# A New Understanding and Modelling of TSP and BP Indices Compared to Safety IMO Ship Requirements
## 1 Introduction
Professional works [26] confirm that tug owners publish little information about steering pull of tugs, so usually this information is estimated by multiplying BP by a figure that depends on the circumstance. In fact, even many tugs classified as Escort by a classification society do not have the TSP available in its ship's particulars. Then, the lack of reliable information about the performance of escort tugs is evident from the genesis of this assistance mode. This can be observed in the simulation study carried out by [PERSON] (2002) [27] in order to assess, through dynamic models, the scheme escorting in Prince William Sound (Alaska), precisely the location where \(m/v\) Exvon Valdez suffered the accident. This work confirms the difficulty of reaching an agreement about the purpose of escort tugs after more than 8 years of discussions with all locally involved interests. More precisely, it was proposed that in some areas where two escort tugs were being employed, afterwards, only one escort tug was needed, but no performance characteristics of the involved tugs were specified. In other research works about models and simulations with tugs [6], the efficiency of escort tugs assisting the biggest vessels is connected only with BP and the size of tugs. Following [PERSON] et al. [28], and taking into account their length, escort tugs operate in off-design hydrodynamic conditions. For that, there is much research with model experiments focused on predicting the total forces generated and the limits of safe working. Furthermore, in the literature review, the reader can find many relevant studies concerning numerical simulations and proposed mathematical models of manoeuvring ships and tugs' motion [29; 30; 31]. Regarding the tethered vessel, the manoeuvring performance, including the zigzag manoeuvres with different rudder angles, was also simulated in previous works research [26; 32] with the objective of obtaining the yaw angle and lateral hull force, among others. Regarding the influence of forces between the tug and the tethered ship, recent studies [3] use mathematical models to analyse the problems of forces generated in dynamic operations between both vessels considering the effects of drift and wind force, but in the port environment, i.e., under very low-speed conditions. Although the research with mathematical models of forces affecting ships covers areas that are very different (hydrodynamic, hydrostatic, propulsion, and external forces), recent papers propose to develop synthetic and simple models of forces for its application to a ship in real-time mode [33]. This intention is in connection with the present work, that it is to find the simplest mathematical model able to relate the performance characteristics of tugs with the assisted vessels. Tethered vessels must be always operated within the performance capabilities of escort tugs [26], and, therefore, the capabilities and requirements of both should be studied as a whole. However, as it can be observed in the literature revision, there are few research works that analyse, from a safety point of view, the requirements of assisted vessels together with capabilities and performance of tugs working as a unique element at high speeds. Furthermore, there is a lack of information about TSP in tugs, and BP is generally employed. Then, as the problem of relating ships' characteristics and tugs' performance has not been approached, the present paper aims to connect the tugs' efficiency at different speeds (TSP of tugs) in relation to their BP, starting from the ships' requirement (TSP of ships) and using different mathematical models. With this objective, this study proposes providing the shipping industry with new information to be used from two points of view. On one side, ship owners benefit from considering the BP tugs (always available) to know the TSP that a tug can exert to their vessels; as a consequence, they can select the type and the appropriate number of tugs to be employed.