were used to process experimental data on the distribution of FFS on sections of the road network of the city of Kharkiv. Statistical analysis of data samples involves calculating the characteristics of the distribution center, and its structure and evaluating the degree of variation and differentiation. Indicators of the main tendency of the distributions are the arithmetic mean, mode, and median. The main indicators of distribution variation are variance, standard deviation, and coefficient of variation. Indicators such as median, quartiles, deciles, and other percentiles are used to characterize the distribution structure. Studying the shape of the distribution involves assessing asymmetry and kurtosis. The listed indicators have an independent analytical value and allow obtaining a complex characteristic of the empirical distribution.

The calculated statistical characteristics of free traffic speed distributions for the four studied sections of the road network of the city of Kharkiv are shown in

Table 2. Statistical characteristics of free-flowing speed distributions.

Station Number	Stat. 1	Stat. 2	Stat. 3	Stat. 4
Station location	Ave. Heroiv Kharkova	Ave. Traktorobudivnykiv	St. 12-ho Kvitnia	St. Myru
Valid N	50	50	50	50
Mean	71.74	70.40	58.38	65.04
Confidence 95.000%	68.94	67.28	55.55	62.34
Geometric mean	74.54	73.52	61.21	67.74
Median	71.10	69.57	57.56	64.36
Mode	70.00	70.50	57.50	65.50
Minimum	51	50	37	46
Maximum	96	97	86	85
15th percentile	62.00	58.00	48.00	55.00
50th percentile	70.00	70.50	57.50	65.50
85th percentile	81.00	81.00	68.00	76.00
95th percentile	91.00	90.00	76.00	80.00
Range	45.00	47.00	49.00	39.00
Quartile range	12.00	14.00	12.00	17.00
Std. Dev.	9.84	10.96	9.97	9.49
Coef. Var.	13.71	15.57	17.08	14.59
Standard error	1.39	1.55	1.41	1.34

.

Since the mean values $m$ as well as the standard deviation $\sigma$ of the FFS values are now known for all four measuring sites, we can assess the representativeness of the resulting samples. One method is to calculate the minimum sample size $n_{\min }$ for a given value of the maximum acceptable error $\eta$ as follows:

$$n_{\min }={\displaystyle \frac{t_{\alpha }^2\cdot {\sigma }^2}{{\eta }^2}}$$

(1)

where $t_{\alpha }$ is the confidence probability function, which for the standard in statistical studies value of the probability function α = 0.95 will be $t_{\alpha }$=1.96. The value $\eta$ can be represented as the product of the mean $m$ and the relative required accuracy $\Delta$. If we take $\Delta$ = 5%, then, for the first sample (stat. 1), we obtain the following:

$$n_{\min 1}={\displaystyle \frac{{1.96}^2\cdot {9.84}^2}{{\left(0.1\cdot 71.74\right)}^2}}=29$$

(2)

We obtain similar values for samples at other measuring stations: $n_{\min 2}=37$, $n_{\min 3}=45$, and $n_{\min 3}=33$. Since the actual number of observations $n_{\min 3}=50$ exceeds $n_{\min }$, the sample size is sufficient to ensure the required reliability of the results.

Table 2 shows the close location of the modal and median values, which is a sign of normal distribution and can also be an indirect confirmation of the representativeness of the samples.

The “Box-and-Whisker Plot” is an effective tool for a compact image and analysis of a statistical population, which provides both diagnostic and descriptive information about the studied distributions (Figure 2).

The lower and upper bounds of the boxes in Figure 2 correspond to the lower and upper quartiles of the distributions. The length of the straight lines above and below the boxes is defined as 1.5 of the interquartile range.

It can be immediately noted that a significant percentage of these speeds are located in the zone of more than 50 km/h, which indicates that the majority of drivers exceed the speed limit established in the city. This is also facilitated by the relatively high value of speeding tolerance in Ukraine, which is 20 km/h. However, a large number of drivers exceeded the speed limit of 70 km/h, especially on main streets. This result confirms the well-known thesis that in the absence of speed control, drivers rely on the actual road conditions on a specific section of the road when choosing a speed.

