# Modeling of Vessel Traffic Flow for Waterway Design-Port of Swinoujscie Case Study
## 2 Methodology  
### Methods Used  
The methods presented here are directed toward finding the relation between the standard deviation of ship position with waterway length and width. The variability in the route planning by navigators results mainly from their individual preferences, mainly concerning distances to the dangers. With the assumption that the route consists of the straight leg of length \(L_{tr}\), its variability is the result of the start and end waypoint selection. The navigator's choice of given waypoints can be treated as random, so it can be described utilizing one- or two-dimensional random variables, as shown in Figure 1. A variable describing the starting and endpoint of the route can be determined through the expert opinion carried out by a group of navigators or by radar or AIS observations. Parameters of distributions depend especially on water area type, availability of position fixing systems, size, type of ship, and length of route. Studies of the available literature were carried out by [PERSON] [5], based on expert opinions, and his observations and models showed that the distribution of the distance of ships from the average route depends especially on the route's length. This led to the linear model of standard deviation dependence from the route length for routes up to 600 NMs of length in the following form:  
\[\sigma_{tr}=-0.027+0.0073L_{tr} \tag{1}\]  
where:  
* \(\sigma_{tr}\) is the total standard deviation from the mean route due to the route planning, means of ship control, and external factors;
* \(L_{tr}\) is the length of the route (NM). The standard deviation obtained in such a way enables us to estimate the geometrical probability by a normal distribution. [PERSON] [25] introduced dependence of the geometrical probability estimation in the following form:  
\[P_{G}=\frac{D_{p}+B}{D} \tag{2}\]  
where:  
* \(Dp\) is the available width between the bridge piers'
* \(B\) is the the breadth of the ship;
* \(D\) is the width of the waterway. The above formula is very simplified and assumes the uniform distribution of ship positions in the waterway. [PERSON]'s [2] studies enabled evaluation of simplified relations for the determination of geometrical probability in the following form:  
\[P_{G}=\frac{4l_{s}}{\pi D} \tag{3}\]  
where:  
* \(l_{s}\) is the distance to stop a ship;
* \(D\) is the available waterway width. [PERSON]'s [26] research enabled determination of the relations between the parameters of normal distribution and the position of a ship on the waterway, as follows:  
\[m=aD;\;\;\;\sigma=0.105D \tag{4}\]  
where:  
* \(m\), \(\sigma\) are the normal distribution parameters;
* \(D\) is the available width of the waterway;
* \(a\) is the parameter assumed as 0.2 for canals with central marking and 0.1 without such marking. AASHTO [27] is the recommended application for the normal distribution of parameters:  
\[m=0;\;\;\;\sigma=L \tag{5}\]  
where:  
* \(m\), \(\sigma\) are the normal distribution parameters;  
Figure 1: Influence of the distribution of initial and final points of the ship’s route. \(L\) is the length of the ship. The calculations are partially performed using the mathematical software tool IWRAP MK2 [28], recommended by the International Association of Marine Aids to Navigation and Lighthouse Authorities (IALA). The main objective of the research was to search for the dependencies between the main parameters of the ship traffic stream in the waterway system. Such dependencies are necessary to build statistical models aiming at a generalization of ship traffic and presenting it in the form of useful models. Therefore, the article has a strong utilitarian character. The authors use such models in later analyses when they need to design anticipated waterway features.