Research on Calculating Traffic Capacity in Extra-Long Subsea Tunnels—A Case Study of the Qingdao Jiaozhou Bay Subsea Tunnel

Abstract

Analyzing the traffic capacity of extra-long tunnels is crucial in assessing their sustainable capacity. However, previous studies on tunnel capacity mainly considered the influence of a single factor, ignoring the interaction between multiple factors, which cannot reflect the actual tunnel capacity. Therefore, considering the influence of multiple factors, this paper constructs an actual capacity calculation model for extra-long tunnels. Firstly, by combining hierarchical analysis and the entropy method, we determined the key factors that influence the capacity of extra-long tunnels. Secondly, based on the constructed traffic simulation model, we constructed an actual capacity model of extra-long tunnels by using multiple non-linear regression equations and tested the goodness of fit with the help of the misfit term. Finally, we determined the key correction coefficients of the model using the difference proportion method. Taking Qingdao Jiaozhou Bay undersea Tunnel as an example, the research results show that the method proposed in this paper can accurately determine the tunnel capacity with an error of less than 4%, providing a theoretical basis and practical guidance for the management and control of the tunnel’s sustainable carrying capacity after traffic congestion.

1. Introduction

Extra-long subsea tunnels are unique pieces of infrastructure that require specific research to optimize their performance. These tunnels present different challenges compared to standard highway tunnels, such as more longitudinal sections and varying slopes, leading to vehicle inefficiency and traffic congestion [1]. Additionally, driving for long distances within these tunnels can increase the risk of driver fatigue and inattention, which could potentially lead to traffic accidents. Therefore, investigating the capacity of extra-long subsea tunnels is crucial to enhancing the efficiency and safety of vehicle traffic within the tunnel and to promoting the sustainable development of these tunnels.

Previous studies have explored various factors that affect capacity in subsea tunnels. For instance, the Highway Capacity Manual has identified road condition, traffic, and technical factors as critical elements influencing highway capacity [2]. Yang et al. [3] investigated the impact of lane-changing behavior on capacity reduction, while Anders et al. [4] studied the effect of public information on highway tunnel capacity. Mostafa et al. [5] investigated how tunnel structure affects capacity, and Cai [6] used numerical simulation to determine the main influencing factors on traffic capacity, including slope, slope length, and the proportion of large vehicles. Marcel et al. [7] noted that lane change rates could lead to decreased road capacity, with a higher rate of lane changes resulting in lower observed traffic flow. Other studies, such as Zhou et al. [8], Hashim et al. [9], Ghiasi et al. [10], Cheng et al. [11], Chang et al. [12], and Hisanaga [13], have examined various factors such as road alignment, design speed, the presence of auxiliary lanes, mixed passenger and freight traffic, road radius, and the number of connected and autonomous vehicle (CAV) lanes, among others, and their impact on tunnel capacity using different analytical methods.

Research on subsea tunnel capacity has a long history dating back to the 1950s. Early studies, such as those by Bressler [14] and Hughes [15], used theoretical analysis to estimate tunnel capacity, but the limited accuracy of their models led many scholars to develop capacity models from different perspectives using various analytical methods. For example, Wada et al. [16] studied tunnel bottleneck capacity using the continuous following model, while Ghiasi et al. [10] analyzed the relationship between CAV lanes and capacity using a Markov chain and proposed a mixed traffic flow capacity model. Yuan et al. [17] developed a capacity calculation model based on time headway and applied it to an underpass tunnel. Dhamaniya et al. [18] established a simultaneous velocity equation for different vehicle classes to determine the capacity of an urban trunk road, while Fu et al. [19] evaluated tunnel section capacity based on the Greenshields traffic flow model and compared it with adjacent ordinary sections. Due to the high data requirements of these methods and the lack of sufficient data, many scholars have turned to simulation-based capacity models. For instance, Beheshtitabar et al. [20] used a VISSIM model to investigate the impact of increased look-ahead distance at the Hampton Road Bridge Tunnel in Virginia, while Fang et al. [21] developed an urban tunnel capacity model based on VISSIM and analyzed the impact of variable lanes on capacity. Ye et al. [22] analyzed the impact of driverless driving on capacity using a road cellular automaton model based on the two-state safety speed model, and Winkler and Fan [23] built a capacity model for high-speed traffic paths based on VISSIM simulations to evaluate the effects of lane restrictions, driver behavior, and interchange density on expressway capacity.

Most tunnel capacity standards currently refer to highway or municipal road standards due to a lack of standardized capacity calculation standards for extra-long subsea tunnels. On the basis of quantifying the relationship between multiple factors and capacity, a tunnel capacity model based on simulation was proposed, which reflected the actual capacity of the tunnel under different routes and control measures in detail, and provided theoretical support for improving the sustainable carrying capacity of the tunnel by optimizing the linear and control conditions of the tunnel. The main contributions of this paper follow:

• This paper aims to examine the impact of various factors related to the road environment and control measures on tunnel traffic capacity. Unlike the existing literature [3,4,5,6,7,8,9,10,11,12,13], which focuses mainly on single factors and neglects the interplay between multiple factors, this paper considers a combination of factors that could potentially influence traffic capacity. By analyzing both single and multiple factors, the paper seeks to provide a more comprehensive understanding of the complex dynamics involved in tunnel traffic capacity. Overall, this approach enhances the existing body of research on this topic and provides valuable insights for policymakers, engineers, and other stakeholders involved in the design and management of tunnels.
• The paper suggests a novel screening method for determining the influencing factors of tunnel capacity, which is based on the analytic hierarchy process (AHP) and the entropy weight method. This approach improves upon the subjective judgment method that has been traditionally used in most of the existing literature [6,8,9,11,12]. By combining AHP with the entropy weight method, the proposed screening method can effectively quantify the relative importance of different factors and minimize the impact of subjective biases. This enables more objective and accurate identification of the factors that have the greatest influence on tunnel capacity. The use of this screening method provides a valuable contribution to the field of tunnel engineering and can help guide the design and management of tunnels in a more effective and efficient manner.
• The paper discusses the various methods that have been used in existing traffic capacity model research and divides them into three categories [14,15,16,17,18,19,20,21,22,23]: theoretical analysis, data mining, and simulation.

  Table 1. Comparison of existing capacity model construction methods.

  	Theoretical Analysis	Data Mining	Simulation
  Pertinent literature	[14,15]	[16,17,18,19]	[20,21,22,23]
  Advantage	The calculation formula is easy to deduce and the calculation speed is fast.	A large amount of data on tunnel transportation systems can be mined, from which key factors can be extracted, and the corresponding mathematical model can be established to evaluate the traffic capacity.	It can take into account the influence of different factors, such as vehicle flow, speed, vehicle type, etc., with high accuracy, and it can simulate the actual operation of the tunnel transportation system.
  Deficiency	Not accurate enough, limited by data quality and sample size, not applicable to all types of tunnels, and unable to take into account the complexity and randomness of traffic flow and other factors.	It requires a large amount of data support and data preprocessing and analysis. The accuracy of the model is limited by the quality and integrity of the data, and the establishment and maintenance of the model require more time and resources.	More data and computing resources are required, and the computational complexity is high, so the parameters and assumptions of the model need to be set in advance.

  provides a comparison of the three methods. Theoretical analysis, although useful in some cases, neglects the interaction between vehicles in traffic flow and can result in traffic capacity results that are far from the actual situation. As a result, its current application is limited. Given the difficulty in collecting actual traffic data in very long tunnels with many factors influencing traffic capacity, this paper proposes the use of simulation methods. Simulation methods can help compensate for the lack of traffic observation data and allow for the reproduction of actual traffic capacity in tunnels under different combinations of influencing factors. Therefore, the use of simulation methods can provide valuable insights for tunnel engineers and other stakeholders involved in the design and management of tunnels, ultimately leading to more effective and efficient tunnel systems.

This paper is structured as follows: Section 2 examines the factors of influencing extra-long tunnels capacity and contructs the capacity model of these tunnels; Section 3 validates the built model; Section 4 presents the conclusions and outlook.

2. Methods

2.1. Overview of the Extra-Long Subsea Tunnel Capacity Model

Since the traffic operating conditions of extra-long subsea tunnels are different from those of general urban roads or highways, the recommended capacity values in the relevant manuals are not adaptable. In this paper, the existing road capacity calculation method is used as the basis, and the geometrical conditions and control measures of the tunnel are combined to establish a capacity calculation model applicable to extra-long subsea tunnels. The main process is shown in Figure 1.

