# A New Understanding and Modelling of TSP and BP Indices Compared to Safety IMO Ship Requirements
## 4 Discussion  
### Models Analysis  
From a safety point of view of twoing operations, when vessels are being escorted, ship operators need to know if the tugs' capabilities employed during the assistance match their needs of steering force. As it was previously mentioned, and unlike BP, the TSP of tugs are not always available, so the present section shows the relationship between both \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Deadweight**} & \multicolumn{3}{c}{**Deadweight**} & \multicolumn{3}{c}{**Deadweight**} \\ \cline{2-13}
**Ship** & \multicolumn{3}{c}{**100,000 tons**} & \multicolumn{3}{c}{**200,000 tons**} & \multicolumn{3}{c}{**300,000 tons**} \\ \cline{2-13}
**Speed** & \multicolumn{3}{c}{**Rudder Angle**} & & & & & & & & & \\ \cline{2-13}  & **10\({}^{\circ}\)** & **15\({}^{\circ}\)** & **25\({}^{\circ}\)** & **35\({}^{\circ}\)** & **10\({}^{\circ}\)** & **15\({}^{\circ}\)** & **25\({}^{\circ}\)** & **35\({}^{\circ}\)** & **10\({}^{\circ}\)** & **15\({}^{\circ}\)** & **25\({}^{\circ}\)** & **35\({}^{\circ}\)** \\ \hline
6 knots & 25 & 30 & 45 & 30 & 30 & 50 & 60 & 50 & 40 & 55 & 80 & 60 \\
8 knots & 35 & 55 & 75 & 60 & 55 & 85 & 115 & 90 & 70 & 100 & 140 & 105 \\
10 knots & 60 & 85 & 120 & 90 & 90 & 130 & 185 & 145 & 110 & 155 & 220 & 165 \\
12 knots & 65 & 120 & 175 & 135 & 130 & 190 & 260 & 205 & 160 & 230 & 320 & 245 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: TSP data in tons of different tanker vessels. Figure 4: Representation of tanker vessels’ real TSP and the corresponding TSP obtained with Model 4. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variable** & **Minimum** & **Maximum** & **Unit** \\ \hline Knots & 0 & 10 & Knots \\ \hline Deadweight & 10,000 & 300,000 & tons \\ \hline Rudder angle & 15 & 25 & degrees \\ \hline Model 4 (TSP) & 18.5 & 341.5 & tons \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Validity range of Model 4.  
variables. Thus, anyone and at any time, without the need to carry out full-scale trials or complex simulations, can assess the tug TSP starting from their BP and vice versa. To achieve this objective, Equation (3) of Model 3 was used to calculate TSP with a tug's characteristics, and Equation (4) of Model 4 was used to calculate TSP with a ship's particulars; then, these were matched with the objective of defining the tug BP variable as follows:  
\[ \begin{array}{l}563+4.93\cdot\text{BP}-113.3\cdot\text{TS}-0.0192\cdot\text {BP}^{2}+4.93\cdot\text{TS}^{2}-0.081\cdot\text{BP}\cdot\text{TS}=\\ =-37.8-23.4\cdot\text{SS}-0.000074\cdot\text{DWT}+11.80\cdot\text{RA}+1.406\cdot \text{SS}^{2}-0.3196\cdot\text{RA}^{2}+\\ +0.000069\cdot\text{SS}\cdot\text{DWT}+0.3847\cdot\text{SS}\cdot\text{RA}+0.00000 5\cdot\text{DWT}\cdot\text{RA}.\end{array}  \tag{5}\]  
Therefore, Equation (5) allows us to define tug BP as a function of the required TSP of a tethered vessel using the deadweight, ship speed, and rudder angle as independent variables. Considering that the literature review indicates that maximum steering forces (TSP) are required from a rudder angle of 25\({}^{\circ}\)[25], BP was calculated for series of assisted ship's rudder angles of 15\({}^{\circ}\) and 25\({}^{\circ}\) according to Equation (5). In this simulation, the tug speed of 10 knots was considered as a constant. In each series, an individualised BP was calculated for each ship's speed, from 10 knots to zero knots; once the escort tug begins to exert force, the ship's speed will decrease progressively until it is under control. Therefore, more than 800 runs were tabulated, as can be seen in Figures 5 and 6. Figure 5 shows the BP needed from tugs to assist tethered tanker vessels sailing throughout the range of escort speeds with a rudder angle of 15\({}^{\circ}\). Figure 6 represents the same but with a rudder angle of 25\({}^{\circ}\). If we now compare Figures 5 and 6, it can be concluded that when the assisted ship is in static condition (zero speed), the BP hardly varies as a function of deadweight.