Research on the impact of ship traffic flow on the restricted channel segment of the middle Yangtze River based on traffic wave theory

On the basis of the influence of dry season on ship traffic flow, the gathering and dissipating process of ship traffic flow was researched with Greenshields linear flow—density relationship model, the intrinsic relationship between the ship traffic congestion state and traffic wave in the unclosed restricted channel segment was emphatically explored when the ship traffic flow in a tributary channel inflows, and the influence law of multiple traffic waves on the ship traffic flow characteristics in unclosed restricted segment is revealed. On this basis, the expressions of traffic wave speed and direction, dissipation time of queued ships and the number of ships affected were provided, and combined with Monte Carlo method, the ship traffic flow simulation model in the restricted channel segment was built. The simulation results show that in closed restricted channel segment the dissipation time of ships queued is mainly related to the ship traffic flow rate of segments A and C, and the total number of ships affected to the ship traffic flow rate of segment A. And in unclosed restricted channel segment, the dissipation time and the total number of ships affected are also determined by the meeting time of the traffic waves in addition to the ship traffic flow rate of segments. The research results can provide the theoretical support for further studying the ship traffic flow in unclosed restricted channel segment with multiple tributaries The inflow of tributaries' ship traffic flows has an obvious impact on the traffic conditions in the unenclosed restricted channel segment. The interaction and influence between multiple ship traffic waves and the mechanism of generating new traffic waves are explained. The expression of both dissipation time of queued ships and the total number of ships affected in the closed and unclosed restricted channel segment are given. The inflow of tributaries' ship traffic flows has an obvious impact on the traffic conditions in the unenclosed restricted channel segment. The interaction and influence between multiple ship traffic waves and the mechanism of generating new traffic waves are explained. The expression of both dissipation time of queued ships and the total number of ships affected in the closed and unclosed restricted channel segment are given.


