Estimating the traction factor and designing the deck gear for the anchor handling tug

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1 Original Article Estimating the traction factor and designing the deck gear for the anchor handling tug Proc IMechE Part M: J Engineering for the Maritime Environment 1 16 Ó IMechE 2016 Reprints and permissions: sagepub.co.uk/journalspermissions.nav DOI: / pim.sagepub.com Luis Carral-Couce 1,SalvadorNaya 2, Carlos Álvarez Feal 3, Miguel Lamas Pardo 4 and Javier Tarrío-Saavedra 2 Abstract In offshore activities, it is necessary for the floating equipment extracting raw materials to reach and maintain a static position along the seabed. Increasingly crucial in these operations is a new type of auxiliary vessel: the anchor handling and supply. While this work is being carried out, the deck gear related to towing and anchor handling secondary and tugger winches, as well as capstans play a key role on the anchor handling and supply. To estimate the vessels capacity for taking part in towing and anchor handling tasks, one can turn to the concept of the pull number as the sum total of the traction for this equipment. Linear regression, non-linear and semi-parametric generalised and additive models are adjusted around a broad data base about state-of-the-art vessels. This makes it possible to estimate their value, using as a starting point the vessels main dimensions and power. From this estimate, the traction values for the main and anchor handling winches can be determined. The next step is to propose a way of calculating the traction needed by the secondary, capstan and towing winches assisting the manoeuvre. This study provides procedures and statistical models that can help determine the traction of the towing and anchor handling winches for the anchor handling and supply. These depend on the traction obtained in Bollard Pull testing. The analysis proposed here involves dimensioning the key equipment used on these kinds of vessels so that valuable design information is obtained. Keywords Offshore, anchor handling tug and supply, anchor handling winch, towing winch, secondary winches, statistical learning Date received: 18 June 2016; accepted: 29 September 2016 Introduction: auxiliary platform vessels (anchor handling tug and anchor handling and supply) activity and winches involved in their manoeuvres Offshore Oil and Gas (O&G) exploration requires specialised technologies. This is particularly the case when there is greater difficulty in locating and extracting these resources. For this reason, the additional work done by the anchor handling tug (AHT) and anchor handling and supply (AHTS) vessels can be extremely complicated and dangerous. 1 To illustrate this point, Figure 1 shows how the anchoring system for a semisubmersible platform, towed by the AHTS Olympic Poseidon, is tested. In the event sequence for extracting natural resources like oil and gas on the high seas, guidelines establish that an Oil Construction and Supply vessel (OCV) is to take part in preliminary tasks. Later in the process, an AHT vessel will be involved with assisting the tug in its initial duties. It also helps with positioning and anchoring the drilling platform, as well as casting subsea lines and other floating devices. Other tasks carried out by the AHT include monitoring how stores are required in the extraction process and removing waste products. Under certain circumstances, guidelines 2,3 specify that a vessel must be on stand-by near the 1 Mixed Engineering Group, Department of Naval and Ocean Engineering, Higher Polytechnic University College, 15403, Universidade da Coruña, Ferrol, Spain 2 MODES Group, Department of Mathematics, Higher Polytechnic University College, 15403, Universidade da Coruña, Ferrol, Spain 3 Mixed Engineering Group, Department of Industrial Engineering II, Higher Polytechnic University College, 15403, Universidade da Coruña, Ferrol, Spain 4 Reyser S.A., 08039, Barcelona. Corresponding author: Javier Tarrío-Saavedra, MODES Group, Department of Mathematics, Higher Polytechnic, 15403, Universidade da Coruña, Ferrol, Spain. javier.tarrio@udc.es

2 2 Proc IMechE Part M: J Engineering for the Maritime Environment Table 1. Comparing the main characteristics of vessels involved in towing, handling anchors and supplying with deck gear. Offshore tug Stern design to facilitate work with pennants and chains. Powerful and rapid twin-drummed winch that can work over extended periods of time and stow a spare tow line. Fenders along the sides that make it possible to tow another vessel broadside. Reinforced with fenders along the stern when nudging exercises are being carried out. Anchor handling Low sides make it easier to handle anchors on deck. Powerful and rapid double-drummed winch to manoeuvre the anchors; can work over long periods of time. Easy to adjust towing functions and anchor handling without the need to change the manoeuvre positioning. Supply High power and Bollar Pull (BP), along with ample cargo space and volume. Combined propulsion systems (stern tubes and driver) to improve handling conditions in rough seas. Stowage on deck for anchors and buoys while the vessel is towing and handling anchors. Powerful winches and cranes for loading and supplying platforms. Deck used for a variety of purposes. Source: Authors own work based on Carral Couce et al. 1 Figure 1. Testing the anchoring system for a semi-submersible platform being towed by an AHTS Olympic Poseidon (May, 2015). Source: Authors own work. platform, ready for emergencies like fire or evacuation. A vessel must be designed specifically to carry out these tasks; she must be able to assist in emergencies and spend prolonged periods at sea. 4 Table 1 shows the main characteristics of vessels involved in towing, handling anchors and supplying with deck gear. In recent years, there has been an increase in offshore operations in deep waters and particularly harsh conditions. This trend has had on impact on the size of the gear used on platforms, so that bigger and more powerful assistance vessels are called upon. 4 This study offers a methodology based on statistical modelling to determine the pull number T t using the dimensions and power of the towing vessel. In this way, design work can be facilitated. At the design stage for the towing vessel project, the first step is to establish the power that is to be installed on board. To a lesser extent, the length must also be considered. As a result, the Bollar Pull (BP) will be determined. With the BP, it is possible to work out the parameters for the towing line, as well as the winch design, including static brake traction, dynamic traction and the paying out and recovering speed. 5 In turn, the type of task carried out by the towing vessel will determine the required number and length of towing lines. At this point, it is also possible to calculate the dimensions for the component that stows these lines: the drum. 6 For this reason, the vessels BP will influence the value for the total tractions of the main and auxiliary winches fitted on board. Towing With the towing operations needed by the platform, the AHT has to operate like an offshore towing vessel. For this reason, it has to undergo certain transformations in order to handle towing line in a suitable way. On one hand, the line must be cast from the deck. At the same time, a range of components like stop pins and gog eyes must control how the tow line moves along the

