As an important component of the urban adaptive traffic control system, subarea partition algorithm divides the road network into some small subareas and then determines the optimal signal control mode for each signalized intersection. Correlation model is the core of subarea partition algorithm because it can quantify the correlation degree of adjacent signalized intersections and decides whether these intersections can be grouped into one subarea. In most cases, there are more than two intersections in one subarea. However, current researches only focus on the correlation model for two adjacent intersections. The objective of this study is to develop a model which can calculate the correlation degree of multiple intersections adaptively. The cycle lengths, link lengths, number of intersections, and path flow between upstream and downstream coordinated phases were selected as the contributing factors of the correlation model. Their jointly impacts on the performance of the coordinated control mode relative to the isolated control mode were further studied using numerical experiments. The paper then proposed a correlation index (CI) as an alternative to relative performance. The relationship between CI and the four contributing factors was established in order to predict the correlation, which determined whether adjacent intersections could be partitioned into one subarea. A value of 0 was set as the threshold of CI. If CI was larger than 0, multiple intersections could be partitioned into one subarea; otherwise, they should be separated. Finally, case studies were conducted in a reallife signalized network to evaluate the performance of the model. The results show that the CI simulates the relative performance well and could be a reliable index for subarea partition.
When the networkwide traffic signal control strategy is implemented to an urban road network, subarea partition must be conducted to divide the network into some small subareas. The subarea can be classified into three types. The first type only includes one signalized intersection. The second type includes several intersections located on one artery. The third type includes several intersections located on some intersected arteries. For the three type subareas, isolated signal control mode, arterial signal coordination mode, and area signal coordination mode should be implemented to the intersection(s), respectively. Accordingly, subarea partition is the basis for networkwide signal control. A well subarea partition algorithm can improve the adaptive level of the signal control system significantly [
In fact, the most important part of subarea partition is calculating the correlation degree among adjacent signalized intersections. The purpose of subarea partition is to improve the whole control performance of the network. If the performance could be improved, then adjacent intersections can be partitioned into one subarea. Otherwise they should be partitioned into different subareas. Thus, a straightforward method for subarea partition is to calculate and compare the control performances under the coordinated control mode and the isolated control mode. However, this method is not easy to implement in practice although it is theoretically straightforward. This is because it would be time consuming to iteratively estimate the performances of hundreds of adjacent intersections in the network. Thus, an effective alternative is to propose an index that reflects the changes in control performance indirectly; this is called the correlation index (CI) in this paper.
Because of the importance of subarea partition, the related researches have been conducted from the 1970s [
As described above, some valuable results have been achieved on the subject of correlation model development. However, there are still some deficiencies that needed to be resolved, which are summarized as follows.
The contributing factors to the correlation index and the model structure are mainly determined according to the past experience of traffic engineers. Most weighting factors and parameters are fixed and cannot vary as the traffic state changes. Thus, the developed models cannot quantify the correlation between adjacent intersections accurately.
The threshold value for the correlation has not been theoretically justified. The threshold value directly determines whether adjacent intersections can be partitioned into one subarea. However, in existing studies, the threshold value is again determined based on the experience of engineers. Accordingly, the subarea partition results may not be optimal.
The existing studies have not yet developed a model that can measure the correlation among multiple intersections. In most cases, one subarea includes more than two intersections. Thus, it is inevitable to calculate the correlation among multiple intersections. However, existing studies only calculate the correlation between two adjacent intersections.
Targeting the realtime traffic control systems, an appropriate correlation index should reflect the dynamic nature of traffic flow and interactions among adjacent intersections. In this paper, we propose an approach that can estimate the dynamic correlation among multiple adjacent intersections for the purpose of subarea partition. The key innovation of this study is that the contributing factors are expressed as variables and the relation model between correlation and the variables is developed. The correlation can be calculated dynamically as the traffic state changes. Moreover, our modeling approach provides a reasonable way to determine the threshold value of correlation at which adjacent intersections should be grouped into one subarea.
The determination of contributing factors for subarea partition is the basis for developing the correlation model. Correlation model is the core of subarea partition, thus the principles of subarea partition must be obeyed when developing the model. If all intersections in the urban network execute isolated signal control mode, it is unnecessary to develop the correlation model because isolated control mode only concerns the intersection itself. However, when we identify whether multiple adjacent intersections can be partitioned together to execute signal coordination mode, the factors that may affect signal coordination should be analyzed. Thus, the factors that can improve signal coordination performance are also those that affect subarea partition. In this paper, the following four factors are selected.
However, if we assume there is no difference between the cycle lengths of adjacent intersections, the link lengths are exactly right, and the path flows are great enough, then these three factors would not have negative impact on signal coordination performance. In such situation, the larger the number of intersections in the subareas, the better the coordination performance would be. Therefore, from this perspective, the number of intersections would be a direct contributing factor to subarea partition.
Let us take two adjacent signalized intersections
In (
If PI is larger than 0, then the signal coordinated mode achieves better control performance than the isolated control mode; accordingly
The performance of the signal coordination is also affected by the coordination type (i.e., onedirectional or twodirectional coordination); different coordination types will produce different partition results. In this paper, we only study the relations between the correlation index and the contributing factors under twodirectional coordination conditions, which is useful to judge whether adjacent intersections can be partitioned into one subarea to execute twodirectional coordination algorithm.
The best way to analyze the impact of a contributing factor on PI is to determine the relationship between them based on theoretical derivation and then calculate the change in PI resulting from a unit change in the contributing factor. However, the relationship is difficult to determine due to the complex nature of vehicle movements between consecutive intersections. There is no universal function that can be used to calculate
An important issue that should be noted is that the traffic control objective is closely related to the traffic state of the intersection. For example, the control objective for an oversaturated intersection is usually set as minimizing the queue length to avoid queue spillovers, while that of an unsaturated intersection is usually set as minimizing vehicle delay. In this paper, we focus on unsaturated intersections and minimizing vehicle delay is selected as the control objective. Thus, in (
Differences in cycle lengths only affect the control performances of the seed intersection and nonseed intersection. The link length and path flow only affect the performances of two adjacent intersections. Therefore, a network composing of two intersections is selected to conduct numerical experiments on these three factors.
Again taking intersections
Results of the experimental scenarios.
Scenario  CCL/s  OCL of 