Introduction
The middle reaches of the Yangtze River, as the throat segment of the golden waterway of the Yangtze River, are characterized by curved channels, numerous shallow shoals, different river types and river bed boundary conditions along the reach, and complicated and changeable navigation environment. Changes in precipitation and temperature caused by climate change will directly or indirectly lead to the changes in river water level, erosion and deposition characteristics, and river morphology, resulting in continuous high or low water level. For example, in the spring of 2007, the middle reaches of the Yangtze River encountered exceptionally low water level in a century, and in July 2012, due to the extreme flood peak of the Yangtze River, the Three Gorges ship lock was suspended and the navigation in some channel segments of the middle Yangtze River was restricted or banned, and thousands of ships were stranded. From 2012 to 2019, the extreme low water level of the middle Yangtze River continued to occur, and in November 2019, the water level in the middle reaches of the Yangtze River reached its lowest record in 116 years. This change in the frequency and duration of extreme hydrological events will have a serious impact on navigation along the Yangtze River.
In the dry season, the channel is further narrowed, which will increase the risk of ships encounter, and ships are easy to get touching the shore and grounding. Shallow water effect will occur because of low water level, resulting in the decline of ship speed and rudder force, the increase of ship control difficulty, traffic jams and even congestion. During the flood period, the scouring force of the water stream becomes smaller, silt and sand are deposited in the channel, and the water level of the channel gradually decreases. The cargo ship with deep draft is likely to be stranded. The frequent occurrence of extreme hydrologic events has a great negative impact on the shipping organization, transportation safety and transportation economy of the Yangtze River.
From the successful experience of foreign basin development, the industrial agglomeration and central position of the countries or regions along the river basin have largely benefited from the improvement of the channel conditions and shipping methods. At present, under the background of China's strategic implementation of the Yangtze River Economic Belt, the research on ship traffic flow and navigation efficiency in the middle reaches of the Yangtze River has become a hot spot. Some scholars focused on the theoretical research of traffic flow. Wang, Ren and Wu studied the relationship among traffic flow rate, density and speed, and simulated traffic flow with related theories of fluid mechanics [1][2][3]. Another part of scholars concentrated on the research of ship traffic flow of inland waterway, divided into the following directions.
The first was to study the characteristics of ship traffic flow in inland waterway. Liu and Lu et al. [4,5] studied the closure measures of navigation adopted in the channel under the extreme water level, and explained the process of gathering and dissipating process of ship traffic flow after the closure of navigation using the traffic wave theory. Liu et al. and Liao [6,7] respectively analyzed the characteristics of ship traffic flow in sea and inland waterway. In order to improve the safety of Marseille Port, Elloumi et al. [8] developed the decision support system to conduct the statistical analysis on the behavior characteristics of medium and small ships in the port. In 2013, Martin et al. [9] drew the diagram of the relationship between "flowspeed", "speed-density" and "speed-flow" according to the vehicle traffic flow rate in Frankfurt, Germany. Kujala studied the distribution of basic characteristic elements such as vessel flow; composition and position in the ship lock waters [10]. Liu investigated the characteristics, influencing factors and evolution mechanism and model of ships under complex navigable conditions [11].
The second was to study the passing capacity or efficiency of inland waterway. Dong et al. [12] studied the passing ability of natural and channelized channels. Based on integrating ship traffic flow theory and the relative analytical formula, Zhang et al. [13] established the calculation method of passing capability under any ship type. Yang et al. [14] proposed some representational parameters of ship traffic congestion status in restricted channel segment, and discussed the internal correlation between ship traffic congestion status and character of ship motion by using Monte Carlo simulation.
The third is the use of Greenshields model in macrotransportation planning. Since the Greenshields model is simple in form and convenient for the derivation and analysis of theoretical formulas, although it has some shortcomings in its application, it is still widely used in the study of road traffic flow and is a very important model [15][16][17][18][19]. The relationship of velocity-density in ship traffic flow is similar to that in road traffic flow, but the influence factors of the former has some difference from the latter. Ship traffic flow is not only affected by the larger volume, lower speed, larger spacing and smaller density of ships, but also by the speed of water flow, and the regulations that generally in a single channel ships are not allowed to exceed. Therefore, the velocity-density model should be selected in the study of ship traffic flow, which is relatively simple in form and can basically guarantee the accuracy of macroscopic analysis. At present, many scholars have explored the application of the Greenshields model in the study of the ship traffic flow of a single waterway, especially in the Yangtze River waterway, and major relevant literatures have been reported and some research results obtained. Such as Zhang et al. [20] he analyzed the inland ship traffic flow organizational characteristics with Greenshields speed-density linear relationship; Lu et al. [4] studied that impact of extreme water levels on characteristics of ship traffic flow in the middle reaches of Yangtze river using Greenshields model; Bai et al. and Zeng [21,22] applied Greenshields model in the investigation of traffic efficiency based on traffic wave theory.
The fourth, various methods were used to conduct simulation research on ship traffic flow of inland waterway, but the research objects were mostly the ship traffic flow of inland waterway under long-term and normal water level [23][24][25][26]. By studying the characteristics of the bridge waters, Zhou analyzed the statistical information of ship attributes with statistical methods, mainly including the fitting of the distribution rules of the length, speed and spacing of ships, and performed the simulation analysis of ship traffic flow track [27]. In 2017, Wang et al. [28] established the complex simulation model of ship traffic flow considering influencing factors such as ship entry and departure rules, navigation rules and anchorage scale, to simulate ship navigation operation system, and to study the influencing factors of passing ability of channel. Asperen carried out the simulation study on ship arrival rules [29]. Based on historical data, Ercan studied the ship traffic behavior under different circumstances in the Istanbul Strait area, as well as the arrival time and waiting time of various types of ships, and established the traffic simulation model in the Istanbul Strait [30]. Kawamura et al. [31] studied ship behavior under severe sea conditions with simulation method. Qi et al. [32] analyzed the complex traffic phenomena of ships. Based on the spatial logic mapping (SLM) model, they proposed the process to solve the spatial discrete waterway rules and ship movement update rules, and established the ship traffic flow model based on cellular automata (CA). Qi et al. proposed a new maritime traffic model based on cellular automata with the characteristics of ship speed variation. The performance of the model was tested under weather and ocean conditions in the Dover Strait, the Changshan Channel and the Qiongzhou Strait [33]. Feng studied the situation of ship diversion or accident under partially closed channel, established the cellular automata model, and analyzed the influence of these conditions on the passing capacity of channel with numerical calculation method [34].
Due to the wide application of ship traffic flow and the great diversity of research objects, most of the literature selectively studies one or more ship traffic flow characteristics, such as the composition, behavior and arrival rules of the ship traffic flow, according to the application requirements. But there is little research on the impact of extreme water level changes on ship traffic flow of inland waterway. This research can enrich the research system of inland ship traffic flow to a certain extent, and provide the reference for Maritime Department in making the navigation plan and improving navigation efficiency, and further research the ship transport organization mode, ship type, port construction, and control mode to reduce the impact of extreme hydrological events in the middle reaches of Yangtze River, enhance the competitiveness of shipping, and develop the Yangtze River Economic Belt. Therefore, the research has certain theoretical significance and practical application value.
In this paper, under extreme water level, the ship traffic flow characteristics were analyzed with traffic wave theory, flow conservation law and Greenshields model. The gathering and dissipating process of ship traffic flow in closed and unclosed restricted channel segments were studied, the impact of the inflow of ship traffic flow in a tributary channel on the ship traffic flow in the unclosed restricted channel segment were discussed, and the relevant mathematical models of the ship traffic flow in the restricted channel segment were established. And the traffic flow characteristics of both closed and unclosed restricted channel segments were studied by using Monte Carlo method.
The contributions of the paper are as follows: (1) The expressions of traffic wave speed and direction, the dissipation time and the total number of affected ships were given. (2) The dissipation time of the queued ship is mainly related to the ship traffic flow rate of relative segments (upstream, downstream and tributary channel segment). Both total numbers of affected ships in closed and unclosed restricted segments are highpositive correlation with the ship traffic flow of the upstream segment. (3) The interaction and influence laws among multiple traffic waves were given. environment, waterway and time periods, which mainly include ship traffic flow rate, composition of ship traffic flow, and law of ship arrival and distribution of waiting time. Microscopic characteristics refer to the characteristics related to the position, direction, speed and ship domain of a single ship. The dry season in the middle Yangtze River is generally from December to March of the following year, because of the low water level and slow water flow, shoals will appear in some channel segments (in the following text, the channel segments are referred to as segments.), so these segments are often curved, and make ship operation more difficult, passing capacity of these segments declined, and ship density relatively increased. At the peak of ship traffic flow, in order to guarantee ship's safe navigation, Maritime Department will take some measures such as reducing ship loading rate, one-way passing in different time periods, diverting cargoes by other transportation modes, and even closuring channel and so on. The changes of characteristics of ship traffic flow are as follows:

Ship traffic flow rate
In the dry season, the traffic conditions in some channels become worse so that the passing capacity of channels is reduced, and ship congested, which leads to the reduction or uneven distribution of ship traffic flow.

Ship composition
In the dry season, because the water depth cannot meet the ship's full draft, the ship may need to reduce its cargoes, and the proportion of smaller tonnage ships in the ship traffic flow may increase.