In terms of tunnel simulation model selection, the traffic simulation models that are currently used more often are VISSIM, SimTraffic, CORSIM, etc. The main characteristics of each model are as follows:

(1)

VISSIM: It can be used to model and analyze the operation of urban traffic and public transportation under various traffic conditions.

(2)

SimTraffic: The companion software of Synchro, which has good adaptability in urban-oriented traffic network simulation.

(3)

CORSIM: It can be used for microscopic simulation models of urban traffic networks and highway networks, and its highway network simulation model is more complete.

(4)

Transmodeler: Combines macro, meso, and microsimulation models.

In this paper, the VISSIM simulation model is selected for the actual capacity simulation experiment. The main reason is that VISSIM can be adjusted according to the actual situation, such as the following model, lane change model, lateral behavior model, etc., which can make the simulation results more accurately reflect the traffic condition of the long cross-harbor tunnel.

2.2. Identification of the Factors Influencing Extra-Long Subsea Tunnel Capacity

2.2.1. Selection of Influencing Factors

The HCM standard [2] in the United States summarizes the factors affecting capacity, which primarily include: road conditions (road alignment, longitudinal slope gradient, slope length, lane width, etc.), traffic conditions (vehicle composition, lane distribution, etc.), other technical conditions (primarily refers to the Intelligent Traffic System), and other aspects. The following results can be obtained from the above analysis of the traffic flow characteristics of extra-long subsea tunnels: traffic marking in extra-long subsea tunnels will result in uneven traffic distribution; changes in the slope and curve radius of the tunnel, as well as the design of the traffic marking, will affect the speed, resulting in large fluctuations in the speed of the section, the operation of extra-long subsea tunnels’ traffic flow and their road conditions, traffic conditions, and traffic control measurability. As a result, the factors considered to affect the capacity of extra-long subsea tunnels are primarily focused on three aspects: road conditions, traffic conditions, and traffic control measures, which are chosen on the basis of the following criteria.

(1)

Road conditions

Geographical constraints lead to complex alignment combinations in extra-long subsea tunnels with clear distinctions from urban roads. The extra-long subsea tunnels’ cavities are on the ground, at the bottom of the sea, or along the coast, and have a double slope with an increasing slope length. As a result, we consider the longitudinal slope, curve radius, lane road, and lane width as the factors that affect the condition of the road.

(2)

Traffic conditions

Large vehicles must drive on the right side of the extra-long subsea tunnel, which makes the uneven distribution of lanes in the tunnel particularly significant. Additionally, there are some differences between extra-long subsea tunnels and urban roads, so the traffic composition and lane distribution are chosen as the factors that influence traffic conditions.

(3)

Traffic management techniques

The speed limit, traffic markings, and lane control are chosen as the factors that influence the traffic control mode because the control mode in extra-long subsea tunnels is relatively simple compared to city roads and mainly includes these features.

In conclusion, this study combines the traits of extra-long subsea tunnels and chooses the variables that affect their capacity, as shown in

Table 2. Control mode switch principles.

Road Conditions	Traffic Conditions	Traffic Control Measures
Longitudinal slope gradient	Traffic Composition	Speed limit
Curve radius	Lane Distribution	Traffic Markings
Number of lanes	-	Lane Control
Lane width	-	-

.

2.2.2. Identification of the Key Influencing Factors

(1)

Evaluation system construction

In this paper, based on the factors affecting the capacity of extra-long subsea tunnels identified in the previous section, an index system for evaluating the factors affecting the capacity of extra-long subsea tunnels is established, as shown in Figure 2.

(2)

Weighting calculation

This paper integrates hierarchical analysis and the entropy power method to determine the key influencing factors of extra-long subsea tunnel capacity. Firstly, a judgment matrix is constructed by combining the analysis of different influencing factors on tunnel traffic flow with expert scoring, and the judgment matrix of each factor is shown in

Table 3. Judgment matrix of road condition.

Road Conditions	Slope	Curve Radius	Number of Lanes	Lane Width	Wi
Slope	1	2	3	3	0.4495
Curve radius	1/2	1	2	2	0.2596
Number of lanes	1/3	1/2	1	2	0.1707
Lane width	1/3	1/2	1	1	0.1202
${\lambda }_{\max }$ = 4.0716		CR = 0.0268 < 0.1
Wi in the table is the relative weights of program level indicators (same below)

,

Table 4. Judgment matrix of traffic condition.

Traffic Conditions	Traffic Composition	Lane Distribution	Wi
Traffic Composition	1	1	0.5000
Lane Distribution	1	1	0.5000
${\lambda }_{\max }$ = 2		CR = 0 < 0.1

,

Table 5. Judgment matrix of traffic control mode.

Traffic Control Method	Speed Limit	Traffic Markings	Lane Control	Wi
Speed limit	1	1/2	2	0.2973
Traffic Markings	2	1	3	0.5390
Lane control	1/2	1/3	1	0.1638
${\lambda }_{\max }$ = 3.0092		CR = 0.0089 < 0.1

and

Table 6. Judgment matrix of influencing factors of traffic capacity.

Traffic Capacity Influencing Factors	Road Conditions	Traffic Conditions	Traffic Control Mode	Wi
Road conditions	1	2	2	0.4905
Traffic conditions	1/2	1	1/2	0.1976
Traffic control mode	1/2	2	1	0.3119
${\lambda }_{\max }$ = 3.0537		CR = 0.0517 < 0.1

.

Through the calculation, it can be seen that the consistency ratio of the judgment matrix of each influence factor is less than 0.1, indicating that each judgment matrix meets the consistency requirement. Therefore, the weights of the influence factors of extra-long subsea tunnel capacity are obtained, as shown in

Table 7. Weighting of influencing factors on traffic capacity of extra-long subsea tunnels.

Guideline Layer Indicators	Criterion Level Indicator Weights	Program Level Indicators	Program Level Indicator Weights	Order
Road conditions	0.4905	Slope	0.2205	1
		Radius of curve	0.1273	3
		Number of lanes	0.0837	7
		Lane width	0.0590	8
Traffic conditions	0.1976	Traffic composition	0.0988	4
		Lane Distribution	0.0988	5
Traffic control mode	0.3119	Speed limit	0.0927	6
		Traffic Markings	0.1681	2
		Lane Control	0.0511	9

.

Since the hierarchical analysis method is too subjective, this paper uses the entropy weighting method to calculate the objective weights of the indicators. To facilitate the calculation, each line of data represents a section of the tunnel evaluation object. The starting and ending pile numbers are shown in the data table. In the table is also shown traffic composition, lane distribution, lane control set to 1, the solid line set to 1, the left change set to 2, and the vehicle convergence set to 3. The impact factor evaluation table is shown in

Table 8. Data evaluation table.

Starting Stake Number	End Stake Number	Slope (%)	Curve Radius (m)	Number of Lanes	Lane Width	Traffic Composition	Lane Distribution	Speed Limit (km/h)	Traffic Markings	Lane Control
ZK9 + 174	ZK9 + 191	−0.703	0	3	3.5	1	1	80	1	1
ZK8 + 953	ZK9 + 174	−0.703	2000	3	3.5	1	1	80	1	1
ZK8 + 786	ZK8 + 953	−4	2000	3	3.5	1	1	80	1	1
ZK8 + 509	ZK8 + 786	−4	10,000	3	3.5	1	1	80	1	1
ZK8 + 156	ZK8 + 509	−4	6500	3	3.5	1	1	80	1	1
ZK7 + 770	ZK8 + 156	−2.47	6500	3	3.5	1	1	80	1	1
ZK7 + 020	ZK7 + 770	−3.9	6500	3	3.5	1	1	80	1	1
ZK6 + 917	ZK7 + 020	−2.88	6500	3	3.5	1	1	80	1	1
ZK6 + 255	ZK6 + 917	−2.88	10,000	3	3.5	1	1	80	1	1
ZK5 + 380	ZK6 + 255	−0.3	-	3	3.5	1	1	80	1	1
ZK4 + 849	ZK5 + 380	2.228	-	3	3.5	1	1	80	1	1
ZK4 + 779	ZK4 + 849	2.228	372.525	3	3.5	1	1	80	1	1
ZK4 + 330	ZK4 + 779	2.228	1982.5	3	3.5	1	1	80	1	1
ZK4 + 260	ZK4 + 330	2.228	372.525	3	3.5	1	1	80	1	1
ZK3 + 870	ZK4 + 260	2.228	-	3	3.5	1	1	80	1	1
ZK3 + 553	ZK3 + 870	3.448	-	3	3.5	1	1	80	1	1
ZK3 + 453	ZK3 + 553	3.448	346.41	3	3.5	1	1	80	1	1
ZK3 + 055	ZK3 + 453	3.448	1200	3	3.5	1	1	80	1	1
ZK2 + 900	ZK3 + 055	0.302	1200	4	3.5	1	1	80	3	1
ZK2 + 636	ZK2 + 900	0.302	1200	3	3.5	1	1	80	1	1
ZK2 + 536	ZK2 + 636	0.302	346.43	3	3.5	1	1	80	1	1
ZK2 + 160	ZK2 + 536	0.302	-	3	3.5	1	1	80	1	1
ZK1 + 755	ZK2 + 160	2.151	-	3	3.5	1	1	80	1	1
ZK1 + 685	ZK1 + 755	2.151	333.619	3	3.5	1	1	80	1	1
ZK1 + 410	ZK1 + 685	2.151	1590	3	3.5	1	1	80	1	1
ZK1 + 134	ZK1 + 410	3.5	1590	3	3.5	1	1	80	2	1
ZK1 + 096	ZK1 + 134	3.5	333.617	3	3.5	1	1	80	2	1

below.

