2.1. Spatial-temporal discretisation

For vehicle traffic flow modelling, one needs to do the spatial-temporal discretisation first. The space of the road is divided into many small cells, the length of each cell generally being taken as equivalent to the length of a vehicle. The update time is discretised, and the time step is usually 1 s. This is because when the traffic is congested, the distance between two vehicles can be very small, which means that the space headway between two adjacent vehicles is approximately equal to the length of one vehicle. Because a vehicle's velocity is relatively higher than a ship's, multiple cells can be crossed in 1 s. Assuming the length of a cell is 5 m and the velocity of the vehicle ranges from 0 km/h to 100 km/h, the number of cells crossed by the vehicle in one time step is about 0 to 6 cells, which is feasible for vehicle traffic simulation.

However, this is inapplicable to ship traffic, because the sizes of different ships could vary considerably, and it is necessary to maintain a large distance between ships for safety. Therefore, the unit for ship's length should be determined first. In marine traffic engineering, due to the different size and tonnage of ships, in order to achieve a scientific and rational representation of the traffic volume in waterways, the standard ship is used to normalise ships of different sizes (Wu and Zhu, Reference Wu and Zhu2004). Therefore, the length of a standard ship is used as the minimum unit to describe the length of ships in a waterway, which can be denoted by the symbol $L_s$. The lengths of other ships are integral multiples of the standard ship length, denoted by $nL_s$, where $n$ is a positive integer coefficient. In this paper, the standard ship length $L_s$ is used as the size of a cell along the direction of traffic flow.

In addition, different from the concept of lanes in road traffic, there is no further division in a waterway for the route on which a single ship should sail, and it can be seen from the ships’ trajectories distribution that ships do not follow a specific lane in a waterway. However, the rules for the navigation of ships in the waterway are usually enacted, like the maximum velocity allowed for a ship, and whether a ship can be overtaken by other ships. If a ship is not allowed to overtake its preceding ship, the waterway can be considered as a single lane. Otherwise, the waterway can be considered as a multi-lane. Generally, the number of lanes can be two or more, which is determined by the width of the waterway and the lateral safe distance between ships. But in busy waterways the number can also be determined by observing how many ships at most are in parallel.

In the CA model, the temporal discretisation is often performed by the equal interval discretisation method, so the time step $\Delta t$ needs to be determined first, which affects the accuracy of the simulation. The time step can be obtained by Equation (1):

(1)\begin{equation} \Delta t=kL_s/v_0, \end{equation}

where $v_0$ is the minimum integer value of a ship's speed, such as 1 kn (the unit of ship's velocity, one knot, 1 kn $\approx 1{\cdot }852$ km/h). In Equation (1) $k$ is a positive integer coefficient, which is used to adjust the simulation step. When $v_0=1$ kn, the distance that the ship sails forward in time $\Delta t$ is the length of $k$ cells, that is, $kL_s$. Thus $k$ determines the simulation accuracy: the smaller $k$ is, the smaller the time step is, the higher the simulation accuracy is, and, accordingly, the more computing resources are consumed.

2.2. Ship safe distance

The surrounding safe area of a ship is the ship domain (Fujii and Tanaka, Reference Fujii and Tanaka1971; Goodwin, Reference Goodwin1973). The shape of the ship domain can be oval, rectangular or others, and the main parameters are the lateral size and the longitudinal size (Lamb, Reference Lamb1983; Zhao et al., Reference Zhao, Wu and Wang1993). Since in Subsection 2.1 the waterway was divided into several lanes, which has already taken the lateral safe distance into account, this section only deals with the longitudinal safe distance between ships in the same lane. Most of the research results show that the longitudinal size of the ship domain is roughly proportional to the length of the ship (Liu et al., Reference Liu, Wang and Wu2011; Qi et al., Reference Qi, Zheng and Li2011; Hansen et al., Reference Hansen, Jensen, Lehn-Schiøler, Melchild, Rasmussen and Ennemark2013; Liu et al., Reference Liu, Zhou, Li, Wang and Liu2016; Hörteborn et al., Reference Hörteborn, Ringsberg, Svanberg and Holm2019; Zhou and Zheng, Reference Zhou and Zheng2019). As mentioned above, the length of a ship can be denoted by $nL_s$; then, based on the ship domain, the safe distance between a ship and its preceding ship in the same lane can be calculated by Equation (2):

(2)\begin{equation} D_{\textrm{safe}}=mnL_s, \end{equation}

where $m$ is the multiple of safe distance relative to ship length. The value of $m$ is related to the selected ship domain model, and it can also be obtained by ship traffic data statistics.

