2.1. Head-on and overtaking collisions for uniform distribution

A head-on collision is a collision of ships where the front ends of two ships hit each other. In this case the difference between courses is 180°(or close to 180°) (Figure 1). In an overtaking situation, the difference between courses is zero (or close to zero).

Figure 1. Head-on collisions for uniform distribution.

In Figure 1, the following symbology is used: Qi, Qj are the number of class “i” and class “j”ships passing the fairway in a unit of time, respectively; i, j are the indices for ship class (all ships in one class have same size, speed and belong to same type); W is the width of the fairway, D is the sailing distance; B _i, B_j are the beams of the involved ships of class “i” and class “j”, respectively; L _i, L_j are the lengths of the involved ships of class “i” and class “j”, respectively and v _i, v_j are the speeds of the involved ships of class “i” and class “j” respectively.

For two opposing traffic flows⁽¹⁾ and ⁽²⁾, the potential number of collisions Na, for ships belonging to different classes i, j (during the time Δt), may be expressed as follows (Kristiansen, Reference Kristiansen2005):

(4)$$Na = \displaystyle{{D \cdot \Delta t} \over W}\sum\limits_{i,\;j} {\displaystyle{{Q_i^{(1)} \cdot Q_j^{(2)}} \over {v_i^{(1)} \cdot v_j^{(2)}}}} \cdot (B_i^{(1)} + B_j^{(2)} ) \cdot (v_i^{(1)} + v_j^{(2)} )$$

or, in case of overtaking:

(5)$$Na = \displaystyle{{D \cdot \Delta t} \over W}\sum\limits_{i,\;j} {\displaystyle{{Q_i^{(1)} \cdot Q_j^{(2)} \cdot (B_i^{(1)} + B_j^{(2)} )} \over {v_i^{(1)} \cdot v_j^{(2)}}}} \cdot \left\vert {(v_i^{(1)} + v_j^{(2)} )} \right\vert$$

2.2. Crossing collisions for uniform distribution

When two fairways intersect at an angle θ a crossing collision may occur in the overlapping area Ω (Figure 2).

Figure 2. Crossing collisions for uniform distribution. (Ω - Overlapping area, Θ - Angle of course crossing).

For one class of ships in each traffic flow and for 010° ≤ θ ≤ 170° the potential number of collisions Na may be expressed as follows (Kristiansen, Reference Kristiansen2005):

(6)$$Na = d \cdot \displaystyle{{{Q_1} \cdot {Q_2}} \over {{v_1} \cdot {v_2}}} \cdot \left( {\displaystyle{{{v_2}} \over {\sin \,\Theta}} - \displaystyle{{{v_1}} \over {\tan \,\Theta}} + {v_1}} \right)$$

where d represents the impact diameter, i.e. exposed cross-section normal to the direction of the relative speed (can also be simplified by a circle that is equal to the length of the involved ships - circular).

For ships of different classes i, j; all classes i in the first traffic flow ⁽¹⁾ and all classes j in the second traffic flow ⁽²⁾:

(7)$$Na = \sum\nolimits_{i,\;j} {d_{ij}^{(1,\;2)}} \cdot \displaystyle{{Q_i^{(1)} \cdot Q_j^{(2)}} \over {v_i^{(1)} \cdot v_j^{(2)}}} \cdot \left( {\displaystyle{{v_j^{(2)}} \over {\sin \,\Theta}} - \displaystyle{{v_i^{(1)}} \over {\tan \,\Theta}} + v_i^{(1)}} \right)$$

3.1. Head-on and overtaking collisions for normal distribution

Figure 3 shows the head-on situation and normal distribution of ships across the width of the fairway.

Figure 3. Head-on collisions for normal distribution.

In Figure 3, Qi, Qj are the traffic flows i, j; i.e. the number of ships of class “i”,“j” passing along the fairway in a unit of time, f(z_i), f(z_j) are the probabilities of the density function for the ship traffic, z _i, z_j are the transverse coordinates in the direction perpendicular to the route and η is the distance between the mean of two parallel fairways (separation distance).

The cumulative probability of coincident P of two ships as a function of separation distance η and standard deviation σ can be expressed as (Campos and Marques, Reference Campos and Marques2002):

(8)$${\rm P =} \displaystyle{1 \over {\sqrt {2\pi ({{\rm \sigma} _{\rm i}^2} + {{\rm \sigma} _{\rm j}^2})}}} {{\rm e}^{\scriptstyle - {{{{\rm \eta} ^2}} \over {2({\rm \sigma} {{\rm i}^2} + {\rm \sigma} {{\rm j}^2})}}}}$$

The total number of potential collisions for head-on situations for two opposing traffic flows⁽¹⁾ and ⁽²⁾ (Friis-Hansen, Reference Friis-Hansen2000):