Law of ship arrival
Under normal circumstances, the law of ship arrival usually follows Poisson's distribution or Normal distribution. In the dry season, some ships need to wait for the tide to enter the port, and the law of ship arrival changed, which may be achieved by the intermittent traffic flow instead of the steady traffic flow. The equilibrium between busy time and idle time in the port is broken, and the imbalance state is intensified.

Distribution of waiting time
When larger ship traffic flow, poor channel conditions, channel congestion, retention or security risks occur, Maritime Department may control the ships passing through the channel segment according to the actual channel conditions, such as traffic or time restriction, etc. to shunt ships, and even to ban navigation in severe case, which can cause additional time-consuming. In addition, the low water level will also cause the work efficiency of loading and unloading equipment of the port berth to be decreased, so that the waiting time of ships is increased.

Ship speed
The ship's sailing speed is determined by calm water speed of ship and water flow velocity of channel. In the dry season, the water level and the water speed all decreases. At this time, the Maritime Department sometimes requires reducing the ship's cargoes. The decrease of ship load and the influence of water flow velocity of channel will lead to the change of ship speed, which will affect the distribution of the ship speed.

Ship traffic flow theory
Assume that there are two adjacent segments A and B (without other entrances between the two ends) with different density and average speed of ship traffic flow, both density and average speed are ρ A , v A and ρ B , v B respectively, and the direction of ship speed is that of ship traffic flow. When ship traffic flow from segment A inflows into segment B and A < B , the ship traffic flow status changes. Ships line up one after another and gather into high-density queues, causing turbulence, congestion and even blockage; when A > B , the queued ships will start to evacuate into a queue with appropriate density. The vertical line S is used to divide two ship traffic flows of different densities, called wavefront S. The movement of the wavefront S in the ship traffic flow is called traffic wave [1,2]; the speed of the wavefront S moving along the channel is called the wavefront speed, which is the absolute speed relative to the channel, referred to as traffic wave speed v w . It also stipulates that the traffic wave speed is positive when the direction of the ship's traffic flow is the same, as shown in Fig. 1.

Basic mathematical model of ship traffic flow
Assuming that the same ship type is single, the ship speed is uniform, the ship heading is the same, and the channel width is unchanged, the basic equation of ship traffic flow rate can be expressed as: where q is ship traffic flow rate, that is the number of ships passing through a certain cross section per unit time, ves/h; ρ is the density of ship traffic flow, that is the number of ships in per unit length of channel segment, ves/km; and v is average speed of ship traffic flow, km/h.

Flow conservation law
According to mass conservation law in fluid mechanics, the number of ships leaving segment A per unit time t is equal to the number entering segment B.
where v w is traffic wave speed, km/h; v A -v w is the speed of ship traffic flow relative to wavefront S before it enters wavefront S, km/h; v B -v w is the speed of ship traffic flow relative to wavefront S after it enters wavefront S, km/h. So, Substituting Eqs. (1) into (3), traffic wave speed can be expressed as:

Greenshields linear model
According to the linear speed-density relationship model proposed by Greensheilds, the average speed of ship traffic flow in segment i can be expressed as: Then the ship traffic flow rate in segment i is written as: where v f is ship's free flow speed, km/h; ρ j is congestion density, that is the density of ship traffic flow when traffic jams occur and the ship is almost impossible to navigate, ves/km.
The relationship between traffic flow rate and density is a quadratic function as shown in Fig. 2: In Fig. 2, ship traffic flow rate increases with increasing density. When the density reaches ρ m , ship traffic flow rate reaches the maximum flow rate q m , the density at this time is the optimum density ρ m , and the best speed is v m ; when density continues to increase, the flow rate begins to fall, when the ship traffic flow rate arrives blocking density ρ j , ships are almost impossible to navigate, both speed and flow rate is zero. Take the derivative of Eq. (6) and set it equal to zero. And the following formula can be obtained: The maximum traffic flow rate can be expressed as:

Influence of traffic waves on ship traffic flow
Traffic wave describes the transformation process of two traffic flow states, as well as both direction and speed of transformation. When ship traffic flow enters segment B from segment A, and ρ A ˃ ρ B , it indicates that the density decreases, the ship speed is up, and the generated traffic wave is called dissipating wave; when ρ A ˂ ρ B , the density increases, the ship speed is down, and the traffic wave is gathering wave, the speed of traffic wave is represented by v w . When v w ˃ 0, the traffic wave propagates downstream (i.e. forward propagation), consistent with the direction of the ship navigation. The traffic conditions of segment B will not affect the upstream segment A.
(1) When q A ˃ q B and ρ A ˃ ρ B , the ships from the high-flow rate, high-density, low-speed segment A enter the low-flow rate, low-density, high-speed segment B, the traffic conditions become better. Due to the forward propagation of the traffic wave, the dense ship traffic flow of segment A begins to dissipate after entering segment B, forming a dissipating wave, but it will not improve the traffic conditions of upstream segment A, as shown in Fig. 3a.
There are two situations. One is when ρ A ˃ ρ m ˃ ρ B , the ship density in segment A is greater than the optimum density and the segment is crowded. In order to ensure safety, the ship speed slows down and the ship traffic flow rate passing through is small. When the ship enters from segment A to segment B, the traffic conditions become better, the ship speed accelerates and the dense ship traffic flow begins to dissipate; the other is when ρ m ˃ ρ A' ˃ ρ B , the segment A and B are not congestion. However, the traffic conditions of segment B are better than that of segment A, and the ships speed up after entering from segment A to segment B. But, they all fail to reach the optimum density and traffic flow rate.
(2) When q A ˂ q B and ρ A ˂ ρ B , the ships from low-flow rate, low-density, high-speed segment A enter high-flow, high-density, low-speed segment B, traffic conditions become worse. Due to the forward propagation of the traffic wave, the ship traffic flow entering the segment B follows the original ship traffic flow of segment B, ship density increases so that gathering wave is formed, but it will not worsen the traffic conditions of upstream segment A, as shown in Fig. 3b.
There are two situations. One is ρ A ˂ ρ m ˂ ρ B , the ship density in segment B is larger than the optimum density, and the segment is crowded, the ship speed is slow, and the ship traffic flow rate is low; the other is when ρ A ˂ ρ B' ˂ ρ m , segment A and B are not congestion. However, the traffic conditions of segment B are worse than that of segment A, and the ship speed slows down after entering from segment A to segment B, but it does not reach the optimum density and traffic flow rate.