2.3. Collision avoidance timing

In a waterway, when a ship finds itself in collision danger with its preceding ship, it will choose a timing to change its action to avoid the potential risk (Zhuo and Tang, Reference Zhuo and Tang2008; Chen and Huang, Reference Chen and Huang2012). The timing when a ship takes action to avoid collision can be expressed by the time left before the collision, i.e. $T_{\textrm {act}}=\Delta d/\Delta v$, which is also called the time to closest point of approach (TCPA) (Erwin, Reference Erwin2012) in nautical navigation. Here $\Delta d$ is the distance between two adjacent ships in the same lane, which is equal to the distance at closest point of approach (DCPA). The ship's location, which is used to calculate the value of $\Delta d$, is in the centre of the ship; and $\Delta v$ is the relative velocity of two adjacent ships (Zhao et al., Reference Zhao, Price, Wilson and Tan1995; Kang et al., Reference Kang, Lu, Meng, Gao and Wang2019). It is generally believed that $T_{\textrm {act}}$ is relevant to the manoeuvrability of the ship, and the manoeuvrability is relevant to the ship's size and velocity (Bi et al., Reference Bi, Jia and Wu2004a, Reference Bi, Shi, Jia and Wu2004b). Therefore, the collision avoidance timing for ships can be expressed by Equation (3):

(3)\begin{equation} T_{\textrm{act}}=f_2(nL_s,v), \end{equation}

where $nL_s$ represents the length of the ship, and $v$ represents the velocity of the ship. Here $f_2$ represents the functional relationship between collision avoidance timing and the ship's length and velocity.

In addition, the ship density ($\rho$) and environmental factors ($e$) in the waterway can affect the collision avoidance behaviour of the ship (Wu and Zhu, Reference Wu and Zhu2004). Therefore, these factors can also be taken into account, so the collision avoidance timing can be expressed as

(4)\begin{equation} T_{\textrm{act}}=f_3(nL_s,v,\rho,e,\ldots). \end{equation}

However, the relationship between $T_{\textrm {act}}$ and the parameters $nL_s$, $v$, $\rho$ and $e$ is very complicated, and the specific form of $f_3$ has not been found in related research literature. Specific and simplified forms can be expressed according to the specific application scenarios.

2.4. Update rules

Because the model studied in this paper is applicable to a busy waterway in which a ship is allowed to overtake its preceding ship, it is necessary to consider lane-changing behaviour. In addition, when slow ships occupy all the lanes and cause traffic congestion, the following ships can contact a slow ship by very-high-frequency radiotelephone (VHF-RT) to ask the slow ship to give way to them. This is similar to the honk behaviour of a fast vehicle in road traffic, which will be called the VHF-RT effect for the ship traffic in this paper. If the study of this paper needs to be applied to a single-lane waterway, just set all the ships to be not qualified for the lane-change condition in any case.

We use $v_i(t)$ to represent the velocity of ship $i$ at time $t$. The length of ship $i$ is $n_iL_s$, and the maximum velocity of ship $i$ at time $t$ is $V_{\textrm {max},i}(t)$; the preceding ship of $i$ in the same lane is expressed by $j$.

Safe distance: According to Equation (2) in Subsection 2.2, the safe distance between ships $i$ and $j$ is $mnL_s$, where the value of $L_s$ is the same as the value of the cell size. Therefore, the safe distance between $i$ and $j$ can be expressed by $mn$. Let $n_iL_s$ be the length of ship $i$. The safe distance is

(5)\begin{equation} D_{\textrm{safe},i}=m n_i. \end{equation}

Action distance: According to Equation (4) in Subsection 2.3, when ship $i$ is in collision danger with its preceding ship $j$, the action timing of ship $i$, which is $T_{\textrm {act},i}$, can be calculated. Let the velocity of ship $j$ be $v_j (t)$, and $\Delta v_{ij}(t)=v_i(t)-v_j(t)$, with $v_i(t)>v_j(t)$. The action distance from ship $i$ to ship $j$ is

(6)\begin{equation} D_{\textrm{act},i}(t)=T_{\textrm{act},i}\Delta v_{ij}(t). \end{equation}

Lane-change condition: When the ship that needs to change lane is in a new lane, all the other ships should not be in collision danger with it or no ship needs to take action to avoid collision at that time.