It can also be seen that certain differences are observed in the statistical characteristics of the obtained experimental speed distributions; in particular, the average and median values for the main streets were higher than for the streets of local connections. This is a result of more favorable and safer road conditions on arterial streets, which encourages drivers to drive at higher speeds.

For a detailed study of the obtained data sets, it is advisable to present them in the form of a table of variation series, which are built according to the principle of grouping (

Table 3. FFS frequency distribution for main streets.

Station Number			Stat. 1			Stat. 2
Intervals, from to	Count	Cumul. Count	Percent	Cumul. Percent	Count	Cumul. Count	Percent	Cumul. Percent
35 ≤ x < 43	0	0	0.0	0.0	0	0	0.0	0.0
43 ≤ x < 51	0	0	0.0	0.0	1	1	2.0	2.0
51 ≤ x < 59	2	2	4.0	4.0	8	9	16.0	18.0
59 ≤ x < 67	16	18	32.0	36.0	9	18	18.0	36.0
67 ≤ x < 75	13	31	26.0	62.0	15	33	30.0	66.0
75 ≤ x < 83	13	44	26.0	88.0	10	43	20.0	86.0
83 ≤ x < 91	3	47	6.0	94.0	5	48	10.0	96.0
91 ≤ x < 99	3	50	6.0	100.0	2	50	4.0	100.0
99 ≤ x < 107	0	50	0.0	100.0	0	50	0.0	100.0

and

Table 4. FFS frequency distribution for local traffic streets.

Station Number	Stat. 3				Stat. 4
Intervals, from to	Count	Cumul. Count	Percent	Cumul. Percent	Count	Cumul. Count	Percent	Cumul. Percent
35 ≤ x < 43	1	1	2.0	2.0	0	0	0.0	0.0
43 ≤ x < 51	10	11	20.0	22.0	2	2	4.0	4.0
51 ≤ x < 59	16	27	32.0	54.0	13	15	26.0	30.0
59 ≤ x < 67	13	40	26.0	80.0	11	26	22.0	52.0
67 ≤ x < 75	7	47	14.0	94.0	14	40	28.0	80.0
75 ≤ x < 83	2	49	4.0	98.0	9	49	18.0	98.0
83 ≤ x < 91	1	50	2.0	100.0	1	50	2.0	100.0
91 ≤ x < 99	0	50	0.0	100.0	0	50	0.0	100.0
99 ≤ x < 107	0	50	0.0	100.0	0	50	0.0	100.0

). Sturges’s formula for determining the optimal number of groups gives the following result:

$$k=1+1.44\cdot \ln N=1+1.44\cdot \ln 50=6.63\approx 7$$

(3)

Another effective tool for analyzing the shape of distributions is the construction of polygonal graphs and graphs of accumulated percentages (Figure 3).

It can be seen that the mutual location of the FFS distributions for the main streets, as well as their statistical characteristics, are very close. Therefore, it can be stated that the factor of one-way traffic, as well as other features of external conditions, almost did not affect the main characteristics of the distribution of FFS for main streets.

The characteristics and location of the distributions for the local streets revealed greater differences, the reason for which can be both the peculiarities of traffic organization and other factors of road conditions. At the same time, contrary to expectations, the weighted average value of speed for a one-way street turned out to be almost 7 km/h less than for a two-way street. This may indicate a greater weight of other factors of traffic conditions, which require a separate analysis.

4.2. Approximation of Experimental Data by Theoretical Distributions

The procedure for approximating experimental FFS distributions of cars consists of replacing empirical frequencies with theoretical ones determined by mathematical dependencies of the theoretical distribution. The choice of the type of theoretical distribution model is made either on the basis of a meaningful analysis of the previously obtained statistical characteristics of the studied quantity or based on general considerations based on a visual analysis of the obtained distribution graphs.