3 Carral-Couce et al. 3 Figure 2. Layout for the offshore manoeuvre of an AHTS, indicating where the TW and TGW have been placed. Source: p. 48 of Hancox book. 7 deck. Another component is a line that has a shackle on one end 7,8 (Figure 2). In the towing operation, the towing line comes into play 6 as does the towing winch (TW). The TW is responsible for three tasks: it stows the main towing line; it pays out and recovers the line and, using brake action, it carries out the towing operation. 5,9 Anchor handling Once in place, the semi-submerged platforms need an anchoring system that keeps them immobilised at the operational zone. To this end, 8 or 10 anchors in a star distribution are placed at previously established points. With a weight of between 12 and 45 tons, each one of these anchors must be precisely positioned and buried to achieve maximum grip. The winch employed in this work has to be extremely powerful to tow the towing line. 4 In the same way, when the platform has to be moved on completing its mission, the anchor handling winch (AHW) helps extract these anchors. To assist in this operation, auxiliary winches are situated along the stern (Figure 3). When work is done in deep waters, the weight of the anchors and the length of the lines dramatically increase. This means that BP requirements are also greater. Moreover, secondary winches (SWs) are needed because there has to be greater storage capacity for the cable lines used for anchoring. If operational requirements mean the platform is frequently re-positioned, the AHW never stops playing a role. 1,4,10 Supply The vessel involved in supplying a vessel must be capable of storing, transporting and transferring solid bulks and liquid products. 11 When solid loads are being transferred, this cargo may need to be transferred at an earlier stage. For this task, metal lines and auxiliary tugger winches (TGWs) are used. However, with liquid products, discharge pumps do the work; they are connected with the hoses from the discharge manifold.

4 4 Proc IMechE Part M: J Engineering for the Maritime Environment Figure 3. Layout for the offshore manoeuvre of an AHTS, indicating where the AHW, SW and capstan have been placed. Source: p. 338 of Hancox 7 book. Vessel deck equipment needed for towing and anchor handling regulations The deck is where preparations start for towing and anchor handling manoeuvres for the platforms. TWs and AHWs are involved, with the help of SWs, TGW and a capstan. These make it possible to set up necessary manoeuvres. A singular characteristic of the tug (Figure 2) is that it is facilitated by a winch that contains and holds the towing line. This winch boasts an additional drum that stows the tow line in accordance with regulatory guidelines. As a result, when the TW is operated, a reducer is activated to move two identical drums in a cascade formation 1 (Figure 4). When it is time to help with handling the line and control its movements across the deck, two TGWs are used. Similarly, the anchor handling manoeuvre (Figure 3) is carried out by means of the winch unit. According to Gaston, 4 the traction of this unit will exceed that of the low speed of a single drum TW (sometimes two). This drum will hold the cable necessary for the recovery and release of anchors. Towing and handling the anchors is never done simultaneously. For this reason, the two

5 Carral-Couce et al. 5 Figure 4. Layout of the winch components for towing and anchor handling, with a 150 T first layer traction and a stowage capacity for 1330 and 820 m of 60-mm line. Source: Carral design. Figure 5. Longitudinal view of an AHTS vessel, indicating how the towing (TW) and anchor handling winches (AHW) have been positioned. Source: Carral Couce et al. 1 winches are placed together along the centre line just before the control bridge 1,10 (Figure 5). Moreover, at least two more SWs have to be installed. 4 Located near the TW and AHW, these winches hold additional line, assist in handling the heavy line used and make it possible to move the anchors and supplies on the deck. Additionally capstans may be found along the stern (Figure 3).

