## 3 Methodology  
### Anchorage Circle Radius Model  
The anchorage circle radius is not only related to the size of the safety mooring area of ships, but also influences the utilization rate of the anchorage area to a great extent. The anchorage circle radius is related to the length of the anchoring ship outgoing chain,  
Figure 4: Diagram of selected anchor position of ship. (**a**) The diagram of ship anchor position selection under no influence of wind and current. (**b**) The diagram of ship anchor position selection under influence of wind and current. Figure 3: Diagram of vertical movement of anchoring ship. the ship parameters, the loading conditions, the draft and the safety of ships passing through the anchorage area. The extended chain length \(S\) of the anchoring ship includes the following three parts: the length \(k\) of the anchor chain on the deck, the suspended length \(S_{S}\) and the length \(S_{L}\) of the chain cable lying on the seabed. Referring to Figure 1, the anchorage circle radius of the ship can be gained, which can be expressed as Equation (1). \[\left\{ \begin{array}{l}R=L_{1}+L_{2}+L_{4}-L_{5}+\varepsilon\\ L_{4}=L_{S}\times\cos\alpha\\ L_{5}=L_{SA}\times\cos\alpha\end{array} \right. \tag{1}\]  
where \(R\) indicates the radius of the anchorage circle of the ship; \(L_{1}\) indicates the length of the chain cable lied on seabed; \(L_{2}\) indicates the horizontal projected length of the catenary of the anchor chain; \(L_{4}\) indicates the horizontal projected length of the bow and stern of the ship; \(L_{5}\) indicates the horizontal projected length from the anchor chain hole to the front end of the deck; \(\varepsilon\) indicates the positioning error of the position sensor of the ship; \(\alpha\) indicates the ship trim angle, which can be obtained from the state sensor of the ship; \(L_{S}\) indicates the length between the bow and stern of the ship; \(L_{SA}\) indicates the length from the anchor chain hole to the bow of the ship. Referring to Figure 3 with a consideration of the motion limit of the anchoring ship, the horizontal projection of the anchor chain is simplified and can be expressed as Equation (2). \[\left\{ \begin{array}{l}L^{\prime}=L_{1}+L_{2}=\sqrt{\left(S-k\right)^{2}-D^{ 2}}\\ D=D_{W}+D_{A}-d_{F}\end{array} \right. \tag{2}\]  
where, \(D\) indicates the water depth from the anchor chain hole to the seabed; \(D_{W}\) indicates the charted water depth, \(D_{A}\) indicates the ship shape depth at the anchor chain hole; \(d_{F}\) indicates the draft of the bow of the ship. Therefore, the anchoring radius model under general conditions can be derived, which can be expressed as Equation (3):  
\[R=\sqrt{\left(S-k\right)^{2}-\left(D_{W}+D_{A}-d_{F}\right)^{2}}+\left(L_{s}-L _{SA}\right)\times\cos\alpha+\varepsilon \tag{3}\]  
Considering the safety factors of other ships passing through the anchorage area, the anchorage circle radius needs to be reserved for 2-3 times the ship width [43] so as to ensure the anchoring safety. In addition, considering the characteristics of ships carrying hazardous goods, the anchorage circle radius should increase the safety margin by a certain amount compared with general freight anchoring ships. Therefore, the general anchorage circle radius model can be further expressed as Equation (4). \[R=\sigma\bigg{[}\sqrt{\left(S-k\right)^{2}-\left(D_{W}+D_{A}-d_{F}\right)^{2} }+\left(L_{s}-L_{SA}\right)\times\cos\alpha+\varepsilon+\tau B\bigg{]} \tag{4}\]  
where, \(\sigma\) indicates the ship type, \(\sigma\in[1,1.2]\); the \(\sigma\) of general cargo takes the lower limit, and the \(\sigma\) of ships carrying oil, liquefied gas and chemical products takes the upper limit according to the risk; \(\tau\) indicates the ship width coefficient, \(\tau\in[2,3]\); \(B\) represents the width of the ship [44]. The chain length greatly influences the anchorage circle radius.