Modelling the Coupling Relationship between Urban Road Spatial Structure and Traffic Flow

Abstract

In order to promote the sustainable development of urban traffic systems, improve the accuracy of traffic system analysis in the urban planning stage and reduce the possibility of traffic congestion in the operation stage of road networks, the coupling relationship and evolution mechanism between urban road spatial structure and traffic flow were studied, and a model of the relationship between the metrics was established in this study based on real road network and traffic flow data. First, the road spatial structure model of the study area was established from the perspective of road space, and the spatial syntax method was applied to verify the rationality of the spatial structure of the road network. Secondly, the initial OD matrix was determined by OD backpropagation based on the measured traffic flow data. Thirdly, the coupling rule between the spatial structure and the traffic flow of the road network was explored by loading the increment in the OD matrix to the initial OD matrix step by step based on a simulation experiment. Finally, the relationship between the degree of integration of the spatial syntactic feature parameter and the saturation of the traffic flow feature parameter was modelled on the basis of experimental results and verified by an example. This research shows that the spatial structure of urban roads has a significant impact on the characterisation of the traffic flow distribution of road networks, and a strong correlation can be found between the integration degree and saturation degree. An optimal fit, which can be used as a reference for the design of road spatial structure, was explored in this research.

1. Introduction

Urban traffic is an important component of residents’ lives and is an important indicator of a city’s progress towards civilization and modernization. The composition of urban road systems is complex. Much research has been conducted to depict the operational principle of traffic flow and discover the mechanism of traffic congestion [1,2] based on data from actual road systems so as to ensure their smooth operation. In these studies, many traffic flow parameters, such as traffic volume, capacity and degree of saturation [3], have been set up to describe the traffic flow characteristics.

In the urban planning stage, rational urban planning is the precondition for the efficient operation of the road system. Analysing the characteristics of traffic flow parameters is also essential, and it is helpful to evaluate urban planning achievements and forecast the reliability and operation effect of urban road systems [4]. However, this work is insufficient because traffic flow characteristics are difficult to express precisely in the urban planning stage. Although the use of a macroscopic fundamental diagram (MFD) is an important research direction to depict the space characteristics of traffic flow, its main function is to characterize the traffic state in road networks and to evaluate the effect of traffic management [5]. An MFD can potentially play a role in imposing an upper limit on the number of vehicles a road network can serve [6], but it cannot provide a clear relationship between the spatial structure and the traffic volume of a road network, especially between the spatial structure and traffic volume on every road. This results in a disconnect between the urban planning stage and the traffic operation stage. If the relationship between spatial road network structure in the urban planning stage and traffic flow characteristics in the operational stage can be constructed on the basis of the spatial structure of the existing road network and the actual traffic flow characteristics, then the weakness of the insufficient assessment of traffic operation efficiency in the urban planning stage can be mitigated.

This research aims to address the relationship between the feature parameters of spatial syntax in the urban planning stage and the traffic flow parameters in the road traffic operation stage. The purpose of the study is to reveal the inner coupling mechanism between these two stages and construct the corresponding mathematical model in order to realize the optimization and enhancement of the road traffic system in the planning process.

Specifically, the research questions include the following:

(1)

How can the feature parameters of spatial syntax in the urban planning stage be effectively combined with the traffic flow parameters in the road traffic operation stage so as to achieve mutual support and coordinated development between the two?

(2)

What is the inherent coupling mechanism between the two? How can we extract the key factors and describe them quantitatively through analysis of this mechanism?

(3)

Based on this coupling mechanism, how can we construct a mathematical model between the two parameters in order to provide strong theoretical support for road traffic system planning?

(4)

How can we use this mathematical model to provide a new applied perspective and practical method for traffic flow analysis in the urban planning stage?

The research objectives and implications include the following:

(1)

To reveal the intrinsic connection and coupling mechanism between the feature parameters of spatial syntax in the urban planning stage and the traffic flow parameters in the road traffic operation stage;

(2)

To construct mathematical models to effectively combine these two indicators and provide a theoretical basis for the optimal planning of road traffic systems;

(3)

To expand the methods of traffic flow analysis in the urban planning stage, provide new application perspectives and practical tools and help to realize the synergistic development of traffic operation and planning;

(4)

To provide support to improve the level of road traffic system planning, optimize urban planning and design and further improve the travel experience and quality of life of urban residents through professional research.