As already mentioned, most authors use relatively simple one- or two-parameter distributions to describe the FFS distribution. In order to approximate the experimental FFS distributions of cars on the road network of the city of Kharkiv, it was decided to check four standardized laws of distribution: Normal, Log-normal, Gamma, and Chi-square. Dependencies of the functions of these distribution laws can be taken from any reference book on mathematical statistics. For example, the normal distribution function in our case will look like the following:

$$f\left(v\right)={\displaystyle \frac{1}{\sigma \cdot \sqrt{2\pi }}}\cdot e^{-{\displaystyle \frac{{\left(v-m\right)}^2}{2\cdot {\sigma }^2}}}$$

(4)

where $v$ is the value of speed; $m$ is the average speed value; and $\sigma$ is the root mean square deviation. Thus, the parameters of the normal distribution are $m$ and $\sigma$.

The Gamma distribution function is expressed as follows:

$$f\left(v\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\lambda \right)}}\cdot V^{\lambda -1}\cdot e^{-\beta \cdot v}$$

(5)

where $\mathrm{\Gamma }\left(\lambda \right)$ is the Gamma function; $\lambda$ is the shape parameter; and $\beta$ is the scale parameter.

The function of the Log-normal distribution is expressed as follows:

$$f\left(v\right)={\displaystyle \frac{1}{v\cdot s\cdot \sqrt{2\pi }}}\cdot e^{-{\displaystyle \frac{1}{2}}\cdot {\left({\displaystyle \frac{\ln \left(v\right)-\mu }{s}}\right)}^2}$$

(6)

where $\mu$ is the average arithmetic logarithm of the speed and $s$ is the standard deviation of the logarithm of the speed.

The function of the Chi-squared distribution is expressed as follows:

$$f\left(v\right)={\displaystyle \frac{{\left(\frac{1}{2}\right)}^{\frac{k}{2}}}{\mathrm{\Gamma }\left(\frac{k}{2}\right)}}\cdot v^{{\displaystyle \frac{k}{2}}-1}\cdot e^{-{\displaystyle \frac{v}{2}}}$$

(7)

where $k$ is the number of degrees of freedom.

The degree of reliability of the approximation of the experimental FFS distribution by the theoretical distribution law can be established using standard consistency criteria. In this study, the Pearson test was used, the characteristics of which are the discrepancy parameter ${\chi }^2$, the number of degrees of freedom $d_f$, and the calculated value of the p-value $p$. At the same time, the limit on the value of the theoretical frequencies when calculating the Pearson criterion was set at the level of $f\left(i\right)\ge 5$.

FFS distribution histograms with superimposed graphs of theoretical distribution laws for the section of Ave. The heroes of Kharkiv (Stat. 1) are shown in Figure 4. The values of the relevant parameters of the distribution laws and consistency criteria are given in

Table 5. Parameters of the theoretical functions of FFS distribution for the section Ave. Heroiv Kharkova.

Theoretical Distribution	Normal	Gamma	Log-Normal	Chi-Square
Image
Parameters of the distribution law	$m$ = 71.74 $\sigma$ = 9.84	$\lambda$ = 1.28 $\beta$ = 55.85	$\mu$ = 4.26 $s$ = 0.13	$k$ = 71.74
Parameters of the Pearson test	${\chi }^2$ = 1.056 $d_f$ = 1 $p$ = 0.293	${\chi }^2$ = 0.507 $d_f$ = 1 $p$ = 0.476	${\chi }^2$ = 0.327 $d_f$ = 1 $p$ = 0.568	${\chi }^2$ = 2.836 $d_f$ = 2 $p$ = 0.242

.

Conclusions about the validity of the hypothesis about the law of distribution can be made by focusing on the empirical value of the criterion ${\chi }^2$ or on the calculated value of the p-value $p$, which can be compared with the accepted significance level, which in our case can be accepted at the level of $\alpha$ = 0.05. It can be seen in Table 5 that the condition $p>\alpha$ is fulfilled for all distributions, so the hypothesis that the theoretical distributions correspond to the experimental data can be accepted. Thus, for the section of the main street Ave. Heroes of Kharkiv, FFS distribution can be described with sufficient practical accuracy by any of the four proposed distribution laws. Greater accuracy is provided by the Log-normal law, which is asymmetric and has a longer right side compared to the Normal law.