v w < 0
When v w ˂ 0, the traffic wave propagates upstream (i.e. backward propagation), opposite to the direction of the ship speed. The upstream segment A is affected by the traffic condition of channel segment B.
(1) When q A > q B and ρ A < ρ B , the ships from high-flow rate, low-density, high-speed segment A enter lowflow rate, high-density, low-speed segment B, the gathering wave is formed. Among them, when ρ A < ρ m < ρ B , segment B has been congested, the traffic conditions of upstream segment A is affected by the obstruction of segment B, the ships slow down and begin to queue into segment B. When ρ m <ρ A' < ρ B , both segments A and B have been congestion, the traffic conditions of segment B is worse, and the traffic conditions of upstream segment A will also deteriorate further, the queue is longer and more congestion. Both cases are shown in Fig. 3c. (2) When q A < q B and ρ A > ρ B , the ships from the lowflow rate, high-density, low-speed segment A enter the high-flow rate, low-density, high-speed segment B, dissipating wave is formed. Among them, when ρ A ˃ ρ m > ρ B , the traffic conditions of downstream segment B is favorable, the ship traffic flow can be accelerated into segment B, the congestion of segment A can be alleviated. When ρ A > ρ B' > ρ m , although both segments A and B are congestion, the traffic conditions of segment B is better than that of segment A, which can improve the traffic conditions of upstream segment A, and the queue condition of segment A is dissipated, as shown in Fig. 3d.

v w = 0
When v w = 0, only q A = q B , this is a change in the state of the traffic flow of two ships with the same flow but different speeds and densities. The traffic wave does not move and does not have any impact on upstream and downstream. According to the above analysis, when the ship traffic flow changes from low density to high density, gathering wave will be formed, and when the ship traffic flow changes from high density to low density, dissipating wave will be formed. Traffic wave has a great impact on the existing traffic conditions, which should be the focus of the study.

Analysis of ship traffic flow characteristics in closed restricted channel segment
In the dry season, the water level is low and the flow speed is declined, so that some segments are shallow and narrow. As a result, navigation bottlenecks are created, and the passing capacity of the segment is restricted, the density of ships in the segment is relatively increased and the ship speed is declined. The closed segment means that there are no other tributaries on the main channel, and the ship traffic flow rate mainly comes from upstream segment. Related assumptions are as follows.
(1) Assuming that segments A to C are closed, the width of the segments is the same, and there are no other tributaries, that is, all ship traffic flow rate flowing into segment A will eventually flow out of segment C. (2) In the dry season, segment B becomes a restricted one due to the deterioration of navigation conditions in the channel, and its maximum passing capacity q Bm is less than that of segments A and C, that is q Bm ˂ q Am = q Cm , at this time, the restricted segment B reaches the maximum passing capacity, that is q B = q Bm and ρ B = ρ m . (3) In order to ensure safety and reduce congestion of the channel, the Maritime Department usually adopts the control measures of one-way release at different times during peak periods; ship overtaking is prohibited within restricted segment. When the ship of segment A enters segment B, the ship needs to slow down. When the ship leaves segment B and reaches the downstream segment C, the ship starts to accelerate, as shown in Fig. 4. When the ship traffic flow from the low-density segment A enters the high-density segment B, that is, ρ A < ρ B (according to the analysis in Chapter 3, the gathering wave is generated this moment), wavefront S 1 is generated, and the wave speed is v w1 . Ship traffic flow can be smooth sailing in segment A, but due to the limited passing capacity of segment B, the density of ships in the segment B is high and the ship speed is slow. At this time, segment B has reached the maximum passing capacity, that is v A >v B and ρ A < ρ B = ρ m . There are two situations here. One is when q A ˃ q B , v w1 < 0; it indicates that the traffic wave propagates to upstream segment A, and the direction of the traffic wave is opposite to that of ship navigation. Therefore, the traffic conditions of upstream segment A are affected, and the ships need to slow down and queue to enter the restricted segment B. The ships start to queue at wavefront S 1 , and the queue propagates to upstream segment A, so gathering wave is generated. The other is when q A ˂ q B , so v w1 ˃ 0; it indicates that the traffic wave propagates to downstream segment B, and the direction of the traffic wave is consistent with that of the ship navigation. The ship traffic flow of segment A enters segment B, due to the increase in density, the ships will slow down and queue up at the back of the ship traffic flow in the original segment B, forming gathering wave, but will not affect the traffic conditions of upstream segment A, as shown in Fig. 5.

Wavefront S 2
When the ship traffic flow leaves the high-density segment B and enters the low-density segment C, that is, ρ B > ρ C (according to the analysis in Chapter 3, the dissipating wave is generated this moment), wavefront S 2 is generated, and the wave speed is v w2 . Because the traffic condition of downstream segment C is good, that is, v B ˂ v C and ρ B = ρ m > ρ C . There are two situations here. One is when q B > q C , so v w2 ˃ 0; it indicates that the traffic wave propagates to downstream segment C, and the direction of the traffic wave is consistent with to that of ship navigation. So the dissipating wave is formed in channel segment C, but it will not affect the upstream traffic conditions. Second, when q B ˂ q C , then v w2 ˂ 0; it indicates that the traffic wave propagates to upstream segment B, and the direction of the traffic wave is opposite to that of the ship navigation. The dissipating wave propagates upstream, the existing queues begin to dissipate, and the traffic conditions of upstream segment B are improved.