6 6 Proc IMechE Part M: J Engineering for the Maritime Environment Table 2. Line length according to the tug category and the vessels BP for offshore towing. Noble denton IMO Tug category Line length (m310 3 ) Min. value (m) Line length (m310 3 ) ST-ocean going salvage tugs 23BP=MBL 800 1:83BP=MBL U-unrestricted towages 1:83BP=MBL 650 R1-restricted towages R2-benign area towages R3-restricted benign area towages 1:23BP=MBL 500 Source: Denton 3 and Carral Couce et al. 1 IMO: International Maritime Organisation; BP: Bollard Pull; MBL: minimum breaking load. Table 3. Values taken by the coefficients in the pull factor equation according to the layout chosen for a towing and anchor handling manoeuvre. Coefficients for total traction equation (1) a b c d e Offshore tug (towing manoeuvre) Anchor handling vessel (anchor handling manoeuvre) AHTS vessel (both towing and anchor handling) a Source: Authors own work. AHTS: anchor handling and supply. a Additionally, one of the TW drums was added. On the AHT and AHTS, three main winches will be installed. Along with the auxiliary winches, they will have a total traction capacity, the result of adding the tractions for all the winches, T t, which are very high. 1 The insurance company will cover the risk incurred by the vessel while it is doing this work and, for this reason, a class certification must be held. The ship owner therefore has to ensure that the vessel and its equipment is built and later maintained according to classification society (CS) regulations. It is appropriate for the tug owner to decide which society provides the class certificate. On the other hand, the vessels flag country is obliged to comply with other guidelines, in this case those of the International Maritime Organisation (IMO). In the Guidelines for safe ocean towing, 2 the IMO looks at towing vessels and their operation. Chapter 12 of these guidelines deals with towing equipment specifications. As for CS regulations on TWs and gear, Carral Couce et al. 5 and Allan 12 have done an exhaustive study on the scope of these guidelines. According to their findings, operational aspects are clear, but the winches design parameters have not been sufficiently defined. In terms of the towing line, CS guidelines determine the minimum breaking load (MBL). It is calculated as a function of the design loads (DF) that come into play. Although the Bollard Pull (BP) is considered as the parameter, there is discrepancy over how BP is determined. 6,12 The TW drum capacity will be conditioned by the tow line lengths stipulated in the regulations. This value will vary enormously according to the particular circumstances of each tugs category and features. Offshore tugs are regulated in 2 ; the minimum tow line length is expressed in meters L = 1800 BP MBL ð1þ No distinction is made by the IMO between tug categories. However, Denton 3 and Carral et al. 6 lay down the differentiated formulation for calculating the length for the five categories for the tow vessel. They propose a recommended and minimum value for the length. These are found in Table 2. Concept of the pull number for the main (TW and AHW) and auxiliary winches (secondary, capstan and tugger), determining its value Defining the pull number for AHTS vessels The study 1 introduces the concept of pull number for the winches found on the deck of an AHTS. In the kind of manoeuvre that is carried out by this vessel certain demands are placed on the main winches, as well as on the auxiliary ones that work alongside them. Therefore, one must consider the pull number value for the winches that have been installed (T t ). This value will serve as an indicator of the equipments towing and anchor handling capacity. 1 Expression (2) determines the value for this variable in any type of AHT and AHTS vessel T total = a T towing which ++ b anchor handling whiches + + c T secondary winches + d T tugger winches + + e T capstan ð2þ

7 Carral-Couce et al. 7 Table 4. Relationship between, on one hand, the interval and average value for the vessels considered in this study (power, length and displacement) and, on the other, the interval and average BP. No. of vessels Power interval (BHP) Total length interval (m) Dead weight interval (tpm) BP interval (t) 69 15,700 28, Average power (BHP) Total length average (m) Dead weight average (t) BP average (t) 21, Source: Authors own work. Table 5. Signification analysis of the regression model for the BP according to the power and length for each vessel (adjusted R 2 =0:97, BP =0:008 Power +1:382 Length 52:26). Estimates Standard deviation t values p-values Intercept Length Power The different layouts in which the equipment is positioned and installed on deck ready for a manoeuvre comes into play here. There are five possible coefficients a, b, c, d and e that take on the range of values provided in Table 3. Determining the pull factor for AHTS vessels in practical terms The aim here was to work out the variation in the value taken by the pull number function (T t ). A data base with information about 69 vessels served as a starting point. These were state-of-the-art AHTS vessels built between 2005 and In Table 4, one can refer to data on their power intervals, length, displacement and Bollard Pull (BP). Moreover, the average values for these data are represented. First of all, regression statistical analysis makes it possible to model parameters and show how the BP is dependent on the vessels length and power values. A multivariate linear regression model was built. It used a response variable, BP, and two independent variables related to the dimensions and power of the vessel: length and installed propulsion power. No co-linearity was observed between the regression values (the variance inflation factor, VIF \ 4). Table 5 shows the linear regression model parameters and an analysis of the significance. The BP value is significantly dependent on the vessels power and length, in accordance with a linear relationship (p-values \ 0:05). Indeed, this explains 97% of the variance in the BP by means of the proposed linear regression model (adjusted R 2 =0:97). Figure 6(a) outlines the BP estimates for each value pair (length and installed power) as well as the plains situated at two deviations common in adjusted values. If one considers Figure 6(c) and (d), it is possible to observe the linear effects that the power and length variables have on the BP response variable. Both sections reveal the linear relationship between the variables and the degree of accuracy the BP estimates have for such a wide range of length and power values. Turning to Figure 6(b), what stands out is the relative importance of each independent variable in explaining the response variable, in % of the total value for R 2. A bootstrap procedure 13 was employed to obtain the confidence intervals for the relative importance calculated by means of the lmg metric, the R 2 contribution averaged over orderings among regressors. 14 The most explanatory variable appears to be power, representing almost 80% of the variance in the BP. The characteristics of the vessel being towed will be revealed by the BP value. Here, it is important to stress that the installed propulsion power is a determining variable for estimating the BP. The linear correlation between both is strong, with a linear correlation of r =0:97 and R 2 =0:94. At the same time, AHTS work will depend on this variable. This work will also be influenced by how much space is devoted on deck to store anchors and other materials. This must be taken into account when thinking about the length. Go rski and Giernalczyk 15 proposed using the product for the length, beam and moulded depth. However, only the length was used to obtain the results mentioned earlier (r = 0:70 between BP and length). Given the data variability, a proposal was made to compare how the different commonly used adjusted models with the variable dependence model. Figure 7 provides the pull number T t estimates, with estimate and prediction intervals whose confidence rate was 95%. These were obtained by averaging the adjustment of different regression models: semi-parametric generalised additive models (GAMs) using thin plate splines, 16 as well as linear and non-linear regression models with a three- and four-parameter logistic function. 17 The models proposed here provided feasible data for estimating