The research flow chart is shown in Figure 1.

The remainder of this paper is organized as follows. Section 2 examines pertinent previous studies. Section 3 delves into the application of spatial syntax and outlines the process of establishing the abstract road network structure for the study area. In Section 4, we elucidate the coupling relationships between spatial syntax feature parameters and traffic flow parameters and develop models to represent these relationships. Finally, in Section 5, we summarize our findings and suggest future research directions.

2. Literature Review

Road traffic flow exhibits time–space characteristics [7,8,9]. To enhance road traffic efficiency, scholars have conducted extensive research on traffic flow characteristics and developed numerous models, including the cellular automata model [10,11] and the viscoelastic model [12]. Researchers have also explored intelligent traffic management strategies [13,14], big data analysis [15,16] and traffic flow prediction methods [17,18] to alleviate traffic congestion, yielding notable results. However, the crux of traffic problems lies in the rationality of the road network structure. Inadequate road network planning cannot be remedied by traffic management measures. Space syntax attempts to explore the rationality of the road network structure at a macroscopic level and from the perspective of travellers and has achieved notable results.

Spatial syntax is a method used to study the relationship between spatial organisation and human society through the quantitative description of spatial structures that contain landscapes, buildings and urban systems [19,20,21]. Spatial syntax emphasises that the relationship between spaces is important, as an essentially mathematical approach based on graph theory [22,23].

The quantitative spatial analysis method in spatial syntax has been used for a wide range of applications in the field of urban traffic [24,25]. Dario Esposito et al. [26] studied the relationship between the spatial structure of an urban environment and the movement patterns of pedestrians using spatial syntax and how the field of perception of agents impacts pedestrians’ route choice. Given that pedestrian trips are composed of two main components—choice of origin and choice of travel route—the degree of choice and integration of spatial syntactic variables can be used to analyse the road network conditions and even to predict traffic flows. Jiang et al. [27] and Tao et al. [28] applied the spatial syntax method to the study of traffic flow forecasting on the basis of the topological structure of the street. In recent years, the rapid development of GIS technology has combined spatial syntax with GIS technology [29], which provides a stronger operability of spatial syntax in dealing with urban traffic problems [30].

Numerous studies have demonstrated the successful application of spatial syntax theory to traffic flow analysis within road networks, presenting a broad array of potential applications. Nonetheless, further investigation is required to enhance the quantitative analysis aspects of spatial syntax. First, recent studies pay more attention to the relationship between the topological structure of streets and the traffic flow [6]; however, two key issues remain to be solved: identifying how many traffic flows can be served and determining the evolution mechanism of traffic flow under one topological structure of roads. Secondly, the lack of sufficient traffic flow data support leads to limited quantitative analysis results. Thirdly, after abstracting the roads into axes based on spatial syntax, the axes form an abstract map, and the differences in traffic flow characteristics are ignored. From the perspective of traffic analysis, researchers have also attempted to relate the spatial structure of the road network to traffic characteristics on the basis of the MFD, but the MFD only builds a model of average flow, average density and average speed within an area [31]. Even though the MFD can express the capacity of a road network [32] or be used in relation to route choice [33], it can not provide the capacity of one road and reflect the coupling relationship between the spatial structure of a road network and the traffic characteristics of a road. Therefore, the results of the MFD cannot be used to evaluate the efficiency of a road network in the urban planning stage.

To overcome the shortcomings outlined above, in this paper, we analyse the relationship between the spatial structure of roads and traffic flow quantity, especially the coupling relationship based on a real road network and the traffic flow data surveyed in the field. Then, we establish the prediction models between the characteristic parameter of spatial syntax and the urban road traffic flow characteristic parameter.

The primary contributions of this study are as follows: (a) Based on the traffic flow database for the main urban area of Huadu District, Guangzhou, the coupling characteristics between the characteristic parameter of spatial syntax (integration degree) and the road traffic flow characteristic parameter are analysed. (b) The coupling relationship model between the degree of integration and the urban road traffic flow characteristic parameter is established.