The weights of each influencing factor were calculated by the entropy weighting method, as shown in

Table 9. Summary of the results of calculating weights by the entropy value method.

Item	Information Entropy Value e	Information Utility Value d	Weighting Factor w
Slope (%)	0.896	0.104	0.3347
Curve radius (m)	0.8165	0.1835	0.5906
Number of lanes	0.9993	0.0007	0.0024
Lane width	1	0	0.00
Traffic composition	1	0	0.00
Lane distribution	1	0	0.00
Speed limit (km/h)	1	0	0.00
Traffic markings	0.9775	0.0225	0.0723
Lane control	1	0	0.00

below.

The capacity of extra-long subsea tunnels is primarily influenced by slope, traffic marking, curve radius, traffic composition, and lane distribution, according to Table 9. As a result, the effect of traffic patterns and the environment in which vehicles operate on the capacity of exceptionally long subsea tunnels is not taken into account in this paper. The slope, traffic markings, and curve radius are found to be the key influencing factors of extra-long subsea tunnels based on the results of the integrated hierarchical analysis and entropy power method.

2.3. Development of the Capacity Model for Extra-Long Subsea Tunnels

Based on the comparison of capacity models in various manuals, it can be determined that the basic calculation of capacity is based on the basic capacity of the lane and its correction, so the basic form of the capacity calculation model can be summarized as follows:

$$C_r=C_0\times N\times f(i_1,i_2,\cdot \cdot \cdot ,i_n)$$

(1)

C_r—Actual capacity of the extra-long subsea tunnel;

C₀—Basic capacity of a single lane under ideal conditions, vehicles/h/lane;

N—Number of lanes;

I—Tunnel Slope, %;

f(i₁,i₂,…,i_n)—Correction factor for the influence factor of traffic capacity.

(1)

According to the key influencing factors of the extra-long subsea tunnel capacity identified in Section 2.2.2 of this paper, the influence of each factor on the capacity was analyzed, the relationship between the influencing factors and the capacity was established, and the form of each parameter in the capacity model was determined.

(2)

By analyzing the three key influencing factors on the capacity of the extra-long subsea tunnel, including the slope of the longitudinal slope, the radius of the flat curve, and the marking, and after determining the form of each parameter, the capacity model of the extra-long subsea tunnel was constructed, and the multiple non-linear regression equation is as follows, where the dependent variable is the actual capacity, the independent variable is the slope of the longitudinal slope and the radius of the flat curve, and the form of the marking is used as the correction coefficient.

$$C_r=(C_0\times N\times {\beta }_0+f(i)+f(R))\times f_l$$

(2)

R—Radius of the flat curve, km;

f_l—Traffic marking correction factor;

${\beta }_0$—Coefficient to be determined.

(3)

Using VISSIM simulation software, we constructed a traffic simulation model of an extra-long underwater tunnel, conducted a sensitivity analysis on the driving behavior and other parameters in the model, determined the key parameters, and then calibrated them to obtain a traffic simulation model that can accurately reflect the actual operation of the tunnel. Based on the simulation model of the extra-long cross-harbor tunnel, the traffic flow data of the tunnel were obtained, and the actual capacity of the tunnel was obtained by comparing the measured capacity methods.

(4)

Based on the data in the third step, the model was fitted using the multiple nonlinear regression method to determine the parameters of each influencing factor.

(5)

For the test of the multiple nonlinear regression model, the test used was the misfit term test, which was used to judge the reliability of the model.

2.4. Development and Calibration of the Simulation Model for Extra-Long Subsea Tunnels

2.4.1. Simulation Model Construction Process

When using VISSIM for road simulation, model distortion may occur if default driving behavior parameters are used. To solve this problem, the process of constructing and correcting the traffic simulation model of the extra-long subsea tunnel based on VISSIM simulation software used in this paper is shown in Figure 3.

As can be seen from Figure 3, the simulation model construction and calibration methods for the extra-long subsea tunnel in this paper mainly include: data acquisition, traffic element modeling, model element checking, simulation parameter sensitivity analysis, simulation model parameter calibration, and simulation model validation.

(1)

Data acquisition: The data required for modeling the long undersea tunnel mainly include: road network data (road type, road length, lane width, number of lanes, slope, etc.); traffic control data (signs and markings); driver-vehicle data (vehicle class, traffic composition, etc.); model calibration data (cross-sectional flow, etc.).

(2)

Traffic element modeling: Based on the collected traffic elements, they are correctly and reasonably expressed in VISSIM.

(3)

Model element checking: mainly includes basic road network model checking (no cut-off roads, road alignment matching with actual); traffic control checking (whether simulated vehicles comply with control measures); animation checking (whether there is abnormal congestion, vehicle crush collision phenomenon).

(4)

Sensitivity analysis of simulation parameters: test the degree of influence of each driving behavior parameter in the simulation model on the traffic operation status of the extra-long subsea tunnel, statistically analyze the influence of each parameter on the flow and travel time of the extra-long subsea tunnel, and determine the parameters to be calibrated for the simulation model.

(5)

Simulation model parameter calibration: Based on the simulation parameter sensitivity analysis results, the parameters to be calibrated are determined, and the measured data are used as data support to determine the values of key parameters.

(6)

Simulation model validation: Select typical indicators for statistical analysis, such as: section flow, speed, queue length and other indicators of the measured data and simulation results for comparison.

2.4.2. Simulation Parameter Sensitivity Analysis

The models that need to be calibrated based on measured traffic data in VISSIM are the desired speed distribution, driving behavior model (follow-along model), lane change model, and lateral behavior model. In this paper, we focus on the parameters of the desired speed and driving behavior model (Wiedemann 99 follow-along model) for parameter sensitivity analysis.

(1)

Expected speed distribution

Desired speed is the safe driving speed that the driver expects to achieve under the condition that the vehicle is not disturbed or constrained by other vehicles. In the VISSIM simulation model, the desired speed determines the initial speed and acceleration of the vehicle entering the road network, which has an important impact on the vehicle’s lane-changing behavior and queuing.

(2)

Follower model

VISSIM uses the physiological–psychological driving behavior model established by Wiedemann. The basic idea of the model is that the driver of a vehicle traveling at a faster speed perceives a slower vehicle in front of them and starts to decelerate and reaches the desired speed limit to follow the driver because the driver of the car behind cannot accurately judge the speed of the car in front, so their speed will be lower than the speed of the car in front, which triggers them to take accelerating measures to a certain extent until they reach the desired value again. Thus, the iterative process of acceleration and deceleration is repeated.

In the VISSIM simulation software, there are two vehicle-following models: the Wiedemann 74 model and the Wiedemann 99 model. The Wiedemann 74 model is mainly applied to the study of urban road traffic flow characteristics, while the Wiedemann 99 model is optimized on the original Wiedemann 74 model and mainly applied to expressways and highways. Therefore, this paper focuses on the calibration of the relevant parameters in the Wiedemann 99 model.

The Wiedemann 99 model is a following model proposed by Wiedemann in 1999, which refines the impact parameters into nine, including stopping distance, headway time distance, and following variables.

①

CC0 (Parking distance): the desired stopping distance between two vehicles.

②

CC1 (Headway time distance): the distance the driver wants to maintain at a given speed. As the value increases, the driver also has to increase their attention.

③

CC2 (Following variable): the distance difference allowed (longitudinal fluctuation) needs to be limited before the driver becomes aware of approaching the vehicle driving in front.