8 8 Proc IMechE Part M: J Engineering for the Maritime Environment Figure 6. Regression model for the BP according to the power and length for each vessel. (a) Plain for BP estimated according to a vessel power and length with confidence bands at two typical deviations. (b) Relative influence of the power and length on the BP, in % of R2, with bootstrap confidence intervals. (c and d) Effects of power and length variations on the BP according to the estimated multivariate regression model. Source: Authors own work. the Tt according to the BP. They may prove extremely useful when these kinds of vessels are being designed. In Appendix 1, there is a brief description of the statistical models employed and information technology application used to implement them, R statistical. In Figure 7(a), semi-parametrical adjustment can be observed. It is based on thin splines, with bootstrap confidence intervals.16,18 A mild/slight adjustment is obtained, with a clear sigmoid tendency and a determination coefficient R2 = 0:86, an indication of the adjustments high rightness of fit. With the earlier adjustment made to the parametric and semi-parametric regression techniques, it is possible to glean the most suitable parametrical model using an adjusted curve analysis as the starting point. Therefore, using a semi-parametric model for this study, linear regression adjustments are proposed. They are the most straightforward and easiest to interpret. Also used are non-linear regression adjustments with logistic functions, among the most commonly found in statistic modelling. Figure 7(b) shows the Tt in relation to the BP for the vessels and the linear regression adjustment. All of the parameters were significantly different from zero, with a 95% confidence rate. It was possible to see a

9 Carral-Couce et al. 9 carried out for the T t in relation to BP with the following expression 1983:2 T t = ð4þ 1 + exp ð108:58 BPÞ 54:82 Finally, Figure 7(d) shows the non-linear adjustment of a four-parameter logistic model. Again, all are significantly different from zero with a 95% confidence rate. The model accounts for 85% of the variability in T t (R 2 =0:85). With the determination coefficient serving as a rightness of fit criterion, the four-parameter logistic model is more explanatory than the other parametric models. Moreover, it is comparable to the semiparametric one, which has the greatest flexibility. The four-parameter logistic model defines two of the main T t levels, low ( T-f) and high ( T-f). The BP value is around 223 T-f, marking the inflection point of the non-linear model with logistic function. Estimates can be made for the total traction by means of this adjusted expression 1907: :602 T t = 1547:602 + ð5þ 1+exp (223:194 BP) 8:196 Figure 7. Estimating the T t according to the BP. (a) T t according to the BP and semi-parametric regression model with thin plate splines, including a bootstrap confidence interval. (b) T t according to the BP and a linear regression model with confidence levels. (c) T t according to the BP, non-linear regression model with three-parameter logistic function and estimation and prediction confidence intervals. (d) T t according to the BP, non-linear regression model with a four-parameter logistic function and estimation and prediction confidence intervals. Source: Authors own work. strong linear correlation between the variables and a linear regression model was obtained that can account for 75% variability for T t (R 2 =0:75). Once the vessel BP is known, it is possible to have reliable estimates for the its by means of the expression T t = 992:3+3:254BP ð3þ Moving on to Figure 7(c), one can see a non-linear regression adjustment for a logistic model defined by three parameters. They are all significantly different from zero with a 95% confidence level. The model that is obtained is not more explanatory than the linear one, as the determination coefficient is considered the adjustments rightness of fit rate (R 2 =0:75). Estimates can be Dimensioning main and associated winches using the pull number Recent publications by Bjørhovde and Aasen 19 or Carral Couce et al. 5,20 have made progress in determining the design parameters for winches, although further work is needed. In the former study, a statistical analysis provides an expression to determine a winch s weight. On the other hand, in the latter study, the working characteristics are determined for offshore tugs (TW). Table 6 gathers data from this work: the variation intervals for the sum total of working parameters for the full range of equipment (TW, AHW, SW and TW). These were based on the statistical values for the vessel sample, a quantitative consideration. Qualitative aspects, as seen in Table 7, included supervision and monitoring. Similarly, a procedure was proposed to determine the traction for each of these. TW Figure 8(a) and (b) represents the traction value for this winch in relation to the vessels BP value. Moreover, Figure 8(a) and (b) includes the linear and non-linear regression adjustments with estimate and prediction confidence bands at a 95% confidence level. The proposed models are a useful tool for estimating the traction needed for the TW for the obtained BP value. However, one has to take into account the vessels have a wide range of characteristics. Figure 8(a) has the linear regression model with all the statistically parameters adjusted to a 95% confidence level. It is possible to detect a relatively strong linear correlation between two