3. Construction of Spatial Syntactic Model

3.1. Fundamentals of Spatial Syntax

Spatial syntax refers to the computational approach grounded in the axial model that utilizes the principle of ’maximum length and minimum number of axes’ to depict the urban road network. In this representation, the urban road network is illustrated using straight lines, neglecting road width; thus, the entire road is symbolized by a single straight line to convey the spatial context in which the road exists [19]. The construction of the axial model is presented as follows. First, the spatial structure of the road is summarised with the fewest and longest straight lines, as shown in Figure 2a. Secondly, the straight lines are abstracted as traffic axes to form a traffic axis map, as shown in Figure 2b. Finally, the axis relationship is transformed into a topological structure composed of points and lines through the topological metric (i.e., each axis is replaced by a node, and the nodes are directly connected by line segments between axes with intersections, as shown in Figure 2c). In this topology, the number of line segments between nodes is the number of steps, and the number of steps between two adjacent nodes is one. The path with the fewest line segments that connect two nodes is defined as the shortest path, and the number of steps of this path is the topological distance between these two points.

In the axial model, node-to-node link characteristics are taken into account to determine the accessibility and reachability of each node within the road system. The spatial syntactic model further abstracts relevant connections between spaces as topological connection diagrams, which are based on network accessibility and correlation. Moreover, the accessibility of axial nodes is topologically analysed according to the principles of graph theory to finally derive a series of morphological analysis variables.

3.2. Morphological Variables of Spatial Syntax

(1) The depth value (D): refers to the sum of the minimum number of steps from a node to other nodes in an axial system, that is, the sum of topological distances. The topological distance from a node in space to any other node is set as d, and the maximum topological distance of this node in the spatial system is set as S. The depth value [19] of this point can be expressed as

$$D=\sum _{d=1}^{S}d$$

(1)

When $1<d<S$, the computed depth value is the local depth value, and when $d=S$, the computed depth value is the global depth value. In general network analysis, the average depth value ($\overline{D}$) parameter [19] is often used:

$$\overline{D}=D/(n-1)$$

(2)

where n is the total number of nodes in the spatial system, and $(n-1)$ indicates that there are, at most, $(n-1)$ nodes connected to the specified node in an axis diagram with a total of n nodes. Similarly, when $1<d<S$, the average depth value is the local average depth value, and when $d=S$, the average depth value is the average global depth value. (2) Integration (R) reflects the accessibility and convenience of the road, as represented by the line segment over the entire space, which can be used to measure the degree of spatial agglomeration and dispersion. Places with high integration are often locations in cities with high pedestrian and traffic flow and high land use. Corresponding to the depth value, integration is also divided into local integration and global integration. When analysing intelligence, the local integration degree is usually calculated by taking the value of S in Equation (1) as 3, that is, each node reaches the node with the farthest distance of three steps from its own topology. The calculation formula for integration [29] is presented as

$$R=\frac{n\left[{log}_2(\frac{n+2}{3}-1)+1\right]}{(\overline{D}-1)(n-1)}$$

(3)

All parameters in the formula have the same meaning as before. (3) Intelligence refers to whether the position of local space in the whole system and its relationship with the surrounding space are related and unified, reflecting the connectivity and perceptibility in local space. The numerical value is the correlation ($C{R}^2$) between local integration and global integration. The larger the calculated value, the more intelligent the whole space and the better the ability to perceive and distinguish the whole from the local. The space is more correlated at $C{R}^2⩾0.5$ and less correlated at $C{R}^2⩽0.5$ [34].

3.3. Construction of the Abstract Road Network of the Study Area

In this paper, the road network in the central city of Huadu District, Guangzhou City, is selected as the research object. Based on the findings of Wang et al. [35], the study area should typically encompass a large region that includes the subject area under investigation and should be delineated by sections exhibiting the most significant barrier effects. Therefore, the area enclosed by expressways and highways in the periphery of the central city of Huadu District is delineated as the study area, including Huadu Avenue, National Highway 106, Commercial Avenue and Xuguang Highway. Then, the road network axis map within the study area is drawn according to the drawing standard of the axis map. The abstract road network is shown in Figure 3.