Analysis of gathering and dissipating waves
There are two situations in the gathering wave and the dissipating wave. One is that the gathering wave and the dissipating wave will never meet, and the other is that that the gathering wave and the dissipating wave will meet.
(1) The gathering wave and the dissipating wave will never meet There are two situations: One is when v w2 ˃ 0, no matter if v w1 ˃ 0 or v w1 < 0, the gathering wave and the dissipating wave will not meet, that is, the queuing situation always exists in this case. The other is when v w2 ˃ 0, v w1 < 0 and , the queuing speed of ships is greater than the dissipating speed, and the queuing will continue to increase until the off-peak period, when the traffic flow rate decreases, the queue slowly dissipates on its own. These two situations are not discussed in this paper.
(2) The gathering wave and the dissipating wave will meet There are two situations. One is when v w2 < 0, v w1 < 0 and ; the other is when v w2 < 0 and v w1 > 0, both the gathering wave and the dissipating wave will meet, which is the focus of this paper.
According to as shown in Fig. 4, in the peak period the Maritime Department uses control measures of one-way release at different time. The ship overtaking is prohibited in the restricted segment B; when ρ A < ρ B, the ships in segment A need to slow down entering into the segment B, and line up behind the ship traffic flow of the original segment B at wave speed v w1 , so the density of the queue is ρ B , which generates gathering wave. When ρ B > ρ C , the ship that leaves segment B and reaches downstream segment C begins to accelerate, and dissipate the queue at wave speed v w2 , which generates dissipating wave. According to Eq. (3), the wave speed v w1 and v w2 can be expressed as: The front end of the ship traffic flow in the restricted segment B is dissipated at the wave speed v w2 , and that of the back is still gathering at the wave speed v w1 . The dissipation time T is the time for the dissipating wave to catch up with the gathering wave. The moment, all queued ships are completely dispersed, so T can be expressed as: Where T is dissipation time of the queue, h; L B is the length of the restricted segment B, km.
The total numbers of ships affected by closed restricted segment N B can be expressed as:

Analysis of ship traffic flow characteristics in unclosed restricted segment
The  As can be seen from Fig. 6, when ∆q D outflows from segment B 2 , that is, ∆q D < 0, the density of segment B 2 decreases and the traffic condition improves. When ∆q D inflows into segment B 1 , namely ∆q D > 0, ship traffic flow rate in upstream segment B 2 allowed to enter segment B 1 is q Bm -∆q D , but at this time the ship traffic flow rate in segment B 2 is q Bm , therefore the ships with ∆q D will be stuck, while the ships in segment A will continue to inflow. As a result, the density of segment B 2 rises, the channel congests and the traffic condition deteriorates. At the moment, the wavefront S 3 with wave speed v w3 is formed between channel segment B 1 and B 2 , the wavefront S 1 with wave speed v w1 is formed between channel segment B 2 and A, and the wavefront S 2 with wave speed v w2 is formed between channel segment B 1 and C.

Wavefront S 3
Since segment B 2 has only ship traffic flow rate q B -Δq D that can be inflowed into segment B 1 , which indicates that maximum passing capacity of segment B 2 has changed from the original largest passing capacity of restricted segment B ( q B 1 m = q B 2 m = q Bm ) to q B -Δq D . Therefore, segment B 2 occurs congested, traffic flow density increases and speed decreases, it can be seen that segment B 2 becomes the bottleneck section in segment B. The ship traffic flow status changes from B 2 of the traffic flow ratedensity curve to B 2 ′ , that is, the maximum passing capacity of segment B 2 is equal to that of the original restricted segment B, which exceeds the maximum passing capacity. As a result, the density increases from ρ m to ρ B2' . Ship speed declines from v B2 to v B2' , ship traffic flow rate drops to as q Bm -∆q D , namely q B2' = q B2m = q Bm -∆q D , as shown in Fig. 7. Since ρ m < ρ B2' , dissipating wave is generated, and the wave speed v w3 of wavefront S 3 can be expressed as: Due to ∆q D > 0 and ρ m < ρ B2' , so v w3 < 0, the dissipating wave propagates from S 3 to upstream segment B 2 .

Wavefront S 1
In unclosed restricted segment, the wave speed v w1 of wavefront S 1 will be different from the previous analysis of closed restricted segment, can be expressed as: Since ρ A < ρ B2' , gathering wave is generated. There are two situations: (1) if q A > q Bm --∆q D , then v w1 < 0, the gathering wave propagates from S 1 to upstream segment A, and ships queue up in segment A with the density of ρ B2' ; (2) if q A < q Bm -∆q D , then v w1 > 0, the gathering wave is propagates from S 1 to the downstream section B 2 , and ships queue up in the section B 2 with the density of ρ B2' .

Wavefront S 2
In unclosed restricted segment, the wave speed v w2 of wavefront S 2 is exactly the same as the analysis of the closed restricted segment, see the analysis in 4.1.2.