10 10 Proc IMechE Part M: J Engineering for the Maritime Environment Table 6. Interval and average traction are given for the winches on deck (anchor handling, towing and service) and the total traction value as the number corresponding to the sum total of the equipment traction. Anchor handling winch (AHW) Towing winch (TW) Secondary winch (SW) Tugger winch (TGW) Capstan Number Average value number , , Line capacity (m) Cable diameter (mm) Traction on braking (t) Rendering and recovering speed (m/s) Source: Authors own work. 1 Table 7. Common characteristics of the AHT and AHTS according to Hancox 7,21. Towing winch Anchor handling winch Withstands the effort of the towing itself and the additional force present during the operation Recovers the towing line while sailing under 50% of Usually operates in a similar way to the towing winch the propulsion power of the tug Operation and local and remote control: Brakes (on/off and emergency) Clutches (in/out, sliding) Operating (paying out rapid, hauling-slow and Similar traction characteristics and capacity as towing free wheel) winch Instrumentation: Length of paid out line Pull on line Braking traction Location of the clutches In terms of speed, there are differences between the Power applied two, given that in this case the range of speeds is Automatic line stowing feature that can withstand greater. High speeds are used for recovery and an lateral loads above the stall load when the winch is especially low speed for higher traction levels starting up Can pay out the line without needing to disengage With the emergency pay out system, it should be possible to release the brake on and disengage the winch when the power source fails The gears traction value can be adjusted so that the One end of the towing line is attached to the drum anchor is recovered by means of combining winch and with a clip vessel traction, without the risk of breaking the line Common features of the towing winch: Cascade configuration One or two drums with wildcats at the ends of their axles for anchor handling All of these drums operate by means of one set Remote control can be used for all of the functions of clutches Proximity and similarity between the drums for anchor handling and towing Source: Authors own work based on Carral Couce et al. 1 variables. The result is a linear regression model that can explain 71% of the variability for the TW traction (R 2 =0:71). For this reason, the vessels BP can help provide value estimates for the TWs traction value (6) T total =1:5763 BP + 533:86 ð6þ In Figure 8(b), one can see the non-linear regression for a logistic model. All of the parameters are significantly different from zero with a 95% confidence rate. The result is a model that is slightly more explanatory than the linear model. The determination coefficient is considered the confidence rate for the adjustment (R 2 =0:75). Two main traction levels are established for the TW, low (798 T-f) and high (970 T-f), with a BP of approximately 220 T-f, marking the inflection point for the non-linear model with a logistic function. Estimates can be made for the TW traction by means of this expression T t = 798: :98 798:11 ð7þ 1+exp (220:05 BP) 4:835

11 Carral-Couce et al. 11 with statistically significant parameters at a 95% confidence level. It is possible to see a strong linear correlation between the AHW traction and the BP. A linear regression model is obtained; it can explain 75% of the variability in the AHW traction (R 2 =0:75). In this way, once the vessels BP is known, it is possible to get the traction value estimate for the AHW (8) T total =0:8973 BP + 237:52 ð8þ In Figure 8(d), one finds the non-linear regression adjustment with a four-parameter logistic function. All of its parameters are significantly different from zero and have a 95% confidence level. This model is more explanatory that the linear one, with the determination serving as the adjustments rightness of fit (R 2 =0:88). Two main levels of AHW traction are established, low (399 T-f) and high (488 T-f), with a BP value of around 239 T-f. The point of inflection is marked for the nonlinear model with a logistic function. Changes in the lower and higher level of the AHW and TW can be observed producing the same BP value (239 and 220 T-f, respectively). Estimates for the AHW traction can be made with this expression (9) 488:24 399:31 T t = 399:31 + ð9þ 1+exp ð239:01 BPÞ 4:598 Figure 8. Determining the traction for the TW and AHW using the BP as a starting point. (a) TW traction according to the BP and the linear regression model with estimate confidence intervals. (b) TW traction according to the BP and non-linear regression model with a logistic function and estimate and prediction confidence intervals. (c) AHW traction according to the BP and linear regression model with estimate confidence intervals. (d) AHW traction according to the BP and linear regression model with a logistic function and estimate and prediction confidence intervals. Source: Authors own work. AHW Figure 8(c) and (d) gives the AHW traction value in relation to the vessels BP value. Along similar lines to the TW, Figure 8(c) and (d) includes the linear and non-linear regression adjustments with estimate and prediction confidence bands at a 95% confidence level. Although one must remember that the vessels are extremely varied, the regression models obtained can be a useful option in vessel design for estimating the traction for the AHW based on the value given for the BP. Figure 7(c) shows the linear regression model adjusted TGW, SW and capstan Values related to the traction of the secondary and TGWs and capstan that help carry out the manoeuvres rely on the traction for the main equipment. Nevertheless, this value does not have such a close relationship with the vessels BP given that regulations fail to define their dimensions. For this reason, there is a weak correlation between these tractions and the vessels BP (0.53 and 0.46). In these cases, the shipyard and owners criteria and experience of the come into play when establishing a value. In practical terms, it is possible to obtain a traction value for all of the auxiliary winches once the AHW and TW tractions are known. It is simply a question of subtracting these values from the one initially obtained and related to the traction factor value based on the expression (2) d T tugger winghes + c T secundary winches + d T capstan = T total a T towing winches + b anchorhandlingwinches After one has determined the value taken by the sum total of tractions for these auxiliary winches, a unitary value can be obtained for each group. Here, the breakdown coefficients play a role. These are found by analysing the data base under consideration, and knowing the number of installed winches that intervenes (values c, d and e). Break-down coefficient secondary winches = Break-down coefficient tugger winches =