4. Construction of Coupling Relationship Model

Numerous variables can be utilized for morphological analysis based on spatial syntax, which can be categorized into local and global variables. Local variables represent specific characteristics within a defined spatial scale, whereas global variables are more appropriate to examine the comprehensive situation of the region. This research on the relationship between road spatial structure and traffic flow is mainly analysed from the overall perspective. Therefore, in this paper, the concept of integration degree is distinguished as the overall integration degree and local integration degree only in intelligent degree analysis, while the rest of the relevant content and the integration degree refer to the global integration degree.

4.1. Integration Degree Analysis of Abstract Road Network

The abstract road network is imported into UCLDepthmap 1.0 software to establish the axial model. Then, the integration degree of the abstract road network is calculated as shown in Figure 4. The integration index is reflected by the chromatography in the road network axis map. The warm colour indicates higher index values, the cold colour indicates lower index values and the specific integration value is shown in

Table 1. Integration degree and measured peak hourly flow statistics for major road sections.

Road Name	Road Grade	Evening Peak Traffic Flow (pcu/h)	Saturation	Integration
Huadu Avenue	Expressway	3952	0.76	1.0600
G106	Trunk road	1701	0.63	0.8628
Commercial Avenue	Trunk road	2052	0.76	0.9780
Xu-Guang highway	Highway	3672	0.51	0.9539
Ziwei Road	Subsidiary road	1428	0.68	1.5434
Sandong Avenue	Trunk road	1872	0.52	1.1320
Gongyi Road	Subsidiary road	1736	0.62	1.6200
Tiangui Road	Trunk road	1755	0.65	1.1840
Shuguang Road	Subsidiary road	1491	0.71	1.1890
Fenghuang Road	Trunk road	1809	0.67	1.2740
Yingbin Avenue	Trunk road	2808	0.78	1.1622
North Jianshe Road	Trunk road	1944	0.72	1.2349

. Overall, the roads with high integration are concentrated in the central location of the city, thereby reflecting the locational advantage of the central area, which also corresponds to the location of the future CBD identified in the latest planning documents of Huadu District (Guangzhou Municipal Planning Bureau 2013) and conforms to the overall circular radial stretching structure; that is, the degree of integration is gradually decreased from the inside to the outside. The integration degree values of many roads, such as North Jianshe Road, Gongyi Road, Tiangui Road and North Fenghuang Road, and Huadu Avenue, Ziwei Road, Sandong Avenue and Yingbin Road, exceed 1.0, making them the first echelon of traffic axes in the main urban area of Huadu District. The interlocking enclosures of urban spaces constitute the integrated nucleus of the whole metropolitan area, which has a strong penetration and integration ability and represents the most dynamic location in the centre of the metropolitan area.

Further analysis of the road network structure in the main urban area of Huadu District shows that the traffic axes with a high degree of integration do not rely on urban expressways, main roads and other high-level road networks that gather a large amount of traffic. Instead, roads such as Ziwei Road, Gongyi Road, etc., are subsidiary roads in the urban centre. This finding shows that the hierarchy of the road network topology connections is very important. The central area’s road network is designed with high integration, which does not necessarily result in increased traffic. The outer road network should be of a high level to solve the problem of transit traffic. The internal and external road networks are connected effectively, compounded functionally and shared reasonably to constitute an efficient road system.

4.2. Intelligence Degree Analysis of Abstract Road Network

According to the principle of statistical hypothesis testing, the null hypothesis is that local and global integration variables are independent, and the alternative hypothesis is that local and global integration variables are correlated. Tests were performed using the Pearson statistic, with p being the significance level, to evaluate whether the correlation between the two variables is statistically significant. Generally, $p<0.05$ indicates a significant correlation between the two datasets. In this paper, the test result is $p<0.001$, indicating that the null hypothesis is rejected; that is, the two variables are not independent. Then, the correlation degree between the two is calculated to be $C{R}^2=0.746$, indicating a strong correlation, which shows the high spatial intelligence of the road network system. This also means that the urban spatial layout of the area is better, thereby reflecting that the existing road network planning in the main city of Huadu District is reasonable. Therefore, the study based on this road network is highly representative and scientific.

The above analysis is conducted from the perspective of spatial syntax. Therefore, higher integration means that the road network can carry more traffic flow and can define its maximum carrying capacity; these questions are studied in the next subsection.