Improvement ratio 

1  125  90  351576  20044  5.70% 
95  358915  25438  7.09%  
100  371277  34919  9.41%  
105  378642  40453  10.68%  
110  390619  50311  12.88%  
115  399471  57848  14.48%  
120  408876  65726  16.07%  
125  416925  71878  17.24%  
 
2  110  90  322504  22407  6.95% 
95  331007  28399  8.58%  
100  342658  37425  10.92%  
105  351739  44573  12.67%  
110  360691  51658  14.32% 
For example, in scenario 1 when the OCL of
The data shown in Table
As can be seen from Table
Scatter diagrams between CI and
Scatter diagrams between CI and the normalized difference in cycle lengths.
There is a strong linear relationship between CI and
Linear functions can fit the scatter diagrams very well, as indicated by the high
Theoretically,
In the function,
Numerical experiments are conducted to establish the relationship between the link length
Scatter diagrams between
In Figure
In numerical experiments,
When PI achieves its maximum at
Following detailed analysis, it was concluded that a piecewise linear function may be suitable to fit the curve. Moreover, the differences between adjacent minimal values of CI are nearly constant, while the differences between each maximal value of CI and its previous minimal value are also nearly constant. For scenarios 1 and 2, the extreme values of CI corresponding to integer multiples of
Values of CI corresponding to the extreme points.
Scenario  First minimum  First maximum  Second minimum  Second maximum  Third minimum 

1  0.663  1  0.533  0.858  0.4 
2  0.674  1  0.527  0.816  0.372 
In scenario 1, the difference between the first maximum and the first minimum equals 0.337. The difference between the second maximum and the second minimum equals 0.325. The two numbers are very close. The difference between the first minimum and the second minimum equals 0.130, while the difference between the second and third minimum equals 0.133. These numbers are also very close. In scenario 2, the same phenomenon is observed. The extreme points of the
Scatter diagram between
In the numerical experiments,
To fit the curve in Figure
The regression functions are shown in (
Then,
The
For scenario 1, the function between
The results of the
Let
Again taking intersections
Table
Impact of the difference between
Scenario 








1  0%  265103  77701  0  0  0  0.00% 
5%  271860  72611  5090  6757  1667  3.08%  
10%  276639  69391  8310  11536  3226  5.95%  
15%  283045  67884  9817  17942  8124  14.99%  
20%  287528  64228  13473  22425  8951  16.52%  
25%  297323  59695  18006  32220  14214  26.23%  
30%  299637  57025  20676  34534  13858  25.57%  
35%  305460  53310  24391  40357  15965  29.46%  
40%  309357  49541  28160  44254  16094  29.69%  
 