12 12 Proc IMechE Part M: J Engineering for the Maritime Environment Table 8. An analysis of the significance of the regression model for the Tt in relation to the AHW and SW, TGW and capstan traction for each vessel (adjusted R 2 =0:92). Estimates Standard deviation t values p-values Intercept AHW SW Capstan TGW AHW: anchor handling winch; SW: secondary winch; TGW: auxiliary tugger winches. Break-down coefficient capstan = In this study, an empirical alternative is proposed for estimating the traction for auxiliary winches by adjusting the multiple linear regression model. In particular, the proposal is for a linear model that relates the pull number for the tug in relation to the tractions for the TW and AHW. Also taken into account is the sum total of the tractions for the auxiliary winches. Once the ordinary squared minimums have been adjusted, the following equation is obtained (10) T t =36:78 + 1:582 AHW +0:558 TW + + 1:061 Auxiliary winches R 2 =0:91 ð10þ Figure 9 shows the characteristics of the model that is obtained. Figure 9(a) indicates the relative importance of the AHW, TW and auxiliary winch traction on the T t, in % of the R 2, with bootstrap confidence intervals. 13,22 One can observe that the model is highly explanatory and the relationship between the variables is strongly linear (R 2 =0:91). The AHW and TW tractions are of similar importance in the T t response (around 36% of the R 2 ). However, the auxiliary winches have an importance of 26% of the R 2. Figure 9(b) shows the partial linear effects with standard error bands for the AHW, TW and auxiliary winches traction over the T t. The relationship is directly linear in every case. Finally, Figure 9(c) has a prediction graph based on the values observed for T t. A leave-one-out crossvalidation procedure was employed to assess the predictive power of the model for the remaining observations. The step that immediately comes after is to estimate the pull number for the observation that had been left aside. As the prediction-observation pairs lie along the bi-sector of the first quadrant, this shows how reliable the model is for providing predictions for the T t.with the expression for the multiple linear model proposed and by knowing the T t, as well as AHW and TW tractions, it is possible to estimate the traction for the group of auxiliary winches. If one would like to estimate the traction for each type of auxiliary winch separately, it is necessary to apply a logarithmic transformation to the T t before adjusting the multivariable linear model. This operation makes it possible to transform the non-linear relationship between the response and the regressor variables in a linear fashion. In this way, one can estimate the models parameters that are related to each kind of auxiliary winch. These will be significantly different from zero at 0.05 significance rate (p-values \ 0:05). Table 8 provides the parameters for the model and an analysis of its significance. As a result, nearly every log change (T t )(R 2 =0:92) can be explained by the linear addition of the variations in AHW and auxiliary tractions. Thus, if the values for the T t and AHW are known, the tractions for the auxiliary winches can be obtained (11) logðt t Þ=6:14 + 0:00141 AHW +0:00114 SW + + 0:00062 TWG + 0:00925 Capstan Conclusion ð11þ Considering the pull number (T t ) for the winches fitted on an AHT/AHTS helps indicate the vessels capacity for towing operations and anchor handling. In this study, a relationship has been determined between the variables or the winches defining characteristics. A methodology is proposed: statistical regression models are applied to determine the T t, as well as the AHW, TW and auxiliary tractions on an AHT/AHTS, The T t value has a reasonable relationship with the Bollard Pull (BP) of the vessel. This is because the BP value intervenes to determine the characteristics for towing line and its accessories, as well as the TW traction. In a similar way, this characteristic of the vessel will determine its working capacity in handling anchors. Thus, the BP can be estimated by means of the linear regression and multiple linear regression models according to the vessel s length and, above all, the power that is desired for it, which represents almost 80% of the BP variability, explained by the model. As a result, a highly explanatory and statistically significant model was estimated (R 2 =0:97). By studying the data base and carrying out a regression analysis, it was possible to show a significant relationship of dependence between the T t, as well as the AHW and TW tractions, and the BP. To make it possible to estimate the T t, the first step entailed modelling the relationship of dependence of the T t in relation to the BP. Here, flexible semi-parametric regression models came into play. These are linear and logistic, with

13 Carral-Couce et al. 13 Figure 9. Characteristics of the linear regression model relating T t with AHW, TW and auxiliary winch tractions. (a) Estimate of the relative influence (in % of R 2 ) for each regressor variable applying bootstrap re-sampling. (b) Partially linear effects of the AHW, TW and auxiliary winch tractions on the T t (including standard error). (c) T t values estimating according to the ones observed by applying a leave-one-out cross-validation procedure. Source: Authors own work.