Similar results shown in Figure 5 and

Table 6. Parameters of the theoretical functions of free speed distribution for the section Ave. Traktorbudivnykiv.

Theoretical Distribution	Normal	Gamma	Log-Normal	Chi-Square
Image
Parameters of the distribution law	$m$ = 70.4 $\sigma$ = 10.96	$\lambda$ = 1.664 $\beta$ = 10.96	$\mu$ = 4.24 $s$ = 0.16	$k$ = 70.4
Parameters of the Pearson test	${\chi }^2$ = 0.649 $d_f$ = 1 $p$ = 0.420	${\chi }^2$ = 0.959 $d_f$ = 1 $p$ = 0.327	${\chi }^2$ = 1.241 $d_f$ = 1 $p$ = 0.265	${\chi }^2$ = 1.503 $d_f$ = 2 $p$ = 0.472

were obtained for the second section of the main street, Ave. Traktorobodivnykiv (Stat. 2).

It can be seen that for the section of the main street of Ave. Traktorbudivniki, the FFS distribution can also be described by any of the four distribution laws. But, in this case, the Chi-square distribution has a slightly higher accuracy, which indicates a more symmetrical arrangement of experimental data.

The results of the approximation for the section of the street of the local connection, St. 12-ho Kvitnia (Stat. 3), are shown in Figure 6 and

Table 7. Parameters of the theoretical functions of the free flow speed distribution for St. 12-ho Kvitnia.

Theoretical Distribution	Normal	Gamma	Log-Normal	Chi-Square
Image
Parameters of the distribution law	$m$ = 58.38 $\sigma$ = 9.97	$\lambda$ = 1.642 $\beta$ = 35.56	$\mu$ = 4.05 $s$ = 0.17	$k$ = 58.38
Parameters of the Pearson test	${\chi }^2$ = 0.139 $d_f$ = 1 $p$ = 0.709	${\chi }^2$ = 0.257 $d_f$ = 1 $p$ = 0.612	${\chi }^2$ = 0.328 $d_f$ = 1 $p$ = 0.567	${\chi }^2$ = 0.250 $d_f$ = 2 $p$ = 0.882

.

For the local St. 12-ho Kvitnia street section, the FFS distribution is also approximated with reasonable accuracy by any of the four distribution laws. A Chi-square distribution provides a little more accuracy, which indicates a symmetrical arrangement of experimental data and the absence of a clearly defined peak in the distribution of experimental data.

Figure 7 and

Table 8. Parameters of the theoretical functions of the free flow speed distribution for St. Myru.

Theoretical Distribution	Normal	Gamma	Log-Normal	Chi-Square
Image
Parameters of the distribution law	$m$ = 65.04 $\sigma$ = 9.49	$\lambda$ = 1.36 $\beta$ = 47.67	$\mu$ = 4.16 $s$ = 0.15	$k$ = 65.04
Parameters of the Pearson test	${\chi }^2$ = 2.629 $d_f$ = 2 $p$ = 0.269	${\chi }^2$ = 1.922 $d_f$ = 2 $p$ = 0.382	${\chi }^2$ = 1.569 $d_f$ = 2 $p$ = 0.486	${\chi }^2$ = 1.596 $d_f$ = 3 $p$ = 0.660

show the results of approximation for the section of the local flow street, St. Myru (Stat. 4).

Thus, for the Myru local flow street section, the FFS distribution can also be approximated by any of the four proposed distribution laws. A relatively larger value of the Pearson test parameter also indicates the absence of a pronounced peak of the FFS distribution.

5. Discussion

Many studies recognize the difficulty in controlling all factors in real-world driving conditions on specific street segments, which can influence FFS measurements. Therefore, the measured FFS values will be different for each site. In our opinion, in order to obtain reliable results, it is more important not so much the number of measurement points but their correspondence to free traffic conditions, as well as the availability of reference road conditions. Isolating any one factor, such as one-way traffic, requires, if possible, eliminating all other factors and taking measurements under identical conditions.