④

CC3 (Threshold for entering the following state): defines, in seconds, the beginning of the deceleration process before reaching a safe distance. At this point in time, the driver will detect the slower vehicle ahead.

⑤

CC4 (Negative following threshold): defines the negative speed difference during following. Lower values cause the driver to react more sensitively to the acceleration or deceleration of the vehicle moving ahead.

⑥

CC5 (Threshold for active following state): defines the positive speed difference during following.

⑦

CC6 (Vehicle speed vibration): the effect of distance on speed fluctuation during following a vehicle value 0; fluctuations are independent of distance. As the distance increases, the speed fluctuation also becomes larger.

⑧

CC7 (Acceleration vibration amplitude): acceleration during fluctuation.

⑨

CC8 (Acceleration at stopping): desired acceleration when starting from a stop by the maximum acceleration limit of the acceleration curve.

⑩

CC9 (Acceleration at vehicle speed of 80 Km/h): Acceleration at a speed of 80 km/h limited by the maximum acceleration of the acceleration curve.

3. Results and Discussion

3.1. Qingdao Jiaozhou Subsea Tunnel

3.1.1. Tunnel General Information

The Qingdao Jiaozhou Bay Subsea Tunnel is about 7797 m long overall, and the left and right lines are set up separately in the main tunnel. The Jiaozhou Bay Subsea Tunnel is designed as an urban expressway; the main road has six two-way lanes, and the total width of the travel lane is 10.75 m. Among them, the headroom in the main tunnel is about 8.2 m, and the width is about 14.3 m; it has a designated speed of 80 km/h. The right tunnel (Qingdao–Huangdao) has a maximum longitudinal slope of 3.9%, an 830 m slope length, nine variable slope points, and a minimum turning radius of 1000 m. The left tunnel (Huangdao–Qingdao) has a maximum longitudinal slope of 4.0%, a slope length of 797 m, nine side slope points, and a minimum turning radius of 1200 m. The subsea tunnel is a V-shaped, multi-bore, one-way subsea tunnel that is particularly long, has a steep longitudinal slope, and has a road alignment that generally slopes continuously from downhill to gently uphill throughout the tunnel.

The road conditions following the second alignment of the Qingdao Jiaozhou Bay Subsea Tunnel were chosen. After the second delineation, the design of the tunnel markings was determined, and the VISSIM simulation model was adjusted to obtain the traffic flow parameters to determine the actual capacity of the tunnel section; then, the parameters corresponding to the influencing factors such as tunnel slope, curve radius, and marking form were brought into the actual capacity calculation model of Jiaozhou Bay Subsea Tunnel constructed in this paper to obtain the calculated valence.

Figure 4 depicts the tunnel’s road plan following the second round of delineation of the Qingdao Jiaozhou Bay Tunnel.

Table 10. Statistical table of marking forms after the second marking of the left line of Jiaozhou Bay Crossing.

Starting Stake Number	End Stake Number	Slope (%)	Curve Radius (m)	Marking Form	Number of Lanes
ZK9 + 174	ZK9 + 191	−0.703	0	Solid Line	3
ZK8 + 953	ZK9 + 174	−0.703	2000	Solid Line
ZK8 + 786	ZK8 + 953	−4	2000	Solid Line
ZK8 + 509	ZK8 + 786	−4	10,000	Solid Line
ZK8 + 156	ZK8 + 509	−4	6500	Solid Line
ZK7 + 770	ZK8 + 156	−2.47	6500	Solid Line
ZK7 + 020	ZK7 + 770	−3.9	6500	Solid Line
ZK6 + 917	ZK7 + 020	−2.88	6500	Solid Line
ZK6 + 255	ZK6 + 917	−2.88	10,000	Solid Line
ZK6 + 115	ZK6 + 255	−0.3	-	Solid line
ZK5 + 480	ZK6 + 115	−0.3	-	Dashed line
ZK5 + 380	ZK5 + 480	−0.3	-	Solid Line
ZK4 + 849	ZK5 + 380	2.228	-	Solid Line
ZK4 + 779	ZK4 + 849	2.228	372.525	Solid Line
ZK4 + 355	ZK4 + 779	2.228	1982.5	Solid line
ZK4 + 330	ZK4 + 355	2.228	1982.5	Dashed line
ZK4 + 260	ZK4 + 330	2.228	372.525	Dashed line
ZK3 + 870	ZK4 + 260	2.228	-	dashed line
ZK3 + 705	ZK3 + 870	3.448	-	Dashed line
ZK3 + 553	ZK3 + 705	3.448	-	solid line
ZK3 + 453	ZK3 + 553	3.448	346.41	left dashed and right solid
ZK3 + 055	ZK3 + 453	3.448	1200	left dashed right solid
ZK2 + 900	ZK3 + 055	0.302	1200	Vehicle convergence out	4
ZK2 + 636	ZK2 + 900	0.302	1200	Solid line	3
ZK2 + 536	ZK2 + 636	0.302	346.43	Solid line
ZK2 + 160	ZK2 + 536	0.302	-	solid line
ZK1 + 755	ZK2 + 160	2.151	-	Solid line
ZK1 + 685	ZK1 + 755	2.151	333.619	Solid line
ZK1 + 410	ZK1 + 685	2.151	1590	Dashed line
ZK1 + 134	ZK1 + 410	3.5	1590	Dashed line
ZK1 + 096	ZK1 + 134	3.5	333.617	Dashed line

and

Table 11. Statistical table of the form of the right line marking after the second delineation of the Jiaozhou Bay Crossing.

Starting Stake Number	End Stake Number	Slope (%)	Curve Radius (m)	Marking Form	Number of Lanes
YK1 + 200	YK1 + 462	−3.5	10,000	Solid Line	3
YK1 + 462	YK1 + 562	−0.801	10,000	Solid Line
YK1 + 562	YK1 + 632	−0.801	-	Solid Line
YK1 + 632	YK1 + 835	−0.801	1000	Solid Line
YK1 + 835	YK1 + 905	−0.801	-	Solid Line
YK1 + 905	YK2 + 100	−0.801	10,000	Solid Line
YK2 + 100	YK2 + 214	−0.4	10,000	Solid Line
YK2 + 214	YK2 + 284	−0.4	-	Solid Line
YK2 + 284	YK2 + 553	−0.4	1000	Solid Line
YK2 + 553	YK2 + 623	−0.4	-	Solid Line
YK2 + 623	YK2 + 783	−0.4	10,000	Solid Line
YK2 + 783	YK2 + 853	−0.4	-	Solid Line
YK2 + 853	YK3 + 045	−0.4	1000	Solid Line
YK3 + 045	YK3 + 162	−3.5	1000	Vehicle convergence	4
YK3 + 162	YK3 + 232	−3.5	-	Solid line	3
YK3 + 232	YK3 + 541	−3.5	1500	Solid line
YK3 + 541	YK3 + 611	−3.5	-	Dashed line
YK3 + 611	YK3 + 800	−3.5	10,000	Dashed line
YK3 + 800	YK3 + 980	−2.2	10,000	Dashed line
YK3 + 980	YK4 + 263	−2.2	10,000	solid line
YK4 + 263	YK4 + 333	−2.2	-	Solid Line
YK4 + 333	YK4 + 796	−2.2	2038	Solid Line
YK4 + 796	YK4 + 866	−2.2	-	Solid Line
YK4 + 866	YK5 + 400	−2.2	10,000	Solid Line
YK5 + 400	YK5 + 785	0.3	-	Solid line
YK5 + 785	YK6 + 200	0.3	-	Dashed line
YK6 + 200	YK6 + 900	2.5	--	Dashed line
YK6 + 900	YK7	3.9	-	Dashed line
YK7	YK7 + 745	3.9	4945	solid line
YK7 + 745	YK7 + 875	2.605	4945	dashed line
YK7 + 875	YK8 + 130	2.605	-	dashed line
YK8 + 130	YK8 + 600	3.9	-	Dashed line
YK8 + 600	YK8 + 770	3.9	-	Solid Line
YK8 + 770	YK8 + 960	3.9	2500	Solid Line
YK8 + 960	YK9 + 085	−0.7	2500	Solid Line
YK9 + 085	YK9 + 200	−0.7	-	Solid line

show the road data.

According to the above data, the VISSIM simulation model of Jiaozhou Bay Crossing was modified, and some of the model modifications are shown in Figure 5 (the red box is where the marking form changed after the second crossing).