The update rules can be described as follows:

1. (i) Acceleration: Let $\textrm {gap}_i(t)$ denote the distance from $i$ to $j$ at time $t$. If $\textrm {gap}_i (t)>D_{\textrm {act},i}(t)$, then

   (7)\begin{equation} v_i(t) \rightarrow \min[v_i(t)+1,V_{\textrm{max},i}(t)]. \end{equation}

2. (ii) Lane-change: If $gap_i (t)\leq D_{\textrm {act},i}(t)$, and all the other ships in the neighbouring lane are far away from $i$, i.e. the distances are all greater than $D_{\textrm {act},i}(t)$, $i$ should change to the neighbouring lane, and update the velocity of $i$ by Equation (7).

3. (iii) VHF-RT effect: If $\textrm {gap}_i(t)\leq D_{\textrm {act},i}(t)$, but there is a ship in the neighbouring lane and its distance to $i$ is less than $D_{\textrm {act},i}(t)$, so $i$ cannot change lane, then the VHF-RT effect needs to be considered, and $j$ should try to give way to $i$. If $j$ is qualified for the lane-change condition, $j$ should change to another lane. Then, find out whether there is another ship in front of ship $i$. If there is such a ship, the front ship should be named as ship $j$, and recalculate the motion relationship between $i$ and $j$, and go back to item (i). Otherwise, directly update the velocity of $i$ by Equation (7). If $j$ is not qualified for the lane-change condition, $i$ should follow $j$; then go to the next item.

4. (iv) Deceleration: If $\textrm {gap}_i(t)\leq D_{\textrm {act},i}(t)$, and $i$ cannot overtake $j$, and $j$ cannot give way to $i$, then $i$ must decelerate to follow $j$, so

   (8)\begin{equation} v_i(t) \rightarrow \min[v_i(t),\textrm{gap}_i(t)-D_{\textrm{safe},i}]. \end{equation}

5. (v) Update velocity $v_i(t+1)=v_i(t)$, location $x_i(t+1)=x_i(t)+v_i(t+1)$ and maximum velocity $V_{\textrm {max},i}(t+1)$ by referring to the previous research of Qi et al. (Reference Qi, Zheng and Gang2017b).

In addition, if a ship is not allowed to overtake other ships by the traffic rules, in the situation of item (ii) and item (iii), $i$ must follow $j$ and update its velocity by Equation (8).

3.1. Statistics and analysis of ship traffic

The ship traffic data were collected by the Automatic Identification System (AIS). The sample data are from the AIS message records of the down-bound ships in a segment of Nancao Waterway at the estuary of the Yangtze River, which is one of the busiest water shipping areas in China. Part of the ships’ trajectories from the AIS data are shown in Figure 1. The parameters of the traffic flow are: $0{\cdot }4$ n mile (width) and 8 n miles (length), southeast (direction). Because the ships should keep a lateral safe distance between each other and only at most three ships are found sailing side by side in the waterway, the number of lanes in the CA model can be set to three.

Figure 1. The map of a segment of waterway in Yangtze River

The ships’ length, which is a very important parameter as mentioned in Section 2, should be analysed for the traffic simulation. Figure 2(a) shows the statistical result of the length of the ships sailing through the waterway. The ships’ length is generally consistent with the generalised Gaussian distribution, and ships of around 100 m account for the largest proportion. The minimum length of the ships is around 30 to 50 m, and the maximum length of the ships is around 320 to 370 m. According to the definition of standard ship length in Subsection 2.1, the value of $L_s$ could be set to 30 m, which is also the cell size along the direction of traffic flow. By referring to the research of Fujii and Tanaka (Reference Fujii and Tanaka1971) for ship traffic in a river, the value of $m$ can be 8.

Figure 2. Statistical results of one segment in the waterway of Yangtze River from AIS data: (a) distribution of ship lengths and its fitting curve; (b) distribution of ship velocities and its fitting curve; (c) the velocity–time curves of ships; and (d) distribution of time intervals between arrival of ships and its fitting curve

Figure 2(b) shows the statistical result of the velocity of the ships, which is also generally consistent with the generalised Gaussian distribution. The velocity of most ships ranges from 3 to 15 kn, with a very small number of ships occasionally exceeding 15 kn. The mean velocity of all the ships is around 10 to 11 kn. According to the update rules analysed in Subsection 2.4, the velocity of ships changes with the influence of the meteorological and hydrological conditions, so the velocity–time curves of ships are analysed and shown in Figure 2(c). The change cycle of ship's velocity is approximately 12.4 h, and the amplitude of the regular change is about 6 kn per cycle; the random change is less than 1 kn. In addition, the time interval should be set for generating ships’ simulation traffic data, which can be approximately equivalent to the interval time of arrival of ships. The distribution of interval time of ships’ arrival is shown in Figure 2(d), which is almost consistent with the exponential distribution.