(9)$$Na = \displaystyle{{D \cdot \Delta t} \over {\sqrt {2\pi}}} \sum\limits_i {\sum\limits_j {\displaystyle{{Q_i^{(1)} \cdot Q_j^{(2)}} \over {vi^{(1)} \cdot v\,j^{(2)}}}}} \cdot (v_i^{(1)} + v_j^{(2)}) \cdot (B_i^{(1)} + B_j^{(2)}) \cdot \displaystyle{1 \over {\sqrt {(\sigma_i^2 + \sigma_j^2)}}} e^{\scriptstyle{{\eta ^2} \over { - 2(\sigma _{i}^2 + \sigma _{j}^2)}}}$$

And for overtaking:

(10)$$Na = \displaystyle{{D \cdot \Delta t} \over {\sqrt {2\pi}}} \sum\limits_i {\sum\limits_j {\displaystyle{{Q_i^{(1)} \cdot Q_j^{(2)}} \over {vi^{(1)} \cdot vj^{(2)}}}}} \cdot \!\left\vert {(vi^{(1)} \!-\! vj^{(2)})} \right\vert\!\! \cdot (B_{i}^{(1)} + B_j^{(2)}) \cdot \displaystyle{1 \over {\sqrt {(\sigma _{i}^2 + \sigma_{j}^2)}}} e^{\scriptstyle{{\eta ^2} \over { - 2(\sigma_i^2 + \sigma_j^2)}}}$$

3.2. Crossing collisions for normal distribution

Figure 4 shows two crossing waterways. To obtain the potential number of collision, it is first necessary to obtain the relative speed v_ij and collision diameter D_ij – model based on the research carried out by Pedersen (Reference Pedersen1995) and cited in Montewka et al. (Reference Montewka, Goerlandt, Hanninen, Ylitalo and Seppala2011), Ylitalo (Reference Ylitalo2009) and Pedersen et al. (Reference Pedersen and Zhang1999). See also research carried out later by Friis-Hansen (Reference Friis-Hansen2000) and Friis-Hansen and Engberg (Reference Friis-Hansen and Engberg2008).

Figure 4. Crossing collisions for normal distribution.

Relative speed v_ij:

(11)$${\rm v}_{\rm ij}^{(1,\;2)} = \sqrt {{{\left( {{\rm v}{_{\rm i}^{(1)}}} \right)}^2} + {{\left( {{\rm v}{_{\rm j}^{(2)}}} \right)}^2} - 2 \cdot {\rm v}{_{\rm i}^{(1)}} \cdot {\rm v}{_{\rm j}^{(2)}} \cdot \cos \,\Theta} $$

Collision diameter D_ij:

(12)$$\eqalign{{\rm D}{_{\rm ij}^{(1,\;2)}} & = \displaystyle{{{\rm L}_{\rm j}^{(2)} {\rm v}{{\rm i}^{(1)}}} \over {{\rm v}{_{\rm ij}^{(1,2)}}}}\sin \Theta + \displaystyle{{{\rm L}_{\rm i}^{(1)} {\rm v}{_{\rm j}^{(2)}}} \over {{\rm v}{_{\rm ij}^{(1,2)}}}}\sin \Theta + {\rm B}_{\rm j}^{(2)} {\left( {1 - {{\left( {\displaystyle{{{\rm v}{_{\rm i}^{(1)}}\sin \Theta} \over {{\rm v}{_{\rm ij}^{(1,\;2)}}}}} \right)}^2}} \right)^{\displaystyle{1 \over 2}}} \cr & \quad + {\rm B}_{\rm i}^{(1)} {\left( {1 - {{\left( {\displaystyle{{{\rm v}{_{\rm j}^{(2)}}\sin \Theta} \over {{\rm v}{_{\rm ij}^{(1,\;2)}}}}} \right)}^2}} \right)^{\displaystyle{1 \over 2}}}}$$

The total number of potential collisions in crossing fairways (0° ≤ θ ≤ 170°):

(13)$${\eqalign{Na = \displaystyle{{\Delta t} \over {\sin \,\Theta}} \cdot \sum\limits_i {\sum\limits_j \;{\displaystyle{{Q_i^{(1)} Q_j^{(2)}} \over {v{_i^{(1)}}v{_j^{(2)}}}}}} v{ij^{(1,2)}}Di{j^{(1,\;2)}}\int\!\!\!\int\limits\hskip2pt_{\Omega \left( {z{i^{(1)}},z{\,j^{(2)}}} \right)} {\,f\,{{(Z_{i})}^{(1)}}} f\,{(Z_{j})^{(2)}}dZ_{i}^{(1)}}dZ_{\,j}^{(2)}}$$

In the case where the considered accident area is extended to infinity $\left( {\int\limits_{ - \infty} ^{ + \infty} {f(z)dz}} \right)$ Equation (13) can be simplified to (Friis-Hansen, Reference Friis-Hansen2000):