Analysis of gathering and dissipating waves
According to the above analysis, three wavefronts S 1 , S 2 and S 3 will be formed in the unclosed restricted segment, and their wave speeds are v w1 , v w2 and v w3 respectively. The v w1 is the wave speed of the gathering wave, and v w2 and v w3 are the wave speed of the dissipating wave. When two traffic waves meet, a new traffic wave appears and sequentially propagates. According to the analysis of 4.1.3, the paper does not discuss the case of v w2 > 0. When v w2 < 0, there are two situations One is when v w2 and v w3 meet first (i.e. |v w2 | >|v w3 |, the meeting time is T 23 ) to produce a new dissipating wave that propagates to upstream at the wave speed v w4 . When the v w4 meets v w1 , queued ships are disappeared. The other is when v w3 and v w1 meet first (i.e. |v w3 | >|v w1 |, the meeting time is T 31 ), queued ships have all disappeared. Both T 23 and T 31 can be respectively expressed as: When T 23 > T 31 , it means that the gathering wave(wave speed is v w1 ) and the dissipating wave (wave speed is v w3 ) meet first, the front end of the ship traffic flow in the segment B 2 is dissipating at wave speed v w3 , and the back end is still gathering at wave speed v w1 . The dissipation time T is the encounter time of the dissipating wave and the gathering wave. At this time, all the queued ships due to the gathering wave are completely dissipated, and the dissipation time T can be expressed as: The total numbers of ships affected by unclosed restricted segment N B can be expressed as: (2) T 23 < T 31 When T 23 < T 31 , it means that two dissipating waves meet first, the wave speed is v w2 and v w3 respectively, and a new dissipating wave is generated at wave speed v w4 , which continues to propagate upstream. The dissipating wave speed v w4 is expressed as: When the new dissipating wave is generated at wave speed v w4 . The front end of the ship traffic flow in segment B 2 is dissipating at wave speed v w4 , while the back end is still gathering at wave speed v w1 . When the new dissipating wave speed v w4 and the gathering wave speed v w1 meet, at this time, all queued ships will dissipate. The dissipation time T is divided into two parts: one is the meeting time of two dissipating waves speed v w2 and v w3 , i.e. T 23 ; the other is the meeting time of dissipating wave v w4 and gathering wave v w1 , i.e. T 14 . T 14 and T are expressed as: The total numbers of ships affected by unclosed restricted segment N B can be expressed as:

Research on vessel traffic flow characteristics in the middle reaches of the Yangtze River
The middle reaches of the Yangtze River starts from Yichang, Hubei Province, and ends at the Hukou, Jiangxi Province, with a total length of 927 km. The most troublesome segment in the middle reaches of the Yangtze River is the Zhicheng-Chenglingji segment, also known as the Jingjiang segment, which is a "bottleneck" on the Yangtze's golden waterway. This paper selected the 2013 annual monitoring data of Jingzhou Yangtze River Highway Bridge located in the lower segment of Taipingkou waterway (the latest data cannot be obtained, so 2013 data was used for research).

Law of ship arrival
According to the monitoring data of Jingzhou Yangtze River Highway Bridge in 2013, the average monthly daily ship traffic flow rate is shown in Table 1.
According to season and month, the middle reaches of the Yangtze River can be divided into three periods: flood, moderate and dry periods. In general, the flood period is from May to October; the dry period from December to March of the following year; and the moderate period from April to November. As can be seen from Table 1, the ship traffic flow generally conforms to the variation of the river reach in the flood, moderate and dry periods, but this change is not very obvious because the overall ship traffic flow is not large.
The number of ships arriving in different time periods is observed continuously for 24 h, as shown in Table 2. The researches of the ship arrival law in some Chinese seaports indicate that the law generally obeys the Poisson distribution or Normal distribution. In order to further verify these results, SPSS software was used to fit the data in Table 2 to analyze the law of ship arrival in different time periods within a day, and carry out Kolmogorov-Smirnov test. The results are shown in Tables 3 and 4.
It can be seen from Tables 3 and 4 that the law of upstream and downstream ships arrival at different time periods within a day do not conform to the Poisson distribution characteristics, but all conform to the Normal distribution characteristics, and Sig. values are all greater than 0.05 (The Sig. value represents the test value of significant difference. Sig. > 0.05 means to accept the null hypothesis), indicating good fitting degree, so the law of ship arrival follow Normal distribution.

Ship speed distribution
Ship speed distribution refers to the distribution range and law of the speed of all ships in the water area. The speed distribution of ships generally obeys to Normal, Weibull or Lognormal distributions. The ship speed distribution characteristics in the middle reaches of the Yangtze River are shown in Table 5.
The data in Table 5 were fitted with SPSS software (Statistical Product and Service Solutions) to analyze the speed distribution of upstream and downstream ships, and conduct Kolmogorov-Smirnov test. The results are shown in Tables 3 and 4. Both speed distributions of upstream and downstream ships obey to Normal and Poisson distributions, and both Sig. values are greater than 0.05, indicating a good fitting degree. As the Sig. value of Kolmogorov-Smirnov test of Normal distribution is larger than that of Poisson distribution, it shows that the data are more consistent with Normal distribution. The mean speed of upstream ships is 7 kn and that of downstream ships is 8 kn.

Density distribution characteristics
The density distribution of ship traffic flow reflects the congestion degree of ship flow in the water area. The density distribution of ship traffic flow is affected by channel width, geographical factors, water level change, pilot quality and many other factors. The density and its distribution of ship traffic flow are different concepts. The density of ship traffic flow in a certain area is relatively high, but the ship traffic flow may not be blocked. On the contrary, blockage or collision accidents may occur in the water area where the density of ship traffic flow is not too high, which depends on whether the density distribution of ship traffic flow is reasonable. If most of the same numbers of ships in a certain area are concentrated in a local water area, the possibility of blockage accident is much greater than that of the evenly distributed number of ships. These characteristics cannot be described by statistical laws.