3.1.2. Environmental Characteristics

(1)

Spatial structure characteristics

The key components of a cross-harbor tunnel’s structure are the tunnel door, the tunnel body, and the internal systems necessary to maintain traffic safety. To ensure the comfort and visual safety of the driver’s eyes, the tunnel door can moderate the contrast between the light and dark inside and outside the tunnel entrance. The topography, geology, and engineering technology of the tunnel place restrictions on the tunnel body’s structure, resulting in a confined space inside the tunnel.

The tunnel’s enclosed space creates a single environment with poor lighting, which makes for a poor driving environment and makes it simple for drivers to underestimate their vehicle’s speed and go over the posted limit. The driver in the entrance and exit section of the cross-harbor tunnel will quickly complete the conversion of strong and weak visual reference environments despite the high illumination outside, the open field of view, and the presence of some safety hazards. Additionally, as cars are moving through the tunnel, the monotonous side walls will make the driver tense, which will cause cars to veer to the left. Additionally, when a driver is in a long-term state of anxiety, their operational effectiveness decreases, which affects the stability of the tunnel traffic flow and lowers the efficiency of tunnel traffic.

(2)

Travel environment characteristics

Cross-harbor tunnels have advantages over regular highways in that they are stable, disaster-resistant, and weather-unaffected, ensuring traffic flow on both sides of the strait. At the same time, cross-harbor tunnels’ driving conditions differ greatly from those of other highways due to their unique geographic location and intricate structural features.

①

Lighting conditions

To ensure traffic safety, the interior of the cross-harbor tunnel is illuminated by an artificial lighting system to achieve all-weather lighting, while the illumination outside the tunnel is similar to that of a regular open road, with natural lighting during the day and artificial lighting at night.

②

Air quality, ventilation, and noise

The noise generated by vehicles and facilities in the tunnel is difficult to dissipate quickly due to its closed space, making drivers and passengers easily bored and anxious, affecting driving safety to some extent; the longitudinal slope of the tunnel affects ventilation conditions, and the deposited vehicle exhaust will reduce the air quality inside the tunnel, posing a potential threat to drivers and passengers.

③

Auxiliary services

The cross-harbor tunnel is outfitted with a central control system, signs, markings, ventilation system, lighting system, traffic monitoring system, fire detection and alarm facilities, fire-fighting facilities, and passageways, among other things, to ensure traffic safety and comfort. These complex ancillary facilities require long-term maintenance to ensure normal operation.

3.1.3. Traffic Flow Characteristics

(1)

Uneven traffic distribution

The statistical results of the all-day traffic flow of the right lane of the Qingdao Jiaozhou Bay Tunnel (Qingdao–Huangdao) for a week in January 2021 for the left center and right lanes are shown in Figure 6.

As shown in Figure 6, the middle lane traffic is slightly higher than the left lane traffic, while the right lane traffic accounts for less than 25%, and there is an obvious lane unevenness phenomenon in the traffic distribution. The analysis of its causes focuses on three key points: ① Large vehicles (buses and coaches) must drive on the right side of the Qingdao Jiaozhou Bay Subsea Tunnel, while the remaining two lanes are small car lanes. The subsea tunnel’s right lane is a mix of large and small cars, and the slow speed of buses and coaches reduces the rate of small cars driving, resulting in low utilization of the right lane. ② Because the tunnel markings are mostly solid lines, vehicles entering the tunnel are limited in their ability to choose a lane, resulting in low utilization of the right lane. ③ Due to the side wall effect, which the tunnel’s side wall will have on the driver’s psychology, the driver will be more inclined to choose the middle lane.

(2)

Excessive head spacing

The average headway corresponding to different slopes in the subsea tunnel was counted separately using Qingdao Jiaozhou Bay Subsea Tunnel inside-loading video images for a total of 11 cross-sections in the left lane and 10 cross-sections in the right lane, as shown in Figure 7 (Figure: 1—downhill, 2—flat slope, 3—uphill).

As shown in Figure 7, the change in slope in the tunnel has an effect on vehicle headway, and there is an inverse relationship between headway and traffic flow; the larger the average headway with the same car ratio, the lower the traffic flow through, so the change in slope will have an effect on road capacity.

(3)

Significant fluctuations in operating speed

The section speed data of different slopes are collected from the subsea tunnel side-loading video, and the speed variation of vehicles in different sections of each lane is plotted, as shown in Figure 8.

As can be seen from Figure 8, the tunnel speed has lane unevenness and the speed of each section of the tunnel fluctuates widely, which is due to the following reasons. ① The driver’s own factors. ② Traffic composition—the mix of different types of vehicles on the road and the proportion of different vehicle types are important factors affecting the distribution of speed; the subsea tunnel provides for large vehicles to drive on the right, making the right lane speed significantly lower than the left lane and the middle lane. ③ Road factors, changes in slope and curve radius in the subsea tunnel, and the design of traffic markings will all have an impact on vehicle speed. ④ Environmental factors—bright and dark adaptation of drivers at tunnel entrances and exits leads to fluctuations in vehicle speed.

3.1.4. Key Factors and Characteristics Affecting Tunnel Capacity

(1)

Longitudinal slope gradient

In this paper, we use the video data collected from the side-mounted video of the Qingdao Jiaozhou Bay Tunnel to obtain the section flow of different slope sections and analyze the influence of longitudinal slope on the section flow of the tunnel. The statistical results are shown in Figure 9 and Figure 10 (section 1—downhill, 2—flat slope, 3—uphill in the figure).

As seen in the above figure, there is a relationship between the tunnel flow and the slope of the longitudinal slope; in the downhill section, as the slope decreases, the tunnel cross-sectional flow increases, but in the flat section and the uphill section, the relationship is not significant. This is primarily because changes in the tunnel’s slope cause speed fluctuations, such as when a vehicle is driving a downhill section immediately after ascending.

(2)

Curve radius

In this paper, we use the video data collected from the side-mounted video of the Qingdao Jiaozhou Bay Tunnel to obtain the section flow of different curve radius sections and analyze the influence of curve radius on the tunnel section flow. The statistical results are shown in Figure 11 (section 1—downhill, 2—flat slope, 3—uphill).

The left lane of the subsea tunnel increases with the radius of the tunnel curve in the downhill and flat slope sections, as shown in Figure 11, and the tunnel cross-sectional flow increases with the radius of the curve. However, the relationship between the curve radius and flow rate is not significant in the uphill section; in the right lane of the tunnel, the tunnel cross-sectional flow rate increases with increasing curve radius in the flat-slope section, but not in the other sections. The main reason for this could be that the main influencing factor for the change in tunnel section flow is the stability of the traffic flow, i.e., the speed of the vehicles in the section before and after a certain section, whereas the longitudinal slope has a greater impact on the speed of the moving vehicles, so the relationship is not significant when analyzing the relationship between the curve radius and the flow.

(3)

Traffic Markings

This paper chooses a tunnel marking before and after the application of tunnel cross-sectional flow data for comparative analysis, and for workdays, cross-sectional hourly traffic flow statistics before and after the marking of the roadway, as shown by the solid line; the statistical results are shown in Figure 12.

According to the above graph, the overall situation demonstrates an increase in cross-sectional traffic following the marking of the line, as well as a slight increase in cross-sectional traffic at all times, with an average growth rate of 4.95%. As a result, the marking will have an impact on the tunnel section’s effectiveness.

3.2. Simulation Model for Qingdao Jiaozhou Bay Tunnel

3.2.1. Development of Simulation Models

Given the complex geometric alignment of the Jiaozhou Bay Subsea Tunnel, and in order to analyze the impact of the single factors of tunnel slope and curve radius on capacity, this section builds a traffic simulation model of the Jiaozhou Bay Subsea Tunnel based on the VISSIM simulation model to lay the groundwork for the capacity model.

In order to accurately simulate the actual operation of the Jiaozhou Bay Subsea Tunnel under different traffic operating conditions, this paper constructs a microscopic simulation model of the Jiaozhou Bay Subsea Tunnel based on the VISSIM simulation software, including the establishment of the basic road sections, where the tunnel slope is achieved based on the Z coordinate settings, as shown in Figure 13 (since the depth of the Jiaozhou Bay Subsea Tunnel is about 80 m, the horizontal elevation is set to 100 m for the convenience of model adjustment). The traffic input and traffic composition are set in the upstream lane of the tunnel to achieve accurate adjustment of lane level traffic, vehicle composition, and lane transformation, which can simulate the operation of the Jiaozhou Bay Subsea Tunnel under different traffic conditions more accurately. The section of the Jiaozhou Bay Subsea Tunnel is drawn in the VISSIM simulation software as shown in Figure 14.

