# A New Understanding and Modelling of TSP and BP Indices Compared to Safety IMO Ship Requirements
## 4 Discussion
### Models Analysis
This means that unlike the safety policy of many port authorities, which require excessive BP regardless of the type of traffic, it would not be necessary to employ tugs in ship-handling towing with BP as high as those currently used in many cases. In this circumstance, at zero speed, BP and TSP concur. Furthermore, comparing both figures and at higher speeds, it is observed that the BP required is higher with a higher rudder angle (25\({}^{\circ}\)). This aspect would be in line with the literature stating that the maximum TSP required by an assisted vessel is produced with a rudder angle of 25\({}^{\circ}\), and as a consequence, the corresponding BP needed from tugs. Figure 5: Tug BP needed as a function of ship’s characteristics (constant rudder angle of 15\({}^{\circ}\)). On the other hand, taking into consideration the tendency of curves at higher speeds, especially with a rudder angle of 25\({}^{\circ}\), it could be concluded that at present, there are no escort tugs available in the market with that BP required. Nevertheless, this circumstance is solved in real life using two tugs working in tandem [39], so the BP of each escort tug is added as it is regulated in Los Angeles/Long Beach port [40,41]. Another particular case of interest observed in both Figures 5 and 6 is that except for 10 knots, in the range of lower deadweight, the BP required is lower than the BP required in static condition. Although this result could be considered illogical, it should be noted that these vessels are not sailing straight ahead, so the greater the rudder angle, the greater the yaw angle, and as a consequence, the greater the lateral surface of the underwater hull working against the flow of water. This hydrodynamic force will reduce the shipping speed and, therefore, tugs can control the TSP demanded with a lower BP. In order to know if our models satisfy the tools used by the towing industry at present, Figure 7 represents the IMO recommendations about upper and lower TSP limits required by a tanker vessel of different deadweight sailing at 10 knots. As there are no published equations that define the IMO TSP limits, once we obtained several values of IMO graphs, they were used in our Model 3 in order to calculate the BP limits corresponding to IMO TSP limits. The results are also represented in Figure 7, which shows a good relationship between both variables. Figure 6: Tug BP needed as a function of ship’s characteristics (constant rudder angle of 25\({}^{\circ}\)). Figure 7: IMO TSP limits and the corresponding BP limits as per Model 3. In this sense, considering that the TSP provided by escort tugs are not always available, it is interesting to understand that starting from the TSP demanded by assisted vessels according to IMO, it is possible to assess the minimum BP (which are always available) that tugs would have to have to satisfy the TSP required. At the same time, with the objective of analysing the relation between IMO TSP and TSP demanded by a tanker vessel according to our Model 4, we developed a simulation of 160 runs for rudder angles of 10\({}^{\circ}\), 15\({}^{\circ}\), 25\({}^{\circ}\), and 35\({}^{\circ}\), where the ship speed variable was considered as constant (10 knots). From Figure 8, it can be concluded that IMO TSP limits correspond very well with our Model 4 when the tanker vessel is working with a rudder angle of 10\({}^{\circ}\). Although in IMO TSP no information is published about rudder angles, this conclusion can be considered as acceptable taking into account that, calculating the correct wheel-over point, merchant vessels rarely steer with large angles of the rudder. From Figure 8, it is also interesting to observe that at higher rudder angles, the TSP needed according to Model 4 is considerably higher than the upper limits of TSP required by IMO. This conclusion could be related to previous simulations and manoeuvring studies [26], where it is stated that the needed steering pull is 50% higher than upper IMO guidelines.