4.3. Coupling Characteristic Analysis of Integration and Traffic Flow Characteristic Parameters

4.3.1. Selection of Traffic Flow Characteristic Parameters

In the analysis of the relationship between integration and traffic flow characteristic parameters, representative parameters should be selected for analysis. Traffic volume is the preferred traffic flow characteristic parameter because of its ease of collection and its ability to reflect changes in traffic demand. However, the traffic volume for various levels of road lacks comparability. This phenomenon occurs because the traffic capacity that can be accommodated is influenced by factors such as road conditions, traffic flow, control measures and environmental conditions. Consequently, the amount of traffic that can be supported varies under different levels of road operating conditions. Therefore, the choice of traffic volume parameters as a characteristic parameter is not reasonable enough, so the saturation is used as a characteristic parameter in this study. So-called saturation refers to the number of standard cars that pass through a road section in an hour and the ratio of the road capacity. The formula is expressed as follows:

$$q_i=Q_i/C_i$$

(4)

where $q_i$ represents the saturation of segment i, $Q_i$ represents the traffic volume of segment i and $C_i$ represents the capacity of the segment. According to Wardrop’s first equilibrium principle, road users consider the influence of road impedance when selecting a travel route. Greater impedance leads to a smaller likelihood of a particular road being chosen. Therefore, changes in impedance directly determine the changes in traffic flow on the road. Road impedance is a function of road traffic running saturation. An increase in saturation increases impedance, and a decline in saturation decreases impedance. Therefore, the saturation, as a characteristic parameter of traffic flow, not only makes the traffic flow characteristics between different levels of roads comparable but can also respond to changes in road traffic flow.

4.3.2. Data Sources and Processing Methods

On 6 August 2021, an evening peak traffic flow survey was conducted within the study area, obtaining the traffic volume of different vehicle types on different road segments. Then, we converted the traffic volume of different vehicle types into that of a passenger car unit (pcu) according to the passenger car equivalents for different types of vehicles. Moreover, the capacity of different road segments can be calculated in accordance with the design parameters. Therefore, the saturation of different road segments can be calculated on the basis of Equation (4). The traffic volume of different segments selected for analysis, along with their corresponding saturation levels, are displayed in Table 1.

4.3.3. Coupling Characteristic Analysis of Integration and Saturation

According to the data in Table 1, a scatter plot of the relationship between integration and saturation is plotted, as shown in Figure 5. The scatter distribution in Figure 5 reveals that no rules can be found between integration and saturation. Furthermore, after calculating the correlation between the two, the results show that the correlation coefficients are all very small, with the biggest being only 0.286, further indicating that the correlation in terms of the current traffic flow situation between the two is not obvious.

However, in Figure 5, the distribution of saturation values is more concentrated when the integration is between 1.1 and 1.3, and the maximum saturation value can reach 0.8. In the case of other integration values, the saturation values are more discrete and smaller. This result suggests that some intrinsic connection may be found between integration and saturation, which is just not fully reflected in the current traffic flow conditions.

4.4. Model Analysis of the Relationship between Integration and Saturation Coupling

To further study the coupling relationship between integration degree and saturation degree, the traffic volume is loaded step by step on the basis of the existing abstract road network to observe the variation pattern between them. To ensure that the spatial layout within the study area is not changed during the step-by-step loading of traffic volume, this study draws on the four-stage method used in traffic planning.

(1) According to the existing spatial layout, the study area is divided into 28 subzones, of which 12 subzones are within the study area and 16 subzones are at the periphery. The results of the subzone division are shown in Figure 6.

(2) The Origin–destination (OD) traffic volume between subzones is projected by applying the OD estimation method, resulting in an OD matrix. OD estimation is the inverse process of traffic assignment, which is based on the following principle:

$V_a$ is the traffic volume of the road segment (a), and $a=1,2,3\dots m$, where m is the number of road segments investigated for traffic volume).

$T_{ij}$ is the trip demand between subzone i and j, $i,j=1,2,3\dots n$ ( n is the number of subzones).

$P_{ij}^a$ is the proportion of trips from i to j through $a,i,j=1,2,3\dots n$.