According to Subsection 2.3, the collision avoidance timing $T_{\textrm {act}}$ is related to the parameters $nL_s$, $v$, $\rho$ and $e$, and is very complicated. But in this test, Equation (4) can be simplified. Because the environmental factor has been considered above, as shown in Figure 2(c), and the ship density does not change very much during the simulation, $\rho$ and $e$ can be ignored in Equation (4). Because of the ships’ hydrodynamic characteristics, the larger the ship is, the worse the ship's manoeuvrability will be, which needs a longer time to take action to avoid collision, and vice versa. Therefore, Equation (4) can be simplified to $T_{\textrm {act}}=k_1nL_s-k_2v+k_3$, where $k_1$ and $k_2$ reflect the influence of a ship's length and speed on the collision avoidance timing; here $k_3$ is a harmonic constant.

In order to determine the parameters $k_1$, $k_2$ and $k_3$, the avoidance actions of the ships in the waterway are observed; then the action timing, velocity and length of the ship are recorded at each sample point. The data samples were fitted based on a regression method, with $k_1\approx 0{\cdot }0174$, $k_2\approx 0{\cdot }8944$ and $k_3\approx 18{\cdot }2821$. The sample points and fitting surface are shown in Figure 3(a). Figure 3(b) shows the confidence intervals on the residuals from the regression. During the simulation, if the position, length and velocity of two adjacent ships are known, as $T_{\textrm {act}}$ can be calculated, the action distance $D_{\textrm {act}}$ can be calculated according to Equation (6) in Subsection 2.4, which is a very important dynamic variable in update rules.

Figure 3. The relationship between $T_{\textrm {act}}$, ships’ velocity and ships’ length: (a) the sample points and fitting surface by the regression; and (b) the confidence intervals on the residuals from the regression

3.2. Analysis of simulation results

In this subsection, the simulation results are compared with the real ships’ trajectories data in the waterway; then the performance of the proposed model is analysed.

Because the ship traffic flow is modelled mainly by considering the safe distance and collision avoidance, the lane-changing actions for a ship to overtake its preceding ship or to give way to its following ship, and the deceleration behaviour, are analysed. In order to analyse the characteristics of these three types of ships’ collision avoidance phenomena, it is necessary to combine the ships’ trajectories map with the ships’ spatial-temporal diagram. The trajectories map can describe the position change of the ships during movement, and the spatial-temporal diagram can reflect the change of the ships’ velocity and the positional relationship between the ships. Therefore, based on Figure 1, the time axis is added, and the ships’ three-dimensional spatial-temporal trajectories are drawn to analyse these three types of ships’ collision avoidance phenomena. The three coordinate axes are longitude, latitude and time.

In order to analyse the phenomenon of a fast ship overtaking its preceding slow ship by lane-changing, the three-dimensional spatial-temporal trajectories of ships is drawn, as shown in Figure 4(a). The projection of the three-dimensional trajectories on the latitude–longitude plane is the two-dimensional trajectories of the ships on a map. It can be seen from Figure 4(a) that a faster ship $i_1$ overtook its preceding ship $j_1$. The trajectories of the ships during the overtaking action is indicated with broken lines, as shown on the latitude–longitude plane in Figure 4(a). Figure 4(b) shows the trajectories in the selected area. Ship $i_1$ sailed on the left side of the waterway at first, and then sailed to the middle of the waterway in order to overtake ship $j_1$. Figure 4(c) shows the overtaking action from simulation results. Ship $i_1'$ overtook ship $j_1'$. As shown in Figure 4(d), ship $i_1'$ sailed in lane 2 at first, and then, in order to overtake ship $j_1'$, it changed to lane 1 and overtook ship $j_1'$. Compared with Figures 4(a) and 4(b), respectively, the simulation results in Figures 4(c) and 4(d) are consistent with the observed ship lane-changing behaviour, which shows that the model can simulate the overtaking phenomenon very well.