Distribution of ship types
According to the statistical analysis of the collected data, there are three main types of ships that pass the Jingzhou Yangtze River Highway Bridge: general cargo ships, dangerous goods ships and container ships. Among upstream ships, the proportions of these three types of ships are 78.01%, 11.77% and 10.21% respectively. In downstream ships, the proportions of that are 77.11%, 11.57% and 14.99% respectively. According to China's the Navigation Standard of Inland Waterway, small ships with a length of less than 90 m in the segment account for more than 90%.

Traffic flow simulation test
Crystal ball software was used to perform Monte Carlo simulation, the simulation assumptions are as follows: (1) The arrival state of ships is the mutually independent random process and obeys Normal distribution; (2) Ship speed obeys Normal distribution; (3) In the peak period, the Maritime Department adopts one-way traffic for this segment (downstream ships are released); (4) Irrespective of ship type; (5) In the case of extreme low water level, ship overtaking is prohibited after the ship enters the restricted segment in order to ensure safety. (6) Ship traffic flow meets Greenshields' linear model; (7) The width of upstream segments A, restricted segment B and downstream segment C is the same; Table 5 The Research Article SN Applied Sciences (2021) 3:748 | https://doi.org/10.1007/s42452-021-04727-w (8) The psychology of the ship's pilot and maneuvering behavior of ships when they merge are not considered; (9) Taking the Taipingkou channel as the research object, which is located in the Jingjiang channel segment of the middle reaches of the Yangtze River. It has always been the key navigation obstacle, namely the segment B, whose with length L B is 17.5 km. Small ships with the length of less than 90 m in the segment account for more than 90%. The distribution of ship types is not considered here, and the representative ship type CJB(2004)H1000-I in the middle reaches of the Yangtze River is taken as the research object.

Parameter settings
The To unity the unit, the unit k n should be conver ted to k m/h (1kn = 1.852 km/h). The simulated object is the ship traffic flow in the continuous segments A-B-C and the restricted segment B. For this traffic process, there are many parameters that will affect the congestion state of the segments. In the simulation test, five parameters are considered as input variables such as the ship traffic flow rate and average speed of upstream segment A and downstream segment C, and the difference of traffic flow rate between inflowed from segment D and outflowed from segment B (In the simulation test of closed restricted segment, the first four parameters are selected; and in that of unclosed restricted segment, five parameters are selected). In Crystal Ball software, these parameters are set as Assumption cells. The two parameters, the dissipation time of queuing and the total number of ships affected, are taken as the output variables and set as forecast cells, as shown in Table 6. Monte Carlo method was conducted for the dissipation time of queuing and the total number of ships affected in the closed and unclosed restricted segments respectively, and the simulating times were 1000. The simulation data were summarized, histograms were drawn, and the 10%, 90%, mean and median data of the statistical data were displayed. Finally, the sensitivity analysis table was generated.

Simulation test of the closed restricted segment
After 1000 times of simulations, the data results of 901 times are shown. Considering the randomness of the test results, these errors are deemed to be reasonable. The simulation results and sensitivity analysis of the dissipation time are shown in Fig. 8 and Table 7 separately.
As shown in Fig. 8 and Table 7, the mean dissipation time is 91.87 h, the median is 23.25 h, and the 90% probability of dissipation time is within 120.15 h. According to the sensitivity analysis, the ship traffic flow rate of segments A and C are negatively correlated with the dissipation time. The average speed of ship traffic flow of segments A and C are positively correlated with the dissipation time. A positive correlation coefficient implies that an increase in the hypothesis will result in an increase in the dissipation time, while a negative correlation coefficient implies the opposite.
The simulation results and sensitivity analysis of the number of ships affected are shown in Fig. 9 and Table 8 separately.
As shown in Fig. 9 and Table 8, the mean of the total number of ships affected is 1485.02 ves, the median is 383.64 ves, and the 90% probability of the total number of ships affected is within 2055.93 ves. According to sensitivity analysis, the ship traffic flow rate and the average speed of ship traffic flow of segment A, and the average speed of ship traffic flow of segment C are positively correlated with the total number of ships affected, and the variance contribution of ship traffic flow rate of segment A is as high as 79.4%. But the correlation between ship traffic flow rate in segment C and the total number of ships affected is negative.
The results show that the larger the ship traffic flow rate of segments A and C, the shorter the dissipation time. The larger the average speed of ship traffic flow of segments A and C, the longer the dissipation time. Ship traffic flow rate of segment A is the key factor of the total number of ships affected.

Simulation test of the unclosed restricted segment
When T 23 > T 31 , the simulation results and sensitivity analysis of the dissipation time are shown in Fig. 10 and Table 9 separately.
As shown in Fig. 10 and Table 9, the mean dissipation time is 2.41 h, the median is 2.09 h, and the 90% probability of dissipation time is within 3.23 h. According to the sensitivity analysis, the ship traffic flow rate of segment  Fig. 11 and Table 10 separately.
As shown in Fig. 11 and Table 10, the mean of the total number of ships affected is 64.61 ves, the median is 54.89 ves, and the 90% probability of the total number of ships affected is within 102.12 ves. According to sensitivity analysis, the ship traffic flow rate of segment A is positively correlated with the total number of ships affected, and the variance contribution is as high as 99.4%. The average speed of ship traffic flow of segment A and C are negatively correlated with the total number of affected ships, with the total variance contribution of 0.5%.
The results show that the ship traffic flow rate of downstream segment C and ∆q D will greatly affect the dissipation time, while the ship traffic flow rate of upstream segment A is the decisive factor of the total number of affected ships.
(2) T 23 < T 31 When T 23 < T 31 , the simulation results and sensitivity analysis of the dissipation time are shown in Fig. 12 and Table 11 separately.
As shown in Fig. 12 and Table 11, the mean dissipation time is 117.21 h, the median is 39.23 h, and the 90% probability of dissipation time is within 189.50 h. According to the sensitivity analysis, the ship traffic flow rate and average speed of segment C are negatively correlated with the dissipation time, and the sum of variance contribution is 72.3%. The ship traffic flow rate and average speed of segment A, and ∆q D are positively correlated with dissipation time, and the sum of variance contribution is 27.7%.
When T 23 < T 31 , the simulation results and sensitivity analysis of the number of ships affected are shown in Fig. 13 and Table 12 separately.
As shown in Fig. 13 and Table 12, the mean of the total number of ships affected is 2223.88 ves, the median is 745.15 ves, and the 90% probability of the total number of ships affected is within 3725.99 ves. According to the The results show that the ship traffic flow rate of downstream segment C and ∆q D will greatly affect the dissipation time, while the ship traffic flow rate of upstream