14 14 Proc IMechE Part M: J Engineering for the Maritime Environment three and four parameters. The linear regression model accounts for 75%, R 2 =0:75, of the variability in the T t, a strong linear relationship. However, the most explanatory parametric model was the four-parameter logistic regression one (R 2 =0:85). Indeed, the semi-parametric model indicates that the relationship between the T t and BP is sigmoid (R 2 =0:86). The semi-parametric and logistic adjustments have a low level (about T-f) and a high one as well (around T-t) for the T t. Inflection occurs around the BP = 223 T-f. It is interesting to point out that the inflection point coincides with those obtained for the AHW and TW tractions when these are adjusted according to a logistic model. Moreover, the semi-parametric regression stands out as a useful tool that is much more flexible for estimating the T t. When it comes to estimating the AHW traction according to the BP, one can detect a moderate to strong linear relationship between the two variables. A linear regression model has been adjusted; this can explain 75% of the AHW traction variability (R 2 =0:75). Nevertheless, a four-parameter linear regression model with logistic functions was the most informative adjusted model. It accounts for 88% of the AHW traction variance (R 2 =0:88). After the dependence between the TW traction and the BP was studied, it was possible to observe a moderate to strong linear relationship. Therefore, linear and non-linear regression models can be obtained. These account for most of the variances in TW traction. What is the case is that differences in terms of the variance explained between the linear and logistic models are not so marked (R 2 linear =0:71 and R2 logistic =0:75). In conclusion, this study can help estimate the total necessary traction, T t, as well as the TW and AHW tractions using the BP for each vessel as a starting point. At the same time, by considering the design principles for the TW and AHW traction, it is possible to establish their relationship to the vessels BP. It is therefore possible to obtain the values for the traction those winches should have. To do so, modelling needs to be done on the relationship of dependence between the TW and AHW tractions and the BP. As was the case with the T t vis-à-vis the BP, the linear relationship was relatively strong, especially with the AHW, but the most explanatory model was a four-parameter, nonlinear one. This can be shown in a significance analysis on the parameters and rightness of fit. Secondary and TGW traction, as well as that of the capstan, facilitate the manoeuvre, even though they rely on the main equipments traction. However, they are not so closely related. There is no expression in the guidelines for establishing their dimensions. In practical terms, it is possible to obtain the value for the auxiliary winches being used for the manoeuvres with the expression put forward in the text. Another proposal has been to use two empirical expressions obtained by applying multivariate linear regression models (R 2 =0:91 and R 2 =0:92). With these one can estimate the tractions for the all or just one of the auxiliary winches. Moreover, by studying these models, one can estimate the relative importance of the traction for various kinds of winches on the pull number. In the end, the AHW and TW tractions were significantly higher and equal in importance in terms of the AHW and TW tractions, representing around 36% of the variability explained for the T t. Declaration of Conflicting Interests The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article. Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study has been, in part, financed by the MTM R project (ERDF included). References 1. Carral Couce L, Fraguela Formoso J, Villa Caro R, et al. Estudio del valor total de la traccio n en la maquinaria de cubierta de un buque offshore anchor handling tug. Congreso Internacional Copinaval, Instituto Panamericano de Ingenierı a Naval, pp International Maritime Organization. Guidelines for safe ocean towing MSC/Circ London: International Maritime Organization, Denton N. Guidelines for the approval of towing vessels 0021/no rev. 8. Technical report, GL Group Noble Denton Tools, London, Gaston MJ. The tug book. Doncaster, UK: Haynes Publishing, Carral Couce JA, Carral Couce LM, Fraguela Formoso JA, et al. El chigre de remolque en las maniobras de altura y de escolta: propuesta de armonizacin en sus parmetros de diseo. DYNA Ind En 2013; 88: Carral L, Fraguela JA, De Troya J, et al. Influence of the towline material: steel or high-modulus polyethylene on towing gear design and tug deck fittings. Proc IMechE, Part M: J Engineering. Epub ahead of print 27 October DOI: / Hancox M. Anchor Handling (Oilfield Seamanship Series vol. 3). Houston: Oilfield Publications Ltd, Ter Haar J. Towing manual. STC-Group, Rotterdam: Hensen H. Tug use in port: a practical guide. London: Nautical Institute, Wennersberg LAL. Modeling and simulation of anchor handling vessels, 2009, smash/get/diva2:348896/cover01.pdf 11. Cepeda FS, Maricruz A, da Silva RB, et al. Prediction of delays in supply logistics of offshore platforms. In XXIV Congreso internacional copinaval, Montevideo, Uruguay, October 2015, pp Instituto Panamericano de Ingeniera Naval Copinaval. Montevideo, Uruguay. 12. Allan RG. A proposal for harmonized international regulations for the design and construction of tugboats. In: Ter Haar J (ed.) 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15 Carral-Couce et al. 15 with related shipping matters and opinions. Rotterdam: STC Group, 2010, pp Canty AJ. Resampling methods in R: the boot package. R News 2002; 2(3): Chevan A and Sutherland M. Hierarchical partitioning. Am Stat 1991; 45(2): Go rski Z and Giernalczyk M. Statistic determination of main propulsion power and total power of onboard electric power station on anchor handling tug supply vessels AHTS servicing oil rings. J Pol CIMAC 2012; 7: Wood S. Generalized additive models: an introduction with R. Boca Raton, FL, CRC Press, Tarrío-Saavedra J, Lo pez-beceiro J, Naya S, et al. Simulation study for generalized logistic function in thermal data modeling. J Therm Anal Calorim 2014; 118(2): Efron B and Tibshirani R. Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Stat Sci 1986; 1: Bjørhovde S and Aasen R. Parametric estimation of anchor handling/towing winches. In: 71st international conference on mass properties (Society of Allied Weight Engineers Inc.), Bavaria, 5 10 May 2012, pp Red Hook, NY: Curran Associates Inc. 20. Carral Couce L, Carral Couce JC and Fraguela Formoso JN. Operation and handling in escort tugboat manoeuvres with the aid of automatic towing winch systems. J Navigation 2015; 68: Hancox M. The Oilfield Seamanship Series: volume 4 towing (The Oilfield Seamanship Series). Houston: Oilfield Publications Ltd, Grömping U. Relative importance for linear regression in R: the package relaimpo. J Stat Softw 2006; 17(1): Venables W and Smith D. An Introduction to R: Notes on R, A Programming Environment for Data Analysis and Graphics, R Core Team, Electronic edition. lookupid?key=olbp Ríos-Fachal M, Tarrío-Saavedra J, Lo pez-beceiro J, et al. Optimizing fitting parameters in thermogravimetry. J Therm Anal Calorim 2014; 116(3): Elzhov TV, Mullen KM, Spiess AN, et al. minpack.lm: r interface to the Levenberg-Marquardt nonlinear leastsquares algorithm found in MINPACK, Plus Support for Bounds (R package version 1.2-0), 2015, CRAN.R-project.org/package=minpack.lm 26. Mullen KM, Ardia D, Gil DL, et al. Deoptim: an r package for global optimization by differential evolution. J Stat Softw 2011; 40(6): Baty F, Ritz C, Charles S, et al. A toolbox for nonlinear regression in r: the package nlstools. J Stat Softw 2014; 66: Harrell FE Jr. rms: regression modeling strategies (R package version 4.5-0), 2016, org/package=rms 29. Davies C, Hyde J, Bangdiwala S, et al. An example of dependencies among variables in a conditional logistic regression. In: Moolgavkar SH and Prentice RL (eds) Modern statistical methods in chronic disease epidemiology. New York: Willey, Appendix 1 This section briefly describes and provides references for the statistical techniques applied in the article and the information technology tools used to put them into practice. In particular, R software program was useful; it is now the most complete, wide reaching and flexible tool for carrying out the statistical analysis of data. 23 In statistics, a regression model can be defined according to following expression y i = mx ð i, uþ+ e i i =1,2,..., n ð12þ Within the expression where the response variable is y i, the independent variable is x i, m(x i, u) is the linear or non-linear parametric regression model, u is the vector for the adjustment parameters estimated in the squared minimums and e i = y i m(x i, u), i =1,2,..., n are errors found in the model, assuming that they are distributed in the normal way with a zero average and constant variance. Model parameters u are estimated by minimising the sum of the squares for the errors, SSE(u)= P i (y i m(x i, u)) 2. When the model is neither linear nor can be linearised, one needs an optimisation algorithm application to obtain the parameters, as is the case with Nelder Mead, Newton Raphson, and Levenberg Marquart algorithms along with more recent methods like evolutionary algorithms. 24 In this study, a combination of algorithms comes into play: (1) an evolutionary one called differential evolution to obtain a valid initial solution for the parameter vector Iˆ, so that, as far as possible, one does not fall into local minimums and (2) once the initial solution has been found, the parameter vector that is obtained minimises the SSE by applying the Levenberg Marquart algorithm. 24,25 To apply this procedure, R statistical software was employed, particularly DEoptim 26 and minpack. 25 The utilities of the nls and nlstools packages also make it possible to carry out a significance analysis for the estimated parameters. 27 At this point, expressions will be shown for the nonlinear models used here, focusing on those based on three- and four-parameter logistic functions. Non-linear model with a three-parameter logistic function Asym y i = ð13þ 1 + exp ðxmid xþ scal Here, Asym is the saturation asymptote, xmid is the value for x for which a greater rate of growth or slope and scale is a parameter related to the speed of growth or decline for the logistic model. Four-parameter non-linear model with a logistic function