Figure 4. The phenomenon of a fast ship overtaking its preceding slow ship by lane-changing: (a) the three-dimensional spatial-temporal trajectories of ships in the waterway; (b) the trajectories of ship $i_1$ overtaking its preceding ship $j_1$; (c) the three-dimensional spatial-temporal trajectories of ships in the simulation; and (d) the trajectories of ship $i_1'$ overtaking its preceding ship $j_1'$

In order to analyse the phenomenon of a slow ship giving way to its following ship, the three-dimensional spatial-temporal trajectories of ships is drawn, as shown in Figure 5(a). The velocity of ship $i_2$ was faster than that of its preceding ship $j_2$. When ship $i_2$ overtook ship $j_2$, ship $j_2$ gave way to ship $i_2$. As can be seen from Figure 5(b), ship $j_2$ was near the left side of the waterway at first, and then sailed to its right to give way to the following ship $i_2$, which is called the VHF-RT effect, as mentioned above. Figure 5(c) shows the process of giving way action from the simulation results. The slow ship $j_2'$ gave way to the fast ship $i_2'$. As shown in Figure 5(d), ship $j_2'$ sailed in lane 2 at first, and then, in order to give this lane to ship $i_2'$, it changed to lane 1. Compared with Figures 5(a) and 5(b), respectively, the simulation results in Figures 5(c) and 5(d) are consistent with the observed ship lane-changing behaviour, which shows that the model can simulate the giving way phenomenon very well.

Figure 5. The phenomenon of a slow ship giving way to its following fast ship by lane-changing: (a) the three-dimensional spatial-temporal trajectories of ships in the waterway; (b) the trajectories of ship $j_2$ giving way to its following ship $i_2$; (c) the three-dimensional spatial-temporal trajectories of ships in the simulation; and (d) the trajectories of ship $j_2'$ giving way to its following ship $i_2'$

In order to analyse the phenomenon of a fast ship following its preceding slow ship by deceleration, the three-dimensional spatial-temporal trajectories of ships is drawn, as shown in Figure 6(a). The velocity of ship $i_3$ was faster than that of its preceding ship $j_3$. When ship $i_3$ sailed near ship $j_3$, there was traffic congestion; ship $j_3$ could not give way to ship $i_3$. Ship $i_3$ had to follow $j_3$ by deceleration. In order to show the deceleration of ship $i_3$, the projection of the three-dimensional trajectories on the latitude–longitude plane is drawn, as shown in Figure 6(a). Figure 6(b) shows the trajectories in the selected area with broken lines in Figure 6(a). The velocity of ship $i_3$ was fast at first, but when it sailed near ship $j_3$, its velocity slowed down, to be similar to the velocity of ship $j_3$. Figure 6(c) shows the process of the following action from simulation results. The fast ship $i_3'$ followed its preceding slow ship $j_3'$ in lane 3 by deceleration, because there were ships in the other two lanes, lanes 1 and 3, which resulted in traffic congestion. As shown in Figure 6(d), the velocity of ship $i_3'$ was faster than that of ship $j_3'$, but when ship $i_3'$ sailed near ship $j_3'$, its velocity slowed down. Compared with Figures 6(a) and 6(b), respectively, the simulation results in Figures 6(c) and 6(d) are consistent with the observed ship deceleration behaviour, which shows that the model can simulate the following phenomenon very well.

Figure 6. The phenomenon of a fast ship following its preceding slow ship by deceleration: (a) the three-dimensional spatial-temporal trajectories of ships in the waterway; (b) the trajectories of ship $i_3$ following its preceding ship $j_3$; (c) the three-dimensional spatial-temporal trajectories of ships in the simulation; and (d) the trajectories of ship $i_3'$ following its preceding ship $j_3'$

To study the traffic capacity of the waterway under conditions when overtaking is allowed or prohibited, simulations of ship traffic flow under these conditions were conducted. Ship traffic flow under different traffic densities of the waterway are statistically analysed, and the results are presented in Figure 7. When overtaking action is allowed in the waterway, the ship traffic capacity is larger than it is under the condition that overtaking action is prohibited. This illustrates that when the fast ship is prohibited to overtake its preceding slow ship by lane-changing, the speed of the ships in the waterway is greatly constrained by the slow ships, and the capacity of the waterway declined significantly in this situation.

Figure 7. The simulation results of ship traffic flow at different ship densities under the conditions when overtaking is allowed or prohibited in the waterway