2  0%  198840  59700  0  0  0  0.00% 
5%  202996  58878  613  4156  3543  5.95%  
10%  206948  57892  3032  8107  5075  8.52%  
15%  210717  56741  5926  11877  5950  9.99%  
20%  212268  56461  7176  13428  6253  10.49%  
25%  215648  54948  9405  16808  7403  12.43%  
30%  218853  53261  11658  20013  8354  14.02%  
35%  221884  51394  12925  23044  10118  16.98%  
40%  224744  49342  14916  25903  10988  18.44% 
In Table
Scatter diagrams between normalized path flow and CI.
As can be seen from Figure
The problem is that, as in the earlier case, the slopes of the two functions are different. The slope may again be affected by our previously used set of factors, namely the CCL, the green split and the
As in the earlier case,
A total of 10 groups of data are used in the regression and the result is shown in
In the above numerical experiments, the path flows in the two directions are assumed to be the same. However, in reality they may be different. Further numerical experiments are conducted and show that
The number of intersections in the subarea affects the number of coordinated traffic streams. As was shown in Figure
Sketch of three adjacent signalized intersections.
In the numerical experiments, the values of the three other contributing factors are fixed. The difference in cycle lengths is set to 0. The link lengths between adjacent intersections are
Results for the four experimental scenarios.
Scenario 1  Scenario 2  Scenario 3  Scenario 4  


Improvement ratio 

Improvement ratio 

Improvement ratio 

Improvement ratio 
2  8.81%  2  15.35%  2  13.70%  2  17.10% 
3  12.35%  3  20.30%  3  17.39%  3  20.39% 
4  14.92%  4  23.37%  4  18.21%  4  21.65% 
5  15.96%  5  24.56%  5  19.76%  5  22.36% 
6  16.34%  6  25.40%  6  20.55%  6  23.16% 
7  17.20%  7  26.20%  7  21.38%  7  23.90% 
In Table
Since there is no difference between the cycle lengths in each scenario, the lengths of the links are optimal and the path flows are equal to their maximum historical values. Therefore, when
The correlations between the intersections shown in Table
Scatter diagram between number of intersections in the subarea and CI.
From the figure, we can see that, with an increase in
The
CI equals 1 when
The
In Sections
In (
As analyzed above, the difference in cycle lengths, the path flow, and the link length only affect the correlation of two intersections. Therefore, the three related correlations are subtracted from 1. However,
PI is the improved network performance when signal coordination is implemented. When PI is larger than 0, the coordination of adjacent intersections can obtain better performance than the isolated control mode. In such situations, the two can be grouped into the same subarea; otherwise, they must be placed in different subareas.
In this paper, the CI is developed in order to simulate PI. When PI is larger than 0, CI is also larger than 0. Therefore, 0 is set as the threshold value of CI, to determine whether adjacent intersections should be placed in the same subarea.
In this section, field experiments are conducted to validate the correlation model developed in this paper. The experimental setup and then the results are described in the following subsections.
A small road network composed of four signalized intersections in Harberin, China, is selected as the experimental network. The four intersections are all located in the central area of the city. A sketch and phasing diagram of the network are shown in Figure
Sketch and phasing diagram of the investigated network.
An excellent CI should reflect the performance of the coordinated control mode relative to the isolated control mode precisely. Namely, the correlation should vary as the relative performance varies. Thus, five scenarios are tested. Because the signal coordinated mode is not suitable for those intersections facing oversaturated or close to oversaturated conditions, the five scenarios are all selected during offpeak hours. The duration of each scenario is 15 minutes. Table
Survey times of the five experimental scenarios.
Scenario  Survey time 

1  9:01 am~09:15 am 
2  9:16 am~09:30 am 
3  9:31 am~09:45 am 
4  9:46 am~10:00 am 
5  10:01 am~10:15 am 
Traffic data are collected every 5 minutes and the timing schemes are optimized and updated every 15 minutes. Thus, the correlation among the four intersections is calculated once in each scenario. The calculation of the correlation requires some basic traffic parameters, such as cycle lengths, green splits of coordinated phases, and saturation degrees. The values of the parameters are shown in Table
Parameters used to calculate the correlations among the intersections.
Scenarios  Intersections 