According to the conservation of the road network traffic flow, for non-zero OD pairs, the inflow and outflow traffic volume are equal; in addition, the traffic volumes of different sections on the same road segment are equal. It is concluded that

$$V_a=\sum T_{ij}{P}_{ij}^a.$$

(5)

Equation (5) is the most basic relationship when estimating the OD matrix using roadway traffic volume. When performing OD estimation, $V_a$ and $P_{ij}^a$ are regarded as known quantities, and $T_{ij}$ is the unknown quantity to be found. Equation (5) constitutes a linear system of equations linking traffic volume to the OD matrix. Theoretically, obtaining $T_{ij}$ by solving a system of simultaneous linear equations is possible as long as one has a sufficient amount of traffic volume on the road network. However, for an actual road network, the road segment traffic volume is often much smaller than the number of OD pairs to be found. Therefore, the OD matrix cannot be determined based on road segment traffic alone. An OD matrix that best reflects the actual situation is currently selected using structured and unstructured methods among multiple sets of feasible solutions. Given that the optimal solution of $T_{ij}$ cannot be provided by Equation (5), the optimal solution of $T_{ij}$ is generated by introducing a convex objective function with the linear system of equations in Equation (5) as the constraint to form a mathematical programming problem with the optimal estimate as the starting point. Usually, the objective function is in the form of

$$minf(T,t,V,v)$$

(6)

where T is the OD matrix obtained by OD estimation, V is the observed roadway flow, t is the a priori OD matrix and v is the traffic volume of the roadway assigned by T. In this paper, TRANSCAD 6.0 software is applied for OD estimation. Some of the results are listed in

Table 2. Partial OD matrix calculated by applying the OD backpropagation method.

Subzone Number	1	2	3	4	5	6	7	8	9
1	0	53	60	203	122	136	200	153	93
2	53	0	52	60	932	190	60	60	64
3	60	52	0	326	205	136	326	156	205
4	198	60	325	0	300	58	60	869	300
5	86	977	217	311	0	125	311	144	60
6	134	199	134	59	115	0	56	52	188
7	188	60	325	60	300	55	0	60	110
8	149	60	156	886	138	52	60	0	138
9	99	65	217	311	60	164	121	144	0

.

(3) Overall expansion of the original OD matrix and traffic reassignment.

Under the assumption that no changes are found in the original spatial layout, the original OD matrix as a whole is expanded by multiplying it by a certain multiple. The traffic assignment is made on the basis of this expanded OD matrix to obtain the traffic volume of the road segment. In this study, expansion multipliers of 1.2 times, 1.4 times, 1.6 times and 1.8 times are used to obtain the traffic volumes of the road segments for different expansion multiplier cases, which, in turn, allows for the calculation of the corresponding saturation. The results are shown in

Table 3. Saturation and integration of each road segment after expansion.

Road Name	Road Grade	Saturation				Integration
		1.2 Times	1.4 Times	1.6 Times	1.8 Times
Huadu Avenue	Expressway	0.802	0.810	0.873	0.959	1.0600
G106	Trunk road	0.789	0.803	0.893	0.947	0.8628
Commercial Avenue	Trunk road	0.814	0.822	0.894	0.955	0.9780
Xu-Guang highway	Highway	0.747	0.812	0.833	0.854	0.9539
Ziwei Road	Subsidiary road	0.872	0.910	0.911	0.918	1.5434
Sandong Avenue	Trunk road	0.768	0.842	0.866	0.899	1.1320
Gongyi Road	Subsidiary road	0.852	0.900	0.912	0.923	1.6200
Tiangui Road	Trunk road	0.808	0.835	0.847	0.877	1.1840
Shuguang Road	Subsidiary road	0.811	0.833	0.868	0.874	1.1890
Fenghuang Road	Trunk road	0.793	0.865	0.879	0.884	1.2740
Yingbin Avenue	Trunk road	0.835	0.844	0.855	0.892	1.1622
North Jianshe Road	Trunk road	0.805	0.866	0.873	0.883	1.2350

.

The saturation in Table 3 reflects the evolution of the saturation of each road segment in the road network under different OD matrices. Evidently, the saturation of the road network increases in accordance with the OD traffic volume, which is consistent with the actual situation. The variation pattern of integration with saturation is further analysed, and the coupling between the two can be found by plotting the relationship between integration and saturation, as shown in Figure 7.