This paper uses additional lane-level trail sections as the starting point and end point of lane change path decisions before and after the tunnel markings to simulate the operating characteristics of vehicles in the tunnel at different markings, as shown in Figure 15. Using existing lane change data for proportional input, the left lane level traffic error is 5% at maximum and 1.6% at minimum, the total traffic difference is 3.3%, and the right lane level traffic error is 3.3% at maximum.

Table 12. Simulation model lane change behavior-flow error (left line).

	Lanes	Not Set	Path Decision Setting	Actual
Left line	Left lane	1174	1262	1282
	Middle Lane	1152	1291	1342
	Right lane	1108	882	928
Error	Left lane	8.4%	1.6%	3.3%
	Middle lane	14.2%	3.8%
	Right lane	19.4%	5.0%

and

Table 13. Simulation model channel change behavior-flow error (right line).

	Lanes	Not Set	Path Decision Setting	Actual
Right line	Left lane	870	1005	1036
	Middle Lane	1153	1270	1230
	Right lane	1298	1047	1080
Error	Left lane	16.0%	3.0%	0.7%
	Middle lane	6.3%	3.3%
	Right lane	20.2%	3.1%

show the specific values.

3.2.2. Test Simulation Parameter Sensitivity

The set of parameters to be calibrated is determined based on the parameters in the VISSIM simulation, and a parameter sensitivity analysis is performed using the control single variable method, with the other parameters taking default values when one of the parameters is selected for testing. The following parameters were chosen for testing for the Jiaozhou Bay Subsea Tunnel, and the parameter sensitivity analysis scheme is shown in

Table 14. Parameter sensitivity analysis scheme.

Parameters	Parameter Meaning	Default Value	Range of Values	Test Program
Desired speed distribution		-	20–120	Step length 10
CC0	Parking Distance	1.5	0.5–2.5	Step length 0.5
CC1	Headway time distance	0.9	0.7–1.7	-
CC2	Following variable	4	2–8	Step length 2
CC3	Threshold for entering the following state	−8	−10–5	-
CC4	Negative following threshold	−0.35	0.05–1.05	Step length 0.2
CC5	Threshold for active following state	0.35	0.7–1.11	Step length 0.2
CC6	Vehicle speed vibration	11.44	0–20	Step length 4
CC7	Acceleration vibration amplitude	0.25	0.1–2.1	Step length 0.2
CC8	Acceleration at stopping	3.5	2–3.5	-
CC9	Acceleration at vehicle speed of 80 Km/h	1.5	0.5–2	Step length 0.5
Safety distance discount factor		0.6	0.1–0.6	Step length 0.1
Minimum headroom		0.5	0.3–2.1	Step length 0.2

.

For the purpose of testing the sensitivity of the simulation parameters, the morning peak data of the right (Qingdao–Huangdao) and left (Huangdao–Qingdao) test sections of the Jiaozhou Bay Subsea Tunnel were chosen. The measured traffic flow data of the tunnel, including the flow through the tunnel entrance, the ratio of vehicles changing lanes, and the composition of the vehicles, were entered into the simulation model, and the results are shown in

Table 15. Jiaozhou Bay Crossing input data sheet.

Category	Lanes	Upstream Flow (Vehicles/h)	Downstream Flow (Vehicles/h)	Bus Ratio
Huangdao–Qingdao	Left lane	1251	1282	-
	Middle Lane	1387	1342	-
	Right lane	929	928	9.29%
Qingdao–Huangdao	Left lane	1129	1036	-
	Middle lane	1166	1230	-
	Right lane	1060	1080	8.65%

.

Dual indicators are used in this paper to assess the sensitivity of each parameter, and the specific results are shown in

Table 16. Comparison of sensitivity test results for the left lane of the Jiaozhou Bay Crossing.

Parameters		Maximum Flow Rate (Vehicles/h)	Minimum Flow Rate (Vehicles/h)	Flow Differential Ratio	Maximum Travel Time (s)	Minimum Travel Time (s)	Trip Time Difference Ratio
Expected speed		3263	3238	0.77%	66.21	13.53	79.57%
CC0	Parking Distance	4653	4590	1.35%	38.41	27.19	29.21%
CC1	Headway time distance	3242	1064	67.18%	17.82	16.25	8.81%
CC2	Following variable	4601	3902	15.19%	49.45	24.63	50.19%
CC4&5	Threshold value of following state	3242	3242	0.00%	16.32	16.31	0.06%
CC6	Vehicle speed vibration	3242	3242	0.00%	16.32	16.31	0.06%
CC7	Acceleration vibration amplitude	3242	3242	0.00%	16.32	16.31	0.06%
Safety distance discount factor		4671	4539	2.83%	33.49	32.14	4.03%

and

Table 17. Comparison of sensitivity test results for the right lane of the Jiaozhou Bay Crossing.

Parameters		Maximum Flow Rate (Vehicles/h)	Minimum Flow Rate (Vehicles/h)	Flow Differential Ratio	Maximum Travel Time (s)	Minimum Travel Time (s)	Trip Time Difference Ratio
Expected speed		3566	3544	0.62%	60.24	12.95	78.50%
CC0	Parking Distance	5301	4891	7.73%	23.94	22.29	6.89%
CC1	Headway time distance	3568	502	85.93%	355.56	15.03	95.77%
CC2	Following variable	5630	3745	33.48%	24.71	22.71	8.09%
CC4&5	Threshold value of following state	3567	3566	0.03%	15.15	15.12	0.20%
CC6	Vehicle speed vibration	3568	3566	0.06%	15.14	15.13	0.07%
CC7	Acceleration vibration amplitude	3568	3566	0.06%	15.15	15.12	0.20%
Safety distance discount factor		5093	5054	0.77%	23.20	22.95	1.08%

below. The table below shows that the expected speed, CC0 stopping distance, CC1 headway time distance, and CC2 following variables have a large impact on the simulation model’s flow and travel time results, with a difference ratio of greater than 7%; the other parameters have a smaller impact on the simulation model, with a difference ratio of less than 5%. As a result, the key influencing parameters in the Jiaozhou Bay Subsea Tunnel simulation model are as follows: desired speed, CC0 stopping distance, CC1 headway time distance, and CC2 following variables.

3.2.3. Calibration of Simulation Model Parameters

For the key parameters of the model (desired speed, CC0 stopping distance, CC1 headway time distance, and CC2 following variables), the simulation parameters were calibrated and the results were determined as shown below.

(1)

CC0 stopping distance: 2.3 m.

(2)

CC1 headway time distance: as shown in

Table 18. Tunnel headway values.

Camera Number	Headway (s)	Camera Number	Headway (s)
v85	1.56	v5	1.74
v89	1.60	v9	1.74
v93	1.70	v14	1.47
v102	1.81	v27	1.38
v111	1.56	v32	1.41
v114	1.44	v45	1.38
v125	1.52	v54	1.43
v134	1.42	v62	1.54
v140	1.61	v72	1.60
v150	1.64	v76	1.62
v158	1.56	-	-

.

(3)

CC2 following variables: tunnel left lane (Huangdao–Qingdao)—4.094; tunnel right lane (Qingdao–Huangdao)—3.520.

After optimization according to the above parameters, a comparative analysis was performed for the optimized simulation model and the actual measured tunnel data, comparing the flow rates obtained from Wiedemann 99 (default parameters), Wiedemann 99 (optimized) and the actual measured data; the data results are shown in

Table 19. Comparison of simulation data and actual measurement data.

Flow Solutions	Actual Flow Rate	Wiedemann 99 (after Optimization)	Wiedemann 99 (Default)	Wiedemann 99 (after Optimization)	Wiedemann 99 (Default)
Scenario 1	3420	3268	3076	4.44%	10.06%
Program 2	3265	3198	2976	2.05%	8.85%
Scenario 3	2914	2822	2846	3.16%	2.33%
Scenario 4	2871	2794	2633	2.68%	8.29%

.

In Table 19, it can be seen that the matching effect of the traffic operation parameters of the calibrated and optimized Jiaozhou Bay Subsea Tunnel VISSIM simulation model is better than the simulation results when the default parameters are set, and the relative error of its traffic is controlled below 5%, which can reflect the actual operation of the tunnel vehicles.

3.3. Relationship between Key Influencing Factors and Capacity

3.3.1. Longitudinal Slope Gradient

There are differences in the capacity obtained at different longitudinal slope gradients, and the capacity varies with the slope, as shown in

Table 20. Traffic capacity under different longitudinal gradients.

Slope (%)	0	1	2	3	4	5	6	−1	−2	−3	−4	−5	−6
Passage capacity	4834	4777	4800	4722	4658	4567	4521	4798	4806	4775	4796	4791	4893

.