Summary of simulation tests
The simulation results of ship traffic flow of closed and unclosed restricted segment show that the total number of ships affected is highly positively correlated with the ship traffic flow rate of upstream segment. During the peak period of the ship traffic flow, the total number of ships affected will enhance as the traffic flow rate of upstream segment increases. Dissipation time is different between closed and unclosed restricted segments. In closed restricted segment, dissipation time is mainly related to ship traffic flow rate of segment A and C, while in unclosed restricted segment, dissipation time is mainly related to ship traffic flow rate of segment C and ∆q D . In closed restricted segment, the total number of ships affected is mainly related to ship traffic flow rate of segment A. And in unclosed restricted channel segment, the dissipation time and the total number of ships affected are also determined by the meeting time of the traffic waves in addition to the ship traffic flow rate of segments. The values of dissipation time and the total number of ships affected at T 23 > T 31 are significantly smaller than those at T 23 < T 31 .

Main research results and significance
(1) The formation mechanism of multiple traffic waves (≥ 3) and the correlation between traffic wave speed  and ship traffic flow rate, ship speed and density of each segment are given in the unclosed restricted segment (i.e., the restricted segment with tributaries). The influence of speed and direction of traffic wave on the ship traffic flow is more complex. Under certain conditions, the ship on the voyage will produce multiple gathering and dissipating waves at the same time. When two traffic waves of the same nature meet, a new traffic wave will be generated. When two traffic waves of different nature meet, the queued ships will dissipate and the channel blockage will be relieved. The research results can enrich the theoretical study of inland ship traffic flow to a certain extent, and provide theoretical support for making navigation control strategy scientifically. The research has a certain theoretical significance and practical application value for reducing the influence of the extreme hydrological events in the middle reaches of Yangtze River on the shipping, improving navigation efficiency and competitiveness of shipping, and developing Yangtze River Belt.

Research features
Based on the ship traffic flow theory and the Greenshields linear model, this study focuses on the formation of multiple traffic waves (≥ 3) in the unclosed restricted segment of inland river under extreme water level conditions, as well as the influence laws of the corresponding gathering wave and dissipating wave on ship traffic flow, and gives the relevant parameter expressions. This research results have certain characteristics, which is different from the related research in China and abroad. Many scholars study traffic flow theory in normal water level, including the passing capacity or passing efficiency of ships in inland channel segment, characteristics of traffic flow, traffic congestion state, traffic flow rate, density and speed, and related simulation tests, as well as gathering and dissipation process of ship traffic flow based on traffic wave theory, etc. From the perspective of domestic and foreign researches, there are few studies on the impact of extreme water level changes on the ship traffic flow of inland river, especially the formation of multiple traffic waves and their effects on both gathering and dissipation waves of unclosed and unclosed restricted segments.

Existing deficiencies
The ship traffic flow discussed in this paper is only limited to the closed and unclosed restricted segments with oneway traffic in peak periods, as well as the meeting situation of two traffic waves. There are still many complex issues that have not been covered and need to be further studied in the future. For example, the unsteady ship traffic flow, the two-way ship traffic flow, the ship traffic flow with different ship types and crossing the main segment, the influence of the location of the tributary on the ship traffic flow of the main segment, and the situation where multiple tributaries merge into restricted segment, etc.

Conclusions
This paper focuses on the study of the internal relationship between ship traffic congestion and traffic waves in the closed and unclosed restricted segments and the following conclusions are drawn: (1) The expressions of traffic wave speed and direction, the dissipation time and the total number of ships affected were given, which provides theoretical support for the simulation calculation of ship traffic flow in such channel segment. (2) The dissipation time of the queued ship of closed restricted segment is mainly related to the ship traffic flow rate of segments A and C, and the total number of ships affected is highly positively correlated with the ship traffic flow of the upstream segment A. (3) The dissipation time of the queued ship of unclosed restricted segment is mainly related to the ship traffic flow rate of segment C and the difference of traffic flow rate between inflowed from segment D and outflowed from segment B. The total number of ships affected is positively correlated with the ship traffic flow rate of the upstream segment A, and when T 23 > T 31 , both the dissipation time and the total num-ber of ships affected are significantly less than those when T 23 < T 31 . (4) The interaction and influence laws between multiple traffic waves were given, as well as the process of the two traffic waves meeting and the formation rule of the new traffic wave, including the speed and direction of the new traffic wave and its influence on the ship traffic flow.
The ship traffic flow discussed in this paper is only limited to closed and unclosed restricted segments of oneway traffic in peak periods, and many problems have not been dealt with yet, which are worthy of further study, are shown in 6.2.

Declarations
Conflict of interest On behalf of all authors, the corresponding author states that there is no conflict of interest.
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