Scenario 1  1  108  0.31  0.89  0.79  0.27  0.51 
2  114  0.32  0.89  0.80  0.29  0.51  
3  102  0.29  0.88  0.77  0.25  0.52  
4  106  0.27  0.88  0.78  0.24  0.54  
 
Scenario 2  1  105  0.31  0.88  0.78  0.27  0.51 
2  113  0.33  0.89  0.80  0.29  0.50  
3  105  0.3  0.88  0.78  0.26  0.52  
4  97  0.28  0.87  0.76  0.24  0.52  
 
Scenario 3  1  103  0.3  0.88  0.78  0.26  0.51 
2  110  0.29  0.89  0.79  0.26  0.53  
3  101  0.31  0.88  0.77  0.27  0.50  
4  95  0.29  0.87  0.76  0.25  0.51  
 
Scenario 4  1  98  0.28  0.87  0.77  0.24  0.52 
2  101  0.3  0.88  0.77  0.26  0.51  
3  94  0.27  0.87  0.76  0.23  0.52  
4  92  0.31  0.86  0.75  0.27  0.48  
 
Scenario 5  1  95  0.32  0.87  0.76  0.28  0.48 
2  99  0.33  0.87  0.77  0.29  0.48  
3  93  0.31  0.86  0.75  0.27  0.48  
4  88  0.31  0.86  0.74  0.27  0.47 
As can be seen from Table
Table
Results for the five experimental scenarios.
Parameters  Scenario 1  Scenario 2  Scenario 3  Scenario 4  Scenario 5 


0.86  0.81  0.82  0.91  0.89 

0.71  0.81  0.76  0.80  0.83 

0.81  0.62  0.61  0.74  0.69 

0.71  0.74  0.73  0.72  0.74 

0.86  0.84  0.87  0.74  0.82 

0.83  0.88  0.80  0.82  0.82 

0.84  0.78  0.82  0.87  0.85 

0.79  0.75  0.78  0.86  0.88 

0.77  0.84  0.81  0.86  0.91 

1.07  1.08  1.100  1.100  1.08 

0.42  0.38  0.37  0.48  0.51 
PI/s  229740  212245  236980  196572  184634 
Improvement ratio  26.3%  22.5%  20.7%  33.8%  34.4% 
From scenario 1 to scenario 5, the integrated correlations of the four intersections are 0.42, 0.38, 0.34, 0.51, and 0.55, respectively. Because the threshold value is 0, in the five scenarios, the four intersections can be grouped into one subarea. To test whether CI reflects PI accurately, we also give the values of PI and the improvement ratio of the signal coordination mode to the isolated control mode.
However, CI and the improvement ratio of PI have different dimensions and cannot be compared directly. Thus, the data for CI and the improvement ratio in Table
Normalized data for CI and the improvement ratio.
Normalized CI  0.357  0.071  0.000  0.786  1.000 

Normalized improvement ratio  0.409  0.131  0.000  0.956  1.000 
The scatter diagrams for the scenarios between the normalized CIs and improvement ratios are shown in Figure
Comparison between normalized CI and improvement ratio.
From Figure
An
Statistical results of
Sample size  Significance level 







 
5  0.95  0.16  0.90  2.31  −0.20 
From the experimental results, we can see that the CI is closely related to the change in the PI, which indicates that the CI can be used as an alternative to the PI. Therefore, the correlation model developed in this paper is suitable for subarea partition.
This paper developed a correlation model for multiple adjacent intersections by taking into consideration four contributing factors: the difference between cycle lengths, the link length, the path flow between the upstream and downstream coordinated phases, and the number of intersections. A case study was used to explain the model and the results show that it is reliable for subarea partition.
In this paper, we selected four typical signal intersections located on an arterial road under moderately congested traffic conditions for our experiment. Although the traffic and signal timing conditions of the selected sites are pretty typical, the conclusions may still not be transferable to other intersections due to the complexity and diversity of reallife situations. Future research is necessary to develop statistical models to quantitatively assess the generality of these findings.
The authors are grateful to LiangTay Lin and ShouMin Tsao. Their paper helped us to find some useful related literature and provided us with some meaningful research ideas. This study is supported by the China Postdoctoral Science Foundation funded project (no. 2013M530159) and National Natural Science Foundation of China (no. 61304198).