16 16 Proc IMechE Part M: J Engineering for the Maritime Environment mðu; xþ= A + B A ð14þ 1+exp ðxmid xþ This model also includes the three-parameter logistic model and two parameters: A, an initial horizontal asymptote horizontal, and B, a final horizontal asymptote. In Tarrı o-saavedra et al., 17 further information can be found on the regression models with a logistic function. On the other hand, this work proposes an application that uses a semi-parametric regression model based on adjusting the base of a thin plate spline, especially ideal when used in regression. This model is applied within the general structure of the generalised additive models (GAMs), an extension of the generalised linear models (GLMs). The GAM makes it possible to estimate a quantitative variable according to the continuous soft effects (unknowns that do not have a parametric expression) for one or more co-variables. 16,17 This is expressed as follows scal gðm x Þ= b 0 + f 1 ðx 1 Þ+ f 2 ðx 2 Þ+ f 12 ðx 1 ; X 2 Þ+ X 3 ð15þ In which f() are the soft effects for the regressor variables, estimated by means of a base spline, in this case, thin plate splines. With this model, factors (X 3 ) and iterations f 12 (X 1 ; X 2 ) can also be included between regression variables. For the study, the R mgcv package was used to implement this kind of semi-parametric tool. 16 On the other hand, bootstrap re-sampling offered the possibility of estimating the regression curve without having to know in advance the probability distribution for the response variable. 18 Bootstrap resampling is a statistical and computational method for estimating in terms of averages, variance and percentiles. Data are randomly taken or resampled from the original set. For validity regression or classification models, subsets are resampled. To assess and analyse the linear multivariate regression models with two or more independent variables or regressors, different statistic techniques have come into play. One is the variance inflection factor (VIF), used to detect multiple co-linearity in a multiple linear regression. Another step is to estimate the relative importance of each independent variable in the response. The VIF was estimated with the rms package, 28 implementing the method of Davies et al. 29 It is normally the assumption that if VIF \ 4, the degree of multiple co-linearity between regressor variables does not have a significant effect on how the model parameters are estimated. There is also the reliampo package 22 which can estimate the relative importance of each independent variable in the response, measured in % of the global coefficient of determination R 2. To this end, with the reliampo package, one can use the various metrics, including the lmg, the R 2 contribution averaged over orderings among regressors. 14 It is based on calculating from what each regressor is capable of explaining, equivalent to the squared correlation coefficients. Furthermore, the boot package makes it possible to estimate each relative importance through confidence intervals obtained with re-sampling techniques. 13