Figure 7 shows that with the increase in OD traffic volume and with the constant spatial structure of roads being unchanged, the coupling relationship between the spatial structure and the traffic volume of major roads within the study area evolves. The relationship between saturation and integration takes on a more complex structural form with the increase in OD traffic volume. As shown in Figure 7a, when the road network carries 1.2 times the original OD traffic volume, the dispersion of the scatter plot is decreased relative to that shown in Figure 5. That is, with the increase in OD traffic volume, the traffic volume on each road segment in the road network not only gradually increases but also tends to gather. In Figure 7b, the relationship between integration and saturation is more pronounced when the road network carries 1.4 times the original OD traffic volume, with an almost linear correlation. In Figure 7c,d, the relationship between saturation and integration is changed significantly when carrying 1.6 times and 1.8 times the OD traffic volume, and the scatter plot shows dissociation and clustering effects, which are clearly divided into multiple clusters.

From Figure 7, we also can see that the saturation of different road segments increases with the increase in OD traffic volume. Because of the influence of the road impedance function, a road segment with large road resistance allocates relatively less traffic— and vice versa—thereby achieving overall road network traffic distribution results that gradually tend toward balance. To illustrate the saturation–variation relationship further, the saturation heat map of the road network is drawn in Figure 8, with the red colour representing the highest saturation. Figure 8a,b are heat maps for the 1.4× and 1.8× original OD traffic volume, respectively. These figures clearly show that with the increase in traffic demand, the saturation of all segments continues to increase, and the saturation of the whole road network continues to increase.

However, during this evolutionary process, the level of increase in saturation differs depending on the road segment, and the degree of integration has a considerable impact on the traffic flow distribution of the road network. For example, in Figure 7a–c, the two roads with the highest integration, namely Ziwei Road and Gongyi Road, carry the highest traffic volume loads. These two roads are precisely the roads within the core area. Although both of these roads are subsidiary road grades, they play a key role in the road network. However, in Figure 7d, the three roads with integrations less than 1.2 have the highest saturation.

Moreover, as shown in Figure 7c,d, the integration of the peripheral road network is relatively low, whereas the saturation level increases sharply when the OD matrices are 1.6 and 1.8 times the original OD matrix, with the saturation of Huadu Avenue, G106 National Road and Commercial Avenue reaching 0.95. The increase in the saturation of Xuguang Highway is relatively moderate because of its large capacity, but it also reaches 0.85, which is close to the limit of the service level for the highway.

Further analysis of the saturation in the central area reveals that as the OD traffic increases from 1.6 times to 1.8 times, the saturation of each segment does not change a great deal and remains at approximately 0.88. The two roads in the most central area, Ziwei Road and Gongyi Road, remain largely saturated at around 0.92, with no significant increase. Thus, in the case of relatively high traffic volume, the peripheral roads exert an obvious role in decongestion, and attention should be paid to improving the peripheral road grade and increasing the capacity. When the internal roads reach a saturation level of approximately 0.88, their saturation level becomes less feasible to be increased further. At this point, increasing the density of the road network is an appropriate means to decongest traffic flow.

Model Analysis of the Relationship between Integration and Saturation

The relationship diagram in Figure 7 shows that although integration and saturation are coupled, their relationship may be modelled in different ways, including by approximate linear, linear and curvilinear models, because the OD traffic volume varies. Then, which kind of model is reasonable and fit for use in traffic plans should be analysed further.

First, the relationship models depicted in Figure 7c,d are not expected to occur in real road systems, as the saturation of the segments is too high, and the use of these two states as planning objectives in the road network planning process is not possible. Secondly, in the relationship model depicted in Figure 7a, the saturation is between 0.75 and 0.85, the saturation of most segments is lower than 0.8 and the road network function is not fully utilised. Therefore, it is also not the goal of road network planning. In contrast, in Figure 7b, the saturation of most segments is between 0.80 and 0.9, which is a rather ideal state, although the road system network operation is in a critical state. A slight disturbance may cause traffic congestion, and the traffic function of the road can be given full play. The critical state at this time can be used as the goal of road network planning. Furthermore, the relationship between integration and saturation for different OD traffic volume cases is fitted. The degree of fit is calculated, and the curve is plotted in Figure 9 to show the effect of the fit.