(14)$$Na = \displaystyle{{\Delta t} \over {\sin \,\Theta}} \cdot \sum\limits_i {\sum\limits_j {\displaystyle{{Q_i^{(1)} Q_j^{(2)}} \over {v_{i}^{(1)}v_{j}^{(2)}}}}} v_{ij}^{(1,2)}D_{ij}^{(1,2)}$$

4.1. The proposed simulation model

The proposed model has been developed as an intuitive graphical application using modern object oriented language – C#. It is based on a discrete – event simulation (Nance, Reference Nance1993) and aircraft collision models (Endoh, Reference Endoh1982). It calculates the number of vessel collision candidates when no anti-collision procedure is performed. The model operates by following three different scenarios: head-on meeting, overtaking and crossing of the vessels. Since the simulation model has been created as an object oriented application, everything is considered to be an object (in terms of programming languages). Objects interact with each other as the simulation progresses. The simplification of the programming process while obtaining the accurate number of vessel collision candidates is achieved through the replacement of the actual vessels and their characteristics with geometric circles (Čorić et al., Reference Čorić, Gudelj and Krčum2013). A geometric circle is an object that carries all information about the state of the vessel (dimension, position, direction, velocity) in a simulated time period. Therefore the vessel's motion is represented through a discretised circle movement in a simulated time period. The sailing area is represented by the coordinate system. Both the sailing area (coordinate system) and the vessels (circles) are graphically and interactively illustrated in a simulation process.

An overlap of two circles in simulated time is considered to be a collision situation. In other words, a collision occurs if the distance between two centres of the circles (d) is less or equal to the sum of both radius (r1 + r2), where r1 is the radius of the first circle, and r2 is the radius of the second circle (Figure 5). The model considers two different groups of vessels. Each group is given its own set of parameters as follows:

• – Number of vessels in a group.

• – Width and length of the vessels in a group – approximation of the width and length of the vessels is achieved by grouping more circles into one appropriate form (Figure 6).

• – Width and length of the sailing route for a particular group of vessels.

• – Velocity for a particular group of vessels.

• – Normal (Gaussian) distribution for a particular group of vessels – used in order to distribute vessels on their initial positions in a coordinate system.

The common parameter for both groups of vessels is the simulated time period. Vessels (circles) from both groups are generated linearly in the simulated time period – the time period is divided by the number of vessels in a particular group in order to get the period of time between generation of two vessels for that particular group.

Figure 5. Graphical illustration of the simulation process.

Figure 6. Simulation of the ship's width and length.

4.2. Overtaking and head-on meeting – basic modelling logic

In these two particular scenarios the model provides flexibility when selecting suitable normal (Gaussian) distribution. This means that the mean value μ and the standard deviation σ of the normal distribution may be custom-chosen. This distribution is used for the initial placement of the vessels (circles) across the width of the route and for the simulation of their movement path along the route as well. Therefore, the mean value μ represents the central (ideal) sailing line for all the vessels (circles) using this distribution, and the standard deviation σ represents an average deviation of the vessels (circles) from the mentioned central (ideal) sailing line.

In the head-on meeting scenario, the vessels (circles) from one direction (e.g. vessels from port B) are generated on one side of the route with their own custom parameters, while the vessels (circles) from the opposing direction (e.g. vessels from port A) are generated on the other side of the route using their own custom parameters (Figure 7 left). The parameters (desired normal distribution, velocity, dimensions of the vessels, etc.) are arbitrarily selected for each group of the vessels (e.g. vessels from port A and vessels from port B), and are independent of each other. All the vessels (circles) are generated linearly in a simulated time period as mentioned before. In the process of generation, their starting positions are normally distributed across the width of the route. In a simulated time period, the vessels (circles) have rectilinear movement that is parallel to the route. The model provides a custom selection of the width and length of the sailing route.The same modelling logic is applied to the overtaking scenario, except that in this scenario all the vessels (circles) are generated on the same side of the route (Figure 7, right). The model assumes perfectly parallel courses of all ships within a group, as in the analytical model, and the issue can be partly corrected by increased breadth of ship. More realistic simulation of ship movements (with the appropriate deviations from the general course) is for future research on this subject.

Figure 7. Simulation of head-on (left) and overtaking collisions (right).

4.3. Crossing scenario – basic modelling logic

This scenario uses most of the modelling logic from the above two scenarios, with a few changes. Instead of one route, two routes are used within the model. The width of each route is arbitrarily selected, and the angle of intersection between two routes is custom-specified as well. Each route is used by its own group of vessels (e.g. vessels from port A and vessels from port B). Each group of vessels uses its own parameters. Only the crossing area of the two routes is observed in the model, while the rest of the area of the routes is neglected (Figure 8).

Figure 8. Simulation of crossing collisions.