Although in our work, when choosing experimental road sections, we tried to ensure that all unstudied factors remained identical, it is not possible to completely solve this problem. So, for example, for the main streets in the site 1 section, there were three lanes, while in the site 2 section, there were four lanes. In terms of local traffic streets, site 3, unlike site 4, had sidewalks on both sides of the road and slightly heavier pedestrian traffic. The question of how much these differences could introduce disturbances into the FFS measurement results can be the subject of a separate discussion.

It should be noted that the characteristics of the adjacent territory are rarely considered as a factor influencing speed. The effect of the number of lanes on FFS appears to be significant only for one- and two-lane urban roads [7], which is typical for local streets. In our case, the differences concern main streets with three and four lanes. The work [15] notes a more significant FFS correlation for the position of the vehicle in the lane than for the number of lanes. At the same time, in our opinion, the position of the car on the lane is a consequence of the speed chosen by the driver and not its cause.

It should also be noted that transport flows in different countries and regions differ both in composition and in the quality of vehicles. The use of data only for passenger cars in this study makes the results more universal since traffic flows reduced to a passenger car can be used when solving traffic modeling problems. But still, the generalizability of the results may vary depending on the geographical and cultural context, which requires taking this factor into account in the practical sphere of traffic management on the street network of a particular country or region.

6. Conclusions

The FFS of vehicles on sections of the city road network is an important indicator that is widely used in traffic management tasks. Since the driver’s choice of driving speed in conditions of low road traffic is influenced by many factors of not only a technical but also a psychophysiological and social nature, the issue of studying FFS distributions on road sections for different cities and regions of the world is relevant. Reliable results can be obtained by field observations of the movement of cars in real conditions.

Such experimental studies were conducted for four sections of the road network of the city of Kharkiv (Ukraine), where a portable radar was used to record the speed of cars. The processing of the received data made it possible to obtain certain practical results.

It was found that a large number of drivers were exceeding the established speed limits, especially on sections of main streets.

The functional class of the street was found to be an important factor affecting the value of FFS. The average values of FFS on sections of main streets, compared to streets of local connections, turned out to be 19 km/h higher on average, which indicates a higher speed regime. At the same time, one-way traffic has less influence on the FFS distribution, especially for arterial streets.

No unequivocal influence on the shape of the FFS distribution of the functional class of the street and one-way traffic was found. A non-trivial result turned out to be a smaller value of the average speed for a street of local flow with one-way traffic, which can indicate both the ambiguity or low significance of this factor or the presence of other factors that influenced the choice of speed by drivers.

Both the functional class of the street and the one-way traffic factor did not significantly affect the shape of the FFS distribution curves, although the distributions for arterial streets had more pronounced peaks.

The slight difference in the characteristics of the FFS distribution for sections of roads with one-way and two-way traffic indicates a low correlation of this factor. This means, in particular, that when modeling traffic on streets of the same functional class, it is possible to use identical FFS values for both one-way and two-way traffic, whereas to simulate traffic on streets of different functional classes, different values of the FFS parameter should be used.

When choosing the optimal number of intervals according to Sturges’s formula, experimental distributions of FFS were approximated with sufficient accuracy by the following distribution laws: Normal, Gamma, Log-normal, and Chi-square. At the same time, for a main street with one-way traffic, due to the presence of a more pronounced peak and a relatively greater asymmetry of the distribution, the Log-normal distribution showed greater accuracy. For a main street with two-way traffic, the Normal and Chi-square distributions showed better accuracy. For the local streets, due to the relatively greater uniformity of the distributions, the Chi-square distribution showed better accuracy.

The obtained results can be useful for solving the problems of modeling and forecasting the parameters of traffic flows on the streets and road networks of large cities, as well as the preliminary results when conducting further scientific research on this problem.

Further studies may include analyzing factors such as road geometrics, intersections, and pedestrian circulation separately for each type of street such as city thoroughfares, local thoroughfares, and intra-district streets. Summarizing the results of this and similar studies could result in the development of a reference manual or regulatory document that provides FFS values for all types of urban streets. Such a document would be very useful for creating models of the functioning of the street network of cities both in Ukraine and other Eastern European countries with similar topologies of the street and road network.