The curve in Figure 9 varies as a single-peak function. On the left side of the peak, the correlation between saturation and integration increases from weak to strong with the increase in OD traffic volume. On the right side of the peak, the correlation between saturation and integration weakensed as the volume of OD traffic increases. At the peak of the curve, the correlation coefficient between integration and saturation exceeds 0.95, indicating that the traffic volume condition at this time fully reflects the intrinsic relationship between the spatial layout of the road system and the road saturation. Therefore, taking the OD traffic volume condition at the peak of the curve as the prerequisite for constructing the relationship model between saturation and integration is reasonable.

To this end, a linear relationship between road saturation and integration is modelled on the basis of 1.4 times the OD traffic volume, as shown in Equation (7), with a model correlation coefficient of $R^2=0.957$.

$$y=0.67113+0.14713x$$

(7)

where y is the road saturation, the value of which should theoretically be more than 0.8, and x is the road integration.

4.5. Decision of Threshold of Integration

During the traffic planning stage, the topological structure of the road network is determined by planners or designers based on design specifications, and adopting different saturations as the design basis for different classes of roads is the requirement of these specifications. In Equation (7), once the saturation of a segment is given, reasonable integration can be decided using Equation (8).

$$x=(y-0.67113)/0.14713$$

(8)

For example, if the saturation of one segment is designated as 0.85, then the reasonable integration threshold is 1.22. After comparing the actual integration of the planned road with the integration threshold, if the actual integration is less than the threshold, there is room for increased road network density. Conversely, if the actual integration is more than the threshold, the road network density may be too high, and the saturation of road network may not have enough space to increase, with the loss even outweighing the gain.

5. Conclusions

In this paper, selecting integration as the typical parameter of the spatial structure of the road network and saturation as the typical parameter of the traffic flow characteristics, the intrinsic relationship between the spatial structure of the road network and road traffic flow was studied in depth, and the following findings were obtained:

(1)

The spatial structure of the road network and traffic flow show a coupled relationship. The coupling pattern in the core area is different from that in the peripheral area. In the core area, the integration of segments is high, with excessive saturation in these segments with an increase in the OD demand, followed by stable saturation. In the peripheral regions, the integration of segments remains minimal, and the saturation of these segments gradually increases. However, when the OD demand becomes significantly high, the saturation in the peripheral areas may surpass that in the core area. Therefore, a dense road network is reasonable in the core area, and high capacity is important in the peripheral area.

(2)

When the saturation of the road network is low, for example, less than 0.75, as in this paper, the relationship between integration and saturation is not obvious. When the saturation of the road network is high, for example, between 0.75 and 0.9, as in this paper, the relationship between integration and saturation is evident. Therefore, in the traffic planning stage, high OD demand should be used to test the toughness of the road network.

(3)

When the saturation of the road network takes different values, the theoretical models between integration and saturation are different. Among these models, the linear model has the best fit. The linear model discovers the nature of the relationship between integration and saturation, and according to the model, the threshold of integration for one segment can be obtained. This threshold can supply the critical value of integration for traffic planning.

Hiller explained that long streets and roads form foreground street networks; these streets are attractive for high levels of through movement. The background network is largely made up of short streets that tend to intersect with other streets, generally located in residential areas [19]. Many other researchers have attempted to verify the relationship between the spatial structure of road networks and the traffic flow [29,30]. However, there are very few studies that attempt to connect the spatial structure characteristics with the traffic flow characteristics. Although preliminary studies were conducted to quantify the relationship between the spatial structure of road networks and road traffic flow, as reported in this paper, many areas are still unexplored. First, the impact of changes in the road network structure on traffic flow distribution was not considered in this paper. Secondly, the OD traffic loading process considered in this paper, which adopts the same expansion multiplier, is an ideal situation and lacks analysis of unbalanced loading. Thirdly, the integration degree used in this paper is the overall integration degree, and analysis derived from the local integration degree may lead to deeper patterns. Moreover, the rule of the relationship between urban road spatial structure and traffic flow only depends on the data in a city, which are not enough. These weaknesses represent directions in which the authors of this paper can continue this research in the future.