# Navigational Risk Assessment in Offshore Wind Farms Using Spatial Ship Domain Models
## 2 Materials and Methods
### Spatial Models of the Ship Domain  
This paper focuses on three spatial models of the ship domain: the PIANC [28], the 2D domain by [PERSON] [20], and the ship's 3D domain model by [PERSON] [8; 23]. The three domain models were selected based on their relevance and frequency of use in maritime risk assessments, each representing a different level of complexity. The PIANC model is widely recognized as a regulatory reference, the [PERSON] 2D domain offers maneuverability-based insights, and the [PERSON] 3D model enables dynamic spatial risk analysis. In the following sections, these models are applied to three representative vessel types under defined environmental conditions to assess and compare navigational risk (R\({}_{\text{N}}\)) values within offshore wind farm areas. As outlined in the PIANC guidelines, the minimum distance necessary for safe navigation is linked to the COLREG regulations (1972) and is defined according to IMO resolutions, specifically MSC.137(76) [29] and MSC/Circ.1053 [30]. These resolutions address vessel maneuverability, particularly concerning the turning circle and emergency stopping distance. The PIANC guidelines stipulate that the minimum safe distance (d\({}_{\text{N}\,\text{min}}\)) from a navigational obstacle should be calculated using the following formulas:  
\[SD_{LF}=d_{NF\,\,min}=5\,\,\text{Nm}=9260\,\,\text{m}\,\,[\text{m}] \tag{1}\]  
\[SD_{WP}=d_{NP\,\,min}=6\cdot\text{L}+500\,\,\text{m}\,\,[\text{m}] \tag{2}\]  
\[SD_{WS}=d_{NS}=d_{NP\,\,min}+0.3\,\,\text{Mm}\approx 6\cdot\text{L}+1056\,\, \text{m}\,\,[\text{m}] \tag{3}\] \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Average Conditions** & **Deteriorated Conditions** \\ \hline Visibility & At least 5 NM & Reduced to 2 NM \\ Wave height & _hf_\(\approx\) 1.0 m & _hf_\(\approx\) 3.0 m \\ Wind & 3–4 \({}^{\circ}\)B & 5–6 \({}^{\circ}\)B \\ Permanent surface & \(Vp\leq\) 0.2 kn & _Vp_\(\leq\) 0.4 kn \\ current velocity & In line with the direction of the vessel & Perpendicular to the direction of the \\ Current direction (Kp) & traffic flow within the TSS & vessel traffic flow within the TSS \\ Water level vertical oscillations & \(\pm\)0.3 m & Not more than \(\pm\)0.60 m \\ referred to chart datum (Chart & \(\pm\)0.3 m & 1.0066 g cm\({}^{-3}\) \\ Datum = MSL) & Not more than \(\pm\)1\({}^{\circ}\) & Not more than \(\pm\)2\({}^{\circ}\) \\ Water density (\(\rho\)) & 1.0066 g cm\({}^{-3}\) & 1.0066 g cm\({}^{-3}\) \\ Ship drift angle (a) & Not more than \(\pm\)1\({}^{\circ}\) & Not more than \(\pm\)2\({}^{\circ}\) \\ Maximum yawing (\(\Delta\)) & Up to \(\pm\)2\({}^{\circ}\) & Up to \(\pm\)4\({}^{\circ}\) \\ Roll angle (a) & Up to \(\pm\)4\({}^{\circ}\) & Up to \(\pm\)8\({}^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 3: The average and deteriorated conditions, table from [8], source: Sailing Directions. Polish Coast, 2016 [27] and “Admiralty Sailing Directions: Baltic Pilot Vol. 2 (Np19), 18 th Edition 2022” [26]. According to [PERSON]'s 2D domain model, the length of the ship's domain is measured horizontally in both the forward and atf directions, as well as to the port and starboard sides from the center of the ship's layout:  
\[SD_{LF}=6.1\cdot\text{L [m]} \tag{4}\]  
\[SD_{LA}=3.9\cdot\text{L [m]} \tag{5}\]  
\[SD_{WP}=1.75\cdot\text{L [m]} \tag{6}\]  
\[SD_{WS}=3.25\cdot\text{L [m]} \tag{7}\]  
The 3D domain by [PERSON] was described with reference to the XYZ coordinate system by [PERSON] in 2000-2021 [23; 24; 25; 31; 32]. The [PERSON] 3D domain model integrates spatial safety margins in three planes: horizontal (OX and OY) and vertical (OZ). The model takes into account ship-specific dimensions, maneuvering characteristics (e.g., stopping distance, turning radius), and external conditions such as wave height, current speed, and wind force. Key assumptions include the requirement to maintain minimum under-keel clearance (UKC) and over-head clearance (OHC), as well as directional safety margins (forward, astern, port, starboard), computed dynamically from simulator data.