A polynomial regression fit was performed based on the simulation data in Table 20 to establish the relationship between slope and capacity, as shown in Figure 16.

The goodness-of-fit R-sq of the model is close to 1, the fit is good, and the regression equation is obtained as

$$C=4809-1685i-52535i^2$$

(3)

Figure 16 clearly shows that as the slope increases, the road capacity decreases. When the longitudinal slope reaches 3%, the road capacity decreases significantly, and this value is more significant in the tunnel’s uphill section.

3.3.2. Curve Radius

Given the issue of experimental data unit conversion, the unit of curve radius was set to km. Because the curve radius of the straight section of the tunnel cannot be directly expressed, the reciprocal of the curve radius, i.e., 1/R, was used to represent the variable of curve radius when analyzing the relationship between the curve radius and the tunnel capacity, and the statistical results are shown in

Table 21. Capacity under different curve radii.

Curve Radius (km)	1/r	Passing Capacity
∞	0	4834
10	0.1	4795
6.5	0.153846	4764
4.945	0.202224	4795
2.5	0.4	4786
2.038	0.490677	4763
2	0.5	4731
1.9825	0.504414	4769
1.59	0.628931	4806
1.5	0.666667	4775
1.2	0.833333	4741
1	1	4722

.

A polynomial fit regression was performed to establish the relationship between curve radius and capacity based on the simulation data in Table 21, as shown in Figure 17. The model fits with average goodness of fit and the regression equation is obtained:

$$C=4807-\frac{74.46}{R}$$

(4)

Overall, the subsea tunnel capacity is linearly negatively correlated with 1/R, and the tunnel capacity shows a decreasing trend with the increase of 1/R, i.e., the capacity increases with the increase of curve radius.

3.3.3. Traffic Markings

Fifteen cross-sections in the tunnel were selected to count the capacity under different traffic markings, and the statistical results are shown in Figure 18.

From the above figure, it can be seen that the capacity of the traffic marking is increased when the dashed line is compared with the solid line, but there is no significant pattern, which can be used as a correction factor to optimize the actual capacity calculation model of the subsea tunnel in the subsequent model construction process.

3.4. Qingdao Jiaozhou Subsea Tunnel Capacity Model

3.4.1. Development of the Capacity Model

Based on the relationship between slope, radius and marking, and tunnel capacity established above, a multiple non-linear regression analysis is used to establish a tunnel capacity model, where the dependent variable is the actual capacity, the independent variables are the slope and radius of the flat curve, and the traffic marking is used as a correction factor. The multiple non-linear regression equation is:

$${\mathrm{C}}_r=(C_0\times N\times {\beta }_0+{\beta }_1{i}^2+{\beta }_2{R}^{-1})\times f_l$$

(5)

C_r—Actual capacity of the Qingdao Jiaozhou Bay Subsea Tunnel;

C₀—Basic capacity of a single lane under ideal conditions, vehicles/h/lane;

N—Number of lanes;

i—Tunnel slope, %;

R—Radius of the flat curve, km;

f_l—Traffic marking correction factor;

${\beta }_0$,${\beta }_1$,${\beta }_2$—Coefficient to be determined.

3.4.2. Obtaining Actual Tunnel Capacity

The actual capacities of the left lane (Huangdao–Qingdao) and right lane (Qingdao–Huangdao) of the Qingdao Jiaozhou Bay Subsea Tunnel were obtained and are shown in

Table 22. Measured traffic capacity of the Qingdao Jiaozhou Bay Subsea Tunnel.

Left Line (Huangdao–Qingdao)				Right Line (Qingdao–Huangdao)
Section Number	i (%)	R−1	Passage Capacity	Section Number	i (%)	R−1	Passage Capacity
Z1	−0.703	0	4716	Y1	−3.5	0.1	4713
Z2	−4	0.5	4608	Y2	−0.801	0.1	4762
Z3	−4	0.1	4668	Y3	−0.801	0	4772
Z4	−4	0.15	4653	Y4	−0.801	1	4735
Z5	−2.47	0.15	4688	Y5	−0.801	0	4772
Z6	−3.9	0.15	4672	Y6	−0.801	0.1	4785
Z7	−2.88	0.153	4704	Y7	−0.4	0.1	4702
Z8	−2.88	0.1	4712	Y8	−0.4	0	4726
Z9	−0.3	0	4704	Y9	−0.4	1	4693
Z10	2.228	0	4622	Y10	−0.4	0	4768
Z11	2.228	0.5	4704	Y11	−0.4	0.1	4788
Z12	2.228	0	4584	Y12	−0.4	1	4804
Z13	2.228	0	4560	Y13	−3.5	1	4609
Z14	3.448	0	4573	Y14	−3.5	0	4623
Z15	3.448	0	4572	Y15	−3.5	0.67	4715
Z16	3.448	0.83	4553	Y16	−3.5	0	4808
Z17	0.302	0.83	4641	Y17	−3.5	0.1	4791
Z18	0.302	0	4684	Y18	−2.2	0.1	4777
Z19	2.15	0	4608	Y19	−2.2	0	4804
Z20	2.151	0.63	4596	Y20	−2.2	0.49	4804
Z21	3.5	0.63	4620	Y21	−2.2	0.1	4740
				Y22	0.3	0	4840
				Y23	2.5	0	4772
				Y24	3.9	0	4732
				Y25	3.9	0.2	4656
				Y26	2.605	0.2	4620
				Y27	2.605	0	4704
				Y28	3.9	0	4705
				Y29	3.9	0.4	4608

.

3.4.3. Fitting the Nonlinear Regression Equation

Based on the data obtained from Table 22 above, a multivariate nonlinear regression was fitted and the calculated results were obtained as shown in

Table 23. Calculation table of nonlinear regression parameters of the Qingdao Jiaozhou Bay Subsea Tunnel.

Category	Estimated Variables	βEstimated Value	95% Confidence Interval
Tunnel left lane (Huangdao–Qingdao)	Constants ${\beta }_0$	0.7406	(0.733, 0.7481)
	Longitudinal slope gradient ${\beta }_1$	−2.4254	(−7.124, 2.2728)
	Curve radius ${\beta }_2$	−30.8482	(−119.426, 57.7291)
Tunnel right lane (Qingdao–Huangdao)	Constants ${\beta }_0$	0.7591	(0.754, 0.7645)
	Longitudinal slope gradient ${\beta }_1$	−5.7040	(−9.349, −2.0589)
	Curve radius ${\beta }_2$	−50.6497	(−111.698, 10.3982)

.

According to Table 23, the nonlinear regression equation can be obtained as:

(1)

Left line (Huangdao–Qingdao)

$$C_r=C_0\times N\times 0.74063-2.42536i^2-30.8482R^{-1}$$

(6)

(2)

Right line (Qingdao–Huangdao)

$$C_r=C_0\times N\times 0.759083-5.70397i^2-50.6497R^{-1}$$

(7)

C_r—Actual capacity of the Qingdao Jiaozhou Bay Subsea Tunnel;

C₀—The basic capacity of a single lane, pcu/h/lane, under ideal conditions, is taken as 2100 pcu/h/lane;

N—Number of lanes, taken as 3;

i—Slope of the tunnel, %;

R—Radius of the flat curve, km.

3.4.4. Checking the Goodness of Fit

The misfit term test was used to determine whether the regression model is acceptable for the goodness-of-fit test for the affordability model established in this paper. The misfit test’s key parameters are DF (degrees of freedom), SS (different sums of squared errors), MS (mean square of different errors), F (chi-square test), and p-value (significance).

Table 24. Checklist for mismatch of calculation model of actual traffic capacity of the Qingdao Jiaozhou Bay Subsea Tunnel.

Category	Source	Degree of Freedom	SS	MS	F	p
Tunnel Left Line (Huangdao–Qingdao)	Error	18	54,602.8	3033.49	5.39	0.096
	Misfit	15	52,647.7	3509.85
	Pure error	3	1955.2	651.72
Tunnel right line (Qingdao–Huangdao)	Error	26	78,428.4	3016.48	0.76	0.701
	Misfit	17	46,219.9	2718.82
	Pure Error	9	32,208.5	3578.72

shows the misfit test results for the models used in this paper to calculate the actual capacity of the left and right lanes of the Qingdao Jiaozhou Bay Subsea Tunnel. The table shows that the misfit test parameter F is small for both models, and the p-values are greater than 0.05 for the left line of the tunnel and 0.701 for the right line of the tunnel. If the p-value is greater than the significance level, no model misfit is detected, and the regression equation is proven to be reliable.

3.4.5. Determining the Marking form Correction Factor

On the basis of the geometric condition-based Qingdao Jiaozhou Bay Subsea Tunnel capacity model developed above, the correction factors for traffic markings were determined using the difference scaling method, as shown in

Table 25. Data table of traffic capacity corresponding to marking form.

	Solid Line	Dotted Line	Difference	Difference Ratio
Passage capacity	4620	4644	24	0.52%
	4608	4644	36	0.78%
	4608	4620	12	0.26%
	4584	4608	24	0.52%
	4560	4584	24	0.53%
	4548	4548	0	0.00%
	4536	4548	12	0.26%
	4536	4548	12	0.26%
	4584	4596	12	0.26%
	4680	4692	12	0.26%
	4656	4716	60	1.29%
	4620	4656	36	0.78%
	4572	4668	96	2.10%
	4644	4692	48	1.03%
	4644	4740	96	2.07%
Average value	4600	4633.6	33.6	0.73%

.

According to the above table, the average difference in capacity after traffic marking changes from solid lines to dashed lines is 33.6, and the percentage difference is 0.73%, so the correction factor is 1.0073. To summarize, the actual capacity model of the Qingdao Jiaozhou Bay Subsea Tunnel built in this paper is as follows:

(1)

Left line (Huangdao–Qingdao)

$$C_r=(C_0\times N\times 0.74063-2.42536i^2-30.8482R^{-1})\times f_l$$

(8)

(2)

Right line (Qingdao–Huangdao)

$$C_r=(C_0\times N\times 0.759083-5.70397i^2-50.6497R^{-1})\times f_l$$

(9)

C_r—Actual capacity of the Qingdao Jiaozhou Bay Subsea Tunnel;

C₀—Under ideal conditions, the basic capacity of a single lane, taken as 2100 pcu/h/lane;

N—Number of lanes, number of lanes in the Qingdao Jiaozhou Bay Subsea Tunnel, taken as 3;

i—Slope of the tunnel, %;

R—Radius of flat curve, km;

f_l—Traffic marking correction factor, solid lines take 1, dashed lines take 1.0073.

3.5. Comparative Analysis of Passage Capacity

For model validation, the Qingdao Jiaozhou Bay Tunnel’s road conditions following the second marking were chosen. The design of the tunnel markings after the second alignment, the traffic flow parameters, and the parameters corresponding to the tunnel slope, curve radius, and traffic markings were then added to the actual capacity calculation model of the subsea tunnel built in this paper, the calculated values were obtained, and the two were then compared for analysis.

The actual capacity of eight cross-sections before and after delineation was compared with the calculated capacity of the model to validate the capacity calculation model of the subsea tunnel based on geometric conditions and control measures established in this paper. Because the majority of the delineated places in the tunnel are concentrated on flat or uphill slopes, six uphill slopes, one flat slope, and one downhill slope were chosen, and traffic data were extracted in the VISSIM simulation model. The section selections are shown in

Table 26. Selection table of instance verification section.

Number	Starting Stake Number	End Stake Number	Slope i (%)	Curve Radius R (km)	R−1	Traffic Markings
1	ZK8 + 953	ZK9 + 174	−0.703	2000	0.5000	Solid line
2	ZK4 + 260	ZK4 + 330	2.228	-	0	Dashed line
3	ZK3 + 705	ZK3 + 870	3.448	-	0	Dashed line
4	ZK1 + 410	ZK1 + 685	2.151	1590	0.6289	dashed line
5	YK5 + 785	YK6 + 200	0.3	-	0	dashed line
6	YK6 + 900	YK7	3.9	-	0.0000	dashed line
7	YK7 + 745	YK7 + 875	2.605	4945	0.2022	Dashed Line
8	YK8 + 770	YK8 + 960	3.9	2500	0.4000	solid line

below, and the results of the Jiaozhou Bay Tunnel example verification section capacity comparison are shown in

Table 27. Comparison table of traffic capacity of cross-sections verified by example in Jiaozhou Bay.

Number	Slope (%)	R−1	Traffic Markings	Calculation of Capacity	Actual Capacity	Error	Error Ratio
1	−0.703	0.50	solid line	4649	4716	67	1.41%
2	2.228	0	dashed line	4687	4638	−49	−1.05%
3	3.448	0	dashed line	4670	4529	−141	−3.10%
4	2.151	0.63	dashed line	4668	4596	−72	−1.56%
5	0.3	0	dashed line	4815	4840	25	0.51%
6	3.9	0	dashed line	4728	4798	70	1.45%
7	2.605	0.20	dashed Line	4766	4620	−146	−3.17%
8	3.9	0.40	solid line	4708	4537	−171	−3.77%

and Figure 19.

In the previous article, the factors affecting the tunnel capacity were determined, and the slope, marking form, etc. have a greater impact on the capacity. For No. 5, the slope is 0.3, the vehicle driving experience is better, the traffic marking form is dashed, and the vehicle can change driving lanes in the tunnel safely, avoiding the existence of tunnel congestion caused by improper driving behavior of the previous vehicle, thus improving the capacity, so the capacity of No. 5 is higher. As mentioned above, the slopes of No. 3 and No. 8 are larger, and the marking in No. 8 is in the form of solid lines, vehicles cannot change lanes, vehicle driving behavior is restricted, and in some specific cases, the tunnel capacity will be reduced.

By comparing the actual capacity and the calculated capacity of the model in Table 25, it can be seen that the calculated results of the tunnel capacity model constructed in this paper have a small error compared to the actual capacity value, where the error between the calculated capacity and the actual capacity is only 0.51% at No. 5, which proves that the tunnel capacity model constructed in this paper has a smaller error compared to the actual one and can calculate the tunnel capacity better. Moreover, by comparing the data, we can conclude that the calculation results of the Jiaozhou Bay Subsea Tunnel capacity model based on geometric conditions and control measures constructed in this paper are within 4% error of the actual capacity, which is within the acceptable range and can calculate the actual capacity of the subsea tunnel more accurately.

4. Conclusions

The effects of longitudinal slope gradient, the radius of the flat curve, and control measures on the capacity of extra-long subsea tunnels are studied in this paper, and the multiple linear regression model idea is used to establish a calculation model of the capacity of Jiaozhou Bay’s extra-long subsea tunnel based on geometric conditions and control measures. The following are the key conclusions:

(1)

Road conditions, traffic conditions, traffic control measures, and tunnel traffic flow operation are all closely related to the tunnel and have an effect on its capacity. The traffic marking in extra-long subsea tunnels will result in uneven traffic distribution; the change of slope will affect the headway between vehicles, affecting road capacity; the change of slope and curve radius in the tunnel, as well as the design of traffic marking, will affect speed, resulting in large fluctuations in speed in the section, which provides a new idea for the improvement of the continuous carrying capacity of the tunnel.

(2)

As the slope increases, the capacity of the tunnel road decreases. When the longitudinal slope reaches 3%, the road capacity decreases significantly as the slope increases.

(3)

Tunnel capacity is linearly and negatively correlated with the reciprocal of the curve radius (1/R), with tunnel capacity decreasing as 1/R increases, i.e., the capacity increases as the curve radius increases.

(4)

Although there is no significant relationship between traffic marking and capacity growth, the difference ratio method can be used to optimize the actual capacity calculation model of the Qingdao Jiaozhou Bay Subsea Tunnel.

(5)

The capacity of the Qingdao Jiaozhou Bay Subsea Tunnel is calculated using a multiple non-linear regression model with an error of less than 4% when compared to measured data, indicating that the method is practical and scientific, and has important significance for evaluating the sustainable traffic capacity of the tunnel.

This paper examines the factors that influence the capacity of extra-long subsea tunnels, develops a model for calculating actual capacity, and validates it using the Jiaozhou Bay Subsea Tunnel as a case study. Due to time and resource constraints, more research on the capacity of subsea tunnels is required, and the following aspects are suggested.

(1)

Because of the cross-harbor tunnel’s unique spatial structure and road alignment, only the impact factors of its capacity, such as the slope of the longitudinal slope, the radius of the flat curve, and the design of traffic markings, are examined in this paper.

(2)

The lane change characteristics and driver behavior characteristics are not verified and calibrated when constructing the simulation model of the cross-harbor tunnel, which can be further analyzed in the future.

(3)

In this paper, the actual capacity calculation model for the basic section of extra-long subsea tunnels is established, and the capacity calculation model for the intertwined area and ramps of the subsea tunnel can be considered later.